LMFDB - Embedded newform 1875.2.b.h.1249.11

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			1249.11
Root			$-0.0898194i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08982i q^{2} +1.00000i q^{3} +0.812294 q^{4} -1.08982 q^{6} -3.08724i q^{7} +3.06489i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.08982i q^{2} +1.00000i q^{3} +0.812294 q^{4} -1.08982 q^{6} -3.08724i q^{7} +3.06489i q^{8} -1.00000 q^{9} -1.14831 q^{11} +0.812294i q^{12} -4.07954i q^{13} +3.36454 q^{14} -1.71559 q^{16} +4.62758i q^{17} -1.08982i q^{18} +5.96899 q^{19} +3.08724 q^{21} -1.25145i q^{22} +2.32568i q^{23} -3.06489 q^{24} +4.44596 q^{26} -1.00000i q^{27} -2.50775i q^{28} +5.28417 q^{29} -0.589279 q^{31} +4.26010i q^{32} -1.14831i q^{33} -5.04322 q^{34} -0.812294 q^{36} +11.3997i q^{37} +6.50512i q^{38} +4.07954 q^{39} +9.49200 q^{41} +3.36454i q^{42} -2.42954i q^{43} -0.932764 q^{44} -2.53458 q^{46} +6.04998i q^{47} -1.71559i q^{48} -2.53108 q^{49} -4.62758 q^{51} -3.31379i q^{52} +3.24380i q^{53} +1.08982 q^{54} +9.46207 q^{56} +5.96899i q^{57} +5.75879i q^{58} -3.18640 q^{59} +13.7452 q^{61} -0.642208i q^{62} +3.08724i q^{63} -8.07392 q^{64} +1.25145 q^{66} +3.15873i q^{67} +3.75895i q^{68} -2.32568 q^{69} +6.46551 q^{71} -3.06489i q^{72} -7.20998i q^{73} -12.4236 q^{74} +4.84857 q^{76} +3.54511i q^{77} +4.44596i q^{78} +12.3374 q^{79} +1.00000 q^{81} +10.3446i q^{82} -12.3941i q^{83} +2.50775 q^{84} +2.64776 q^{86} +5.28417i q^{87} -3.51944i q^{88} -1.08404 q^{89} -12.5945 q^{91} +1.88914i q^{92} -0.589279i q^{93} -6.59339 q^{94} -4.26010 q^{96} +4.52132i q^{97} -2.75842i q^{98} +1.14831 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.08982i	0.770619i	0.922787	+	0.385309i	$0.125905\pi$
$2$	1.08982i	0.770619i	−0.922787	+	0.385309i	$0.874095\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0.812294	0.406147
$5$	0	0
$5$	0	0
$6$	−1.08982	−0.444917
$7$	− 3.08724i	− 1.16687i	−0.812160	−	0.583434i	$-0.801709\pi$
$7$	− 3.08724i	− 1.16687i	0.812160	−	0.583434i	$-0.198291\pi$
$8$	3.06489i	1.08360i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.14831	−0.346228	−0.173114	−	0.984902i	$-0.555383\pi$
$11$	−1.14831	−0.346228	−0.173114	+	0.984902i	$0.555383\pi$
$12$	0.812294i	0.234489i
$13$	− 4.07954i	− 1.13146i	−0.824590	−	0.565731i	$-0.808594\pi$
$13$	− 4.07954i	− 1.13146i	0.824590	−	0.565731i	$-0.191406\pi$
$14$	3.36454	0.899211
$15$	0	0
$16$	−1.71559	−0.428898
$17$	4.62758i	1.12235i	0.827696	+	0.561176i	$0.189651\pi$
$17$	4.62758i	1.12235i	−0.827696	+	0.561176i	$0.810349\pi$
$18$	− 1.08982i	− 0.256873i
$19$	5.96899	1.36938	0.684690	−	0.728834i	$-0.259938\pi$
$19$	5.96899	1.36938	0.684690	+	0.728834i	$0.259938\pi$
$20$	0	0
$21$	3.08724	0.673692
$22$	− 1.25145i	− 0.266810i
$23$	2.32568i	0.484939i	0.970159	+	0.242469i	$0.0779574\pi$
$23$	2.32568i	0.484939i	−0.970159	+	0.242469i	$0.922043\pi$
$24$	−3.06489	−0.625619
$25$	0	0
$26$	4.44596	0.871925
$27$	− 1.00000i	− 0.192450i
$28$	− 2.50775i	− 0.473920i
$29$	5.28417	0.981245	0.490623	−	0.871372i	$-0.336769\pi$
$29$	5.28417	0.981245	0.490623	+	0.871372i	$0.336769\pi$
$30$	0	0
$31$	−0.589279	−0.105838	−0.0529188	−	0.998599i	$-0.516852\pi$
$31$	−0.589279	−0.105838	−0.0529188	+	0.998599i	$0.516852\pi$
$32$	4.26010i	0.753086i
$33$	− 1.14831i	− 0.199895i
$34$	−5.04322	−0.864906
$35$	0	0
$36$	−0.812294	−0.135382
$37$	11.3997i	1.87409i	0.349204	+	0.937047i	$0.386452\pi$
$37$	11.3997i	1.87409i	−0.349204	+	0.937047i	$0.613548\pi$
$38$	6.50512i	1.05527i
$39$	4.07954	0.653250
$40$	0	0
$41$	9.49200	1.48240	0.741201	−	0.671283i	$-0.234257\pi$
$41$	9.49200	1.48240	0.741201	+	0.671283i	$0.234257\pi$
$42$	3.36454i	0.519160i
$43$	− 2.42954i	− 0.370501i	−0.982691	−	0.185250i	$-0.940690\pi$
$43$	− 2.42954i	− 0.370501i	0.982691	−	0.185250i	$-0.0593096\pi$
$44$	−0.932764	−0.140619
$45$	0	0
$46$	−2.53458	−0.373703
$47$	6.04998i	0.882480i	0.897389	+	0.441240i	$0.145461\pi$
$47$	6.04998i	0.882480i	−0.897389	+	0.441240i	$0.854539\pi$
$48$	− 1.71559i	− 0.247624i
$49$	−2.53108	−0.361583
$50$	0	0
$51$	−4.62758	−0.647990
$52$	− 3.31379i	− 0.459539i
$53$	3.24380i	0.445570i	0.974868	+	0.222785i	$0.0715148\pi$
$53$	3.24380i	0.445570i	−0.974868	+	0.222785i	$0.928485\pi$
$54$	1.08982	0.148306
$55$	0	0
$56$	9.46207	1.26442
$57$	5.96899i	0.790612i
$58$	5.75879i	0.756166i
$59$	−3.18640	−0.414833	−0.207417	−	0.978253i	$-0.566506\pi$
$59$	−3.18640	−0.414833	−0.207417	+	0.978253i	$0.566506\pi$
$60$	0	0
$61$	13.7452	1.75989	0.879946	−	0.475074i	$-0.157579\pi$
$61$	13.7452	1.75989	0.879946	+	0.475074i	$0.157579\pi$
$62$	− 0.642208i	− 0.0815605i
$63$	3.08724i	0.388956i
$64$	−8.07392	−1.00924
$65$	0	0
$66$	1.25145	0.154043
$67$	3.15873i	0.385901i	0.981209	+	0.192950i	$0.0618057\pi$
$67$	3.15873i	0.385901i	−0.981209	+	0.192950i	$0.938194\pi$
$68$	3.75895i	0.455840i
$69$	−2.32568	−0.279979
$70$	0	0
$71$	6.46551	0.767315	0.383657	−	0.923475i	$-0.374664\pi$
$71$	6.46551	0.767315	0.383657	+	0.923475i	$0.374664\pi$
$72$	− 3.06489i	− 0.361201i
$73$	− 7.20998i	− 0.843864i	−0.906627	−	0.421932i	$-0.861352\pi$
$73$	− 7.20998i	− 0.843864i	0.906627	−	0.421932i	$-0.138648\pi$
$74$	−12.4236	−1.44421
$75$	0	0
$76$	4.84857	0.556169
$77$	3.54511i	0.404003i
$78$	4.44596i	0.503406i
$79$	12.3374	1.38807	0.694033	−	0.719944i	$-0.255832\pi$
$79$	12.3374	1.38807	0.694033	+	0.719944i	$0.255832\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.3446i	1.14237i
$83$	− 12.3941i	− 1.36043i	−0.733013	−	0.680214i	$-0.761887\pi$
