LMFDB - Embedded newform 3971.2.a.k.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3971,2,Mod(1,3971)] 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("3971.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3971, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3971 = 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3971.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.7085946427$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 13x^{7} + 12x^{6} + 51x^{5} - 38x^{4} - 70x^{3} + 30x^{2} + 27x + 3 $ x^9 - x^8 - 13x^7 + 12x^6 + 51x^5 - 38x^4 - 70x^3 + 30x^2 + 27*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.138860$ of defining polynomial
Character	$\chi$	$=$	3971.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.138860 q^{2} -1.49679 q^{3} -1.98072 q^{4} -3.51650 q^{5} -0.207844 q^{6} -1.46902 q^{7} -0.552764 q^{8} -0.759634 q^{9} -0.488302 q^{10} -1.00000 q^{11} +2.96471 q^{12} +2.39255 q^{13} -0.203989 q^{14} +5.26344 q^{15} +3.88468 q^{16} -3.24041 q^{17} -0.105483 q^{18} +6.96519 q^{20} +2.19881 q^{21} -0.138860 q^{22} -7.38177 q^{23} +0.827369 q^{24} +7.36576 q^{25} +0.332231 q^{26} +5.62736 q^{27} +2.90971 q^{28} +6.40954 q^{29} +0.730884 q^{30} +0.866600 q^{31} +1.64496 q^{32} +1.49679 q^{33} -0.449965 q^{34} +5.16581 q^{35} +1.50462 q^{36} -2.78244 q^{37} -3.58113 q^{39} +1.94379 q^{40} +9.85358 q^{41} +0.305327 q^{42} +7.30504 q^{43} +1.98072 q^{44} +2.67125 q^{45} -1.02504 q^{46} +3.35902 q^{47} -5.81453 q^{48} -4.84198 q^{49} +1.02281 q^{50} +4.85020 q^{51} -4.73897 q^{52} -3.78360 q^{53} +0.781418 q^{54} +3.51650 q^{55} +0.812022 q^{56} +0.890031 q^{58} +8.40207 q^{59} -10.4254 q^{60} -1.43767 q^{61} +0.120336 q^{62} +1.11592 q^{63} -7.54094 q^{64} -8.41340 q^{65} +0.207844 q^{66} +6.48612 q^{67} +6.41834 q^{68} +11.0489 q^{69} +0.717326 q^{70} -3.41568 q^{71} +0.419898 q^{72} +2.81088 q^{73} -0.386370 q^{74} -11.0250 q^{75} +1.46902 q^{77} -0.497278 q^{78} -17.7188 q^{79} -13.6605 q^{80} -6.14405 q^{81} +1.36827 q^{82} +17.7129 q^{83} -4.35522 q^{84} +11.3949 q^{85} +1.01438 q^{86} -9.59370 q^{87} +0.552764 q^{88} +12.8724 q^{89} +0.370931 q^{90} -3.51471 q^{91} +14.6212 q^{92} -1.29711 q^{93} +0.466435 q^{94} -2.46215 q^{96} -15.7259 q^{97} -0.672359 q^{98} +0.759634 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 9 q^{4} + 3 q^{6} - 3 q^{7} + 11 q^{9} - 21 q^{10} - 9 q^{11} + 8 q^{12} - 11 q^{13} - q^{14} - 7 q^{15} + 13 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{20} - 24 q^{21} + q^{22} + q^{23} + 3 q^{24}+ \cdots - 11 q^{99}+O(q^{100}) $ 9 * q - q^2 + 9 * q^4 + 3 * q^6 - 3 * q^7 + 11 * q^9 - 21 * q^10 - 9 * q^11 + 8 * q^12 - 11 * q^13 - q^14 - 7 * q^15 + 13 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^20 - 24 * q^21 + q^22 + q^23 + 3 * q^24 + 7 * q^25 - 2 * q^26 + 3 * q^27 + 2 * q^28 - 13 * q^29 - 32 * q^30 - 16 * q^31 + 14 * q^32 - 8 * q^34 + 12 * q^35 - 10 * q^36 - 18 * q^37 + 10 * q^39 - 32 * q^40 - 4 * q^41 - 11 * q^42 - q^43 - 9 * q^44 + 11 * q^45 - 7 * q^46 + 14 * q^47 - 37 * q^48 - 6 * q^49 + 15 * q^50 + 16 * q^51 - 57 * q^52 + 4 * q^53 + 33 * q^54 - 17 * q^56 + 4 * q^58 - 31 * q^59 - 9 * q^60 - 3 * q^61 - 5 * q^62 - 50 * q^63 + 32 * q^64 - 28 * q^65 - 3 * q^66 + 2 * q^67 - 48 * q^68 + 58 * q^70 + 12 * q^71 - 5 * q^72 + 26 * q^73 - 35 * q^74 + 25 * q^75 + 3 * q^77 - 3 * q^78 - 31 * q^79 - 31 * q^80 + 21 * q^81 - 25 * q^82 + 18 * q^83 - 51 * q^84 + 11 * q^85 - 41 * q^86 - 62 * q^87 + 11 * q^89 - 51 * q^90 + 20 * q^91 + 33 * q^92 - 20 * q^93 - 43 * q^94 + 30 * q^96 - 16 * q^97 - 4 * q^98 - 11 * 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 $

$2$	0.138860	0.0981891	0.0490946	−	0.998794i	$-0.484366\pi$
$2$	0.138860	0.0981891	0.0490946	+	0.998794i	$0.484366\pi$
$3$	−1.49679	−0.864169	−0.432085	−	0.901833i	$-0.642222\pi$
$3$	−1.49679	−0.864169	−0.432085	+	0.901833i	$0.642222\pi$
$4$	−1.98072	−0.990359
$5$	−3.51650	−1.57263	−0.786313	−	0.617828i	$-0.788013\pi$
$5$	−3.51650	−1.57263	−0.786313	+	0.617828i	$0.788013\pi$
$6$	−0.207844	−0.0848520
$7$	−1.46902	−0.555237	−0.277619	−	0.960691i	$-0.589545\pi$
$7$	−1.46902	−0.555237	−0.277619	+	0.960691i	$0.589545\pi$
$8$	−0.552764	−0.195432
$9$	−0.759634	−0.253211
$10$	−0.488302	−0.154415
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.96471	0.855838
$13$	2.39255	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$13$	2.39255	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$14$	−0.203989	−0.0545183
$15$	5.26344	1.35902
$16$	3.88468	0.971170
$17$	−3.24041	−0.785915	−0.392957	−	0.919557i	$-0.628548\pi$
$17$	−3.24041	−0.785915	−0.392957	+	0.919557i	$0.628548\pi$
$18$	−0.105483	−0.0248626
$19$	0	0
$19$	0	0
$20$	6.96519	1.55746
$21$	2.19881	0.479819
$22$	−0.138860	−0.0296051
$23$	−7.38177	−1.53921	−0.769603	−	0.638523i	$-0.779546\pi$
$23$	−7.38177	−1.53921	−0.769603	+	0.638523i	$0.779546\pi$
$24$	0.827369	0.168886
$25$	7.36576	1.47315
$26$	0.332231	0.0651558
$27$	5.62736	1.08299
$28$	2.90971	0.549884
$29$	6.40954	1.19022	0.595111	−	0.803644i	$-0.297108\pi$
$29$	6.40954	1.19022	0.595111	+	0.803644i	$0.297108\pi$
$30$	0.730884	0.133441
$31$	0.866600	0.155646	0.0778230	−	0.996967i	$-0.475203\pi$
$31$	0.866600	0.155646	0.0778230	+	0.996967i	$0.475203\pi$
$32$	1.64496	0.290790
$33$	1.49679	0.260557
$34$	−0.449965	−0.0771683
$35$	5.16581	0.873181
$36$	1.50462	0.250770
$37$	−2.78244	−0.457430	−0.228715	−	0.973493i	$-0.573452\pi$
$37$	−2.78244	−0.457430	−0.228715	+	0.973493i	$0.573452\pi$
$38$	0	0
$39$	−3.58113	−0.573440
$40$	1.94379	0.307341
$41$	9.85358	1.53887	0.769435	−	0.638725i	$-0.220538\pi$
$41$	9.85358	1.53887	0.769435	+	0.638725i	$0.220538\pi$
$42$	0.305327	0.0471130
$43$	7.30504	1.11401	0.557004	−	0.830510i	$-0.311951\pi$
$43$	7.30504	1.11401	0.557004	+	0.830510i	$0.311951\pi$
$44$	1.98072	0.298604
$45$	2.67125	0.398207
$46$	−1.02504	−0.151133
