LMFDB - Embedded newform 8619.2.a.bp.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$68.8230615021$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.575048 q^{2} +1.00000 q^{3} -1.66932 q^{4} -1.38919 q^{5} -0.575048 q^{6} +4.30260 q^{7} +2.11004 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.575048 q^{2} +1.00000 q^{3} -1.66932 q^{4} -1.38919 q^{5} -0.575048 q^{6} +4.30260 q^{7} +2.11004 q^{8} +1.00000 q^{9} +0.798850 q^{10} -0.556120 q^{11} -1.66932 q^{12} -2.47420 q^{14} -1.38919 q^{15} +2.12527 q^{16} -1.00000 q^{17} -0.575048 q^{18} +7.33062 q^{19} +2.31900 q^{20} +4.30260 q^{21} +0.319796 q^{22} -9.41329 q^{23} +2.11004 q^{24} -3.07016 q^{25} +1.00000 q^{27} -7.18242 q^{28} +6.13375 q^{29} +0.798850 q^{30} -5.57311 q^{31} -5.44220 q^{32} -0.556120 q^{33} +0.575048 q^{34} -5.97713 q^{35} -1.66932 q^{36} +2.07677 q^{37} -4.21546 q^{38} -2.93124 q^{40} -5.45346 q^{41} -2.47420 q^{42} -8.29324 q^{43} +0.928343 q^{44} -1.38919 q^{45} +5.41309 q^{46} -7.47038 q^{47} +2.12527 q^{48} +11.5124 q^{49} +1.76549 q^{50} -1.00000 q^{51} -5.69124 q^{53} -0.575048 q^{54} +0.772556 q^{55} +9.07865 q^{56} +7.33062 q^{57} -3.52720 q^{58} -0.523791 q^{59} +2.31900 q^{60} -0.333468 q^{61} +3.20481 q^{62} +4.30260 q^{63} -1.12101 q^{64} +0.319796 q^{66} -5.02874 q^{67} +1.66932 q^{68} -9.41329 q^{69} +3.43714 q^{70} -7.85716 q^{71} +2.11004 q^{72} -12.9237 q^{73} -1.19424 q^{74} -3.07016 q^{75} -12.2371 q^{76} -2.39277 q^{77} +11.3004 q^{79} -2.95240 q^{80} +1.00000 q^{81} +3.13600 q^{82} +3.66884 q^{83} -7.18242 q^{84} +1.38919 q^{85} +4.76901 q^{86} +6.13375 q^{87} -1.17343 q^{88} -9.80657 q^{89} +0.798850 q^{90} +15.7138 q^{92} -5.57311 q^{93} +4.29583 q^{94} -10.1836 q^{95} -5.44220 q^{96} -0.527132 q^{97} -6.62019 q^{98} -0.556120 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9}+O(q^{10}) $ 21 * q - 9 * q^2 + 21 * q^3 + 29 * q^4 - 26 * q^5 - 9 * q^6 + 5 * q^7 - 24 * q^8 + 21 * q^9	\( 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9} + 12 q^{10} - 37 q^{11} + 29 q^{12} - 6 q^{14} - 26 q^{15} + 33 q^{16} - 21 q^{17} - 9 q^{18} + 6 q^{19} - 53 q^{20} + 5 q^{21} + 17 q^{22} - 8 q^{23} - 24 q^{24} + 57 q^{25} + 21 q^{27} - 5 q^{28} - 6 q^{29} + 12 q^{30} - 9 q^{31} - 37 q^{32} - 37 q^{33} + 9 q^{34} - 8 q^{35} + 29 q^{36} - 9 q^{37} + 11 q^{38} - 8 q^{40} - 50 q^{41} - 6 q^{42} + 18 q^{43} - 67 q^{44} - 26 q^{45} - 23 q^{46} - 71 q^{47} + 33 q^{48} + 46 q^{49} - 2 q^{50} - 21 q^{51} - 14 q^{53} - 9 q^{54} + 29 q^{55} - 17 q^{56} + 6 q^{57} - 37 q^{58} - 59 q^{59} - 53 q^{60} + 44 q^{61} + 2 q^{62} + 5 q^{63} + 44 q^{64} + 17 q^{66} - 8 q^{67} - 29 q^{68} - 8 q^{69} - 15 q^{70} - 60 q^{71} - 24 q^{72} - 13 q^{73} - 14 q^{74} + 57 q^{75} - 15 q^{76} - 28 q^{77} + 51 q^{79} - 131 q^{80} + 21 q^{81} + 18 q^{82} - 30 q^{83} - 5 q^{84} + 26 q^{85} + 15 q^{86} - 6 q^{87} + 71 q^{88} - 62 q^{89} + 12 q^{90} + 20 q^{92} - 9 q^{93} - 72 q^{94} - 16 q^{95} - 37 q^{96} + 6 q^{97} - 58 q^{98} - 37 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 + 21 * q^3 + 29 * q^4 - 26 * q^5 - 9 * q^6 + 5 * q^7 - 24 * q^8 + 21 * q^9 + 12 * q^10 - 37 * q^11 + 29 * q^12 - 6 * q^14 - 26 * q^15 + 33 * q^16 - 21 * q^17 - 9 * q^18 + 6 * q^19 - 53 * q^20 + 5 * q^21 + 17 * q^22 - 8 * q^23 - 24 * q^24 + 57 * q^25 + 21 * q^27 - 5 * q^28 - 6 * q^29 + 12 * q^30 - 9 * q^31 - 37 * q^32 - 37 * q^33 + 9 * q^34 - 8 * q^35 + 29 * q^36 - 9 * q^37 + 11 * q^38 - 8 * q^40 - 50 * q^41 - 6 * q^42 + 18 * q^43 - 67 * q^44 - 26 * q^45 - 23 * q^46 - 71 * q^47 + 33 * q^48 + 46 * q^49 - 2 * q^50 - 21 * q^51 - 14 * q^53 - 9 * q^54 + 29 * q^55 - 17 * q^56 + 6 * q^57 - 37 * q^58 - 59 * q^59 - 53 * q^60 + 44 * q^61 + 2 * q^62 + 5 * q^63 + 44 * q^64 + 17 * q^66 - 8 * q^67 - 29 * q^68 - 8 * q^69 - 15 * q^70 - 60 * q^71 - 24 * q^72 - 13 * q^73 - 14 * q^74 + 57 * q^75 - 15 * q^76 - 28 * q^77 + 51 * q^79 - 131 * q^80 + 21 * q^81 + 18 * q^82 - 30 * q^83 - 5 * q^84 + 26 * q^85 + 15 * q^86 - 6 * q^87 + 71 * q^88 - 62 * q^89 + 12 * q^90 + 20 * q^92 - 9 * q^93 - 72 * q^94 - 16 * q^95 - 37 * q^96 + 6 * q^97 - 58 * q^98 - 37 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.575048	−0.406620	−0.203310	−	0.979114i	$-0.565170\pi$
$2$	−0.575048	−0.406620	−0.203310	+	0.979114i	$0.565170\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.66932	−0.834660
$5$	−1.38919	−0.621264	−0.310632	−	0.950530i	$-0.600541\pi$
$5$	−1.38919	−0.621264	−0.310632	+	0.950530i	$0.600541\pi$
$6$	−0.575048	−0.234762
$7$	4.30260	1.62623	0.813116	−	0.582102i	$-0.197770\pi$
$7$	4.30260	1.62623	0.813116	+	0.582102i	$0.197770\pi$
$8$	2.11004	0.746010
$9$	1.00000	0.333333
$10$	0.798850	0.252619
$11$	−0.556120	−0.167677	−0.0838383	−	0.996479i	$-0.526718\pi$
$11$	−0.556120	−0.167677	−0.0838383	+	0.996479i	$0.526718\pi$
$12$	−1.66932	−0.481891
$13$	0	0
$13$	0	0
$14$	−2.47420	−0.661259
$15$	−1.38919	−0.358687
$16$	2.12527	0.531317
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.575048	−0.135540
$19$	7.33062	1.68176	0.840880	−	0.541222i	$-0.182038\pi$
$19$	7.33062	1.68176	0.840880	+	0.541222i	$0.182038\pi$
$20$	2.31900	0.518544
$21$	4.30260	0.938905
$22$	0.319796	0.0681807
$23$	−9.41329	−1.96281	−0.981403	−	0.191958i	$-0.938516\pi$
$23$	−9.41329	−1.96281	−0.981403	+	0.191958i	$0.938516\pi$
$24$	2.11004	0.430709
$25$	−3.07016	−0.614031
$26$	0	0
$27$	1.00000	0.192450
$28$	−7.18242	−1.35735
$29$	6.13375	1.13901	0.569504	−	0.821988i	$-0.307135\pi$
$29$	6.13375	1.13901	0.569504	+	0.821988i	$0.307135\pi$
$30$	0.798850	0.145849
$31$	−5.57311	−1.00096	−0.500480	−	0.865748i	$-0.666843\pi$
$31$	−5.57311	−1.00096	−0.500480	+	0.865748i	$0.666843\pi$
$32$	−5.44220	−0.962054
$33$	−0.556120	−0.0968082
$34$	0.575048	0.0986199
$35$	−5.97713	−1.01032
$36$	−1.66932	−0.278220
$37$	2.07677	0.341419	0.170709	−	0.985321i	$-0.445394\pi$
$37$	2.07677	0.341419	0.170709	+	0.985321i	$0.445394\pi$
$38$	−4.21546	−0.683838
$39$	0	0
$40$	−2.93124	−0.463469
$41$	−5.45346	−0.851688	−0.425844	−	0.904797i	$-0.640023\pi$
