LMFDB - Embedded newform 1875.2.b.h.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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 $ x^16 + 20x^14 + 156x^12 + 610x^10 + 1286x^8 + 1440x^6 + 761x^4 + 130*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.12
Root			$-2.35083i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35083i q^{2} -1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} -1.59580i q^{7} +2.93840i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.35083i q^{2} -1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} -1.59580i q^{7} +2.93840i q^{8} -1.00000 q^{9} +3.33277 q^{11} -0.175259i q^{12} +7.05132i q^{13} +2.15565 q^{14} -3.61877 q^{16} -4.09625i q^{17} -1.35083i q^{18} -0.567535 q^{19} -1.59580 q^{21} +4.50200i q^{22} +6.30400i q^{23} +2.93840 q^{24} -9.52513 q^{26} +1.00000i q^{27} -0.279678i q^{28} +2.78357 q^{29} -0.995824 q^{31} +0.988473i q^{32} -3.33277i q^{33} +5.53333 q^{34} -0.175259 q^{36} +3.55334i q^{37} -0.766643i q^{38} +7.05132 q^{39} +1.16293 q^{41} -2.15565i q^{42} +0.117022i q^{43} +0.584098 q^{44} -8.51563 q^{46} -7.64173i q^{47} +3.61877i q^{48} +4.45343 q^{49} -4.09625 q^{51} +1.23581i q^{52} +0.523635i q^{53} -1.35083 q^{54} +4.68910 q^{56} +0.567535i q^{57} +3.76013i q^{58} -0.983998 q^{59} +10.6137 q^{61} -1.34519i q^{62} +1.59580i q^{63} -8.57279 q^{64} +4.50200 q^{66} +15.2159i q^{67} -0.717905i q^{68} +6.30400 q^{69} -10.6639 q^{71} -2.93840i q^{72} +5.55832i q^{73} -4.79996 q^{74} -0.0994657 q^{76} -5.31842i q^{77} +9.52513i q^{78} +14.5969 q^{79} +1.00000 q^{81} +1.57091i q^{82} +5.02398i q^{83} -0.279678 q^{84} -0.158076 q^{86} -2.78357i q^{87} +9.79302i q^{88} -2.82350 q^{89} +11.2525 q^{91} +1.10483i q^{92} +0.995824i q^{93} +10.3227 q^{94} +0.988473 q^{96} +1.70592i q^{97} +6.01583i q^{98} -3.33277 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * 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 $

$2$	1.35083i	0.955181i	0.878583	+	0.477590i	$0.158490\pi$
$2$	1.35083i	0.955181i	−0.878583	+	0.477590i	$0.841510\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0.175259	0.0876296
$5$	0	0
$5$	0	0
$6$	1.35083	0.551474
$7$	− 1.59580i	− 0.603155i	−0.953442	−	0.301577i	$-0.902487\pi$
$7$	− 1.59580i	− 0.603155i	0.953442	−	0.301577i	$-0.0975131\pi$
$8$	2.93840i	1.03888i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.33277	1.00487	0.502434	−	0.864616i	$-0.332438\pi$
$11$	3.33277	1.00487	0.502434	+	0.864616i	$0.332438\pi$
$12$	− 0.175259i	− 0.0505930i
$13$	7.05132i	1.95568i	0.209345	+	0.977842i	$0.432867\pi$
$13$	7.05132i	1.95568i	−0.209345	+	0.977842i	$0.567133\pi$
$14$	2.15565	0.576122
$15$	0	0
$16$	−3.61877	−0.904691
$17$	− 4.09625i	− 0.993486i	−0.867898	−	0.496743i	$-0.834529\pi$
$17$	− 4.09625i	− 0.993486i	0.867898	−	0.496743i	$-0.165471\pi$
$18$	− 1.35083i	− 0.318394i
$19$	−0.567535	−0.130201	−0.0651007	−	0.997879i	$-0.520737\pi$
$19$	−0.567535	−0.130201	−0.0651007	+	0.997879i	$0.520737\pi$
$20$	0	0
$21$	−1.59580	−0.348231
$22$	4.50200i	0.959830i
$23$	6.30400i	1.31448i	0.753683	+	0.657238i	$0.228275\pi$
$23$	6.30400i	1.31448i	−0.753683	+	0.657238i	$0.771725\pi$
$24$	2.93840	0.599799
$25$	0	0
$26$	−9.52513	−1.86803
$27$	1.00000i	0.192450i
$28$	− 0.279678i	− 0.0528542i
$29$	2.78357	0.516897	0.258448	−	0.966025i	$-0.416789\pi$
$29$	2.78357	0.516897	0.258448	+	0.966025i	$0.416789\pi$
$30$	0	0
$31$	−0.995824	−0.178855	−0.0894276	−	0.995993i	$-0.528504\pi$
$31$	−0.995824	−0.178855	−0.0894276	+	0.995993i	$0.528504\pi$
$32$	0.988473i	0.174739i
$33$	− 3.33277i	− 0.580161i
$34$	5.53333	0.948959
$35$	0	0
$36$	−0.175259	−0.0292099
$37$	3.55334i	0.584165i	0.956393	+	0.292083i	$0.0943483\pi$
$37$	3.55334i	0.584165i	−0.956393	+	0.292083i	$0.905652\pi$
$38$	− 0.766643i	− 0.124366i
$39$	7.05132	1.12911
$40$	0	0
$41$	1.16293	0.181618	0.0908092	−	0.995868i	$-0.471055\pi$
$41$	1.16293	0.181618	0.0908092	+	0.995868i	$0.471055\pi$
$42$	− 2.15565i	− 0.332624i
$43$	0.117022i	0.0178456i	0.999960	+	0.00892281i	$0.00284026\pi$
$43$	0.117022i	0.0178456i	−0.999960	+	0.00892281i	$0.997160\pi$
$44$	0.584098	0.0880561
$45$	0	0
$46$	−8.51563	−1.25556
$47$	− 7.64173i	− 1.11466i	−0.830291	−	0.557331i	$-0.811826\pi$
$47$	− 7.64173i	− 1.11466i	0.830291	−	0.557331i	$-0.188174\pi$
$48$	3.61877i	0.522324i
$49$	4.45343	0.636205
$50$	0	0
$51$	−4.09625	−0.573589
$52$	1.23581i	0.171376i
$53$	0.523635i	0.0719268i	0.999353	+	0.0359634i	$0.0114500\pi$
$53$	0.523635i	0.0719268i	−0.999353	+	0.0359634i	$0.988550\pi$
$54$	−1.35083	−0.183825
$55$	0	0
$56$	4.68910	0.626607
$57$	0.567535i	0.0751718i
$58$	3.76013i	0.493730i
$59$	−0.983998	−0.128106	−0.0640528	−	0.997947i	$-0.520403\pi$
$59$	−0.983998	−0.128106	−0.0640528	+	0.997947i	$0.520403\pi$
$60$	0	0
$61$	10.6137	1.35895	0.679473	−	0.733701i	$-0.262208\pi$
$61$	10.6137	1.35895	0.679473	+	0.733701i	$0.262208\pi$
$62$	− 1.34519i	− 0.170839i
$63$	1.59580i	0.201052i
$64$	−8.57279	−1.07160
$65$	0	0
$66$	4.50200	0.554158
$67$	15.2159i	1.85892i	0.368920	+	0.929461i	$0.379728\pi$
$67$	15.2159i	1.85892i	−0.368920	+	0.929461i	$0.620272\pi$
$68$	− 0.717905i	− 0.0870588i
$69$	6.30400	0.758913
$70$	0	0
$71$	−10.6639	−1.26558	−0.632788	−	0.774325i	$-0.718090\pi$
$71$	−10.6639	−1.26558	−0.632788	+	0.774325i	$0.718090\pi$
$72$	− 2.93840i	− 0.346294i
$73$	5.55832i	0.650552i	0.945619	+	0.325276i	$0.105457\pi$
$73$	5.55832i	0.650552i	−0.945619	+	0.325276i	$0.894543\pi$
$74$	−4.79996	−0.557984
$75$	0	0
$76$	−0.0994657	−0.0114095
$77$	− 5.31842i	− 0.606090i