$83$	− 12.3941i	− 1.36043i	0.733013	−	0.680214i	$-0.238113\pi$
$84$	2.50775	0.273618
$85$	0	0
$86$	2.64776	0.285515
$87$	5.28417i	0.566522i
$88$	− 3.51944i	− 0.375174i
$89$	−1.08404	−0.114907	−0.0574537	−	0.998348i	$-0.518298\pi$
$89$	−1.08404	−0.114907	−0.0574537	+	0.998348i	$0.518298\pi$
$90$	0	0
$91$	−12.5945	−1.32027
$92$	1.88914i	0.196956i
$93$	− 0.589279i	− 0.0611054i
$94$	−6.59339	−0.680056
$95$	0	0
$96$	−4.26010	−0.434795
$97$	4.52132i	0.459071i	0.973300	+	0.229535i	$0.0737206\pi$
$97$	4.52132i	0.459071i	−0.973300	+	0.229535i	$0.926279\pi$
$98$	− 2.75842i	− 0.278642i
$99$	1.14831	0.115409
$100$	0	0
$101$	−6.61332	−0.658050	−0.329025	−	0.944321i	$-0.606720\pi$
$101$	−6.61332	−0.658050	−0.329025	+	0.944321i	$0.606720\pi$
$102$	− 5.04322i	− 0.499354i
$103$	− 4.20634i	− 0.414463i	−0.978292	−	0.207231i	$-0.933555\pi$
$103$	− 4.20634i	− 0.414463i	0.978292	−	0.207231i	$-0.0664453\pi$
$104$	12.5034	1.22606
$105$	0	0
$106$	−3.53515	−0.343364
$107$	− 4.01195i	− 0.387849i	−0.981016	−	0.193925i	$-0.937878\pi$
$107$	− 4.01195i	− 0.387849i	0.981016	−	0.193925i	$-0.0621218\pi$
$108$	− 0.812294i	− 0.0781630i
$109$	−9.09364	−0.871013	−0.435506	−	0.900186i	$-0.643431\pi$
$109$	−9.09364	−0.871013	−0.435506	+	0.900186i	$0.643431\pi$
$110$	0	0
$111$	−11.3997	−1.08201
$112$	5.29645i	0.500468i
$113$	− 3.75465i	− 0.353208i	−0.984282	−	0.176604i	$-0.943489\pi$
$113$	− 3.75465i	− 0.353208i	0.984282	−	0.176604i	$-0.0565111\pi$
$114$	−6.50512	−0.609260
$115$	0	0
$116$	4.29230	0.398530
$117$	4.07954i	0.377154i
$118$	− 3.47260i	− 0.319678i
$119$	14.2865	1.30964
$120$	0	0
$121$	−9.68139	−0.880126
$122$	14.9798i	1.35621i
$123$	9.49200i	0.855865i
$124$	−0.478668	−0.0429856
$125$	0	0
$126$	−3.36454	−0.299737
$127$	11.3583i	1.00788i	0.863738	+	0.503942i	$0.168117\pi$
$127$	11.3583i	1.00788i	−0.863738	+	0.503942i	$0.831883\pi$
$128$	− 0.278920i	− 0.0246532i
$129$	2.42954	0.213909
$130$	0	0
$131$	−2.07849	−0.181599	−0.0907993	−	0.995869i	$-0.528942\pi$
$131$	−2.07849	−0.181599	−0.0907993	+	0.995869i	$0.528942\pi$
$132$	− 0.932764i	− 0.0811867i
$133$	− 18.4277i	− 1.59789i
$134$	−3.44245	−0.297382
$135$	0	0
$136$	−14.1830	−1.21618
$137$	− 19.4032i	− 1.65773i	−0.559452	−	0.828863i	$-0.688988\pi$
$137$	− 19.4032i	− 1.65773i	0.559452	−	0.828863i	$-0.311012\pi$
$138$	− 2.53458i	− 0.215757i
$139$	17.1603	1.45552	0.727758	−	0.685834i	$-0.240562\pi$
$139$	17.1603	1.45552	0.727758	+	0.685834i	$0.240562\pi$
$140$	0	0
$141$	−6.04998	−0.509500
$142$	7.04624i	0.591307i
$143$	4.68458i	0.391744i
$144$	1.71559	0.142966
$145$	0	0
$146$	7.85758	0.650298
$147$	− 2.53108i	− 0.208760i
$148$	9.25988i	0.761157i
$149$	−0.210127	−0.0172143	−0.00860714	−	0.999963i	$-0.502740\pi$
$149$	−0.210127	−0.0172143	−0.00860714	+	0.999963i	$0.502740\pi$
$150$	0	0
$151$	−4.05924	−0.330336	−0.165168	−	0.986265i	$-0.552817\pi$
$151$	−4.05924	−0.330336	−0.165168	+	0.986265i	$0.552817\pi$
$152$	18.2943i	1.48386i
$153$	− 4.62758i	− 0.374117i
$154$	−3.86353	−0.311332
$155$	0	0
$156$	3.31379	0.265315
$157$	0.440336i	0.0351426i	0.999846	+	0.0175713i	$0.00559341\pi$
$157$	0.440336i	0.0351426i	−0.999846	+	0.0175713i	$0.994407\pi$
$158$	13.4455i	1.06967i
$159$	−3.24380	−0.257250
$160$	0	0
$161$	7.17996	0.565860
$162$	1.08982i	0.0856243i
$163$	− 3.24109i	− 0.253862i	−0.991912	−	0.126931i	$-0.959487\pi$
$163$	− 3.24109i	− 0.253862i	0.991912	−	0.126931i	$-0.0405126\pi$
$164$	7.71029	0.602073
$165$	0	0
$166$	13.5073	1.04837
$167$	− 15.9605i	− 1.23506i	−0.786547	−	0.617530i	$-0.788133\pi$
$167$	− 15.9605i	− 1.23506i	0.786547	−	0.617530i	$-0.211867\pi$
$168$	9.46207i	0.730015i
$169$	−3.64267	−0.280205
$170$	0	0
$171$	−5.96899	−0.456460
$172$	− 1.97350i	− 0.150478i
$173$	− 0.561789i	− 0.0427121i	−0.999772	−	0.0213560i	$-0.993202\pi$
$173$	− 0.561789i	− 0.0427121i	0.999772	−	0.0213560i	$-0.00679835\pi$
$174$	−5.75879	−0.436573
$175$	0	0
$176$	1.97003	0.148497
$177$	− 3.18640i	− 0.239504i
$178$	− 1.18140i	− 0.0885499i
$179$	−15.9387	−1.19131	−0.595656	−	0.803239i	$-0.703108\pi$
$179$	−15.9387	−1.19131	−0.595656	+	0.803239i	$0.703108\pi$
$180$	0	0
$181$	−21.4875	−1.59715	−0.798577	−	0.601892i	$-0.794414\pi$
$181$	−21.4875	−1.59715	−0.798577	+	0.601892i	$0.794414\pi$
$182$	− 13.7258i	− 1.01742i
$183$	13.7452i	1.01607i
$184$	−7.12797	−0.525481
$185$	0	0
$186$	0.642208	0.0470890
$187$	− 5.31389i	− 0.388590i
$188$	4.91436i	0.358417i
$189$	−3.08724	−0.224564
$190$	0	0
$191$	−18.3939	−1.33093	−0.665467	−	0.746427i	$-0.731768\pi$
$191$	−18.3939	−1.33093	−0.665467	+	0.746427i	$0.731768\pi$
$192$	− 8.07392i	− 0.582685i
$193$	2.02523i	0.145780i	0.997340	+	0.0728898i	$0.0232221\pi$
$193$	2.02523i	0.145780i	−0.997340	+	0.0728898i	$0.976778\pi$
$194$	−4.92742	−0.353768
$195$	0	0
$196$	−2.05598	−0.146856
$197$	11.5454i	0.822578i	0.911505	+	0.411289i	$0.134921\pi$
$197$	11.5454i	0.822578i	−0.911505	+	0.411289i	$0.865079\pi$
$198$	1.25145i	0.0889367i
$199$	−22.9779	−1.62886	−0.814431	−	0.580260i	$-0.802951\pi$
$199$	−22.9779	−1.62886	−0.814431	+	0.580260i	$0.802951\pi$
$200$	0	0
$201$	−3.15873	−0.222800
$202$	− 7.20733i	− 0.507106i
$203$	− 16.3135i	− 1.14498i
$204$	−3.75895	−0.263179
$205$	0	0
$206$	4.58415	0.319393
$207$	− 2.32568i	− 0.161646i
$208$	6.99883i	0.485282i
$209$	−6.85425	−0.474118
$210$	0	0
$211$	−6.88140	−0.473735	−0.236868	−	0.971542i	$-0.576121\pi$
$211$	−6.88140	−0.473735	−0.236868	+	0.971542i	$0.576121\pi$
$212$	2.63492i	0.180967i
$213$	6.46551i	0.443010i
