LMFDB - Embedded newform 129.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [129,2,Mod(1,129)] 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("129.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(129, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 129 = 3 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	129.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.03007018607$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} -4.12489 q^{5} +2.12489 q^{6} +1.48486 q^{7} +1.09461 q^{8} +1.00000 q^{9} -8.76491 q^{10} -2.60975 q^{11} +2.51514 q^{12} +3.00000 q^{13} +3.15516 q^{14} -4.12489 q^{15} -2.70436 q^{16} +0.484862 q^{17} +2.12489 q^{18} -6.76491 q^{19} -10.3747 q^{20} +1.48486 q^{21} -5.54541 q^{22} +8.79518 q^{23} +1.09461 q^{24} +12.0147 q^{25} +6.37466 q^{26} +1.00000 q^{27} +3.73463 q^{28} -2.12489 q^{29} -8.76491 q^{30} +3.76491 q^{31} -7.93567 q^{32} -2.60975 q^{33} +1.03028 q^{34} -6.12489 q^{35} +2.51514 q^{36} +5.28005 q^{37} -14.3747 q^{38} +3.00000 q^{39} -4.51514 q^{40} +0.734633 q^{41} +3.15516 q^{42} -1.00000 q^{43} -6.56387 q^{44} -4.12489 q^{45} +18.6888 q^{46} -8.43521 q^{47} -2.70436 q^{48} -4.79518 q^{49} +25.5298 q^{50} +0.484862 q^{51} +7.54541 q^{52} +3.76491 q^{53} +2.12489 q^{54} +10.7649 q^{55} +1.62534 q^{56} -6.76491 q^{57} -4.51514 q^{58} +2.96972 q^{59} -10.3747 q^{60} -11.2195 q^{61} +8.00000 q^{62} +1.48486 q^{63} -11.4537 q^{64} -12.3747 q^{65} -5.54541 q^{66} +7.04496 q^{67} +1.21949 q^{68} +8.79518 q^{69} -13.0147 q^{70} +13.2195 q^{71} +1.09461 q^{72} -5.03028 q^{73} +11.2195 q^{74} +12.0147 q^{75} -17.0147 q^{76} -3.87511 q^{77} +6.37466 q^{78} +2.71995 q^{79} +11.1552 q^{80} +1.00000 q^{81} +1.56101 q^{82} -5.57947 q^{83} +3.73463 q^{84} -2.00000 q^{85} -2.12489 q^{86} -2.12489 q^{87} -2.85665 q^{88} -18.2498 q^{89} -8.76491 q^{90} +4.45459 q^{91} +22.1211 q^{92} +3.76491 q^{93} -17.9239 q^{94} +27.9045 q^{95} -7.93567 q^{96} +6.21949 q^{97} -10.1892 q^{98} -2.60975 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 3 q^{9} - 10 q^{10} + q^{11} + 8 q^{12} + 9 q^{13} + 2 q^{14} - 4 q^{15} + 10 q^{16} + q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{20}+ \cdots + q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^7 - 6 * q^8 + 3 * q^9 - 10 * q^10 + q^11 + 8 * q^12 + 9 * q^13 + 2 * q^14 - 4 * q^15 + 10 * q^16 + q^17 - 2 * q^18 - 4 * q^19 - 6 * q^20 + 4 * q^21 - 18 * q^22 + 11 * q^23 - 6 * q^24 + 3 * q^25 - 6 * q^26 + 3 * q^27 - 6 * q^28 + 2 * q^29 - 10 * q^30 - 5 * q^31 - 34 * q^32 + q^33 + 4 * q^34 - 10 * q^35 + 8 * q^36 - 18 * q^38 + 9 * q^39 - 14 * q^40 - 15 * q^41 + 2 * q^42 - 3 * q^43 + 24 * q^44 - 4 * q^45 + 4 * q^46 - 2 * q^47 + 10 * q^48 + q^49 + 44 * q^50 + q^51 + 24 * q^52 - 5 * q^53 - 2 * q^54 + 16 * q^55 + 30 * q^56 - 4 * q^57 - 14 * q^58 + 8 * q^59 - 6 * q^60 - 16 * q^61 + 24 * q^62 + 4 * q^63 + 34 * q^64 - 12 * q^65 - 18 * q^66 - 11 * q^67 - 14 * q^68 + 11 * q^69 - 6 * q^70 + 22 * q^71 - 6 * q^72 - 16 * q^73 + 16 * q^74 + 3 * q^75 - 18 * q^76 - 20 * q^77 - 6 * q^78 + 24 * q^79 + 26 * q^80 + 3 * q^81 + 40 * q^82 - 7 * q^83 - 6 * q^84 - 6 * q^85 + 2 * q^86 + 2 * q^87 - 62 * q^88 - 38 * q^89 - 10 * q^90 + 12 * q^91 + 70 * q^92 - 5 * q^93 - 18 * q^94 + 26 * q^95 - 34 * q^96 + q^97 - 12 * q^98 + 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 $

$2$	2.12489	1.50252	0.751260	−	0.660006i	$-0.229446\pi$
$2$	2.12489	1.50252	0.751260	+	0.660006i	$0.229446\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.51514	1.25757
$5$	−4.12489	−1.84470	−0.922352	−	0.386350i	$-0.873736\pi$
$5$	−4.12489	−1.84470	−0.922352	+	0.386350i	$0.873736\pi$
$6$	2.12489	0.867481
$7$	1.48486	0.561225	0.280613	−	0.959821i	$-0.409462\pi$
$7$	1.48486	0.561225	0.280613	+	0.959821i	$0.409462\pi$
$8$	1.09461	0.387003
$9$	1.00000	0.333333
$10$	−8.76491	−2.77171
$11$	−2.60975	−0.786868	−0.393434	−	0.919353i	$-0.628713\pi$
$11$	−2.60975	−0.786868	−0.393434	+	0.919353i	$0.628713\pi$
$12$	2.51514	0.726058
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	3.15516	0.843252
$15$	−4.12489	−1.06504
$16$	−2.70436	−0.676089
$17$	0.484862	0.117596	0.0587981	−	0.998270i	$-0.481273\pi$
$17$	0.484862	0.117596	0.0587981	+	0.998270i	$0.481273\pi$
$18$	2.12489	0.500840
$19$	−6.76491	−1.55198	−0.775988	−	0.630747i	$-0.782748\pi$
$19$	−6.76491	−1.55198	−0.775988	+	0.630747i	$0.782748\pi$
$20$	−10.3747	−2.31984
$21$	1.48486	0.324023
$22$	−5.54541	−1.18229
$23$	8.79518	1.83392	0.916961	−	0.398976i	$-0.130634\pi$
$23$	8.79518	1.83392	0.916961	+	0.398976i	$0.130634\pi$
$24$	1.09461	0.223436
$25$	12.0147	2.40294
$26$	6.37466	1.25017
$27$	1.00000	0.192450
$28$	3.73463	0.705779
$29$	−2.12489	−0.394581	−0.197291	−	0.980345i	$-0.563214\pi$
$29$	−2.12489	−0.394581	−0.197291	+	0.980345i	$0.563214\pi$
$30$	−8.76491	−1.60025
$31$	3.76491	0.676198	0.338099	−	0.941111i	$-0.390216\pi$
$31$	3.76491	0.676198	0.338099	+	0.941111i	$0.390216\pi$
$32$	−7.93567	−1.40284
$33$	−2.60975	−0.454299
$34$	1.03028	0.176691
$35$	−6.12489	−1.03529
$36$	2.51514	0.419190
$37$	5.28005	0.868034	0.434017	−	0.900905i	$-0.357096\pi$
$37$	5.28005	0.868034	0.434017	+	0.900905i	$0.357096\pi$
$38$	−14.3747	−2.33188
$39$	3.00000	0.480384
$40$	−4.51514	−0.713906
$41$	0.734633	0.114730	0.0573652	−	0.998353i	$-0.481730\pi$
$41$	0.734633	0.114730	0.0573652	+	0.998353i	$0.481730\pi$
$42$	3.15516	0.486852
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−6.56387	−0.989541
$45$	−4.12489	−0.614902
$46$	18.6888	2.75551
$47$	−8.43521	−1.23040	−0.615201	−	0.788370i	$-0.710925\pi$
$47$	−8.43521	−1.23040	−0.615201	+	0.788370i	$0.710925\pi$
$48$	−2.70436	−0.390340
$49$	−4.79518	−0.685026
$50$	25.5298	3.61046
$51$	0.484862	0.0678943
$52$	7.54541	1.04636
$53$	3.76491	0.517150	0.258575	−	0.965991i	$-0.416747\pi$
$53$	3.76491	0.517150	0.258575	+	0.965991i	$0.416747\pi$
$54$	2.12489	0.289160
$55$	10.7649	1.45154
$56$	1.62534	0.217196
