LMFDB - Embedded newform 375.2.b.b.124.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	375.124
Dual form			375.2.b.b.124.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +0.618034i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +0.618034i q^{7} +2.23607i q^{8} -1.00000 q^{9} -0.236068 q^{11} +1.61803i q^{12} +6.23607i q^{13} -0.381966 q^{14} +1.85410 q^{16} -6.61803i q^{17} -0.618034i q^{18} +5.00000 q^{19} -0.618034 q^{21} -0.145898i q^{22} +3.47214i q^{23} -2.23607 q^{24} -3.85410 q^{26} -1.00000i q^{27} +1.00000i q^{28} -6.70820 q^{29} -2.14590 q^{31} +5.61803i q^{32} -0.236068i q^{33} +4.09017 q^{34} -1.61803 q^{36} -0.236068i q^{37} +3.09017i q^{38} -6.23607 q^{39} +5.09017 q^{41} -0.381966i q^{42} -8.56231i q^{43} -0.381966 q^{44} -2.14590 q^{46} -3.00000i q^{47} +1.85410i q^{48} +6.61803 q^{49} +6.61803 q^{51} +10.0902i q^{52} -4.09017i q^{53} +0.618034 q^{54} -1.38197 q^{56} +5.00000i q^{57} -4.14590i q^{58} -8.61803 q^{59} -6.61803 q^{61} -1.32624i q^{62} -0.618034i q^{63} +0.236068 q^{64} +0.145898 q^{66} -11.4164i q^{67} -10.7082i q^{68} -3.47214 q^{69} +8.38197 q^{71} -2.23607i q^{72} +4.85410i q^{73} +0.145898 q^{74} +8.09017 q^{76} -0.145898i q^{77} -3.85410i q^{78} +11.7082 q^{79} +1.00000 q^{81} +3.14590i q^{82} -18.0344i q^{83} -1.00000 q^{84} +5.29180 q^{86} -6.70820i q^{87} -0.527864i q^{88} +13.9443 q^{89} -3.85410 q^{91} +5.61803i q^{92} -2.14590i q^{93} +1.85410 q^{94} -5.61803 q^{96} -12.1459i q^{97} +4.09017i q^{98} +0.236068 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9	$ 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} + 20 q^{19} + 2 q^{21} - 2 q^{26} - 22 q^{31} - 6 q^{34} - 2 q^{36} - 16 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{46} + 22 q^{49} + 22 q^{51} - 2 q^{54} - 10 q^{56} - 30 q^{59} - 22 q^{61} - 8 q^{64} + 14 q^{66} + 4 q^{69} + 38 q^{71} + 14 q^{74} + 10 q^{76} + 20 q^{79} + 4 q^{81} - 4 q^{84} + 48 q^{86} + 20 q^{89} - 2 q^{91} - 6 q^{94} - 18 q^{96} - 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 + 20 * q^19 + 2 * q^21 - 2 * q^26 - 22 * q^31 - 6 * q^34 - 2 * q^36 - 16 * q^39 - 2 * q^41 - 6 * q^44 - 22 * q^46 + 22 * q^49 + 22 * q^51 - 2 * q^54 - 10 * q^56 - 30 * q^59 - 22 * q^61 - 8 * q^64 + 14 * q^66 + 4 * q^69 + 38 * q^71 + 14 * q^74 + 10 * q^76 + 20 * q^79 + 4 * q^81 - 4 * q^84 + 48 * q^86 + 20 * q^89 - 2 * q^91 - 6 * q^94 - 18 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0.618034i	0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$	0.618034i	0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	0.618034i	0.233595i	0.993156	+	0.116797i	$0.0372628\pi$
$7$	0.618034i	0.233595i	−0.993156	+	0.116797i	$0.962737\pi$
$8$	2.23607i	0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	1.61803i	0.467086i
$13$	6.23607i	1.72957i	0.502139	+	0.864787i	$0.332547\pi$
$13$	6.23607i	1.72957i	−0.502139	+	0.864787i	$0.667453\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	1.85410	0.463525
$17$	− 6.61803i	− 1.60511i	−0.596579	−	0.802555i	$-0.703474\pi$
$17$	− 6.61803i	− 1.60511i	0.596579	−	0.802555i	$-0.296526\pi$
$18$	− 0.618034i	− 0.145672i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	−0.618034	−0.134866
$22$	− 0.145898i	− 0.0311056i
$23$	3.47214i	0.723990i	0.932180	+	0.361995i	$0.117904\pi$
$23$	3.47214i	0.723990i	−0.932180	+	0.361995i	$0.882096\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−3.85410	−0.755852
$27$	− 1.00000i	− 0.192450i
$28$	1.00000i	0.188982i
$29$	−6.70820	−1.24568	−0.622841	−	0.782348i	$-0.714022\pi$
$29$	−6.70820	−1.24568	−0.622841	+	0.782348i	$0.714022\pi$
$30$	0	0
$31$	−2.14590	−0.385415	−0.192707	−	0.981256i	$-0.561727\pi$
$31$	−2.14590	−0.385415	−0.192707	+	0.981256i	$0.561727\pi$
$32$	5.61803i	0.993137i
$33$	− 0.236068i	− 0.0410942i
$34$	4.09017	0.701458
$35$	0	0
$36$	−1.61803	−0.269672
$37$	− 0.236068i	− 0.0388093i	−0.999812	−	0.0194047i	$-0.993823\pi$
$37$	− 0.236068i	− 0.0388093i	0.999812	−	0.0194047i	$-0.00617709\pi$
$38$	3.09017i	0.501292i
$39$	−6.23607	−0.998570
$40$	0	0
$41$	5.09017	0.794951	0.397475	−	0.917613i	$-0.369886\pi$
$41$	5.09017	0.794951	0.397475	+	0.917613i	$0.369886\pi$
$42$	− 0.381966i	− 0.0589386i
$43$	− 8.56231i	− 1.30574i	−0.757470	−	0.652870i	$-0.773565\pi$
$43$	− 8.56231i	− 1.30574i	0.757470	−	0.652870i	$-0.226435\pi$
$44$	−0.381966	−0.0575835
$45$	0	0
$46$	−2.14590	−0.316395
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	1.85410i	0.267617i
$49$	6.61803	0.945433
$50$	0	0
$51$	6.61803	0.926710
$52$	10.0902i	1.39925i
$53$	− 4.09017i	− 0.561828i	−0.959733	−	0.280914i	$-0.909362\pi$
$53$	− 4.09017i	− 0.561828i	0.959733	−	0.280914i	$-0.0906376\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	−1.38197	−0.184673
$57$	5.00000i	0.662266i
$58$	− 4.14590i	− 0.544383i
$59$	−8.61803	−1.12197	−0.560986	−	0.827825i	$-0.689578\pi$
$59$	−8.61803	−1.12197	−0.560986	+	0.827825i	$0.689578\pi$
$60$	0	0
$61$	−6.61803	−0.847352	−0.423676	−	0.905814i	$-0.639261\pi$
$61$	−6.61803	−0.847352	−0.423676	+	0.905814i	$0.639261\pi$
$62$	− 1.32624i	− 0.168432i
$63$	− 0.618034i	− 0.0778650i
$64$	0.236068	0.0295085
$65$	0	0
$66$	0.145898	0.0179588
$67$	− 11.4164i	− 1.39474i	−0.716713	−	0.697368i	$-0.754354\pi$
$67$	− 11.4164i	− 1.39474i	0.716713	−	0.697368i	$-0.245646\pi$
$68$	− 10.7082i	− 1.29856i
$69$	−3.47214	−0.417996
$70$	0	0
$71$	8.38197	0.994756	0.497378	−	0.867534i	$-0.334296\pi$
$71$	8.38197	0.994756	0.497378	+	0.867534i	$0.334296\pi$
$72$	− 2.23607i	− 0.263523i
$73$	4.85410i	0.568130i	0.958805	+	0.284065i	$0.0916831\pi$
$73$	4.85410i	0.568130i	−0.958805	+	0.284065i	$0.908317\pi$