$78$	9.52513i	1.07851i
$79$	14.5969	1.64227	0.821137	−	0.570731i	$-0.193340\pi$
$79$	14.5969	1.64227	0.821137	+	0.570731i	$0.193340\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.57091i	0.173478i
$83$	5.02398i	0.551453i	0.961236	+	0.275727i	$0.0889184\pi$
$83$	5.02398i	0.551453i	−0.961236	+	0.275727i	$0.911082\pi$
$84$	−0.279678	−0.0305154
$85$	0	0
$86$	−0.158076	−0.0170458
$87$	− 2.78357i	− 0.298430i
$88$	9.79302i	1.04394i
$89$	−2.82350	−0.299291	−0.149645	−	0.988740i	$-0.547813\pi$
$89$	−2.82350	−0.299291	−0.149645	+	0.988740i	$0.547813\pi$
$90$	0	0
$91$	11.2525	1.17958
$92$	1.10483i	0.115187i
$93$	0.995824i	0.103262i
$94$	10.3227	1.06470
$95$	0	0
$96$	0.988473	0.100886
$97$	1.70592i	0.173210i	0.996243	+	0.0866049i	$0.0276018\pi$
$97$	1.70592i	0.173210i	−0.996243	+	0.0866049i	$0.972398\pi$
$98$	6.01583i	0.607690i
$99$	−3.33277	−0.334956
$100$	0	0
$101$	13.1747	1.31093	0.655464	−	0.755226i	$-0.272473\pi$
$101$	13.1747	1.31093	0.655464	+	0.755226i	$0.272473\pi$
$102$	− 5.53333i	− 0.547882i
$103$	10.8720i	1.07125i	0.844456	+	0.535624i	$0.179924\pi$
$103$	10.8720i	1.07125i	−0.844456	+	0.535624i	$0.820076\pi$
$104$	−20.7196	−2.03173
$105$	0	0
$106$	−0.707341	−0.0687031
$107$	− 9.37236i	− 0.906060i	−0.891495	−	0.453030i	$-0.850343\pi$
$107$	− 9.37236i	− 0.906060i	0.891495	−	0.453030i	$-0.149657\pi$
$108$	0.175259i	0.0168643i
$109$	15.5899	1.49324	0.746622	−	0.665249i	$-0.231674\pi$
$109$	15.5899	1.49324	0.746622	+	0.665249i	$0.231674\pi$
$110$	0	0
$111$	3.55334	0.337268
$112$	5.77482i	0.545669i
$113$	− 10.0481i	− 0.945243i	−0.881265	−	0.472622i	$-0.843308\pi$
$113$	− 10.0481i	− 0.945243i	0.881265	−	0.472622i	$-0.156692\pi$
$114$	−0.766643	−0.0718027
$115$	0	0
$116$	0.487847	0.0452954
$117$	− 7.05132i	− 0.651895i
$118$	− 1.32921i	− 0.122364i
$119$	−6.53678	−0.599226
$120$	0	0
$121$	0.107347	0.00975880
$122$	14.3373i	1.29804i
$123$	− 1.16293i	− 0.104857i
$124$	−0.174527	−0.0156730
$125$	0	0
$126$	−2.15565	−0.192041
$127$	− 0.976784i	− 0.0866756i	−0.999060	−	0.0433378i	$-0.986201\pi$
$127$	− 0.976784i	− 0.0866756i	0.999060	−	0.0433378i	$-0.0137992\pi$
$128$	− 9.60343i	− 0.848832i
$129$	0.117022	0.0103032
$130$	0	0
$131$	10.0616	0.879086	0.439543	−	0.898221i	$-0.355140\pi$
$131$	10.0616	0.879086	0.439543	+	0.898221i	$0.355140\pi$
$132$	− 0.584098i	− 0.0508392i
$133$	0.905670i	0.0785316i
$134$	−20.5541	−1.77561
$135$	0	0
$136$	12.0364	1.03212
$137$	− 4.87244i	− 0.416281i	−0.978099	−	0.208140i	$-0.933259\pi$
$137$	− 4.87244i	− 0.416281i	0.978099	−	0.208140i	$-0.0667411\pi$
$138$	8.51563i	0.724899i
$139$	−0.185784	−0.0157580	−0.00787898	−	0.999969i	$-0.502508\pi$
$139$	−0.185784	−0.0157580	−0.00787898	+	0.999969i	$0.502508\pi$
$140$	0	0
$141$	−7.64173	−0.643550
$142$	− 14.4052i	− 1.20885i
$143$	23.5004i	1.96520i
$144$	3.61877	0.301564
$145$	0	0
$146$	−7.50834	−0.621395
$147$	− 4.45343i	− 0.367313i
$148$	0.622755i	0.0511902i
$149$	−3.88889	−0.318590	−0.159295	−	0.987231i	$-0.550922\pi$
$149$	−3.88889	−0.318590	−0.159295	+	0.987231i	$0.550922\pi$
$150$	0	0
$151$	−22.1146	−1.79966	−0.899829	−	0.436242i	$-0.856309\pi$
$151$	−22.1146	−1.79966	−0.899829	+	0.436242i	$0.856309\pi$
$152$	− 1.66765i	− 0.135264i
$153$	4.09625i	0.331162i
$154$	7.18428	0.578926
$155$	0	0
$156$	1.23581	0.0989438
$157$	13.6058i	1.08586i	0.839777	+	0.542931i	$0.182686\pi$
$157$	13.6058i	1.08586i	−0.839777	+	0.542931i	$0.817314\pi$
$158$	19.7179i	1.56867i
$159$	0.523635	0.0415269
$160$	0	0
$161$	10.0599	0.792832
$162$	1.35083i	0.106131i
$163$	− 8.62895i	− 0.675871i	−0.941169	−	0.337936i	$-0.890271\pi$
$163$	− 8.62895i	− 0.675871i	0.941169	−	0.337936i	$-0.109729\pi$
$164$	0.203813	0.0159151
$165$	0	0
$166$	−6.78654	−0.526737
$167$	6.59891i	0.510639i	0.966857	+	0.255319i	$0.0821806\pi$
$167$	6.59891i	0.510639i	−0.966857	+	0.255319i	$0.917819\pi$
$168$	− 4.68910i	− 0.361772i
$169$	−36.7211	−2.82470
$170$	0	0
$171$	0.567535	0.0434005
$172$	0.0205091i	0.00156381i
$173$	− 13.6595i	− 1.03851i	−0.854618	−	0.519257i	$-0.826209\pi$
$173$	− 13.6595i	− 1.03851i	0.854618	−	0.519257i	$-0.173791\pi$
$174$	3.76013	0.285055
$175$	0	0
$176$	−12.0605	−0.909095
$177$	0.983998i	0.0739618i
$178$	− 3.81407i	− 0.285877i
$179$	−9.82880	−0.734639	−0.367320	−	0.930095i	$-0.619724\pi$
$179$	−9.82880	−0.734639	−0.367320	+	0.930095i	$0.619724\pi$
$180$	0	0
$181$	17.8687	1.32817	0.664085	−	0.747657i	$-0.268821\pi$
$181$	17.8687	1.32817	0.664085	+	0.747657i	$0.268821\pi$
$182$	15.2002i	1.12671i
$183$	− 10.6137i	− 0.784588i
$184$	−18.5237	−1.36559
$185$	0	0
$186$	−1.34519	−0.0986340
$187$	− 13.6518i	− 0.998322i
$188$	− 1.33928i	− 0.0976773i
$189$	1.59580	0.116077
$190$	0	0
$191$	0.325813	0.0235750	0.0117875	−	0.999931i	$-0.496248\pi$
$191$	0.325813	0.0235750	0.0117875	+	0.999931i	$0.496248\pi$
$192$	8.57279i	0.618688i
$193$	2.90187i	0.208881i	0.994531	+	0.104441i	$0.0333052\pi$
$193$	2.90187i	0.208881i	−0.994531	+	0.104441i	$0.966695\pi$
$194$	−2.30441	−0.165447
$195$	0	0
$196$	0.780505	0.0557503
$197$	− 18.1220i	− 1.29114i	−0.763702	−	0.645568i	$-0.776620\pi$
$197$	− 18.1220i	− 1.29114i	0.763702	−	0.645568i	$-0.223380\pi$
$198$	− 4.50200i	− 0.319943i
$199$	−1.53256	−0.108640	−0.0543201	−	0.998524i	$-0.517299\pi$
$199$	−1.53256	−0.108640	−0.0543201	+	0.998524i	$0.517299\pi$
$200$	0	0
$201$	15.2159	1.07325
$202$	17.7967i	1.25217i
$203$	− 4.44202i	− 0.311769i
$204$	−0.717905	−0.0502634
$205$	0	0
$206$	−14.6862	−1.02324
$207$	− 6.30400i	− 0.438158i
$208$	− 25.5171i	− 1.76929i
$209$	−1.89146	−0.130835
