LMFDB - Embedded newform 8470.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +1.00000 q^{10} -0.618034 q^{12} -0.763932 q^{13} -1.00000 q^{14} +0.618034 q^{15} +1.00000 q^{16} +0.381966 q^{17} +2.61803 q^{18} +1.14590 q^{19} -1.00000 q^{20} -0.618034 q^{21} +8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +0.763932 q^{26} +3.47214 q^{27} +1.00000 q^{28} -4.47214 q^{29} -0.618034 q^{30} -6.47214 q^{31} -1.00000 q^{32} -0.381966 q^{34} -1.00000 q^{35} -2.61803 q^{36} -6.76393 q^{37} -1.14590 q^{38} +0.472136 q^{39} +1.00000 q^{40} -3.85410 q^{41} +0.618034 q^{42} +9.09017 q^{43} +2.61803 q^{45} -8.47214 q^{46} +9.23607 q^{47} -0.618034 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.236068 q^{51} -0.763932 q^{52} -7.23607 q^{53} -3.47214 q^{54} -1.00000 q^{56} -0.708204 q^{57} +4.47214 q^{58} +10.8541 q^{59} +0.618034 q^{60} +0.291796 q^{61} +6.47214 q^{62} -2.61803 q^{63} +1.00000 q^{64} +0.763932 q^{65} -5.61803 q^{67} +0.381966 q^{68} -5.23607 q^{69} +1.00000 q^{70} -6.94427 q^{71} +2.61803 q^{72} -3.85410 q^{73} +6.76393 q^{74} -0.618034 q^{75} +1.14590 q^{76} -0.472136 q^{78} +4.00000 q^{79} -1.00000 q^{80} +5.70820 q^{81} +3.85410 q^{82} -2.14590 q^{83} -0.618034 q^{84} -0.381966 q^{85} -9.09017 q^{86} +2.76393 q^{87} -8.14590 q^{89} -2.61803 q^{90} -0.763932 q^{91} +8.47214 q^{92} +4.00000 q^{93} -9.23607 q^{94} -1.14590 q^{95} +0.618034 q^{96} -10.6180 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{12} - 6 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 9 q^{19} - 2 q^{20} + q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} - 2 q^{27} + 2 q^{28} + q^{30} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 2 q^{35} - 3 q^{36} - 18 q^{37} - 9 q^{38} - 8 q^{39} + 2 q^{40} - q^{41} - q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} + 14 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 6 q^{52} - 10 q^{53} + 2 q^{54} - 2 q^{56} + 12 q^{57} + 15 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 6 q^{65} - 9 q^{67} + 3 q^{68} - 6 q^{69} + 2 q^{70} + 4 q^{71} + 3 q^{72} - q^{73} + 18 q^{74} + q^{75} + 9 q^{76} + 8 q^{78} + 8 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} - 11 q^{83} + q^{84} - 3 q^{85} - 7 q^{86} + 10 q^{87} - 23 q^{89} - 3 q^{90} - 6 q^{91} + 8 q^{92} + 8 q^{93} - 14 q^{94} - 9 q^{95} - q^{96} - 19 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 + 2 * q^10 + q^12 - 6 * q^13 - 2 * q^14 - q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 9 * q^19 - 2 * q^20 + q^21 + 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 - 2 * q^27 + 2 * q^28 + q^30 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 2 * q^35 - 3 * q^36 - 18 * q^37 - 9 * q^38 - 8 * q^39 + 2 * q^40 - q^41 - q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 + 14 * q^47 + q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 6 * q^52 - 10 * q^53 + 2 * q^54 - 2 * q^56 + 12 * q^57 + 15 * q^59 - q^60 + 14 * q^61 + 4 * q^62 - 3 * q^63 + 2 * q^64 + 6 * q^65 - 9 * q^67 + 3 * q^68 - 6 * q^69 + 2 * q^70 + 4 * q^71 + 3 * q^72 - q^73 + 18 * q^74 + q^75 + 9 * q^76 + 8 * q^78 + 8 * q^79 - 2 * q^80 - 2 * q^81 + q^82 - 11 * q^83 + q^84 - 3 * q^85 - 7 * q^86 + 10 * q^87 - 23 * q^89 - 3 * q^90 - 6 * q^91 + 8 * q^92 + 8 * q^93 - 14 * q^94 - 9 * q^95 - q^96 - 19 * q^97 - 2 * q^98

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
$2$	−1.00000	−0.707107
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.618034	0.252311
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.61803	−0.872678
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−0.618034	−0.178411
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	−1.00000	−0.267261
$15$	0.618034	0.159576
$16$	1.00000	0.250000
$17$	0.381966	0.0926404	0.0463202	−	0.998927i	$-0.485251\pi$
$17$	0.381966	0.0926404	0.0463202	+	0.998927i	$0.485251\pi$
$18$	2.61803	0.617077
$19$	1.14590	0.262887	0.131444	−	0.991324i	$-0.458039\pi$
$19$	1.14590	0.262887	0.131444	+	0.991324i	$0.458039\pi$
$20$	−1.00000	−0.223607
$21$	−0.618034	−0.134866
$22$	0	0
$23$	8.47214	1.76656	0.883281	−	0.468844i	$-0.155329\pi$
$23$	8.47214	1.76656	0.883281	+	0.468844i	$0.155329\pi$
$24$	0.618034	0.126156
$25$	1.00000	0.200000
$26$	0.763932	0.149819
$27$	3.47214	0.668213
$28$	1.00000	0.188982
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	−0.618034	−0.112837
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.381966	−0.0655066
$35$	−1.00000	−0.169031
$36$	−2.61803	−0.436339
$37$	−6.76393	−1.11198	−0.555992	−	0.831188i	$-0.687661\pi$
$37$	−6.76393	−1.11198	−0.555992	+	0.831188i	$0.687661\pi$
$38$	−1.14590	−0.185889
$39$	0.472136	0.0756023
$40$	1.00000	0.158114
$41$	−3.85410	−0.601910	−0.300955	−	0.953638i	$-0.597305\pi$
$41$	−3.85410	−0.601910	−0.300955	+	0.953638i	$0.597305\pi$
$42$	0.618034	0.0953647
$43$	9.09017	1.38624	0.693119	−	0.720823i	$-0.256236\pi$
$43$	9.09017	1.38624	0.693119	+	0.720823i	$0.256236\pi$
$44$	0	0
$45$	2.61803	0.390273
$46$	−8.47214	−1.24915
$47$	9.23607	1.34722	0.673609	−	0.739087i	$-0.264743\pi$
$47$	9.23607	1.34722	0.673609	+	0.739087i	$0.264743\pi$
$48$	−0.618034	−0.0892055
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−0.236068	−0.0330561
$52$	−0.763932	−0.105938
$53$	−7.23607	−0.993950	−0.496975	−	0.867765i	$-0.665556\pi$
$53$	−7.23607	−0.993950	−0.496975	+	0.867765i	$0.665556\pi$
$54$	−3.47214	−0.472498
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−0.708204	−0.0938039
$58$	4.47214	0.587220
$59$	10.8541	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$59$	10.8541	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$60$	0.618034	0.0797878
$61$	0.291796	0.0373607	0.0186803	−	0.999826i	$-0.494054\pi$
