LMFDB - Embedded newform 375.2.b.c.124.2

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			124.2
Root			$-0.777484i$ of defining polynomial
Character	$\chi$	$=$	375.124
Dual form			375.2.b.c.124.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25800i q^{2} +1.00000i q^{3} -3.09855 q^{4} +2.25800 q^{6} -1.55497i q^{7} +2.48051i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.25800i q^{2} +1.00000i q^{3} -3.09855 q^{4} +2.25800 q^{6} -1.55497i q^{7} +2.48051i q^{8} -1.00000 q^{9} -3.75206 q^{11} -3.09855i q^{12} -6.79104i q^{13} -3.51111 q^{14} -0.596106 q^{16} -3.85410i q^{17} +2.25800i q^{18} +2.13403 q^{19} +1.55497 q^{21} +8.47214i q^{22} +0.420943i q^{23} -2.48051 q^{24} -15.3341 q^{26} -1.00000i q^{27} +4.81814i q^{28} -9.30703 q^{29} +9.92506 q^{31} +6.30703i q^{32} -3.75206i q^{33} -8.70255 q^{34} +3.09855 q^{36} +6.83489i q^{37} -4.81862i q^{38} +6.79104 q^{39} +5.68110 q^{41} -3.51111i q^{42} -3.59882i q^{43} +11.6259 q^{44} +0.950487 q^{46} +4.13403i q^{47} -0.596106i q^{48} +4.58207 q^{49} +3.85410 q^{51} +21.0423i q^{52} -6.72797i q^{53} -2.25800 q^{54} +3.85712 q^{56} +2.13403i q^{57} +21.0152i q^{58} +3.19221 q^{59} -1.49190 q^{61} -22.4108i q^{62} +1.55497i q^{63} +13.0490 q^{64} -8.47214 q^{66} -1.47702i q^{67} +11.9421i q^{68} -0.420943 q^{69} +7.18522 q^{71} -2.48051i q^{72} +2.51599i q^{73} +15.4332 q^{74} -6.61238 q^{76} +5.83433i q^{77} -15.3341i q^{78} -2.00789 q^{79} +1.00000 q^{81} -12.8279i q^{82} -2.60616i q^{83} -4.81814 q^{84} -8.12613 q^{86} -9.30703i q^{87} -9.30703i q^{88} -14.3460 q^{89} -10.5598 q^{91} -1.30431i q^{92} +9.92506i q^{93} +9.33461 q^{94} -6.30703 q^{96} -12.5702i q^{97} -10.3463i q^{98} +3.75206 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.25800i	− 1.59664i	−0.602231	−	0.798322i	$-0.705721\pi$
$2$	− 2.25800i	− 1.59664i	0.602231	−	0.798322i	$-0.294279\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−3.09855	−1.54927
$5$	0	0
$5$	0	0
$6$	2.25800	0.921823
$7$	− 1.55497i	− 0.587723i	−0.955848	−	0.293861i	$-0.905060\pi$
$7$	− 1.55497i	− 0.587723i	0.955848	−	0.293861i	$-0.0949404\pi$
$8$	2.48051i	0.876993i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.75206	−1.13129	−0.565644	−	0.824649i	$-0.691372\pi$
$11$	−3.75206	−1.13129	−0.565644	+	0.824649i	$0.691372\pi$
$12$	− 3.09855i	− 0.894473i
$13$	− 6.79104i	− 1.88349i	−0.336321	−	0.941747i	$-0.609183\pi$
$13$	− 6.79104i	− 1.88349i	0.336321	−	0.941747i	$-0.390817\pi$
$14$	−3.51111	−0.938384
$15$	0	0
$16$	−0.596106	−0.149027
$17$	− 3.85410i	− 0.934757i	−0.884057	−	0.467379i	$-0.845199\pi$
$17$	− 3.85410i	− 0.934757i	0.884057	−	0.467379i	$-0.154801\pi$
$18$	2.25800i	0.532215i
$19$	2.13403	0.489579	0.244790	−	0.969576i	$-0.421281\pi$
$19$	2.13403	0.489579	0.244790	+	0.969576i	$0.421281\pi$
$20$	0	0
$21$	1.55497	0.339322
$22$	8.47214i	1.80627i
$23$	0.420943i	0.0877726i	0.999037	+	0.0438863i	$0.0139739\pi$
$23$	0.420943i	0.0877726i	−0.999037	+	0.0438863i	$0.986026\pi$
$24$	−2.48051	−0.506332
$25$	0	0
$26$	−15.3341	−3.00727
$27$	− 1.00000i	− 0.192450i
$28$	4.81814i	0.910543i
$29$	−9.30703	−1.72827	−0.864136	−	0.503259i	$-0.832134\pi$
$29$	−9.30703	−1.72827	−0.864136	+	0.503259i	$0.832134\pi$
$30$	0	0
$31$	9.92506	1.78259	0.891297	−	0.453420i	$-0.149796\pi$
$31$	9.92506	1.78259	0.891297	+	0.453420i	$0.149796\pi$
$32$	6.30703i	1.11494i
$33$	− 3.75206i	− 0.653150i
$34$	−8.70255	−1.49247
$35$	0	0
$36$	3.09855	0.516424
$37$	6.83489i	1.12365i	0.827256	+	0.561825i	$0.189900\pi$
$37$	6.83489i	1.12365i	−0.827256	+	0.561825i	$0.810100\pi$
$38$	− 4.81862i	− 0.781684i
$39$	6.79104	1.08744
$40$	0	0
$41$	5.68110	0.887239	0.443619	−	0.896215i	$-0.353694\pi$
$41$	5.68110	0.887239	0.443619	+	0.896215i	$0.353694\pi$
$42$	− 3.51111i	− 0.541776i
$43$	− 3.59882i	− 0.548816i	−0.961613	−	0.274408i	$-0.911518\pi$
$43$	− 3.59882i	− 0.548816i	0.961613	−	0.274408i	$-0.0884818\pi$
$44$	11.6259	1.75267
$45$	0	0
$46$	0.950487	0.140142
$47$	4.13403i	0.603010i	0.953465	+	0.301505i	$0.0974889\pi$
$47$	4.13403i	0.603010i	−0.953465	+	0.301505i	$0.902511\pi$
$48$	− 0.596106i	− 0.0860405i
$49$	4.58207	0.654582
$50$	0	0
$51$	3.85410	0.539682
$52$	21.0423i	2.91805i
$53$	− 6.72797i	− 0.924158i	−0.886839	−	0.462079i	$-0.847104\pi$
$53$	− 6.72797i	− 0.924158i	0.886839	−	0.462079i	$-0.152896\pi$
$54$	−2.25800	−0.307274
$55$	0	0
$56$	3.85712	0.515429
$57$	2.13403i	0.282659i
$58$	21.0152i	2.75944i
$59$	3.19221	0.415591	0.207795	−	0.978172i	$-0.433371\pi$
$59$	3.19221	0.415591	0.207795	+	0.978172i	$0.433371\pi$
$60$	0	0
$61$	−1.49190	−0.191019	−0.0955093	−	0.995429i	$-0.530448\pi$
$61$	−1.49190	−0.191019	−0.0955093	+	0.995429i	$0.530448\pi$
$62$	− 22.4108i	− 2.84617i
$63$	1.55497i	0.195908i
$64$	13.0490	1.63113
$65$	0	0
$66$	−8.47214	−1.04285
$67$	− 1.47702i	− 0.180446i	−0.995922	−	0.0902231i	$-0.971242\pi$
$67$	− 1.47702i	− 0.180446i	0.995922	−	0.0902231i	$-0.0287580\pi$
$68$	11.9421i	1.44819i
$69$	−0.420943	−0.0506755
$70$	0	0
$71$	7.18522	0.852729	0.426364	−	0.904552i	$-0.359794\pi$
$71$	7.18522	0.852729	0.426364	+	0.904552i	$0.359794\pi$
$72$	− 2.48051i	− 0.292331i
$73$	2.51599i	0.294475i	0.989101	+	0.147237i	$0.0470381\pi$
$73$	2.51599i	0.294475i	−0.989101	+	0.147237i	$0.952962\pi$
$74$	15.4332	1.79407
$75$	0	0
$76$	−6.61238	−0.758492
$77$	5.83433i	0.664884i
$78$	− 15.3341i	− 1.73625i
$79$	−2.00789	−0.225906	−0.112953	−	0.993600i	$-0.536031\pi$