$214$	4.37230	0.298884
$215$	0	0
$216$	3.06489	0.208540
$217$	1.81925i	0.123499i
$218$	− 9.91043i	− 0.671219i
$219$	7.20998	0.487205
$220$	0	0
$221$	18.8784	1.26990
$222$	− 12.4236i	− 0.833816i
$223$	− 8.12631i	− 0.544178i	−0.962272	−	0.272089i	$-0.912286\pi$
$223$	− 8.12631i	− 0.544178i	0.962272	−	0.272089i	$-0.0877145\pi$
$224$	13.1520	0.878753
$225$	0	0
$226$	4.09189	0.272188
$227$	25.6300i	1.70112i	0.525876	+	0.850561i	$0.323738\pi$
$227$	25.6300i	1.70112i	−0.525876	+	0.850561i	$0.676262\pi$
$228$	4.84857i	0.321105i
$229$	−1.23314	−0.0814884	−0.0407442	−	0.999170i	$-0.512973\pi$
$229$	−1.23314	−0.0814884	−0.0407442	+	0.999170i	$0.512973\pi$
$230$	0	0
$231$	−3.54511	−0.233251
$232$	16.1954i	1.06328i
$233$	2.99987i	0.196528i	0.995160	+	0.0982641i	$0.0313290\pi$
$233$	2.99987i	0.196528i	−0.995160	+	0.0982641i	$0.968671\pi$
$234$	−4.44596	−0.290642
$235$	0	0
$236$	−2.58829	−0.168483
$237$	12.3374i	0.801400i
$238$	15.5697i	1.00923i
$239$	−22.1341	−1.43174	−0.715869	−	0.698235i	$-0.753969\pi$
$239$	−22.1341	−1.43174	−0.715869	+	0.698235i	$0.753969\pi$
$240$	0	0
$241$	−13.4403	−0.865764	−0.432882	−	0.901451i	$-0.642503\pi$
$241$	−13.4403	−0.865764	−0.432882	+	0.901451i	$0.642503\pi$
$242$	− 10.5510i	− 0.678242i
$243$	1.00000i	0.0641500i
$244$	11.1651	0.714774
$245$	0	0
$246$	−10.3446	−0.659546
$247$	− 24.3507i	− 1.54940i
$248$	− 1.80608i	− 0.114686i
$249$	12.3941	0.785444
$250$	0	0
$251$	18.8799	1.19169	0.595843	−	0.803101i	$-0.296818\pi$
$251$	18.8799	1.19169	0.595843	+	0.803101i	$0.296818\pi$
$252$	2.50775i	0.157973i
$253$	− 2.67060i	− 0.167899i
$254$	−12.3785	−0.776694
$255$	0	0
$256$	−15.8439	−0.990242
$257$	7.06320i	0.440590i	0.975433	+	0.220295i	$0.0707020\pi$
$257$	7.06320i	0.440590i	−0.975433	+	0.220295i	$0.929298\pi$
$258$	2.64776i	0.164842i
$259$	35.1936	2.18682
$260$	0	0
$261$	−5.28417	−0.327082
$262$	− 2.26518i	− 0.139943i
$263$	− 20.4356i	− 1.26011i	−0.776549	−	0.630057i	$-0.783031\pi$
$263$	− 20.4356i	− 1.26011i	0.776549	−	0.630057i	$-0.216969\pi$
$264$	3.51944	0.216607
$265$	0	0
$266$	20.0829	1.23136
$267$	− 1.08404i	− 0.0663419i
$268$	2.56582i	0.156732i
$269$	−21.1923	−1.29212	−0.646060	−	0.763287i	$-0.723584\pi$
$269$	−21.1923	−1.29212	−0.646060	+	0.763287i	$0.723584\pi$
$270$	0	0
$271$	9.95317	0.604612	0.302306	−	0.953211i	$-0.402244\pi$
$271$	9.95317	0.604612	0.302306	+	0.953211i	$0.402244\pi$
$272$	− 7.93903i	− 0.481375i
$273$	− 12.5945i	− 0.762257i
$274$	21.1460	1.27747
$275$	0	0
$276$	−1.88914	−0.113713
$277$	− 18.7349i	− 1.12567i	−0.826568	−	0.562837i	$-0.809710\pi$
$277$	− 18.7349i	− 1.12567i	0.826568	−	0.562837i	$-0.190290\pi$
$278$	18.7016i	1.12165i
$279$	0.589279	0.0352792
$280$	0	0
$281$	29.0916	1.73546	0.867730	−	0.497035i	$-0.165578\pi$
$281$	29.0916	1.73546	0.867730	+	0.497035i	$0.165578\pi$
$282$	− 6.59339i	− 0.392630i
$283$	− 7.18007i	− 0.426811i	−0.976964	−	0.213405i	$-0.931545\pi$
$283$	− 7.18007i	− 0.426811i	0.976964	−	0.213405i	$-0.0684555\pi$
$284$	5.25189	0.311643
$285$	0	0
$286$	−5.10534	−0.301885
$287$	− 29.3041i	− 1.72977i
$288$	− 4.26010i	− 0.251029i
$289$	−4.41447	−0.259675
$290$	0	0
$291$	−4.52132	−0.265045
$292$	− 5.85662i	− 0.342733i
$293$	1.79825i	0.105055i	0.998619	+	0.0525276i	$0.0167277\pi$
$293$	1.79825i	0.105055i	−0.998619	+	0.0525276i	$0.983272\pi$
$294$	2.75842	0.160874
$295$	0	0
$296$	−34.9388	−2.03077
$297$	1.14831i	0.0666317i
$298$	− 0.229001i	− 0.0132656i
$299$	9.48773	0.548689
$300$	0	0
$301$	−7.50057	−0.432326
$302$	− 4.42384i	− 0.254563i
$303$	− 6.61332i	− 0.379926i
$304$	−10.2404	−0.587324
$305$	0	0
$306$	5.04322	0.288302
$307$	− 5.98864i	− 0.341790i	−0.985289	−	0.170895i	$-0.945334\pi$
$307$	− 5.98864i	− 0.341790i	0.985289	−	0.170895i	$-0.0546659\pi$
$308$	2.87967i	0.164085i
$309$	4.20634	0.239290
$310$	0	0
$311$	23.8684	1.35345	0.676726	−	0.736235i	$-0.263398\pi$
$311$	23.8684	1.35345	0.676726	+	0.736235i	$0.263398\pi$
$312$	12.5034i	0.707863i
$313$	5.75913i	0.325525i	0.986665	+	0.162763i	$0.0520405\pi$
$313$	5.75913i	0.325525i	−0.986665	+	0.162763i	$0.947959\pi$
$314$	−0.479887	−0.0270816
$315$	0	0
$316$	10.0216	0.563758
$317$	12.2481i	0.687922i	0.938984	+	0.343961i	$0.111769\pi$
$317$	12.2481i	0.687922i	−0.938984	+	0.343961i	$0.888231\pi$
$318$	− 3.53515i	− 0.198242i
$319$	−6.06786	−0.339735
$320$	0	0
$321$	4.01195	0.223925
$322$	7.82486i	0.436062i
$323$	27.6220i	1.53693i
$324$	0.812294	0.0451274
$325$	0	0
$326$	3.53220	0.195631
$327$	− 9.09364i	− 0.502880i
$328$	29.0920i	1.60633i
$329$	18.6778	1.02974
$330$	0	0
$331$	−6.02899	−0.331383	−0.165692	−	0.986178i	$-0.552986\pi$
$331$	−6.02899	−0.331383	−0.165692	+	0.986178i	$0.552986\pi$
$332$	− 10.0676i	− 0.552534i
$333$	− 11.3997i	− 0.624698i
$334$	17.3941	0.951761
$335$	0	0
$336$	−5.29645	−0.288945
$337$	− 5.53083i	− 0.301284i	−0.988588	−	0.150642i	$-0.951866\pi$
$337$	− 5.53083i	− 0.301284i	0.988588	−	0.150642i	$-0.0481340\pi$
$338$	− 3.96985i	− 0.215931i
$339$	3.75465	0.203925
$340$	0	0
$341$	0.676675	0.0366440
$342$	− 6.50512i	− 0.351757i
$343$	− 13.7967i	− 0.744949i
$344$	7.44626	0.401476
$345$	0	0
$346$	0.612249	0.0329147
$347$	25.9445i	1.39277i	0.717666	+	0.696387i	$0.245210\pi$
$347$	25.9445i	1.39277i	−0.717666	+	0.696387i	$0.754790\pi$
$348$	4.29230i	0.230091i
$349$	19.0025	1.01718	0.508591	−	0.861008i	$-0.330167\pi$
$349$	19.0025	1.01718	0.508591	+	0.861008i	$0.330167\pi$
$350$	0	0
$351$	−4.07954	−0.217750
$352$	− 4.89191i	− 0.260740i
$353$	8.19134i	0.435981i	0.975951	+	0.217990i	$0.0699502\pi$
