LMFDB - Embedded newform 1875.2.b.f.1249.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 $ x^12 + 22x^10 + 179x^8 + 641x^6 + 869x^4 + 67*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.12
Root			$2.68704i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.f.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68704i q^{2} -1.00000i q^{3} -5.22020 q^{4} +2.68704 q^{6} -1.68704i q^{7} -8.65280i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.68704i q^{2} -1.00000i q^{3} -5.22020 q^{4} +2.68704 q^{6} -1.68704i q^{7} -8.65280i q^{8} -1.00000 q^{9} +1.07882 q^{11} +5.22020i q^{12} +2.67723i q^{13} +4.53315 q^{14} +12.8101 q^{16} +3.93167i q^{17} -2.68704i q^{18} +1.17755 q^{19} -1.68704 q^{21} +2.89883i q^{22} -4.06295i q^{23} -8.65280 q^{24} -7.19384 q^{26} +1.00000i q^{27} +8.80669i q^{28} -5.95595 q^{29} -7.10310 q^{31} +17.1155i q^{32} -1.07882i q^{33} -10.5646 q^{34} +5.22020 q^{36} +4.58187i q^{37} +3.16412i q^{38} +2.67723 q^{39} -11.2614 q^{41} -4.53315i q^{42} +2.58587i q^{43} -5.63164 q^{44} +10.9173 q^{46} +1.91782i q^{47} -12.8101i q^{48} +4.15389 q^{49} +3.93167 q^{51} -13.9757i q^{52} -2.54861i q^{53} -2.68704 q^{54} -14.5976 q^{56} -1.17755i q^{57} -16.0039i q^{58} -1.33599 q^{59} -7.28106 q^{61} -19.0863i q^{62} +1.68704i q^{63} -20.3701 q^{64} +2.89883 q^{66} -12.4451i q^{67} -20.5241i q^{68} -4.06295 q^{69} -5.98480 q^{71} +8.65280i q^{72} -3.30065i q^{73} -12.3117 q^{74} -6.14702 q^{76} -1.82001i q^{77} +7.19384i q^{78} +4.00404 q^{79} +1.00000 q^{81} -30.2600i q^{82} +8.87482i q^{83} +8.80669 q^{84} -6.94834 q^{86} +5.95595i q^{87} -9.33480i q^{88} -15.4436 q^{89} +4.51660 q^{91} +21.2094i q^{92} +7.10310i q^{93} -5.15327 q^{94} +17.1155 q^{96} -10.7682i q^{97} +11.1617i q^{98} -1.07882 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) $ 12 * q - 20 * q^4 - 12 * q^9	$ 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) $ 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.68704i	1.90003i	0.312211	+	0.950013i	$0.398931\pi$
$2$	2.68704i	1.90003i	−0.312211	+	0.950013i	$0.601069\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.22020	−2.61010
$5$	0	0
$5$	0	0
$6$	2.68704	1.09698
$7$	− 1.68704i	− 0.637642i	−0.947815	−	0.318821i	$-0.896713\pi$
$7$	− 1.68704i	− 0.637642i	0.947815	−	0.318821i	$-0.103287\pi$
$8$	− 8.65280i	− 3.05923i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.07882	0.325276	0.162638	−	0.986686i	$-0.448000\pi$
$11$	1.07882	0.325276	0.162638	+	0.986686i	$0.448000\pi$
$12$	5.22020i	1.50694i
$13$	2.67723i	0.742531i	0.928527	+	0.371265i	$0.121076\pi$
$13$	2.67723i	0.742531i	−0.928527	+	0.371265i	$0.878924\pi$
$14$	4.53315	1.21154
$15$	0	0
$16$	12.8101	3.20251
$17$	3.93167i	0.953570i	0.879020	+	0.476785i	$0.158198\pi$
$17$	3.93167i	0.953570i	−0.879020	+	0.476785i	$0.841802\pi$
$18$	− 2.68704i	− 0.633342i
$19$	1.17755	0.270148	0.135074	−	0.990836i	$-0.456873\pi$
$19$	1.17755	0.270148	0.135074	+	0.990836i	$0.456873\pi$
$20$	0	0
$21$	−1.68704	−0.368143
$22$	2.89883i	0.618032i
$23$	− 4.06295i	− 0.847183i	−0.905853	−	0.423591i	$-0.860769\pi$
$23$	− 4.06295i	− 0.847183i	0.905853	−	0.423591i	$-0.139231\pi$
$24$	−8.65280	−1.76625
$25$	0	0
$26$	−7.19384	−1.41083
$27$	1.00000i	0.192450i
$28$	8.80669i	1.66431i
$29$	−5.95595	−1.10599	−0.552996	−	0.833184i	$-0.686516\pi$
$29$	−5.95595	−1.10599	−0.552996	+	0.833184i	$0.686516\pi$
$30$	0	0
$31$	−7.10310	−1.27575	−0.637877	−	0.770138i	$-0.720187\pi$
$31$	−7.10310	−1.27575	−0.637877	+	0.770138i	$0.720187\pi$
$32$	17.1155i	3.02563i
$33$	− 1.07882i	− 0.187798i
$34$	−10.5646	−1.81181
$35$	0	0
$36$	5.22020	0.870033
$37$	4.58187i	0.753254i	0.926365	+	0.376627i	$0.122916\pi$
$37$	4.58187i	0.753254i	−0.926365	+	0.376627i	$0.877084\pi$
$38$	3.16412i	0.513287i
$39$	2.67723	0.428700
$40$	0	0
$41$	−11.2614	−1.75874	−0.879371	−	0.476137i	$-0.842037\pi$
$41$	−11.2614	−1.75874	−0.879371	+	0.476137i	$0.842037\pi$
$42$	− 4.53315i	− 0.699481i
$43$	2.58587i	0.394342i	0.980369	+	0.197171i	$0.0631754\pi$
$43$	2.58587i	0.394342i	−0.980369	+	0.197171i	$0.936825\pi$
$44$	−5.63164	−0.849002
$45$	0	0
$46$	10.9173	1.60967
$47$	1.91782i	0.279743i	0.990170	+	0.139871i	$0.0446689\pi$
$47$	1.91782i	0.279743i	−0.990170	+	0.139871i	$0.955331\pi$
$48$	− 12.8101i	− 1.84897i
$49$	4.15389	0.593413
$50$	0	0
$51$	3.93167	0.550544
$52$	− 13.9757i	− 1.93808i
$53$	− 2.54861i	− 0.350078i	−0.984561	−	0.175039i	$-0.943995\pi$
$53$	− 2.54861i	− 0.350078i	0.984561	−	0.175039i	$-0.0560052\pi$
$54$	−2.68704	−0.365660
$55$	0	0
$56$	−14.5976	−1.95069
$57$	− 1.17755i	− 0.155970i
$58$	− 16.0039i	− 2.10141i
$59$	−1.33599	−0.173931	−0.0869654	−	0.996211i	$-0.527717\pi$
$59$	−1.33599	−0.173931	−0.0869654	+	0.996211i	$0.527717\pi$
$60$	0	0
$61$	−7.28106	−0.932245	−0.466122	−	0.884720i	$-0.654349\pi$
$61$	−7.28106	−0.932245	−0.466122	+	0.884720i	$0.654349\pi$
$62$	− 19.0863i	− 2.42397i
$63$	1.68704i	0.212547i
$64$	−20.3701	−2.54626
$65$	0	0
$66$	2.89883	0.356821
$67$	− 12.4451i	− 1.52041i	−0.649686	−	0.760203i	$-0.725100\pi$
$67$	− 12.4451i	− 1.52041i	0.649686	−	0.760203i	$-0.274900\pi$
$68$	− 20.5241i	− 2.48891i
$69$	−4.06295	−0.489121
$70$	0	0
$71$	−5.98480	−0.710266	−0.355133	−	0.934816i	$-0.615564\pi$
$71$	−5.98480	−0.710266	−0.355133	+	0.934816i	$0.615564\pi$
$72$	8.65280i	1.01974i
$73$	− 3.30065i	− 0.386312i	−0.981168	−	0.193156i	$-0.938128\pi$
$73$	− 3.30065i	− 0.386312i	0.981168	−	0.193156i	$-0.0618724\pi$
$74$	−12.3117	−1.43120
$75$	0	0
