LMFDB - Embedded newform 2445.2.a.g.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2445,2,Mod(1,2445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2445 = 3 \cdot 5 \cdot 163 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-7,11,9,11,-7,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.5234232942$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 6x^{9} + 33x^{8} + 11x^{7} - 95x^{6} - 10x^{5} + 108x^{4} + 15x^{3} - 39x^{2} - 14x - 1 $ x^11 - 4x^10 - 6x^9 + 33x^8 + 11x^7 - 95x^6 - 10x^5 + 108x^4 + 15x^3 - 39x^2 - 14x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.93918$ of defining polynomial
Character	$\chi$	$=$	2445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.93918 q^{2} +1.00000 q^{3} +1.76043 q^{4} +1.00000 q^{5} +1.93918 q^{6} -5.18066 q^{7} -0.464570 q^{8} +1.00000 q^{9} +1.93918 q^{10} -1.57611 q^{11} +1.76043 q^{12} +0.0444470 q^{13} -10.0463 q^{14} +1.00000 q^{15} -4.42175 q^{16} -4.10373 q^{17} +1.93918 q^{18} -2.67946 q^{19} +1.76043 q^{20} -5.18066 q^{21} -3.05636 q^{22} -8.71738 q^{23} -0.464570 q^{24} +1.00000 q^{25} +0.0861909 q^{26} +1.00000 q^{27} -9.12020 q^{28} -3.21964 q^{29} +1.93918 q^{30} -4.51578 q^{31} -7.64543 q^{32} -1.57611 q^{33} -7.95788 q^{34} -5.18066 q^{35} +1.76043 q^{36} +9.88589 q^{37} -5.19596 q^{38} +0.0444470 q^{39} -0.464570 q^{40} +4.75304 q^{41} -10.0463 q^{42} +0.0236316 q^{43} -2.77463 q^{44} +1.00000 q^{45} -16.9046 q^{46} +11.5866 q^{47} -4.42175 q^{48} +19.8393 q^{49} +1.93918 q^{50} -4.10373 q^{51} +0.0782459 q^{52} +12.1869 q^{53} +1.93918 q^{54} -1.57611 q^{55} +2.40678 q^{56} -2.67946 q^{57} -6.24347 q^{58} -6.16261 q^{59} +1.76043 q^{60} +1.41603 q^{61} -8.75693 q^{62} -5.18066 q^{63} -5.98240 q^{64} +0.0444470 q^{65} -3.05636 q^{66} +6.52939 q^{67} -7.22433 q^{68} -8.71738 q^{69} -10.0463 q^{70} +2.30407 q^{71} -0.464570 q^{72} -12.4986 q^{73} +19.1705 q^{74} +1.00000 q^{75} -4.71700 q^{76} +8.16529 q^{77} +0.0861909 q^{78} -8.96920 q^{79} -4.42175 q^{80} +1.00000 q^{81} +9.21702 q^{82} -10.3732 q^{83} -9.12020 q^{84} -4.10373 q^{85} +0.0458260 q^{86} -3.21964 q^{87} +0.732212 q^{88} +2.55458 q^{89} +1.93918 q^{90} -0.230265 q^{91} -15.3463 q^{92} -4.51578 q^{93} +22.4685 q^{94} -2.67946 q^{95} -7.64543 q^{96} -0.225465 q^{97} +38.4720 q^{98} -1.57611 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 7 q^{2} + 11 q^{3} + 9 q^{4} + 11 q^{5} - 7 q^{6} - 16 q^{7} - 18 q^{8} + 11 q^{9} - 7 q^{10} - 12 q^{11} + 9 q^{12} - 11 q^{13} + 3 q^{14} + 11 q^{15} + 13 q^{16} - 22 q^{17} - 7 q^{18} - 19 q^{19}+ \cdots - 12 q^{99}+O(q^{100}) $ 11 * q - 7 * q^2 + 11 * q^3 + 9 * q^4 + 11 * q^5 - 7 * q^6 - 16 * q^7 - 18 * q^8 + 11 * q^9 - 7 * q^10 - 12 * q^11 + 9 * q^12 - 11 * q^13 + 3 * q^14 + 11 * q^15 + 13 * q^16 - 22 * q^17 - 7 * q^18 - 19 * q^19 + 9 * q^20 - 16 * q^21 - 6 * q^22 - 21 * q^23 - 18 * q^24 + 11 * q^25 + 4 * q^26 + 11 * q^27 - 10 * q^28 - 19 * q^29 - 7 * q^30 - 8 * q^31 - 28 * q^32 - 12 * q^33 + 12 * q^34 - 16 * q^35 + 9 * q^36 - q^37 - 5 * q^38 - 11 * q^39 - 18 * q^40 - 18 * q^41 + 3 * q^42 - 30 * q^43 + 7 * q^44 + 11 * q^45 + 2 * q^46 - 21 * q^47 + 13 * q^48 + 19 * q^49 - 7 * q^50 - 22 * q^51 - 20 * q^52 - 21 * q^53 - 7 * q^54 - 12 * q^55 - 3 * q^56 - 19 * q^57 + 17 * q^58 - 9 * q^59 + 9 * q^60 - 27 * q^61 - 11 * q^62 - 16 * q^63 + 46 * q^64 - 11 * q^65 - 6 * q^66 - 32 * q^67 - 53 * q^68 - 21 * q^69 + 3 * q^70 - 8 * q^71 - 18 * q^72 - 25 * q^73 + 24 * q^74 + 11 * q^75 - 7 * q^76 - 31 * q^77 + 4 * q^78 + 8 * q^79 + 13 * q^80 + 11 * q^81 + 9 * q^82 - 54 * q^83 - 10 * q^84 - 22 * q^85 - 2 * q^86 - 19 * q^87 + 16 * q^88 + 7 * q^89 - 7 * q^90 - 22 * q^91 - 37 * q^92 - 8 * q^93 + 53 * q^94 - 19 * q^95 - 28 * q^96 + 21 * q^97 - 24 * q^98 - 12 * q^99

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 $

$2$	1.93918	1.37121	0.685605	−	0.727974i	$-0.259538\pi$
$2$	1.93918	1.37121	0.685605	+	0.727974i	$0.259538\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.76043	0.880215
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.93918	0.791668
$7$	−5.18066	−1.95811	−0.979053	−	0.203604i	$-0.934735\pi$
$7$	−5.18066	−1.95811	−0.979053	+	0.203604i	$0.934735\pi$
$8$	−0.464570	−0.164250
$9$	1.00000	0.333333
$10$	1.93918	0.613223
$11$	−1.57611	−0.475215	−0.237607	−	0.971361i	$-0.576363\pi$
$11$	−1.57611	−0.475215	−0.237607	+	0.971361i	$0.576363\pi$
$12$	1.76043	0.508192
$13$	0.0444470	0.0123274	0.00616369	−	0.999981i	$-0.498038\pi$
$13$	0.0444470	0.0123274	0.00616369	+	0.999981i	$0.498038\pi$
$14$	−10.0463	−2.68497
$15$	1.00000	0.258199
$16$	−4.42175	−1.10544
$17$	−4.10373	−0.995301	−0.497650	−	0.867378i	$-0.665804\pi$
$17$	−4.10373	−0.995301	−0.497650	+	0.867378i	$0.665804\pi$
$18$	1.93918	0.457070
$19$	−2.67946	−0.614710	−0.307355	−	0.951595i	$-0.599444\pi$
$19$	−2.67946	−0.614710	−0.307355	+	0.951595i	$0.599444\pi$
$20$	1.76043	0.393644
$21$	−5.18066	−1.13051
$22$	−3.05636	−0.651619
$23$	−8.71738	−1.81770	−0.908850	−	0.417123i	$-0.863038\pi$
$23$	−8.71738	−1.81770	−0.908850	+	0.417123i	$0.863038\pi$
$24$	−0.464570	−0.0948299
$25$	1.00000	0.200000
$26$	0.0861909	0.0169034
$27$	1.00000	0.192450
$28$	−9.12020	−1.72356
$29$	−3.21964	−0.597872	−0.298936	−	0.954273i	$-0.596632\pi$
$29$	−3.21964	−0.597872	−0.298936	+	0.954273i	$0.596632\pi$
$30$	1.93918	0.354045
$31$	−4.51578	−0.811059	−0.405529	−	0.914082i	$-0.632913\pi$
$31$	−4.51578	−0.811059	−0.405529	+	0.914082i	$0.632913\pi$
$32$	−7.64543	−1.35153
$33$	−1.57611	−0.274365
$34$	−7.95788	−1.36477
$35$	−5.18066	−0.875692
$36$	1.76043	0.293405
$37$	9.88589	1.62523	0.812615	−	0.582801i	$-0.198043\pi$
$37$	9.88589	1.62523	0.812615	+	0.582801i	$0.198043\pi$
$38$	−5.19596	−0.842896
$39$	0.0444470	0.00711722
$40$	−0.464570	−0.0734549
$41$	4.75304	0.742300	0.371150	−	0.928573i	$-0.378963\pi$
$41$	4.75304	0.742300	0.371150	+	0.928573i	$0.378963\pi$
$42$	−10.0463	−1.55017
$43$	0.0236316	0.00360378	0.00180189	−	0.999998i	$-0.499426\pi$