$41$	−5.45346	−0.851688	−0.425844	+	0.904797i	$0.640023\pi$
$42$	−2.47420	−0.381778
$43$	−8.29324	−1.26471	−0.632354	−	0.774680i	$-0.717911\pi$
$43$	−8.29324	−1.26471	−0.632354	+	0.774680i	$0.717911\pi$
$44$	0.928343	0.139953
$45$	−1.38919	−0.207088
$46$	5.41309	0.798117
$47$	−7.47038	−1.08967	−0.544833	−	0.838544i	$-0.683407\pi$
$47$	−7.47038	−1.08967	−0.544833	+	0.838544i	$0.683407\pi$
$48$	2.12527	0.306756
$49$	11.5124	1.64463
$50$	1.76549	0.249678
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−5.69124	−0.781752	−0.390876	−	0.920443i	$-0.627828\pi$
$53$	−5.69124	−0.781752	−0.390876	+	0.920443i	$0.627828\pi$
$54$	−0.575048	−0.0782541
$55$	0.772556	0.104171
$56$	9.07865	1.21319
$57$	7.33062	0.970964
$58$	−3.52720	−0.463144
$59$	−0.523791	−0.0681918	−0.0340959	−	0.999419i	$-0.510855\pi$
$59$	−0.523791	−0.0681918	−0.0340959	+	0.999419i	$0.510855\pi$
$60$	2.31900	0.299382
$61$	−0.333468	−0.0426962	−0.0213481	−	0.999772i	$-0.506796\pi$
$61$	−0.333468	−0.0426962	−0.0213481	+	0.999772i	$0.506796\pi$
$62$	3.20481	0.407011
$63$	4.30260	0.542077
$64$	−1.12101	−0.140126
$65$	0	0
$66$	0.319796	0.0393642
$67$	−5.02874	−0.614358	−0.307179	−	0.951652i	$-0.599385\pi$
$67$	−5.02874	−0.614358	−0.307179	+	0.951652i	$0.599385\pi$
$68$	1.66932	0.202435
$69$	−9.41329	−1.13323
$70$	3.43714	0.410816
$71$	−7.85716	−0.932474	−0.466237	−	0.884660i	$-0.654391\pi$
$71$	−7.85716	−0.932474	−0.466237	+	0.884660i	$0.654391\pi$
$72$	2.11004	0.248670
$73$	−12.9237	−1.51261	−0.756303	−	0.654221i	$-0.772997\pi$
$73$	−12.9237	−1.51261	−0.756303	+	0.654221i	$0.772997\pi$
$74$	−1.19424	−0.138828
$75$	−3.07016	−0.354511
$76$	−12.2371	−1.40370
$77$	−2.39277	−0.272681
$78$	0	0
$79$	11.3004	1.27139	0.635695	−	0.771940i	$-0.280714\pi$
$79$	11.3004	1.27139	0.635695	+	0.771940i	$0.280714\pi$
$80$	−2.95240	−0.330088
$81$	1.00000	0.111111
$82$	3.13600	0.346314
$83$	3.66884	0.402708	0.201354	−	0.979519i	$-0.435466\pi$
$83$	3.66884	0.402708	0.201354	+	0.979519i	$0.435466\pi$
$84$	−7.18242	−0.783667
$85$	1.38919	0.150679
$86$	4.76901	0.514256
$87$	6.13375	0.657607
$88$	−1.17343	−0.125088
$89$	−9.80657	−1.03949	−0.519747	−	0.854320i	$-0.673974\pi$
$89$	−9.80657	−1.03949	−0.519747	+	0.854320i	$0.673974\pi$
$90$	0.798850	0.0842062
$91$	0	0
$92$	15.7138	1.63828
$93$	−5.57311	−0.577905
$94$	4.29583	0.443081
$95$	−10.1836	−1.04482
$96$	−5.44220	−0.555442
$97$	−0.527132	−0.0535222	−0.0267611	−	0.999642i	$-0.508519\pi$
$97$	−0.527132	−0.0535222	−0.0267611	+	0.999642i	$0.508519\pi$
$98$	−6.62019	−0.668740
$99$	−0.556120	−0.0558922
$100$	5.12507	0.512507
$101$	7.17027	0.713469	0.356734	−	0.934206i	$-0.383890\pi$
$101$	7.17027	0.713469	0.356734	+	0.934206i	$0.383890\pi$
$102$	0.575048	0.0569382
$103$	5.76258	0.567804	0.283902	−	0.958853i	$-0.408371\pi$
$103$	5.76258	0.567804	0.283902	+	0.958853i	$0.408371\pi$
$104$	0	0
$105$	−5.97713	−0.583308
$106$	3.27274	0.317876
$107$	−9.66909	−0.934747	−0.467373	−	0.884060i	$-0.654800\pi$
$107$	−9.66909	−0.934747	−0.467373	+	0.884060i	$0.654800\pi$
$108$	−1.66932	−0.160630
$109$	12.3646	1.18431	0.592156	−	0.805823i	$-0.298277\pi$
$109$	12.3646	1.18431	0.592156	+	0.805823i	$0.298277\pi$
$110$	−0.444257	−0.0423582
$111$	2.07677	0.197118
$112$	9.14419	0.864045
$113$	−7.09878	−0.667797	−0.333898	−	0.942609i	$-0.608364\pi$
$113$	−7.09878	−0.667797	−0.333898	+	0.942609i	$0.608364\pi$
$114$	−4.21546	−0.394814
$115$	13.0768	1.21942
$116$	−10.2392	−0.950685
$117$	0	0
$118$	0.301205	0.0277282
$119$	−4.30260	−0.394419
$120$	−2.93124	−0.267584
$121$	−10.6907	−0.971885
$122$	0.191760	0.0173611
$123$	−5.45346	−0.491722
$124$	9.30331	0.835462
$125$	11.2110	1.00274
$126$	−2.47420	−0.220420
$127$	18.4645	1.63846	0.819231	−	0.573463i	$-0.194401\pi$
$127$	18.4645	1.63846	0.819231	+	0.573463i	$0.194401\pi$
$128$	11.5290	1.01903
$129$	−8.29324	−0.730179
$130$	0	0
$131$	−3.23991	−0.283072	−0.141536	−	0.989933i	$-0.545204\pi$
$131$	−3.23991	−0.283072	−0.141536	+	0.989933i	$0.545204\pi$
$132$	0.928343	0.0808019
$133$	31.5408	2.73493
$134$	2.89177	0.249811
$135$	−1.38919	−0.119562
$136$	−2.11004	−0.180934
$137$	−13.4040	−1.14518	−0.572592	−	0.819840i	$-0.694062\pi$
$137$	−13.4040	−1.14518	−0.572592	+	0.819840i	$0.694062\pi$
$138$	5.41309	0.460793
$139$	11.6895	0.991494	0.495747	−	0.868467i	$-0.334894\pi$
$139$	11.6895	0.991494	0.495747	+	0.868467i	$0.334894\pi$
$140$	9.97774	0.843273
$141$	−7.47038	−0.629119
$142$	4.51825	0.379163
$143$	0	0
$144$	2.12527	0.177106
$145$	−8.52093	−0.707625
$146$	7.43176	0.615057
$147$	11.5124	0.949527
$148$	−3.46679	−0.284968
$149$	−15.8060	−1.29488	−0.647438	−	0.762118i	$-0.724159\pi$
$149$	−15.8060	−1.29488	−0.647438	+	0.762118i	$0.724159\pi$
$150$	1.76549	0.144151
$151$	5.26099	0.428133	0.214067	−	0.976819i	$-0.431329\pi$
$151$	5.26099	0.428133	0.214067	+	0.976819i	$0.431329\pi$
$152$	15.4679	1.25461
$153$	−1.00000	−0.0808452
$154$	1.37596	0.110878
$155$	7.74211	0.621861
$156$	0	0
$157$	−11.9972	−0.957483	−0.478741	−	0.877956i	$-0.658907\pi$
$157$	−11.9972	−0.957483	−0.478741	+	0.877956i	$0.658907\pi$
$158$	−6.49825	−0.516973
$159$	−5.69124	−0.451345
$160$	7.56024	0.597690
$161$	−40.5017	−3.19198
$162$	−0.575048	−0.0451800
$163$	−18.3521	−1.43745	−0.718725	−	0.695294i	$-0.755274\pi$
$163$	−18.3521	−1.43745	−0.718725	+	0.695294i	$0.755274\pi$
$164$	9.10358	0.710870
$165$	0.772556	0.0601434
$166$	−2.10976	−0.163749
$167$	18.3480	1.41981	0.709905	−	0.704297i	$-0.248738\pi$
$167$	18.3480	1.41981	0.709905	+	0.704297i	$0.248738\pi$
$168$	9.07865	0.700433
$169$	0	0
$170$	−0.798850	−0.0612690
$171$	7.33062	0.560586
$172$	13.8441	1.05560
$173$	0.155919	0.0118543	0.00592715	−	0.999982i	$-0.498113\pi$
$173$	0.155919	0.0118543	0.00592715	+	0.999982i	$0.498113\pi$
$174$	−3.52720	−0.267396
$175$	−13.2097	−0.998557
$176$	−1.18191	−0.0890894
$177$	−0.523791	−0.0393705
$178$	5.63925	0.422680
$179$	17.5244	1.30983	0.654916	−	0.755702i	$-0.272704\pi$
$179$	17.5244	1.30983	0.654916	+	0.755702i	$0.272704\pi$
