LMFDB - Embedded newform 3750.2.c.k.1249.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1249,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, 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("3750.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.c (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:	$29.9439007580$
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} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 $ x^16 - 4x^15 - 24x^14 + 94x^13 + 262x^12 - 936x^11 - 1584x^10 + 4642x^9 + 6259x^8 - 11958x^7 - 15752x^6 + 14670x^5 + 18271x^4 - 10440x^3 + 1135x^2 + 21080*x + 11105
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.15
Root			$-0.705457 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.k.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.23143i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.23143i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.30029 q^{11} +1.00000i q^{12} -5.90441i q^{13} -3.23143 q^{14} +1.00000 q^{16} +1.02469i q^{17} -1.00000i q^{18} +4.50718 q^{19} +3.23143 q^{21} +5.30029i q^{22} -1.92827i q^{23} -1.00000 q^{24} +5.90441 q^{26} +1.00000i q^{27} -3.23143i q^{28} +0.260762 q^{29} -2.24205 q^{31} +1.00000i q^{32} -5.30029i q^{33} -1.02469 q^{34} +1.00000 q^{36} -9.55354i q^{37} +4.50718i q^{38} -5.90441 q^{39} -9.96112 q^{41} +3.23143i q^{42} +5.30164i q^{43} -5.30029 q^{44} +1.92827 q^{46} -10.5282i q^{47} -1.00000i q^{48} -3.44213 q^{49} +1.02469 q^{51} +5.90441i q^{52} +2.31107i q^{53} -1.00000 q^{54} +3.23143 q^{56} -4.50718i q^{57} +0.260762i q^{58} +6.49202 q^{59} -3.94494 q^{61} -2.24205i q^{62} -3.23143i q^{63} -1.00000 q^{64} +5.30029 q^{66} +9.29210i q^{67} -1.02469i q^{68} -1.92827 q^{69} +8.53771 q^{71} +1.00000i q^{72} -3.95422i q^{73} +9.55354 q^{74} -4.50718 q^{76} +17.1275i q^{77} -5.90441i q^{78} +13.3714 q^{79} +1.00000 q^{81} -9.96112i q^{82} -7.48698i q^{83} -3.23143 q^{84} -5.30164 q^{86} -0.260762i q^{87} -5.30029i q^{88} +0.733574 q^{89} +19.0797 q^{91} +1.92827i q^{92} +2.24205i q^{93} +10.5282 q^{94} +1.00000 q^{96} -17.2567i q^{97} -3.44213i q^{98} -5.30029 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	$ 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$.

$n$	$2501$	$3127$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	3.23143i	1.22136i	0.791876	+	0.610682i	$0.209105\pi$
$7$	3.23143i	1.22136i	−0.791876	+	0.610682i	$0.790895\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.30029	1.59810	0.799048	−	0.601267i	$-0.205337\pi$
$11$	5.30029	1.59810	0.799048	+	0.601267i	$0.205337\pi$
$12$	1.00000i	0.288675i
$13$	− 5.90441i	− 1.63759i	−0.574087	−	0.818795i	$-0.694643\pi$
$13$	− 5.90441i	− 1.63759i	0.574087	−	0.818795i	$-0.305357\pi$
$14$	−3.23143	−0.863635
$15$	0	0
$16$	1.00000	0.250000
$17$	1.02469i	0.248525i	0.992249	+	0.124262i	$0.0396564\pi$
$17$	1.02469i	0.248525i	−0.992249	+	0.124262i	$0.960344\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.50718	1.03402	0.517010	−	0.855980i	$-0.327045\pi$
$19$	4.50718	1.03402	0.517010	+	0.855980i	$0.327045\pi$
$20$	0	0
$21$	3.23143	0.705155
$22$	5.30029i	1.13002i
$23$	− 1.92827i	− 0.402073i	−0.979584	−	0.201037i	$-0.935569\pi$
$23$	− 1.92827i	− 0.402073i	0.979584	−	0.201037i	$-0.0644310\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	5.90441	1.15795
$27$	1.00000i	0.192450i
$28$	− 3.23143i	− 0.610682i
$29$	0.260762	0.0484222	0.0242111	−	0.999707i	$-0.492293\pi$
$29$	0.260762	0.0484222	0.0242111	+	0.999707i	$0.492293\pi$
$30$	0	0
$31$	−2.24205	−0.402684	−0.201342	−	0.979521i	$-0.564530\pi$
$31$	−2.24205	−0.402684	−0.201342	+	0.979521i	$0.564530\pi$
$32$	1.00000i	0.176777i
$33$	− 5.30029i	− 0.922661i
$34$	−1.02469	−0.175734
$35$	0	0
$36$	1.00000	0.166667
$37$	− 9.55354i	− 1.57059i	−0.619120	−	0.785296i	$-0.712511\pi$
$37$	− 9.55354i	− 1.57059i	0.619120	−	0.785296i	$-0.287489\pi$
$38$	4.50718i	0.731162i
$39$	−5.90441	−0.945463
$40$	0	0
$41$	−9.96112	−1.55567	−0.777833	−	0.628471i	$-0.783681\pi$
$41$	−9.96112	−1.55567	−0.777833	+	0.628471i	$0.783681\pi$
$42$	3.23143i	0.498620i
$43$	5.30164i	0.808492i	0.914650	+	0.404246i	$0.132466\pi$
$43$	5.30164i	0.808492i	−0.914650	+	0.404246i	$0.867534\pi$
$44$	−5.30029	−0.799048
$45$	0	0
$46$	1.92827	0.284309
$47$	− 10.5282i	− 1.53569i	−0.640635	−	0.767846i	$-0.721329\pi$
$47$	− 10.5282i	− 1.53569i	0.640635	−	0.767846i	$-0.278671\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−3.44213	−0.491732
$50$	0	0
$51$	1.02469	0.143486
$52$	5.90441i	0.818795i
$53$	2.31107i	0.317450i	0.987323	+	0.158725i	$0.0507383\pi$
$53$	2.31107i	0.317450i	−0.987323	+	0.158725i	$0.949262\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.23143	0.431818
$57$	− 4.50718i	− 0.596991i
$58$	0.260762i	0.0342397i
$59$	6.49202	0.845190	0.422595	−	0.906319i	$-0.361119\pi$
$59$	6.49202	0.845190	0.422595	+	0.906319i	$0.361119\pi$
$60$	0	0
$61$	−3.94494	−0.505098	−0.252549	−	0.967584i	$-0.581269\pi$
$61$	−3.94494	−0.505098	−0.252549	+	0.967584i	$0.581269\pi$
$62$	− 2.24205i	− 0.284741i
$63$	− 3.23143i	− 0.407122i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.30029	0.652420
$67$	9.29210i	1.13521i	0.823301	+	0.567605i	$0.192130\pi$
$67$	9.29210i	1.13521i	−0.823301	+	0.567605i	$0.807870\pi$
$68$	− 1.02469i	− 0.124262i
$69$	−1.92827	−0.232137
$70$	0	0
$71$	8.53771	1.01324	0.506620	−	0.862170i	$-0.330895\pi$
$71$	8.53771	1.01324	0.506620	+	0.862170i	$0.330895\pi$
$72$	1.00000i	0.117851i
$73$	− 3.95422i	− 0.462806i	−0.972858	−	0.231403i	$-0.925668\pi$
