LMFDB - Embedded newform 6015.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.a (trivial)

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:	yes
Analytic conductor:	$48.0300168158$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.82843 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.82843 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +5.41421 q^{13} +4.82843 q^{14} +1.00000 q^{15} -1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} +2.24264 q^{19} -1.00000 q^{20} +4.82843 q^{21} -4.00000 q^{22} +1.17157 q^{23} -3.00000 q^{24} +1.00000 q^{25} +5.41421 q^{26} +1.00000 q^{27} -4.82843 q^{28} -1.65685 q^{29} +1.00000 q^{30} +5.07107 q^{31} +5.00000 q^{32} -4.00000 q^{33} +1.41421 q^{34} +4.82843 q^{35} -1.00000 q^{36} -0.242641 q^{37} +2.24264 q^{38} +5.41421 q^{39} -3.00000 q^{40} +0.343146 q^{41} +4.82843 q^{42} -9.65685 q^{43} +4.00000 q^{44} +1.00000 q^{45} +1.17157 q^{46} -5.65685 q^{47} -1.00000 q^{48} +16.3137 q^{49} +1.00000 q^{50} +1.41421 q^{51} -5.41421 q^{52} +9.89949 q^{53} +1.00000 q^{54} -4.00000 q^{55} -14.4853 q^{56} +2.24264 q^{57} -1.65685 q^{58} -7.41421 q^{59} -1.00000 q^{60} -0.343146 q^{61} +5.07107 q^{62} +4.82843 q^{63} +7.00000 q^{64} +5.41421 q^{65} -4.00000 q^{66} -4.48528 q^{67} -1.41421 q^{68} +1.17157 q^{69} +4.82843 q^{70} -7.89949 q^{71} -3.00000 q^{72} +2.00000 q^{73} -0.242641 q^{74} +1.00000 q^{75} -2.24264 q^{76} -19.3137 q^{77} +5.41421 q^{78} +7.41421 q^{79} -1.00000 q^{80} +1.00000 q^{81} +0.343146 q^{82} +12.8284 q^{83} -4.82843 q^{84} +1.41421 q^{85} -9.65685 q^{86} -1.65685 q^{87} +12.0000 q^{88} +7.31371 q^{89} +1.00000 q^{90} +26.1421 q^{91} -1.17157 q^{92} +5.07107 q^{93} -5.65685 q^{94} +2.24264 q^{95} +5.00000 q^{96} +14.5858 q^{97} +16.3137 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^5 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{12} + 8 q^{13} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{18} - 4 q^{19} - 2 q^{20} + 4 q^{21} - 8 q^{22} + 8 q^{23} - 6 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} + 2 q^{30} - 4 q^{31} + 10 q^{32} - 8 q^{33} + 4 q^{35} - 2 q^{36} + 8 q^{37} - 4 q^{38} + 8 q^{39} - 6 q^{40} + 12 q^{41} + 4 q^{42} - 8 q^{43} + 8 q^{44} + 2 q^{45} + 8 q^{46} - 2 q^{48} + 10 q^{49} + 2 q^{50} - 8 q^{52} + 2 q^{54} - 8 q^{55} - 12 q^{56} - 4 q^{57} + 8 q^{58} - 12 q^{59} - 2 q^{60} - 12 q^{61} - 4 q^{62} + 4 q^{63} + 14 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} + 8 q^{69} + 4 q^{70} + 4 q^{71} - 6 q^{72} + 4 q^{73} + 8 q^{74} + 2 q^{75} + 4 q^{76} - 16 q^{77} + 8 q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 12 q^{82} + 20 q^{83} - 4 q^{84} - 8 q^{86} + 8 q^{87} + 24 q^{88} - 8 q^{89} + 2 q^{90} + 24 q^{91} - 8 q^{92} - 4 q^{93} - 4 q^{95} + 10 q^{96} + 32 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^5 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 8 * q^11 - 2 * q^12 + 8 * q^13 + 4 * q^14 + 2 * q^15 - 2 * q^16 + 2 * q^18 - 4 * q^19 - 2 * q^20 + 4 * q^21 - 8 * q^22 + 8 * q^23 - 6 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 + 2 * q^30 - 4 * q^31 + 10 * q^32 - 8 * q^33 + 4 * q^35 - 2 * q^36 + 8 * q^37 - 4 * q^38 + 8 * q^39 - 6 * q^40 + 12 * q^41 + 4 * q^42 - 8 * q^43 + 8 * q^44 + 2 * q^45 + 8 * q^46 - 2 * q^48 + 10 * q^49 + 2 * q^50 - 8 * q^52 + 2 * q^54 - 8 * q^55 - 12 * q^56 - 4 * q^57 + 8 * q^58 - 12 * q^59 - 2 * q^60 - 12 * q^61 - 4 * q^62 + 4 * q^63 + 14 * q^64 + 8 * q^65 - 8 * q^66 + 8 * q^67 + 8 * q^69 + 4 * q^70 + 4 * q^71 - 6 * q^72 + 4 * q^73 + 8 * q^74 + 2 * q^75 + 4 * q^76 - 16 * q^77 + 8 * q^78 + 12 * q^79 - 2 * q^80 + 2 * q^81 + 12 * q^82 + 20 * q^83 - 4 * q^84 - 8 * q^86 + 8 * q^87 + 24 * q^88 - 8 * q^89 + 2 * q^90 + 24 * q^91 - 8 * q^92 - 4 * q^93 - 4 * q^95 + 10 * q^96 + 32 * q^97 + 10 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	5.41421	1.50163	0.750816	−	0.660511i	$-0.229660\pi$
$13$	5.41421	1.50163	0.750816	+	0.660511i	$0.229660\pi$
$14$	4.82843	1.29045
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	1.00000	0.235702
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	−1.00000	−0.223607
$21$	4.82843	1.05365
$22$	−4.00000	−0.852803
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	5.41421	1.06181
$27$	1.00000	0.192450
$28$	−4.82843	−0.912487
$29$	−1.65685	−0.307670	−0.153835	−	0.988097i	$-0.549162\pi$
$29$	−1.65685	−0.307670	−0.153835	+	0.988097i	$0.549162\pi$
$30$	1.00000	0.182574
$31$	5.07107	0.910791	0.455395	−	0.890289i	$-0.349498\pi$
$31$	5.07107	0.910791	0.455395	+	0.890289i	$0.349498\pi$
$32$	5.00000	0.883883
$33$	−4.00000	−0.696311
$34$	1.41421	0.242536
$35$	4.82843	0.816153
$36$	−1.00000	−0.166667
$37$	−0.242641	−0.0398899	−0.0199449	−	0.999801i	$-0.506349\pi$
$37$	−0.242641	−0.0398899	−0.0199449	+	0.999801i	$0.506349\pi$
$38$	2.24264	0.363804
$39$	5.41421	0.866968
$40$	−3.00000	−0.474342
$41$	0.343146	0.0535904	0.0267952	−	0.999641i	$-0.491470\pi$
$41$	0.343146	0.0535904	0.0267952	+	0.999641i	$0.491470\pi$
$42$	4.82843	0.745042
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	1.17157	0.172739
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	−1.00000	−0.144338
$49$	16.3137	2.33053
$50$	1.00000	0.141421
$51$	1.41421	0.198030
$52$	−5.41421	−0.750816
$53$	9.89949	1.35980	0.679900	−	0.733305i	$-0.262023\pi$
$53$	9.89949	1.35980	0.679900	+	0.733305i	$0.262023\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	−14.4853	−1.93568
$57$	2.24264	0.297045
$58$	−1.65685	−0.217556
$59$	−7.41421	−0.965248	−0.482624	−	0.875828i	$-0.660316\pi$
$59$	−7.41421	−0.965248	−0.482624	+	0.875828i	$0.660316\pi$
$60$	−1.00000	−0.129099
$61$	−0.343146	−0.0439353	−0.0219677	−	0.999759i	$-0.506993\pi$