$79$	−2.00789	−0.225906	−0.112953	+	0.993600i	$0.536031\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 12.8279i	− 1.41660i
$83$	− 2.60616i	− 0.286063i	−0.989718	−	0.143032i	$-0.954315\pi$
$83$	− 2.60616i	− 0.286063i	0.989718	−	0.143032i	$-0.0456851\pi$
$84$	−4.81814	−0.525702
$85$	0	0
$86$	−8.12613	−0.876263
$87$	− 9.30703i	− 0.997818i
$88$	− 9.30703i	− 0.992133i
$89$	−14.3460	−1.52067	−0.760337	−	0.649529i	$-0.774966\pi$
$89$	−14.3460	−1.52067	−0.760337	+	0.649529i	$0.774966\pi$
$90$	0	0
$91$	−10.5598	−1.10697
$92$	− 1.30431i	− 0.135984i
$93$	9.92506i	1.02918i
$94$	9.33461	0.962792
$95$	0	0
$96$	−6.30703	−0.643708
$97$	− 12.5702i	− 1.27631i	−0.769908	−	0.638155i	$-0.779698\pi$
$97$	− 12.5702i	− 1.27631i	0.769908	−	0.638155i	$-0.220302\pi$
$98$	− 10.3463i	− 1.04513i
$99$	3.75206	0.377096
$100$	0	0
$101$	8.11482	0.807454	0.403727	−	0.914879i	$-0.367714\pi$
$101$	8.11482	0.807454	0.403727	+	0.914879i	$0.367714\pi$
$102$	− 8.70255i	− 0.861681i
$103$	− 1.73195i	− 0.170654i	−0.996353	−	0.0853270i	$-0.972807\pi$
$103$	− 1.73195i	− 0.170654i	0.996353	−	0.0853270i	$-0.0271935\pi$
$104$	16.8452	1.65181
$105$	0	0
$106$	−15.1917	−1.47555
$107$	− 11.5791i	− 1.11939i	−0.828699	−	0.559695i	$-0.810918\pi$
$107$	− 11.5791i	− 1.11939i	0.828699	−	0.559695i	$-0.189082\pi$
$108$	3.09855i	0.298158i
$109$	12.7600	1.22218	0.611091	−	0.791560i	$-0.290731\pi$
$109$	12.7600	1.22218	0.611091	+	0.791560i	$0.290731\pi$
$110$	0	0
$111$	−6.83489	−0.648739
$112$	0.926926i	0.0875863i
$113$	9.20011i	0.865473i	0.901520	+	0.432737i	$0.142452\pi$
$113$	9.20011i	0.865473i	−0.901520	+	0.432737i	$0.857548\pi$
$114$	4.81862	0.451305
$115$	0	0
$116$	28.8383	2.67756
$117$	6.79104i	0.627832i
$118$	− 7.20800i	− 0.663550i
$119$	−5.99301	−0.549378
$120$	0	0
$121$	3.07795	0.279814
$122$	3.36871i	0.304989i
$123$	5.68110i	0.512247i
$124$	−30.7533	−2.76172
$125$	0	0
$126$	3.51111	0.312795
$127$	17.1852i	1.52494i	0.647023	+	0.762471i	$0.276014\pi$
$127$	17.1852i	1.52494i	−0.647023	+	0.762471i	$0.723986\pi$
$128$	− 16.8506i	− 1.48940i
$129$	3.59882	0.316859
$130$	0	0
$131$	10.0753	0.880281	0.440141	−	0.897929i	$-0.354929\pi$
$131$	10.0753	0.880281	0.440141	+	0.897929i	$0.354929\pi$
$132$	11.6259i	1.01191i
$133$	− 3.31834i	− 0.287737i
$134$	−3.33509	−0.288108
$135$	0	0
$136$	9.56014	0.819776
$137$	− 2.39418i	− 0.204549i	−0.994756	−	0.102274i	$-0.967388\pi$
$137$	− 2.39418i	− 0.204549i	0.994756	−	0.102274i	$-0.0326120\pi$
$138$	0.950487i	0.0809108i
$139$	8.05119	0.682893	0.341447	−	0.939901i	$-0.389083\pi$
$139$	8.05119	0.682893	0.341447	+	0.939901i	$0.389083\pi$
$140$	0	0
$141$	−4.13403	−0.348148
$142$	− 16.2242i	− 1.36150i
$143$	25.4804i	2.13078i
$144$	0.596106	0.0496755
$145$	0	0
$146$	5.68110	0.470171
$147$	4.58207i	0.377923i
$148$	− 21.1782i	− 1.74084i
$149$	−3.11426	−0.255130	−0.127565	−	0.991830i	$-0.540716\pi$
$149$	−3.11426	−0.255130	−0.127565	+	0.991830i	$0.540716\pi$
$150$	0	0
$151$	15.7161	1.27896	0.639479	−	0.768809i	$-0.279150\pi$
$151$	15.7161	1.27896	0.639479	+	0.768809i	$0.279150\pi$
$152$	5.29348i	0.429358i
$153$	3.85410i	0.311586i
$154$	13.1739	1.06158
$155$	0	0
$156$	−21.0423	−1.68474
$157$	− 4.44503i	− 0.354752i	−0.984143	−	0.177376i	$-0.943239\pi$
$157$	− 4.44503i	− 0.354752i	0.984143	−	0.177376i	$-0.0567609\pi$
$158$	4.53382i	0.360691i
$159$	6.72797	0.533563
$160$	0	0
$161$	0.654553	0.0515860
$162$	− 2.25800i	− 0.177405i
$163$	− 12.1803i	− 0.954038i	−0.878893	−	0.477019i	$-0.841717\pi$
$163$	− 12.1803i	− 0.954038i	0.878893	−	0.477019i	$-0.158283\pi$
$164$	−17.6031	−1.37457
$165$	0	0
$166$	−5.88470	−0.456742
$167$	17.3463i	1.34230i	0.741321	+	0.671150i	$0.234200\pi$
$167$	17.3463i	1.34230i	−0.741321	+	0.671150i	$0.765800\pi$
$168$	3.85712i	0.297583i
$169$	−33.1182	−2.54755
$170$	0	0
$171$	−2.13403	−0.163193
$172$	11.1511i	0.850265i
$173$	10.0542i	0.764407i	0.924078	+	0.382204i	$0.124835\pi$
$173$	10.0542i	0.764407i	−0.924078	+	0.382204i	$0.875165\pi$
$174$	−21.0152	−1.59316
$175$	0	0
$176$	2.23663	0.168592
$177$	3.19221i	0.239941i
$178$	32.3932i	2.42797i
$179$	9.55929	0.714495	0.357247	−	0.934010i	$-0.383715\pi$
$179$	9.55929	0.714495	0.357247	+	0.934010i	$0.383715\pi$
$180$	0	0
$181$	−14.8304	−1.10233	−0.551166	−	0.834396i	$-0.685817\pi$
$181$	−14.8304	−1.10233	−0.551166	+	0.834396i	$0.685817\pi$
$182$	23.8441i	1.76744i
$183$	− 1.49190i	− 0.110285i
$184$	−1.04415	−0.0769760
$185$	0	0
$186$	22.4108	1.64324
$187$	14.4608i	1.05748i
$188$	− 12.8095i	− 0.934226i
$189$	−1.55497	−0.113107
$190$	0	0
$191$	−17.2291	−1.24665	−0.623326	−	0.781962i	$-0.714219\pi$
$191$	−17.2291	−1.24665	−0.623326	+	0.781962i	$0.714219\pi$
$192$	13.0490i	0.941733i
$193$	− 16.9373i	− 1.21917i	−0.792720	−	0.609586i	$-0.791336\pi$
$193$	− 16.9373i	− 1.21917i	0.792720	−	0.609586i	$-0.208664\pi$
$194$	−28.3835	−2.03781
$195$	0	0
$196$	−14.1978	−1.01413
$197$	0.491903i	0.0350466i	0.999846	+	0.0175233i	$0.00557813\pi$
$197$	0.491903i	0.0350466i	−0.999846	+	0.0175233i	$0.994422\pi$
$198$	− 8.47214i	− 0.602088i
$199$	22.1152	1.56770	0.783851	−	0.620949i	$-0.213253\pi$
$199$	22.1152	1.56770	0.783851	+	0.620949i	$0.213253\pi$
$200$	0	0
$201$	1.47702	0.104181
$202$	− 18.3232i	− 1.28922i
$203$	14.4721i	1.01574i
$204$	−11.9421	−0.836115
$205$	0	0
$206$	−3.91073	−0.272474
$207$	− 0.420943i	− 0.0292575i
$208$	4.04818i	0.280691i
$209$	−8.00699	−0.553855
$210$	0	0
$211$	−4.54707	−0.313033	−0.156517	−	0.987675i	$-0.550027\pi$
