LMFDB - Embedded newform 1875.2.b.h.1249.14

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			1249.14
Root			$-2.53767i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53767i q^{2} -1.00000i q^{3} -0.364440 q^{4} +1.53767 q^{6} -1.68601i q^{7} +2.51496i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.53767i q^{2} -1.00000i q^{3} -0.364440 q^{4} +1.53767 q^{6} -1.68601i q^{7} +2.51496i q^{8} -1.00000 q^{9} -2.97757 q^{11} +0.364440i q^{12} +0.232892i q^{13} +2.59253 q^{14} -4.59606 q^{16} -7.45901i q^{17} -1.53767i q^{18} -0.753527 q^{19} -1.68601 q^{21} -4.57853i q^{22} +0.872721i q^{23} +2.51496 q^{24} -0.358112 q^{26} +1.00000i q^{27} +0.614448i q^{28} -6.87482 q^{29} -9.81929 q^{31} -2.03733i q^{32} +2.97757i q^{33} +11.4695 q^{34} +0.364440 q^{36} -10.1272i q^{37} -1.15868i q^{38} +0.232892 q^{39} +3.79732 q^{41} -2.59253i q^{42} -5.27322i q^{43} +1.08514 q^{44} -1.34196 q^{46} +8.56747i q^{47} +4.59606i q^{48} +4.15738 q^{49} -7.45901 q^{51} -0.0848751i q^{52} +5.97876i q^{53} -1.53767 q^{54} +4.24024 q^{56} +0.753527i q^{57} -10.5712i q^{58} +3.85114 q^{59} -4.39643 q^{61} -15.0989i q^{62} +1.68601i q^{63} -6.05938 q^{64} -4.57853 q^{66} -1.79282i q^{67} +2.71836i q^{68} +0.872721 q^{69} -4.37450 q^{71} -2.51496i q^{72} -15.0528i q^{73} +15.5723 q^{74} +0.274615 q^{76} +5.02021i q^{77} +0.358112i q^{78} -7.37584 q^{79} +1.00000 q^{81} +5.83904i q^{82} +4.34451i q^{83} +0.614448 q^{84} +8.10849 q^{86} +6.87482i q^{87} -7.48847i q^{88} -12.1032 q^{89} +0.392657 q^{91} -0.318054i q^{92} +9.81929i q^{93} -13.1740 q^{94} -2.03733 q^{96} -9.47426i q^{97} +6.39269i q^{98} +2.97757 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.53767i	1.08730i	0.839312	+	0.543650i	$0.182958\pi$
$2$	1.53767i	1.08730i	−0.839312	+	0.543650i	$0.817042\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−0.364440	−0.182220
$5$	0	0
$5$	0	0
$6$	1.53767	0.627753
$7$	− 1.68601i	− 0.637251i	−0.947881	−	0.318625i	$-0.896779\pi$
$7$	− 1.68601i	− 0.637251i	0.947881	−	0.318625i	$-0.103221\pi$
$8$	2.51496i	0.889172i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.97757	−0.897772	−0.448886	−	0.893589i	$-0.648179\pi$
$11$	−2.97757	−0.897772	−0.448886	+	0.893589i	$0.648179\pi$
$12$	0.364440i	0.105205i
$13$	0.232892i	0.0645926i	0.999478	+	0.0322963i	$0.0102820\pi$
$13$	0.232892i	0.0645926i	−0.999478	+	0.0322963i	$0.989718\pi$
$14$	2.59253	0.692882
$15$	0	0
$16$	−4.59606	−1.14902
$17$	− 7.45901i	− 1.80908i	−0.426392	−	0.904538i	$-0.640216\pi$
$17$	− 7.45901i	− 1.80908i	0.426392	−	0.904538i	$-0.359784\pi$
$18$	− 1.53767i	− 0.362433i
$19$	−0.753527	−0.172871	−0.0864354	−	0.996257i	$-0.527548\pi$
$19$	−0.753527	−0.172871	−0.0864354	+	0.996257i	$0.527548\pi$
$20$	0	0
$21$	−1.68601	−0.367917
$22$	− 4.57853i	− 0.976146i
$23$	0.872721i	0.181975i	0.995852	+	0.0909874i	$0.0290023\pi$
$23$	0.872721i	0.181975i	−0.995852	+	0.0909874i	$0.970998\pi$
$24$	2.51496	0.513364
$25$	0	0
$26$	−0.358112	−0.0702315
$27$	1.00000i	0.192450i
$28$	0.614448i	0.116120i
$29$	−6.87482	−1.27662	−0.638311	−	0.769778i	$-0.720367\pi$
$29$	−6.87482	−1.27662	−0.638311	+	0.769778i	$0.720367\pi$
$30$	0	0
$31$	−9.81929	−1.76360	−0.881798	−	0.471627i	$-0.843667\pi$
$31$	−9.81929	−1.76360	−0.881798	+	0.471627i	$0.843667\pi$
$32$	− 2.03733i	− 0.360152i
$33$	2.97757i	0.518329i
$34$	11.4695	1.96701
$35$	0	0
$36$	0.364440	0.0607399
$37$	− 10.1272i	− 1.66489i	−0.554105	−	0.832447i	$-0.686939\pi$
$37$	− 10.1272i	− 1.66489i	0.554105	−	0.832447i	$-0.313061\pi$
$38$	− 1.15868i	− 0.187962i
$39$	0.232892	0.0372926
$40$	0	0
$41$	3.79732	0.593042	0.296521	−	0.955026i	$-0.404174\pi$
$41$	3.79732	0.593042	0.296521	+	0.955026i	$0.404174\pi$
$42$	− 2.59253i	− 0.400036i
$43$	− 5.27322i	− 0.804159i	−0.915605	−	0.402079i	$-0.868288\pi$
$43$	− 5.27322i	− 0.804159i	0.915605	−	0.402079i	$-0.131712\pi$
$44$	1.08514	0.163592
$45$	0	0
$46$	−1.34196	−0.197861
$47$	8.56747i	1.24969i	0.780747	+	0.624847i	$0.214839\pi$
$47$	8.56747i	1.24969i	−0.780747	+	0.624847i	$0.785161\pi$
$48$	4.59606i	0.663385i
$49$	4.15738	0.593911
$50$	0	0
$51$	−7.45901	−1.04447
$52$	− 0.0848751i	− 0.0117701i
$53$	5.97876i	0.821246i	0.911805	+	0.410623i	$0.134689\pi$
$53$	5.97876i	0.821246i	−0.911805	+	0.410623i	$0.865311\pi$
$54$	−1.53767	−0.209251
$55$	0	0
$56$	4.24024	0.566625
$57$	0.753527i	0.0998070i
$58$	− 10.5712i	− 1.38807i
$59$	3.85114	0.501376	0.250688	−	0.968068i	$-0.419343\pi$
$59$	3.85114	0.501376	0.250688	+	0.968068i	$0.419343\pi$
$60$	0	0
$61$	−4.39643	−0.562905	−0.281452	−	0.959575i	$-0.590816\pi$
$61$	−4.39643	−0.562905	−0.281452	+	0.959575i	$0.590816\pi$
$62$	− 15.0989i	− 1.91756i
$63$	1.68601i	0.212417i
$64$	−6.05938	−0.757423
$65$	0	0
$66$	−4.57853	−0.563578
$67$	− 1.79282i	− 0.219028i	−0.993985	−	0.109514i	$-0.965070\pi$
$67$	− 1.79282i	− 0.219028i	0.993985	−	0.109514i	$-0.0349295\pi$
$68$	2.71836i	0.329650i
$69$	0.872721	0.105063
$70$	0	0
$71$	−4.37450	−0.519157	−0.259579	−	0.965722i	$-0.583584\pi$
$71$	−4.37450	−0.519157	−0.259579	+	0.965722i	$0.583584\pi$
$72$	− 2.51496i	− 0.296391i
$73$	− 15.0528i	− 1.76180i	−0.473303	−	0.880900i	$-0.656938\pi$
$73$	− 15.0528i	− 1.76180i	0.473303	−	0.880900i	$-0.343062\pi$
$74$	15.5723	1.81024
$75$	0	0
$76$	0.274615	0.0315005
$77$	5.02021i	0.572106i
$78$	0.358112i	0.0405482i
$79$	−7.37584	−0.829847	−0.414924	−	0.909856i	$-0.636192\pi$
$79$	−7.37584	−0.829847	−0.414924	+	0.909856i	$0.636192\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.83904i	0.644814i