$74$	0.145898	0.0169603
$75$	0	0
$76$	8.09017	0.928006
$77$	− 0.145898i	− 0.0166266i
$78$	− 3.85410i	− 0.436391i
$79$	11.7082	1.31728	0.658638	−	0.752460i	$-0.271133\pi$
$79$	11.7082	1.31728	0.658638	+	0.752460i	$0.271133\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.14590i	0.347406i
$83$	− 18.0344i	− 1.97954i	−0.142682	−	0.989769i	$-0.545573\pi$
$83$	− 18.0344i	− 1.97954i	0.142682	−	0.989769i	$-0.454427\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	5.29180	0.570629
$87$	− 6.70820i	− 0.719195i
$88$	− 0.527864i	− 0.0562705i
$89$	13.9443	1.47809	0.739045	−	0.673656i	$-0.235277\pi$
$89$	13.9443	1.47809	0.739045	+	0.673656i	$0.235277\pi$
$90$	0	0
$91$	−3.85410	−0.404020
$92$	5.61803i	0.585721i
$93$	− 2.14590i	− 0.222519i
$94$	1.85410	0.191236
$95$	0	0
$96$	−5.61803	−0.573388
$97$	− 12.1459i	− 1.23323i	−0.787265	−	0.616615i	$-0.788504\pi$
$97$	− 12.1459i	− 1.23323i	0.787265	−	0.616615i	$-0.211496\pi$
$98$	4.09017i	0.413170i
$99$	0.236068	0.0237257
$100$	0	0
$101$	−16.9443	−1.68602	−0.843009	−	0.537899i	$-0.819218\pi$
$101$	−16.9443	−1.68602	−0.843009	+	0.537899i	$0.819218\pi$
$102$	4.09017i	0.404987i
$103$	− 14.0902i	− 1.38835i	−0.719808	−	0.694173i	$-0.755770\pi$
$103$	− 14.0902i	− 1.38835i	0.719808	−	0.694173i	$-0.244230\pi$
$104$	−13.9443	−1.36735
$105$	0	0
$106$	2.52786	0.245528
$107$	8.38197i	0.810315i	0.914247	+	0.405158i	$0.132783\pi$
$107$	8.38197i	0.810315i	−0.914247	+	0.405158i	$0.867217\pi$
$108$	− 1.61803i	− 0.155695i
$109$	0.527864	0.0505602	0.0252801	−	0.999680i	$-0.491952\pi$
$109$	0.527864	0.0505602	0.0252801	+	0.999680i	$0.491952\pi$
$110$	0	0
$111$	0.236068	0.0224066
$112$	1.14590i	0.108277i
$113$	6.76393i	0.636297i	0.948041	+	0.318149i	$0.103061\pi$
$113$	6.76393i	0.636297i	−0.948041	+	0.318149i	$0.896939\pi$
$114$	−3.09017	−0.289421
$115$	0	0
$116$	−10.8541	−1.00778
$117$	− 6.23607i	− 0.576525i
$118$	− 5.32624i	− 0.490320i
$119$	4.09017	0.374945
$120$	0	0
$121$	−10.9443	−0.994934
$122$	− 4.09017i	− 0.370307i
$123$	5.09017i	0.458965i
$124$	−3.47214	−0.311807
$125$	0	0
$126$	0.381966	0.0340282
$127$	5.29180i	0.469571i	0.972047	+	0.234785i	$0.0754388\pi$
$127$	5.29180i	0.469571i	−0.972047	+	0.234785i	$0.924561\pi$
$128$	11.3820i	1.00603i
$129$	8.56231	0.753869
$130$	0	0
$131$	16.7984	1.46768	0.733840	−	0.679322i	$-0.237726\pi$
$131$	16.7984	1.46768	0.733840	+	0.679322i	$0.237726\pi$
$132$	− 0.381966i	− 0.0332459i
$133$	3.09017i	0.267952i
$134$	7.05573	0.609522
$135$	0	0
$136$	14.7984	1.26895
$137$	1.47214i	0.125773i	0.998021	+	0.0628865i	$0.0200306\pi$
$137$	1.47214i	0.125773i	−0.998021	+	0.0628865i	$0.979969\pi$
$138$	− 2.14590i	− 0.182671i
$139$	−6.70820	−0.568982	−0.284491	−	0.958679i	$-0.591825\pi$
$139$	−6.70820	−0.568982	−0.284491	+	0.958679i	$0.591825\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	5.18034i	0.434724i
$143$	− 1.47214i	− 0.123106i
$144$	−1.85410	−0.154508
$145$	0	0
$146$	−3.00000	−0.248282
$147$	6.61803i	0.545846i
$148$	− 0.381966i	− 0.0313974i
$149$	−6.05573	−0.496105	−0.248052	−	0.968747i	$-0.579790\pi$
$149$	−6.05573	−0.496105	−0.248052	+	0.968747i	$0.579790\pi$
$150$	0	0
$151$	3.70820	0.301769	0.150885	−	0.988551i	$-0.451788\pi$
$151$	3.70820	0.301769	0.150885	+	0.988551i	$0.451788\pi$
$152$	11.1803i	0.906845i
$153$	6.61803i	0.535036i
$154$	0.0901699	0.00726610
$155$	0	0
$156$	−10.0902	−0.807860
$157$	9.56231i	0.763155i	0.924337	+	0.381578i	$0.124619\pi$
$157$	9.56231i	0.763155i	−0.924337	+	0.381578i	$0.875381\pi$
$158$	7.23607i	0.575671i
$159$	4.09017	0.324372
$160$	0	0
$161$	−2.14590	−0.169120
$162$	0.618034i	0.0485573i
$163$	15.1803i	1.18902i	0.804090	+	0.594508i	$0.202653\pi$
$163$	15.1803i	1.18902i	−0.804090	+	0.594508i	$0.797347\pi$
$164$	8.23607	0.643129
$165$	0	0
$166$	11.1459	0.865089
$167$	− 12.4721i	− 0.965123i	−0.875862	−	0.482561i	$-0.839707\pi$
$167$	− 12.4721i	− 0.965123i	0.875862	−	0.482561i	$-0.160293\pi$
$168$	− 1.38197i	− 0.106621i
$169$	−25.8885	−1.99143
$170$	0	0
$171$	−5.00000	−0.382360
$172$	− 13.8541i	− 1.05637i
$173$	11.5623i	0.879066i	0.898226	+	0.439533i	$0.144856\pi$
$173$	11.5623i	0.879066i	−0.898226	+	0.439533i	$0.855144\pi$
$174$	4.14590	0.314300
$175$	0	0
$176$	−0.437694	−0.0329924
$177$	− 8.61803i	− 0.647771i
$178$	8.61803i	0.645949i
$179$	−2.23607	−0.167132	−0.0835658	−	0.996502i	$-0.526631\pi$
$179$	−2.23607	−0.167132	−0.0835658	+	0.996502i	$0.526631\pi$
$180$	0	0
$181$	−6.94427	−0.516164	−0.258082	−	0.966123i	$-0.583090\pi$
$181$	−6.94427	−0.516164	−0.258082	+	0.966123i	$0.583090\pi$
$182$	− 2.38197i	− 0.176563i
$183$	− 6.61803i	− 0.489219i
$184$	−7.76393	−0.572365
$185$	0	0
$186$	1.32624	0.0972445
$187$	1.56231i	0.114247i
$188$	− 4.85410i	− 0.354022i
$189$	0.618034	0.0449554
$190$	0	0
$191$	−17.4721	−1.26424	−0.632120	−	0.774871i	$-0.717815\pi$
$191$	−17.4721	−1.26424	−0.632120	+	0.774871i	$0.717815\pi$
$192$	0.236068i	0.0170367i
$193$	23.4721i	1.68956i	0.535113	+	0.844781i	$0.320269\pi$
$193$	23.4721i	1.68956i	−0.535113	+	0.844781i	$0.679731\pi$
$194$	7.50658	0.538941
$195$	0	0
$196$	10.7082	0.764872
$197$	14.8885i	1.06076i	0.847759	+	0.530382i	$0.177952\pi$
$197$	14.8885i	1.06076i	−0.847759	+	0.530382i	$0.822048\pi$
$198$	0.145898i	0.0103685i
$199$	−3.29180	−0.233349	−0.116675	−	0.993170i	$-0.537223\pi$
$199$	−3.29180	−0.233349	−0.116675	+	0.993170i	$0.537223\pi$
$200$	0	0
$201$	11.4164	0.805251
$202$	− 10.4721i	− 0.736817i
$203$	− 4.14590i	− 0.290985i
$204$	10.7082	0.749724
$205$	0	0
$206$	8.70820	0.606729
$207$	− 3.47214i	− 0.241330i
$208$	11.5623i	0.801702i
$209$	−1.18034	−0.0816458
