LMFDB - Embedded newform 375.2.b.c.124.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1632160000.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25 $ x^8 + 12x^6 + 46x^4 + 65*x^2 + 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.1
Root			$2.46673i$ of defining polynomial
Character	$\chi$	$=$	375.124
Dual form			375.2.b.c.124.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52452i q^{2} +1.00000i q^{3} -4.37322 q^{4} +2.52452 q^{6} +4.93346i q^{7} +5.99126i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.52452i q^{2} +1.00000i q^{3} -4.37322 q^{4} +2.52452 q^{6} +4.93346i q^{7} +5.99126i q^{8} -1.00000 q^{9} +0.187020 q^{11} -4.37322i q^{12} +4.16953i q^{13} +12.4546 q^{14} +6.37863 q^{16} +2.85410i q^{17} +2.52452i q^{18} +0.431014 q^{19} -4.93346 q^{21} -0.472136i q^{22} -4.36448i q^{23} -5.99126 q^{24} +10.5261 q^{26} -1.00000i q^{27} -21.5751i q^{28} +1.12048 q^{29} -2.73852 q^{31} -4.12048i q^{32} +0.187020i q^{33} +7.20525 q^{34} +4.37322 q^{36} +5.35165i q^{37} -1.08811i q^{38} -4.16953 q^{39} +7.69740 q^{41} +12.4546i q^{42} -6.58772i q^{43} -0.817879 q^{44} -11.0182 q^{46} +2.43101i q^{47} +6.37863i q^{48} -17.3391 q^{49} -2.85410 q^{51} -18.2343i q^{52} +8.48496i q^{53} -2.52452 q^{54} -29.5576 q^{56} +0.431014i q^{57} -2.82869i q^{58} -10.7573 q^{59} +9.24889 q^{61} +6.91345i q^{62} -4.93346i q^{63} +2.35499 q^{64} +0.472136 q^{66} -9.03156i q^{67} -12.4816i q^{68} +4.36448 q^{69} +1.32336 q^{71} -5.99126i q^{72} +3.04905i q^{73} +13.5104 q^{74} -1.88492 q^{76} +0.922655i q^{77} +10.5261i q^{78} +8.19985 q^{79} +1.00000 q^{81} -19.4323i q^{82} +8.04112i q^{83} +21.5751 q^{84} -16.6309 q^{86} +1.12048i q^{87} +1.12048i q^{88} +3.10300 q^{89} -20.5702 q^{91} +19.0868i q^{92} -2.73852i q^{93} +6.13715 q^{94} +4.12048 q^{96} +17.7622i q^{97} +43.7729i q^{98} -0.187020 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9	$ 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9} + 12 q^{11} + 4 q^{14} + 10 q^{16} - 16 q^{19} - 8 q^{21} - 18 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 12 q^{34} + 14 q^{36} + 16 q^{39} + 48 q^{41} + 28 q^{44} - 32 q^{46} - 40 q^{49} + 4 q^{51} - 6 q^{54} - 60 q^{56} - 4 q^{59} + 20 q^{61} + 54 q^{64} - 32 q^{66} - 16 q^{69} - 24 q^{71} + 84 q^{74} - 16 q^{76} + 40 q^{79} + 8 q^{81} + 56 q^{84} - 88 q^{86} - 56 q^{89} - 72 q^{91} + 46 q^{94} + 12 q^{96} - 12 q^{99}+O(q^{100}) $ 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9 + 12 * q^11 + 4 * q^14 + 10 * q^16 - 16 * q^19 - 8 * q^21 - 18 * q^24 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 12 * q^34 + 14 * q^36 + 16 * q^39 + 48 * q^41 + 28 * q^44 - 32 * q^46 - 40 * q^49 + 4 * q^51 - 6 * q^54 - 60 * q^56 - 4 * q^59 + 20 * q^61 + 54 * q^64 - 32 * q^66 - 16 * q^69 - 24 * q^71 + 84 * q^74 - 16 * q^76 + 40 * q^79 + 8 * q^81 + 56 * q^84 - 88 * q^86 - 56 * q^89 - 72 * q^91 + 46 * q^94 + 12 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 2.52452i	− 1.78511i	−0.450940	−	0.892554i	$-0.648911\pi$
$2$	− 2.52452i	− 1.78511i	0.450940	−	0.892554i	$-0.351089\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.37322	−2.18661
$5$	0	0
$5$	0	0
$6$	2.52452	1.03063
$7$	4.93346i	1.86467i	0.361591	+	0.932337i	$0.382234\pi$
$7$	4.93346i	1.86467i	−0.361591	+	0.932337i	$0.617766\pi$
$8$	5.99126i	2.11823i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.187020	0.0563886	0.0281943	−	0.999602i	$-0.491024\pi$
$11$	0.187020	0.0563886	0.0281943	+	0.999602i	$0.491024\pi$
$12$	− 4.37322i	− 1.26244i
$13$	4.16953i	1.15642i	0.815888	+	0.578210i	$0.196249\pi$
$13$	4.16953i	1.15642i	−0.815888	+	0.578210i	$0.803751\pi$
$14$	12.4546	3.32864
$15$	0	0
$16$	6.37863	1.59466
$17$	2.85410i	0.692221i	0.938194	+	0.346111i	$0.112498\pi$
$17$	2.85410i	0.692221i	−0.938194	+	0.346111i	$0.887502\pi$
$18$	2.52452i	0.595036i
$19$	0.431014	0.0988814	0.0494407	−	0.998777i	$-0.484256\pi$
$19$	0.431014	0.0988814	0.0494407	+	0.998777i	$0.484256\pi$
$20$	0	0
$21$	−4.93346	−1.07657
$22$	− 0.472136i	− 0.100660i
$23$	− 4.36448i	− 0.910057i	−0.890477	−	0.455028i	$-0.849629\pi$
$23$	− 4.36448i	− 0.910057i	0.890477	−	0.455028i	$-0.150371\pi$
$24$	−5.99126	−1.22296
$25$	0	0
$26$	10.5261	2.06433
$27$	− 1.00000i	− 0.192450i
$28$	− 21.5751i	− 4.07732i
$29$	1.12048	0.208069	0.104034	−	0.994574i	$-0.466825\pi$
$29$	1.12048	0.208069	0.104034	+	0.994574i	$0.466825\pi$
$30$	0	0
$31$	−2.73852	−0.491852	−0.245926	−	0.969289i	$-0.579092\pi$
$31$	−2.73852	−0.491852	−0.245926	+	0.969289i	$0.579092\pi$
$32$	− 4.12048i	− 0.728405i
$33$	0.187020i	0.0325560i
$34$	7.20525	1.23569
$35$	0	0
$36$	4.37322	0.728870
$37$	5.35165i	0.879806i	0.898045	+	0.439903i	$0.144987\pi$
$37$	5.35165i	0.879806i	−0.898045	+	0.439903i	$0.855013\pi$
$38$	− 1.08811i	− 0.176514i
$39$	−4.16953	−0.667659
$40$	0	0
$41$	7.69740	1.20213	0.601066	−	0.799200i	$-0.294743\pi$
$41$	7.69740	1.20213	0.601066	+	0.799200i	$0.294743\pi$
$42$	12.4546i	1.92179i
$43$	− 6.58772i	− 1.00462i	−0.864688	−	0.502309i	$-0.832484\pi$
$43$	− 6.58772i	− 1.00462i	0.864688	−	0.502309i	$-0.167516\pi$
$44$	−0.817879	−0.123300
$45$	0	0
$46$	−11.0182	−1.62455
$47$	2.43101i	0.354600i	0.984157	+	0.177300i	$0.0567363\pi$
$47$	2.43101i	0.354600i	−0.984157	+	0.177300i	$0.943264\pi$
$48$	6.37863i	0.920675i
$49$	−17.3391	−2.47701
$50$	0	0
$51$	−2.85410	−0.399654
$52$	− 18.2343i	− 2.52864i
$53$	8.48496i	1.16550i	0.812652	+	0.582750i	$0.198023\pi$
$53$	8.48496i	1.16550i	−0.812652	+	0.582750i	$0.801977\pi$
$54$	−2.52452	−0.343544
$55$	0	0
$56$	−29.5576	−3.94981
$57$	0.431014i	0.0570892i
$58$	− 2.82869i	− 0.371425i
$59$	−10.7573	−1.40047	−0.700237	−	0.713910i	$-0.746923\pi$