$76$	−6.14702	−0.705112
$77$	− 1.82001i	− 0.207410i
$78$	7.19384i	0.814542i
$79$	4.00404	0.450490	0.225245	−	0.974302i	$-0.427682\pi$
$79$	4.00404	0.450490	0.225245	+	0.974302i	$0.427682\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 30.2600i	− 3.34166i
$83$	8.87482i	0.974138i	0.873364	+	0.487069i	$0.161934\pi$
$83$	8.87482i	0.974138i	−0.873364	+	0.487069i	$0.838066\pi$
$84$	8.80669	0.960889
$85$	0	0
$86$	−6.94834	−0.749259
$87$	5.95595i	0.638545i
$88$	− 9.33480i	− 0.995093i
$89$	−15.4436	−1.63702	−0.818509	−	0.574493i	$-0.805199\pi$
$89$	−15.4436	−1.63702	−0.818509	+	0.574493i	$0.805199\pi$
$90$	0	0
$91$	4.51660	0.473469
$92$	21.2094i	2.21123i
$93$	7.10310i	0.736557i
$94$	−5.15327	−0.531519
$95$	0	0
$96$	17.1155	1.74685
$97$	− 10.7682i	− 1.09335i	−0.837346	−	0.546673i	$-0.815894\pi$
$97$	− 10.7682i	− 1.09335i	0.837346	−	0.546673i	$-0.184106\pi$
$98$	11.1617i	1.12750i
$99$	−1.07882	−0.108425
$100$	0	0
$101$	−13.5654	−1.34981	−0.674905	−	0.737905i	$-0.735815\pi$
$101$	−13.5654	−1.34981	−0.674905	+	0.737905i	$0.735815\pi$
$102$	10.5646i	1.04605i
$103$	1.36904i	0.134895i	0.997723	+	0.0674476i	$0.0214856\pi$
$103$	1.36904i	0.134895i	−0.997723	+	0.0674476i	$0.978514\pi$
$104$	23.1656	2.27157
$105$	0	0
$106$	6.84822	0.665158
$107$	− 11.0115i	− 1.06452i	−0.846579	−	0.532262i	$-0.821342\pi$
$107$	− 11.0115i	− 1.06452i	0.846579	−	0.532262i	$-0.178658\pi$
$108$	− 5.22020i	− 0.502314i
$109$	−3.44890	−0.330344	−0.165172	−	0.986265i	$-0.552818\pi$
$109$	−3.44890	−0.330344	−0.165172	+	0.986265i	$0.552818\pi$
$110$	0	0
$111$	4.58187	0.434891
$112$	− 21.6111i	− 2.04206i
$113$	6.05194i	0.569319i	0.958629	+	0.284660i	$0.0918805\pi$
$113$	6.05194i	0.569319i	−0.958629	+	0.284660i	$0.908119\pi$
$114$	3.16412	0.296347
$115$	0	0
$116$	31.0912	2.88675
$117$	− 2.67723i	− 0.247510i
$118$	− 3.58986i	− 0.330473i
$119$	6.63289	0.608036
$120$	0	0
$121$	−9.83615	−0.894196
$122$	− 19.5645i	− 1.77129i
$123$	11.2614i	1.01541i
$124$	37.0796	3.32984
$125$	0	0
$126$	−4.53315	−0.403845
$127$	20.2362i	1.79567i	0.440332	+	0.897835i	$0.354861\pi$
$127$	20.2362i	1.79567i	−0.440332	+	0.897835i	$0.645139\pi$
$128$	− 20.5042i	− 1.81233i
$129$	2.58587	0.227673
$130$	0	0
$131$	−3.05011	−0.266490	−0.133245	−	0.991083i	$-0.542540\pi$
$131$	−3.05011	−0.266490	−0.133245	+	0.991083i	$0.542540\pi$
$132$	5.63164i	0.490171i
$133$	− 1.98657i	− 0.172257i
$134$	33.4404	2.88881
$135$	0	0
$136$	34.0200	2.91719
$137$	20.5100i	1.75229i	0.482048	+	0.876145i	$0.339893\pi$
$137$	20.5100i	1.75229i	−0.482048	+	0.876145i	$0.660107\pi$
$138$	− 10.9173i	− 0.929343i
$139$	−18.3184	−1.55374	−0.776871	−	0.629659i	$-0.783194\pi$
$139$	−18.3184	−1.55374	−0.776871	+	0.629659i	$0.783194\pi$
$140$	0	0
$141$	1.91782	0.161510
$142$	− 16.0814i	− 1.34952i
$143$	2.88825i	0.241527i
$144$	−12.8101	−1.06750
$145$	0	0
$146$	8.86899	0.734003
$147$	− 4.15389i	− 0.342607i
$148$	− 23.9182i	− 1.96607i
$149$	−0.705938	−0.0578327	−0.0289163	−	0.999582i	$-0.509206\pi$
$149$	−0.705938	−0.0578327	−0.0289163	+	0.999582i	$0.509206\pi$
$150$	0	0
$151$	7.04538	0.573345	0.286673	−	0.958029i	$-0.407451\pi$
$151$	7.04538	0.573345	0.286673	+	0.958029i	$0.407451\pi$
$152$	− 10.1891i	− 0.826443i
$153$	− 3.93167i	− 0.317857i
$154$	4.89045	0.394083
$155$	0	0
$156$	−13.9757	−1.11895
$157$	5.12880i	0.409323i	0.978833	+	0.204662i	$0.0656094\pi$
$157$	5.12880i	0.409323i	−0.978833	+	0.204662i	$0.934391\pi$
$158$	10.7590i	0.855942i
$159$	−2.54861	−0.202118
$160$	0	0
$161$	−6.85436	−0.540199
$162$	2.68704i	0.211114i
$163$	− 24.2613i	− 1.90029i	−0.311806	−	0.950146i	$-0.600934\pi$
$163$	− 24.2613i	− 1.90029i	0.311806	−	0.950146i	$-0.399066\pi$
$164$	58.7869	4.59049
$165$	0	0
$166$	−23.8470	−1.85089
$167$	− 4.54890i	− 0.352004i	−0.984390	−	0.176002i	$-0.943683\pi$
$167$	− 4.54890i	− 0.352004i	0.984390	−	0.176002i	$-0.0563165\pi$
$168$	14.5976i	1.12623i
$169$	5.83243	0.448648
$170$	0	0
$171$	−1.17755	−0.0900492
$172$	− 13.4988i	− 1.02927i
$173$	− 2.67729i	− 0.203550i	−0.994807	−	0.101775i	$-0.967548\pi$
$173$	− 2.67729i	− 0.203550i	0.994807	−	0.101775i	$-0.0324522\pi$
$174$	−16.0039	−1.21325
$175$	0	0
$176$	13.8197	1.04170
$177$	1.33599i	0.100419i
$178$	− 41.4976i	− 3.11038i
$179$	15.6266	1.16799	0.583995	−	0.811757i	$-0.301489\pi$
$179$	15.6266	1.16799	0.583995	+	0.811757i	$0.301489\pi$
$180$	0	0
$181$	0.202843	0.0150772	0.00753859	−	0.999972i	$-0.497600\pi$
$181$	0.202843	0.0150772	0.00753859	+	0.999972i	$0.497600\pi$
$182$	12.1363i	0.899603i
$183$	7.28106i	0.538232i
$184$	−35.1559	−2.59172
$185$	0	0
$186$	−19.0863	−1.39948
$187$	4.24155i	0.310173i
$188$	− 10.0114i	− 0.730156i
$189$	1.68704	0.122714
$190$	0	0
$191$	−7.91975	−0.573053	−0.286526	−	0.958072i	$-0.592501\pi$
$191$	−7.91975	−0.573053	−0.286526	+	0.958072i	$0.592501\pi$
$192$	20.3701i	1.47008i
$193$	− 16.4269i	− 1.18243i	−0.806513	−	0.591216i	$-0.798648\pi$
$193$	− 16.4269i	− 1.18243i	0.806513	−	0.591216i	$-0.201352\pi$
$194$	28.9346	2.07738
$195$	0	0
$196$	−21.6841	−1.54887
$197$	17.1369i	1.22095i	0.792035	+	0.610475i	$0.209022\pi$
$197$	17.1369i	1.22095i	−0.792035	+	0.610475i	$0.790978\pi$
$198$	− 2.89883i	− 0.206011i
$199$	2.01848	0.143086	0.0715430	−	0.997438i	$-0.477208\pi$
$199$	2.01848	0.143086	0.0715430	+	0.997438i	$0.477208\pi$
$200$	0	0
$201$	−12.4451	−0.877806
$202$	− 36.4509i	− 2.56467i
$203$	10.0479i	0.705227i
$204$	−20.5241	−1.43697
$205$	0	0
$206$	−3.67866	−0.256304
$207$	4.06295i	0.282394i
$208$	34.2955i	2.37796i