$43$	0.0236316	0.00360378	0.00180189	+	0.999998i	$0.499426\pi$
$44$	−2.77463	−0.418291
$45$	1.00000	0.149071
$46$	−16.9046	−2.49245
$47$	11.5866	1.69007	0.845036	−	0.534709i	$-0.179579\pi$
$47$	11.5866	1.69007	0.845036	+	0.534709i	$0.179579\pi$
$48$	−4.42175	−0.638224
$49$	19.8393	2.83418
$50$	1.93918	0.274242
$51$	−4.10373	−0.574637
$52$	0.0782459	0.0108507
$53$	12.1869	1.67400	0.836999	−	0.547204i	$-0.184308\pi$
$53$	12.1869	1.67400	0.836999	+	0.547204i	$0.184308\pi$
$54$	1.93918	0.263889
$55$	−1.57611	−0.212522
$56$	2.40678	0.321619
$57$	−2.67946	−0.354903
$58$	−6.24347	−0.819807
$59$	−6.16261	−0.802303	−0.401151	−	0.916012i	$-0.631390\pi$
$59$	−6.16261	−0.802303	−0.401151	+	0.916012i	$0.631390\pi$
$60$	1.76043	0.227271
$61$	1.41603	0.181304	0.0906520	−	0.995883i	$-0.471105\pi$
$61$	1.41603	0.181304	0.0906520	+	0.995883i	$0.471105\pi$
$62$	−8.75693	−1.11213
$63$	−5.18066	−0.652702
$64$	−5.98240	−0.747800
$65$	0.0444470	0.00551297
$66$	−3.05636	−0.376212
$67$	6.52939	0.797692	0.398846	−	0.917018i	$-0.369411\pi$
$67$	6.52939	0.797692	0.398846	+	0.917018i	$0.369411\pi$
$68$	−7.22433	−0.876078
$69$	−8.71738	−1.04945
$70$	−10.0463	−1.20076
$71$	2.30407	0.273443	0.136722	−	0.990610i	$-0.456343\pi$
$71$	2.30407	0.273443	0.136722	+	0.990610i	$0.456343\pi$
$72$	−0.464570	−0.0547501
$73$	−12.4986	−1.46285	−0.731426	−	0.681921i	$-0.761145\pi$
$73$	−12.4986	−1.46285	−0.731426	+	0.681921i	$0.761145\pi$
$74$	19.1705	2.22853
$75$	1.00000	0.115470
$76$	−4.71700	−0.541077
$77$	8.16529	0.930521
$78$	0.0861909	0.00975920
$79$	−8.96920	−1.00911	−0.504557	−	0.863379i	$-0.668344\pi$
$79$	−8.96920	−1.00911	−0.504557	+	0.863379i	$0.668344\pi$
$80$	−4.42175	−0.494366
$81$	1.00000	0.111111
$82$	9.21702	1.01785
$83$	−10.3732	−1.13861	−0.569305	−	0.822126i	$-0.692788\pi$
$83$	−10.3732	−1.13861	−0.569305	+	0.822126i	$0.692788\pi$
$84$	−9.12020	−0.995095
$85$	−4.10373	−0.445112
$86$	0.0458260	0.00494154
$87$	−3.21964	−0.345181
$88$	0.732212	0.0780541
$89$	2.55458	0.270785	0.135393	−	0.990792i	$-0.456770\pi$
$89$	2.55458	0.270785	0.135393	+	0.990792i	$0.456770\pi$
$90$	1.93918	0.204408
$91$	−0.230265	−0.0241383
$92$	−15.3463	−1.59997
$93$	−4.51578	−0.468265
$94$	22.4685	2.31744
$95$	−2.67946	−0.274907
$96$	−7.64543	−0.780309
$97$	−0.225465	−0.0228925	−0.0114463	−	0.999934i	$-0.503644\pi$
$97$	−0.225465	−0.0228925	−0.0114463	+	0.999934i	$0.503644\pi$
$98$	38.4720	3.88626
$99$	−1.57611	−0.158405
$100$	1.76043	0.176043
$101$	5.51515	0.548778	0.274389	−	0.961619i	$-0.411524\pi$
$101$	5.51515	0.548778	0.274389	+	0.961619i	$0.411524\pi$
$102$	−7.95788	−0.787948
$103$	−5.37365	−0.529481	−0.264741	−	0.964320i	$-0.585286\pi$
$103$	−5.37365	−0.529481	−0.264741	+	0.964320i	$0.585286\pi$
$104$	−0.0206487	−0.00202478
$105$	−5.18066	−0.505581
$106$	23.6326	2.29540
$107$	−14.3287	−1.38520	−0.692602	−	0.721320i	$-0.743536\pi$
$107$	−14.3287	−1.38520	−0.692602	+	0.721320i	$0.743536\pi$
$108$	1.76043	0.169397
$109$	−10.5174	−1.00738	−0.503692	−	0.863883i	$-0.668025\pi$
$109$	−10.5174	−1.00738	−0.503692	+	0.863883i	$0.668025\pi$
$110$	−3.05636	−0.291413
$111$	9.88589	0.938327
$112$	22.9076	2.16456
$113$	−0.561508	−0.0528223	−0.0264111	−	0.999651i	$-0.508408\pi$
$113$	−0.561508	−0.0528223	−0.0264111	+	0.999651i	$0.508408\pi$
$114$	−5.19596	−0.486646
$115$	−8.71738	−0.812900
$116$	−5.66795	−0.526256
$117$	0.0444470	0.00410913
$118$	−11.9504	−1.10013
$119$	21.2600	1.94890
$120$	−0.464570	−0.0424092
$121$	−8.51588	−0.774171
$122$	2.74594	0.248606
$123$	4.75304	0.428567
$124$	−7.94972	−0.713906
$125$	1.00000	0.0894427
$126$	−10.0463	−0.894991
$127$	−17.2507	−1.53075	−0.765377	−	0.643582i	$-0.777448\pi$
$127$	−17.2507	−1.53075	−0.765377	+	0.643582i	$0.777448\pi$
$128$	3.68990	0.326144
$129$	0.0236316	0.00208065
$130$	0.0861909	0.00755944
$131$	6.75454	0.590147	0.295073	−	0.955475i	$-0.404656\pi$
$131$	6.75454	0.590147	0.295073	+	0.955475i	$0.404656\pi$
$132$	−2.77463	−0.241500
$133$	13.8814	1.20367
$134$	12.6617	1.09380
$135$	1.00000	0.0860663
$136$	1.90647	0.163478
$137$	−10.8580	−0.927660	−0.463830	−	0.885924i	$-0.653525\pi$
$137$	−10.8580	−0.927660	−0.463830	+	0.885924i	$0.653525\pi$
$138$	−16.9046	−1.43902
$139$	−5.52221	−0.468388	−0.234194	−	0.972190i	$-0.575245\pi$
$139$	−5.52221	−0.468388	−0.234194	+	0.972190i	$0.575245\pi$
$140$	−9.12020	−0.770797
$141$	11.5866	0.975764
$142$	4.46802	0.374948
$143$	−0.0700533	−0.00585815
$144$	−4.42175	−0.368479
$145$	−3.21964	−0.267376
$146$	−24.2371	−2.00588
$147$	19.8393	1.63632
$148$	17.4034	1.43055
$149$	−8.93430	−0.731927	−0.365963	−	0.930629i	$-0.619260\pi$
$149$	−8.93430	−0.731927	−0.365963	+	0.930629i	$0.619260\pi$
$150$	1.93918	0.158334
$151$	−3.28509	−0.267337	−0.133668	−	0.991026i	$-0.542676\pi$
$151$	−3.28509	−0.267337	−0.133668	+	0.991026i	$0.542676\pi$
$152$	1.24479	0.100966
$153$	−4.10373	−0.331767
$154$	15.8340	1.27594
$155$	−4.51578	−0.362716
$156$	0.0782459	0.00626468
$157$	0.193490	0.0154422	0.00772110	−	0.999970i	$-0.497542\pi$
$157$	0.193490	0.0154422	0.00772110	+	0.999970i	$0.497542\pi$
$158$	−17.3929	−1.38371
$159$	12.1869	0.966483
$160$	−7.64543	−0.604425
$161$	45.1618	3.55925
$162$	1.93918	0.152357
$163$	−1.00000	−0.0783260
$163$	−1.00000	−0.0783260
$164$	8.36740	0.653384
$165$	−1.57611	−0.122700
$166$	−20.1156	−1.56127
$167$	−5.19711	−0.402165	−0.201082	−	0.979574i	$-0.564446\pi$
$167$	−5.19711	−0.402165	−0.201082	+	0.979574i	$0.564446\pi$
$168$	2.40678	0.185687
$169$	−12.9980	−0.999848
$170$	−7.95788	−0.610342
$171$	−2.67946	−0.204903
$172$	0.0416018	0.00317211
$173$	−14.6300	−1.11230	−0.556151	−	0.831081i	$-0.687722\pi$
$173$	−14.6300	−1.11230	−0.556151	+	0.831081i	$0.687722\pi$
$174$	−6.24347	−0.473316
$175$	−5.18066	−0.391621
$176$	6.96915	0.525320
$177$	−6.16261	−0.463210
$178$	4.95381	0.371303
$179$	17.2523	1.28949	0.644747	−	0.764396i	$-0.276963\pi$
$179$	17.2523	1.28949	0.644747	+	0.764396i	$0.276963\pi$
$180$	1.76043	0.131215
$181$	10.0732	0.748732	0.374366	−	0.927281i	$-0.377860\pi$