$59$	−10.7573	−1.40047	−0.700237	+	0.713910i	$0.746923\pi$
$60$	0	0
$61$	9.24889	1.18420	0.592100	−	0.805865i	$-0.298299\pi$
$61$	9.24889	1.18420	0.592100	+	0.805865i	$0.298299\pi$
$62$	6.91345i	0.878009i
$63$	− 4.93346i	− 0.621558i
$64$	2.35499	0.294374
$65$	0	0
$66$	0.472136	0.0581159
$67$	− 9.03156i	− 1.10338i	−0.834049	−	0.551690i	$-0.813983\pi$
$67$	− 9.03156i	− 1.10338i	0.834049	−	0.551690i	$-0.186017\pi$
$68$	− 12.4816i	− 1.51362i
$69$	4.36448	0.525421
$70$	0	0
$71$	1.32336	0.157053	0.0785267	−	0.996912i	$-0.474978\pi$
$71$	1.32336	0.157053	0.0785267	+	0.996912i	$0.474978\pi$
$72$	− 5.99126i	− 0.706076i
$73$	3.04905i	0.356864i	0.983952	+	0.178432i	$0.0571025\pi$
$73$	3.04905i	0.356864i	−0.983952	+	0.178432i	$0.942898\pi$
$74$	13.5104	1.57055
$75$	0	0
$76$	−1.88492	−0.216215
$77$	0.922655i	0.105146i
$78$	10.5261i	1.19184i
$79$	8.19985	0.922555	0.461277	−	0.887256i	$-0.347391\pi$
$79$	8.19985	0.922555	0.461277	+	0.887256i	$0.347391\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 19.4323i	− 2.14593i
$83$	8.04112i	0.882628i	0.897353	+	0.441314i	$0.145487\pi$
$83$	8.04112i	0.882628i	−0.897353	+	0.441314i	$0.854513\pi$
$84$	21.5751	2.35404
$85$	0	0
$86$	−16.6309	−1.79335
$87$	1.12048i	0.120128i
$88$	1.12048i	0.119444i
$89$	3.10300	0.328917	0.164458	−	0.986384i	$-0.447412\pi$
$89$	3.10300	0.328917	0.164458	+	0.986384i	$0.447412\pi$
$90$	0	0
$91$	−20.5702	−2.15635
$92$	19.0868i	1.98994i
$93$	− 2.73852i	− 0.283971i
$94$	6.13715	0.632999
$95$	0	0
$96$	4.12048	0.420545
$97$	17.7622i	1.80347i	0.432286	+	0.901737i	$0.357707\pi$
$97$	17.7622i	1.80347i	−0.432286	+	0.901737i	$0.642293\pi$
$98$	43.7729i	4.42173i
$99$	−0.187020	−0.0187962
$100$	0	0
$101$	11.6368	1.15790	0.578951	−	0.815362i	$-0.303462\pi$
$101$	11.6368	1.15790	0.578951	+	0.815362i	$0.303462\pi$
$102$	7.20525i	0.713426i
$103$	− 5.13797i	− 0.506259i	−0.967432	−	0.253130i	$-0.918540\pi$
$103$	− 5.13797i	− 0.506259i	0.967432	−	0.253130i	$-0.0814599\pi$
$104$	−24.9807	−2.44956
$105$	0	0
$106$	21.4205	2.08054
$107$	− 16.3645i	− 1.58201i	−0.611807	−	0.791007i	$-0.709557\pi$
$107$	− 16.3645i	− 1.58201i	0.611807	−	0.791007i	$-0.290443\pi$
$108$	4.37322i	0.420813i
$109$	−1.38686	−0.132838	−0.0664188	−	0.997792i	$-0.521157\pi$
$109$	−1.38686	−0.132838	−0.0664188	+	0.997792i	$0.521157\pi$
$110$	0	0
$111$	−5.35165	−0.507956
$112$	31.4687i	2.97351i
$113$	− 14.9571i	− 1.40705i	−0.710673	−	0.703523i	$-0.751609\pi$
$113$	− 14.9571i	− 1.40705i	0.710673	−	0.703523i	$-0.248391\pi$
$114$	1.08811	0.101910
$115$	0	0
$116$	−4.90012	−0.454965
$117$	− 4.16953i	− 0.385473i
$118$	27.1569i	2.50000i
$119$	−14.0806	−1.29077
$120$	0	0
$121$	−10.9650	−0.996820
$122$	− 23.3491i	− 2.11392i
$123$	7.69740i	0.694051i
$124$	11.9761	1.07549
$125$	0	0
$126$	−12.4546	−1.10955
$127$	11.3234i	1.00479i	0.864640	+	0.502393i	$0.167547\pi$
$127$	11.3234i	1.00479i	−0.864640	+	0.502393i	$0.832453\pi$
$128$	− 14.1862i	− 1.25389i
$129$	6.58772	0.580016
$130$	0	0
$131$	17.1903	1.50192	0.750961	−	0.660347i	$-0.229591\pi$
$131$	17.1903	1.50192	0.750961	+	0.660347i	$0.229591\pi$
$132$	− 0.817879i	− 0.0711872i
$133$	2.12639i	0.184382i
$134$	−22.8004	−1.96965
$135$	0	0
$136$	−17.0997	−1.46628
$137$	− 7.49289i	− 0.640161i	−0.947390	−	0.320080i	$-0.896290\pi$
$137$	− 7.49289i	− 0.640161i	0.947390	−	0.320080i	$-0.103710\pi$
$138$	− 11.0182i	− 0.937934i
$139$	3.89234	0.330144	0.165072	−	0.986282i	$-0.447214\pi$
$139$	3.89234	0.330144	0.165072	+	0.986282i	$0.447214\pi$
$140$	0	0
$141$	−2.43101	−0.204728
$142$	− 3.34084i	− 0.280357i
$143$	0.779785i	0.0652089i
$144$	−6.37863	−0.531552
$145$	0	0
$146$	7.69740	0.637041
$147$	− 17.3391i	− 1.43010i
$148$	− 23.4040i	− 1.92379i
$149$	−3.20777	−0.262791	−0.131395	−	0.991330i	$-0.541946\pi$
$149$	−3.20777	−0.262791	−0.131395	+	0.991330i	$0.541946\pi$
$150$	0	0
$151$	−7.90805	−0.643548	−0.321774	−	0.946817i	$-0.604279\pi$
$151$	−7.90805	−0.643548	−0.321774	+	0.946817i	$0.604279\pi$
$152$	2.58232i	0.209454i
$153$	− 2.85410i	− 0.230740i
$154$	2.32927	0.187698
$155$	0	0
$156$	18.2343	1.45991
$157$	− 10.9335i	− 0.872585i	−0.899805	−	0.436293i	$-0.856291\pi$
$157$	− 10.9335i	− 0.872585i	0.899805	−	0.436293i	$-0.143709\pi$
$158$	− 20.7007i	− 1.64686i
$159$	−8.48496	−0.672901
$160$	0	0
$161$	21.5320	1.69696
$162$	− 2.52452i	− 0.198345i
$163$	10.1803i	0.797386i	0.917085	+	0.398693i	$0.130536\pi$
$163$	10.1803i	0.797386i	−0.917085	+	0.398693i	$0.869464\pi$
$164$	−33.6624	−2.62859
$165$	0	0
$166$	20.3000	1.57559
$167$	− 5.65123i	− 0.437305i	−0.975803	−	0.218653i	$-0.929834\pi$
$167$	− 5.65123i	− 0.437305i	0.975803	−	0.218653i	$-0.0701662\pi$
$168$	− 29.5576i	− 2.28042i
$169$	−4.38499	−0.337307
$170$	0	0
$171$	−0.431014	−0.0329605
$172$	28.8096i	2.19671i
$173$	− 20.8112i	− 1.58225i	−0.611657	−	0.791123i	$-0.709497\pi$
$173$	− 20.8112i	− 1.58225i	0.611657	−	0.791123i	$-0.290503\pi$
$174$	2.82869	0.214442
$175$	0	0
$176$	1.19293	0.0899204
$177$	− 10.7573i	− 0.808565i
$178$	− 7.83359i	− 0.587152i
$179$	16.1412	1.20645	0.603226	−	0.797570i	$-0.293882\pi$
$179$	16.1412	1.20645	0.603226	+	0.797570i	$0.293882\pi$
$180$	0	0
$181$	8.70028	0.646687	0.323343	−	0.946282i	$-0.395193\pi$
$181$	8.70028	0.646687	0.323343	+	0.946282i	$0.395193\pi$
$182$	51.9300i	3.84931i
$183$	9.24889i	0.683698i
$184$	26.1487	1.92771
$185$	0	0
$186$	−6.91345	−0.506919
$187$	0.533774i	0.0390334i
$188$	− 10.6314i	− 0.775372i
$189$	4.93346	0.358857
$190$	0	0
$191$	−20.8445	−1.50826	−0.754129	−	0.656726i	$-0.771941\pi$
$191$	−20.8445	−1.50826	−0.754129	+	0.656726i	$0.771941\pi$
$192$	2.35499i	0.169957i
$193$	− 7.13634i	− 0.513685i	−0.966453	−	0.256842i	$-0.917318\pi$
$193$	− 7.13634i	− 0.513685i	0.966453	−	0.256842i	$-0.0826821\pi$
