LMFDB - Embedded newform 3750.2.c.k.1249.6

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			1249.6
Root			$-1.16141 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.k.1249.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.533559i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.533559i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.43425 q^{11} -1.00000i q^{12} +6.57392i q^{13} +0.533559 q^{14} +1.00000 q^{16} +0.958413i q^{17} +1.00000i q^{18} -0.212889 q^{19} -0.533559 q^{21} +1.43425i q^{22} +3.76401i q^{23} -1.00000 q^{24} +6.57392 q^{26} -1.00000i q^{27} -0.533559i q^{28} -6.19448 q^{29} -2.33773 q^{31} -1.00000i q^{32} -1.43425i q^{33} +0.958413 q^{34} +1.00000 q^{36} -4.06291i q^{37} +0.212889i q^{38} -6.57392 q^{39} -7.94156 q^{41} +0.533559i q^{42} -11.3607i q^{43} +1.43425 q^{44} +3.76401 q^{46} -10.1489i q^{47} +1.00000i q^{48} +6.71531 q^{49} -0.958413 q^{51} -6.57392i q^{52} -3.23354i q^{53} -1.00000 q^{54} -0.533559 q^{56} -0.212889i q^{57} +6.19448i q^{58} -7.52455 q^{59} +12.5721 q^{61} +2.33773i q^{62} -0.533559i q^{63} -1.00000 q^{64} -1.43425 q^{66} +6.91285i q^{67} -0.958413i q^{68} -3.76401 q^{69} +10.1247 q^{71} -1.00000i q^{72} -13.9771i q^{73} -4.06291 q^{74} +0.212889 q^{76} -0.765259i q^{77} +6.57392i q^{78} -15.5279 q^{79} +1.00000 q^{81} +7.94156i q^{82} +16.3308i q^{83} +0.533559 q^{84} -11.3607 q^{86} -6.19448i q^{87} -1.43425i q^{88} -5.62220 q^{89} -3.50758 q^{91} -3.76401i q^{92} -2.33773i q^{93} -10.1489 q^{94} +1.00000 q^{96} +5.56558i q^{97} -6.71531i q^{98} +1.43425 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.533559i	0.201666i	0.994903	+	0.100833i	$0.0321508\pi$
$7$	0.533559i	0.201666i	−0.994903	+	0.100833i	$0.967849\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.43425	−0.432444	−0.216222	−	0.976344i	$-0.569373\pi$
$11$	−1.43425	−0.432444	−0.216222	+	0.976344i	$0.569373\pi$
$12$	− 1.00000i	− 0.288675i
$13$	6.57392i	1.82328i	0.410991	+	0.911639i	$0.365183\pi$
$13$	6.57392i	1.82328i	−0.410991	+	0.911639i	$0.634817\pi$
$14$	0.533559	0.142600
$15$	0	0
$16$	1.00000	0.250000
$17$	0.958413i	0.232449i	0.993223	+	0.116225i	$0.0370793\pi$
$17$	0.958413i	0.232449i	−0.993223	+	0.116225i	$0.962921\pi$
$18$	1.00000i	0.235702i
$19$	−0.212889	−0.0488401	−0.0244201	−	0.999702i	$-0.507774\pi$
$19$	−0.212889	−0.0488401	−0.0244201	+	0.999702i	$0.507774\pi$
$20$	0	0
$21$	−0.533559	−0.116432
$22$	1.43425i	0.305784i
$23$	3.76401i	0.784850i	0.919784	+	0.392425i	$0.128364\pi$
$23$	3.76401i	0.784850i	−0.919784	+	0.392425i	$0.871636\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.57392	1.28925
$27$	− 1.00000i	− 0.192450i
$28$	− 0.533559i	− 0.100833i
$29$	−6.19448	−1.15029	−0.575143	−	0.818053i	$-0.695054\pi$
$29$	−6.19448	−1.15029	−0.575143	+	0.818053i	$0.695054\pi$
$30$	0	0
$31$	−2.33773	−0.419869	−0.209934	−	0.977716i	$-0.567325\pi$
$31$	−2.33773	−0.419869	−0.209934	+	0.977716i	$0.567325\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.43425i	− 0.249671i
$34$	0.958413	0.164367
$35$	0	0
$36$	1.00000	0.166667
$37$	− 4.06291i	− 0.667938i	−0.942584	−	0.333969i	$-0.891612\pi$
$37$	− 4.06291i	− 0.667938i	0.942584	−	0.333969i	$-0.108388\pi$
$38$	0.212889i	0.0345352i
$39$	−6.57392	−1.05267
$40$	0	0
$41$	−7.94156	−1.24026	−0.620132	−	0.784497i	$-0.712921\pi$
$41$	−7.94156	−1.24026	−0.620132	+	0.784497i	$0.712921\pi$
$42$	0.533559i	0.0823300i
$43$	− 11.3607i	− 1.73250i	−0.499614	−	0.866248i	$-0.666525\pi$
$43$	− 11.3607i	− 1.73250i	0.499614	−	0.866248i	$-0.333475\pi$
$44$	1.43425	0.216222
$45$	0	0
$46$	3.76401	0.554973
$47$	− 10.1489i	− 1.48037i	−0.672402	−	0.740186i	$-0.734737\pi$
$47$	− 10.1489i	− 1.48037i	0.672402	−	0.740186i	$-0.265263\pi$
$48$	1.00000i	0.144338i
$49$	6.71531	0.959331
$50$	0	0
$51$	−0.958413	−0.134205
$52$	− 6.57392i	− 0.911639i
$53$	− 3.23354i	− 0.444162i	−0.975028	−	0.222081i	$-0.928715\pi$
$53$	− 3.23354i	− 0.444162i	0.975028	−	0.222081i	$-0.0712849\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−0.533559	−0.0712998
$57$	− 0.212889i	− 0.0281978i
$58$	6.19448i	0.813375i
$59$	−7.52455	−0.979613	−0.489806	−	0.871831i	$-0.662933\pi$
$59$	−7.52455	−0.979613	−0.489806	+	0.871831i	$0.662933\pi$
$60$	0	0