$181$	10.0732	0.748732	0.374366	+	0.927281i	$0.377860\pi$
$182$	−0.446526	−0.0330987
$183$	1.41603	0.104676
$184$	4.04983	0.298558
$185$	9.88589	0.726825
$186$	−8.75693	−0.642089
$187$	6.46792	0.472981
$188$	20.3973	1.48763
$189$	−5.18066	−0.376838
$190$	−5.19596	−0.376954
$191$	17.6445	1.27671	0.638357	−	0.769740i	$-0.279614\pi$
$191$	17.6445	1.27671	0.638357	+	0.769740i	$0.279614\pi$
$192$	−5.98240	−0.431743
$193$	7.05381	0.507744	0.253872	−	0.967238i	$-0.418296\pi$
$193$	7.05381	0.507744	0.253872	+	0.967238i	$0.418296\pi$
$194$	−0.437219	−0.0313905
$195$	0.0444470	0.00318292
$196$	34.9257	2.49469
$197$	−12.5712	−0.895662	−0.447831	−	0.894118i	$-0.647803\pi$
$197$	−12.5712	−0.895662	−0.447831	+	0.894118i	$0.647803\pi$
$198$	−3.05636	−0.217206
$199$	10.3587	0.734308	0.367154	−	0.930160i	$-0.380332\pi$
$199$	10.3587	0.734308	0.367154	+	0.930160i	$0.380332\pi$
$200$	−0.464570	−0.0328500
$201$	6.52939	0.460548
$202$	10.6949	0.752489
$203$	16.6799	1.17070
$204$	−7.22433	−0.505804
$205$	4.75304	0.331967
$206$	−10.4205	−0.726029
$207$	−8.71738	−0.605900
$208$	−0.196533	−0.0136271
$209$	4.22312	0.292119
$210$	−10.0463	−0.693257
$211$	17.9368	1.23482	0.617410	−	0.786641i	$-0.288182\pi$
$211$	17.9368	1.23482	0.617410	+	0.786641i	$0.288182\pi$
$212$	21.4542	1.47348
$213$	2.30407	0.157873
$214$	−27.7859	−1.89940
$215$	0.0236316	0.00161166
$216$	−0.464570	−0.0316100
$217$	23.3948	1.58814
$218$	−20.3951	−1.38133
$219$	−12.4986	−0.844578
$220$	−2.77463	−0.187065
$221$	−0.182399	−0.0122695
$222$	19.1705	1.28664
$223$	−21.0153	−1.40729	−0.703643	−	0.710553i	$-0.748445\pi$
$223$	−21.0153	−1.40729	−0.703643	+	0.710553i	$0.748445\pi$
$224$	39.6084	2.64645
$225$	1.00000	0.0666667
$226$	−1.08887	−0.0724304
$227$	−21.9526	−1.45704	−0.728522	−	0.685022i	$-0.759793\pi$
$227$	−21.9526	−1.45704	−0.728522	+	0.685022i	$0.759793\pi$
$228$	−4.71700	−0.312391
$229$	10.2433	0.676895	0.338448	−	0.940985i	$-0.390098\pi$
$229$	10.2433	0.676895	0.338448	+	0.940985i	$0.390098\pi$
$230$	−16.9046	−1.11466
$231$	8.16529	0.537236
$232$	1.49575	0.0982006
$233$	−23.7336	−1.55484	−0.777419	−	0.628983i	$-0.783471\pi$
$233$	−23.7336	−1.55484	−0.777419	+	0.628983i	$0.783471\pi$
$234$	0.0861909	0.00563448
$235$	11.5866	0.755824
$236$	−10.8488	−0.706199
$237$	−8.96920	−0.582612
$238$	41.2271	2.67236
$239$	−6.94283	−0.449094	−0.224547	−	0.974463i	$-0.572090\pi$
$239$	−6.94283	−0.449094	−0.224547	+	0.974463i	$0.572090\pi$
$240$	−4.42175	−0.285422
$241$	14.6735	0.945205	0.472602	−	0.881276i	$-0.343315\pi$
$241$	14.6735	0.945205	0.472602	+	0.881276i	$0.343315\pi$
$242$	−16.5139	−1.06155
$243$	1.00000	0.0641500
$244$	2.49282	0.159587
$245$	19.8393	1.26748
$246$	9.21702	0.587656
$247$	−0.119094	−0.00757776
$248$	2.09790	0.133217
$249$	−10.3732	−0.657377
$250$	1.93918	0.122645
$251$	9.56051	0.603454	0.301727	−	0.953394i	$-0.402437\pi$
$251$	9.56051	0.603454	0.301727	+	0.953394i	$0.402437\pi$
$252$	−9.12020	−0.574518
$253$	13.7395	0.863798
$254$	−33.4523	−2.09899
$255$	−4.10373	−0.256985
$256$	19.1202	1.19501
$257$	−14.5028	−0.904663	−0.452331	−	0.891850i	$-0.649408\pi$
$257$	−14.5028	−0.904663	−0.452331	+	0.891850i	$0.649408\pi$
$258$	0.0458260	0.00285300
$259$	−51.2155	−3.18237
$260$	0.0782459	0.00485260
$261$	−3.21964	−0.199291
$262$	13.0983	0.809215
$263$	10.3921	0.640806	0.320403	−	0.947281i	$-0.396182\pi$
$263$	10.3921	0.640806	0.320403	+	0.947281i	$0.396182\pi$
$264$	0.732212	0.0450646
$265$	12.1869	0.748635
$266$	26.9185	1.65048
$267$	2.55458	0.156338
$268$	11.4945	0.702140
$269$	10.8014	0.658576	0.329288	−	0.944230i	$-0.393191\pi$
$269$	10.8014	0.658576	0.329288	+	0.944230i	$0.393191\pi$
$270$	1.93918	0.118015
$271$	2.75197	0.167170	0.0835851	−	0.996501i	$-0.473363\pi$
$271$	2.75197	0.167170	0.0835851	+	0.996501i	$0.473363\pi$
$272$	18.1456	1.10024
$273$	−0.230265	−0.0139363
$274$	−21.0556	−1.27202
$275$	−1.57611	−0.0950429
$276$	−15.3463	−0.923741
$277$	5.27160	0.316740	0.158370	−	0.987380i	$-0.449376\pi$
$277$	5.27160	0.316740	0.158370	+	0.987380i	$0.449376\pi$
$278$	−10.7086	−0.642257
$279$	−4.51578	−0.270353
$280$	2.40678	0.143833
$281$	12.6226	0.753000	0.376500	−	0.926417i	$-0.377128\pi$
$281$	12.6226	0.753000	0.376500	+	0.926417i	$0.377128\pi$
$282$	22.4685	1.33798
$283$	21.6402	1.28638	0.643188	−	0.765709i	$-0.277612\pi$
$283$	21.6402	1.28638	0.643188	+	0.765709i	$0.277612\pi$
$284$	4.05616	0.240689
$285$	−2.67946	−0.158717
$286$	−0.135846	−0.00803275
$287$	−24.6239	−1.45350
$288$	−7.64543	−0.450512
$289$	−0.159407	−0.00937689
$290$	−6.24347	−0.366629
$291$	−0.225465	−0.0132170
$292$	−22.0029	−1.28762
$293$	2.86643	0.167458	0.0837292	−	0.996489i	$-0.473317\pi$
$293$	2.86643	0.167458	0.0837292	+	0.996489i	$0.473317\pi$
$294$	38.4720	2.24373
$295$	−6.16261	−0.358801
$296$	−4.59269	−0.266944
$297$	−1.57611	−0.0914551
$298$	−17.3252	−1.00362
$299$	−0.387462	−0.0224075
$300$	1.76043	0.101638
$301$	−0.122427	−0.00705659
$302$	−6.37039	−0.366575
$303$	5.51515	0.316837
$304$	11.8479	0.679523
$305$	1.41603	0.0810816
$306$	−7.95788	−0.454922
$307$	12.7886	0.729883	0.364941	−	0.931031i	$-0.381089\pi$
$307$	12.7886	0.729883	0.364941	+	0.931031i	$0.381089\pi$
$308$	14.3744	0.819058
$309$	−5.37365	−0.305696
$310$	−8.75693	−0.497360
$311$	3.46833	0.196671	0.0983356	−	0.995153i	$-0.468648\pi$
$311$	3.46833	0.196671	0.0983356	+	0.995153i	$0.468648\pi$
$312$	−0.0206487	−0.00116900
$313$	−17.2996	−0.977833	−0.488916	−	0.872331i	$-0.662608\pi$
$313$	−17.2996	−0.977833	−0.488916	+	0.872331i	$0.662608\pi$
$314$	0.375213	0.0211745
$315$	−5.18066	−0.291897
$316$	−15.7896	−0.888237
$317$	−11.2304	−0.630763	−0.315381	−	0.948965i	$-0.602132\pi$
$317$	−11.2304	−0.630763	−0.315381	+	0.948965i	$0.602132\pi$
$318$	23.6326	1.32525
$319$	5.07450	0.284117
$320$	−5.98240	−0.334427
$321$	−14.3287	−0.799747
$322$	87.5770	4.88048
$323$	10.9958	0.611821
$324$	1.76043	0.0978017
$325$	0.0444470	0.00246548
$326$	−1.93918	−0.107401
$327$	−10.5174	−0.581613
$328$	−2.20812	−0.121923