$209$	1.27036	0.0878725
$210$	0	0
$211$	−23.8213	−1.63993	−0.819963	−	0.572416i	$-0.806006\pi$
$211$	−23.8213	−1.63993	−0.819963	+	0.572416i	$0.806006\pi$
$212$	13.3042i	0.913738i
$213$	5.98480i	0.410072i
$214$	29.5884	2.02262
$215$	0	0
$216$	8.65280	0.588749
$217$	11.9832i	0.813475i
$218$	− 9.26733i	− 0.627663i
$219$	−3.30065	−0.223037
$220$	0	0
$221$	−10.5260	−0.708055
$222$	12.3117i	0.826305i
$223$	13.1112i	0.877989i	0.898490	+	0.438995i	$0.144665\pi$
$223$	13.1112i	0.877989i	−0.898490	+	0.438995i	$0.855335\pi$
$224$	28.8746	1.92927
$225$	0	0
$226$	−16.2618	−1.08172
$227$	26.2807i	1.74431i	0.489229	+	0.872155i	$0.337278\pi$
$227$	26.2807i	1.74431i	−0.489229	+	0.872155i	$0.662722\pi$
$228$	6.14702i	0.407096i
$229$	−1.99605	−0.131903	−0.0659513	−	0.997823i	$-0.521008\pi$
$229$	−1.99605	−0.131903	−0.0659513	+	0.997823i	$0.521008\pi$
$230$	0	0
$231$	−1.82001	−0.119748
$232$	51.5357i	3.38348i
$233$	− 0.525548i	− 0.0344298i	−0.999852	−	0.0172149i	$-0.994520\pi$
$233$	− 0.525548i	− 0.0344298i	0.999852	−	0.0172149i	$-0.00547995\pi$
$234$	7.19384	0.470276
$235$	0	0
$236$	6.97412	0.453976
$237$	− 4.00404i	− 0.260090i
$238$	17.8229i	1.15528i
$239$	7.00971	0.453421	0.226710	−	0.973962i	$-0.427203\pi$
$239$	7.00971	0.453421	0.226710	+	0.973962i	$0.427203\pi$
$240$	0	0
$241$	2.31554	0.149157	0.0745786	−	0.997215i	$-0.476239\pi$
$241$	2.31554	0.149157	0.0745786	+	0.997215i	$0.476239\pi$
$242$	− 26.4302i	− 1.69899i
$243$	− 1.00000i	− 0.0641500i
$244$	38.0086	2.43325
$245$	0	0
$246$	−30.2600	−1.92931
$247$	3.15257i	0.200593i
$248$	61.4617i	3.90282i
$249$	8.87482	0.562419
$250$	0	0
$251$	−6.17885	−0.390005	−0.195003	−	0.980803i	$-0.562472\pi$
$251$	−6.17885	−0.390005	−0.195003	+	0.980803i	$0.562472\pi$
$252$	− 8.80669i	− 0.554769i
$253$	− 4.38318i	− 0.275568i
$254$	−54.3755	−3.41182
$255$	0	0
$256$	14.3555	0.897217
$257$	− 13.2239i	− 0.824882i	−0.910984	−	0.412441i	$-0.864676\pi$
$257$	− 13.2239i	− 0.824882i	0.910984	−	0.412441i	$-0.135324\pi$
$258$	6.94834i	0.432585i
$259$	7.72980	0.480306
$260$	0	0
$261$	5.95595	0.368664
$262$	− 8.19578i	− 0.506337i
$263$	− 6.97643i	− 0.430185i	−0.976594	−	0.215092i	$-0.930995\pi$
$263$	− 6.97643i	− 0.430185i	0.976594	−	0.215092i	$-0.0690053\pi$
$264$	−9.33480	−0.574517
$265$	0	0
$266$	5.33800	0.327294
$267$	15.4436i	0.945133i
$268$	64.9656i	3.96841i
$269$	−23.7199	−1.44623	−0.723113	−	0.690730i	$-0.757289\pi$
$269$	−23.7199	−1.44623	−0.723113	+	0.690730i	$0.757289\pi$
$270$	0	0
$271$	28.9910	1.76108	0.880539	−	0.473975i	$-0.157181\pi$
$271$	28.9910	1.76108	0.880539	+	0.473975i	$0.157181\pi$
$272$	50.3649i	3.05382i
$273$	− 4.51660i	− 0.273357i
$274$	−55.1113	−3.32940
$275$	0	0
$276$	21.2094	1.27665
$277$	8.58667i	0.515923i	0.966155	+	0.257962i	$0.0830508\pi$
$277$	8.58667i	0.515923i	−0.966155	+	0.257962i	$0.916949\pi$
$278$	− 49.2222i	− 2.95215i
$279$	7.10310	0.425251
$280$	0	0
$281$	17.1661	1.02405	0.512023	−	0.858972i	$-0.328896\pi$
$281$	17.1661	1.02405	0.512023	+	0.858972i	$0.328896\pi$
$282$	5.15327i	0.306872i
$283$	− 9.62005i	− 0.571853i	−0.958252	−	0.285926i	$-0.907699\pi$
$283$	− 9.62005i	− 0.571853i	0.958252	−	0.285926i	$-0.0923012\pi$
$284$	31.2419	1.85386
$285$	0	0
$286$	−7.76084	−0.458908
$287$	18.9985i	1.12145i
$288$	− 17.1155i	− 1.00854i
$289$	1.54198	0.0907048
$290$	0	0
$291$	−10.7682	−0.631243
$292$	17.2301i	1.00831i
$293$	16.8202i	0.982649i	0.870977	+	0.491324i	$0.163487\pi$
$293$	16.8202i	0.982649i	−0.870977	+	0.491324i	$0.836513\pi$
$294$	11.1617	0.650962
$295$	0	0
$296$	39.6460	2.30438
$297$	1.07882i	0.0625994i
$298$	− 1.89688i	− 0.109884i
$299$	10.8775	0.629059
$300$	0	0
$301$	4.36247	0.251449
$302$	18.9312i	1.08937i
$303$	13.5654i	0.779313i
$304$	15.0844	0.865151
$305$	0	0
$306$	10.5646	0.603936
$307$	− 11.3212i	− 0.646134i	−0.946376	−	0.323067i	$-0.895286\pi$
$307$	− 11.3212i	− 0.646134i	0.946376	−	0.323067i	$-0.104714\pi$
$308$	9.50081i	0.541359i
$309$	1.36904	0.0778818
$310$	0	0
$311$	18.5635	1.05264	0.526318	−	0.850288i	$-0.323572\pi$
$311$	18.5635	1.05264	0.526318	+	0.850288i	$0.323572\pi$
$312$	− 23.1656i	− 1.31149i
$313$	− 9.69951i	− 0.548249i	−0.961694	−	0.274124i	$-0.911612\pi$
$313$	− 9.69951i	− 0.548249i	0.961694	−	0.274124i	$-0.0883880\pi$
$314$	−13.7813	−0.777724
$315$	0	0
$316$	−20.9019	−1.17582
$317$	− 7.47647i	− 0.419920i	−0.977710	−	0.209960i	$-0.932667\pi$
$317$	− 7.47647i	− 0.419920i	0.977710	−	0.209960i	$-0.0673334\pi$
$318$	− 6.84822i	− 0.384029i
$319$	−6.42538	−0.359752
$320$	0	0
$321$	−11.0115	−0.614604
$322$	− 18.4180i	− 1.02639i
$323$	4.62972i	0.257605i
$324$	−5.22020	−0.290011
$325$	0	0
$326$	65.1911	3.61060
$327$	3.44890i	0.190724i
$328$	97.4430i	5.38039i
$329$	3.23544	0.178376
$330$	0	0
$331$	−25.2570	−1.38825	−0.694125	−	0.719854i	$-0.744209\pi$
$331$	−25.2570	−1.38825	−0.694125	+	0.719854i	$0.744209\pi$
$332$	− 46.3283i	− 2.54259i
$333$	− 4.58187i	− 0.251085i
$334$	12.2231	0.668817
$335$	0	0
$336$	−21.6111	−1.17898
$337$	13.0240i	0.709463i	0.934968	+	0.354731i	$0.115428\pi$
$337$	13.0240i	0.709463i	−0.934968	+	0.354731i	$0.884572\pi$
$338$	15.6720i	0.852443i
$339$	6.05194	0.328697
$340$	0	0
$341$	−7.66295	−0.414972
$342$	− 3.16412i	− 0.171096i
$343$	− 18.8171i	− 1.01603i
$344$	22.3750	1.20638
$345$	0	0
$346$	7.19398	0.386751
$347$	6.00149i	0.322177i	0.986940	+	0.161089i	$0.0515005\pi$
$347$	6.00149i	0.322177i	−0.986940	+	0.161089i	$0.948500\pi$
$348$	− 31.0912i	− 1.66666i
$349$	22.7183	1.21608	0.608042	−	0.793905i	$-0.291955\pi$