$61$	12.5721	1.60969	0.804844	−	0.593486i	$-0.202249\pi$
$61$	12.5721	1.60969	0.804844	+	0.593486i	$0.202249\pi$
$62$	2.33773i	0.296892i
$63$	− 0.533559i	− 0.0672221i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.43425	−0.176544
$67$	6.91285i	0.844539i	0.906470	+	0.422269i	$0.138766\pi$
$67$	6.91285i	0.844539i	−0.906470	+	0.422269i	$0.861234\pi$
$68$	− 0.958413i	− 0.116225i
$69$	−3.76401	−0.453134
$70$	0	0
$71$	10.1247	1.20158	0.600789	−	0.799408i	$-0.294853\pi$
$71$	10.1247	1.20158	0.600789	+	0.799408i	$0.294853\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 13.9771i	− 1.63589i	−0.575294	−	0.817947i	$-0.695112\pi$
$73$	− 13.9771i	− 1.63589i	0.575294	−	0.817947i	$-0.304888\pi$
$74$	−4.06291	−0.472304
$75$	0	0
$76$	0.212889	0.0244201
$77$	− 0.765259i	− 0.0872093i
$78$	6.57392i	0.744350i
$79$	−15.5279	−1.74703	−0.873515	−	0.486798i	$-0.838165\pi$
$79$	−15.5279	−1.74703	−0.873515	+	0.486798i	$0.838165\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.94156i	0.876999i
$83$	16.3308i	1.79254i	0.443508	+	0.896270i	$0.353734\pi$
$83$	16.3308i	1.79254i	−0.443508	+	0.896270i	$0.646266\pi$
$84$	0.533559	0.0582161
$85$	0	0
$86$	−11.3607	−1.22506
$87$	− 6.19448i	− 0.664118i
$88$	− 1.43425i	− 0.152892i
$89$	−5.62220	−0.595951	−0.297976	−	0.954573i	$-0.596311\pi$
$89$	−5.62220	−0.595951	−0.297976	+	0.954573i	$0.596311\pi$
$90$	0	0
$91$	−3.50758	−0.367694
$92$	− 3.76401i	− 0.392425i
$93$	− 2.33773i	− 0.242411i
$94$	−10.1489	−1.04678
$95$	0	0
$96$	1.00000	0.102062
$97$	5.56558i	0.565099i	0.959253	+	0.282549i	$0.0911801\pi$
$97$	5.56558i	0.565099i	−0.959253	+	0.282549i	$0.908820\pi$
$98$	− 6.71531i	− 0.678349i
$99$	1.43425	0.144148
$100$	0	0
$101$	−9.42708	−0.938029	−0.469015	−	0.883190i	$-0.655391\pi$
$101$	−9.42708	−0.938029	−0.469015	+	0.883190i	$0.655391\pi$
$102$	0.958413i	0.0948971i
$103$	7.99871i	0.788137i	0.919081	+	0.394068i	$0.128933\pi$
$103$	7.99871i	0.788137i	−0.919081	+	0.394068i	$0.871067\pi$
$104$	−6.57392	−0.644626
$105$	0	0
$106$	−3.23354	−0.314070
$107$	− 18.9260i	− 1.82964i	−0.403857	−	0.914822i	$-0.632331\pi$
$107$	− 18.9260i	− 1.82964i	0.403857	−	0.914822i	$-0.367669\pi$
$108$	1.00000i	0.0962250i
$109$	2.65524	0.254326	0.127163	−	0.991882i	$-0.459413\pi$
$109$	2.65524	0.254326	0.127163	+	0.991882i	$0.459413\pi$
$110$	0	0
$111$	4.06291	0.385634
$112$	0.533559i	0.0504166i
$113$	3.06283i	0.288127i	0.989568	+	0.144063i	$0.0460169\pi$
$113$	3.06283i	0.288127i	−0.989568	+	0.144063i	$0.953983\pi$
$114$	−0.212889	−0.0199389
$115$	0	0
$116$	6.19448	0.575143
$117$	− 6.57392i	− 0.607759i
$118$	7.52455i	0.692691i
$119$	−0.511370	−0.0468772
$120$	0	0
$121$	−8.94292	−0.812993
$122$	− 12.5721i	− 1.13822i
$123$	− 7.94156i	− 0.716067i
$124$	2.33773	0.209934
$125$	0	0
$126$	−0.533559	−0.0475332
$127$	6.48257i	0.575235i	0.957745	+	0.287618i	$0.0928632\pi$
$127$	6.48257i	0.575235i	−0.957745	+	0.287618i	$0.907137\pi$
$128$	1.00000i	0.0883883i
$129$	11.3607	1.00026
$130$	0	0
$131$	−9.45794	−0.826344	−0.413172	−	0.910653i	$-0.635579\pi$
$131$	−9.45794	−0.826344	−0.413172	+	0.910653i	$0.635579\pi$
$132$	1.43425i	0.124836i
$133$	− 0.113589i	− 0.00984941i
$134$	6.91285	0.597179
$135$	0	0
$136$	−0.958413	−0.0821833
$137$	12.3361i	1.05394i	0.849883	+	0.526971i	$0.176672\pi$
$137$	12.3361i	1.05394i	−0.849883	+	0.526971i	$0.823328\pi$
$138$	3.76401i	0.320414i
$139$	−4.53918	−0.385008	−0.192504	−	0.981296i	$-0.561661\pi$
$139$	−4.53918	−0.385008	−0.192504	+	0.981296i	$0.561661\pi$
$140$	0	0
$141$	10.1489	0.854693
$142$	− 10.1247i	− 0.849643i
$143$	− 9.42867i	− 0.788465i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−13.9771	−1.15675
$147$	6.71531i	0.553870i
$148$	4.06291i	0.333969i
$149$	−11.0750	−0.907303	−0.453651	−	0.891179i	$-0.649879\pi$
$149$	−11.0750	−0.907303	−0.453651	+	0.891179i	$0.649879\pi$
$150$	0	0
$151$	−1.63387	−0.132962	−0.0664812	−	0.997788i	$-0.521177\pi$
$151$	−1.63387	−0.132962	−0.0664812	+	0.997788i	$0.521177\pi$
$152$	− 0.212889i	− 0.0172676i
$153$	− 0.958413i	− 0.0774831i
$154$	−0.765259	−0.0616663
$155$	0	0
$156$	6.57392	0.526335
$157$	− 6.64544i	− 0.530364i	−0.964198	−	0.265182i	$-0.914568\pi$
$157$	− 6.64544i	− 0.530364i	0.964198	−	0.265182i	$-0.0854320\pi$
$158$	15.5279i	1.23534i