$194$	44.8410	3.21939
$195$	0	0
$196$	75.8276	5.41626
$197$	− 10.2489i	− 0.730203i	−0.930968	−	0.365102i	$-0.881034\pi$
$197$	− 10.2489i	− 0.730203i	0.930968	−	0.365102i	$-0.118966\pi$
$198$	0.472136i	0.0335532i
$199$	20.0885	1.42404	0.712019	−	0.702160i	$-0.247781\pi$
$199$	20.0885	1.42404	0.712019	+	0.702160i	$0.247781\pi$
$200$	0	0
$201$	9.03156	0.637037
$202$	− 29.3773i	− 2.06698i
$203$	5.52786i	0.387980i
$204$	12.4816	0.873888
$205$	0	0
$206$	−12.9709	−0.903728
$207$	4.36448i	0.303352i
$208$	26.5959i	1.84409i
$209$	0.0806082	0.00557578
$210$	0	0
$211$	−8.26638	−0.569081	−0.284541	−	0.958664i	$-0.591841\pi$
$211$	−8.26638	−0.569081	−0.284541	+	0.958664i	$0.591841\pi$
$212$	− 37.1066i	− 2.54849i
$213$	1.32336i	0.0906749i
$214$	−41.3125	−2.82407
$215$	0	0
$216$	5.99126	0.407653
$217$	− 13.5104i	− 0.917144i
$218$	3.50117i	0.237129i
$219$	−3.04905	−0.206036
$220$	0	0
$221$	−11.9003	−0.800499
$222$	13.5104i	0.906757i
$223$	− 1.59262i	− 0.106650i	−0.998577	−	0.0533248i	$-0.983018\pi$
$223$	− 1.59262i	− 0.106650i	0.998577	−	0.0533248i	$-0.0169819\pi$
$224$	20.3283	1.35824
$225$	0	0
$226$	−37.7596	−2.51173
$227$	− 18.6213i	− 1.23594i	−0.786202	−	0.617969i	$-0.787956\pi$
$227$	− 18.6213i	− 1.23594i	0.786202	−	0.617969i	$-0.212044\pi$
$228$	− 1.88492i	− 0.124832i
$229$	5.63965	0.372679	0.186339	−	0.982485i	$-0.440338\pi$
$229$	5.63965	0.372679	0.186339	+	0.982485i	$0.440338\pi$
$230$	0	0
$231$	−0.922655	−0.0607063
$232$	6.71310i	0.440737i
$233$	6.57023i	0.430430i	0.976567	+	0.215215i	$0.0690453\pi$
$233$	6.57023i	0.430430i	−0.976567	+	0.215215i	$0.930955\pi$
$234$	−10.5261	−0.688112
$235$	0	0
$236$	47.0439	3.06229
$237$	8.19985i	0.532637i
$238$	35.5468i	2.30416i
$239$	−4.32825	−0.279972	−0.139986	−	0.990154i	$-0.544706\pi$
$239$	−4.32825	−0.279972	−0.139986	+	0.990154i	$0.544706\pi$
$240$	0	0
$241$	6.98717	0.450083	0.225042	−	0.974349i	$-0.427748\pi$
$241$	6.98717	0.450083	0.225042	+	0.974349i	$0.427748\pi$
$242$	27.6815i	1.77943i
$243$	1.00000i	0.0641500i
$244$	−40.4475	−2.58938
$245$	0	0
$246$	19.4323	1.23896
$247$	1.79713i	0.114348i
$248$	− 16.4072i	− 1.04186i
$249$	−8.04112	−0.509585
$250$	0	0
$251$	−20.3216	−1.28269	−0.641343	−	0.767254i	$-0.721623\pi$
$251$	−20.3216	−1.28269	−0.641343	+	0.767254i	$0.721623\pi$
$252$	21.5751i	1.35911i
$253$	− 0.816244i	− 0.0513168i
$254$	28.5861	1.79365
$255$	0	0
$256$	−31.1034	−1.94396
$257$	− 6.93145i	− 0.432372i	−0.976352	−	0.216186i	$-0.930638\pi$
$257$	− 6.93145i	− 0.432372i	0.976352	−	0.216186i	$-0.0693617\pi$
$258$	− 16.6309i	− 1.03539i
$259$	−26.4022	−1.64055
$260$	0	0
$261$	−1.12048	−0.0693562
$262$	− 43.3973i	− 2.68109i
$263$	21.6478i	1.33486i	0.744672	+	0.667431i	$0.232606\pi$
$263$	21.6478i	1.33486i	−0.744672	+	0.667431i	$0.767394\pi$
$264$	−1.12048	−0.0689610
$265$	0	0
$266$	5.36813	0.329141
$267$	3.10300i	0.189900i
$268$	39.4970i	2.41266i
$269$	18.2518	1.11283	0.556415	−	0.830904i	$-0.312176\pi$
$269$	18.2518	1.11283	0.556415	+	0.830904i	$0.312176\pi$
$270$	0	0
$271$	−4.10766	−0.249522	−0.124761	−	0.992187i	$-0.539816\pi$
$271$	−4.10766	−0.249522	−0.124761	+	0.992187i	$0.539816\pi$
$272$	18.2052i	1.10386i
$273$	− 20.5702i	− 1.24497i
$274$	−18.9160	−1.14276
$275$	0	0
$276$	−19.0868	−1.14889
$277$	20.8395i	1.25212i	0.779773	+	0.626062i	$0.215334\pi$
$277$	20.8395i	1.25212i	−0.779773	+	0.626062i	$0.784666\pi$
$278$	− 9.82631i	− 0.589343i
$279$	2.73852	0.163951
$280$	0	0
$281$	21.3490	1.27357	0.636787	−	0.771039i	$-0.280263\pi$
$281$	21.3490	1.27357	0.636787	+	0.771039i	$0.280263\pi$
$282$	6.13715i	0.365462i
$283$	− 2.23016i	− 0.132569i	−0.997801	−	0.0662846i	$-0.978885\pi$
$283$	− 2.23016i	− 0.132569i	0.997801	−	0.0662846i	$-0.0211145\pi$
$284$	−5.78733	−0.343415
$285$	0	0
$286$	1.96859	0.116405
$287$	37.9748i	2.24158i
$288$	4.12048i	0.242802i
$289$	8.85410	0.520830
$290$	0	0
$291$	−17.7622	−1.04124
$292$	− 13.3342i	− 0.780323i
$293$	14.0177i	0.818924i	0.912327	+	0.409462i	$0.134284\pi$
$293$	14.0177i	0.818924i	−0.912327	+	0.409462i	$0.865716\pi$
$294$	−43.7729	−2.55289
$295$	0	0
$296$	−32.0631	−1.86363
$297$	− 0.187020i	− 0.0108520i
$298$	8.09810i	0.469110i
$299$	18.1978	1.05241
$300$	0	0
$301$	32.5003	1.87328
$302$	19.9641i	1.14880i
$303$	11.6368i	0.668515i
$304$	2.74928	0.157682
$305$	0	0
$306$	−7.20525	−0.411897
$307$	17.2294i	0.983333i	0.870784	+	0.491667i	$0.163612\pi$
$307$	17.2294i	0.983333i	−0.870784	+	0.491667i	$0.836388\pi$
$308$	− 4.03498i	− 0.229914i
$309$	5.13797	0.292289
$310$	0	0
$311$	1.47703	0.0837550	0.0418775	−	0.999123i	$-0.486666\pi$
$311$	1.47703	0.0837550	0.0418775	+	0.999123i	$0.486666\pi$
$312$	− 24.9807i	− 1.41426i
$313$	30.7937i	1.74056i	0.492554	+	0.870282i	$0.336063\pi$
$313$	30.7937i	1.74056i	−0.492554	+	0.870282i	$0.663937\pi$
$314$	−27.6018	−1.55766
$315$	0	0
$316$	−35.8597	−2.01727
$317$	7.86693i	0.441851i	0.975291	+	0.220925i	$0.0709077\pi$
$317$	7.86693i	0.441851i	−0.975291	+	0.220925i	$0.929092\pi$
$318$	21.4205i	1.20120i
$319$	0.209553	0.0117327
$320$	0	0
$321$	16.3645	0.913376
$322$	− 54.3580i	− 3.02925i
$323$	1.23016i	0.0684478i
$324$	−4.37322	−0.242957
$325$	0	0
$326$	25.7005	1.42342
$327$	− 1.38686i	− 0.0766938i
$328$	46.1171i	2.54639i
$329$	−11.9933	−0.661213
$330$	0	0
$331$	25.6794	1.41147	0.705733	−	0.708478i	$-0.250618\pi$
$331$	25.6794	1.41147	0.705733	+	0.708478i	$0.250618\pi$
$332$	− 35.1656i	− 1.92996i
$333$	− 5.35165i	− 0.293269i
$334$	−14.2667	−0.780637
$335$	0	0
$336$	−31.4687	−1.71676
$337$	13.3059i	0.724817i	0.932019	+	0.362408i	$0.118045\pi$
$337$	13.3059i	0.724817i	−0.932019	+	0.362408i	$0.881955\pi$
$338$	11.0700i	0.602130i
$339$	14.9571	0.812358