$47$	3.35902	0.489964	0.244982	−	0.969528i	$-0.421218\pi$
$47$	3.35902	0.489964	0.244982	+	0.969528i	$0.421218\pi$
$48$	−5.81453	−0.839255
$49$	−4.84198	−0.691711
$50$	1.02281	0.144648
$51$	4.85020	0.679164
$52$	−4.73897	−0.657177
$53$	−3.78360	−0.519718	−0.259859	−	0.965647i	$-0.583676\pi$
$53$	−3.78360	−0.519718	−0.259859	+	0.965647i	$0.583676\pi$
$54$	0.781418	0.106338
$55$	3.51650	0.474165
$56$	0.812022	0.108511
$57$	0	0
$58$	0.890031	0.116867
$59$	8.40207	1.09386	0.546928	−	0.837180i	$-0.315797\pi$
$59$	8.40207	1.09386	0.546928	+	0.837180i	$0.315797\pi$
$60$	−10.4254	−1.34591
$61$	−1.43767	−0.184075	−0.0920375	−	0.995756i	$-0.529338\pi$
$61$	−1.43767	−0.184075	−0.0920375	+	0.995756i	$0.529338\pi$
$62$	0.120336	0.0152827
$63$	1.11592	0.140592
$64$	−7.54094	−0.942617
$65$	−8.41340	−1.04355
$66$	0.207844	0.0255839
$67$	6.48612	0.792406	0.396203	−	0.918163i	$-0.370328\pi$
$67$	6.48612	0.792406	0.396203	+	0.918163i	$0.370328\pi$
$68$	6.41834	0.778338
$69$	11.0489	1.33013
$70$	0.717326	0.0857369
$71$	−3.41568	−0.405367	−0.202683	−	0.979244i	$-0.564966\pi$
$71$	−3.41568	−0.405367	−0.202683	+	0.979244i	$0.564966\pi$
$72$	0.419898	0.0494855
$73$	2.81088	0.328989	0.164495	−	0.986378i	$-0.447401\pi$
$73$	2.81088	0.328989	0.164495	+	0.986378i	$0.447401\pi$
$74$	−0.386370	−0.0449146
$75$	−11.0250	−1.27305
$76$	0	0
$77$	1.46902	0.167410
$78$	−0.497278	−0.0563056
$79$	−17.7188	−1.99353	−0.996763	−	0.0803939i	$-0.974382\pi$
$79$	−17.7188	−1.99353	−0.996763	+	0.0803939i	$0.974382\pi$
$80$	−13.6605	−1.52729
$81$	−6.14405	−0.682673
$82$	1.36827	0.151100
$83$	17.7129	1.94425	0.972124	−	0.234467i	$-0.0753345\pi$
$83$	17.7129	1.94425	0.972124	+	0.234467i	$0.0753345\pi$
$84$	−4.35522	−0.475193
$85$	11.3949	1.23595
$86$	1.01438	0.109383
$87$	−9.59370	−1.02855
$88$	0.552764	0.0589249
$89$	12.8724	1.36447	0.682236	−	0.731132i	$-0.261007\pi$
$89$	12.8724	1.36447	0.682236	+	0.731132i	$0.261007\pi$
$90$	0.370931	0.0390996
$91$	−3.51471	−0.368441
$92$	14.6212	1.52437
$93$	−1.29711	−0.134505
$94$	0.466435	0.0481091
$95$	0	0
$96$	−2.46215	−0.251292
$97$	−15.7259	−1.59673	−0.798364	−	0.602175i	$-0.794301\pi$
$97$	−15.7259	−1.59673	−0.798364	+	0.602175i	$0.794301\pi$
$98$	−0.672359	−0.0679185
$99$	0.759634	0.0763461
$100$	−14.5895	−1.45895
$101$	−7.83751	−0.779861	−0.389931	−	0.920844i	$-0.627501\pi$
$101$	−7.83751	−0.779861	−0.389931	+	0.920844i	$0.627501\pi$
$102$	0.673501	0.0666865
$103$	−7.27715	−0.717039	−0.358519	−	0.933522i	$-0.616718\pi$
$103$	−7.27715	−0.717039	−0.358519	+	0.933522i	$0.616718\pi$
$104$	−1.32252	−0.129683
$105$	−7.73211	−0.754576
$106$	−0.525392	−0.0510306
$107$	6.23745	0.602997	0.301498	−	0.953467i	$-0.402513\pi$
$107$	6.23745	0.602997	0.301498	+	0.953467i	$0.402513\pi$
$108$	−11.1462	−1.07255
$109$	2.71842	0.260377	0.130189	−	0.991489i	$-0.458442\pi$
$109$	2.71842	0.260377	0.130189	+	0.991489i	$0.458442\pi$
$110$	0.488302	0.0465578
$111$	4.16471	0.395297
$112$	−5.70667	−0.539230
$113$	0.486637	0.0457789	0.0228895	−	0.999738i	$-0.492713\pi$
$113$	0.486637	0.0457789	0.0228895	+	0.999738i	$0.492713\pi$
$114$	0	0
$115$	25.9580	2.42060
$116$	−12.6955	−1.17875
$117$	−1.81746	−0.168025
$118$	1.16671	0.107405
$119$	4.76023	0.436369
$120$	−2.90944	−0.265595
$121$	1.00000	0.0909091
$122$	−0.199636	−0.0180742
$123$	−14.7487	−1.32984
$124$	−1.71649	−0.154145
$125$	−8.31921	−0.744093
$126$	0.154957	0.0138046
$127$	3.68254	0.326773	0.163386	−	0.986562i	$-0.447758\pi$
$127$	3.68254	0.326773	0.163386	+	0.986562i	$0.447758\pi$
$128$	−4.33705	−0.383345
$129$	−10.9341	−0.962691
$130$	−1.16829	−0.102466
$131$	0.0541158	0.00472812	0.00236406	−	0.999997i	$-0.499247\pi$
$131$	0.0541158	0.00472812	0.00236406	+	0.999997i	$0.499247\pi$
$132$	−2.96471	−0.258045
$133$	0	0
$134$	0.900665	0.0778056
$135$	−19.7886	−1.70313
$136$	1.79118	0.153593
$137$	16.9727	1.45008	0.725038	−	0.688709i	$-0.241822\pi$
$137$	16.9727	1.45008	0.725038	+	0.688709i	$0.241822\pi$
$138$	1.53426	0.130605
$139$	−18.0362	−1.52981	−0.764907	−	0.644141i	$-0.777215\pi$
$139$	−18.0362	−1.52981	−0.764907	+	0.644141i	$0.777215\pi$
$140$	−10.2320	−0.864763
$141$	−5.02773	−0.423412
$142$	−0.474303	−0.0398026
$143$	−2.39255	−0.200075
$144$	−2.95093	−0.245911
$145$	−22.5391	−1.87177
$146$	0.390320	0.0323031
$147$	7.24740	0.597756
$148$	5.51122	0.453020
$149$	−5.30375	−0.434501	−0.217250	−	0.976116i	$-0.569709\pi$
$149$	−5.30375	−0.434501	−0.217250	+	0.976116i	$0.569709\pi$
$150$	−1.53093	−0.125000
$151$	−10.8143	−0.880057	−0.440029	−	0.897984i	$-0.645032\pi$
$151$	−10.8143	−0.880057	−0.440029	+	0.897984i	$0.645032\pi$
$152$	0	0
$153$	2.46153	0.199003
$154$	0.203989	0.0164379
$155$	−3.04740	−0.244773
$156$	7.09322	0.567912
$157$	3.19528	0.255011	0.127506	−	0.991838i	$-0.459303\pi$
$157$	3.19528	0.255011	0.127506	+	0.991838i	$0.459303\pi$
$158$	−2.46045	−0.195743
$159$	5.66324	0.449124
$160$	−5.78449	−0.457304
$161$	10.8440	0.854625
$162$	−0.853166	−0.0670310
$163$	−14.3438	−1.12349	−0.561747	−	0.827309i	$-0.689871\pi$
$163$	−14.3438	−1.12349	−0.561747	+	0.827309i	$0.689871\pi$
$164$	−19.5172	−1.52403
$165$	−5.26344	−0.409759
$166$	2.45963	0.190904
$167$	15.2567	1.18060	0.590298	−	0.807186i	$-0.299010\pi$
$167$	15.2567	1.18060	0.590298	+	0.807186i	$0.299010\pi$
$168$	−1.21542	−0.0937718
$169$	−7.27570	−0.559669
$170$	1.58230	0.121357
$171$	0	0
$172$	−14.4692	−1.10327
$173$	−21.6749	−1.64791	−0.823954	−	0.566656i	$-0.808237\pi$
$173$	−21.6749	−1.64791	−0.823954	+	0.566656i	$0.808237\pi$
$174$	−1.33219	−0.100993
$175$	−10.8205	−0.817950
$176$	−3.88468	−0.292819
$177$	−12.5761	−0.945277
$178$	1.78747	0.133976
$179$	4.19425	0.313493	0.156746	−	0.987639i	$-0.449899\pi$
$179$	4.19425	0.313493	0.156746	+	0.987639i	$0.449899\pi$
$180$	−5.29100	−0.394368
$181$	−24.9278	−1.85287	−0.926436	−	0.376453i	$-0.877144\pi$
$181$	−24.9278	−1.85287	−0.926436	+	0.376453i	$0.877144\pi$