$61$	0.291796	0.0373607	0.0186803	+	0.999826i	$0.494054\pi$
$62$	6.47214	0.821962
$63$	−2.61803	−0.329841
$64$	1.00000	0.125000
$65$	0.763932	0.0947541
$66$	0	0
$67$	−5.61803	−0.686352	−0.343176	−	0.939271i	$-0.611503\pi$
$67$	−5.61803	−0.686352	−0.343176	+	0.939271i	$0.611503\pi$
$68$	0.381966	0.0463202
$69$	−5.23607	−0.630349
$70$	1.00000	0.119523
$71$	−6.94427	−0.824133	−0.412067	−	0.911154i	$-0.635193\pi$
$71$	−6.94427	−0.824133	−0.412067	+	0.911154i	$0.635193\pi$
$72$	2.61803	0.308538
$73$	−3.85410	−0.451089	−0.225544	−	0.974233i	$-0.572416\pi$
$73$	−3.85410	−0.451089	−0.225544	+	0.974233i	$0.572416\pi$
$74$	6.76393	0.786291
$75$	−0.618034	−0.0713644
$76$	1.14590	0.131444
$77$	0	0
$78$	−0.472136	−0.0534589
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	5.70820	0.634245
$82$	3.85410	0.425614
$83$	−2.14590	−0.235543	−0.117771	−	0.993041i	$-0.537575\pi$
$83$	−2.14590	−0.235543	−0.117771	+	0.993041i	$0.537575\pi$
$84$	−0.618034	−0.0674330
$85$	−0.381966	−0.0414300
$86$	−9.09017	−0.980218
$87$	2.76393	0.296325
$88$	0	0
$89$	−8.14590	−0.863463	−0.431732	−	0.902002i	$-0.642097\pi$
$89$	−8.14590	−0.863463	−0.431732	+	0.902002i	$0.642097\pi$
$90$	−2.61803	−0.275965
$91$	−0.763932	−0.0800818
$92$	8.47214	0.883281
$93$	4.00000	0.414781
$94$	−9.23607	−0.952628
$95$	−1.14590	−0.117567
$96$	0.618034	0.0630778
$97$	−10.6180	−1.07810	−0.539049	−	0.842274i	$-0.681216\pi$
$97$	−10.6180	−1.07810	−0.539049	+	0.842274i	$0.681216\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	2.94427	0.292966	0.146483	−	0.989213i	$-0.453205\pi$
$101$	2.94427	0.292966	0.146483	+	0.989213i	$0.453205\pi$
$102$	0.236068	0.0233742
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0.763932	0.0749097
$105$	0.618034	0.0603139
$106$	7.23607	0.702829
$107$	−3.85410	−0.372590	−0.186295	−	0.982494i	$-0.559648\pi$
$107$	−3.85410	−0.372590	−0.186295	+	0.982494i	$0.559648\pi$
$108$	3.47214	0.334106
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	4.18034	0.396780
$112$	1.00000	0.0944911
$113$	−11.3262	−1.06548	−0.532741	−	0.846278i	$-0.678838\pi$
$113$	−11.3262	−1.06548	−0.532741	+	0.846278i	$0.678838\pi$
$114$	0.708204	0.0663294
$115$	−8.47214	−0.790031
$116$	−4.47214	−0.415227
$117$	2.00000	0.184900
$118$	−10.8541	−0.999201
$119$	0.381966	0.0350148
$120$	−0.618034	−0.0564185
$121$	0	0
$122$	−0.291796	−0.0264180
$123$	2.38197	0.214775
$124$	−6.47214	−0.581215
$125$	−1.00000	−0.0894427
$126$	2.61803	0.233233
$127$	3.52786	0.313047	0.156524	−	0.987674i	$-0.449971\pi$
$127$	3.52786	0.313047	0.156524	+	0.987674i	$0.449971\pi$
$128$	−1.00000	−0.0883883
$129$	−5.61803	−0.494640
$130$	−0.763932	−0.0670013
$131$	−6.09017	−0.532101	−0.266050	−	0.963959i	$-0.585719\pi$
$131$	−6.09017	−0.532101	−0.266050	+	0.963959i	$0.585719\pi$
$132$	0	0
$133$	1.14590	0.0993620
$134$	5.61803	0.485324
$135$	−3.47214	−0.298834
$136$	−0.381966	−0.0327533
$137$	−1.85410	−0.158407	−0.0792033	−	0.996858i	$-0.525238\pi$
$137$	−1.85410	−0.158407	−0.0792033	+	0.996858i	$0.525238\pi$
$138$	5.23607	0.445724
$139$	12.9443	1.09792	0.548959	−	0.835849i	$-0.315024\pi$
$139$	12.9443	1.09792	0.548959	+	0.835849i	$0.315024\pi$
$140$	−1.00000	−0.0845154
$141$	−5.70820	−0.480717
$142$	6.94427	0.582750
$143$	0	0
$144$	−2.61803	−0.218169
$145$	4.47214	0.371391
$146$	3.85410	0.318968
$147$	−0.618034	−0.0509746
$148$	−6.76393	−0.555992
$149$	−1.70820	−0.139942	−0.0699708	−	0.997549i	$-0.522291\pi$
$149$	−1.70820	−0.139942	−0.0699708	+	0.997549i	$0.522291\pi$
$150$	0.618034	0.0504623
$151$	9.52786	0.775367	0.387683	−	0.921793i	$-0.373275\pi$
$151$	9.52786	0.775367	0.387683	+	0.921793i	$0.373275\pi$
$152$	−1.14590	−0.0929446
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	6.47214	0.519854
$156$	0.472136	0.0378011
$157$	2.94427	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$157$	2.94427	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$158$	−4.00000	−0.318223
$159$	4.47214	0.354663
$160$	1.00000	0.0790569
$161$	8.47214	0.667698
$162$	−5.70820	−0.448479
$163$	−1.79837	−0.140860	−0.0704298	−	0.997517i	$-0.522437\pi$
$163$	−1.79837	−0.140860	−0.0704298	+	0.997517i	$0.522437\pi$
$164$	−3.85410	−0.300955
$165$	0	0
$166$	2.14590	0.166554
$167$	21.7082	1.67983	0.839916	−	0.542717i	$-0.182604\pi$
$167$	21.7082	1.67983	0.839916	+	0.542717i	$0.182604\pi$
$168$	0.618034	0.0476824
$169$	−12.4164	−0.955108
$170$	0.381966	0.0292955
$171$	−3.00000	−0.229416
$172$	9.09017	0.693119
$173$	−15.8885	−1.20798	−0.603992	−	0.796991i	$-0.706424\pi$
$173$	−15.8885	−1.20798	−0.603992	+	0.796991i	$0.706424\pi$
$174$	−2.76393	−0.209533
$175$	1.00000	0.0755929
$176$	0	0
$177$	−6.70820	−0.504219
$178$	8.14590	0.610561
$179$	8.85410	0.661787	0.330893	−	0.943668i	$-0.392650\pi$
$179$	8.85410	0.661787	0.330893	+	0.943668i	$0.392650\pi$
$180$	2.61803	0.195137
$181$	3.05573	0.227130	0.113565	−	0.993531i	$-0.463773\pi$
$181$	3.05573	0.227130	0.113565	+	0.993531i	$0.463773\pi$
$182$	0.763932	0.0566264
$183$	−0.180340	−0.0133311
$184$	−8.47214	−0.624574
$185$	6.76393	0.497294
$186$	−4.00000	−0.293294
$187$	0	0
$188$	9.23607	0.673609
$189$	3.47214	0.252561
$190$	1.14590	0.0831322
$191$	3.23607	0.234154	0.117077	−	0.993123i	$-0.462648\pi$
$191$	3.23607	0.234154	0.117077	+	0.993123i	$0.462648\pi$
$192$	−0.618034	−0.0446028
$193$	5.41641	0.389882	0.194941	−	0.980815i	$-0.437549\pi$
$193$	5.41641	0.389882	0.194941	+	0.980815i	$0.437549\pi$
$194$	10.6180	0.762330
$195$	−0.472136	−0.0338104
$196$	1.00000	0.0714286
$197$	20.4721	1.45858	0.729290	−	0.684205i	$-0.239851\pi$
$197$	20.4721	1.45858	0.729290	+	0.684205i	$0.239851\pi$
$198$	0	0
$199$	22.9443	1.62648	0.813238	−	0.581931i	$-0.197703\pi$
