LMFDB - Embedded newform 546.8.c.a.337.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(337,546)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("546.337");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$170.562223914$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.16
Character	$\chi$	$=$	546.337
Dual form			546.8.c.a.337.9

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000i q^{2} -27.0000 q^{3} -64.0000 q^{4} -176.597i q^{5} -216.000i q^{6} -343.000i q^{7} -512.000i q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000i q^{2} -27.0000 q^{3} -64.0000 q^{4} -176.597i q^{5} -216.000i q^{6} -343.000i q^{7} -512.000i q^{8} +729.000 q^{9} +1412.78 q^{10} -6905.52i q^{11} +1728.00 q^{12} +(7647.39 + 2065.40i) q^{13} +2744.00 q^{14} +4768.13i q^{15} +4096.00 q^{16} +17886.3 q^{17} +5832.00i q^{18} -3373.06i q^{19} +11302.2i q^{20} +9261.00i q^{21} +55244.1 q^{22} +11915.5 q^{23} +13824.0i q^{24} +46938.3 q^{25} +(-16523.2 + 61179.1i) q^{26} -19683.0 q^{27} +21952.0i q^{28} +69516.1 q^{29} -38145.0 q^{30} +201566. i q^{31} +32768.0i q^{32} +186449. i q^{33} +143090. i q^{34} -60572.9 q^{35} -46656.0 q^{36} -403514. i q^{37} +26984.4 q^{38} +(-206480. - 55765.9i) q^{39} -90417.9 q^{40} +31360.1i q^{41} -74088.0 q^{42} +709040. q^{43} +441953. i q^{44} -128740. i q^{45} +95324.2i q^{46} +1.26222e6i q^{47} -110592. q^{48} -117649. q^{49} +375507. i q^{50} -482930. q^{51} +(-489433. - 132186. i) q^{52} +25435.1 q^{53} -157464. i q^{54} -1.21950e6 q^{55} -175616. q^{56} +91072.5i q^{57} +556129. i q^{58} -1.26067e6i q^{59} -305160. i q^{60} +748748. q^{61} -1.61253e6 q^{62} -250047. i q^{63} -262144. q^{64} +(364745. - 1.35051e6i) q^{65} -1.49159e6 q^{66} -955833. i q^{67} -1.14472e6 q^{68} -321719. q^{69} -484583. i q^{70} +1.19719e6i q^{71} -373248. i q^{72} +467672. i q^{73} +3.22811e6 q^{74} -1.26734e6 q^{75} +215876. i q^{76} -2.36859e6 q^{77} +(446127. - 1.65184e6i) q^{78} +2.78700e6 q^{79} -723343. i q^{80} +531441. q^{81} -250881. q^{82} +8.61365e6i q^{83} -592704. i q^{84} -3.15867e6i q^{85} +5.67232e6i q^{86} -1.87693e6 q^{87} -3.53563e6 q^{88} +6.49378e6i q^{89} +1.02992e6 q^{90} +(708434. - 2.62306e6i) q^{91} -762593. q^{92} -5.44229e6i q^{93} -1.00978e7 q^{94} -595673. q^{95} -884736. i q^{96} -1.62174e7i q^{97} -941192. i q^{98} -5.03412e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9}+O(q^{10}) $ 24 * q - 648 * q^3 - 1536 * q^4 + 17496 * q^9	$ 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9} + 1520 q^{10} + 41472 q^{12} + 13718 q^{13} + 65856 q^{14} + 98304 q^{16} + 13586 q^{17} - 11824 q^{22} + 76574 q^{23} - 357850 q^{25} + 132848 q^{26} - 472392 q^{27} + 85198 q^{29} - 41040 q^{30} - 65170 q^{35} - 1119744 q^{36} + 295696 q^{38} - 370386 q^{39} - 97280 q^{40} - 1778112 q^{42} - 91370 q^{43} - 2654208 q^{48} - 2823576 q^{49} - 366822 q^{51} - 877952 q^{52} - 360000 q^{53} - 2519718 q^{55} - 4214784 q^{56} + 874734 q^{61} + 1984480 q^{62} - 6291456 q^{64} - 12450388 q^{65} + 319248 q^{66} - 869504 q^{68} - 2067498 q^{69} - 1405616 q^{74} + 9661950 q^{75} + 506954 q^{77} - 3586896 q^{78} - 2141176 q^{79} + 12754584 q^{81} + 14087488 q^{82} - 2300346 q^{87} + 756736 q^{88} + 1108080 q^{90} - 5695858 q^{91} - 4900736 q^{92} + 8123808 q^{94} + 19191246 q^{95}+O(q^{100}) $ 24 * q - 648 * q^3 - 1536 * q^4 + 17496 * q^9 + 1520 * q^10 + 41472 * q^12 + 13718 * q^13 + 65856 * q^14 + 98304 * q^16 + 13586 * q^17 - 11824 * q^22 + 76574 * q^23 - 357850 * q^25 + 132848 * q^26 - 472392 * q^27 + 85198 * q^29 - 41040 * q^30 - 65170 * q^35 - 1119744 * q^36 + 295696 * q^38 - 370386 * q^39 - 97280 * q^40 - 1778112 * q^42 - 91370 * q^43 - 2654208 * q^48 - 2823576 * q^49 - 366822 * q^51 - 877952 * q^52 - 360000 * q^53 - 2519718 * q^55 - 4214784 * q^56 + 874734 * q^61 + 1984480 * q^62 - 6291456 * q^64 - 12450388 * q^65 + 319248 * q^66 - 869504 * q^68 - 2067498 * q^69 - 1405616 * q^74 + 9661950 * q^75 + 506954 * q^77 - 3586896 * q^78 - 2141176 * q^79 + 12754584 * q^81 + 14087488 * q^82 - 2300346 * q^87 + 756736 * q^88 + 1108080 * q^90 - 5695858 * q^91 - 4900736 * q^92 + 8123808 * q^94 + 19191246 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$.

$n$	$157$	$365$	$379$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			8.00000i			0.707107i
$2$			8.00000i			0.707107i
$3$	−27.0000			−0.577350
$3$	−27.0000			−0.577350
$4$	−64.0000			−0.500000
$5$		−	176.597i		−	0.631814i	−0.948790	−	0.315907i	$-0.897691\pi$
$5$		−	176.597i		−	0.631814i	0.948790	−	0.315907i	$-0.102309\pi$
$6$		−	216.000i		−	0.408248i
$7$		−	343.000i		−	0.377964i
$7$		−	343.000i		−	0.377964i
$8$		−	512.000i		−	0.353553i
$9$	729.000			0.333333
$10$	1412.78			0.446760
$11$		−	6905.52i		−	1.56431i	−0.623086	−	0.782153i	$-0.714121\pi$
$11$		−	6905.52i		−	1.56431i	0.623086	−	0.782153i	$-0.285879\pi$
$12$	1728.00			0.288675
$13$	7647.39	+	2065.40i	0.965410	+	0.260737i
$13$	7647.39	+	2065.40i	0.965410	+	0.260737i
$14$	2744.00			0.267261
$15$			4768.13i			0.364778i
$16$	4096.00			0.250000
$17$	17886.3			0.882976			0.441488	−	0.897267i	$-0.354451\pi$
$17$	17886.3			0.882976			0.441488	+	0.897267i	$0.354451\pi$
$18$			5832.00i			0.235702i
$19$		−	3373.06i		−	0.112820i	−0.998408	−	0.0564100i	$-0.982035\pi$
$19$		−	3373.06i		−	0.112820i	0.998408	−	0.0564100i	$-0.0179654\pi$
$20$			11302.2i			0.315907i
$21$			9261.00i			0.218218i
$22$	55244.1			1.10613
$23$	11915.5			0.204205			0.102102	−	0.994774i	$-0.467443\pi$
$23$	11915.5			0.204205			0.102102	+	0.994774i	$0.467443\pi$
$24$			13824.0i			0.204124i
$25$	46938.3			0.600811
$26$	−16523.2	+	61179.1i	−0.184369	+	0.682648i
$27$	−19683.0			−0.192450
$28$			21952.0i			0.188982i
$29$	69516.1			0.529288			0.264644	−	0.964346i	$-0.414746\pi$
$29$	69516.1			0.529288			0.264644	+	0.964346i	$0.414746\pi$
$30$	−38145.0			−0.257937
$31$			201566.i			1.21521i	0.794239	+	0.607605i	$0.207870\pi$
$31$			201566.i			1.21521i	−0.794239	+	0.607605i	$0.792130\pi$
$32$			32768.0i			0.176777i
$33$			186449.i			0.903153i
$34$			143090.i			0.624358i
$35$	−60572.9			−0.238803
$36$	−46656.0			−0.166667
$37$		−	403514.i		−	1.30964i	−0.755784	−	0.654821i	$-0.772744\pi$
$37$		−	403514.i		−	1.30964i	0.755784	−	0.654821i	$-0.227256\pi$
$38$	26984.4			0.0797758
$39$	−206480.	−	55765.9i	−0.557380	−	0.150537i
$40$	−90417.9			−0.223380
$41$			31360.1i			0.0710614i	0.999369	+	0.0355307i	$0.0113122\pi$
$41$			31360.1i			0.0710614i	−0.999369	+	0.0355307i	$0.988688\pi$
$42$	−74088.0			−0.154303
$43$	709040.			1.35997			0.679987	−	0.733224i	$-0.261985\pi$
$43$	709040.			1.35997			0.679987	+	0.733224i	$0.261985\pi$
$44$			441953.i			0.782153i
$45$		−	128740.i		−	0.210605i
$46$			95324.2i			0.144395i
$47$			1.26222e6i			1.77334i	0.462402	+	0.886671i	$0.346988\pi$
$47$			1.26222e6i			1.77334i	−0.462402	+	0.886671i	$0.653012\pi$
$48$	−110592.			−0.144338
$49$	−117649.			−0.142857
$50$			375507.i			0.424837i
$51$	−482930.			−0.509786
$52$	−489433.	−	132186.i	−0.482705	−	0.130369i
$53$	25435.1			0.0234675			0.0117338	−	0.999931i	$-0.496265\pi$
$53$	25435.1			0.0234675			0.0117338	+	0.999931i	$0.496265\pi$
$54$		−	157464.i		−	0.136083i
$55$	−1.21950e6			−0.988351
$56$	−175616.			−0.133631