$340$	0	0
$341$	−0.512157	−0.0277349
$342$	1.08811i	0.0588380i
$343$	− 51.0074i	− 2.75414i
$344$	39.4687	2.12801
$345$	0	0
$346$	−52.5384	−2.82448
$347$	12.2630i	0.658310i	0.944276	+	0.329155i	$0.106764\pi$
$347$	12.2630i	0.658310i	−0.944276	+	0.329155i	$0.893236\pi$
$348$	− 4.90012i	− 0.262674i
$349$	12.5880	0.673818	0.336909	−	0.941537i	$-0.390619\pi$
$349$	12.5880	0.673818	0.336909	+	0.941537i	$0.390619\pi$
$350$	0	0
$351$	4.16953	0.222553
$352$	− 0.770612i	− 0.0410738i
$353$	17.9473i	0.955238i	0.878567	+	0.477619i	$0.158500\pi$
$353$	17.9473i	0.955238i	−0.878567	+	0.477619i	$0.841500\pi$
$354$	−27.1569	−1.44338
$355$	0	0
$356$	−13.5701	−0.719213
$357$	− 14.0806i	− 0.745225i
$358$	− 40.7489i	− 2.15365i
$359$	36.8096	1.94273	0.971367	−	0.237583i	$-0.0763552\pi$
$359$	36.8096	1.94273	0.971367	+	0.237583i	$0.0763552\pi$
$360$	0	0
$361$	−18.8142	−0.990222
$362$	− 21.9641i	− 1.15441i
$363$	− 10.9650i	− 0.575514i
$364$	89.9582	4.71509
$365$	0	0
$366$	23.3491	1.22047
$367$	− 1.81874i	− 0.0949377i	−0.998873	−	0.0474688i	$-0.984885\pi$
$367$	− 1.81874i	− 0.0949377i	0.998873	−	0.0474688i	$-0.0151155\pi$
$368$	− 27.8394i	− 1.45123i
$369$	−7.69740	−0.400710
$370$	0	0
$371$	−41.8602	−2.17328
$372$	11.9761i	0.620934i
$373$	22.7115i	1.17596i	0.808877	+	0.587978i	$0.200076\pi$
$373$	22.7115i	1.17596i	−0.808877	+	0.587978i	$0.799924\pi$
$374$	1.34752	0.0696788
$375$	0	0
$376$	−14.5648	−0.751124
$377$	4.67189i	0.240615i
$378$	− 12.4546i	− 0.640598i
$379$	23.4164	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$379$	23.4164	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$380$	0	0
$381$	−11.3234	−0.580113
$382$	52.6225i	2.69240i
$383$	− 9.35958i	− 0.478252i	−0.970989	−	0.239126i	$-0.923139\pi$
$383$	− 9.35958i	− 0.478252i	0.970989	−	0.239126i	$-0.0768609\pi$
$384$	14.1862	0.723937
$385$	0	0
$386$	−18.0159	−0.916983
$387$	6.58772i	0.334873i
$388$	− 77.6778i	− 3.94349i
$389$	23.4323	1.18806	0.594031	−	0.804442i	$-0.297535\pi$
$389$	23.4323	1.18806	0.594031	+	0.804442i	$0.297535\pi$
$390$	0	0
$391$	12.4567	0.629961
$392$	− 103.883i	− 5.24687i
$393$	17.1903i	0.867135i
$394$	−25.8736	−1.30349
$395$	0	0
$396$	0.817879	0.0411000
$397$	22.9915i	1.15391i	0.816775	+	0.576956i	$0.195760\pi$
$397$	22.9915i	1.15391i	−0.816775	+	0.576956i	$0.804240\pi$
$398$	− 50.7140i	− 2.54206i
$399$	−2.12639	−0.106453
$400$	0	0
$401$	−25.4862	−1.27272	−0.636360	−	0.771392i	$-0.719561\pi$
$401$	−25.4862	−1.27272	−0.636360	+	0.771392i	$0.719561\pi$
$402$	− 22.8004i	− 1.13718i
$403$	− 11.4183i	− 0.568788i
$404$	−50.8902	−2.53188
$405$	0	0
$406$	13.9552	0.692586
$407$	1.00086i	0.0496110i
$408$	− 17.0997i	− 0.846559i
$409$	−9.14100	−0.451993	−0.225997	−	0.974128i	$-0.572564\pi$
$409$	−9.14100	−0.451993	−0.225997	+	0.974128i	$0.572564\pi$
$410$	0	0
$411$	7.49289	0.369597
$412$	22.4695i	1.10699i
$413$	− 53.0705i	− 2.61143i
$414$	11.0182	0.541516
$415$	0	0
$416$	17.1805	0.842343
$417$	3.89234i	0.190609i
$418$	− 0.203497i	− 0.00995338i
$419$	−4.62581	−0.225986	−0.112993	−	0.993596i	$-0.536044\pi$
$419$	−4.62581	−0.225986	−0.112993	+	0.993596i	$0.536044\pi$
$420$	0	0
$421$	22.1325	1.07867	0.539337	−	0.842090i	$-0.318675\pi$
$421$	22.1325	1.07867	0.539337	+	0.842090i	$0.318675\pi$
$422$	20.8687i	1.01587i
$423$	− 2.43101i	− 0.118200i
$424$	−50.8356	−2.46879
$425$	0	0
$426$	3.34084	0.161864
$427$	45.6291i	2.20815i
$428$	71.5655i	3.45925i
$429$	−0.779785	−0.0376484
$430$	0	0
$431$	−33.1503	−1.59679	−0.798396	−	0.602133i	$-0.794318\pi$
$431$	−33.1503	−1.59679	−0.798396	+	0.602133i	$0.794318\pi$
$432$	− 6.37863i	− 0.306892i
$433$	13.5576i	0.651539i	0.945449	+	0.325769i	$0.105623\pi$
$433$	13.5576i	0.651539i	−0.945449	+	0.325769i	$0.894377\pi$
$434$	−34.1073	−1.63720
$435$	0	0
$436$	6.06507	0.290464
$437$	− 1.88115i	− 0.0899877i
$438$	7.69740i	0.367796i
$439$	−13.1188	−0.626128	−0.313064	−	0.949732i	$-0.601356\pi$
$439$	−13.1188	−0.626128	−0.313064	+	0.949732i	$0.601356\pi$
$440$	0	0
$441$	17.3391	0.825670
$442$	30.0425i	1.42898i
$443$	− 0.589737i	− 0.0280193i	−0.999902	−	0.0140096i	$-0.995540\pi$
$443$	− 0.589737i	− 0.0280193i	0.999902	−	0.0140096i	$-0.00445955\pi$
$444$	23.4040	1.11070
$445$	0	0
$446$	−4.02061	−0.190381
$447$	− 3.20777i	− 0.151722i
$448$	11.6183i	0.548912i
$449$	−25.7878	−1.21700	−0.608501	−	0.793553i	$-0.708229\pi$
$449$	−25.7878	−1.21700	−0.608501	+	0.793553i	$0.708229\pi$
$450$	0	0
$451$	1.43957	0.0677865
$452$	65.4107i	3.07666i
$453$	− 7.90805i	− 0.371553i
$454$	−47.0099	−2.20628
$455$	0	0
$456$	−2.58232	−0.120928
$457$	− 23.1744i	− 1.08405i	−0.840361	−	0.542027i	$-0.817657\pi$
$457$	− 23.1744i	− 1.08405i	0.840361	−	0.542027i	$-0.182343\pi$
$458$	− 14.2374i	− 0.665272i
$459$	2.85410	0.133218
$460$	0	0
$461$	7.94605	0.370085	0.185042	−	0.982731i	$-0.440758\pi$
$461$	7.94605	0.370085	0.185042	+	0.982731i	$0.440758\pi$
$462$	2.32927i	0.108367i
$463$	− 22.0914i	− 1.02668i	−0.858187	−	0.513338i	$-0.828409\pi$
$463$	− 22.0914i	− 1.02668i	0.858187	−	0.513338i	$-0.171591\pi$
$464$	7.14714	0.331798
$465$	0	0
$466$	16.5867	0.768364
$467$	− 25.8124i	− 1.19446i	−0.802071	−	0.597229i	$-0.796268\pi$
$467$	− 25.8124i	− 1.19446i	0.802071	−	0.597229i	$-0.203732\pi$
$468$	18.2343i	0.842880i
$469$	44.5569	2.05745
$470$	0	0
$471$	10.9335	0.503787
$472$	− 64.4495i	− 2.96653i
$473$	− 1.23203i	− 0.0566490i
$474$	20.7007	0.950815
$475$	0	0
$476$	61.5776	2.82241
$477$	− 8.48496i	− 0.388500i
$478$	10.9268i	0.499779i
$479$	−38.9201	−1.77830	−0.889152	−	0.457611i	$-0.848705\pi$
$479$	−38.9201	−1.77830	−0.889152	+	0.457611i	$0.848705\pi$
$480$	0	0
$481$	−22.3139	−1.01743
$482$	− 17.6393i	− 0.803448i
$483$	21.5320i	0.979740i
$484$	47.9525	2.17966