$73$	− 3.95422i	− 0.462806i	0.972858	−	0.231403i	$-0.0743316\pi$
$74$	9.55354	1.11058
$75$	0	0
$76$	−4.50718	−0.517010
$77$	17.1275i	1.95186i
$78$	− 5.90441i	− 0.668543i
$79$	13.3714	1.50440	0.752199	−	0.658936i	$-0.228993\pi$
$79$	13.3714	1.50440	0.752199	+	0.658936i	$0.228993\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 9.96112i	− 1.10002i
$83$	− 7.48698i	− 0.821803i	−0.911680	−	0.410901i	$-0.865214\pi$
$83$	− 7.48698i	− 0.821803i	0.911680	−	0.410901i	$-0.134786\pi$
$84$	−3.23143	−0.352578
$85$	0	0
$86$	−5.30164	−0.571691
$87$	− 0.260762i	− 0.0279566i
$88$	− 5.30029i	− 0.565012i
$89$	0.733574	0.0777587	0.0388794	−	0.999244i	$-0.487621\pi$
$89$	0.733574	0.0777587	0.0388794	+	0.999244i	$0.487621\pi$
$90$	0	0
$91$	19.0797	2.00009
$92$	1.92827i	0.201037i
$93$	2.24205i	0.232490i
$94$	10.5282	1.08590
$95$	0	0
$96$	1.00000	0.102062
$97$	− 17.2567i	− 1.75215i	−0.482174	−	0.876075i	$-0.660153\pi$
$97$	− 17.2567i	− 1.75215i	0.482174	−	0.876075i	$-0.339847\pi$
$98$	− 3.44213i	− 0.347707i
$99$	−5.30029	−0.532699
$100$	0	0
$101$	11.8454	1.17866	0.589331	−	0.807892i	$-0.299391\pi$
$101$	11.8454	1.17866	0.589331	+	0.807892i	$0.299391\pi$
$102$	1.02469i	0.101460i
$103$	7.15084i	0.704593i	0.935888	+	0.352297i	$0.114599\pi$
$103$	7.15084i	0.704593i	−0.935888	+	0.352297i	$0.885401\pi$
$104$	−5.90441	−0.578975
$105$	0	0
$106$	−2.31107	−0.224471
$107$	3.20058i	0.309412i	0.987961	+	0.154706i	$0.0494430\pi$
$107$	3.20058i	0.309412i	−0.987961	+	0.154706i	$0.950557\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−9.89555	−0.947822	−0.473911	−	0.880573i	$-0.657158\pi$
$109$	−9.89555	−0.947822	−0.473911	+	0.880573i	$0.657158\pi$
$110$	0	0
$111$	−9.55354	−0.906782
$112$	3.23143i	0.305341i
$113$	4.24575i	0.399406i	0.979856	+	0.199703i	$0.0639978\pi$
$113$	4.24575i	0.399406i	−0.979856	+	0.199703i	$0.936002\pi$
$114$	4.50718	0.422137
$115$	0	0
$116$	−0.260762	−0.0242111
$117$	5.90441i	0.545863i
$118$	6.49202i	0.597639i
$119$	−3.31122	−0.303539
$120$	0	0
$121$	17.0930	1.55391
$122$	− 3.94494i	− 0.357158i
$123$	9.96112i	0.898164i
$124$	2.24205	0.201342
$125$	0	0
$126$	3.23143	0.287878
$127$	− 9.25090i	− 0.820884i	−0.911887	−	0.410442i	$-0.865374\pi$
$127$	− 9.25090i	− 0.820884i	0.911887	−	0.410442i	$-0.134626\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	5.30164	0.466783
$130$	0	0
$131$	−4.86497	−0.425054	−0.212527	−	0.977155i	$-0.568169\pi$
$131$	−4.86497	−0.425054	−0.212527	+	0.977155i	$0.568169\pi$
$132$	5.30029i	0.461331i
$133$	14.5646i	1.26291i
$134$	−9.29210	−0.802715
$135$	0	0
$136$	1.02469	0.0878668
$137$	6.01666i	0.514038i	0.966406	+	0.257019i	$0.0827403\pi$
$137$	6.01666i	0.514038i	−0.966406	+	0.257019i	$0.917260\pi$
$138$	− 1.92827i	− 0.164146i
$139$	5.62720	0.477293	0.238646	−	0.971107i	$-0.423296\pi$
$139$	5.62720	0.477293	0.238646	+	0.971107i	$0.423296\pi$
$140$	0	0
$141$	−10.5282	−0.886632
$142$	8.53771i	0.716469i
$143$	− 31.2951i	− 2.61703i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	3.95422	0.327253
$147$	3.44213i	0.283902i
$148$	9.55354i	0.785296i
$149$	11.6556	0.954863	0.477432	−	0.878669i	$-0.341568\pi$
$149$	11.6556	0.954863	0.477432	+	0.878669i	$0.341568\pi$
$150$	0	0
$151$	5.52354	0.449499	0.224750	−	0.974417i	$-0.427844\pi$
$151$	5.52354	0.449499	0.224750	+	0.974417i	$0.427844\pi$
$152$	− 4.50718i	− 0.365581i
$153$	− 1.02469i	− 0.0828416i
$154$	−17.1275	−1.38017
$155$	0	0
$156$	5.90441	0.472731
$157$	− 6.84307i	− 0.546136i	−0.961995	−	0.273068i	$-0.911962\pi$
$157$	− 6.84307i	− 0.546136i	0.961995	−	0.273068i	$-0.0880384\pi$
$158$	13.3714i	1.06377i
$159$	2.31107	0.183280
$160$	0	0
$161$	6.23108	0.491078
$162$	1.00000i	0.0785674i
$163$	9.38717i	0.735260i	0.929972	+	0.367630i	$0.119831\pi$
$163$	9.38717i	0.735260i	−0.929972	+	0.367630i	$0.880169\pi$
$164$	9.96112	0.777833
$165$	0	0
$166$	7.48698	0.581102
$167$	6.82883i	0.528431i	0.964464	+	0.264215i	$0.0851129\pi$
$167$	6.82883i	0.528431i	−0.964464	+	0.264215i	$0.914887\pi$
$168$	− 3.23143i	− 0.249310i
$169$	−21.8621	−1.68170
$170$	0	0
$171$	−4.50718	−0.344673
$172$	− 5.30164i	− 0.404246i
$173$	− 18.8604i	− 1.43393i	−0.697108	−	0.716966i	$-0.745530\pi$
$173$	− 18.8604i	− 1.43393i	0.697108	−	0.716966i	$-0.254470\pi$
$174$	0.260762	0.0197683
$175$	0	0
$176$	5.30029	0.399524
$177$	− 6.49202i	− 0.487970i
$178$	0.733574i	0.0549837i
$179$	−0.941336	−0.0703588	−0.0351794	−	0.999381i	$-0.511200\pi$
$179$	−0.941336	−0.0703588	−0.0351794	+	0.999381i	$0.511200\pi$
$180$	0	0
$181$	14.1057	1.04847	0.524235	−	0.851573i	$-0.324351\pi$
$181$	14.1057	1.04847	0.524235	+	0.851573i	$0.324351\pi$
$182$	19.0797i	1.41428i
$183$	3.94494i	0.291618i
$184$	−1.92827	−0.142154
$185$	0	0
$186$	−2.24205	−0.164395
$187$	5.43117i	0.397166i
$188$	10.5282i	0.767846i
$189$	−3.23143	−0.235052
$190$	0	0
$191$	25.4719	1.84308	0.921541	−	0.388280i	$-0.126931\pi$
$191$	25.4719	1.84308	0.921541	+	0.388280i	$0.126931\pi$
$192$	1.00000i	0.0721688i
$193$	6.63439i	0.477554i	0.971074	+	0.238777i	$0.0767465\pi$
$193$	6.63439i	0.477554i	−0.971074	+	0.238777i	$0.923254\pi$
$194$	17.2567	1.23896
$195$	0	0
$196$	3.44213	0.245866
$197$	− 10.1777i	− 0.725133i	−0.931958	−	0.362567i	$-0.881900\pi$
$197$	− 10.1777i	− 0.725133i	0.931958	−	0.362567i	$-0.118100\pi$
$198$	− 5.30029i	− 0.376675i
$199$	2.52238	0.178807	0.0894035	−	0.995995i	$-0.471504\pi$
$199$	2.52238	0.178807	0.0894035	+	0.995995i	$0.471504\pi$
$200$	0	0
$201$	9.29210	0.655414
$202$	11.8454i	0.833440i
$203$	0.842633i	0.0591412i
$204$	−1.02469	−0.0717429
$205$	0	0
$206$	−7.15084	−0.498223
$207$	1.92827i	0.134024i