$353$	8.19134i	0.435981i	−0.975951	+	0.217990i	$0.930050\pi$
$354$	3.47260	0.184566
$355$	0	0
$356$	−0.880555	−0.0466693
$357$	14.2865i	0.756120i
$358$	− 17.3703i	− 0.918048i
$359$	26.7857	1.41369	0.706847	−	0.707367i	$-0.250117\pi$
$359$	26.7857	1.41369	0.706847	+	0.707367i	$0.250117\pi$
$360$	0	0
$361$	16.6288	0.875202
$362$	− 23.4175i	− 1.23080i
$363$	− 9.68139i	− 0.508141i
$364$	−10.2305	−0.536222
$365$	0	0
$366$	−14.9798	−0.783006
$367$	− 18.8533i	− 0.984135i	−0.870557	−	0.492068i	$-0.836241\pi$
$367$	− 18.8533i	− 0.984135i	0.870557	−	0.492068i	$-0.163759\pi$
$368$	− 3.98992i	− 0.207989i
$369$	−9.49200	−0.494134
$370$	0	0
$371$	10.0144	0.519921
$372$	− 0.478668i	− 0.0248178i
$373$	− 17.8477i	− 0.924118i	−0.886849	−	0.462059i	$-0.847111\pi$
$373$	− 17.8477i	− 0.924118i	0.886849	−	0.462059i	$-0.152889\pi$
$374$	5.79118	0.299455
$375$	0	0
$376$	−18.5425	−0.956259
$377$	− 21.5570i	− 1.11024i
$378$	− 3.36454i	− 0.173053i
$379$	−12.5653	−0.645438	−0.322719	−	0.946495i	$-0.604597\pi$
$379$	−12.5653	−0.645438	−0.322719	+	0.946495i	$0.604597\pi$
$380$	0	0
$381$	−11.3583	−0.581902
$382$	− 20.0460i	− 1.02564i
$383$	7.53739i	0.385143i	0.981283	+	0.192571i	$0.0616827\pi$
$383$	7.53739i	0.385143i	−0.981283	+	0.192571i	$0.938317\pi$
$384$	0.278920	0.0142336
$385$	0	0
$386$	−2.20714	−0.112340
$387$	2.42954i	0.123500i
$388$	3.67264i	0.186450i
$389$	13.7331	0.696296	0.348148	−	0.937440i	$-0.386811\pi$
$389$	13.7331	0.696296	0.348148	+	0.937440i	$0.386811\pi$
$390$	0	0
$391$	−10.7623	−0.544272
$392$	− 7.75749i	− 0.391812i
$393$	− 2.07849i	− 0.104846i
$394$	−12.5824	−0.633894
$395$	0	0
$396$	0.932764	0.0468732
$397$	− 13.0022i	− 0.652562i	−0.945273	−	0.326281i	$-0.894204\pi$
$397$	− 13.0022i	− 0.652562i	0.945273	−	0.326281i	$-0.105796\pi$
$398$	− 25.0418i	− 1.25523i
$399$	18.4277	0.922540
$400$	0	0
$401$	−6.47047	−0.323120	−0.161560	−	0.986863i	$-0.551653\pi$
$401$	−6.47047	−0.323120	−0.161560	+	0.986863i	$0.551653\pi$
$402$	− 3.44245i	− 0.171694i
$403$	2.40399i	0.119751i
$404$	−5.37196	−0.267265
$405$	0	0
$406$	17.7788	0.882346
$407$	− 13.0903i	− 0.648864i
$408$	− 14.1830i	− 0.702164i
$409$	1.11200	0.0549850	0.0274925	−	0.999622i	$-0.491248\pi$
$409$	1.11200	0.0549850	0.0274925	+	0.999622i	$0.491248\pi$
$410$	0	0
$411$	19.4032	0.957088
$412$	− 3.41678i	− 0.168333i
$413$	9.83718i	0.484056i
$414$	2.53458	0.124568
$415$	0	0
$416$	17.3793	0.852088
$417$	17.1603i	0.840343i
$418$	− 7.46989i	− 0.365364i
$419$	−11.7932	−0.576136	−0.288068	−	0.957610i	$-0.593013\pi$
$419$	−11.7932	−0.576136	−0.288068	+	0.957610i	$0.593013\pi$
$420$	0	0
$421$	−10.1257	−0.493494	−0.246747	−	0.969080i	$-0.579362\pi$
$421$	−10.1257	−0.493494	−0.246747	+	0.969080i	$0.579362\pi$
$422$	− 7.49949i	− 0.365069i
$423$	− 6.04998i	− 0.294160i
$424$	−9.94189	−0.482821
$425$	0	0
$426$	−7.04624	−0.341391
$427$	− 42.4348i	− 2.05356i
$428$	− 3.25888i	− 0.157524i
$429$	−4.68458	−0.226173
$430$	0	0
$431$	−5.69259	−0.274202	−0.137101	−	0.990557i	$-0.543779\pi$
$431$	−5.69259	−0.274202	−0.137101	+	0.990557i	$0.543779\pi$
$432$	1.71559i	0.0825415i
$433$	33.0488i	1.58822i	0.607771	+	0.794112i	$0.292064\pi$
$433$	33.0488i	1.58822i	−0.607771	+	0.794112i	$0.707936\pi$
$434$	−1.98265	−0.0951704
$435$	0	0
$436$	−7.38671	−0.353759
$437$	13.8820i	0.664065i
$438$	7.85758i	0.375449i
$439$	−11.4637	−0.547131	−0.273566	−	0.961853i	$-0.588203\pi$
$439$	−11.4637	−0.547131	−0.273566	+	0.961853i	$0.588203\pi$
$440$	0	0
$441$	2.53108	0.120528
$442$	20.5740i	0.978607i
$443$	− 17.8993i	− 0.850422i	−0.905094	−	0.425211i	$-0.860200\pi$
$443$	− 17.8993i	− 0.850422i	0.905094	−	0.425211i	$-0.139800\pi$
$444$	−9.25988	−0.439454
$445$	0	0
$446$	8.85621	0.419353
$447$	− 0.210127i	− 0.00993867i
$448$	24.9262i	1.17765i
$449$	6.82040	0.321874	0.160937	−	0.986965i	$-0.448548\pi$
$449$	6.82040	0.321874	0.160937	+	0.986965i	$0.448548\pi$
$450$	0	0
$451$	−10.8998	−0.513249
$452$	− 3.04988i	− 0.143454i
$453$	− 4.05924i	− 0.190720i
$454$	−27.9321	−1.31092
$455$	0	0
$456$	−18.2943	−0.856710
$457$	2.76381i	0.129286i	0.997908	+	0.0646429i	$0.0205908\pi$
$457$	2.76381i	0.129286i	−0.997908	+	0.0646429i	$0.979409\pi$
$458$	− 1.34390i	− 0.0627965i
$459$	4.62758	0.215997
$460$	0	0
$461$	−8.96463	−0.417525	−0.208762	−	0.977966i	$-0.566943\pi$
$461$	−8.96463	−0.417525	−0.208762	+	0.977966i	$0.566943\pi$
$462$	− 3.86353i	− 0.179748i
$463$	− 13.1617i	− 0.611675i	−0.952084	−	0.305837i	$-0.901064\pi$
$463$	− 13.1617i	− 0.611675i	0.952084	−	0.305837i	$-0.0989364\pi$
$464$	−9.06547	−0.420854
$465$	0	0
$466$	−3.26932	−0.151448
$467$	− 21.8005i	− 1.00880i	−0.863469	−	0.504402i	$-0.831713\pi$
$467$	− 21.8005i	− 1.00880i	0.863469	−	0.504402i	$-0.168287\pi$
$468$	3.31379i	0.153180i
$469$	9.75179	0.450296
$470$	0	0
$471$	−0.440336	−0.0202896
$472$	− 9.76596i	− 0.449515i
$473$	2.78986i	0.128278i
$474$	−13.4455	−0.617574
$475$	0	0
$476$	11.6048	0.531905
$477$	− 3.24380i	− 0.148523i
$478$	− 24.1222i	− 1.10332i
$479$	−42.7490	−1.95325	−0.976626	−	0.214947i	$-0.931042\pi$
$479$	−42.7490	−1.95325	−0.976626	+	0.214947i	$0.931042\pi$
$480$	0	0
$481$	46.5054	2.12046
$482$	− 14.6475i	− 0.667174i
$483$	7.17996i	0.326699i
$484$	−7.86413	−0.357460
$485$	0	0
$486$	−1.08982	−0.0494352
$487$	11.3497i	0.514305i	0.966371	+	0.257153i	$0.0827843\pi$
$487$	11.3497i	0.514305i	−0.966371	+	0.257153i	$0.917216\pi$
$488$	42.1275i	1.90702i
$489$	3.24109	0.146567
$490$	0	0
$491$	16.6546	0.751614	0.375807	−	0.926698i	$-0.377366\pi$