$485$	0	0
$486$	2.52452	0.114515
$487$	− 27.5335i	− 1.24766i	−0.781560	−	0.623830i	$-0.785576\pi$
$487$	− 27.5335i	− 1.24766i	0.781560	−	0.623830i	$-0.214424\pi$
$488$	55.4125i	2.50841i
$489$	−10.1803	−0.460371
$490$	0	0
$491$	10.1531	0.458201	0.229100	−	0.973403i	$-0.426422\pi$
$491$	10.1531	0.458201	0.229100	+	0.973403i	$0.426422\pi$
$492$	− 33.6624i	− 1.51762i
$493$	3.19797i	0.144029i
$494$	4.53689	0.204124
$495$	0	0
$496$	−17.4680	−0.784335
$497$	6.52873i	0.292854i
$498$	20.3000i	0.909665i
$499$	−40.9313	−1.83234	−0.916168	−	0.400794i	$-0.868734\pi$
$499$	−40.9313	−1.83234	−0.916168	+	0.400794i	$0.868734\pi$
$500$	0	0
$501$	5.65123	0.252478
$502$	51.3023i	2.28973i
$503$	− 40.8527i	− 1.82153i	−0.412923	−	0.910766i	$-0.635492\pi$
$503$	− 40.8527i	− 1.82153i	0.412923	−	0.910766i	$-0.364508\pi$
$504$	29.5576	1.31660
$505$	0	0
$506$	−2.06063	−0.0916060
$507$	− 4.38499i	− 0.194744i
$508$	− 49.5195i	− 2.19707i
$509$	5.04400	0.223572	0.111786	−	0.993732i	$-0.464343\pi$
$509$	5.04400	0.223572	0.111786	+	0.993732i	$0.464343\pi$
$510$	0	0
$511$	−15.0424	−0.665435
$512$	50.1489i	2.21629i
$513$	− 0.431014i	− 0.0190297i
$514$	−17.4986	−0.771830
$515$	0	0
$516$	−28.8096	−1.26827
$517$	0.454648i	0.0199954i
$518$	66.6529i	2.92856i
$519$	20.8112	0.913510
$520$	0	0
$521$	22.9426	1.00514	0.502568	−	0.864538i	$-0.332389\pi$
$521$	22.9426	1.00514	0.502568	+	0.864538i	$0.332389\pi$
$522$	2.82869i	0.123808i
$523$	10.7448i	0.469838i	0.972015	+	0.234919i	$0.0754824\pi$
$523$	10.7448i	0.469838i	−0.972015	+	0.234919i	$0.924518\pi$
$524$	−75.1769	−3.28412
$525$	0	0
$526$	54.6504	2.38287
$527$	− 7.81601i	− 0.340471i
$528$	1.19293i	0.0519156i
$529$	3.95133	0.171797
$530$	0	0
$531$	10.7573	0.466825
$532$	− 9.29919i	− 0.403171i
$533$	32.0945i	1.39017i
$534$	7.83359	0.338992
$535$	0	0
$536$	54.1104	2.33721
$537$	16.1412i	0.696546i
$538$	− 46.0770i	− 1.98652i
$539$	−3.24275	−0.139675
$540$	0	0
$541$	41.9437	1.80330	0.901651	−	0.432464i	$-0.142356\pi$
$541$	41.9437	1.80330	0.901651	+	0.432464i	$0.142356\pi$
$542$	10.3699i	0.445425i
$543$	8.70028i	0.373365i
$544$	11.7603	0.504218
$545$	0	0
$546$	−51.9300	−2.22240
$547$	1.09887i	0.0469842i	0.999724	+	0.0234921i	$0.00747845\pi$
$547$	1.09887i	0.0469842i	−0.999724	+	0.0234921i	$0.992522\pi$
$548$	32.7681i	1.39978i
$549$	−9.24889	−0.394733
$550$	0	0
$551$	0.482944	0.0205741
$552$	26.1487i	1.11296i
$553$	40.4536i	1.72026i
$554$	52.6098	2.23518
$555$	0	0
$556$	−17.0221	−0.721897
$557$	− 2.25279i	− 0.0954536i	−0.998860	−	0.0477268i	$-0.984802\pi$
$557$	− 2.25279i	− 0.0954536i	0.998860	−	0.0477268i	$-0.0151977\pi$
$558$	− 6.91345i	− 0.292670i
$559$	27.4677	1.16176
$560$	0	0
$561$	−0.533774	−0.0225359
$562$	− 53.8961i	− 2.27347i
$563$	29.7361i	1.25323i	0.779330	+	0.626614i	$0.215560\pi$
$563$	29.7361i	1.25323i	−0.779330	+	0.626614i	$0.784440\pi$
$564$	10.6314	0.447661
$565$	0	0
$566$	−5.63009	−0.236650
$567$	4.93346i	0.207186i
$568$	7.92856i	0.332675i
$569$	−27.1667	−1.13889	−0.569444	−	0.822030i	$-0.692842\pi$
$569$	−27.1667	−1.13889	−0.569444	+	0.822030i	$0.692842\pi$
$570$	0	0
$571$	−1.97482	−0.0826437	−0.0413218	−	0.999146i	$-0.513157\pi$
$571$	−1.97482	−0.0826437	−0.0413218	+	0.999146i	$0.513157\pi$
$572$	− 3.41017i	− 0.142586i
$573$	− 20.8445i	− 0.870793i
$574$	95.8684	4.00147
$575$	0	0
$576$	−2.35499	−0.0981247
$577$	− 13.6634i	− 0.568816i	−0.958703	−	0.284408i	$-0.908203\pi$
$577$	− 13.6634i	− 0.568816i	0.958703	−	0.284408i	$-0.0917970\pi$
$578$	− 22.3524i	− 0.929737i
$579$	7.13634	0.296576
$580$	0	0
$581$	−39.6706	−1.64581
$582$	44.8410i	1.85872i
$583$	1.58686i	0.0657208i
$584$	−18.2676	−0.755920
$585$	0	0
$586$	35.3881	1.46187
$587$	36.3511i	1.50037i	0.661227	+	0.750186i	$0.270036\pi$
$587$	36.3511i	1.50037i	−0.661227	+	0.750186i	$0.729964\pi$
$588$	75.8276i	3.12708i
$589$	−1.18034	−0.0486351
$590$	0	0
$591$	10.2489	0.421583
$592$	34.1362i	1.40299i
$593$	− 37.7467i	− 1.55007i	−0.631918	−	0.775035i	$-0.717732\pi$
$593$	− 37.7467i	− 1.55007i	0.631918	−	0.775035i	$-0.282268\pi$
$594$	−0.472136	−0.0193720
$595$	0	0
$596$	14.0283	0.574621
$597$	20.0885i	0.822169i
$598$	− 45.9409i	− 1.87866i
$599$	17.3814	0.710186	0.355093	−	0.934831i	$-0.384449\pi$
$599$	17.3814	0.710186	0.355093	+	0.934831i	$0.384449\pi$
$600$	0	0
$601$	−22.3120	−0.910126	−0.455063	−	0.890459i	$-0.650383\pi$
$601$	−22.3120	−0.910126	−0.455063	+	0.890459i	$0.650383\pi$
$602$	− 82.0477i	− 3.34402i
$603$	9.03156i	0.367794i
$604$	34.5837	1.40719
$605$	0	0
$606$	29.3773	1.19337
$607$	− 13.6024i	− 0.552105i	−0.961143	−	0.276053i	$-0.910974\pi$
$607$	− 13.6024i	− 0.552105i	0.961143	−	0.276053i	$-0.0890264\pi$
$608$	− 1.77599i	− 0.0720258i
$609$	−5.52786	−0.224000
$610$	0	0
$611$	−10.1362	−0.410066
$612$	12.4816i	0.504540i
$613$	22.2835i	0.900021i	0.893023	+	0.450011i	$0.148580\pi$
$613$	22.2835i	0.900021i	−0.893023	+	0.450011i	$0.851420\pi$
$614$	43.4960	1.75536
$615$	0	0
$616$	−5.52786	−0.222724
$617$	0.115820i	0.00466276i	0.999997	+	0.00233138i	$0.000742101\pi$
$617$	0.115820i	0.00466276i	−0.999997	+	0.00233138i	$0.999258\pi$
$618$	− 12.9709i	− 0.521767i
$619$	0.0726817	0.00292133	0.00146066	−	0.999999i	$-0.499535\pi$
$619$	0.0726817	0.00292133	0.00146066	+	0.999999i	$0.499535\pi$
$620$	0	0
$621$	−4.36448	−0.175140
$622$	− 3.72881i	− 0.149512i
$623$	15.3085i	0.613323i
$624$	−26.5959	−1.06469
$625$	0	0
$626$	77.7395	3.10709
$627$	0.0806082i	0.00321918i
$628$	47.8145i	1.90800i
$629$	−15.2742	−0.609021
$630$	0	0
$631$	−29.5462	−1.17622	−0.588108	−	0.808782i	$-0.700127\pi$
$631$	−29.5462	−1.17622	−0.588108	+	0.808782i	$0.700127\pi$
$632$	49.1274i	1.95418i
$633$	− 8.26638i	− 0.328559i
$634$	19.8602	0.788751