$208$	− 5.90441i	− 0.409397i
$209$	23.8894	1.65246
$210$	0	0
$211$	15.7638	1.08523	0.542613	−	0.839983i	$-0.317435\pi$
$211$	15.7638	1.08523	0.542613	+	0.839983i	$0.317435\pi$
$212$	− 2.31107i	− 0.158725i
$213$	− 8.53771i	− 0.584994i
$214$	−3.20058	−0.218787
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 7.24502i	− 0.491824i
$218$	− 9.89555i	− 0.670211i
$219$	−3.95422	−0.267201
$220$	0	0
$221$	6.05021	0.406981
$222$	− 9.55354i	− 0.641192i
$223$	9.29534i	0.622462i	0.950334	+	0.311231i	$0.100741\pi$
$223$	9.29534i	0.622462i	−0.950334	+	0.311231i	$0.899259\pi$
$224$	−3.23143	−0.215909
$225$	0	0
$226$	−4.24575	−0.282423
$227$	− 3.63809i	− 0.241468i	−0.992685	−	0.120734i	$-0.961475\pi$
$227$	− 3.63809i	− 0.241468i	0.992685	−	0.120734i	$-0.0385249\pi$
$228$	4.50718i	0.298496i
$229$	−9.93181	−0.656312	−0.328156	−	0.944623i	$-0.606427\pi$
$229$	−9.93181	−0.656312	−0.328156	+	0.944623i	$0.606427\pi$
$230$	0	0
$231$	17.1275	1.12691
$232$	− 0.260762i	− 0.0171198i
$233$	− 18.7893i	− 1.23093i	−0.788164	−	0.615465i	$-0.788968\pi$
$233$	− 18.7893i	− 1.23093i	0.788164	−	0.615465i	$-0.211032\pi$
$234$	−5.90441	−0.385984
$235$	0	0
$236$	−6.49202	−0.422595
$237$	− 13.3714i	− 0.868565i
$238$	− 3.31122i	− 0.214635i
$239$	9.75186	0.630795	0.315398	−	0.948960i	$-0.397862\pi$
$239$	9.75186	0.630795	0.315398	+	0.948960i	$0.397862\pi$
$240$	0	0
$241$	18.9211	1.21881	0.609407	−	0.792858i	$-0.291408\pi$
$241$	18.9211	1.21881	0.609407	+	0.792858i	$0.291408\pi$
$242$	17.0930i	1.09878i
$243$	− 1.00000i	− 0.0641500i
$244$	3.94494	0.252549
$245$	0	0
$246$	−9.96112	−0.635098
$247$	− 26.6123i	− 1.69330i
$248$	2.24205i	0.142370i
$249$	−7.48698	−0.474468
$250$	0	0
$251$	−1.98608	−0.125360	−0.0626801	−	0.998034i	$-0.519965\pi$
$251$	−1.98608	−0.125360	−0.0626801	+	0.998034i	$0.519965\pi$
$252$	3.23143i	0.203561i
$253$	− 10.2204i	− 0.642551i
$254$	9.25090	0.580453
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 19.2797i	− 1.20263i	−0.799010	−	0.601317i	$-0.794643\pi$
$257$	− 19.2797i	− 1.20263i	0.799010	−	0.601317i	$-0.205357\pi$
$258$	5.30164i	0.330066i
$259$	30.8716	1.91827
$260$	0	0
$261$	−0.260762	−0.0161407
$262$	− 4.86497i	− 0.300559i
$263$	− 12.5297i	− 0.772613i	−0.922370	−	0.386307i	$-0.873751\pi$
$263$	− 12.5297i	− 0.772613i	0.922370	−	0.386307i	$-0.126249\pi$
$264$	−5.30029	−0.326210
$265$	0	0
$266$	−14.5646	−0.893016
$267$	− 0.733574i	− 0.0448940i
$268$	− 9.29210i	− 0.567605i
$269$	−13.4929	−0.822680	−0.411340	−	0.911482i	$-0.634939\pi$
$269$	−13.4929	−0.822680	−0.411340	+	0.911482i	$0.634939\pi$
$270$	0	0
$271$	−26.8135	−1.62880	−0.814402	−	0.580301i	$-0.802935\pi$
$271$	−26.8135	−1.62880	−0.814402	+	0.580301i	$0.802935\pi$
$272$	1.02469i	0.0621312i
$273$	− 19.0797i	− 1.15475i
$274$	−6.01666	−0.363480
$275$	0	0
$276$	1.92827	0.116068
$277$	8.16724i	0.490722i	0.969432	+	0.245361i	$0.0789065\pi$
$277$	8.16724i	0.490722i	−0.969432	+	0.245361i	$0.921094\pi$
$278$	5.62720i	0.337497i
$279$	2.24205	0.134228
$280$	0	0
$281$	31.3418	1.86970	0.934848	−	0.355049i	$-0.115536\pi$
$281$	31.3418	1.86970	0.934848	+	0.355049i	$0.115536\pi$
$282$	− 10.5282i	− 0.626943i
$283$	16.3503i	0.971927i	0.873979	+	0.485964i	$0.161531\pi$
$283$	16.3503i	0.971927i	−0.873979	+	0.485964i	$0.838469\pi$
$284$	−8.53771	−0.506620
$285$	0	0
$286$	31.2951	1.85052
$287$	− 32.1886i	− 1.90004i
$288$	− 1.00000i	− 0.0589256i
$289$	15.9500	0.938235
$290$	0	0
$291$	−17.2567	−1.01160
$292$	3.95422i	0.231403i
$293$	28.1867i	1.64669i	0.567545	+	0.823343i	$0.307893\pi$
$293$	28.1867i	1.64669i	−0.567545	+	0.823343i	$0.692107\pi$
$294$	−3.44213	−0.200749
$295$	0	0
$296$	−9.55354	−0.555288
$297$	5.30029i	0.307554i
$298$	11.6556i	0.675190i
$299$	−11.3853	−0.658431
$300$	0	0
$301$	−17.1319	−0.987464
$302$	5.52354i	0.317844i
$303$	− 11.8454i	− 0.680501i
$304$	4.50718	0.258505
$305$	0	0
$306$	1.02469	0.0585778
$307$	− 14.5372i	− 0.829680i	−0.909894	−	0.414840i	$-0.863837\pi$
$307$	− 14.5372i	− 0.829680i	0.909894	−	0.414840i	$-0.136163\pi$
$308$	− 17.1275i	− 0.975929i
$309$	7.15084	0.406797
$310$	0	0
$311$	14.4924	0.821791	0.410896	−	0.911682i	$-0.365216\pi$
$311$	14.4924	0.821791	0.410896	+	0.911682i	$0.365216\pi$
$312$	5.90441i	0.334272i
$313$	31.5288i	1.78211i	0.453892	+	0.891057i	$0.350035\pi$
$313$	31.5288i	1.78211i	−0.453892	+	0.891057i	$0.649965\pi$
$314$	6.84307	0.386177
$315$	0	0
$316$	−13.3714	−0.752199
$317$	− 14.9810i	− 0.841419i	−0.907195	−	0.420710i	$-0.861781\pi$
$317$	− 14.9810i	− 0.841419i	0.907195	−	0.420710i	$-0.138219\pi$
$318$	2.31107i	0.129598i
$319$	1.38211	0.0773834
$320$	0	0
$321$	3.20058	0.178639
$322$	6.23108i	0.347245i
$323$	4.61848i	0.256979i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−9.38717	−0.519907
$327$	9.89555i	0.547225i
$328$	9.96112i	0.550011i
$329$	34.0210	1.87564
$330$	0	0
$331$	−21.6174	−1.18820	−0.594100	−	0.804391i	$-0.702492\pi$
$331$	−21.6174	−1.18820	−0.594100	+	0.804391i	$0.702492\pi$
$332$	7.48698i	0.410901i
$333$	9.55354i	0.523531i
$334$	−6.82883	−0.373657
$335$	0	0
$336$	3.23143	0.176289
$337$	− 7.59495i	− 0.413723i	−0.978370	−	0.206862i	$-0.933675\pi$
$337$	− 7.59495i	− 0.413723i	0.978370	−	0.206862i	$-0.0663250\pi$
$338$	− 21.8621i	− 1.18914i
$339$	4.24575	0.230597
$340$	0	0
$341$	−11.8835	−0.643528
$342$	− 4.50718i	− 0.243721i
$343$	11.4970i	0.620780i
$344$	5.30164	0.285845
$345$	0	0
$346$	18.8604	1.01394
$347$	− 7.53304i	− 0.404395i	−0.979345	−	0.202197i	$-0.935192\pi$
$347$	− 7.53304i	− 0.404395i	0.979345	−	0.202197i	$-0.0648083\pi$
$348$	0.260762i	0.0139783i