$83$	4.34451i	0.476872i	0.971158	+	0.238436i	$0.0766347\pi$
$83$	4.34451i	0.476872i	−0.971158	+	0.238436i	$0.923365\pi$
$84$	0.614448	0.0670417
$85$	0	0
$86$	8.10849	0.874361
$87$	6.87482i	0.737058i
$88$	− 7.48847i	− 0.798273i
$89$	−12.1032	−1.28294	−0.641469	−	0.767149i	$-0.721675\pi$
$89$	−12.1032	−1.28294	−0.641469	+	0.767149i	$0.721675\pi$
$90$	0	0
$91$	0.392657	0.0411617
$92$	− 0.318054i	− 0.0331594i
$93$	9.81929i	1.01821i
$94$	−13.1740	−1.35879
$95$	0	0
$96$	−2.03733	−0.207934
$97$	− 9.47426i	− 0.961965i	−0.876730	−	0.480982i	$-0.840280\pi$
$97$	− 9.47426i	− 0.961965i	0.876730	−	0.480982i	$-0.159720\pi$
$98$	6.39269i	0.645760i
$99$	2.97757	0.299257
$100$	0	0
$101$	−6.54468	−0.651220	−0.325610	−	0.945504i	$-0.605570\pi$
$101$	−6.54468	−0.651220	−0.325610	+	0.945504i	$0.605570\pi$
$102$	− 11.4695i	− 1.13565i
$103$	− 0.700804i	− 0.0690522i	−0.999404	−	0.0345261i	$-0.989008\pi$
$103$	− 0.700804i	− 0.0690522i	0.999404	−	0.0345261i	$-0.0109922\pi$
$104$	−0.585713	−0.0574339
$105$	0	0
$106$	−9.19338	−0.892940
$107$	− 12.5288i	− 1.21120i	−0.795768	−	0.605602i	$-0.792933\pi$
$107$	− 12.5288i	− 1.21120i	0.795768	−	0.605602i	$-0.207067\pi$
$108$	− 0.364440i	− 0.0350682i
$109$	−4.28348	−0.410283	−0.205141	−	0.978732i	$-0.565765\pi$
$109$	−4.28348	−0.410283	−0.205141	+	0.978732i	$0.565765\pi$
$110$	0	0
$111$	−10.1272	−0.961227
$112$	7.74899i	0.732211i
$113$	8.22761i	0.773988i	0.922082	+	0.386994i	$0.126487\pi$
$113$	8.22761i	0.773988i	−0.922082	+	0.386994i	$0.873513\pi$
$114$	−1.15868	−0.108520
$115$	0	0
$116$	2.50546	0.232626
$117$	− 0.232892i	− 0.0215309i
$118$	5.92180i	0.545146i
$119$	−12.5760	−1.15284
$120$	0	0
$121$	−2.13407	−0.194006
$122$	− 6.76027i	− 0.612046i
$123$	− 3.79732i	− 0.342393i
$124$	3.57854	0.321362
$125$	0	0
$126$	−2.59253	−0.230961
$127$	− 11.9482i	− 1.06023i	−0.847926	−	0.530114i	$-0.822149\pi$
$127$	− 11.9482i	− 1.06023i	0.847926	−	0.530114i	$-0.177851\pi$
$128$	− 13.3920i	− 1.18370i
$129$	−5.27322	−0.464281
$130$	0	0
$131$	21.4573	1.87474	0.937369	−	0.348339i	$-0.113254\pi$
$131$	21.4573	1.87474	0.937369	+	0.348339i	$0.113254\pi$
$132$	− 1.08514i	− 0.0944497i
$133$	1.27045i	0.110162i
$134$	2.75678	0.238149
$135$	0	0
$136$	18.7591	1.60858
$137$	− 10.0192i	− 0.855998i	−0.903779	−	0.427999i	$-0.859219\pi$
$137$	− 10.0192i	− 0.855998i	0.903779	−	0.427999i	$-0.140781\pi$
$138$	1.34196i	0.114235i
$139$	6.57467	0.557656	0.278828	−	0.960341i	$-0.410054\pi$
$139$	6.57467	0.557656	0.278828	+	0.960341i	$0.410054\pi$
$140$	0	0
$141$	8.56747	0.721511
$142$	− 6.72655i	− 0.564479i
$143$	− 0.693452i	− 0.0579894i
$144$	4.59606	0.383005
$145$	0	0
$146$	23.1463	1.91560
$147$	− 4.15738i	− 0.342895i
$148$	3.69074i	0.303377i
$149$	−10.9143	−0.894132	−0.447066	−	0.894501i	$-0.647531\pi$
$149$	−10.9143	−0.894132	−0.447066	+	0.894501i	$0.647531\pi$
$150$	0	0
$151$	20.4128	1.66117	0.830584	−	0.556894i	$-0.188007\pi$
$151$	20.4128	1.66117	0.830584	+	0.556894i	$0.188007\pi$
$152$	− 1.89509i	− 0.153712i
$153$	7.45901i	0.603026i
$154$	−7.71944	−0.622050
$155$	0	0
$156$	−0.0848751	−0.00679544
$157$	− 3.49944i	− 0.279286i	−0.990202	−	0.139643i	$-0.955405\pi$
$157$	− 3.49944i	− 0.279286i	0.990202	−	0.139643i	$-0.0445954\pi$
$158$	− 11.3416i	− 0.902292i
$159$	5.97876	0.474146
$160$	0	0
$161$	1.47141	0.115964
$162$	1.53767i	0.120811i
$163$	5.97357i	0.467886i	0.972250	+	0.233943i	$0.0751630\pi$
$163$	5.97357i	0.467886i	−0.972250	+	0.233943i	$0.924837\pi$
$164$	−1.38389	−0.108064
$165$	0	0
$166$	−6.68044	−0.518503
$167$	2.33767i	0.180895i	0.995901	+	0.0904473i	$0.0288297\pi$
$167$	2.33767i	0.180895i	−0.995901	+	0.0904473i	$0.971170\pi$
$168$	− 4.24024i	− 0.327141i
$169$	12.9458	0.995828
$170$	0	0
$171$	0.753527	0.0576236
$172$	1.92177i	0.146534i
$173$	6.07099i	0.461569i	0.973005	+	0.230784i	$0.0741292\pi$
$173$	6.07099i	0.461569i	−0.973005	+	0.230784i	$0.925871\pi$
$174$	−10.5712	−0.801403
$175$	0	0
$176$	13.6851	1.03155
$177$	− 3.85114i	− 0.289470i
$178$	− 18.6108i	− 1.39494i
$179$	−11.4235	−0.853835	−0.426917	−	0.904291i	$-0.640401\pi$
$179$	−11.4235	−0.853835	−0.426917	+	0.904291i	$0.640401\pi$
$180$	0	0
$181$	−7.85007	−0.583491	−0.291746	−	0.956496i	$-0.594236\pi$
$181$	−7.85007	−0.583491	−0.291746	+	0.956496i	$0.594236\pi$
$182$	0.603779i	0.0447551i
$183$	4.39643i	0.324993i
$184$	−2.19486	−0.161807
$185$	0	0
$186$	−15.0989	−1.10710
$187$	22.2097i	1.62414i
$188$	− 3.12232i	− 0.227719i
$189$	1.68601	0.122639
$190$	0	0
$191$	12.5641	0.909110	0.454555	−	0.890719i	$-0.349798\pi$
$191$	12.5641	0.909110	0.454555	+	0.890719i	$0.349798\pi$
$192$	6.05938i	0.437298i
$193$	− 10.1437i	− 0.730161i	−0.930976	−	0.365081i	$-0.881041\pi$
$193$	− 10.1437i	− 0.730161i	0.930976	−	0.365081i	$-0.118959\pi$
$194$	14.5683	1.04594
$195$	0	0
$196$	−1.51511	−0.108222
$197$	2.04703i	0.145845i	0.997338	+	0.0729223i	$0.0232325\pi$
$197$	2.04703i	0.145845i	−0.997338	+	0.0729223i	$0.976767\pi$
$198$	4.57853i	0.325382i
$199$	−3.57125	−0.253159	−0.126580	−	0.991956i	$-0.540400\pi$
$199$	−3.57125	−0.253159	−0.126580	+	0.991956i	$0.540400\pi$
$200$	0	0
$201$	−1.79282	−0.126456
$202$	− 10.0636i	− 0.708071i
$203$	11.5910i	0.813529i
$204$	2.71836	0.190323
$205$	0	0
$206$	1.07761	0.0750805
$207$	− 0.872721i	− 0.0606583i
$208$	− 1.07039i	− 0.0742179i
$209$	2.24368	0.155199
$210$	0	0
$211$	3.72643	0.256538	0.128269	−	0.991739i	$-0.459058\pi$
$211$	3.72643	0.256538	0.128269	+	0.991739i	$0.459058\pi$
$212$	− 2.17890i	− 0.149647i