$635$	0	0
$636$	37.1066	1.47137
$637$	− 72.2958i	− 2.86446i
$638$	− 0.529020i	− 0.0209441i
$639$	−1.32336	−0.0523512
$640$	0	0
$641$	4.28099	0.169089	0.0845444	−	0.996420i	$-0.473057\pi$
$641$	4.28099	0.169089	0.0845444	+	0.996420i	$0.473057\pi$
$642$	− 41.3125i	− 1.63048i
$643$	− 38.7887i	− 1.52968i	−0.644223	−	0.764838i	$-0.722819\pi$
$643$	− 38.7887i	− 1.52968i	0.644223	−	0.764838i	$-0.277181\pi$
$644$	−94.1642	−3.71059
$645$	0	0
$646$	3.10556	0.122187
$647$	− 19.5243i	− 0.767580i	−0.923421	−	0.383790i	$-0.874619\pi$
$647$	− 19.5243i	− 0.767580i	0.923421	−	0.383790i	$-0.125381\pi$
$648$	5.99126i	0.235359i
$649$	−2.01182	−0.0789708
$650$	0	0
$651$	13.5104	0.529513
$652$	− 44.5209i	− 1.74357i
$653$	16.5733i	0.648562i	0.945961	+	0.324281i	$0.105122\pi$
$653$	16.5733i	0.648562i	−0.945961	+	0.324281i	$0.894878\pi$
$654$	−3.50117	−0.136907
$655$	0	0
$656$	49.0988	1.91699
$657$	− 3.04905i	− 0.118955i
$658$	30.2774i	1.18034i
$659$	−6.79371	−0.264645	−0.132323	−	0.991207i	$-0.542244\pi$
$659$	−6.79371	−0.264645	−0.132323	+	0.991207i	$0.542244\pi$
$660$	0	0
$661$	13.2090	0.513771	0.256886	−	0.966442i	$-0.417304\pi$
$661$	13.2090	0.513771	0.256886	+	0.966442i	$0.417304\pi$
$662$	− 64.8282i	− 2.51962i
$663$	− 11.9003i	− 0.462168i
$664$	−48.1764	−1.86961
$665$	0	0
$666$	−13.5104	−0.523516
$667$	− 4.89032i	− 0.189354i
$668$	24.7141i	0.956216i
$669$	1.59262	0.0615742
$670$	0	0
$671$	1.72973	0.0667753
$672$	20.3283i	0.784179i
$673$	− 47.7353i	− 1.84006i	−0.391846	−	0.920031i	$-0.628163\pi$
$673$	− 47.7353i	− 1.84006i	0.391846	−	0.920031i	$-0.371837\pi$
$674$	33.5910	1.29388
$675$	0	0
$676$	19.1766	0.737560
$677$	− 11.6486i	− 0.447691i	−0.974625	−	0.223846i	$-0.928139\pi$
$677$	− 11.6486i	− 0.447691i	0.974625	−	0.223846i	$-0.0718612\pi$
$678$	− 37.7596i	− 1.45015i
$679$	−87.6289	−3.36289
$680$	0	0
$681$	18.6213	0.713570
$682$	1.29295i	0.0495097i
$683$	30.1235i	1.15264i	0.817223	+	0.576322i	$0.195513\pi$
$683$	30.1235i	1.15264i	−0.817223	+	0.576322i	$0.804487\pi$
$684$	1.88492	0.0720717
$685$	0	0
$686$	−128.769	−4.91644
$687$	5.63965i	0.215166i
$688$	− 42.0206i	− 1.60202i
$689$	−35.3783	−1.34781
$690$	0	0
$691$	−34.2787	−1.30402	−0.652011	−	0.758209i	$-0.726074\pi$
$691$	−34.2787	−1.30402	−0.652011	+	0.758209i	$0.726074\pi$
$692$	91.0120i	3.45976i
$693$	− 0.922655i	− 0.0350488i
$694$	30.9582	1.17516
$695$	0	0
$696$	−6.71310	−0.254460
$697$	21.9692i	0.832141i
$698$	− 31.7786i	− 1.20284i
$699$	−6.57023	−0.248509
$700$	0	0
$701$	−45.2669	−1.70971	−0.854853	−	0.518871i	$-0.826353\pi$
$701$	−45.2669	−1.70971	−0.854853	+	0.518871i	$0.826353\pi$
$702$	− 10.5261i	− 0.397281i
$703$	2.30664i	0.0869965i
$704$	0.440430	0.0165993
$705$	0	0
$706$	45.3084	1.70520
$707$	57.4096i	2.15911i
$708$	47.0439i	1.76802i
$709$	27.0295	1.01512	0.507558	−	0.861618i	$-0.330548\pi$
$709$	27.0295	1.01512	0.507558	+	0.861618i	$0.330548\pi$
$710$	0	0
$711$	−8.19985	−0.307518
$712$	18.5908i	0.696721i
$713$	11.9522i	0.447613i
$714$	−35.5468	−1.33031
$715$	0	0
$716$	−70.5892	−2.63804
$717$	− 4.32825i	− 0.161642i
$718$	− 92.9266i	− 3.46799i
$719$	−4.57600	−0.170656	−0.0853279	−	0.996353i	$-0.527194\pi$
$719$	−4.57600	−0.170656	−0.0853279	+	0.996353i	$0.527194\pi$
$720$	0	0
$721$	25.3480	0.944009
$722$	47.4970i	1.76765i
$723$	6.98717i	0.259856i
$724$	−38.0482	−1.41405
$725$	0	0
$726$	−27.6815	−1.02736
$727$	− 44.0914i	− 1.63526i	−0.575744	−	0.817630i	$-0.695287\pi$
$727$	− 44.0914i	− 1.63526i	0.575744	−	0.817630i	$-0.304713\pi$
$728$	− 123.242i	− 4.56763i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	18.8020	0.695418
$732$	− 40.4475i	− 1.49498i
$733$	− 21.3757i	− 0.789528i	−0.918782	−	0.394764i	$-0.870826\pi$
$733$	− 21.3757i	− 0.789528i	0.918782	−	0.394764i	$-0.129174\pi$
$734$	−4.59146	−0.169474
$735$	0	0
$736$	−17.9838	−0.662890
$737$	− 1.68908i	− 0.0622181i
$738$	19.4323i	0.715311i
$739$	5.60055	0.206019	0.103010	−	0.994680i	$-0.467153\pi$
$739$	5.60055	0.206019	0.103010	+	0.994680i	$0.467153\pi$
$740$	0	0
$741$	−1.79713	−0.0660191
$742$	105.677i	3.87953i
$743$	21.2617i	0.780017i	0.920811	+	0.390008i	$0.127528\pi$
$743$	21.2617i	0.780017i	−0.920811	+	0.390008i	$0.872472\pi$
$744$	16.4072	0.601516
$745$	0	0
$746$	57.3356	2.09921
$747$	− 8.04112i	− 0.294209i
$748$	− 2.33431i	− 0.0853508i
$749$	80.7336	2.94994
$750$	0	0
$751$	14.4685	0.527962	0.263981	−	0.964528i	$-0.414964\pi$
$751$	14.4685	0.527962	0.263981	+	0.964528i	$0.414964\pi$
$752$	15.5065i	0.565465i
$753$	− 20.3216i	− 0.740559i
$754$	11.7943	0.429523
$755$	0	0
$756$	−21.5751	−0.784680
$757$	− 12.5210i	− 0.455085i	−0.973768	−	0.227542i	$-0.926931\pi$
$757$	− 12.5210i	− 0.455085i	0.973768	−	0.227542i	$-0.0730690\pi$
$758$	− 59.1153i	− 2.14716i
$759$	0.816244	0.0296278
$760$	0	0
$761$	−19.1769	−0.695163	−0.347581	−	0.937650i	$-0.612997\pi$
$761$	−19.1769	−0.695163	−0.347581	+	0.937650i	$0.612997\pi$
$762$	28.5861i	1.03556i
$763$	− 6.84205i	− 0.247699i
$764$	91.1578	3.29797
$765$	0	0
$766$	−23.6285	−0.853732
$767$	− 44.8527i	− 1.61954i
$768$	− 31.1034i	− 1.12235i
$769$	−41.8407	−1.50882	−0.754408	−	0.656406i	$-0.772076\pi$
$769$	−41.8407	−1.50882	−0.754408	+	0.656406i	$0.772076\pi$
$770$	0	0
$771$	6.93145	0.249630
$772$	31.2088i	1.12323i
$773$	− 22.9098i	− 0.824009i	−0.911182	−	0.412005i	$-0.864829\pi$
$773$	− 22.9098i	− 0.824009i	0.911182	−	0.412005i	$-0.135171\pi$
$774$	16.6309	0.597784
$775$	0	0
$776$	−106.418	−3.82017
$777$	− 26.4022i	− 0.947173i
$778$	− 59.1553i	− 2.12082i
$779$	3.31769	0.118868
$780$	0	0
$781$	0.247494	0.00885602
$782$	− 31.4472i	− 1.12455i
$783$	− 1.12048i	− 0.0400428i
$784$	−110.599	−3.94998
$785$	0	0
$786$	43.3973	1.54793