$159$	3.23354	0.256437
$160$	0	0
$161$	−2.00832	−0.158278
$162$	− 1.00000i	− 0.0785674i
$163$	− 10.1134i	− 0.792142i	−0.918220	−	0.396071i	$-0.870373\pi$
$163$	− 10.1134i	− 0.792142i	0.918220	−	0.396071i	$-0.129627\pi$
$164$	7.94156	0.620132
$165$	0	0
$166$	16.3308	1.26752
$167$	− 12.7885i	− 0.989600i	−0.869007	−	0.494800i	$-0.835241\pi$
$167$	− 12.7885i	− 0.989600i	0.869007	−	0.494800i	$-0.164759\pi$
$168$	− 0.533559i	− 0.0411650i
$169$	−30.2165	−2.32434
$170$	0	0
$171$	0.212889	0.0162800
$172$	11.3607i	0.866248i
$173$	− 13.6575i	− 1.03836i	−0.854664	−	0.519182i	$-0.826237\pi$
$173$	− 13.6575i	− 1.03836i	0.854664	−	0.519182i	$-0.173763\pi$
$174$	−6.19448	−0.469602
$175$	0	0
$176$	−1.43425	−0.108111
$177$	− 7.52455i	− 0.565580i
$178$	5.62220i	0.421401i
$179$	−2.99142	−0.223589	−0.111795	−	0.993731i	$-0.535660\pi$
$179$	−2.99142	−0.223589	−0.111795	+	0.993731i	$0.535660\pi$
$180$	0	0
$181$	7.62285	0.566602	0.283301	−	0.959031i	$-0.408570\pi$
$181$	7.62285	0.566602	0.283301	+	0.959031i	$0.408570\pi$
$182$	3.50758i	0.259999i
$183$	12.5721i	0.929354i
$184$	−3.76401	−0.277487
$185$	0	0
$186$	−2.33773	−0.171411
$187$	− 1.37461i	− 0.100521i
$188$	10.1489i	0.740186i
$189$	0.533559	0.0388107
$190$	0	0
$191$	−8.40045	−0.607835	−0.303917	−	0.952698i	$-0.598295\pi$
$191$	−8.40045	−0.607835	−0.303917	+	0.952698i	$0.598295\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 10.5266i	− 0.757723i	−0.925453	−	0.378861i	$-0.876316\pi$
$193$	− 10.5266i	− 0.757723i	0.925453	−	0.378861i	$-0.123684\pi$
$194$	5.56558	0.399585
$195$	0	0
$196$	−6.71531	−0.479665
$197$	10.2080i	0.727293i	0.931537	+	0.363647i	$0.118468\pi$
$197$	10.2080i	0.727293i	−0.931537	+	0.363647i	$0.881532\pi$
$198$	− 1.43425i	− 0.101928i
$199$	−3.84318	−0.272436	−0.136218	−	0.990679i	$-0.543495\pi$
$199$	−3.84318	−0.272436	−0.136218	+	0.990679i	$0.543495\pi$
$200$	0	0
$201$	−6.91285	−0.487595
$202$	9.42708i	0.663287i
$203$	− 3.30512i	− 0.231974i
$204$	0.958413	0.0671023
$205$	0	0
$206$	7.99871	0.557297
$207$	− 3.76401i	− 0.261617i
$208$	6.57392i	0.455820i
$209$	0.305337	0.0211206
$210$	0	0
$211$	5.24920	0.361370	0.180685	−	0.983541i	$-0.442169\pi$
$211$	5.24920	0.361370	0.180685	+	0.983541i	$0.442169\pi$
$212$	3.23354i	0.222081i
$213$	10.1247i	0.693731i
$214$	−18.9260	−1.29375
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 1.24732i	− 0.0846734i
$218$	− 2.65524i	− 0.179836i
$219$	13.9771	0.944484
$220$	0	0
$221$	−6.30054	−0.423820
$222$	− 4.06291i	− 0.272685i
$223$	− 19.2782i	− 1.29097i	−0.763774	−	0.645483i	$-0.776656\pi$
$223$	− 19.2782i	− 1.29097i	0.763774	−	0.645483i	$-0.223344\pi$
$224$	0.533559	0.0356499
$225$	0	0
$226$	3.06283	0.203736
$227$	− 8.43957i	− 0.560154i	−0.959978	−	0.280077i	$-0.909640\pi$
$227$	− 8.43957i	− 0.560154i	0.959978	−	0.280077i	$-0.0903600\pi$
$228$	0.212889i	0.0140989i
$229$	−5.95568	−0.393563	−0.196781	−	0.980447i	$-0.563049\pi$
$229$	−5.95568	−0.393563	−0.196781	+	0.980447i	$0.563049\pi$
$230$	0	0
$231$	0.765259	0.0503503
$232$	− 6.19448i	− 0.406688i
$233$	− 2.09293i	− 0.137112i	−0.997647	−	0.0685561i	$-0.978161\pi$
$233$	− 2.09293i	− 0.137112i	0.997647	−	0.0685561i	$-0.0218392\pi$
$234$	−6.57392	−0.429751
$235$	0	0
$236$	7.52455	0.489806
$237$	− 15.5279i	− 1.00865i
$238$	0.511370i	0.0331472i
$239$	−9.89840	−0.640274	−0.320137	−	0.947371i	$-0.603729\pi$
$239$	−9.89840	−0.640274	−0.320137	+	0.947371i	$0.603729\pi$
$240$	0	0
$241$	−21.4567	−1.38215	−0.691074	−	0.722784i	$-0.742862\pi$
$241$	−21.4567	−1.38215	−0.691074	+	0.722784i	$0.742862\pi$
$242$	8.94292i	0.574873i
$243$	1.00000i	0.0641500i
$244$	−12.5721	−0.804844
$245$	0	0
$246$	−7.94156	−0.506336
$247$	− 1.39952i	− 0.0890491i
$248$	− 2.33773i	− 0.148446i
$249$	−16.3308	−1.03492
$250$	0	0
$251$	−4.10753	−0.259265	−0.129632	−	0.991562i	$-0.541380\pi$
$251$	−4.10753	−0.259265	−0.129632	+	0.991562i	$0.541380\pi$
$252$	0.533559i	0.0336111i
$253$	− 5.39854i	− 0.339404i
$254$	6.48257	0.406753
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.7748i	1.91968i	0.280552	+	0.959839i	$0.409482\pi$
$257$	30.7748i	1.91968i	−0.280552	+	0.959839i	$0.590518\pi$
$258$	− 11.3607i	− 0.707289i
$259$	2.16780	0.134701
$260$	0	0
$261$	6.19448	0.383429
$262$	9.45794i	0.584313i