$61$	−0.343146	−0.0439353	−0.0219677	+	0.999759i	$0.506993\pi$
$62$	5.07107	0.644026
$63$	4.82843	0.608325
$64$	7.00000	0.875000
$65$	5.41421	0.671551
$66$	−4.00000	−0.492366
$67$	−4.48528	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$68$	−1.41421	−0.171499
$69$	1.17157	0.141041
$70$	4.82843	0.577107
$71$	−7.89949	−0.937498	−0.468749	−	0.883332i	$-0.655295\pi$
$71$	−7.89949	−0.937498	−0.468749	+	0.883332i	$0.655295\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−0.242641	−0.0282064
$75$	1.00000	0.115470
$76$	−2.24264	−0.257249
$77$	−19.3137	−2.20100
$78$	5.41421	0.613039
$79$	7.41421	0.834164	0.417082	−	0.908869i	$-0.363053\pi$
$79$	7.41421	0.834164	0.417082	+	0.908869i	$0.363053\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0.343146	0.0378941
$83$	12.8284	1.40810	0.704051	−	0.710149i	$-0.251372\pi$
$83$	12.8284	1.40810	0.704051	+	0.710149i	$0.251372\pi$
$84$	−4.82843	−0.526825
$85$	1.41421	0.153393
$86$	−9.65685	−1.04133
$87$	−1.65685	−0.177633
$88$	12.0000	1.27920
$89$	7.31371	0.775252	0.387626	−	0.921817i	$-0.373295\pi$
$89$	7.31371	0.775252	0.387626	+	0.921817i	$0.373295\pi$
$90$	1.00000	0.105409
$91$	26.1421	2.74044
$92$	−1.17157	−0.122145
$93$	5.07107	0.525845
$94$	−5.65685	−0.583460
$95$	2.24264	0.230090
$96$	5.00000	0.510310
$97$	14.5858	1.48096	0.740481	−	0.672077i	$-0.234598\pi$
$97$	14.5858	1.48096	0.740481	+	0.672077i	$0.234598\pi$
$98$	16.3137	1.64793
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−2.48528	−0.247295	−0.123647	−	0.992326i	$-0.539459\pi$
$101$	−2.48528	−0.247295	−0.123647	+	0.992326i	$0.539459\pi$
$102$	1.41421	0.140028
$103$	11.1716	1.10077	0.550384	−	0.834912i	$-0.314481\pi$
$103$	11.1716	1.10077	0.550384	+	0.834912i	$0.314481\pi$
$104$	−16.2426	−1.59272
$105$	4.82843	0.471206
$106$	9.89949	0.961524
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	−1.00000	−0.0962250
$109$	−1.65685	−0.158698	−0.0793489	−	0.996847i	$-0.525284\pi$
$109$	−1.65685	−0.158698	−0.0793489	+	0.996847i	$0.525284\pi$
$110$	−4.00000	−0.381385
$111$	−0.242641	−0.0230304
$112$	−4.82843	−0.456243
$113$	−8.14214	−0.765948	−0.382974	−	0.923759i	$-0.625100\pi$
$113$	−8.14214	−0.765948	−0.382974	+	0.923759i	$0.625100\pi$
$114$	2.24264	0.210043
$115$	1.17157	0.109250
$116$	1.65685	0.153835
$117$	5.41421	0.500544
$118$	−7.41421	−0.682534
$119$	6.82843	0.625961
$120$	−3.00000	−0.273861
$121$	5.00000	0.454545
$122$	−0.343146	−0.0310670
$123$	0.343146	0.0309404
$124$	−5.07107	−0.455395
$125$	1.00000	0.0894427
$126$	4.82843	0.430150
$127$	−1.65685	−0.147022	−0.0735110	−	0.997294i	$-0.523420\pi$
$127$	−1.65685	−0.147022	−0.0735110	+	0.997294i	$0.523420\pi$
$128$	−3.00000	−0.265165
$129$	−9.65685	−0.850239
$130$	5.41421	0.474858
$131$	−6.24264	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$131$	−6.24264	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$132$	4.00000	0.348155
$133$	10.8284	0.938944
$134$	−4.48528	−0.387469
$135$	1.00000	0.0860663
$136$	−4.24264	−0.363803
$137$	1.41421	0.120824	0.0604122	−	0.998174i	$-0.480758\pi$
$137$	1.41421	0.120824	0.0604122	+	0.998174i	$0.480758\pi$
$138$	1.17157	0.0997309
$139$	−0.100505	−0.00852473	−0.00426236	−	0.999991i	$-0.501357\pi$
$139$	−0.100505	−0.00852473	−0.00426236	+	0.999991i	$0.501357\pi$
$140$	−4.82843	−0.408077
$141$	−5.65685	−0.476393
$142$	−7.89949	−0.662911
$143$	−21.6569	−1.81104
$144$	−1.00000	−0.0833333
$145$	−1.65685	−0.137594
$146$	2.00000	0.165521
$147$	16.3137	1.34553
$148$	0.242641	0.0199449
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	1.00000	0.0816497
$151$	−13.6569	−1.11138	−0.555690	−	0.831390i	$-0.687546\pi$
$151$	−13.6569	−1.11138	−0.555690	+	0.831390i	$0.687546\pi$
$152$	−6.72792	−0.545707
$153$	1.41421	0.114332
$154$	−19.3137	−1.55634
$155$	5.07107	0.407318
$156$	−5.41421	−0.433484
$157$	5.41421	0.432101	0.216051	−	0.976382i	$-0.430682\pi$
$157$	5.41421	0.432101	0.216051	+	0.976382i	$0.430682\pi$
$158$	7.41421	0.589843
$159$	9.89949	0.785081
$160$	5.00000	0.395285
$161$	5.65685	0.445823
$162$	1.00000	0.0785674
$163$	23.7990	1.86408	0.932040	−	0.362354i	$-0.118027\pi$
$163$	23.7990	1.86408	0.932040	+	0.362354i	$0.118027\pi$
$164$	−0.343146	−0.0267952
$165$	−4.00000	−0.311400
$166$	12.8284	0.995679
$167$	20.4853	1.58520	0.792599	−	0.609743i	$-0.208727\pi$
$167$	20.4853	1.58520	0.792599	+	0.609743i	$0.208727\pi$
$168$	−14.4853	−1.11756
$169$	16.3137	1.25490
$170$	1.41421	0.108465
$171$	2.24264	0.171499
$172$	9.65685	0.736328
$173$	11.1716	0.849359	0.424679	−	0.905344i	$-0.360387\pi$
$173$	11.1716	0.849359	0.424679	+	0.905344i	$0.360387\pi$
$174$	−1.65685	−0.125606
$175$	4.82843	0.364995
$176$	4.00000	0.301511
$177$	−7.41421	−0.557286
$178$	7.31371	0.548186
$179$	−10.8284	−0.809355	−0.404677	−	0.914460i	$-0.632616\pi$
$179$	−10.8284	−0.809355	−0.404677	+	0.914460i	$0.632616\pi$
$180$	−1.00000	−0.0745356
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	26.1421	1.93778
$183$	−0.343146	−0.0253661
$184$	−3.51472	−0.259108
$185$	−0.242641	−0.0178393
$186$	5.07107	0.371829
$187$	−5.65685	−0.413670
$188$	5.65685	0.412568
$189$	4.82843	0.351216
$190$	2.24264	0.162698
$191$	3.41421	0.247044	0.123522	−	0.992342i	$-0.460581\pi$
$191$	3.41421	0.247044	0.123522	+	0.992342i	$0.460581\pi$
$192$	7.00000	0.505181
$193$	4.24264	0.305392	0.152696	−	0.988273i	$-0.451204\pi$
$193$	4.24264	0.305392	0.152696	+	0.988273i	$0.451204\pi$
$194$	14.5858	1.04720
$195$	5.41421	0.387720
$196$	−16.3137	−1.16526
$197$	13.7990	0.983137	0.491569	−	0.870839i	$-0.336424\pi$
$197$	13.7990	0.983137	0.491569	+	0.870839i	$0.336424\pi$
$198$	−4.00000	−0.284268
$199$	1.75736	0.124576	0.0622879	−	0.998058i	$-0.480160\pi$
$199$	1.75736	0.124576	0.0622879	+	0.998058i	$0.480160\pi$