$349$	22.7183	1.21608	0.608042	+	0.793905i	$0.291955\pi$
$350$	0	0
$351$	−2.67723	−0.142900
$352$	18.4646i	0.984164i
$353$	− 4.85493i	− 0.258402i	−0.991618	−	0.129201i	$-0.958759\pi$
$353$	− 4.85493i	− 0.258402i	0.991618	−	0.129201i	$-0.0412413\pi$
$354$	−3.58986	−0.190799
$355$	0	0
$356$	80.6186	4.27278
$357$	− 6.63289i	− 0.351050i
$358$	41.9895i	2.21921i
$359$	−28.8910	−1.52481	−0.762403	−	0.647102i	$-0.775981\pi$
$359$	−28.8910	−1.52481	−0.762403	+	0.647102i	$0.775981\pi$
$360$	0	0
$361$	−17.6134	−0.927020
$362$	0.545047i	0.0286470i
$363$	9.83615i	0.516264i
$364$	−23.5776	−1.23580
$365$	0	0
$366$	−19.5645	−1.02265
$367$	14.7089i	0.767800i	0.923375	+	0.383900i	$0.125419\pi$
$367$	14.7089i	0.767800i	−0.923375	+	0.383900i	$0.874581\pi$
$368$	− 52.0465i	− 2.71311i
$369$	11.2614	0.586247
$370$	0	0
$371$	−4.29961	−0.223225
$372$	− 37.0796i	− 1.92249i
$373$	− 33.0391i	− 1.71070i	−0.518051	−	0.855350i	$-0.673342\pi$
$373$	− 33.0391i	− 1.71070i	0.518051	−	0.855350i	$-0.326658\pi$
$374$	−11.3972	−0.589337
$375$	0	0
$376$	16.5945	0.855797
$377$	− 15.9455i	− 0.821233i
$378$	4.53315i	0.233160i
$379$	15.1556	0.778493	0.389247	−	0.921134i	$-0.372735\pi$
$379$	15.1556	0.778493	0.389247	+	0.921134i	$0.372735\pi$
$380$	0	0
$381$	20.2362	1.03673
$382$	− 21.2807i	− 1.08882i
$383$	− 8.00811i	− 0.409195i	−0.978846	−	0.204598i	$-0.934411\pi$
$383$	− 8.00811i	− 0.409195i	0.978846	−	0.204598i	$-0.0655886\pi$
$384$	−20.5042	−1.04635
$385$	0	0
$386$	44.1397	2.24665
$387$	− 2.58587i	− 0.131447i
$388$	56.2121i	2.85374i
$389$	−1.07463	−0.0544858	−0.0272429	−	0.999629i	$-0.508673\pi$
$389$	−1.07463	−0.0544858	−0.0272429	+	0.999629i	$0.508673\pi$
$390$	0	0
$391$	15.9742	0.807848
$392$	− 35.9428i	− 1.81538i
$393$	3.05011i	0.153858i
$394$	−46.0475	−2.31984
$395$	0	0
$396$	5.63164	0.283001
$397$	24.2139i	1.21526i	0.794220	+	0.607630i	$0.207880\pi$
$397$	24.2139i	1.21526i	−0.794220	+	0.607630i	$0.792120\pi$
$398$	5.42373i	0.271867i
$399$	−1.98657	−0.0994529
$400$	0	0
$401$	−1.99317	−0.0995340	−0.0497670	−	0.998761i	$-0.515848\pi$
$401$	−1.99317	−0.0995340	−0.0497670	+	0.998761i	$0.515848\pi$
$402$	− 33.4404i	− 1.66785i
$403$	− 19.0166i	− 0.947287i
$404$	70.8142	3.52314
$405$	0	0
$406$	−26.9992	−1.33995
$407$	4.94300i	0.245015i
$408$	− 34.0200i	− 1.68424i
$409$	−6.66478	−0.329552	−0.164776	−	0.986331i	$-0.552690\pi$
$409$	−6.66478	−0.329552	−0.164776	+	0.986331i	$0.552690\pi$
$410$	0	0
$411$	20.5100	1.01168
$412$	− 7.14664i	− 0.352090i
$413$	2.25387i	0.110906i
$414$	−10.9173	−0.536556
$415$	0	0
$416$	−45.8223	−2.24662
$417$	18.3184i	0.897054i
$418$	3.41351i	0.166960i
$419$	−21.3540	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$419$	−21.3540	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$420$	0	0
$421$	22.3367	1.08863	0.544313	−	0.838882i	$-0.316790\pi$
$421$	22.3367	1.08863	0.544313	+	0.838882i	$0.316790\pi$
$422$	− 64.0088i	− 3.11590i
$423$	− 1.91782i	− 0.0932476i
$424$	−22.0526	−1.07097
$425$	0	0
$426$	−16.0814	−0.779147
$427$	12.2835i	0.594438i
$428$	57.4823i	2.77851i
$429$	2.88825	0.139446
$430$	0	0
$431$	−30.4129	−1.46494	−0.732469	−	0.680800i	$-0.761632\pi$
$431$	−30.4129	−1.46494	−0.732469	+	0.680800i	$0.761632\pi$
$432$	12.8101i	0.616324i
$433$	0.576639i	0.0277115i	0.999904	+	0.0138558i	$0.00441057\pi$
$433$	0.576639i	0.0277115i	−0.999904	+	0.0138558i	$0.995589\pi$
$434$	−32.1994	−1.54562
$435$	0	0
$436$	18.0039	0.862231
$437$	− 4.78431i	− 0.228864i
$438$	− 8.86899i	− 0.423777i
$439$	−12.8045	−0.611125	−0.305563	−	0.952172i	$-0.598845\pi$
$439$	−12.8045	−0.611125	−0.305563	+	0.952172i	$0.598845\pi$
$440$	0	0
$441$	−4.15389	−0.197804
$442$	− 28.2838i	− 1.34532i
$443$	14.3147i	0.680110i	0.940405	+	0.340055i	$0.110446\pi$
$443$	14.3147i	0.680110i	−0.940405	+	0.340055i	$0.889554\pi$
$444$	−23.9182	−1.13511
$445$	0	0
$446$	−35.2303	−1.66820
$447$	0.705938i	0.0333897i
$448$	34.3652i	1.62360i
$449$	11.9663	0.564724	0.282362	−	0.959308i	$-0.408882\pi$
$449$	11.9663	0.564724	0.282362	+	0.959308i	$0.408882\pi$
$450$	0	0
$451$	−12.1490	−0.572076
$452$	− 31.5923i	− 1.48598i
$453$	− 7.04538i	− 0.331021i
$454$	−70.6173	−3.31424
$455$	0	0
$456$	−10.1891	−0.477147
$457$	8.22154i	0.384587i	0.981337	+	0.192294i	$0.0615926\pi$
$457$	8.22154i	0.384587i	−0.981337	+	0.192294i	$0.938407\pi$
$458$	− 5.36346i	− 0.250618i
$459$	−3.93167	−0.183515
$460$	0	0
$461$	1.27216	0.0592505	0.0296253	−	0.999561i	$-0.490569\pi$
$461$	1.27216	0.0592505	0.0296253	+	0.999561i	$0.490569\pi$
$462$	− 4.89045i	− 0.227524i
$463$	32.6764i	1.51860i	0.650740	+	0.759300i	$0.274459\pi$
$463$	32.6764i	1.51860i	−0.650740	+	0.759300i	$0.725541\pi$
$464$	−76.2960	−3.54195
$465$	0	0
$466$	1.41217	0.0654175
$467$	− 14.4347i	− 0.667958i	−0.942581	−	0.333979i	$-0.891609\pi$
$467$	− 14.4347i	− 0.667958i	0.942581	−	0.333979i	$-0.108391\pi$
$468$	13.9757i	0.646026i
$469$	−20.9953	−0.969474
$470$	0	0
$471$	5.12880	0.236323
$472$	11.5600i	0.532094i
$473$	2.78968i	0.128270i
$474$	10.7590	0.494178
$475$	0	0
$476$	−34.6250	−1.58703
$477$	2.54861i	0.116693i
$478$	18.8354i	0.861511i
$479$	16.8618	0.770436	0.385218	−	0.922826i	$-0.374126\pi$
$479$	16.8618	0.770436	0.385218	+	0.922826i	$0.374126\pi$
$480$	0	0
$481$	−12.2667	−0.559314
$482$	6.22196i	0.283403i
$483$	6.85436i	0.311884i
$484$	51.3466	2.33394
$485$	0	0
$486$	2.68704	0.121887
$487$	− 39.8235i	− 1.80457i	−0.431135	−	0.902287i	$-0.641887\pi$
$487$	− 39.8235i	− 1.80457i	0.431135	−	0.902287i	$-0.358113\pi$