$211$	−4.54707	−0.313033	−0.156517	+	0.987675i	$0.550027\pi$
$212$	20.8469i	1.43177i
$213$	7.18522i	0.492323i
$214$	−26.1455	−1.78727
$215$	0	0
$216$	2.48051	0.168777
$217$	− 15.4332i	− 1.04767i
$218$	− 28.8119i	− 1.95139i
$219$	−2.51599	−0.170015
$220$	0	0
$221$	−26.1733	−1.76061
$222$	15.4332i	1.03581i
$223$	17.7792i	1.19058i	0.803511	+	0.595290i	$0.202963\pi$
$223$	17.7792i	1.19058i	−0.803511	+	0.595290i	$0.797037\pi$
$224$	9.80723	0.655273
$225$	0	0
$226$	20.7738	1.38185
$227$	− 13.2093i	− 0.876733i	−0.898796	−	0.438366i	$-0.855557\pi$
$227$	− 13.2093i	− 0.876733i	0.898796	−	0.438366i	$-0.144443\pi$
$228$	− 6.61238i	− 0.437915i
$229$	−19.3966	−1.28177	−0.640883	−	0.767639i	$-0.721432\pi$
$229$	−19.3966	−1.28177	−0.640883	+	0.767639i	$0.721432\pi$
$230$	0	0
$231$	−5.83433	−0.383871
$232$	− 23.0862i	− 1.51568i
$233$	− 3.44015i	− 0.225372i	−0.993631	−	0.112686i	$-0.964055\pi$
$233$	− 3.44015i	− 0.225372i	0.993631	−	0.112686i	$-0.0359454\pi$
$234$	15.3341	1.00242
$235$	0	0
$236$	−9.89121	−0.643863
$237$	− 2.00789i	− 0.130427i
$238$	13.5322i	0.877161i
$239$	6.19277	0.400577	0.200288	−	0.979737i	$-0.435812\pi$
$239$	6.19277	0.400577	0.200288	+	0.979737i	$0.435812\pi$
$240$	0	0
$241$	13.2558	0.853883	0.426942	−	0.904279i	$-0.359591\pi$
$241$	13.2558	0.853883	0.426942	+	0.904279i	$0.359591\pi$
$242$	− 6.95001i	− 0.446763i
$243$	1.00000i	0.0641500i
$244$	4.62273	0.295940
$245$	0	0
$246$	12.8279	0.817877
$247$	− 14.4922i	− 0.922120i
$248$	24.6192i	1.56332i
$249$	2.60616	0.165159
$250$	0	0
$251$	8.62105	0.544156	0.272078	−	0.962275i	$-0.412289\pi$
$251$	8.62105	0.544156	0.272078	+	0.962275i	$0.412289\pi$
$252$	− 4.81814i	− 0.303514i
$253$	− 1.57940i	− 0.0992961i
$254$	38.8042	2.43479
$255$	0	0
$256$	−11.9505	−0.746908
$257$	4.68844i	0.292457i	0.989251	+	0.146228i	$0.0467134\pi$
$257$	4.68844i	0.292457i	−0.989251	+	0.146228i	$0.953287\pi$
$258$	− 8.12613i	− 0.505911i
$259$	10.6280	0.660394
$260$	0	0
$261$	9.30703	0.576091
$262$	− 22.7499i	− 1.40550i
$263$	− 22.9473i	− 1.41499i	−0.706718	−	0.707495i	$-0.749825\pi$
$263$	− 22.9473i	− 1.41499i	0.706718	−	0.707495i	$-0.250175\pi$
$264$	9.30703	0.572808
$265$	0	0
$266$	−7.49281	−0.459413
$267$	− 14.3460i	− 0.877961i
$268$	4.57660i	0.279560i
$269$	−14.0034	−0.853800	−0.426900	−	0.904299i	$-0.640394\pi$
$269$	−14.0034	−0.853800	−0.426900	+	0.904299i	$0.640394\pi$
$270$	0	0
$271$	0.0511933	0.00310977	0.00155489	−	0.999999i	$-0.499505\pi$
$271$	0.0511933	0.00310977	0.00155489	+	0.999999i	$0.499505\pi$
$272$	2.29745i	0.139304i
$273$	− 10.5598i	− 0.639111i
$274$	−5.40605	−0.326592
$275$	0	0
$276$	1.30431	0.0785102
$277$	− 14.4045i	− 0.865485i	−0.901518	−	0.432742i	$-0.857546\pi$
$277$	− 14.4045i	− 0.865485i	0.901518	−	0.432742i	$-0.142454\pi$
$278$	− 18.1796i	− 1.09034i
$279$	−9.92506	−0.594198
$280$	0	0
$281$	14.6735	0.875351	0.437675	−	0.899133i	$-0.355802\pi$
$281$	14.6735	0.875351	0.437675	+	0.899133i	$0.355802\pi$
$282$	9.33461i	0.555868i
$283$	7.22475i	0.429467i	0.976673	+	0.214734i	$0.0688883\pi$
$283$	7.22475i	0.429467i	−0.976673	+	0.214734i	$0.931112\pi$
$284$	−22.2637	−1.32111
$285$	0	0
$286$	57.5346	3.40209
$287$	− 8.83393i	− 0.521450i
$288$	− 6.30703i	− 0.371645i
$289$	2.14590	0.126229
$290$	0	0
$291$	12.5702	0.736878
$292$	− 7.79592i	− 0.456221i
$293$	− 8.63382i	− 0.504393i	−0.967676	−	0.252197i	$-0.918847\pi$
$293$	− 8.63382i	− 0.504393i	0.967676	−	0.252197i	$-0.0811530\pi$
$294$	10.3463	0.603409
$295$	0	0
$296$	−16.9540	−0.985433
$297$	3.75206i	0.217717i
$298$	7.03198i	0.407352i
$299$	2.85864	0.165319
$300$	0	0
$301$	−5.59606	−0.322551
$302$	− 35.4869i	− 2.04204i
$303$	8.11482i	0.466184i
$304$	−1.27211	−0.0729603
$305$	0	0
$306$	8.70255	0.497491
$307$	− 5.66435i	− 0.323281i	−0.986850	−	0.161641i	$-0.948321\pi$
$307$	− 5.66435i	− 0.323281i	0.986850	−	0.161641i	$-0.0516786\pi$
$308$	− 18.0780i	− 1.03009i
$309$	1.73195	0.0985271
$310$	0	0
$311$	−23.8501	−1.35242	−0.676208	−	0.736711i	$-0.736378\pi$
$311$	−23.8501	−1.35242	−0.676208	+	0.736711i	$0.736378\pi$
$312$	16.8452i	0.953674i
$313$	− 7.09319i	− 0.400931i	−0.979701	−	0.200465i	$-0.935755\pi$
$313$	− 7.09319i	− 0.400931i	0.979701	−	0.200465i	$-0.0642454\pi$
$314$	−10.0369	−0.566413
$315$	0	0
$316$	6.22155	0.349990
$317$	− 5.10994i	− 0.287003i	−0.989650	−	0.143501i	$-0.954164\pi$
$317$	− 5.10994i	− 0.287003i	0.989650	−	0.143501i	$-0.0458361\pi$
$318$	− 15.1917i	− 0.851910i
$319$	34.9205	1.95517
$320$	0	0
$321$	11.5791	0.646280
$322$	− 1.47798i	− 0.0823644i
$323$	− 8.22475i	− 0.457638i
$324$	−3.09855	−0.172141
$325$	0	0
$326$	−27.5032	−1.52326
$327$	12.7600i	0.705627i
$328$	14.0920i	0.778102i
$329$	6.42828	0.354403
$330$	0	0
$331$	−26.4703	−1.45494	−0.727469	−	0.686141i	$-0.759303\pi$
$331$	−26.4703	−1.45494	−0.727469	+	0.686141i	$0.759303\pi$
$332$	8.07531i	0.443190i
$333$	− 6.83489i	− 0.374550i
$334$	39.1680	2.14318
$335$	0	0
$336$	−0.926926	−0.0505680
$337$	12.1462i	0.661648i	0.943692	+	0.330824i	$0.107327\pi$
$337$	12.1462i	0.661648i	−0.943692	+	0.330824i	$0.892673\pi$
$338$	74.7807i	4.06753i
$339$	−9.20011	−0.499681
$340$	0	0
$341$	−37.2394	−2.01663
$342$	4.81862i	0.260561i
$343$	− 18.0098i	− 0.972436i
$344$	8.92693	0.481308
$345$	0	0
$346$	22.7024	1.22049
$347$	− 13.0539i	− 0.700768i	−0.936606	−	0.350384i	$-0.886051\pi$
$347$	− 13.0539i	− 0.700768i	0.936606	−	0.350384i	$-0.113949\pi$
$348$	28.8383i	1.54589i