$263$	− 28.3729i	− 1.74955i	−0.484533	−	0.874773i	$-0.661011\pi$
$263$	− 28.3729i	− 1.74955i	0.484533	−	0.874773i	$-0.338989\pi$
$264$	1.43425	0.0882722
$265$	0	0
$266$	−0.113589	−0.00696458
$267$	− 5.62220i	− 0.344073i
$268$	− 6.91285i	− 0.422269i
$269$	−10.6668	−0.650364	−0.325182	−	0.945651i	$-0.605426\pi$
$269$	−10.6668	−0.650364	−0.325182	+	0.945651i	$0.605426\pi$
$270$	0	0
$271$	22.8439	1.38767	0.693835	−	0.720134i	$-0.255920\pi$
$271$	22.8439	1.38767	0.693835	+	0.720134i	$0.255920\pi$
$272$	0.958413i	0.0581123i
$273$	− 3.50758i	− 0.212288i
$274$	12.3361	0.745250
$275$	0	0
$276$	3.76401	0.226567
$277$	31.4592i	1.89020i	0.326784	+	0.945099i	$0.394035\pi$
$277$	31.4592i	1.89020i	−0.326784	+	0.945099i	$0.605965\pi$
$278$	4.53918i	0.272242i
$279$	2.33773	0.139956
$280$	0	0
$281$	−10.7785	−0.642992	−0.321496	−	0.946911i	$-0.604186\pi$
$281$	−10.7785	−0.642992	−0.321496	+	0.946911i	$0.604186\pi$
$282$	− 10.1489i	− 0.604359i
$283$	− 4.16428i	− 0.247541i	−0.992311	−	0.123770i	$-0.960501\pi$
$283$	− 4.16428i	− 0.247541i	0.992311	−	0.123770i	$-0.0394986\pi$
$284$	−10.1247	−0.600789
$285$	0	0
$286$	−9.42867	−0.557529
$287$	− 4.23729i	− 0.250120i
$288$	1.00000i	0.0589256i
$289$	16.0814	0.945967
$290$	0	0
$291$	−5.56558	−0.326260
$292$	13.9771i	0.817947i
$293$	− 17.9603i	− 1.04925i	−0.851333	−	0.524626i	$-0.824205\pi$
$293$	− 17.9603i	− 1.04925i	0.851333	−	0.524626i	$-0.175795\pi$
$294$	6.71531	0.391645
$295$	0	0
$296$	4.06291	0.236152
$297$	1.43425i	0.0832238i
$298$	11.0750i	0.641560i
$299$	−24.7443	−1.43100
$300$	0	0
$301$	6.06163	0.349386
$302$	1.63387i	0.0940187i
$303$	− 9.42708i	− 0.541571i
$304$	−0.212889	−0.0122100
$305$	0	0
$306$	−0.958413	−0.0547888
$307$	20.8174i	1.18811i	0.804423	+	0.594056i	$0.202474\pi$
$307$	20.8174i	1.18811i	−0.804423	+	0.594056i	$0.797526\pi$
$308$	0.765259i	0.0436047i
$309$	−7.99871	−0.455031
$310$	0	0
$311$	−19.4526	−1.10305	−0.551527	−	0.834157i	$-0.685955\pi$
$311$	−19.4526	−1.10305	−0.551527	+	0.834157i	$0.685955\pi$
$312$	− 6.57392i	− 0.372175i
$313$	5.88310i	0.332532i	0.986081	+	0.166266i	$0.0531711\pi$
$313$	5.88310i	0.332532i	−0.986081	+	0.166266i	$0.946829\pi$
$314$	−6.64544	−0.375024
$315$	0	0
$316$	15.5279	0.873515
$317$	10.7610i	0.604400i	0.953245	+	0.302200i	$0.0977210\pi$
$317$	10.7610i	0.604400i	−0.953245	+	0.302200i	$0.902279\pi$
$318$	− 3.23354i	− 0.181328i
$319$	8.88445	0.497434
$320$	0	0
$321$	18.9260	1.05635
$322$	2.00832i	0.111919i
$323$	− 0.204036i	− 0.0113528i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−10.1134	−0.560129
$327$	2.65524i	0.146835i
$328$	− 7.94156i	− 0.438500i
$329$	5.41505	0.298541
$330$	0	0
$331$	−0.617092	−0.0339184	−0.0169592	−	0.999856i	$-0.505399\pi$
$331$	−0.617092	−0.0339184	−0.0169592	+	0.999856i	$0.505399\pi$
$332$	− 16.3308i	− 0.896270i
$333$	4.06291i	0.222646i
$334$	−12.7885	−0.699753
$335$	0	0
$336$	−0.533559	−0.0291080
$337$	15.4106i	0.839471i	0.907647	+	0.419735i	$0.137877\pi$
$337$	15.4106i	0.839471i	−0.907647	+	0.419735i	$0.862123\pi$
$338$	30.2165i	1.64356i
$339$	−3.06283	−0.166350
$340$	0	0
$341$	3.35289	0.181569
$342$	− 0.212889i	− 0.0115117i
$343$	7.31793i	0.395131i
$344$	11.3607	0.612530
$345$	0	0
$346$	−13.6575	−0.734234
$347$	− 22.6565i	− 1.21627i	−0.793835	−	0.608133i	$-0.791919\pi$
$347$	− 22.6565i	− 1.21627i	0.793835	−	0.608133i	$-0.208081\pi$
$348$	6.19448i	0.332059i
$349$	16.9543	0.907544	0.453772	−	0.891118i	$-0.350078\pi$
$349$	16.9543	0.907544	0.453772	+	0.891118i	$0.350078\pi$
$350$	0	0
$351$	6.57392	0.350890
$352$	1.43425i	0.0764459i
$353$	6.06255i	0.322677i	0.986899	+	0.161339i	$0.0515811\pi$
$353$	6.06255i	0.322677i	−0.986899	+	0.161339i	$0.948419\pi$
$354$	−7.52455	−0.399925
$355$	0	0
$356$	5.62220	0.297976
$357$	− 0.511370i	− 0.0270646i
$358$	2.99142i	0.158101i
$359$	−18.8987	−0.997437	−0.498718	−	0.866764i	$-0.666196\pi$
$359$	−18.8987	−0.997437	−0.498718	+	0.866764i	$0.666196\pi$
$360$	0	0
$361$	−18.9547	−0.997615
$362$	− 7.62285i	− 0.400648i
$363$	− 8.94292i	− 0.469381i
$364$	3.50758	0.183847
$365$	0	0
$366$	12.5721	0.657152
$367$	25.0483i	1.30751i	0.756705	+	0.653756i	$0.226808\pi$
$367$	25.0483i	1.30751i	−0.756705	+	0.653756i	$0.773192\pi$
$368$	3.76401i	0.196213i