$213$	4.37450i	0.299736i
$214$	19.2652	1.31694
$215$	0	0
$216$	−2.51496	−0.171121
$217$	16.5554i	1.12385i
$218$	− 6.58659i	− 0.446100i
$219$	−15.0528	−1.01718
$220$	0	0
$221$	1.73714	0.116853
$222$	− 15.5723i	− 1.04514i
$223$	26.2488i	1.75775i	0.477051	+	0.878875i	$0.341706\pi$
$223$	26.2488i	1.75775i	−0.477051	+	0.878875i	$0.658294\pi$
$224$	−3.43495	−0.229507
$225$	0	0
$226$	−12.6514	−0.841557
$227$	4.49111i	0.298085i	0.988831	+	0.149043i	$0.0476191\pi$
$227$	4.49111i	0.298085i	−0.988831	+	0.149043i	$0.952381\pi$
$228$	− 0.274615i	− 0.0181868i
$229$	7.53935	0.498214	0.249107	−	0.968476i	$-0.419863\pi$
$229$	7.53935	0.498214	0.249107	+	0.968476i	$0.419863\pi$
$230$	0	0
$231$	5.02021	0.330305
$232$	− 17.2899i	− 1.13514i
$233$	10.3640i	0.678971i	0.940611	+	0.339485i	$0.110253\pi$
$233$	10.3640i	0.678971i	−0.940611	+	0.339485i	$0.889747\pi$
$234$	0.358112	0.0234105
$235$	0	0
$236$	−1.40351	−0.0913607
$237$	7.37584i	0.479112i
$238$	− 19.3377i	− 1.25348i
$239$	18.5738	1.20144	0.600719	−	0.799460i	$-0.294881\pi$
$239$	18.5738	1.20144	0.600719	+	0.799460i	$0.294881\pi$
$240$	0	0
$241$	−5.43543	−0.350127	−0.175063	−	0.984557i	$-0.556013\pi$
$241$	−5.43543	−0.350127	−0.175063	+	0.984557i	$0.556013\pi$
$242$	− 3.28150i	− 0.210943i
$243$	− 1.00000i	− 0.0641500i
$244$	1.60223	0.102572
$245$	0	0
$246$	5.83904	0.372283
$247$	− 0.175490i	− 0.0111662i
$248$	− 24.6951i	− 1.56814i
$249$	4.34451	0.275322
$250$	0	0
$251$	23.3577	1.47432	0.737162	−	0.675716i	$-0.236166\pi$
$251$	23.3577	1.47432	0.737162	+	0.675716i	$0.236166\pi$
$252$	− 0.614448i	− 0.0387066i
$253$	− 2.59859i	− 0.163372i
$254$	18.3724	1.15279
$255$	0	0
$256$	8.47377	0.529611
$257$	− 4.48380i	− 0.279692i	−0.990173	−	0.139846i	$-0.955339\pi$
$257$	− 4.48380i	− 0.279692i	0.990173	−	0.139846i	$-0.0446607\pi$
$258$	− 8.10849i	− 0.504813i
$259$	−17.0745	−1.06095
$260$	0	0
$261$	6.87482	0.425541
$262$	32.9944i	2.03840i
$263$	− 7.72550i	− 0.476375i	−0.971219	−	0.238187i	$-0.923447\pi$
$263$	− 7.72550i	− 0.476375i	0.971219	−	0.238187i	$-0.0765532\pi$
$264$	−7.48847	−0.460883
$265$	0	0
$266$	−1.95354	−0.119779
$267$	12.1032i	0.740704i
$268$	0.653376i	0.0399113i
$269$	−10.7394	−0.654791	−0.327396	−	0.944887i	$-0.606171\pi$
$269$	−10.7394	−0.654791	−0.327396	+	0.944887i	$0.606171\pi$
$270$	0	0
$271$	−25.7126	−1.56193	−0.780964	−	0.624576i	$-0.785272\pi$
$271$	−25.7126	−1.56193	−0.780964	+	0.624576i	$0.785272\pi$
$272$	34.2821i	2.07866i
$273$	− 0.392657i	− 0.0237647i
$274$	15.4063	0.930726
$275$	0	0
$276$	−0.318054	−0.0191446
$277$	− 9.62446i	− 0.578278i	−0.957287	−	0.289139i	$-0.906631\pi$
$277$	− 9.62446i	− 0.578278i	0.957287	−	0.289139i	$-0.0933689\pi$
$278$	10.1097i	0.606340i
$279$	9.81929	0.587865
$280$	0	0
$281$	−2.83275	−0.168988	−0.0844939	−	0.996424i	$-0.526927\pi$
$281$	−2.83275	−0.168988	−0.0844939	+	0.996424i	$0.526927\pi$
$282$	13.1740i	0.784498i
$283$	23.6634i	1.40664i	0.710873	+	0.703321i	$0.248300\pi$
$283$	23.6634i	1.40664i	−0.710873	+	0.703321i	$0.751700\pi$
$284$	1.59424	0.0946007
$285$	0	0
$286$	1.06630	0.0630518
$287$	− 6.40231i	− 0.377916i
$288$	2.03733i	0.120051i
$289$	−38.6369	−2.27276
$290$	0	0
$291$	−9.47426	−0.555391
$292$	5.48584i	0.321035i
$293$	23.2376i	1.35755i	0.734345	+	0.678777i	$0.237490\pi$
$293$	23.2376i	1.35755i	−0.734345	+	0.678777i	$0.762510\pi$
$294$	6.39269	0.372829
$295$	0	0
$296$	25.4694	1.48038
$297$	− 2.97757i	− 0.172776i
$298$	− 16.7826i	− 0.972189i
$299$	−0.203250	−0.0117542
$300$	0	0
$301$	−8.89069	−0.512451
$302$	31.3882i	1.80619i
$303$	6.54468i	0.375982i
$304$	3.46326	0.198631
$305$	0	0
$306$	−11.4695	−0.655669
$307$	21.8131i	1.24494i	0.782643	+	0.622470i	$0.213871\pi$
$307$	21.8131i	1.24494i	−0.782643	+	0.622470i	$0.786129\pi$
$308$	− 1.82956i	− 0.104249i
$309$	−0.700804	−0.0398673
$310$	0	0
$311$	10.0414	0.569394	0.284697	−	0.958618i	$-0.408107\pi$
$311$	10.0414	0.569394	0.284697	+	0.958618i	$0.408107\pi$
$312$	0.585713i	0.0331595i
$313$	6.05366i	0.342173i	0.985256	+	0.171087i	$0.0547278\pi$
$313$	6.05366i	0.342173i	−0.985256	+	0.171087i	$0.945272\pi$
$314$	5.38099	0.303667
$315$	0	0
$316$	2.68805	0.151215
$317$	8.49907i	0.477355i	0.971099	+	0.238678i	$0.0767139\pi$
$317$	8.49907i	0.477355i	−0.971099	+	0.238678i	$0.923286\pi$
$318$	9.19338i	0.515539i
$319$	20.4703	1.14612
$320$	0	0
$321$	−12.5288	−0.699289
$322$	2.26255i	0.126087i
$323$	5.62057i	0.312737i
$324$	−0.364440	−0.0202466
$325$	0	0
$326$	−9.18540	−0.508733
$327$	4.28348i	0.236877i
$328$	9.55010i	0.527316i
$329$	14.4448	0.796368
$330$	0	0
$331$	16.2945	0.895624	0.447812	−	0.894128i	$-0.352203\pi$
$331$	16.2945	0.895624	0.447812	+	0.894128i	$0.352203\pi$
$332$	− 1.58331i	− 0.0868955i
$333$	10.1272i	0.554965i
$334$	−3.59458	−0.196687
$335$	0	0
$336$	7.74899	0.422742
$337$	− 24.6962i	− 1.34529i	−0.739966	−	0.672644i	$-0.765159\pi$
$337$	− 24.6962i	− 1.34529i	0.739966	−	0.672644i	$-0.234841\pi$
$338$	19.9064i	1.08276i
$339$	8.22761	0.446862
$340$	0	0
$341$	29.2376	1.58331
$342$	1.15868i	0.0626541i
$343$	− 18.8114i	− 1.01572i
$344$	13.2619	0.715035
$345$	0	0
$346$	−9.33520	−0.501864
$347$	13.9466i	0.748695i	0.927289	+	0.374347i	$0.122133\pi$
$347$	13.9466i	0.748695i	−0.927289	+	0.374347i	$0.877867\pi$
$348$	− 2.50546i	− 0.134307i
$349$	−18.4966	−0.990099	−0.495049	−	0.868865i	$-0.664850\pi$
$349$	−18.4966	−0.990099	−0.495049	+	0.868865i	$0.664850\pi$
$350$	0	0
$351$	−0.232892	−0.0124309
$352$	6.06629i	0.323334i
$353$	− 0.441733i	− 0.0235111i	−0.999931	−	0.0117555i	$-0.996258\pi$