$349$	−20.0740	−1.07454	−0.537268	−	0.843412i	$-0.680543\pi$
$349$	−20.0740	−1.07454	−0.537268	+	0.843412i	$0.680543\pi$
$350$	0	0
$351$	−6.79104	−0.362479
$352$	− 23.6643i	− 1.26131i
$353$	26.5559i	1.41343i	0.707500	+	0.706713i	$0.249823\pi$
$353$	26.5559i	1.41343i	−0.707500	+	0.706713i	$0.750177\pi$
$354$	7.20800	0.383101
$355$	0	0
$356$	44.4518	2.35594
$357$	− 5.99301i	− 0.317184i
$358$	− 21.5848i	− 1.14079i
$359$	19.1511	1.01076	0.505379	−	0.862898i	$-0.331353\pi$
$359$	19.1511	1.01076	0.505379	+	0.862898i	$0.331353\pi$
$360$	0	0
$361$	−14.4459	−0.760312
$362$	33.4869i	1.76003i
$363$	3.07795i	0.161551i
$364$	32.7202	1.71500
$365$	0	0
$366$	−3.36871	−0.176085
$367$	37.2709i	1.94552i	0.231809	+	0.972761i	$0.425536\pi$
$367$	37.2709i	1.94552i	−0.231809	+	0.972761i	$0.574464\pi$
$368$	− 0.250926i	− 0.0130804i
$369$	−5.68110	−0.295746
$370$	0	0
$371$	−10.4618	−0.543149
$372$	− 30.7533i	− 1.59448i
$373$	6.11914i	0.316837i	0.987372	+	0.158418i	$0.0506395\pi$
$373$	6.11914i	0.316837i	−0.987372	+	0.158418i	$0.949360\pi$
$374$	32.6525	1.68842
$375$	0	0
$376$	−10.2545	−0.528835
$377$	63.2044i	3.25519i
$378$	3.51111i	0.180592i
$379$	−3.41641	−0.175489	−0.0877445	−	0.996143i	$-0.527966\pi$
$379$	−3.41641	−0.175489	−0.0877445	+	0.996143i	$0.527966\pi$
$380$	0	0
$381$	−17.1852	−0.880425
$382$	38.9032i	1.99046i
$383$	− 20.9570i	− 1.07086i	−0.844581	−	0.535428i	$-0.820151\pi$
$383$	− 20.9570i	− 1.07086i	0.844581	−	0.535428i	$-0.179849\pi$
$384$	16.8506	0.859904
$385$	0	0
$386$	−38.2443	−1.94658
$387$	3.59882i	0.182939i
$388$	38.9493i	1.97735i
$389$	16.8279	0.853208	0.426604	−	0.904438i	$-0.359710\pi$
$389$	16.8279	0.853208	0.426604	+	0.904438i	$0.359710\pi$
$390$	0	0
$391$	1.62236	0.0820461
$392$	11.3659i	0.574064i
$393$	10.0753i	0.508231i
$394$	1.11072	0.0559570
$395$	0	0
$396$	−11.6259	−0.584225
$397$	− 30.2345i	− 1.51743i	−0.651424	−	0.758714i	$-0.725828\pi$
$397$	− 30.2345i	− 1.51743i	0.651424	−	0.758714i	$-0.274172\pi$
$398$	− 49.9359i	− 2.50306i
$399$	3.31834	0.166125
$400$	0	0
$401$	−1.96590	−0.0981725	−0.0490862	−	0.998795i	$-0.515631\pi$
$401$	−1.96590	−0.0981725	−0.0490862	+	0.998795i	$0.515631\pi$
$402$	− 3.33509i	− 0.166339i
$403$	− 67.4015i	− 3.35751i
$404$	−25.1441	−1.25097
$405$	0	0
$406$	32.6780	1.62178
$407$	− 25.6449i	− 1.27117i
$408$	9.56014i	0.473298i
$409$	−32.2321	−1.59377	−0.796887	−	0.604128i	$-0.793521\pi$
$409$	−32.2321	−1.59377	−0.796887	+	0.604128i	$0.793521\pi$
$410$	0	0
$411$	2.39418	0.118096
$412$	5.36652i	0.264390i
$413$	− 4.96379i	− 0.244252i
$414$	−0.950487	−0.0467139
$415$	0	0
$416$	42.8313	2.09998
$417$	8.05119i	0.394269i
$418$	18.0798i	0.884310i
$419$	35.5075	1.73465	0.867327	−	0.497739i	$-0.165836\pi$
$419$	35.5075	1.73465	0.867327	+	0.497739i	$0.165836\pi$
$420$	0	0
$421$	−8.00246	−0.390016	−0.195008	−	0.980802i	$-0.562473\pi$
$421$	−8.00246	−0.390016	−0.195008	+	0.980802i	$0.562473\pi$
$422$	10.2673i	0.499803i
$423$	− 4.13403i	− 0.201003i
$424$	16.6888	0.810480
$425$	0	0
$426$	16.2242	0.786065
$427$	2.31986i	0.112266i
$428$	35.8782i	1.73424i
$429$	−25.4804	−1.23020
$430$	0	0
$431$	19.6363	0.945846	0.472923	−	0.881104i	$-0.343199\pi$
$431$	19.6363	0.945846	0.472923	+	0.881104i	$0.343199\pi$
$432$	0.596106i	0.0286802i
$433$	− 19.8571i	− 0.954272i	−0.878829	−	0.477136i	$-0.841675\pi$
$433$	− 19.8571i	− 0.954272i	0.878829	−	0.477136i	$-0.158325\pi$
$434$	−34.8480	−1.67276
$435$	0	0
$436$	−39.5373	−1.89349
$437$	0.898302i	0.0429716i
$438$	5.68110i	0.271453i
$439$	−15.8983	−0.758785	−0.379392	−	0.925236i	$-0.623867\pi$
$439$	−15.8983	−0.758785	−0.379392	+	0.925236i	$0.623867\pi$
$440$	0	0
$441$	−4.58207	−0.218194
$442$	59.0993i	2.81107i
$443$	− 2.73229i	− 0.129815i	−0.997891	−	0.0649076i	$-0.979325\pi$
$443$	− 2.73229i	− 0.129815i	0.997891	−	0.0649076i	$-0.0206753\pi$
$444$	21.1782	1.00507
$445$	0	0
$446$	40.1453	1.90093
$447$	− 3.11426i	− 0.147299i
$448$	− 20.2908i	− 0.958652i
$449$	17.0819	0.806143	0.403072	−	0.915168i	$-0.367943\pi$
$449$	17.0819	0.806143	0.403072	+	0.915168i	$0.367943\pi$
$450$	0	0
$451$	−21.3158	−1.00372
$452$	− 28.5070i	− 1.34085i
$453$	15.7161i	0.738407i
$454$	−29.8266	−1.39983
$455$	0	0
$456$	−5.29348	−0.247890
$457$	4.16903i	0.195019i	0.995235	+	0.0975094i	$0.0310876\pi$
$457$	4.16903i	0.195019i	−0.995235	+	0.0975094i	$0.968912\pi$
$458$	43.7975i	2.04652i
$459$	−3.85410	−0.179894
$460$	0	0
$461$	24.8620	1.15794	0.578969	−	0.815349i	$-0.303455\pi$
$461$	24.8620	1.15794	0.578969	+	0.815349i	$0.303455\pi$
$462$	13.1739i	0.612906i
$463$	− 2.60370i	− 0.121004i	−0.998168	−	0.0605022i	$-0.980730\pi$
$463$	− 2.60370i	− 0.121004i	0.998168	−	0.0605022i	$-0.0192702\pi$
$464$	5.54798	0.257558
$465$	0	0
$466$	−7.76785	−0.359839
$467$	13.3603i	0.618243i	0.951023	+	0.309121i	$0.100035\pi$
$467$	13.3603i	0.618243i	−0.951023	+	0.309121i	$0.899965\pi$
$468$	− 21.0423i	− 0.972682i
$469$	−2.29671	−0.106052
$470$	0	0
$471$	4.44503	0.204816
$472$	7.91832i	0.364470i
$473$	13.5030i	0.620869i
$474$	−4.53382	−0.208245
$475$	0	0
$476$	18.5696	0.851137
$477$	6.72797i	0.308053i
$478$	− 13.9832i	− 0.639579i
$479$	4.41153	0.201568	0.100784	−	0.994908i	$-0.467865\pi$
$479$	4.41153	0.201568	0.100784	+	0.994908i	$0.467865\pi$
$480$	0	0
$481$	46.4160	2.11639
$482$	− 29.9316i	− 1.36335i
$483$	0.654553i	0.0297832i
$484$	−9.53718	−0.433508
$485$	0	0
$486$	2.25800	0.102425
$487$	31.3244i	1.41944i	0.704482	+	0.709721i	$0.251179\pi$