$182$	−0.488053	−0.0361769
$183$	2.15189	0.159072
$184$	4.08038	0.300810
$185$	9.78444	0.719366
$186$	−0.180118	−0.0132069
$187$	3.24041	0.236962
$188$	−6.65327	−0.485240
$189$	−8.26671	−0.601315
$190$	0	0
$191$	−7.74450	−0.560372	−0.280186	−	0.959946i	$-0.590396\pi$
$191$	−7.74450	−0.560372	−0.280186	+	0.959946i	$0.590396\pi$
$192$	11.2872	0.814581
$193$	13.9096	1.00123	0.500617	−	0.865669i	$-0.333106\pi$
$193$	13.9096	1.00123	0.500617	+	0.865669i	$0.333106\pi$
$194$	−2.18371	−0.156781
$195$	12.5931	0.901807
$196$	9.59059	0.685042
$197$	20.3595	1.45056	0.725279	−	0.688455i	$-0.241711\pi$
$197$	20.3595	1.45056	0.725279	+	0.688455i	$0.241711\pi$
$198$	0.105483	0.00749636
$199$	3.03426	0.215093	0.107546	−	0.994200i	$-0.465701\pi$
$199$	3.03426	0.215093	0.107546	+	0.994200i	$0.465701\pi$
$200$	−4.07153	−0.287901
$201$	−9.70833	−0.684773
$202$	−1.08832	−0.0765739
$203$	−9.41574	−0.660855
$204$	−9.60687	−0.672616
$205$	−34.6501	−2.42007
$206$	−1.01051	−0.0704054
$207$	5.60745	0.389744
$208$	9.29429	0.644443
$209$	0	0
$210$	−1.07368	−0.0740912
$211$	5.01758	0.345425	0.172712	−	0.984972i	$-0.444747\pi$
$211$	5.01758	0.345425	0.172712	+	0.984972i	$0.444747\pi$
$212$	7.49425	0.514707
$213$	5.11254	0.350305
$214$	0.866134	0.0592077
$215$	−25.6882	−1.75192
$216$	−3.11061	−0.211650
$217$	−1.27305	−0.0864205
$218$	0.377481	0.0255662
$219$	−4.20729	−0.284302
$220$	−6.96519	−0.469593
$221$	−7.75285	−0.521513
$222$	0.578313	0.0388139
$223$	−11.8616	−0.794314	−0.397157	−	0.917751i	$-0.630003\pi$
$223$	−11.8616	−0.794314	−0.397157	+	0.917751i	$0.630003\pi$
$224$	−2.41647	−0.161457
$225$	−5.59529	−0.373019
$226$	0.0675746	0.00449499
$227$	−1.40838	−0.0934777	−0.0467388	−	0.998907i	$-0.514883\pi$
$227$	−1.40838	−0.0934777	−0.0467388	+	0.998907i	$0.514883\pi$
$228$	0	0
$229$	11.3642	0.750971	0.375485	−	0.926828i	$-0.377476\pi$
$229$	11.3642	0.750971	0.375485	+	0.926828i	$0.377476\pi$
$230$	3.60454	0.237676
$231$	−2.19881	−0.144671
$232$	−3.54296	−0.232607
$233$	−12.8001	−0.838563	−0.419281	−	0.907856i	$-0.637718\pi$
$233$	−12.8001	−0.838563	−0.419281	+	0.907856i	$0.637718\pi$
$234$	−0.252374	−0.0164982
$235$	−11.8120	−0.770530
$236$	−16.6421	−1.08331
$237$	26.5213	1.72274
$238$	0.661007	0.0428467
$239$	−3.62977	−0.234790	−0.117395	−	0.993085i	$-0.537454\pi$
$239$	−3.62977	−0.234790	−0.117395	+	0.993085i	$0.537454\pi$
$240$	20.4468	1.31983
$241$	19.3216	1.24461	0.622306	−	0.782774i	$-0.286196\pi$
$241$	19.3216	1.24461	0.622306	+	0.782774i	$0.286196\pi$
$242$	0.138860	0.00892629
$243$	−7.68576	−0.493042
$244$	2.84762	0.182300
$245$	17.0268	1.08780
$246$	−2.04801	−0.130576
$247$	0	0
$248$	−0.479026	−0.0304182
$249$	−26.5125	−1.68016
$250$	−1.15521	−0.0730618
$251$	18.1298	1.14434	0.572172	−	0.820133i	$-0.306101\pi$
$251$	18.1298	1.14434	0.572172	+	0.820133i	$0.306101\pi$
$252$	−2.21032	−0.139237
$253$	7.38177	0.464088
$254$	0.511360	0.0320856
$255$	−17.0557	−1.06807
$256$	14.4796	0.904977
$257$	1.55938	0.0972713	0.0486357	−	0.998817i	$-0.484513\pi$
$257$	1.55938	0.0972713	0.0486357	+	0.998817i	$0.484513\pi$
$258$	−1.51831	−0.0945258
$259$	4.08746	0.253982
$260$	16.6646	1.03349
$261$	−4.86890	−0.301378
$262$	0.00751455	0.000464250 0
$263$	17.2721	1.06504	0.532522	−	0.846416i	$-0.321244\pi$
$263$	17.2721	1.06504	0.532522	+	0.846416i	$0.321244\pi$
$264$	−0.827369	−0.0509211
$265$	13.3050	0.817321
$266$	0	0
$267$	−19.2672	−1.17914
$268$	−12.8472	−0.784766
$269$	−13.9309	−0.849384	−0.424692	−	0.905338i	$-0.639618\pi$
$269$	−13.9309	−0.849384	−0.424692	+	0.905338i	$0.639618\pi$
$270$	−2.74786	−0.167229
$271$	6.15305	0.373771	0.186886	−	0.982382i	$-0.440161\pi$
$271$	6.15305	0.373771	0.186886	+	0.982382i	$0.440161\pi$
$272$	−12.5880	−0.763257
$273$	5.26076	0.318396
$274$	2.35684	0.142382
$275$	−7.36576	−0.444172
$276$	−21.8848	−1.31731
$277$	−18.4147	−1.10643	−0.553215	−	0.833039i	$-0.686599\pi$
$277$	−18.4147	−1.10643	−0.553215	+	0.833039i	$0.686599\pi$
$278$	−2.50452	−0.150211
$279$	−0.658299	−0.0394113
$280$	−2.85547	−0.170647
$281$	24.8330	1.48141	0.740706	−	0.671830i	$-0.234491\pi$
$281$	24.8330	1.48141	0.740706	+	0.671830i	$0.234491\pi$
$282$	−0.698153	−0.0415744
$283$	17.1761	1.02101	0.510506	−	0.859874i	$-0.329458\pi$
$283$	17.1761	1.02101	0.510506	+	0.859874i	$0.329458\pi$
$284$	6.76550	0.401458
$285$	0	0
$286$	−0.332231	−0.0196452
$287$	−14.4751	−0.854438
$288$	−1.24956	−0.0736313
$289$	−6.49974	−0.382338
$290$	−3.12979	−0.183788
$291$	23.5384	1.37984
$292$	−5.56757	−0.325817
$293$	23.1770	1.35402	0.677008	−	0.735976i	$-0.263276\pi$
$293$	23.1770	1.35402	0.677008	+	0.735976i	$0.263276\pi$
$294$	1.00638	0.0586931
$295$	−29.5459	−1.72023
$296$	1.53803	0.0893963
$297$	−5.62736	−0.326533
$298$	−0.736482	−0.0426632
$299$	−17.6613	−1.02138
$300$	21.8373	1.26078
$301$	−10.7312	−0.618539
$302$	−1.50168	−0.0864121
$303$	11.7311	0.673932
$304$	0	0
$305$	5.05557	0.289481
$306$	0.341809	0.0195399
$307$	26.3151	1.50188	0.750940	−	0.660370i	$-0.229601\pi$
$307$	26.3151	1.50188	0.750940	+	0.660370i	$0.229601\pi$
$308$	−2.90971	−0.165796
$309$	10.8923	0.619643
$310$	−0.423163	−0.0240341
$311$	25.2900	1.43407	0.717033	−	0.697039i	$-0.245500\pi$
$311$	25.2900	1.43407	0.717033	+	0.697039i	$0.245500\pi$
$312$	1.97952	0.112068
$313$	−27.6880	−1.56502	−0.782509	−	0.622640i	$-0.786060\pi$
$313$	−27.6880	−1.56502	−0.782509	+	0.622640i	$0.786060\pi$
$314$	0.443698	0.0250393
$315$	−3.92412	−0.221099
$316$	35.0960	1.97431
$317$	10.1236	0.568599	0.284299	−	0.958736i	$-0.408239\pi$
$317$	10.1236	0.568599	0.284299	+	0.958736i	$0.408239\pi$
$318$	0.786400	0.0440991
$319$	−6.40954	−0.358865
$320$	26.5177	1.48238
$321$	−9.33612	−0.521091
$322$	1.50580	0.0839149
$323$	0	0
$324$	12.1696	0.676091
$325$	17.6230	0.977546
$326$	−1.99179	−0.110315
$327$	−4.06889	−0.225010
$328$	−5.44670	−0.300744
$329$	−4.93447	−0.272046