$369$	7.94156	0.413421
$370$	0	0
$371$	1.72529	0.0895725
$372$	2.33773i	0.121206i
$373$	− 11.8954i	− 0.615922i	−0.951399	−	0.307961i	$-0.900353\pi$
$373$	− 11.8954i	− 0.615922i	0.951399	−	0.307961i	$-0.0996466\pi$
$374$	−1.37461	−0.0710792
$375$	0	0
$376$	10.1489	0.523390
$377$	− 40.7220i	− 2.09729i
$378$	− 0.533559i	− 0.0274433i
$379$	−27.5184	−1.41353	−0.706763	−	0.707450i	$-0.749845\pi$
$379$	−27.5184	−1.41353	−0.706763	+	0.707450i	$0.749845\pi$
$380$	0	0
$381$	−6.48257	−0.332112
$382$	8.40045i	0.429804i
$383$	− 22.0314i	− 1.12575i	−0.826541	−	0.562877i	$-0.809695\pi$
$383$	− 22.0314i	− 1.12575i	0.826541	−	0.562877i	$-0.190305\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.5266	−0.535791
$387$	11.3607i	0.577499i
$388$	− 5.56558i	− 0.282549i
$389$	−17.4892	−0.886738	−0.443369	−	0.896339i	$-0.646217\pi$
$389$	−17.4892	−0.886738	−0.443369	+	0.896339i	$0.646217\pi$
$390$	0	0
$391$	−3.60748	−0.182438
$392$	6.71531i	0.339175i
$393$	− 9.45794i	− 0.477090i
$394$	10.2080	0.514274
$395$	0	0
$396$	−1.43425	−0.0720739
$397$	24.4990i	1.22957i	0.788694	+	0.614786i	$0.210758\pi$
$397$	24.4990i	1.22957i	−0.788694	+	0.614786i	$0.789242\pi$
$398$	3.84318i	0.192641i
$399$	0.113589	0.00568656
$400$	0	0
$401$	1.04105	0.0519875	0.0259937	−	0.999662i	$-0.491725\pi$
$401$	1.04105	0.0519875	0.0259937	+	0.999662i	$0.491725\pi$
$402$	6.91285i	0.344782i
$403$	− 15.3681i	− 0.765537i
$404$	9.42708	0.469015
$405$	0	0
$406$	−3.30512	−0.164030
$407$	5.82724i	0.288845i
$408$	− 0.958413i	− 0.0474485i
$409$	−22.2568	−1.10053	−0.550263	−	0.834991i	$-0.685473\pi$
$409$	−22.2568	−1.10053	−0.550263	+	0.834991i	$0.685473\pi$
$410$	0	0
$411$	−12.3361	−0.608494
$412$	− 7.99871i	− 0.394068i
$413$	− 4.01479i	− 0.197555i
$414$	−3.76401	−0.184991
$415$	0	0
$416$	6.57392	0.322313
$417$	− 4.53918i	− 0.222284i
$418$	− 0.305337i	− 0.0149345i
$419$	23.1294	1.12994	0.564972	−	0.825110i	$-0.308887\pi$
$419$	23.1294	1.12994	0.564972	+	0.825110i	$0.308887\pi$
$420$	0	0
$421$	9.76139	0.475741	0.237870	−	0.971297i	$-0.423551\pi$
$421$	9.76139	0.475741	0.237870	+	0.971297i	$0.423551\pi$
$422$	− 5.24920i	− 0.255527i
$423$	10.1489i	0.493457i
$424$	3.23354	0.157035
$425$	0	0
$426$	10.1247	0.490542
$427$	6.70794i	0.324620i
$428$	18.9260i	0.914822i
$429$	9.42867	0.455220
$430$	0	0
$431$	8.49298	0.409093	0.204546	−	0.978857i	$-0.434428\pi$
$431$	8.49298	0.409093	0.204546	+	0.978857i	$0.434428\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	22.0042i	1.05745i	0.848792	+	0.528727i	$0.177330\pi$
$433$	22.0042i	1.05745i	−0.848792	+	0.528727i	$0.822670\pi$
$434$	−1.24732	−0.0598731
$435$	0	0
$436$	−2.65524	−0.127163
$437$	− 0.801317i	− 0.0383322i
$438$	− 13.9771i	− 0.667851i
$439$	38.3582	1.83074	0.915369	−	0.402615i	$-0.131899\pi$
$439$	38.3582	1.83074	0.915369	+	0.402615i	$0.131899\pi$
$440$	0	0
$441$	−6.71531	−0.319777
$442$	6.30054i	0.299686i
$443$	12.7478i	0.605666i	0.953044	+	0.302833i	$0.0979324\pi$
$443$	12.7478i	0.605666i	−0.953044	+	0.302833i	$0.902068\pi$
$444$	−4.06291	−0.192817
$445$	0	0
$446$	−19.2782	−0.912851
$447$	− 11.0750i	− 0.523831i
$448$	− 0.533559i	− 0.0252083i
$449$	−18.0358	−0.851161	−0.425580	−	0.904921i	$-0.639930\pi$
$449$	−18.0358	−0.851161	−0.425580	+	0.904921i	$0.639930\pi$
$450$	0	0
$451$	11.3902	0.536344
$452$	− 3.06283i	− 0.144063i
$453$	− 1.63387i	− 0.0767659i
$454$	−8.43957	−0.396089
$455$	0	0
$456$	0.212889	0.00996944
$457$	21.1495i	0.989334i	0.869083	+	0.494667i	$0.164710\pi$
$457$	21.1495i	0.989334i	−0.869083	+	0.494667i	$0.835290\pi$
$458$	5.95568i	0.278291i
$459$	0.958413	0.0447349
$460$	0	0
$461$	32.0085	1.49078	0.745391	−	0.666627i	$-0.232263\pi$
$461$	32.0085	1.49078	0.745391	+	0.666627i	$0.232263\pi$
$462$	− 0.765259i	− 0.0356031i
$463$	28.7776i	1.33741i	0.743529	+	0.668703i	$0.233150\pi$
$463$	28.7776i	1.33741i	−0.743529	+	0.668703i	$0.766850\pi$
$464$	−6.19448	−0.287572
$465$	0	0
$466$	−2.09293	−0.0969529
$467$	− 5.20326i	− 0.240778i	−0.992727	−	0.120389i	$-0.961586\pi$
$467$	− 5.20326i	− 0.240778i	0.992727	−	0.120389i	$-0.0384142\pi$
$468$	6.57392i	0.303880i
$469$	−3.68841	−0.170315
$470$	0	0
$471$	6.64544	0.306206
$472$	− 7.52455i	− 0.346345i
$473$	16.2942i	0.749207i
$474$	−15.5279	−0.713222
$475$	0	0