$210$	0	0
$211$	−11.3698	−0.782727	−0.391363	−	0.920236i	$-0.627996\pi$
$211$	−11.3698	−0.782727	−0.391363	+	0.920236i	$0.627996\pi$
$212$	0.0917718i	0.00630291i
$213$	10.6639i	0.730681i
$214$	12.6605	0.865451
$215$	0	0
$216$	−2.93840	−0.199933
$217$	1.58913i	0.107877i
$218$	21.0593i	1.42632i
$219$	5.55832	0.375596
$220$	0	0
$221$	28.8839	1.94294
$222$	4.79996i	0.322152i
$223$	− 16.9507i	− 1.13510i	−0.823339	−	0.567550i	$-0.807891\pi$
$223$	− 16.9507i	− 1.13510i	0.823339	−	0.567550i	$-0.192109\pi$
$224$	1.57740	0.105395
$225$	0	0
$226$	13.5732	0.902878
$227$	14.1117i	0.936627i	0.883562	+	0.468314i	$0.155138\pi$
$227$	14.1117i	0.936627i	−0.883562	+	0.468314i	$0.844862\pi$
$228$	0.0994657i	0.00658728i
$229$	−0.0619945	−0.00409671	−0.00204835	−	0.999998i	$-0.500652\pi$
$229$	−0.0619945	−0.00409671	−0.00204835	+	0.999998i	$0.500652\pi$
$230$	0	0
$231$	−5.31842	−0.349926
$232$	8.17927i	0.536995i
$233$	26.0191i	1.70457i	0.523081	+	0.852283i	$0.324782\pi$
$233$	26.0191i	1.70457i	−0.523081	+	0.852283i	$0.675218\pi$
$234$	9.52513	0.622677
$235$	0	0
$236$	−0.172455	−0.0112258
$237$	− 14.5969i	− 0.948168i
$238$	− 8.83008i	− 0.572369i
$239$	−19.4970	−1.26116	−0.630579	−	0.776125i	$-0.717183\pi$
$239$	−19.4970	−1.26116	−0.630579	+	0.776125i	$0.717183\pi$
$240$	0	0
$241$	−4.09860	−0.264014	−0.132007	−	0.991249i	$-0.542142\pi$
$241$	−4.09860	−0.264014	−0.132007	+	0.991249i	$0.542142\pi$
$242$	0.145007i	0.00932141i
$243$	− 1.00000i	− 0.0641500i
$244$	1.86015	0.119084
$245$	0	0
$246$	1.57091	0.100158
$247$	− 4.00187i	− 0.254633i
$248$	− 2.92613i	− 0.185810i
$249$	5.02398	0.318382
$250$	0	0
$251$	−1.02933	−0.0649704	−0.0324852	−	0.999472i	$-0.510342\pi$
$251$	−1.02933	−0.0649704	−0.0324852	+	0.999472i	$0.510342\pi$
$252$	0.279678i	0.0176181i
$253$	21.0098i	1.32087i
$254$	1.31947	0.0827909
$255$	0	0
$256$	−4.17298	−0.260811
$257$	− 18.5597i	− 1.15772i	−0.815426	−	0.578862i	$-0.803497\pi$
$257$	− 18.5597i	− 1.15772i	0.815426	−	0.578862i	$-0.196503\pi$
$258$	0.158076i	0.00984140i
$259$	5.67041	0.352342
$260$	0	0
$261$	−2.78357	−0.172299
$262$	13.5915i	0.839686i
$263$	− 12.3938i	− 0.764231i	−0.924114	−	0.382116i	$-0.875196\pi$
$263$	− 12.3938i	− 0.764231i	0.924114	−	0.382116i	$-0.124804\pi$
$264$	9.79302	0.602719
$265$	0	0
$266$	−1.22341	−0.0750119
$267$	2.82350i	0.172796i
$268$	2.66673i	0.162897i
$269$	5.30032	0.323166	0.161583	−	0.986859i	$-0.448340\pi$
$269$	5.30032	0.323166	0.161583	+	0.986859i	$0.448340\pi$
$270$	0	0
$271$	0.797428	0.0484403	0.0242201	−	0.999707i	$-0.492290\pi$
$271$	0.797428	0.0484403	0.0242201	+	0.999707i	$0.492290\pi$
$272$	14.8234i	0.898798i
$273$	− 11.2525i	− 0.681031i
$274$	6.58184	0.397624
$275$	0	0
$276$	1.10483	0.0665032
$277$	− 4.74425i	− 0.285054i	−0.989791	−	0.142527i	$-0.954477\pi$
$277$	− 4.74425i	− 0.285054i	0.989791	−	0.142527i	$-0.0455228\pi$
$278$	− 0.250962i	− 0.0150517i
$279$	0.995824	0.0596184
$280$	0	0
$281$	−18.7398	−1.11792	−0.558961	−	0.829194i	$-0.688800\pi$
$281$	−18.7398	−1.11792	−0.558961	+	0.829194i	$0.688800\pi$
$282$	− 10.3227i	− 0.614707i
$283$	− 11.7612i	− 0.699130i	−0.936912	−	0.349565i	$-0.886329\pi$
$283$	− 11.7612i	− 0.699130i	0.936912	−	0.349565i	$-0.113671\pi$
$284$	−1.86895	−0.110902
$285$	0	0
$286$	−31.7451	−1.87712
$287$	− 1.85579i	− 0.109544i
$288$	− 0.988473i	− 0.0582463i
$289$	0.220750	0.0129853
$290$	0	0
$291$	1.70592	0.100003
$292$	0.974146i	0.0570076i
$293$	− 22.2819i	− 1.30172i	−0.759198	−	0.650860i	$-0.774408\pi$
$293$	− 22.2819i	− 1.30172i	0.759198	−	0.650860i	$-0.225592\pi$
$294$	6.01583	0.350850
$295$	0	0
$296$	−10.4411	−0.606879
$297$	3.33277i	0.193387i
$298$	− 5.25322i	− 0.304311i
$299$	−44.4515	−2.57070
$300$	0	0
$301$	0.186743	0.0107637
$302$	− 29.8730i	− 1.71900i
$303$	− 13.1747i	− 0.756865i
$304$	2.05378	0.117792
$305$	0	0
$306$	−5.53333	−0.316320
$307$	− 15.3063i	− 0.873574i	−0.899565	−	0.436787i	$-0.856116\pi$
$307$	− 15.3063i	− 0.873574i	0.899565	−	0.436787i	$-0.143884\pi$
$308$	− 0.932102i	− 0.0531115i
$309$	10.8720	0.618486
$310$	0	0
$311$	−12.8545	−0.728913	−0.364456	−	0.931220i	$-0.618745\pi$
$311$	−12.8545	−0.728913	−0.364456	+	0.931220i	$0.618745\pi$
$312$	20.7196i	1.17302i
$313$	10.5072i	0.593901i	0.954893	+	0.296951i	$0.0959697\pi$
$313$	10.5072i	0.593901i	−0.954893	+	0.296951i	$0.904030\pi$
$314$	−18.3791	−1.03720
$315$	0	0
$316$	2.55823	0.143912
$317$	− 19.4806i	− 1.09414i	−0.837086	−	0.547071i	$-0.815743\pi$
$317$	− 19.4806i	− 1.09414i	0.837086	−	0.547071i	$-0.184257\pi$
$318$	0.707341i	0.0396657i
$319$	9.27701	0.519413
$320$	0	0
$321$	−9.37236	−0.523114
$322$	13.5892i	0.757298i
$323$	2.32476i	0.129353i
$324$	0.175259	0.00973662
$325$	0	0
$326$	11.6562	0.645579
$327$	− 15.5899i	− 0.862125i
$328$	3.41714i	0.188680i
$329$	−12.1947	−0.672313
$330$	0	0
$331$	−14.4925	−0.796579	−0.398289	−	0.917260i	$-0.630396\pi$
$331$	−14.4925	−0.796579	−0.398289	+	0.917260i	$0.630396\pi$
$332$	0.880498i	0.0483236i
$333$	− 3.55334i	− 0.194722i
$334$	−8.91400	−0.487753
$335$	0	0
$336$	5.77482	0.315042
$337$	9.33225i	0.508360i	0.967157	+	0.254180i	$0.0818056\pi$
$337$	9.33225i	0.508360i	−0.967157	+	0.254180i	$0.918194\pi$
$338$	− 49.6039i	− 2.69810i
$339$	−10.0481	−0.545737
$340$	0	0
$341$	−3.31885	−0.179726
$342$	0.766643i	0.0414553i
$343$	− 18.2774i	− 0.986884i
$344$	−0.343857	−0.0185395
$345$	0	0
$346$	18.4517	0.991968
$347$	− 1.05341i	− 0.0565499i	−0.999600	−	0.0282750i	$-0.990999\pi$
$347$	− 1.05341i	− 0.0565499i	0.999600	−	0.0282750i	$-0.00900140\pi$
$348$	− 0.487847i	− 0.0261513i