$57$			91072.5i			0.0651366i
$58$			556129.i			0.374263i
$59$		−	1.26067e6i		−	0.799132i	−0.916704	−	0.399566i	$-0.869161\pi$
$59$		−	1.26067e6i		−	0.799132i	0.916704	−	0.399566i	$-0.130839\pi$
$60$		−	305160.i		−	0.182389i
$61$	748748.			0.422359			0.211179	−	0.977447i	$-0.432270\pi$
$61$	748748.			0.422359			0.211179	+	0.977447i	$0.432270\pi$
$62$	−1.61253e6			−0.859284
$63$		−	250047.i		−	0.125988i
$64$	−262144.			−0.125000
$65$	364745.	−	1.35051e6i	0.164738	−	0.609960i
$66$	−1.49159e6			−0.638625
$67$		−	955833.i		−	0.388258i	−0.980976	−	0.194129i	$-0.937812\pi$
$67$		−	955833.i		−	0.388258i	0.980976	−	0.194129i	$-0.0621880\pi$
$68$	−1.14472e6			−0.441488
$69$	−321719.			−0.117898
$70$		−	484583.i		−	0.168859i
$71$			1.19719e6i			0.396972i	0.980104	+	0.198486i	$0.0636024\pi$
$71$			1.19719e6i			0.396972i	−0.980104	+	0.198486i	$0.936398\pi$
$72$		−	373248.i		−	0.117851i
$73$			467672.i			0.140705i	0.997522	+	0.0703527i	$0.0224125\pi$
$73$			467672.i			0.140705i	−0.997522	+	0.0703527i	$0.977588\pi$
$74$	3.22811e6			0.926056
$75$	−1.26734e6			−0.346878
$76$			215876.i			0.0564100i
$77$	−2.36859e6			−0.591252
$78$	446127.	−	1.65184e6i	0.106446	−	0.394127i
$79$	2.78700e6			0.635979			0.317989	−	0.948094i	$-0.396992\pi$
$79$	2.78700e6			0.635979			0.317989	+	0.948094i	$0.396992\pi$
$80$		−	723343.i		−	0.157954i
$81$	531441.			0.111111
$82$	−250881.			−0.0502480
$83$			8.61365e6i			1.65354i	0.562543	+	0.826768i	$0.309823\pi$
$83$			8.61365e6i			1.65354i	−0.562543	+	0.826768i	$0.690177\pi$
$84$		−	592704.i		−	0.109109i
$85$		−	3.15867e6i		−	0.557877i
$86$			5.67232e6i			0.961647i
$87$	−1.87693e6			−0.305585
$88$	−3.53563e6			−0.553066
$89$			6.49378e6i			0.976411i	0.872729	+	0.488205i	$0.162348\pi$
$89$			6.49378e6i			0.976411i	−0.872729	+	0.488205i	$0.837652\pi$
$90$	1.02992e6			0.148920
$91$	708434.	−	2.62306e6i	0.0985495	−	0.364891i
$92$	−762593.			−0.102102
$93$		−	5.44229e6i		−	0.701602i
$94$	−1.00978e7			−1.25394
$95$	−595673.			−0.0712813
$96$		−	884736.i		−	0.102062i
$97$		−	1.62174e7i		−	1.80418i	−0.431551	−	0.902089i	$-0.642033\pi$
$97$		−	1.62174e7i		−	1.80418i	0.431551	−	0.902089i	$-0.357967\pi$
$98$		−	941192.i		−	0.101015i
$99$		−	5.03412e6i		−	0.521436i
$100$	−3.00405e6			−0.300405
$101$	−301619.			−0.0291296			−0.0145648	−	0.999894i	$-0.504636\pi$
$101$	−301619.			−0.0291296			−0.0145648	+	0.999894i	$0.504636\pi$
$102$		−	3.86344e6i		−	0.360473i
$103$	1.59512e6			0.143834			0.0719171	−	0.997411i	$-0.477088\pi$
$103$	1.59512e6			0.143834			0.0719171	+	0.997411i	$0.477088\pi$
$104$	1.05749e6	−	3.91547e6i	0.0921846	−	0.341324i
$105$	1.63547e6			0.137873
$106$			203481.i			0.0165941i
$107$	6.66267e6			0.525781			0.262890	−	0.964826i	$-0.415324\pi$
$107$	6.66267e6			0.525781			0.262890	+	0.964826i	$0.415324\pi$
$108$	1.25971e6			0.0962250
$109$			7.29733e6i			0.539723i	0.962899	+	0.269862i	$0.0869779\pi$
$109$			7.29733e6i			0.539723i	−0.962899	+	0.269862i	$0.913022\pi$
$110$		−	9.75597e6i		−	0.698870i
$111$			1.08949e7i			0.756122i
$112$		−	1.40493e6i		−	0.0944911i
$113$	−1.69452e7			−1.10477			−0.552386	−	0.833588i	$-0.686283\pi$
$113$	−1.69452e7			−1.10477			−0.552386	+	0.833588i	$0.686283\pi$
$114$	−728580.			−0.0460586
$115$		−	2.10425e6i		−	0.129019i
$116$	−4.44903e6			−0.264644
$117$	5.57495e6	+	1.50568e6i	0.321803	+	0.0869125i
$118$	1.00853e7			0.565072
$119$		−	6.13499e6i		−	0.333734i
$120$	2.44128e6			0.128969
$121$	−2.81990e7			−1.44706
$122$			5.98999e6i			0.298653i
$123$		−	846723.i		−	0.0410273i
$124$		−	1.29002e7i		−	0.607605i
$125$		−	2.20859e7i		−	1.01142i
$126$	2.00038e6			0.0890871
$127$	1.48594e7			0.643706			0.321853	−	0.946790i	$-0.395694\pi$
$127$	1.48594e7			0.643706			0.321853	+	0.946790i	$0.395694\pi$
$128$		−	2.09715e6i		−	0.0883883i
$129$	−1.91441e7			−0.785182
$130$	1.08041e7	+	2.91796e6i	0.431307	+	0.116487i
$131$	−7.93957e6			−0.308565			−0.154283	−	0.988027i	$-0.549307\pi$
$131$	−7.93957e6			−0.308565			−0.154283	+	0.988027i	$0.549307\pi$
$132$		−	1.19327e7i		−	0.451576i
$133$	−1.15696e6			−0.0426419
$134$	7.64667e6			0.274540
$135$			3.47597e6i			0.121593i
$136$		−	9.15778e6i		−	0.312179i
$137$		−	5.08138e6i		−	0.168834i	−0.996431	−	0.0844169i	$-0.973097\pi$
$137$		−	5.08138e6i		−	0.168834i	0.996431	−	0.0844169i	$-0.0269028\pi$
$138$		−	2.57375e6i		−	0.0833662i
$139$	4.29710e7			1.35714			0.678569	−	0.734537i	$-0.262600\pi$
$139$	4.29710e7			1.35714			0.678569	+	0.734537i	$0.262600\pi$
$140$	3.87667e6			0.119402
$141$		−	3.40799e7i		−	1.02384i
$142$	−9.57754e6			−0.280702
$143$	1.42627e7	−	5.28092e7i	0.407873	−	1.51020i
$144$	2.98598e6			0.0833333
$145$		−	1.22764e7i		−	0.334412i
$146$	−3.74137e6			−0.0994938
$147$	3.17652e6			0.0824786
$148$			2.58249e7i			0.654821i
$149$		−	2.90268e7i		−	0.718865i	−0.933171	−	0.359433i	$-0.882970\pi$
$149$		−	2.90268e7i		−	0.718865i	0.933171	−	0.359433i	$-0.117030\pi$
$150$		−	1.01387e7i		−	0.245280i
$151$		−	149215.i		−	0.00352691i	−0.999998	−	0.00176345i	$-0.999439\pi$
$151$		−	149215.i		−	0.00352691i	0.999998	−	0.00176345i	$-0.000561325\pi$
$152$	−1.72700e6			−0.0398879
$153$	1.30391e7			0.294325
$154$		−	1.89487e7i		−	0.418079i
$155$	3.55961e7			0.767787
$156$	1.32147e7	+	3.56902e6i	0.278690	+	0.0752684i
$157$	−1.65903e7			−0.342142			−0.171071	−	0.985259i	$-0.554723\pi$
$157$	−1.65903e7			−0.342142			−0.171071	+	0.985259i	$0.554723\pi$
$158$			2.22960e7i			0.449705i
$159$	−686747.			−0.0135490
$160$	5.78674e6			0.111690
$161$		−	4.08702e6i		−	0.0771821i
$162$			4.25153e6i			0.0785674i
$163$			6.72115e7i			1.21559i	0.794094	+	0.607794i	$0.207946\pi$
$163$			6.72115e7i			1.21559i	−0.794094	+	0.607794i	$0.792054\pi$
$164$		−	2.00705e6i		−	0.0355307i
$165$	3.29264e7			0.570625
$166$	−6.89092e7			−1.16923
$167$			7.62130e7i			1.26626i	0.774047	+	0.633128i	$0.218229\pi$
$167$			7.62130e7i			1.26626i	−0.774047	+	0.633128i	$0.781771\pi$
$168$	4.74163e6			0.0771517
$169$	5.42167e7	+	3.15899e7i	0.864032	+	0.503437i
$170$	2.52694e7			0.394478
$171$		−	2.45896e6i		−	0.0376067i
$172$	−4.53785e7			−0.679987
$173$	3.34239e7			0.490790			0.245395	−	0.969423i	$-0.421082\pi$
$173$	3.34239e7			0.490790			0.245395	+	0.969423i	$0.421082\pi$
$174$		−	1.50155e7i		−	0.216081i
$175$		−	1.60999e7i		−	0.227085i
$176$		−	2.82850e7i		−	0.391077i
$177$			3.40380e7i			0.461379i
$178$	−5.19502e7			−0.690427
$179$	1.21806e8			1.58739			0.793696	−	0.608315i	$-0.208154\pi$
$179$	1.21806e8			1.58739			0.793696	+	0.608315i	$0.208154\pi$
$180$			8.23933e6i			0.105302i
$181$	8.43983e7			1.05793			0.528967	−	0.848642i	$-0.322579\pi$
$181$	8.43983e7			1.05793			0.528967	+	0.848642i	$0.322579\pi$
$182$	2.09844e7	+	5.66747e6i	0.258017	+	0.0696850i
$183$	−2.02162e7			−0.243849
$184$		−	6.10075e6i		−	0.0721973i
$185$	−7.12595e7			−0.827450
$186$	4.35383e7			0.496108
$187$		−	1.23514e8i		−	1.38125i
$188$		−	8.07820e7i		−	0.886671i
$189$			6.75127e6i			0.0727393i
$190$		−	4.76538e6i		−	0.0504035i
$191$	9.89820e7			1.02787			0.513937	−	0.857828i	$-0.328187\pi$
$191$	9.89820e7			1.02787			0.513937	+	0.857828i	$0.328187\pi$
$192$	7.07789e6			0.0721688
$193$			3.80093e7i			0.380574i	0.981728	+	0.190287i	$0.0609419\pi$
$193$			3.80093e7i			0.380574i	−0.981728	+	0.190287i	$0.939058\pi$