$330$	−0.730884	−0.0402338
$331$	8.45572	0.464769	0.232384	−	0.972624i	$-0.425347\pi$
$331$	8.45572	0.464769	0.232384	+	0.972624i	$0.425347\pi$
$332$	−35.0843	−1.92550
$333$	2.11363	0.115826
$334$	2.11855	0.115922
$335$	−22.8084	−1.24616
$336$	8.54166	0.465986
$337$	0.0173781	0.000946646 0	0.000473323	−	1.00000i	$-0.499849\pi$
$337$	0.0173781	0.000946646 0	0.000473323		1.00000i	$0.499849\pi$
$338$	−1.01031	−0.0549534
$339$	−0.728391	−0.0395608
$340$	−22.5701	−1.22403
$341$	−0.866600	−0.0469290
$342$	0	0
$343$	17.3961	0.939302
$344$	−4.03796	−0.217712
$345$	−38.8535	−2.09180
$346$	−3.00978	−0.161807
$347$	8.63862	0.463746	0.231873	−	0.972746i	$-0.425515\pi$
$347$	8.63862	0.463746	0.231873	+	0.972746i	$0.425515\pi$
$348$	19.0024	1.01864
$349$	−3.41607	−0.182858	−0.0914289	−	0.995812i	$-0.529143\pi$
$349$	−3.41607	−0.182858	−0.0914289	+	0.995812i	$0.529143\pi$
$350$	−1.50253	−0.0803138
$351$	13.4638	0.718642
$352$	−1.64496	−0.0876765
$353$	−10.9807	−0.584443	−0.292221	−	0.956351i	$-0.594394\pi$
$353$	−10.9807	−0.584443	−0.292221	+	0.956351i	$0.594394\pi$
$354$	−1.74632	−0.0928159
$355$	12.0112	0.637490
$356$	−25.4966	−1.35132
$357$	−7.12504	−0.377097
$358$	0.582415	0.0307816
$359$	7.56251	0.399134	0.199567	−	0.979884i	$-0.436046\pi$
$359$	7.56251	0.399134	0.199567	+	0.979884i	$0.436046\pi$
$360$	−1.47657	−0.0778222
$361$	0	0
$362$	−3.46149	−0.181932
$363$	−1.49679	−0.0785609
$364$	6.96164	0.364889
$365$	−9.88447	−0.517377
$366$	0.298812	0.0156191
$367$	2.97462	0.155274	0.0776369	−	0.996982i	$-0.475263\pi$
$367$	2.97462	0.155274	0.0776369	+	0.996982i	$0.475263\pi$
$368$	−28.6758	−1.49483
$369$	−7.48511	−0.389659
$370$	1.35867	0.0706339
$371$	5.55819	0.288567
$372$	2.56922	0.133208
$373$	29.9852	1.55258	0.776288	−	0.630378i	$-0.217100\pi$
$373$	29.9852	1.55258	0.776288	+	0.630378i	$0.217100\pi$
$374$	0.449965	0.0232671
$375$	12.4521	0.643022
$376$	−1.85675	−0.0957544
$377$	15.3351	0.789800
$378$	−1.14792	−0.0590426
$379$	−30.6010	−1.57187	−0.785933	−	0.618312i	$-0.787817\pi$
$379$	−30.6010	−1.57187	−0.785933	+	0.618312i	$0.787817\pi$
$380$	0	0
$381$	−5.51198	−0.282387
$382$	−1.07540	−0.0550225
$383$	−1.69358	−0.0865379	−0.0432690	−	0.999063i	$-0.513777\pi$
$383$	−1.69358	−0.0865379	−0.0432690	+	0.999063i	$0.513777\pi$
$384$	6.49163	0.331275
$385$	−5.16581	−0.263274
$386$	1.93149	0.0983104
$387$	−5.54915	−0.282079
$388$	31.1487	1.58133
$389$	−23.0617	−1.16928	−0.584638	−	0.811294i	$-0.698764\pi$
$389$	−23.0617	−1.16928	−0.584638	+	0.811294i	$0.698764\pi$
$390$	1.74868	0.0885477
$391$	23.9200	1.20969
$392$	2.67647	0.135182
$393$	−0.0809998	−0.00408590
$394$	2.82713	0.142429
$395$	62.3083	3.13507
$396$	−1.50462	−0.0756100
$397$	−19.0356	−0.955369	−0.477685	−	0.878531i	$-0.658524\pi$
$397$	−19.0356	−0.955369	−0.477685	+	0.878531i	$0.658524\pi$
$398$	0.421338	0.0211198
$399$	0	0
$400$	28.6136	1.43068
$401$	27.9015	1.39334	0.696668	−	0.717394i	$-0.254665\pi$
$401$	27.9015	1.39334	0.696668	+	0.717394i	$0.254665\pi$
$402$	−1.34810	−0.0672372
$403$	2.07339	0.103283
$404$	15.5239	0.772343
$405$	21.6056	1.07359
$406$	−1.30747	−0.0648888
$407$	2.78244	0.137920
$408$	−2.68102	−0.132730
$409$	−7.32038	−0.361970	−0.180985	−	0.983486i	$-0.557928\pi$
$409$	−7.32038	−0.361970	−0.180985	+	0.983486i	$0.557928\pi$
$410$	−4.81153	−0.237624
$411$	−25.4045	−1.25311
$412$	14.4140	0.710126
$413$	−12.3428	−0.607350
$414$	0.778652	0.0382687
$415$	−62.2875	−3.05758
$416$	3.93564	0.192961
$417$	26.9964	1.32202
$418$	0	0
$419$	−37.5029	−1.83214	−0.916069	−	0.401021i	$-0.868655\pi$
$419$	−37.5029	−1.83214	−0.916069	+	0.401021i	$0.868655\pi$
$420$	15.3151	0.747301
$421$	36.8529	1.79610	0.898050	−	0.439893i	$-0.144983\pi$
$421$	36.8529	1.79610	0.898050	+	0.439893i	$0.144983\pi$
$422$	0.696744	0.0339170
$423$	−2.55163	−0.124064
$424$	2.09144	0.101569
$425$	−23.8681	−1.15777
$426$	0.709929	0.0343962
$427$	2.11197	0.102205
$428$	−12.3546	−0.597183
$429$	3.58113	0.172899
$430$	−3.56707	−0.172019
$431$	−25.7897	−1.24225	−0.621124	−	0.783712i	$-0.713324\pi$
$431$	−25.7897	−1.24225	−0.621124	+	0.783712i	$0.713324\pi$
$432$	21.8605	1.05176
$433$	−31.0028	−1.48990	−0.744949	−	0.667121i	$-0.767526\pi$
$433$	−31.0028	−1.48990	−0.744949	+	0.667121i	$0.767526\pi$
$434$	−0.176777	−0.00848556
$435$	33.7362	1.61753
$436$	−5.38442	−0.257867
$437$	0	0
$438$	−0.584226	−0.0279154
$439$	18.5779	0.886674	0.443337	−	0.896355i	$-0.353794\pi$
$439$	18.5779	0.886674	0.443337	+	0.896355i	$0.353794\pi$
$440$	−1.94379	−0.0926668
$441$	3.67813	0.175149
$442$	−1.07656	−0.0512069
$443$	−26.6129	−1.26442	−0.632208	−	0.774799i	$-0.717851\pi$
$443$	−26.6129	−1.26442	−0.632208	+	0.774799i	$0.717851\pi$
$444$	−8.24912	−0.391486
$445$	−45.2658	−2.14581
$446$	−1.64711	−0.0779930
$447$	7.93858	0.375482
$448$	11.0778	0.523376
$449$	7.83267	0.369647	0.184823	−	0.982772i	$-0.440829\pi$
$449$	7.83267	0.369647	0.184823	+	0.982772i	$0.440829\pi$
$450$	−0.776964	−0.0366264
$451$	−9.85358	−0.463987
$452$	−0.963890	−0.0453376
$453$	16.1867	0.760519
$454$	−0.195569	−0.00917849
$455$	12.3595	0.579420
$456$	0	0
$457$	2.02478	0.0947150	0.0473575	−	0.998878i	$-0.484920\pi$
$457$	2.02478	0.0947150	0.0473575	+	0.998878i	$0.484920\pi$
$458$	1.57804	0.0737372
$459$	−18.2350	−0.851136
$460$	−51.4155	−2.39726
$461$	−38.1417	−1.77644	−0.888219	−	0.459421i	$-0.848057\pi$
$461$	−38.1417	−1.77644	−0.888219	+	0.459421i	$0.848057\pi$
$462$	−0.305327	−0.0142051
$463$	−27.6957	−1.28713	−0.643564	−	0.765393i	$-0.722545\pi$
$463$	−27.6957	−1.28713	−0.643564	+	0.765393i	$0.722545\pi$
$464$	24.8990	1.15591
$465$	4.56130	0.211525
$466$	−1.77743	−0.0823378
$467$	−3.91063	−0.180962	−0.0904811	−	0.995898i	$-0.528840\pi$
$467$	−3.91063	−0.180962	−0.0904811	+	0.995898i	$0.528840\pi$
$468$	3.59988	0.166405
$469$	−9.52824	−0.439973
$470$	−1.64022	−0.0756576
$471$	−4.78265	−0.220373
$472$	−4.64436	−0.213774