$487$	31.3244i	1.41944i	−0.704482	+	0.709721i	$0.748821\pi$
$488$	− 3.70068i	− 0.167522i
$489$	12.1803	0.550814
$490$	0	0
$491$	13.5367	0.610901	0.305450	−	0.952208i	$-0.401193\pi$
$491$	13.5367	0.610901	0.305450	+	0.952208i	$0.401193\pi$
$492$	− 17.6031i	− 0.793611i
$493$	35.8702i	1.61551i
$494$	−32.7234	−1.47230
$495$	0	0
$496$	−5.91639	−0.265654
$497$	− 11.1728i	− 0.501168i
$498$	− 5.88470i	− 0.263700i
$499$	−4.53797	−0.203147	−0.101574	−	0.994828i	$-0.532388\pi$
$499$	−4.53797	−0.203147	−0.101574	+	0.994828i	$0.532388\pi$
$500$	0	0
$501$	−17.3463	−0.774978
$502$	− 19.4663i	− 0.868823i
$503$	− 17.6784i	− 0.788242i	−0.919059	−	0.394121i	$-0.871049\pi$
$503$	− 17.6784i	− 0.788242i	0.919059	−	0.394121i	$-0.128951\pi$
$504$	−3.85712	−0.171810
$505$	0	0
$506$	−3.56628	−0.158541
$507$	− 33.1182i	− 1.47083i
$508$	− 53.2492i	− 2.36255i
$509$	−27.1176	−1.20197	−0.600984	−	0.799261i	$-0.705224\pi$
$509$	−27.1176	−1.20197	−0.600984	+	0.799261i	$0.705224\pi$
$510$	0	0
$511$	3.91229	0.173069
$512$	− 6.71695i	− 0.296850i
$513$	− 2.13403i	− 0.0942195i
$514$	10.5865	0.466949
$515$	0	0
$516$	−11.1511	−0.490901
$517$	− 15.5111i	− 0.682178i
$518$	− 23.9981i	− 1.05441i
$519$	−10.0542	−0.441331
$520$	0	0
$521$	18.2611	0.800032	0.400016	−	0.916508i	$-0.369005\pi$
$521$	18.2611	0.800032	0.400016	+	0.916508i	$0.369005\pi$
$522$	− 21.0152i	− 0.919812i
$523$	21.4024i	0.935863i	0.883765	+	0.467931i	$0.155001\pi$
$523$	21.4024i	0.935863i	−0.883765	+	0.467931i	$0.844999\pi$
$524$	−31.2187	−1.36380
$525$	0	0
$526$	−51.8149	−2.25924
$527$	− 38.2522i	− 1.66629i
$528$	2.23663i	0.0973366i
$529$	22.8228	0.992296
$530$	0	0
$531$	−3.19221	−0.138530
$532$	10.2820i	0.445783i
$533$	− 38.5806i	− 1.67111i
$534$	−32.3932	−1.40179
$535$	0	0
$536$	3.66375	0.158250
$537$	9.55929i	0.412514i
$538$	31.6195i	1.36321i
$539$	−17.1922	−0.740521
$540$	0	0
$541$	−19.0567	−0.819310	−0.409655	−	0.912241i	$-0.634351\pi$
$541$	−19.0567	−0.819310	−0.409655	+	0.912241i	$0.634351\pi$
$542$	− 0.115594i	− 0.00496520i
$543$	− 14.8304i	− 0.636432i
$544$	24.3079	1.04219
$545$	0	0
$546$	−23.8441	−1.02043
$547$	13.4716i	0.576003i	0.957630	+	0.288002i	$0.0929908\pi$
$547$	13.4716i	0.576003i	−0.957630	+	0.288002i	$0.907009\pi$
$548$	7.41848i	0.316902i
$549$	1.49190	0.0636729
$550$	0	0
$551$	−19.8614	−0.846126
$552$	− 1.04415i	− 0.0444421i
$553$	3.12221i	0.132770i
$554$	−32.5254	−1.38187
$555$	0	0
$556$	−24.9470	−1.05799
$557$	8.63669i	0.365948i	0.983118	+	0.182974i	$0.0585724\pi$
$557$	8.63669i	0.365948i	−0.983118	+	0.182974i	$0.941428\pi$
$558$	22.4108i	0.948723i
$559$	−24.4397	−1.03369
$560$	0	0
$561$	−14.4608	−0.610536
$562$	− 33.1328i	− 1.39762i
$563$	16.8407i	0.709750i	0.934914	+	0.354875i	$0.115477\pi$
$563$	16.8407i	0.709750i	−0.934914	+	0.354875i	$0.884523\pi$
$564$	12.8095	0.539376
$565$	0	0
$566$	16.3135	0.685706
$567$	− 1.55497i	− 0.0653025i
$568$	17.8230i	0.747837i
$569$	39.9640	1.67538	0.837689	−	0.546148i	$-0.183906\pi$
$569$	39.9640	1.67538	0.837689	+	0.546148i	$0.183906\pi$
$570$	0	0
$571$	44.8339	1.87624	0.938121	−	0.346308i	$-0.112565\pi$
$571$	44.8339	1.87624	0.938121	+	0.346308i	$0.112565\pi$
$572$	− 78.9521i	− 3.30115i
$573$	− 17.2291i	− 0.719755i
$574$	−19.9470	−0.832571
$575$	0	0
$576$	−13.0490	−0.543710
$577$	− 18.9698i	− 0.789724i	−0.918740	−	0.394862i	$-0.870792\pi$
$577$	− 18.9698i	− 0.789724i	0.918740	−	0.394862i	$-0.129208\pi$
$578$	− 4.84543i	− 0.201543i
$579$	16.9373	0.703889
$580$	0	0
$581$	−4.05250	−0.168126
$582$	− 28.3835i	− 1.17653i
$583$	25.2437i	1.04549i
$584$	−6.24095	−0.258252
$585$	0	0
$586$	−19.4951	−0.805337
$587$	− 5.27750i	− 0.217826i	−0.994051	−	0.108913i	$-0.965263\pi$
$587$	− 5.27750i	− 0.217826i	0.994051	−	0.108913i	$-0.0347370\pi$
$588$	− 14.1978i	− 0.585506i
$589$	21.1803	0.872721
$590$	0	0
$591$	−0.491903	−0.0202342
$592$	− 4.07432i	− 0.167454i
$593$	− 5.52429i	− 0.226855i	−0.993546	−	0.113428i	$-0.963817\pi$
$593$	− 5.52429i	− 0.226855i	0.993546	−	0.113428i	$-0.0361830\pi$
$594$	8.47214	0.347616
$595$	0	0
$596$	9.64967	0.395266
$597$	22.1152i	0.905113i
$598$	− 6.45479i	− 0.263956i
$599$	−23.4944	−0.959954	−0.479977	−	0.877281i	$-0.659355\pi$
$599$	−23.4944	−0.959954	−0.479977	+	0.877281i	$0.659355\pi$
$600$	0	0
$601$	3.53787	0.144313	0.0721564	−	0.997393i	$-0.477012\pi$
$601$	3.53787	0.144313	0.0721564	+	0.997393i	$0.477012\pi$
$602$	12.6359i	0.515000i
$603$	1.47702i	0.0601487i
$604$	−48.6971	−1.98145
$605$	0	0
$606$	18.3232	0.744330
$607$	38.5351i	1.56409i	0.623220	+	0.782047i	$0.285824\pi$
$607$	38.5351i	1.56409i	−0.623220	+	0.782047i	$0.714176\pi$
$608$	13.4594i	0.545849i
$609$	−14.4721	−0.586441
$610$	0	0
$611$	28.0743	1.13577
$612$	− 11.9421i	− 0.482731i
$613$	30.4853i	1.23129i	0.788024	+	0.615644i	$0.211104\pi$
$613$	30.4853i	1.23129i	−0.788024	+	0.615644i	$0.788896\pi$
$614$	−12.7901	−0.516165
$615$	0	0
$616$	−14.4721	−0.583099
$617$	− 23.6018i	− 0.950174i	−0.879939	−	0.475087i	$-0.842417\pi$
$617$	− 23.6018i	− 0.950174i	0.879939	−	0.475087i	$-0.157583\pi$
$618$	− 3.91073i	− 0.157313i
$619$	−18.1291	−0.728672	−0.364336	−	0.931268i	$-0.618704\pi$
$619$	−18.1291	−0.728672	−0.364336	+	0.931268i	$0.618704\pi$
$620$	0	0
$621$	0.420943	0.0168918
$622$	53.8535i	2.15933i
$623$	22.3076i	0.893735i
$624$	−4.04818	−0.162057
$625$	0	0
$626$	−16.0164	−0.640143
$627$	− 8.00699i	− 0.319768i
$628$	13.7731i	0.549608i
$629$	26.3424	1.05034