$349$	13.0715	0.699700	0.349850	−	0.936806i	$-0.386233\pi$
$349$	13.0715	0.699700	0.349850	+	0.936806i	$0.386233\pi$
$350$	0	0
$351$	−7.05132	−0.376371
$352$	3.29435i	0.175590i
$353$	− 33.9473i	− 1.80683i	−0.428767	−	0.903415i	$-0.641052\pi$
$353$	− 33.9473i	− 1.80683i	0.428767	−	0.903415i	$-0.358948\pi$
$354$	−1.32921	−0.0706469
$355$	0	0
$356$	−0.494845	−0.0262267
$357$	6.53678i	0.345963i
$358$	− 13.2770i	− 0.701713i
$359$	6.09450	0.321656	0.160828	−	0.986982i	$-0.448584\pi$
$359$	6.09450	0.321656	0.160828	+	0.986982i	$0.448584\pi$
$360$	0	0
$361$	−18.6779	−0.983048
$362$	24.1376i	1.26864i
$363$	− 0.107347i	− 0.00563424i
$364$	1.97210	0.103366
$365$	0	0
$366$	14.3373	0.749423
$367$	− 21.8636i	− 1.14127i	−0.821203	−	0.570636i	$-0.806697\pi$
$367$	− 21.8636i	− 1.14127i	0.821203	−	0.570636i	$-0.193303\pi$
$368$	− 22.8127i	− 1.18919i
$369$	−1.16293	−0.0605395
$370$	0	0
$371$	0.835615	0.0433830
$372$	0.174527i	0.00904882i
$373$	24.4559i	1.26628i	0.774037	+	0.633140i	$0.218234\pi$
$373$	24.4559i	1.26628i	−0.774037	+	0.633140i	$0.781766\pi$
$374$	18.4413	0.953578
$375$	0	0
$376$	22.4545	1.15800
$377$	19.6279i	1.01089i
$378$	2.15565i	0.110875i
$379$	6.27821	0.322490	0.161245	−	0.986914i	$-0.448449\pi$
$379$	6.27821	0.322490	0.161245	+	0.986914i	$0.448449\pi$
$380$	0	0
$381$	−0.976784	−0.0500422
$382$	0.440118i	0.0225184i
$383$	− 24.6876i	− 1.26148i	−0.775996	−	0.630738i	$-0.782752\pi$
$383$	− 24.6876i	− 1.26148i	0.775996	−	0.630738i	$-0.217248\pi$
$384$	−9.60343	−0.490073
$385$	0	0
$386$	−3.91993	−0.199519
$387$	− 0.117022i	− 0.00594854i
$388$	0.298978i	0.0151783i
$389$	12.9236	0.655251	0.327626	−	0.944808i	$-0.393752\pi$
$389$	12.9236	0.655251	0.327626	+	0.944808i	$0.393752\pi$
$390$	0	0
$391$	25.8228	1.30591
$392$	13.0860i	0.660942i
$393$	− 10.0616i	− 0.507541i
$394$	24.4797	1.23327
$395$	0	0
$396$	−0.584098	−0.0293520
$397$	− 29.0214i	− 1.45654i	−0.685288	−	0.728272i	$-0.740324\pi$
$397$	− 29.0214i	− 1.45654i	0.685288	−	0.728272i	$-0.259676\pi$
$398$	− 2.07022i	− 0.103771i
$399$	0.905670	0.0453402
$400$	0	0
$401$	23.3926	1.16817	0.584084	−	0.811693i	$-0.301454\pi$
$401$	23.3926	1.16817	0.584084	+	0.811693i	$0.301454\pi$
$402$	20.5541i	1.02515i
$403$	− 7.02187i	− 0.349784i
$404$	2.30898	0.114876
$405$	0	0
$406$	6.00041	0.297795
$407$	11.8425i	0.587009i
$408$	− 12.0364i	− 0.595892i
$409$	15.9918	0.790742	0.395371	−	0.918521i	$-0.370616\pi$
$409$	15.9918	0.790742	0.395371	+	0.918521i	$0.370616\pi$
$410$	0	0
$411$	−4.87244	−0.240340
$412$	1.90542i	0.0938731i
$413$	1.57026i	0.0772675i
$414$	8.51563	0.418521
$415$	0	0
$416$	−6.97004	−0.341734
$417$	0.185784i	0.00909787i
$418$	− 2.55504i	− 0.124971i
$419$	−32.3769	−1.58172	−0.790858	−	0.611999i	$-0.790366\pi$
$419$	−32.3769	−1.58172	−0.790858	+	0.611999i	$0.790366\pi$
$420$	0	0
$421$	18.0520	0.879801	0.439900	−	0.898047i	$-0.355014\pi$
$421$	18.0520	0.879801	0.439900	+	0.898047i	$0.355014\pi$
$422$	− 15.3586i	− 0.747646i
$423$	7.64173i	0.371554i
$424$	−1.53865	−0.0747235
$425$	0	0
$426$	−14.4052	−0.697932
$427$	− 16.9373i	− 0.819654i
$428$	− 1.64259i	− 0.0793977i
$429$	23.5004	1.13461
$430$	0	0
$431$	33.1353	1.59607	0.798035	−	0.602612i	$-0.205873\pi$
$431$	33.1353	1.59607	0.798035	+	0.602612i	$0.205873\pi$
$432$	− 3.61877i	− 0.174108i
$433$	22.6653i	1.08922i	0.838688	+	0.544612i	$0.183323\pi$
$433$	22.6653i	1.08922i	−0.838688	+	0.544612i	$0.816677\pi$
$434$	−2.14665	−0.103042
$435$	0	0
$436$	2.73228	0.130852
$437$	− 3.57774i	− 0.171147i
$438$	7.50834i	0.358762i
$439$	8.50436	0.405891	0.202945	−	0.979190i	$-0.434949\pi$
$439$	8.50436	0.405891	0.202945	+	0.979190i	$0.434949\pi$
$440$	0	0
$441$	−4.45343	−0.212068
$442$	39.0173i	1.85586i
$443$	− 6.35768i	− 0.302063i	−0.988529	−	0.151031i	$-0.951741\pi$
$443$	− 6.35768i	− 0.302063i	0.988529	−	0.151031i	$-0.0482594\pi$
$444$	0.622755	0.0295547
$445$	0	0
$446$	22.8975	1.08423
$447$	3.88889i	0.183938i
$448$	13.6804i	0.646340i
$449$	6.25726	0.295298	0.147649	−	0.989040i	$-0.452829\pi$
$449$	6.25726	0.295298	0.147649	+	0.989040i	$0.452829\pi$
$450$	0	0
$451$	3.87576	0.182502
$452$	− 1.76102i	− 0.0828313i
$453$	22.1146i	1.03903i
$454$	−19.0625	−0.894648
$455$	0	0
$456$	−1.66765	−0.0780947
$457$	− 11.0441i	− 0.516620i	−0.966062	−	0.258310i	$-0.916834\pi$
$457$	− 11.0441i	− 0.516620i	0.966062	−	0.258310i	$-0.0831657\pi$
$458$	− 0.0837440i	− 0.00391310i
$459$	4.09625	0.191196
$460$	0	0
$461$	−23.6622	−1.10206	−0.551029	−	0.834486i	$-0.685765\pi$
$461$	−23.6622	−1.10206	−0.551029	+	0.834486i	$0.685765\pi$
$462$	− 7.18428i	− 0.334243i
$463$	6.22442i	0.289273i	0.989485	+	0.144637i	$0.0462013\pi$
$463$	6.22442i	0.289273i	−0.989485	+	0.144637i	$0.953799\pi$
$464$	−10.0731	−0.467632
$465$	0	0
$466$	−35.1473	−1.62817
$467$	4.94679i	0.228910i	0.993428	+	0.114455i	$0.0365122\pi$
$467$	4.94679i	0.228910i	−0.993428	+	0.114455i	$0.963488\pi$
$468$	− 1.23581i	− 0.0571252i
$469$	24.2815	1.12122
$470$	0	0
$471$	13.6058	0.626923
$472$	− 2.89138i	− 0.133087i
$473$	0.390006i	0.0179325i
$474$	19.7179	0.905671
$475$	0	0
$476$	−1.14563	−0.0525099
$477$	− 0.523635i	− 0.0239756i
$478$	− 26.3372i	− 1.20463i
$479$	30.0898	1.37484	0.687419	−	0.726261i	$-0.258744\pi$
$479$	30.0898	1.37484	0.687419	+	0.726261i	$0.258744\pi$
$480$	0	0
$481$	−25.0557	−1.14244
$482$	− 5.53651i	− 0.252181i
$483$	− 10.0599i	− 0.457742i
$484$	0.0188135	0.000855159 0
$485$	0	0
$486$	1.35083	0.0612749
$487$	− 34.2499i	− 1.55201i	−0.630726	−	0.776006i	$-0.717243\pi$