$476$	0.511370	0.0234386
$477$	3.23354i	0.148054i
$478$	9.89840i	0.452742i
$479$	39.6385	1.81113	0.905565	−	0.424206i	$-0.139447\pi$
$479$	39.6385	1.81113	0.905565	+	0.424206i	$0.139447\pi$
$480$	0	0
$481$	26.7093	1.21784
$482$	21.4567i	0.977326i
$483$	− 2.00832i	− 0.0913818i
$484$	8.94292	0.406496
$485$	0	0
$486$	1.00000	0.0453609
$487$	5.44308i	0.246649i	0.992366	+	0.123325i	$0.0393557\pi$
$487$	5.44308i	0.246649i	−0.992366	+	0.123325i	$0.960644\pi$
$488$	12.5721i	0.569111i
$489$	10.1134	0.457344
$490$	0	0
$491$	0.873965	0.0394415	0.0197207	−	0.999806i	$-0.493722\pi$
$491$	0.873965	0.0394415	0.0197207	+	0.999806i	$0.493722\pi$
$492$	7.94156i	0.358033i
$493$	− 5.93687i	− 0.267383i
$494$	−1.39952	−0.0629672
$495$	0	0
$496$	−2.33773	−0.104967
$497$	5.40211i	0.242318i
$498$	16.3308i	0.731802i
$499$	34.9604	1.56504	0.782522	−	0.622623i	$-0.213933\pi$
$499$	34.9604	1.56504	0.782522	+	0.622623i	$0.213933\pi$
$500$	0	0
$501$	12.7885	0.571346
$502$	4.10753i	0.183328i
$503$	− 0.319724i	− 0.0142558i	−0.999975	−	0.00712790i	$-0.997731\pi$
$503$	− 0.319724i	− 0.0142558i	0.999975	−	0.00712790i	$-0.00226890\pi$
$504$	0.533559	0.0237666
$505$	0	0
$506$	−5.39854	−0.239995
$507$	− 30.2165i	− 1.34196i
$508$	− 6.48257i	− 0.287618i
$509$	−20.6259	−0.914225	−0.457113	−	0.889409i	$-0.651116\pi$
$509$	−20.6259	−0.914225	−0.457113	+	0.889409i	$0.651116\pi$
$510$	0	0
$511$	7.45760	0.329905
$512$	− 1.00000i	− 0.0441942i
$513$	0.212889i	0.00939928i
$514$	30.7748	1.35742
$515$	0	0
$516$	−11.3607	−0.500129
$517$	14.5561i	0.640177i
$518$	− 2.16780i	− 0.0952477i
$519$	13.6575	0.599500
$520$	0	0
$521$	−4.22039	−0.184899	−0.0924493	−	0.995717i	$-0.529470\pi$
$521$	−4.22039	−0.184899	−0.0924493	+	0.995717i	$0.529470\pi$
$522$	− 6.19448i	− 0.271125i
$523$	21.5579i	0.942659i	0.881957	+	0.471330i	$0.156226\pi$
$523$	21.5579i	0.942659i	−0.881957	+	0.471330i	$0.843774\pi$
$524$	9.45794	0.413172
$525$	0	0
$526$	−28.3729	−1.23712
$527$	− 2.24051i	− 0.0975982i
$528$	− 1.43425i	− 0.0624178i
$529$	8.83222	0.384010
$530$	0	0
$531$	7.52455	0.326538
$532$	0.113589i	0.00492470i
$533$	− 52.2072i	− 2.26135i
$534$	−5.62220	−0.243296
$535$	0	0
$536$	−6.91285	−0.298590
$537$	− 2.99142i	− 0.129089i
$538$	10.6668i	0.459877i
$539$	−9.63146	−0.414856
$540$	0	0
$541$	−8.93662	−0.384215	−0.192108	−	0.981374i	$-0.561532\pi$
$541$	−8.93662	−0.384215	−0.192108	+	0.981374i	$0.561532\pi$
$542$	− 22.8439i	− 0.981231i
$543$	7.62285i	0.327128i
$544$	0.958413	0.0410916
$545$	0	0
$546$	−3.50758	−0.150110
$547$	12.0786i	0.516443i	0.966086	+	0.258222i	$0.0831365\pi$
$547$	12.0786i	0.516443i	−0.966086	+	0.258222i	$0.916864\pi$
$548$	− 12.3361i	− 0.526971i
$549$	−12.5721	−0.536563
$550$	0	0
$551$	1.31874	0.0561801
$552$	− 3.76401i	− 0.160207i
$553$	− 8.28507i	− 0.352317i
$554$	31.4592	1.33657
$555$	0	0
$556$	4.53918	0.192504
$557$	− 8.23596i	− 0.348969i	−0.984660	−	0.174484i	$-0.944174\pi$
$557$	− 8.23596i	− 0.348969i	0.984660	−	0.174484i	$-0.0558259\pi$
$558$	− 2.33773i	− 0.0989640i
$559$	74.6846	3.15882
$560$	0	0
$561$	1.37461	0.0580360
$562$	10.7785i	0.454664i
$563$	− 14.6358i	− 0.616825i	−0.951253	−	0.308413i	$-0.900202\pi$
$563$	− 14.6358i	− 0.616825i	0.951253	−	0.308413i	$-0.0997978\pi$
$564$	−10.1489	−0.427347
$565$	0	0
$566$	−4.16428	−0.175038
$567$	0.533559i	0.0224074i
$568$	10.1247i	0.424822i
$569$	−31.2471	−1.30995	−0.654974	−	0.755651i	$-0.727321\pi$
$569$	−31.2471	−1.30995	−0.654974	+	0.755651i	$0.727321\pi$
$570$	0	0
$571$	26.2261	1.09753	0.548763	−	0.835978i	$-0.315099\pi$
$571$	26.2261	1.09753	0.548763	+	0.835978i	$0.315099\pi$
$572$	9.42867i	0.394233i
$573$	− 8.40045i	− 0.350934i
$574$	−4.23729	−0.176861
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 8.93340i	− 0.371902i	−0.982559	−	0.185951i	$-0.940463\pi$
$577$	− 8.93340i	− 0.371902i	0.982559	−	0.185951i	$-0.0595366\pi$
$578$	− 16.0814i	− 0.668900i
$579$	10.5266	0.437471
$580$	0	0
$581$	−8.71346	−0.361495
$582$	5.56558i	0.230701i
$583$	4.63772i	0.192075i
$584$	13.9771	0.578376
$585$	0	0
$586$	−17.9603	−0.741933
$587$	− 3.99360i	− 0.164833i	−0.996598	−	0.0824167i	$-0.973736\pi$
$587$	− 3.99360i	− 0.164833i	0.996598	−	0.0824167i	$-0.0262638\pi$
$588$	− 6.71531i	− 0.276935i