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	− 6.61803i	− 0.454528i
$213$	8.38197i	0.574323i
$214$	−5.18034	−0.354121
$215$	0	0
$216$	2.23607	0.152145
$217$	− 1.32624i	− 0.0900309i
$218$	0.326238i	0.0220956i
$219$	−4.85410	−0.328010
$220$	0	0
$221$	41.2705	2.77615
$222$	0.145898i	0.00979203i
$223$	− 12.7082i	− 0.851004i	−0.904957	−	0.425502i	$-0.860097\pi$
$223$	− 12.7082i	− 0.851004i	0.904957	−	0.425502i	$-0.139903\pi$
$224$	−3.47214	−0.231992
$225$	0	0
$226$	−4.18034	−0.278072
$227$	− 9.70820i	− 0.644356i	−0.946679	−	0.322178i	$-0.895585\pi$
$227$	− 9.70820i	− 0.644356i	0.946679	−	0.322178i	$-0.104415\pi$
$228$	8.09017i	0.535785i
$229$	−11.7082	−0.773700	−0.386850	−	0.922143i	$-0.626437\pi$
$229$	−11.7082	−0.773700	−0.386850	+	0.922143i	$0.626437\pi$
$230$	0	0
$231$	0.145898	0.00959939
$232$	− 15.0000i	− 0.984798i
$233$	21.0344i	1.37801i	0.724756	+	0.689006i	$0.241953\pi$
$233$	21.0344i	1.37801i	−0.724756	+	0.689006i	$0.758047\pi$
$234$	3.85410	0.251951
$235$	0	0
$236$	−13.9443	−0.907695
$237$	11.7082i	0.760530i
$238$	2.52786i	0.163857i
$239$	17.5623	1.13601	0.568006	−	0.823025i	$-0.307715\pi$
$239$	17.5623	1.13601	0.568006	+	0.823025i	$0.307715\pi$
$240$	0	0
$241$	−13.1246	−0.845431	−0.422715	−	0.906263i	$-0.638923\pi$
$241$	−13.1246	−0.845431	−0.422715	+	0.906263i	$0.638923\pi$
$242$	− 6.76393i	− 0.434802i
$243$	1.00000i	0.0641500i
$244$	−10.7082	−0.685523
$245$	0	0
$246$	−3.14590	−0.200575
$247$	31.1803i	1.98396i
$248$	− 4.79837i	− 0.304697i
$249$	18.0344	1.14289
$250$	0	0
$251$	−15.0344	−0.948966	−0.474483	−	0.880265i	$-0.657365\pi$
$251$	−15.0344	−0.948966	−0.474483	+	0.880265i	$0.657365\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	− 0.819660i	− 0.0515316i
$254$	−3.27051	−0.205210
$255$	0	0
$256$	−6.56231	−0.410144
$257$	1.47214i	0.0918293i	0.998945	+	0.0459147i	$0.0146202\pi$
$257$	1.47214i	0.0918293i	−0.998945	+	0.0459147i	$0.985380\pi$
$258$	5.29180i	0.329453i
$259$	0.145898	0.00906566
$260$	0	0
$261$	6.70820	0.415227
$262$	10.3820i	0.641400i
$263$	− 4.61803i	− 0.284760i	−0.989812	−	0.142380i	$-0.954524\pi$
$263$	− 4.61803i	− 0.284760i	0.989812	−	0.142380i	$-0.0454755\pi$
$264$	0.527864	0.0324878
$265$	0	0
$266$	−1.90983	−0.117099
$267$	13.9443i	0.853376i
$268$	− 18.4721i	− 1.12837i
$269$	30.9787	1.88881	0.944403	−	0.328791i	$-0.106641\pi$
$269$	30.9787	1.88881	0.944403	+	0.328791i	$0.106641\pi$
$270$	0	0
$271$	−17.7984	−1.08117	−0.540587	−	0.841288i	$-0.681798\pi$
$271$	−17.7984	−1.08117	−0.540587	+	0.841288i	$0.681798\pi$
$272$	− 12.2705i	− 0.744009i
$273$	− 3.85410i	− 0.233261i
$274$	−0.909830	−0.0549648
$275$	0	0
$276$	−5.61803	−0.338166
$277$	0.0901699i	0.00541779i	0.999996	+	0.00270889i	$0.000862269\pi$
$277$	0.0901699i	0.00541779i	−0.999996	+	0.00270889i	$0.999138\pi$
$278$	− 4.14590i	− 0.248654i
$279$	2.14590	0.128472
$280$	0	0
$281$	−6.09017	−0.363309	−0.181655	−	0.983362i	$-0.558145\pi$
$281$	−6.09017	−0.363309	−0.181655	+	0.983362i	$0.558145\pi$
$282$	1.85410i	0.110410i
$283$	− 7.18034i	− 0.426827i	−0.976962	−	0.213413i	$-0.931542\pi$
$283$	− 7.18034i	− 0.426827i	0.976962	−	0.213413i	$-0.0684581\pi$
$284$	13.5623	0.804775
$285$	0	0
$286$	0.909830	0.0537994
$287$	3.14590i	0.185696i
$288$	− 5.61803i	− 0.331046i
$289$	−26.7984	−1.57637
$290$	0	0
$291$	12.1459	0.712005
$292$	7.85410i	0.459627i
$293$	16.3607i	0.955801i	0.878414	+	0.477901i	$0.158602\pi$
$293$	16.3607i	0.955801i	−0.878414	+	0.477901i	$0.841398\pi$
$294$	−4.09017	−0.238544
$295$	0	0
$296$	0.527864	0.0306815
$297$	0.236068i	0.0136981i
$298$	− 3.74265i	− 0.216806i
$299$	−21.6525	−1.25220
$300$	0	0
$301$	5.29180	0.305014
$302$	2.29180i	0.131878i
$303$	− 16.9443i	− 0.973423i
$304$	9.27051	0.531700
$305$	0	0
$306$	−4.09017	−0.233819
$307$	− 16.4164i	− 0.936934i	−0.883481	−	0.468467i	$-0.844807\pi$
$307$	− 16.4164i	− 0.936934i	0.883481	−	0.468467i	$-0.155193\pi$
$308$	− 0.236068i	− 0.0134512i
$309$	14.0902	0.801562
$310$	0	0
$311$	14.5623	0.825753	0.412876	−	0.910787i	$-0.364524\pi$
$311$	14.5623	0.825753	0.412876	+	0.910787i	$0.364524\pi$
$312$	− 13.9443i	− 0.789439i
$313$	− 25.2705i	− 1.42837i	−0.699955	−	0.714187i	$-0.746797\pi$
$313$	− 25.2705i	− 1.42837i	0.699955	−	0.714187i	$-0.253203\pi$
$314$	−5.90983	−0.333511
$315$	0	0
$316$	18.9443	1.06570
$317$	24.2361i	1.36123i	0.732640	+	0.680617i	$0.238288\pi$
$317$	24.2361i	1.36123i	−0.732640	+	0.680617i	$0.761712\pi$
$318$	2.52786i	0.141756i
$319$	1.58359	0.0886641
$320$	0	0
$321$	−8.38197	−0.467836
$322$	− 1.32624i	− 0.0739083i
$323$	− 33.0902i	− 1.84119i
$324$	1.61803	0.0898908
$325$	0	0
$326$	−9.38197	−0.519619
$327$	0.527864i	0.0291909i
$328$	11.3820i	0.628464i
$329$	1.85410	0.102220
$330$	0	0
$331$	−10.5623	−0.580557	−0.290278	−	0.956942i	$-0.593748\pi$
$331$	−10.5623	−0.580557	−0.290278	+	0.956942i	$0.593748\pi$
$332$	− 29.1803i	− 1.60148i
$333$	0.236068i	0.0129364i
$334$	7.70820	0.421774
$335$	0	0
$336$	−1.14590	−0.0625139
$337$	0.944272i	0.0514378i	0.999669	+	0.0257189i	$0.00818748\pi$
$337$	0.944272i	0.0514378i	−0.999669	+	0.0257189i	$0.991813\pi$
$338$	− 16.0000i	− 0.870285i
$339$	−6.76393	−0.367366
$340$	0	0
$341$	0.506578	0.0274327
$342$	− 3.09017i	− 0.167097i
$343$	8.41641i	0.454443i
$344$	19.1459	1.03228
$345$	0	0
$346$	−7.14590	−0.384166
$347$	13.9098i	0.746719i	0.927687	+	0.373359i	$0.121794\pi$
$347$	13.9098i	0.746719i	−0.927687	+	0.373359i	$0.878206\pi$
$348$	− 10.8541i	− 0.581841i
$349$	−5.32624	−0.285107	−0.142553	−	0.989787i	$-0.545531\pi$
$349$	−5.32624	−0.285107	−0.142553	+	0.989787i	$0.545531\pi$
$350$	0	0
$351$	6.23607	0.332857
