LMFDB - Embedded newform 3381.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69353$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69353 q^{2} +1.00000 q^{3} +0.868028 q^{4} -3.51256 q^{5} -1.69353 q^{6} +1.91702 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.69353 q^{2} +1.00000 q^{3} +0.868028 q^{4} -3.51256 q^{5} -1.69353 q^{6} +1.91702 q^{8} +1.00000 q^{9} +5.94860 q^{10} -1.74252 q^{11} +0.868028 q^{12} -6.33805 q^{13} -3.51256 q^{15} -4.98258 q^{16} +5.94860 q^{17} -1.69353 q^{18} -1.74252 q^{19} -3.04900 q^{20} +2.95100 q^{22} +1.00000 q^{23} +1.91702 q^{24} +7.33805 q^{25} +10.7337 q^{26} +1.00000 q^{27} -5.68466 q^{29} +5.94860 q^{30} -7.94860 q^{31} +4.60408 q^{32} -1.74252 q^{33} -10.0741 q^{34} +0.868028 q^{36} +1.53644 q^{37} +2.95100 q^{38} -6.33805 q^{39} -6.73365 q^{40} -12.1547 q^{41} +6.43605 q^{43} -1.51256 q^{44} -3.51256 q^{45} -1.69353 q^{46} -3.59313 q^{47} -4.98258 q^{48} -12.4272 q^{50} +5.94860 q^{51} -5.50161 q^{52} +12.9397 q^{53} -1.69353 q^{54} +6.12071 q^{55} -1.74252 q^{57} +9.62711 q^{58} +8.69353 q^{59} -3.04900 q^{60} -8.47618 q^{61} +13.4612 q^{62} +2.16804 q^{64} +22.2628 q^{65} +2.95100 q^{66} -4.46971 q^{67} +5.16355 q^{68} +1.00000 q^{69} -13.4612 q^{71} +1.91702 q^{72} -4.29761 q^{73} -2.60200 q^{74} +7.33805 q^{75} -1.51256 q^{76} +10.7337 q^{78} -7.42071 q^{79} +17.5016 q^{80} +1.00000 q^{81} +20.5843 q^{82} +7.75262 q^{83} -20.8948 q^{85} -10.8996 q^{86} -5.68466 q^{87} -3.34045 q^{88} -4.18503 q^{89} +5.94860 q^{90} +0.868028 q^{92} -7.94860 q^{93} +6.08506 q^{94} +6.12071 q^{95} +4.60408 q^{96} -5.53644 q^{97} -1.74252 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9	$ 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9} - 2 q^{10} + q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 2 q^{17} + 2 q^{18} + q^{19} - 13 q^{20} + 11 q^{22} + 4 q^{23} + 9 q^{24} + 11 q^{25} + 19 q^{26} + 4 q^{27} + 2 q^{29} - 2 q^{30} - 6 q^{31} + 20 q^{32} + q^{33} - 23 q^{34} + 4 q^{36} + 16 q^{37} + 11 q^{38} - 7 q^{39} - 3 q^{40} - 5 q^{41} + 9 q^{43} + 3 q^{44} - 5 q^{45} + 2 q^{46} + 21 q^{47} + 8 q^{48} - 17 q^{50} - 2 q^{51} + 24 q^{52} + 10 q^{53} + 2 q^{54} - 17 q^{55} + q^{57} + q^{58} + 26 q^{59} - 13 q^{60} - 2 q^{61} + 19 q^{62} + 27 q^{64} + 26 q^{65} + 11 q^{66} + 5 q^{67} - 7 q^{68} + 4 q^{69} - 19 q^{71} + 9 q^{72} - 10 q^{73} + 9 q^{74} + 11 q^{75} + 3 q^{76} + 19 q^{78} - 6 q^{79} + 24 q^{80} + 4 q^{81} + 31 q^{82} + 2 q^{83} - 7 q^{85} - 17 q^{86} + 2 q^{87} - 20 q^{88} + 17 q^{89} - 2 q^{90} + 4 q^{92} - 6 q^{93} + 44 q^{94} - 17 q^{95} + 20 q^{96} - 32 q^{97} + q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9 - 2 * q^10 + q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 2 * q^17 + 2 * q^18 + q^19 - 13 * q^20 + 11 * q^22 + 4 * q^23 + 9 * q^24 + 11 * q^25 + 19 * q^26 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 6 * q^31 + 20 * q^32 + q^33 - 23 * q^34 + 4 * q^36 + 16 * q^37 + 11 * q^38 - 7 * q^39 - 3 * q^40 - 5 * q^41 + 9 * q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + 21 * q^47 + 8 * q^48 - 17 * q^50 - 2 * q^51 + 24 * q^52 + 10 * q^53 + 2 * q^54 - 17 * q^55 + q^57 + q^58 + 26 * q^59 - 13 * q^60 - 2 * q^61 + 19 * q^62 + 27 * q^64 + 26 * q^65 + 11 * q^66 + 5 * q^67 - 7 * q^68 + 4 * q^69 - 19 * q^71 + 9 * q^72 - 10 * q^73 + 9 * q^74 + 11 * q^75 + 3 * q^76 + 19 * q^78 - 6 * q^79 + 24 * q^80 + 4 * q^81 + 31 * q^82 + 2 * q^83 - 7 * q^85 - 17 * q^86 + 2 * q^87 - 20 * q^88 + 17 * q^89 - 2 * q^90 + 4 * q^92 - 6 * q^93 + 44 * q^94 - 17 * q^95 + 20 * q^96 - 32 * q^97 + 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 $

$2$	−1.69353	−1.19750	−0.598752	−	0.800935i	$-0.704336\pi$
$2$	−1.69353	−1.19750	−0.598752	+	0.800935i	$0.704336\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.868028	0.434014
$5$	−3.51256	−1.57086	−0.785432	−	0.618949i	$-0.787559\pi$
$5$	−3.51256	−1.57086	−0.785432	+	0.618949i	$0.787559\pi$
$6$	−1.69353	−0.691379
$7$	0	0
$7$	0	0
$8$	1.91702	0.677770
$9$	1.00000	0.333333
$10$	5.94860	1.88111
$11$	−1.74252	−0.525390	−0.262695	−	0.964879i	$-0.584611\pi$
$11$	−1.74252	−0.525390	−0.262695	+	0.964879i	$0.584611\pi$
$12$	0.868028	0.250578
$13$	−6.33805	−1.75786	−0.878930	−	0.476951i	$-0.841742\pi$
$13$	−6.33805	−1.75786	−0.878930	+	0.476951i	$0.841742\pi$
$14$	0	0
$15$	−3.51256	−0.906938
$16$	−4.98258	−1.24565
$17$	5.94860	1.44275	0.721374	−	0.692546i	$-0.243511\pi$
$17$	5.94860	1.44275	0.721374	+	0.692546i	$0.243511\pi$
$18$	−1.69353	−0.399168
$19$	−1.74252	−0.399762	−0.199881	−	0.979820i	$-0.564056\pi$
$19$	−1.74252	−0.399762	−0.199881	+	0.979820i	$0.564056\pi$
$20$	−3.04900	−0.681776
$21$	0	0
$22$	2.95100	0.629156
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.91702	0.391311
$25$	7.33805	1.46761
$26$	10.7337	2.10504
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	5.94860	1.08606
$31$	−7.94860	−1.42761	−0.713806	−	0.700344i	$-0.753030\pi$
$31$	−7.94860	−1.42761	−0.713806	+	0.700344i	$0.753030\pi$
$32$	4.60408	0.813895
$33$	−1.74252	−0.303334
$34$	−10.0741	−1.72770
$35$	0	0
$36$	0.868028	0.144671
$37$	1.53644	0.252589	0.126295	−	0.991993i	$-0.459692\pi$
$37$	1.53644	0.252589	0.126295	+	0.991993i	$0.459692\pi$
$38$	2.95100	0.478716
$39$	−6.33805	−1.01490
$40$	−6.73365	−1.06468
$41$	−12.1547	−1.89824	−0.949121	−	0.314910i	$-0.898026\pi$
$41$	−12.1547	−1.89824	−0.949121	+	0.314910i	$0.898026\pi$
$42$	0	0
$43$	6.43605	0.981488	0.490744	−	0.871304i	$-0.336725\pi$
$43$	6.43605	0.981488	0.490744	+	0.871304i	$0.336725\pi$
$44$	−1.51256	−0.228026
$45$	−3.51256	−0.523621
$46$	−1.69353	−0.249697
$47$	−3.59313	−0.524112	−0.262056	−	0.965053i	$-0.584401\pi$
$47$	−3.59313	−0.524112	−0.262056	+	0.965053i	$0.584401\pi$