$199$	22.9443	1.62648	0.813238	+	0.581931i	$0.197703\pi$
$200$	−1.00000	−0.0707107
$201$	3.47214	0.244906
$202$	−2.94427	−0.207158
$203$	−4.47214	−0.313882
$204$	−0.236068	−0.0165281
$205$	3.85410	0.269182
$206$	6.00000	0.418040
$207$	−22.1803	−1.54164
$208$	−0.763932	−0.0529692
$209$	0	0
$210$	−0.618034	−0.0426484
$211$	26.3262	1.81237	0.906186	−	0.422878i	$-0.138980\pi$
$211$	26.3262	1.81237	0.906186	+	0.422878i	$0.138980\pi$
$212$	−7.23607	−0.496975
$213$	4.29180	0.294069
$214$	3.85410	0.263461
$215$	−9.09017	−0.619944
$216$	−3.47214	−0.236249
$217$	−6.47214	−0.439357
$218$	−4.00000	−0.270914
$219$	2.38197	0.160958
$220$	0	0
$221$	−0.291796	−0.0196283
$222$	−4.18034	−0.280566
$223$	−27.1246	−1.81640	−0.908199	−	0.418538i	$-0.862543\pi$
$223$	−27.1246	−1.81640	−0.908199	+	0.418538i	$0.862543\pi$
$224$	−1.00000	−0.0668153
$225$	−2.61803	−0.174536
$226$	11.3262	0.753410
$227$	29.3262	1.94645	0.973225	−	0.229853i	$-0.0738245\pi$
$227$	29.3262	1.94645	0.973225	+	0.229853i	$0.0738245\pi$
$228$	−0.708204	−0.0469020
$229$	−22.3607	−1.47764	−0.738818	−	0.673905i	$-0.764616\pi$
$229$	−22.3607	−1.47764	−0.738818	+	0.673905i	$0.764616\pi$
$230$	8.47214	0.558636
$231$	0	0
$232$	4.47214	0.293610
$233$	16.3262	1.06957	0.534784	−	0.844989i	$-0.320393\pi$
$233$	16.3262	1.06957	0.534784	+	0.844989i	$0.320393\pi$
$234$	−2.00000	−0.130744
$235$	−9.23607	−0.602495
$236$	10.8541	0.706542
$237$	−2.47214	−0.160582
$238$	−0.381966	−0.0247592
$239$	19.7082	1.27482	0.637409	−	0.770526i	$-0.280006\pi$
$239$	19.7082	1.27482	0.637409	+	0.770526i	$0.280006\pi$
$240$	0.618034	0.0398939
$241$	15.9787	1.02928	0.514640	−	0.857407i	$-0.327926\pi$
$241$	15.9787	1.02928	0.514640	+	0.857407i	$0.327926\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0.291796	0.0186803
$245$	−1.00000	−0.0638877
$246$	−2.38197	−0.151869
$247$	−0.875388	−0.0556996
$248$	6.47214	0.410981
$249$	1.32624	0.0840469
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−2.61803	−0.164921
$253$	0	0
$254$	−3.52786	−0.221358
$255$	0.236068	0.0147832
$256$	1.00000	0.0625000
$257$	8.14590	0.508127	0.254064	−	0.967188i	$-0.418233\pi$
$257$	8.14590	0.508127	0.254064	+	0.967188i	$0.418233\pi$
$258$	5.61803	0.349764
$259$	−6.76393	−0.420290
$260$	0.763932	0.0473771
$261$	11.7082	0.724720
$262$	6.09017	0.376252
$263$	−18.3607	−1.13217	−0.566084	−	0.824348i	$-0.691542\pi$
$263$	−18.3607	−1.13217	−0.566084	+	0.824348i	$0.691542\pi$
$264$	0	0
$265$	7.23607	0.444508
$266$	−1.14590	−0.0702595
$267$	5.03444	0.308103
$268$	−5.61803	−0.343176
$269$	12.6525	0.771435	0.385718	−	0.922617i	$-0.373954\pi$
$269$	12.6525	0.771435	0.385718	+	0.922617i	$0.373954\pi$
$270$	3.47214	0.211307
$271$	16.1803	0.982886	0.491443	−	0.870910i	$-0.336470\pi$
$271$	16.1803	0.982886	0.491443	+	0.870910i	$0.336470\pi$
$272$	0.381966	0.0231601
$273$	0.472136	0.0285750
$274$	1.85410	0.112010
$275$	0	0
$276$	−5.23607	−0.315174
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	−12.9443	−0.776346
$279$	16.9443	1.01443
$280$	1.00000	0.0597614
$281$	−4.32624	−0.258082	−0.129041	−	0.991639i	$-0.541190\pi$
$281$	−4.32624	−0.258082	−0.129041	+	0.991639i	$0.541190\pi$
$282$	5.70820	0.339919
$283$	−28.3607	−1.68587	−0.842934	−	0.538017i	$-0.819173\pi$
$283$	−28.3607	−1.68587	−0.842934	+	0.538017i	$0.819173\pi$
$284$	−6.94427	−0.412067
$285$	0.708204	0.0419504
$286$	0	0
$287$	−3.85410	−0.227500
$288$	2.61803	0.154269
$289$	−16.8541	−0.991418
$290$	−4.47214	−0.262613
$291$	6.56231	0.384689
$292$	−3.85410	−0.225544
$293$	9.52786	0.556624	0.278312	−	0.960491i	$-0.410225\pi$
$293$	9.52786	0.556624	0.278312	+	0.960491i	$0.410225\pi$
$294$	0.618034	0.0360445
$295$	−10.8541	−0.631950
$296$	6.76393	0.393146
$297$	0	0
$298$	1.70820	0.0989536
$299$	−6.47214	−0.374293
$300$	−0.618034	−0.0356822
$301$	9.09017	0.523949
$302$	−9.52786	−0.548267
$303$	−1.81966	−0.104537
$304$	1.14590	0.0657218
$305$	−0.291796	−0.0167082
$306$	1.00000	0.0571662
$307$	−1.56231	−0.0891655	−0.0445828	−	0.999006i	$-0.514196\pi$
$307$	−1.56231	−0.0891655	−0.0445828	+	0.999006i	$0.514196\pi$
$308$	0	0
$309$	3.70820	0.210952
$310$	−6.47214	−0.367593
$311$	20.1803	1.14432	0.572161	−	0.820141i	$-0.306105\pi$
$311$	20.1803	1.14432	0.572161	+	0.820141i	$0.306105\pi$
$312$	−0.472136	−0.0267294
$313$	2.27051	0.128337	0.0641684	−	0.997939i	$-0.479561\pi$
$313$	2.27051	0.128337	0.0641684	+	0.997939i	$0.479561\pi$
$314$	−2.94427	−0.166155
$315$	2.61803	0.147510
$316$	4.00000	0.225018
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	−4.47214	−0.250785
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	2.38197	0.132948
$322$	−8.47214	−0.472134
$323$	0.437694	0.0243540
$324$	5.70820	0.317122
$325$	−0.763932	−0.0423753
$326$	1.79837	0.0996027
$327$	−2.47214	−0.136709
$328$	3.85410	0.212807
$329$	9.23607	0.509201
$330$	0	0
$331$	−30.0344	−1.65084	−0.825421	−	0.564517i	$-0.809062\pi$
$331$	−30.0344	−1.65084	−0.825421	+	0.564517i	$0.809062\pi$
$332$	−2.14590	−0.117771
$333$	17.7082	0.970404
$334$	−21.7082	−1.18782
$335$	5.61803	0.306946
$336$	−0.618034	−0.0337165
$337$	−25.7984	−1.40533	−0.702663	−	0.711522i	$-0.748006\pi$
$337$	−25.7984	−1.40533	−0.702663	+	0.711522i	$0.748006\pi$
$338$	12.4164	0.675364
$339$	7.00000	0.380188
$340$	−0.381966	−0.0207150
$341$	0	0
$342$	3.00000	0.162221
$343$	1.00000	0.0539949
$344$	−9.09017	−0.490109
$345$	5.23607	0.281900
$346$	15.8885	0.854173
$347$	3.90983	0.209891	0.104945	−	0.994478i	$-0.466533\pi$
$347$	3.90983	0.209891	0.104945	+	0.994478i	$0.466533\pi$
$348$	2.76393	0.148162
$349$	7.41641	0.396991	0.198496	−	0.980102i	$-0.436394\pi$
$349$	7.41641	0.396991	0.198496	+	0.980102i	$0.436394\pi$
$350$	−1.00000	−0.0534522