$352$	− 1.32624i	− 0.0706887i
$353$	− 10.1459i	− 0.540012i	−0.962859	−	0.270006i	$-0.912974\pi$
$353$	− 10.1459i	− 0.540012i	0.962859	−	0.270006i	$-0.0870256\pi$
$354$	5.32624	0.283086
$355$	0	0
$356$	22.5623	1.19580
$357$	4.09017i	0.216475i
$358$	− 1.38197i	− 0.0730392i
$359$	−18.9443	−0.999840	−0.499920	−	0.866071i	$-0.666637\pi$
$359$	−18.9443	−0.999840	−0.499920	+	0.866071i	$0.666637\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	− 4.29180i	− 0.225572i
$363$	− 10.9443i	− 0.574425i
$364$	−6.23607	−0.326859
$365$	0	0
$366$	4.09017	0.213797
$367$	8.38197i	0.437535i	0.975777	+	0.218768i	$0.0702036\pi$
$367$	8.38197i	0.437535i	−0.975777	+	0.218768i	$0.929796\pi$
$368$	6.43769i	0.335588i
$369$	−5.09017	−0.264984
$370$	0	0
$371$	2.52786	0.131240
$372$	− 3.47214i	− 0.180022i
$373$	− 2.38197i	− 0.123334i	−0.998097	−	0.0616668i	$-0.980358\pi$
$373$	− 2.38197i	− 0.123334i	0.998097	−	0.0616668i	$-0.0196416\pi$
$374$	−0.965558	−0.0499278
$375$	0	0
$376$	6.70820	0.345949
$377$	− 41.8328i	− 2.15450i
$378$	0.381966i	0.0196462i
$379$	2.43769	0.125216	0.0626080	−	0.998038i	$-0.480058\pi$
$379$	2.43769	0.125216	0.0626080	+	0.998038i	$0.480058\pi$
$380$	0	0
$381$	−5.29180	−0.271107
$382$	− 10.7984i	− 0.552493i
$383$	− 23.5623i	− 1.20398i	−0.798505	−	0.601989i	$-0.794375\pi$
$383$	− 23.5623i	− 1.20398i	0.798505	−	0.601989i	$-0.205625\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−14.5066	−0.738365
$387$	8.56231i	0.435246i
$388$	− 19.6525i	− 0.997703i
$389$	10.8541	0.550325	0.275162	−	0.961398i	$-0.411268\pi$
$389$	10.8541	0.550325	0.275162	+	0.961398i	$0.411268\pi$
$390$	0	0
$391$	22.9787	1.16208
$392$	14.7984i	0.747431i
$393$	16.7984i	0.847366i
$394$	−9.20163	−0.463571
$395$	0	0
$396$	0.381966	0.0191945
$397$	17.5279i	0.879698i	0.898072	+	0.439849i	$0.144968\pi$
$397$	17.5279i	0.879698i	−0.898072	+	0.439849i	$0.855032\pi$
$398$	− 2.03444i	− 0.101977i
$399$	−3.09017	−0.154702
$400$	0	0
$401$	14.2361	0.710915	0.355458	−	0.934692i	$-0.384325\pi$
$401$	14.2361	0.710915	0.355458	+	0.934692i	$0.384325\pi$
$402$	7.05573i	0.351908i
$403$	− 13.3820i	− 0.666603i
$404$	−27.4164	−1.36402
$405$	0	0
$406$	2.56231	0.127165
$407$	0.0557281i	0.00276234i
$408$	14.7984i	0.732629i
$409$	−8.29180	−0.410003	−0.205001	−	0.978762i	$-0.565720\pi$
$409$	−8.29180	−0.410003	−0.205001	+	0.978762i	$0.565720\pi$
$410$	0	0
$411$	−1.47214	−0.0726151
$412$	− 22.7984i	− 1.12320i
$413$	− 5.32624i	− 0.262087i
$414$	2.14590	0.105465
$415$	0	0
$416$	−35.0344	−1.71770
$417$	− 6.70820i	− 0.328502i
$418$	− 0.729490i	− 0.0356805i
$419$	7.03444	0.343655	0.171827	−	0.985127i	$-0.445033\pi$
$419$	7.03444	0.343655	0.171827	+	0.985127i	$0.445033\pi$
$420$	0	0
$421$	14.3607	0.699897	0.349948	−	0.936769i	$-0.386199\pi$
$421$	14.3607	0.699897	0.349948	+	0.936769i	$0.386199\pi$
$422$	− 4.94427i	− 0.240683i
$423$	3.00000i	0.145865i
$424$	9.14590	0.444164
$425$	0	0
$426$	−5.18034	−0.250988
$427$	− 4.09017i	− 0.197937i
$428$	13.5623i	0.655559i
$429$	1.47214	0.0710754
$430$	0	0
$431$	−6.81966	−0.328491	−0.164246	−	0.986419i	$-0.552519\pi$
$431$	−6.81966	−0.328491	−0.164246	+	0.986419i	$0.552519\pi$
$432$	− 1.85410i	− 0.0892055i
$433$	− 3.88854i	− 0.186871i	−0.995625	−	0.0934357i	$-0.970215\pi$
$433$	− 3.88854i	− 0.186871i	0.995625	−	0.0934357i	$-0.0297850\pi$
$434$	0.819660	0.0393449
$435$	0	0
$436$	0.854102	0.0409041
$437$	17.3607i	0.830474i
$438$	− 3.00000i	− 0.143346i
$439$	34.2705	1.63564	0.817821	−	0.575473i	$-0.195182\pi$
$439$	34.2705	1.63564	0.817821	+	0.575473i	$0.195182\pi$
$440$	0	0
$441$	−6.61803	−0.315144
$442$	25.5066i	1.21322i
$443$	18.7984i	0.893138i	0.894749	+	0.446569i	$0.147354\pi$
$443$	18.7984i	0.893138i	−0.894749	+	0.446569i	$0.852646\pi$
$444$	0.381966	0.0181273
$445$	0	0
$446$	7.85410	0.371903
$447$	− 6.05573i	− 0.286426i
$448$	0.145898i	0.00689303i
$449$	−8.74265	−0.412591	−0.206295	−	0.978490i	$-0.566141\pi$
$449$	−8.74265	−0.412591	−0.206295	+	0.978490i	$0.566141\pi$
$450$	0	0
$451$	−1.20163	−0.0565824
$452$	10.9443i	0.514775i
$453$	3.70820i	0.174227i
$454$	6.00000	0.281594
$455$	0	0
$456$	−11.1803	−0.523567
$457$	12.9787i	0.607119i	0.952813	+	0.303559i	$0.0981751\pi$
$457$	12.9787i	0.607119i	−0.952813	+	0.303559i	$0.901825\pi$
$458$	− 7.23607i	− 0.338119i
$459$	−6.61803	−0.308903
$460$	0	0
$461$	−25.3607	−1.18116	−0.590582	−	0.806977i	$-0.701102\pi$
$461$	−25.3607	−1.18116	−0.590582	+	0.806977i	$0.701102\pi$
$462$	0.0901699i	0.00419509i
$463$	− 11.8541i	− 0.550907i	−0.961314	−	0.275453i	$-0.911172\pi$
$463$	− 11.8541i	− 0.550907i	0.961314	−	0.275453i	$-0.0888280\pi$
$464$	−12.4377	−0.577405
$465$	0	0
$466$	−13.0000	−0.602213
$467$	31.7984i	1.47145i	0.677279	+	0.735727i	$0.263159\pi$
$467$	31.7984i	1.47145i	−0.677279	+	0.735727i	$0.736841\pi$
$468$	− 10.0902i	− 0.466418i
$469$	7.05573	0.325803
$470$	0	0
$471$	−9.56231	−0.440608
$472$	− 19.2705i	− 0.886997i
$473$	2.02129i	0.0929388i
$474$	−7.23607	−0.332364
$475$	0	0
$476$	6.61803	0.303337
$477$	4.09017i	0.187276i
$478$	10.8541i	0.496455i
$479$	−6.18034	−0.282387	−0.141193	−	0.989982i	$-0.545094\pi$
$479$	−6.18034	−0.282387	−0.141193	+	0.989982i	$0.545094\pi$
$480$	0	0
$481$	1.47214	0.0671236
$482$	− 8.11146i	− 0.369467i
$483$	− 2.14590i	− 0.0976417i
$484$	−17.7082	−0.804918
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	− 18.0000i	− 0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$	− 18.0000i	− 0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$	− 14.7984i	− 0.669891i
$489$	−15.1803	−0.686479
$490$	0	0
$491$	30.0902	1.35795	0.678975	−	0.734161i	$-0.262424\pi$