$349$	−18.7527	−1.00381	−0.501904	−	0.864923i	$-0.667367\pi$
$349$	−18.7527	−1.00381	−0.501904	+	0.864923i	$0.667367\pi$
$350$	0	0
$351$	5.90441	0.315154
$352$	5.30029i	0.282506i
$353$	− 4.36607i	− 0.232383i	−0.993227	−	0.116191i	$-0.962931\pi$
$353$	− 4.36607i	− 0.232383i	0.993227	−	0.116191i	$-0.0370686\pi$
$354$	6.49202	0.345047
$355$	0	0
$356$	−0.733574	−0.0388794
$357$	3.31122i	0.175249i
$358$	− 0.941336i	− 0.0497512i
$359$	−0.737212	−0.0389086	−0.0194543	−	0.999811i	$-0.506193\pi$
$359$	−0.737212	−0.0389086	−0.0194543	+	0.999811i	$0.506193\pi$
$360$	0	0
$361$	1.31471	0.0691954
$362$	14.1057i	0.741381i
$363$	− 17.0930i	− 0.897151i
$364$	−19.0797	−1.00005
$365$	0	0
$366$	−3.94494	−0.206205
$367$	18.8600i	0.984486i	0.870458	+	0.492243i	$0.163823\pi$
$367$	18.8600i	0.984486i	−0.870458	+	0.492243i	$0.836177\pi$
$368$	− 1.92827i	− 0.100518i
$369$	9.96112	0.518555
$370$	0	0
$371$	−7.46806	−0.387722
$372$	− 2.24205i	− 0.116245i
$373$	29.8366i	1.54488i	0.635086	+	0.772441i	$0.280965\pi$
$373$	29.8366i	1.54488i	−0.635086	+	0.772441i	$0.719035\pi$
$374$	−5.43117	−0.280839
$375$	0	0
$376$	−10.5282	−0.542949
$377$	− 1.53964i	− 0.0792958i
$378$	− 3.23143i	− 0.166207i
$379$	−25.4826	−1.30895	−0.654477	−	0.756082i	$-0.727111\pi$
$379$	−25.4826	−1.30895	−0.654477	+	0.756082i	$0.727111\pi$
$380$	0	0
$381$	−9.25090	−0.473938
$382$	25.4719i	1.30326i
$383$	− 31.6433i	− 1.61690i	−0.588566	−	0.808449i	$-0.700307\pi$
$383$	− 31.6433i	− 1.61690i	0.588566	−	0.808449i	$-0.299693\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.63439	−0.337682
$387$	− 5.30164i	− 0.269497i
$388$	17.2567i	0.876075i
$389$	−12.0480	−0.610858	−0.305429	−	0.952215i	$-0.598800\pi$
$389$	−12.0480	−0.610858	−0.305429	+	0.952215i	$0.598800\pi$
$390$	0	0
$391$	1.97589	0.0999251
$392$	3.44213i	0.173854i
$393$	4.86497i	0.245405i
$394$	10.1777	0.512747
$395$	0	0
$396$	5.30029	0.266349
$397$	25.7565i	1.29268i	0.763049	+	0.646341i	$0.223702\pi$
$397$	25.7565i	1.29268i	−0.763049	+	0.646341i	$0.776298\pi$
$398$	2.52238i	0.126436i
$399$	14.5646	0.729144
$400$	0	0
$401$	26.3196	1.31434	0.657169	−	0.753743i	$-0.271754\pi$
$401$	26.3196	1.31434	0.657169	+	0.753743i	$0.271754\pi$
$402$	9.29210i	0.463448i
$403$	13.2380i	0.659431i
$404$	−11.8454	−0.589331
$405$	0	0
$406$	−0.842633	−0.0418192
$407$	− 50.6365i	− 2.50996i
$408$	− 1.02469i	− 0.0507299i
$409$	−36.7714	−1.81823	−0.909113	−	0.416549i	$-0.863239\pi$
$409$	−36.7714	−1.81823	−0.909113	+	0.416549i	$0.863239\pi$
$410$	0	0
$411$	6.01666	0.296780
$412$	− 7.15084i	− 0.352297i
$413$	20.9785i	1.03229i
$414$	−1.92827	−0.0947695
$415$	0	0
$416$	5.90441	0.289488
$417$	− 5.62720i	− 0.275565i
$418$	23.8894i	1.16847i
$419$	31.7245	1.54984	0.774922	−	0.632057i	$-0.217789\pi$
$419$	31.7245	1.54984	0.774922	+	0.632057i	$0.217789\pi$
$420$	0	0
$421$	8.81726	0.429727	0.214864	−	0.976644i	$-0.431069\pi$
$421$	8.81726	0.429727	0.214864	+	0.976644i	$0.431069\pi$
$422$	15.7638i	0.767371i
$423$	10.5282i	0.511897i
$424$	2.31107	0.112236
$425$	0	0
$426$	8.53771	0.413653
$427$	− 12.7478i	− 0.616908i
$428$	− 3.20058i	− 0.154706i
$429$	−31.2951	−1.51094
$430$	0	0
$431$	23.5065	1.13227	0.566135	−	0.824313i	$-0.308438\pi$
$431$	23.5065	1.13227	0.566135	+	0.824313i	$0.308438\pi$
$432$	1.00000i	0.0481125i
$433$	8.40312i	0.403828i	0.979403	+	0.201914i	$0.0647162\pi$
$433$	8.40312i	0.403828i	−0.979403	+	0.201914i	$0.935284\pi$
$434$	7.24502	0.347772
$435$	0	0
$436$	9.89555	0.473911
$437$	− 8.69109i	− 0.415751i
$438$	− 3.95422i	− 0.188940i
$439$	−35.6325	−1.70065	−0.850324	−	0.526260i	$-0.823594\pi$
$439$	−35.6325	−1.70065	−0.850324	+	0.526260i	$0.823594\pi$
$440$	0	0
$441$	3.44213	0.163911
$442$	6.05021i	0.287779i
$443$	6.79550i	0.322864i	0.986884	+	0.161432i	$0.0516112\pi$
$443$	6.79550i	0.322864i	−0.986884	+	0.161432i	$0.948389\pi$
$444$	9.55354	0.453391
$445$	0	0
$446$	−9.29534	−0.440147
$447$	− 11.6556i	− 0.551291i
$448$	− 3.23143i	− 0.152671i
$449$	−8.75011	−0.412943	−0.206472	−	0.978453i	$-0.566198\pi$
$449$	−8.75011	−0.412943	−0.206472	+	0.978453i	$0.566198\pi$
$450$	0	0
$451$	−52.7968	−2.48610
$452$	− 4.24575i	− 0.199703i
$453$	− 5.52354i	− 0.259519i
$454$	3.63809	0.170744
$455$	0	0
$456$	−4.50718	−0.211068
$457$	38.1474i	1.78446i	0.451581	+	0.892230i	$0.350860\pi$
$457$	38.1474i	1.78446i	−0.451581	+	0.892230i	$0.649140\pi$
$458$	− 9.93181i	− 0.464083i
$459$	−1.02469	−0.0478286
$460$	0	0
$461$	−1.27077	−0.0591856	−0.0295928	−	0.999562i	$-0.509421\pi$
$461$	−1.27077	−0.0591856	−0.0295928	+	0.999562i	$0.509421\pi$
$462$	17.1275i	0.796843i
$463$	8.93349i	0.415175i	0.978216	+	0.207587i	$0.0665611\pi$
$463$	8.93349i	0.415175i	−0.978216	+	0.207587i	$0.933439\pi$
$464$	0.260762	0.0121056
$465$	0	0
$466$	18.7893	0.870399
$467$	5.75682i	0.266394i	0.991090	+	0.133197i	$0.0425243\pi$
$467$	5.75682i	0.266394i	−0.991090	+	0.133197i	$0.957476\pi$
$468$	− 5.90441i	− 0.272932i
$469$	−30.0268	−1.38651
$470$	0	0
$471$	−6.84307	−0.315312
$472$	− 6.49202i	− 0.298820i
$473$	28.1002i	1.29205i
$474$	13.3714	0.614168
$475$	0	0
$476$	3.31122	0.151770
$477$	− 2.31107i	− 0.105817i
$478$	9.75186i	0.446040i
$479$	4.63051	0.211573	0.105787	−	0.994389i	$-0.466264\pi$
$479$	4.63051	0.211573	0.105787	+	0.994389i	$0.466264\pi$
$480$	0	0
$481$	−56.4080	−2.57199
$482$	18.9211i	0.861832i
$483$	− 6.23108i	− 0.283524i
$484$	−17.0930	−0.776956
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 15.9235i	− 0.721564i	−0.932650	−	0.360782i	$-0.882510\pi$
$487$	− 15.9235i	− 0.721564i	0.932650	−	0.360782i	$-0.117490\pi$
$488$	3.94494i	0.178579i