$200$	−3.00000	−0.212132
$201$	−4.48528	−0.316367
$202$	−2.48528	−0.174864
$203$	−8.00000	−0.561490
$204$	−1.41421	−0.0990148
$205$	0.343146	0.0239663
$206$	11.1716	0.778360
$207$	1.17157	0.0814299
$208$	−5.41421	−0.375408
$209$	−8.97056	−0.620507
$210$	4.82843	0.333193
$211$	−17.0711	−1.17522	−0.587610	−	0.809144i	$-0.699931\pi$
$211$	−17.0711	−1.17522	−0.587610	+	0.809144i	$0.699931\pi$
$212$	−9.89949	−0.679900
$213$	−7.89949	−0.541264
$214$	−9.65685	−0.660129
$215$	−9.65685	−0.658592
$216$	−3.00000	−0.204124
$217$	24.4853	1.66217
$218$	−1.65685	−0.112216
$219$	2.00000	0.135147
$220$	4.00000	0.269680
$221$	7.65685	0.515056
$222$	−0.242641	−0.0162850
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	24.1421	1.61306
$225$	1.00000	0.0666667
$226$	−8.14214	−0.541607
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−2.24264	−0.148523
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	1.17157	0.0772512
$231$	−19.3137	−1.27075
$232$	4.97056	0.326333
$233$	22.5858	1.47964	0.739822	−	0.672803i	$-0.234910\pi$
$233$	22.5858	1.47964	0.739822	+	0.672803i	$0.234910\pi$
$234$	5.41421	0.353938
$235$	−5.65685	−0.369012
$236$	7.41421	0.482624
$237$	7.41421	0.481605
$238$	6.82843	0.442621
$239$	−14.3431	−0.927781	−0.463890	−	0.885893i	$-0.653547\pi$
$239$	−14.3431	−0.927781	−0.463890	+	0.885893i	$0.653547\pi$
$240$	−1.00000	−0.0645497
$241$	17.6569	1.13738	0.568689	−	0.822553i	$-0.307451\pi$
$241$	17.6569	1.13738	0.568689	+	0.822553i	$0.307451\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	0.343146	0.0219677
$245$	16.3137	1.04224
$246$	0.343146	0.0218782
$247$	12.1421	0.772586
$248$	−15.2132	−0.966039
$249$	12.8284	0.812969
$250$	1.00000	0.0632456
$251$	−10.9289	−0.689828	−0.344914	−	0.938634i	$-0.612092\pi$
$251$	−10.9289	−0.689828	−0.344914	+	0.938634i	$0.612092\pi$
$252$	−4.82843	−0.304162
$253$	−4.68629	−0.294625
$254$	−1.65685	−0.103960
$255$	1.41421	0.0885615
$256$	−17.0000	−1.06250
$257$	−7.17157	−0.447350	−0.223675	−	0.974664i	$-0.571805\pi$
$257$	−7.17157	−0.447350	−0.223675	+	0.974664i	$0.571805\pi$
$258$	−9.65685	−0.601209
$259$	−1.17157	−0.0727980
$260$	−5.41421	−0.335775
$261$	−1.65685	−0.102557
$262$	−6.24264	−0.385672
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	12.0000	0.738549
$265$	9.89949	0.608121
$266$	10.8284	0.663933
$267$	7.31371	0.447592
$268$	4.48528	0.273982
$269$	−12.1421	−0.740319	−0.370160	−	0.928968i	$-0.620697\pi$
$269$	−12.1421	−0.740319	−0.370160	+	0.928968i	$0.620697\pi$
$270$	1.00000	0.0608581
$271$	−22.0416	−1.33893	−0.669467	−	0.742842i	$-0.733477\pi$
$271$	−22.0416	−1.33893	−0.669467	+	0.742842i	$0.733477\pi$
$272$	−1.41421	−0.0857493
$273$	26.1421	1.58219
$274$	1.41421	0.0854358
$275$	−4.00000	−0.241209
$276$	−1.17157	−0.0705204
$277$	9.41421	0.565645	0.282823	−	0.959172i	$-0.408729\pi$
$277$	9.41421	0.565645	0.282823	+	0.959172i	$0.408729\pi$
$278$	−0.100505	−0.00602789
$279$	5.07107	0.303597
$280$	−14.4853	−0.865661
$281$	1.51472	0.0903605	0.0451803	−	0.998979i	$-0.485614\pi$
$281$	1.51472	0.0903605	0.0451803	+	0.998979i	$0.485614\pi$
$282$	−5.65685	−0.336861
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	7.89949	0.468749
$285$	2.24264	0.132843
$286$	−21.6569	−1.28060
$287$	1.65685	0.0978010
$288$	5.00000	0.294628
$289$	−15.0000	−0.882353
$290$	−1.65685	−0.0972938
$291$	14.5858	0.855034
$292$	−2.00000	−0.117041
$293$	7.27208	0.424839	0.212420	−	0.977179i	$-0.431866\pi$
$293$	7.27208	0.424839	0.212420	+	0.977179i	$0.431866\pi$
$294$	16.3137	0.951435
$295$	−7.41421	−0.431672
$296$	0.727922	0.0423096
$297$	−4.00000	−0.232104
$298$	−12.0000	−0.695141
$299$	6.34315	0.366834
$300$	−1.00000	−0.0577350
$301$	−46.6274	−2.68756
$302$	−13.6569	−0.785864
$303$	−2.48528	−0.142776
$304$	−2.24264	−0.128624
$305$	−0.343146	−0.0196485
$306$	1.41421	0.0808452
$307$	29.7990	1.70072	0.850359	−	0.526203i	$-0.176385\pi$
$307$	29.7990	1.70072	0.850359	+	0.526203i	$0.176385\pi$
$308$	19.3137	1.10050
$309$	11.1716	0.635529
$310$	5.07107	0.288017
$311$	15.3137	0.868361	0.434180	−	0.900826i	$-0.357038\pi$
$311$	15.3137	0.868361	0.434180	+	0.900826i	$0.357038\pi$
$312$	−16.2426	−0.919558
$313$	−25.7990	−1.45825	−0.729123	−	0.684383i	$-0.760072\pi$
$313$	−25.7990	−1.45825	−0.729123	+	0.684383i	$0.760072\pi$
$314$	5.41421	0.305542
$315$	4.82843	0.272051
$316$	−7.41421	−0.417082
$317$	−23.5563	−1.32306	−0.661528	−	0.749920i	$-0.730092\pi$
$317$	−23.5563	−1.32306	−0.661528	+	0.749920i	$0.730092\pi$
$318$	9.89949	0.555136
$319$	6.62742	0.371064
$320$	7.00000	0.391312
$321$	−9.65685	−0.538993
$322$	5.65685	0.315244
$323$	3.17157	0.176471
$324$	−1.00000	−0.0555556
$325$	5.41421	0.300327
$326$	23.7990	1.31810
$327$	−1.65685	−0.0916242
$328$	−1.02944	−0.0568412
$329$	−27.3137	−1.50585
$330$	−4.00000	−0.220193
$331$	0.485281	0.0266735	0.0133367	−	0.999911i	$-0.495755\pi$
$331$	0.485281	0.0266735	0.0133367	+	0.999911i	$0.495755\pi$
$332$	−12.8284	−0.704051
$333$	−0.242641	−0.0132966
$334$	20.4853	1.12090
$335$	−4.48528	−0.245057
$336$	−4.82843	−0.263412
$337$	16.6274	0.905753	0.452877	−	0.891573i	$-0.350398\pi$
$337$	16.6274	0.905753	0.452877	+	0.891573i	$0.350398\pi$
$338$	16.3137	0.887349
$339$	−8.14214	−0.442220
$340$	−1.41421	−0.0766965
$341$	−20.2843	−1.09845
$342$	2.24264	0.121268
$343$	44.9706	2.42818
$344$	28.9706	1.56199
$345$	1.17157	0.0630754
$346$	11.1716	0.600587
$347$	−28.4853	−1.52917	−0.764585	−	0.644523i	$-0.777056\pi$
$347$	−28.4853	−1.52917	−0.764585	+	0.644523i	$0.777056\pi$
$348$	1.65685	0.0888167
$349$	17.3137	0.926782	0.463391	−	0.886154i	$-0.346633\pi$
$349$	17.3137	0.926782	0.463391	+	0.886154i	$0.346633\pi$
$350$	4.82843	0.258090
$351$	5.41421	0.288989
$352$	−20.0000	−1.06600