$491$	30.0902	1.35795	0.678975	+	0.734161i	$0.262424\pi$
$492$	8.23607i	0.371311i
$493$	44.3951i	1.99946i
$494$	−19.2705	−0.867021
$495$	0	0
$496$	−3.97871	−0.178650
$497$	5.18034i	0.232370i
$498$	11.1459i	0.499460i
$499$	36.7082	1.64328	0.821642	−	0.570003i	$-0.193058\pi$
$499$	36.7082	1.64328	0.821642	+	0.570003i	$0.193058\pi$
$500$	0	0
$501$	12.4721	0.557214
$502$	− 9.29180i	− 0.414713i
$503$	27.6180i	1.23143i	0.787970	+	0.615714i	$0.211132\pi$
$503$	27.6180i	1.23143i	−0.787970	+	0.615714i	$0.788868\pi$
$504$	1.38197	0.0615577
$505$	0	0
$506$	0.506578	0.0225201
$507$	− 25.8885i	− 1.14975i
$508$	8.56231i	0.379891i
$509$	0.652476	0.0289205	0.0144602	−	0.999895i	$-0.495397\pi$
$509$	0.652476	0.0289205	0.0144602	+	0.999895i	$0.495397\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	18.7082i	0.826794i
$513$	− 5.00000i	− 0.220755i
$514$	−0.909830	−0.0401309
$515$	0	0
$516$	13.8541	0.609893
$517$	0.708204i	0.0311468i
$518$	0.0901699i	0.00396184i
$519$	−11.5623	−0.507529
$520$	0	0
$521$	−16.0902	−0.704923	−0.352462	−	0.935826i	$-0.614655\pi$
$521$	−16.0902	−0.704923	−0.352462	+	0.935826i	$0.614655\pi$
$522$	4.14590i	0.181461i
$523$	10.8328i	0.473686i	0.971548	+	0.236843i	$0.0761127\pi$
$523$	10.8328i	0.473686i	−0.971548	+	0.236843i	$0.923887\pi$
$524$	27.1803	1.18738
$525$	0	0
$526$	2.85410	0.124445
$527$	14.2016i	0.618633i
$528$	− 0.437694i	− 0.0190482i
$529$	10.9443	0.475838
$530$	0	0
$531$	8.61803	0.373991
$532$	5.00000i	0.216777i
$533$	31.7426i	1.37493i
$534$	−8.61803	−0.372939
$535$	0	0
$536$	25.5279	1.10264
$537$	− 2.23607i	− 0.0964935i
$538$	19.1459i	0.825438i
$539$	−1.56231	−0.0672933
$540$	0	0
$541$	−5.56231	−0.239142	−0.119571	−	0.992826i	$-0.538152\pi$
$541$	−5.56231	−0.239142	−0.119571	+	0.992826i	$0.538152\pi$
$542$	− 11.0000i	− 0.472490i
$543$	− 6.94427i	− 0.298007i
$544$	37.1803	1.59409
$545$	0	0
$546$	2.38197	0.101939
$547$	17.8541i	0.763386i	0.924289	+	0.381693i	$0.124659\pi$
$547$	17.8541i	0.763386i	−0.924289	+	0.381693i	$0.875341\pi$
$548$	2.38197i	0.101753i
$549$	6.61803	0.282451
$550$	0	0
$551$	−33.5410	−1.42890
$552$	− 7.76393i	− 0.330455i
$553$	7.23607i	0.307709i
$554$	−0.0557281	−0.00236766
$555$	0	0
$556$	−10.8541	−0.460316
$557$	− 16.0902i	− 0.681762i	−0.940106	−	0.340881i	$-0.889275\pi$
$557$	− 16.0902i	− 0.681762i	0.940106	−	0.340881i	$-0.110725\pi$
$558$	1.32624i	0.0561441i
$559$	53.3951	2.25837
$560$	0	0
$561$	−1.56231	−0.0659606
$562$	− 3.76393i	− 0.158772i
$563$	27.6180i	1.16396i	0.813203	+	0.581981i	$0.197722\pi$
$563$	27.6180i	1.16396i	−0.813203	+	0.581981i	$0.802278\pi$
$564$	4.85410	0.204395
$565$	0	0
$566$	4.43769	0.186530
$567$	0.618034i	0.0259550i
$568$	18.7426i	0.786424i
$569$	−22.2361	−0.932184	−0.466092	−	0.884736i	$-0.654339\pi$
$569$	−22.2361	−0.932184	−0.466092	+	0.884736i	$0.654339\pi$
$570$	0	0
$571$	−39.5066	−1.65330	−0.826649	−	0.562717i	$-0.809756\pi$
$571$	−39.5066	−1.65330	−0.826649	+	0.562717i	$0.809756\pi$
$572$	− 2.38197i	− 0.0995950i
$573$	− 17.4721i	− 0.729909i
$574$	−1.94427	−0.0811523
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	− 40.2361i	− 1.67505i	−0.546399	−	0.837525i	$-0.684002\pi$
$577$	− 40.2361i	− 1.67505i	0.546399	−	0.837525i	$-0.315998\pi$
$578$	− 16.5623i	− 0.688901i
$579$	−23.4721	−0.975469
$580$	0	0
$581$	11.1459	0.462410
$582$	7.50658i	0.311158i
$583$	0.965558i	0.0399893i
$584$	−10.8541	−0.449146
$585$	0	0
$586$	−10.1115	−0.417700
$587$	− 29.1803i	− 1.20440i	−0.798345	−	0.602201i	$-0.794291\pi$
$587$	− 29.1803i	− 1.20440i	0.798345	−	0.602201i	$-0.205709\pi$
$588$	10.7082i	0.441599i
$589$	−10.7295	−0.442101
$590$	0	0
$591$	−14.8885	−0.612433
$592$	− 0.437694i	− 0.0179891i
$593$	21.7639i	0.893738i	0.894600	+	0.446869i	$0.147461\pi$
$593$	21.7639i	0.893738i	−0.894600	+	0.446869i	$0.852539\pi$
$594$	−0.145898	−0.00598627
$595$	0	0
$596$	−9.79837	−0.401357
$597$	− 3.29180i	− 0.134724i
$598$	− 13.3820i	− 0.547229i
$599$	−4.79837	−0.196056	−0.0980281	−	0.995184i	$-0.531254\pi$
$599$	−4.79837	−0.196056	−0.0980281	+	0.995184i	$0.531254\pi$
$600$	0	0
$601$	30.4164	1.24071	0.620356	−	0.784321i	$-0.286988\pi$
$601$	30.4164	1.24071	0.620356	+	0.784321i	$0.286988\pi$
$602$	3.27051i	0.133296i
$603$	11.4164i	0.464912i
$604$	6.00000	0.244137
$605$	0	0
$606$	10.4721	0.425401
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	28.0902i	1.13921i
$609$	4.14590	0.168000
$610$	0	0
$611$	18.7082	0.756853
$612$	10.7082i	0.432853i
$613$	− 11.0000i	− 0.444286i	−0.975014	−	0.222143i	$-0.928695\pi$
$613$	− 11.0000i	− 0.444286i	0.975014	−	0.222143i	$-0.0713052\pi$
$614$	10.1459	0.409455
$615$	0	0
$616$	0.326238	0.0131445
$617$	− 29.8328i	− 1.20102i	−0.799616	−	0.600512i	$-0.794963\pi$
$617$	− 29.8328i	− 1.20102i	0.799616	−	0.600512i	$-0.205037\pi$
$618$	8.70820i	0.350295i
$619$	6.83282	0.274634	0.137317	−	0.990527i	$-0.456152\pi$
$619$	6.83282	0.274634	0.137317	+	0.990527i	$0.456152\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	9.00000i	0.360867i
$623$	8.61803i	0.345274i
$624$	−11.5623	−0.462863
$625$	0	0
$626$	15.6180	0.624222
$627$	− 1.18034i	− 0.0471382i
$628$	15.4721i	0.617405i
$629$	−1.56231	−0.0622932
$630$	0	0
$631$	30.0132	1.19480	0.597402	−	0.801942i	$-0.296200\pi$
$631$	30.0132	1.19480	0.597402	+	0.801942i	$0.296200\pi$
$632$	26.1803i	1.04140i
$633$	− 8.00000i	− 0.317971i
$634$	−14.9787	−0.594881
$635$	0	0
$636$	6.61803	0.262422
$637$	41.2705i	1.63520i
$638$	0.978714i	0.0387476i
$639$	−8.38197	−0.331585
$640$	0	0
$641$	22.6525	0.894719	0.447360	−	0.894354i	$-0.352364\pi$
$641$	22.6525	0.894719	0.447360	+	0.894354i	$0.352364\pi$