$488$	63.0016i	2.85195i
$489$	−24.2613	−1.09713
$490$	0	0
$491$	14.0255	0.632961	0.316480	−	0.948599i	$-0.397499\pi$
$491$	14.0255	0.632961	0.316480	+	0.948599i	$0.397499\pi$
$492$	− 58.7869i	− 2.65032i
$493$	− 23.4168i	− 1.05464i
$494$	−8.47108	−0.381132
$495$	0	0
$496$	−90.9911	−4.08562
$497$	10.0966i	0.452895i
$498$	23.8470i	1.06861i
$499$	−15.2315	−0.681857	−0.340929	−	0.940089i	$-0.610741\pi$
$499$	−15.2315	−0.681857	−0.340929	+	0.940089i	$0.610741\pi$
$500$	0	0
$501$	−4.54890	−0.203230
$502$	− 16.6028i	− 0.741020i
$503$	23.7330i	1.05820i	0.848559	+	0.529101i	$0.177471\pi$
$503$	23.7330i	1.05820i	−0.848559	+	0.529101i	$0.822529\pi$
$504$	14.5976	0.650231
$505$	0	0
$506$	11.7778	0.523586
$507$	− 5.83243i	− 0.259027i
$508$	− 105.637i	− 4.68688i
$509$	12.5097	0.554482	0.277241	−	0.960800i	$-0.410580\pi$
$509$	12.5097	0.554482	0.277241	+	0.960800i	$0.410580\pi$
$510$	0	0
$511$	−5.56834	−0.246329
$512$	− 2.43464i	− 0.107597i
$513$	1.17755i	0.0519899i
$514$	35.5331	1.56730
$515$	0	0
$516$	−13.4988	−0.594249
$517$	2.06898i	0.0909936i
$518$	20.7703i	0.912595i
$519$	−2.67729	−0.117520
$520$	0	0
$521$	−27.9638	−1.22512	−0.612559	−	0.790425i	$-0.709860\pi$
$521$	−27.9638	−1.22512	−0.612559	+	0.790425i	$0.709860\pi$
$522$	16.0039i	0.700471i
$523$	15.7013i	0.686569i	0.939231	+	0.343285i	$0.111540\pi$
$523$	15.7013i	0.686569i	−0.939231	+	0.343285i	$0.888460\pi$
$524$	15.9222	0.695564
$525$	0	0
$526$	18.7460	0.817362
$527$	− 27.9270i	− 1.21652i
$528$	− 13.8197i	− 0.601426i
$529$	6.49247	0.282281
$530$	0	0
$531$	1.33599	0.0579769
$532$	10.3703i	0.449609i
$533$	− 30.1495i	− 1.30592i
$534$	−41.4976	−1.79578
$535$	0	0
$536$	−107.685	−4.65127
$537$	− 15.6266i	− 0.674340i
$538$	− 63.7363i	− 2.74787i
$539$	4.48129	0.193023
$540$	0	0
$541$	−23.9783	−1.03091	−0.515455	−	0.856917i	$-0.672377\pi$
$541$	−23.9783	−1.03091	−0.515455	+	0.856917i	$0.672377\pi$
$542$	77.9000i	3.34609i
$543$	− 0.202843i	− 0.00870482i
$544$	−67.2927	−2.88515
$545$	0	0
$546$	12.1363	0.519386
$547$	5.62681i	0.240585i	0.992738	+	0.120293i	$0.0383833\pi$
$547$	5.62681i	0.240585i	−0.992738	+	0.120293i	$0.961617\pi$
$548$	− 107.066i	− 4.57365i
$549$	7.28106	0.310748
$550$	0	0
$551$	−7.01341	−0.298781
$552$	35.1559i	1.49633i
$553$	− 6.75498i	− 0.287251i
$554$	−23.0728	−0.980268
$555$	0	0
$556$	95.6254	4.05542
$557$	42.4247i	1.79759i	0.438366	+	0.898796i	$0.355557\pi$
$557$	42.4247i	1.79759i	−0.438366	+	0.898796i	$0.644443\pi$
$558$	19.0863i	0.807989i
$559$	−6.92298	−0.292811
$560$	0	0
$561$	4.24155	0.179079
$562$	46.1261i	1.94571i
$563$	5.78597i	0.243850i	0.992539	+	0.121925i	$0.0389067\pi$
$563$	5.78597i	0.243850i	−0.992539	+	0.121925i	$0.961093\pi$
$564$	−10.0114	−0.421556
$565$	0	0
$566$	25.8495	1.08653
$567$	− 1.68704i	− 0.0708491i
$568$	51.7853i	2.17286i
$569$	22.0056	0.922522	0.461261	−	0.887265i	$-0.347397\pi$
$569$	22.0056	0.922522	0.461261	+	0.887265i	$0.347397\pi$
$570$	0	0
$571$	33.6143	1.40672	0.703358	−	0.710835i	$-0.251683\pi$
$571$	33.6143	1.40672	0.703358	+	0.710835i	$0.251683\pi$
$572$	− 15.0772i	− 0.630410i
$573$	7.91975i	0.330852i
$574$	−51.0499	−2.13078
$575$	0	0
$576$	20.3701	0.848754
$577$	− 40.2350i	− 1.67501i	−0.546433	−	0.837503i	$-0.684015\pi$
$577$	− 40.2350i	− 1.67501i	0.546433	−	0.837503i	$-0.315985\pi$
$578$	4.14337i	0.172341i
$579$	−16.4269	−0.682677
$580$	0	0
$581$	14.9722	0.621151
$582$	− 28.9346i	− 1.19938i
$583$	− 2.74948i	− 0.113872i
$584$	−28.5599	−1.18182
$585$	0	0
$586$	−45.1967	−1.86706
$587$	2.82299i	0.116517i	0.998302	+	0.0582587i	$0.0185548\pi$
$587$	2.82299i	0.116517i	−0.998302	+	0.0582587i	$0.981445\pi$
$588$	21.6841i	0.894238i
$589$	−8.36423	−0.344642
$590$	0	0
$591$	17.1369	0.704916
$592$	58.6939i	2.41231i
$593$	− 24.6805i	− 1.01351i	−0.862091	−	0.506754i	$-0.830845\pi$
$593$	− 24.6805i	− 1.01351i	0.862091	−	0.506754i	$-0.169155\pi$
$594$	−2.89883	−0.118940
$595$	0	0
$596$	3.68513	0.150949
$597$	− 2.01848i	− 0.0826108i
$598$	29.2282i	1.19523i
$599$	7.71547	0.315245	0.157623	−	0.987499i	$-0.449617\pi$
$599$	7.71547	0.315245	0.157623	+	0.987499i	$0.449617\pi$
$600$	0	0
$601$	22.6506	0.923938	0.461969	−	0.886896i	$-0.347143\pi$
$601$	22.6506	0.923938	0.461969	+	0.886896i	$0.347143\pi$
$602$	11.7222i	0.477759i
$603$	12.4451i	0.506802i
$604$	−36.7783	−1.49649
$605$	0	0
$606$	−36.4509	−1.48072
$607$	24.6423i	1.00020i	0.865968	+	0.500099i	$0.166703\pi$
$607$	24.6423i	1.00020i	−0.865968	+	0.500099i	$0.833297\pi$
$608$	20.1543i	0.817367i
$609$	10.0479	0.407163
$610$	0	0
$611$	−5.13445	−0.207718
$612$	20.5241i	0.829637i
$613$	− 41.8034i	− 1.68842i	−0.536011	−	0.844211i	$-0.680069\pi$
$613$	− 41.8034i	− 1.68842i	0.536011	−	0.844211i	$-0.319931\pi$
$614$	30.4205	1.22767
$615$	0	0
$616$	−15.7482	−0.634513
$617$	− 15.9390i	− 0.641680i	−0.947133	−	0.320840i	$-0.896035\pi$
$617$	− 15.9390i	− 0.641680i	0.947133	−	0.320840i	$-0.103965\pi$
$618$	3.67866i	0.147977i
$619$	47.1700	1.89592	0.947961	−	0.318385i	$-0.103141\pi$
$619$	47.1700	1.89592	0.947961	+	0.318385i	$0.103141\pi$
$620$	0	0
$621$	4.06295	0.163040
$622$	49.8808i	2.00004i
$623$	26.0540i	1.04383i
$624$	34.2955	1.37292
$625$	0	0
$626$	26.0630	1.04169
$627$	− 1.27036i	− 0.0507332i
$628$	− 26.7734i	− 1.06837i
$629$	−18.0144	−0.718280
$630$	0	0
$631$	32.3634	1.28836	0.644182	−	0.764872i	$-0.277198\pi$
$631$	32.3634	1.28836	0.644182	+	0.764872i	$0.277198\pi$
$632$	− 34.6462i	− 1.37815i
$633$	23.8213i	0.946812i
$634$	20.0896	0.797859