$180$	2.31900	0.172848
$181$	6.66347	0.495291	0.247646	−	0.968851i	$-0.420343\pi$
$181$	6.66347	0.495291	0.247646	+	0.968851i	$0.420343\pi$
$182$	0	0
$183$	−0.333468	−0.0246506
$184$	−19.8624	−1.46427
$185$	−2.88502	−0.212111
$186$	3.20481	0.234988
$187$	0.556120	0.0406676
$188$	12.4705	0.909501
$189$	4.30260	0.312968
$190$	5.85607	0.424844
$191$	−8.21408	−0.594350	−0.297175	−	0.954823i	$-0.596045\pi$
$191$	−8.21408	−0.594350	−0.297175	+	0.954823i	$0.596045\pi$
$192$	−1.12101	−0.0809018
$193$	−4.39726	−0.316522	−0.158261	−	0.987397i	$-0.550589\pi$
$193$	−4.39726	−0.316522	−0.158261	+	0.987397i	$0.550589\pi$
$194$	0.303126	0.0217632
$195$	0	0
$196$	−19.2179	−1.37271
$197$	17.5911	1.25331	0.626656	−	0.779296i	$-0.284423\pi$
$197$	17.5911	1.25331	0.626656	+	0.779296i	$0.284423\pi$
$198$	0.319796	0.0227269
$199$	−26.7645	−1.89729	−0.948643	−	0.316349i	$-0.897543\pi$
$199$	−26.7645	−1.89729	−0.948643	+	0.316349i	$0.897543\pi$
$200$	−6.47814	−0.458073
$201$	−5.02874	−0.354700
$202$	−4.12325	−0.290111
$203$	26.3911	1.85229
$204$	1.66932	0.116876
$205$	7.57589	0.529123
$206$	−3.31376	−0.230881
$207$	−9.41329	−0.654269
$208$	0	0
$209$	−4.07671	−0.281992
$210$	3.43714	0.237185
$211$	−15.2569	−1.05033	−0.525164	−	0.851001i	$-0.675996\pi$
$211$	−15.2569	−1.05033	−0.525164	+	0.851001i	$0.675996\pi$
$212$	9.50050	0.652497
$213$	−7.85716	−0.538364
$214$	5.56019	0.380087
$215$	11.5209	0.785717
$216$	2.11004	0.143570
$217$	−23.9789	−1.62779
$218$	−7.11023	−0.481565
$219$	−12.9237	−0.873304
$220$	−1.28964	−0.0869477
$221$	0	0
$222$	−1.19424	−0.0801522
$223$	2.75963	0.184798	0.0923992	−	0.995722i	$-0.470546\pi$
$223$	2.75963	0.184798	0.0923992	+	0.995722i	$0.470546\pi$
$224$	−23.4156	−1.56452
$225$	−3.07016	−0.204677
$226$	4.08214	0.271540
$227$	12.1227	0.804609	0.402305	−	0.915506i	$-0.368209\pi$
$227$	12.1227	0.804609	0.402305	+	0.915506i	$0.368209\pi$
$228$	−12.2371	−0.810425
$229$	−22.1475	−1.46355	−0.731774	−	0.681548i	$-0.761307\pi$
$229$	−22.1475	−1.46355	−0.731774	+	0.681548i	$0.761307\pi$
$230$	−7.51981	−0.495841
$231$	−2.39277	−0.157432
$232$	12.9424	0.849712
$233$	−22.3005	−1.46096	−0.730478	−	0.682936i	$-0.760703\pi$
$233$	−22.3005	−1.46096	−0.730478	+	0.682936i	$0.760703\pi$
$234$	0	0
$235$	10.3778	0.676971
$236$	0.874374	0.0569169
$237$	11.3004	0.734037
$238$	2.47420	0.160379
$239$	−13.6415	−0.882396	−0.441198	−	0.897410i	$-0.645446\pi$
$239$	−13.6415	−0.882396	−0.441198	+	0.897410i	$0.645446\pi$
$240$	−2.95240	−0.190576
$241$	−15.9632	−1.02828	−0.514140	−	0.857706i	$-0.671889\pi$
$241$	−15.9632	−1.02828	−0.514140	+	0.857706i	$0.671889\pi$
$242$	6.14768	0.395188
$243$	1.00000	0.0641500
$244$	0.556664	0.0356368
$245$	−15.9929	−1.02175
$246$	3.13600	0.199944
$247$	0	0
$248$	−11.7595	−0.746727
$249$	3.66884	0.232504
$250$	−6.44684	−0.407734
$251$	13.3503	0.842664	0.421332	−	0.906906i	$-0.361563\pi$
$251$	13.3503	0.842664	0.421332	+	0.906906i	$0.361563\pi$
$252$	−7.18242	−0.452450
$253$	5.23492	0.329117
$254$	−10.6180	−0.666232
$255$	1.38919	0.0869944
$256$	−4.38773	−0.274233
$257$	16.4554	1.02646	0.513231	−	0.858251i	$-0.328448\pi$
$257$	16.4554	1.02646	0.513231	+	0.858251i	$0.328448\pi$
$258$	4.76901	0.296906
$259$	8.93551	0.555226
$260$	0	0
$261$	6.13375	0.379670
$262$	1.86310	0.115103
$263$	−22.8587	−1.40953	−0.704764	−	0.709442i	$-0.748947\pi$
$263$	−22.8587	−1.40953	−0.704764	+	0.709442i	$0.748947\pi$
$264$	−1.17343	−0.0722199
$265$	7.90621	0.485675
$266$	−18.1374	−1.11208
$267$	−9.80657	−0.600153
$268$	8.39457	0.512780
$269$	0.521231	0.0317800	0.0158900	−	0.999874i	$-0.494942\pi$
$269$	0.521231	0.0317800	0.0158900	+	0.999874i	$0.494942\pi$
$270$	0.798850	0.0486165
$271$	−12.7522	−0.774640	−0.387320	−	0.921945i	$-0.626599\pi$
$271$	−12.7522	−0.774640	−0.387320	+	0.921945i	$0.626599\pi$
$272$	−2.12527	−0.128863
$273$	0	0
$274$	7.70797	0.465655
$275$	1.70738	0.102959
$276$	15.7138	0.945859
$277$	28.7261	1.72598	0.862992	−	0.505217i	$-0.168588\pi$
$277$	28.7261	1.72598	0.862992	+	0.505217i	$0.168588\pi$
$278$	−6.72205	−0.403162
$279$	−5.57311	−0.333654
$280$	−12.6120	−0.753708
$281$	−10.1207	−0.603752	−0.301876	−	0.953347i	$-0.597613\pi$
$281$	−10.1207	−0.603752	−0.301876	+	0.953347i	$0.597613\pi$
$282$	4.29583	0.255813
$283$	32.2916	1.91953	0.959767	−	0.280797i	$-0.0905989\pi$
$283$	32.2916	1.91953	0.959767	+	0.280797i	$0.0905989\pi$
$284$	13.1161	0.778298
$285$	−10.1836	−0.603225
$286$	0	0
$287$	−23.4641	−1.38504
$288$	−5.44220	−0.320685
$289$	1.00000	0.0588235
$290$	4.89995	0.287735
$291$	−0.527132	−0.0309010
$292$	21.5738	1.26251
$293$	21.0648	1.23062	0.615310	−	0.788285i	$-0.289031\pi$
$293$	21.0648	1.23062	0.615310	+	0.788285i	$0.289031\pi$
$294$	−6.62019	−0.386097
$295$	0.727644	0.0423651
$296$	4.38205	0.254702
$297$	−0.556120	−0.0322694
$298$	9.08919	0.526523
$299$	0	0
$300$	5.12507	0.295896
$301$	−35.6825	−2.05671
$302$	−3.02532	−0.174088
$303$	7.17027	0.411921
$304$	15.5795	0.893547
$305$	0.463250	0.0265256
$306$	0.575048	0.0328733
$307$	−9.77755	−0.558034	−0.279017	−	0.960286i	$-0.590009\pi$
$307$	−9.77755	−0.558034	−0.279017	+	0.960286i	$0.590009\pi$
$308$	3.99429	0.227596
$309$	5.76258	0.327822
$310$	−4.45208	−0.252861
$311$	7.09378	0.402251	0.201126	−	0.979565i	$-0.435540\pi$
$311$	7.09378	0.402251	0.201126	+	0.979565i	$0.435540\pi$
$312$	0	0
$313$	7.46344	0.421858	0.210929	−	0.977501i	$-0.432351\pi$
$313$	7.46344	0.421858	0.210929	+	0.977501i	$0.432351\pi$
$314$	6.89898	0.389332
$315$	−5.97713	−0.336773
$316$	−18.8639	−1.06118
$317$	−33.6602	−1.89055	−0.945273	−	0.326280i	$-0.894205\pi$
$317$	−33.6602	−1.89055	−0.945273	+	0.326280i	$0.894205\pi$
$318$	3.27274	0.183526
$319$	−3.41110	−0.190985
$320$	1.55729	0.0870553
$321$	−9.66909	−0.539676
$322$	23.2904	1.29792
$323$	−7.33062	−0.407887
$324$	−1.66932	−0.0927400
$325$	0	0
$326$	10.5534	0.584497
$327$	12.3646	0.683763
$328$	−11.5070	−0.635368
$329$	−32.1421	−1.77205
$330$	−0.444257	−0.0244555