$642$	− 5.18034i	− 0.204452i
$643$	− 36.2492i	− 1.42953i	−0.699365	−	0.714765i	$-0.746534\pi$
$643$	− 36.2492i	− 1.42953i	0.699365	−	0.714765i	$-0.253466\pi$
$644$	−3.47214	−0.136821
$645$	0	0
$646$	20.4508	0.804628
$647$	21.9230i	0.861882i	0.902380	+	0.430941i	$0.141818\pi$
$647$	21.9230i	0.861882i	−0.902380	+	0.430941i	$0.858182\pi$
$648$	2.23607i	0.0878410i
$649$	2.03444	0.0798588
$650$	0	0
$651$	1.32624	0.0519794
$652$	24.5623i	0.961934i
$653$	− 28.7639i	− 1.12562i	−0.826586	−	0.562810i	$-0.809720\pi$
$653$	− 28.7639i	− 1.12562i	0.826586	−	0.562810i	$-0.190280\pi$
$654$	−0.326238	−0.0127569
$655$	0	0
$656$	9.43769	0.368480
$657$	− 4.85410i	− 0.189377i
$658$	1.14590i	0.0446718i
$659$	−31.1803	−1.21461	−0.607307	−	0.794467i	$-0.707750\pi$
$659$	−31.1803	−1.21461	−0.607307	+	0.794467i	$0.707750\pi$
$660$	0	0
$661$	2.72949	0.106165	0.0530824	−	0.998590i	$-0.483095\pi$
$661$	2.72949	0.106165	0.0530824	+	0.998590i	$0.483095\pi$
$662$	− 6.52786i	− 0.253713i
$663$	41.2705i	1.60281i
$664$	40.3262	1.56496
$665$	0	0
$666$	−0.145898	−0.00565343
$667$	− 23.2918i	− 0.901862i
$668$	− 20.1803i	− 0.780801i
$669$	12.7082	0.491328
$670$	0	0
$671$	1.56231	0.0603122
$672$	− 3.47214i	− 0.133941i
$673$	18.4721i	0.712049i	0.934477	+	0.356024i	$0.115868\pi$
$673$	18.4721i	0.712049i	−0.934477	+	0.356024i	$0.884132\pi$
$674$	−0.583592	−0.0224791
$675$	0	0
$676$	−41.8885	−1.61110
$677$	32.6525i	1.25494i	0.778642	+	0.627468i	$0.215909\pi$
$677$	32.6525i	1.25494i	−0.778642	+	0.627468i	$0.784091\pi$
$678$	− 4.18034i	− 0.160545i
$679$	7.50658	0.288076
$680$	0	0
$681$	9.70820	0.372019
$682$	0.313082i	0.0119885i
$683$	− 8.23607i	− 0.315144i	−0.987507	−	0.157572i	$-0.949633\pi$
$683$	− 8.23607i	− 0.315144i	0.987507	−	0.157572i	$-0.0503667\pi$
$684$	−8.09017	−0.309335
$685$	0	0
$686$	−5.20163	−0.198599
$687$	− 11.7082i	− 0.446696i
$688$	− 15.8754i	− 0.605244i
$689$	25.5066	0.971723
$690$	0	0
$691$	−39.1803	−1.49049	−0.745245	−	0.666791i	$-0.767668\pi$
$691$	−39.1803	−1.49049	−0.745245	+	0.666791i	$0.767668\pi$
$692$	18.7082i	0.711179i
$693$	0.145898i	0.00554221i
$694$	−8.59675	−0.326328
$695$	0	0
$696$	15.0000	0.568574
$697$	− 33.6869i	− 1.27598i
$698$	− 3.29180i	− 0.124596i
$699$	−21.0344	−0.795596
$700$	0	0
$701$	−17.3475	−0.655207	−0.327603	−	0.944815i	$-0.606241\pi$
$701$	−17.3475	−0.655207	−0.327603	+	0.944815i	$0.606241\pi$
$702$	3.85410i	0.145464i
$703$	− 1.18034i	− 0.0445174i
$704$	−0.0557281	−0.00210033
$705$	0	0
$706$	6.27051	0.235994
$707$	− 10.4721i	− 0.393845i
$708$	− 13.9443i	− 0.524058i
$709$	−10.1246	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$709$	−10.1246	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$710$	0	0
$711$	−11.7082	−0.439092
$712$	31.1803i	1.16853i
$713$	− 7.45085i	− 0.279037i
$714$	−2.52786	−0.0946029
$715$	0	0
$716$	−3.61803	−0.135212
$717$	17.5623i	0.655876i
$718$	− 11.7082i	− 0.436946i
$719$	−28.4164	−1.05975	−0.529877	−	0.848075i	$-0.677762\pi$
$719$	−28.4164	−1.05975	−0.529877	+	0.848075i	$0.677762\pi$
$720$	0	0
$721$	8.70820	0.324310
$722$	3.70820i	0.138005i
$723$	− 13.1246i	− 0.488110i
$724$	−11.2361	−0.417585
$725$	0	0
$726$	6.76393	0.251033
$727$	− 28.1246i	− 1.04308i	−0.853226	−	0.521542i	$-0.825357\pi$
$727$	− 28.1246i	− 1.04308i	0.853226	−	0.521542i	$-0.174643\pi$
$728$	− 8.61803i	− 0.319406i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−56.6656	−2.09585
$732$	− 10.7082i	− 0.395787i
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	−5.18034	−0.191210
$735$	0	0
$736$	−19.5066	−0.719022
$737$	2.69505i	0.0992734i
$738$	− 3.14590i	− 0.115802i
$739$	−12.0344	−0.442694	−0.221347	−	0.975195i	$-0.571045\pi$
$739$	−12.0344	−0.442694	−0.221347	+	0.975195i	$0.571045\pi$
$740$	0	0
$741$	−31.1803	−1.14544
$742$	1.56231i	0.0573541i
$743$	− 41.1246i	− 1.50872i	−0.656463	−	0.754358i	$-0.727948\pi$
$743$	− 41.1246i	− 1.50872i	0.656463	−	0.754358i	$-0.272052\pi$
$744$	4.79837	0.175917
$745$	0	0
$746$	1.47214	0.0538987
$747$	18.0344i	0.659846i
$748$	2.52786i	0.0924279i
$749$	−5.18034	−0.189285
$750$	0	0
$751$	14.0344	0.512124	0.256062	−	0.966660i	$-0.417575\pi$
$751$	14.0344	0.512124	0.256062	+	0.966660i	$0.417575\pi$
$752$	− 5.56231i	− 0.202836i
$753$	− 15.0344i	− 0.547886i
$754$	25.8541	0.941551
$755$	0	0
$756$	1.00000	0.0363696
$757$	30.4164i	1.10550i	0.833346	+	0.552752i	$0.186422\pi$
$757$	30.4164i	1.10550i	−0.833346	+	0.552752i	$0.813578\pi$
$758$	1.50658i	0.0547214i
$759$	0.819660	0.0297518
$760$	0	0
$761$	15.5410	0.563362	0.281681	−	0.959508i	$-0.409108\pi$
$761$	15.5410	0.563362	0.281681	+	0.959508i	$0.409108\pi$
$762$	− 3.27051i	− 0.118478i
$763$	0.326238i	0.0118106i
$764$	−28.2705	−1.02279
$765$	0	0
$766$	14.5623	0.526157
$767$	− 53.7426i	− 1.94053i
$768$	− 6.56231i	− 0.236797i
$769$	3.41641	0.123199	0.0615994	−	0.998101i	$-0.480380\pi$
$769$	3.41641	0.123199	0.0615994	+	0.998101i	$0.480380\pi$
$770$	0	0
$771$	−1.47214	−0.0530177
$772$	37.9787i	1.36688i
$773$	− 24.8197i	− 0.892701i	−0.894858	−	0.446351i	$-0.852723\pi$
$773$	− 24.8197i	− 0.892701i	0.894858	−	0.446351i	$-0.147277\pi$
$774$	−5.29180	−0.190210
$775$	0	0
$776$	27.1591	0.974953
$777$	0.145898i	0.00523406i
$778$	6.70820i	0.240501i
$779$	25.4508	0.911871
$780$	0	0
$781$	−1.97871	−0.0708039
$782$	14.2016i	0.507849i
$783$	6.70820i	0.239732i
$784$	12.2705	0.438232
$785$	0	0
$786$	−10.3820	−0.370312
$787$	− 23.9787i	− 0.854749i	−0.904075	−	0.427374i	$-0.859439\pi$
$787$	− 23.9787i	− 0.854749i	0.904075	−	0.427374i	$-0.140561\pi$
$788$	24.0902i	0.858177i
$789$	4.61803	0.164406
$790$	0	0
$791$	−4.18034	−0.148636