$487$	− 34.2499i	− 1.55201i	0.630726	−	0.776006i	$-0.282757\pi$
$488$	31.1874i	1.41179i
$489$	−8.62895	−0.390215
$490$	0	0
$491$	−10.4193	−0.470218	−0.235109	−	0.971969i	$-0.575545\pi$
$491$	−10.4193	−0.470218	−0.235109	+	0.971969i	$0.575545\pi$
$492$	− 0.203813i	− 0.00918861i
$493$	− 11.4022i	− 0.513530i
$494$	5.40584	0.243220
$495$	0	0
$496$	3.60365	0.161809
$497$	17.0175i	0.763338i
$498$	6.78654i	0.304112i
$499$	−8.83514	−0.395515	−0.197757	−	0.980251i	$-0.563366\pi$
$499$	−8.83514	−0.395515	−0.197757	+	0.980251i	$0.563366\pi$
$500$	0	0
$501$	6.59891	0.294818
$502$	− 1.39044i	− 0.0620585i
$503$	− 21.3734i	− 0.952994i	−0.879176	−	0.476497i	$-0.841906\pi$
$503$	− 21.3734i	− 0.952994i	0.879176	−	0.476497i	$-0.158094\pi$
$504$	−4.68910	−0.208869
$505$	0	0
$506$	−28.3806	−1.26167
$507$	36.7211i	1.63084i
$508$	− 0.171190i	− 0.00759535i
$509$	16.7800	0.743759	0.371879	−	0.928281i	$-0.378714\pi$
$509$	16.7800	0.743759	0.371879	+	0.928281i	$0.378714\pi$
$510$	0	0
$511$	8.86994	0.392383
$512$	− 24.8438i	− 1.09795i
$513$	− 0.567535i	− 0.0250573i
$514$	25.0710	1.10584
$515$	0	0
$516$	0.0205091	0.000902863 0
$517$	− 25.4681i	− 1.12009i
$518$	7.65976i	0.336550i
$519$	−13.6595	−0.599586
$520$	0	0
$521$	−4.60508	−0.201752	−0.100876	−	0.994899i	$-0.532165\pi$
$521$	−4.60508	−0.201752	−0.100876	+	0.994899i	$0.532165\pi$
$522$	− 3.76013i	− 0.164577i
$523$	− 12.9634i	− 0.566852i	−0.958994	−	0.283426i	$-0.908529\pi$
$523$	− 12.9634i	− 0.566852i	0.958994	−	0.283426i	$-0.0914710\pi$
$524$	1.76339	0.0770340
$525$	0	0
$526$	16.7418	0.729979
$527$	4.07914i	0.177690i
$528$	12.0605i	0.524866i
$529$	−16.7405	−0.727846
$530$	0	0
$531$	0.983998	0.0427019
$532$	0.158727i	0.00688169i
$533$	8.20015i	0.355188i
$534$	−3.81407	−0.165051
$535$	0	0
$536$	−44.7106	−1.93120
$537$	9.82880i	0.424144i
$538$	7.15983i	0.308682i
$539$	14.8423	0.639301
$540$	0	0
$541$	−27.3641	−1.17647	−0.588237	−	0.808688i	$-0.700178\pi$
$541$	−27.3641	−1.17647	−0.588237	+	0.808688i	$0.700178\pi$
$542$	1.07719i	0.0462692i
$543$	− 17.8687i	− 0.766819i
$544$	4.04903	0.173601
$545$	0	0
$546$	15.2002	0.650507
$547$	27.6453i	1.18203i	0.806661	+	0.591015i	$0.201272\pi$
$547$	27.6453i	1.18203i	−0.806661	+	0.591015i	$0.798728\pi$
$548$	− 0.853941i	− 0.0364785i
$549$	−10.6137	−0.452982
$550$	0	0
$551$	−1.57978	−0.0673007
$552$	18.5237i	0.788422i
$553$	− 23.2936i	− 0.990545i
$554$	6.40868	0.272279
$555$	0	0
$556$	−0.0325603	−0.00138086
$557$	− 6.17333i	− 0.261572i	−0.991411	−	0.130786i	$-0.958250\pi$
$557$	− 6.17333i	− 0.261572i	0.991411	−	0.130786i	$-0.0417501\pi$
$558$	1.34519i	0.0569464i
$559$	−0.825156	−0.0349004
$560$	0	0
$561$	−13.6518	−0.576381
$562$	− 25.3143i	− 1.06782i
$563$	− 5.69934i	− 0.240198i	−0.992762	−	0.120099i	$-0.961679\pi$
$563$	− 5.69934i	− 0.240198i	0.992762	−	0.120099i	$-0.0383213\pi$
$564$	−1.33928	−0.0563940
$565$	0	0
$566$	15.8874	0.667796
$567$	− 1.59580i	− 0.0670172i
$568$	− 31.3350i	− 1.31479i
$569$	−20.1708	−0.845605	−0.422802	−	0.906222i	$-0.638954\pi$
$569$	−20.1708	−0.845605	−0.422802	+	0.906222i	$0.638954\pi$
$570$	0	0
$571$	−15.7554	−0.659341	−0.329671	−	0.944096i	$-0.606938\pi$
$571$	−15.7554	−0.659341	−0.329671	+	0.944096i	$0.606938\pi$
$572$	4.11866i	0.172210i
$573$	− 0.325813i	− 0.0136111i
$574$	2.50686	0.104634
$575$	0	0
$576$	8.57279	0.357200
$577$	− 16.1062i	− 0.670509i	−0.942128	−	0.335255i	$-0.891178\pi$
$577$	− 16.1062i	− 0.670509i	0.942128	−	0.335255i	$-0.108822\pi$
$578$	0.298195i	0.0124033i
$579$	2.90187	0.120598
$580$	0	0
$581$	8.01725	0.332611
$582$	2.30441i	0.0955207i
$583$	1.74515i	0.0722769i
$584$	−16.3326	−0.675847
$585$	0	0
$586$	30.0990	1.24338
$587$	2.00072i	0.0825786i	0.999147	+	0.0412893i	$0.0131465\pi$
$587$	2.00072i	0.0825786i	−0.999147	+	0.0412893i	$0.986853\pi$
$588$	− 0.780505i	− 0.0321875i
$589$	0.565165	0.0232872
$590$	0	0
$591$	−18.1220	−0.745438
$592$	− 12.8587i	− 0.528489i
$593$	26.8231i	1.10149i	0.834672	+	0.550747i	$0.185657\pi$
$593$	26.8231i	1.10149i	−0.834672	+	0.550747i	$0.814343\pi$
$594$	−4.50200	−0.184719
$595$	0	0
$596$	−0.681563	−0.0279179
$597$	1.53256i	0.0627234i
$598$	− 60.0464i	− 2.45548i
$599$	44.8025	1.83058	0.915290	−	0.402796i	$-0.131962\pi$
$599$	44.8025	1.83058	0.915290	+	0.402796i	$0.131962\pi$
$600$	0	0
$601$	−14.2298	−0.580446	−0.290223	−	0.956959i	$-0.593729\pi$
$601$	−14.2298	−0.580446	−0.290223	+	0.956959i	$0.593729\pi$
$602$	0.252258i	0.0102813i
$603$	− 15.2159i	− 0.619641i
$604$	−3.87578	−0.157703
$605$	0	0
$606$	17.7967	0.722943
$607$	− 20.3346i	− 0.825356i	−0.910877	−	0.412678i	$-0.864594\pi$
$607$	− 20.3346i	− 0.825356i	0.910877	−	0.412678i	$-0.135406\pi$
$608$	− 0.560993i	− 0.0227513i
$609$	−4.44202	−0.180000
$610$	0	0
$611$	53.8843	2.17993
$612$	0.717905i	0.0290196i
$613$	20.7921i	0.839783i	0.907574	+	0.419892i	$0.137932\pi$
$613$	20.7921i	0.839783i	−0.907574	+	0.419892i	$0.862068\pi$
$614$	20.6761	0.834421
$615$	0	0
$616$	15.6277	0.629657
$617$	4.94634i	0.199132i	0.995031	+	0.0995660i	$0.0317454\pi$
$617$	4.94634i	0.199132i	−0.995031	+	0.0995660i	$0.968255\pi$
$618$	14.6862i	0.590766i
$619$	45.9527	1.84700	0.923498	−	0.383603i	$-0.125317\pi$
$619$	45.9527	1.84700	0.923498	+	0.383603i	$0.125317\pi$
$620$	0	0
$621$	−6.30400	−0.252971
$622$	− 17.3643i	− 0.696243i
$623$	4.50574i	0.180519i
$624$	−25.5171	−1.02150
$625$	0	0
$626$	−14.1934	−0.567283
$627$	1.89146i	0.0755377i
$628$	2.38454i	0.0951537i
$629$	14.5554	0.580360
$630$	0	0
$631$	−21.6636	−0.862414	−0.431207	−	0.902253i	$-0.641912\pi$
$631$	−21.6636	−0.862414	−0.431207	+	0.902253i	$0.641912\pi$