$353$	11.0711	0.589253	0.294627	−	0.955612i	$-0.404805\pi$
$353$	11.0711	0.589253	0.294627	+	0.955612i	$0.404805\pi$
$354$	−7.41421	−0.394061
$355$	−7.89949	−0.419262
$356$	−7.31371	−0.387626
$357$	6.82843	0.361399
$358$	−10.8284	−0.572300
$359$	6.72792	0.355086	0.177543	−	0.984113i	$-0.443185\pi$
$359$	6.72792	0.355086	0.177543	+	0.984113i	$0.443185\pi$
$360$	−3.00000	−0.158114
$361$	−13.9706	−0.735293
$362$	0	0
$363$	5.00000	0.262432
$364$	−26.1421	−1.37022
$365$	2.00000	0.104685
$366$	−0.343146	−0.0179365
$367$	−17.6569	−0.921680	−0.460840	−	0.887483i	$-0.652452\pi$
$367$	−17.6569	−0.921680	−0.460840	+	0.887483i	$0.652452\pi$
$368$	−1.17157	−0.0610725
$369$	0.343146	0.0178635
$370$	−0.242641	−0.0126143
$371$	47.7990	2.48160
$372$	−5.07107	−0.262923
$373$	2.68629	0.139091	0.0695455	−	0.997579i	$-0.477845\pi$
$373$	2.68629	0.139091	0.0695455	+	0.997579i	$0.477845\pi$
$374$	−5.65685	−0.292509
$375$	1.00000	0.0516398
$376$	16.9706	0.875190
$377$	−8.97056	−0.462007
$378$	4.82843	0.248347
$379$	−31.7990	−1.63340	−0.816702	−	0.577059i	$-0.804200\pi$
$379$	−31.7990	−1.63340	−0.816702	+	0.577059i	$0.804200\pi$
$380$	−2.24264	−0.115045
$381$	−1.65685	−0.0848832
$382$	3.41421	0.174686
$383$	16.8284	0.859892	0.429946	−	0.902854i	$-0.358533\pi$
$383$	16.8284	0.859892	0.429946	+	0.902854i	$0.358533\pi$
$384$	−3.00000	−0.153093
$385$	−19.3137	−0.984318
$386$	4.24264	0.215945
$387$	−9.65685	−0.490885
$388$	−14.5858	−0.740481
$389$	17.5147	0.888031	0.444016	−	0.896019i	$-0.353553\pi$
$389$	17.5147	0.888031	0.444016	+	0.896019i	$0.353553\pi$
$390$	5.41421	0.274159
$391$	1.65685	0.0837907
$392$	−48.9411	−2.47190
$393$	−6.24264	−0.314900
$394$	13.7990	0.695183
$395$	7.41421	0.373050
$396$	4.00000	0.201008
$397$	−12.1421	−0.609396	−0.304698	−	0.952449i	$-0.598556\pi$
$397$	−12.1421	−0.609396	−0.304698	+	0.952449i	$0.598556\pi$
$398$	1.75736	0.0880885
$399$	10.8284	0.542099
$400$	−1.00000	−0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−4.48528	−0.223706
$403$	27.4558	1.36767
$404$	2.48528	0.123647
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	0.970563	0.0481090
$408$	−4.24264	−0.210042
$409$	−32.2843	−1.59635	−0.798177	−	0.602423i	$-0.794202\pi$
$409$	−32.2843	−1.59635	−0.798177	+	0.602423i	$0.794202\pi$
$410$	0.343146	0.0169468
$411$	1.41421	0.0697580
$412$	−11.1716	−0.550384
$413$	−35.7990	−1.76155
$414$	1.17157	0.0575797
$415$	12.8284	0.629723
$416$	27.0711	1.32727
$417$	−0.100505	−0.00492175
$418$	−8.97056	−0.438765
$419$	−28.4853	−1.39160	−0.695799	−	0.718237i	$-0.744949\pi$
$419$	−28.4853	−1.39160	−0.695799	+	0.718237i	$0.744949\pi$
$420$	−4.82843	−0.235603
$421$	36.6274	1.78511	0.892556	−	0.450937i	$-0.148910\pi$
$421$	36.6274	1.78511	0.892556	+	0.450937i	$0.148910\pi$
$422$	−17.0711	−0.831007
$423$	−5.65685	−0.275046
$424$	−29.6985	−1.44229
$425$	1.41421	0.0685994
$426$	−7.89949	−0.382732
$427$	−1.65685	−0.0801808
$428$	9.65685	0.466782
$429$	−21.6569	−1.04560
$430$	−9.65685	−0.465695
$431$	−22.7279	−1.09477	−0.547383	−	0.836882i	$-0.684376\pi$
$431$	−22.7279	−1.09477	−0.547383	+	0.836882i	$0.684376\pi$
$432$	−1.00000	−0.0481125
$433$	14.4853	0.696118	0.348059	−	0.937473i	$-0.386841\pi$
$433$	14.4853	0.696118	0.348059	+	0.937473i	$0.386841\pi$
$434$	24.4853	1.17533
$435$	−1.65685	−0.0794401
$436$	1.65685	0.0793489
$437$	2.62742	0.125686
$438$	2.00000	0.0955637
$439$	10.9289	0.521609	0.260805	−	0.965392i	$-0.416012\pi$
$439$	10.9289	0.521609	0.260805	+	0.965392i	$0.416012\pi$
$440$	12.0000	0.572078
$441$	16.3137	0.776843
$442$	7.65685	0.364199
$443$	18.8284	0.894566	0.447283	−	0.894393i	$-0.352392\pi$
$443$	18.8284	0.894566	0.447283	+	0.894393i	$0.352392\pi$
$444$	0.242641	0.0115152
$445$	7.31371	0.346703
$446$	0	0
$447$	−12.0000	−0.567581
$448$	33.7990	1.59685
$449$	−4.34315	−0.204966	−0.102483	−	0.994735i	$-0.532679\pi$
$449$	−4.34315	−0.204966	−0.102483	+	0.994735i	$0.532679\pi$
$450$	1.00000	0.0471405
$451$	−1.37258	−0.0646324
$452$	8.14214	0.382974
$453$	−13.6569	−0.641655
$454$	−24.0000	−1.12638
$455$	26.1421	1.22556
$456$	−6.72792	−0.315064
$457$	−38.4853	−1.80027	−0.900133	−	0.435616i	$-0.856531\pi$
$457$	−38.4853	−1.80027	−0.900133	+	0.435616i	$0.856531\pi$
$458$	−1.31371	−0.0613856
$459$	1.41421	0.0660098
$460$	−1.17157	−0.0546249
$461$	28.1421	1.31071	0.655355	−	0.755321i	$-0.272519\pi$
$461$	28.1421	1.31071	0.655355	+	0.755321i	$0.272519\pi$
$462$	−19.3137	−0.898555
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	1.65685	0.0769175
$465$	5.07107	0.235165
$466$	22.5858	1.04627
$467$	−1.65685	−0.0766701	−0.0383350	−	0.999265i	$-0.512205\pi$
$467$	−1.65685	−0.0766701	−0.0383350	+	0.999265i	$0.512205\pi$
$468$	−5.41421	−0.250272
$469$	−21.6569	−1.00002
$470$	−5.65685	−0.260931
$471$	5.41421	0.249474
$472$	22.2426	1.02380
$473$	38.6274	1.77609
$474$	7.41421	0.340546
$475$	2.24264	0.102899
$476$	−6.82843	−0.312980
$477$	9.89949	0.453267
$478$	−14.3431	−0.656040
$479$	39.1127	1.78710	0.893552	−	0.448959i	$-0.148205\pi$
$479$	39.1127	1.78710	0.893552	+	0.448959i	$0.148205\pi$
$480$	5.00000	0.228218
$481$	−1.31371	−0.0599000
$482$	17.6569	0.804248
$483$	5.65685	0.257396
$484$	−5.00000	−0.227273
$485$	14.5858	0.662306
$486$	1.00000	0.0453609
$487$	26.4853	1.20016	0.600081	−	0.799939i	$-0.295135\pi$
$487$	26.4853	1.20016	0.600081	+	0.799939i	$0.295135\pi$
$488$	1.02944	0.0466004
$489$	23.7990	1.07623
$490$	16.3137	0.736978
$491$	16.9706	0.765871	0.382935	−	0.923775i	$-0.374913\pi$
$491$	16.9706	0.765871	0.382935	+	0.923775i	$0.374913\pi$
$492$	−0.343146	−0.0154702
$493$	−2.34315	−0.105530
$494$	12.1421	0.546301
$495$	−4.00000	−0.179787
$496$	−5.07107	−0.227698
$497$	−38.1421	−1.71091
$498$	12.8284	0.574856