$331$	−13.0770	−0.718779	−0.359389	−	0.933188i	$-0.617015\pi$
$331$	−13.0770	−0.718779	−0.359389	+	0.933188i	$0.617015\pi$
$332$	−6.12447	−0.336124
$333$	2.07677	0.113806
$334$	−10.5510	−0.577324
$335$	6.98586	0.381679
$336$	9.14419	0.498856
$337$	24.5919	1.33961	0.669803	−	0.742538i	$-0.266378\pi$
$337$	24.5919	1.33961	0.669803	+	0.742538i	$0.266378\pi$
$338$	0	0
$339$	−7.09878	−0.385553
$340$	−2.31900	−0.125765
$341$	3.09932	0.167838
$342$	−4.21546	−0.227946
$343$	19.4151	1.04832
$344$	−17.4990	−0.943485
$345$	13.0768	0.704033
$346$	−0.0896609	−0.00482020
$347$	8.60823	0.462114	0.231057	−	0.972940i	$-0.425782\pi$
$347$	8.60823	0.462114	0.231057	+	0.972940i	$0.425782\pi$
$348$	−10.2392	−0.548878
$349$	2.18913	0.117181	0.0585907	−	0.998282i	$-0.481339\pi$
$349$	2.18913	0.117181	0.0585907	+	0.998282i	$0.481339\pi$
$350$	7.59619	0.406033
$351$	0	0
$352$	3.02652	0.161314
$353$	3.30506	0.175910	0.0879552	−	0.996124i	$-0.471967\pi$
$353$	3.30506	0.175910	0.0879552	+	0.996124i	$0.471967\pi$
$354$	0.301205	0.0160089
$355$	10.9151	0.579312
$356$	16.3703	0.867625
$357$	−4.30260	−0.227718
$358$	−10.0773	−0.532604
$359$	5.76319	0.304169	0.152085	−	0.988367i	$-0.451401\pi$
$359$	5.76319	0.304169	0.152085	+	0.988367i	$0.451401\pi$
$360$	−2.93124	−0.154490
$361$	34.7380	1.82831
$362$	−3.83181	−0.201396
$363$	−10.6907	−0.561118
$364$	0	0
$365$	17.9535	0.939728
$366$	0.191760	0.0100235
$367$	−7.36006	−0.384192	−0.192096	−	0.981376i	$-0.561528\pi$
$367$	−7.36006	−0.384192	−0.192096	+	0.981376i	$0.561528\pi$
$368$	−20.0058	−1.04287
$369$	−5.45346	−0.283896
$370$	1.65903	0.0862487
$371$	−24.4872	−1.27131
$372$	9.30331	0.482354
$373$	34.2792	1.77491	0.887455	−	0.460894i	$-0.152471\pi$
$373$	34.2792	1.77491	0.887455	+	0.460894i	$0.152471\pi$
$374$	−0.319796	−0.0165363
$375$	11.2110	0.578932
$376$	−15.7628	−0.812902
$377$	0	0
$378$	−2.47420	−0.127259
$379$	10.9325	0.561566	0.280783	−	0.959771i	$-0.409406\pi$
$379$	10.9325	0.561566	0.280783	+	0.959771i	$0.409406\pi$
$380$	16.9997	0.872066
$381$	18.4645	0.945967
$382$	4.72349	0.241675
$383$	19.9948	1.02168	0.510842	−	0.859674i	$-0.329334\pi$
$383$	19.9948	1.02168	0.510842	+	0.859674i	$0.329334\pi$
$384$	11.5290	0.588339
$385$	3.32400	0.169407
$386$	2.52864	0.128704
$387$	−8.29324	−0.421569
$388$	0.879952	0.0446728
$389$	13.5876	0.688917	0.344458	−	0.938802i	$-0.388063\pi$
$389$	13.5876	0.688917	0.344458	+	0.938802i	$0.388063\pi$
$390$	0	0
$391$	9.41329	0.476050
$392$	24.2916	1.22691
$393$	−3.23991	−0.163432
$394$	−10.1157	−0.509622
$395$	−15.6983	−0.789869
$396$	0.928343	0.0466510
$397$	−23.0805	−1.15838	−0.579189	−	0.815193i	$-0.696631\pi$
$397$	−23.0805	−1.15838	−0.579189	+	0.815193i	$0.696631\pi$
$398$	15.3909	0.771475
$399$	31.5408	1.57901
$400$	−6.52490	−0.326245
$401$	3.00330	0.149978	0.0749888	−	0.997184i	$-0.476108\pi$
$401$	3.00330	0.149978	0.0749888	+	0.997184i	$0.476108\pi$
$402$	2.89177	0.144228
$403$	0	0
$404$	−11.9695	−0.595504
$405$	−1.38919	−0.0690293
$406$	−15.1761	−0.753180
$407$	−1.15493	−0.0572479
$408$	−2.11004	−0.104462
$409$	−19.8611	−0.982067	−0.491033	−	0.871141i	$-0.663381\pi$
$409$	−19.8611	−0.982067	−0.491033	+	0.871141i	$0.663381\pi$
$410$	−4.35650	−0.215152
$411$	−13.4040	−0.661173
$412$	−9.61959	−0.473923
$413$	−2.25366	−0.110896
$414$	5.41309	0.266039
$415$	−5.09672	−0.250188
$416$	0	0
$417$	11.6895	0.572440
$418$	2.34430	0.114664
$419$	−14.9899	−0.732305	−0.366152	−	0.930555i	$-0.619325\pi$
$419$	−14.9899	−0.732305	−0.366152	+	0.930555i	$0.619325\pi$
$420$	9.97774	0.486864
$421$	−31.8609	−1.55281	−0.776403	−	0.630236i	$-0.782958\pi$
$421$	−31.8609	−1.55281	−0.776403	+	0.630236i	$0.782958\pi$
$422$	8.77344	0.427084
$423$	−7.47038	−0.363222
$424$	−12.0087	−0.583195
$425$	3.07016	0.148924
$426$	4.51825	0.218910
$427$	−1.43478	−0.0694339
$428$	16.1408	0.780195
$429$	0	0
$430$	−6.62506	−0.319489
$431$	−30.5921	−1.47357	−0.736784	−	0.676128i	$-0.763657\pi$
$431$	−30.5921	−1.47357	−0.736784	+	0.676128i	$0.763657\pi$
$432$	2.12527	0.102252
$433$	−31.1966	−1.49921	−0.749606	−	0.661884i	$-0.769757\pi$
$433$	−31.1966	−1.49921	−0.749606	+	0.661884i	$0.769757\pi$
$434$	13.7890	0.661894
$435$	−8.52093	−0.408548
$436$	−20.6404	−0.988498
$437$	−69.0052	−3.30097
$438$	7.43176	0.355103
$439$	−20.2675	−0.967315	−0.483658	−	0.875257i	$-0.660692\pi$
$439$	−20.2675	−0.967315	−0.483658	+	0.875257i	$0.660692\pi$
$440$	1.63012	0.0777130
$441$	11.5124	0.548210
$442$	0	0
$443$	2.17345	0.103264	0.0516319	−	0.998666i	$-0.483558\pi$
$443$	2.17345	0.103264	0.0516319	+	0.998666i	$0.483558\pi$
$444$	−3.46679	−0.164527
$445$	13.6232	0.645801
$446$	−1.58692	−0.0751428
$447$	−15.8060	−0.747597
$448$	−4.82326	−0.227877
$449$	4.44916	0.209969	0.104984	−	0.994474i	$-0.466521\pi$
$449$	4.44916	0.209969	0.104984	+	0.994474i	$0.466521\pi$
$450$	1.76549	0.0832258
$451$	3.03278	0.142808
$452$	11.8501	0.557383
$453$	5.26099	0.247183
$454$	−6.97111	−0.327171
$455$	0	0
$456$	15.4679	0.724349
$457$	27.1141	1.26835	0.634173	−	0.773191i	$-0.281341\pi$
$457$	27.1141	1.26835	0.634173	+	0.773191i	$0.281341\pi$
$458$	12.7359	0.595108
$459$	−1.00000	−0.0466760
$460$	−21.8294	−1.01780
$461$	−13.0690	−0.608683	−0.304342	−	0.952563i	$-0.598436\pi$
$461$	−13.0690	−0.608683	−0.304342	+	0.952563i	$0.598436\pi$
$462$	1.37596	0.0640153
$463$	0.164751	0.00765665	0.00382832	−	0.999993i	$-0.498781\pi$
$463$	0.164751	0.00765665	0.00382832	+	0.999993i	$0.498781\pi$
$464$	13.0359	0.605175
$465$	7.74211	0.359032
$466$	12.8239	0.594055
$467$	−10.4486	−0.483502	−0.241751	−	0.970338i	$-0.577722\pi$
$467$	−10.4486	−0.483502	−0.241751	+	0.970338i	$0.577722\pi$
$468$	0	0
$469$	−21.6367	−0.999089
$470$	−5.96771	−0.275270
$471$	−11.9972	−0.552803
$472$	−1.10522	−0.0508717
$473$	4.61204	0.212062
$474$	−6.49825	−0.298475
$475$	−22.5061	−1.03265
$476$	7.18242	0.329206
$477$	−5.69124	−0.260584
$478$	7.84452	0.358800
$479$	−5.26860	−0.240728	−0.120364	−	0.992730i	$-0.538406\pi$
$479$	−5.26860	−0.240728	−0.120364	+	0.992730i	$0.538406\pi$