$194$	1.29739e8			1.27575
$195$	−9.84812e6	+	3.64638e7i	−0.0951113	+	0.352160i
$196$	7.52954e6			0.0714286
$197$		−	8.42313e7i		−	0.784949i	−0.919763	−	0.392475i	$-0.871619\pi$
$197$		−	8.42313e7i		−	0.784949i	0.919763	−	0.392475i	$-0.128381\pi$
$198$	4.02730e7			0.368711
$199$	−6.94671e7			−0.624876			−0.312438	−	0.949938i	$-0.601146\pi$
$199$	−6.94671e7			−0.624876			−0.312438	+	0.949938i	$0.601146\pi$
$200$		−	2.40324e7i		−	0.212419i
$201$			2.58075e7i			0.224161i
$202$		−	2.41296e6i		−	0.0205977i
$203$		−	2.38440e7i		−	0.200052i
$204$	3.09075e7			0.254893
$205$	5.53811e6			0.0448976
$206$			1.27609e7i			0.101706i
$207$	8.68642e6			0.0680682
$208$	3.13237e7	+	8.45990e6i	0.241352	+	0.0651844i
$209$	−2.32927e7			−0.176485
$210$			1.30838e7i			0.0974910i
$211$	1.37760e8			1.00957			0.504783	−	0.863246i	$-0.331573\pi$
$211$	1.37760e8			1.00957			0.504783	+	0.863246i	$0.331573\pi$
$212$	−1.62784e6			−0.0117338
$213$		−	3.23242e7i		−	0.229192i
$214$			5.33013e7i			0.371783i
$215$		−	1.25215e8i		−	0.859251i
$216$			1.00777e7i			0.0680414i
$217$	6.91372e7			0.459307
$218$	−5.83786e7			−0.381642
$219$		−	1.26271e7i		−	0.0812363i
$220$	7.80478e7			0.494176
$221$	1.36783e8	+	3.69424e7i	0.852434	+	0.230225i
$222$	−8.71590e7			−0.534659
$223$		−	1.28724e7i		−	0.0777310i	−0.999244	−	0.0388655i	$-0.987626\pi$
$223$		−	1.28724e7i		−	0.0777310i	0.999244	−	0.0388655i	$-0.0123744\pi$
$224$	1.12394e7			0.0668153
$225$	3.42181e7			0.200270
$226$		−	1.35562e8i		−	0.781192i
$227$			1.59386e7i			0.0904401i	0.998977	+	0.0452200i	$0.0143989\pi$
$227$			1.59386e7i			0.0904401i	−0.998977	+	0.0452200i	$0.985601\pi$
$228$		−	5.82864e6i		−	0.0325683i
$229$		−	1.99898e8i		−	1.09998i	−0.835171	−	0.549990i	$-0.814632\pi$
$229$		−	1.99898e8i		−	1.09998i	0.835171	−	0.549990i	$-0.185368\pi$
$230$	1.68340e7			0.0912305
$231$	6.39520e7			0.341360
$232$		−	3.55922e7i		−	0.187132i
$233$	−2.27464e8			−1.17806			−0.589029	−	0.808112i	$-0.700490\pi$
$233$	−2.27464e8			−1.17806			−0.589029	+	0.808112i	$0.700490\pi$
$234$	−1.20454e7	+	4.45996e7i	−0.0614564	+	0.227549i
$235$	2.22905e8			1.12042
$236$			8.06827e7i			0.399566i
$237$	−7.52491e7			−0.367183
$238$	4.90800e7			0.235985
$239$			1.90600e8i			0.903091i	0.892248	+	0.451545i	$0.149127\pi$
$239$			1.90600e8i			0.903091i	−0.892248	+	0.451545i	$0.850873\pi$
$240$			1.95303e7i			0.0911945i
$241$			9.54077e7i			0.439060i	0.975606	+	0.219530i	$0.0704524\pi$
$241$			9.54077e7i			0.439060i	−0.975606	+	0.219530i	$0.929548\pi$
$242$		−	2.25592e8i		−	1.02322i
$243$	−1.43489e7			−0.0641500
$244$	−4.79199e7			−0.211179
$245$			2.07765e7i			0.0902592i
$246$	6.77378e6			0.0290107
$247$	6.96673e6	−	2.57951e7i	0.0294164	−	0.108917i
$248$	1.03202e8			0.429642
$249$		−	2.32568e8i		−	0.954669i
$250$	1.76687e8			0.715178
$251$	2.85884e8			1.14112			0.570560	−	0.821256i	$-0.306726\pi$
$251$	2.85884e8			1.14112			0.570560	+	0.821256i	$0.306726\pi$
$252$			1.60030e7i			0.0629941i
$253$		−	8.22829e7i		−	0.319439i
$254$			1.18875e8i			0.455169i
$255$			8.52841e7i			0.322090i
$256$	1.67772e7			0.0625000
$257$	−2.48361e8			−0.912677			−0.456338	−	0.889806i	$-0.650839\pi$
$257$	−2.48361e8			−0.912677			−0.456338	+	0.889806i	$0.650839\pi$
$258$		−	1.53153e8i		−	0.555207i
$259$	−1.38405e8			−0.494998
$260$	−2.33437e7	+	8.64326e7i	−0.0823688	+	0.304980i
$261$	5.06772e7			0.176429
$262$		−	6.35165e7i		−	0.218189i
$263$	2.14466e8			0.726965			0.363482	−	0.931601i	$-0.381588\pi$
$263$	2.14466e8			0.726965			0.363482	+	0.931601i	$0.381588\pi$
$264$	9.54619e7			0.319313
$265$		−	4.49177e6i		−	0.0148271i
$266$		−	9.25567e6i		−	0.0301524i
$267$		−	1.75332e8i		−	0.563731i
$268$			6.11733e7i			0.194129i
$269$	1.55789e8			0.487983			0.243991	−	0.969777i	$-0.421543\pi$
$269$	1.55789e8			0.487983			0.243991	+	0.969777i	$0.421543\pi$
$270$	−2.78077e7			−0.0859790
$271$			3.68412e8i			1.12445i	0.826983	+	0.562227i	$0.190055\pi$
$271$			3.68412e8i			1.12445i	−0.826983	+	0.562227i	$0.809945\pi$
$272$	7.32622e7			0.220744
$273$	−1.91277e7	+	7.08225e7i	−0.0568976	+	0.210670i
$274$	4.06510e7			0.119384
$275$		−	3.24134e8i		−	0.939852i
$276$	2.05900e7			0.0589488
$277$	3.44212e7			0.0973075			0.0486537	−	0.998816i	$-0.484507\pi$
$277$	3.44212e7			0.0973075			0.0486537	+	0.998816i	$0.484507\pi$
$278$			3.43768e8i			0.959641i
$279$			1.46942e8i			0.405070i
$280$			3.10133e7i			0.0844297i
$281$			2.84906e8i			0.766002i	0.923748	+	0.383001i	$0.125109\pi$
$281$			2.84906e8i			0.766002i	−0.923748	+	0.383001i	$0.874891\pi$
$282$	2.72639e8			0.723964
$283$	1.48929e8			0.390594			0.195297	−	0.980744i	$-0.437433\pi$
$283$	1.48929e8			0.390594			0.195297	+	0.980744i	$0.437433\pi$
$284$		−	7.66203e7i		−	0.198486i
$285$	1.60832e7			0.0411543
$286$	4.22474e8	+	1.14102e8i	1.06787	+	0.288410i
$287$	1.07565e7			0.0268587
$288$			2.38879e7i			0.0589256i
$289$	−9.04196e7			−0.220354
$290$	9.82109e7			0.236465
$291$			4.37869e8i			1.04164i
$292$		−	2.99310e7i		−	0.0703527i
$293$			1.03780e8i			0.241034i	0.992711	+	0.120517i	$0.0384552\pi$
$293$			1.03780e8i			0.241034i	−0.992711	+	0.120517i	$0.961545\pi$
$294$			2.54122e7i			0.0583212i
$295$	−2.22631e8			−0.504903
$296$	−2.06599e8			−0.463028
$297$			1.35921e8i			0.301051i
$298$	2.32215e8			0.508314
$299$	9.11227e7	+	2.46104e7i	0.197141	+	0.0532438i
$300$	8.11095e7			0.173439
$301$		−	2.43201e8i		−	0.514022i
$302$	1.19372e6			0.00249390
$303$	8.14373e6			0.0168180
$304$		−	1.38160e7i		−	0.0282050i
$305$		−	1.32227e8i		−	0.266852i
$306$			1.04313e8i			0.208119i
$307$			3.54422e8i			0.699094i	0.936919	+	0.349547i	$0.113665\pi$
$307$			3.54422e8i			0.699094i	−0.936919	+	0.349547i	$0.886335\pi$
$308$	1.51590e8			0.295626
$309$	−4.30681e7			−0.0830427
$310$			2.84769e8i			0.542908i
$311$	2.59043e8			0.488326			0.244163	−	0.969734i	$-0.421487\pi$
$311$	2.59043e8			0.488326			0.244163	+	0.969734i	$0.421487\pi$
$312$	−2.85522e7	+	1.05718e8i	−0.0532228	+	0.197063i
$313$	5.75038e8			1.05996			0.529982	−	0.848009i	$-0.322199\pi$
$313$	5.75038e8			1.05996			0.529982	+	0.848009i	$0.322199\pi$
$314$		−	1.32723e8i		−	0.241931i
$315$	−4.41577e7			−0.0796011
$316$	−1.78368e8			−0.317989
$317$		−	1.35492e8i		−	0.238895i	−0.992841	−	0.119447i	$-0.961888\pi$
$317$		−	1.35492e8i		−	0.238895i	0.992841	−	0.119447i	$-0.0381122\pi$
$318$		−	5.49398e6i		−	0.00958058i
$319$		−	4.80045e8i		−	0.827969i
$320$			4.62940e7i			0.0789768i
$321$	−1.79892e8			−0.303560
$322$	3.26962e7			0.0545760
$323$		−	6.03314e7i		−	0.0996173i
$324$	−3.40122e7			−0.0555556
$325$	3.58956e8	+	9.69467e7i	0.580029	+	0.156654i
$326$	−5.37692e8			−0.859551
$327$		−	1.97028e8i		−	0.311609i
$328$	1.60564e7			0.0251240
$329$	4.32941e8			0.670260
$330$			2.63411e8i			0.403493i
$331$		−	1.05430e9i		−	1.59795i	−0.601361	−	0.798977i	$-0.705375\pi$
$331$		−	1.05430e9i		−	1.59795i	0.601361	−	0.798977i	$-0.294625\pi$
$332$		−	5.51273e8i		−	0.826768i
$333$		−	2.94162e8i		−	0.436547i
$334$	−6.09704e8			−0.895378
$335$	−1.68798e8			−0.245307
$336$			3.79331e7i			0.0545545i
$337$	−5.29021e8			−0.752953			−0.376477	−	0.926426i	$-0.622865\pi$
$337$	−5.29021e8			−0.752953			−0.376477	+	0.926426i	$0.622865\pi$
$338$	−2.52719e8	+	4.33734e8i	−0.355984	+	0.610963i