$489$	9.38717	0.424503
$490$	0	0
$491$	−6.38076	−0.287959	−0.143980	−	0.989581i	$-0.545990\pi$
$491$	−6.38076	−0.287959	−0.143980	+	0.989581i	$0.545990\pi$
$492$	− 9.96112i	− 0.449082i
$493$	0.267201i	0.0120341i
$494$	26.6123	1.19734
$495$	0	0
$496$	−2.24205	−0.100671
$497$	27.5890i	1.23754i
$498$	− 7.48698i	− 0.335500i
$499$	−12.9135	−0.578087	−0.289044	−	0.957316i	$-0.593337\pi$
$499$	−12.9135	−0.578087	−0.289044	+	0.957316i	$0.593337\pi$
$500$	0	0
$501$	6.82883	0.305090
$502$	− 1.98608i	− 0.0886431i
$503$	− 1.45203i	− 0.0647429i	−0.999476	−	0.0323714i	$-0.989694\pi$
$503$	− 1.45203i	− 0.0647429i	0.999476	−	0.0323714i	$-0.0103059\pi$
$504$	−3.23143	−0.143939
$505$	0	0
$506$	10.2204	0.454352
$507$	21.8621i	0.970929i
$508$	9.25090i	0.410442i
$509$	12.4097	0.550051	0.275026	−	0.961437i	$-0.411314\pi$
$509$	12.4097	0.550051	0.275026	+	0.961437i	$0.411314\pi$
$510$	0	0
$511$	12.7778	0.565255
$512$	1.00000i	0.0441942i
$513$	4.50718i	0.198997i
$514$	19.2797	0.850391
$515$	0	0
$516$	−5.30164	−0.233392
$517$	− 55.8023i	− 2.45418i
$518$	30.8716i	1.35642i
$519$	−18.8604	−0.827881
$520$	0	0
$521$	7.94615	0.348127	0.174064	−	0.984734i	$-0.444310\pi$
$521$	7.94615	0.348127	0.174064	+	0.984734i	$0.444310\pi$
$522$	− 0.260762i	− 0.0114132i
$523$	− 34.8305i	− 1.52303i	−0.648145	−	0.761517i	$-0.724455\pi$
$523$	− 34.8305i	− 1.52303i	0.648145	−	0.761517i	$-0.275545\pi$
$524$	4.86497	0.212527
$525$	0	0
$526$	12.5297	0.546320
$527$	− 2.29741i	− 0.100077i
$528$	− 5.30029i	− 0.230665i
$529$	19.2818	0.838337
$530$	0	0
$531$	−6.49202	−0.281730
$532$	− 14.5646i	− 0.631457i
$533$	58.8146i	2.54754i
$534$	0.733574	0.0317449
$535$	0	0
$536$	9.29210	0.401358
$537$	0.941336i	0.0406216i
$538$	− 13.4929i	− 0.581722i
$539$	−18.2443	−0.785836
$540$	0	0
$541$	28.1723	1.21122	0.605610	−	0.795762i	$-0.292929\pi$
$541$	28.1723	1.21122	0.605610	+	0.795762i	$0.292929\pi$
$542$	− 26.8135i	− 1.15174i
$543$	− 14.1057i	− 0.605335i
$544$	−1.02469	−0.0439334
$545$	0	0
$546$	19.0797	0.816535
$547$	27.1778i	1.16204i	0.813890	+	0.581019i	$0.197346\pi$
$547$	27.1778i	1.16204i	−0.813890	+	0.581019i	$0.802654\pi$
$548$	− 6.01666i	− 0.257019i
$549$	3.94494	0.168366
$550$	0	0
$551$	1.17530	0.0500695
$552$	1.92827i	0.0820728i
$553$	43.2087i	1.83742i
$554$	−8.16724	−0.346993
$555$	0	0
$556$	−5.62720	−0.238646
$557$	− 6.74857i	− 0.285946i	−0.989727	−	0.142973i	$-0.954334\pi$
$557$	− 6.74857i	− 0.285946i	0.989727	−	0.142973i	$-0.0456663\pi$
$558$	2.24205i	0.0949135i
$559$	31.3031	1.32398
$560$	0	0
$561$	5.43117	0.229304
$562$	31.3418i	1.32207i
$563$	− 11.3076i	− 0.476561i	−0.971196	−	0.238280i	$-0.923416\pi$
$563$	− 11.3076i	− 0.476561i	0.971196	−	0.238280i	$-0.0765837\pi$
$564$	10.5282	0.443316
$565$	0	0
$566$	−16.3503	−0.687256
$567$	3.23143i	0.135707i
$568$	− 8.53771i	− 0.358234i
$569$	−25.2185	−1.05721	−0.528607	−	0.848867i	$-0.677286\pi$
$569$	−25.2185	−1.05721	−0.528607	+	0.848867i	$0.677286\pi$
$570$	0	0
$571$	−27.8001	−1.16340	−0.581699	−	0.813404i	$-0.697612\pi$
$571$	−27.8001	−1.16340	−0.581699	+	0.813404i	$0.697612\pi$
$572$	31.2951i	1.30851i
$573$	− 25.4719i	− 1.06410i
$574$	32.1886	1.34353
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 11.4935i	− 0.478479i	−0.970961	−	0.239240i	$-0.923102\pi$
$577$	− 11.4935i	− 0.478479i	0.970961	−	0.239240i	$-0.0768982\pi$
$578$	15.9500i	0.663433i
$579$	6.63439	0.275716
$580$	0	0
$581$	24.1936	1.00372
$582$	− 17.2567i	− 0.715313i
$583$	12.2493i	0.507316i
$584$	−3.95422	−0.163627
$585$	0	0
$586$	−28.1867	−1.16438
$587$	42.2281i	1.74294i	0.490450	+	0.871469i	$0.336832\pi$
$587$	42.2281i	1.74294i	−0.490450	+	0.871469i	$0.663168\pi$
$588$	− 3.44213i	− 0.141951i
$589$	−10.1053	−0.416383
$590$	0	0
$591$	−10.1777	−0.418656
$592$	− 9.55354i	− 0.392648i
$593$	0.0131121i	0 0.000538451i	1.00000		0.000269225i	$8.56971e-5\pi$
$593$	0.0131121i	0 0.000538451i	−1.00000		0.000269225i	$0.999914\pi$
$594$	−5.30029	−0.217473
$595$	0	0
$596$	−11.6556	−0.477432
$597$	− 2.52238i	− 0.103234i
$598$	− 11.3853i	− 0.465581i
$599$	−2.38072	−0.0972738	−0.0486369	−	0.998817i	$-0.515488\pi$
$599$	−2.38072	−0.0972738	−0.0486369	+	0.998817i	$0.515488\pi$
$600$	0	0
$601$	−19.6020	−0.799581	−0.399790	−	0.916607i	$-0.630917\pi$
$601$	−19.6020	−0.799581	−0.399790	+	0.916607i	$0.630917\pi$
$602$	− 17.1319i	− 0.698243i
$603$	− 9.29210i	− 0.378404i
$604$	−5.52354	−0.224750
$605$	0	0
$606$	11.8454	0.481187
$607$	10.7155i	0.434930i	0.976068	+	0.217465i	$0.0697788\pi$
$607$	10.7155i	0.434930i	−0.976068	+	0.217465i	$0.930221\pi$
$608$	4.50718i	0.182790i
$609$	0.842633	0.0341452
$610$	0	0
$611$	−62.1627	−2.51483
$612$	1.02469i	0.0414208i
$613$	− 1.16434i	− 0.0470273i	−0.999724	−	0.0235137i	$-0.992515\pi$
$613$	− 1.16434i	− 0.0470273i	0.999724	−	0.0235137i	$-0.00748532\pi$
$614$	14.5372	0.586673
$615$	0	0
$616$	17.1275	0.690086
$617$	22.6435i	0.911594i	0.890084	+	0.455797i	$0.150646\pi$
$617$	22.6435i	0.911594i	−0.890084	+	0.455797i	$0.849354\pi$
$618$	7.15084i	0.287649i
$619$	−24.7725	−0.995689	−0.497844	−	0.867266i	$-0.665875\pi$
$619$	−24.7725	−0.995689	−0.497844	+	0.867266i	$0.665875\pi$
$620$	0	0
$621$	1.92827	0.0773790
$622$	14.4924i	0.581094i
$623$	2.37049i	0.0949718i
$624$	−5.90441	−0.236366
$625$	0	0
$626$	−31.5288	−1.26014
$627$	− 23.8894i	− 0.954049i
$628$	6.84307i	0.273068i
$629$	9.78945	0.390331
$630$	0	0
$631$	−32.6515	−1.29983	−0.649917	−	0.760005i	$-0.725196\pi$
$631$	−32.6515	−1.29983	−0.649917	+	0.760005i	$0.725196\pi$
$632$	− 13.3714i	− 0.531885i
$633$	− 15.7638i	− 0.626556i
$634$	14.9810	0.594973