$480$	7.56024	0.345076
$481$	0	0
$482$	9.17961	0.418120
$483$	−40.5017	−1.84289
$484$	17.8462	0.811193
$485$	0.732286	0.0332514
$486$	−0.575048	−0.0260847
$487$	−4.95998	−0.224758	−0.112379	−	0.993665i	$-0.535847\pi$
$487$	−4.95998	−0.224758	−0.112379	+	0.993665i	$0.535847\pi$
$488$	−0.703629	−0.0318518
$489$	−18.3521	−0.829913
$490$	9.19669	0.415464
$491$	−13.8599	−0.625488	−0.312744	−	0.949837i	$-0.601248\pi$
$491$	−13.8599	−0.625488	−0.312744	+	0.949837i	$0.601248\pi$
$492$	9.10358	0.410421
$493$	−6.13375	−0.276250
$494$	0	0
$495$	0.772556	0.0347238
$496$	−11.8444	−0.531827
$497$	−33.8063	−1.51642
$498$	−2.10976	−0.0945407
$499$	−16.4165	−0.734902	−0.367451	−	0.930043i	$-0.619770\pi$
$499$	−16.4165	−0.734902	−0.367451	+	0.930043i	$0.619770\pi$
$500$	−18.7147	−0.836946
$501$	18.3480	0.819728
$502$	−7.67707	−0.342645
$503$	−24.2095	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$503$	−24.2095	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$504$	9.07865	0.404395
$505$	−9.96086	−0.443252
$506$	−3.01033	−0.133826
$507$	0	0
$508$	−30.8232	−1.36756
$509$	−18.5740	−0.823276	−0.411638	−	0.911347i	$-0.635043\pi$
$509$	−18.5740	−0.823276	−0.411638	+	0.911347i	$0.635043\pi$
$510$	−0.798850	−0.0353737
$511$	−55.6056	−2.45985
$512$	−20.5349	−0.907524
$513$	7.33062	0.323655
$514$	−9.46267	−0.417380
$515$	−8.00531	−0.352756
$516$	13.8441	0.609451
$517$	4.15443	0.182712
$518$	−5.13835	−0.225766
$519$	0.155919	0.00684409
$520$	0	0
$521$	−29.3147	−1.28430	−0.642151	−	0.766578i	$-0.721958\pi$
$521$	−29.3147	−1.28430	−0.642151	+	0.766578i	$0.721958\pi$
$522$	−3.52720	−0.154381
$523$	−28.5384	−1.24790	−0.623949	−	0.781465i	$-0.714473\pi$
$523$	−28.5384	−1.24790	−0.623949	+	0.781465i	$0.714473\pi$
$524$	5.40845	0.236269
$525$	−13.2097	−0.576517
$526$	13.1448	0.573143
$527$	5.57311	0.242769
$528$	−1.18191	−0.0514358
$529$	65.6100	2.85261
$530$	−4.54645	−0.197485
$531$	−0.523791	−0.0227306
$532$	−52.6516	−2.28274
$533$	0	0
$534$	5.63925	0.244034
$535$	13.4322	0.580724
$536$	−10.6108	−0.458317
$537$	17.5244	0.756232
$538$	−0.299733	−0.0129224
$539$	−6.40229	−0.275766
$540$	2.31900	0.0997939
$541$	24.5062	1.05360	0.526801	−	0.849989i	$-0.323391\pi$
$541$	24.5062	1.05360	0.526801	+	0.849989i	$0.323391\pi$
$542$	7.33312	0.314984
$543$	6.66347	0.285957
$544$	5.44220	0.233332
$545$	−17.1767	−0.735770
$546$	0	0
$547$	−8.29262	−0.354567	−0.177284	−	0.984160i	$-0.556731\pi$
$547$	−8.29262	−0.354567	−0.177284	+	0.984160i	$0.556731\pi$
$548$	22.3756	0.955840
$549$	−0.333468	−0.0142321
$550$	−0.981823	−0.0418651
$551$	44.9642	1.91554
$552$	−19.8624	−0.845398
$553$	48.6210	2.06757
$554$	−16.5189	−0.701821
$555$	−2.88502	−0.122462
$556$	−19.5136	−0.827561
$557$	41.7827	1.77039	0.885195	−	0.465221i	$-0.154025\pi$
$557$	41.7827	1.77039	0.885195	+	0.465221i	$0.154025\pi$
$558$	3.20481	0.135670
$559$	0	0
$560$	−12.7030	−0.536800
$561$	0.556120	0.0234794
$562$	5.81990	0.245498
$563$	21.5098	0.906530	0.453265	−	0.891376i	$-0.350259\pi$
$563$	21.5098	0.906530	0.453265	+	0.891376i	$0.350259\pi$
$564$	12.4705	0.525101
$565$	9.86154	0.414878
$566$	−18.5692	−0.780522
$567$	4.30260	0.180692
$568$	−16.5789	−0.695635
$569$	17.6313	0.739143	0.369572	−	0.929202i	$-0.379504\pi$
$569$	17.6313	0.739143	0.369572	+	0.929202i	$0.379504\pi$
$570$	5.85607	0.245284
$571$	−16.2098	−0.678360	−0.339180	−	0.940722i	$-0.610149\pi$
$571$	−16.2098	−0.678360	−0.339180	+	0.940722i	$0.610149\pi$
$572$	0	0
$573$	−8.21408	−0.343148
$574$	13.4930	0.563186
$575$	28.9003	1.20522
$576$	−1.12101	−0.0467087
$577$	32.4596	1.35131	0.675656	−	0.737217i	$-0.263860\pi$
$577$	32.4596	1.35131	0.675656	+	0.737217i	$0.263860\pi$
$578$	−0.575048	−0.0239188
$579$	−4.39726	−0.182744
$580$	14.2242	0.590626
$581$	15.7856	0.654896
$582$	0.303126	0.0125650
$583$	3.16502	0.131082
$584$	−27.2695	−1.12842
$585$	0	0
$586$	−12.1133	−0.500395
$587$	11.5581	0.477053	0.238527	−	0.971136i	$-0.423336\pi$
$587$	11.5581	0.477053	0.238527	+	0.971136i	$0.423336\pi$
$588$	−19.2179	−0.792532
$589$	−40.8544	−1.68338
$590$	−0.418430	−0.0172265
$591$	17.5911	0.723600
$592$	4.41369	0.181401
$593$	−44.0984	−1.81091	−0.905453	−	0.424447i	$-0.860469\pi$
$593$	−44.0984	−1.81091	−0.905453	+	0.424447i	$0.860469\pi$
$594$	0.319796	0.0131214
$595$	5.97713	0.245038
$596$	26.3852	1.08078
$597$	−26.7645	−1.09540
$598$	0	0
$599$	−42.6781	−1.74378	−0.871891	−	0.489701i	$-0.837106\pi$
$599$	−42.6781	−1.74378	−0.871891	+	0.489701i	$0.837106\pi$
$600$	−6.47814	−0.264469
$601$	0.0941182	0.00383916	0.00191958	−	0.999998i	$-0.499389\pi$
$601$	0.0941182	0.00383916	0.00191958	+	0.999998i	$0.499389\pi$
$602$	20.5192	0.836299
$603$	−5.02874	−0.204786
$604$	−8.78227	−0.357346
$605$	14.8514	0.603797
$606$	−4.12325	−0.167496
$607$	−24.7098	−1.00294	−0.501471	−	0.865175i	$-0.667207\pi$
$607$	−24.7098	−1.00294	−0.501471	+	0.865175i	$0.667207\pi$
$608$	−39.8947	−1.61794
$609$	26.3911	1.06942
$610$	−0.266391	−0.0107858
$611$	0	0
$612$	1.66932	0.0674783
$613$	−25.8186	−1.04280	−0.521402	−	0.853311i	$-0.674591\pi$
$613$	−25.8186	−1.04280	−0.521402	+	0.853311i	$0.674591\pi$
$614$	5.62256	0.226908
$615$	7.57589	0.305489
$616$	−5.04882	−0.203423
$617$	−29.0951	−1.17133	−0.585663	−	0.810555i	$-0.699166\pi$
$617$	−29.0951	−1.17133	−0.585663	+	0.810555i	$0.699166\pi$
$618$	−3.31376	−0.133299
$619$	−4.26680	−0.171497	−0.0857486	−	0.996317i	$-0.527328\pi$
$619$	−4.26680	−0.171497	−0.0857486	+	0.996317i	$0.527328\pi$
$620$	−12.9241	−0.519042
$621$	−9.41329	−0.377742
$622$	−4.07926	−0.163564
$623$	−42.1938	−1.69046
$624$	0	0
$625$	−0.223373	−0.00893492
$626$	−4.29183	−0.171536
$627$	−4.07671	−0.162808
$628$	20.0272	0.799172
$629$	−2.07677	−0.0828062
$630$	3.43714	0.136939
$631$	−28.5398	−1.13615	−0.568076	−	0.822976i	$-0.692312\pi$
$631$	−28.5398	−1.13615	−0.568076	+	0.822976i	$0.692312\pi$
$632$	23.8442	0.948470
$633$	−15.2569	−0.606407
$634$	19.3562	0.768735
$635$	−25.6507	−1.01792
$636$	9.50050	0.376719
$637$	0	0
$638$	1.96155	0.0776584