$351$	−2.65248	−0.141579
$352$	0	0
$353$	8.56231	0.455726	0.227863	−	0.973693i	$-0.426826\pi$
$353$	8.56231	0.455726	0.227863	+	0.973693i	$0.426826\pi$
$354$	6.70820	0.356537
$355$	6.94427	0.368564
$356$	−8.14590	−0.431732
$357$	−0.236068	−0.0124940
$358$	−8.85410	−0.467954
$359$	−19.5967	−1.03428	−0.517138	−	0.855902i	$-0.673003\pi$
$359$	−19.5967	−1.03428	−0.517138	+	0.855902i	$0.673003\pi$
$360$	−2.61803	−0.137983
$361$	−17.6869	−0.930890
$362$	−3.05573	−0.160606
$363$	0	0
$364$	−0.763932	−0.0400409
$365$	3.85410	0.201733
$366$	0.180340	0.00942652
$367$	−19.8885	−1.03817	−0.519087	−	0.854722i	$-0.673728\pi$
$367$	−19.8885	−1.03817	−0.519087	+	0.854722i	$0.673728\pi$
$368$	8.47214	0.441641
$369$	10.0902	0.525273
$370$	−6.76393	−0.351640
$371$	−7.23607	−0.375678
$372$	4.00000	0.207390
$373$	−16.6525	−0.862233	−0.431116	−	0.902296i	$-0.641880\pi$
$373$	−16.6525	−0.862233	−0.431116	+	0.902296i	$0.641880\pi$
$374$	0	0
$375$	0.618034	0.0319151
$376$	−9.23607	−0.476314
$377$	3.41641	0.175954
$378$	−3.47214	−0.178587
$379$	13.3820	0.687385	0.343693	−	0.939082i	$-0.388322\pi$
$379$	13.3820	0.687385	0.343693	+	0.939082i	$0.388322\pi$
$380$	−1.14590	−0.0587833
$381$	−2.18034	−0.111702
$382$	−3.23607	−0.165572
$383$	2.18034	0.111410	0.0557051	−	0.998447i	$-0.482259\pi$
$383$	2.18034	0.111410	0.0557051	+	0.998447i	$0.482259\pi$
$384$	0.618034	0.0315389
$385$	0	0
$386$	−5.41641	−0.275688
$387$	−23.7984	−1.20974
$388$	−10.6180	−0.539049
$389$	12.2918	0.623219	0.311609	−	0.950210i	$-0.399132\pi$
$389$	12.2918	0.623219	0.311609	+	0.950210i	$0.399132\pi$
$390$	0.472136	0.0239075
$391$	3.23607	0.163655
$392$	−1.00000	−0.0505076
$393$	3.76393	0.189865
$394$	−20.4721	−1.03137
$395$	−4.00000	−0.201262
$396$	0	0
$397$	35.8885	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$397$	35.8885	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$398$	−22.9443	−1.15009
$399$	−0.708204	−0.0354545
$400$	1.00000	0.0500000
$401$	−17.5623	−0.877020	−0.438510	−	0.898726i	$-0.644494\pi$
$401$	−17.5623	−0.877020	−0.438510	+	0.898726i	$0.644494\pi$
$402$	−3.47214	−0.173174
$403$	4.94427	0.246292
$404$	2.94427	0.146483
$405$	−5.70820	−0.283643
$406$	4.47214	0.221948
$407$	0	0
$408$	0.236068	0.0116871
$409$	−5.05573	−0.249990	−0.124995	−	0.992157i	$-0.539891\pi$
$409$	−5.05573	−0.249990	−0.124995	+	0.992157i	$0.539891\pi$
$410$	−3.85410	−0.190341
$411$	1.14590	0.0565230
$412$	−6.00000	−0.295599
$413$	10.8541	0.534095
$414$	22.1803	1.09010
$415$	2.14590	0.105338
$416$	0.763932	0.0374548
$417$	−8.00000	−0.391762
$418$	0	0
$419$	2.43769	0.119089	0.0595446	−	0.998226i	$-0.481035\pi$
$419$	2.43769	0.119089	0.0595446	+	0.998226i	$0.481035\pi$
$420$	0.618034	0.0301570
$421$	16.8328	0.820381	0.410191	−	0.912000i	$-0.365462\pi$
$421$	16.8328	0.820381	0.410191	+	0.912000i	$0.365462\pi$
$422$	−26.3262	−1.28154
$423$	−24.1803	−1.17569
$424$	7.23607	0.351415
$425$	0.381966	0.0185281
$426$	−4.29180	−0.207938
$427$	0.291796	0.0141210
$428$	−3.85410	−0.186295
$429$	0	0
$430$	9.09017	0.438367
$431$	30.6525	1.47648	0.738239	−	0.674539i	$-0.235658\pi$
$431$	30.6525	1.47648	0.738239	+	0.674539i	$0.235658\pi$
$432$	3.47214	0.167053
$433$	15.7984	0.759221	0.379611	−	0.925146i	$-0.376058\pi$
$433$	15.7984	0.759221	0.379611	+	0.925146i	$0.376058\pi$
$434$	6.47214	0.310672
$435$	−2.76393	−0.132520
$436$	4.00000	0.191565
$437$	9.70820	0.464406
$438$	−2.38197	−0.113815
$439$	−36.5410	−1.74401	−0.872004	−	0.489499i	$-0.837180\pi$
$439$	−36.5410	−1.74401	−0.872004	+	0.489499i	$0.837180\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	0.291796	0.0138793
$443$	−8.43769	−0.400887	−0.200443	−	0.979705i	$-0.564238\pi$
$443$	−8.43769	−0.400887	−0.200443	+	0.979705i	$0.564238\pi$
$444$	4.18034	0.198390
$445$	8.14590	0.386153
$446$	27.1246	1.28439
$447$	1.05573	0.0499342
$448$	1.00000	0.0472456
$449$	9.03444	0.426362	0.213181	−	0.977013i	$-0.431618\pi$
$449$	9.03444	0.426362	0.213181	+	0.977013i	$0.431618\pi$
$450$	2.61803	0.123415
$451$	0	0
$452$	−11.3262	−0.532741
$453$	−5.88854	−0.276668
$454$	−29.3262	−1.37635
$455$	0.763932	0.0358137
$456$	0.708204	0.0331647
$457$	20.9787	0.981343	0.490671	−	0.871345i	$-0.336752\pi$
$457$	20.9787	0.981343	0.490671	+	0.871345i	$0.336752\pi$
$458$	22.3607	1.04485
$459$	1.32624	0.0619035
$460$	−8.47214	−0.395015
$461$	40.9443	1.90696	0.953482	−	0.301449i	$-0.0974702\pi$
$461$	40.9443	1.90696	0.953482	+	0.301449i	$0.0974702\pi$
$462$	0	0
$463$	6.94427	0.322728	0.161364	−	0.986895i	$-0.448411\pi$
$463$	6.94427	0.322728	0.161364	+	0.986895i	$0.448411\pi$
$464$	−4.47214	−0.207614
$465$	−4.00000	−0.185496
$466$	−16.3262	−0.756298
$467$	−4.94427	−0.228794	−0.114397	−	0.993435i	$-0.536494\pi$
$467$	−4.94427	−0.228794	−0.114397	+	0.993435i	$0.536494\pi$
$468$	2.00000	0.0924500
$469$	−5.61803	−0.259417
$470$	9.23607	0.426028
$471$	−1.81966	−0.0838455
$472$	−10.8541	−0.499601
$473$	0	0
$474$	2.47214	0.113549
$475$	1.14590	0.0525774
$476$	0.381966	0.0175074
$477$	18.9443	0.867399
$478$	−19.7082	−0.901432
$479$	35.7082	1.63155	0.815775	−	0.578370i	$-0.196311\pi$
$479$	35.7082	1.63155	0.815775	+	0.578370i	$0.196311\pi$
$480$	−0.618034	−0.0282093
$481$	5.16718	0.235603
$482$	−15.9787	−0.727810
$483$	−5.23607	−0.238249
$484$	0	0
$485$	10.6180	0.482140
$486$	13.9443	0.632525
$487$	15.8197	0.716857	0.358429	−	0.933557i	$-0.383313\pi$
$487$	15.8197	0.716857	0.358429	+	0.933557i	$0.383313\pi$
$488$	−0.291796	−0.0132090
$489$	1.11146	0.0502618
$490$	1.00000	0.0451754
$491$	32.2148	1.45383	0.726916	−	0.686726i	$-0.240953\pi$
$491$	32.2148	1.45383	0.726916	+	0.686726i	$0.240953\pi$
$492$	2.38197	0.107387
$493$	−1.70820	−0.0769336