$57$	−6.76491	−0.896034
$58$	−4.51514	−0.592867
$59$	2.96972	0.386625	0.193313	−	0.981137i	$-0.438077\pi$
$59$	2.96972	0.386625	0.193313	+	0.981137i	$0.438077\pi$
$60$	−10.3747	−1.33936
$61$	−11.2195	−1.43651	−0.718255	−	0.695780i	$-0.755059\pi$
$61$	−11.2195	−1.43651	−0.718255	+	0.695780i	$0.755059\pi$
$62$	8.00000	1.01600
$63$	1.48486	0.187075
$64$	−11.4537	−1.43171
$65$	−12.3747	−1.53489
$66$	−5.54541	−0.682593
$67$	7.04496	0.860678	0.430339	−	0.902667i	$-0.358394\pi$
$67$	7.04496	0.860678	0.430339	+	0.902667i	$0.358394\pi$
$68$	1.21949	0.147885
$69$	8.79518	1.05882
$70$	−13.0147	−1.55555
$71$	13.2195	1.56887	0.784433	−	0.620214i	$-0.212954\pi$
$71$	13.2195	1.56887	0.784433	+	0.620214i	$0.212954\pi$
$72$	1.09461	0.129001
$73$	−5.03028	−0.588749	−0.294375	−	0.955690i	$-0.595111\pi$
$73$	−5.03028	−0.588749	−0.294375	+	0.955690i	$0.595111\pi$
$74$	11.2195	1.30424
$75$	12.0147	1.38734
$76$	−17.0147	−1.95172
$77$	−3.87511	−0.441610
$78$	6.37466	0.721788
$79$	2.71995	0.306019	0.153009	−	0.988225i	$-0.451104\pi$
$79$	2.71995	0.306019	0.153009	+	0.988225i	$0.451104\pi$
$80$	11.1552	1.24718
$81$	1.00000	0.111111
$82$	1.56101	0.172385
$83$	−5.57947	−0.612427	−0.306213	−	0.951963i	$-0.599062\pi$
$83$	−5.57947	−0.612427	−0.306213	+	0.951963i	$0.599062\pi$
$84$	3.73463	0.407482
$85$	−2.00000	−0.216930
$86$	−2.12489	−0.229132
$87$	−2.12489	−0.227812
$88$	−2.85665	−0.304520
$89$	−18.2498	−1.93447	−0.967236	−	0.253879i	$-0.918293\pi$
$89$	−18.2498	−1.93447	−0.967236	+	0.253879i	$0.918293\pi$
$90$	−8.76491	−0.923903
$91$	4.45459	0.466967
$92$	22.1211	2.30628
$93$	3.76491	0.390403
$94$	−17.9239	−1.84870
$95$	27.9045	2.86294
$96$	−7.93567	−0.809931
$97$	6.21949	0.631494	0.315747	−	0.948843i	$-0.397745\pi$
$97$	6.21949	0.631494	0.315747	+	0.948843i	$0.397745\pi$
$98$	−10.1892	−1.02927
$99$	−2.60975	−0.262289
$100$	30.2186	3.02186
$101$	7.76491	0.772637	0.386319	−	0.922365i	$-0.373747\pi$
$101$	7.76491	0.772637	0.386319	+	0.922365i	$0.373747\pi$
$102$	1.03028	0.102013
$103$	2.54541	0.250807	0.125404	−	0.992106i	$-0.459977\pi$
$103$	2.54541	0.250807	0.125404	+	0.992106i	$0.459977\pi$
$104$	3.28383	0.322006
$105$	−6.12489	−0.597728
$106$	8.00000	0.777029
$107$	−3.15516	−0.305021	−0.152510	−	0.988302i	$-0.548736\pi$
$107$	−3.15516	−0.305021	−0.152510	+	0.988302i	$0.548736\pi$
$108$	2.51514	0.242019
$109$	−9.24977	−0.885967	−0.442984	−	0.896530i	$-0.646080\pi$
$109$	−9.24977	−0.885967	−0.442984	+	0.896530i	$0.646080\pi$
$110$	22.8742	2.18097
$111$	5.28005	0.501160
$112$	−4.01560	−0.379438
$113$	3.40493	0.320309	0.160155	−	0.987092i	$-0.448801\pi$
$113$	3.40493	0.320309	0.160155	+	0.987092i	$0.448801\pi$
$114$	−14.3747	−1.34631
$115$	−36.2791	−3.38305
$116$	−5.34438	−0.496213
$117$	3.00000	0.277350
$118$	6.31032	0.580912
$119$	0.719953	0.0659980
$120$	−4.51514	−0.412174
$121$	−4.18922	−0.380838
$122$	−23.8401	−2.15838
$123$	0.734633	0.0662396
$124$	9.46927	0.850365
$125$	−28.9348	−2.58800
$126$	3.15516	0.281084
$127$	−9.04496	−0.802610	−0.401305	−	0.915944i	$-0.631443\pi$
$127$	−9.04496	−0.802610	−0.401305	+	0.915944i	$0.631443\pi$
$128$	−8.46640	−0.748331
$129$	−1.00000	−0.0880451
$130$	−26.2947	−2.30620
$131$	19.0596	1.66525	0.832624	−	0.553839i	$-0.186838\pi$
$131$	19.0596	1.66525	0.832624	+	0.553839i	$0.186838\pi$
$132$	−6.56387	−0.571312
$133$	−10.0450	−0.871008
$134$	14.9697	1.29319
$135$	−4.12489	−0.355014
$136$	0.530734	0.0455101
$137$	−20.8742	−1.78340	−0.891702	−	0.452624i	$-0.850488\pi$
$137$	−20.8742	−1.78340	−0.891702	+	0.452624i	$0.850488\pi$
$138$	18.6888	1.59089
$139$	−13.2342	−1.12251	−0.561254	−	0.827644i	$-0.689681\pi$
$139$	−13.2342	−1.12251	−0.561254	+	0.827644i	$0.689681\pi$
$140$	−15.4049	−1.30195
$141$	−8.43521	−0.710373
$142$	28.0899	2.35725
$143$	−7.82924	−0.654714
$144$	−2.70436	−0.225363
$145$	8.76491	0.727886
$146$	−10.6888	−0.884608
$147$	−4.79518	−0.395500
$148$	13.2800	1.09161
$149$	−11.2195	−0.919137	−0.459568	−	0.888142i	$-0.651996\pi$
$149$	−11.2195	−0.919137	−0.459568	+	0.888142i	$0.651996\pi$
$150$	25.5298	2.08450
$151$	7.52982	0.612768	0.306384	−	0.951908i	$-0.400881\pi$
$151$	7.52982	0.612768	0.306384	+	0.951908i	$0.400881\pi$
$152$	−7.40493	−0.600619
$153$	0.484862	0.0391988
$154$	−8.23417	−0.663529
$155$	−15.5298	−1.24738
$156$	7.54541	0.604117
$157$	24.8099	1.98004	0.990021	−	0.140917i	$-0.0450051\pi$
$157$	24.8099	1.98004	0.990021	+	0.140917i	$0.0450051\pi$
$158$	5.77959	0.459799
$159$	3.76491	0.298577
$160$	32.7337	2.58783
$161$	13.0596	1.02924
$162$	2.12489	0.166947
$163$	−1.42431	−0.111561	−0.0557803	−	0.998443i	$-0.517765\pi$
$163$	−1.42431	−0.111561	−0.0557803	+	0.998443i	$0.517765\pi$
$164$	1.84770	0.144281
$165$	10.7649	0.838047
$166$	−11.8557	−0.920184
$167$	8.35998	0.646914	0.323457	−	0.946243i	$-0.395155\pi$
$167$	8.35998	0.646914	0.323457	+	0.946243i	$0.395155\pi$
$168$	1.62534	0.125398
$169$	−4.00000	−0.307692
$170$	−4.24977	−0.325943
$171$	−6.76491	−0.517326
$172$	−2.51514	−0.191777
$173$	12.4390	0.945719	0.472859	−	0.881138i	$-0.343222\pi$
$173$	12.4390	0.945719	0.472859	+	0.881138i	$0.343222\pi$
$174$	−4.51514	−0.342292
$175$	17.8401	1.34859
$176$	7.05769	0.531993
$177$	2.96972	0.223218
$178$	−38.7787	−2.90658
$179$	2.24977	0.168156	0.0840779	−	0.996459i	$-0.473206\pi$
$179$	2.24977	0.168156	0.0840779	+	0.996459i	$0.473206\pi$
$180$	−10.3747	−0.773281
$181$	−6.88601	−0.511833	−0.255917	−	0.966699i	$-0.582377\pi$
$181$	−6.88601	−0.511833	−0.255917	+	0.966699i	$0.582377\pi$
$182$	9.46548	0.701628
$183$	−11.2195	−0.829369
$184$	9.62729	0.709733
$185$	−21.7796	−1.60127
$186$	8.00000	0.586588
$187$	−1.26537	−0.0925328
$188$	−21.2157	−1.54731
$189$	1.48486	0.108008
$190$	59.2938	4.30163
$191$	6.47018	0.468166	0.234083	−	0.972217i	$-0.424791\pi$
$191$	6.47018	0.468166	0.234083	+	0.972217i	$0.424791\pi$
$192$	−11.4537	−0.826597
$193$	2.23509	0.160885	0.0804427	−	0.996759i	$-0.474367\pi$