$589$	0.497677	0.0205064
$590$	0	0
$591$	−10.2080	−0.419903
$592$	− 4.06291i	− 0.166985i
$593$	11.2114i	0.460396i	0.973144	+	0.230198i	$0.0739374\pi$
$593$	11.2114i	0.460396i	−0.973144	+	0.230198i	$0.926063\pi$
$594$	1.43425	0.0588481
$595$	0	0
$596$	11.0750	0.453651
$597$	− 3.84318i	− 0.157291i
$598$	24.7443i	1.01187i
$599$	6.64762	0.271614	0.135807	−	0.990735i	$-0.456637\pi$
$599$	6.64762	0.271614	0.135807	+	0.990735i	$0.456637\pi$
$600$	0	0
$601$	−10.7465	−0.438359	−0.219179	−	0.975685i	$-0.570338\pi$
$601$	−10.7465	−0.438359	−0.219179	+	0.975685i	$0.570338\pi$
$602$	− 6.06163i	− 0.247053i
$603$	− 6.91285i	− 0.281513i
$604$	1.63387	0.0664812
$605$	0	0
$606$	−9.42708	−0.382949
$607$	− 15.4591i	− 0.627466i	−0.949511	−	0.313733i	$-0.898420\pi$
$607$	− 15.4591i	− 0.627466i	0.949511	−	0.313733i	$-0.101580\pi$
$608$	0.212889i	0.00863379i
$609$	3.30512	0.133930
$610$	0	0
$611$	66.7182	2.69913
$612$	0.958413i	0.0387416i
$613$	− 24.4867i	− 0.989007i	−0.869176	−	0.494503i	$-0.835350\pi$
$613$	− 24.4867i	− 0.989007i	0.869176	−	0.494503i	$-0.164650\pi$
$614$	20.8174	0.840122
$615$	0	0
$616$	0.765259	0.0308332
$617$	18.9157i	0.761515i	0.924675	+	0.380758i	$0.124337\pi$
$617$	18.9157i	0.761515i	−0.924675	+	0.380758i	$0.875663\pi$
$618$	7.99871i	0.321755i
$619$	−2.55005	−0.102495	−0.0512476	−	0.998686i	$-0.516320\pi$
$619$	−2.55005	−0.102495	−0.0512476	+	0.998686i	$0.516320\pi$
$620$	0	0
$621$	3.76401	0.151045
$622$	19.4526i	0.779978i
$623$	− 2.99977i	− 0.120183i
$624$	−6.57392	−0.263168
$625$	0	0
$626$	5.88310	0.235136
$627$	0.305337i	0.0121940i
$628$	6.64544i	0.265182i
$629$	3.89395	0.155262
$630$	0	0
$631$	−36.4530	−1.45117	−0.725585	−	0.688133i	$-0.758431\pi$
$631$	−36.4530	−1.45117	−0.725585	+	0.688133i	$0.758431\pi$
$632$	− 15.5279i	− 0.617668i
$633$	5.24920i	0.208637i
$634$	10.7610	0.427375
$635$	0	0
$636$	−3.23354	−0.128218
$637$	44.1460i	1.74913i
$638$	− 8.88445i	− 0.351739i
$639$	−10.1247	−0.400526
$640$	0	0
$641$	−20.3669	−0.804444	−0.402222	−	0.915542i	$-0.631762\pi$
$641$	−20.3669	−0.804444	−0.402222	+	0.915542i	$0.631762\pi$
$642$	− 18.9260i	− 0.746949i
$643$	− 11.4218i	− 0.450433i	−0.974309	−	0.225217i	$-0.927691\pi$
$643$	− 11.4218i	− 0.450433i	0.974309	−	0.225217i	$-0.0723090\pi$
$644$	2.00832	0.0791390
$645$	0	0
$646$	−0.204036	−0.00802768
$647$	− 5.89217i	− 0.231645i	−0.993270	−	0.115823i	$-0.963050\pi$
$647$	− 5.89217i	− 0.231645i	0.993270	−	0.115823i	$-0.0369504\pi$
$648$	1.00000i	0.0392837i
$649$	10.7921	0.423627
$650$	0	0
$651$	1.24732	0.0488862
$652$	10.1134i	0.396071i
$653$	17.1063i	0.669422i	0.942321	+	0.334711i	$0.108639\pi$
$653$	17.1063i	0.669422i	−0.942321	+	0.334711i	$0.891361\pi$
$654$	2.65524	0.103828
$655$	0	0
$656$	−7.94156	−0.310066
$657$	13.9771i	0.545298i
$658$	− 5.41505i	− 0.211101i
$659$	29.7191	1.15769	0.578846	−	0.815437i	$-0.303503\pi$
$659$	29.7191	1.15769	0.578846	+	0.815437i	$0.303503\pi$
$660$	0	0
$661$	46.3036	1.80100	0.900501	−	0.434854i	$-0.143200\pi$
$661$	46.3036	1.80100	0.900501	+	0.434854i	$0.143200\pi$
$662$	0.617092i	0.0239840i
$663$	− 6.30054i	− 0.244693i
$664$	−16.3308	−0.633759
$665$	0	0
$666$	4.06291	0.157435
$667$	− 23.3161i	− 0.902803i
$668$	12.7885i	0.494800i
$669$	19.2782	0.745340
$670$	0	0
$671$	−18.0315	−0.696099
$672$	0.533559i	0.0205825i
$673$	19.7080i	0.759689i	0.925050	+	0.379845i	$0.124023\pi$
$673$	19.7080i	0.759689i	−0.925050	+	0.379845i	$0.875977\pi$
$674$	15.4106	0.593595
$675$	0	0
$676$	30.2165	1.16217
$677$	24.7774i	0.952273i	0.879371	+	0.476137i	$0.157963\pi$
$677$	24.7774i	0.952273i	−0.879371	+	0.476137i	$0.842037\pi$
$678$	3.06283i	0.117627i
$679$	−2.96957	−0.113961
$680$	0	0
$681$	8.43957	0.323405
$682$	− 3.35289i	− 0.128389i
$683$	− 16.5874i	− 0.634699i	−0.948309	−	0.317349i	$-0.897207\pi$
$683$	− 16.5874i	− 0.634699i	0.948309	−	0.317349i	$-0.102793\pi$
$684$	−0.212889	−0.00814002
$685$	0	0
$686$	7.31793	0.279400
$687$	− 5.95568i	− 0.227224i
$688$	− 11.3607i	− 0.433124i
$689$	21.2571	0.809830
$690$	0	0
$691$	42.2893	1.60876	0.804381	−	0.594114i	$-0.202497\pi$
$691$	42.2893	1.60876	0.804381	+	0.594114i	$0.202497\pi$
$692$	13.6575i	0.519182i
$693$	0.765259i	0.0290698i
$694$	−22.6565	−0.860031
$695$	0	0
$696$	6.19448	0.234801