$339$	4.57521e8			0.637841
$340$			2.02155e8i			0.278938i
$341$	1.39192e9			1.90096
$342$	1.96717e7			0.0265919
$343$			4.03536e7i			0.0539949i
$344$		−	3.63028e8i		−	0.480824i
$345$			5.68148e7i			0.0744894i
$346$			2.67391e8i			0.347041i
$347$	−1.71111e7			−0.0219849			−0.0109924	−	0.999940i	$-0.503499\pi$
$347$	−1.71111e7			−0.0219849			−0.0109924	+	0.999940i	$0.503499\pi$
$348$	1.20124e8			0.152792
$349$			1.06368e9i			1.33944i	0.742615	+	0.669718i	$0.233585\pi$
$349$			1.06368e9i			1.33944i	−0.742615	+	0.669718i	$0.766415\pi$
$350$	1.28799e8			0.160573
$351$	−1.50524e8	−	4.06534e7i	−0.185793	−	0.0501789i
$352$	2.26280e8			0.276533
$353$		−	1.06384e9i		−	1.28726i	−0.765339	−	0.643628i	$-0.777429\pi$
$353$		−	1.06384e9i		−	1.28726i	0.765339	−	0.643628i	$-0.222571\pi$
$354$	−2.72304e8			−0.326244
$355$	2.11421e8			0.250812
$356$		−	4.15602e8i		−	0.488205i
$357$			1.65645e8i			0.192681i
$358$			9.74451e8i			1.12246i
$359$		−	1.29862e9i		−	1.48133i	−0.671877	−	0.740663i	$-0.734512\pi$
$359$		−	1.29862e9i		−	1.48133i	0.671877	−	0.740663i	$-0.265488\pi$
$360$	−6.59146e7			−0.0744600
$361$	8.82494e8			0.987272
$362$			6.75186e8i			0.748073i
$363$	7.61373e8			0.835458
$364$	−4.53398e7	+	1.67876e8i	−0.0492747	+	0.182445i
$365$	8.25896e7			0.0888997
$366$		−	1.61730e8i		−	0.172427i
$367$	2.28160e8			0.240940			0.120470	−	0.992717i	$-0.461560\pi$
$367$	2.28160e8			0.240940			0.120470	+	0.992717i	$0.461560\pi$
$368$	4.88060e7			0.0510512
$369$			2.28615e7i			0.0236871i
$370$		−	5.70076e8i		−	0.585095i
$371$		−	8.72423e6i		−	0.00886989i
$372$			3.48306e8i			0.350801i
$373$	1.36139e9			1.35831			0.679157	−	0.733993i	$-0.262346\pi$
$373$	1.36139e9			1.35831			0.679157	+	0.733993i	$0.262346\pi$
$374$	9.88112e8			0.976688
$375$			5.96318e8i			0.583941i
$376$	6.46256e8			0.626971
$377$	5.31617e8	+	1.43579e8i	0.510980	+	0.138005i
$378$	−5.40102e7			−0.0514344
$379$			5.39775e8i			0.509302i	0.967033	+	0.254651i	$0.0819606\pi$
$379$			5.39775e8i			0.509302i	−0.967033	+	0.254651i	$0.918039\pi$
$380$	3.81231e7			0.0356406
$381$	−4.01203e8			−0.371644
$382$			7.91856e8i			0.726816i
$383$		−	1.58641e9i		−	1.44285i	−0.692495	−	0.721423i	$-0.743488\pi$
$383$		−	1.58641e9i		−	1.44285i	0.692495	−	0.721423i	$-0.256512\pi$
$384$			5.66231e7i			0.0510310i
$385$			4.18287e8i			0.373562i
$386$	−3.04075e8			−0.269107
$387$	5.16890e8			0.453325
$388$			1.03791e9i			0.902089i
$389$	−8.66697e8			−0.746524			−0.373262	−	0.927726i	$-0.621761\pi$
$389$	−8.66697e8			−0.746524			−0.373262	+	0.927726i	$0.621761\pi$
$390$	−2.91710e8	−	7.87850e7i	−0.249015	−	0.0672539i
$391$	2.13124e8			0.180308
$392$			6.02363e7i			0.0505076i
$393$	2.14368e8			0.178150
$394$	6.73850e8			0.555043
$395$		−	4.92178e8i		−	0.401820i
$396$			3.22184e8i			0.260718i
$397$		−	7.88951e8i		−	0.632824i	−0.948622	−	0.316412i	$-0.897522\pi$
$397$		−	7.88951e8i		−	0.632824i	0.948622	−	0.316412i	$-0.102478\pi$
$398$		−	5.55737e8i		−	0.441854i
$399$	3.12379e7			0.0246193
$400$	1.92259e8			0.150203
$401$		−	7.06048e8i		−	0.546800i	−0.961900	−	0.273400i	$-0.911852\pi$
$401$		−	7.06048e8i		−	0.546800i	0.961900	−	0.273400i	$-0.0881483\pi$
$402$	−2.06460e8			−0.158506
$403$	−4.16316e8	+	1.54146e9i	−0.316851	+	1.17318i
$404$	1.93036e7			0.0145648
$405$		−	9.38511e7i		−	0.0702016i
$406$	1.90752e8			0.141458
$407$	−2.78647e9			−2.04868
$408$			2.47260e8i			0.180237i
$409$		−	1.17364e9i		−	0.848212i	−0.905612	−	0.424106i	$-0.860588\pi$
$409$		−	1.17364e9i		−	0.848212i	0.905612	−	0.424106i	$-0.139412\pi$
$410$			4.43049e7i			0.0317474i
$411$			1.37197e8i			0.0974763i
$412$	−1.02087e8			−0.0719171
$413$	−4.32409e8			−0.302044
$414$			6.94913e7i			0.0481315i
$415$	1.52115e9			1.04473
$416$	−6.76792e7	+	2.50590e8i	−0.0460923	+	0.170662i
$417$	−1.16022e9			−0.783544
$418$		−	1.86342e8i		−	0.124794i
$419$	−1.19621e9			−0.794432			−0.397216	−	0.917725i	$-0.630024\pi$
$419$	−1.19621e9			−0.794432			−0.397216	+	0.917725i	$0.630024\pi$
$420$	−1.04670e8			−0.0689366
$421$		−	9.87704e8i		−	0.645118i	−0.946549	−	0.322559i	$-0.895457\pi$
$421$		−	9.87704e8i		−	0.645118i	0.946549	−	0.322559i	$-0.104543\pi$
$422$			1.10208e9i			0.713871i
$423$			9.20158e8i			0.591114i
$424$		−	1.30228e7i		−	0.00829703i
$425$	8.39552e8			0.530501
$426$	2.58594e8			0.162063
$427$		−	2.56821e8i		−	0.159637i
$428$	−4.26411e8			−0.262890
$429$	−3.85093e8	+	1.42585e9i	−0.235486	+	0.871913i
$430$	1.00172e9			0.607583
$431$		−	1.31251e9i		−	0.789648i	−0.918757	−	0.394824i	$-0.870805\pi$
$431$		−	1.31251e9i		−	0.789648i	0.918757	−	0.394824i	$-0.129195\pi$
$432$	−8.06216e7			−0.0481125
$433$	4.82709e8			0.285744			0.142872	−	0.989741i	$-0.454366\pi$
$433$	4.82709e8			0.285744			0.142872	+	0.989741i	$0.454366\pi$
$434$			5.53097e8i			0.324779i
$435$			3.31462e8i			0.193073i
$436$		−	4.67029e8i		−	0.269862i
$437$		−	4.01917e7i		−	0.0230384i
$438$	1.01017e8			0.0574428
$439$	9.93187e7			0.0560280			0.0280140	−	0.999608i	$-0.491082\pi$
$439$	9.93187e7			0.0560280			0.0280140	+	0.999608i	$0.491082\pi$
$440$			6.24382e8i			0.349435i
$441$	−8.57661e7			−0.0476190
$442$	−2.95539e8	+	1.09427e9i	−0.162794	+	0.602762i
$443$	−3.46991e9			−1.89629			−0.948145	−	0.317839i	$-0.897043\pi$
$443$	−3.46991e9			−1.89629			−0.948145	+	0.317839i	$0.897043\pi$
$444$		−	6.97272e8i		−	0.378061i
$445$	1.14678e9			0.616910
$446$	1.02980e8			0.0549641
$447$			7.83724e8i			0.415037i
$448$			8.99154e7i			0.0472456i
$449$			1.08211e8i			0.0564168i	0.999602	+	0.0282084i	$0.00898020\pi$
$449$			1.08211e8i			0.0564168i	−0.999602	+	0.0282084i	$0.991020\pi$
$450$			2.73744e8i			0.141612i
$451$	2.16558e8			0.111162
$452$	1.08449e9			0.552386
$453$			4.02881e6i			0.00203626i
$454$	−1.27509e8			−0.0639508
$455$	−4.63225e8	−	1.25108e8i	−0.230543	−	0.0622650i
$456$	4.66291e7			0.0230293
$457$		−	3.09717e8i		−	0.151795i	−0.997116	−	0.0758977i	$-0.975818\pi$
$457$		−	3.09717e8i		−	0.151795i	0.997116	−	0.0758977i	$-0.0241822\pi$
$458$	1.59919e9			0.777803
$459$	−3.52056e8			−0.169929
$460$			1.34672e8i			0.0645097i
$461$			1.24186e8i			0.0590363i	0.999564	+	0.0295181i	$0.00939728\pi$
$461$			1.24186e8i			0.0590363i	−0.999564	+	0.0295181i	$0.990603\pi$
$462$			5.11616e8i			0.241378i
$463$		−	8.06172e8i		−	0.377480i	−0.982027	−	0.188740i	$-0.939560\pi$
$463$		−	8.06172e8i		−	0.377480i	0.982027	−	0.188740i	$-0.0604404\pi$
$464$	2.84738e8			0.132322
$465$	−9.61094e8			−0.443282
$466$		−	1.81971e9i		−	0.833013i
$467$	2.46403e9			1.11953			0.559767	−	0.828650i	$-0.310891\pi$
$467$	2.46403e9			1.11953			0.559767	+	0.828650i	$0.310891\pi$
$468$	−3.56797e8	−	9.63635e7i	−0.160902	−	0.0434562i
$469$	−3.27851e8			−0.146748
$470$			1.78324e9i			0.792258i
$471$	4.47939e8			0.197536
$472$	−6.45462e8			−0.282536
$473$		−	4.89629e9i		−	2.12742i
$474$		−	6.01993e8i		−	0.259637i
$475$		−	1.58326e8i		−	0.0677835i
$476$			3.92640e8i			0.166867i
$477$	1.85422e7			0.00782251
$478$	−1.52480e9			−0.638582
$479$		−	3.35817e9i		−	1.39614i	−0.716031	−	0.698069i	$-0.754043\pi$
$479$		−	3.35817e9i		−	1.39614i	0.716031	−	0.698069i	$-0.245957\pi$
$480$	−1.56242e8			−0.0644843
$481$	8.33419e8	−	3.08583e9i	0.341473	−	1.26434i
$482$	−7.63261e8			−0.310462