$635$	0	0
$636$	−2.31107	−0.0916400
$637$	20.3237i	0.805256i
$638$	1.38211i	0.0547183i
$639$	−8.53771	−0.337747
$640$	0	0
$641$	−26.8543	−1.06068	−0.530340	−	0.847785i	$-0.677936\pi$
$641$	−26.8543	−1.06068	−0.530340	+	0.847785i	$0.677936\pi$
$642$	3.20058i	0.126317i
$643$	− 17.1051i	− 0.674559i	−0.941405	−	0.337279i	$-0.890493\pi$
$643$	− 17.1051i	− 0.674559i	0.941405	−	0.337279i	$-0.109507\pi$
$644$	−6.23108	−0.245539
$645$	0	0
$646$	−4.61848	−0.181712
$647$	− 45.8135i	− 1.80112i	−0.434736	−	0.900558i	$-0.643158\pi$
$647$	− 45.8135i	− 1.80112i	0.434736	−	0.900558i	$-0.356842\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	34.4096	1.35069
$650$	0	0
$651$	−7.24502	−0.283955
$652$	− 9.38717i	− 0.367630i
$653$	11.4555i	0.448288i	0.974556	+	0.224144i	$0.0719587\pi$
$653$	11.4555i	0.448288i	−0.974556	+	0.224144i	$0.928041\pi$
$654$	−9.89555	−0.386947
$655$	0	0
$656$	−9.96112	−0.388916
$657$	3.95422i	0.154269i
$658$	34.0210i	1.32628i
$659$	−39.9982	−1.55811	−0.779054	−	0.626957i	$-0.784300\pi$
$659$	−39.9982	−1.55811	−0.779054	+	0.626957i	$0.784300\pi$
$660$	0	0
$661$	−5.69132	−0.221367	−0.110683	−	0.993856i	$-0.535304\pi$
$661$	−5.69132	−0.221367	−0.110683	+	0.993856i	$0.535304\pi$
$662$	− 21.6174i	− 0.840184i
$663$	− 6.05021i	− 0.234971i
$664$	−7.48698	−0.290551
$665$	0	0
$666$	−9.55354	−0.370192
$667$	− 0.502820i	− 0.0194693i
$668$	− 6.82883i	− 0.264215i
$669$	9.29534	0.359379
$670$	0	0
$671$	−20.9093	−0.807194
$672$	3.23143i	0.124655i
$673$	48.4527i	1.86772i	0.357645	+	0.933858i	$0.383580\pi$
$673$	48.4527i	1.86772i	−0.357645	+	0.933858i	$0.616420\pi$
$674$	7.59495	0.292547
$675$	0	0
$676$	21.8621	0.840849
$677$	− 18.5092i	− 0.711367i	−0.934607	−	0.355683i	$-0.884248\pi$
$677$	− 18.5092i	− 0.711367i	0.934607	−	0.355683i	$-0.115752\pi$
$678$	4.24575i	0.163057i
$679$	55.7637	2.14002
$680$	0	0
$681$	−3.63809	−0.139412
$682$	− 11.8835i	− 0.455043i
$683$	− 36.0242i	− 1.37843i	−0.724558	−	0.689214i	$-0.757956\pi$
$683$	− 36.0242i	− 1.37843i	0.724558	−	0.689214i	$-0.242044\pi$
$684$	4.50718	0.172337
$685$	0	0
$686$	−11.4970	−0.438958
$687$	9.93181i	0.378922i
$688$	5.30164i	0.202123i
$689$	13.6455	0.519853
$690$	0	0
$691$	−14.4695	−0.550446	−0.275223	−	0.961380i	$-0.588752\pi$
$691$	−14.4695	−0.550446	−0.275223	+	0.961380i	$0.588752\pi$
$692$	18.8604i	0.716966i
$693$	− 17.1275i	− 0.650620i
$694$	7.53304	0.285950
$695$	0	0
$696$	−0.260762	−0.00988415
$697$	− 10.2071i	− 0.386621i
$698$	− 18.7527i	− 0.709799i
$699$	−18.7893	−0.710678
$700$	0	0
$701$	8.31284	0.313972	0.156986	−	0.987601i	$-0.449822\pi$
$701$	8.31284	0.313972	0.156986	+	0.987601i	$0.449822\pi$
$702$	5.90441i	0.222848i
$703$	− 43.0596i	− 1.62402i
$704$	−5.30029	−0.199762
$705$	0	0
$706$	4.36607	0.164319
$707$	38.2776i	1.43958i
$708$	6.49202i	0.243985i
$709$	8.97215	0.336956	0.168478	−	0.985705i	$-0.446115\pi$
$709$	8.97215	0.336956	0.168478	+	0.985705i	$0.446115\pi$
$710$	0	0
$711$	−13.3714	−0.501466
$712$	− 0.733574i	− 0.0274919i
$713$	4.32329i	0.161908i
$714$	−3.31122	−0.123919
$715$	0	0
$716$	0.941336	0.0351794
$717$	− 9.75186i	− 0.364190i
$718$	− 0.737212i	− 0.0275125i
$719$	−28.1138	−1.04847	−0.524234	−	0.851574i	$-0.675648\pi$
$719$	−28.1138	−1.04847	−0.524234	+	0.851574i	$0.675648\pi$
$720$	0	0
$721$	−23.1074	−0.860566
$722$	1.31471i	0.0489286i
$723$	− 18.9211i	− 0.703683i
$724$	−14.1057	−0.524235
$725$	0	0
$726$	17.0930	0.634382
$727$	24.8767i	0.922627i	0.887237	+	0.461314i	$0.152622\pi$
$727$	24.8767i	0.922627i	−0.887237	+	0.461314i	$0.847378\pi$
$728$	− 19.0797i	− 0.707140i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−5.43256	−0.200930
$732$	− 3.94494i	− 0.145809i
$733$	6.75601i	0.249539i	0.992186	+	0.124769i	$0.0398191\pi$
$733$	6.75601i	0.249539i	−0.992186	+	0.124769i	$0.960181\pi$
$734$	−18.8600	−0.696136
$735$	0	0
$736$	1.92827	0.0710771
$737$	49.2508i	1.81418i
$738$	9.96112i	0.366674i
$739$	5.09161	0.187298	0.0936490	−	0.995605i	$-0.470147\pi$
$739$	5.09161	0.187298	0.0936490	+	0.995605i	$0.470147\pi$
$740$	0	0
$741$	−26.6123	−0.977626
$742$	− 7.46806i	− 0.274161i
$743$	− 16.0999i	− 0.590647i	−0.955397	−	0.295324i	$-0.904573\pi$
$743$	− 16.0999i	− 0.590647i	0.955397	−	0.295324i	$-0.0954274\pi$
$744$	2.24205	0.0821975
$745$	0	0
$746$	−29.8366	−1.09240
$747$	7.48698i	0.273934i
$748$	− 5.43117i	− 0.198583i
$749$	−10.3424	−0.377905
$750$	0	0
$751$	−25.4366	−0.928196	−0.464098	−	0.885784i	$-0.653622\pi$
$751$	−25.4366	−0.928196	−0.464098	+	0.885784i	$0.653622\pi$
$752$	− 10.5282i	− 0.383923i
$753$	1.98608i	0.0723768i
$754$	1.53964	0.0560706
$755$	0	0
$756$	3.23143	0.117526
$757$	− 12.4947i	− 0.454129i	−0.973880	−	0.227064i	$-0.927087\pi$
$757$	− 12.4947i	− 0.454129i	0.973880	−	0.227064i	$-0.0729128\pi$
$758$	− 25.4826i	− 0.925570i
$759$	−10.2204	−0.370977
$760$	0	0
$761$	−11.5890	−0.420101	−0.210051	−	0.977691i	$-0.567363\pi$
$761$	−11.5890	−0.420101	−0.210051	+	0.977691i	$0.567363\pi$
$762$	− 9.25090i	− 0.335125i
$763$	− 31.9768i	− 1.15764i
$764$	−25.4719	−0.921541
$765$	0	0
$766$	31.6433	1.14332
$767$	− 38.3316i	− 1.38407i
$768$	− 1.00000i	− 0.0360844i
$769$	23.7516	0.856505	0.428253	−	0.903659i	$-0.359129\pi$
$769$	23.7516	0.856505	0.428253	+	0.903659i	$0.359129\pi$
$770$	0	0
$771$	−19.2797	−0.694341
$772$	− 6.63439i	− 0.238777i
$773$	12.0949i	0.435024i	0.976058	+	0.217512i	$0.0697941\pi$
$773$	12.0949i	0.435024i	−0.976058	+	0.217512i	$0.930206\pi$
$774$	5.30164	0.190564
$775$	0	0
$776$	−17.2567	−0.619479
$777$	− 30.8716i	− 1.10751i
$778$	− 12.0480i	− 0.431942i
$779$	−44.8966	−1.60859