$329$	−60.0260	−3.30934
$330$	−3.05636	−0.168247
$331$	−24.0717	−1.32310	−0.661550	−	0.749901i	$-0.730101\pi$
$331$	−24.0717	−1.32310	−0.661550	+	0.749901i	$0.730101\pi$
$332$	−18.2614	−1.00222
$333$	9.88589	0.541743
$334$	−10.0782	−0.551452
$335$	6.52939	0.356739
$336$	22.9076	1.24971
$337$	−4.79946	−0.261443	−0.130722	−	0.991419i	$-0.541729\pi$
$337$	−4.79946	−0.261443	−0.130722	+	0.991419i	$0.541729\pi$
$338$	−25.2055	−1.37100
$339$	−0.561508	−0.0304970
$340$	−7.22433	−0.391794
$341$	7.11736	0.385427
$342$	−5.19596	−0.280965
$343$	−66.5160	−3.59152
$344$	−0.0109785	−0.000591922 0
$345$	−8.71738	−0.469328
$346$	−28.3703	−1.52520
$347$	−32.4862	−1.74395	−0.871975	−	0.489551i	$-0.837161\pi$
$347$	−32.4862	−1.74395	−0.871975	+	0.489551i	$0.837161\pi$
$348$	−5.66795	−0.303834
$349$	13.2238	0.707855	0.353928	−	0.935273i	$-0.384846\pi$
$349$	13.2238	0.707855	0.353928	+	0.935273i	$0.384846\pi$
$350$	−10.0463	−0.536995
$351$	0.0444470	0.00237241
$352$	12.0500	0.642269
$353$	−24.9429	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$353$	−24.9429	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$354$	−11.9504	−0.635158
$355$	2.30407	0.122288
$356$	4.49717	0.238349
$357$	21.2600	1.12520
$358$	33.4553	1.76817
$359$	3.71034	0.195824	0.0979121	−	0.995195i	$-0.468784\pi$
$359$	3.71034	0.195824	0.0979121	+	0.995195i	$0.468784\pi$
$360$	−0.464570	−0.0244850
$361$	−11.8205	−0.622132
$362$	19.5337	1.02667
$363$	−8.51588	−0.446968
$364$	−0.405366	−0.0212469
$365$	−12.4986	−0.654207
$366$	2.74594	0.143533
$367$	3.82903	0.199874	0.0999369	−	0.994994i	$-0.468136\pi$
$367$	3.82903	0.199874	0.0999369	+	0.994994i	$0.468136\pi$
$368$	38.5461	2.00935
$369$	4.75304	0.247433
$370$	19.1705	0.996629
$371$	−63.1362	−3.27787
$372$	−7.94972	−0.412174
$373$	35.0822	1.81649	0.908244	−	0.418441i	$-0.137423\pi$
$373$	35.0822	1.81649	0.908244	+	0.418441i	$0.137423\pi$
$374$	12.5425	0.648556
$375$	1.00000	0.0516398
$376$	−5.38276	−0.277595
$377$	−0.143103	−0.00737019
$378$	−10.0463	−0.516724
$379$	5.92389	0.304290	0.152145	−	0.988358i	$-0.451382\pi$
$379$	5.92389	0.304290	0.152145	+	0.988358i	$0.451382\pi$
$380$	−4.71700	−0.241977
$381$	−17.2507	−0.883782
$382$	34.2160	1.75064
$383$	26.6377	1.36112	0.680561	−	0.732692i	$-0.261736\pi$
$383$	26.6377	1.36112	0.680561	+	0.732692i	$0.261736\pi$
$384$	3.68990	0.188299
$385$	8.16529	0.416142
$386$	13.6786	0.696224
$387$	0.0236316	0.00120126
$388$	−0.396916	−0.0201504
$389$	−14.1939	−0.719657	−0.359829	−	0.933018i	$-0.617165\pi$
$389$	−14.1939	−0.719657	−0.359829	+	0.933018i	$0.617165\pi$
$390$	0.0861909	0.00436445
$391$	35.7738	1.80916
$392$	−9.21673	−0.465515
$393$	6.75454	0.340722
$394$	−24.3779	−1.22814
$395$	−8.96920	−0.451289
$396$	−2.77463	−0.139430
$397$	−2.52633	−0.126793	−0.0633965	−	0.997988i	$-0.520193\pi$
$397$	−2.52633	−0.126793	−0.0633965	+	0.997988i	$0.520193\pi$
$398$	20.0874	1.00689
$399$	13.8814	0.694938
$400$	−4.42175	−0.221087
$401$	13.1483	0.656596	0.328298	−	0.944574i	$-0.393525\pi$
$401$	13.1483	0.656596	0.328298	+	0.944574i	$0.393525\pi$
$402$	12.6617	0.631507
$403$	−0.200713	−0.00999823
$404$	9.70903	0.483042
$405$	1.00000	0.0496904
$406$	32.3453	1.60527
$407$	−15.5812	−0.772333
$408$	1.90647	0.0943843
$409$	−19.4698	−0.962718	−0.481359	−	0.876524i	$-0.659857\pi$
$409$	−19.4698	−0.962718	−0.481359	+	0.876524i	$0.659857\pi$
$410$	9.21702	0.455196
$411$	−10.8580	−0.535585
$412$	−9.45993	−0.466057
$413$	31.9264	1.57099
$414$	−16.9046	−0.830816
$415$	−10.3732	−0.509202
$416$	−0.339817	−0.0166609
$417$	−5.52221	−0.270424
$418$	8.18939	0.400556
$419$	−17.6804	−0.863745	−0.431873	−	0.901935i	$-0.642147\pi$
$419$	−17.6804	−0.863745	−0.431873	+	0.901935i	$0.642147\pi$
$420$	−9.12020	−0.445020
$421$	0.488674	0.0238165	0.0119082	−	0.999929i	$-0.496209\pi$
$421$	0.488674	0.0238165	0.0119082	+	0.999929i	$0.496209\pi$
$422$	34.7827	1.69320
$423$	11.5866	0.563358
$424$	−5.66166	−0.274955
$425$	−4.10373	−0.199060
$426$	4.46802	0.216476
$427$	−7.33597	−0.355013
$428$	−25.2246	−1.21928
$429$	−0.0700533	−0.00338221
$430$	0.0458260	0.00220992
$431$	18.4726	0.889794	0.444897	−	0.895582i	$-0.353240\pi$
$431$	18.4726	0.889794	0.444897	+	0.895582i	$0.353240\pi$
$432$	−4.42175	−0.212741
$433$	20.6868	0.994142	0.497071	−	0.867710i	$-0.334409\pi$
$433$	20.6868	0.994142	0.497071	+	0.867710i	$0.334409\pi$
$434$	45.3667	2.17767
$435$	−3.21964	−0.154370
$436$	−18.5151	−0.886714
$437$	23.3579	1.11736
$438$	−24.2371	−1.15809
$439$	−22.3798	−1.06813	−0.534065	−	0.845444i	$-0.679336\pi$
$439$	−22.3798	−1.06813	−0.534065	+	0.845444i	$0.679336\pi$
$440$	0.732212	0.0349069
$441$	19.8393	0.944727
$442$	−0.353704	−0.0168240
$443$	10.5686	0.502130	0.251065	−	0.967970i	$-0.419219\pi$
$443$	10.5686	0.502130	0.251065	+	0.967970i	$0.419219\pi$
$444$	17.4034	0.825930
$445$	2.55458	0.121099
$446$	−40.7525	−1.92968
$447$	−8.93430	−0.422578
$448$	30.9928	1.46427
$449$	0.557781	0.0263233	0.0131617	−	0.999913i	$-0.495810\pi$
$449$	0.557781	0.0263233	0.0131617	+	0.999913i	$0.495810\pi$
$450$	1.93918	0.0914140
$451$	−7.49131	−0.352752
$452$	−0.988496	−0.0464950
$453$	−3.28509	−0.154347
$454$	−42.5701	−1.99791
$455$	−0.230265	−0.0107950
$456$	1.24479	0.0582929
$457$	36.2041	1.69355	0.846777	−	0.531948i	$-0.178540\pi$
$457$	36.2041	1.69355	0.846777	+	0.531948i	$0.178540\pi$
$458$	19.8636	0.928165
$459$	−4.10373	−0.191546
$460$	−15.3463	−0.715527
$461$	22.1064	1.02960	0.514799	−	0.857311i	$-0.327867\pi$
$461$	22.1064	1.02960	0.514799	+	0.857311i	$0.327867\pi$
$462$	15.8340	0.736664
$463$	−29.1365	−1.35409	−0.677044	−	0.735943i	$-0.736739\pi$
$463$	−29.1365	−1.35409	−0.677044	+	0.735943i	$0.736739\pi$
$464$	14.2364	0.660909
$465$	−4.51578	−0.209414
$466$	−46.0237	−2.13201
$467$	3.96605	0.183527	0.0917634	−	0.995781i	$-0.470750\pi$
$467$	3.96605	0.183527	0.0917634	+	0.995781i	$0.470750\pi$
$468$	0.0782459	0.00361692
$469$	−33.8266	−1.56197
$470$	22.4685	1.03639
$471$	0.193490	0.00891556
$472$	2.86296	0.131778