$494$	0.875388	0.0393856
$495$	0	0
$496$	−6.47214	−0.290607
$497$	−6.94427	−0.311493
$498$	−1.32624	−0.0594301
$499$	26.9787	1.20773	0.603867	−	0.797085i	$-0.293626\pi$
$499$	26.9787	1.20773	0.603867	+	0.797085i	$0.293626\pi$
$500$	−1.00000	−0.0447214
$501$	−13.4164	−0.599401
$502$	−20.0000	−0.892644
$503$	10.6525	0.474970	0.237485	−	0.971391i	$-0.423677\pi$
$503$	10.6525	0.474970	0.237485	+	0.971391i	$0.423677\pi$
$504$	2.61803	0.116617
$505$	−2.94427	−0.131018
$506$	0	0
$507$	7.67376	0.340804
$508$	3.52786	0.156524
$509$	−27.1246	−1.20228	−0.601139	−	0.799145i	$-0.705286\pi$
$509$	−27.1246	−1.20228	−0.601139	+	0.799145i	$0.705286\pi$
$510$	−0.236068	−0.0104533
$511$	−3.85410	−0.170495
$512$	−1.00000	−0.0441942
$513$	3.97871	0.175665
$514$	−8.14590	−0.359300
$515$	6.00000	0.264392
$516$	−5.61803	−0.247320
$517$	0	0
$518$	6.76393	0.297190
$519$	9.81966	0.431035
$520$	−0.763932	−0.0335006
$521$	−25.3262	−1.10956	−0.554781	−	0.831996i	$-0.687198\pi$
$521$	−25.3262	−1.10956	−0.554781	+	0.831996i	$0.687198\pi$
$522$	−11.7082	−0.512454
$523$	3.79837	0.166091	0.0830456	−	0.996546i	$-0.473535\pi$
$523$	3.79837	0.166091	0.0830456	+	0.996546i	$0.473535\pi$
$524$	−6.09017	−0.266050
$525$	−0.618034	−0.0269732
$526$	18.3607	0.800564
$527$	−2.47214	−0.107688
$528$	0	0
$529$	48.7771	2.12074
$530$	−7.23607	−0.314315
$531$	−28.4164	−1.23317
$532$	1.14590	0.0496810
$533$	2.94427	0.127531
$534$	−5.03444	−0.217862
$535$	3.85410	0.166627
$536$	5.61803	0.242662
$537$	−5.47214	−0.236140
$538$	−12.6525	−0.545487
$539$	0	0
$540$	−3.47214	−0.149417
$541$	33.0132	1.41935	0.709673	−	0.704531i	$-0.248843\pi$
$541$	33.0132	1.41935	0.709673	+	0.704531i	$0.248843\pi$
$542$	−16.1803	−0.695005
$543$	−1.88854	−0.0810452
$544$	−0.381966	−0.0163767
$545$	−4.00000	−0.171341
$546$	−0.472136	−0.0202056
$547$	−28.8541	−1.23371	−0.616856	−	0.787076i	$-0.711594\pi$
$547$	−28.8541	−1.23371	−0.616856	+	0.787076i	$0.711594\pi$
$548$	−1.85410	−0.0792033
$549$	−0.763932	−0.0326038
$550$	0	0
$551$	−5.12461	−0.218316
$552$	5.23607	0.222862
$553$	4.00000	0.170097
$554$	12.0000	0.509831
$555$	−4.18034	−0.177446
$556$	12.9443	0.548959
$557$	23.2361	0.984544	0.492272	−	0.870441i	$-0.336167\pi$
$557$	23.2361	0.984544	0.492272	+	0.870441i	$0.336167\pi$
$558$	−16.9443	−0.717308
$559$	−6.94427	−0.293711
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	4.32624	0.182491
$563$	27.3820	1.15401	0.577006	−	0.816740i	$-0.304221\pi$
$563$	27.3820	1.15401	0.577006	+	0.816740i	$0.304221\pi$
$564$	−5.70820	−0.240359
$565$	11.3262	0.476498
$566$	28.3607	1.19209
$567$	5.70820	0.239722
$568$	6.94427	0.291375
$569$	28.6738	1.20207	0.601033	−	0.799224i	$-0.294756\pi$
$569$	28.6738	1.20207	0.601033	+	0.799224i	$0.294756\pi$
$570$	−0.708204	−0.0296634
$571$	18.8328	0.788129	0.394064	−	0.919083i	$-0.371069\pi$
$571$	18.8328	0.788129	0.394064	+	0.919083i	$0.371069\pi$
$572$	0	0
$573$	−2.00000	−0.0835512
$574$	3.85410	0.160867
$575$	8.47214	0.353312
$576$	−2.61803	−0.109085
$577$	47.3951	1.97308	0.986542	−	0.163506i	$-0.0522803\pi$
$577$	47.3951	1.97308	0.986542	+	0.163506i	$0.0522803\pi$
$578$	16.8541	0.701038
$579$	−3.34752	−0.139118
$580$	4.47214	0.185695
$581$	−2.14590	−0.0890269
$582$	−6.56231	−0.272016
$583$	0	0
$584$	3.85410	0.159484
$585$	−2.00000	−0.0826898
$586$	−9.52786	−0.393592
$587$	43.2148	1.78366	0.891832	−	0.452366i	$-0.149420\pi$
$587$	43.2148	1.78366	0.891832	+	0.452366i	$0.149420\pi$
$588$	−0.618034	−0.0254873
$589$	−7.41641	−0.305588
$590$	10.8541	0.446856
$591$	−12.6525	−0.520453
$592$	−6.76393	−0.277996
$593$	−4.96556	−0.203911	−0.101956	−	0.994789i	$-0.532510\pi$
$593$	−4.96556	−0.203911	−0.101956	+	0.994789i	$0.532510\pi$
$594$	0	0
$595$	−0.381966	−0.0156591
$596$	−1.70820	−0.0699708
$597$	−14.1803	−0.580363
$598$	6.47214	0.264665
$599$	22.0689	0.901710	0.450855	−	0.892597i	$-0.351119\pi$
$599$	22.0689	0.901710	0.450855	+	0.892597i	$0.351119\pi$
$600$	0.618034	0.0252311
$601$	−17.6738	−0.720928	−0.360464	−	0.932773i	$-0.617382\pi$
$601$	−17.6738	−0.720928	−0.360464	+	0.932773i	$0.617382\pi$
$602$	−9.09017	−0.370488
$603$	14.7082	0.598964
$604$	9.52786	0.387683
$605$	0	0
$606$	1.81966	0.0739186
$607$	46.8328	1.90089	0.950443	−	0.310900i	$-0.100630\pi$
$607$	46.8328	1.90089	0.950443	+	0.310900i	$0.100630\pi$
$608$	−1.14590	−0.0464723
$609$	2.76393	0.112000
$610$	0.291796	0.0118145
$611$	−7.05573	−0.285444
$612$	−1.00000	−0.0404226
$613$	−1.52786	−0.0617098	−0.0308549	−	0.999524i	$-0.509823\pi$
$613$	−1.52786	−0.0617098	−0.0308549	+	0.999524i	$0.509823\pi$
$614$	1.56231	0.0630495
$615$	−2.38197	−0.0960501
$616$	0	0
$617$	14.5623	0.586256	0.293128	−	0.956073i	$-0.405304\pi$
$617$	14.5623	0.586256	0.293128	+	0.956073i	$0.405304\pi$
$618$	−3.70820	−0.149166
$619$	9.72949	0.391061	0.195531	−	0.980698i	$-0.437357\pi$
$619$	9.72949	0.391061	0.195531	+	0.980698i	$0.437357\pi$
$620$	6.47214	0.259927
$621$	29.4164	1.18044
$622$	−20.1803	−0.809158
$623$	−8.14590	−0.326359
$624$	0.472136	0.0189006
$625$	1.00000	0.0400000
$626$	−2.27051	−0.0907478
$627$	0	0
$628$	2.94427	0.117489
$629$	−2.58359	−0.103015
$630$	−2.61803	−0.104305
$631$	−15.1246	−0.602101	−0.301051	−	0.953608i	$-0.597337\pi$
$631$	−15.1246	−0.602101	−0.301051	+	0.953608i	$0.597337\pi$
$632$	−4.00000	−0.159111
$633$	−16.2705	−0.646695
$634$	14.3607	0.570335
$635$	−3.52786	−0.139999
$636$	4.47214	0.177332
$637$	−0.763932	−0.0302681
$638$	0	0
$639$	18.1803	0.719203
$640$	1.00000	0.0395285
$641$	−34.1459	−1.34868	−0.674341	−	0.738420i	$-0.735572\pi$
$641$	−34.1459	−1.34868	−0.674341	+	0.738420i	$0.735572\pi$
$642$	−2.38197	−0.0940087
$643$	−38.7984	−1.53006	−0.765029	−	0.643996i	$-0.777276\pi$