$483$			1.10350e8i			0.0445611i
$484$	1.80474e9			0.723528
$485$	−2.86394e9			−1.13990
$486$		−	1.14791e8i		−	0.0453609i
$487$			8.19629e8i			0.321563i	0.986990	+	0.160782i	$0.0514014\pi$
$487$			8.19629e8i			0.321563i	−0.986990	+	0.160782i	$0.948599\pi$
$488$		−	3.83359e8i		−	0.149326i
$489$		−	1.81471e9i		−	0.701820i
$490$	−1.66212e8			−0.0638229
$491$	−2.04832e9			−0.780932			−0.390466	−	0.920617i	$-0.627686\pi$
$491$	−2.04832e9			−0.780932			−0.390466	+	0.920617i	$0.627686\pi$
$492$			5.41903e7i			0.0205137i
$493$	1.24338e9			0.467349
$494$	2.06361e8	+	5.57338e7i	0.0770163	+	0.0208005i
$495$	−8.89013e8			−0.329450
$496$			8.25615e8i			0.303803i
$497$	4.10637e8			0.150041
$498$	1.86055e9			0.675053
$499$		−	1.98085e9i		−	0.713672i	−0.934167	−	0.356836i	$-0.883855\pi$
$499$		−	1.98085e9i		−	0.713672i	0.934167	−	0.356836i	$-0.116145\pi$
$500$			1.41350e9i			0.505708i
$501$		−	2.05775e9i		−	0.731073i
$502$			2.28707e9i			0.806894i
$503$	−1.74006e9			−0.609646			−0.304823	−	0.952409i	$-0.598597\pi$
$503$	−1.74006e9			−0.609646			−0.304823	+	0.952409i	$0.598597\pi$
$504$	−1.28024e8			−0.0445435
$505$			5.32652e7i			0.0184045i
$506$	6.58263e8			0.225877
$507$	−1.46385e9	−	8.52928e8i	−0.498849	−	0.290659i
$508$	−9.51000e8			−0.321853
$509$		−	4.53438e9i		−	1.52407i	−0.647534	−	0.762036i	$-0.724200\pi$
$509$		−	4.53438e9i		−	1.52407i	0.647534	−	0.762036i	$-0.275800\pi$
$510$	−6.82273e8			−0.227752
$511$	1.60411e8			0.0531817
$512$			1.34218e8i			0.0441942i
$513$			6.63919e7i			0.0217122i
$514$		−	1.98689e9i		−	0.645360i
$515$		−	2.81694e8i		−	0.0908765i
$516$	1.22522e9			0.392591
$517$	8.71628e9			2.77405
$518$		−	1.10724e9i		−	0.350016i
$519$	−9.02446e8			−0.283358
$520$	−6.91461e8	−	1.86750e8i	−0.215653	−	0.0582435i
$521$	−1.08406e9			−0.335831			−0.167915	−	0.985801i	$-0.553703\pi$
$521$	−1.08406e9			−0.335831			−0.167915	+	0.985801i	$0.553703\pi$
$522$			4.05418e8i			0.124754i
$523$	−6.08641e9			−1.86040			−0.930198	−	0.367057i	$-0.880365\pi$
$523$	−6.08641e9			−1.86040			−0.930198	+	0.367057i	$0.880365\pi$
$524$	5.08132e8			0.154283
$525$			4.34696e8i			0.131108i
$526$			1.71573e9i			0.514042i
$527$			3.60527e9i			1.07300i
$528$			7.63695e8i			0.225788i
$529$	−3.26285e9			−0.958300
$530$	3.59342e7			0.0104844
$531$		−	9.19027e8i		−	0.266377i
$532$	7.40453e7			0.0213210
$533$	−6.47713e7	+	2.39823e8i	−0.0185284	+	0.0686034i
$534$	1.40266e9			0.398618
$535$		−	1.17661e9i		−	0.332196i
$536$	−4.89387e8			−0.137270
$537$	−3.28877e9			−0.916481
$538$			1.24631e9i			0.345056i
$539$			8.12427e8i			0.223472i
$540$		−	2.22462e8i		−	0.0607964i
$541$		−	4.98049e9i		−	1.35233i	−0.736752	−	0.676163i	$-0.763641\pi$
$541$		−	4.98049e9i		−	1.35233i	0.736752	−	0.676163i	$-0.236359\pi$
$542$	−2.94730e9			−0.795109
$543$	−2.27875e9			−0.610799
$544$			5.86098e8i			0.156090i
$545$	1.28869e9			0.341005
$546$	−5.66580e8	−	1.53022e8i	−0.148966	−	0.0402327i
$547$	−2.82586e9			−0.738236			−0.369118	−	0.929383i	$-0.620340\pi$
$547$	−2.82586e9			−0.738236			−0.369118	+	0.929383i	$0.620340\pi$
$548$			3.25208e8i			0.0844169i
$549$	5.45838e8			0.140786
$550$	2.59307e9			0.664576
$551$		−	2.34482e8i		−	0.0597143i
$552$			1.64720e8i			0.0416831i
$553$		−	9.55942e8i		−	0.240377i
$554$			2.75369e8i			0.0688068i
$555$	1.92401e9			0.477728
$556$	−2.75015e9			−0.678569
$557$		−	3.63680e8i		−	0.0891716i	−0.999006	−	0.0445858i	$-0.985803\pi$
$557$		−	3.63680e8i		−	0.0891716i	0.999006	−	0.0445858i	$-0.0141968\pi$
$558$	−1.17553e9			−0.286428
$559$	5.42230e9	+	1.46445e9i	1.31293	+	0.354596i
$560$	−2.48107e8			−0.0597008
$561$			3.33488e9i			0.797462i
$562$	−2.27925e9			−0.541645
$563$	−7.13702e9			−1.68553			−0.842767	−	0.538278i	$-0.819075\pi$
$563$	−7.13702e9			−1.68553			−0.842767	+	0.538278i	$0.819075\pi$
$564$			2.18111e9i			0.511920i
$565$			2.99248e9i			0.698011i
$566$			1.19143e9i			0.276192i
$567$		−	1.82284e8i		−	0.0419961i
$568$	6.12963e8			0.140351
$569$	−7.72495e9			−1.75794			−0.878968	−	0.476881i	$-0.841767\pi$
$569$	−7.72495e9			−1.75794			−0.878968	+	0.476881i	$0.841767\pi$
$570$			1.28665e8i			0.0291005i
$571$	−3.54635e9			−0.797177			−0.398588	−	0.917130i	$-0.630500\pi$
$571$	−3.54635e9			−0.797177			−0.398588	+	0.917130i	$0.630500\pi$
$572$	−9.12812e8	+	3.37979e9i	−0.203937	+	0.755098i
$573$	−2.67251e9			−0.593443
$574$			8.60521e7i			0.0189920i
$575$	5.59295e8			0.122688
$576$	−1.91103e8			−0.0416667
$577$			4.43244e9i			0.960566i	0.877114	+	0.480283i	$0.159466\pi$
$577$			4.43244e9i			0.960566i	−0.877114	+	0.480283i	$0.840534\pi$
$578$		−	7.23357e8i		−	0.155813i
$579$		−	1.02625e9i		−	0.219725i
$580$			7.85687e8i			0.167206i
$581$	2.95448e9			0.624978
$582$	−3.50295e9			−0.736552
$583$		−	1.75642e8i		−	0.0367104i
$584$	2.39448e8			0.0497469
$585$	2.65899e8	−	9.84522e8i	0.0549125	−	0.203320i
$586$	−8.30241e8			−0.170437
$587$			4.64786e9i			0.948462i	0.880401	+	0.474231i	$0.157274\pi$
$587$			4.64786e9i			0.948462i	−0.880401	+	0.474231i	$0.842726\pi$
$588$	−2.03297e8			−0.0412393
$589$	6.79894e8			0.137100
$590$		−	1.78105e9i		−	0.357020i
$591$			2.27424e9i			0.453191i
$592$		−	1.65279e9i		−	0.327410i
$593$		−	3.48179e9i		−	0.685664i	−0.939397	−	0.342832i	$-0.888614\pi$
$593$		−	3.48179e9i		−	0.685664i	0.939397	−	0.342832i	$-0.111386\pi$
$594$	−1.08737e9			−0.212875
$595$	−1.08342e9			−0.210858
$596$			1.85772e9i			0.359433i
$597$	1.87561e9			0.360772
$598$	−1.96883e8	+	7.28981e8i	−0.0376491	+	0.139400i
$599$	3.40784e8			0.0647867			0.0323933	−	0.999475i	$-0.489687\pi$
$599$	3.40784e8			0.0647867			0.0323933	+	0.999475i	$0.489687\pi$
$600$			6.48876e8i			0.122640i
$601$	−2.51584e9			−0.472741			−0.236370	−	0.971663i	$-0.575958\pi$
$601$	−2.51584e9			−0.472741			−0.236370	+	0.971663i	$0.575958\pi$
$602$	1.94560e9			0.363469
$603$		−	6.96802e8i		−	0.129419i
$604$			9.54978e6i			0.00176345i
$605$			4.97987e9i			0.914270i
$606$			6.51498e7i			0.0118921i
$607$	9.40908e9			1.70760			0.853801	−	0.520600i	$-0.174292\pi$
$607$	9.40908e9			1.70760			0.853801	+	0.520600i	$0.174292\pi$
$608$	1.10528e8			0.0199439
$609$			6.43788e8i			0.115500i
$610$	1.05782e9			0.188693
$611$	−2.60699e9	+	9.65269e9i	−0.462376	+	1.71200i
$612$	−8.34502e8			−0.147163
$613$		−	7.23342e9i		−	1.26833i	−0.773198	−	0.634165i	$-0.781344\pi$
$613$		−	7.23342e9i		−	1.26833i	0.773198	−	0.634165i	$-0.218656\pi$
$614$	−2.83537e9			−0.494334
$615$	−1.49529e8			−0.0259217
$616$			1.21272e9i			0.209039i
$617$			7.55189e9i			1.29437i	0.762334	+	0.647184i	$0.224053\pi$
$617$			7.55189e9i			1.29437i	−0.762334	+	0.647184i	$0.775947\pi$
$618$		−	3.44545e8i		−	0.0587201i
$619$		−	6.37601e9i		−	1.08052i	−0.841499	−	0.540259i	$-0.818326\pi$
$619$		−	6.37601e9i		−	1.08052i	0.841499	−	0.540259i	$-0.181674\pi$
$620$	−2.27815e9			−0.383894
$621$	−2.34533e8			−0.0392992
$622$			2.07234e9i			0.345299i
$623$	2.22737e9			0.369049
$624$	−8.45740e8	−	2.28417e8i	−0.139345	−	0.0376342i
$625$	−2.33249e8			−0.0382156
$626$			4.60030e9i			0.749508i
$627$	6.28903e8			0.101894
$628$	1.06178e9			0.171071
$629$		−	7.21736e9i		−	1.15638i
$630$		−	3.53261e8i		−	0.0562865i
$631$		−	5.79105e8i		−	0.0917602i	−0.998947	−	0.0458801i	$-0.985391\pi$
$631$		−	5.79105e8i		−	0.0917602i	0.998947	−	0.0458801i	$-0.0146092\pi$