$491$	16.6546	0.751614	0.375807	+	0.926698i	$0.377366\pi$
$492$	7.71029i	0.347607i
$493$	24.4529i	1.10130i
$494$	26.5379	1.19400
$495$	0	0
$496$	1.01096	0.0453936
$497$	− 19.9606i	− 0.895356i
$498$	13.5073i	0.605278i
$499$	12.2321	0.547584	0.273792	−	0.961789i	$-0.411722\pi$
$499$	12.2321	0.547584	0.273792	+	0.961789i	$0.411722\pi$
$500$	0	0
$501$	15.9605	0.713063
$502$	20.5756i	0.918336i
$503$	− 1.30967i	− 0.0583951i	−0.999574	−	0.0291976i	$-0.990705\pi$
$503$	− 1.30967i	− 0.0583951i	0.999574	−	0.0291976i	$-0.00929519\pi$
$504$	−9.46207	−0.421474
$505$	0	0
$506$	2.91048	0.129386
$507$	− 3.64267i	− 0.161777i
$508$	9.22625i	0.409349i
$509$	−17.7318	−0.785946	−0.392973	−	0.919550i	$-0.628553\pi$
$509$	−17.7318	−0.785946	−0.392973	+	0.919550i	$0.628553\pi$
$510$	0	0
$511$	−22.2590	−0.984679
$512$	− 17.8248i	− 0.787752i
$513$	− 5.96899i	− 0.263537i
$514$	−7.69761	−0.339527
$515$	0	0
$516$	1.97350	0.0868783
$517$	− 6.94725i	− 0.305540i
$518$	38.3546i	1.68521i
$519$	0.561789	0.0246598
$520$	0	0
$521$	26.2616	1.15054	0.575270	−	0.817964i	$-0.304897\pi$
$521$	26.2616	1.15054	0.575270	+	0.817964i	$0.304897\pi$
$522$	− 5.75879i	− 0.252055i
$523$	− 2.55674i	− 0.111798i	−0.998436	−	0.0558991i	$-0.982197\pi$
$523$	− 2.55674i	− 0.111798i	0.998436	−	0.0558991i	$-0.0178025\pi$
$524$	−1.68834	−0.0737557
$525$	0	0
$526$	22.2711	0.971068
$527$	− 2.72693i	− 0.118787i
$528$	1.97003i	0.0857346i
$529$	17.5912	0.764835
$530$	0	0
$531$	3.18640	0.138278
$532$	− 14.9687i	− 0.648977i
$533$	− 38.7230i	− 1.67728i
$534$	1.18140	0.0511243
$535$	0	0
$536$	−9.68118	−0.418163
$537$	− 15.9387i	− 0.687805i
$538$	− 23.0958i	− 0.995732i
$539$	2.90646	0.125190
$540$	0	0
$541$	9.38916	0.403672	0.201836	−	0.979419i	$-0.435309\pi$
$541$	9.38916	0.403672	0.201836	+	0.979419i	$0.435309\pi$
$542$	10.8472i	0.465925i
$543$	− 21.4875i	− 0.922118i
$544$	−19.7139	−0.845228
$545$	0	0
$546$	13.7258	0.587409
$547$	− 34.8549i	− 1.49029i	−0.666903	−	0.745144i	$-0.732381\pi$
$547$	− 34.8549i	− 1.49029i	0.666903	−	0.745144i	$-0.267619\pi$
$548$	− 15.7611i	− 0.673280i
$549$	−13.7452	−0.586631
$550$	0	0
$551$	31.5411	1.34370
$552$	− 7.12797i	− 0.303387i
$553$	− 38.0886i	− 1.61969i
$554$	20.4177	0.867465
$555$	0	0
$556$	13.9392	0.591153
$557$	− 14.1466i	− 0.599411i	−0.954032	−	0.299705i	$-0.903112\pi$
$557$	− 14.1466i	− 0.599411i	0.954032	−	0.299705i	$-0.0968884\pi$
$558$	0.642208i	0.0271868i
$559$	−9.91139	−0.419207
$560$	0	0
$561$	5.31389	0.224353
$562$	31.7046i	1.33738i
$563$	− 38.1854i	− 1.60932i	−0.593734	−	0.804661i	$-0.702347\pi$
$563$	− 38.1854i	− 1.60932i	0.593734	−	0.804661i	$-0.297653\pi$
$564$	−4.91436	−0.206932
$565$	0	0
$566$	7.82498	0.328908
$567$	− 3.08724i	− 0.129652i
$568$	19.8161i	0.831465i
$569$	−29.6301	−1.24216	−0.621080	−	0.783747i	$-0.713306\pi$
$569$	−29.6301	−1.24216	−0.621080	+	0.783747i	$0.713306\pi$
$570$	0	0
$571$	32.6504	1.36638	0.683188	−	0.730242i	$-0.260593\pi$
$571$	32.6504	1.36638	0.683188	+	0.730242i	$0.260593\pi$
$572$	3.80525i	0.159106i
$573$	− 18.3939i	− 0.768415i
$574$	31.9362	1.33299
$575$	0	0
$576$	8.07392	0.336413
$577$	22.6433i	0.942653i	0.881959	+	0.471326i	$0.156225\pi$
$577$	22.6433i	0.942653i	−0.881959	+	0.471326i	$0.843775\pi$
$578$	− 4.81097i	− 0.200110i
$579$	−2.02523	−0.0841659
$580$	0	0
$581$	−38.2636	−1.58744
$582$	− 4.92742i	− 0.204248i
$583$	− 3.72488i	− 0.154269i
$584$	22.0978	0.914414
$585$	0	0
$586$	−1.95977	−0.0809575
$587$	41.5667i	1.71564i	0.513950	+	0.857820i	$0.328182\pi$
$587$	41.5667i	1.71564i	−0.513950	+	0.857820i	$0.671818\pi$
$588$	− 2.05598i	− 0.0847872i
$589$	−3.51740	−0.144932
$590$	0	0
$591$	−11.5454	−0.474916
$592$	− 19.5572i	− 0.803795i
$593$	− 2.09050i	− 0.0858465i	−0.999078	−	0.0429233i	$-0.986333\pi$
$593$	− 2.09050i	− 0.0858465i	0.999078	−	0.0429233i	$-0.0136671\pi$
$594$	−1.25145	−0.0513476
$595$	0	0
$596$	−0.170685	−0.00699152
$597$	− 22.9779i	− 0.940424i
$598$	10.3399i	0.422830i
$599$	−17.9768	−0.734511	−0.367255	−	0.930120i	$-0.619702\pi$
$599$	−17.9768	−0.734511	−0.367255	+	0.930120i	$0.619702\pi$
$600$	0	0
$601$	−1.11000	−0.0452778	−0.0226389	−	0.999744i	$-0.507207\pi$
$601$	−1.11000	−0.0452778	−0.0226389	+	0.999744i	$0.507207\pi$
$602$	− 8.17427i	− 0.333158i
$603$	− 3.15873i	− 0.128634i
$604$	−3.29730	−0.134165
$605$	0	0
$606$	7.20733	0.292778
$607$	− 12.2310i	− 0.496441i	−0.968704	−	0.248220i	$-0.920154\pi$
$607$	− 12.2310i	− 0.496441i	0.968704	−	0.248220i	$-0.0798457\pi$
$608$	25.4285i	1.03126i
$609$	16.3135	0.661057
$610$	0	0
$611$	24.6812	0.998493
$612$	− 3.75895i	− 0.151947i
$613$	12.9776i	0.524160i	0.965046	+	0.262080i	$0.0844085\pi$
$613$	12.9776i	0.524160i	−0.965046	+	0.262080i	$0.915591\pi$
$614$	6.52654	0.263390
$615$	0	0
$616$	−10.8654	−0.437779
$617$	20.4921i	0.824981i	0.910962	+	0.412490i	$0.135341\pi$
$617$	20.4921i	0.824981i	−0.910962	+	0.412490i	$0.864659\pi$
$618$	4.58415i	0.184402i
$619$	41.0456	1.64976	0.824881	−	0.565307i	$-0.191242\pi$
$619$	41.0456	1.64976	0.824881	+	0.565307i	$0.191242\pi$
$620$	0	0
$621$	2.32568	0.0933265
$622$	26.0122i	1.04300i
$623$	3.34668i	0.134082i
$624$	−6.99883	−0.280177
$625$	0	0
$626$	−6.27641	−0.250856
$627$	− 6.85425i	− 0.273732i
$628$	0.357682i	0.0142731i
$629$	−52.7528	−2.10339
$630$	0	0
$631$	14.0872	0.560804	0.280402	−	0.959883i	$-0.409532\pi$
$631$	14.0872	0.560804	0.280402	+	0.959883i	$0.409532\pi$
$632$	37.8128i	1.50411i
$633$	− 6.88140i	− 0.273511i
$634$	−13.3482	−0.530126
$635$	0	0