$643$	−38.7984	−1.53006	−0.765029	+	0.643996i	$0.777276\pi$
$644$	8.47214	0.333849
$645$	5.61803	0.221210
$646$	−0.437694	−0.0172208
$647$	24.2918	0.955009	0.477505	−	0.878629i	$-0.341541\pi$
$647$	24.2918	0.955009	0.477505	+	0.878629i	$0.341541\pi$
$648$	−5.70820	−0.224239
$649$	0	0
$650$	0.763932	0.0299639
$651$	4.00000	0.156772
$652$	−1.79837	−0.0704298
$653$	13.3475	0.522329	0.261164	−	0.965294i	$-0.415893\pi$
$653$	13.3475	0.522329	0.261164	+	0.965294i	$0.415893\pi$
$654$	2.47214	0.0966682
$655$	6.09017	0.237963
$656$	−3.85410	−0.150477
$657$	10.0902	0.393655
$658$	−9.23607	−0.360059
$659$	25.9787	1.01199	0.505994	−	0.862537i	$-0.331126\pi$
$659$	25.9787	1.01199	0.505994	+	0.862537i	$0.331126\pi$
$660$	0	0
$661$	24.1803	0.940506	0.470253	−	0.882532i	$-0.344163\pi$
$661$	24.1803	0.940506	0.470253	+	0.882532i	$0.344163\pi$
$662$	30.0344	1.16732
$663$	0.180340	0.00700382
$664$	2.14590	0.0832770
$665$	−1.14590	−0.0444360
$666$	−17.7082	−0.686179
$667$	−37.8885	−1.46705
$668$	21.7082	0.839916
$669$	16.7639	0.648131
$670$	−5.61803	−0.217044
$671$	0	0
$672$	0.618034	0.0238412
$673$	−40.4508	−1.55927	−0.779633	−	0.626237i	$-0.784594\pi$
$673$	−40.4508	−1.55927	−0.779633	+	0.626237i	$0.784594\pi$
$674$	25.7984	0.993716
$675$	3.47214	0.133643
$676$	−12.4164	−0.477554
$677$	−15.4164	−0.592501	−0.296250	−	0.955110i	$-0.595736\pi$
$677$	−15.4164	−0.592501	−0.296250	+	0.955110i	$0.595736\pi$
$678$	−7.00000	−0.268833
$679$	−10.6180	−0.407483
$680$	0.381966	0.0146477
$681$	−18.1246	−0.694537
$682$	0	0
$683$	30.4721	1.16598	0.582992	−	0.812478i	$-0.301882\pi$
$683$	30.4721	1.16598	0.582992	+	0.812478i	$0.301882\pi$
$684$	−3.00000	−0.114708
$685$	1.85410	0.0708416
$686$	−1.00000	−0.0381802
$687$	13.8197	0.527253
$688$	9.09017	0.346559
$689$	5.52786	0.210595
$690$	−5.23607	−0.199334
$691$	12.5623	0.477893	0.238946	−	0.971033i	$-0.423198\pi$
$691$	12.5623	0.477893	0.238946	+	0.971033i	$0.423198\pi$
$692$	−15.8885	−0.603992
$693$	0	0
$694$	−3.90983	−0.148415
$695$	−12.9443	−0.491004
$696$	−2.76393	−0.104767
$697$	−1.47214	−0.0557611
$698$	−7.41641	−0.280715
$699$	−10.0902	−0.381645
$700$	1.00000	0.0377964
$701$	18.8328	0.711306	0.355653	−	0.934618i	$-0.384259\pi$
$701$	18.8328	0.711306	0.355653	+	0.934618i	$0.384259\pi$
$702$	2.65248	0.100111
$703$	−7.75078	−0.292326
$704$	0	0
$705$	5.70820	0.214983
$706$	−8.56231	−0.322247
$707$	2.94427	0.110731
$708$	−6.70820	−0.252110
$709$	−37.7771	−1.41875	−0.709374	−	0.704832i	$-0.751022\pi$
$709$	−37.7771	−1.41875	−0.709374	+	0.704832i	$0.751022\pi$
$710$	−6.94427	−0.260614
$711$	−10.4721	−0.392736
$712$	8.14590	0.305280
$713$	−54.8328	−2.05351
$714$	0.236068	0.00883462
$715$	0	0
$716$	8.85410	0.330893
$717$	−12.1803	−0.454883
$718$	19.5967	0.731344
$719$	−34.0689	−1.27055	−0.635277	−	0.772284i	$-0.719114\pi$
$719$	−34.0689	−1.27055	−0.635277	+	0.772284i	$0.719114\pi$
$720$	2.61803	0.0975684
$721$	−6.00000	−0.223452
$722$	17.6869	0.658239
$723$	−9.87539	−0.367270
$724$	3.05573	0.113565
$725$	−4.47214	−0.166091
$726$	0	0
$727$	−30.2492	−1.12188	−0.560941	−	0.827856i	$-0.689560\pi$
$727$	−30.2492	−1.12188	−0.560941	+	0.827856i	$0.689560\pi$
$728$	0.763932	0.0283132
$729$	−8.50658	−0.315058
$730$	−3.85410	−0.142647
$731$	3.47214	0.128422
$732$	−0.180340	−0.00666555
$733$	13.5967	0.502207	0.251104	−	0.967960i	$-0.419206\pi$
$733$	13.5967	0.502207	0.251104	+	0.967960i	$0.419206\pi$
$734$	19.8885	0.734100
$735$	0.618034	0.0227965
$736$	−8.47214	−0.312287
$737$	0	0
$738$	−10.0902	−0.371424
$739$	16.8541	0.619988	0.309994	−	0.950738i	$-0.399673\pi$
$739$	16.8541	0.619988	0.309994	+	0.950738i	$0.399673\pi$
$740$	6.76393	0.248647
$741$	0.541020	0.0198749
$742$	7.23607	0.265644
$743$	40.2492	1.47660	0.738300	−	0.674472i	$-0.235629\pi$
$743$	40.2492	1.47660	0.738300	+	0.674472i	$0.235629\pi$
$744$	−4.00000	−0.146647
$745$	1.70820	0.0625837
$746$	16.6525	0.609690
$747$	5.61803	0.205553
$748$	0	0
$749$	−3.85410	−0.140826
$750$	−0.618034	−0.0225674
$751$	17.3050	0.631467	0.315733	−	0.948848i	$-0.397750\pi$
$751$	17.3050	0.631467	0.315733	+	0.948848i	$0.397750\pi$
$752$	9.23607	0.336805
$753$	−12.3607	−0.450448
$754$	−3.41641	−0.124418
$755$	−9.52786	−0.346754
$756$	3.47214	0.126280
$757$	13.3475	0.485124	0.242562	−	0.970136i	$-0.422012\pi$
$757$	13.3475	0.485124	0.242562	+	0.970136i	$0.422012\pi$
$758$	−13.3820	−0.486055
$759$	0	0
$760$	1.14590	0.0415661
$761$	50.8115	1.84192	0.920958	−	0.389661i	$-0.127408\pi$
$761$	50.8115	1.84192	0.920958	+	0.389661i	$0.127408\pi$
$762$	2.18034	0.0789854
$763$	4.00000	0.144810
$764$	3.23607	0.117077
$765$	1.00000	0.0361551
$766$	−2.18034	−0.0787789
$767$	−8.29180	−0.299399
$768$	−0.618034	−0.0223014
$769$	−31.3050	−1.12889	−0.564443	−	0.825472i	$-0.690909\pi$
$769$	−31.3050	−1.12889	−0.564443	+	0.825472i	$0.690909\pi$
$770$	0	0
$771$	−5.03444	−0.181311
$772$	5.41641	0.194941
$773$	39.2361	1.41122	0.705612	−	0.708599i	$-0.250672\pi$
$773$	39.2361	1.41122	0.705612	+	0.708599i	$0.250672\pi$
$774$	23.7984	0.855415
$775$	−6.47214	−0.232486
$776$	10.6180	0.381165
$777$	4.18034	0.149969
$778$	−12.2918	−0.440682
$779$	−4.41641	−0.158234
$780$	−0.472136	−0.0169052
$781$	0	0
$782$	−3.23607	−0.115722
$783$	−15.5279	−0.554921
$784$	1.00000	0.0357143
$785$	−2.94427	−0.105086
$786$	−3.76393	−0.134255
$787$	9.72949	0.346819	0.173409	−	0.984850i	$-0.444522\pi$
$787$	9.72949	0.346819	0.173409	+	0.984850i	$0.444522\pi$
$788$	20.4721	0.729290
$789$	11.3475	0.403983
$790$	4.00000	0.142314
$791$	−11.3262	−0.402715
$792$	0	0
$793$	−0.222912	−0.00791585
$794$	−35.8885	−1.27364
$795$	−4.47214	−0.158610
$796$	22.9443	0.813238
$797$	14.1803	0.502293	0.251147	−	0.967949i	$-0.419192\pi$