$48$	−4.98258	−0.719174
$49$	0	0
$50$	−12.4272	−1.75747
$51$	5.94860	0.832971
$52$	−5.50161	−0.762935
$53$	12.9397	1.77741	0.888705	−	0.458480i	$-0.151606\pi$
$53$	12.9397	1.77741	0.888705	+	0.458480i	$0.151606\pi$
$54$	−1.69353	−0.230460
$55$	6.12071	0.825316
$56$	0	0
$57$	−1.74252	−0.230803
$58$	9.62711	1.26410
$59$	8.69353	1.13180	0.565900	−	0.824474i	$-0.308529\pi$
$59$	8.69353	1.13180	0.565900	+	0.824474i	$0.308529\pi$
$60$	−3.04900	−0.393624
$61$	−8.47618	−1.08526	−0.542632	−	0.839971i	$-0.682572\pi$
$61$	−8.47618	−1.08526	−0.542632	+	0.839971i	$0.682572\pi$
$62$	13.4612	1.70957
$63$	0	0
$64$	2.16804	0.271005
$65$	22.2628	2.76136
$66$	2.95100	0.363243
$67$	−4.46971	−0.546062	−0.273031	−	0.962005i	$-0.588026\pi$
$67$	−4.46971	−0.546062	−0.273031	+	0.962005i	$0.588026\pi$
$68$	5.16355	0.626173
$69$	1.00000	0.120386
$70$	0	0
$71$	−13.4612	−1.59755	−0.798773	−	0.601633i	$-0.794517\pi$
$71$	−13.4612	−1.59755	−0.798773	+	0.601633i	$0.794517\pi$
$72$	1.91702	0.225923
$73$	−4.29761	−0.502997	−0.251498	−	0.967858i	$-0.580923\pi$
$73$	−4.29761	−0.502997	−0.251498	+	0.967858i	$0.580923\pi$
$74$	−2.60200	−0.302476
$75$	7.33805	0.847326
$76$	−1.51256	−0.173502
$77$	0	0
$78$	10.7337	1.21535
$79$	−7.42071	−0.834896	−0.417448	−	0.908701i	$-0.637075\pi$
$79$	−7.42071	−0.834896	−0.417448	+	0.908701i	$0.637075\pi$
$80$	17.5016	1.95674
$81$	1.00000	0.111111
$82$	20.5843	2.27315
$83$	7.75262	0.850960	0.425480	−	0.904968i	$-0.360105\pi$
$83$	7.75262	0.850960	0.425480	+	0.904968i	$0.360105\pi$
$84$	0	0
$85$	−20.8948	−2.26636
$86$	−10.8996	−1.17533
$87$	−5.68466	−0.609459
$88$	−3.34045	−0.356094
$89$	−4.18503	−0.443613	−0.221806	−	0.975091i	$-0.571195\pi$
$89$	−4.18503	−0.443613	−0.221806	+	0.975091i	$0.571195\pi$
$90$	5.94860	0.627038
$91$	0	0
$92$	0.868028	0.0904981
$93$	−7.94860	−0.824232
$94$	6.08506	0.627626
$95$	6.12071	0.627971
$96$	4.60408	0.469902
$97$	−5.53644	−0.562140	−0.281070	−	0.959687i	$-0.590689\pi$
$97$	−5.53644	−0.562140	−0.281070	+	0.959687i	$0.590689\pi$
$98$	0	0
$99$	−1.74252	−0.175130
$100$	6.36963	0.636963
$101$	15.9778	1.58985	0.794924	−	0.606709i	$-0.207510\pi$
$101$	15.9778	1.58985	0.794924	+	0.606709i	$0.207510\pi$
$102$	−10.0741	−0.997485
$103$	−8.96725	−0.883569	−0.441785	−	0.897121i	$-0.645654\pi$
$103$	−8.96725	−0.883569	−0.441785	+	0.897121i	$0.645654\pi$
$104$	−12.1502	−1.19143
$105$	0	0
$106$	−21.9138	−2.12845
$107$	20.2353	1.95622	0.978108	−	0.208097i	$-0.0667269\pi$
$107$	20.2353	1.95622	0.978108	+	0.208097i	$0.0667269\pi$
$108$	0.868028	0.0835260
$109$	−3.31657	−0.317670	−0.158835	−	0.987305i	$-0.550774\pi$
$109$	−3.31657	−0.317670	−0.158835	+	0.987305i	$0.550774\pi$
$110$	−10.3656	−0.988318
$111$	1.53644	0.145832
$112$	0	0
$113$	16.1207	1.51651	0.758254	−	0.651959i	$-0.226053\pi$
$113$	16.1207	1.51651	0.758254	+	0.651959i	$0.226053\pi$
$114$	2.95100	0.276387
$115$	−3.51256	−0.327548
$116$	−4.93444	−0.458151
$117$	−6.33805	−0.585953
$118$	−14.7227	−1.35533
$119$	0	0
$120$	−6.73365	−0.614696
$121$	−7.96362	−0.723965
$122$	14.3546	1.29961
$123$	−12.1547	−1.09595
$124$	−6.89961	−0.619603
$125$	−8.21255	−0.734553
$126$	0	0
$127$	0.480977	0.0426798	0.0213399	−	0.999772i	$-0.493207\pi$
$127$	0.480977	0.0426798	0.0213399	+	0.999772i	$0.493207\pi$
$128$	−12.8798	−1.13842
$129$	6.43605	0.566662
$130$	−37.7026	−3.30673
$131$	6.71741	0.586903	0.293451	−	0.955974i	$-0.405196\pi$
$131$	6.71741	0.586903	0.293451	+	0.955974i	$0.405196\pi$
$132$	−1.51256	−0.131651
$133$	0	0
$134$	7.56957	0.653911
$135$	−3.51256	−0.302313
$136$	11.4036	0.977852
$137$	−2.96362	−0.253199	−0.126600	−	0.991954i	$-0.540406\pi$
$137$	−2.96362	−0.253199	−0.126600	+	0.991954i	$0.540406\pi$
$138$	−1.69353	−0.144162
$139$	8.80161	0.746543	0.373272	−	0.927722i	$-0.378236\pi$
$139$	8.80161	0.746543	0.373272	+	0.927722i	$0.378236\pi$
$140$	0	0
$141$	−3.59313	−0.302596
$142$	22.7968	1.91307
$143$	11.0442	0.923562
$144$	−4.98258	−0.415215
$145$	19.9677	1.65823
$146$	7.27811	0.602340
$147$	0	0
$148$	1.33367	0.109627
$149$	−11.2441	−0.921155	−0.460577	−	0.887620i	$-0.652358\pi$
$149$	−11.2441	−0.921155	−0.460577	+	0.887620i	$0.652358\pi$
$150$	−12.4272	−1.01467
$151$	5.01502	0.408116	0.204058	−	0.978959i	$-0.434587\pi$
$151$	5.01502	0.408116	0.204058	+	0.978959i	$0.434587\pi$
$152$	−3.34045	−0.270947
$153$	5.94860	0.480916
$154$	0	0
$155$	27.9199	2.24258
$156$	−5.50161	−0.440481
$157$	−14.8190	−1.18269	−0.591344	−	0.806420i	$-0.701402\pi$
$157$	−14.8190	−1.18269	−0.591344	+	0.806420i	$0.701402\pi$
$158$	12.5672	0.999790
$159$	12.9397	1.02619
$160$	−16.1721	−1.27852
$161$	0	0
$162$	−1.69353	−0.133056
$163$	2.11541	0.165692	0.0828458	−	0.996562i	$-0.473599\pi$
$163$	2.11541	0.165692	0.0828458	+	0.996562i	$0.473599\pi$
$164$	−10.5506	−0.823864
$165$	6.12071	0.476496
$166$	−13.1293	−1.01903
$167$	14.4186	1.11575	0.557874	−	0.829926i	$-0.311617\pi$
$167$	14.4186	1.11575	0.557874	+	0.829926i	$0.311617\pi$
$168$	0	0
$169$	27.1709	2.09007
$170$	35.3859	2.71397
$171$	−1.74252	−0.133254
$172$	5.58667	0.425979
$173$	9.73969	0.740495	0.370247	−	0.928933i	$-0.379273\pi$
$173$	9.73969	0.740495	0.370247	+	0.928933i	$0.379273\pi$
$174$	9.62711	0.729830
$175$	0	0
$176$	8.68226	0.654450
$177$	8.69353	0.653445
$178$	7.08746	0.531228
$179$	−0.376519	−0.0281423	−0.0140712	−	0.999901i	$-0.504479\pi$
$179$	−0.376519	−0.0281423	−0.0140712	+	0.999901i	$0.504479\pi$
$180$	−3.04900	−0.227259
$181$	22.2580	1.65442	0.827211	−	0.561891i	$-0.189926\pi$
$181$	22.2580	1.65442	0.827211	+	0.561891i	$0.189926\pi$
$182$	0	0
$183$	−8.47618	−0.626577
$184$	1.91702	0.141325
$185$	−5.39683	−0.396783
$186$	13.4612	0.987020
$187$	−10.3656	−0.758005