$193$	2.23509	0.160885	0.0804427	+	0.996759i	$0.474367\pi$
$194$	13.2157	0.948833
$195$	−12.3747	−0.886168
$196$	−12.0606	−0.861468
$197$	7.77959	0.554273	0.277136	−	0.960831i	$-0.410615\pi$
$197$	7.77959	0.554273	0.277136	+	0.960831i	$0.410615\pi$
$198$	−5.54541	−0.394095
$199$	6.90917	0.489778	0.244889	−	0.969551i	$-0.421248\pi$
$199$	6.90917	0.489778	0.244889	+	0.969551i	$0.421248\pi$
$200$	13.1514	0.929943
$201$	7.04496	0.496913
$202$	16.4995	1.16090
$203$	−3.15516	−0.221449
$204$	1.21949	0.0853817
$205$	−3.03028	−0.211644
$206$	5.40871	0.376843
$207$	8.79518	0.611308
$208$	−8.11307	−0.562540
$209$	17.6547	1.22120
$210$	−13.0147	−0.898098
$211$	3.42431	0.235739	0.117870	−	0.993029i	$-0.462394\pi$
$211$	3.42431	0.235739	0.117870	+	0.993029i	$0.462394\pi$
$212$	9.46927	0.650352
$213$	13.2195	0.905785
$214$	−6.70436	−0.458300
$215$	4.12489	0.281315
$216$	1.09461	0.0744787
$217$	5.59037	0.379499
$218$	−19.6547	−1.33118
$219$	−5.03028	−0.339915
$220$	27.0752	1.82541
$221$	1.45459	0.0978460
$222$	11.2195	0.753003
$223$	−9.73463	−0.651879	−0.325940	−	0.945391i	$-0.605681\pi$
$223$	−9.73463	−0.651879	−0.325940	+	0.945391i	$0.605681\pi$
$224$	−11.7834	−0.787310
$225$	12.0147	0.800979
$226$	7.23509	0.481271
$227$	3.09083	0.205145	0.102573	−	0.994726i	$-0.467293\pi$
$227$	3.09083	0.205145	0.102573	+	0.994726i	$0.467293\pi$
$228$	−17.0147	−1.12682
$229$	3.20482	0.211780	0.105890	−	0.994378i	$-0.466231\pi$
$229$	3.20482	0.211780	0.105890	+	0.994378i	$0.466231\pi$
$230$	−77.0890	−5.08310
$231$	−3.87511	−0.254964
$232$	−2.32592	−0.152704
$233$	11.0946	0.726832	0.363416	−	0.931627i	$-0.381610\pi$
$233$	11.0946	0.726832	0.363416	+	0.931627i	$0.381610\pi$
$234$	6.37466	0.416724
$235$	34.7943	2.26973
$236$	7.46927	0.486208
$237$	2.71995	0.176680
$238$	1.52982	0.0991634
$239$	17.0946	1.10576	0.552879	−	0.833261i	$-0.313529\pi$
$239$	17.0946	1.10576	0.552879	+	0.833261i	$0.313529\pi$
$240$	11.1552	0.720063
$241$	22.5601	1.45322	0.726612	−	0.687048i	$-0.241094\pi$
$241$	22.5601	1.45322	0.726612	+	0.687048i	$0.241094\pi$
$242$	−8.90161	−0.572217
$243$	1.00000	0.0641500
$244$	−28.2186	−1.80651
$245$	19.7796	1.26367
$246$	1.56101	0.0995264
$247$	−20.2947	−1.29132
$248$	4.12110	0.261690
$249$	−5.57947	−0.353585
$250$	−61.4830	−3.88853
$251$	15.6400	0.987190	0.493595	−	0.869692i	$-0.335683\pi$
$251$	15.6400	0.987190	0.493595	+	0.869692i	$0.335683\pi$
$252$	3.73463	0.235260
$253$	−22.9532	−1.44306
$254$	−19.2195	−1.20594
$255$	−2.00000	−0.125245
$256$	4.91721	0.307325
$257$	−17.5336	−1.09372	−0.546858	−	0.837225i	$-0.684176\pi$
$257$	−17.5336	−1.09372	−0.546858	+	0.837225i	$0.684176\pi$
$258$	−2.12489	−0.132290
$259$	7.84014	0.487163
$260$	−31.1240	−1.93023
$261$	−2.12489	−0.131527
$262$	40.4995	2.50207
$263$	−6.37088	−0.392845	−0.196422	−	0.980519i	$-0.562932\pi$
$263$	−6.37088	−0.392845	−0.196422	+	0.980519i	$0.562932\pi$
$264$	−2.85665	−0.175815
$265$	−15.5298	−0.953989
$266$	−21.3444	−1.30871
$267$	−18.2498	−1.11687
$268$	17.7190	1.08236
$269$	29.5445	1.80136	0.900680	−	0.434483i	$-0.143069\pi$
$269$	29.5445	1.80136	0.900680	+	0.434483i	$0.143069\pi$
$270$	−8.76491	−0.533415
$271$	−11.8255	−0.718346	−0.359173	−	0.933271i	$-0.616941\pi$
$271$	−11.8255	−0.718346	−0.359173	+	0.933271i	$0.616941\pi$
$272$	−1.31124	−0.0795056
$273$	4.45459	0.269604
$274$	−44.3553	−2.67960
$275$	−31.3553	−1.89079
$276$	22.1211	1.33153
$277$	−15.7796	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$277$	−15.7796	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$278$	−28.1211	−1.68659
$279$	3.76491	0.225399
$280$	−6.70436	−0.400662
$281$	−18.0147	−1.07467	−0.537333	−	0.843370i	$-0.680568\pi$
$281$	−18.0147	−1.07467	−0.537333	+	0.843370i	$0.680568\pi$
$282$	−17.9239	−1.06735
$283$	5.64380	0.335489	0.167745	−	0.985830i	$-0.446352\pi$
$283$	5.64380	0.335489	0.167745	+	0.985830i	$0.446352\pi$
$284$	33.2489	1.97296
$285$	27.9045	1.65292
$286$	−16.6362	−0.983722
$287$	1.09083	0.0643896
$288$	−7.93567	−0.467614
$289$	−16.7649	−0.986171
$290$	18.6244	1.09366
$291$	6.21949	0.364593
$292$	−12.6518	−0.740393
$293$	1.34060	0.0783186	0.0391593	−	0.999233i	$-0.487532\pi$
$293$	1.34060	0.0783186	0.0391593	+	0.999233i	$0.487532\pi$
$294$	−10.1892	−0.594247
$295$	−12.2498	−0.713209
$296$	5.77959	0.335932
$297$	−2.60975	−0.151433
$298$	−23.8401	−1.38102
$299$	26.3856	1.52592
$300$	30.2186	1.74467
$301$	−1.48486	−0.0855860
$302$	16.0000	0.920697
$303$	7.76491	0.446082
$304$	18.2947	1.04927
$305$	46.2791	2.64994
$306$	1.03028	0.0588970
$307$	−0.424310	−0.0242166	−0.0121083	−	0.999927i	$-0.503854\pi$
$307$	−0.424310	−0.0242166	−0.0121083	+	0.999927i	$0.503854\pi$
$308$	−9.74645	−0.555355
$309$	2.54541	0.144804
$310$	−32.9991	−1.87422
$311$	0.110206	0.00624919	0.00312460	−	0.999995i	$-0.499005\pi$
$311$	0.110206	0.00624919	0.00312460	+	0.999995i	$0.499005\pi$
$312$	3.28383	0.185910
$313$	−2.96972	−0.167859	−0.0839294	−	0.996472i	$-0.526747\pi$
$313$	−2.96972	−0.167859	−0.0839294	+	0.996472i	$0.526747\pi$
$314$	52.7181	2.97506
$315$	−6.12489	−0.345098
$316$	6.84106	0.384840
$317$	−12.4243	−0.697819	−0.348909	−	0.937156i	$-0.613448\pi$
$317$	−12.4243	−0.697819	−0.348909	+	0.937156i	$0.613448\pi$
$318$	8.00000	0.448618
$319$	5.54541	0.310484
$320$	47.2451	2.64108
$321$	−3.15516	−0.176104
$322$	27.7502	1.54646
$323$	−3.28005	−0.182507
$324$	2.51514	0.139730
$325$	36.0440	1.99936
$326$	−3.02650	−0.167622
$327$	−9.24977	−0.511513
$328$	0.804136	0.0444010
$329$	−12.5251	−0.690532
$330$	22.8742	1.25918
$331$	31.2938	1.72006	0.860032	−	0.510241i	$-0.170444\pi$
$331$	31.2938	1.72006	0.860032	+	0.510241i	$0.170444\pi$
$332$	−14.0331	−0.770169
$333$	5.28005	0.289345
$334$	17.7640	0.972002
$335$	−29.0596	−1.58770
$336$	−4.01560	−0.219069
$337$	−14.8401	−0.808394	−0.404197	−	0.914672i	$-0.632449\pi$
$337$	−14.8401	−0.808394	−0.404197	+	0.914672i	$0.632449\pi$
$338$	−8.49954	−0.462314
$339$	3.40493	0.184931