$632$	42.8915i	1.70613i
$633$	11.3698i	0.451908i
$634$	26.3150	1.04510
$635$	0	0
$636$	0.0917718	0.00363899
$637$	31.4026i	1.24421i
$638$	12.5317i	0.496133i
$639$	10.6639	0.421859
$640$	0	0
$641$	36.3870	1.43720	0.718600	−	0.695424i	$-0.244783\pi$
$641$	36.3870	1.43720	0.718600	+	0.695424i	$0.244783\pi$
$642$	− 12.6605i	− 0.499668i
$643$	1.01349i	0.0399682i	0.999800	+	0.0199841i	$0.00636156\pi$
$643$	1.01349i	0.0399682i	−0.999800	+	0.0199841i	$0.993638\pi$
$644$	1.76309	0.0694755
$645$	0	0
$646$	−3.14036	−0.123556
$647$	13.7007i	0.538629i	0.963052	+	0.269315i	$0.0867972\pi$
$647$	13.7007i	0.538629i	−0.963052	+	0.269315i	$0.913203\pi$
$648$	2.93840i	0.115431i
$649$	−3.27944	−0.128729
$650$	0	0
$651$	1.58913	0.0622830
$652$	− 1.51230i	− 0.0592263i
$653$	23.8372i	0.932822i	0.884568	+	0.466411i	$0.154453\pi$
$653$	23.8372i	0.932822i	−0.884568	+	0.466411i	$0.845547\pi$
$654$	21.0593	0.823485
$655$	0	0
$656$	−4.20835	−0.164309
$657$	− 5.55832i	− 0.216851i
$658$	− 16.4729i	− 0.642181i
$659$	−14.7758	−0.575583	−0.287791	−	0.957693i	$-0.592921\pi$
$659$	−14.7758	−0.575583	−0.287791	+	0.957693i	$0.592921\pi$
$660$	0	0
$661$	7.55471	0.293844	0.146922	−	0.989148i	$-0.453063\pi$
$661$	7.55471	0.293844	0.146922	+	0.989148i	$0.453063\pi$
$662$	− 19.5769i	− 0.760877i
$663$	− 28.8839i	− 1.12176i
$664$	−14.7625	−0.572895
$665$	0	0
$666$	4.79996	0.185995
$667$	17.5477i	0.679448i
$668$	1.15652i	0.0447471i
$669$	−16.9507	−0.655351
$670$	0	0
$671$	35.3730	1.36556
$672$	− 1.57740i	− 0.0608496i
$673$	12.1947i	0.470069i	0.971987	+	0.235035i	$0.0755204\pi$
$673$	12.1947i	0.470069i	−0.971987	+	0.235035i	$0.924480\pi$
$674$	−12.6063	−0.485576
$675$	0	0
$676$	−6.43571	−0.247527
$677$	17.9507i	0.689900i	0.938621	+	0.344950i	$0.112104\pi$
$677$	17.9507i	0.689900i	−0.938621	+	0.344950i	$0.887896\pi$
$678$	− 13.5732i	− 0.521277i
$679$	2.72230	0.104472
$680$	0	0
$681$	14.1117	0.540762
$682$	− 4.48320i	− 0.171671i
$683$	− 10.0273i	− 0.383685i	−0.981426	−	0.191842i	$-0.938554\pi$
$683$	− 10.0273i	− 0.383685i	0.981426	−	0.191842i	$-0.0614462\pi$
$684$	0.0994657	0.00380317
$685$	0	0
$686$	24.6896	0.942653
$687$	0.0619945i	0.00236524i
$688$	− 0.423474i	− 0.0161448i
$689$	−3.69231	−0.140666
$690$	0	0
$691$	19.1296	0.727722	0.363861	−	0.931453i	$-0.381458\pi$
$691$	19.1296	0.727722	0.363861	+	0.931453i	$0.381458\pi$
$692$	− 2.39396i	− 0.0910045i
$693$	5.31842i	0.202030i
$694$	1.42297	0.0540154
$695$	0	0
$696$	8.17927	0.310034
$697$	− 4.76363i	− 0.180435i
$698$	17.6573i	0.668340i
$699$	26.0191	0.984131
$700$	0	0
$701$	−3.81920	−0.144249	−0.0721246	−	0.997396i	$-0.522978\pi$
$701$	−3.81920	−0.144249	−0.0721246	+	0.997396i	$0.522978\pi$
$702$	− 9.52513i	− 0.359503i
$703$	− 2.01664i	− 0.0760592i
$704$	−28.5711	−1.07681
$705$	0	0
$706$	45.8570	1.72585
$707$	− 21.0241i	− 0.790692i
$708$	0.172455i	0.00648124i
$709$	−38.0535	−1.42913	−0.714565	−	0.699569i	$-0.753375\pi$
$709$	−38.0535	−1.42913	−0.714565	+	0.699569i	$0.753375\pi$
$710$	0	0
$711$	−14.5969	−0.547425
$712$	− 8.29660i	− 0.310928i
$713$	− 6.27768i	− 0.235101i
$714$	−8.83008	−0.330457
$715$	0	0
$716$	−1.72259	−0.0643761
$717$	19.4970i	0.728130i
$718$	8.23264i	0.307239i
$719$	−19.4340	−0.724766	−0.362383	−	0.932029i	$-0.618037\pi$
$719$	−19.4340	−0.724766	−0.362383	+	0.932029i	$0.618037\pi$
$720$	0	0
$721$	17.3495	0.646129
$722$	− 25.2307i	− 0.938988i
$723$	4.09860i	0.152429i
$724$	3.13165	0.116387
$725$	0	0
$726$	0.145007	0.00538172
$727$	− 38.0739i	− 1.41208i	−0.708170	−	0.706042i	$-0.750479\pi$
$727$	− 38.0739i	− 1.41208i	0.708170	−	0.706042i	$-0.249521\pi$
$728$	33.0643i	1.22544i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0.479350	0.0177294
$732$	− 1.86015i	− 0.0687531i
$733$	− 16.9591i	− 0.626400i	−0.949687	−	0.313200i	$-0.898599\pi$
$733$	− 16.9591i	− 0.626400i	0.949687	−	0.313200i	$-0.101401\pi$
$734$	29.5341	1.09012
$735$	0	0
$736$	−6.23134	−0.229690
$737$	50.7112i	1.86797i
$738$	− 1.57091i	− 0.0578261i
$739$	−13.2040	−0.485716	−0.242858	−	0.970062i	$-0.578085\pi$
$739$	−13.2040	−0.485716	−0.242858	+	0.970062i	$0.578085\pi$
$740$	0	0
$741$	−4.00187	−0.147012
$742$	1.12877i	0.0414386i
$743$	− 42.2364i	− 1.54950i	−0.632265	−	0.774752i	$-0.717875\pi$
$743$	− 42.2364i	− 1.54950i	0.632265	−	0.774752i	$-0.282125\pi$
$744$	−2.92613	−0.107277
$745$	0	0
$746$	−33.0358	−1.20953
$747$	− 5.02398i	− 0.183818i
$748$	− 2.39261i	− 0.0874825i
$749$	−14.9564	−0.546494
$750$	0	0
$751$	−1.04801	−0.0382426	−0.0191213	−	0.999817i	$-0.506087\pi$
$751$	−1.04801	−0.0382426	−0.0191213	+	0.999817i	$0.506087\pi$
$752$	27.6536i	1.00842i
$753$	1.02933i	0.0375107i
$754$	−26.5139	−0.965579
$755$	0	0
$756$	0.279678	0.0101718
$757$	17.0074i	0.618146i	0.951038	+	0.309073i	$0.100019\pi$
$757$	17.0074i	0.618146i	−0.951038	+	0.309073i	$0.899981\pi$
$758$	8.48079i	0.308036i
$759$	21.0098	0.762607
$760$	0	0
$761$	−24.1244	−0.874508	−0.437254	−	0.899338i	$-0.644049\pi$
$761$	−24.1244	−0.874508	−0.437254	+	0.899338i	$0.644049\pi$
$762$	− 1.31947i	− 0.0477993i
$763$	− 24.8783i	− 0.900657i
$764$	0.0571018	0.00206587
$765$	0	0
$766$	33.3487	1.20494
$767$	− 6.93848i	− 0.250534i
$768$	4.17298i	0.150579i
$769$	−49.7819	−1.79518	−0.897591	−	0.440830i	$-0.854684\pi$
$769$	−49.7819	−1.79518	−0.897591	+	0.440830i	$0.854684\pi$
$770$	0	0
$771$	−18.5597	−0.668412
$772$	0.508580i	0.0183042i
$773$	31.4743i	1.13205i	0.824388	+	0.566025i	$0.191520\pi$
$773$	31.4743i	1.13205i	−0.824388	+	0.566025i	$0.808480\pi$
$774$	0.158076	0.00568193