$630$	0	0
$631$	−42.1042	−1.67614	−0.838071	−	0.545562i	$-0.816316\pi$
$631$	−42.1042	−1.67614	−0.838071	+	0.545562i	$0.816316\pi$
$632$	− 4.98061i	− 0.198118i
$633$	− 4.54707i	− 0.180730i
$634$	−11.5382	−0.458241
$635$	0	0
$636$	−20.8469	−0.826634
$637$	− 31.1170i	− 1.23290i
$638$	− 78.8504i	− 3.12172i
$639$	−7.18522	−0.284243
$640$	0	0
$641$	29.0975	1.14928	0.574641	−	0.818405i	$-0.305142\pi$
$641$	29.0975	1.14928	0.574641	+	0.818405i	$0.305142\pi$
$642$	− 26.1455i	− 1.03188i
$643$	30.7268i	1.21175i	0.795561	+	0.605873i	$0.207176\pi$
$643$	30.7268i	1.21175i	−0.795561	+	0.605873i	$0.792824\pi$
$644$	−2.02816	−0.0799207
$645$	0	0
$646$	−18.5715	−0.730684
$647$	41.1404i	1.61740i	0.588224	+	0.808698i	$0.299827\pi$
$647$	41.1404i	1.61740i	−0.588224	+	0.808698i	$0.700173\pi$
$648$	2.48051i	0.0974437i
$649$	−11.9774	−0.470153
$650$	0	0
$651$	15.4332	0.604873
$652$	37.7413i	1.47806i
$653$	33.0600i	1.29374i	0.762601	+	0.646869i	$0.223922\pi$
$653$	33.0600i	1.29374i	−0.762601	+	0.646869i	$0.776078\pi$
$654$	28.8119	1.12664
$655$	0	0
$656$	−3.38654	−0.132222
$657$	− 2.51599i	− 0.0981582i
$658$	− 14.5150i	− 0.565855i
$659$	31.0932	1.21122	0.605609	−	0.795762i	$-0.292929\pi$
$659$	31.0932	1.21122	0.605609	+	0.795762i	$0.292929\pi$
$660$	0	0
$661$	4.80813	0.187015	0.0935073	−	0.995619i	$-0.470192\pi$
$661$	4.80813	0.187015	0.0935073	+	0.995619i	$0.470192\pi$
$662$	59.7698i	2.32302i
$663$	− 26.1733i	− 1.01649i
$664$	6.46461	0.250876
$665$	0	0
$666$	−15.4332	−0.598023
$667$	− 3.91772i	− 0.151695i
$668$	− 53.7485i	− 2.07959i
$669$	−17.7792	−0.687382
$670$	0	0
$671$	5.59771	0.216097
$672$	9.80723i	0.378322i
$673$	16.1988i	0.624418i	0.950013	+	0.312209i	$0.101069\pi$
$673$	16.1988i	0.624418i	−0.950013	+	0.312209i	$0.898931\pi$
$674$	27.4262	1.05642
$675$	0	0
$676$	102.618	3.94685
$677$	19.5077i	0.749742i	0.927077	+	0.374871i	$0.122313\pi$
$677$	19.5077i	0.749742i	−0.927077	+	0.374871i	$0.877687\pi$
$678$	20.7738i	0.797813i
$679$	−19.5463	−0.750117
$680$	0	0
$681$	13.2093	0.506182
$682$	84.0865i	3.21984i
$683$	46.1931i	1.76753i	0.467931	+	0.883765i	$0.345000\pi$
$683$	46.1931i	1.76753i	−0.467931	+	0.883765i	$0.655000\pi$
$684$	6.61238	0.252831
$685$	0	0
$686$	−40.6660	−1.55263
$687$	− 19.3966i	− 0.740028i
$688$	2.14528i	0.0817881i
$689$	−45.6899	−1.74065
$690$	0	0
$691$	18.8212	0.715991	0.357995	−	0.933723i	$-0.383460\pi$
$691$	18.8212	0.715991	0.357995	+	0.933723i	$0.383460\pi$
$692$	− 31.1534i	− 1.18428i
$693$	− 5.83433i	− 0.221628i
$694$	−29.4756	−1.11888
$695$	0	0
$696$	23.0862	0.875080
$697$	− 21.8955i	− 0.829353i
$698$	45.3270i	1.71565i
$699$	3.44015	0.130119
$700$	0	0
$701$	−19.8014	−0.747887	−0.373943	−	0.927452i	$-0.621995\pi$
$701$	−19.8014	−0.747887	−0.373943	+	0.927452i	$0.621995\pi$
$702$	15.3341i	0.578750i
$703$	14.5858i	0.550115i
$704$	−48.9608	−1.84528
$705$	0	0
$706$	59.9630	2.25674
$707$	− 12.6183i	− 0.474559i
$708$	− 9.89121i	− 0.371735i
$709$	14.3435	0.538683	0.269342	−	0.963045i	$-0.413194\pi$
$709$	14.3435	0.538683	0.269342	+	0.963045i	$0.413194\pi$
$710$	0	0
$711$	2.00789	0.0753019
$712$	− 35.5854i	− 1.33362i
$713$	4.17788i	0.156463i
$714$	−13.5322	−0.506429
$715$	0	0
$716$	−29.6199	−1.10695
$717$	6.19277i	0.231273i
$718$	− 43.2432i	− 1.61382i
$719$	48.4631	1.80737	0.903684	−	0.428201i	$-0.140852\pi$
$719$	48.4631	1.80737	0.903684	+	0.428201i	$0.140852\pi$
$720$	0	0
$721$	−2.69312	−0.100297
$722$	32.6189i	1.21395i
$723$	13.2558i	0.492990i
$724$	45.9525	1.70781
$725$	0	0
$726$	6.95001	0.257939
$727$	− 24.6037i	− 0.912501i	−0.889851	−	0.456250i	$-0.849192\pi$
$727$	− 24.6037i	− 0.912501i	0.889851	−	0.456250i	$-0.150808\pi$
$728$	− 26.1938i	− 0.970808i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−13.8702	−0.513009
$732$	4.62273i	0.170861i
$733$	− 23.5286i	− 0.869047i	−0.900661	−	0.434523i	$-0.856917\pi$
$733$	− 23.5286i	− 0.869047i	0.900661	−	0.434523i	$-0.143083\pi$
$734$	84.1574	3.10631
$735$	0	0
$736$	−2.65490	−0.0978608
$737$	5.54185i	0.204137i
$738$	12.8279i	0.472202i
$739$	−3.65701	−0.134525	−0.0672627	−	0.997735i	$-0.521427\pi$
$739$	−3.65701	−0.134525	−0.0672627	+	0.997735i	$0.521427\pi$
$740$	0	0
$741$	14.4922	0.532386
$742$	23.6227i	0.867215i
$743$	4.25226i	0.156000i	0.996953	+	0.0780002i	$0.0248535\pi$
$743$	4.25226i	0.156000i	−0.996953	+	0.0780002i	$0.975147\pi$
$744$	−24.6192	−0.902585
$745$	0	0
$746$	13.8170	0.505876
$747$	2.60616i	0.0953545i
$748$	− 44.8075i	− 1.63833i
$749$	−18.0051	−0.657891
$750$	0	0
$751$	13.5997	0.496261	0.248131	−	0.968727i	$-0.420184\pi$
$751$	13.5997	0.496261	0.248131	+	0.968727i	$0.420184\pi$
$752$	− 2.46432i	− 0.0898644i
$753$	8.62105i	0.314169i
$754$	142.715	5.19738
$755$	0	0
$756$	4.81814	0.175234
$757$	44.9677i	1.63438i	0.576368	+	0.817190i	$0.304469\pi$
$757$	44.9677i	1.63438i	−0.576368	+	0.817190i	$0.695531\pi$
$758$	7.71424i	0.280194i
$759$	1.57940	0.0573287
$760$	0	0
$761$	24.7813	0.898321	0.449160	−	0.893451i	$-0.351723\pi$
$761$	24.7813	0.898321	0.449160	+	0.893451i	$0.351723\pi$
$762$	38.8042i	1.40573i
$763$	− 19.8413i	− 0.718304i
$764$	53.3851	1.93140
$765$	0	0
$766$	−47.3209	−1.70977
$767$	− 21.6784i	− 0.782763i
$768$	− 11.9505i	− 0.431228i
$769$	1.71066	0.0616880	0.0308440	−	0.999524i	$-0.490180\pi$
$769$	1.71066	0.0616880	0.0308440	+	0.999524i	$0.490180\pi$
$770$	0	0
$771$	−4.68844	−0.168850
$772$	52.4809i	1.88883i
$773$	− 34.0902i	− 1.22614i	−0.790029	−	0.613069i	$-0.789935\pi$