$473$	−0.0372460	−0.00171257
$474$	−17.3929	−0.798883
$475$	−2.67946	−0.122942
$476$	37.4268	1.71546
$477$	12.1869	0.557999
$478$	−13.4634	−0.615802
$479$	−34.7821	−1.58923	−0.794617	−	0.607111i	$-0.792328\pi$
$479$	−34.7821	−1.58923	−0.794617	+	0.607111i	$0.792328\pi$
$480$	−7.64543	−0.348965
$481$	0.439398	0.0200348
$482$	28.4547	1.29607
$483$	45.1618	2.05493
$484$	−14.9916	−0.681437
$485$	−0.225465	−0.0102379
$486$	1.93918	0.0879631
$487$	24.3559	1.10367	0.551835	−	0.833953i	$-0.313928\pi$
$487$	24.3559	1.10367	0.551835	+	0.833953i	$0.313928\pi$
$488$	−0.657845	−0.0297792
$489$	−1.00000	−0.0452216
$490$	38.4720	1.73799
$491$	8.87990	0.400744	0.200372	−	0.979720i	$-0.435785\pi$
$491$	8.87990	0.400744	0.200372	+	0.979720i	$0.435785\pi$
$492$	8.36740	0.377231
$493$	13.2125	0.595062
$494$	−0.230945	−0.0103907
$495$	−1.57611	−0.0708408
$496$	19.9676	0.896574
$497$	−11.9366	−0.535431
$498$	−20.1156	−0.901402
$499$	−35.8001	−1.60263	−0.801317	−	0.598240i	$-0.795867\pi$
$499$	−35.8001	−1.60263	−0.801317	+	0.598240i	$0.795867\pi$
$500$	1.76043	0.0787288
$501$	−5.19711	−0.232190
$502$	18.5396	0.827462
$503$	15.3142	0.682824	0.341412	−	0.939914i	$-0.389095\pi$
$503$	15.3142	0.682824	0.341412	+	0.939914i	$0.389095\pi$
$504$	2.40678	0.107206
$505$	5.51515	0.245421
$506$	26.6435	1.18445
$507$	−12.9980	−0.577263
$508$	−30.3687	−1.34739
$509$	23.7794	1.05400	0.527002	−	0.849864i	$-0.323316\pi$
$509$	23.7794	1.05400	0.527002	+	0.849864i	$0.323316\pi$
$510$	−7.95788	−0.352381
$511$	64.7511	2.86442
$512$	29.6978	1.31247
$513$	−2.67946	−0.118301
$514$	−28.1237	−1.24048
$515$	−5.37365	−0.236791
$516$	0.0416018	0.00183142
$517$	−18.2617	−0.803147
$518$	−99.3162	−4.36370
$519$	−14.6300	−0.642188
$520$	−0.0206487	−0.000905507 0
$521$	1.12265	0.0491840	0.0245920	−	0.999698i	$-0.492171\pi$
$521$	1.12265	0.0491840	0.0245920	+	0.999698i	$0.492171\pi$
$522$	−6.24347	−0.273269
$523$	13.9560	0.610252	0.305126	−	0.952312i	$-0.401301\pi$
$523$	13.9560	0.610252	0.305126	+	0.952312i	$0.401301\pi$
$524$	11.8909	0.519456
$525$	−5.18066	−0.226103
$526$	20.1522	0.878679
$527$	18.5316	0.807247
$528$	6.96915	0.303293
$529$	52.9928	2.30403
$530$	23.6326	1.02654
$531$	−6.16261	−0.267434
$532$	24.4372	1.05949
$533$	0.211259	0.00915062
$534$	4.95381	0.214372
$535$	−14.3287	−0.619482
$536$	−3.03336	−0.131021
$537$	17.2523	0.744490
$538$	20.9460	0.903045
$539$	−31.2688	−1.34684
$540$	1.76043	0.0757569
$541$	35.1710	1.51212	0.756060	−	0.654502i	$-0.227122\pi$
$541$	35.1710	1.51212	0.756060	+	0.654502i	$0.227122\pi$
$542$	5.33657	0.229225
$543$	10.0732	0.432281
$544$	31.3748	1.34518
$545$	−10.5174	−0.450516
$546$	−0.446526	−0.0191096
$547$	−8.60822	−0.368061	−0.184031	−	0.982921i	$-0.558915\pi$
$547$	−8.60822	−0.368061	−0.184031	+	0.982921i	$0.558915\pi$
$548$	−19.1147	−0.816540
$549$	1.41603	0.0604347
$550$	−3.05636	−0.130324
$551$	8.62688	0.367517
$552$	4.04983	0.172372
$553$	46.4664	1.97595
$554$	10.2226	0.434316
$555$	9.88589	0.419633
$556$	−9.72146	−0.412282
$557$	34.1877	1.44858	0.724289	−	0.689496i	$-0.242168\pi$
$557$	34.1877	1.44858	0.724289	+	0.689496i	$0.242168\pi$
$558$	−8.75693	−0.370710
$559$	0.00105035	4.44252e−5 0
$560$	22.9076	0.968022
$561$	6.46792	0.273076
$562$	24.4775	1.03252
$563$	−21.0137	−0.885622	−0.442811	−	0.896615i	$-0.646019\pi$
$563$	−21.0137	−0.885622	−0.442811	+	0.896615i	$0.646019\pi$
$564$	20.3973	0.858882
$565$	−0.561508	−0.0236228
$566$	41.9643	1.76389
$567$	−5.18066	−0.217567
$568$	−1.07040	−0.0449131
$569$	−44.2178	−1.85371	−0.926854	−	0.375423i	$-0.877497\pi$
$569$	−44.2178	−1.85371	−0.926854	+	0.375423i	$0.877497\pi$
$570$	−5.19596	−0.217635
$571$	6.07991	0.254436	0.127218	−	0.991875i	$-0.459395\pi$
$571$	6.07991	0.254436	0.127218	+	0.991875i	$0.459395\pi$
$572$	−0.123324	−0.00515643
$573$	17.6445	0.737111
$574$	−47.7503	−1.99306
$575$	−8.71738	−0.363540
$576$	−5.98240	−0.249267
$577$	−38.2203	−1.59113	−0.795566	−	0.605867i	$-0.792826\pi$
$577$	−38.2203	−1.59113	−0.795566	+	0.605867i	$0.792826\pi$
$578$	−0.309120	−0.0128577
$579$	7.05381	0.293146
$580$	−5.66795	−0.235349
$581$	53.7403	2.22952
$582$	−0.437219	−0.0181233
$583$	−19.2079	−0.795508
$584$	5.80648	0.240274
$585$	0.0444470	0.00183766
$586$	5.55853	0.229621
$587$	29.9346	1.23553	0.617767	−	0.786361i	$-0.288038\pi$
$587$	29.9346	1.23553	0.617767	+	0.786361i	$0.288038\pi$
$588$	34.9257	1.44031
$589$	12.0999	0.498566
$590$	−11.9504	−0.491991
$591$	−12.5712	−0.517110
$592$	−43.7129	−1.79659
$593$	−44.8253	−1.84075	−0.920377	−	0.391032i	$-0.872118\pi$
$593$	−44.8253	−1.84075	−0.920377	+	0.391032i	$0.872118\pi$
$594$	−3.05636	−0.125404
$595$	21.2600	0.871577
$596$	−15.7282	−0.644253
$597$	10.3587	0.423953
$598$	−0.751359	−0.0307254
$599$	−22.3231	−0.912098	−0.456049	−	0.889955i	$-0.650736\pi$
$599$	−22.3231	−0.912098	−0.456049	+	0.889955i	$0.650736\pi$
$600$	−0.464570	−0.0189660
$601$	−37.8141	−1.54247	−0.771234	−	0.636551i	$-0.780360\pi$
$601$	−37.8141	−1.54247	−0.771234	+	0.636551i	$0.780360\pi$
$602$	−0.237409	−0.00967607
$603$	6.52939	0.265897
$604$	−5.78317	−0.235314
$605$	−8.51588	−0.346220
$606$	10.6949	0.434450
$607$	24.1032	0.978317	0.489159	−	0.872195i	$-0.337304\pi$
$607$	24.1032	0.978317	0.489159	+	0.872195i	$0.337304\pi$
$608$	20.4856	0.830801
$609$	16.6799	0.675902
$610$	2.74594	0.111180
$611$	0.514988	0.0208342
$612$	−7.22433	−0.292026
$613$	−35.3614	−1.42823	−0.714117	−	0.700026i	$-0.753172\pi$
$613$	−35.3614	−1.42823	−0.714117	+	0.700026i	$0.753172\pi$
$614$	24.7994	1.00082
$615$	4.75304	0.191661
$616$	−3.79335	−0.152838
$617$	7.91557	0.318669	0.159335	−	0.987225i	$-0.449065\pi$
$617$	7.91557	0.318669	0.159335	+	0.987225i	$0.449065\pi$
$618$	−10.4205	−0.419173
$619$	18.3992	0.739525	0.369763	−	0.929126i	$-0.379439\pi$
$619$	18.3992	0.739525	0.369763	+	0.929126i	$0.379439\pi$
$620$	−7.94972	−0.319269
$621$	−8.71738	−0.349817
$622$	6.72574	0.269677
$623$	−13.2344	−0.530227
$624$	−0.196533	−0.00786763
$625$	1.00000	0.0400000