$340$	−5.03028	−0.272805
$341$	−9.82546	−0.532079
$342$	−14.3747	−0.777292
$343$	−17.5142	−0.945679
$344$	−1.09461	−0.0590174
$345$	−36.2791	−1.95320
$346$	26.4314	1.42096
$347$	17.0984	0.917890	0.458945	−	0.888465i	$-0.348228\pi$
$347$	17.0984	0.917890	0.458945	+	0.888465i	$0.348228\pi$
$348$	−5.34438	−0.286489
$349$	−12.3784	−0.662603	−0.331301	−	0.943525i	$-0.607488\pi$
$349$	−12.3784	−0.662603	−0.331301	+	0.943525i	$0.607488\pi$
$350$	37.9083	2.02628
$351$	3.00000	0.160128
$352$	20.7101	1.10385
$353$	15.4546	0.822565	0.411282	−	0.911508i	$-0.365081\pi$
$353$	15.4546	0.822565	0.411282	+	0.911508i	$0.365081\pi$
$354$	6.31032	0.335390
$355$	−54.5289	−2.89409
$356$	−45.9007	−2.43273
$357$	0.719953	0.0381040
$358$	4.78051	0.252658
$359$	15.3553	0.810421	0.405210	−	0.914223i	$-0.367198\pi$
$359$	15.3553	0.810421	0.405210	+	0.914223i	$0.367198\pi$
$360$	−4.51514	−0.237969
$361$	26.7640	1.40863
$362$	−14.6320	−0.769040
$363$	−4.18922	−0.219877
$364$	11.2039	0.587244
$365$	20.7493	1.08607
$366$	−23.8401	−1.24614
$367$	−0.530734	−0.0277041	−0.0138521	−	0.999904i	$-0.504409\pi$
$367$	−0.530734	−0.0277041	−0.0138521	+	0.999904i	$0.504409\pi$
$368$	−23.7853	−1.23990
$369$	0.734633	0.0382435
$370$	−46.2791	−2.40594
$371$	5.59037	0.290238
$372$	9.46927	0.490959
$373$	−6.02936	−0.312188	−0.156094	−	0.987742i	$-0.549890\pi$
$373$	−6.02936	−0.312188	−0.156094	+	0.987742i	$0.549890\pi$
$374$	−2.68876	−0.139032
$375$	−28.9348	−1.49418
$376$	−9.23326	−0.476169
$377$	−6.37466	−0.328312
$378$	3.15516	0.162284
$379$	11.8936	0.610932	0.305466	−	0.952203i	$-0.401188\pi$
$379$	11.8936	0.610932	0.305466	+	0.952203i	$0.401188\pi$
$380$	70.1836	3.60034
$381$	−9.04496	−0.463387
$382$	13.7484	0.703429
$383$	17.7796	0.908495	0.454247	−	0.890876i	$-0.349908\pi$
$383$	17.7796	0.908495	0.454247	+	0.890876i	$0.349908\pi$
$384$	−8.46640	−0.432049
$385$	15.9844	0.814641
$386$	4.74931	0.241734
$387$	−1.00000	−0.0508329
$388$	15.6429	0.794147
$389$	−24.4646	−1.24040	−0.620201	−	0.784443i	$-0.712949\pi$
$389$	−24.4646	−1.24040	−0.620201	+	0.784443i	$0.712949\pi$
$390$	−26.2947	−1.33149
$391$	4.26445	0.215663
$392$	−5.24885	−0.265107
$393$	19.0596	0.961431
$394$	16.5307	0.832806
$395$	−11.2195	−0.564514
$396$	−6.56387	−0.329847
$397$	−38.5739	−1.93597	−0.967983	−	0.251015i	$-0.919236\pi$
$397$	−38.5739	−1.93597	−0.967983	+	0.251015i	$0.919236\pi$
$398$	14.6812	0.735902
$399$	−10.0450	−0.502877
$400$	−32.4920	−1.62460
$401$	−4.51422	−0.225429	−0.112715	−	0.993627i	$-0.535955\pi$
$401$	−4.51422	−0.225429	−0.112715	+	0.993627i	$0.535955\pi$
$402$	14.9697	0.746622
$403$	11.2947	0.562630
$404$	19.5298	0.971645
$405$	−4.12489	−0.204967
$406$	−6.70436	−0.332732
$407$	−13.7796	−0.683029
$408$	0.530734	0.0262753
$409$	17.7190	0.876150	0.438075	−	0.898938i	$-0.355660\pi$
$409$	17.7190	0.876150	0.438075	+	0.898938i	$0.355660\pi$
$410$	−6.43899	−0.317999
$411$	−20.8742	−1.02965
$412$	6.40207	0.315407
$413$	4.40963	0.216984
$414$	18.6888	0.918502
$415$	23.0147	1.12975
$416$	−23.8070	−1.16723
$417$	−13.2342	−0.648080
$418$	37.5142	1.83488
$419$	16.4390	0.803097	0.401549	−	0.915838i	$-0.368472\pi$
$419$	16.4390	0.803097	0.401549	+	0.915838i	$0.368472\pi$
$420$	−15.4049	−0.751684
$421$	23.9083	1.16522	0.582609	−	0.812753i	$-0.302032\pi$
$421$	23.9083	1.16522	0.582609	+	0.812753i	$0.302032\pi$
$422$	7.27627	0.354203
$423$	−8.43521	−0.410134
$424$	4.12110	0.200139
$425$	5.82546	0.282576
$426$	28.0899	1.36096
$427$	−16.6594	−0.806205
$428$	−7.93567	−0.383585
$429$	−7.82924	−0.377999
$430$	8.76491	0.422681
$431$	−15.6088	−0.751851	−0.375925	−	0.926650i	$-0.622675\pi$
$431$	−15.6088	−0.751851	−0.375925	+	0.926650i	$0.622675\pi$
$432$	−2.70436	−0.130113
$433$	−10.9092	−0.524261	−0.262131	−	0.965032i	$-0.584425\pi$
$433$	−10.9092	−0.524261	−0.262131	+	0.965032i	$0.584425\pi$
$434$	11.8789	0.570205
$435$	8.76491	0.420245
$436$	−23.2645	−1.11417
$437$	−59.4986	−2.84621
$438$	−10.6888	−0.510729
$439$	19.8255	0.946218	0.473109	−	0.881004i	$-0.343132\pi$
$439$	19.8255	0.946218	0.473109	+	0.881004i	$0.343132\pi$
$440$	11.7834	0.561750
$441$	−4.79518	−0.228342
$442$	3.09083	0.147016
$443$	−0.314104	−0.0149235	−0.00746177	−	0.999972i	$-0.502375\pi$
$443$	−0.314104	−0.0149235	−0.00746177	+	0.999972i	$0.502375\pi$
$444$	13.2800	0.630243
$445$	75.2782	3.56853
$446$	−20.6850	−0.979462
$447$	−11.2195	−0.530664
$448$	−17.0071	−0.803511
$449$	10.2536	0.483895	0.241948	−	0.970289i	$-0.422214\pi$
$449$	10.2536	0.483895	0.241948	+	0.970289i	$0.422214\pi$
$450$	25.5298	1.20349
$451$	−1.91721	−0.0902777
$452$	8.56387	0.402811
$453$	7.52982	0.353782
$454$	6.56766	0.308235
$455$	−18.3747	−0.861417
$456$	−7.40493	−0.346768
$457$	3.21949	0.150602	0.0753008	−	0.997161i	$-0.476008\pi$
$457$	3.21949	0.150602	0.0753008	+	0.997161i	$0.476008\pi$
$458$	6.80986	0.318204
$459$	0.484862	0.0226314
$460$	−91.2470	−4.25441
$461$	−16.1892	−0.754007	−0.377004	−	0.926212i	$-0.623046\pi$
$461$	−16.1892	−0.754007	−0.377004	+	0.926212i	$0.623046\pi$
$462$	−8.23417	−0.383088
$463$	14.8860	0.691812	0.345906	−	0.938269i	$-0.387572\pi$
$463$	14.8860	0.691812	0.345906	+	0.938269i	$0.387572\pi$
$464$	5.74645	0.266772
$465$	−15.5298	−0.720178
$466$	23.5748	1.09208
$467$	−8.22041	−0.380395	−0.190198	−	0.981746i	$-0.560913\pi$
$467$	−8.22041	−0.380395	−0.190198	+	0.981746i	$0.560913\pi$
$468$	7.54541	0.348787
$469$	10.4608	0.483034
$470$	73.9338	3.41031
$471$	24.8099	1.14318
$472$	3.25069	0.149625
$473$	2.60975	0.119996
$474$	5.77959	0.265465
$475$	−81.2782	−3.72930
$476$	1.81078	0.0829970
$477$	3.76491	0.172383
$478$	36.3241	1.66143
$479$	9.41961	0.430393	0.215197	−	0.976571i	$-0.430961\pi$
$479$	9.41961	0.430393	0.215197	+	0.976571i	$0.430961\pi$
$480$	32.7337	1.49408
$481$	15.8401	0.722248
$482$	47.9376	2.18350
$483$	13.0596	0.594234
$484$	−10.5365	−0.478930
$485$	−25.6547	−1.16492
$486$	2.12489	0.0963868
$487$	−6.71995	−0.304510	−0.152255	−	0.988341i	$-0.548654\pi$