$353$	− 0.441733i	− 0.0235111i	0.999931	−	0.0117555i	$-0.00374199\pi$
$354$	5.92180	0.314740
$355$	0	0
$356$	4.41089	0.233777
$357$	12.5760i	0.665590i
$358$	− 17.5657i	− 0.928374i
$359$	−4.62127	−0.243901	−0.121951	−	0.992536i	$-0.538915\pi$
$359$	−4.62127	−0.243901	−0.121951	+	0.992536i	$0.538915\pi$
$360$	0	0
$361$	−18.4322	−0.970116
$362$	− 12.0708i	− 0.634429i
$363$	2.13407i	0.112010i
$364$	−0.143100	−0.00750047
$365$	0	0
$366$	−6.76027	−0.353365
$367$	− 8.77696i	− 0.458153i	−0.973408	−	0.229077i	$-0.926429\pi$
$367$	− 8.77696i	− 0.458153i	0.973408	−	0.229077i	$-0.0735707\pi$
$368$	− 4.01108i	− 0.209092i
$369$	−3.79732	−0.197681
$370$	0	0
$371$	10.0802	0.523339
$372$	− 3.57854i	− 0.185538i
$373$	− 15.4955i	− 0.802328i	−0.916006	−	0.401164i	$-0.868606\pi$
$373$	− 15.4955i	− 0.802328i	0.916006	−	0.401164i	$-0.131394\pi$
$374$	−34.1513	−1.76592
$375$	0	0
$376$	−21.5468	−1.11119
$377$	− 1.60109i	− 0.0824604i
$378$	2.59253i	0.133345i
$379$	27.2931	1.40195	0.700977	−	0.713184i	$-0.252748\pi$
$379$	27.2931	1.40195	0.700977	+	0.713184i	$0.252748\pi$
$380$	0	0
$381$	−11.9482	−0.612123
$382$	19.3196i	0.988474i
$383$	− 13.1042i	− 0.669592i	−0.942291	−	0.334796i	$-0.891333\pi$
$383$	− 13.1042i	− 0.669592i	0.942291	−	0.334796i	$-0.108667\pi$
$384$	−13.3920	−0.683408
$385$	0	0
$386$	15.5977	0.793904
$387$	5.27322i	0.268053i
$388$	3.45279i	0.175289i
$389$	−16.9056	−0.857147	−0.428574	−	0.903507i	$-0.640984\pi$
$389$	−16.9056	−0.857147	−0.428574	+	0.903507i	$0.640984\pi$
$390$	0	0
$391$	6.50964	0.329207
$392$	10.4556i	0.528089i
$393$	− 21.4573i	− 1.08238i
$394$	−3.14766	−0.158577
$395$	0	0
$396$	−1.08514	−0.0545306
$397$	− 10.4078i	− 0.522353i	−0.965291	−	0.261177i	$-0.915890\pi$
$397$	− 10.4078i	− 0.522353i	0.965291	−	0.261177i	$-0.0841105\pi$
$398$	− 5.49142i	− 0.275260i
$399$	1.27045	0.0636021
$400$	0	0
$401$	−0.694800	−0.0346967	−0.0173483	−	0.999850i	$-0.505522\pi$
$401$	−0.694800	−0.0346967	−0.0173483	+	0.999850i	$0.505522\pi$
$402$	− 2.75678i	− 0.137496i
$403$	− 2.28683i	− 0.113915i
$404$	2.38514	0.118665
$405$	0	0
$406$	−17.8232	−0.884549
$407$	30.1543i	1.49469i
$408$	− 18.7591i	− 0.928714i
$409$	1.18910	0.0587972	0.0293986	−	0.999568i	$-0.490641\pi$
$409$	1.18910	0.0587972	0.0293986	+	0.999568i	$0.490641\pi$
$410$	0	0
$411$	−10.0192	−0.494211
$412$	0.255401i	0.0125827i
$413$	− 6.49306i	− 0.319502i
$414$	1.34196	0.0659537
$415$	0	0
$416$	0.474477	0.0232632
$417$	− 6.57467i	− 0.321963i
$418$	3.45005i	0.168747i
$419$	−1.97841	−0.0966515	−0.0483258	−	0.998832i	$-0.515389\pi$
$419$	−1.97841	−0.0966515	−0.0483258	+	0.998832i	$0.515389\pi$
$420$	0	0
$421$	11.0825	0.540126	0.270063	−	0.962843i	$-0.412955\pi$
$421$	11.0825	0.540126	0.270063	+	0.962843i	$0.412955\pi$
$422$	5.73003i	0.278933i
$423$	− 8.56747i	− 0.416564i
$424$	−15.0363	−0.730229
$425$	0	0
$426$	−6.72655	−0.325902
$427$	7.41241i	0.358712i
$428$	4.56599i	0.220705i
$429$	−0.693452	−0.0334802
$430$	0	0
$431$	−10.4137	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$431$	−10.4137	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$432$	− 4.59606i	− 0.221128i
$433$	10.2643i	0.493272i	0.969108	+	0.246636i	$0.0793251\pi$
$433$	10.2643i	0.493272i	−0.969108	+	0.246636i	$0.920675\pi$
$434$	−25.4568	−1.22196
$435$	0	0
$436$	1.56107	0.0747616
$437$	− 0.657618i	− 0.0314582i
$438$	− 23.1463i	− 1.10597i
$439$	24.7115	1.17942	0.589708	−	0.807616i	$-0.299243\pi$
$439$	24.7115	1.17942	0.589708	+	0.807616i	$0.299243\pi$
$440$	0	0
$441$	−4.15738	−0.197970
$442$	2.67116i	0.127054i
$443$	− 7.52935i	− 0.357730i	−0.983874	−	0.178865i	$-0.942757\pi$
$443$	− 7.52935i	− 0.357730i	0.983874	−	0.178865i	$-0.0572426\pi$
$444$	3.69074	0.175155
$445$	0	0
$446$	−40.3621	−1.91120
$447$	10.9143i	0.516227i
$448$	10.2162i	0.482668i
$449$	31.6627	1.49426	0.747128	−	0.664681i	$-0.231432\pi$
$449$	31.6627	1.49426	0.747128	+	0.664681i	$0.231432\pi$
$450$	0	0
$451$	−11.3068	−0.532416
$452$	− 2.99847i	− 0.141036i
$453$	− 20.4128i	− 0.959075i
$454$	−6.90585	−0.324108
$455$	0	0
$456$	−1.89509	−0.0887456
$457$	2.95742i	0.138342i	0.997605	+	0.0691712i	$0.0220355\pi$
$457$	2.95742i	0.138342i	−0.997605	+	0.0691712i	$0.977965\pi$
$458$	11.5931i	0.541708i
$459$	7.45901	0.348157
$460$	0	0
$461$	−17.6011	−0.819765	−0.409883	−	0.912138i	$-0.634430\pi$
$461$	−17.6011	−0.819765	−0.409883	+	0.912138i	$0.634430\pi$
$462$	7.71944i	0.359141i
$463$	26.3421i	1.22422i	0.790773	+	0.612110i	$0.209679\pi$
$463$	26.3421i	1.22422i	−0.790773	+	0.612110i	$0.790321\pi$
$464$	31.5971	1.46686
$465$	0	0
$466$	−15.9365	−0.738244
$467$	− 7.35906i	− 0.340537i	−0.985398	−	0.170268i	$-0.945537\pi$
$467$	− 7.35906i	− 0.340537i	0.985398	−	0.170268i	$-0.0544635\pi$
$468$	0.0848751i	0.00392335i
$469$	−3.02271	−0.139576
$470$	0	0
$471$	−3.49944	−0.161246
$472$	9.68547i	0.445810i
$473$	15.7014i	0.721951i
$474$	−11.3416	−0.520939
$475$	0	0
$476$	4.58317	0.210069
$477$	− 5.97876i	− 0.273749i
$478$	28.5604i	1.30632i
$479$	28.4670	1.30069	0.650346	−	0.759638i	$-0.274624\pi$
$479$	28.4670	1.30069	0.650346	+	0.759638i	$0.274624\pi$
$480$	0	0
$481$	2.35853	0.107540
$482$	− 8.35791i	− 0.380692i
$483$	− 1.47141i	− 0.0669516i
$484$	0.777739	0.0353518
$485$	0	0
$486$	1.53767	0.0697503
$487$	− 2.38406i	− 0.108032i	−0.998540	−	0.0540161i	$-0.982798\pi$
$487$	− 2.38406i	− 0.108032i	0.998540	−	0.0540161i	$-0.0172022\pi$
$488$	− 11.0568i	− 0.500519i
$489$	5.97357	0.270134
$490$	0	0
$491$	−25.7231	−1.16087	−0.580434	−	0.814307i	$-0.697117\pi$