$473$	−7.30504	−0.335886
$474$	3.68276	0.169155
$475$	0	0
$476$	−9.42867	−0.432162
$477$	2.87415	0.131598
$478$	−0.504032	−0.0230539
$479$	6.77505	0.309560	0.154780	−	0.987949i	$-0.450533\pi$
$479$	6.77505	0.309560	0.154780	+	0.987949i	$0.450533\pi$
$480$	8.65813	0.395188
$481$	−6.65712	−0.303539
$482$	2.68300	0.122207
$483$	−16.2311	−0.738541
$484$	−1.98072	−0.0900326
$485$	55.3003	2.51106
$486$	−1.06725	−0.0484114
$487$	15.8303	0.717340	0.358670	−	0.933464i	$-0.383230\pi$
$487$	15.8303	0.717340	0.358670	+	0.933464i	$0.383230\pi$
$488$	0.794694	0.0359741
$489$	21.4696	0.970889
$490$	2.36435	0.106810
$491$	−2.66404	−0.120226	−0.0601132	−	0.998192i	$-0.519146\pi$
$491$	−2.66404	−0.120226	−0.0601132	+	0.998192i	$0.519146\pi$
$492$	29.2130	1.31702
$493$	−20.7695	−0.935413
$494$	0	0
$495$	−2.67125	−0.120064
$496$	3.36646	0.151159
$497$	5.01770	0.225075
$498$	−3.68153	−0.164973
$499$	10.9165	0.488690	0.244345	−	0.969688i	$-0.421427\pi$
$499$	10.9165	0.488690	0.244345	+	0.969688i	$0.421427\pi$
$500$	16.4780	0.736919
$501$	−22.8359	−1.02023
$502$	2.51752	0.112362
$503$	30.6519	1.36670	0.683350	−	0.730091i	$-0.260522\pi$
$503$	30.6519	1.36670	0.683350	+	0.730091i	$0.260522\pi$
$504$	−0.616839	−0.0274762
$505$	27.5606	1.22643
$506$	1.02504	0.0455684
$507$	10.8902	0.483649
$508$	−7.29408	−0.323622
$509$	−6.69649	−0.296817	−0.148408	−	0.988926i	$-0.547415\pi$
$509$	−6.69649	−0.296817	−0.148408	+	0.988926i	$0.547415\pi$
$510$	−2.36836	−0.104873
$511$	−4.12924	−0.182667
$512$	10.6847	0.472204
$513$	0	0
$514$	0.216536	0.00955099
$515$	25.5901	1.12763
$516$	21.6573	0.953410
$517$	−3.35902	−0.147730
$518$	0.567586	0.0249383
$519$	32.4426	1.42407
$520$	4.65063	0.203943
$521$	−23.4833	−1.02882	−0.514412	−	0.857543i	$-0.671990\pi$
$521$	−23.4833	−1.02882	−0.514412	+	0.857543i	$0.671990\pi$
$522$	−0.676098	−0.0295920
$523$	35.0640	1.53324	0.766622	−	0.642099i	$-0.221936\pi$
$523$	35.0640	1.53324	0.766622	+	0.642099i	$0.221936\pi$
$524$	−0.107188	−0.00468254
$525$	16.1959	0.706847
$526$	2.39841	0.104576
$527$	−2.80814	−0.122325
$528$	5.81453	0.253045
$529$	31.4906	1.36916
$530$	1.84754	0.0802521
$531$	−6.38249	−0.276977
$532$	0	0
$533$	23.5752	1.02115
$534$	−2.67546	−0.115778
$535$	−21.9340	−0.948288
$536$	−3.58529	−0.154861
$537$	−6.27788	−0.270911
$538$	−1.93445	−0.0834002
$539$	4.84198	0.208559
$540$	39.1957	1.68671
$541$	31.0702	1.33581	0.667906	−	0.744245i	$-0.267191\pi$
$541$	31.0702	1.33581	0.667906	+	0.744245i	$0.267191\pi$
$542$	0.854415	0.0367003
$543$	37.3116	1.60119
$544$	−5.33033	−0.228536
$545$	−9.55931	−0.409476
$546$	0.730511	0.0312630
$547$	−19.1794	−0.820051	−0.410025	−	0.912074i	$-0.634480\pi$
$547$	−19.1794	−0.820051	−0.410025	+	0.912074i	$0.634480\pi$
$548$	−33.6181	−1.43610
$549$	1.09210	0.0466099
$550$	−1.02281	−0.0436129
$551$	0	0
$552$	−6.10745	−0.259950
$553$	26.0293	1.10688
$554$	−2.55707	−0.108639
$555$	−14.6452	−0.621654
$556$	35.7247	1.51506
$557$	−9.68999	−0.410578	−0.205289	−	0.978701i	$-0.565813\pi$
$557$	−9.68999	−0.410578	−0.205289	+	0.978701i	$0.565813\pi$
$558$	−0.0914117	−0.00386977
$559$	17.4777	0.739227
$560$	20.0675	0.848007
$561$	−4.85020	−0.204776
$562$	3.44832	0.145458
$563$	−24.4092	−1.02873	−0.514363	−	0.857573i	$-0.671972\pi$
$563$	−24.4092	−1.02873	−0.514363	+	0.857573i	$0.671972\pi$
$564$	9.95852	0.419329
$565$	−1.71126	−0.0719932
$566$	2.38508	0.100252
$567$	9.02574	0.379045
$568$	1.88807	0.0792214
$569$	34.3379	1.43952	0.719761	−	0.694222i	$-0.244251\pi$
$569$	34.3379	1.43952	0.719761	+	0.694222i	$0.244251\pi$
$570$	0	0
$571$	43.4564	1.81859	0.909296	−	0.416149i	$-0.136621\pi$
$571$	43.4564	1.81859	0.909296	+	0.416149i	$0.136621\pi$
$572$	4.73897	0.198146
$573$	11.5919	0.484257
$574$	−2.01002	−0.0838966
$575$	−54.3724	−2.26749
$576$	5.72835	0.238681
$577$	18.1629	0.756131	0.378066	−	0.925779i	$-0.376589\pi$
$577$	18.1629	0.756131	0.378066	+	0.925779i	$0.376589\pi$
$578$	−0.902557	−0.0375414
$579$	−20.8197	−0.865236
$580$	44.6437	1.85373
$581$	−26.0207	−1.07952
$582$	3.26855	0.135486
$583$	3.78360	0.156701
$584$	−1.55376	−0.0642949
$585$	6.39111	0.264240
$586$	3.21837	0.132950
$587$	−11.5578	−0.477040	−0.238520	−	0.971138i	$-0.576662\pi$
$587$	−11.5578	−0.477040	−0.238520	+	0.971138i	$0.576662\pi$
$588$	−14.3551	−0.591993
$589$	0	0
$590$	−4.10275	−0.168908
$591$	−30.4739	−1.25353
$592$	−10.8089	−0.444242
$593$	30.8182	1.26555	0.632776	−	0.774335i	$-0.281916\pi$
$593$	30.8182	1.26555	0.632776	+	0.774335i	$0.281916\pi$
$594$	−0.781418	−0.0320620
$595$	−16.7393	−0.686246
$596$	10.5052	0.430311
$597$	−4.54163	−0.185877
$598$	−2.45245	−0.100288
$599$	−22.9029	−0.935788	−0.467894	−	0.883785i	$-0.654987\pi$
$599$	−22.9029	−0.935788	−0.467894	+	0.883785i	$0.654987\pi$
$600$	6.09421	0.248795
$601$	−5.27555	−0.215194	−0.107597	−	0.994195i	$-0.534316\pi$
$601$	−5.27555	−0.215194	−0.107597	+	0.994195i	$0.534316\pi$
$602$	−1.49015	−0.0607338
$603$	−4.92708	−0.200646
$604$	21.4201	0.871573
$605$	−3.51650	−0.142966
$606$	1.62898	0.0661728
$607$	−22.3060	−0.905372	−0.452686	−	0.891670i	$-0.649534\pi$
$607$	−22.3060	−0.905372	−0.452686	+	0.891670i	$0.649534\pi$
$608$	0	0
$609$	14.0933	0.571091
$610$	0.702019	0.0284239
$611$	8.03663	0.325127
$612$	−4.87559	−0.197084
$613$	37.7943	1.52650	0.763249	−	0.646104i	$-0.223603\pi$
$613$	37.7943	1.52650	0.763249	+	0.646104i	$0.223603\pi$
$614$	3.65412	0.147468
$615$	51.8637	2.09135
$616$	−0.812022	−0.0327173
$617$	9.69316	0.390232	0.195116	−	0.980780i	$-0.437492\pi$
$617$	9.69316	0.390232	0.195116	+	0.980780i	$0.437492\pi$
$618$	1.51251	0.0608422
$619$	−18.4686	−0.742314	−0.371157	−	0.928570i	$-0.621039\pi$
$619$	−18.4686	−0.742314	−0.371157	+	0.928570i	$0.621039\pi$
$620$	6.03604	0.242413
$621$	−41.5399	−1.66694
$622$	3.51178	0.140810
$623$	−18.9098	−0.757606
$624$	−13.9116	−0.556908
$625$	−7.57434	−0.302973
$626$	−3.84476	−0.153668