$626$	−33.5471	−1.34081
$627$	4.22312	0.168655
$628$	0.340626	0.0135925
$629$	−40.5690	−1.61759
$630$	−10.0463	−0.400252
$631$	19.7660	0.786871	0.393435	−	0.919352i	$-0.371286\pi$
$631$	19.7660	0.786871	0.393435	+	0.919352i	$0.371286\pi$
$632$	4.16682	0.165747
$633$	17.9368	0.712924
$634$	−21.7778	−0.864908
$635$	−17.2507	−0.684574
$636$	21.4542	0.850713
$637$	0.881797	0.0349381
$638$	9.84038	0.389584
$639$	2.30407	0.0911477
$640$	3.68990	0.145856
$641$	−37.5435	−1.48288	−0.741440	−	0.671020i	$-0.765857\pi$
$641$	−37.5435	−1.48288	−0.741440	+	0.671020i	$0.765857\pi$
$642$	−27.7859	−1.09662
$643$	−35.7525	−1.40994	−0.704971	−	0.709236i	$-0.749040\pi$
$643$	−35.7525	−1.40994	−0.704971	+	0.709236i	$0.749040\pi$
$644$	79.5042	3.13291
$645$	0.0236316	0.000930493 0
$646$	21.3228	0.838934
$647$	−18.9559	−0.745235	−0.372617	−	0.927985i	$-0.621540\pi$
$647$	−18.9559	−0.745235	−0.372617	+	0.927985i	$0.621540\pi$
$648$	−0.464570	−0.0182500
$649$	9.71293	0.381266
$650$	0.0861909	0.00338069
$651$	23.3948	0.916913
$652$	−1.76043	−0.0689438
$653$	−46.9188	−1.83608	−0.918038	−	0.396494i	$-0.870227\pi$
$653$	−46.9188	−1.83608	−0.918038	+	0.396494i	$0.870227\pi$
$654$	−20.3951	−0.797513
$655$	6.75454	0.263922
$656$	−21.0167	−0.820566
$657$	−12.4986	−0.487617
$658$	−116.401	−4.53780
$659$	20.2429	0.788550	0.394275	−	0.918993i	$-0.370996\pi$
$659$	20.2429	0.788550	0.394275	+	0.918993i	$0.370996\pi$
$660$	−2.77463	−0.108002
$661$	37.9084	1.47447	0.737233	−	0.675638i	$-0.236132\pi$
$661$	37.9084	1.47447	0.737233	+	0.675638i	$0.236132\pi$
$662$	−46.6794	−1.81425
$663$	−0.182399	−0.00708377
$664$	4.81909	0.187017
$665$	13.8814	0.538296
$666$	19.1705	0.742844
$667$	28.0668	1.08675
$668$	−9.14916	−0.353991
$669$	−21.0153	−0.812497
$670$	12.6617	0.489163
$671$	−2.23182	−0.0861583
$672$	39.6084	1.52793
$673$	3.33580	0.128586	0.0642929	−	0.997931i	$-0.479521\pi$
$673$	3.33580	0.128586	0.0642929	+	0.997931i	$0.479521\pi$
$674$	−9.30704	−0.358494
$675$	1.00000	0.0384900
$676$	−22.8821	−0.880081
$677$	−20.3244	−0.781130	−0.390565	−	0.920575i	$-0.627720\pi$
$677$	−20.3244	−0.781130	−0.390565	+	0.920575i	$0.627720\pi$
$678$	−1.08887	−0.0418177
$679$	1.16806	0.0448261
$680$	1.90647	0.0731097
$681$	−21.9526	−0.841225
$682$	13.8019	0.528501
$683$	−11.5056	−0.440250	−0.220125	−	0.975472i	$-0.570646\pi$
$683$	−11.5056	−0.440250	−0.220125	+	0.975472i	$0.570646\pi$
$684$	−4.71700	−0.180359
$685$	−10.8580	−0.414862
$686$	−128.987	−4.92473
$687$	10.2433	0.390806
$688$	−0.104493	−0.00398375
$689$	0.541671	0.0206360
$690$	−16.9046	−0.643547
$691$	−9.55742	−0.363581	−0.181791	−	0.983337i	$-0.558189\pi$
$691$	−9.55742	−0.363581	−0.181791	+	0.983337i	$0.558189\pi$
$692$	−25.7552	−0.979065
$693$	8.16529	0.310174
$694$	−62.9966	−2.39132
$695$	−5.52221	−0.209469
$696$	1.49575	0.0566961
$697$	−19.5052	−0.738812
$698$	25.6434	0.970618
$699$	−23.7336	−0.897686
$700$	−9.12020	−0.344711
$701$	−42.0050	−1.58651	−0.793253	−	0.608892i	$-0.791614\pi$
$701$	−42.0050	−1.58651	−0.793253	+	0.608892i	$0.791614\pi$
$702$	0.0861909	0.00325307
$703$	−26.4888	−0.999045
$704$	9.42892	0.355366
$705$	11.5866	0.436375
$706$	−48.3688	−1.82038
$707$	−28.5721	−1.07457
$708$	−10.8488	−0.407724
$709$	45.7476	1.71809	0.859044	−	0.511901i	$-0.171059\pi$
$709$	45.7476	1.71809	0.859044	+	0.511901i	$0.171059\pi$
$710$	4.46802	0.167682
$711$	−8.96920	−0.336371
$712$	−1.18678	−0.0444766
$713$	39.3658	1.47426
$714$	41.2271	1.54289
$715$	−0.0700533	−0.00261985
$716$	30.3714	1.13503
$717$	−6.94283	−0.259285
$718$	7.19503	0.268516
$719$	−24.8397	−0.926364	−0.463182	−	0.886263i	$-0.653292\pi$
$719$	−24.8397	−0.926364	−0.463182	+	0.886263i	$0.653292\pi$
$720$	−4.42175	−0.164789
$721$	27.8391	1.03678
$722$	−22.9221	−0.853073
$723$	14.6735	0.545714
$724$	17.7331	0.659046
$725$	−3.21964	−0.119574
$726$	−16.5139	−0.612887
$727$	−45.8001	−1.69863	−0.849317	−	0.527884i	$-0.822986\pi$
$727$	−45.8001	−1.69863	−0.849317	+	0.527884i	$0.822986\pi$
$728$	0.106974	0.00396473
$729$	1.00000	0.0370370
$730$	−24.2371	−0.897055
$731$	−0.0969777	−0.00358685
$732$	2.49282	0.0921373
$733$	0.848127	0.0313263	0.0156632	−	0.999877i	$-0.495014\pi$
$733$	0.848127	0.0313263	0.0156632	+	0.999877i	$0.495014\pi$
$734$	7.42519	0.274069
$735$	19.8393	0.731783
$736$	66.6482	2.45668
$737$	−10.2910	−0.379075
$738$	9.21702	0.339283
$739$	51.0064	1.87630	0.938150	−	0.346230i	$-0.112538\pi$
$739$	51.0064	1.87630	0.938150	+	0.346230i	$0.112538\pi$
$740$	17.4034	0.639762
$741$	−0.119094	−0.00437502
$742$	−122.433	−4.49464
$743$	−37.6333	−1.38063	−0.690316	−	0.723508i	$-0.742528\pi$
$743$	−37.6333	−1.38063	−0.690316	+	0.723508i	$0.742528\pi$
$744$	2.09790	0.0769126
$745$	−8.93430	−0.327328
$746$	68.0308	2.49079
$747$	−10.3732	−0.379537
$748$	11.3863	0.416325
$749$	74.2319	2.71238
$750$	1.93918	0.0708089
$751$	39.4155	1.43829	0.719146	−	0.694859i	$-0.244533\pi$
$751$	39.4155	1.43829	0.719146	+	0.694859i	$0.244533\pi$
$752$	−51.2328	−1.86827
$753$	9.56051	0.348404
$754$	−0.277503	−0.0101061
$755$	−3.28509	−0.119557
$756$	−9.12020	−0.331698
$757$	−21.6971	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$757$	−21.6971	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$758$	11.4875	0.417245
$759$	13.7395	0.498714
$760$	1.24479	0.0451535
$761$	21.2526	0.770407	0.385203	−	0.922832i	$-0.374131\pi$
$761$	21.2526	0.770407	0.385203	+	0.922832i	$0.374131\pi$
$762$	−33.4523	−1.21185
$763$	54.4871	1.97256
$764$	31.0620	1.12378
$765$	−4.10373	−0.148371
$766$	51.6553	1.86638
$767$	−0.273909	−0.00989030
$768$	19.1202	0.689940
$769$	9.73798	0.351160	0.175580	−	0.984465i	$-0.443820\pi$
$769$	9.73798	0.351160	0.175580	+	0.984465i	$0.443820\pi$
$770$	15.8340	0.570617
$771$	−14.5028	−0.522307
$772$	12.4177	0.446924
$773$	54.5312	1.96135	0.980675	−	0.195646i	$-0.0626803\pi$
$773$	54.5312	1.96135	0.980675	+	0.195646i	$0.0626803\pi$
$774$	0.0458260	0.00164718
$775$	−4.51578	−0.162212
$776$	0.104744	0.00376011
$777$	−51.2155	−1.83734