$491$	−25.7231	−1.16087	−0.580434	+	0.814307i	$0.697117\pi$
$492$	1.38389i	0.0623907i
$493$	51.2794i	2.30951i
$494$	0.269847	0.0121410
$495$	0	0
$496$	45.1301	2.02640
$497$	7.37543i	0.330833i
$498$	6.68044i	0.299358i
$499$	−29.9989	−1.34293	−0.671467	−	0.741035i	$-0.734335\pi$
$499$	−29.9989	−1.34293	−0.671467	+	0.741035i	$0.734335\pi$
$500$	0	0
$501$	2.33767	0.104440
$502$	35.9165i	1.60303i
$503$	− 13.8250i	− 0.616426i	−0.951317	−	0.308213i	$-0.900269\pi$
$503$	− 13.8250i	− 0.616426i	0.951317	−	0.308213i	$-0.0997309\pi$
$504$	−4.24024	−0.188875
$505$	0	0
$506$	3.99578	0.177634
$507$	− 12.9458i	− 0.574941i
$508$	4.35439i	0.193195i
$509$	−31.9537	−1.41632	−0.708161	−	0.706051i	$-0.750475\pi$
$509$	−31.9537	−1.41632	−0.708161	+	0.706051i	$0.750475\pi$
$510$	0	0
$511$	−25.3792	−1.12271
$512$	− 13.7541i	− 0.607852i
$513$	− 0.753527i	− 0.0332690i
$514$	6.89462	0.304109
$515$	0	0
$516$	1.92177	0.0846012
$517$	− 25.5102i	− 1.12194i
$518$	− 26.2549i	− 1.15358i
$519$	6.07099	0.266487
$520$	0	0
$521$	−38.7968	−1.69972	−0.849859	−	0.527010i	$-0.823313\pi$
$521$	−38.7968	−1.69972	−0.849859	+	0.527010i	$0.823313\pi$
$522$	10.5712i	0.462690i
$523$	23.7143i	1.03695i	0.855091	+	0.518477i	$0.173501\pi$
$523$	23.7143i	1.03695i	−0.855091	+	0.518477i	$0.826499\pi$
$524$	−7.81991	−0.341614
$525$	0	0
$526$	11.8793	0.517962
$527$	73.2422i	3.19048i
$528$	− 13.6851i	− 0.595568i
$529$	22.2384	0.966885
$530$	0	0
$531$	−3.85114	−0.167125
$532$	− 0.463003i	− 0.0200737i
$533$	0.884365i	0.0383061i
$534$	−18.6108	−0.805367
$535$	0	0
$536$	4.50888	0.194754
$537$	11.4235i	0.492962i
$538$	− 16.5137i	− 0.711954i
$539$	−12.3789	−0.533197
$540$	0	0
$541$	2.81765	0.121140	0.0605702	−	0.998164i	$-0.480708\pi$
$541$	2.81765	0.121140	0.0605702	+	0.998164i	$0.480708\pi$
$542$	− 39.5376i	− 1.69828i
$543$	7.85007i	0.336879i
$544$	−15.1965	−0.651543
$545$	0	0
$546$	0.603779	0.0258394
$547$	4.42379i	0.189148i	0.995518	+	0.0945739i	$0.0301489\pi$
$547$	4.42379i	0.189148i	−0.995518	+	0.0945739i	$0.969851\pi$
$548$	3.65139i	0.155980i
$549$	4.39643	0.187635
$550$	0	0
$551$	5.18036	0.220691
$552$	2.19486i	0.0934193i
$553$	12.4357i	0.528821i
$554$	14.7993	0.628761
$555$	0	0
$556$	−2.39607	−0.101616
$557$	− 7.20182i	− 0.305151i	−0.988292	−	0.152575i	$-0.951243\pi$
$557$	− 7.20182i	− 0.305151i	0.988292	−	0.152575i	$-0.0487567\pi$
$558$	15.0989i	0.639185i
$559$	1.22809	0.0519427
$560$	0	0
$561$	22.2097	0.937696
$562$	− 4.35585i	− 0.183740i
$563$	− 23.2509i	− 0.979907i	−0.871749	−	0.489954i	$-0.837014\pi$
$563$	− 23.2509i	− 0.979907i	0.871749	−	0.489954i	$-0.162986\pi$
$564$	−3.12232	−0.131474
$565$	0	0
$566$	−36.3865	−1.52944
$567$	− 1.68601i	− 0.0708056i
$568$	− 11.0017i	− 0.461620i
$569$	20.6237	0.864589	0.432295	−	0.901732i	$-0.357704\pi$
$569$	20.6237	0.864589	0.432295	+	0.901732i	$0.357704\pi$
$570$	0	0
$571$	1.80372	0.0754834	0.0377417	−	0.999288i	$-0.487984\pi$
$571$	1.80372	0.0754834	0.0377417	+	0.999288i	$0.487984\pi$
$572$	0.252722i	0.0105668i
$573$	− 12.5641i	− 0.524875i
$574$	9.84466	0.410908
$575$	0	0
$576$	6.05938	0.252474
$577$	16.4863i	0.686335i	0.939274	+	0.343168i	$0.111500\pi$
$577$	16.4863i	0.686335i	−0.939274	+	0.343168i	$0.888500\pi$
$578$	− 59.4109i	− 2.47117i
$579$	−10.1437	−0.421559
$580$	0	0
$581$	7.32488	0.303887
$582$	− 14.5683i	− 0.603876i
$583$	− 17.8022i	− 0.737291i
$584$	37.8572	1.56654
$585$	0	0
$586$	−35.7318	−1.47607
$587$	7.27161i	0.300131i	0.988676	+	0.150066i	$0.0479485\pi$
$587$	7.27161i	0.300131i	−0.988676	+	0.150066i	$0.952051\pi$
$588$	1.51511i	0.0624822i
$589$	7.39909	0.304874
$590$	0	0
$591$	2.04703	0.0842034
$592$	46.5450i	1.91299i
$593$	2.47898i	0.101800i	0.998704	+	0.0508998i	$0.0162089\pi$
$593$	2.47898i	0.101800i	−0.998704	+	0.0508998i	$0.983791\pi$
$594$	4.57853	0.187859
$595$	0	0
$596$	3.97759	0.162928
$597$	3.57125i	0.146162i
$598$	− 0.312532i	− 0.0127804i
$599$	−30.2951	−1.23782	−0.618912	−	0.785460i	$-0.712426\pi$
$599$	−30.2951	−1.23782	−0.618912	+	0.785460i	$0.712426\pi$
$600$	0	0
$601$	4.46130	0.181980	0.0909900	−	0.995852i	$-0.470997\pi$
$601$	4.46130	0.181980	0.0909900	+	0.995852i	$0.470997\pi$
$602$	− 13.6710i	− 0.557187i
$603$	1.79282i	0.0730094i
$604$	−7.43922	−0.302698
$605$	0	0
$606$	−10.0636	−0.408805
$607$	− 17.2931i	− 0.701906i	−0.936393	−	0.350953i	$-0.885858\pi$
$607$	− 17.2931i	− 0.701906i	0.936393	−	0.350953i	$-0.114142\pi$
$608$	1.53518i	0.0622598i
$609$	11.5910	0.469691
$610$	0	0
$611$	−1.99529	−0.0807210
$612$	− 2.71836i	− 0.109883i
$613$	14.3129i	0.578094i	0.957315	+	0.289047i	$0.0933383\pi$
$613$	14.3129i	0.578094i	−0.957315	+	0.289047i	$0.906662\pi$
$614$	−33.5415	−1.35362
$615$	0	0
$616$	−12.6256	−0.508700
$617$	26.3569i	1.06109i	0.847658	+	0.530544i	$0.178012\pi$
$617$	26.3569i	1.06109i	−0.847658	+	0.530544i	$0.821988\pi$
$618$	− 1.07761i	− 0.0433477i
$619$	2.79825	0.112471	0.0562356	−	0.998418i	$-0.482090\pi$
$619$	2.79825	0.112471	0.0562356	+	0.998418i	$0.482090\pi$
$620$	0	0
$621$	−0.872721	−0.0350211
$622$	15.4403i	0.619101i
$623$	20.4061i	0.817553i
$624$	−1.07039	−0.0428497
$625$	0	0
$626$	−9.30856	−0.372045
$627$	− 2.24368i	− 0.0896039i
$628$	1.27533i	0.0508914i
$629$	−75.5386	−3.01192
$630$	0	0
$631$	35.4035	1.40939	0.704696	−	0.709510i	$-0.251084\pi$
$631$	35.4035	1.40939	0.704696	+	0.709510i	$0.251084\pi$
$632$	− 18.5499i	− 0.737877i
$633$	− 3.72643i	− 0.148112i
$634$	−13.0688	−0.519028
$635$	0	0
$636$	−2.17890	−0.0863989
$637$	0.968220i	0.0383623i