$627$	0	0
$628$	−6.32896	−0.252553
$629$	9.01624	0.359501
$630$	−0.544905	−0.0217096
$631$	12.8342	0.510921	0.255460	−	0.966820i	$-0.417773\pi$
$631$	12.8342	0.510921	0.255460	+	0.966820i	$0.417773\pi$
$632$	9.79434	0.389598
$633$	−7.51025	−0.298505
$634$	1.40577	0.0558302
$635$	−12.9497	−0.513892
$636$	−11.2173	−0.444794
$637$	−11.5847	−0.459002
$638$	−0.890031	−0.0352367
$639$	2.59467	0.102643
$640$	15.2512	0.602858
$641$	41.2232	1.62822	0.814110	−	0.580711i	$-0.197225\pi$
$641$	41.2232	1.62822	0.814110	+	0.580711i	$0.197225\pi$
$642$	−1.29642	−0.0511655
$643$	−15.6032	−0.615332	−0.307666	−	0.951494i	$-0.599548\pi$
$643$	−15.6032	−0.615332	−0.307666	+	0.951494i	$0.599548\pi$
$644$	−21.4789	−0.846385
$645$	38.4497	1.51395
$646$	0	0
$647$	−12.7748	−0.502228	−0.251114	−	0.967958i	$-0.580797\pi$
$647$	−12.7748	−0.502228	−0.251114	+	0.967958i	$0.580797\pi$
$648$	3.39621	0.133416
$649$	−8.40207	−0.329810
$650$	2.44713	0.0959844
$651$	1.90549	0.0746820
$652$	28.4110	1.11266
$653$	−5.05990	−0.198009	−0.0990045	−	0.995087i	$-0.531566\pi$
$653$	−5.05990	−0.198009	−0.0990045	+	0.995087i	$0.531566\pi$
$654$	−0.565007	−0.0220935
$655$	−0.190298	−0.00743557
$656$	38.2780	1.49450
$657$	−2.13524	−0.0833037
$658$	−0.685203	−0.0267120
$659$	−29.3796	−1.14447	−0.572233	−	0.820091i	$-0.693923\pi$
$659$	−29.3796	−1.14447	−0.572233	+	0.820091i	$0.693923\pi$
$660$	10.4254	0.405808
$661$	−32.5399	−1.26565	−0.632827	−	0.774293i	$-0.718106\pi$
$661$	−32.5399	−1.26565	−0.632827	+	0.774293i	$0.718106\pi$
$662$	1.17417	0.0456352
$663$	11.6043	0.450675
$664$	−9.79108	−0.379968
$665$	0	0
$666$	0.293500	0.0113729
$667$	−47.3138	−1.83200
$668$	−30.2191	−1.16921
$669$	17.7543	0.686422
$670$	−3.16719	−0.122359
$671$	1.43767	0.0555007
$672$	3.61694	0.139527
$673$	−37.8116	−1.45753	−0.728765	−	0.684764i	$-0.759905\pi$
$673$	−37.8116	−1.45753	−0.728765	+	0.684764i	$0.759905\pi$
$674$	0.00241313	9.29504e−5 0
$675$	41.4498	1.59541
$676$	14.4111	0.554273
$677$	−1.66524	−0.0640004	−0.0320002	−	0.999488i	$-0.510188\pi$
$677$	−1.66524	−0.0640004	−0.0320002	+	0.999488i	$0.510188\pi$
$678$	−0.101145	−0.00388444
$679$	23.1017	0.886563
$680$	−6.29869	−0.241544
$681$	2.10805	0.0807805
$682$	−0.120336	−0.00460792
$683$	−22.3652	−0.855779	−0.427890	−	0.903831i	$-0.640743\pi$
$683$	−22.3652	−0.855779	−0.427890	+	0.903831i	$0.640743\pi$
$684$	0	0
$685$	−59.6845	−2.28043
$686$	2.41563	0.0922292
$687$	−17.0098	−0.648966
$688$	28.3777	1.08189
$689$	−9.05246	−0.344871
$690$	−5.39522	−0.205392
$691$	22.7083	0.863865	0.431932	−	0.901906i	$-0.357832\pi$
$691$	22.7083	0.863865	0.431932	+	0.901906i	$0.357832\pi$
$692$	42.9318	1.63202
$693$	−1.11592	−0.0423902
$694$	1.19956	0.0455348
$695$	63.4244	2.40582
$696$	5.30305	0.201012
$697$	−31.9296	−1.20942
$698$	−0.474356	−0.0179547
$699$	19.1590	0.724660
$700$	21.4323	0.810064
$701$	−27.0490	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$701$	−27.0490	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$702$	1.86958	0.0705629
$703$	0	0
$704$	7.54094	0.284210
$705$	17.6800	0.665868
$706$	−1.52478	−0.0573860
$707$	11.5135	0.433008
$708$	24.9097	0.936163
$709$	−31.2772	−1.17464	−0.587320	−	0.809355i	$-0.699817\pi$
$709$	−31.2772	−1.17464	−0.587320	+	0.809355i	$0.699817\pi$
$710$	1.66788	0.0625946
$711$	13.4598	0.504783
$712$	−7.11541	−0.266661
$713$	−6.39705	−0.239571
$714$	−0.989386	−0.0370268
$715$	8.41340	0.314643
$716$	−8.30762	−0.310470
$717$	5.43299	0.202899
$718$	1.05013	0.0391906
$719$	−10.4902	−0.391219	−0.195609	−	0.980682i	$-0.562668\pi$
$719$	−10.4902	−0.391219	−0.195609	+	0.980682i	$0.562668\pi$
$720$	10.3770	0.386726
$721$	10.6903	0.398127
$722$	0	0
$723$	−28.9203	−1.07556
$724$	49.3750	1.83501
$725$	47.2111	1.75338
$726$	−0.207844	−0.00771382
$727$	−43.4181	−1.61029	−0.805145	−	0.593078i	$-0.797913\pi$
$727$	−43.4181	−1.61029	−0.805145	+	0.593078i	$0.797913\pi$
$728$	1.94280	0.0720051
$729$	29.9361	1.10874
$730$	−1.37256	−0.0508008
$731$	−23.6713	−0.875515
$732$	−4.26228	−0.157538
$733$	37.7254	1.39342	0.696710	−	0.717353i	$-0.254646\pi$
$733$	37.7254	1.39342	0.696710	+	0.717353i	$0.254646\pi$
$734$	0.413057	0.0152462
$735$	−25.4855	−0.940046
$736$	−12.1427	−0.447586
$737$	−6.48612	−0.238919
$738$	−1.03939	−0.0382603
$739$	10.5904	0.389575	0.194788	−	0.980845i	$-0.437598\pi$
$739$	10.5904	0.389575	0.194788	+	0.980845i	$0.437598\pi$
$740$	−19.3802	−0.712431
$741$	0	0
$742$	0.771812	0.0283341
$743$	−2.32562	−0.0853187	−0.0426593	−	0.999090i	$-0.513583\pi$
$743$	−2.32562	−0.0853187	−0.0426593	+	0.999090i	$0.513583\pi$
$744$	0.716998	0.0262864
$745$	18.6506	0.683307
$746$	4.16376	0.152446
$747$	−13.4554	−0.492306
$748$	−6.41834	−0.234678
$749$	−9.16294	−0.334806
$750$	1.72910	0.0631378
$751$	−7.37463	−0.269104	−0.134552	−	0.990907i	$-0.542960\pi$
$751$	−7.37463	−0.269104	−0.134552	+	0.990907i	$0.542960\pi$
$752$	13.0487	0.475838
$753$	−27.1365	−0.988907
$754$	2.12944	0.0775498
$755$	38.0286	1.38400
$756$	16.3740	0.595518
$757$	−36.2300	−1.31680	−0.658401	−	0.752667i	$-0.728767\pi$
$757$	−36.2300	−1.31680	−0.658401	+	0.752667i	$0.728767\pi$
$758$	−4.24926	−0.154340
$759$	−11.0489	−0.401051
$760$	0	0
$761$	8.48528	0.307591	0.153796	−	0.988103i	$-0.450850\pi$
$761$	8.48528	0.307591	0.153796	+	0.988103i	$0.450850\pi$
$762$	−0.765395	−0.0277273
$763$	−3.99341	−0.144571
$764$	15.3397	0.554970
$765$	−8.65595	−0.312957
$766$	−0.235171	−0.00849708
$767$	20.1024	0.725854
$768$	−21.6729	−0.782053
$769$	−8.18651	−0.295213	−0.147607	−	0.989046i	$-0.547157\pi$
$769$	−8.18651	−0.295213	−0.147607	+	0.989046i	$0.547157\pi$
$770$	−0.717326	−0.0258506
$771$	−2.33405	−0.0840589
$772$	−27.5510	−0.991582
$773$	−35.4306	−1.27435	−0.637175	−	0.770720i	$-0.719897\pi$
$773$	−35.4306	−1.27435	−0.637175	+	0.770720i	$0.719897\pi$
$774$	−0.770558	−0.0276971
$775$	6.38317	0.229290
$776$	8.69274	0.312051