$188$	−3.11894	−0.227472
$189$	0	0
$190$	−10.3656	−0.751997
$191$	2.98498	0.215986	0.107993	−	0.994152i	$-0.465558\pi$
$191$	2.98498	0.215986	0.107993	+	0.994152i	$0.465558\pi$
$192$	2.16804	0.156465
$193$	4.98498	0.358827	0.179413	−	0.983774i	$-0.442580\pi$
$193$	4.98498	0.358827	0.179413	+	0.983774i	$0.442580\pi$
$194$	9.37610	0.673165
$195$	22.2628	1.59427
$196$	0	0
$197$	3.41573	0.243361	0.121681	−	0.992569i	$-0.461172\pi$
$197$	3.41573	0.243361	0.121681	+	0.992569i	$0.461172\pi$
$198$	2.95100	0.209719
$199$	3.91462	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$199$	3.91462	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$200$	14.0672	0.994703
$201$	−4.46971	−0.315269
$202$	−27.0588	−1.90385
$203$	0	0
$204$	5.16355	0.361521
$205$	42.6940	2.98188
$206$	15.1863	1.05808
$207$	1.00000	0.0695048
$208$	31.5799	2.18967
$209$	3.03638	0.210031
$210$	0	0
$211$	−1.59430	−0.109756	−0.0548782	−	0.998493i	$-0.517477\pi$
$211$	−1.59430	−0.109756	−0.0548782	+	0.998493i	$0.517477\pi$
$212$	11.2320	0.771420
$213$	−13.4612	−0.922343
$214$	−34.2689	−2.34258
$215$	−22.6070	−1.54178
$216$	1.91702	0.130437
$217$	0	0
$218$	5.61670	0.380411
$219$	−4.29761	−0.290405
$220$	5.31294	0.358198
$221$	−37.7026	−2.53615
$222$	−2.60200	−0.174635
$223$	−2.88667	−0.193306	−0.0966530	−	0.995318i	$-0.530814\pi$
$223$	−2.88667	−0.193306	−0.0966530	+	0.995318i	$0.530814\pi$
$224$	0	0
$225$	7.33805	0.489204
$226$	−27.3008	−1.81602
$227$	−2.14324	−0.142252	−0.0711259	−	0.997467i	$-0.522659\pi$
$227$	−2.14324	−0.142252	−0.0711259	+	0.997467i	$0.522659\pi$
$228$	−1.51256	−0.100172
$229$	14.1308	0.933790	0.466895	−	0.884313i	$-0.345373\pi$
$229$	14.1308	0.933790	0.466895	+	0.884313i	$0.345373\pi$
$230$	5.94860	0.392239
$231$	0	0
$232$	−10.8976	−0.715464
$233$	−2.88187	−0.188798	−0.0943989	−	0.995534i	$-0.530093\pi$
$233$	−2.88187	−0.188798	−0.0943989	+	0.995534i	$0.530093\pi$
$234$	10.7337	0.701681
$235$	12.6211	0.823309
$236$	7.54622	0.491217
$237$	−7.42071	−0.482027
$238$	0	0
$239$	16.4956	1.06701	0.533505	−	0.845797i	$-0.320875\pi$
$239$	16.4956	1.06701	0.533505	+	0.845797i	$0.320875\pi$
$240$	17.5016	1.12972
$241$	18.0017	1.15959	0.579795	−	0.814763i	$-0.303133\pi$
$241$	18.0017	1.15959	0.579795	+	0.814763i	$0.303133\pi$
$242$	13.4866	0.866951
$243$	1.00000	0.0641500
$244$	−7.35755	−0.471019
$245$	0	0
$246$	20.5843	1.31240
$247$	11.0442	0.702725
$248$	−15.2377	−0.967592
$249$	7.75262	0.491302
$250$	13.9082	0.879629
$251$	−11.4373	−0.721915	−0.360957	−	0.932582i	$-0.617550\pi$
$251$	−11.4373	−0.721915	−0.360957	+	0.932582i	$0.617550\pi$
$252$	0	0
$253$	−1.74252	−0.109551
$254$	−0.814547	−0.0511092
$255$	−20.8948	−1.30848
$256$	17.4762	1.09226
$257$	30.4045	1.89658	0.948291	−	0.317402i	$-0.102810\pi$
$257$	30.4045	1.89658	0.948291	+	0.317402i	$0.102810\pi$
$258$	−10.8996	−0.678580
$259$	0	0
$260$	19.3247	1.19847
$261$	−5.68466	−0.351872
$262$	−11.3761	−0.702818
$263$	−18.2226	−1.12366	−0.561828	−	0.827254i	$-0.689902\pi$
$263$	−18.2226	−1.12366	−0.561828	+	0.827254i	$0.689902\pi$
$264$	−3.34045	−0.205591
$265$	−45.4516	−2.79207
$266$	0	0
$267$	−4.18503	−0.256120
$268$	−3.87983	−0.236998
$269$	−7.65714	−0.466864	−0.233432	−	0.972373i	$-0.574996\pi$
$269$	−7.65714	−0.466864	−0.233432	+	0.972373i	$0.574996\pi$
$270$	5.94860	0.362020
$271$	−10.2628	−0.623419	−0.311710	−	0.950177i	$-0.600902\pi$
$271$	−10.2628	−0.623419	−0.311710	+	0.950177i	$0.600902\pi$
$272$	−29.6394	−1.79715
$273$	0	0
$274$	5.01896	0.303207
$275$	−12.7867	−0.771068
$276$	0.868028	0.0522491
$277$	7.83319	0.470651	0.235326	−	0.971917i	$-0.424384\pi$
$277$	7.83319	0.470651	0.235326	+	0.971917i	$0.424384\pi$
$278$	−14.9058	−0.893988
$279$	−7.94860	−0.475870
$280$	0	0
$281$	−2.34420	−0.139843	−0.0699217	−	0.997552i	$-0.522275\pi$
$281$	−2.34420	−0.139843	−0.0699217	+	0.997552i	$0.522275\pi$
$282$	6.08506	0.362360
$283$	−0.517476	−0.0307607	−0.0153804	−	0.999882i	$-0.504896\pi$
$283$	−0.517476	−0.0307607	−0.0153804	+	0.999882i	$0.504896\pi$
$284$	−11.6847	−0.693357
$285$	6.12071	0.362559
$286$	−18.7036	−1.10597
$287$	0	0
$288$	4.60408	0.271298
$289$	18.3859	1.08152
$290$	−33.8158	−1.98573
$291$	−5.53644	−0.324552
$292$	−3.73044	−0.218308
$293$	9.15314	0.534732	0.267366	−	0.963595i	$-0.413847\pi$
$293$	9.15314	0.534732	0.267366	+	0.963595i	$0.413847\pi$
$294$	0	0
$295$	−30.5365	−1.77790
$296$	2.94539	0.171197
$297$	−1.74252	−0.101111
$298$	19.0422	1.10309
$299$	−6.33805	−0.366539
$300$	6.36963	0.367751
$301$	0	0
$302$	−8.49306	−0.488720
$303$	15.9778	0.917900
$304$	8.68226	0.497962
$305$	29.7730	1.70480
$306$	−10.0741	−0.575899
$307$	33.0492	1.88622	0.943108	−	0.332487i	$-0.107888\pi$
$307$	33.0492	1.88622	0.943108	+	0.332487i	$0.107888\pi$
$308$	0	0
$309$	−8.96725	−0.510129
$310$	−47.2831	−2.68550
$311$	28.4591	1.61377	0.806883	−	0.590711i	$-0.201153\pi$
$311$	28.4591	1.61377	0.806883	+	0.590711i	$0.201153\pi$
$312$	−12.1502	−0.687870
$313$	−0.0903560	−0.00510722	−0.00255361	−	0.999997i	$-0.500813\pi$
$313$	−0.0903560	−0.00510722	−0.00255361	+	0.999997i	$0.500813\pi$
$314$	25.0964	1.41627
$315$	0	0
$316$	−6.44138	−0.362356
$317$	−13.1029	−0.735930	−0.367965	−	0.929840i	$-0.619945\pi$
$317$	−13.1029	−0.735930	−0.367965	+	0.929840i	$0.619945\pi$
$318$	−21.9138	−1.22886
$319$	9.90564	0.554609
$320$	−7.61535	−0.425711
$321$	20.2353	1.12942
$322$	0	0
$323$	−10.3656	−0.576756
$324$	0.868028	0.0482238
$325$	−46.5090	−2.57985
$326$	−3.58250	−0.198416
$327$	−3.31657	−0.183407
$328$	−23.3008	−1.28657
$329$	0	0
$330$	−10.3656	−0.570606
$331$	4.04013	0.222066	0.111033	−	0.993817i	$-0.464584\pi$
$331$	4.04013	0.222066	0.111033	+	0.993817i	$0.464584\pi$
$332$	6.72949	0.369329
$333$	1.53644	0.0841964