$487$	−6.71995	−0.304510	−0.152255	+	0.988341i	$0.548654\pi$
$488$	−12.2810	−0.555933
$489$	−1.42431	−0.0644095
$490$	42.0294	1.89869
$491$	−36.2186	−1.63452	−0.817261	−	0.576268i	$-0.804508\pi$
$491$	−36.2186	−1.63452	−0.817261	+	0.576268i	$0.804508\pi$
$492$	1.84770	0.0833009
$493$	−1.03028	−0.0464013
$494$	−43.1240	−1.94024
$495$	10.7649	0.483847
$496$	−10.1817	−0.457170
$497$	19.6291	0.880487
$498$	−11.8557	−0.531268
$499$	−29.8245	−1.33513	−0.667565	−	0.744552i	$-0.732663\pi$
$499$	−29.8245	−1.33513	−0.667565	+	0.744552i	$0.732663\pi$
$500$	−72.7749	−3.25459
$501$	8.35998	0.373496
$502$	33.2333	1.48327
$503$	−22.2498	−0.992068	−0.496034	−	0.868303i	$-0.665211\pi$
$503$	−22.2498	−0.992068	−0.496034	+	0.868303i	$0.665211\pi$
$504$	1.62534	0.0723986
$505$	−32.0294	−1.42529
$506$	−48.7729	−2.16822
$507$	−4.00000	−0.177646
$508$	−22.7493	−1.00934
$509$	1.26537	0.0560864	0.0280432	−	0.999607i	$-0.491072\pi$
$509$	1.26537	0.0560864	0.0280432	+	0.999607i	$0.491072\pi$
$510$	−4.24977	−0.188183
$511$	−7.46927	−0.330421
$512$	27.3813	1.21009
$513$	−6.76491	−0.298678
$514$	−37.2569	−1.64333
$515$	−10.4995	−0.462665
$516$	−2.51514	−0.110723
$517$	22.0138	0.968164
$518$	16.6594	0.731972
$519$	12.4390	0.546011
$520$	−13.5454	−0.594006
$521$	21.4343	0.939053	0.469527	−	0.882918i	$-0.344425\pi$
$521$	21.4343	0.939053	0.469527	+	0.882918i	$0.344425\pi$
$522$	−4.51514	−0.197622
$523$	5.11399	0.223619	0.111810	−	0.993730i	$-0.464335\pi$
$523$	5.11399	0.223619	0.111810	+	0.993730i	$0.464335\pi$
$524$	47.9376	2.09416
$525$	17.8401	0.778608
$526$	−13.5374	−0.590258
$527$	1.82546	0.0795183
$528$	7.05769	0.307146
$529$	54.3553	2.36327
$530$	−32.9991	−1.43339
$531$	2.96972	0.128875
$532$	−25.2645	−1.09535
$533$	2.20390	0.0954614
$534$	−38.7787	−1.67812
$535$	13.0147	0.562674
$536$	7.71147	0.333085
$537$	2.24977	0.0970848
$538$	62.7787	2.70658
$539$	12.5142	0.539026
$540$	−10.3747	−0.446454
$541$	−44.2333	−1.90174	−0.950868	−	0.309596i	$-0.899806\pi$
$541$	−44.2333	−1.90174	−0.950868	+	0.309596i	$0.899806\pi$
$542$	−25.1277	−1.07933
$543$	−6.88601	−0.295507
$544$	−3.84770	−0.164969
$545$	38.1542	1.63435
$546$	9.46548	0.405085
$547$	31.0743	1.32864	0.664321	−	0.747447i	$-0.268721\pi$
$547$	31.0743	1.32864	0.664321	+	0.747447i	$0.268721\pi$
$548$	−52.5015	−2.24275
$549$	−11.2195	−0.478836
$550$	−66.6264	−2.84096
$551$	14.3747	0.612381
$552$	9.62729	0.409765
$553$	4.03875	0.171745
$554$	−33.5298	−1.42455
$555$	−21.7796	−0.924492
$556$	−33.2858	−1.41163
$557$	−20.7952	−0.881120	−0.440560	−	0.897723i	$-0.645220\pi$
$557$	−20.7952	−0.881120	−0.440560	+	0.897723i	$0.645220\pi$
$558$	8.00000	0.338667
$559$	−3.00000	−0.126886
$560$	16.5639	0.699951
$561$	−1.26537	−0.0534238
$562$	−38.2791	−1.61471
$563$	4.82454	0.203330	0.101665	−	0.994819i	$-0.467583\pi$
$563$	4.82454	0.203330	0.101665	+	0.994819i	$0.467583\pi$
$564$	−21.2157	−0.893343
$565$	−14.0450	−0.590876
$566$	11.9924	0.504080
$567$	1.48486	0.0623583
$568$	14.4702	0.607155
$569$	25.0450	1.04994	0.524970	−	0.851121i	$-0.324077\pi$
$569$	25.0450	1.04994	0.524970	+	0.851121i	$0.324077\pi$
$570$	59.2938	2.48354
$571$	15.4849	0.648021	0.324011	−	0.946053i	$-0.394969\pi$
$571$	15.4849	0.648021	0.324011	+	0.946053i	$0.394969\pi$
$572$	−19.6916	−0.823348
$573$	6.47018	0.270296
$574$	2.31789	0.0967467
$575$	105.671	4.40680
$576$	−11.4537	−0.477236
$577$	−18.8704	−0.785586	−0.392793	−	0.919627i	$-0.628491\pi$
$577$	−18.8704	−0.785586	−0.392793	+	0.919627i	$0.628491\pi$
$578$	−35.6235	−1.48174
$579$	2.23509	0.0928872
$580$	22.0450	0.915367
$581$	−8.28474	−0.343709
$582$	13.2157	0.547809
$583$	−9.82546	−0.406929
$584$	−5.50619	−0.227848
$585$	−12.3747	−0.511629
$586$	2.84862	0.117675
$587$	−37.1883	−1.53493	−0.767463	−	0.641094i	$-0.778481\pi$
$587$	−37.1883	−1.53493	−0.767463	+	0.641094i	$0.778481\pi$
$588$	−12.0606	−0.497369
$589$	−25.4693	−1.04944
$590$	−26.0294	−1.07161
$591$	7.77959	0.320010
$592$	−14.2791	−0.586869
$593$	9.87133	0.405367	0.202684	−	0.979244i	$-0.435034\pi$
$593$	9.87133	0.405367	0.202684	+	0.979244i	$0.435034\pi$
$594$	−5.54541	−0.227531
$595$	−2.96972	−0.121747
$596$	−28.2186	−1.15588
$597$	6.90917	0.282774
$598$	56.0663	2.29272
$599$	−41.2598	−1.68583	−0.842914	−	0.538048i	$-0.819162\pi$
$599$	−41.2598	−1.68583	−0.842914	+	0.538048i	$0.819162\pi$
$600$	13.1514	0.536903
$601$	−44.3397	−1.80865	−0.904327	−	0.426841i	$-0.859626\pi$
$601$	−44.3397	−1.80865	−0.904327	+	0.426841i	$0.859626\pi$
$602$	−3.15516	−0.128595
$603$	7.04496	0.286893
$604$	18.9385	0.770598
$605$	17.2800	0.702534
$606$	16.4995	0.670248
$607$	−5.76399	−0.233953	−0.116977	−	0.993135i	$-0.537320\pi$
$607$	−5.76399	−0.233953	−0.116977	+	0.993135i	$0.537320\pi$
$608$	53.6841	2.17718
$609$	−3.15516	−0.127854
$610$	98.3378	3.98158
$611$	−25.3056	−1.02376
$612$	1.21949	0.0492952
$613$	26.8780	1.08559	0.542796	−	0.839865i	$-0.317366\pi$
$613$	26.8780	1.08559	0.542796	+	0.839865i	$0.317366\pi$
$614$	−0.901610	−0.0363860
$615$	−3.03028	−0.122193
$616$	−4.24174	−0.170904
$617$	18.3250	0.737737	0.368868	−	0.929482i	$-0.379745\pi$
$617$	18.3250	0.737737	0.368868	+	0.929482i	$0.379745\pi$
$618$	5.40871	0.217570
$619$	−26.9697	−1.08400	−0.542002	−	0.840377i	$-0.682334\pi$
$619$	−26.9697	−1.08400	−0.542002	+	0.840377i	$0.682334\pi$
$620$	−39.0596	−1.56867
$621$	8.79518	0.352939
$622$	0.234174	0.00938954
$623$	−27.0984	−1.08567
$624$	−8.11307	−0.324783
$625$	59.2791	2.37117
$626$	−6.31032	−0.252211
$627$	17.6547	0.705061
$628$	62.4002	2.49004
$629$	2.56009	0.102078
$630$	−13.0147	−0.518517
$631$	15.1514	0.603167	0.301583	−	0.953440i	$-0.402485\pi$
$631$	15.1514	0.603167	0.301583	+	0.953440i	$0.402485\pi$
$632$	2.97729	0.118430
$633$	3.42431	0.136104
$634$	−26.4002	−1.04849
$635$	37.3094	1.48058
$636$	9.46927	0.375481
$637$	−14.3856	−0.569976
$638$	11.7834	0.466508
$639$	13.2195	0.522955
$640$	34.9229	1.38045