$499$	−30.8284	−1.38007	−0.690035	−	0.723776i	$-0.742405\pi$
$499$	−30.8284	−1.38007	−0.690035	+	0.723776i	$0.742405\pi$
$500$	−1.00000	−0.0447214
$501$	20.4853	0.915215
$502$	−10.9289	−0.487782
$503$	19.3137	0.861156	0.430578	−	0.902553i	$-0.358310\pi$
$503$	19.3137	0.861156	0.430578	+	0.902553i	$0.358310\pi$
$504$	−14.4853	−0.645226
$505$	−2.48528	−0.110594
$506$	−4.68629	−0.208331
$507$	16.3137	0.724517
$508$	1.65685	0.0735110
$509$	−16.3431	−0.724397	−0.362199	−	0.932101i	$-0.617974\pi$
$509$	−16.3431	−0.724397	−0.362199	+	0.932101i	$0.617974\pi$
$510$	1.41421	0.0626224
$511$	9.65685	0.427194
$512$	−11.0000	−0.486136
$513$	2.24264	0.0990150
$514$	−7.17157	−0.316325
$515$	11.1716	0.492278
$516$	9.65685	0.425119
$517$	22.6274	0.995153
$518$	−1.17157	−0.0514760
$519$	11.1716	0.490378
$520$	−16.2426	−0.712287
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−1.65685	−0.0725185
$523$	22.8284	0.998217	0.499109	−	0.866539i	$-0.333661\pi$
$523$	22.8284	0.998217	0.499109	+	0.866539i	$0.333661\pi$
$524$	6.24264	0.272711
$525$	4.82843	0.210730
$526$	24.0000	1.04645
$527$	7.17157	0.312399
$528$	4.00000	0.174078
$529$	−21.6274	−0.940322
$530$	9.89949	0.430007
$531$	−7.41421	−0.321749
$532$	−10.8284	−0.469472
$533$	1.85786	0.0804730
$534$	7.31371	0.316495
$535$	−9.65685	−0.417502
$536$	13.4558	0.581204
$537$	−10.8284	−0.467281
$538$	−12.1421	−0.523485
$539$	−65.2548	−2.81072
$540$	−1.00000	−0.0430331
$541$	38.2843	1.64597	0.822985	−	0.568064i	$-0.192307\pi$
$541$	38.2843	1.64597	0.822985	+	0.568064i	$0.192307\pi$
$542$	−22.0416	−0.946769
$543$	0	0
$544$	7.07107	0.303170
$545$	−1.65685	−0.0709718
$546$	26.1421	1.11878
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−1.41421	−0.0604122
$549$	−0.343146	−0.0146451
$550$	−4.00000	−0.170561
$551$	−3.71573	−0.158295
$552$	−3.51472	−0.149596
$553$	35.7990	1.52233
$554$	9.41421	0.399972
$555$	−0.242641	−0.0102995
$556$	0.100505	0.00426236
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	5.07107	0.214675
$559$	−52.2843	−2.21139
$560$	−4.82843	−0.204038
$561$	−5.65685	−0.238833
$562$	1.51472	0.0638945
$563$	−38.4853	−1.62196	−0.810981	−	0.585073i	$-0.801066\pi$
$563$	−38.4853	−1.62196	−0.810981	+	0.585073i	$0.801066\pi$
$564$	5.65685	0.238197
$565$	−8.14214	−0.342542
$566$	−8.48528	−0.356663
$567$	4.82843	0.202775
$568$	23.6985	0.994366
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	2.24264	0.0939339
$571$	1.07107	0.0448228	0.0224114	−	0.999749i	$-0.492866\pi$
$571$	1.07107	0.0448228	0.0224114	+	0.999749i	$0.492866\pi$
$572$	21.6569	0.905519
$573$	3.41421	0.142631
$574$	1.65685	0.0691558
$575$	1.17157	0.0488580
$576$	7.00000	0.291667
$577$	8.14214	0.338962	0.169481	−	0.985533i	$-0.445791\pi$
$577$	8.14214	0.338962	0.169481	+	0.985533i	$0.445791\pi$
$578$	−15.0000	−0.623918
$579$	4.24264	0.176318
$580$	1.65685	0.0687971
$581$	61.9411	2.56975
$582$	14.5858	0.604600
$583$	−39.5980	−1.63998
$584$	−6.00000	−0.248282
$585$	5.41421	0.223850
$586$	7.27208	0.300407
$587$	−17.7990	−0.734643	−0.367321	−	0.930094i	$-0.619725\pi$
$587$	−17.7990	−0.734643	−0.367321	+	0.930094i	$0.619725\pi$
$588$	−16.3137	−0.672766
$589$	11.3726	0.468599
$590$	−7.41421	−0.305238
$591$	13.7990	0.567615
$592$	0.242641	0.00997247
$593$	20.7279	0.851194	0.425597	−	0.904913i	$-0.360064\pi$
$593$	20.7279	0.851194	0.425597	+	0.904913i	$0.360064\pi$
$594$	−4.00000	−0.164122
$595$	6.82843	0.279938
$596$	12.0000	0.491539
$597$	1.75736	0.0719239
$598$	6.34315	0.259391
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−3.00000	−0.122474
$601$	−32.2843	−1.31690	−0.658451	−	0.752623i	$-0.728788\pi$
$601$	−32.2843	−1.31690	−0.658451	+	0.752623i	$0.728788\pi$
$602$	−46.6274	−1.90039
$603$	−4.48528	−0.182655
$604$	13.6569	0.555690
$605$	5.00000	0.203279
$606$	−2.48528	−0.100958
$607$	3.85786	0.156586	0.0782929	−	0.996930i	$-0.475053\pi$
$607$	3.85786	0.156586	0.0782929	+	0.996930i	$0.475053\pi$
$608$	11.2132	0.454755
$609$	−8.00000	−0.324176
$610$	−0.343146	−0.0138936
$611$	−30.6274	−1.23905
$612$	−1.41421	−0.0571662
$613$	−31.3553	−1.26643	−0.633215	−	0.773976i	$-0.718265\pi$
$613$	−31.3553	−1.26643	−0.633215	+	0.773976i	$0.718265\pi$
$614$	29.7990	1.20259
$615$	0.343146	0.0138370
$616$	57.9411	2.33451
$617$	32.5269	1.30948	0.654742	−	0.755852i	$-0.272777\pi$
$617$	32.5269	1.30948	0.654742	+	0.755852i	$0.272777\pi$
$618$	11.1716	0.449387
$619$	−17.1716	−0.690184	−0.345092	−	0.938569i	$-0.612152\pi$
$619$	−17.1716	−0.690184	−0.345092	+	0.938569i	$0.612152\pi$
$620$	−5.07107	−0.203659
$621$	1.17157	0.0470136
$622$	15.3137	0.614024
$623$	35.3137	1.41481
$624$	−5.41421	−0.216742
$625$	1.00000	0.0400000
$626$	−25.7990	−1.03114
$627$	−8.97056	−0.358250
$628$	−5.41421	−0.216051
$629$	−0.343146	−0.0136821
$630$	4.82843	0.192369
$631$	21.0711	0.838826	0.419413	−	0.907796i	$-0.362236\pi$
$631$	21.0711	0.838826	0.419413	+	0.907796i	$0.362236\pi$
$632$	−22.2426	−0.884765
$633$	−17.0711	−0.678514
$634$	−23.5563	−0.935542
$635$	−1.65685	−0.0657503
$636$	−9.89949	−0.392541
$637$	88.3259	3.49960
$638$	6.62742	0.262382
$639$	−7.89949	−0.312499
$640$	−3.00000	−0.118585
$641$	3.85786	0.152376	0.0761882	−	0.997093i	$-0.475725\pi$
$641$	3.85786	0.152376	0.0761882	+	0.997093i	$0.475725\pi$
$642$	−9.65685	−0.381126
$643$	−18.6274	−0.734594	−0.367297	−	0.930104i	$-0.619717\pi$
$643$	−18.6274	−0.734594	−0.367297	+	0.930104i	$0.619717\pi$
$644$	−5.65685	−0.222911
$645$	−9.65685	−0.380238
$646$	3.17157	0.124784
$647$	22.3431	0.878400	0.439200	−	0.898389i	$-0.355262\pi$
$647$	22.3431	0.878400	0.439200	+	0.898389i	$0.355262\pi$
$648$	−3.00000	−0.117851
$649$	29.6569	1.16413
$650$	5.41421	0.212363
$651$	24.4853	0.959654
$652$	−23.7990	−0.932040