$334$	−24.4183	−1.33611
$335$	15.7001	0.857789
$336$	0	0
$337$	−2.24486	−0.122285	−0.0611427	−	0.998129i	$-0.519474\pi$
$337$	−2.24486	−0.122285	−0.0611427	+	0.998129i	$0.519474\pi$
$338$	−46.0147	−2.50287
$339$	16.1207	0.875557
$340$	−18.1373	−0.983631
$341$	13.8506	0.750053
$342$	2.95100	0.159572
$343$	0	0
$344$	12.3381	0.665223
$345$	−3.51256	−0.189110
$346$	−16.4944	−0.886745
$347$	14.4458	0.775493	0.387746	−	0.921766i	$-0.373254\pi$
$347$	14.4458	0.775493	0.387746	+	0.921766i	$0.373254\pi$
$348$	−4.93444	−0.264514
$349$	29.2438	1.56539	0.782693	−	0.622408i	$-0.213846\pi$
$349$	29.2438	1.56539	0.782693	+	0.622408i	$0.213846\pi$
$350$	0	0
$351$	−6.33805	−0.338300
$352$	−8.02271	−0.427612
$353$	−10.6595	−0.567350	−0.283675	−	0.958920i	$-0.591554\pi$
$353$	−10.6595	−0.567350	−0.283675	+	0.958920i	$0.591554\pi$
$354$	−14.7227	−0.782503
$355$	47.2831	2.50953
$356$	−3.63273	−0.192534
$357$	0	0
$358$	0.637644	0.0337005
$359$	−5.44460	−0.287355	−0.143677	−	0.989625i	$-0.545893\pi$
$359$	−5.44460	−0.287355	−0.143677	+	0.989625i	$0.545893\pi$
$360$	−6.73365	−0.354895
$361$	−15.9636	−0.840190
$362$	−37.6944	−1.98118
$363$	−7.96362	−0.417982
$364$	0	0
$365$	15.0956	0.790139
$366$	14.3546	0.750328
$367$	−11.4640	−0.598416	−0.299208	−	0.954188i	$-0.596722\pi$
$367$	−11.4640	−0.598416	−0.299208	+	0.954188i	$0.596722\pi$
$368$	−4.98258	−0.259735
$369$	−12.1547	−0.632748
$370$	9.13967	0.475149
$371$	0	0
$372$	−6.89961	−0.357728
$373$	−27.9859	−1.44905	−0.724527	−	0.689246i	$-0.757942\pi$
$373$	−27.9859	−1.44905	−0.724527	+	0.689246i	$0.757942\pi$
$374$	17.5544	0.907714
$375$	−8.21255	−0.424094
$376$	−6.88812	−0.355228
$377$	36.0297	1.85562
$378$	0	0
$379$	11.1604	0.573271	0.286636	−	0.958040i	$-0.407463\pi$
$379$	11.1604	0.573271	0.286636	+	0.958040i	$0.407463\pi$
$380$	5.31294	0.272548
$381$	0.480977	0.0246412
$382$	−5.05515	−0.258644
$383$	17.2677	0.882338	0.441169	−	0.897424i	$-0.354564\pi$
$383$	17.2677	0.882338	0.441169	+	0.897424i	$0.354564\pi$
$384$	−12.8798	−0.657269
$385$	0	0
$386$	−8.44220	−0.429696
$387$	6.43605	0.327163
$388$	−4.80578	−0.243977
$389$	27.8158	1.41032	0.705158	−	0.709050i	$-0.250876\pi$
$389$	27.8158	1.41032	0.705158	+	0.709050i	$0.250876\pi$
$390$	−37.7026	−1.90914
$391$	5.94860	0.300834
$392$	0	0
$393$	6.71741	0.338848
$394$	−5.78463	−0.291426
$395$	26.0657	1.31151
$396$	−1.51256	−0.0760088
$397$	36.4276	1.82825	0.914123	−	0.405436i	$-0.132880\pi$
$397$	36.4276	1.82825	0.914123	+	0.405436i	$0.132880\pi$
$398$	−6.62951	−0.332307
$399$	0	0
$400$	−36.5625	−1.82812
$401$	−1.41605	−0.0707142	−0.0353571	−	0.999375i	$-0.511257\pi$
$401$	−1.41605	−0.0707142	−0.0353571	+	0.999375i	$0.511257\pi$
$402$	7.56957	0.377536
$403$	50.3787	2.50954
$404$	13.8692	0.690016
$405$	−3.51256	−0.174540
$406$	0	0
$407$	−2.67728	−0.132708
$408$	11.4036	0.564563
$409$	0.339704	0.0167973	0.00839863	−	0.999965i	$-0.497327\pi$
$409$	0.339704	0.0167973	0.00839863	+	0.999965i	$0.497327\pi$
$410$	−72.3034	−3.57081
$411$	−2.96362	−0.146185
$412$	−7.78382	−0.383481
$413$	0	0
$414$	−1.69353	−0.0832322
$415$	−27.2315	−1.33674
$416$	−29.1809	−1.43071
$417$	8.80161	0.431017
$418$	−5.14219	−0.251513
$419$	−36.7313	−1.79444	−0.897221	−	0.441582i	$-0.854418\pi$
$419$	−36.7313	−1.79444	−0.897221	+	0.441582i	$0.854418\pi$
$420$	0	0
$421$	19.6701	0.958661	0.479330	−	0.877635i	$-0.340880\pi$
$421$	19.6701	0.958661	0.479330	+	0.877635i	$0.340880\pi$
$422$	2.69999	0.131434
$423$	−3.59313	−0.174704
$424$	24.8058	1.20468
$425$	43.6512	2.11739
$426$	22.7968	1.10451
$427$	0	0
$428$	17.5648	0.849025
$429$	11.0442	0.533219
$430$	38.2855	1.84629
$431$	−2.51827	−0.121301	−0.0606504	−	0.998159i	$-0.519317\pi$
$431$	−2.51827	−0.121301	−0.0606504	+	0.998159i	$0.519317\pi$
$432$	−4.98258	−0.239725
$433$	27.8393	1.33787	0.668937	−	0.743319i	$-0.266750\pi$
$433$	27.8393	1.33787	0.668937	+	0.743319i	$0.266750\pi$
$434$	0	0
$435$	19.9677	0.957377
$436$	−2.87888	−0.137873
$437$	−1.74252	−0.0833561
$438$	7.27811	0.347761
$439$	28.7859	1.37387	0.686937	−	0.726717i	$-0.258955\pi$
$439$	28.7859	1.37387	0.686937	+	0.726717i	$0.258955\pi$
$440$	11.7335	0.559374
$441$	0	0
$442$	63.8503	3.03705
$443$	−12.3920	−0.588760	−0.294380	−	0.955688i	$-0.595113\pi$
$443$	−12.3920	−0.588760	−0.294380	+	0.955688i	$0.595113\pi$
$444$	1.33367	0.0632933
$445$	14.7002	0.696855
$446$	4.88866	0.231485
$447$	−11.2441	−0.531829
$448$	0	0
$449$	0.359419	0.0169620	0.00848101	−	0.999964i	$-0.497300\pi$
$449$	0.359419	0.0169620	0.00848101	+	0.999964i	$0.497300\pi$
$450$	−12.4272	−0.585823
$451$	21.1798	0.997318
$452$	13.9932	0.658186
$453$	5.01502	0.235626
$454$	3.62963	0.170347
$455$	0	0
$456$	−3.34045	−0.156431
$457$	−27.3677	−1.28021	−0.640103	−	0.768289i	$-0.721108\pi$
$457$	−27.3677	−1.28021	−0.640103	+	0.768289i	$0.721108\pi$
$458$	−23.9309	−1.11822
$459$	5.94860	0.277657
$460$	−3.04900	−0.142160
$461$	7.43230	0.346157	0.173078	−	0.984908i	$-0.444629\pi$
$461$	7.43230	0.346157	0.173078	+	0.984908i	$0.444629\pi$
$462$	0	0
$463$	6.45346	0.299918	0.149959	−	0.988692i	$-0.452086\pi$
$463$	6.45346	0.299918	0.149959	+	0.988692i	$0.452086\pi$
$464$	28.3243	1.31492
$465$	27.9199	1.29476
$466$	4.88053	0.226086
$467$	−42.2953	−1.95719	−0.978596	−	0.205792i	$-0.934023\pi$
$467$	−42.2953	−1.95719	−0.978596	+	0.205792i	$0.934023\pi$
$468$	−5.50161	−0.254312
$469$	0	0
$470$	−21.3741	−0.985915
$471$	−14.8190	−0.682825
$472$	16.6657	0.767101
$473$	−11.2149	−0.515664
$474$	12.5672	0.577229
$475$	−12.7867	−0.586695
$476$	0	0
$477$	12.9397	0.592470
$478$	−27.9357	−1.27775
$479$	21.6417	0.988834	0.494417	−	0.869225i	$-0.335382\pi$
$479$	21.6417	0.988834	0.494417	+	0.869225i	$0.335382\pi$