$636$	−2.63492	−0.104481
$637$	10.3256i	0.409117i
$638$	− 6.61287i	− 0.261806i
$639$	−6.46551	−0.255772
$640$	0	0
$641$	−28.4358	−1.12315	−0.561573	−	0.827427i	$-0.689803\pi$
$641$	−28.4358	−1.12315	−0.561573	+	0.827427i	$0.689803\pi$
$642$	4.37230i	0.172561i
$643$	10.8408i	0.427521i	0.976886	+	0.213761i	$0.0685713\pi$
$643$	10.8408i	0.427521i	−0.976886	+	0.213761i	$0.931429\pi$
$644$	5.83223	0.229822
$645$	0	0
$646$	−30.1029	−1.18438
$647$	− 34.6388i	− 1.36179i	−0.732380	−	0.680896i	$-0.761590\pi$
$647$	− 34.6388i	− 1.36179i	0.732380	−	0.680896i	$-0.238410\pi$
$648$	3.06489i	0.120400i
$649$	3.65897	0.143627
$650$	0	0
$651$	−1.81925	−0.0713020
$652$	− 2.63272i	− 0.103105i
$653$	4.15506i	0.162600i	0.996690	+	0.0813000i	$0.0259072\pi$
$653$	4.15506i	0.162600i	−0.996690	+	0.0813000i	$0.974093\pi$
$654$	9.91043	0.387528
$655$	0	0
$656$	−16.2844	−0.635799
$657$	7.20998i	0.281288i
$658$	20.3554i	0.793536i
$659$	4.01424	0.156373	0.0781863	−	0.996939i	$-0.475087\pi$
$659$	4.01424	0.156373	0.0781863	+	0.996939i	$0.475087\pi$
$660$	0	0
$661$	19.6667	0.764944	0.382472	−	0.923967i	$-0.375073\pi$
$661$	19.6667	0.764944	0.382472	+	0.923967i	$0.375073\pi$
$662$	− 6.57051i	− 0.255370i
$663$	18.8784i	0.733176i
$664$	37.9866	1.47416
$665$	0	0
$666$	12.4236	0.481404
$667$	12.2893i	0.475844i
$668$	− 12.9646i	− 0.501616i
$669$	8.12631	0.314181
$670$	0	0
$671$	−15.7837	−0.609324
$672$	13.1520i	0.507348i
$673$	41.4463i	1.59764i	0.601571	+	0.798819i	$0.294542\pi$
$673$	41.4463i	1.59764i	−0.601571	+	0.798819i	$0.705458\pi$
$674$	6.02761	0.232175
$675$	0	0
$676$	−2.95892	−0.113804
$677$	18.1837i	0.698856i	0.936963	+	0.349428i	$0.113624\pi$
$677$	18.1837i	0.698856i	−0.936963	+	0.349428i	$0.886376\pi$
$678$	4.09189i	0.157148i
$679$	13.9584	0.535675
$680$	0	0
$681$	−25.6300	−0.982144
$682$	0.737453i	0.0282385i
$683$	− 28.4523i	− 1.08870i	−0.838859	−	0.544348i	$-0.816777\pi$
$683$	− 28.4523i	− 1.08870i	0.838859	−	0.544348i	$-0.183223\pi$
$684$	−4.84857	−0.185390
$685$	0	0
$686$	15.0359	0.574072
$687$	− 1.23314i	− 0.0470474i
$688$	4.16809i	0.158907i
$689$	13.2332	0.504145
$690$	0	0
$691$	24.4175	0.928884	0.464442	−	0.885603i	$-0.346255\pi$
$691$	24.4175	0.928884	0.464442	+	0.885603i	$0.346255\pi$
$692$	− 0.456338i	− 0.0173474i
$693$	− 3.54511i	− 0.134668i
$694$	−28.2748	−1.07330
$695$	0	0
$696$	−16.1954	−0.613885
$697$	43.9250i	1.66378i
$698$	20.7093i	0.783859i
$699$	−2.99987	−0.113466
$700$	0	0
$701$	−25.0371	−0.945638	−0.472819	−	0.881159i	$-0.656764\pi$
$701$	−25.0371	−0.945638	−0.472819	+	0.881159i	$0.656764\pi$
$702$	− 4.44596i	− 0.167802i
$703$	68.0445i	2.56635i
$704$	9.27136	0.349428
$705$	0	0
$706$	−8.92708	−0.335975
$707$	20.4170i	0.767858i
$708$	− 2.58829i	− 0.0972738i
$709$	−36.5633	−1.37316	−0.686582	−	0.727052i	$-0.740890\pi$
$709$	−36.5633	−1.37316	−0.686582	+	0.727052i	$0.740890\pi$
$710$	0	0
$711$	−12.3374	−0.462688
$712$	− 3.32245i	− 0.124514i
$713$	− 1.37048i	− 0.0513248i
$714$	−15.5697	−0.582680
$715$	0	0
$716$	−12.9469	−0.483848
$717$	− 22.1341i	− 0.826614i
$718$	29.1915i	1.08942i
$719$	7.23571	0.269847	0.134923	−	0.990856i	$-0.456921\pi$
$719$	7.23571	0.269847	0.134923	+	0.990856i	$0.456921\pi$
$720$	0	0
$721$	−12.9860	−0.483624
$722$	18.1224i	0.674447i
$723$	− 13.4403i	− 0.499849i
$724$	−17.4542	−0.648679
$725$	0	0
$726$	10.5510	0.391583
$727$	13.3099i	0.493638i	0.969062	+	0.246819i	$0.0793852\pi$
$727$	13.3099i	0.493638i	−0.969062	+	0.246819i	$0.920615\pi$
$728$	− 38.6009i	− 1.43065i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	11.2429	0.415832
$732$	11.1651i	0.412675i
$733$	28.7794i	1.06299i	0.847061	+	0.531496i	$0.178370\pi$
$733$	28.7794i	1.06299i	−0.847061	+	0.531496i	$0.821630\pi$
$734$	20.5467	0.758393
$735$	0	0
$736$	−9.90764	−0.365201
$737$	− 3.62720i	− 0.133610i
$738$	− 10.3446i	− 0.380789i
$739$	−40.3208	−1.48322	−0.741612	−	0.670829i	$-0.765939\pi$
$739$	−40.3208	−1.48322	−0.741612	+	0.670829i	$0.765939\pi$
$740$	0	0
$741$	24.3507	0.894547
$742$	10.9139i	0.400661i
$743$	− 27.8114i	− 1.02030i	−0.860085	−	0.510151i	$-0.829589\pi$
$743$	− 27.8114i	− 1.02030i	0.860085	−	0.510151i	$-0.170411\pi$
$744$	1.80608	0.0662140
$745$	0	0
$746$	19.4507	0.712142
$747$	12.3941i	0.453476i
$748$	− 4.31644i	− 0.157825i
$749$	−12.3859	−0.452569
$750$	0	0
$751$	−11.4124	−0.416443	−0.208221	−	0.978082i	$-0.566767\pi$
$751$	−11.4124	−0.416443	−0.208221	+	0.978082i	$0.566767\pi$
$752$	− 10.3793i	− 0.378494i
$753$	18.8799i	0.688021i
$754$	23.4932	0.855573
$755$	0	0
$756$	−2.50775	−0.0912060
$757$	13.7637i	0.500251i	0.968213	+	0.250126i	$0.0804719\pi$
$757$	13.7637i	0.500251i	−0.968213	+	0.250126i	$0.919528\pi$
$758$	− 13.6939i	− 0.497386i
$759$	2.67060	0.0969368
$760$	0	0
$761$	−14.7409	−0.534359	−0.267179	−	0.963647i	$-0.586092\pi$
$761$	−14.7409	−0.534359	−0.267179	+	0.963647i	$0.586092\pi$
$762$	− 12.3785i	− 0.448425i
$763$	28.0743i	1.01636i
$764$	−14.9412	−0.540555
$765$	0	0
$766$	−8.21440	−0.296798
$767$	12.9990i	0.469368i
$768$	− 15.8439i	− 0.571717i
$769$	2.43753	0.0878997	0.0439498	−	0.999034i	$-0.486006\pi$
$769$	2.43753	0.0878997	0.0439498	+	0.999034i	$0.486006\pi$
$770$	0	0
$771$	−7.06320	−0.254375
$772$	1.64509i	0.0592079i
$773$	− 25.3341i	− 0.911204i	−0.890184	−	0.455602i	$-0.849424\pi$
$773$	− 25.3341i	− 0.911204i	0.890184	−	0.455602i	$-0.150576\pi$
$774$	−2.64776	−0.0951716
$775$	0	0
$776$	−13.8574	−0.497450
$777$	35.1936i	1.26256i
$778$	14.9666i	0.536579i
$779$	56.6577	2.02997
$780$	0	0