$639$	−7.85716	−0.310825
$640$	−16.0160	−0.633088
$641$	24.8751	0.982508	0.491254	−	0.871016i	$-0.336539\pi$
$641$	24.8751	0.982508	0.491254	+	0.871016i	$0.336539\pi$
$642$	5.56019	0.219443
$643$	29.3109	1.15591	0.577954	−	0.816069i	$-0.303851\pi$
$643$	29.3109	1.15591	0.577954	+	0.816069i	$0.303851\pi$
$644$	67.6102	2.66422
$645$	11.5209	0.453634
$646$	4.21546	0.165855
$647$	43.7852	1.72137	0.860686	−	0.509136i	$-0.170035\pi$
$647$	43.7852	1.72137	0.860686	+	0.509136i	$0.170035\pi$
$648$	2.11004	0.0828900
$649$	0.291291	0.0114342
$650$	0	0
$651$	−23.9789	−0.939807
$652$	30.6356	1.19978
$653$	14.0175	0.548546	0.274273	−	0.961652i	$-0.411563\pi$
$653$	14.0175	0.548546	0.274273	+	0.961652i	$0.411563\pi$
$654$	−7.11023	−0.278032
$655$	4.50085	0.175863
$656$	−11.5901	−0.452516
$657$	−12.9237	−0.504202
$658$	18.4832	0.720552
$659$	10.1133	0.393960	0.196980	−	0.980408i	$-0.436887\pi$
$659$	10.1133	0.393960	0.196980	+	0.980408i	$0.436887\pi$
$660$	−1.28964	−0.0501993
$661$	−15.6871	−0.610158	−0.305079	−	0.952327i	$-0.598683\pi$
$661$	−15.6871	−0.610158	−0.305079	+	0.952327i	$0.598683\pi$
$662$	7.51992	0.292270
$663$	0	0
$664$	7.74139	0.300424
$665$	−43.8161	−1.69911
$666$	−1.19424	−0.0462759
$667$	−57.7387	−2.23565
$668$	−30.6287	−1.18506
$669$	2.75963	0.106693
$670$	−4.01721	−0.155198
$671$	0.185448	0.00715915
$672$	−23.4156	−0.903278
$673$	−9.80168	−0.377827	−0.188913	−	0.981994i	$-0.560497\pi$
$673$	−9.80168	−0.377827	−0.188913	+	0.981994i	$0.560497\pi$
$674$	−14.1415	−0.544711
$675$	−3.07016	−0.118170
$676$	0	0
$677$	−11.7994	−0.453488	−0.226744	−	0.973954i	$-0.572808\pi$
$677$	−11.7994	−0.453488	−0.226744	+	0.973954i	$0.572808\pi$
$678$	4.08214	0.156774
$679$	−2.26804	−0.0870395
$680$	2.93124	0.112408
$681$	12.1227	0.464541
$682$	−1.78226	−0.0682462
$683$	−40.8473	−1.56298	−0.781489	−	0.623920i	$-0.785539\pi$
$683$	−40.8473	−1.56298	−0.781489	+	0.623920i	$0.785539\pi$
$684$	−12.2371	−0.467899
$685$	18.6207	0.711462
$686$	−11.1646	−0.426267
$687$	−22.1475	−0.844980
$688$	−17.6254	−0.671961
$689$	0	0
$690$	−7.51981	−0.286274
$691$	1.67686	0.0637908	0.0318954	−	0.999491i	$-0.489846\pi$
$691$	1.67686	0.0637908	0.0318954	+	0.999491i	$0.489846\pi$
$692$	−0.260279	−0.00989431
$693$	−2.39277	−0.0908937
$694$	−4.95015	−0.187905
$695$	−16.2390	−0.615980
$696$	12.9424	0.490581
$697$	5.45346	0.206565
$698$	−1.25886	−0.0476484
$699$	−22.3005	−0.843483
$700$	22.0512	0.833455
$701$	−26.4573	−0.999280	−0.499640	−	0.866233i	$-0.666534\pi$
$701$	−26.4573	−0.999280	−0.499640	+	0.866233i	$0.666534\pi$
$702$	0	0
$703$	15.2240	0.574184
$704$	0.623416	0.0234959
$705$	10.3778	0.390849
$706$	−1.90057	−0.0715288
$707$	30.8508	1.16027
$708$	0.874374	0.0328610
$709$	−43.5783	−1.63662	−0.818309	−	0.574779i	$-0.805088\pi$
$709$	−43.5783	−1.63662	−0.818309	+	0.574779i	$0.805088\pi$
$710$	−6.27670	−0.235560
$711$	11.3004	0.423797
$712$	−20.6922	−0.775474
$713$	52.4613	1.96469
$714$	2.47420	0.0925948
$715$	0	0
$716$	−29.2537	−1.09326
$717$	−13.6415	−0.509451
$718$	−3.31411	−0.123681
$719$	−46.3757	−1.72952	−0.864760	−	0.502185i	$-0.832530\pi$
$719$	−46.3757	−1.72952	−0.864760	+	0.502185i	$0.832530\pi$
$720$	−2.95240	−0.110029
$721$	24.7941	0.923381
$722$	−19.9760	−0.743430
$723$	−15.9632	−0.593678
$724$	−11.1235	−0.413400
$725$	−18.8316	−0.699387
$726$	6.14768	0.228162
$727$	8.99556	0.333627	0.166813	−	0.985988i	$-0.446652\pi$
$727$	8.99556	0.333627	0.166813	+	0.985988i	$0.446652\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−10.3241	−0.382113
$731$	8.29324	0.306737
$732$	0.556664	0.0205749
$733$	39.8697	1.47262	0.736311	−	0.676643i	$-0.236566\pi$
$733$	39.8697	1.47262	0.736311	+	0.676643i	$0.236566\pi$
$734$	4.23239	0.156220
$735$	−15.9929	−0.589907
$736$	51.2290	1.88833
$737$	2.79658	0.103013
$738$	3.13600	0.115438
$739$	30.1421	1.10880	0.554398	−	0.832252i	$-0.312948\pi$
$739$	30.1421	1.10880	0.554398	+	0.832252i	$0.312948\pi$
$740$	4.81602	0.177041
$741$	0	0
$742$	14.0813	0.516941
$743$	−25.8926	−0.949906	−0.474953	−	0.880011i	$-0.657535\pi$
$743$	−25.8926	−0.949906	−0.474953	+	0.880011i	$0.657535\pi$
$744$	−11.7595	−0.431123
$745$	21.9575	0.804459
$746$	−19.7122	−0.721715
$747$	3.66884	0.134236
$748$	−0.928343	−0.0339436
$749$	−41.6023	−1.52011
$750$	−6.44684	−0.235405
$751$	−39.9725	−1.45862	−0.729309	−	0.684185i	$-0.760158\pi$
$751$	−39.9725	−1.45862	−0.729309	+	0.684185i	$0.760158\pi$
$752$	−15.8766	−0.578958
$753$	13.3503	0.486513
$754$	0	0
$755$	−7.30851	−0.265984
$756$	−7.18242	−0.261222
$757$	19.3730	0.704123	0.352062	−	0.935977i	$-0.385481\pi$
$757$	19.3730	0.704123	0.352062	+	0.935977i	$0.385481\pi$
$758$	−6.28672	−0.228344
$759$	5.23492	0.190016
$760$	−21.4878	−0.779444
$761$	24.1713	0.876208	0.438104	−	0.898924i	$-0.355650\pi$
$761$	24.1713	0.876208	0.438104	+	0.898924i	$0.355650\pi$
$762$	−10.6180	−0.384649
$763$	53.1999	1.92597
$764$	13.7119	0.496080
$765$	1.38919	0.0502262
$766$	−11.4979	−0.415438
$767$	0	0
$768$	−4.38773	−0.158329
$769$	20.9923	0.757000	0.378500	−	0.925601i	$-0.376440\pi$
$769$	20.9923	0.757000	0.378500	+	0.925601i	$0.376440\pi$
$770$	−1.91146	−0.0688843
$771$	16.4554	0.592628
$772$	7.34044	0.264188
$773$	54.0765	1.94500	0.972499	−	0.232908i	$-0.0748242\pi$
$773$	54.0765	1.94500	0.972499	+	0.232908i	$0.0748242\pi$
$774$	4.76901	0.171419
$775$	17.1103	0.614621
$776$	−1.11227	−0.0399281
$777$	8.93551	0.320560
$778$	−7.81350	−0.280128
$779$	−39.9773	−1.43233
$780$	0	0
$781$	4.36953	0.156354
$782$	−5.41309	−0.193572
$783$	6.13375	0.219202
$784$	24.4669	0.873820
$785$	16.6664	0.594850
$786$	1.86310	0.0664547
$787$	−18.4306	−0.656980	−0.328490	−	0.944508i	$-0.606540\pi$
$787$	−18.4306	−0.656980	−0.328490	+	0.944508i	$0.606540\pi$
$788$	−29.3651	−1.04609
$789$	−22.8587	−0.813791
$790$	9.02730	0.321177
$791$	−30.5432	−1.08599
$792$	−1.17343	−0.0416962
$793$	0	0
$794$	13.2724	0.471020
$795$	7.90621	0.280404
$796$	44.6785	1.58359
$797$	−36.3059	−1.28602	−0.643010	−	0.765858i	$-0.722315\pi$