$480$	−16.1721	−0.738152
$481$	−9.73804	−0.444016
$482$	−30.4863	−1.38861
$483$	0	0
$484$	−6.91264	−0.314211
$485$	19.4471	0.883045
$486$	−1.69353	−0.0768199
$487$	7.21180	0.326798	0.163399	−	0.986560i	$-0.447754\pi$
$487$	7.21180	0.326798	0.163399	+	0.986560i	$0.447754\pi$
$488$	−16.2490	−0.735559
$489$	2.11541	0.0956621
$490$	0	0
$491$	−7.42387	−0.335034	−0.167517	−	0.985869i	$-0.553575\pi$
$491$	−7.42387	−0.335034	−0.167517	+	0.985869i	$0.553575\pi$
$492$	−10.5506	−0.475658
$493$	−33.8158	−1.52299
$494$	−18.7036	−0.841516
$495$	6.12071	0.275105
$496$	39.6046	1.77830
$497$	0	0
$498$	−13.1293	−0.588336
$499$	−29.8147	−1.33469	−0.667344	−	0.744750i	$-0.732569\pi$
$499$	−29.8147	−1.33469	−0.667344	+	0.744750i	$0.732569\pi$
$500$	−7.12872	−0.318806
$501$	14.4186	0.644177
$502$	19.3693	0.864495
$503$	−32.1036	−1.43143	−0.715715	−	0.698393i	$-0.753899\pi$
$503$	−32.1036	−1.43143	−0.715715	+	0.698393i	$0.753899\pi$
$504$	0	0
$505$	−56.1229	−2.49743
$506$	2.95100	0.131188
$507$	27.1709	1.20670
$508$	0.417501	0.0185236
$509$	−19.4401	−0.861668	−0.430834	−	0.902431i	$-0.641781\pi$
$509$	−19.4401	−0.861668	−0.430834	+	0.902431i	$0.641781\pi$
$510$	35.3859	1.56691
$511$	0	0
$512$	−3.83676	−0.169563
$513$	−1.74252	−0.0769342
$514$	−51.4908	−2.27116
$515$	31.4980	1.38797
$516$	5.58667	0.245939
$517$	6.26111	0.275363
$518$	0	0
$519$	9.73969	0.427525
$520$	42.6783	1.87157
$521$	9.59843	0.420515	0.210257	−	0.977646i	$-0.432570\pi$
$521$	9.59843	0.420515	0.210257	+	0.977646i	$0.432570\pi$
$522$	9.62711	0.421367
$523$	−14.6384	−0.640094	−0.320047	−	0.947402i	$-0.603699\pi$
$523$	−14.6384	−0.640094	−0.320047	+	0.947402i	$0.603699\pi$
$524$	5.83090	0.254724
$525$	0	0
$526$	30.8605	1.34558
$527$	−47.2831	−2.05968
$528$	8.68226	0.377847
$529$	1.00000	0.0434783
$530$	76.9734	3.34351
$531$	8.69353	0.377267
$532$	0	0
$533$	77.0371	3.33685
$534$	7.08746	0.306704
$535$	−71.0775	−3.07295
$536$	−8.56854	−0.370105
$537$	−0.376519	−0.0162480
$538$	12.9676	0.559072
$539$	0	0
$540$	−3.04900	−0.131208
$541$	5.49994	0.236461	0.118230	−	0.992986i	$-0.462278\pi$
$541$	5.49994	0.236461	0.118230	+	0.992986i	$0.462278\pi$
$542$	17.3803	0.746546
$543$	22.2580	0.955181
$544$	27.3879	1.17424
$545$	11.6496	0.499016
$546$	0	0
$547$	6.59357	0.281921	0.140960	−	0.990015i	$-0.454981\pi$
$547$	6.59357	0.281921	0.140960	+	0.990015i	$0.454981\pi$
$548$	−2.57250	−0.109892
$549$	−8.47618	−0.361754
$550$	21.6546	0.923356
$551$	9.90564	0.421994
$552$	1.91702	0.0815940
$553$	0	0
$554$	−13.2657	−0.563606
$555$	−5.39683	−0.229083
$556$	7.64004	0.324010
$557$	−5.45021	−0.230933	−0.115466	−	0.993311i	$-0.536836\pi$
$557$	−5.45021	−0.230933	−0.115466	+	0.993311i	$0.536836\pi$
$558$	13.4612	0.569856
$559$	−40.7920	−1.72532
$560$	0	0
$561$	−10.3656	−0.437635
$562$	3.96997	0.167463
$563$	19.3154	0.814047	0.407024	−	0.913418i	$-0.366567\pi$
$563$	19.3154	0.814047	0.407024	+	0.913418i	$0.366567\pi$
$564$	−3.11894	−0.131331
$565$	−56.6249	−2.38223
$566$	0.876358	0.0368361
$567$	0	0
$568$	−25.8054	−1.08277
$569$	−20.2263	−0.847930	−0.423965	−	0.905679i	$-0.639362\pi$
$569$	−20.2263	−0.847930	−0.423965	+	0.905679i	$0.639362\pi$
$570$	−10.3656	−0.434166
$571$	34.6328	1.44934	0.724670	−	0.689096i	$-0.241992\pi$
$571$	34.6328	1.44934	0.724670	+	0.689096i	$0.241992\pi$
$572$	9.58667	0.400839
$573$	2.98498	0.124699
$574$	0	0
$575$	7.33805	0.306018
$576$	2.16804	0.0903348
$577$	36.7645	1.53053	0.765263	−	0.643718i	$-0.222609\pi$
$577$	36.7645	1.53053	0.765263	+	0.643718i	$0.222609\pi$
$578$	−31.1370	−1.29513
$579$	4.98498	0.207169
$580$	17.3325	0.719693
$581$	0	0
$582$	9.37610	0.388652
$583$	−22.5478	−0.933833
$584$	−8.23862	−0.340916
$585$	22.2628	0.920452
$586$	−15.5011	−0.640343
$587$	−4.37818	−0.180707	−0.0903535	−	0.995910i	$-0.528800\pi$
$587$	−4.37818	−0.180707	−0.0903535	+	0.995910i	$0.528800\pi$
$588$	0	0
$589$	13.8506	0.570704
$590$	51.7143	2.12905
$591$	3.41573	0.140505
$592$	−7.65544	−0.314637
$593$	23.5701	0.967908	0.483954	−	0.875093i	$-0.339200\pi$
$593$	23.5701	0.967908	0.483954	+	0.875093i	$0.339200\pi$
$594$	2.95100	0.121081
$595$	0	0
$596$	−9.76021	−0.399794
$597$	3.91462	0.160215
$598$	10.7337	0.438932
$599$	−22.1721	−0.905928	−0.452964	−	0.891529i	$-0.649633\pi$
$599$	−22.1721	−0.905928	−0.452964	+	0.891529i	$0.649633\pi$
$600$	14.0672	0.574292
$601$	−34.7190	−1.41622	−0.708109	−	0.706103i	$-0.750452\pi$
$601$	−34.7190	−1.41622	−0.708109	+	0.706103i	$0.750452\pi$
$602$	0	0
$603$	−4.46971	−0.182021
$604$	4.35317	0.177128
$605$	27.9727	1.13725
$606$	−27.0588	−1.09919
$607$	5.60938	0.227678	0.113839	−	0.993499i	$-0.463685\pi$
$607$	5.60938	0.227678	0.113839	+	0.993499i	$0.463685\pi$
$608$	−8.02271	−0.325364
$609$	0	0
$610$	−50.4214	−2.04150
$611$	22.7735	0.921316
$612$	5.16355	0.208724
$613$	14.1194	0.570275	0.285138	−	0.958487i	$-0.407961\pi$
$613$	14.1194	0.570275	0.285138	+	0.958487i	$0.407961\pi$
$614$	−55.9696	−2.25875
$615$	42.6940	1.72159
$616$	0	0
$617$	37.0094	1.48994	0.744970	−	0.667098i	$-0.232464\pi$
$617$	37.0094	1.48994	0.744970	+	0.667098i	$0.232464\pi$
$618$	15.1863	0.610881
$619$	−42.2855	−1.69960	−0.849799	−	0.527107i	$-0.823277\pi$
$619$	−42.2855	−1.69960	−0.849799	+	0.527107i	$0.823277\pi$
$620$	24.2353	0.973311
$621$	1.00000	0.0401286
$622$	−48.1962	−1.93249
$623$	0	0
$624$	31.5799	1.26421
$625$	−7.84323	−0.313729
$626$	0.153020	0.00611592
$627$	3.03638	0.121261
$628$	−12.8633	−0.513303
$629$	9.13967	0.364422
$630$	0	0
$631$	31.1770	1.24114	0.620568	−	0.784153i	$-0.286902\pi$
$631$	31.1770	1.24114	0.620568	+	0.784153i	$0.286902\pi$
$632$	−14.2257	−0.565867
$633$	−1.59430	−0.0633678
$634$	22.1900	0.881278
$635$	−1.68946	−0.0670442
$636$	11.2320	0.445380