$780$	0	0
$781$	45.2523	1.61925
$782$	1.97589i	0.0706577i
$783$	0.260762i	0.00931887i
$784$	−3.44213	−0.122933
$785$	0	0
$786$	−4.86497	−0.173528
$787$	20.3022i	0.723694i	0.932237	+	0.361847i	$0.117854\pi$
$787$	20.3022i	0.723694i	−0.932237	+	0.361847i	$0.882146\pi$
$788$	10.1777i	0.362567i
$789$	−12.5297	−0.446068
$790$	0	0
$791$	−13.7198	−0.487821
$792$	5.30029i	0.188337i
$793$	23.2925i	0.827142i
$794$	−25.7565	−0.914064
$795$	0	0
$796$	−2.52238	−0.0894035
$797$	31.7025i	1.12296i	0.827490	+	0.561480i	$0.189768\pi$
$797$	31.7025i	1.12296i	−0.827490	+	0.561480i	$0.810232\pi$
$798$	14.5646i	0.515583i
$799$	10.7881	0.381657
$800$	0	0
$801$	−0.733574	−0.0259196
$802$	26.3196i	0.929378i
$803$	− 20.9585i	− 0.739609i
$804$	−9.29210	−0.327707
$805$	0	0
$806$	−13.2380	−0.466288
$807$	13.4929i	0.474974i
$808$	− 11.8454i	− 0.416720i
$809$	−8.72562	−0.306777	−0.153388	−	0.988166i	$-0.549019\pi$
$809$	−8.72562	−0.306777	−0.153388	+	0.988166i	$0.549019\pi$
$810$	0	0
$811$	15.2022	0.533821	0.266911	−	0.963721i	$-0.413997\pi$
$811$	15.2022	0.533821	0.266911	+	0.963721i	$0.413997\pi$
$812$	− 0.842633i	− 0.0295706i
$813$	26.8135i	0.940391i
$814$	50.6365	1.77481
$815$	0	0
$816$	1.02469	0.0358715
$817$	23.8955i	0.835997i
$818$	− 36.7714i	− 1.28568i
$819$	−19.0797	−0.666698
$820$	0	0
$821$	16.4595	0.574441	0.287220	−	0.957865i	$-0.407269\pi$
$821$	16.4595	0.574441	0.287220	+	0.957865i	$0.407269\pi$
$822$	6.01666i	0.209855i
$823$	− 22.9985i	− 0.801676i	−0.916149	−	0.400838i	$-0.868719\pi$
$823$	− 22.9985i	− 0.801676i	0.916149	−	0.400838i	$-0.131281\pi$
$824$	7.15084	0.249111
$825$	0	0
$826$	−20.9785	−0.729936
$827$	24.7319i	0.860011i	0.902826	+	0.430006i	$0.141488\pi$
$827$	24.7319i	0.860011i	−0.902826	+	0.430006i	$0.858512\pi$
$828$	− 1.92827i	− 0.0670122i
$829$	−20.8634	−0.724617	−0.362308	−	0.932058i	$-0.618011\pi$
$829$	−20.8634	−0.724617	−0.362308	+	0.932058i	$0.618011\pi$
$830$	0	0
$831$	8.16724	0.283318
$832$	5.90441i	0.204699i
$833$	− 3.52713i	− 0.122208i
$834$	5.62720	0.194854
$835$	0	0
$836$	−23.8894	−0.826231
$837$	− 2.24205i	− 0.0774966i
$838$	31.7245i	1.09591i
$839$	23.5963	0.814635	0.407317	−	0.913287i	$-0.366464\pi$
$839$	23.5963	0.814635	0.407317	+	0.913287i	$0.366464\pi$
$840$	0	0
$841$	−28.9320	−0.997655
$842$	8.81726i	0.303863i
$843$	− 31.3418i	− 1.07947i
$844$	−15.7638	−0.542613
$845$	0	0
$846$	−10.5282	−0.361966
$847$	55.2349i	1.89789i
$848$	2.31107i	0.0793625i
$849$	16.3503	0.561142
$850$	0	0
$851$	−18.4218	−0.631493
$852$	8.53771i	0.292497i
$853$	26.3629i	0.902647i	0.892360	+	0.451324i	$0.149048\pi$
$853$	26.3629i	0.902647i	−0.892360	+	0.451324i	$0.850952\pi$
$854$	12.7478	0.436220
$855$	0	0
$856$	3.20058	0.109394
$857$	− 16.0891i	− 0.549593i	−0.961502	−	0.274797i	$-0.911389\pi$
$857$	− 16.0891i	− 0.549593i	0.961502	−	0.274797i	$-0.0886105\pi$
$858$	− 31.2951i	− 1.06840i
$859$	−10.5770	−0.360882	−0.180441	−	0.983586i	$-0.557752\pi$
$859$	−10.5770	−0.360882	−0.180441	+	0.983586i	$0.557752\pi$
$860$	0	0
$861$	−32.1886	−1.09699
$862$	23.5065i	0.800636i
$863$	− 41.7617i	− 1.42158i	−0.703403	−	0.710792i	$-0.748337\pi$
$863$	− 41.7617i	− 1.42158i	0.703403	−	0.710792i	$-0.251663\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−8.40312	−0.285550
$867$	− 15.9500i	− 0.541690i
$868$	7.24502i	0.245912i
$869$	70.8721	2.40417
$870$	0	0
$871$	54.8644	1.85901
$872$	9.89555i	0.335106i
$873$	17.2567i	0.584050i
$874$	8.69109	0.293980
$875$	0	0
$876$	3.95422	0.133601
$877$	− 31.8267i	− 1.07471i	−0.843356	−	0.537355i	$-0.819423\pi$
$877$	− 31.8267i	− 1.07471i	0.843356	−	0.537355i	$-0.180577\pi$
$878$	− 35.6325i	− 1.20254i
$879$	28.1867	0.950714
$880$	0	0
$881$	−49.6324	−1.67216	−0.836079	−	0.548609i	$-0.815158\pi$
$881$	−49.6324	−1.67216	−0.836079	+	0.548609i	$0.815158\pi$
$882$	3.44213i	0.115902i
$883$	5.06098i	0.170316i	0.996367	+	0.0851578i	$0.0271394\pi$
$883$	5.06098i	0.170316i	−0.996367	+	0.0851578i	$0.972861\pi$
$884$	−6.05021	−0.203491
$885$	0	0
$886$	−6.79550	−0.228299
$887$	11.0542i	0.371164i	0.982629	+	0.185582i	$0.0594171\pi$
$887$	11.0542i	0.371164i	−0.982629	+	0.185582i	$0.940583\pi$
$888$	9.55354i	0.320596i
$889$	29.8936	1.00260
$890$	0	0
$891$	5.30029	0.177566
$892$	− 9.29534i	− 0.311231i
$893$	− 47.4524i	− 1.58793i
$894$	11.6556	0.389821
$895$	0	0
$896$	3.23143	0.107954
$897$	11.3853i	0.380145i
$898$	− 8.75011i	− 0.291995i
$899$	−0.584641	−0.0194989
$900$	0	0
$901$	−2.36814	−0.0788942
$902$	− 52.7968i	− 1.75794i
$903$	17.1319i	0.570113i
$904$	4.24575	0.141211
$905$	0	0
$906$	5.52354	0.183507
$907$	− 7.57086i	− 0.251386i	−0.992069	−	0.125693i	$-0.959885\pi$
$907$	− 7.57086i	− 0.251386i	0.992069	−	0.125693i	$-0.0401155\pi$
$908$	3.63809i	0.120734i
$909$	−11.8454	−0.392887
$910$	0	0
$911$	35.7684	1.18506	0.592529	−	0.805549i	$-0.298129\pi$
$911$	35.7684	1.18506	0.592529	+	0.805549i	$0.298129\pi$
$912$	− 4.50718i	− 0.149248i
$913$	− 39.6831i	− 1.31332i
$914$	−38.1474	−1.26180
$915$	0	0
$916$	9.93181	0.328156
$917$	− 15.7208i	− 0.519147i
$918$	− 1.02469i	− 0.0338199i
$919$	49.6236	1.63693	0.818466	−	0.574555i	$-0.194825\pi$
$919$	49.6236	1.63693	0.818466	+	0.574555i	$0.194825\pi$
$920$	0	0
$921$	−14.5372	−0.479016
$922$	− 1.27077i	− 0.0418505i
$923$	− 50.4101i	− 1.65927i
$924$	−17.1275	−0.563453
$925$	0	0
$926$	−8.93349	−0.293573
$927$	− 7.15084i	− 0.234864i
$928$	0.260762i	0.00855992i
$929$	16.2616	0.533527	0.266764	−	0.963762i	$-0.414046\pi$
$929$	16.2616	0.533527	0.266764	+	0.963762i	$0.414046\pi$
$930$	0	0
$931$	−15.5143	−0.508461
$932$	18.7893i	0.615465i