$632$		−	1.42695e9i		−	0.224852i
$633$	−3.71952e9			−0.582873
$634$	1.08394e9			0.168924
$635$		−	2.62413e9i		−	0.406703i
$636$	4.39518e7			0.00677449
$637$	−8.99708e8	−	2.42993e8i	−0.137916	−	0.0372482i
$638$	3.84036e9			0.585463
$639$			8.72753e8i			0.132324i
$640$	−3.70352e8			−0.0558450
$641$	−2.68485e9			−0.402640			−0.201320	−	0.979526i	$-0.564523\pi$
$641$	−2.68485e9			−0.402640			−0.201320	+	0.979526i	$0.564523\pi$
$642$		−	1.43914e9i		−	0.214649i
$643$			5.98355e9i			0.887607i	0.896124	+	0.443803i	$0.146371\pi$
$643$			5.98355e9i			0.887607i	−0.896124	+	0.443803i	$0.853629\pi$
$644$			2.61570e8i			0.0385911i
$645$			3.38079e9i			0.496089i
$646$	4.82651e8			0.0704401
$647$	−1.14784e10			−1.66615			−0.833077	−	0.553157i	$-0.813423\pi$
$647$	−1.14784e10			−1.66615			−0.833077	+	0.553157i	$0.813423\pi$
$648$		−	2.72098e8i		−	0.0392837i
$649$	−8.70557e9			−1.25009
$650$	−7.75573e8	+	2.87165e9i	−0.110771	+	0.410142i
$651$	−1.86670e9			−0.265181
$652$		−	4.30153e9i		−	0.607794i
$653$	−1.27744e9			−0.179533			−0.0897666	−	0.995963i	$-0.528612\pi$
$653$	−1.27744e9			−0.179533			−0.0897666	+	0.995963i	$0.528612\pi$
$654$	1.57622e9			0.220341
$655$			1.40211e9i			0.194956i
$656$			1.28451e8i			0.0177654i
$657$			3.40933e8i			0.0469018i
$658$			3.46353e9i			0.473945i
$659$	8.85257e9			1.20495			0.602477	−	0.798136i	$-0.294181\pi$
$659$	8.85257e9			1.20495			0.602477	+	0.798136i	$0.294181\pi$
$660$	−2.10729e9			−0.285312
$661$		−	5.42787e9i		−	0.731012i	−0.930809	−	0.365506i	$-0.880896\pi$
$661$		−	5.42787e9i		−	0.731012i	0.930809	−	0.365506i	$-0.119104\pi$
$662$	8.43437e9			1.12992
$663$	−3.69315e9	−	9.97445e8i	−0.492153	−	0.132920i
$664$	4.41019e9			0.584613
$665$			2.04316e8i			0.0269418i
$666$	2.35329e9			0.308685
$667$	8.28320e8			0.108083
$668$		−	4.87763e9i		−	0.633128i
$669$			3.47556e8i			0.0448780i
$670$		−	1.35038e9i		−	0.173458i
$671$		−	5.17049e9i		−	0.660698i
$672$	−3.03464e8			−0.0385758
$673$	9.78322e9			1.23717			0.618584	−	0.785718i	$-0.287706\pi$
$673$	9.78322e9			1.23717			0.618584	+	0.785718i	$0.287706\pi$
$674$		−	4.23217e9i		−	0.532418i
$675$	−9.23887e8			−0.115626
$676$	−3.46987e9	−	2.02176e9i	−0.432016	−	0.251718i
$677$	−1.25006e10			−1.54836			−0.774180	−	0.632965i	$-0.781838\pi$
$677$	−1.25006e10			−1.54836			−0.774180	+	0.632965i	$0.781838\pi$
$678$			3.66017e9i			0.451021i
$679$	−5.56256e9			−0.681915
$680$	−1.61724e9			−0.197239
$681$		−	4.30343e8i		−	0.0522156i
$682$			1.11353e10i			1.34418i
$683$			1.23048e9i			0.147776i	0.997267	+	0.0738880i	$0.0235407\pi$
$683$			1.23048e9i			0.147776i	−0.997267	+	0.0738880i	$0.976459\pi$
$684$			1.57373e8i			0.0188033i
$685$	−8.97358e8			−0.106672
$686$	−3.22829e8			−0.0381802
$687$			5.39725e9i			0.635074i
$688$	2.90423e9			0.339994
$689$	1.94512e8	+	5.25337e7i	0.0226558	+	0.00611887i
$690$	−4.54518e8			−0.0526720
$691$			1.22932e10i			1.41740i	0.705510	+	0.708700i	$0.250718\pi$
$691$			1.22932e10i			1.41740i	−0.705510	+	0.708700i	$0.749282\pi$
$692$	−2.13913e9			−0.245395
$693$	−1.72670e9			−0.197084
$694$		−	1.36889e8i		−	0.0155456i
$695$		−	7.58857e9i		−	0.857459i
$696$			9.60990e8i			0.108041i
$697$			5.60916e8i			0.0627455i
$698$	−8.50944e9			−0.947124
$699$	6.14152e9			0.680152
$700$			1.03039e9i			0.113543i
$701$	−1.77591e10			−1.94718			−0.973592	−	0.228294i	$-0.926685\pi$
$701$	−1.77591e10			−1.94718			−0.973592	+	0.228294i	$0.926685\pi$
$702$	3.25227e8	−	1.20419e9i	0.0354819	−	0.131376i
$703$	−1.36107e9			−0.147754
$704$			1.81024e9i			0.195538i
$705$	−6.01843e9			−0.646876
$706$	8.51072e9			0.910227
$707$			1.03455e8i			0.0110100i
$708$		−	2.17843e9i		−	0.230690i
$709$			8.81919e8i			0.0929324i	0.998920	+	0.0464662i	$0.0147960\pi$
$709$			8.81919e8i			0.0929324i	−0.998920	+	0.0464662i	$0.985204\pi$
$710$			1.69137e9i			0.177351i
$711$	2.03173e9			0.211993
$712$	3.32482e9			0.345213
$713$			2.40177e9i			0.248152i
$714$	−1.32516e9			−0.136246
$715$	−9.32597e9	−	2.51875e9i	−0.954164	−	0.257700i
$716$	−7.79560e9			−0.793696
$717$		−	5.14621e9i		−	0.521400i
$718$	1.03889e10			1.04746
$719$	6.89262e8			0.0691565			0.0345783	−	0.999402i	$-0.488991\pi$
$719$	6.89262e8			0.0691565			0.0345783	+	0.999402i	$0.488991\pi$
$720$		−	5.27317e8i		−	0.0526512i
$721$		−	5.47125e8i		−	0.0543642i
$722$			7.05995e9i			0.698106i
$723$		−	2.57601e9i		−	0.253491i
$724$	−5.40149e9			−0.528967
$725$	3.26297e9			0.318002
$726$			6.09099e9i			0.590758i
$727$	2.29272e9			0.221299			0.110650	−	0.993859i	$-0.464707\pi$
$727$	2.29272e9			0.221299			0.110650	+	0.993859i	$0.464707\pi$
$728$	−1.34300e9	−	3.62718e8i	−0.129008	−	0.0348425i
$729$	3.87420e8			0.0370370
$730$			6.60717e8i			0.0628616i
$731$	1.26821e10			1.20083
$732$	1.29384e9			0.121924
$733$			1.17209e9i			0.109926i	0.998488	+	0.0549628i	$0.0175040\pi$
$733$			1.17209e9i			0.109926i	−0.998488	+	0.0549628i	$0.982496\pi$
$734$			1.82528e9i			0.170370i
$735$		−	5.60966e8i		−	0.0521112i
$736$			3.90448e8i			0.0360986i
$737$	−6.60052e9			−0.607354
$738$	−1.82892e8			−0.0167493
$739$		−	7.90683e9i		−	0.720687i	−0.932820	−	0.360344i	$-0.882659\pi$
$739$		−	7.90683e9i		−	0.720687i	0.932820	−	0.360344i	$-0.117341\pi$
$740$	4.56061e9			0.413725
$741$	−1.88102e8	+	6.96467e8i	−0.0169836	+	0.0628835i
$742$	6.97938e7			0.00627196
$743$			5.80762e9i			0.519442i	0.965684	+	0.259721i	$0.0836306\pi$
$743$			5.80762e9i			0.519442i	−0.965684	+	0.259721i	$0.916369\pi$
$744$	−2.78645e9			−0.248054
$745$	−5.12606e9			−0.454189
$746$			1.08911e10i			0.960473i
$747$			6.27935e9i			0.551179i
$748$			7.90490e9i			0.690623i
$749$		−	2.28529e9i		−	0.198726i
$750$	−4.77055e9			−0.412908
$751$	−1.11972e10			−0.964648			−0.482324	−	0.875993i	$-0.660207\pi$
$751$	−1.11972e10			−0.964648			−0.482324	+	0.875993i	$0.660207\pi$
$752$			5.17005e9i			0.443335i
$753$	−7.71886e9			−0.658826
$754$	−1.14863e9	+	4.25293e9i	−0.0975845	+	0.361317i
$755$	−2.63510e7			−0.00222835
$756$		−	4.32081e8i		−	0.0363696i
$757$	6.44640e9			0.540109			0.270055	−	0.962845i	$-0.412958\pi$
$757$	6.44640e9			0.540109			0.270055	+	0.962845i	$0.412958\pi$
$758$	−4.31820e9			−0.360131
$759$			2.22164e9i			0.184428i
$760$			3.04985e8i			0.0252017i
$761$			4.88082e9i			0.401464i	0.979646	+	0.200732i	$0.0643321\pi$
$761$			4.88082e9i			0.401464i	−0.979646	+	0.200732i	$0.935668\pi$
$762$		−	3.20962e9i		−	0.262792i
$763$	2.50298e9			0.203996
$764$	−6.33485e9			−0.513937
$765$		−	2.30267e9i		−	0.185959i
$766$	1.26913e10			1.02025
$767$	2.60379e9	−	9.64082e9i	0.208364	−	0.771490i
$768$	−4.52985e8			−0.0360844
$769$		−	1.54643e10i		−	1.22628i	−0.789976	−	0.613138i	$-0.789907\pi$
$769$		−	1.54643e10i		−	1.22628i	0.789976	−	0.613138i	$-0.210093\pi$
$770$	−3.34630e9			−0.264148
$771$	6.70574e9			0.526934
$772$		−	2.43260e9i		−	0.190287i
$773$			3.65255e9i			0.284425i	0.989836	+	0.142212i	$0.0454216\pi$
$773$			3.65255e9i			0.284425i	−0.989836	+	0.142212i	$0.954578\pi$
$774$			4.13512e9i			0.320549i
$775$			9.46118e9i			0.730112i
$776$	−8.30329e9			−0.637873
$777$	3.73694e9			0.285787
$778$		−	6.93358e9i		−	0.527872i
$779$	1.05779e8			0.00801715
$780$	6.30280e8	−	2.33368e9i	0.0475557	−	0.176080i
$781$	8.26723e9			0.620986