$637$	0	0
$638$	−16.7754	−0.664146
$639$	−13.4612	−0.532515
$640$	45.2410	1.78831
$641$	35.9135	1.41850	0.709248	−	0.704959i	$-0.249035\pi$
$641$	35.9135	1.41850	0.709248	+	0.704959i	$0.249035\pi$
$642$	−34.2689	−1.35249
$643$	23.3808	0.922047	0.461024	−	0.887388i	$-0.347482\pi$
$643$	23.3808	0.922047	0.461024	+	0.887388i	$0.347482\pi$
$644$	0	0
$645$	−22.6070	−0.890149
$646$	17.5544	0.690667
$647$	3.09559	0.121700	0.0608501	−	0.998147i	$-0.480619\pi$
$647$	3.09559	0.121700	0.0608501	+	0.998147i	$0.480619\pi$
$648$	1.91702	0.0753078
$649$	−15.1487	−0.594637
$650$	78.7641	3.08938
$651$	0	0
$652$	1.83623	0.0719125
$653$	31.9927	1.25197	0.625985	−	0.779835i	$-0.284697\pi$
$653$	31.9927	1.25197	0.625985	+	0.779835i	$0.284697\pi$
$654$	5.61670	0.219630
$655$	−23.5953	−0.921944
$656$	60.5617	2.36454
$657$	−4.29761	−0.167666
$658$	0	0
$659$	−4.45396	−0.173502	−0.0867508	−	0.996230i	$-0.527648\pi$
$659$	−4.45396	−0.173502	−0.0867508	+	0.996230i	$0.527648\pi$
$660$	5.31294	0.206806
$661$	18.8131	0.731743	0.365872	−	0.930665i	$-0.380771\pi$
$661$	18.8131	0.731743	0.365872	+	0.930665i	$0.380771\pi$
$662$	−6.84206	−0.265924
$663$	−37.7026	−1.46425
$664$	14.8620	0.576756
$665$	0	0
$666$	−2.60200	−0.100825
$667$	−5.68466	−0.220111
$668$	12.5158	0.484250
$669$	−2.88667	−0.111605
$670$	−26.5885	−1.02720
$671$	14.7699	0.570186
$672$	0	0
$673$	−34.6850	−1.33701	−0.668505	−	0.743708i	$-0.733065\pi$
$673$	−34.6850	−1.33701	−0.668505	+	0.743708i	$0.733065\pi$
$674$	3.80173	0.146437
$675$	7.33805	0.282442
$676$	23.5851	0.907120
$677$	−12.9841	−0.499021	−0.249510	−	0.968372i	$-0.580270\pi$
$677$	−12.9841	−0.499021	−0.249510	+	0.968372i	$0.580270\pi$
$678$	−27.3008	−1.04848
$679$	0	0
$680$	−40.0558	−1.53607
$681$	−2.14324	−0.0821291
$682$	−23.4564	−0.898190
$683$	8.41814	0.322111	0.161055	−	0.986945i	$-0.448510\pi$
$683$	8.41814	0.322111	0.161055	+	0.986945i	$0.448510\pi$
$684$	−1.51256	−0.0578340
$685$	10.4099	0.397741
$686$	0	0
$687$	14.1308	0.539124
$688$	−32.0681	−1.22259
$689$	−82.0128	−3.12444
$690$	5.94860	0.226459
$691$	−37.8312	−1.43917	−0.719584	−	0.694406i	$-0.755667\pi$
$691$	−37.8312	−1.43917	−0.719584	+	0.694406i	$0.755667\pi$
$692$	8.45432	0.321385
$693$	0	0
$694$	−24.4644	−0.928655
$695$	−30.9162	−1.17272
$696$	−10.8976	−0.413073
$697$	−72.3034	−2.73869
$698$	−49.5251	−1.87455
$699$	−2.88187	−0.109002
$700$	0	0
$701$	7.93899	0.299851	0.149926	−	0.988697i	$-0.452097\pi$
$701$	7.93899	0.299851	0.149926	+	0.988697i	$0.452097\pi$
$702$	10.7337	0.405116
$703$	−2.67728	−0.100975
$704$	−3.77785	−0.142383
$705$	12.6211	0.475337
$706$	18.0522	0.679404
$707$	0	0
$708$	7.54622	0.283604
$709$	1.33730	0.0502235	0.0251117	−	0.999685i	$-0.492006\pi$
$709$	1.33730	0.0502235	0.0251117	+	0.999685i	$0.492006\pi$
$710$	−80.0751	−3.00516
$711$	−7.42071	−0.278299
$712$	−8.02281	−0.300668
$713$	−7.94860	−0.297678
$714$	0	0
$715$	−38.7934	−1.45079
$716$	−0.326829	−0.0122142
$717$	16.4956	0.616039
$718$	9.22056	0.344108
$719$	−43.6532	−1.62799	−0.813994	−	0.580873i	$-0.802711\pi$
$719$	−43.6532	−1.62799	−0.813994	+	0.580873i	$0.802711\pi$
$720$	17.5016	0.652246
$721$	0	0
$722$	27.0348	1.00613
$723$	18.0017	0.669489
$724$	19.3205	0.718042
$725$	−41.7143	−1.54923
$726$	13.4866	0.500534
$727$	−13.1675	−0.488356	−0.244178	−	0.969730i	$-0.578518\pi$
$727$	−13.1675	−0.488356	−0.244178	+	0.969730i	$0.578518\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−25.5648	−0.946194
$731$	38.2855	1.41604
$732$	−7.35755	−0.271943
$733$	34.4447	1.27224	0.636121	−	0.771589i	$-0.280538\pi$
$733$	34.4447	1.27224	0.636121	+	0.771589i	$0.280538\pi$
$734$	19.4146	0.716605
$735$	0	0
$736$	4.60408	0.169709
$737$	7.78856	0.286895
$738$	20.5843	0.757717
$739$	−7.05398	−0.259485	−0.129742	−	0.991548i	$-0.541415\pi$
$739$	−7.05398	−0.259485	−0.129742	+	0.991548i	$0.541415\pi$
$740$	−4.68460	−0.172209
$741$	11.0442	0.405719
$742$	0	0
$743$	−6.49211	−0.238172	−0.119086	−	0.992884i	$-0.537996\pi$
$743$	−6.49211	−0.238172	−0.119086	+	0.992884i	$0.537996\pi$
$744$	−15.2377	−0.558640
$745$	39.4956	1.44701
$746$	47.3948	1.73525
$747$	7.75262	0.283653
$748$	−8.99760	−0.328985
$749$	0	0
$750$	13.9082	0.507854
$751$	2.31669	0.0845372	0.0422686	−	0.999106i	$-0.486541\pi$
$751$	2.31669	0.0845372	0.0422686	+	0.999106i	$0.486541\pi$
$752$	17.9031	0.652858
$753$	−11.4373	−0.416798
$754$	−61.0172	−2.22211
$755$	−17.6155	−0.641095
$756$	0	0
$757$	−10.8685	−0.395021	−0.197510	−	0.980301i	$-0.563286\pi$
$757$	−10.8685	−0.395021	−0.197510	+	0.980301i	$0.563286\pi$
$758$	−18.9004	−0.686494
$759$	−1.74252	−0.0632495
$760$	11.7335	0.425620
$761$	14.1760	0.513881	0.256941	−	0.966427i	$-0.417285\pi$
$761$	14.1760	0.513881	0.256941	+	0.966427i	$0.417285\pi$
$762$	−0.814547	−0.0295079
$763$	0	0
$764$	2.59105	0.0937408
$765$	−20.8948	−0.755453
$766$	−29.2433	−1.05660
$767$	−55.1000	−1.98955
$768$	17.4762	0.630617
$769$	−11.0939	−0.400058	−0.200029	−	0.979790i	$-0.564104\pi$
$769$	−11.0939	−0.400058	−0.200029	+	0.979790i	$0.564104\pi$
$770$	0	0
$771$	30.4045	1.09499
$772$	4.32710	0.155736
$773$	−26.3956	−0.949384	−0.474692	−	0.880152i	$-0.657441\pi$
$773$	−26.3956	−0.949384	−0.474692	+	0.880152i	$0.657441\pi$
$774$	−10.8996	−0.391778
$775$	−58.3273	−2.09518
$776$	−10.6135	−0.381002
$777$	0	0
$778$	−47.1067	−1.68886
$779$	21.1798	0.758845
$780$	19.3247	0.691935
$781$	23.4564	0.839335
$782$	−10.0741	−0.360249
$783$	−5.68466	−0.203153
$784$	0	0
$785$	52.0527	1.85784
$786$	−11.3761	−0.405772
$787$	−21.0254	−0.749476	−0.374738	−	0.927131i	$-0.622267\pi$
$787$	−21.0254	−0.749476	−0.374738	+	0.927131i	$0.622267\pi$
$788$	2.96495	0.105622
$789$	−18.2226	−0.648743
$790$	−44.1429	−1.57053
$791$	0	0