$787$	10.7906i	0.384643i	0.981332	+	0.192322i	$0.0616017\pi$
$787$	10.7906i	0.384643i	−0.981332	+	0.192322i	$0.938398\pi$
$788$	44.8207i	1.59667i
$789$	−21.6478	−0.770683
$790$	0	0
$791$	73.7903	2.62368
$792$	− 1.12048i	− 0.0398146i
$793$	38.5636i	1.36943i
$794$	58.0427	2.05986
$795$	0	0
$796$	−87.8516	−3.11382
$797$	− 45.2213i	− 1.60182i	−0.598784	−	0.800910i	$-0.704349\pi$
$797$	− 45.2213i	− 1.60182i	0.598784	−	0.800910i	$-0.295651\pi$
$798$	5.36813i	0.190030i
$799$	−6.93836	−0.245462
$800$	0	0
$801$	−3.10300	−0.109639
$802$	64.3405i	2.27194i
$803$	0.570232i	0.0201231i
$804$	−39.4970	−1.39295
$805$	0	0
$806$	−28.8259	−1.01535
$807$	18.2518i	0.642493i
$808$	69.7189i	2.45270i
$809$	−24.1895	−0.850458	−0.425229	−	0.905086i	$-0.639806\pi$
$809$	−24.1895	−0.850458	−0.425229	+	0.905086i	$0.639806\pi$
$810$	0	0
$811$	25.8633	0.908182	0.454091	−	0.890955i	$-0.349964\pi$
$811$	25.8633	0.908182	0.454091	+	0.890955i	$0.349964\pi$
$812$	− 24.1746i	− 0.848361i
$813$	− 4.10766i	− 0.144062i
$814$	2.52671	0.0885611
$815$	0	0
$816$	−18.2052	−0.637311
$817$	− 2.83940i	− 0.0993381i
$818$	23.0767i	0.806857i
$819$	20.5702	0.718782
$820$	0	0
$821$	34.3622	1.19925	0.599624	−	0.800282i	$-0.295317\pi$
$821$	34.3622	1.19925	0.599624	+	0.800282i	$0.295317\pi$
$822$	− 18.9160i	− 0.659770i
$823$	46.7556i	1.62980i	0.579603	+	0.814899i	$0.303208\pi$
$823$	46.7556i	1.62980i	−0.579603	+	0.814899i	$0.696792\pi$
$824$	30.7829	1.07237
$825$	0	0
$826$	−133.978	−4.66168
$827$	− 30.1332i	− 1.04783i	−0.851770	−	0.523916i	$-0.824470\pi$
$827$	− 30.1332i	− 1.04783i	0.851770	−	0.523916i	$-0.175530\pi$
$828$	− 19.0868i	− 0.663313i
$829$	37.5623	1.30459	0.652296	−	0.757964i	$-0.273806\pi$
$829$	37.5623	1.30459	0.652296	+	0.757964i	$0.273806\pi$
$830$	0	0
$831$	−20.8395	−0.722914
$832$	9.81922i	0.340420i
$833$	− 49.4875i	− 1.71464i
$834$	9.82631	0.340257
$835$	0	0
$836$	−0.352517	−0.0121921
$837$	2.73852i	0.0946570i
$838$	11.6780i	0.403409i
$839$	15.7230	0.542820	0.271410	−	0.962464i	$-0.412510\pi$
$839$	15.7230	0.542820	0.271410	+	0.962464i	$0.412510\pi$
$840$	0	0
$841$	−27.7445	−0.956707
$842$	− 55.8741i	− 1.92555i
$843$	21.3490i	0.735299i
$844$	36.1507	1.24436
$845$	0	0
$846$	−6.13715	−0.211000
$847$	− 54.0955i	− 1.85874i
$848$	54.1224i	1.85857i
$849$	2.23016	0.0765388
$850$	0	0
$851$	23.3572	0.800673
$852$	− 5.78733i	− 0.198271i
$853$	37.6116i	1.28780i	0.765111	+	0.643898i	$0.222684\pi$
$853$	37.6116i	1.28780i	−0.765111	+	0.643898i	$0.777316\pi$
$854$	115.192	3.94178
$855$	0	0
$856$	98.0438	3.35107
$857$	− 23.6506i	− 0.807889i	−0.914783	−	0.403945i	$-0.867639\pi$
$857$	− 23.6506i	− 0.807889i	0.914783	−	0.403945i	$-0.132361\pi$
$858$	1.96859i	0.0672064i
$859$	2.06312	0.0703927	0.0351964	−	0.999380i	$-0.488794\pi$
$859$	2.06312	0.0703927	0.0351964	+	0.999380i	$0.488794\pi$
$860$	0	0
$861$	−37.9748	−1.29418
$862$	83.6886i	2.85045i
$863$	23.8417i	0.811583i	0.913966	+	0.405791i	$0.133004\pi$
$863$	23.8417i	0.811583i	−0.913966	+	0.405791i	$0.866996\pi$
$864$	−4.12048	−0.140182
$865$	0	0
$866$	34.2266	1.16307
$867$	8.85410i	0.300701i
$868$	59.0839i	2.00544i
$869$	1.53353	0.0520216
$870$	0	0
$871$	37.6574	1.27597
$872$	− 8.30906i	− 0.281380i
$873$	− 17.7622i	− 0.601158i
$874$	−4.74901	−0.160638
$875$	0	0
$876$	13.3342	0.450520
$877$	24.0338i	0.811564i	0.913970	+	0.405782i	$0.133001\pi$
$877$	24.0338i	0.811564i	−0.913970	+	0.405782i	$0.866999\pi$
$878$	33.1188i	1.11771i
$879$	−14.0177	−0.472806
$880$	0	0
$881$	−10.4557	−0.352260	−0.176130	−	0.984367i	$-0.556358\pi$
$881$	−10.4557	−0.352260	−0.176130	+	0.984367i	$0.556358\pi$
$882$	− 43.7729i	− 1.47391i
$883$	− 34.6714i	− 1.16679i	−0.812190	−	0.583394i	$-0.801725\pi$
$883$	− 34.6714i	− 1.16679i	0.812190	−	0.583394i	$-0.198275\pi$
$884$	52.0425	1.75038
$885$	0	0
$886$	−1.48881	−0.0500174
$887$	3.26739i	0.109708i	0.998494	+	0.0548541i	$0.0174694\pi$
$887$	3.26739i	0.109708i	−0.998494	+	0.0548541i	$0.982531\pi$
$888$	− 32.0631i	− 1.07597i
$889$	−55.8634	−1.87360
$890$	0	0
$891$	0.187020	0.00626540
$892$	6.96488i	0.233201i
$893$	1.04780i	0.0350633i
$894$	−8.09810	−0.270841
$895$	0	0
$896$	69.9871	2.33811
$897$	18.1978i	0.607608i
$898$	65.1019i	2.17248i
$899$	−3.06846	−0.102339
$900$	0	0
$901$	−24.2169	−0.806783
$902$	− 3.63422i	− 0.121006i
$903$	32.5003i	1.08154i
$904$	89.6118	2.98044
$905$	0	0
$906$	−19.9641	−0.663261
$907$	23.2742i	0.772806i	0.922330	+	0.386403i	$0.126283\pi$
$907$	23.2742i	0.772806i	−0.922330	+	0.386403i	$0.873717\pi$
$908$	81.4351i	2.70252i
$909$	−11.6368	−0.385967
$910$	0	0
$911$	−19.4852	−0.645573	−0.322787	−	0.946472i	$-0.604620\pi$
$911$	−19.4852	−0.645573	−0.322787	+	0.946472i	$0.604620\pi$
$912$	2.74928i	0.0910377i
$913$	1.50385i	0.0497701i
$914$	−58.5044	−1.93515
$915$	0	0
$916$	−24.6634	−0.814903
$917$	84.8076i	2.80059i
$918$	− 7.20525i	− 0.237809i
$919$	−5.82595	−0.192180	−0.0960902	−	0.995373i	$-0.530634\pi$
$919$	−5.82595	−0.192180	−0.0960902	+	0.995373i	$0.530634\pi$
$920$	0	0
$921$	−17.2294	−0.567728
$922$	− 20.0600i	− 0.660641i
$923$	5.51777i	0.181620i
$924$	4.03498	0.132741
$925$	0	0
$926$	−55.7703	−1.83273
$927$	5.13797i	0.168753i
$928$	− 4.61693i	− 0.151558i
$929$	−54.9566	−1.80307	−0.901533	−	0.432710i	$-0.857557\pi$
$929$	−54.9566	−1.80307	−0.901533	+	0.432710i	$0.857557\pi$
$930$	0	0
$931$	−7.47338	−0.244930
$932$	− 28.7331i	− 0.941183i
$933$	1.47703i	0.0483559i
$934$	−65.1641	−2.13224
$935$	0	0
$936$	24.9807	0.816521
$937$	7.75384i	0.253307i	0.991947	+	0.126653i	$0.0404236\pi$
$937$	7.75384i	0.253307i	−0.991947	+	0.126653i	$0.959576\pi$
$938$	− 112.485i	− 3.67276i
$939$	−30.7937	−1.00491
$940$	0	0
$941$	−11.0480	−0.360156	−0.180078	−	0.983652i	$-0.557635\pi$