$782$			1.70500e9i			0.127497i
$783$	−1.36828e9			−0.101862
$784$	−4.81890e8			−0.0357143
$785$			2.92981e9i			0.216170i
$786$			1.71495e9i			0.125971i
$787$		−	9.64709e9i		−	0.705480i	−0.935721	−	0.352740i	$-0.885250\pi$
$787$		−	9.64709e9i		−	0.705480i	0.935721	−	0.352740i	$-0.114750\pi$
$788$			5.39080e9i			0.392475i
$789$	−5.79058e9			−0.419713
$790$	3.93742e9			0.284130
$791$			5.81221e9i			0.417565i
$792$	−2.57747e9			−0.184355
$793$	5.72597e9	+	1.54647e9i	0.407749	+	0.110125i
$794$	6.31161e9			0.447474
$795$			1.21278e8i			0.00856044i
$796$	4.44590e9			0.312438
$797$	1.91810e10			1.34205			0.671023	−	0.741436i	$-0.265855\pi$
$797$	1.91810e10			1.34205			0.671023	+	0.741436i	$0.265855\pi$
$798$			2.49903e8i			0.0174085i
$799$			2.25764e10i			1.56582i
$800$			1.53808e9i			0.106209i
$801$			4.73397e9i			0.325470i
$802$	5.64838e9			0.386646
$803$	3.22951e9			0.220106
$804$		−	1.65168e9i		−	0.112080i
$805$	−7.21758e8			−0.0487648
$806$	−1.23316e10	−	3.33053e9i	−0.829561	−	0.224047i
$807$	−4.20631e9			−0.281737
$808$			1.54429e8i			0.0102989i
$809$	−1.53607e9			−0.101998			−0.0509988	−	0.998699i	$-0.516240\pi$
$809$	−1.53607e9			−0.101998			−0.0509988	+	0.998699i	$0.516240\pi$
$810$	7.50809e8			0.0496400
$811$		−	1.89255e9i		−	0.124587i	−0.998058	−	0.0622937i	$-0.980158\pi$
$811$		−	1.89255e9i		−	0.124587i	0.998058	−	0.0622937i	$-0.0198416\pi$
$812$			1.52602e9i			0.100026i
$813$		−	9.94714e9i		−	0.649204i
$814$		−	2.22918e10i		−	1.44864i
$815$	1.18694e10			0.768026
$816$	−1.97808e9			−0.127447
$817$		−	2.39163e9i		−	0.153432i
$818$	9.38915e9			0.599777
$819$	5.16448e8	−	1.91221e9i	0.0328498	−	0.121630i
$820$	−3.54439e8			−0.0224488
$821$			2.33753e10i			1.47420i	0.675786	+	0.737098i	$0.263804\pi$
$821$			2.33753e10i			1.47420i	−0.675786	+	0.737098i	$0.736196\pi$
$822$	−1.09758e9			−0.0689261
$823$	3.73519e8			0.0233568			0.0116784	−	0.999932i	$-0.496283\pi$
$823$	3.73519e8			0.0233568			0.0116784	+	0.999932i	$0.496283\pi$
$824$		−	8.16700e8i		−	0.0508531i
$825$			8.75161e9i			0.542624i
$826$		−	3.45927e9i		−	0.213577i
$827$		−	2.37062e10i		−	1.45744i	−0.684810	−	0.728722i	$-0.740115\pi$
$827$		−	2.37062e10i		−	1.45744i	0.684810	−	0.728722i	$-0.259885\pi$
$828$	−5.55931e8			−0.0340341
$829$	1.64534e10			1.00303			0.501516	−	0.865148i	$-0.332776\pi$
$829$	1.64534e10			1.00303			0.501516	+	0.865148i	$0.332776\pi$
$830$			1.21692e10i			0.738734i
$831$	−9.29371e8			−0.0561805
$832$	−2.00472e9	−	5.41433e8i	−0.120676	−	0.0325922i
$833$	−2.10430e9			−0.126139
$834$		−	9.28174e9i		−	0.554049i
$835$	1.34590e10			0.800038
$836$	1.49073e9			0.0882425
$837$		−	3.96743e9i		−	0.233867i
$838$		−	9.56964e9i		−	0.561748i
$839$			1.13600e10i			0.664069i	0.943267	+	0.332034i	$0.107735\pi$
$839$			1.13600e10i			0.664069i	−0.943267	+	0.332034i	$0.892265\pi$
$840$		−	8.37360e8i		−	0.0487455i
$841$	−1.24174e10			−0.719854
$842$	7.90163e9			0.456168
$843$		−	7.69247e9i		−	0.442251i
$844$	−8.81664e9			−0.504783
$845$	5.57870e9	−	9.57453e9i	0.318079	−	0.545908i
$846$	−7.36126e9			−0.417981
$847$			9.67226e9i			0.546935i
$848$	1.04182e8			0.00586688
$849$	−4.02107e9			−0.225510
$850$			6.71642e9i			0.375121i
$851$		−	4.80808e9i		−	0.267435i
$852$			2.06875e9i			0.114596i
$853$		−	6.63311e9i		−	0.365928i	−0.983120	−	0.182964i	$-0.941431\pi$
$853$		−	6.63311e9i		−	0.365928i	0.983120	−	0.182964i	$-0.0585692\pi$
$854$	2.05457e9			0.112880
$855$	−4.34246e8			−0.0237604
$856$		−	3.41128e9i		−	0.185892i
$857$	−1.63219e9			−0.0885807			−0.0442904	−	0.999019i	$-0.514103\pi$
$857$	−1.63219e9			−0.0885807			−0.0442904	+	0.999019i	$0.514103\pi$
$858$	−1.14068e10	−	3.08074e9i	−0.616535	−	0.166514i
$859$	8.58358e9			0.462054			0.231027	−	0.972947i	$-0.425791\pi$
$859$	8.58358e9			0.462054			0.231027	+	0.972947i	$0.425791\pi$
$860$			8.01373e9i			0.429626i
$861$	−2.90426e8			−0.0155069
$862$	1.05001e10			0.558366
$863$		−	1.70558e10i		−	0.903305i	−0.892194	−	0.451653i	$-0.850835\pi$
$863$		−	1.70558e10i		−	0.903305i	0.892194	−	0.451653i	$-0.149165\pi$
$864$		−	6.44973e8i		−	0.0340207i
$865$		−	5.90258e9i		−	0.310088i
$866$			3.86167e9i			0.202052i
$867$	2.44133e9			0.127221
$868$	−4.42478e9			−0.229653
$869$		−	1.92457e10i		−	0.994866i
$870$	−2.65169e9			−0.136523
$871$	1.97418e9	−	7.30963e9i	0.101233	−	0.374828i
$872$	3.73623e9			0.190821
$873$		−	1.18225e10i		−	0.601392i
$874$	3.21534e8			0.0162906
$875$	−7.57545e9			−0.382279
$876$			8.08136e8i			0.0406182i
$877$			6.11838e9i			0.306294i	0.988203	+	0.153147i	$0.0489407\pi$
$877$			6.11838e9i			0.306294i	−0.988203	+	0.153147i	$0.951059\pi$
$878$			7.94549e8i			0.0396178i
$879$		−	2.80206e9i		−	0.139161i
$880$	−4.99506e9			−0.247088
$881$	4.04834e9			0.199462			0.0997312	−	0.995014i	$-0.468202\pi$
$881$	4.04834e9			0.199462			0.0997312	+	0.995014i	$0.468202\pi$
$882$		−	6.86129e8i		−	0.0336718i
$883$	2.06451e10			1.00915			0.504574	−	0.863368i	$-0.331649\pi$
$883$	2.06451e10			1.00915			0.504574	+	0.863368i	$0.331649\pi$
$884$	−8.75414e9	−	2.36431e9i	−0.426217	−	0.115112i
$885$	6.01103e9			0.291506
$886$		−	2.77592e10i		−	1.34088i
$887$	−3.93993e8			−0.0189564			−0.00947819	−	0.999955i	$-0.503017\pi$
$887$	−3.93993e8			−0.0189564			−0.00947819	+	0.999955i	$0.503017\pi$
$888$	5.57817e9			0.267329
$889$		−	5.09677e9i		−	0.243298i
$890$			9.17428e9i			0.436221i
$891$		−	3.66988e9i		−	0.173812i
$892$			8.23837e8i			0.0388655i
$893$	4.25754e9			0.200068
$894$	−6.26979e9			−0.293476
$895$		−	2.15107e10i		−	1.00294i
$896$	−7.19323e8			−0.0334077
$897$	−2.46031e9	−	6.64480e8i	−0.113820	−	0.0307403i
$898$	−8.65686e8			−0.0398927
$899$			1.40121e10i			0.643197i
$900$	−2.18996e9			−0.100135
$901$	4.54939e8			0.0207213
$902$			1.73246e9i			0.0786033i
$903$			6.56642e9i			0.296771i
$904$			8.67596e9i			0.390596i
$905$		−	1.49045e10i		−	0.668418i
$906$	−3.22305e7			−0.00143985
$907$	2.29692e10			1.02216			0.511082	−	0.859532i	$-0.329245\pi$
$907$	2.29692e10			1.02216			0.511082	+	0.859532i	$0.329245\pi$
$908$		−	1.02007e9i		−	0.0452200i
$909$	−2.19881e8			−0.00970987
$910$	1.00086e9	−	3.70580e9i	0.0440280	−	0.163019i
$911$	1.75669e10			0.769804			0.384902	−	0.922957i	$-0.374235\pi$
$911$	1.75669e10			0.769804			0.384902	+	0.922957i	$0.374235\pi$
$912$			3.73033e8i			0.0162842i
$913$	5.94817e10			2.58664
$914$	2.47774e9			0.107336
$915$			3.57013e9i			0.154067i
$916$			1.27935e10i			0.549990i
$917$			2.72327e9i			0.116627i
$918$		−	2.81645e9i		−	0.120158i
$919$	4.26104e10			1.81097			0.905485	−	0.424378i	$-0.139507\pi$
$919$	4.26104e10			1.81097			0.905485	+	0.424378i	$0.139507\pi$
$920$	−1.07738e9			−0.0456153
$921$		−	9.56938e9i		−	0.403622i
$922$	−9.93487e8			−0.0417449
$923$	−2.47269e9	+	9.15540e9i	−0.103505	+	0.383241i
$924$	−4.09293e9			−0.170680
$925$		−	1.89403e10i		−	0.786847i
$926$	6.44938e9			0.266919
$927$	1.16284e9			0.0479447
$928$			2.27790e9i			0.0935658i
$929$			8.49461e9i			0.347607i	0.984780	+	0.173803i	$0.0556057\pi$
$929$			8.49461e9i			0.347607i	−0.984780	+	0.173803i	$0.944394\pi$
$930$		−	7.68875e9i		−	0.313448i
$931$			3.96837e8i			0.0161171i
$932$	1.45577e10			0.589029
$933$	−6.99415e9			−0.281935
$934$			1.97123e10i			0.791630i
$935$	−2.18123e10			−0.872690