$792$	−3.34045	−0.118698
$793$	53.7225	1.90774
$794$	−61.6910	−2.18933
$795$	−45.4516	−1.61200
$796$	3.39800	0.120439
$797$	3.60323	0.127633	0.0638165	−	0.997962i	$-0.479673\pi$
$797$	3.60323	0.127633	0.0638165	+	0.997962i	$0.479673\pi$
$798$	0	0
$799$	−21.3741	−0.756162
$800$	33.7850	1.19448
$801$	−4.18503	−0.147871
$802$	2.39812	0.0846805
$803$	7.48867	0.264270
$804$	−3.87983	−0.136831
$805$	0	0
$806$	−85.3176	−3.00518
$807$	−7.65714	−0.269544
$808$	30.6298	1.07755
$809$	−46.2734	−1.62689	−0.813443	−	0.581644i	$-0.802410\pi$
$809$	−46.2734	−1.62689	−0.813443	+	0.581644i	$0.802410\pi$
$810$	5.94860	0.209013
$811$	−23.3657	−0.820480	−0.410240	−	0.911978i	$-0.634555\pi$
$811$	−23.3657	−0.820480	−0.410240	+	0.911978i	$0.634555\pi$
$812$	0	0
$813$	−10.2628	−0.359931
$814$	4.53404	0.158918
$815$	−7.43049	−0.260279
$816$	−29.6394	−1.03759
$817$	−11.2149	−0.392361
$818$	−0.575297	−0.0201148
$819$	0	0
$820$	37.0596	1.29418
$821$	−29.6333	−1.03421	−0.517104	−	0.855923i	$-0.672990\pi$
$821$	−29.6333	−1.03421	−0.517104	+	0.855923i	$0.672990\pi$
$822$	5.01896	0.175056
$823$	3.93716	0.137241	0.0686203	−	0.997643i	$-0.478140\pi$
$823$	3.93716	0.137241	0.0686203	+	0.997643i	$0.478140\pi$
$824$	−17.1904	−0.598857
$825$	−12.7867	−0.445176
$826$	0	0
$827$	−21.6408	−0.752526	−0.376263	−	0.926513i	$-0.622791\pi$
$827$	−21.6408	−0.752526	−0.376263	+	0.926513i	$0.622791\pi$
$828$	0.868028	0.0301660
$829$	−20.5616	−0.714132	−0.357066	−	0.934079i	$-0.616223\pi$
$829$	−20.5616	−0.714132	−0.357066	+	0.934079i	$0.616223\pi$
$830$	46.1173	1.60075
$831$	7.83319	0.271730
$832$	−13.7411	−0.476388
$833$	0	0
$834$	−14.9058	−0.516144
$835$	−50.6463	−1.75269
$836$	2.63566	0.0911563
$837$	−7.94860	−0.274744
$838$	62.2054	2.14885
$839$	−14.3203	−0.494392	−0.247196	−	0.968965i	$-0.579509\pi$
$839$	−14.3203	−0.494392	−0.247196	+	0.968965i	$0.579509\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−33.3118	−1.14800
$843$	−2.34420	−0.0807387
$844$	−1.38390	−0.0476358
$845$	−95.4394	−3.28322
$846$	6.08506	0.209209
$847$	0	0
$848$	−64.4733	−2.21402
$849$	−0.517476	−0.0177597
$850$	−73.9244	−2.53558
$851$	1.53644	0.0526685
$852$	−11.6847	−0.400310
$853$	−6.04210	−0.206877	−0.103439	−	0.994636i	$-0.532985\pi$
$853$	−6.04210	−0.206877	−0.103439	+	0.994636i	$0.532985\pi$
$854$	0	0
$855$	6.12071	0.209324
$856$	38.7915	1.32587
$857$	−7.86521	−0.268670	−0.134335	−	0.990936i	$-0.542890\pi$
$857$	−7.86521	−0.268670	−0.134335	+	0.990936i	$0.542890\pi$
$858$	−18.7036	−0.638531
$859$	54.5758	1.86210	0.931051	−	0.364890i	$-0.118893\pi$
$859$	54.5758	1.86210	0.931051	+	0.364890i	$0.118893\pi$
$860$	−19.6235	−0.669155
$861$	0	0
$862$	4.26476	0.145258
$863$	−41.3174	−1.40646	−0.703230	−	0.710962i	$-0.748260\pi$
$863$	−41.3174	−1.40646	−0.703230	+	0.710962i	$0.748260\pi$
$864$	4.60408	0.156634
$865$	−34.2112	−1.16322
$866$	−47.1466	−1.60211
$867$	18.3859	0.624417
$868$	0	0
$869$	12.9308	0.438646
$870$	−33.8158	−1.14646
$871$	28.3293	0.959900
$872$	−6.35795	−0.215307
$873$	−5.53644	−0.187380
$874$	2.95100	0.0998192
$875$	0	0
$876$	−3.73044	−0.126040
$877$	22.5939	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$877$	22.5939	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$878$	−48.7496	−1.64522
$879$	9.15314	0.308728
$880$	−30.4969	−1.02805
$881$	32.4203	1.09227	0.546134	−	0.837698i	$-0.316099\pi$
$881$	32.4203	1.09227	0.546134	+	0.837698i	$0.316099\pi$
$882$	0	0
$883$	46.9261	1.57919	0.789595	−	0.613628i	$-0.210291\pi$
$883$	46.9261	1.57919	0.789595	+	0.613628i	$0.210291\pi$
$884$	−32.7269	−1.10072
$885$	−30.5365	−1.02647
$886$	20.9861	0.705042
$887$	−37.9121	−1.27296	−0.636482	−	0.771291i	$-0.719611\pi$
$887$	−37.9121	−1.27296	−0.636482	+	0.771291i	$0.719611\pi$
$888$	2.94539	0.0988409
$889$	0	0
$890$	−24.8951	−0.834486
$891$	−1.74252	−0.0583767
$892$	−2.50571	−0.0838975
$893$	6.26111	0.209520
$894$	19.0422	0.636867
$895$	1.32254	0.0442078
$896$	0	0
$897$	−6.33805	−0.211621
$898$	−0.608685	−0.0203121
$899$	45.1851	1.50701
$900$	6.36963	0.212321
$901$	76.9734	2.56435
$902$	−35.8685	−1.19429
$903$	0	0
$904$	30.9038	1.02784
$905$	−78.1824	−2.59887
$906$	−8.49306	−0.282163
$907$	−44.4382	−1.47555	−0.737773	−	0.675049i	$-0.764123\pi$
$907$	−44.4382	−1.47555	−0.737773	+	0.675049i	$0.764123\pi$
$908$	−1.86039	−0.0617393
$909$	15.9778	0.529950
$910$	0	0
$911$	−32.9725	−1.09243	−0.546215	−	0.837645i	$-0.683932\pi$
$911$	−32.9725	−1.09243	−0.546215	+	0.837645i	$0.683932\pi$
$912$	8.68226	0.287498
$913$	−13.5091	−0.447086
$914$	46.3478	1.53305
$915$	29.7730	0.984267
$916$	12.2659	0.405278
$917$	0	0
$918$	−10.0741	−0.332495
$919$	−41.8741	−1.38130	−0.690650	−	0.723189i	$-0.742676\pi$
$919$	−41.8741	−1.38130	−0.690650	+	0.723189i	$0.742676\pi$
$920$	−6.73365	−0.222002
$921$	33.0492	1.08901
$922$	−12.5868	−0.414524
$923$	85.3176	2.80826
$924$	0	0
$925$	11.2745	0.370703
$926$	−10.9291	−0.359153
$927$	−8.96725	−0.294523
$928$	−26.1726	−0.859159
$929$	−21.7717	−0.714306	−0.357153	−	0.934046i	$-0.616253\pi$
$929$	−21.7717	−0.714306	−0.357153	+	0.934046i	$0.616253\pi$
$930$	−47.2831	−1.55047
$931$	0	0
$932$	−2.50155	−0.0819409
$933$	28.4591	0.931708
$934$	71.6281	2.34374
$935$	36.4096	1.19072
$936$	−12.1502	−0.397142
$937$	−51.2836	−1.67536	−0.837681	−	0.546160i	$-0.816089\pi$
$937$	−51.2836	−1.67536	−0.837681	+	0.546160i	$0.816089\pi$
$938$	0	0
$939$	−0.0903560	−0.00294866
$940$	10.9554	0.357327
$941$	50.8018	1.65609	0.828046	−	0.560661i	$-0.189453\pi$
$941$	50.8018	1.65609	0.828046	+	0.560661i	$0.189453\pi$
$942$	25.0964	0.817685
$943$	−12.1547	−0.395811
$944$	−43.3162	−1.40982
$945$	0	0
$946$	18.9928	0.617509
$947$	40.5255	1.31690	0.658452	−	0.752623i	$-0.271212\pi$