$797$	14.1803	0.502293	0.251147	+	0.967949i	$0.419192\pi$
$798$	0.708204	0.0250701
$799$	3.52786	0.124807
$800$	−1.00000	−0.0353553
$801$	21.3262	0.753526
$802$	17.5623	0.620147
$803$	0	0
$804$	3.47214	0.122453
$805$	−8.47214	−0.298604
$806$	−4.94427	−0.174155
$807$	−7.81966	−0.275265
$808$	−2.94427	−0.103579
$809$	−42.1591	−1.48223	−0.741117	−	0.671376i	$-0.765703\pi$
$809$	−42.1591	−1.48223	−0.741117	+	0.671376i	$0.765703\pi$
$810$	5.70820	0.200566
$811$	−24.6869	−0.866875	−0.433437	−	0.901184i	$-0.642699\pi$
$811$	−24.6869	−0.866875	−0.433437	+	0.901184i	$0.642699\pi$
$812$	−4.47214	−0.156941
$813$	−10.0000	−0.350715
$814$	0	0
$815$	1.79837	0.0629943
$816$	−0.236068	−0.00826403
$817$	10.4164	0.364424
$818$	5.05573	0.176769
$819$	2.00000	0.0698857
$820$	3.85410	0.134591
$821$	−35.8885	−1.25252	−0.626259	−	0.779615i	$-0.715415\pi$
$821$	−35.8885	−1.25252	−0.626259	+	0.779615i	$0.715415\pi$
$822$	−1.14590	−0.0399678
$823$	−26.1803	−0.912589	−0.456295	−	0.889829i	$-0.650824\pi$
$823$	−26.1803	−0.912589	−0.456295	+	0.889829i	$0.650824\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	−10.8541	−0.377663
$827$	36.6738	1.27527	0.637636	−	0.770338i	$-0.279912\pi$
$827$	36.6738	1.27527	0.637636	+	0.770338i	$0.279912\pi$
$828$	−22.1803	−0.770820
$829$	42.4721	1.47512	0.737559	−	0.675283i	$-0.235978\pi$
$829$	42.4721	1.47512	0.737559	+	0.675283i	$0.235978\pi$
$830$	−2.14590	−0.0744852
$831$	7.41641	0.257272
$832$	−0.763932	−0.0264846
$833$	0.381966	0.0132343
$834$	8.00000	0.277017
$835$	−21.7082	−0.751243
$836$	0	0
$837$	−22.4721	−0.776751
$838$	−2.43769	−0.0842087
$839$	49.7082	1.71612	0.858059	−	0.513551i	$-0.171670\pi$
$839$	49.7082	1.71612	0.858059	+	0.513551i	$0.171670\pi$
$840$	−0.618034	−0.0213242
$841$	−9.00000	−0.310345
$842$	−16.8328	−0.580097
$843$	2.67376	0.0920893
$844$	26.3262	0.906186
$845$	12.4164	0.427137
$846$	24.1803	0.831337
$847$	0	0
$848$	−7.23607	−0.248488
$849$	17.5279	0.601555
$850$	−0.381966	−0.0131013
$851$	−57.3050	−1.96439
$852$	4.29180	0.147035
$853$	13.4164	0.459369	0.229685	−	0.973265i	$-0.426231\pi$
$853$	13.4164	0.459369	0.229685	+	0.973265i	$0.426231\pi$
$854$	−0.291796	−0.00998506
$855$	3.00000	0.102598
$856$	3.85410	0.131730
$857$	4.43769	0.151589	0.0757944	−	0.997123i	$-0.475851\pi$
$857$	4.43769	0.151589	0.0757944	+	0.997123i	$0.475851\pi$
$858$	0	0
$859$	−1.32624	−0.0452507	−0.0226253	−	0.999744i	$-0.507202\pi$
$859$	−1.32624	−0.0452507	−0.0226253	+	0.999744i	$0.507202\pi$
$860$	−9.09017	−0.309972
$861$	2.38197	0.0811772
$862$	−30.6525	−1.04403
$863$	−49.8885	−1.69823	−0.849113	−	0.528211i	$-0.822863\pi$
$863$	−49.8885	−1.69823	−0.849113	+	0.528211i	$0.822863\pi$
$864$	−3.47214	−0.118124
$865$	15.8885	0.540227
$866$	−15.7984	−0.536851
$867$	10.4164	0.353760
$868$	−6.47214	−0.219679
$869$	0	0
$870$	2.76393	0.0937061
$871$	4.29180	0.145422
$872$	−4.00000	−0.135457
$873$	27.7984	0.940832
$874$	−9.70820	−0.328385
$875$	−1.00000	−0.0338062
$876$	2.38197	0.0804792
$877$	−7.23607	−0.244345	−0.122172	−	0.992509i	$-0.538986\pi$
$877$	−7.23607	−0.244345	−0.122172	+	0.992509i	$0.538986\pi$
$878$	36.5410	1.23320
$879$	−5.88854	−0.198616
$880$	0	0
$881$	30.3262	1.02172	0.510858	−	0.859665i	$-0.329328\pi$
$881$	30.3262	1.02172	0.510858	+	0.859665i	$0.329328\pi$
$882$	2.61803	0.0881538
$883$	8.21478	0.276449	0.138225	−	0.990401i	$-0.455860\pi$
$883$	8.21478	0.276449	0.138225	+	0.990401i	$0.455860\pi$
$884$	−0.291796	−0.00981416
$885$	6.70820	0.225494
$886$	8.43769	0.283470
$887$	−40.5410	−1.36123	−0.680617	−	0.732639i	$-0.738288\pi$
$887$	−40.5410	−1.36123	−0.680617	+	0.732639i	$0.738288\pi$
$888$	−4.18034	−0.140283
$889$	3.52786	0.118321
$890$	−8.14590	−0.273051
$891$	0	0
$892$	−27.1246	−0.908199
$893$	10.5836	0.354166
$894$	−1.05573	−0.0353088
$895$	−8.85410	−0.295960
$896$	−1.00000	−0.0334077
$897$	4.00000	0.133556
$898$	−9.03444	−0.301483
$899$	28.9443	0.965346
$900$	−2.61803	−0.0872678
$901$	−2.76393	−0.0920799
$902$	0	0
$903$	−5.61803	−0.186956
$904$	11.3262	0.376705
$905$	−3.05573	−0.101576
$906$	5.88854	0.195634
$907$	19.1459	0.635729	0.317865	−	0.948136i	$-0.397034\pi$
$907$	19.1459	0.635729	0.317865	+	0.948136i	$0.397034\pi$
$908$	29.3262	0.973225
$909$	−7.70820	−0.255665
$910$	−0.763932	−0.0253241
$911$	18.4721	0.612009	0.306005	−	0.952030i	$-0.401008\pi$
$911$	18.4721	0.612009	0.306005	+	0.952030i	$0.401008\pi$
$912$	−0.708204	−0.0234510
$913$	0	0
$914$	−20.9787	−0.693914
$915$	0.180340	0.00596185
$916$	−22.3607	−0.738818
$917$	−6.09017	−0.201115
$918$	−1.32624	−0.0437724
$919$	24.9443	0.822836	0.411418	−	0.911447i	$-0.365034\pi$
$919$	24.9443	0.822836	0.411418	+	0.911447i	$0.365034\pi$
$920$	8.47214	0.279318
$921$	0.965558	0.0318162
$922$	−40.9443	−1.34843
$923$	5.30495	0.174615
$924$	0	0
$925$	−6.76393	−0.222397
$926$	−6.94427	−0.228203
$927$	15.7082	0.515925
$928$	4.47214	0.146805
$929$	22.0902	0.724755	0.362377	−	0.932031i	$-0.381965\pi$
$929$	22.0902	0.724755	0.362377	+	0.932031i	$0.381965\pi$
$930$	4.00000	0.131165
$931$	1.14590	0.0375553
$932$	16.3262	0.534784
$933$	−12.4721	−0.408319
$934$	4.94427	0.161782
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	9.50658	0.310566	0.155283	−	0.987870i	$-0.450371\pi$
$937$	9.50658	0.310566	0.155283	+	0.987870i	$0.450371\pi$
$938$	5.61803	0.183435
$939$	−1.40325	−0.0457934
$940$	−9.23607	−0.301247
$941$	20.3607	0.663739	0.331870	−	0.943325i	$-0.392321\pi$
$941$	20.3607	0.663739	0.331870	+	0.943325i	$0.392321\pi$
$942$	1.81966	0.0592877
$943$	−32.6525	−1.06331
$944$	10.8541	0.353271
$945$	−3.47214	−0.112949
$946$	0	0
$947$	24.6312	0.800406	0.400203	−	0.916426i	$-0.368940\pi$
$947$	24.6312	0.800406	0.400203	+	0.916426i	$0.368940\pi$