$641$	−4.06811	−0.160681	−0.0803404	−	0.996767i	$-0.525601\pi$
$641$	−4.06811	−0.160681	−0.0803404	+	0.996767i	$0.525601\pi$
$642$	−6.70436	−0.264600
$643$	12.6594	0.499238	0.249619	−	0.968344i	$-0.419695\pi$
$643$	12.6594	0.499238	0.249619	+	0.968344i	$0.419695\pi$
$644$	32.8468	1.29434
$645$	4.12489	0.162417
$646$	−6.96972	−0.274220
$647$	0.128666	0.00505840	0.00252920	−	0.999997i	$-0.499195\pi$
$647$	0.128666	0.00505840	0.00252920	+	0.999997i	$0.499195\pi$
$648$	1.09461	0.0430003
$649$	−7.75023	−0.304223
$650$	76.5895	3.00409
$651$	5.59037	0.219104
$652$	−3.58234	−0.140295
$653$	−37.7834	−1.47858	−0.739289	−	0.673389i	$-0.764838\pi$
$653$	−37.7834	−1.47858	−0.739289	+	0.673389i	$0.764838\pi$
$654$	−19.6547	−0.768560
$655$	−78.6188	−3.07189
$656$	−1.98671	−0.0775680
$657$	−5.03028	−0.196250
$658$	−26.6144	−1.03754
$659$	−29.9154	−1.16534	−0.582669	−	0.812710i	$-0.697991\pi$
$659$	−29.9154	−1.16534	−0.582669	+	0.812710i	$0.697991\pi$
$660$	27.0752	1.05390
$661$	20.7796	0.808232	0.404116	−	0.914708i	$-0.367579\pi$
$661$	20.7796	0.808232	0.404116	+	0.914708i	$0.367579\pi$
$662$	66.4958	2.58443
$663$	1.45459	0.0564914
$664$	−6.10734	−0.237011
$665$	41.4343	1.60675
$666$	11.2195	0.434747
$667$	−18.6888	−0.723632
$668$	21.0265	0.813540
$669$	−9.73463	−0.376363
$670$	−61.7484	−2.38555
$671$	29.2800	1.13034
$672$	−11.7834	−0.454553
$673$	−48.7181	−1.87795	−0.938973	−	0.343992i	$-0.888221\pi$
$673$	−48.7181	−1.87795	−0.938973	+	0.343992i	$0.888221\pi$
$674$	−31.5336	−1.21463
$675$	12.0147	0.462445
$676$	−10.0606	−0.386944
$677$	−46.2498	−1.77752	−0.888762	−	0.458370i	$-0.848434\pi$
$677$	−46.2498	−1.77752	−0.888762	+	0.458370i	$0.848434\pi$
$678$	7.23509	0.277862
$679$	9.23509	0.354410
$680$	−2.18922	−0.0839527
$681$	3.09083	0.118441
$682$	−20.8780	−0.799459
$683$	4.17454	0.159734	0.0798671	−	0.996806i	$-0.474550\pi$
$683$	4.17454	0.159734	0.0798671	+	0.996806i	$0.474550\pi$
$684$	−17.0147	−0.650573
$685$	86.1037	3.28985
$686$	−37.2157	−1.42090
$687$	3.20482	0.122271
$688$	2.70436	0.103103
$689$	11.2947	0.430295
$690$	−77.0890	−2.93473
$691$	−25.5904	−0.973504	−0.486752	−	0.873540i	$-0.661818\pi$
$691$	−25.5904	−0.973504	−0.486752	+	0.873540i	$0.661818\pi$
$692$	31.2858	1.18931
$693$	−3.87511	−0.147203
$694$	36.3321	1.37915
$695$	54.5895	2.07070
$696$	−2.32592	−0.0881637
$697$	0.356195	0.0134919
$698$	−26.3028	−0.995574
$699$	11.0946	0.419637
$700$	44.8704	1.69594
$701$	−15.0303	−0.567686	−0.283843	−	0.958871i	$-0.591609\pi$
$701$	−15.0303	−0.567686	−0.283843	+	0.958871i	$0.591609\pi$
$702$	6.37466	0.240596
$703$	−35.7190	−1.34717
$704$	29.8912	1.12657
$705$	34.7943	1.31043
$706$	32.8392	1.23592
$707$	11.5298	0.433623
$708$	7.46927	0.280712
$709$	3.12110	0.117216	0.0586078	−	0.998281i	$-0.481334\pi$
$709$	3.12110	0.117216	0.0586078	+	0.998281i	$0.481334\pi$
$710$	−115.868	−4.34844
$711$	2.71995	0.102006
$712$	−19.9764	−0.748646
$713$	33.1131	1.24009
$714$	1.52982	0.0572520
$715$	32.2947	1.20775
$716$	5.65848	0.211467
$717$	17.0946	0.638410
$718$	32.6282	1.21767
$719$	−12.5895	−0.469507	−0.234754	−	0.972055i	$-0.575428\pi$
$719$	−12.5895	−0.469507	−0.234754	+	0.972055i	$0.575428\pi$
$720$	11.1552	0.415728
$721$	3.77959	0.140759
$722$	56.8704	2.11650
$723$	22.5601	0.839019
$724$	−17.3193	−0.643666
$725$	−25.5298	−0.948154
$726$	−8.90161	−0.330370
$727$	19.4537	0.721497	0.360748	−	0.932663i	$-0.382521\pi$
$727$	19.4537	0.721497	0.360748	+	0.932663i	$0.382521\pi$
$728$	4.87603	0.180718
$729$	1.00000	0.0370370
$730$	44.0899	1.63184
$731$	−0.484862	−0.0179333
$732$	−28.2186	−1.04299
$733$	25.9394	0.958095	0.479048	−	0.877789i	$-0.340982\pi$
$733$	25.9394	0.958095	0.479048	+	0.877789i	$0.340982\pi$
$734$	−1.12775	−0.0416260
$735$	19.7796	0.729581
$736$	−69.7957	−2.57270
$737$	−18.3856	−0.677241
$738$	1.56101	0.0574616
$739$	−49.4225	−1.81804	−0.909018	−	0.416758i	$-0.863166\pi$
$739$	−49.4225	−1.81804	−0.909018	+	0.416758i	$0.863166\pi$
$740$	−54.7787	−2.01370
$741$	−20.2947	−0.745545
$742$	11.8789	0.436088
$743$	−30.5677	−1.12142	−0.560709	−	0.828013i	$-0.689471\pi$
$743$	−30.5677	−1.12142	−0.560709	+	0.828013i	$0.689471\pi$
$744$	4.12110	0.151087
$745$	46.2791	1.69554
$746$	−12.8117	−0.469070
$747$	−5.57947	−0.204142
$748$	−3.18257	−0.116366
$749$	−4.68498	−0.171185
$750$	−61.4830	−2.24504
$751$	2.73555	0.0998216	0.0499108	−	0.998754i	$-0.484106\pi$
$751$	2.73555	0.0998216	0.0499108	+	0.998754i	$0.484106\pi$
$752$	22.8118	0.831861
$753$	15.6400	0.569954
$754$	−13.5454	−0.493295
$755$	−31.0596	−1.13038
$756$	3.73463	0.135827
$757$	−2.68876	−0.0977247	−0.0488623	−	0.998806i	$-0.515560\pi$
$757$	−2.68876	−0.0977247	−0.0488623	+	0.998806i	$0.515560\pi$
$758$	25.2725	0.917938
$759$	−22.9532	−0.833149
$760$	30.5445	1.10797
$761$	−42.4002	−1.53701	−0.768504	−	0.639845i	$-0.778998\pi$
$761$	−42.4002	−1.53701	−0.768504	+	0.639845i	$0.778998\pi$
$762$	−19.2195	−0.696249
$763$	−13.7346	−0.497227
$764$	16.2734	0.588751
$765$	−2.00000	−0.0723102
$766$	37.7796	1.36503
$767$	8.90917	0.321692
$768$	4.91721	0.177434
$769$	−35.1046	−1.26590	−0.632952	−	0.774191i	$-0.718157\pi$
$769$	−35.1046	−1.26590	−0.632952	+	0.774191i	$0.718157\pi$
$770$	33.9650	1.22401
$771$	−17.5336	−0.631457
$772$	5.62156	0.202324
$773$	−15.0946	−0.542915	−0.271458	−	0.962450i	$-0.587506\pi$
$773$	−15.0946	−0.542915	−0.271458	+	0.962450i	$0.587506\pi$
$774$	−2.12489	−0.0763774
$775$	45.2342	1.62486
$776$	6.80792	0.244390
$777$	7.84014	0.281263
$778$	−51.9844	−1.86373
$779$	−4.96972	−0.178059
$780$	−31.1240	−1.11442
$781$	−34.4995	−1.23449
$782$	9.06147	0.324037
$783$	−2.12489	−0.0759372
$784$	12.9679	0.463139
$785$	−102.338	−3.65259
$786$	40.4995	1.44457
$787$	41.1202	1.46578	0.732888	−	0.680349i	$-0.238172\pi$
$787$	41.1202	1.46578	0.732888	+	0.680349i	$0.238172\pi$
$788$	19.5667	0.697036
$789$	−6.37088	−0.226809
$790$	−23.8401	−0.848194
$791$	5.05585	0.179765
$792$	−2.85665	−0.101507