$778$	−27.5245	−0.986801
$779$	−12.7356	−0.456299
$780$	0.0782459	0.00280165
$781$	−3.63147	−0.129944
$782$	69.3719	2.48073
$783$	−3.21964	−0.115060
$784$	−87.7242	−3.13301
$785$	0.193490	0.00690596
$786$	13.0983	0.467201
$787$	16.6509	0.593542	0.296771	−	0.954949i	$-0.404090\pi$
$787$	16.6509	0.593542	0.296771	+	0.954949i	$0.404090\pi$
$788$	−22.1307	−0.788375
$789$	10.3921	0.369969
$790$	−17.3929	−0.618812
$791$	2.90899	0.103432
$792$	0.732212	0.0260180
$793$	0.0629383	0.00223500
$794$	−4.89902	−0.173860
$795$	12.1869	0.432225
$796$	18.2358	0.646349
$797$	−39.5128	−1.39962	−0.699808	−	0.714331i	$-0.746731\pi$
$797$	−39.5128	−1.39962	−0.699808	+	0.714331i	$0.746731\pi$
$798$	26.9185	0.952905
$799$	−47.5481	−1.68213
$800$	−7.64543	−0.270307
$801$	2.55458	0.0902618
$802$	25.4970	0.900330
$803$	19.6992	0.695168
$804$	11.4945	0.405381
$805$	45.1618	1.59175
$806$	−0.389219	−0.0137097
$807$	10.8014	0.380229
$808$	−2.56217	−0.0901369
$809$	−19.7986	−0.696081	−0.348041	−	0.937479i	$-0.613153\pi$
$809$	−19.7986	−0.696081	−0.348041	+	0.937479i	$0.613153\pi$
$810$	1.93918	0.0681359
$811$	32.7971	1.15166	0.575831	−	0.817569i	$-0.304679\pi$
$811$	32.7971	1.15166	0.575831	+	0.817569i	$0.304679\pi$
$812$	29.3637	1.03046
$813$	2.75197	0.0965157
$814$	−30.2149	−1.05903
$815$	−1.00000	−0.0350285
$816$	18.1456	0.635225
$817$	−0.0633198	−0.00221528
$818$	−37.7554	−1.32009
$819$	−0.230265	−0.00804611
$820$	8.36740	0.292202
$821$	18.8684	0.658511	0.329256	−	0.944241i	$-0.393202\pi$
$821$	18.8684	0.658511	0.329256	+	0.944241i	$0.393202\pi$
$822$	−21.0556	−0.734399
$823$	−4.89629	−0.170674	−0.0853370	−	0.996352i	$-0.527197\pi$
$823$	−4.89629	−0.170674	−0.0853370	+	0.996352i	$0.527197\pi$
$824$	2.49643	0.0869674
$825$	−1.57611	−0.0548731
$826$	61.9111	2.15416
$827$	49.9098	1.73553	0.867766	−	0.496973i	$-0.165555\pi$
$827$	49.9098	1.73553	0.867766	+	0.496973i	$0.165555\pi$
$828$	−15.3463	−0.533322
$829$	35.7140	1.24040	0.620199	−	0.784444i	$-0.287052\pi$
$829$	35.7140	1.24040	0.620199	+	0.784444i	$0.287052\pi$
$830$	−20.1156	−0.698223
$831$	5.27160	0.182870
$832$	−0.265900	−0.00921842
$833$	−81.4150	−2.82086
$834$	−10.7086	−0.370808
$835$	−5.19711	−0.179854
$836$	7.43450	0.257128
$837$	−4.51578	−0.156088
$838$	−34.2856	−1.18438
$839$	−6.75186	−0.233100	−0.116550	−	0.993185i	$-0.537184\pi$
$839$	−6.75186	−0.233100	−0.116550	+	0.993185i	$0.537184\pi$
$840$	2.40678	0.0830418
$841$	−18.6339	−0.642550
$842$	0.947627	0.0326574
$843$	12.6226	0.434745
$844$	31.5765	1.08691
$845$	−12.9980	−0.447146
$846$	22.4685	0.772481
$847$	44.1179	1.51591
$848$	−53.8873	−1.85050
$849$	21.6402	0.742689
$850$	−7.95788	−0.272953
$851$	−86.1791	−2.95418
$852$	4.05616	0.138962
$853$	−42.1953	−1.44474	−0.722370	−	0.691507i	$-0.756947\pi$
$853$	−42.1953	−1.44474	−0.722370	+	0.691507i	$0.756947\pi$
$854$	−14.2258	−0.486797
$855$	−2.67946	−0.0916355
$856$	6.65666	0.227520
$857$	−20.5464	−0.701851	−0.350926	−	0.936403i	$-0.614133\pi$
$857$	−20.5464	−0.701851	−0.350926	+	0.936403i	$0.614133\pi$
$858$	−0.135846	−0.00463771
$859$	−7.40852	−0.252775	−0.126388	−	0.991981i	$-0.540338\pi$
$859$	−7.40852	−0.252775	−0.126388	+	0.991981i	$0.540338\pi$
$860$	0.0416018	0.00141861
$861$	−24.6239	−0.839181
$862$	35.8218	1.22009
$863$	11.4437	0.389548	0.194774	−	0.980848i	$-0.437603\pi$
$863$	11.4437	0.389548	0.194774	+	0.980848i	$0.437603\pi$
$864$	−7.64543	−0.260103
$865$	−14.6300	−0.497436
$866$	40.1154	1.36318
$867$	−0.159407	−0.00541375
$868$	41.1848	1.39790
$869$	14.1364	0.479545
$870$	−6.24347	−0.211673
$871$	0.290212	0.00983346
$872$	4.88606	0.165463
$873$	−0.225465	−0.00763085
$874$	45.2952	1.53213
$875$	−5.18066	−0.175138
$876$	−22.0029	−0.743410
$877$	−15.9201	−0.537585	−0.268793	−	0.963198i	$-0.586625\pi$
$877$	−15.9201	−0.537585	−0.268793	+	0.963198i	$0.586625\pi$
$878$	−43.3985	−1.46463
$879$	2.86643	0.0966822
$880$	6.96915	0.234930
$881$	−56.8974	−1.91692	−0.958461	−	0.285225i	$-0.907932\pi$
$881$	−56.8974	−1.91692	−0.958461	+	0.285225i	$0.907932\pi$
$882$	38.4720	1.29542
$883$	44.4449	1.49569	0.747845	−	0.663874i	$-0.231089\pi$
$883$	44.4449	1.49569	0.747845	+	0.663874i	$0.231089\pi$
$884$	−0.321100	−0.0107998
$885$	−6.16261	−0.207154
$886$	20.4945	0.688525
$887$	−38.2747	−1.28514	−0.642570	−	0.766227i	$-0.722132\pi$
$887$	−38.2747	−1.28514	−0.642570	+	0.766227i	$0.722132\pi$
$888$	−4.59269	−0.154120
$889$	89.3702	2.99738
$890$	4.95381	0.166052
$891$	−1.57611	−0.0528016
$892$	−36.9959	−1.23872
$893$	−31.0457	−1.03890
$894$	−17.3252	−0.579443
$895$	17.2523	0.576679
$896$	−19.1161	−0.638624
$897$	−0.387462	−0.0129370
$898$	1.08164	0.0360948
$899$	14.5392	0.484909
$900$	1.76043	0.0586810
$901$	−50.0117	−1.66613
$902$	−14.5270	−0.483697
$903$	−0.122427	−0.00407413
$904$	0.260860	0.00867607
$905$	10.0732	0.334843
$906$	−6.37039	−0.211642
$907$	47.2631	1.56935	0.784673	−	0.619910i	$-0.212831\pi$
$907$	47.2631	1.56935	0.784673	+	0.619910i	$0.212831\pi$
$908$	−38.6460	−1.28251
$909$	5.51515	0.182926
$910$	−0.446526	−0.0148022
$911$	33.2688	1.10225	0.551123	−	0.834424i	$-0.314200\pi$
$911$	33.2688	1.10225	0.551123	+	0.834424i	$0.314200\pi$
$912$	11.8479	0.392322
$913$	16.3493	0.541085
$914$	70.2063	2.32222
$915$	1.41603	0.0468125
$916$	18.0326	0.595813
$917$	−34.9930	−1.15557
$918$	−7.95788	−0.262649
$919$	16.2123	0.534795	0.267397	−	0.963586i	$-0.413836\pi$
$919$	16.2123	0.534795	0.267397	+	0.963586i	$0.413836\pi$
$920$	4.04983	0.133519
$921$	12.7886	0.421398
$922$	42.8683	1.41179
$923$	0.102409	0.00337084
$924$	14.3744	0.472884
$925$	9.88589	0.325046
$926$	−56.5010	−1.85674
$927$	−5.37365	−0.176494
$928$	24.6155	0.808044
$929$	0.361086	0.0118469	0.00592343	−	0.999982i	$-0.498115\pi$
$929$	0.361086	0.0118469	0.00592343	+	0.999982i	$0.498115\pi$
$930$	−8.75693	−0.287151
$931$	−53.1585	−1.74220
$932$	−41.7813	−1.36859
$933$	3.46833	0.113548
$934$	7.69089	0.251654
$935$	6.46792	0.211524
$936$	−0.0206487	−0.000674925 0