$638$	31.4766i	1.24617i
$639$	4.37450	0.173052
$640$	0	0
$641$	16.9334	0.668829	0.334415	−	0.942426i	$-0.391461\pi$
$641$	16.9334	0.668829	0.334415	+	0.942426i	$0.391461\pi$
$642$	− 19.2652i	− 0.760336i
$643$	25.9118i	1.02186i	0.859622	+	0.510931i	$0.170699\pi$
$643$	25.9118i	1.02186i	−0.859622	+	0.510931i	$0.829301\pi$
$644$	−0.536241	−0.0211309
$645$	0	0
$646$	−8.64260	−0.340038
$647$	− 10.9259i	− 0.429542i	−0.976664	−	0.214771i	$-0.931100\pi$
$647$	− 10.9259i	− 0.429542i	0.976664	−	0.214771i	$-0.0689004\pi$
$648$	2.51496i	0.0987969i
$649$	−11.4671	−0.450121
$650$	0	0
$651$	16.5554	0.648857
$652$	− 2.17701i	− 0.0852582i
$653$	14.2580i	0.557958i	0.960297	+	0.278979i	$0.0899960\pi$
$653$	14.2580i	0.557958i	−0.960297	+	0.278979i	$0.910004\pi$
$654$	−6.58659	−0.257556
$655$	0	0
$656$	−17.4527	−0.681414
$657$	15.0528i	0.587267i
$658$	22.2114i	0.865890i
$659$	−22.8894	−0.891644	−0.445822	−	0.895122i	$-0.647088\pi$
$659$	−22.8894	−0.891644	−0.445822	+	0.895122i	$0.647088\pi$
$660$	0	0
$661$	−39.9050	−1.55213	−0.776063	−	0.630656i	$-0.782786\pi$
$661$	−39.9050	−1.55213	−0.776063	+	0.630656i	$0.782786\pi$
$662$	25.0555i	0.973811i
$663$	− 1.73714i	− 0.0674651i
$664$	−10.9263	−0.424021
$665$	0	0
$666$	−15.5723	−0.603413
$667$	− 5.99980i	− 0.232313i
$668$	− 0.851941i	− 0.0329626i
$669$	26.2488	1.01484
$670$	0	0
$671$	13.0907	0.505360
$672$	3.43495i	0.132506i
$673$	− 2.15847i	− 0.0832029i	−0.999134	−	0.0416015i	$-0.986754\pi$
$673$	− 2.15847i	− 0.0832029i	0.999134	−	0.0416015i	$-0.0132460\pi$
$674$	37.9747	1.46273
$675$	0	0
$676$	−4.71795	−0.181460
$677$	− 41.9740i	− 1.61319i	−0.591103	−	0.806596i	$-0.701307\pi$
$677$	− 41.9740i	− 1.61319i	0.591103	−	0.806596i	$-0.298693\pi$
$678$	12.6514i	0.485873i
$679$	−15.9737	−0.613013
$680$	0	0
$681$	4.49111	0.172100
$682$	44.9579i	1.72153i
$683$	− 29.2062i	− 1.11754i	−0.829321	−	0.558772i	$-0.811273\pi$
$683$	− 29.2062i	− 1.11754i	0.829321	−	0.558772i	$-0.188727\pi$
$684$	−0.274615	−0.0105002
$685$	0	0
$686$	28.9258	1.10439
$687$	− 7.53935i	− 0.287644i
$688$	24.2361i	0.923991i
$689$	−1.39240	−0.0530464
$690$	0	0
$691$	44.5257	1.69384	0.846919	−	0.531721i	$-0.178455\pi$
$691$	44.5257	1.69384	0.846919	+	0.531721i	$0.178455\pi$
$692$	− 2.21251i	− 0.0841070i
$693$	− 5.02021i	− 0.190702i
$694$	−21.4454	−0.814055
$695$	0	0
$696$	−17.2899	−0.655372
$697$	− 28.3243i	− 1.07286i
$698$	− 28.4417i	− 1.07653i
$699$	10.3640	0.392004
$700$	0	0
$701$	−4.50567	−0.170177	−0.0850884	−	0.996373i	$-0.527117\pi$
$701$	−4.50567	−0.170177	−0.0850884	+	0.996373i	$0.527117\pi$
$702$	− 0.358112i	− 0.0135161i
$703$	7.63108i	0.287812i
$704$	18.0422	0.679992
$705$	0	0
$706$	0.679241	0.0255636
$707$	11.0344i	0.414990i
$708$	1.40351i	0.0527471i
$709$	51.2706	1.92551	0.962754	−	0.270380i	$-0.0871492\pi$
$709$	51.2706	1.92551	0.962754	+	0.270380i	$0.0871492\pi$
$710$	0	0
$711$	7.37584	0.276616
$712$	− 30.4390i	− 1.14075i
$713$	− 8.56950i	− 0.320930i
$714$	−19.3377	−0.723695
$715$	0	0
$716$	4.16319	0.155586
$717$	− 18.5738i	− 0.693651i
$718$	− 7.10600i	− 0.265194i
$719$	−35.2653	−1.31517	−0.657586	−	0.753379i	$-0.728422\pi$
$719$	−35.2653	−1.31517	−0.657586	+	0.753379i	$0.728422\pi$
$720$	0	0
$721$	−1.18156	−0.0440036
$722$	− 28.3427i	− 1.05481i
$723$	5.43543i	0.202146i
$724$	2.86088	0.106324
$725$	0	0
$726$	−3.28150	−0.121788
$727$	− 44.0038i	− 1.63201i	−0.578045	−	0.816005i	$-0.696184\pi$
$727$	− 44.0038i	− 1.63201i	0.578045	−	0.816005i	$-0.303816\pi$
$728$	0.987517i	0.0365998i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−39.3330	−1.45478
$732$	− 1.60223i	− 0.0592202i
$733$	− 27.8741i	− 1.02955i	−0.857325	−	0.514776i	$-0.827875\pi$
$733$	− 27.8741i	− 1.02955i	0.857325	−	0.514776i	$-0.172125\pi$
$734$	13.4961	0.498150
$735$	0	0
$736$	1.77802	0.0655386
$737$	5.33826i	0.196637i
$738$	− 5.83904i	− 0.214938i
$739$	32.4413	1.19337	0.596687	−	0.802474i	$-0.296484\pi$
$739$	32.4413	1.19337	0.596687	+	0.802474i	$0.296484\pi$
$740$	0	0
$741$	−0.175490	−0.00644680
$742$	15.5001i	0.569027i
$743$	9.09256i	0.333574i	0.985993	+	0.166787i	$0.0533392\pi$
$743$	9.09256i	0.333574i	−0.985993	+	0.166787i	$0.946661\pi$
$744$	−24.6951	−0.905366
$745$	0	0
$746$	23.8271	0.872370
$747$	− 4.34451i	− 0.158957i
$748$	− 8.09411i	− 0.295950i
$749$	−21.1236	−0.771840
$750$	0	0
$751$	−49.8861	−1.82037	−0.910185	−	0.414202i	$-0.864061\pi$
$751$	−49.8861	−1.82037	−0.910185	+	0.414202i	$0.864061\pi$
$752$	− 39.3766i	− 1.43592i
$753$	− 23.3577i	− 0.851201i
$754$	2.46196	0.0896591
$755$	0	0
$756$	−0.614448	−0.0223472
$757$	− 31.1239i	− 1.13122i	−0.824674	−	0.565608i	$-0.808642\pi$
$757$	− 31.1239i	− 1.13122i	0.824674	−	0.565608i	$-0.191358\pi$
$758$	41.9679i	1.52434i
$759$	−2.59859	−0.0943228
$760$	0	0
$761$	39.1269	1.41835	0.709175	−	0.705032i	$-0.249067\pi$
$761$	39.1269	1.41835	0.709175	+	0.705032i	$0.249067\pi$
$762$	− 18.3724i	− 0.665561i
$763$	7.22197i	0.261453i
$764$	−4.57887	−0.165658
$765$	0	0
$766$	20.1499	0.728047
$767$	0.896901i	0.0323852i
$768$	− 8.47377i	− 0.305771i
$769$	−14.3382	−0.517050	−0.258525	−	0.966005i	$-0.583236\pi$
$769$	−14.3382	−0.517050	−0.258525	+	0.966005i	$0.583236\pi$
$770$	0	0
$771$	−4.48380	−0.161480
$772$	3.69677i	0.133050i
$773$	− 4.24997i	− 0.152861i	−0.997075	−	0.0764304i	$-0.975648\pi$
$773$	− 4.24997i	− 0.152861i	0.997075	−	0.0764304i	$-0.0243523\pi$
$774$	−8.10849	−0.291454
$775$	0	0
$776$	23.8274	0.855352
$777$	17.0745i	0.612543i
$778$	− 25.9953i	− 0.931975i
$779$	−2.86138	−0.102520
$780$	0	0