$635$	0	0
$636$	13.3042	0.527547
$637$	11.1209i	0.440627i
$638$	− 17.2653i	− 0.683539i
$639$	5.98480	0.236755
$640$	0	0
$641$	18.0042	0.711122	0.355561	−	0.934653i	$-0.384290\pi$
$641$	18.0042	0.711122	0.355561	+	0.934653i	$0.384290\pi$
$642$	− 29.5884i	− 1.16776i
$643$	21.8891i	0.863224i	0.902059	+	0.431612i	$0.142055\pi$
$643$	21.8891i	0.863224i	−0.902059	+	0.431612i	$0.857945\pi$
$644$	35.7811	1.40997
$645$	0	0
$646$	−12.4403	−0.489455
$647$	− 19.0263i	− 0.748003i	−0.927428	−	0.374001i	$-0.877985\pi$
$647$	− 19.0263i	− 0.748003i	0.927428	−	0.374001i	$-0.122015\pi$
$648$	− 8.65280i	− 0.339914i
$649$	−1.44129	−0.0565755
$650$	0	0
$651$	11.9832	0.469660
$652$	126.649i	4.95995i
$653$	− 2.31971i	− 0.0907774i	−0.998969	−	0.0453887i	$-0.985547\pi$
$653$	− 2.31971i	− 0.0907774i	0.998969	−	0.0453887i	$-0.0144526\pi$
$654$	−9.26733	−0.362381
$655$	0	0
$656$	−144.260	−5.63239
$657$	3.30065i	0.128771i
$658$	8.69378i	0.338919i
$659$	−7.19180	−0.280153	−0.140076	−	0.990141i	$-0.544735\pi$
$659$	−7.19180	−0.280153	−0.140076	+	0.990141i	$0.544735\pi$
$660$	0	0
$661$	43.0214	1.67334	0.836669	−	0.547709i	$-0.184500\pi$
$661$	43.0214	1.67334	0.836669	+	0.547709i	$0.184500\pi$
$662$	− 67.8666i	− 2.63771i
$663$	10.5260i	0.408796i
$664$	76.7920	2.98011
$665$	0	0
$666$	12.3117	0.477067
$667$	24.1987i	0.936977i
$668$	23.7461i	0.918765i
$669$	13.1112	0.506907
$670$	0	0
$671$	−7.85494	−0.303237
$672$	− 28.8746i	− 1.11386i
$673$	− 29.8031i	− 1.14882i	−0.818566	−	0.574412i	$-0.805231\pi$
$673$	− 29.8031i	− 1.14882i	0.818566	−	0.574412i	$-0.194769\pi$
$674$	−34.9961	−1.34800
$675$	0	0
$676$	−30.4464	−1.17102
$677$	26.7346i	1.02749i	0.857942	+	0.513746i	$0.171743\pi$
$677$	26.7346i	1.02749i	−0.857942	+	0.513746i	$0.828257\pi$
$678$	16.2618i	0.624532i
$679$	−18.1664	−0.697163
$680$	0	0
$681$	26.2807	1.00708
$682$	− 20.5907i	− 0.788457i
$683$	28.4950i	1.09033i	0.838329	+	0.545165i	$0.183533\pi$
$683$	28.4950i	1.09033i	−0.838329	+	0.545165i	$0.816467\pi$
$684$	6.14702	0.235037
$685$	0	0
$686$	50.5623	1.93048
$687$	1.99605i	0.0761540i
$688$	33.1251i	1.26288i
$689$	6.82321	0.259944
$690$	0	0
$691$	34.6469	1.31803	0.659015	−	0.752130i	$-0.270973\pi$
$691$	34.6469	1.31803	0.659015	+	0.752130i	$0.270973\pi$
$692$	13.9760i	0.531286i
$693$	1.82001i	0.0691365i
$694$	−16.1263	−0.612145
$695$	0	0
$696$	51.5357	1.95345
$697$	− 44.2763i	− 1.67708i
$698$	61.0451i	2.31059i
$699$	−0.525548	−0.0198781
$700$	0	0
$701$	0.973305	0.0367612	0.0183806	−	0.999831i	$-0.494149\pi$
$701$	0.973305	0.0367612	0.0183806	+	0.999831i	$0.494149\pi$
$702$	− 7.19384i	− 0.271514i
$703$	5.39536i	0.203490i
$704$	−21.9756	−0.828237
$705$	0	0
$706$	13.0454	0.490971
$707$	22.8854i	0.860696i
$708$	− 6.97412i	− 0.262103i
$709$	11.3242	0.425291	0.212645	−	0.977129i	$-0.431792\pi$
$709$	11.3242	0.425291	0.212645	+	0.977129i	$0.431792\pi$
$710$	0	0
$711$	−4.00404	−0.150163
$712$	133.630i	5.00801i
$713$	28.8595i	1.08080i
$714$	17.8229	0.667004
$715$	0	0
$716$	−81.5742	−3.04857
$717$	− 7.00971i	− 0.261783i
$718$	− 77.6312i	− 2.89717i
$719$	30.7320	1.14611	0.573056	−	0.819516i	$-0.305758\pi$
$719$	30.7320	1.14611	0.573056	+	0.819516i	$0.305758\pi$
$720$	0	0
$721$	2.30962	0.0860149
$722$	− 47.3279i	− 1.76136i
$723$	− 2.31554i	− 0.0861159i
$724$	−1.05888	−0.0393529
$725$	0	0
$726$	−26.4302	−0.980915
$727$	6.56836i	0.243607i	0.992554	+	0.121803i	$0.0388678\pi$
$727$	6.56836i	0.243607i	−0.992554	+	0.121803i	$0.961132\pi$
$728$	− 39.0813i	− 1.44845i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−10.1668	−0.376032
$732$	− 38.0086i	− 1.40484i
$733$	− 31.9276i	− 1.17927i	−0.807669	−	0.589636i	$-0.799271\pi$
$733$	− 31.9276i	− 1.17927i	0.807669	−	0.589636i	$-0.200729\pi$
$734$	−39.5235	−1.45884
$735$	0	0
$736$	69.5395	2.56326
$737$	− 13.4259i	− 0.494551i
$738$	30.2600i	1.11389i
$739$	15.3072	0.563083	0.281542	−	0.959549i	$-0.409154\pi$
$739$	15.3072	0.563083	0.281542	+	0.959549i	$0.409154\pi$
$740$	0	0
$741$	3.15257	0.115812
$742$	− 11.5532i	− 0.424132i
$743$	16.5455i	0.606995i	0.952832	+	0.303498i	$0.0981545\pi$
$743$	16.5455i	0.606995i	−0.952832	+	0.303498i	$0.901846\pi$
$744$	61.4617	2.25330
$745$	0	0
$746$	88.7774	3.25037
$747$	− 8.87482i	− 0.324713i
$748$	− 22.1417i	− 0.809582i
$749$	−18.5769	−0.678786
$750$	0	0
$751$	46.0279	1.67958	0.839791	−	0.542911i	$-0.182678\pi$
$751$	46.0279	1.67958	0.839791	+	0.542911i	$0.182678\pi$
$752$	24.5674i	0.895880i
$753$	6.17885i	0.225170i
$754$	42.8461	1.56036
$755$	0	0
$756$	−8.80669	−0.320296
$757$	26.5282i	0.964184i	0.876121	+	0.482092i	$0.160123\pi$
$757$	26.5282i	0.964184i	−0.876121	+	0.482092i	$0.839877\pi$
$758$	40.7239i	1.47916i
$759$	−4.38318	−0.159099
$760$	0	0
$761$	−28.4241	−1.03037	−0.515187	−	0.857078i	$-0.672277\pi$
$761$	−28.4241	−1.03037	−0.515187	+	0.857078i	$0.672277\pi$
$762$	54.3755i	1.96982i
$763$	5.81843i	0.210641i
$764$	41.3426	1.49572
$765$	0	0
$766$	21.5181	0.777481
$767$	− 3.57675i	− 0.129149i
$768$	− 14.3555i	− 0.518009i
$769$	8.95952	0.323089	0.161544	−	0.986865i	$-0.448353\pi$
$769$	8.95952	0.323089	0.161544	+	0.986865i	$0.448353\pi$
$770$	0	0
$771$	−13.2239	−0.476246
$772$	85.7515i	3.08626i
$773$	23.1042i	0.831001i	0.909593	+	0.415500i	$0.136393\pi$
$773$	23.1042i	0.831001i	−0.909593	+	0.415500i	$0.863607\pi$
$774$	6.94834	0.249753
$775$	0	0
$776$	−93.1752	−3.34479
$777$	− 7.72980i	− 0.277305i
$778$	− 2.88757i	− 0.103524i
$779$	−13.2609	−0.475120
$780$	0	0
$781$	−6.45651	−0.231032
$782$	42.9232i	1.53493i