$775$	0	0
$776$	−5.01268	−0.179945
$777$	− 5.67041i	− 0.203425i
$778$	17.4575i	0.625883i
$779$	−0.660001	−0.0236470
$780$	0	0
$781$	−35.5404	−1.27174
$782$	34.8822i	1.24738i
$783$	2.78357i	0.0994768i
$784$	−16.1159	−0.575569
$785$	0	0
$786$	13.5915	0.484793
$787$	− 19.1415i	− 0.682321i	−0.940005	−	0.341161i	$-0.889180\pi$
$787$	− 19.1415i	− 0.682321i	0.940005	−	0.341161i	$-0.110820\pi$
$788$	− 3.17604i	− 0.113142i
$789$	−12.3938	−0.441229
$790$	0	0
$791$	−16.0347	−0.570128
$792$	− 9.79302i	− 0.347980i
$793$	74.8406i	2.65767i
$794$	39.2030	1.39126
$795$	0	0
$796$	−0.268595	−0.00952009
$797$	− 7.21437i	− 0.255546i	−0.991803	−	0.127773i	$-0.959217\pi$
$797$	− 7.21437i	− 0.255546i	0.991803	−	0.127773i	$-0.0407829\pi$
$798$	1.22341i	0.0433081i
$799$	−31.3024	−1.10740
$800$	0	0
$801$	2.82350	0.0997636
$802$	31.5994i	1.11581i
$803$	18.5246i	0.653718i
$804$	2.66673	0.0940484
$805$	0	0
$806$	9.48535	0.334107
$807$	− 5.30032i	− 0.186580i
$808$	38.7125i	1.36190i
$809$	−1.05941	−0.0372469	−0.0186234	−	0.999827i	$-0.505928\pi$
$809$	−1.05941	−0.0372469	−0.0186234	+	0.999827i	$0.505928\pi$
$810$	0	0
$811$	21.6400	0.759883	0.379941	−	0.925011i	$-0.375944\pi$
$811$	21.6400	0.759883	0.379941	+	0.925011i	$0.375944\pi$
$812$	− 0.778504i	− 0.0273202i
$813$	− 0.797428i	− 0.0279670i
$814$	−15.9971	−0.560700
$815$	0	0
$816$	14.8234	0.518922
$817$	− 0.0664138i	− 0.00232353i
$818$	21.6022i	0.755302i
$819$	−11.2525	−0.393193
$820$	0	0
$821$	−23.1828	−0.809085	−0.404543	−	0.914519i	$-0.632569\pi$
$821$	−23.1828	−0.809085	−0.404543	+	0.914519i	$0.632569\pi$
$822$	− 6.58184i	− 0.229568i
$823$	27.9769i	0.975214i	0.873063	+	0.487607i	$0.162130\pi$
$823$	27.9769i	0.975214i	−0.873063	+	0.487607i	$0.837870\pi$
$824$	−31.9463	−1.11290
$825$	0	0
$826$	−2.12116	−0.0738044
$827$	− 41.3045i	− 1.43630i	−0.695889	−	0.718150i	$-0.744989\pi$
$827$	− 41.3045i	− 1.43630i	0.695889	−	0.718150i	$-0.255011\pi$
$828$	− 1.10483i	− 0.0383956i
$829$	−47.5687	−1.65213	−0.826064	−	0.563577i	$-0.809425\pi$
$829$	−47.5687	−1.65213	−0.826064	+	0.563577i	$0.809425\pi$
$830$	0	0
$831$	−4.74425	−0.164576
$832$	− 60.4495i	− 2.09571i
$833$	− 18.2424i	− 0.632060i
$834$	−0.250962	−0.00869011
$835$	0	0
$836$	−0.331496	−0.0114650
$837$	− 0.995824i	− 0.0344207i
$838$	− 43.7357i	− 1.51083i
$839$	−49.0322	−1.69278	−0.846389	−	0.532565i	$-0.821228\pi$
$839$	−49.0322	−1.69278	−0.846389	+	0.532565i	$0.821228\pi$
$840$	0	0
$841$	−21.2517	−0.732818
$842$	24.3852i	0.840369i
$843$	18.7398i	0.645433i
$844$	−1.99266	−0.0685900
$845$	0	0
$846$	−10.3227	−0.354901
$847$	− 0.171304i	− 0.00588606i
$848$	− 1.89491i	− 0.0650715i
$849$	−11.7612	−0.403643
$850$	0	0
$851$	−22.4003	−0.767871
$852$	1.86895i	0.0640293i
$853$	32.5376i	1.11407i	0.830490	+	0.557033i	$0.188061\pi$
$853$	32.5376i	1.11407i	−0.830490	+	0.557033i	$0.811939\pi$
$854$	22.8794	0.782918
$855$	0	0
$856$	27.5398	0.941290
$857$	− 30.4813i	− 1.04122i	−0.853794	−	0.520610i	$-0.825704\pi$
$857$	− 30.4813i	− 1.04122i	0.853794	−	0.520610i	$-0.174296\pi$
$858$	31.7451i	1.08376i
$859$	−4.12215	−0.140646	−0.0703231	−	0.997524i	$-0.522403\pi$
$859$	−4.12215	−0.140646	−0.0703231	+	0.997524i	$0.522403\pi$
$860$	0	0
$861$	−1.85579	−0.0632452
$862$	44.7601i	1.52453i
$863$	18.4184i	0.626969i	0.949593	+	0.313485i	$0.101496\pi$
$863$	18.4184i	0.626969i	−0.949593	+	0.313485i	$0.898504\pi$
$864$	−0.988473	−0.0336285
$865$	0	0
$866$	−30.6169	−1.04041
$867$	− 0.220750i	− 0.00749705i
$868$	0.278510i	0.00945325i
$869$	48.6479	1.65027
$870$	0	0
$871$	−107.292	−3.63546
$872$	45.8095i	1.55131i
$873$	− 1.70592i	− 0.0577366i
$874$	4.83292	0.163476
$875$	0	0
$876$	0.974146	0.0329133
$877$	− 18.9126i	− 0.638634i	−0.947648	−	0.319317i	$-0.896547\pi$
$877$	− 18.9126i	− 0.638634i	0.947648	−	0.319317i	$-0.103453\pi$
$878$	11.4879i	0.387699i
$879$	−22.2819	−0.751548
$880$	0	0
$881$	34.6336	1.16684	0.583418	−	0.812172i	$-0.301715\pi$
$881$	34.6336	1.16684	0.583418	+	0.812172i	$0.301715\pi$
$882$	− 6.01583i	− 0.202563i
$883$	32.6874i	1.10002i	0.835158	+	0.550009i	$0.185376\pi$
$883$	32.6874i	1.10002i	−0.835158	+	0.550009i	$0.814624\pi$
$884$	5.06218	0.170259
$885$	0	0
$886$	8.58814	0.288524
$887$	48.7918i	1.63827i	0.573602	+	0.819134i	$0.305545\pi$
$887$	48.7918i	1.63827i	−0.573602	+	0.819134i	$0.694455\pi$
$888$	10.4411i	0.350382i
$889$	−1.55875	−0.0522788
$890$	0	0
$891$	3.33277	0.111652
$892$	− 2.97076i	− 0.0994684i
$893$	4.33695i	0.145131i
$894$	−5.25322	−0.175694
$895$	0	0
$896$	−15.3251	−0.511977
$897$	44.4515i	1.48419i
$898$	8.45249i	0.282063i
$899$	−2.77195	−0.0924497
$900$	0	0
$901$	2.14494	0.0714582
$902$	5.23549i	0.174323i
$903$	− 0.186743i	− 0.00621441i
$904$	29.5253	0.981997
$905$	0	0
$906$	−29.8730	−0.992465
$907$	45.0367i	1.49542i	0.664025	+	0.747710i	$0.268847\pi$
$907$	45.0367i	1.49542i	−0.664025	+	0.747710i	$0.731153\pi$
$908$	2.47321i	0.0820762i
$909$	−13.1747	−0.436976
$910$	0	0
$911$	12.9449	0.428882	0.214441	−	0.976737i	$-0.431207\pi$
$911$	12.9449	0.428882	0.214441	+	0.976737i	$0.431207\pi$
$912$	− 2.05378i	− 0.0680073i
$913$	16.7438i	0.554137i
$914$	14.9187	0.493466
$915$	0	0
$916$	−0.0108651	−0.000358993 0
$917$	− 16.0563i	− 0.530225i
$918$	5.53333i	0.182627i
$919$	−14.5719	−0.480682	−0.240341	−	0.970689i	$-0.577259\pi$
$919$	−14.5719	−0.480682	−0.240341	+	0.970689i	$0.577259\pi$
$920$	0	0
$921$	−15.3063	−0.504358
$922$	− 31.9636i	− 1.05266i
$923$	− 75.1948i	− 2.47507i
$924$	−0.932102	−0.0306639
$925$	0	0
$926$	−8.40813	−0.276308
$927$	− 10.8720i	− 0.357083i