$777$	−6.11804	−0.219484
$778$	−3.20236	−0.114810
$779$	0	0
$780$	−24.9433	−0.893113
$781$	3.41568	0.122223
$782$	3.32154	0.118778
$783$	36.0688	1.28899
$784$	−18.8095	−0.671769
$785$	−11.2362	−0.401038
$786$	−0.0112477	−0.000401191 0
$787$	−2.32432	−0.0828529	−0.0414264	−	0.999142i	$-0.513190\pi$
$787$	−2.32432	−0.0828529	−0.0414264	+	0.999142i	$0.513190\pi$
$788$	−40.3265	−1.43657
$789$	−25.8526	−0.920378
$790$	8.65216	0.307830
$791$	−0.714879	−0.0254182
$792$	−0.419898	−0.0149204
$793$	−3.43970	−0.122147
$794$	−2.64329	−0.0938069
$795$	−19.9148	−0.706304
$796$	−6.01000	−0.213019
$797$	7.98937	0.282998	0.141499	−	0.989938i	$-0.454808\pi$
$797$	7.98937	0.282998	0.141499	+	0.989938i	$0.454808\pi$
$798$	0	0
$799$	−10.8846	−0.385070
$800$	12.1164	0.428378
$801$	−9.77832	−0.345500
$802$	3.87442	0.136810
$803$	−2.81088	−0.0991939
$804$	19.2295	0.678171
$805$	−38.1328	−1.34401
$806$	0.287911	0.0101412
$807$	20.8516	0.734011
$808$	4.33229	0.152410
$809$	−3.66337	−0.128797	−0.0643987	−	0.997924i	$-0.520513\pi$
$809$	−3.66337	−0.128797	−0.0643987	+	0.997924i	$0.520513\pi$
$810$	3.00016	0.105415
$811$	−38.8311	−1.36355	−0.681773	−	0.731564i	$-0.738791\pi$
$811$	−38.8311	−1.36355	−0.681773	+	0.731564i	$0.738791\pi$
$812$	18.6499	0.654484
$813$	−9.20980	−0.323002
$814$	0.386370	0.0135423
$815$	50.4400	1.76684
$816$	18.8415	0.659583
$817$	0	0
$818$	−1.01651	−0.0355415
$819$	2.66989	0.0932935
$820$	68.6320	2.39673
$821$	−21.9475	−0.765975	−0.382987	−	0.923754i	$-0.625105\pi$
$821$	−21.9475	−0.765975	−0.382987	+	0.923754i	$0.625105\pi$
$822$	−3.52768	−0.123042
$823$	8.08878	0.281957	0.140979	−	0.990013i	$-0.454975\pi$
$823$	8.08878	0.281957	0.140979	+	0.990013i	$0.454975\pi$
$824$	4.02255	0.140132
$825$	11.0250	0.383840
$826$	−1.71393	−0.0596351
$827$	28.5117	0.991449	0.495724	−	0.868480i	$-0.334903\pi$
$827$	28.5117	0.991449	0.495724	+	0.868480i	$0.334903\pi$
$828$	−11.1068	−0.385987
$829$	−52.4728	−1.82246	−0.911228	−	0.411901i	$-0.864865\pi$
$829$	−52.4728	−1.82246	−0.911228	+	0.411901i	$0.864865\pi$
$830$	−8.64927	−0.300221
$831$	27.5628	0.956143
$832$	−18.0421	−0.625496
$833$	15.6900	0.543626
$834$	3.74873	0.129808
$835$	−53.6500	−1.85664
$836$	0	0
$837$	4.87668	0.168563
$838$	−5.20767	−0.179896
$839$	10.9975	0.379677	0.189839	−	0.981815i	$-0.439203\pi$
$839$	10.9975	0.379677	0.189839	+	0.981815i	$0.439203\pi$
$840$	4.27403	0.147468
$841$	12.0822	0.416627
$842$	5.11741	0.176358
$843$	−37.1696	−1.28019
$844$	−9.93842	−0.342094
$845$	25.5850	0.880151
$846$	−0.354320	−0.0121818
$847$	−1.46902	−0.0504761
$848$	−14.6981	−0.504734
$849$	−25.7089	−0.882327
$850$	−3.31433	−0.113681
$851$	20.5393	0.704079
$852$	−10.1265	−0.346928
$853$	−34.0760	−1.16674	−0.583369	−	0.812207i	$-0.698266\pi$
$853$	−34.0760	−1.16674	−0.583369	+	0.812207i	$0.698266\pi$
$854$	0.293269	0.0100355
$855$	0	0
$856$	−3.44784	−0.117845
$857$	19.4670	0.664981	0.332490	−	0.943107i	$-0.392111\pi$
$857$	19.4670	0.664981	0.332490	+	0.943107i	$0.392111\pi$
$858$	0.497278	0.0169768
$859$	−41.7466	−1.42437	−0.712187	−	0.701990i	$-0.752295\pi$
$859$	−41.7466	−1.42437	−0.712187	+	0.701990i	$0.752295\pi$
$860$	50.8810	1.73503
$861$	21.6661	0.738379
$862$	−3.58117	−0.121975
$863$	20.0106	0.681169	0.340584	−	0.940214i	$-0.389375\pi$
$863$	20.0106	0.681169	0.340584	+	0.940214i	$0.389375\pi$
$864$	9.25677	0.314922
$865$	76.2196	2.59154
$866$	−4.30506	−0.146292
$867$	9.72871	0.330404
$868$	2.52156	0.0855873
$869$	17.7188	0.601071
$870$	4.68463	0.158824
$871$	15.5184	0.525820
$872$	−1.50264	−0.0508859
$873$	11.9460	0.404310
$874$	0	0
$875$	12.2211	0.413148
$876$	8.33345	0.281561
$877$	14.6756	0.495560	0.247780	−	0.968816i	$-0.420299\pi$
$877$	14.6756	0.495560	0.247780	+	0.968816i	$0.420299\pi$
$878$	2.57973	0.0870618
$879$	−34.6910	−1.17010
$880$	13.6605	0.460494
$881$	17.9508	0.604777	0.302388	−	0.953185i	$-0.402216\pi$
$881$	17.9508	0.604777	0.302388	+	0.953185i	$0.402216\pi$
$882$	0.510747	0.0171977
$883$	23.0794	0.776684	0.388342	−	0.921515i	$-0.373048\pi$
$883$	23.0794	0.776684	0.388342	+	0.921515i	$0.373048\pi$
$884$	15.3562	0.516485
$885$	44.2238	1.48657
$886$	−3.69547	−0.124152
$887$	2.24450	0.0753629	0.0376814	−	0.999290i	$-0.488003\pi$
$887$	2.24450	0.0753629	0.0376814	+	0.999290i	$0.488003\pi$
$888$	−2.30210	−0.0772535
$889$	−5.40973	−0.181437
$890$	−6.28563	−0.210695
$891$	6.14405	0.205834
$892$	23.4945	0.786656
$893$	0	0
$894$	1.10235	0.0368683
$895$	−14.7491	−0.493007
$896$	6.37121	0.212847
$897$	26.4351	0.882643
$898$	1.08765	0.0362953
$899$	5.55451	0.185253
$900$	11.0827	0.369423
$901$	12.2604	0.408454
$902$	−1.36827	−0.0455585
$903$	16.0624	0.534522
$904$	−0.268995	−0.00894665
$905$	87.6587	2.91387
$906$	2.24770	0.0746747
$907$	−28.1759	−0.935565	−0.467783	−	0.883844i	$-0.654947\pi$
$907$	−28.1759	−0.935565	−0.467783	+	0.883844i	$0.654947\pi$
$908$	2.78961	0.0925764
$909$	5.95364	0.197470
$910$	1.71624	0.0568928
$911$	−32.9161	−1.09056	−0.545280	−	0.838254i	$-0.683577\pi$
$911$	−32.9161	−1.09056	−0.545280	+	0.838254i	$0.683577\pi$
$912$	0	0
$913$	−17.7129	−0.586213
$914$	0.281161	0.00929998
$915$	−7.56711	−0.250161
$916$	−22.5094	−0.743730
$917$	−0.0794973	−0.00262523
$918$	−2.53212	−0.0835723
$919$	54.3280	1.79211	0.896057	−	0.443938i	$-0.146419\pi$
$919$	54.3280	1.79211	0.896057	+	0.443938i	$0.146419\pi$
$920$	−14.3486	−0.473061
$921$	−39.3880	−1.29788
$922$	−5.29638	−0.174427
$923$	−8.17219	−0.268991
$924$	4.35522	0.143276
$925$	−20.4948	−0.673864
$926$	−3.84583	−0.126382
$927$	5.52797	0.181562
$928$	10.5434	0.346104
$929$	−17.8175	−0.584574	−0.292287	−	0.956331i	$-0.594416\pi$
$929$	−17.8175	−0.584574	−0.292287	+	0.956331i	$0.594416\pi$
$930$	0.633384	0.0207695
$931$	0	0
$932$	25.3534	0.830478
$933$	−37.8537	−1.23928
$934$	−0.543031	−0.0177685
$935$	−11.3949	−0.372653
$936$	1.00463	0.0328373
$937$	−52.7066	−1.72185	−0.860925	−	0.508732i	$-0.830115\pi$