$773$	− 34.0902i	− 1.22614i	0.790029	−	0.613069i	$-0.210065\pi$
$774$	8.12613	0.292088
$775$	0	0
$776$	31.1805	1.11932
$777$	10.6280i	0.381279i
$778$	− 37.9973i	− 1.36227i
$779$	12.1236	0.434374
$780$	0	0
$781$	−26.9594	−0.964682
$782$	− 3.66327i	− 0.130998i
$783$	9.30703i	0.332606i
$784$	−2.73140	−0.0975500
$785$	0	0
$786$	22.7499	0.811464
$787$	24.0911i	0.858754i	0.903125	+	0.429377i	$0.141267\pi$
$787$	24.0911i	0.858754i	−0.903125	+	0.429377i	$0.858733\pi$
$788$	− 1.52418i	− 0.0542968i
$789$	22.9473	0.816945
$790$	0	0
$791$	14.3059	0.508658
$792$	9.30703i	0.330711i
$793$	10.1316i	0.359783i
$794$	−68.2695	−2.42279
$795$	0	0
$796$	−68.5248	−2.42880
$797$	12.5601i	0.444901i	0.974944	+	0.222451i	$0.0714057\pi$
$797$	12.5601i	0.444901i	−0.974944	+	0.222451i	$0.928594\pi$
$798$	− 7.49281i	− 0.265242i
$799$	15.9330	0.563668
$800$	0	0
$801$	14.3460	0.506891
$802$	4.43900i	0.156747i
$803$	− 9.44015i	− 0.333136i
$804$	−4.57660	−0.161404
$805$	0	0
$806$	−152.192	−5.36074
$807$	− 14.0034i	− 0.492942i
$808$	20.1289i	0.708132i
$809$	−3.63569	−0.127824	−0.0639120	−	0.997956i	$-0.520358\pi$
$809$	−3.63569	−0.127824	−0.0639120	+	0.997956i	$0.520358\pi$
$810$	0	0
$811$	20.9619	0.736073	0.368036	−	0.929811i	$-0.380030\pi$
$811$	20.9619	0.736073	0.368036	+	0.929811i	$0.380030\pi$
$812$	− 44.8426i	− 1.57367i
$813$	0.0511933i	0.00179543i
$814$	−57.9061	−2.02961
$815$	0	0
$816$	−2.29745	−0.0804270
$817$	− 7.67998i	− 0.268689i
$818$	72.7799i	2.54469i
$819$	10.5598	0.368991
$820$	0	0
$821$	−48.3396	−1.68706	−0.843532	−	0.537079i	$-0.819528\pi$
$821$	−48.3396	−1.68706	−0.843532	+	0.537079i	$0.819528\pi$
$822$	− 5.40605i	− 0.188558i
$823$	46.0131i	1.60392i	0.597380	+	0.801958i	$0.296208\pi$
$823$	46.0131i	1.60392i	−0.597380	+	0.801958i	$0.703792\pi$
$824$	4.29612	0.149662
$825$	0	0
$826$	−11.2082	−0.389984
$827$	34.5745i	1.20227i	0.799147	+	0.601136i	$0.205285\pi$
$827$	34.5745i	1.20227i	−0.799147	+	0.601136i	$0.794715\pi$
$828$	1.30431i	0.0453279i
$829$	17.4377	0.605636	0.302818	−	0.953048i	$-0.402073\pi$
$829$	17.4377	0.605636	0.302818	+	0.953048i	$0.402073\pi$
$830$	0	0
$831$	14.4045	0.499688
$832$	− 88.6165i	− 3.07222i
$833$	− 17.6598i	− 0.611875i
$834$	18.1796	0.629507
$835$	0	0
$836$	24.8100	0.858073
$837$	− 9.92506i	− 0.343060i
$838$	− 80.1757i	− 2.76963i
$839$	1.16943	0.0403732	0.0201866	−	0.999796i	$-0.493574\pi$
$839$	1.16943	0.0403732	0.0201866	+	0.999796i	$0.493574\pi$
$840$	0	0
$841$	57.6208	1.98692
$842$	18.0695i	0.622717i
$843$	14.6735i	0.505384i
$844$	14.0893	0.484974
$845$	0	0
$846$	−9.33461	−0.320931
$847$	− 4.78612i	− 0.164453i
$848$	4.01058i	0.137724i
$849$	−7.22475	−0.247953
$850$	0	0
$851$	−2.87710	−0.0986256
$852$	− 22.2637i	− 0.762743i
$853$	− 12.7191i	− 0.435494i	−0.976005	−	0.217747i	$-0.930129\pi$
$853$	− 12.7191i	− 0.435494i	0.976005	−	0.217747i	$-0.0698708\pi$
$854$	5.23824	0.179249
$855$	0	0
$856$	28.7220	0.981697
$857$	− 35.2257i	− 1.20329i	−0.798765	−	0.601643i	$-0.794513\pi$
$857$	− 35.2257i	− 1.20329i	0.798765	−	0.601643i	$-0.205487\pi$
$858$	57.5346i	1.96420i
$859$	−13.0460	−0.445123	−0.222561	−	0.974919i	$-0.571442\pi$
$859$	−13.0460	−0.445123	−0.222561	+	0.974919i	$0.571442\pi$
$860$	0	0
$861$	8.83393	0.301060
$862$	− 44.3386i	− 1.51018i
$863$	− 35.9439i	− 1.22354i	−0.791034	−	0.611772i	$-0.790457\pi$
$863$	− 35.9439i	− 1.22354i	0.791034	−	0.611772i	$-0.209543\pi$
$864$	6.30703	0.214569
$865$	0	0
$866$	−44.8373	−1.52363
$867$	2.14590i	0.0728785i
$868$	47.8203i	1.62313i
$869$	7.53374	0.255565
$870$	0	0
$871$	−10.0305	−0.339869
$872$	31.6512i	1.07185i
$873$	12.5702i	0.425437i
$874$	2.02836	0.0686104
$875$	0	0
$876$	7.79592	0.263400
$877$	− 8.06232i	− 0.272245i	−0.990692	−	0.136123i	$-0.956536\pi$
$877$	− 8.06232i	− 0.272245i	0.990692	−	0.136123i	$-0.0434641\pi$
$878$	35.8983i	1.21151i
$879$	8.63382	0.291212
$880$	0	0
$881$	−15.8556	−0.534187	−0.267094	−	0.963671i	$-0.586063\pi$
$881$	−15.8556	−0.534187	−0.267094	+	0.963671i	$0.586063\pi$
$882$	10.3463i	0.348378i
$883$	27.5924i	0.928559i	0.885689	+	0.464280i	$0.153687\pi$
$883$	27.5924i	0.928559i	−0.885689	+	0.464280i	$0.846313\pi$
$884$	81.0993	2.72767
$885$	0	0
$886$	−6.16951	−0.207269
$887$	20.9137i	0.702215i	0.936335	+	0.351107i	$0.114195\pi$
$887$	20.9137i	0.702215i	−0.936335	+	0.351107i	$0.885805\pi$
$888$	− 16.9540i	− 0.568940i
$889$	26.7225	0.896243
$890$	0	0
$891$	−3.75206	−0.125699
$892$	− 55.0896i	− 1.84453i
$893$	8.82212i	0.295221i
$894$	−7.03198	−0.235185
$895$	0	0
$896$	−26.2022	−0.875353
$897$	2.85864i	0.0954471i
$898$	− 38.5708i	− 1.28712i
$899$	−92.3728	−3.08081
$900$	0	0
$901$	−25.9303	−0.863863
$902$	48.1310i	1.60259i
$903$	− 5.59606i	− 0.186225i
$904$	−22.8210	−0.759014
$905$	0	0
$906$	35.4869	1.17897
$907$	− 18.3424i	− 0.609048i	−0.952505	−	0.304524i	$-0.901503\pi$
$907$	− 18.3424i	− 0.609048i	0.952505	−	0.304524i	$-0.0984975\pi$
$908$	40.9296i	1.35830i
$909$	−8.11482	−0.269151
$910$	0	0
$911$	25.4008	0.841565	0.420783	−	0.907162i	$-0.361756\pi$
$911$	25.4008	0.841565	0.420783	+	0.907162i	$0.361756\pi$
$912$	− 1.27211i	− 0.0421236i
$913$	9.77848i	0.323620i
$914$	9.41364	0.311376
$915$	0	0
$916$	60.1014	1.98581
$917$	− 15.6667i	− 0.517362i
$918$	8.70255i	0.287227i
$919$	−51.5078	−1.69909	−0.849543	−	0.527519i	$-0.823122\pi$
$919$	−51.5078	−1.69909	−0.849543	+	0.527519i	$0.823122\pi$
$920$	0	0
$921$	5.66435	0.186647