$793$	−33.6585	−1.19525
$794$	−81.9650	−2.90883
$795$	−15.5298	−0.550786
$796$	17.3775	0.615930
$797$	2.78807	0.0987584	0.0493792	−	0.998780i	$-0.484276\pi$
$797$	2.78807	0.0987584	0.0493792	+	0.998780i	$0.484276\pi$
$798$	−21.3444	−0.755583
$799$	−4.08991	−0.144691
$800$	−95.3445	−3.37094
$801$	−18.2498	−0.644824
$802$	−9.59220	−0.338712
$803$	13.1277	0.463268
$804$	17.7190	0.624902
$805$	−53.8695	−1.89865
$806$	24.0000	0.845364
$807$	29.5445	1.04002
$808$	8.49954	0.299013
$809$	−8.27913	−0.291079	−0.145539	−	0.989352i	$-0.546492\pi$
$809$	−8.27913	−0.291079	−0.145539	+	0.989352i	$0.546492\pi$
$810$	−8.76491	−0.307968
$811$	47.1807	1.65674	0.828370	−	0.560181i	$-0.189269\pi$
$811$	47.1807	1.65674	0.828370	+	0.560181i	$0.189269\pi$
$812$	−7.93567	−0.278487
$813$	−11.8255	−0.414737
$814$	−29.2800	−1.02627
$815$	5.87511	0.205796
$816$	−1.31124	−0.0459026
$817$	6.76491	0.236674
$818$	37.6509	1.31643
$819$	4.45459	0.155656
$820$	−7.62156	−0.266157
$821$	38.7640	1.35287	0.676436	−	0.736501i	$-0.263523\pi$
$821$	38.7640	1.35287	0.676436	+	0.736501i	$0.263523\pi$
$822$	−44.3553	−1.54707
$823$	36.1433	1.25988	0.629939	−	0.776645i	$-0.283080\pi$
$823$	36.1433	1.25988	0.629939	+	0.776645i	$0.283080\pi$
$824$	2.78623	0.0970631
$825$	−31.3553	−1.09165
$826$	9.36996	0.326023
$827$	30.4958	1.06044	0.530221	−	0.847860i	$-0.322109\pi$
$827$	30.4958	1.06044	0.530221	+	0.847860i	$0.322109\pi$
$828$	22.1211	0.768762
$829$	38.5601	1.33925	0.669624	−	0.742701i	$-0.266455\pi$
$829$	38.5601	1.33925	0.669624	+	0.742701i	$0.266455\pi$
$830$	48.9036	1.69747
$831$	−15.7796	−0.547388
$832$	−34.3610	−1.19125
$833$	−2.32500	−0.0805566
$834$	−28.1211	−0.973754
$835$	−34.4839	−1.19337
$836$	44.4040	1.53574
$837$	3.76491	0.130134
$838$	34.9310	1.20667
$839$	−44.2791	−1.52869	−0.764343	−	0.644810i	$-0.776936\pi$
$839$	−44.2791	−1.52869	−0.764343	+	0.644810i	$0.776936\pi$
$840$	−6.70436	−0.231322
$841$	−24.4849	−0.844306
$842$	50.8023	1.75076
$843$	−18.0147	−0.620459
$844$	8.61261	0.296458
$845$	16.4995	0.567601
$846$	−17.9239	−0.616235
$847$	−6.22041	−0.213736
$848$	−10.1817	−0.349640
$849$	5.64380	0.193695
$850$	12.3784	0.424577
$851$	46.4390	1.59191
$852$	33.2489	1.13909
$853$	29.2716	1.00224	0.501120	−	0.865378i	$-0.332922\pi$
$853$	29.2716	1.00224	0.501120	+	0.865378i	$0.332922\pi$
$854$	−35.3993	−1.21134
$855$	27.9045	0.954313
$856$	−3.45367	−0.118044
$857$	6.37088	0.217625	0.108812	−	0.994062i	$-0.465295\pi$
$857$	6.37088	0.217625	0.108812	+	0.994062i	$0.465295\pi$
$858$	−16.6362	−0.567952
$859$	32.4839	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$859$	32.4839	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$860$	10.3747	0.353773
$861$	1.09083	0.0371753
$862$	−33.1670	−1.12967
$863$	32.2791	1.09879	0.549397	−	0.835561i	$-0.314857\pi$
$863$	32.2791	1.09879	0.549397	+	0.835561i	$0.314857\pi$
$864$	−7.93567	−0.269977
$865$	−51.3094	−1.74457
$866$	−23.1807	−0.787714
$867$	−16.7649	−0.569366
$868$	14.0606	0.477246
$869$	−7.09839	−0.240796
$870$	18.6244	0.631427
$871$	21.1349	0.716128
$872$	−10.1249	−0.342872
$873$	6.21949	0.210498
$874$	−126.428	−4.27648
$875$	−42.9641	−1.45245
$876$	−12.6518	−0.427466
$877$	−0.674081	−0.0227621	−0.0113810	−	0.999935i	$-0.503623\pi$
$877$	−0.674081	−0.0227621	−0.0113810	+	0.999935i	$0.503623\pi$
$878$	42.1268	1.42171
$879$	1.34060	0.0452173
$880$	−29.1122	−0.981370
$881$	1.01560	0.0342163	0.0171082	−	0.999854i	$-0.494554\pi$
$881$	1.01560	0.0342163	0.0171082	+	0.999854i	$0.494554\pi$
$882$	−10.1892	−0.343089
$883$	−31.6732	−1.06589	−0.532943	−	0.846151i	$-0.678914\pi$
$883$	−31.6732	−1.06589	−0.532943	+	0.846151i	$0.678914\pi$
$884$	3.65848	0.123048
$885$	−12.2498	−0.411772
$886$	−0.667435	−0.0224229
$887$	40.8686	1.37223	0.686116	−	0.727492i	$-0.259314\pi$
$887$	40.8686	1.37223	0.686116	+	0.727492i	$0.259314\pi$
$888$	5.77959	0.193950
$889$	−13.4305	−0.450445
$890$	159.958	5.36179
$891$	−2.60975	−0.0874298
$892$	−24.4839	−0.819783
$893$	57.0634	1.90955
$894$	−23.8401	−0.797334
$895$	−9.28005	−0.310198
$896$	−12.5714	−0.419982
$897$	26.3856	0.880988
$898$	21.7876	0.727062
$899$	−8.00000	−0.266815
$900$	30.2186	1.00729
$901$	1.82546	0.0608149
$902$	−4.07384	−0.135644
$903$	−1.48486	−0.0494131
$904$	3.72707	0.123961
$905$	28.4040	0.944181
$906$	16.0000	0.531564
$907$	−16.0828	−0.534020	−0.267010	−	0.963694i	$-0.586036\pi$
$907$	−16.0828	−0.534020	−0.267010	+	0.963694i	$0.586036\pi$
$908$	7.77386	0.257985
$909$	7.76491	0.257546
$910$	−39.0440	−1.29430
$911$	37.1277	1.23010	0.615049	−	0.788489i	$-0.289136\pi$
$911$	37.1277	1.23010	0.615049	+	0.788489i	$0.289136\pi$
$912$	18.2947	0.605799
$913$	14.5610	0.481899
$914$	6.84106	0.226282
$915$	46.2791	1.52994
$916$	8.06055	0.266328
$917$	28.3009	0.934579
$918$	1.03028	0.0340042
$919$	27.3940	0.903646	0.451823	−	0.892108i	$-0.350774\pi$
$919$	27.3940	0.903646	0.451823	+	0.892108i	$0.350774\pi$
$920$	−39.7115	−1.30925
$921$	−0.424310	−0.0139815
$922$	−34.4002	−1.13291
$923$	39.6585	1.30537
$924$	−9.74645	−0.320635
$925$	63.4381	2.08583
$926$	31.6311	1.03946
$927$	2.54541	0.0836024
$928$	16.8624	0.553535
$929$	30.5639	1.00277	0.501384	−	0.865225i	$-0.332824\pi$
$929$	30.5639	1.00277	0.501384	+	0.865225i	$0.332824\pi$
$930$	−32.9991	−1.08208
$931$	32.4390	1.06314
$932$	27.9045	0.914041
$933$	0.110206	0.00360797
$934$	−17.4674	−0.571552
$935$	5.21949	0.170696
$936$	3.28383	0.107335
$937$	−20.4702	−0.668732	−0.334366	−	0.942443i	$-0.608522\pi$
$937$	−20.4702	−0.668732	−0.334366	+	0.942443i	$0.608522\pi$
$938$	22.2280	0.725769
$939$	−2.96972	−0.0969133
$940$	87.5124	2.85434
$941$	1.89358	0.0617288	0.0308644	−	0.999524i	$-0.490174\pi$
$941$	1.89358	0.0617288	0.0308644	+	0.999524i	$0.490174\pi$
$942$	52.7181	1.71765
$943$	6.46123	0.210407
$944$	−8.03119	−0.261393
$945$	−6.12489	−0.199243
$946$	5.54541	0.180297
$947$	−33.1311	−1.07662	−0.538308	−	0.842748i	$-0.680936\pi$