$937$	−15.9783	−0.521990	−0.260995	−	0.965340i	$-0.584051\pi$
$937$	−15.9783	−0.521990	−0.260995	+	0.965340i	$0.584051\pi$
$938$	−65.5959	−2.14178
$939$	−17.2996	−0.564552
$940$	20.3973	0.665287
$941$	30.6532	0.999265	0.499633	−	0.866237i	$-0.333468\pi$
$941$	30.6532	0.999265	0.499633	+	0.866237i	$0.333468\pi$
$942$	0.375213	0.0122251
$943$	−41.4341	−1.34928
$944$	27.2495	0.886895
$945$	−5.18066	−0.168527
$946$	−0.0722267	−0.00234829
$947$	−13.1178	−0.426273	−0.213136	−	0.977022i	$-0.568368\pi$
$947$	−13.1178	−0.426273	−0.213136	+	0.977022i	$0.568368\pi$
$948$	−15.7896	−0.512824
$949$	−0.555526	−0.0180331
$950$	−5.19596	−0.168579
$951$	−11.2304	−0.364171
$952$	−9.87677	−0.320108
$953$	−37.1989	−1.20499	−0.602495	−	0.798122i	$-0.705827\pi$
$953$	−37.1989	−1.20499	−0.602495	+	0.798122i	$0.705827\pi$
$954$	23.6326	0.765134
$955$	17.6445	0.570964
$956$	−12.2224	−0.395300
$957$	5.07450	0.164035
$958$	−67.4489	−2.17917
$959$	56.2516	1.81646
$960$	−5.98240	−0.193081
$961$	−10.6077	−0.342184
$962$	0.852074	0.0274720
$963$	−14.3287	−0.461734
$964$	25.8317	0.831984
$965$	7.05381	0.227070
$966$	87.5770	2.81775
$967$	29.8845	0.961021	0.480510	−	0.876989i	$-0.340451\pi$
$967$	29.8845	0.961021	0.480510	+	0.876989i	$0.340451\pi$
$968$	3.95622	0.127158
$969$	10.9958	0.353235
$970$	−0.437219	−0.0140382
$971$	−4.10860	−0.131851	−0.0659257	−	0.997825i	$-0.521000\pi$
$971$	−4.10860	−0.131851	−0.0659257	+	0.997825i	$0.521000\pi$
$972$	1.76043	0.0564658
$973$	28.6087	0.917153
$974$	47.2305	1.51336
$975$	0.0444470	0.00142344
$976$	−6.26132	−0.200420
$977$	−4.51636	−0.144491	−0.0722456	−	0.997387i	$-0.523017\pi$
$977$	−4.51636	−0.144491	−0.0722456	+	0.997387i	$0.523017\pi$
$978$	−1.93918	−0.0620082
$979$	−4.02630	−0.128681
$980$	34.9257	1.11566
$981$	−10.5174	−0.335794
$982$	17.2197	0.549504
$983$	−26.8259	−0.855613	−0.427807	−	0.903870i	$-0.640714\pi$
$983$	−26.8259	−0.855613	−0.427807	+	0.903870i	$0.640714\pi$
$984$	−2.20812	−0.0703923
$985$	−12.5712	−0.400552
$986$	25.6215	0.815954
$987$	−60.0260	−1.91065
$988$	−0.209656	−0.00667006
$989$	−0.206006	−0.00655060
$990$	−3.05636	−0.0971376
$991$	35.5374	1.12888	0.564442	−	0.825473i	$-0.309092\pi$
$991$	35.5374	1.12888	0.564442	+	0.825473i	$0.309092\pi$
$992$	34.5251	1.09617
$993$	−24.0717	−0.763892
$994$	−23.1473	−0.734188
$995$	10.3587	0.328393
$996$	−18.2614	−0.578633
$997$	43.2768	1.37059	0.685295	−	0.728266i	$-0.259673\pi$
$997$	43.2768	1.37059	0.685295	+	0.728266i	$0.259673\pi$
$998$	−69.4230	−2.19755
$999$	9.88589	0.312776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2445.2.a.g.1.11	✓	11
3.2	odd	2		7335.2.a.l.1.1		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2445.2.a.g.1.11	✓	11	1.1	even	1	trivial
7335.2.a.l.1.1		11	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2445 = 3 \cdot 5 \cdot 163 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93918 q^{2} +1.00000 q^{3} +1.76043 q^{4} +1.00000 q^{5} +1.93918 q^{6} -5.18066 q^{7} -0.464570 q^{8} +1.00000 q^{9} +1.93918 q^{10} -1.57611 q^{11} +1.76043 q^{12} +0.0444470 q^{13} -10.0463 q^{14} +1.00000 q^{15} -4.42175 q^{16} -4.10373 q^{17} +1.93918 q^{18} -2.67946 q^{19} +1.76043 q^{20} -5.18066 q^{21} -3.05636 q^{22} -8.71738 q^{23} -0.464570 q^{24} +1.00000 q^{25} +0.0861909 q^{26} +1.00000 q^{27} -9.12020 q^{28} -3.21964 q^{29} +1.93918 q^{30} -4.51578 q^{31} -7.64543 q^{32} -1.57611 q^{33} -7.95788 q^{34} -5.18066 q^{35} +1.76043 q^{36} +9.88589 q^{37} -5.19596 q^{38} +0.0444470 q^{39} -0.464570 q^{40} +4.75304 q^{41} -10.0463 q^{42} +0.0236316 q^{43} -2.77463 q^{44} +1.00000 q^{45} -16.9046 q^{46} +11.5866 q^{47} -4.42175 q^{48} +19.8393 q^{49} +1.93918 q^{50} -4.10373 q^{51} +0.0782459 q^{52} +12.1869 q^{53} +1.93918 q^{54} -1.57611 q^{55} +2.40678 q^{56} -2.67946 q^{57} -6.24347 q^{58} -6.16261 q^{59} +1.76043 q^{60} +1.41603 q^{61} -8.75693 q^{62} -5.18066 q^{63} -5.98240 q^{64} +0.0444470 q^{65} -3.05636 q^{66} +6.52939 q^{67} -7.22433 q^{68} -8.71738 q^{69} -10.0463 q^{70} +2.30407 q^{71} -0.464570 q^{72} -12.4986 q^{73} +19.1705 q^{74} +1.00000 q^{75} -4.71700 q^{76} +8.16529 q^{77} +0.0861909 q^{78} -8.96920 q^{79} -4.42175 q^{80} +1.00000 q^{81} +9.21702 q^{82} -10.3732 q^{83} -9.12020 q^{84} -4.10373 q^{85} +0.0458260 q^{86} -3.21964 q^{87} +0.732212 q^{88} +2.55458 q^{89} +1.93918 q^{90} -0.230265 q^{91} -15.3463 q^{92} -4.51578 q^{93} +22.4685 q^{94} -2.67946 q^{95} -7.64543 q^{96} -0.225465 q^{97} +38.4720 q^{98} -1.57611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 7 q^{2} + 11 q^{3} + 9 q^{4} + 11 q^{5} - 7 q^{6} - 16 q^{7} - 18 q^{8} + 11 q^{9} - 7 q^{10} - 12 q^{11} + 9 q^{12} - 11 q^{13} + 3 q^{14} + 11 q^{15} + 13 q^{16} - 22 q^{17} - 7 q^{18} - 19 q^{19}+ \cdots - 12 q^{99}+O(q^{100}) \) 11 * q - 7 * q^2 + 11 * q^3 + 9 * q^4 + 11 * q^5 - 7 * q^6 - 16 * q^7 - 18 * q^8 + 11 * q^9 - 7 * q^10 - 12 * q^11 + 9 * q^12 - 11 * q^13 + 3 * q^14 + 11 * q^15 + 13 * q^16 - 22 * q^17 - 7 * q^18 - 19 * q^19 + 9 * q^20 - 16 * q^21 - 6 * q^22 - 21 * q^23 - 18 * q^24 + 11 * q^25 + 4 * q^26 + 11 * q^27 - 10 * q^28 - 19 * q^29 - 7 * q^30 - 8 * q^31 - 28 * q^32 - 12 * q^33 + 12 * q^34 - 16 * q^35 + 9 * q^36 - q^37 - 5 * q^38 - 11 * q^39 - 18 * q^40 - 18 * q^41 + 3 * q^42 - 30 * q^43 + 7 * q^44 + 11 * q^45 + 2 * q^46 - 21 * q^47 + 13 * q^48 + 19 * q^49 - 7 * q^50 - 22 * q^51 - 20 * q^52 - 21 * q^53 - 7 * q^54 - 12 * q^55 - 3 * q^56 - 19 * q^57 + 17 * q^58 - 9 * q^59 + 9 * q^60 - 27 * q^61 - 11 * q^62 - 16 * q^63 + 46 * q^64 - 11 * q^65 - 6 * q^66 - 32 * q^67 - 53 * q^68 - 21 * q^69 + 3 * q^70 - 8 * q^71 - 18 * q^72 - 25 * q^73 + 24 * q^74 + 11 * q^75 - 7 * q^76 - 31 * q^77 + 4 * q^78 + 8 * q^79 + 13 * q^80 + 11 * q^81 + 9 * q^82 - 54 * q^83 - 10 * q^84 - 22 * q^85 - 2 * q^86 - 19 * q^87 + 16 * q^88 + 7 * q^89 - 7 * q^90 - 22 * q^91 - 37 * q^92 - 8 * q^93 + 53 * q^94 - 19 * q^95 - 28 * q^96 + 21 * q^97 - 24 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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