$948$	−2.47214	−0.0802912
$949$	2.94427	0.0955751
$950$	−1.14590	−0.0371778
$951$	8.87539	0.287804
$952$	−0.381966	−0.0123796
$953$	−5.85410	−0.189633	−0.0948165	−	0.995495i	$-0.530226\pi$
$953$	−5.85410	−0.189633	−0.0948165	+	0.995495i	$0.530226\pi$
$954$	−18.9443	−0.613343
$955$	−3.23607	−0.104717
$956$	19.7082	0.637409
$957$	0	0
$958$	−35.7082	−1.15368
$959$	−1.85410	−0.0598721
$960$	0.618034	0.0199470
$961$	10.8885	0.351243
$962$	−5.16718	−0.166597
$963$	10.0902	0.325151
$964$	15.9787	0.514640
$965$	−5.41641	−0.174360
$966$	5.23607	0.168468
$967$	48.6525	1.56456	0.782279	−	0.622928i	$-0.214057\pi$
$967$	48.6525	1.56456	0.782279	+	0.622928i	$0.214057\pi$
$968$	0	0
$969$	−0.270510	−0.00869003
$970$	−10.6180	−0.340925
$971$	18.8328	0.604374	0.302187	−	0.953249i	$-0.402283\pi$
$971$	18.8328	0.604374	0.302187	+	0.953249i	$0.402283\pi$
$972$	−13.9443	−0.447263
$973$	12.9443	0.414974
$974$	−15.8197	−0.506895
$975$	0.472136	0.0151205
$976$	0.291796	0.00934016
$977$	61.1935	1.95775	0.978877	−	0.204452i	$-0.0655411\pi$
$977$	61.1935	1.95775	0.978877	+	0.204452i	$0.0655411\pi$
$978$	−1.11146	−0.0355404
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−10.4721	−0.334350
$982$	−32.2148	−1.02801
$983$	8.94427	0.285278	0.142639	−	0.989775i	$-0.454441\pi$
$983$	8.94427	0.285278	0.142639	+	0.989775i	$0.454441\pi$
$984$	−2.38197	−0.0759343
$985$	−20.4721	−0.652296
$986$	1.70820	0.0544003
$987$	−5.70820	−0.181694
$988$	−0.875388	−0.0278498
$989$	77.0132	2.44888
$990$	0	0
$991$	−10.3607	−0.329118	−0.164559	−	0.986367i	$-0.552620\pi$
$991$	−10.3607	−0.329118	−0.164559	+	0.986367i	$0.552620\pi$
$992$	6.47214	0.205491
$993$	18.5623	0.589057
$994$	6.94427	0.220259
$995$	−22.9443	−0.727382
$996$	1.32624	0.0420235
$997$	4.83282	0.153057	0.0765284	−	0.997067i	$-0.475616\pi$
$997$	4.83282	0.153057	0.0765284	+	0.997067i	$0.475616\pi$
$998$	−26.9787	−0.853996
$999$	−23.4853	−0.743042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bp.1.1		2
11.3	even	5		770.2.n.d.141.1	yes	4
11.4	even	5		770.2.n.d.71.1	✓	4
11.10	odd	2		8470.2.a.cb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.d.71.1	✓	4	11.4	even	5
770.2.n.d.141.1	yes	4	11.3	even	5
8470.2.a.bp.1.1		2	1.1	even	1	trivial
8470.2.a.cb.1.1		2	11.10	odd	2

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +1.00000 q^{10} -0.618034 q^{12} -0.763932 q^{13} -1.00000 q^{14} +0.618034 q^{15} +1.00000 q^{16} +0.381966 q^{17} +2.61803 q^{18} +1.14590 q^{19} -1.00000 q^{20} -0.618034 q^{21} +8.47214 q^{23} +0.618034 q^{24} +1.00000 q^{25} +0.763932 q^{26} +3.47214 q^{27} +1.00000 q^{28} -4.47214 q^{29} -0.618034 q^{30} -6.47214 q^{31} -1.00000 q^{32} -0.381966 q^{34} -1.00000 q^{35} -2.61803 q^{36} -6.76393 q^{37} -1.14590 q^{38} +0.472136 q^{39} +1.00000 q^{40} -3.85410 q^{41} +0.618034 q^{42} +9.09017 q^{43} +2.61803 q^{45} -8.47214 q^{46} +9.23607 q^{47} -0.618034 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.236068 q^{51} -0.763932 q^{52} -7.23607 q^{53} -3.47214 q^{54} -1.00000 q^{56} -0.708204 q^{57} +4.47214 q^{58} +10.8541 q^{59} +0.618034 q^{60} +0.291796 q^{61} +6.47214 q^{62} -2.61803 q^{63} +1.00000 q^{64} +0.763932 q^{65} -5.61803 q^{67} +0.381966 q^{68} -5.23607 q^{69} +1.00000 q^{70} -6.94427 q^{71} +2.61803 q^{72} -3.85410 q^{73} +6.76393 q^{74} -0.618034 q^{75} +1.14590 q^{76} -0.472136 q^{78} +4.00000 q^{79} -1.00000 q^{80} +5.70820 q^{81} +3.85410 q^{82} -2.14590 q^{83} -0.618034 q^{84} -0.381966 q^{85} -9.09017 q^{86} +2.76393 q^{87} -8.14590 q^{89} -2.61803 q^{90} -0.763932 q^{91} +8.47214 q^{92} +4.00000 q^{93} -9.23607 q^{94} -1.14590 q^{95} +0.618034 q^{96} -10.6180 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 2 q^{10} + q^{12} - 6 q^{13} - 2 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} + 3 q^{18} + 9 q^{19} - 2 q^{20} + q^{21} + 8 q^{23} - q^{24} + 2 q^{25} + 6 q^{26} - 2 q^{27} + 2 q^{28} + q^{30} - 4 q^{31} - 2 q^{32} - 3 q^{34} - 2 q^{35} - 3 q^{36} - 18 q^{37} - 9 q^{38} - 8 q^{39} + 2 q^{40} - q^{41} - q^{42} + 7 q^{43} + 3 q^{45} - 8 q^{46} + 14 q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} - 6 q^{52} - 10 q^{53} + 2 q^{54} - 2 q^{56} + 12 q^{57} + 15 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} - 3 q^{63} + 2 q^{64} + 6 q^{65} - 9 q^{67} + 3 q^{68} - 6 q^{69} + 2 q^{70} + 4 q^{71} + 3 q^{72} - q^{73} + 18 q^{74} + q^{75} + 9 q^{76} + 8 q^{78} + 8 q^{79} - 2 q^{80} - 2 q^{81} + q^{82} - 11 q^{83} + q^{84} - 3 q^{85} - 7 q^{86} + 10 q^{87} - 23 q^{89} - 3 q^{90} - 6 q^{91} + 8 q^{92} + 8 q^{93} - 14 q^{94} - 9 q^{95} - q^{96} - 19 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 2 * q^7 - 2 * q^8 - 3 * q^9 + 2 * q^10 + q^12 - 6 * q^13 - 2 * q^14 - q^15 + 2 * q^16 + 3 * q^17 + 3 * q^18 + 9 * q^19 - 2 * q^20 + q^21 + 8 * q^23 - q^24 + 2 * q^25 + 6 * q^26 - 2 * q^27 + 2 * q^28 + q^30 - 4 * q^31 - 2 * q^32 - 3 * q^34 - 2 * q^35 - 3 * q^36 - 18 * q^37 - 9 * q^38 - 8 * q^39 + 2 * q^40 - q^41 - q^42 + 7 * q^43 + 3 * q^45 - 8 * q^46 + 14 * q^47 + q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 - 6 * q^52 - 10 * q^53 + 2 * q^54 - 2 * q^56 + 12 * q^57 + 15 * q^59 - q^60 + 14 * q^61 + 4 * q^62 - 3 * q^63 + 2 * q^64 + 6 * q^65 - 9 * q^67 + 3 * q^68 - 6 * q^69 + 2 * q^70 + 4 * q^71 + 3 * q^72 - q^73 + 18 * q^74 + q^75 + 9 * q^76 + 8 * q^78 + 8 * q^79 - 2 * q^80 - 2 * q^81 + q^82 - 11 * q^83 + q^84 - 3 * q^85 - 7 * q^86 + 10 * q^87 - 23 * q^89 - 3 * q^90 - 6 * q^91 + 8 * q^92 + 8 * q^93 - 14 * q^94 - 9 * q^95 - q^96 - 19 * q^97 - 2 * q^98

Introduction

L-functions

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Database

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