$653$	12.8284	0.502015	0.251008	−	0.967985i	$-0.419238\pi$
$653$	12.8284	0.502015	0.251008	+	0.967985i	$0.419238\pi$
$654$	−1.65685	−0.0647881
$655$	−6.24264	−0.243920
$656$	−0.343146	−0.0133976
$657$	2.00000	0.0780274
$658$	−27.3137	−1.06480
$659$	15.2132	0.592622	0.296311	−	0.955091i	$-0.404243\pi$
$659$	15.2132	0.592622	0.296311	+	0.955091i	$0.404243\pi$
$660$	4.00000	0.155700
$661$	−20.3431	−0.791257	−0.395628	−	0.918411i	$-0.629473\pi$
$661$	−20.3431	−0.791257	−0.395628	+	0.918411i	$0.629473\pi$
$662$	0.485281	0.0188610
$663$	7.65685	0.297368
$664$	−38.4853	−1.49352
$665$	10.8284	0.419908
$666$	−0.242641	−0.00940214
$667$	−1.94113	−0.0751607
$668$	−20.4853	−0.792599
$669$	0	0
$670$	−4.48528	−0.173282
$671$	1.37258	0.0529880
$672$	24.1421	0.931303
$673$	32.2426	1.24286	0.621431	−	0.783469i	$-0.286552\pi$
$673$	32.2426	1.24286	0.621431	+	0.783469i	$0.286552\pi$
$674$	16.6274	0.640464
$675$	1.00000	0.0384900
$676$	−16.3137	−0.627450
$677$	−7.45584	−0.286551	−0.143276	−	0.989683i	$-0.545764\pi$
$677$	−7.45584	−0.286551	−0.143276	+	0.989683i	$0.545764\pi$
$678$	−8.14214	−0.312697
$679$	70.4264	2.70272
$680$	−4.24264	−0.162698
$681$	−24.0000	−0.919682
$682$	−20.2843	−0.776725
$683$	31.5980	1.20906	0.604532	−	0.796581i	$-0.293360\pi$
$683$	31.5980	1.20906	0.604532	+	0.796581i	$0.293360\pi$
$684$	−2.24264	−0.0857495
$685$	1.41421	0.0540343
$686$	44.9706	1.71698
$687$	−1.31371	−0.0501211
$688$	9.65685	0.368164
$689$	53.5980	2.04192
$690$	1.17157	0.0446010
$691$	−41.4558	−1.57705	−0.788527	−	0.615000i	$-0.789156\pi$
$691$	−41.4558	−1.57705	−0.788527	+	0.615000i	$0.789156\pi$
$692$	−11.1716	−0.424679
$693$	−19.3137	−0.733667
$694$	−28.4853	−1.08129
$695$	−0.100505	−0.00381237
$696$	4.97056	0.188409
$697$	0.485281	0.0183813
$698$	17.3137	0.655334
$699$	22.5858	0.854273
$700$	−4.82843	−0.182497
$701$	−13.3137	−0.502852	−0.251426	−	0.967877i	$-0.580899\pi$
$701$	−13.3137	−0.502852	−0.251426	+	0.967877i	$0.580899\pi$
$702$	5.41421	0.204346
$703$	−0.544156	−0.0205232
$704$	−28.0000	−1.05529
$705$	−5.65685	−0.213049
$706$	11.0711	0.416665
$707$	−12.0000	−0.451306
$708$	7.41421	0.278643
$709$	−11.3137	−0.424895	−0.212448	−	0.977172i	$-0.568143\pi$
$709$	−11.3137	−0.424895	−0.212448	+	0.977172i	$0.568143\pi$
$710$	−7.89949	−0.296463
$711$	7.41421	0.278055
$712$	−21.9411	−0.822278
$713$	5.94113	0.222497
$714$	6.82843	0.255547
$715$	−21.6569	−0.809920
$716$	10.8284	0.404677
$717$	−14.3431	−0.535655
$718$	6.72792	0.251084
$719$	−52.0833	−1.94238	−0.971189	−	0.238311i	$-0.923406\pi$
$719$	−52.0833	−1.94238	−0.971189	+	0.238311i	$0.923406\pi$
$720$	−1.00000	−0.0372678
$721$	53.9411	2.00887
$722$	−13.9706	−0.519931
$723$	17.6569	0.656665
$724$	0	0
$725$	−1.65685	−0.0615340
$726$	5.00000	0.185567
$727$	−43.3137	−1.60642	−0.803208	−	0.595698i	$-0.796875\pi$
$727$	−43.3137	−1.60642	−0.803208	+	0.595698i	$0.796875\pi$
$728$	−78.4264	−2.90668
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−13.6569	−0.505117
$732$	0.343146	0.0126830
$733$	−27.1716	−1.00360	−0.501802	−	0.864982i	$-0.667330\pi$
$733$	−27.1716	−1.00360	−0.501802	+	0.864982i	$0.667330\pi$
$734$	−17.6569	−0.651726
$735$	16.3137	0.601740
$736$	5.85786	0.215924
$737$	17.9411	0.660870
$738$	0.343146	0.0126314
$739$	−6.82843	−0.251188	−0.125594	−	0.992082i	$-0.540084\pi$
$739$	−6.82843	−0.251188	−0.125594	+	0.992082i	$0.540084\pi$
$740$	0.242641	0.00891965
$741$	12.1421	0.446052
$742$	47.7990	1.75476
$743$	−6.34315	−0.232707	−0.116354	−	0.993208i	$-0.537121\pi$
$743$	−6.34315	−0.232707	−0.116354	+	0.993208i	$0.537121\pi$
$744$	−15.2132	−0.557743
$745$	−12.0000	−0.439646
$746$	2.68629	0.0983521
$747$	12.8284	0.469368
$748$	5.65685	0.206835
$749$	−46.6274	−1.70373
$750$	1.00000	0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	5.65685	0.206284
$753$	−10.9289	−0.398272
$754$	−8.97056	−0.326689
$755$	−13.6569	−0.497024
$756$	−4.82843	−0.175608
$757$	−24.1421	−0.877461	−0.438730	−	0.898619i	$-0.644572\pi$
$757$	−24.1421	−0.877461	−0.438730	+	0.898619i	$0.644572\pi$
$758$	−31.7990	−1.15499
$759$	−4.68629	−0.170102
$760$	−6.72792	−0.244047
$761$	−25.9411	−0.940365	−0.470183	−	0.882569i	$-0.655812\pi$
$761$	−25.9411	−0.940365	−0.470183	+	0.882569i	$0.655812\pi$
$762$	−1.65685	−0.0600215
$763$	−8.00000	−0.289619
$764$	−3.41421	−0.123522
$765$	1.41421	0.0511310
$766$	16.8284	0.608036
$767$	−40.1421	−1.44945
$768$	−17.0000	−0.613435
$769$	8.82843	0.318361	0.159181	−	0.987249i	$-0.449115\pi$
$769$	8.82843	0.318361	0.159181	+	0.987249i	$0.449115\pi$
$770$	−19.3137	−0.696018
$771$	−7.17157	−0.258278
$772$	−4.24264	−0.152696
$773$	−19.9411	−0.717232	−0.358616	−	0.933485i	$-0.616751\pi$
$773$	−19.9411	−0.717232	−0.358616	+	0.933485i	$0.616751\pi$
$774$	−9.65685	−0.347108
$775$	5.07107	0.182158
$776$	−43.7574	−1.57080
$777$	−1.17157	−0.0420299
$778$	17.5147	0.627933
$779$	0.769553	0.0275721
$780$	−5.41421	−0.193860
$781$	31.5980	1.13066
$782$	1.65685	0.0592490
$783$	−1.65685	−0.0592111
$784$	−16.3137	−0.582632
$785$	5.41421	0.193242
$786$	−6.24264	−0.222668
$787$	−45.2548	−1.61316	−0.806580	−	0.591125i	$-0.798684\pi$
$787$	−45.2548	−1.61316	−0.806580	+	0.591125i	$0.798684\pi$
$788$	−13.7990	−0.491569
$789$	24.0000	0.854423
$790$	7.41421	0.263786
$791$	−39.3137	−1.39783
$792$	12.0000	0.426401
$793$	−1.85786	−0.0659747
$794$	−12.1421	−0.430908
$795$	9.89949	0.351099
$796$	−1.75736	−0.0622879
$797$	−21.1127	−0.747850	−0.373925	−	0.927459i	$-0.621988\pi$
$797$	−21.1127	−0.747850	−0.373925	+	0.927459i	$0.621988\pi$
$798$	10.8284	0.383322
$799$	−8.00000	−0.283020
$800$	5.00000	0.176777
$801$	7.31371	0.258417
$802$	1.00000	0.0353112
$803$	−8.00000	−0.282314
$804$	4.48528	0.158184
$805$	5.65685	0.199378
$806$	27.4558	0.967091