$797$	−36.3059	−1.28602	−0.643010	+	0.765858i	$0.722315\pi$
$798$	−18.1374	−0.642059
$799$	7.47038	0.264283
$800$	16.7084	0.590731
$801$	−9.80657	−0.346498
$802$	−1.72704	−0.0609839
$803$	7.18714	0.253629
$804$	8.39457	0.296054
$805$	56.2644	1.98306
$806$	0	0
$807$	0.521231	0.0183482
$808$	15.1295	0.532255
$809$	−38.5250	−1.35447	−0.677233	−	0.735769i	$-0.736821\pi$
$809$	−38.5250	−1.35447	−0.677233	+	0.735769i	$0.736821\pi$
$810$	0.798850	0.0280687
$811$	5.87217	0.206200	0.103100	−	0.994671i	$-0.467124\pi$
$811$	5.87217	0.206200	0.103100	+	0.994671i	$0.467124\pi$
$812$	−44.0552	−1.54603
$813$	−12.7522	−0.447238
$814$	0.664142	0.0232782
$815$	25.4946	0.893036
$816$	−2.12527	−0.0743993
$817$	−60.7946	−2.12693
$818$	11.4211	0.399328
$819$	0	0
$820$	−12.6466	−0.441638
$821$	31.4377	1.09718	0.548591	−	0.836091i	$-0.315164\pi$
$821$	31.4377	1.09718	0.548591	+	0.836091i	$0.315164\pi$
$822$	7.70797	0.268846
$823$	−27.0765	−0.943826	−0.471913	−	0.881645i	$-0.656436\pi$
$823$	−27.0765	−0.943826	−0.471913	+	0.881645i	$0.656436\pi$
$824$	12.1592	0.423587
$825$	1.70738	0.0594432
$826$	1.29597	0.0450924
$827$	−30.7203	−1.06825	−0.534126	−	0.845405i	$-0.679359\pi$
$827$	−30.7203	−1.06825	−0.534126	+	0.845405i	$0.679359\pi$
$828$	15.7138	0.546092
$829$	−14.4126	−0.500569	−0.250285	−	0.968172i	$-0.580524\pi$
$829$	−14.4126	−0.500569	−0.250285	+	0.968172i	$0.580524\pi$
$830$	2.93086	0.101732
$831$	28.7261	0.996498
$832$	0	0
$833$	−11.5124	−0.398881
$834$	−6.72205	−0.232766
$835$	−25.4888	−0.882077
$836$	6.80533	0.235367
$837$	−5.57311	−0.192635
$838$	8.61992	0.297770
$839$	3.30496	0.114100	0.0570500	−	0.998371i	$-0.481831\pi$
$839$	3.30496	0.114100	0.0570500	+	0.998371i	$0.481831\pi$
$840$	−12.6120	−0.435154
$841$	8.62287	0.297341
$842$	18.3216	0.631403
$843$	−10.1207	−0.348576
$844$	25.4686	0.876666
$845$	0	0
$846$	4.29583	0.147694
$847$	−45.9980	−1.58051
$848$	−12.0954	−0.415358
$849$	32.2916	1.10824
$850$	−1.76549	−0.0605557
$851$	−19.5492	−0.670138
$852$	13.1161	0.449351
$853$	27.6394	0.946354	0.473177	−	0.880967i	$-0.343107\pi$
$853$	27.6394	0.946354	0.473177	+	0.880967i	$0.343107\pi$
$854$	0.825067	0.0282332
$855$	−10.1836	−0.348272
$856$	−20.4021	−0.697330
$857$	13.5048	0.461316	0.230658	−	0.973035i	$-0.425912\pi$
$857$	13.5048	0.461316	0.230658	+	0.973035i	$0.425912\pi$
$858$	0	0
$859$	42.8913	1.46343	0.731716	−	0.681609i	$-0.238720\pi$
$859$	42.8913	1.46343	0.731716	+	0.681609i	$0.238720\pi$
$860$	−19.2320	−0.655807
$861$	−23.4641	−0.799654
$862$	17.5919	0.599183
$863$	12.0310	0.409540	0.204770	−	0.978810i	$-0.434355\pi$
$863$	12.0310	0.409540	0.204770	+	0.978810i	$0.434355\pi$
$864$	−5.44220	−0.185147
$865$	−0.216601	−0.00736465
$866$	17.9395	0.609610
$867$	1.00000	0.0339618
$868$	40.0285	1.35865
$869$	−6.28436	−0.213182
$870$	4.89995	0.166124
$871$	0	0
$872$	26.0897	0.883509
$873$	−0.527132	−0.0178407
$874$	39.6813	1.34224
$875$	48.2364	1.63069
$876$	21.5738	0.728912
$877$	−24.0313	−0.811480	−0.405740	−	0.913989i	$-0.632986\pi$
$877$	−24.0313	−0.811480	−0.405740	+	0.913989i	$0.632986\pi$
$878$	11.6548	0.393330
$879$	21.0648	0.710499
$880$	1.64189	0.0553481
$881$	29.1093	0.980716	0.490358	−	0.871521i	$-0.336866\pi$
$881$	29.1093	0.980716	0.490358	+	0.871521i	$0.336866\pi$
$882$	−6.62019	−0.222913
$883$	−39.8535	−1.34118	−0.670589	−	0.741829i	$-0.733959\pi$
$883$	−39.8535	−1.34118	−0.670589	+	0.741829i	$0.733959\pi$
$884$	0	0
$885$	0.727644	0.0244595
$886$	−1.24984	−0.0419892
$887$	38.0628	1.27802	0.639012	−	0.769196i	$-0.279343\pi$
$887$	38.0628	1.27802	0.639012	+	0.769196i	$0.279343\pi$
$888$	4.38205	0.147052
$889$	79.4456	2.66452
$890$	−7.83398	−0.262596
$891$	−0.556120	−0.0186307
$892$	−4.60670	−0.154244
$893$	−54.7625	−1.83256
$894$	9.08919	0.303988
$895$	−24.3446	−0.813751
$896$	49.6049	1.65718
$897$	0	0
$898$	−2.55848	−0.0853775
$899$	−34.1841	−1.14010
$900$	5.12507	0.170836
$901$	5.69124	0.189603
$902$	−1.74400	−0.0580687
$903$	−35.6825	−1.18744
$904$	−14.9787	−0.498183
$905$	−9.25681	−0.307707
$906$	−3.02532	−0.100510
$907$	−4.18496	−0.138959	−0.0694797	−	0.997583i	$-0.522134\pi$
$907$	−4.18496	−0.138959	−0.0694797	+	0.997583i	$0.522134\pi$
$908$	−20.2366	−0.671575
$909$	7.17027	0.237823
$910$	0	0
$911$	39.7556	1.31716	0.658581	−	0.752510i	$-0.271157\pi$
$911$	39.7556	1.31716	0.658581	+	0.752510i	$0.271157\pi$
$912$	15.5795	0.515890
$913$	−2.04032	−0.0675247
$914$	−15.5919	−0.515735
$915$	0.463250	0.0153146
$916$	36.9712	1.22156
$917$	−13.9401	−0.460341
$918$	0.575048	0.0189794
$919$	0.264249	0.00871677	0.00435839	−	0.999991i	$-0.498613\pi$
$919$	0.264249	0.00871677	0.00435839	+	0.999991i	$0.498613\pi$
$920$	27.5926	0.909700
$921$	−9.77755	−0.322181
$922$	7.51530	0.247503
$923$	0	0
$924$	3.99429	0.131403
$925$	−6.37600	−0.209642
$926$	−0.0947400	−0.00311335
$927$	5.76258	0.189268
$928$	−33.3811	−1.09579
$929$	−31.0637	−1.01917	−0.509583	−	0.860421i	$-0.670200\pi$
$929$	−31.0637	−1.01917	−0.509583	+	0.860421i	$0.670200\pi$
$930$	−4.45208	−0.145990
$931$	84.3931	2.76587
$932$	37.2267	1.21940
$933$	7.09378	0.232240
$934$	6.00843	0.196602
$935$	−0.772556	−0.0252653
$936$	0	0
$937$	51.7978	1.69216	0.846081	−	0.533054i	$-0.178956\pi$
$937$	51.7978	1.69216	0.846081	+	0.533054i	$0.178956\pi$
$938$	12.4421	0.406250
$939$	7.46344	0.243560
$940$	−17.3238	−0.565040
$941$	−6.56945	−0.214158	−0.107079	−	0.994251i	$-0.534150\pi$
$941$	−6.56945	−0.214158	−0.107079	+	0.994251i	$0.534150\pi$
$942$	6.89898	0.224781
$943$	51.3350	1.67170
$944$	−1.11320	−0.0362314
$945$	−5.97713	−0.194436
$946$	−2.65215	−0.0862287
$947$	24.1500	0.784768	0.392384	−	0.919801i	$-0.371650\pi$
$947$	24.1500	0.784768	0.392384	+	0.919801i	$0.371650\pi$
$948$	−18.8639	−0.612671
$949$	0	0
$950$	12.9421	0.419897
$951$	−33.6602	−1.09151
$952$	−9.07865	−0.294241
$953$	30.1570	0.976882	0.488441	−	0.872597i	$-0.337566\pi$
$953$	30.1570	0.976882	0.488441	+	0.872597i	$0.337566\pi$
$954$	3.27274	0.105959
$955$	11.4109	0.369248