$781$	−7.42441	−0.265666
$782$	− 11.7289i	− 0.419426i
$783$	− 5.28417i	− 0.188841i
$784$	4.34230	0.155082
$785$	0	0
$786$	2.26518	0.0807963
$787$	− 38.6480i	− 1.37765i	−0.724927	−	0.688826i	$-0.758127\pi$
$787$	− 38.6480i	− 1.37765i	0.724927	−	0.688826i	$-0.241873\pi$
$788$	9.37829i	0.334088i
$789$	20.4356	0.727527
$790$	0	0
$791$	−11.5915	−0.412147
$792$	3.51944i	0.125058i
$793$	− 56.0741i	− 1.99125i
$794$	14.1700	0.502876
$795$	0	0
$796$	−18.6648	−0.661557
$797$	52.1020i	1.84555i	0.385342	+	0.922774i	$0.374083\pi$
$797$	52.1020i	1.84555i	−0.385342	+	0.922774i	$0.625917\pi$
$798$	20.0829i	0.710927i
$799$	−27.9968	−0.990454
$800$	0	0
$801$	1.08404	0.0383025
$802$	− 7.05165i	− 0.249002i
$803$	8.27929i	0.292170i
$804$	−2.56582	−0.0904895
$805$	0	0
$806$	−2.61991	−0.0922825
$807$	− 21.1923i	− 0.746006i
$808$	− 20.2691i	− 0.713065i
$809$	−31.7938	−1.11781	−0.558905	−	0.829231i	$-0.688778\pi$
$809$	−31.7938	−1.11781	−0.558905	+	0.829231i	$0.688778\pi$
$810$	0	0
$811$	−31.1213	−1.09282	−0.546409	−	0.837518i	$-0.684006\pi$
$811$	−31.1213	−1.09282	−0.546409	+	0.837518i	$0.684006\pi$
$812$	− 13.2514i	− 0.465032i
$813$	9.95317i	0.349073i
$814$	14.2661	0.500027
$815$	0	0
$816$	7.93903	0.277922
$817$	− 14.5019i	− 0.507356i
$818$	1.21188i	0.0423725i
$819$	12.5945	0.440089
$820$	0	0
$821$	−47.0263	−1.64123	−0.820615	−	0.571482i	$-0.806369\pi$
$821$	−47.0263	−1.64123	−0.820615	+	0.571482i	$0.806369\pi$
$822$	21.1460i	0.737550i
$823$	− 28.3425i	− 0.987959i	−0.869474	−	0.493979i	$-0.835542\pi$
$823$	− 28.3425i	− 0.987959i	0.869474	−	0.493979i	$-0.164458\pi$
$824$	12.8920	0.449113
$825$	0	0
$826$	−10.7208	−0.373023
$827$	− 14.1680i	− 0.492670i	−0.969185	−	0.246335i	$-0.920774\pi$
$827$	− 14.1680i	− 0.492670i	0.969185	−	0.246335i	$-0.0792264\pi$
$828$	− 1.88914i	− 0.0656521i
$829$	−1.51657	−0.0526728	−0.0263364	−	0.999653i	$-0.508384\pi$
$829$	−1.51657	−0.0526728	−0.0263364	+	0.999653i	$0.508384\pi$
$830$	0	0
$831$	18.7349	0.649908
$832$	32.9379i	1.14192i
$833$	− 11.7128i	− 0.405823i
$834$	−18.7016	−0.647584
$835$	0	0
$836$	−5.56766	−0.192562
$837$	0.589279i	0.0203685i
$838$	− 12.8525i	− 0.443981i
$839$	−10.4919	−0.362220	−0.181110	−	0.983463i	$-0.557969\pi$
$839$	−10.4919	−0.362220	−0.181110	+	0.983463i	$0.557969\pi$
$840$	0	0
$841$	−1.07758	−0.0371578
$842$	− 11.0351i	− 0.380296i
$843$	29.0916i	1.00197i
$844$	−5.58972	−0.192406
$845$	0	0
$846$	6.59339	0.226685
$847$	29.8888i	1.02699i
$848$	− 5.56503i	− 0.191104i
$849$	7.18007	0.246419
$850$	0	0
$851$	−26.5120	−0.908820
$852$	5.25189i	0.179927i
$853$	− 14.8571i	− 0.508698i	−0.967112	−	0.254349i	$-0.918139\pi$
$853$	− 14.8571i	− 0.508698i	0.967112	−	0.254349i	$-0.0818612\pi$
$854$	46.2462	1.58251
$855$	0	0
$856$	12.2962	0.420275
$857$	26.4088i	0.902109i	0.892496	+	0.451054i	$0.148952\pi$
$857$	26.4088i	0.902109i	−0.892496	+	0.451054i	$0.851048\pi$
$858$	− 5.10534i	− 0.174294i
$859$	9.13013	0.311516	0.155758	−	0.987795i	$-0.450218\pi$
$859$	9.13013	0.311516	0.155758	+	0.987795i	$0.450218\pi$
$860$	0	0
$861$	29.3041	0.998682
$862$	− 6.20389i	− 0.211305i
$863$	12.4796i	0.424810i	0.977182	+	0.212405i	$0.0681297\pi$
$863$	12.4796i	0.424810i	−0.977182	+	0.212405i	$0.931870\pi$
$864$	4.26010	0.144932
$865$	0	0
$866$	−36.0172	−1.22392
$867$	− 4.41447i	− 0.149923i
$868$	1.47776i	0.0501586i
$869$	−14.1671	−0.480587
$870$	0	0
$871$	12.8862	0.436632
$872$	− 27.8710i	− 0.943832i
$873$	− 4.52132i	− 0.153024i
$874$	−15.1289	−0.511741
$875$	0	0
$876$	5.85662	0.197877
$877$	40.8121i	1.37813i	0.724701	+	0.689064i	$0.241978\pi$
$877$	40.8121i	1.37813i	−0.724701	+	0.689064i	$0.758022\pi$
$878$	− 12.4933i	− 0.421629i
$879$	−1.79825	−0.0606536
$880$	0	0
$881$	−3.64199	−0.122702	−0.0613509	−	0.998116i	$-0.519541\pi$
$881$	−3.64199	−0.122702	−0.0613509	+	0.998116i	$0.519541\pi$
$882$	2.75842i	0.0928808i
$883$	− 0.332855i	− 0.0112015i	−0.999984	−	0.00560073i	$-0.998217\pi$
$883$	− 0.332855i	− 0.0112015i	0.999984	−	0.00560073i	$-0.00178278\pi$
$884$	15.3348	0.515765
$885$	0	0
$886$	19.5070	0.655351
$887$	− 1.14274i	− 0.0383694i	−0.999816	−	0.0191847i	$-0.993893\pi$
$887$	− 1.14274i	− 0.0383694i	0.999816	−	0.0191847i	$-0.00610706\pi$
$888$	− 34.9388i	− 1.17247i
$889$	35.0658	1.17607
$890$	0	0
$891$	−1.14831	−0.0384698
$892$	− 6.60095i	− 0.221016i
$893$	36.1123i	1.20845i
$894$	0.229001	0.00765892
$895$	0	0
$896$	−0.861093	−0.0287671
$897$	9.48773i	0.316786i
$898$	7.43300i	0.248042i
$899$	−3.11385	−0.103853
$900$	0	0
$901$	−15.0109	−0.500086
$902$	− 11.8788i	− 0.395520i
$903$	− 7.50057i	− 0.249603i
$904$	11.5076	0.382737
$905$	0	0
$906$	4.42384	0.146972
$907$	− 18.4202i	− 0.611632i	−0.952091	−	0.305816i	$-0.901071\pi$
$907$	− 18.4202i	− 0.611632i	0.952091	−	0.305816i	$-0.0989293\pi$
$908$	20.8191i	0.690906i
$909$	6.61332	0.219350
$910$	0	0
$911$	18.2506	0.604670	0.302335	−	0.953202i	$-0.402234\pi$
$911$	18.2506	0.604670	0.302335	+	0.953202i	$0.402234\pi$
$912$	− 10.2404i	− 0.339092i
$913$	14.2323i	0.471019i
$914$	−3.01206	−0.0996300
$915$	0	0
$916$	−1.00167	−0.0330963
$917$	6.41681i	0.211902i
$918$	5.04322i	0.166451i
$919$	−10.8382	−0.357520	−0.178760	−	0.983893i	$-0.557209\pi$
$919$	−10.8382	−0.357520	−0.178760	+	0.983893i	$0.557209\pi$
$920$	0	0
$921$	5.98864	0.197333
$922$	− 9.76983i	− 0.321752i
$923$	− 26.3763i	− 0.868187i
$924$	−2.87967	−0.0947342
$925$	0	0
$926$	14.3438	0.471368
$927$	4.20634i	0.138154i
$928$	22.5111i	0.738962i
$929$	52.3491	1.71752	0.858759	−	0.512379i	$-0.171236\pi$