$781$	13.0254	0.466085
$782$	10.0097i	0.357946i
$783$	− 6.87482i	− 0.245686i
$784$	−19.1076	−0.682414
$785$	0	0
$786$	32.9944	1.17687
$787$	27.1296i	0.967067i	0.875326	+	0.483534i	$0.160647\pi$
$787$	27.1296i	0.967067i	−0.875326	+	0.483534i	$0.839353\pi$
$788$	− 0.746017i	− 0.0265758i
$789$	−7.72550	−0.275035
$790$	0	0
$791$	13.8718	0.493225
$792$	7.48847i	0.266091i
$793$	− 1.02389i	− 0.0363595i
$794$	16.0038	0.567954
$795$	0	0
$796$	1.30151	0.0461306
$797$	12.6175i	0.446935i	0.974711	+	0.223468i	$0.0717377\pi$
$797$	12.6175i	0.446935i	−0.974711	+	0.223468i	$0.928262\pi$
$798$	1.95354i	0.0691545i
$799$	63.9049	2.26079
$800$	0	0
$801$	12.1032	0.427646
$802$	− 1.06838i	− 0.0377257i
$803$	44.8209i	1.58169i
$804$	0.653376	0.0230428
$805$	0	0
$806$	3.51640	0.123860
$807$	10.7394i	0.378044i
$808$	− 16.4596i	− 0.579047i
$809$	37.4138	1.31540	0.657699	−	0.753280i	$-0.271530\pi$
$809$	37.4138	1.31540	0.657699	+	0.753280i	$0.271530\pi$
$810$	0	0
$811$	37.1992	1.30624	0.653120	−	0.757254i	$-0.273460\pi$
$811$	37.1992	1.30624	0.653120	+	0.757254i	$0.273460\pi$
$812$	− 4.22422i	− 0.148241i
$813$	25.7126i	0.901780i
$814$	−46.3675	−1.62518
$815$	0	0
$816$	34.2821	1.20011
$817$	3.97351i	0.139016i
$818$	1.82845i	0.0639302i
$819$	−0.392657	−0.0137206
$820$	0	0
$821$	27.4740	0.958850	0.479425	−	0.877583i	$-0.340845\pi$
$821$	27.4740	0.958850	0.479425	+	0.877583i	$0.340845\pi$
$822$	− 15.4063i	− 0.537355i
$823$	− 23.4267i	− 0.816602i	−0.912847	−	0.408301i	$-0.866121\pi$
$823$	− 23.4267i	− 0.816602i	0.912847	−	0.408301i	$-0.133879\pi$
$824$	1.76249	0.0613993
$825$	0	0
$826$	9.98420	0.347395
$827$	− 32.4570i	− 1.12864i	−0.825556	−	0.564320i	$-0.809138\pi$
$827$	− 32.4570i	− 1.12864i	0.825556	−	0.564320i	$-0.190862\pi$
$828$	0.318054i	0.0110531i
$829$	0.154106	0.00535231	0.00267615	−	0.999996i	$-0.499148\pi$
$829$	0.154106	0.00535231	0.00267615	+	0.999996i	$0.499148\pi$
$830$	0	0
$831$	−9.62446	−0.333869
$832$	− 1.41118i	− 0.0489239i
$833$	− 31.0100i	− 1.07443i
$834$	10.1097	0.350070
$835$	0	0
$836$	−0.817686	−0.0282802
$837$	− 9.81929i	− 0.339404i
$838$	− 3.04214i	− 0.105089i
$839$	35.4217	1.22289	0.611447	−	0.791285i	$-0.290588\pi$
$839$	35.4217	1.22289	0.611447	+	0.791285i	$0.290588\pi$
$840$	0	0
$841$	18.2632	0.629765
$842$	17.0412i	0.587279i
$843$	2.83275i	0.0975651i
$844$	−1.35806	−0.0467462
$845$	0	0
$846$	13.1740	0.452930
$847$	3.59806i	0.123631i
$848$	− 27.4788i	− 0.943624i
$849$	23.6634	0.812125
$850$	0	0
$851$	8.83818	0.302969
$852$	− 1.59424i	− 0.0546178i
$853$	36.6066i	1.25339i	0.779266	+	0.626694i	$0.215592\pi$
$853$	36.6066i	1.25339i	−0.779266	+	0.626694i	$0.784408\pi$
$854$	−11.3979	−0.390027
$855$	0	0
$856$	31.5094	1.07697
$857$	− 2.04867i	− 0.0699813i	−0.999388	−	0.0349907i	$-0.988860\pi$
$857$	− 2.04867i	− 0.0699813i	0.999388	−	0.0349907i	$-0.0111402\pi$
$858$	− 1.06630i	− 0.0364030i
$859$	−14.1821	−0.483888	−0.241944	−	0.970290i	$-0.577785\pi$
$859$	−14.1821	−0.483888	−0.241944	+	0.970290i	$0.577785\pi$
$860$	0	0
$861$	−6.40231	−0.218190
$862$	− 16.0128i	− 0.545398i
$863$	− 43.1358i	− 1.46836i	−0.678955	−	0.734180i	$-0.737567\pi$
$863$	− 43.1358i	− 1.46836i	0.678955	−	0.734180i	$-0.262433\pi$
$864$	2.03733	0.0693113
$865$	0	0
$866$	−15.7832	−0.536334
$867$	38.6369i	1.31218i
$868$	− 6.03344i	− 0.204788i
$869$	21.9621	0.745013
$870$	0	0
$871$	0.417534	0.0141476
$872$	− 10.7728i	− 0.364812i
$873$	9.47426i	0.320655i
$874$	1.01120	0.0342044
$875$	0	0
$876$	5.48584	0.185350
$877$	− 6.74680i	− 0.227823i	−0.993491	−	0.113912i	$-0.963662\pi$
$877$	− 6.74680i	− 0.227823i	0.993491	−	0.113912i	$-0.0363381\pi$
$878$	37.9983i	1.28238i
$879$	23.2376	0.783784
$880$	0	0
$881$	17.1783	0.578752	0.289376	−	0.957216i	$-0.406552\pi$
$881$	17.1783	0.578752	0.289376	+	0.957216i	$0.406552\pi$
$882$	− 6.39269i	− 0.215253i
$883$	56.8617i	1.91355i	0.290831	+	0.956774i	$0.406068\pi$
$883$	56.8617i	1.91355i	−0.290831	+	0.956774i	$0.593932\pi$
$884$	−0.633084	−0.0212929
$885$	0	0
$886$	11.5777	0.388960
$887$	36.9297i	1.23998i	0.784610	+	0.619989i	$0.212863\pi$
$887$	36.9297i	1.23998i	−0.784610	+	0.619989i	$0.787137\pi$
$888$	− 25.4694i	− 0.854696i
$889$	−20.1447	−0.675632
$890$	0	0
$891$	−2.97757	−0.0997524
$892$	− 9.56611i	− 0.320297i
$893$	− 6.45581i	− 0.216036i
$894$	−16.7826	−0.561293
$895$	0	0
$896$	−22.5790	−0.754312
$897$	0.203250i	0.00678631i
$898$	48.6869i	1.62470i
$899$	67.5058	2.25145
$900$	0	0
$901$	44.5956	1.48570
$902$	− 17.3862i	− 0.578895i
$903$	8.89069i	0.295864i
$904$	−20.6921	−0.688209
$905$	0	0
$906$	31.3882	1.04280
$907$	− 25.7833i	− 0.856120i	−0.903750	−	0.428060i	$-0.859197\pi$
$907$	− 25.7833i	− 0.856120i	0.903750	−	0.428060i	$-0.140803\pi$
$908$	− 1.63674i	− 0.0543170i
$909$	6.54468	0.217073
$910$	0	0
$911$	−27.0066	−0.894768	−0.447384	−	0.894342i	$-0.647644\pi$
$911$	−27.0066	−0.894768	−0.447384	+	0.894342i	$0.647644\pi$
$912$	− 3.46326i	− 0.114680i
$913$	− 12.9361i	− 0.428122i
$914$	−4.54755	−0.150420
$915$	0	0
$916$	−2.74764	−0.0907845
$917$	− 36.1772i	− 1.19468i
$918$	11.4695i	0.378551i
$919$	25.9342	0.855492	0.427746	−	0.903899i	$-0.359308\pi$
$919$	25.9342	0.855492	0.427746	+	0.903899i	$0.359308\pi$
$920$	0	0
$921$	21.8131	0.718767
$922$	− 27.0647i	− 0.891330i
$923$	− 1.01879i	− 0.0335337i
$924$	−1.82956	−0.0601882
$925$	0	0
$926$	−40.5055	−1.33109
$927$	0.700804i	0.0230174i
$928$	14.0063i	0.459778i
$929$	−15.0866	−0.494974	−0.247487	−	0.968891i	$-0.579605\pi$
$929$	−15.0866	−0.494974	−0.247487	+	0.968891i	$0.579605\pi$