$937$	−52.7066	−1.72185	−0.860925	+	0.508732i	$0.830115\pi$
$938$	−1.32310	−0.0432006
$939$	41.4430	1.35244
$940$	23.3962	0.763101
$941$	−51.4855	−1.67838	−0.839189	−	0.543839i	$-0.816970\pi$
$941$	−51.4855	−1.67838	−0.839189	+	0.543839i	$0.816970\pi$
$942$	−0.664121	−0.0216382
$943$	−72.7369	−2.36864
$944$	32.6393	1.06232
$945$	29.0699	0.945643
$946$	−1.01438	−0.0329804
$947$	−26.4120	−0.858274	−0.429137	−	0.903239i	$-0.641182\pi$
$947$	−26.4120	−0.858274	−0.429137	+	0.903239i	$0.641182\pi$
$948$	−52.5312	−1.70614
$949$	6.72518	0.218309
$950$	0	0
$951$	−15.1529	−0.491366
$952$	−2.63128	−0.0852804
$953$	−25.4872	−0.825613	−0.412806	−	0.910819i	$-0.635451\pi$
$953$	−25.4872	−0.825613	−0.412806	+	0.910819i	$0.635451\pi$
$954$	0.399106	0.0129215
$955$	27.2335	0.881256
$956$	7.18955	0.232527
$957$	9.59370	0.310120
$958$	0.940787	0.0303954
$959$	−24.9332	−0.805136
$960$	−39.6913	−1.28103
$961$	−30.2490	−0.975774
$962$	−0.924411	−0.0298042
$963$	−4.73818	−0.152686
$964$	−38.2706	−1.23261
$965$	−48.9131	−1.57457
$966$	−2.25386	−0.0725167
$967$	18.1540	0.583793	0.291897	−	0.956450i	$-0.405714\pi$
$967$	18.1540	0.583793	0.291897	+	0.956450i	$0.405714\pi$
$968$	−0.552764	−0.0177665
$969$	0	0
$970$	7.67902	0.246558
$971$	−10.1058	−0.324311	−0.162156	−	0.986765i	$-0.551845\pi$
$971$	−10.1058	−0.324311	−0.162156	+	0.986765i	$0.551845\pi$
$972$	15.2233	0.488289
$973$	26.4956	0.849410
$974$	2.19820	0.0704350
$975$	−26.3778	−0.844765
$976$	−5.58490	−0.178768
$977$	36.3162	1.16186	0.580929	−	0.813954i	$-0.302689\pi$
$977$	36.3162	1.16186	0.580929	+	0.813954i	$0.302689\pi$
$978$	2.98128	0.0953308
$979$	−12.8724	−0.411404
$980$	−33.7253	−1.07732
$981$	−2.06500	−0.0659304
$982$	−0.369930	−0.0118049
$983$	7.72465	0.246378	0.123189	−	0.992383i	$-0.460688\pi$
$983$	7.72465	0.246378	0.123189	+	0.992383i	$0.460688\pi$
$984$	8.15254	0.259894
$985$	−71.5943	−2.28118
$986$	−2.88407	−0.0918474
$987$	7.38584	0.235094
$988$	0	0
$989$	−53.9241	−1.71469
$990$	−0.370931	−0.0117890
$991$	19.1661	0.608830	0.304415	−	0.952539i	$-0.401539\pi$
$991$	19.1661	0.608830	0.304415	+	0.952539i	$0.401539\pi$
$992$	1.42552	0.0452603
$993$	−12.6564	−0.401639
$994$	0.696760	0.0220999
$995$	−10.6700	−0.338260
$996$	52.5137	1.66396
$997$	27.2322	0.862454	0.431227	−	0.902244i	$-0.358081\pi$
$997$	27.2322	0.862454	0.431227	+	0.902244i	$0.358081\pi$
$998$	1.51587	0.0479840
$999$	−15.6578	−0.495390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3971.2.a.k.1.5		9
19.7	even	3		209.2.e.b.144.5	yes	18
19.11	even	3		209.2.e.b.45.5	✓	18
19.18	odd	2		3971.2.a.l.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.e.b.45.5	✓	18	19.11	even	3
209.2.e.b.144.5	yes	18	19.7	even	3
3971.2.a.k.1.5		9	1.1	even	1	trivial
3971.2.a.l.1.5		9	19.18	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:	\( N \)	\(=\)	\( 3971 = 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3971.a (trivial)

\(f(q)\)	\(=\)	\(q+0.138860 q^{2} -1.49679 q^{3} -1.98072 q^{4} -3.51650 q^{5} -0.207844 q^{6} -1.46902 q^{7} -0.552764 q^{8} -0.759634 q^{9} -0.488302 q^{10} -1.00000 q^{11} +2.96471 q^{12} +2.39255 q^{13} -0.203989 q^{14} +5.26344 q^{15} +3.88468 q^{16} -3.24041 q^{17} -0.105483 q^{18} +6.96519 q^{20} +2.19881 q^{21} -0.138860 q^{22} -7.38177 q^{23} +0.827369 q^{24} +7.36576 q^{25} +0.332231 q^{26} +5.62736 q^{27} +2.90971 q^{28} +6.40954 q^{29} +0.730884 q^{30} +0.866600 q^{31} +1.64496 q^{32} +1.49679 q^{33} -0.449965 q^{34} +5.16581 q^{35} +1.50462 q^{36} -2.78244 q^{37} -3.58113 q^{39} +1.94379 q^{40} +9.85358 q^{41} +0.305327 q^{42} +7.30504 q^{43} +1.98072 q^{44} +2.67125 q^{45} -1.02504 q^{46} +3.35902 q^{47} -5.81453 q^{48} -4.84198 q^{49} +1.02281 q^{50} +4.85020 q^{51} -4.73897 q^{52} -3.78360 q^{53} +0.781418 q^{54} +3.51650 q^{55} +0.812022 q^{56} +0.890031 q^{58} +8.40207 q^{59} -10.4254 q^{60} -1.43767 q^{61} +0.120336 q^{62} +1.11592 q^{63} -7.54094 q^{64} -8.41340 q^{65} +0.207844 q^{66} +6.48612 q^{67} +6.41834 q^{68} +11.0489 q^{69} +0.717326 q^{70} -3.41568 q^{71} +0.419898 q^{72} +2.81088 q^{73} -0.386370 q^{74} -11.0250 q^{75} +1.46902 q^{77} -0.497278 q^{78} -17.7188 q^{79} -13.6605 q^{80} -6.14405 q^{81} +1.36827 q^{82} +17.7129 q^{83} -4.35522 q^{84} +11.3949 q^{85} +1.01438 q^{86} -9.59370 q^{87} +0.552764 q^{88} +12.8724 q^{89} +0.370931 q^{90} -3.51471 q^{91} +14.6212 q^{92} -1.29711 q^{93} +0.466435 q^{94} -2.46215 q^{96} -15.7259 q^{97} -0.672359 q^{98} +0.759634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 9 q^{4} + 3 q^{6} - 3 q^{7} + 11 q^{9} - 21 q^{10} - 9 q^{11} + 8 q^{12} - 11 q^{13} - q^{14} - 7 q^{15} + 13 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{20} - 24 q^{21} + q^{22} + q^{23} + 3 q^{24}+ \cdots - 11 q^{99}+O(q^{100}) \) 9 * q - q^2 + 9 * q^4 + 3 * q^6 - 3 * q^7 + 11 * q^9 - 21 * q^10 - 9 * q^11 + 8 * q^12 - 11 * q^13 - q^14 - 7 * q^15 + 13 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^20 - 24 * q^21 + q^22 + q^23 + 3 * q^24 + 7 * q^25 - 2 * q^26 + 3 * q^27 + 2 * q^28 - 13 * q^29 - 32 * q^30 - 16 * q^31 + 14 * q^32 - 8 * q^34 + 12 * q^35 - 10 * q^36 - 18 * q^37 + 10 * q^39 - 32 * q^40 - 4 * q^41 - 11 * q^42 - q^43 - 9 * q^44 + 11 * q^45 - 7 * q^46 + 14 * q^47 - 37 * q^48 - 6 * q^49 + 15 * q^50 + 16 * q^51 - 57 * q^52 + 4 * q^53 + 33 * q^54 - 17 * q^56 + 4 * q^58 - 31 * q^59 - 9 * q^60 - 3 * q^61 - 5 * q^62 - 50 * q^63 + 32 * q^64 - 28 * q^65 - 3 * q^66 + 2 * q^67 - 48 * q^68 + 58 * q^70 + 12 * q^71 - 5 * q^72 + 26 * q^73 - 35 * q^74 + 25 * q^75 + 3 * q^77 - 3 * q^78 - 31 * q^79 - 31 * q^80 + 21 * q^81 - 25 * q^82 + 18 * q^83 - 51 * q^84 + 11 * q^85 - 41 * q^86 - 62 * q^87 + 11 * q^89 - 51 * q^90 + 20 * q^91 + 33 * q^92 - 20 * q^93 - 43 * q^94 + 30 * q^96 - 16 * q^97 - 4 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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