$929$	52.3491	1.71752	0.858759	+	0.512379i	$0.171236\pi$
$930$	0	0
$931$	−15.1080	−0.495144
$932$	2.43678i	0.0798193i
$933$	23.8684i	0.781416i
$934$	23.7586	0.777404
$935$	0	0
$936$	−12.5034	−0.408685
$937$	− 0.00131225i	0 4.28694e-5i	−1.00000		2.14347e-5i	$-0.999993\pi$
$937$	− 0.00131225i	0 4.28694e-5i	1.00000		2.14347e-5i	$-6.82288e-6\pi$
$938$	10.6277i	0.347006i
$939$	−5.75913	−0.187942
$940$	0	0
$941$	−21.7222	−0.708124	−0.354062	−	0.935222i	$-0.615200\pi$
$941$	−21.7222	−0.708124	−0.354062	+	0.935222i	$0.615200\pi$
$942$	− 0.479887i	− 0.0156356i
$943$	22.0754i	0.718874i
$944$	5.46655	0.177921
$945$	0	0
$946$	−3.04044	−0.0988533
$947$	− 33.9266i	− 1.10247i	−0.834351	−	0.551234i	$-0.814157\pi$
$947$	− 33.9266i	− 1.10247i	0.834351	−	0.551234i	$-0.185843\pi$
$948$	10.0216i	0.325486i
$949$	−29.4134	−0.954800
$950$	0	0
$951$	−12.2481	−0.397172
$952$	43.7865i	1.41913i
$953$	8.30308i	0.268963i	0.990916	+	0.134482i	$0.0429369\pi$
$953$	8.30308i	0.268963i	−0.990916	+	0.134482i	$0.957063\pi$
$954$	3.53515	0.114455
$955$	0	0
$956$	−17.9794	−0.581496
$957$	− 6.06786i	− 0.196146i
$958$	− 46.5887i	− 1.50521i
$959$	−59.9024	−1.93435
$960$	0	0
$961$	−30.6528	−0.988798
$962$	50.6825i	1.63407i
$963$	4.01195i	0.129283i
$964$	−10.9174	−0.351627
$965$	0	0
$966$	−7.82486	−0.251761
$967$	27.7915i	0.893714i	0.894605	+	0.446857i	$0.147457\pi$
$967$	27.7915i	0.893714i	−0.894605	+	0.446857i	$0.852543\pi$
$968$	− 29.6724i	− 0.953707i
$969$	−27.6220	−0.887345
$970$	0	0
$971$	−31.0461	−0.996317	−0.498158	−	0.867086i	$-0.665990\pi$
$971$	−31.0461	−0.996317	−0.498158	+	0.867086i	$0.665990\pi$
$972$	0.812294i	0.0260543i
$973$	− 52.9780i	− 1.69840i
$974$	−12.3692	−0.396333
$975$	0	0
$976$	−23.5811	−0.754814
$977$	44.3917i	1.42022i	0.704092	+	0.710109i	$0.251354\pi$
$977$	44.3917i	1.42022i	−0.704092	+	0.710109i	$0.748646\pi$
$978$	3.53220i	0.112947i
$979$	1.24481	0.0397842
$980$	0	0
$981$	9.09364	0.290338
$982$	18.1506i	0.579208i
$983$	43.7857i	1.39655i	0.715831	+	0.698274i	$0.246048\pi$
$983$	43.7857i	1.39655i	−0.715831	+	0.698274i	$0.753952\pi$
$984$	−29.0920	−0.927418
$985$	0	0
$986$	−26.6492	−0.848685
$987$	18.6778i	0.594520i
$988$	− 19.7800i	− 0.629284i
$989$	5.65033	0.179670
$990$	0	0
$991$	34.4351	1.09387	0.546933	−	0.837176i	$-0.315795\pi$
$991$	34.4351	1.09387	0.546933	+	0.837176i	$0.315795\pi$
$992$	− 2.51039i	− 0.0797049i
$993$	− 6.02899i	− 0.191324i
$994$	21.7535	0.689978
$995$	0	0
$996$	10.0676	0.319006
$997$	19.3890i	0.614056i	0.951700	+	0.307028i	$0.0993346\pi$
$997$	19.3890i	0.614056i	−0.951700	+	0.307028i	$0.900665\pi$
$998$	13.3308i	0.421979i
$999$	11.3997	0.360670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.h.1249.11		16
5.2	odd	4		1875.2.a.m.1.4		8
5.3	odd	4		1875.2.a.p.1.5		8
5.4	even	2	inner	1875.2.b.h.1249.6		16
15.2	even	4		5625.2.a.bd.1.5		8
15.8	even	4		5625.2.a.t.1.4		8
25.3	odd	20		375.2.g.d.76.2		16
25.4	even	10		75.2.i.a.34.2	✓	16
25.6	even	5		75.2.i.a.64.2	yes	16
25.8	odd	20		375.2.g.d.301.2		16
25.17	odd	20		375.2.g.e.301.3		16
25.19	even	10		375.2.i.c.199.3		16
25.21	even	5		375.2.i.c.49.3		16
25.22	odd	20		375.2.g.e.76.3		16
75.29	odd	10		225.2.m.b.109.3		16
75.56	odd	10		225.2.m.b.64.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.2	✓	16	25.4	even	10
75.2.i.a.64.2	yes	16	25.6	even	5
225.2.m.b.64.3		16	75.56	odd	10
225.2.m.b.109.3		16	75.29	odd	10
375.2.g.d.76.2		16	25.3	odd	20
375.2.g.d.301.2		16	25.8	odd	20
375.2.g.e.76.3		16	25.22	odd	20
375.2.g.e.301.3		16	25.17	odd	20
375.2.i.c.49.3		16	25.21	even	5
375.2.i.c.199.3		16	25.19	even	10
1875.2.a.m.1.4		8	5.2	odd	4
1875.2.a.p.1.5		8	5.3	odd	4
1875.2.b.h.1249.6		16	5.4	even	2	inner
1875.2.b.h.1249.11		16	1.1	even	1	trivial
5625.2.a.t.1.4		8	15.8	even	4
5625.2.a.bd.1.5		8	15.2	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)

\(f(q)\)	\(=\)	\(q+1.08982i q^{2} +1.00000i q^{3} +0.812294 q^{4} -1.08982 q^{6} -3.08724i q^{7} +3.06489i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.08982i q^{2} +1.00000i q^{3} +0.812294 q^{4} -1.08982 q^{6} -3.08724i q^{7} +3.06489i q^{8} -1.00000 q^{9} -1.14831 q^{11} +0.812294i q^{12} -4.07954i q^{13} +3.36454 q^{14} -1.71559 q^{16} +4.62758i q^{17} -1.08982i q^{18} +5.96899 q^{19} +3.08724 q^{21} -1.25145i q^{22} +2.32568i q^{23} -3.06489 q^{24} +4.44596 q^{26} -1.00000i q^{27} -2.50775i q^{28} +5.28417 q^{29} -0.589279 q^{31} +4.26010i q^{32} -1.14831i q^{33} -5.04322 q^{34} -0.812294 q^{36} +11.3997i q^{37} +6.50512i q^{38} +4.07954 q^{39} +9.49200 q^{41} +3.36454i q^{42} -2.42954i q^{43} -0.932764 q^{44} -2.53458 q^{46} +6.04998i q^{47} -1.71559i q^{48} -2.53108 q^{49} -4.62758 q^{51} -3.31379i q^{52} +3.24380i q^{53} +1.08982 q^{54} +9.46207 q^{56} +5.96899i q^{57} +5.75879i q^{58} -3.18640 q^{59} +13.7452 q^{61} -0.642208i q^{62} +3.08724i q^{63} -8.07392 q^{64} +1.25145 q^{66} +3.15873i q^{67} +3.75895i q^{68} -2.32568 q^{69} +6.46551 q^{71} -3.06489i q^{72} -7.20998i q^{73} -12.4236 q^{74} +4.84857 q^{76} +3.54511i q^{77} +4.44596i q^{78} +12.3374 q^{79} +1.00000 q^{81} +10.3446i q^{82} -12.3941i q^{83} +2.50775 q^{84} +2.64776 q^{86} +5.28417i q^{87} -3.51944i q^{88} -1.08404 q^{89} -12.5945 q^{91} +1.88914i q^{92} -0.589279i q^{93} -6.59339 q^{94} -4.26010 q^{96} +4.52132i q^{97} -2.75842i q^{98} +1.14831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 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

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