$783$	− 5.95595i	− 0.212848i
$784$	53.2115	1.90041
$785$	0	0
$786$	−8.19578	−0.292334
$787$	2.03390i	0.0725009i	0.999343	+	0.0362504i	$0.0115414\pi$
$787$	2.03390i	0.0725009i	−0.999343	+	0.0362504i	$0.988459\pi$
$788$	− 89.4578i	− 3.18680i
$789$	−6.97643	−0.248367
$790$	0	0
$791$	10.2099	0.363022
$792$	9.33480i	0.331698i
$793$	− 19.4931i	− 0.692220i
$794$	−65.0637	−2.30902
$795$	0	0
$796$	−10.5368	−0.373469
$797$	12.1923i	0.431874i	0.976407	+	0.215937i	$0.0692806\pi$
$797$	12.1923i	0.431874i	−0.976407	+	0.215937i	$0.930719\pi$
$798$	− 5.33800i	− 0.188963i
$799$	−7.54024	−0.266754
$800$	0	0
$801$	15.4436	0.545673
$802$	− 5.35572i	− 0.189117i
$803$	− 3.56080i	− 0.125658i
$804$	64.9656	2.29116
$805$	0	0
$806$	51.0985	1.79987
$807$	23.7199i	0.834979i
$808$	117.379i	4.12938i
$809$	−8.83355	−0.310571	−0.155286	−	0.987870i	$-0.549630\pi$
$809$	−8.83355	−0.310571	−0.155286	+	0.987870i	$0.549630\pi$
$810$	0	0
$811$	−13.9184	−0.488741	−0.244370	−	0.969682i	$-0.578581\pi$
$811$	−13.9184	−0.488741	−0.244370	+	0.969682i	$0.578581\pi$
$812$	− 52.4522i	− 1.84071i
$813$	− 28.9910i	− 1.01676i
$814$	−13.2820	−0.465535
$815$	0	0
$816$	50.3649	1.76312
$817$	3.04498i	0.106530i
$818$	− 17.9085i	− 0.626158i
$819$	−4.51660	−0.157823
$820$	0	0
$821$	−29.4930	−1.02931	−0.514656	−	0.857397i	$-0.672080\pi$
$821$	−29.4930	−1.02931	−0.514656	+	0.857397i	$0.672080\pi$
$822$	55.1113i	1.92223i
$823$	− 41.9661i	− 1.46285i	−0.681924	−	0.731423i	$-0.738857\pi$
$823$	− 41.9661i	− 1.46285i	0.681924	−	0.731423i	$-0.261143\pi$
$824$	11.8460	0.412675
$825$	0	0
$826$	−6.05624	−0.210724
$827$	− 37.4674i	− 1.30287i	−0.758704	−	0.651435i	$-0.774167\pi$
$827$	− 37.4674i	− 1.30287i	0.758704	−	0.651435i	$-0.225833\pi$
$828$	− 21.2094i	− 0.737077i
$829$	9.55614	0.331899	0.165949	−	0.986134i	$-0.446931\pi$
$829$	9.55614	0.331899	0.165949	+	0.986134i	$0.446931\pi$
$830$	0	0
$831$	8.58667	0.297868
$832$	− 54.5355i	− 1.89068i
$833$	16.3317i	0.565860i
$834$	−49.2222	−1.70443
$835$	0	0
$836$	−6.63152	−0.229356
$837$	− 7.10310i	− 0.245519i
$838$	− 57.3792i	− 1.98213i
$839$	−54.7310	−1.88952	−0.944761	−	0.327759i	$-0.893707\pi$
$839$	−54.7310	−1.88952	−0.944761	+	0.327759i	$0.893707\pi$
$840$	0	0
$841$	6.47334	0.223219
$842$	60.0197i	2.06842i
$843$	− 17.1661i	− 0.591233i
$844$	124.352	4.28037
$845$	0	0
$846$	5.15327	0.177173
$847$	16.5940i	0.570177i
$848$	− 32.6478i	− 1.12113i
$849$	−9.62005	−0.330159
$850$	0	0
$851$	18.6159	0.638144
$852$	− 31.2419i	− 1.07033i
$853$	− 15.3067i	− 0.524092i	−0.965055	−	0.262046i	$-0.915603\pi$
$853$	− 15.3067i	− 0.524092i	0.965055	−	0.262046i	$-0.0843972\pi$
$854$	−33.0062	−1.12945
$855$	0	0
$856$	−95.2806	−3.25662
$857$	8.66910i	0.296131i	0.988978	+	0.148065i	$0.0473046\pi$
$857$	8.66910i	0.296131i	−0.988978	+	0.148065i	$0.952695\pi$
$858$	7.76084i	0.264951i
$859$	30.4078	1.03750	0.518750	−	0.854926i	$-0.326397\pi$
$859$	30.4078	1.03750	0.518750	+	0.854926i	$0.326397\pi$
$860$	0	0
$861$	18.9985	0.647468
$862$	− 81.7208i	− 2.78342i
$863$	− 18.0425i	− 0.614175i	−0.951681	−	0.307088i	$-0.900646\pi$
$863$	− 18.0425i	− 0.614175i	0.951681	−	0.307088i	$-0.0993545\pi$
$864$	−17.1155	−0.582283
$865$	0	0
$866$	−1.54945	−0.0526526
$867$	− 1.54198i	− 0.0523684i
$868$	− 62.5548i	− 2.12325i
$869$	4.31963	0.146533
$870$	0	0
$871$	33.3183	1.12895
$872$	29.8426i	1.01060i
$873$	10.7682i	0.364449i
$874$	12.8556	0.434848
$875$	0	0
$876$	17.2301	0.582149
$877$	− 22.3057i	− 0.753211i	−0.926374	−	0.376605i	$-0.877091\pi$
$877$	− 22.3057i	− 0.753211i	0.926374	−	0.376605i	$-0.122909\pi$
$878$	− 34.4062i	− 1.16115i
$879$	16.8202	0.567332
$880$	0	0
$881$	7.75956	0.261426	0.130713	−	0.991420i	$-0.458273\pi$
$881$	7.75956	0.261426	0.130713	+	0.991420i	$0.458273\pi$
$882$	− 11.1617i	− 0.375833i
$883$	− 19.9527i	− 0.671462i	−0.941958	−	0.335731i	$-0.891017\pi$
$883$	− 19.9527i	− 0.671462i	0.941958	−	0.335731i	$-0.108983\pi$
$884$	54.9477	1.84809
$885$	0	0
$886$	−38.4641	−1.29223
$887$	− 8.60289i	− 0.288857i	−0.989515	−	0.144428i	$-0.953866\pi$
$887$	− 8.60289i	− 0.288857i	0.989515	−	0.144428i	$-0.0461343\pi$
$888$	− 39.6460i	− 1.33043i
$889$	34.1393	1.14499
$890$	0	0
$891$	1.07882	0.0361418
$892$	− 68.4429i	− 2.29164i
$893$	2.25832i	0.0755719i
$894$	−1.89688	−0.0634413
$895$	0	0
$896$	−34.5915	−1.15562
$897$	− 10.8775i	− 0.363187i
$898$	32.1539i	1.07299i
$899$	42.3057	1.41097
$900$	0	0
$901$	10.0203	0.333824
$902$	− 32.6450i	− 1.08696i
$903$	− 4.36247i	− 0.145174i
$904$	52.3663	1.74168
$905$	0	0
$906$	18.9312	0.628948
$907$	− 6.26125i	− 0.207902i	−0.994582	−	0.103951i	$-0.966852\pi$
$907$	− 6.26125i	− 0.207902i	0.994582	−	0.103951i	$-0.0331484\pi$
$908$	− 137.190i	− 4.55282i
$909$	13.5654	0.449937
$910$	0	0
$911$	0.905575	0.0300030	0.0150015	−	0.999887i	$-0.495225\pi$
$911$	0.905575	0.0300030	0.0150015	+	0.999887i	$0.495225\pi$
$912$	− 15.0844i	− 0.499495i
$913$	9.57431i	0.316863i
$914$	−22.0916	−0.730726
$915$	0	0
$916$	10.4198	0.344279
$917$	5.14567i	0.169925i
$918$	− 10.5646i	− 0.348682i
$919$	−43.9933	−1.45120	−0.725602	−	0.688114i	$-0.758439\pi$
$919$	−43.9933	−1.45120	−0.725602	+	0.688114i	$0.758439\pi$
$920$	0	0
$921$	−11.3212	−0.373046
$922$	3.41836i	0.112578i
$923$	− 16.0227i	− 0.527394i
$924$	9.50081	0.312554
$925$	0	0
$926$	−87.8029	−2.88538
$927$	− 1.36904i	− 0.0449651i
$928$	− 101.939i	− 3.34632i
$929$	36.7447	1.20555	0.602777	−	0.797910i	$-0.294061\pi$
$929$	36.7447	1.20555	0.602777	+	0.797910i	$0.294061\pi$