$922$	− 56.1383i	− 1.84882i
$923$	− 48.7951i	− 1.60611i
$924$	18.0780	0.594721
$925$	0	0
$926$	−5.87915	−0.193201
$927$	1.73195i	0.0568846i
$928$	− 58.6997i	− 1.92691i
$929$	12.3125	0.403960	0.201980	−	0.979390i	$-0.435262\pi$
$929$	12.3125	0.403960	0.201980	+	0.979390i	$0.435262\pi$
$930$	0	0
$931$	9.77826	0.320470
$932$	10.6595i	0.349163i
$933$	− 23.8501i	− 0.780818i
$934$	30.1676	0.987113
$935$	0	0
$936$	−16.8452	−0.550604
$937$	− 27.7932i	− 0.907963i	−0.891011	−	0.453981i	$-0.850003\pi$
$937$	− 27.7932i	− 0.907963i	0.891011	−	0.453981i	$-0.149997\pi$
$938$	5.18597i	0.169328i
$939$	7.09319	0.231477
$940$	0	0
$941$	10.8507	0.353722	0.176861	−	0.984236i	$-0.443406\pi$
$941$	10.8507	0.353722	0.176861	+	0.984236i	$0.443406\pi$
$942$	− 10.0369i	− 0.327019i
$943$	2.39142i	0.0778752i
$944$	−1.90290	−0.0619340
$945$	0	0
$946$	30.4897	0.991307
$947$	− 44.4130i	− 1.44323i	−0.692296	−	0.721614i	$-0.743401\pi$
$947$	− 44.4130i	− 1.44323i	0.692296	−	0.721614i	$-0.256599\pi$
$948$	6.22155i	0.202067i
$949$	17.0862	0.554641
$950$	0	0
$951$	5.10994	0.165701
$952$	− 14.8657i	− 0.481801i
$953$	19.3729i	0.627550i	0.949497	+	0.313775i	$0.101594\pi$
$953$	19.3729i	0.627550i	−0.949497	+	0.313775i	$0.898406\pi$
$954$	15.1917	0.491850
$955$	0	0
$956$	−19.1886	−0.620603
$957$	34.9205i	1.12882i
$958$	− 9.96121i	− 0.321832i
$959$	−3.72288	−0.120218
$960$	0	0
$961$	67.5069	2.17764
$962$	− 104.807i	− 3.37912i
$963$	11.5791i	0.373130i
$964$	−41.0738	−1.32290
$965$	0	0
$966$	1.47798	0.0475531
$967$	37.5047i	1.20607i	0.797715	+	0.603035i	$0.206042\pi$
$967$	37.5047i	1.20607i	−0.797715	+	0.603035i	$0.793958\pi$
$968$	7.63490i	0.245395i
$969$	8.22475	0.264217
$970$	0	0
$971$	−39.6906	−1.27373	−0.636866	−	0.770975i	$-0.719769\pi$
$971$	−39.6906	−1.27373	−0.636866	+	0.770975i	$0.719769\pi$
$972$	− 3.09855i	− 0.0993859i
$973$	− 12.5194i	− 0.401352i
$974$	70.7303	2.26635
$975$	0	0
$976$	0.889332	0.0284668
$977$	− 27.2392i	− 0.871460i	−0.900077	−	0.435730i	$-0.856490\pi$
$977$	− 27.2392i	− 0.871460i	0.900077	−	0.435730i	$-0.143510\pi$
$978$	− 27.5032i	− 0.879454i
$979$	53.8271	1.72032
$980$	0	0
$981$	−12.7600	−0.407394
$982$	− 30.5657i	− 0.975392i
$983$	− 12.7243i	− 0.405843i	−0.979195	−	0.202922i	$-0.934956\pi$
$983$	− 12.7243i	− 0.405843i	0.979195	−	0.202922i	$-0.0650437\pi$
$984$	−14.0920	−0.449238
$985$	0	0
$986$	80.9948	2.57940
$987$	6.42828i	0.204614i
$988$	44.9049i	1.42861i
$989$	1.51490	0.0481710
$990$	0	0
$991$	−3.05386	−0.0970092	−0.0485046	−	0.998823i	$-0.515446\pi$
$991$	−3.05386	−0.0970092	−0.0485046	+	0.998823i	$0.515446\pi$
$992$	62.5976i	1.98748i
$993$	− 26.4703i	− 0.840009i
$994$	−25.2281	−0.800187
$995$	0	0
$996$	−8.07531	−0.255876
$997$	− 21.1641i	− 0.670275i	−0.942169	−	0.335138i	$-0.891217\pi$
$997$	− 21.1641i	− 0.670275i	0.942169	−	0.335138i	$-0.108783\pi$
$998$	10.2467i	0.324354i
$999$	6.83489	0.216246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.b.c.124.2		8
3.2	odd	2		1125.2.b.g.874.7		8
4.3	odd	2		6000.2.f.o.1249.3		8
5.2	odd	4		375.2.a.f.1.3	yes	4
5.3	odd	4		375.2.a.e.1.2	✓	4
5.4	even	2	inner	375.2.b.c.124.7		8
15.2	even	4		1125.2.a.h.1.2		4
15.8	even	4		1125.2.a.l.1.3		4
15.14	odd	2		1125.2.b.g.874.2		8
20.3	even	4		6000.2.a.bh.1.3		4
20.7	even	4		6000.2.a.bg.1.2		4
20.19	odd	2		6000.2.f.o.1249.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.e.1.2	✓	4	5.3	odd	4
375.2.a.f.1.3	yes	4	5.2	odd	4
375.2.b.c.124.2		8	1.1	even	1	trivial
375.2.b.c.124.7		8	5.4	even	2	inner
1125.2.a.h.1.2		4	15.2	even	4
1125.2.a.l.1.3		4	15.8	even	4
1125.2.b.g.874.2		8	15.14	odd	2
1125.2.b.g.874.7		8	3.2	odd	2
6000.2.a.bg.1.2		4	20.7	even	4
6000.2.a.bh.1.3		4	20.3	even	4
6000.2.f.o.1249.3		8	4.3	odd	2
6000.2.f.o.1249.6		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)

\(f(q)\)	\(=\)	\(q-2.25800i q^{2} +1.00000i q^{3} -3.09855 q^{4} +2.25800 q^{6} -1.55497i q^{7} +2.48051i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.25800i q^{2} +1.00000i q^{3} -3.09855 q^{4} +2.25800 q^{6} -1.55497i q^{7} +2.48051i q^{8} -1.00000 q^{9} -3.75206 q^{11} -3.09855i q^{12} -6.79104i q^{13} -3.51111 q^{14} -0.596106 q^{16} -3.85410i q^{17} +2.25800i q^{18} +2.13403 q^{19} +1.55497 q^{21} +8.47214i q^{22} +0.420943i q^{23} -2.48051 q^{24} -15.3341 q^{26} -1.00000i q^{27} +4.81814i q^{28} -9.30703 q^{29} +9.92506 q^{31} +6.30703i q^{32} -3.75206i q^{33} -8.70255 q^{34} +3.09855 q^{36} +6.83489i q^{37} -4.81862i q^{38} +6.79104 q^{39} +5.68110 q^{41} -3.51111i q^{42} -3.59882i q^{43} +11.6259 q^{44} +0.950487 q^{46} +4.13403i q^{47} -0.596106i q^{48} +4.58207 q^{49} +3.85410 q^{51} +21.0423i q^{52} -6.72797i q^{53} -2.25800 q^{54} +3.85712 q^{56} +2.13403i q^{57} +21.0152i q^{58} +3.19221 q^{59} -1.49190 q^{61} -22.4108i q^{62} +1.55497i q^{63} +13.0490 q^{64} -8.47214 q^{66} -1.47702i q^{67} +11.9421i q^{68} -0.420943 q^{69} +7.18522 q^{71} -2.48051i q^{72} +2.51599i q^{73} +15.4332 q^{74} -6.61238 q^{76} +5.83433i q^{77} -15.3341i q^{78} -2.00789 q^{79} +1.00000 q^{81} -12.8279i q^{82} -2.60616i q^{83} -4.81814 q^{84} -8.12613 q^{86} -9.30703i q^{87} -9.30703i q^{88} -14.3460 q^{89} -10.5598 q^{91} -1.30431i q^{92} +9.92506i q^{93} +9.33461 q^{94} -6.30703 q^{96} -12.5702i q^{97} -10.3463i q^{98} +3.75206 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) 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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