$792$	0.527864i	0.0187568i
$793$	− 41.2705i	− 1.46556i
$794$	−10.8328	−0.384442
$795$	0	0
$796$	−5.32624	−0.188783
$797$	− 54.6312i	− 1.93514i	−0.252611	−	0.967568i	$-0.581289\pi$
$797$	− 54.6312i	− 1.93514i	0.252611	−	0.967568i	$-0.418711\pi$
$798$	− 1.90983i	− 0.0676073i
$799$	−19.8541	−0.702388
$800$	0	0
$801$	−13.9443	−0.492697
$802$	8.79837i	0.310681i
$803$	− 1.14590i	− 0.0404379i
$804$	18.4721	0.651462
$805$	0	0
$806$	8.27051	0.291316
$807$	30.9787i	1.09050i
$808$	− 37.8885i	− 1.33291i
$809$	30.9787	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$809$	30.9787	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$810$	0	0
$811$	41.3951	1.45358	0.726790	−	0.686860i	$-0.241011\pi$
$811$	41.3951	1.45358	0.726790	+	0.686860i	$0.241011\pi$
$812$	− 6.70820i	− 0.235412i
$813$	− 17.7984i	− 0.624216i
$814$	−0.0344419	−0.00120719
$815$	0	0
$816$	12.2705	0.429554
$817$	− 42.8115i	− 1.49779i
$818$	− 5.12461i	− 0.179178i
$819$	3.85410	0.134673
$820$	0	0
$821$	−38.1246	−1.33056	−0.665279	−	0.746595i	$-0.731687\pi$
$821$	−38.1246	−1.33056	−0.665279	+	0.746595i	$0.731687\pi$
$822$	− 0.909830i	− 0.0317340i
$823$	17.4164i	0.607098i	0.952816	+	0.303549i	$0.0981716\pi$
$823$	17.4164i	0.607098i	−0.952816	+	0.303549i	$0.901828\pi$
$824$	31.5066	1.09758
$825$	0	0
$826$	3.29180	0.114536
$827$	− 4.90983i	− 0.170732i	−0.996350	−	0.0853658i	$-0.972794\pi$
$827$	− 4.90983i	− 0.170732i	0.996350	−	0.0853658i	$-0.0272059\pi$
$828$	− 5.61803i	− 0.195240i
$829$	−55.1246	−1.91456	−0.957278	−	0.289168i	$-0.906621\pi$
$829$	−55.1246	−1.91456	−0.957278	+	0.289168i	$0.906621\pi$
$830$	0	0
$831$	−0.0901699	−0.00312796
$832$	1.47214i	0.0510371i
$833$	− 43.7984i	− 1.51752i
$834$	4.14590	0.143561
$835$	0	0
$836$	−1.90983	−0.0660529
$837$	2.14590i	0.0741731i
$838$	4.34752i	0.150183i
$839$	−44.1935	−1.52573	−0.762864	−	0.646558i	$-0.776208\pi$
$839$	−44.1935	−1.52573	−0.762864	+	0.646558i	$0.776208\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	8.87539i	0.305866i
$843$	− 6.09017i	− 0.209757i
$844$	−12.9443	−0.445560
$845$	0	0
$846$	−1.85410	−0.0637453
$847$	− 6.76393i	− 0.232411i
$848$	− 7.58359i	− 0.260422i
$849$	7.18034	0.246429
$850$	0	0
$851$	0.819660	0.0280976
$852$	13.5623i	0.464637i
$853$	− 41.2492i	− 1.41235i	−0.708039	−	0.706173i	$-0.750420\pi$
$853$	− 41.2492i	− 1.41235i	0.708039	−	0.706173i	$-0.249580\pi$
$854$	2.52786	0.0865017
$855$	0	0
$856$	−18.7426	−0.640610
$857$	− 4.38197i	− 0.149685i	−0.997195	−	0.0748426i	$-0.976155\pi$
$857$	− 4.38197i	− 0.149685i	0.997195	−	0.0748426i	$-0.0238454\pi$
$858$	0.909830i	0.0310611i
$859$	13.2918	0.453510	0.226755	−	0.973952i	$-0.427188\pi$
$859$	13.2918	0.453510	0.226755	+	0.973952i	$0.427188\pi$
$860$	0	0
$861$	−3.14590	−0.107212
$862$	− 4.21478i	− 0.143556i
$863$	18.9230i	0.644146i	0.946715	+	0.322073i	$0.104380\pi$
$863$	18.9230i	0.644146i	−0.946715	+	0.322073i	$0.895620\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	2.40325	0.0816658
$867$	− 26.7984i	− 0.910120i
$868$	− 2.14590i	− 0.0728365i
$869$	−2.76393	−0.0937600
$870$	0	0
$871$	71.1935	2.41230
$872$	1.18034i	0.0399714i
$873$	12.1459i	0.411076i
$874$	−10.7295	−0.362930
$875$	0	0
$876$	−7.85410	−0.265366
$877$	31.0689i	1.04912i	0.851373	+	0.524561i	$0.175770\pi$
$877$	31.0689i	1.04912i	−0.851373	+	0.524561i	$0.824230\pi$
$878$	21.1803i	0.714802i
$879$	−16.3607	−0.551832
$880$	0	0
$881$	−36.5410	−1.23110	−0.615549	−	0.788099i	$-0.711066\pi$
$881$	−36.5410	−1.23110	−0.615549	+	0.788099i	$0.711066\pi$
$882$	− 4.09017i	− 0.137723i
$883$	− 37.1803i	− 1.25122i	−0.780137	−	0.625609i	$-0.784851\pi$
$883$	− 37.1803i	− 1.25122i	0.780137	−	0.625609i	$-0.215149\pi$
$884$	66.7771	2.24596
$885$	0	0
$886$	−11.6180	−0.390315
$887$	7.65248i	0.256945i	0.991713	+	0.128472i	$0.0410074\pi$
$887$	7.65248i	0.256945i	−0.991713	+	0.128472i	$0.958993\pi$
$888$	0.527864i	0.0177140i
$889$	−3.27051	−0.109689
$890$	0	0
$891$	−0.236068	−0.00790857
$892$	− 20.5623i	− 0.688477i
$893$	− 15.0000i	− 0.501956i
$894$	3.74265	0.125173
$895$	0	0
$896$	−7.03444	−0.235004
$897$	− 21.6525i	− 0.722955i
$898$	− 5.40325i	− 0.180309i
$899$	14.3951	0.480104
$900$	0	0
$901$	−27.0689	−0.901795
$902$	− 0.742646i	− 0.0247274i
$903$	5.29180i	0.176100i
$904$	−15.1246	−0.503037
$905$	0	0
$906$	−2.29180	−0.0761398
$907$	41.4721i	1.37706i	0.725208	+	0.688530i	$0.241744\pi$
$907$	41.4721i	1.37706i	−0.725208	+	0.688530i	$0.758256\pi$
$908$	− 15.7082i	− 0.521295i
$909$	16.9443	0.562006
$910$	0	0
$911$	20.6180	0.683106	0.341553	−	0.939863i	$-0.389047\pi$
$911$	20.6180	0.683106	0.341553	+	0.939863i	$0.389047\pi$
$912$	9.27051i	0.306977i
$913$	4.25735i	0.140898i
$914$	−8.02129	−0.265321
$915$	0	0
$916$	−18.9443	−0.625936
$917$	10.3820i	0.342843i
$918$	− 4.09017i	− 0.134996i
$919$	39.3951	1.29953	0.649763	−	0.760137i	$-0.274868\pi$
$919$	39.3951	1.29953	0.649763	+	0.760137i	$0.274868\pi$
$920$	0	0
$921$	16.4164	0.540939
$922$	− 15.6738i	− 0.516188i
$923$	52.2705i	1.72050i
$924$	0.236068	0.00776607
$925$	0	0
$926$	7.32624	0.240755
$927$	14.0902i	0.462782i
$928$	− 37.6869i	− 1.23713i
$929$	24.7984	0.813608	0.406804	−	0.913515i	$-0.366643\pi$
$929$	24.7984	0.813608	0.406804	+	0.913515i	$0.366643\pi$
$930$	0	0
$931$	33.0902	1.08449
$932$	34.0344i	1.11484i
$933$	14.5623i	0.476748i
$934$	−19.6525	−0.643049
$935$	0	0
$936$	13.9443	0.455783
$937$	− 38.6525i	− 1.26272i	−0.775489	−	0.631361i	$-0.782497\pi$
$937$	− 38.6525i	− 1.26272i	0.775489	−	0.631361i	$-0.217503\pi$
$938$	4.36068i	0.142381i
$939$	25.2705	0.824672
$940$	0	0
$941$	42.9787	1.40107	0.700533	−	0.713620i	$-0.252946\pi$