$928$	2.75149i	0.0903220i
$929$	31.3591	1.02886	0.514430	−	0.857532i	$-0.328004\pi$
$929$	31.3591	1.02886	0.514430	+	0.857532i	$0.328004\pi$
$930$	0	0
$931$	−2.52748	−0.0828347
$932$	4.56008i	0.149370i
$933$	12.8545i	0.420838i
$934$	−6.68227	−0.218651
$935$	0	0
$936$	20.7196	0.677242
$937$	− 48.7726i	− 1.59333i	−0.604420	−	0.796666i	$-0.706595\pi$
$937$	− 48.7726i	− 1.59333i	0.604420	−	0.796666i	$-0.293405\pi$
$938$	32.8002i	1.07097i
$939$	10.5072	0.342889
$940$	0	0
$941$	−32.2359	−1.05086	−0.525431	−	0.850836i	$-0.676096\pi$
$941$	−32.2359	−1.05086	−0.525431	+	0.850836i	$0.676096\pi$
$942$	18.3791i	0.598825i
$943$	7.33108i	0.238733i
$944$	3.56086	0.115896
$945$	0	0
$946$	−0.526832	−0.0171288
$947$	− 51.9235i	− 1.68729i	−0.536903	−	0.843644i	$-0.680406\pi$
$947$	− 51.9235i	− 1.68729i	0.536903	−	0.843644i	$-0.319594\pi$
$948$	− 2.55823i	− 0.0830875i
$949$	−39.1935	−1.27227
$950$	0	0
$951$	−19.4806	−0.631703
$952$	− 19.2077i	− 0.622525i
$953$	39.7788i	1.28856i	0.764789	+	0.644281i	$0.222843\pi$
$953$	39.7788i	1.28856i	−0.764789	+	0.644281i	$0.777157\pi$
$954$	0.707341	0.0229010
$955$	0	0
$956$	−3.41703	−0.110515
$957$	− 9.27701i	− 0.299883i
$958$	40.6462i	1.31322i
$959$	−7.77543	−0.251082
$960$	0	0
$961$	−30.0083	−0.968011
$962$	− 33.8460i	− 1.09124i
$963$	9.37236i	0.302020i
$964$	−0.718317	−0.0231354
$965$	0	0
$966$	13.5892	0.437226
$967$	− 19.8817i	− 0.639353i	−0.947527	−	0.319677i	$-0.896426\pi$
$967$	− 19.8817i	− 0.639353i	0.947527	−	0.319677i	$-0.103574\pi$
$968$	0.315428i	0.0101382i
$969$	2.32476	0.0746822
$970$	0	0
$971$	−41.3841	−1.32808	−0.664039	−	0.747698i	$-0.731159\pi$
$971$	−41.3841	−1.32808	−0.664039	+	0.747698i	$0.731159\pi$
$972$	− 0.175259i	− 0.00562144i
$973$	0.296473i	0.00950449i
$974$	46.2658	1.48245
$975$	0	0
$976$	−38.4085	−1.22943
$977$	− 3.61991i	− 0.115811i	−0.998322	−	0.0579056i	$-0.981558\pi$
$977$	− 3.61991i	− 0.115811i	0.998322	−	0.0579056i	$-0.0184422\pi$
$978$	− 11.6562i	− 0.372725i
$979$	−9.41009	−0.300748
$980$	0	0
$981$	−15.5899	−0.497748
$982$	− 14.0747i	− 0.449143i
$983$	− 20.1099i	− 0.641405i	−0.947180	−	0.320702i	$-0.896081\pi$
$983$	− 20.1099i	− 0.641405i	0.947180	−	0.320702i	$-0.103919\pi$
$984$	3.41714	0.108935
$985$	0	0
$986$	15.4024	0.490514
$987$	12.1947i	0.388160i
$988$	− 0.701364i	− 0.0223134i
$989$	−0.737705	−0.0234576
$990$	0	0
$991$	7.43990	0.236336	0.118168	−	0.992994i	$-0.462298\pi$
$991$	7.43990	0.236336	0.118168	+	0.992994i	$0.462298\pi$
$992$	− 0.984345i	− 0.0312530i
$993$	14.4925i	0.459905i
$994$	−22.9877	−0.729126
$995$	0	0
$996$	0.880498	0.0278996
$997$	46.1472i	1.46150i	0.682648	+	0.730748i	$0.260828\pi$
$997$	46.1472i	1.46150i	−0.682648	+	0.730748i	$0.739172\pi$
$998$	− 11.9348i	− 0.377788i
$999$	−3.55334	−0.112423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.h.1249.12		16
5.2	odd	4		1875.2.a.p.1.2		8
5.3	odd	4		1875.2.a.m.1.7		8
5.4	even	2	inner	1875.2.b.h.1249.5		16
15.2	even	4		5625.2.a.t.1.7		8
15.8	even	4		5625.2.a.bd.1.2		8
25.2	odd	20		375.2.g.d.226.1		16
25.9	even	10		75.2.i.a.19.3	yes	16
25.11	even	5		75.2.i.a.4.3	✓	16
25.12	odd	20		375.2.g.d.151.1		16
25.13	odd	20		375.2.g.e.151.4		16
25.14	even	10		375.2.i.c.274.2		16
25.16	even	5		375.2.i.c.349.2		16
25.23	odd	20		375.2.g.e.226.4		16
75.11	odd	10		225.2.m.b.154.2		16
75.59	odd	10		225.2.m.b.19.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.3	✓	16	25.11	even	5
75.2.i.a.19.3	yes	16	25.9	even	10
225.2.m.b.19.2		16	75.59	odd	10
225.2.m.b.154.2		16	75.11	odd	10
375.2.g.d.151.1		16	25.12	odd	20
375.2.g.d.226.1		16	25.2	odd	20
375.2.g.e.151.4		16	25.13	odd	20
375.2.g.e.226.4		16	25.23	odd	20
375.2.i.c.274.2		16	25.14	even	10
375.2.i.c.349.2		16	25.16	even	5
1875.2.a.m.1.7		8	5.3	odd	4
1875.2.a.p.1.2		8	5.2	odd	4
1875.2.b.h.1249.5		16	5.4	even	2	inner
1875.2.b.h.1249.12		16	1.1	even	1	trivial
5625.2.a.t.1.7		8	15.2	even	4
5625.2.a.bd.1.2		8	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)

\(f(q)\)	\(=\)	\(q+1.35083i q^{2} -1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} -1.59580i q^{7} +2.93840i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.35083i q^{2} -1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} -1.59580i q^{7} +2.93840i q^{8} -1.00000 q^{9} +3.33277 q^{11} -0.175259i q^{12} +7.05132i q^{13} +2.15565 q^{14} -3.61877 q^{16} -4.09625i q^{17} -1.35083i q^{18} -0.567535 q^{19} -1.59580 q^{21} +4.50200i q^{22} +6.30400i q^{23} +2.93840 q^{24} -9.52513 q^{26} +1.00000i q^{27} -0.279678i q^{28} +2.78357 q^{29} -0.995824 q^{31} +0.988473i q^{32} -3.33277i q^{33} +5.53333 q^{34} -0.175259 q^{36} +3.55334i q^{37} -0.766643i q^{38} +7.05132 q^{39} +1.16293 q^{41} -2.15565i q^{42} +0.117022i q^{43} +0.584098 q^{44} -8.51563 q^{46} -7.64173i q^{47} +3.61877i q^{48} +4.45343 q^{49} -4.09625 q^{51} +1.23581i q^{52} +0.523635i q^{53} -1.35083 q^{54} +4.68910 q^{56} +0.567535i q^{57} +3.76013i q^{58} -0.983998 q^{59} +10.6137 q^{61} -1.34519i q^{62} +1.59580i q^{63} -8.57279 q^{64} +4.50200 q^{66} +15.2159i q^{67} -0.717905i q^{68} +6.30400 q^{69} -10.6639 q^{71} -2.93840i q^{72} +5.55832i q^{73} -4.79996 q^{74} -0.0994657 q^{76} -5.31842i q^{77} +9.52513i q^{78} +14.5969 q^{79} +1.00000 q^{81} +1.57091i q^{82} +5.02398i q^{83} -0.279678 q^{84} -0.158076 q^{86} -2.78357i q^{87} +9.79302i q^{88} -2.82350 q^{89} +11.2525 q^{91} +1.10483i q^{92} +0.995824i q^{93} +10.3227 q^{94} +0.988473 q^{96} +1.70592i q^{97} +6.01583i q^{98} -3.33277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more