$936$	7.70908e8	−	2.85437e9i	0.0307282	−	0.113775i
$937$	−2.49416e10			−0.990456			−0.495228	−	0.868763i	$-0.664916\pi$
$937$	−2.49416e10			−0.990456			−0.495228	+	0.868763i	$0.664916\pi$
$938$		−	2.62281e9i		−	0.103766i
$939$	−1.55260e10			−0.611971
$940$	−1.42659e10			−0.560211
$941$		−	4.18175e10i		−	1.63604i	−0.575189	−	0.818020i	$-0.695072\pi$
$941$		−	4.18175e10i		−	1.63604i	0.575189	−	0.818020i	$-0.304928\pi$
$942$			3.58351e9i			0.139679i
$943$			3.73672e8i			0.0145111i
$944$		−	5.16370e9i		−	0.199783i
$945$	1.19226e9			0.0459577
$946$	3.91703e10			1.50431
$947$			3.11006e10i			1.18999i	0.803729	+	0.594995i	$0.202846\pi$
$947$			3.11006e10i			1.18999i	−0.803729	+	0.594995i	$0.797154\pi$
$948$	4.81594e9			0.183591
$949$	−9.65931e8	+	3.57647e9i	−0.0366872	+	0.135838i
$950$	1.26661e9			0.0479301
$951$			3.65828e9i			0.137926i
$952$	−3.14112e9			−0.117993
$953$	−2.72183e10			−1.01867			−0.509337	−	0.860567i	$-0.670109\pi$
$953$	−2.72183e10			−1.01867			−0.509337	+	0.860567i	$0.670109\pi$
$954$			1.48337e8i			0.00553135i
$955$		−	1.74800e10i		−	0.649425i
$956$		−	1.21984e10i		−	0.451545i
$957$			1.29612e10i			0.478028i
$958$	2.68653e10			0.987218
$959$	−1.74291e9			−0.0638132
$960$		−	1.24994e9i		−	0.0455973i
$961$	−1.31163e10			−0.476737
$962$	2.46866e10	+	6.66735e9i	0.894024	+	0.241458i
$963$	4.85708e9			0.175260
$964$		−	6.10609e9i		−	0.219530i
$965$	6.71235e9			0.240452
$966$	−8.82797e8			−0.0315095
$967$			3.68073e10i			1.30900i	0.756060	+	0.654502i	$0.227122\pi$
$967$			3.68073e10i			1.30900i	−0.756060	+	0.654502i	$0.772878\pi$
$968$			1.44379e10i			0.511611i
$969$			1.62895e9i			0.0575141i
$970$		−	2.29116e10i		−	0.806034i
$971$	−4.01221e10			−1.40642			−0.703212	−	0.710981i	$-0.748251\pi$
$971$	−4.01221e10			−1.40642			−0.703212	+	0.710981i	$0.748251\pi$
$972$	9.18330e8			0.0320750
$973$		−	1.47391e10i		−	0.512950i
$974$	−6.55703e9			−0.227379
$975$	−9.69181e9	−	2.61756e9i	−0.334880	−	0.0904442i
$976$	3.06687e9			0.105590
$977$		−	2.03699e10i		−	0.698807i	−0.936972	−	0.349403i	$-0.886384\pi$
$977$		−	2.03699e10i		−	0.698807i	0.936972	−	0.349403i	$-0.113616\pi$
$978$	1.45177e10			0.496262
$979$	4.48429e10			1.52741
$980$		−	1.32970e9i		−	0.0451296i
$981$			5.31975e9i			0.179908i
$982$		−	1.63866e10i		−	0.552202i
$983$			3.29011e10i			1.10477i	0.833588	+	0.552386i	$0.186283\pi$
$983$			3.29011e10i			1.10477i	−0.833588	+	0.552386i	$0.813717\pi$
$984$	−4.33522e8			−0.0145054
$985$	−1.48750e10			−0.495942
$986$			9.94707e9i			0.330465i
$987$	−1.16894e10			−0.386975
$988$	−4.45870e8	+	1.65089e9i	−0.0147082	+	0.0544587i
$989$	8.44858e9			0.277713
$990$		−	7.11211e9i		−	0.232957i
$991$	2.44924e10			0.799416			0.399708	−	0.916642i	$-0.369111\pi$
$991$	2.44924e10			0.799416			0.399708	+	0.916642i	$0.369111\pi$
$992$	−6.60492e9			−0.214821
$993$			2.84660e10i			0.922580i
$994$			3.28510e9i			0.106095i
$995$			1.22677e10i			0.394805i
$996$			1.48844e10i			0.477335i
$997$	2.35208e10			0.751654			0.375827	−	0.926690i	$-0.377359\pi$
$997$	2.35208e10			0.751654			0.375827	+	0.926690i	$0.377359\pi$
$998$	1.58468e10			0.504643
$999$			7.94236e9i			0.252041i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.c.a.337.16	yes	24
13.12	even	2	inner	546.8.c.a.337.9	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.c.a.337.9	✓	24	13.12	even	2	inner
546.8.c.a.337.16	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} -27.0000 q^{3} -64.0000 q^{4} -176.597i q^{5} -216.000i q^{6} -343.000i q^{7} -512.000i q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000i q^{2} -27.0000 q^{3} -64.0000 q^{4} -176.597i q^{5} -216.000i q^{6} -343.000i q^{7} -512.000i q^{8} +729.000 q^{9} +1412.78 q^{10} -6905.52i q^{11} +1728.00 q^{12} +(7647.39 + 2065.40i) q^{13} +2744.00 q^{14} +4768.13i q^{15} +4096.00 q^{16} +17886.3 q^{17} +5832.00i q^{18} -3373.06i q^{19} +11302.2i q^{20} +9261.00i q^{21} +55244.1 q^{22} +11915.5 q^{23} +13824.0i q^{24} +46938.3 q^{25} +(-16523.2 + 61179.1i) q^{26} -19683.0 q^{27} +21952.0i q^{28} +69516.1 q^{29} -38145.0 q^{30} +201566. i q^{31} +32768.0i q^{32} +186449. i q^{33} +143090. i q^{34} -60572.9 q^{35} -46656.0 q^{36} -403514. i q^{37} +26984.4 q^{38} +(-206480. - 55765.9i) q^{39} -90417.9 q^{40} +31360.1i q^{41} -74088.0 q^{42} +709040. q^{43} +441953. i q^{44} -128740. i q^{45} +95324.2i q^{46} +1.26222e6i q^{47} -110592. q^{48} -117649. q^{49} +375507. i q^{50} -482930. q^{51} +(-489433. - 132186. i) q^{52} +25435.1 q^{53} -157464. i q^{54} -1.21950e6 q^{55} -175616. q^{56} +91072.5i q^{57} +556129. i q^{58} -1.26067e6i q^{59} -305160. i q^{60} +748748. q^{61} -1.61253e6 q^{62} -250047. i q^{63} -262144. q^{64} +(364745. - 1.35051e6i) q^{65} -1.49159e6 q^{66} -955833. i q^{67} -1.14472e6 q^{68} -321719. q^{69} -484583. i q^{70} +1.19719e6i q^{71} -373248. i q^{72} +467672. i q^{73} +3.22811e6 q^{74} -1.26734e6 q^{75} +215876. i q^{76} -2.36859e6 q^{77} +(446127. - 1.65184e6i) q^{78} +2.78700e6 q^{79} -723343. i q^{80} +531441. q^{81} -250881. q^{82} +8.61365e6i q^{83} -592704. i q^{84} -3.15867e6i q^{85} +5.67232e6i q^{86} -1.87693e6 q^{87} -3.53563e6 q^{88} +6.49378e6i q^{89} +1.02992e6 q^{90} +(708434. - 2.62306e6i) q^{91} -762593. q^{92} -5.44229e6i q^{93} -1.00978e7 q^{94} -595673. q^{95} -884736. i q^{96} -1.62174e7i q^{97} -941192. i q^{98} -5.03412e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9}+O(q^{10}) \) 24 * q - 648 * q^3 - 1536 * q^4 + 17496 * q^9	\( 24 q - 648 q^{3} - 1536 q^{4} + 17496 q^{9} + 1520 q^{10} + 41472 q^{12} + 13718 q^{13} + 65856 q^{14} + 98304 q^{16} + 13586 q^{17} - 11824 q^{22} + 76574 q^{23} - 357850 q^{25} + 132848 q^{26} - 472392 q^{27} + 85198 q^{29} - 41040 q^{30} - 65170 q^{35} - 1119744 q^{36} + 295696 q^{38} - 370386 q^{39} - 97280 q^{40} - 1778112 q^{42} - 91370 q^{43} - 2654208 q^{48} - 2823576 q^{49} - 366822 q^{51} - 877952 q^{52} - 360000 q^{53} - 2519718 q^{55} - 4214784 q^{56} + 874734 q^{61} + 1984480 q^{62} - 6291456 q^{64} - 12450388 q^{65} + 319248 q^{66} - 869504 q^{68} - 2067498 q^{69} - 1405616 q^{74} + 9661950 q^{75} + 506954 q^{77} - 3586896 q^{78} - 2141176 q^{79} + 12754584 q^{81} + 14087488 q^{82} - 2300346 q^{87} + 756736 q^{88} + 1108080 q^{90} - 5695858 q^{91} - 4900736 q^{92} + 8123808 q^{94} + 19191246 q^{95}+O(q^{100}) \) 24 * q - 648 * q^3 - 1536 * q^4 + 17496 * q^9 + 1520 * q^10 + 41472 * q^12 + 13718 * q^13 + 65856 * q^14 + 98304 * q^16 + 13586 * q^17 - 11824 * q^22 + 76574 * q^23 - 357850 * q^25 + 132848 * q^26 - 472392 * q^27 + 85198 * q^29 - 41040 * q^30 - 65170 * q^35 - 1119744 * q^36 + 295696 * q^38 - 370386 * q^39 - 97280 * q^40 - 1778112 * q^42 - 91370 * q^43 - 2654208 * q^48 - 2823576 * q^49 - 366822 * q^51 - 877952 * q^52 - 360000 * q^53 - 2519718 * q^55 - 4214784 * q^56 + 874734 * q^61 + 1984480 * q^62 - 6291456 * q^64 - 12450388 * q^65 + 319248 * q^66 - 869504 * q^68 - 2067498 * q^69 - 1405616 * q^74 + 9661950 * q^75 + 506954 * q^77 - 3586896 * q^78 - 2141176 * q^79 + 12754584 * q^81 + 14087488 * q^82 - 2300346 * q^87 + 756736 * q^88 + 1108080 * q^90 - 5695858 * q^91 - 4900736 * q^92 + 8123808 * q^94 + 19191246 * q^95

Introduction

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