$947$	−33.1311	−1.07662	−0.538308	+	0.842748i	$0.680936\pi$
$948$	6.84106	0.222187
$949$	−15.0908	−0.489869
$950$	−172.707	−5.60335
$951$	−12.4243	−0.402886
$952$	0.788067	0.0255414
$953$	−29.5942	−0.958649	−0.479324	−	0.877638i	$-0.659118\pi$
$953$	−29.5942	−0.958649	−0.479324	+	0.877638i	$0.659118\pi$
$954$	8.00000	0.259010
$955$	−26.6888	−0.863628
$956$	42.9953	1.39057
$957$	5.54541	0.179258
$958$	20.0156	0.646675
$959$	−30.9953	−1.00089
$960$	47.2451	1.52483
$961$	−16.8255	−0.542757
$962$	33.6585	1.08519
$963$	−3.15516	−0.101674
$964$	56.7418	1.82753
$965$	−9.21949	−0.296786
$966$	27.7502	0.892849
$967$	15.4234	0.495983	0.247991	−	0.968762i	$-0.420230\pi$
$967$	15.4234	0.495983	0.247991	+	0.968762i	$0.420230\pi$
$968$	−4.58556	−0.147385
$969$	−3.28005	−0.105370
$970$	−54.5133	−1.75032
$971$	−25.6438	−0.822949	−0.411474	−	0.911421i	$-0.634986\pi$
$971$	−25.6438	−0.822949	−0.411474	+	0.911421i	$0.634986\pi$
$972$	2.51514	0.0806731
$973$	−19.6509	−0.629980
$974$	−14.2791	−0.457533
$975$	36.0440	1.15433
$976$	30.3415	0.971208
$977$	11.1202	0.355766	0.177883	−	0.984052i	$-0.443075\pi$
$977$	11.1202	0.355766	0.177883	+	0.984052i	$0.443075\pi$
$978$	−3.02650	−0.0967766
$979$	47.6273	1.52217
$980$	49.7484	1.58915
$981$	−9.24977	−0.295322
$982$	−76.9603	−2.45590
$983$	−36.9697	−1.17915	−0.589576	−	0.807713i	$-0.700705\pi$
$983$	−36.9697	−1.17915	−0.589576	+	0.807713i	$0.700705\pi$
$984$	0.804136	0.0256349
$985$	−32.0899	−1.02247
$986$	−2.18922	−0.0697189
$987$	−12.5251	−0.398679
$988$	−51.0440	−1.62393
$989$	−8.79518	−0.279671
$990$	22.8742	0.726990
$991$	9.70527	0.308298	0.154149	−	0.988048i	$-0.450736\pi$
$991$	9.70527	0.308298	0.154149	+	0.988048i	$0.450736\pi$
$992$	−29.8771	−0.948598
$993$	31.2938	0.993079
$994$	41.7096	1.32295
$995$	−28.4995	−0.903496
$996$	−14.0331	−0.444657
$997$	−23.2413	−0.736059	−0.368030	−	0.929814i	$-0.619967\pi$
$997$	−23.2413	−0.736059	−0.368030	+	0.929814i	$0.619967\pi$
$998$	−63.3737	−2.00606
$999$	5.28005	0.167053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	129.2.a.d.1.3	✓	3
3.2	odd	2		387.2.a.i.1.1		3
4.3	odd	2		2064.2.a.x.1.1		3
5.4	even	2		3225.2.a.t.1.1		3
7.6	odd	2		6321.2.a.p.1.3		3
8.3	odd	2		8256.2.a.cu.1.3		3
8.5	even	2		8256.2.a.cr.1.3		3
12.11	even	2		6192.2.a.bw.1.3		3
15.14	odd	2		9675.2.a.bq.1.3		3
43.42	odd	2		5547.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.2.a.d.1.3	✓	3	1.1	even	1	trivial
387.2.a.i.1.1		3	3.2	odd	2
2064.2.a.x.1.1		3	4.3	odd	2
3225.2.a.t.1.1		3	5.4	even	2
5547.2.a.p.1.1		3	43.42	odd	2
6192.2.a.bw.1.3		3	12.11	even	2
6321.2.a.p.1.3		3	7.6	odd	2
8256.2.a.cr.1.3		3	8.5	even	2
8256.2.a.cu.1.3		3	8.3	odd	2
9675.2.a.bq.1.3		3	15.14	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:	\( N \)	\(=\)	\( 129 = 3 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	129.a (trivial)

\(f(q)\)	\(=\)	\(q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} -4.12489 q^{5} +2.12489 q^{6} +1.48486 q^{7} +1.09461 q^{8} +1.00000 q^{9} -8.76491 q^{10} -2.60975 q^{11} +2.51514 q^{12} +3.00000 q^{13} +3.15516 q^{14} -4.12489 q^{15} -2.70436 q^{16} +0.484862 q^{17} +2.12489 q^{18} -6.76491 q^{19} -10.3747 q^{20} +1.48486 q^{21} -5.54541 q^{22} +8.79518 q^{23} +1.09461 q^{24} +12.0147 q^{25} +6.37466 q^{26} +1.00000 q^{27} +3.73463 q^{28} -2.12489 q^{29} -8.76491 q^{30} +3.76491 q^{31} -7.93567 q^{32} -2.60975 q^{33} +1.03028 q^{34} -6.12489 q^{35} +2.51514 q^{36} +5.28005 q^{37} -14.3747 q^{38} +3.00000 q^{39} -4.51514 q^{40} +0.734633 q^{41} +3.15516 q^{42} -1.00000 q^{43} -6.56387 q^{44} -4.12489 q^{45} +18.6888 q^{46} -8.43521 q^{47} -2.70436 q^{48} -4.79518 q^{49} +25.5298 q^{50} +0.484862 q^{51} +7.54541 q^{52} +3.76491 q^{53} +2.12489 q^{54} +10.7649 q^{55} +1.62534 q^{56} -6.76491 q^{57} -4.51514 q^{58} +2.96972 q^{59} -10.3747 q^{60} -11.2195 q^{61} +8.00000 q^{62} +1.48486 q^{63} -11.4537 q^{64} -12.3747 q^{65} -5.54541 q^{66} +7.04496 q^{67} +1.21949 q^{68} +8.79518 q^{69} -13.0147 q^{70} +13.2195 q^{71} +1.09461 q^{72} -5.03028 q^{73} +11.2195 q^{74} +12.0147 q^{75} -17.0147 q^{76} -3.87511 q^{77} +6.37466 q^{78} +2.71995 q^{79} +11.1552 q^{80} +1.00000 q^{81} +1.56101 q^{82} -5.57947 q^{83} +3.73463 q^{84} -2.00000 q^{85} -2.12489 q^{86} -2.12489 q^{87} -2.85665 q^{88} -18.2498 q^{89} -8.76491 q^{90} +4.45459 q^{91} +22.1211 q^{92} +3.76491 q^{93} -17.9239 q^{94} +27.9045 q^{95} -7.93567 q^{96} +6.21949 q^{97} -10.1892 q^{98} -2.60975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 3 q^{9} - 10 q^{10} + q^{11} + 8 q^{12} + 9 q^{13} + 2 q^{14} - 4 q^{15} + 10 q^{16} + q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{20}+ \cdots + q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^7 - 6 * q^8 + 3 * q^9 - 10 * q^10 + q^11 + 8 * q^12 + 9 * q^13 + 2 * q^14 - 4 * q^15 + 10 * q^16 + q^17 - 2 * q^18 - 4 * q^19 - 6 * q^20 + 4 * q^21 - 18 * q^22 + 11 * q^23 - 6 * q^24 + 3 * q^25 - 6 * q^26 + 3 * q^27 - 6 * q^28 + 2 * q^29 - 10 * q^30 - 5 * q^31 - 34 * q^32 + q^33 + 4 * q^34 - 10 * q^35 + 8 * q^36 - 18 * q^38 + 9 * q^39 - 14 * q^40 - 15 * q^41 + 2 * q^42 - 3 * q^43 + 24 * q^44 - 4 * q^45 + 4 * q^46 - 2 * q^47 + 10 * q^48 + q^49 + 44 * q^50 + q^51 + 24 * q^52 - 5 * q^53 - 2 * q^54 + 16 * q^55 + 30 * q^56 - 4 * q^57 - 14 * q^58 + 8 * q^59 - 6 * q^60 - 16 * q^61 + 24 * q^62 + 4 * q^63 + 34 * q^64 - 12 * q^65 - 18 * q^66 - 11 * q^67 - 14 * q^68 + 11 * q^69 - 6 * q^70 + 22 * q^71 - 6 * q^72 - 16 * q^73 + 16 * q^74 + 3 * q^75 - 18 * q^76 - 20 * q^77 - 6 * q^78 + 24 * q^79 + 26 * q^80 + 3 * q^81 + 40 * q^82 - 7 * q^83 - 6 * q^84 - 6 * q^85 + 2 * q^86 + 2 * q^87 - 62 * q^88 - 38 * q^89 - 10 * q^90 + 12 * q^91 + 70 * q^92 - 5 * q^93 - 18 * q^94 + 26 * q^95 - 34 * q^96 + q^97 - 12 * q^98 + q^99

Introduction

L-functions

Modular forms

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Groups

Database

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