$930$	0	0
$931$	−3.13270	−0.102670
$932$	− 3.77707i	− 0.123722i
$933$	− 10.0414i	− 0.328740i
$934$	11.3158	0.370265
$935$	0	0
$936$	0.585713	0.0191446
$937$	0.655563i	0.0214163i	0.999943	+	0.0107082i	$0.00340858\pi$
$937$	0.655563i	0.0214163i	−0.999943	+	0.0107082i	$0.996591\pi$
$938$	− 4.64795i	− 0.151761i
$939$	6.05366	0.197554
$940$	0	0
$941$	−36.4140	−1.18706	−0.593531	−	0.804811i	$-0.702267\pi$
$941$	−36.4140	−1.18706	−0.593531	+	0.804811i	$0.702267\pi$
$942$	− 5.38099i	− 0.175322i
$943$	3.31400i	0.107919i
$944$	−17.7001	−0.576089
$945$	0	0
$946$	−24.1436	−0.784976
$947$	− 45.4127i	− 1.47571i	−0.674957	−	0.737857i	$-0.735838\pi$
$947$	− 45.4127i	− 1.47571i	0.674957	−	0.737857i	$-0.264162\pi$
$948$	− 2.68805i	− 0.0873038i
$949$	3.50568	0.113799
$950$	0	0
$951$	8.49907	0.275601
$952$	− 31.6280i	− 1.02507i
$953$	− 46.2536i	− 1.49830i	−0.662399	−	0.749151i	$-0.730462\pi$
$953$	− 46.2536i	− 1.49830i	0.662399	−	0.749151i	$-0.269538\pi$
$954$	9.19338	0.297647
$955$	0	0
$956$	−6.76903	−0.218926
$957$	− 20.4703i	− 0.661710i
$958$	43.7730i	1.41424i
$959$	−16.8924	−0.545485
$960$	0	0
$961$	65.4184	2.11027
$962$	3.62665i	0.116928i
$963$	12.5288i	0.403734i
$964$	1.98088	0.0638000
$965$	0	0
$966$	2.26255	0.0727965
$967$	2.91740i	0.0938172i	0.998899	+	0.0469086i	$0.0149370\pi$
$967$	2.91740i	0.0938172i	−0.998899	+	0.0469086i	$0.985063\pi$
$968$	− 5.36709i	− 0.172505i
$969$	5.62057	0.180559
$970$	0	0
$971$	−1.21820	−0.0390940	−0.0195470	−	0.999809i	$-0.506222\pi$
$971$	−1.21820	−0.0390940	−0.0195470	+	0.999809i	$0.506222\pi$
$972$	0.364440i	0.0116894i
$973$	− 11.0849i	− 0.355367i
$974$	3.66591	0.117463
$975$	0	0
$976$	20.2063	0.646787
$977$	− 44.6720i	− 1.42918i	−0.699542	−	0.714591i	$-0.746613\pi$
$977$	− 44.6720i	− 1.42918i	0.699542	−	0.714591i	$-0.253387\pi$
$978$	9.18540i	0.293717i
$979$	36.0382	1.15178
$980$	0	0
$981$	4.28348	0.136761
$982$	− 39.5538i	− 1.26221i
$983$	− 36.1310i	− 1.15240i	−0.817308	−	0.576201i	$-0.804535\pi$
$983$	− 36.1310i	− 1.15240i	0.817308	−	0.576201i	$-0.195465\pi$
$984$	9.55010	0.304446
$985$	0	0
$986$	−78.8510	−2.51113
$987$	− 14.4448i	− 0.459783i
$988$	0.0639556i	0.00203470i
$989$	4.60205	0.146337
$990$	0	0
$991$	−40.0195	−1.27126	−0.635631	−	0.771993i	$-0.719260\pi$
$991$	−40.0195	−1.27126	−0.635631	+	0.771993i	$0.719260\pi$
$992$	20.0051i	0.635163i
$993$	− 16.2945i	− 0.517089i
$994$	−11.3410	−0.359715
$995$	0	0
$996$	−1.58331	−0.0501692
$997$	− 3.24338i	− 0.102719i	−0.998680	−	0.0513595i	$-0.983645\pi$
$997$	− 3.24338i	− 0.102719i	0.998680	−	0.0513595i	$-0.0163554\pi$
$998$	− 46.1285i	− 1.46017i
$999$	10.1272	0.320409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.h.1249.14		16
5.2	odd	4		1875.2.a.p.1.1		8
5.3	odd	4		1875.2.a.m.1.8		8
5.4	even	2	inner	1875.2.b.h.1249.3		16
15.2	even	4		5625.2.a.t.1.8		8
15.8	even	4		5625.2.a.bd.1.1		8
25.3	odd	20		375.2.g.e.76.1		16
25.4	even	10		75.2.i.a.34.1	✓	16
25.6	even	5		75.2.i.a.64.1	yes	16
25.8	odd	20		375.2.g.e.301.1		16
25.17	odd	20		375.2.g.d.301.4		16
25.19	even	10		375.2.i.c.199.4		16
25.21	even	5		375.2.i.c.49.4		16
25.22	odd	20		375.2.g.d.76.4		16
75.29	odd	10		225.2.m.b.109.4		16
75.56	odd	10		225.2.m.b.64.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.1	✓	16	25.4	even	10
75.2.i.a.64.1	yes	16	25.6	even	5
225.2.m.b.64.4		16	75.56	odd	10
225.2.m.b.109.4		16	75.29	odd	10
375.2.g.d.76.4		16	25.22	odd	20
375.2.g.d.301.4		16	25.17	odd	20
375.2.g.e.76.1		16	25.3	odd	20
375.2.g.e.301.1		16	25.8	odd	20
375.2.i.c.49.4		16	25.21	even	5
375.2.i.c.199.4		16	25.19	even	10
1875.2.a.m.1.8		8	5.3	odd	4
1875.2.a.p.1.1		8	5.2	odd	4
1875.2.b.h.1249.3		16	5.4	even	2	inner
1875.2.b.h.1249.14		16	1.1	even	1	trivial
5625.2.a.t.1.8		8	15.2	even	4
5625.2.a.bd.1.1		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)

\(f(q)\)	\(=\)	\(q+1.53767i q^{2} -1.00000i q^{3} -0.364440 q^{4} +1.53767 q^{6} -1.68601i q^{7} +2.51496i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.53767i q^{2} -1.00000i q^{3} -0.364440 q^{4} +1.53767 q^{6} -1.68601i q^{7} +2.51496i q^{8} -1.00000 q^{9} -2.97757 q^{11} +0.364440i q^{12} +0.232892i q^{13} +2.59253 q^{14} -4.59606 q^{16} -7.45901i q^{17} -1.53767i q^{18} -0.753527 q^{19} -1.68601 q^{21} -4.57853i q^{22} +0.872721i q^{23} +2.51496 q^{24} -0.358112 q^{26} +1.00000i q^{27} +0.614448i q^{28} -6.87482 q^{29} -9.81929 q^{31} -2.03733i q^{32} +2.97757i q^{33} +11.4695 q^{34} +0.364440 q^{36} -10.1272i q^{37} -1.15868i q^{38} +0.232892 q^{39} +3.79732 q^{41} -2.59253i q^{42} -5.27322i q^{43} +1.08514 q^{44} -1.34196 q^{46} +8.56747i q^{47} +4.59606i q^{48} +4.15738 q^{49} -7.45901 q^{51} -0.0848751i q^{52} +5.97876i q^{53} -1.53767 q^{54} +4.24024 q^{56} +0.753527i q^{57} -10.5712i q^{58} +3.85114 q^{59} -4.39643 q^{61} -15.0989i q^{62} +1.68601i q^{63} -6.05938 q^{64} -4.57853 q^{66} -1.79282i q^{67} +2.71836i q^{68} +0.872721 q^{69} -4.37450 q^{71} -2.51496i q^{72} -15.0528i q^{73} +15.5723 q^{74} +0.274615 q^{76} +5.02021i q^{77} +0.358112i q^{78} -7.37584 q^{79} +1.00000 q^{81} +5.83904i q^{82} +4.34451i q^{83} +0.614448 q^{84} +8.10849 q^{86} +6.87482i q^{87} -7.48847i q^{88} -12.1032 q^{89} +0.392657 q^{91} -0.318054i q^{92} +9.81929i q^{93} -13.1740 q^{94} -2.03733 q^{96} -9.47426i q^{97} +6.39269i q^{98} +2.97757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 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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