$933$	− 14.4924i	− 0.474461i
$934$	−5.75682	−0.188369
$935$	0	0
$936$	5.90441	0.192992
$937$	0.569874i	0.0186170i	0.999957	+	0.00930849i	$0.00296303\pi$
$937$	0.569874i	0.0186170i	−0.999957	+	0.00930849i	$0.997037\pi$
$938$	− 30.0268i	− 0.980408i
$939$	31.5288	1.02890
$940$	0	0
$941$	6.99538	0.228043	0.114021	−	0.993478i	$-0.463627\pi$
$941$	6.99538	0.228043	0.114021	+	0.993478i	$0.463627\pi$
$942$	− 6.84307i	− 0.222959i
$943$	19.2078i	0.625491i
$944$	6.49202	0.211297
$945$	0	0
$946$	−28.1002	−0.913616
$947$	34.4620i	1.11987i	0.828538	+	0.559933i	$0.189173\pi$
$947$	34.4620i	1.11987i	−0.828538	+	0.559933i	$0.810827\pi$
$948$	13.3714i	0.434282i
$949$	−23.3473	−0.757886
$950$	0	0
$951$	−14.9810	−0.485794
$952$	3.31122i	0.107317i
$953$	− 42.0363i	− 1.36169i	−0.732427	−	0.680845i	$-0.761613\pi$
$953$	− 42.0363i	− 1.36169i	0.732427	−	0.680845i	$-0.238387\pi$
$954$	2.31107	0.0748237
$955$	0	0
$956$	−9.75186	−0.315398
$957$	− 1.38211i	− 0.0446773i
$958$	4.63051i	0.149605i
$959$	−19.4424	−0.627828
$960$	0	0
$961$	−25.9732	−0.837846
$962$	− 56.4080i	− 1.81867i
$963$	− 3.20058i	− 0.103137i
$964$	−18.9211	−0.609407
$965$	0	0
$966$	6.23108	0.200482
$967$	− 11.9686i	− 0.384885i	−0.981308	−	0.192443i	$-0.938359\pi$
$967$	− 11.9686i	− 0.384885i	0.981308	−	0.192443i	$-0.0616410\pi$
$968$	− 17.0930i	− 0.549391i
$969$	4.61848	0.148367
$970$	0	0
$971$	1.35194	0.0433858	0.0216929	−	0.999765i	$-0.493094\pi$
$971$	1.35194	0.0433858	0.0216929	+	0.999765i	$0.493094\pi$
$972$	1.00000i	0.0320750i
$973$	18.1839i	0.582949i
$974$	15.9235	0.510223
$975$	0	0
$976$	−3.94494	−0.126274
$977$	18.5448i	0.593300i	0.954986	+	0.296650i	$0.0958695\pi$
$977$	18.5448i	0.593300i	−0.954986	+	0.296650i	$0.904131\pi$
$978$	9.38717i	0.300169i
$979$	3.88815	0.124266
$980$	0	0
$981$	9.89555	0.315941
$982$	− 6.38076i	− 0.203618i
$983$	16.3821i	0.522508i	0.965270	+	0.261254i	$0.0841360\pi$
$983$	16.3821i	0.522508i	−0.965270	+	0.261254i	$0.915864\pi$
$984$	9.96112	0.317549
$985$	0	0
$986$	−0.267201	−0.00850941
$987$	− 34.0210i	− 1.08290i
$988$	26.6123i	0.846649i
$989$	10.2230	0.325073
$990$	0	0
$991$	19.8303	0.629930	0.314965	−	0.949103i	$-0.398007\pi$
$991$	19.8303	0.629930	0.314965	+	0.949103i	$0.398007\pi$
$992$	− 2.24205i	− 0.0711851i
$993$	21.6174i	0.686007i
$994$	−27.5890	−0.875070
$995$	0	0
$996$	7.48698	0.237234
$997$	− 10.4675i	− 0.331508i	−0.986167	−	0.165754i	$-0.946994\pi$
$997$	− 10.4675i	− 0.331508i	0.986167	−	0.165754i	$-0.0530057\pi$
$998$	− 12.9135i	− 0.408769i
$999$	9.55354	0.302261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.k.1249.15		16
5.2	odd	4		3750.2.a.u.1.2		8
5.3	odd	4		3750.2.a.v.1.7		8
5.4	even	2	inner	3750.2.c.k.1249.2		16
25.2	odd	20		750.2.g.g.601.1		16
25.9	even	10		150.2.h.b.19.3	✓	16
25.11	even	5		150.2.h.b.79.3	yes	16
25.12	odd	20		750.2.g.g.151.1		16
25.13	odd	20		750.2.g.f.151.4		16
25.14	even	10		750.2.h.d.649.1		16
25.16	even	5		750.2.h.d.349.2		16
25.23	odd	20		750.2.g.f.601.4		16
75.11	odd	10		450.2.l.c.379.2		16
75.59	odd	10		450.2.l.c.19.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.b.19.3	✓	16	25.9	even	10
150.2.h.b.79.3	yes	16	25.11	even	5
450.2.l.c.19.2		16	75.59	odd	10
450.2.l.c.379.2		16	75.11	odd	10
750.2.g.f.151.4		16	25.13	odd	20
750.2.g.f.601.4		16	25.23	odd	20
750.2.g.g.151.1		16	25.12	odd	20
750.2.g.g.601.1		16	25.2	odd	20
750.2.h.d.349.2		16	25.16	even	5
750.2.h.d.649.1		16	25.14	even	10
3750.2.a.u.1.2		8	5.2	odd	4
3750.2.a.v.1.7		8	5.3	odd	4
3750.2.c.k.1249.2		16	5.4	even	2	inner
3750.2.c.k.1249.15		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.23143i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.23143i q^{7} -1.00000i q^{8} -1.00000 q^{9} +5.30029 q^{11} +1.00000i q^{12} -5.90441i q^{13} -3.23143 q^{14} +1.00000 q^{16} +1.02469i q^{17} -1.00000i q^{18} +4.50718 q^{19} +3.23143 q^{21} +5.30029i q^{22} -1.92827i q^{23} -1.00000 q^{24} +5.90441 q^{26} +1.00000i q^{27} -3.23143i q^{28} +0.260762 q^{29} -2.24205 q^{31} +1.00000i q^{32} -5.30029i q^{33} -1.02469 q^{34} +1.00000 q^{36} -9.55354i q^{37} +4.50718i q^{38} -5.90441 q^{39} -9.96112 q^{41} +3.23143i q^{42} +5.30164i q^{43} -5.30029 q^{44} +1.92827 q^{46} -10.5282i q^{47} -1.00000i q^{48} -3.44213 q^{49} +1.02469 q^{51} +5.90441i q^{52} +2.31107i q^{53} -1.00000 q^{54} +3.23143 q^{56} -4.50718i q^{57} +0.260762i q^{58} +6.49202 q^{59} -3.94494 q^{61} -2.24205i q^{62} -3.23143i q^{63} -1.00000 q^{64} +5.30029 q^{66} +9.29210i q^{67} -1.02469i q^{68} -1.92827 q^{69} +8.53771 q^{71} +1.00000i q^{72} -3.95422i q^{73} +9.55354 q^{74} -4.50718 q^{76} +17.1275i q^{77} -5.90441i q^{78} +13.3714 q^{79} +1.00000 q^{81} -9.96112i q^{82} -7.48698i q^{83} -3.23143 q^{84} -5.30164 q^{86} -0.260762i q^{87} -5.30029i q^{88} +0.733574 q^{89} +19.0797 q^{91} +1.92827i q^{92} +2.24205i q^{93} +10.5282 q^{94} +1.00000 q^{96} -17.2567i q^{97} -3.44213i q^{98} -5.30029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^6 - 16 * q^9 + 12 * q^11 - 8 * q^14 + 16 * q^16 - 20 * q^19 + 8 * q^21 - 16 * q^24 + 4 * q^26 - 20 * q^29 + 32 * q^31 - 28 * q^34 + 16 * q^36 - 4 * q^39 + 12 * q^41 - 12 * q^44 + 24 * q^46 - 52 * q^49 + 28 * q^51 - 16 * q^54 + 8 * q^56 + 32 * q^61 - 16 * q^64 + 12 * q^66 - 24 * q^69 + 12 * q^71 + 12 * q^74 + 20 * q^76 - 20 * q^79 + 16 * q^81 - 8 * q^84 + 4 * q^86 - 40 * q^89 + 12 * q^91 - 28 * q^94 + 16 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

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Database

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