$941$	−11.0480	−0.360156	−0.180078	+	0.983652i	$0.557635\pi$
$942$	− 27.6018i	− 0.899315i
$943$	− 33.5951i	− 1.09401i
$944$	−68.6165	−2.23328
$945$	0	0
$946$	−3.11030	−0.101125
$947$	− 2.38966i	− 0.0776534i	−0.999246	−	0.0388267i	$-0.987638\pi$
$947$	− 2.38966i	− 0.0776534i	0.999246	−	0.0388267i	$-0.0123620\pi$
$948$	− 35.8597i	− 1.16467i
$949$	−12.7131	−0.412685
$950$	0	0
$951$	−7.86693	−0.255103
$952$	− 84.3605i	− 2.73414i
$953$	− 22.4858i	− 0.728387i	−0.931323	−	0.364194i	$-0.881345\pi$
$953$	− 22.4858i	− 0.728387i	0.931323	−	0.364194i	$-0.118655\pi$
$954$	−21.4205	−0.693514
$955$	0	0
$956$	18.9284	0.612189
$957$	0.209553i	0.00677387i
$958$	98.2547i	3.17447i
$959$	36.9659	1.19369
$960$	0	0
$961$	−23.5005	−0.758081
$962$	56.3319i	1.81621i
$963$	16.3645i	0.527338i
$964$	−30.5565	−0.984157
$965$	0	0
$966$	54.3580	1.74894
$967$	− 43.7138i	− 1.40574i	−0.711318	−	0.702871i	$-0.751901\pi$
$967$	− 43.7138i	− 1.40574i	0.711318	−	0.702871i	$-0.248099\pi$
$968$	− 65.6943i	− 2.11149i
$969$	−1.23016	−0.0395184
$970$	0	0
$971$	−18.2818	−0.586692	−0.293346	−	0.956006i	$-0.594769\pi$
$971$	−18.2818	−0.586692	−0.293346	+	0.956006i	$0.594769\pi$
$972$	− 4.37322i	− 0.140271i
$973$	19.2027i	0.615611i
$974$	−69.5089	−2.22721
$975$	0	0
$976$	58.9952	1.88839
$977$	42.4081i	1.35675i	0.734714	+	0.678377i	$0.237317\pi$
$977$	42.4081i	1.35675i	−0.734714	+	0.678377i	$0.762683\pi$
$978$	25.7005i	0.821812i
$979$	0.580321	0.0185472
$980$	0	0
$981$	1.38686	0.0442792
$982$	− 25.6316i	− 0.817938i
$983$	26.6561i	0.850198i	0.905147	+	0.425099i	$0.139761\pi$
$983$	26.6561i	0.850198i	−0.905147	+	0.425099i	$0.860239\pi$
$984$	−46.1171	−1.47016
$985$	0	0
$986$	8.07336	0.257108
$987$	− 11.9933i	− 0.381752i
$988$	− 7.85924i	− 0.250036i
$989$	−28.7520	−0.914259
$990$	0	0
$991$	22.2630	0.707206	0.353603	−	0.935396i	$-0.384956\pi$
$991$	22.2630	0.707206	0.353603	+	0.935396i	$0.384956\pi$
$992$	11.2840i	0.358268i
$993$	25.6794i	0.814910i
$994$	16.4819	0.522775
$995$	0	0
$996$	35.1656	1.11426
$997$	22.6781i	0.718223i	0.933295	+	0.359112i	$0.116920\pi$
$997$	22.6781i	0.718223i	−0.933295	+	0.359112i	$0.883080\pi$
$998$	103.332i	3.27092i
$999$	5.35165	0.169319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.b.c.124.1		8
3.2	odd	2		1125.2.b.g.874.8		8
4.3	odd	2		6000.2.f.o.1249.1		8
5.2	odd	4		375.2.a.f.1.4	yes	4
5.3	odd	4		375.2.a.e.1.1	✓	4
5.4	even	2	inner	375.2.b.c.124.8		8
15.2	even	4		1125.2.a.h.1.1		4
15.8	even	4		1125.2.a.l.1.4		4
15.14	odd	2		1125.2.b.g.874.1		8
20.3	even	4		6000.2.a.bh.1.1		4
20.7	even	4		6000.2.a.bg.1.4		4
20.19	odd	2		6000.2.f.o.1249.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.e.1.1	✓	4	5.3	odd	4
375.2.a.f.1.4	yes	4	5.2	odd	4
375.2.b.c.124.1		8	1.1	even	1	trivial
375.2.b.c.124.8		8	5.4	even	2	inner
1125.2.a.h.1.1		4	15.2	even	4
1125.2.a.l.1.4		4	15.8	even	4
1125.2.b.g.874.1		8	15.14	odd	2
1125.2.b.g.874.8		8	3.2	odd	2
6000.2.a.bg.1.4		4	20.7	even	4
6000.2.a.bh.1.1		4	20.3	even	4
6000.2.f.o.1249.1		8	4.3	odd	2
6000.2.f.o.1249.8		8	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.52452i q^{2} +1.00000i q^{3} -4.37322 q^{4} +2.52452 q^{6} +4.93346i q^{7} +5.99126i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.52452i q^{2} +1.00000i q^{3} -4.37322 q^{4} +2.52452 q^{6} +4.93346i q^{7} +5.99126i q^{8} -1.00000 q^{9} +0.187020 q^{11} -4.37322i q^{12} +4.16953i q^{13} +12.4546 q^{14} +6.37863 q^{16} +2.85410i q^{17} +2.52452i q^{18} +0.431014 q^{19} -4.93346 q^{21} -0.472136i q^{22} -4.36448i q^{23} -5.99126 q^{24} +10.5261 q^{26} -1.00000i q^{27} -21.5751i q^{28} +1.12048 q^{29} -2.73852 q^{31} -4.12048i q^{32} +0.187020i q^{33} +7.20525 q^{34} +4.37322 q^{36} +5.35165i q^{37} -1.08811i q^{38} -4.16953 q^{39} +7.69740 q^{41} +12.4546i q^{42} -6.58772i q^{43} -0.817879 q^{44} -11.0182 q^{46} +2.43101i q^{47} +6.37863i q^{48} -17.3391 q^{49} -2.85410 q^{51} -18.2343i q^{52} +8.48496i q^{53} -2.52452 q^{54} -29.5576 q^{56} +0.431014i q^{57} -2.82869i q^{58} -10.7573 q^{59} +9.24889 q^{61} +6.91345i q^{62} -4.93346i q^{63} +2.35499 q^{64} +0.472136 q^{66} -9.03156i q^{67} -12.4816i q^{68} +4.36448 q^{69} +1.32336 q^{71} -5.99126i q^{72} +3.04905i q^{73} +13.5104 q^{74} -1.88492 q^{76} +0.922655i q^{77} +10.5261i q^{78} +8.19985 q^{79} +1.00000 q^{81} -19.4323i q^{82} +8.04112i q^{83} +21.5751 q^{84} -16.6309 q^{86} +1.12048i q^{87} +1.12048i q^{88} +3.10300 q^{89} -20.5702 q^{91} +19.0868i q^{92} -2.73852i q^{93} +6.13715 q^{94} +4.12048 q^{96} +17.7622i q^{97} +43.7729i q^{98} -0.187020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9	\( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9} + 12 q^{11} + 4 q^{14} + 10 q^{16} - 16 q^{19} - 8 q^{21} - 18 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 12 q^{34} + 14 q^{36} + 16 q^{39} + 48 q^{41} + 28 q^{44} - 32 q^{46} - 40 q^{49} + 4 q^{51} - 6 q^{54} - 60 q^{56} - 4 q^{59} + 20 q^{61} + 54 q^{64} - 32 q^{66} - 16 q^{69} - 24 q^{71} + 84 q^{74} - 16 q^{76} + 40 q^{79} + 8 q^{81} + 56 q^{84} - 88 q^{86} - 56 q^{89} - 72 q^{91} + 46 q^{94} + 12 q^{96} - 12 q^{99}+O(q^{100}) \) 8 * q - 14 * q^4 + 6 * q^6 - 8 * q^9 + 12 * q^11 + 4 * q^14 + 10 * q^16 - 16 * q^19 - 8 * q^21 - 18 * q^24 - 4 * q^26 - 12 * q^29 + 8 * q^31 + 12 * q^34 + 14 * q^36 + 16 * q^39 + 48 * q^41 + 28 * q^44 - 32 * q^46 - 40 * q^49 + 4 * q^51 - 6 * q^54 - 60 * q^56 - 4 * q^59 + 20 * q^61 + 54 * q^64 - 32 * q^66 - 16 * q^69 - 24 * q^71 + 84 * q^74 - 16 * q^76 + 40 * q^79 + 8 * q^81 + 56 * q^84 - 88 * q^86 - 56 * q^89 - 72 * q^91 + 46 * q^94 + 12 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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