$930$	0	0
$931$	4.89140	0.160309
$932$	2.74347i	0.0898652i
$933$	− 18.5635i	− 0.607740i
$934$	38.7866	1.26914
$935$	0	0
$936$	−23.1656	−0.757190
$937$	− 24.6490i	− 0.805248i	−0.915366	−	0.402624i	$-0.868098\pi$
$937$	− 24.6490i	− 0.805248i	0.915366	−	0.402624i	$-0.131902\pi$
$938$	− 56.4153i	− 1.84203i
$939$	−9.69951	−0.316532
$940$	0	0
$941$	−59.1479	−1.92817	−0.964083	−	0.265602i	$-0.914429\pi$
$941$	−59.1479	−1.92817	−0.964083	+	0.265602i	$0.914429\pi$
$942$	13.7813i	0.449019i
$943$	45.7546i	1.48998i
$944$	−17.1141	−0.557016
$945$	0	0
$946$	−7.49600	−0.243716
$947$	30.0960i	0.977989i	0.872287	+	0.488995i	$0.162636\pi$
$947$	30.0960i	0.977989i	−0.872287	+	0.488995i	$0.837364\pi$
$948$	20.9019i	0.678861i
$949$	8.83661	0.286849
$950$	0	0
$951$	−7.47647	−0.242441
$952$	− 57.3931i	− 1.86012i
$953$	− 20.6393i	− 0.668572i	−0.942472	−	0.334286i	$-0.891505\pi$
$953$	− 20.6393i	− 0.668572i	0.942472	−	0.334286i	$-0.108495\pi$
$954$	−6.84822	−0.221719
$955$	0	0
$956$	−36.5921	−1.18347
$957$	6.42538i	0.207703i
$958$	45.3084i	1.46385i
$959$	34.6013	1.11733
$960$	0	0
$961$	19.4540	0.627549
$962$	− 32.9612i	− 1.06271i
$963$	11.0115i	0.354842i
$964$	−12.0876	−0.389315
$965$	0	0
$966$	−18.4180	−0.592588
$967$	22.6016i	0.726817i	0.931630	+	0.363408i	$0.118387\pi$
$967$	22.6016i	0.726817i	−0.931630	+	0.363408i	$0.881613\pi$
$968$	85.1103i	2.73555i
$969$	4.62972	0.148728
$970$	0	0
$971$	24.1381	0.774628	0.387314	−	0.921948i	$-0.373403\pi$
$971$	24.1381	0.774628	0.387314	+	0.921948i	$0.373403\pi$
$972$	5.22020i	0.167438i
$973$	30.9038i	0.990732i
$974$	107.007	3.42874
$975$	0	0
$976$	−93.2708	−2.98553
$977$	− 23.0506i	− 0.737454i	−0.929538	−	0.368727i	$-0.879794\pi$
$977$	− 23.0506i	− 0.737454i	0.929538	−	0.368727i	$-0.120206\pi$
$978$	− 65.1911i	− 2.08458i
$979$	−16.6608	−0.532483
$980$	0	0
$981$	3.44890	0.110115
$982$	37.6870i	1.20264i
$983$	29.8713i	0.952745i	0.879243	+	0.476373i	$0.158049\pi$
$983$	29.8713i	0.952745i	−0.879243	+	0.476373i	$0.841951\pi$
$984$	97.4430	3.10637
$985$	0	0
$986$	62.9220	2.00384
$987$	− 3.23544i	− 0.102985i
$988$	− 16.4570i	− 0.523567i
$989$	10.5063	0.334079
$990$	0	0
$991$	16.6560	0.529096	0.264548	−	0.964372i	$-0.414777\pi$
$991$	16.6560	0.529096	0.264548	+	0.964372i	$0.414777\pi$
$992$	− 121.573i	− 3.85996i
$993$	25.2570i	0.801507i
$994$	−27.1300	−0.860513
$995$	0	0
$996$	−46.3283	−1.46797
$997$	− 12.4934i	− 0.395669i	−0.980235	−	0.197835i	$-0.936609\pi$
$997$	− 12.4934i	− 0.395669i	0.980235	−	0.197835i	$-0.0633909\pi$
$998$	− 40.9278i	− 1.29555i
$999$	−4.58187	−0.144964

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.f.1249.12		12
5.2	odd	4		1875.2.a.j.1.1		6
5.3	odd	4		1875.2.a.k.1.6		6
5.4	even	2	inner	1875.2.b.f.1249.1		12
15.2	even	4		5625.2.a.p.1.6		6
15.8	even	4		5625.2.a.q.1.1		6
25.2	odd	20		75.2.g.c.46.1	yes	12
25.9	even	10		375.2.i.d.349.6		24
25.11	even	5		375.2.i.d.274.6		24
25.12	odd	20		75.2.g.c.31.1	✓	12
25.13	odd	20		375.2.g.c.151.3		12
25.14	even	10		375.2.i.d.274.1		24
25.16	even	5		375.2.i.d.349.1		24
25.23	odd	20		375.2.g.c.226.3		12
75.2	even	20		225.2.h.d.46.3		12
75.62	even	20		225.2.h.d.181.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.31.1	✓	12	25.12	odd	20
75.2.g.c.46.1	yes	12	25.2	odd	20
225.2.h.d.46.3		12	75.2	even	20
225.2.h.d.181.3		12	75.62	even	20
375.2.g.c.151.3		12	25.13	odd	20
375.2.g.c.226.3		12	25.23	odd	20
375.2.i.d.274.1		24	25.14	even	10
375.2.i.d.274.6		24	25.11	even	5
375.2.i.d.349.1		24	25.16	even	5
375.2.i.d.349.6		24	25.9	even	10
1875.2.a.j.1.1		6	5.2	odd	4
1875.2.a.k.1.6		6	5.3	odd	4
1875.2.b.f.1249.1		12	5.4	even	2	inner
1875.2.b.f.1249.12		12	1.1	even	1	trivial
5625.2.a.p.1.6		6	15.2	even	4
5625.2.a.q.1.1		6	15.8	even	4

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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.68704i q^{2} -1.00000i q^{3} -5.22020 q^{4} +2.68704 q^{6} -1.68704i q^{7} -8.65280i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.68704i q^{2} -1.00000i q^{3} -5.22020 q^{4} +2.68704 q^{6} -1.68704i q^{7} -8.65280i q^{8} -1.00000 q^{9} +1.07882 q^{11} +5.22020i q^{12} +2.67723i q^{13} +4.53315 q^{14} +12.8101 q^{16} +3.93167i q^{17} -2.68704i q^{18} +1.17755 q^{19} -1.68704 q^{21} +2.89883i q^{22} -4.06295i q^{23} -8.65280 q^{24} -7.19384 q^{26} +1.00000i q^{27} +8.80669i q^{28} -5.95595 q^{29} -7.10310 q^{31} +17.1155i q^{32} -1.07882i q^{33} -10.5646 q^{34} +5.22020 q^{36} +4.58187i q^{37} +3.16412i q^{38} +2.67723 q^{39} -11.2614 q^{41} -4.53315i q^{42} +2.58587i q^{43} -5.63164 q^{44} +10.9173 q^{46} +1.91782i q^{47} -12.8101i q^{48} +4.15389 q^{49} +3.93167 q^{51} -13.9757i q^{52} -2.54861i q^{53} -2.68704 q^{54} -14.5976 q^{56} -1.17755i q^{57} -16.0039i q^{58} -1.33599 q^{59} -7.28106 q^{61} -19.0863i q^{62} +1.68704i q^{63} -20.3701 q^{64} +2.89883 q^{66} -12.4451i q^{67} -20.5241i q^{68} -4.06295 q^{69} -5.98480 q^{71} +8.65280i q^{72} -3.30065i q^{73} -12.3117 q^{74} -6.14702 q^{76} -1.82001i q^{77} +7.19384i q^{78} +4.00404 q^{79} +1.00000 q^{81} -30.2600i q^{82} +8.87482i q^{83} +8.80669 q^{84} -6.94834 q^{86} +5.95595i q^{87} -9.33480i q^{88} -15.4436 q^{89} +4.51660 q^{91} +21.2094i q^{92} +7.10310i q^{93} -5.15327 q^{94} +17.1155 q^{96} -10.7682i q^{97} +11.1617i q^{98} -1.07882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) 12 * q - 20 * q^4 - 12 * q^9	\( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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