$956$	22.7720	0.736500
$957$	−3.41110	−0.110265
$958$	3.02970	0.0978850
$959$	−57.6723	−1.86234
$960$	1.55729	0.0502614
$961$	0.0595994	0.00192256
$962$	0	0
$963$	−9.66909	−0.311582
$964$	26.6477	0.858264
$965$	6.10863	0.196644
$966$	23.2904	0.749356
$967$	19.0267	0.611856	0.305928	−	0.952055i	$-0.401033\pi$
$967$	19.0267	0.611856	0.305928	+	0.952055i	$0.401033\pi$
$968$	−22.5578	−0.725036
$969$	−7.33062	−0.235493
$970$	−0.421100	−0.0135207
$971$	−7.84570	−0.251780	−0.125890	−	0.992044i	$-0.540179\pi$
$971$	−7.84570	−0.251780	−0.125890	+	0.992044i	$0.540179\pi$
$972$	−1.66932	−0.0535435
$973$	50.2955	1.61240
$974$	2.85223	0.0913912
$975$	0	0
$976$	−0.708708	−0.0226852
$977$	−16.9103	−0.541008	−0.270504	−	0.962719i	$-0.587190\pi$
$977$	−16.9103	−0.541008	−0.270504	+	0.962719i	$0.587190\pi$
$978$	10.5534	0.337459
$979$	5.45364	0.174299
$980$	26.6973	0.852813
$981$	12.3646	0.394771
$982$	7.97011	0.254336
$983$	40.0463	1.27728	0.638639	−	0.769506i	$-0.279498\pi$
$983$	40.0463	1.27728	0.638639	+	0.769506i	$0.279498\pi$
$984$	−11.5070	−0.366830
$985$	−24.4373	−0.778638
$986$	3.52720	0.112329
$987$	−32.1421	−1.02309
$988$	0	0
$989$	78.0667	2.48238
$990$	−0.444257	−0.0141194
$991$	35.4703	1.12675	0.563375	−	0.826201i	$-0.309503\pi$
$991$	35.4703	1.12675	0.563375	+	0.826201i	$0.309503\pi$
$992$	30.3300	0.962979
$993$	−13.0770	−0.414987
$994$	19.4402	0.616607
$995$	37.1810	1.17872
$996$	−6.12447	−0.194061
$997$	−41.9651	−1.32905	−0.664525	−	0.747266i	$-0.731366\pi$
$997$	−41.9651	−1.32905	−0.664525	+	0.747266i	$0.731366\pi$
$998$	9.44026	0.298826
$999$	2.07677	0.0657060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.bp.1.11	✓	21
13.12	even	2		8619.2.a.bq.1.11	yes	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8619.2.a.bp.1.11	✓	21	1.1	even	1	trivial
8619.2.a.bq.1.11	yes	21	13.12	even	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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q-0.575048 q^{2} +1.00000 q^{3} -1.66932 q^{4} -1.38919 q^{5} -0.575048 q^{6} +4.30260 q^{7} +2.11004 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.575048 q^{2} +1.00000 q^{3} -1.66932 q^{4} -1.38919 q^{5} -0.575048 q^{6} +4.30260 q^{7} +2.11004 q^{8} +1.00000 q^{9} +0.798850 q^{10} -0.556120 q^{11} -1.66932 q^{12} -2.47420 q^{14} -1.38919 q^{15} +2.12527 q^{16} -1.00000 q^{17} -0.575048 q^{18} +7.33062 q^{19} +2.31900 q^{20} +4.30260 q^{21} +0.319796 q^{22} -9.41329 q^{23} +2.11004 q^{24} -3.07016 q^{25} +1.00000 q^{27} -7.18242 q^{28} +6.13375 q^{29} +0.798850 q^{30} -5.57311 q^{31} -5.44220 q^{32} -0.556120 q^{33} +0.575048 q^{34} -5.97713 q^{35} -1.66932 q^{36} +2.07677 q^{37} -4.21546 q^{38} -2.93124 q^{40} -5.45346 q^{41} -2.47420 q^{42} -8.29324 q^{43} +0.928343 q^{44} -1.38919 q^{45} +5.41309 q^{46} -7.47038 q^{47} +2.12527 q^{48} +11.5124 q^{49} +1.76549 q^{50} -1.00000 q^{51} -5.69124 q^{53} -0.575048 q^{54} +0.772556 q^{55} +9.07865 q^{56} +7.33062 q^{57} -3.52720 q^{58} -0.523791 q^{59} +2.31900 q^{60} -0.333468 q^{61} +3.20481 q^{62} +4.30260 q^{63} -1.12101 q^{64} +0.319796 q^{66} -5.02874 q^{67} +1.66932 q^{68} -9.41329 q^{69} +3.43714 q^{70} -7.85716 q^{71} +2.11004 q^{72} -12.9237 q^{73} -1.19424 q^{74} -3.07016 q^{75} -12.2371 q^{76} -2.39277 q^{77} +11.3004 q^{79} -2.95240 q^{80} +1.00000 q^{81} +3.13600 q^{82} +3.66884 q^{83} -7.18242 q^{84} +1.38919 q^{85} +4.76901 q^{86} +6.13375 q^{87} -1.17343 q^{88} -9.80657 q^{89} +0.798850 q^{90} +15.7138 q^{92} -5.57311 q^{93} +4.29583 q^{94} -10.1836 q^{95} -5.44220 q^{96} -0.527132 q^{97} -6.62019 q^{98} -0.556120 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9}+O(q^{10}) \) 21 * q - 9 * q^2 + 21 * q^3 + 29 * q^4 - 26 * q^5 - 9 * q^6 + 5 * q^7 - 24 * q^8 + 21 * q^9	\( 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9} + 12 q^{10} - 37 q^{11} + 29 q^{12} - 6 q^{14} - 26 q^{15} + 33 q^{16} - 21 q^{17} - 9 q^{18} + 6 q^{19} - 53 q^{20} + 5 q^{21} + 17 q^{22} - 8 q^{23} - 24 q^{24} + 57 q^{25} + 21 q^{27} - 5 q^{28} - 6 q^{29} + 12 q^{30} - 9 q^{31} - 37 q^{32} - 37 q^{33} + 9 q^{34} - 8 q^{35} + 29 q^{36} - 9 q^{37} + 11 q^{38} - 8 q^{40} - 50 q^{41} - 6 q^{42} + 18 q^{43} - 67 q^{44} - 26 q^{45} - 23 q^{46} - 71 q^{47} + 33 q^{48} + 46 q^{49} - 2 q^{50} - 21 q^{51} - 14 q^{53} - 9 q^{54} + 29 q^{55} - 17 q^{56} + 6 q^{57} - 37 q^{58} - 59 q^{59} - 53 q^{60} + 44 q^{61} + 2 q^{62} + 5 q^{63} + 44 q^{64} + 17 q^{66} - 8 q^{67} - 29 q^{68} - 8 q^{69} - 15 q^{70} - 60 q^{71} - 24 q^{72} - 13 q^{73} - 14 q^{74} + 57 q^{75} - 15 q^{76} - 28 q^{77} + 51 q^{79} - 131 q^{80} + 21 q^{81} + 18 q^{82} - 30 q^{83} - 5 q^{84} + 26 q^{85} + 15 q^{86} - 6 q^{87} + 71 q^{88} - 62 q^{89} + 12 q^{90} + 20 q^{92} - 9 q^{93} - 72 q^{94} - 16 q^{95} - 37 q^{96} + 6 q^{97} - 58 q^{98} - 37 q^{99}+O(q^{100}) \) 21 * q - 9 * q^2 + 21 * q^3 + 29 * q^4 - 26 * q^5 - 9 * q^6 + 5 * q^7 - 24 * q^8 + 21 * q^9 + 12 * q^10 - 37 * q^11 + 29 * q^12 - 6 * q^14 - 26 * q^15 + 33 * q^16 - 21 * q^17 - 9 * q^18 + 6 * q^19 - 53 * q^20 + 5 * q^21 + 17 * q^22 - 8 * q^23 - 24 * q^24 + 57 * q^25 + 21 * q^27 - 5 * q^28 - 6 * q^29 + 12 * q^30 - 9 * q^31 - 37 * q^32 - 37 * q^33 + 9 * q^34 - 8 * q^35 + 29 * q^36 - 9 * q^37 + 11 * q^38 - 8 * q^40 - 50 * q^41 - 6 * q^42 + 18 * q^43 - 67 * q^44 - 26 * q^45 - 23 * q^46 - 71 * q^47 + 33 * q^48 + 46 * q^49 - 2 * q^50 - 21 * q^51 - 14 * q^53 - 9 * q^54 + 29 * q^55 - 17 * q^56 + 6 * q^57 - 37 * q^58 - 59 * q^59 - 53 * q^60 + 44 * q^61 + 2 * q^62 + 5 * q^63 + 44 * q^64 + 17 * q^66 - 8 * q^67 - 29 * q^68 - 8 * q^69 - 15 * q^70 - 60 * q^71 - 24 * q^72 - 13 * q^73 - 14 * q^74 + 57 * q^75 - 15 * q^76 - 28 * q^77 + 51 * q^79 - 131 * q^80 + 21 * q^81 + 18 * q^82 - 30 * q^83 - 5 * q^84 + 26 * q^85 + 15 * q^86 - 6 * q^87 + 71 * q^88 - 62 * q^89 + 12 * q^90 + 20 * q^92 - 9 * q^93 - 72 * q^94 - 16 * q^95 - 37 * q^96 + 6 * q^97 - 58 * q^98 - 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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