$947$	40.5255	1.31690	0.658452	+	0.752623i	$0.271212\pi$
$948$	−6.44138	−0.209206
$949$	27.2385	0.884198
$950$	21.6546	0.702569
$951$	−13.1029	−0.424889
$952$	0	0
$953$	−6.05417	−0.196114	−0.0980570	−	0.995181i	$-0.531263\pi$
$953$	−6.05417	−0.196114	−0.0980570	+	0.995181i	$0.531263\pi$
$954$	−21.9138	−0.709484
$955$	−10.4849	−0.339284
$956$	14.3186	0.463097
$957$	9.90564	0.320204
$958$	−36.6508	−1.18413
$959$	0	0
$960$	−7.61535	−0.245784
$961$	32.1803	1.03807
$962$	16.4916	0.531711
$963$	20.2353	0.652072
$964$	15.6259	0.503278
$965$	−17.5100	−0.563668
$966$	0	0
$967$	40.4901	1.30207	0.651037	−	0.759046i	$-0.274334\pi$
$967$	40.4901	1.30207	0.651037	+	0.759046i	$0.274334\pi$
$968$	−15.2664	−0.490682
$969$	−10.3656	−0.332990
$970$	−32.9341	−1.05745
$971$	17.9470	0.575945	0.287973	−	0.957639i	$-0.407019\pi$
$971$	17.9470	0.575945	0.287973	+	0.957639i	$0.407019\pi$
$972$	0.868028	0.0278420
$973$	0	0
$974$	−12.2134	−0.391341
$975$	−46.5090	−1.48948
$976$	42.2333	1.35185
$977$	15.5013	0.495930	0.247965	−	0.968769i	$-0.420238\pi$
$977$	15.5013	0.495930	0.247965	+	0.968769i	$0.420238\pi$
$978$	−3.58250	−0.114556
$979$	7.29251	0.233070
$980$	0	0
$981$	−3.31657	−0.105890
$982$	12.5725	0.401205
$983$	24.1422	0.770018	0.385009	−	0.922913i	$-0.374198\pi$
$983$	24.1422	0.770018	0.385009	+	0.922913i	$0.374198\pi$
$984$	−23.3008	−0.742803
$985$	−11.9980	−0.382287
$986$	57.2679	1.82378
$987$	0	0
$988$	9.58667	0.304992
$989$	6.43605	0.204654
$990$	−10.3656	−0.329439
$991$	−3.26567	−0.103737	−0.0518687	−	0.998654i	$-0.516518\pi$
$991$	−3.26567	−0.103737	−0.0518687	+	0.998654i	$0.516518\pi$
$992$	−36.5960	−1.16193
$993$	4.04013	0.128210
$994$	0	0
$995$	−13.7503	−0.435915
$996$	6.72949	0.213232
$997$	16.6749	0.528101	0.264050	−	0.964509i	$-0.414942\pi$
$997$	16.6749	0.528101	0.264050	+	0.964509i	$0.414942\pi$
$998$	50.4919	1.59829
$999$	1.53644	0.0486108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.x.1.1		4
7.6	odd	2		483.2.a.j.1.1	✓	4
21.20	even	2		1449.2.a.o.1.4		4
28.27	even	2		7728.2.a.ce.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.1	✓	4	7.6	odd	2
1449.2.a.o.1.4		4	21.20	even	2
3381.2.a.x.1.1		4	1.1	even	1	trivial
7728.2.a.ce.1.4		4	28.27	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69353 q^{2} +1.00000 q^{3} +0.868028 q^{4} -3.51256 q^{5} -1.69353 q^{6} +1.91702 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.69353 q^{2} +1.00000 q^{3} +0.868028 q^{4} -3.51256 q^{5} -1.69353 q^{6} +1.91702 q^{8} +1.00000 q^{9} +5.94860 q^{10} -1.74252 q^{11} +0.868028 q^{12} -6.33805 q^{13} -3.51256 q^{15} -4.98258 q^{16} +5.94860 q^{17} -1.69353 q^{18} -1.74252 q^{19} -3.04900 q^{20} +2.95100 q^{22} +1.00000 q^{23} +1.91702 q^{24} +7.33805 q^{25} +10.7337 q^{26} +1.00000 q^{27} -5.68466 q^{29} +5.94860 q^{30} -7.94860 q^{31} +4.60408 q^{32} -1.74252 q^{33} -10.0741 q^{34} +0.868028 q^{36} +1.53644 q^{37} +2.95100 q^{38} -6.33805 q^{39} -6.73365 q^{40} -12.1547 q^{41} +6.43605 q^{43} -1.51256 q^{44} -3.51256 q^{45} -1.69353 q^{46} -3.59313 q^{47} -4.98258 q^{48} -12.4272 q^{50} +5.94860 q^{51} -5.50161 q^{52} +12.9397 q^{53} -1.69353 q^{54} +6.12071 q^{55} -1.74252 q^{57} +9.62711 q^{58} +8.69353 q^{59} -3.04900 q^{60} -8.47618 q^{61} +13.4612 q^{62} +2.16804 q^{64} +22.2628 q^{65} +2.95100 q^{66} -4.46971 q^{67} +5.16355 q^{68} +1.00000 q^{69} -13.4612 q^{71} +1.91702 q^{72} -4.29761 q^{73} -2.60200 q^{74} +7.33805 q^{75} -1.51256 q^{76} +10.7337 q^{78} -7.42071 q^{79} +17.5016 q^{80} +1.00000 q^{81} +20.5843 q^{82} +7.75262 q^{83} -20.8948 q^{85} -10.8996 q^{86} -5.68466 q^{87} -3.34045 q^{88} -4.18503 q^{89} +5.94860 q^{90} +0.868028 q^{92} -7.94860 q^{93} +6.08506 q^{94} +6.12071 q^{95} +4.60408 q^{96} -5.53644 q^{97} -1.74252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9} - 2 q^{10} + q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 2 q^{17} + 2 q^{18} + q^{19} - 13 q^{20} + 11 q^{22} + 4 q^{23} + 9 q^{24} + 11 q^{25} + 19 q^{26} + 4 q^{27} + 2 q^{29} - 2 q^{30} - 6 q^{31} + 20 q^{32} + q^{33} - 23 q^{34} + 4 q^{36} + 16 q^{37} + 11 q^{38} - 7 q^{39} - 3 q^{40} - 5 q^{41} + 9 q^{43} + 3 q^{44} - 5 q^{45} + 2 q^{46} + 21 q^{47} + 8 q^{48} - 17 q^{50} - 2 q^{51} + 24 q^{52} + 10 q^{53} + 2 q^{54} - 17 q^{55} + q^{57} + q^{58} + 26 q^{59} - 13 q^{60} - 2 q^{61} + 19 q^{62} + 27 q^{64} + 26 q^{65} + 11 q^{66} + 5 q^{67} - 7 q^{68} + 4 q^{69} - 19 q^{71} + 9 q^{72} - 10 q^{73} + 9 q^{74} + 11 q^{75} + 3 q^{76} + 19 q^{78} - 6 q^{79} + 24 q^{80} + 4 q^{81} + 31 q^{82} + 2 q^{83} - 7 q^{85} - 17 q^{86} + 2 q^{87} - 20 q^{88} + 17 q^{89} - 2 q^{90} + 4 q^{92} - 6 q^{93} + 44 q^{94} - 17 q^{95} + 20 q^{96} - 32 q^{97} + q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9 - 2 * q^10 + q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 2 * q^17 + 2 * q^18 + q^19 - 13 * q^20 + 11 * q^22 + 4 * q^23 + 9 * q^24 + 11 * q^25 + 19 * q^26 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 6 * q^31 + 20 * q^32 + q^33 - 23 * q^34 + 4 * q^36 + 16 * q^37 + 11 * q^38 - 7 * q^39 - 3 * q^40 - 5 * q^41 + 9 * q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + 21 * q^47 + 8 * q^48 - 17 * q^50 - 2 * q^51 + 24 * q^52 + 10 * q^53 + 2 * q^54 - 17 * q^55 + q^57 + q^58 + 26 * q^59 - 13 * q^60 - 2 * q^61 + 19 * q^62 + 27 * q^64 + 26 * q^65 + 11 * q^66 + 5 * q^67 - 7 * q^68 + 4 * q^69 - 19 * q^71 + 9 * q^72 - 10 * q^73 + 9 * q^74 + 11 * q^75 + 3 * q^76 + 19 * q^78 - 6 * q^79 + 24 * q^80 + 4 * q^81 + 31 * q^82 + 2 * q^83 - 7 * q^85 - 17 * q^86 + 2 * q^87 - 20 * q^88 + 17 * q^89 - 2 * q^90 + 4 * q^92 - 6 * q^93 + 44 * q^94 - 17 * q^95 + 20 * q^96 - 32 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more