$807$	−12.1421	−0.427423
$808$	7.45584	0.262296
$809$	−9.02944	−0.317458	−0.158729	−	0.987322i	$-0.550740\pi$
$809$	−9.02944	−0.317458	−0.158729	+	0.987322i	$0.550740\pi$
$810$	1.00000	0.0351364
$811$	−27.1127	−0.952056	−0.476028	−	0.879430i	$-0.657924\pi$
$811$	−27.1127	−0.952056	−0.476028	+	0.879430i	$0.657924\pi$
$812$	8.00000	0.280745
$813$	−22.0416	−0.773034
$814$	0.970563	0.0340182
$815$	23.7990	0.833642
$816$	−1.41421	−0.0495074
$817$	−21.6569	−0.757677
$818$	−32.2843	−1.12879
$819$	26.1421	0.913480
$820$	−0.343146	−0.0119832
$821$	0.343146	0.0119759	0.00598793	−	0.999982i	$-0.498094\pi$
$821$	0.343146	0.0119759	0.00598793	+	0.999982i	$0.498094\pi$
$822$	1.41421	0.0493264
$823$	10.8284	0.377455	0.188728	−	0.982029i	$-0.439564\pi$
$823$	10.8284	0.377455	0.188728	+	0.982029i	$0.439564\pi$
$824$	−33.5147	−1.16754
$825$	−4.00000	−0.139262
$826$	−35.7990	−1.24561
$827$	27.5980	0.959676	0.479838	−	0.877357i	$-0.340695\pi$
$827$	27.5980	0.959676	0.479838	+	0.877357i	$0.340695\pi$
$828$	−1.17157	−0.0407150
$829$	12.1421	0.421714	0.210857	−	0.977517i	$-0.432375\pi$
$829$	12.1421	0.421714	0.210857	+	0.977517i	$0.432375\pi$
$830$	12.8284	0.445281
$831$	9.41421	0.326575
$832$	37.8995	1.31393
$833$	23.0711	0.799365
$834$	−0.100505	−0.00348021
$835$	20.4853	0.708922
$836$	8.97056	0.310253
$837$	5.07107	0.175282
$838$	−28.4853	−0.984008
$839$	−37.7574	−1.30353	−0.651764	−	0.758421i	$-0.725971\pi$
$839$	−37.7574	−1.30353	−0.651764	+	0.758421i	$0.725971\pi$
$840$	−14.4853	−0.499790
$841$	−26.2548	−0.905339
$842$	36.6274	1.26226
$843$	1.51472	0.0521697
$844$	17.0711	0.587610
$845$	16.3137	0.561209
$846$	−5.65685	−0.194487
$847$	24.1421	0.829534
$848$	−9.89949	−0.339950
$849$	−8.48528	−0.291214
$850$	1.41421	0.0485071
$851$	−0.284271	−0.00974469
$852$	7.89949	0.270632
$853$	−21.3137	−0.729767	−0.364884	−	0.931053i	$-0.618891\pi$
$853$	−21.3137	−0.729767	−0.364884	+	0.931053i	$0.618891\pi$
$854$	−1.65685	−0.0566964
$855$	2.24264	0.0766967
$856$	28.9706	0.990193
$857$	−0.544156	−0.0185880	−0.00929401	−	0.999957i	$-0.502958\pi$
$857$	−0.544156	−0.0185880	−0.00929401	+	0.999957i	$0.502958\pi$
$858$	−21.6569	−0.739353
$859$	−39.7990	−1.35792	−0.678962	−	0.734173i	$-0.737570\pi$
$859$	−39.7990	−1.35792	−0.678962	+	0.734173i	$0.737570\pi$
$860$	9.65685	0.329296
$861$	1.65685	0.0564654
$862$	−22.7279	−0.774116
$863$	−1.37258	−0.0467233	−0.0233616	−	0.999727i	$-0.507437\pi$
$863$	−1.37258	−0.0467233	−0.0233616	+	0.999727i	$0.507437\pi$
$864$	5.00000	0.170103
$865$	11.1716	0.379845
$866$	14.4853	0.492230
$867$	−15.0000	−0.509427
$868$	−24.4853	−0.831085
$869$	−29.6569	−1.00604
$870$	−1.65685	−0.0561726
$871$	−24.2843	−0.822841
$872$	4.97056	0.168324
$873$	14.5858	0.493654
$874$	2.62742	0.0888737
$875$	4.82843	0.163231
$876$	−2.00000	−0.0675737
$877$	0.443651	0.0149810	0.00749051	−	0.999972i	$-0.497616\pi$
$877$	0.443651	0.0149810	0.00749051	+	0.999972i	$0.497616\pi$
$878$	10.9289	0.368834
$879$	7.27208	0.245281
$880$	4.00000	0.134840
$881$	−1.51472	−0.0510322	−0.0255161	−	0.999674i	$-0.508123\pi$
$881$	−1.51472	−0.0510322	−0.0255161	+	0.999674i	$0.508123\pi$
$882$	16.3137	0.549311
$883$	25.1127	0.845110	0.422555	−	0.906337i	$-0.361133\pi$
$883$	25.1127	0.845110	0.422555	+	0.906337i	$0.361133\pi$
$884$	−7.65685	−0.257528
$885$	−7.41421	−0.249226
$886$	18.8284	0.632553
$887$	−26.6274	−0.894061	−0.447031	−	0.894519i	$-0.647518\pi$
$887$	−26.6274	−0.894061	−0.447031	+	0.894519i	$0.647518\pi$
$888$	0.727922	0.0244275
$889$	−8.00000	−0.268311
$890$	7.31371	0.245156
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−12.6863	−0.424531
$894$	−12.0000	−0.401340
$895$	−10.8284	−0.361954
$896$	−14.4853	−0.483919
$897$	6.34315	0.211791
$898$	−4.34315	−0.144933
$899$	−8.40202	−0.280223
$900$	−1.00000	−0.0333333
$901$	14.0000	0.466408
$902$	−1.37258	−0.0457020
$903$	−46.6274	−1.55166
$904$	24.4264	0.812410
$905$	0	0
$906$	−13.6569	−0.453719
$907$	−0.686292	−0.0227879	−0.0113940	−	0.999935i	$-0.503627\pi$
$907$	−0.686292	−0.0227879	−0.0113940	+	0.999935i	$0.503627\pi$
$908$	24.0000	0.796468
$909$	−2.48528	−0.0824316
$910$	26.1421	0.866603
$911$	39.1127	1.29586	0.647931	−	0.761699i	$-0.275635\pi$
$911$	39.1127	1.29586	0.647931	+	0.761699i	$0.275635\pi$
$912$	−2.24264	−0.0742613
$913$	−51.3137	−1.69824
$914$	−38.4853	−1.27298
$915$	−0.343146	−0.0113440
$916$	1.31371	0.0434062
$917$	−30.1421	−0.995381
$918$	1.41421	0.0466760
$919$	30.0416	0.990982	0.495491	−	0.868613i	$-0.334988\pi$
$919$	30.0416	0.990982	0.495491	+	0.868613i	$0.334988\pi$
$920$	−3.51472	−0.115877
$921$	29.7990	0.981910
$922$	28.1421	0.926812
$923$	−42.7696	−1.40778
$924$	19.3137	0.635374
$925$	−0.242641	−0.00797798
$926$	−6.14214	−0.201843
$927$	11.1716	0.366923
$928$	−8.28427	−0.271945
$929$	−51.6569	−1.69481	−0.847403	−	0.530950i	$-0.821835\pi$
$929$	−51.6569	−1.69481	−0.847403	+	0.530950i	$0.821835\pi$
$930$	5.07107	0.166287
$931$	36.5858	1.19905
$932$	−22.5858	−0.739822
$933$	15.3137	0.501348
$934$	−1.65685	−0.0542139
$935$	−5.65685	−0.184999
$936$	−16.2426	−0.530907
$937$	8.92893	0.291695	0.145848	−	0.989307i	$-0.453409\pi$
$937$	8.92893	0.291695	0.145848	+	0.989307i	$0.453409\pi$
$938$	−21.6569	−0.707121
$939$	−25.7990	−0.841918
$940$	5.65685	0.184506
$941$	−40.1421	−1.30860	−0.654298	−	0.756237i	$-0.727036\pi$
$941$	−40.1421	−1.30860	−0.654298	+	0.756237i	$0.727036\pi$
$942$	5.41421	0.176405
$943$	0.402020	0.0130916
$944$	7.41421	0.241312
$945$	4.82843	0.157069
$946$	38.6274	1.25589
$947$	2.48528	0.0807608	0.0403804	−	0.999184i	$-0.487143\pi$
$947$	2.48528	0.0807608	0.0403804	+	0.999184i	$0.487143\pi$
$948$	−7.41421	−0.240802
$949$	10.8284	0.351506
$950$	2.24264	0.0727609
$951$	−23.5563	−0.763867