$941$	42.9787	1.40107	0.700533	+	0.713620i	$0.252946\pi$
$942$	− 5.90983i	− 0.192553i
$943$	17.6738i	0.575537i
$944$	−15.9787	−0.520063
$945$	0	0
$946$	−1.24922	−0.0406158
$947$	− 11.2918i	− 0.366934i	−0.983026	−	0.183467i	$-0.941268\pi$
$947$	− 11.2918i	− 0.366934i	0.983026	−	0.183467i	$-0.0587321\pi$
$948$	18.9443i	0.615281i
$949$	−30.2705	−0.982622
$950$	0	0
$951$	−24.2361	−0.785908
$952$	9.14590i	0.296420i
$953$	12.4934i	0.404702i	0.979313	+	0.202351i	$0.0648581\pi$
$953$	12.4934i	0.404702i	−0.979313	+	0.202351i	$0.935142\pi$
$954$	−2.52786	−0.0818426
$955$	0	0
$956$	28.4164	0.919052
$957$	1.58359i	0.0511903i
$958$	− 3.81966i	− 0.123408i
$959$	−0.909830	−0.0293799
$960$	0	0
$961$	−26.3951	−0.851456
$962$	0.909830i	0.0293341i
$963$	− 8.38197i	− 0.270105i
$964$	−21.2361	−0.683968
$965$	0	0
$966$	1.32624	0.0426710
$967$	35.5410i	1.14292i	0.820629	+	0.571461i	$0.193623\pi$
$967$	35.5410i	1.14292i	−0.820629	+	0.571461i	$0.806377\pi$
$968$	− 24.4721i	− 0.786564i
$969$	33.0902	1.06301
$970$	0	0
$971$	25.0132	0.802710	0.401355	−	0.915922i	$-0.368539\pi$
$971$	25.0132	0.802710	0.401355	+	0.915922i	$0.368539\pi$
$972$	1.61803i	0.0518985i
$973$	− 4.14590i	− 0.132911i
$974$	11.1246	0.356456
$975$	0	0
$976$	−12.2705	−0.392769
$977$	− 2.02129i	− 0.0646667i	−0.999477	−	0.0323333i	$-0.989706\pi$
$977$	− 2.02129i	− 0.0646667i	0.999477	−	0.0323333i	$-0.0102938\pi$
$978$	− 9.38197i	− 0.300002i
$979$	−3.29180	−0.105206
$980$	0	0
$981$	−0.527864	−0.0168534
$982$	18.5967i	0.593446i
$983$	− 14.7426i	− 0.470217i	−0.971969	−	0.235109i	$-0.924455\pi$
$983$	− 14.7426i	− 0.470217i	0.971969	−	0.235109i	$-0.0755446\pi$
$984$	−11.3820	−0.362844
$985$	0	0
$986$	−27.4377	−0.873794
$987$	1.85410i	0.0590167i
$988$	50.4508i	1.60506i
$989$	29.7295	0.945343
$990$	0	0
$991$	−2.27051	−0.0721251	−0.0360626	−	0.999350i	$-0.511482\pi$
$991$	−2.27051	−0.0721251	−0.0360626	+	0.999350i	$0.511482\pi$
$992$	− 12.0557i	− 0.382770i
$993$	− 10.5623i	− 0.335185i
$994$	−3.20163	−0.101549
$995$	0	0
$996$	29.1803	0.924614
$997$	− 22.5967i	− 0.715646i	−0.933789	−	0.357823i	$-0.883519\pi$
$997$	− 22.5967i	− 0.715646i	0.933789	−	0.357823i	$-0.116481\pi$
$998$	22.6869i	0.718142i
$999$	−0.236068	−0.00746886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.b.b.124.3		4
3.2	odd	2		1125.2.b.e.874.2		4
4.3	odd	2		6000.2.f.b.1249.1		4
5.2	odd	4		375.2.a.c.1.1	yes	2
5.3	odd	4		375.2.a.b.1.2	✓	2
5.4	even	2	inner	375.2.b.b.124.2		4
15.2	even	4		1125.2.a.b.1.2		2
15.8	even	4		1125.2.a.e.1.1		2
15.14	odd	2		1125.2.b.e.874.3		4
20.3	even	4		6000.2.a.v.1.1		2
20.7	even	4		6000.2.a.e.1.2		2
20.19	odd	2		6000.2.f.b.1249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.b.1.2	✓	2	5.3	odd	4
375.2.a.c.1.1	yes	2	5.2	odd	4
375.2.b.b.124.2		4	5.4	even	2	inner
375.2.b.b.124.3		4	1.1	even	1	trivial
1125.2.a.b.1.2		2	15.2	even	4
1125.2.a.e.1.1		2	15.8	even	4
1125.2.b.e.874.2		4	3.2	odd	2
1125.2.b.e.874.3		4	15.14	odd	2
6000.2.a.e.1.2		2	20.7	even	4
6000.2.a.v.1.1		2	20.3	even	4
6000.2.f.b.1249.1		4	4.3	odd	2
6000.2.f.b.1249.4		4	20.19	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:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +0.618034i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} -0.618034 q^{6} +0.618034i q^{7} +2.23607i q^{8} -1.00000 q^{9} -0.236068 q^{11} +1.61803i q^{12} +6.23607i q^{13} -0.381966 q^{14} +1.85410 q^{16} -6.61803i q^{17} -0.618034i q^{18} +5.00000 q^{19} -0.618034 q^{21} -0.145898i q^{22} +3.47214i q^{23} -2.23607 q^{24} -3.85410 q^{26} -1.00000i q^{27} +1.00000i q^{28} -6.70820 q^{29} -2.14590 q^{31} +5.61803i q^{32} -0.236068i q^{33} +4.09017 q^{34} -1.61803 q^{36} -0.236068i q^{37} +3.09017i q^{38} -6.23607 q^{39} +5.09017 q^{41} -0.381966i q^{42} -8.56231i q^{43} -0.381966 q^{44} -2.14590 q^{46} -3.00000i q^{47} +1.85410i q^{48} +6.61803 q^{49} +6.61803 q^{51} +10.0902i q^{52} -4.09017i q^{53} +0.618034 q^{54} -1.38197 q^{56} +5.00000i q^{57} -4.14590i q^{58} -8.61803 q^{59} -6.61803 q^{61} -1.32624i q^{62} -0.618034i q^{63} +0.236068 q^{64} +0.145898 q^{66} -11.4164i q^{67} -10.7082i q^{68} -3.47214 q^{69} +8.38197 q^{71} -2.23607i q^{72} +4.85410i q^{73} +0.145898 q^{74} +8.09017 q^{76} -0.145898i q^{77} -3.85410i q^{78} +11.7082 q^{79} +1.00000 q^{81} +3.14590i q^{82} -18.0344i q^{83} -1.00000 q^{84} +5.29180 q^{86} -6.70820i q^{87} -0.527864i q^{88} +13.9443 q^{89} -3.85410 q^{91} +5.61803i q^{92} -2.14590i q^{93} +1.85410 q^{94} -5.61803 q^{96} -12.1459i q^{97} +4.09017i q^{98} +0.236068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9	\( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} + 8 q^{11} - 6 q^{14} - 6 q^{16} + 20 q^{19} + 2 q^{21} - 2 q^{26} - 22 q^{31} - 6 q^{34} - 2 q^{36} - 16 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{46} + 22 q^{49} + 22 q^{51} - 2 q^{54} - 10 q^{56} - 30 q^{59} - 22 q^{61} - 8 q^{64} + 14 q^{66} + 4 q^{69} + 38 q^{71} + 14 q^{74} + 10 q^{76} + 20 q^{79} + 4 q^{81} - 4 q^{84} + 48 q^{86} + 20 q^{89} - 2 q^{91} - 6 q^{94} - 18 q^{96} - 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 2 * q^6 - 4 * q^9 + 8 * q^11 - 6 * q^14 - 6 * q^16 + 20 * q^19 + 2 * q^21 - 2 * q^26 - 22 * q^31 - 6 * q^34 - 2 * q^36 - 16 * q^39 - 2 * q^41 - 6 * q^44 - 22 * q^46 + 22 * q^49 + 22 * q^51 - 2 * q^54 - 10 * q^56 - 30 * q^59 - 22 * q^61 - 8 * q^64 + 14 * q^66 + 4 * q^69 + 38 * q^71 + 14 * q^74 + 10 * q^76 + 20 * q^79 + 4 * q^81 - 4 * q^84 + 48 * q^86 + 20 * q^89 - 2 * q^91 - 6 * q^94 - 18 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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