$952$	−20.4853	−0.663932
$953$	−39.4558	−1.27810	−0.639050	−	0.769165i	$-0.720672\pi$
$953$	−39.4558	−1.27810	−0.639050	+	0.769165i	$0.720672\pi$
$954$	9.89949	0.320508
$955$	3.41421	0.110481
$956$	14.3431	0.463890
$957$	6.62742	0.214234
$958$	39.1127	1.26367
$959$	6.82843	0.220501
$960$	7.00000	0.225924
$961$	−5.28427	−0.170460
$962$	−1.31371	−0.0423557
$963$	−9.65685	−0.311188
$964$	−17.6569	−0.568689
$965$	4.24264	0.136575
$966$	5.65685	0.182006
$967$	31.7990	1.02259	0.511293	−	0.859406i	$-0.329167\pi$
$967$	31.7990	1.02259	0.511293	+	0.859406i	$0.329167\pi$
$968$	−15.0000	−0.482118
$969$	3.17157	0.101886
$970$	14.5858	0.468321
$971$	−29.4558	−0.945283	−0.472642	−	0.881255i	$-0.656699\pi$
$971$	−29.4558	−0.945283	−0.472642	+	0.881255i	$0.656699\pi$
$972$	−1.00000	−0.0320750
$973$	−0.485281	−0.0155574
$974$	26.4853	0.848643
$975$	5.41421	0.173394
$976$	0.343146	0.0109838
$977$	8.82843	0.282446	0.141223	−	0.989978i	$-0.454896\pi$
$977$	8.82843	0.282446	0.141223	+	0.989978i	$0.454896\pi$
$978$	23.7990	0.761008
$979$	−29.2548	−0.934989
$980$	−16.3137	−0.521122
$981$	−1.65685	−0.0528993
$982$	16.9706	0.541552
$983$	−5.79899	−0.184959	−0.0924795	−	0.995715i	$-0.529479\pi$
$983$	−5.79899	−0.184959	−0.0924795	+	0.995715i	$0.529479\pi$
$984$	−1.02944	−0.0328173
$985$	13.7990	0.439672
$986$	−2.34315	−0.0746210
$987$	−27.3137	−0.869405
$988$	−12.1421	−0.386293
$989$	−11.3137	−0.359755
$990$	−4.00000	−0.127128
$991$	−14.4437	−0.458818	−0.229409	−	0.973330i	$-0.573679\pi$
$991$	−14.4437	−0.458818	−0.229409	+	0.973330i	$0.573679\pi$
$992$	25.3553	0.805033
$993$	0.485281	0.0153999
$994$	−38.1421	−1.20980
$995$	1.75736	0.0557120
$996$	−12.8284	−0.406484
$997$	1.31371	0.0416056	0.0208028	−	0.999784i	$-0.493378\pi$
$997$	1.31371	0.0416056	0.0208028	+	0.999784i	$0.493378\pi$
$998$	−30.8284	−0.975857
$999$	−0.242641	−0.00767681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.a.1.2	✓	2	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:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.82843 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.82843 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +5.41421 q^{13} +4.82843 q^{14} +1.00000 q^{15} -1.00000 q^{16} +1.41421 q^{17} +1.00000 q^{18} +2.24264 q^{19} -1.00000 q^{20} +4.82843 q^{21} -4.00000 q^{22} +1.17157 q^{23} -3.00000 q^{24} +1.00000 q^{25} +5.41421 q^{26} +1.00000 q^{27} -4.82843 q^{28} -1.65685 q^{29} +1.00000 q^{30} +5.07107 q^{31} +5.00000 q^{32} -4.00000 q^{33} +1.41421 q^{34} +4.82843 q^{35} -1.00000 q^{36} -0.242641 q^{37} +2.24264 q^{38} +5.41421 q^{39} -3.00000 q^{40} +0.343146 q^{41} +4.82843 q^{42} -9.65685 q^{43} +4.00000 q^{44} +1.00000 q^{45} +1.17157 q^{46} -5.65685 q^{47} -1.00000 q^{48} +16.3137 q^{49} +1.00000 q^{50} +1.41421 q^{51} -5.41421 q^{52} +9.89949 q^{53} +1.00000 q^{54} -4.00000 q^{55} -14.4853 q^{56} +2.24264 q^{57} -1.65685 q^{58} -7.41421 q^{59} -1.00000 q^{60} -0.343146 q^{61} +5.07107 q^{62} +4.82843 q^{63} +7.00000 q^{64} +5.41421 q^{65} -4.00000 q^{66} -4.48528 q^{67} -1.41421 q^{68} +1.17157 q^{69} +4.82843 q^{70} -7.89949 q^{71} -3.00000 q^{72} +2.00000 q^{73} -0.242641 q^{74} +1.00000 q^{75} -2.24264 q^{76} -19.3137 q^{77} +5.41421 q^{78} +7.41421 q^{79} -1.00000 q^{80} +1.00000 q^{81} +0.343146 q^{82} +12.8284 q^{83} -4.82843 q^{84} +1.41421 q^{85} -9.65685 q^{86} -1.65685 q^{87} +12.0000 q^{88} +7.31371 q^{89} +1.00000 q^{90} +26.1421 q^{91} -1.17157 q^{92} +5.07107 q^{93} -5.65685 q^{94} +2.24264 q^{95} +5.00000 q^{96} +14.5858 q^{97} +16.3137 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^5 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{12} + 8 q^{13} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{18} - 4 q^{19} - 2 q^{20} + 4 q^{21} - 8 q^{22} + 8 q^{23} - 6 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 4 q^{28} + 8 q^{29} + 2 q^{30} - 4 q^{31} + 10 q^{32} - 8 q^{33} + 4 q^{35} - 2 q^{36} + 8 q^{37} - 4 q^{38} + 8 q^{39} - 6 q^{40} + 12 q^{41} + 4 q^{42} - 8 q^{43} + 8 q^{44} + 2 q^{45} + 8 q^{46} - 2 q^{48} + 10 q^{49} + 2 q^{50} - 8 q^{52} + 2 q^{54} - 8 q^{55} - 12 q^{56} - 4 q^{57} + 8 q^{58} - 12 q^{59} - 2 q^{60} - 12 q^{61} - 4 q^{62} + 4 q^{63} + 14 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} + 8 q^{69} + 4 q^{70} + 4 q^{71} - 6 q^{72} + 4 q^{73} + 8 q^{74} + 2 q^{75} + 4 q^{76} - 16 q^{77} + 8 q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 12 q^{82} + 20 q^{83} - 4 q^{84} - 8 q^{86} + 8 q^{87} + 24 q^{88} - 8 q^{89} + 2 q^{90} + 24 q^{91} - 8 q^{92} - 4 q^{93} - 4 q^{95} + 10 q^{96} + 32 q^{97} + 10 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 2 * q^5 + 2 * q^6 + 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 8 * q^11 - 2 * q^12 + 8 * q^13 + 4 * q^14 + 2 * q^15 - 2 * q^16 + 2 * q^18 - 4 * q^19 - 2 * q^20 + 4 * q^21 - 8 * q^22 + 8 * q^23 - 6 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 4 * q^28 + 8 * q^29 + 2 * q^30 - 4 * q^31 + 10 * q^32 - 8 * q^33 + 4 * q^35 - 2 * q^36 + 8 * q^37 - 4 * q^38 + 8 * q^39 - 6 * q^40 + 12 * q^41 + 4 * q^42 - 8 * q^43 + 8 * q^44 + 2 * q^45 + 8 * q^46 - 2 * q^48 + 10 * q^49 + 2 * q^50 - 8 * q^52 + 2 * q^54 - 8 * q^55 - 12 * q^56 - 4 * q^57 + 8 * q^58 - 12 * q^59 - 2 * q^60 - 12 * q^61 - 4 * q^62 + 4 * q^63 + 14 * q^64 + 8 * q^65 - 8 * q^66 + 8 * q^67 + 8 * q^69 + 4 * q^70 + 4 * q^71 - 6 * q^72 + 4 * q^73 + 8 * q^74 + 2 * q^75 + 4 * q^76 - 16 * q^77 + 8 * q^78 + 12 * q^79 - 2 * q^80 + 2 * q^81 + 12 * q^82 + 20 * q^83 - 4 * q^84 - 8 * q^86 + 8 * q^87 + 24 * q^88 - 8 * q^89 + 2 * q^90 + 24 * q^91 - 8 * q^92 - 4 * q^93 - 4 * q^95 + 10 * q^96 + 32 * q^97 + 10 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more