$697$	− 7.61130i	− 0.288299i
$698$	− 16.9543i	− 0.641731i
$699$	2.09293	0.0791617
$700$	0	0
$701$	−22.7240	−0.858273	−0.429137	−	0.903240i	$-0.641182\pi$
$701$	−22.7240	−0.858273	−0.429137	+	0.903240i	$0.641182\pi$
$702$	− 6.57392i	− 0.248117i
$703$	0.864949i	0.0326222i
$704$	1.43425	0.0540554
$705$	0	0
$706$	6.06255	0.228167
$707$	− 5.02990i	− 0.189169i
$708$	7.52455i	0.282790i
$709$	−46.4122	−1.74305	−0.871524	−	0.490353i	$-0.836868\pi$
$709$	−46.4122	−1.74305	−0.871524	+	0.490353i	$0.836868\pi$
$710$	0	0
$711$	15.5279	0.582343
$712$	− 5.62220i	− 0.210701i
$713$	− 8.79924i	− 0.329534i
$714$	−0.511370	−0.0191375
$715$	0	0
$716$	2.99142	0.111795
$717$	− 9.89840i	− 0.369662i
$718$	18.8987i	0.705294i
$719$	28.2066	1.05193	0.525964	−	0.850507i	$-0.323705\pi$
$719$	28.2066	1.05193	0.525964	+	0.850507i	$0.323705\pi$
$720$	0	0
$721$	−4.26779	−0.158941
$722$	18.9547i	0.705420i
$723$	− 21.4567i	− 0.797983i
$724$	−7.62285	−0.283301
$725$	0	0
$726$	−8.94292	−0.331903
$727$	0.851592i	0.0315838i	0.999875	+	0.0157919i	$0.00502693\pi$
$727$	0.851592i	0.0315838i	−0.999875	+	0.0157919i	$0.994973\pi$
$728$	− 3.50758i	− 0.129999i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	10.8883	0.402718
$732$	− 12.5721i	− 0.464677i
$733$	− 30.7299i	− 1.13504i	−0.823361	−	0.567518i	$-0.807904\pi$
$733$	− 30.7299i	− 1.13504i	0.823361	−	0.567518i	$-0.192096\pi$
$734$	25.0483	0.924551
$735$	0	0
$736$	3.76401	0.138743
$737$	− 9.91477i	− 0.365215i
$738$	− 7.94156i	− 0.292333i
$739$	−2.06485	−0.0759566	−0.0379783	−	0.999279i	$-0.512092\pi$
$739$	−2.06485	−0.0759566	−0.0379783	+	0.999279i	$0.512092\pi$
$740$	0	0
$741$	1.39952	0.0514125
$742$	− 1.72529i	− 0.0633373i
$743$	37.8972i	1.39031i	0.718858	+	0.695157i	$0.244665\pi$
$743$	37.8972i	1.39031i	−0.718858	+	0.695157i	$0.755335\pi$
$744$	2.33773	0.0857053
$745$	0	0
$746$	−11.8954	−0.435522
$747$	− 16.3308i	− 0.597514i
$748$	1.37461i	0.0502606i
$749$	10.0981	0.368978
$750$	0	0
$751$	−40.6331	−1.48272	−0.741361	−	0.671106i	$-0.765819\pi$
$751$	−40.6331	−1.48272	−0.741361	+	0.671106i	$0.765819\pi$
$752$	− 10.1489i	− 0.370093i
$753$	− 4.10753i	− 0.149687i
$754$	−40.7220	−1.48301
$755$	0	0
$756$	−0.533559	−0.0194054
$757$	10.0032i	0.363572i	0.983338	+	0.181786i	$0.0581877\pi$
$757$	10.0032i	0.363572i	−0.983338	+	0.181786i	$0.941812\pi$
$758$	27.5184i	0.999514i
$759$	5.39854	0.195955
$760$	0	0
$761$	−30.1813	−1.09407	−0.547035	−	0.837110i	$-0.684244\pi$
$761$	−30.1813	−1.09407	−0.547035	+	0.837110i	$0.684244\pi$
$762$	6.48257i	0.234839i
$763$	1.41673i	0.0512890i
$764$	8.40045	0.303917
$765$	0	0
$766$	−22.0314	−0.796028
$767$	− 49.4658i	− 1.78611i
$768$	1.00000i	0.0360844i
$769$	−21.1468	−0.762571	−0.381286	−	0.924457i	$-0.624519\pi$
$769$	−21.1468	−0.762571	−0.381286	+	0.924457i	$0.624519\pi$
$770$	0	0
$771$	−30.7748	−1.10833
$772$	10.5266i	0.378861i
$773$	29.4470i	1.05913i	0.848268	+	0.529567i	$0.177646\pi$
$773$	29.4470i	1.05913i	−0.848268	+	0.529567i	$0.822354\pi$
$774$	11.3607	0.408353
$775$	0	0
$776$	−5.56558	−0.199793
$777$	2.16780i	0.0777695i
$778$	17.4892i	0.627019i
$779$	1.69067	0.0605746
$780$	0	0
$781$	−14.5213	−0.519614
$782$	3.60748i	0.129003i
$783$	6.19448i	0.221373i
$784$	6.71531	0.239833
$785$	0	0
$786$	−9.45794	−0.337353
$787$	− 5.16875i	− 0.184246i	−0.995748	−	0.0921230i	$-0.970635\pi$
$787$	− 5.16875i	− 0.184246i	0.995748	−	0.0921230i	$-0.0293653\pi$
$788$	− 10.2080i	− 0.363647i
$789$	28.3729	1.01010
$790$	0	0
$791$	−1.63420	−0.0581055
$792$	1.43425i	0.0509640i
$793$	82.6478i	2.93491i
$794$	24.4990	0.869439
$795$	0	0
$796$	3.84318	0.136218
$797$	− 3.75807i	− 0.133118i	−0.997783	−	0.0665588i	$-0.978798\pi$
$797$	− 3.75807i	− 0.133118i	0.997783	−	0.0665588i	$-0.0212020\pi$
$798$	− 0.113589i	− 0.00402100i
$799$	9.72686	0.344111
$800$	0	0
$801$	5.62220	0.198650
$802$	− 1.04105i	− 0.0367607i
$803$	20.0467i	0.707432i
$804$	6.91285	0.243797
$805$	0	0
$806$	−15.3681	−0.541317
$807$	− 10.6668i	− 0.375488i
$808$	− 9.42708i	− 0.331643i
$809$	20.8847	0.734266	0.367133	−	0.930168i	$-0.380339\pi$
$809$	20.8847	0.734266	0.367133	+	0.930168i	$0.380339\pi$
$810$	0	0
$811$	−36.5032	−1.28180	−0.640900	−	0.767625i	$-0.721439\pi$
$811$	−36.5032	−1.28180	−0.640900	+	0.767625i	$0.721439\pi$
$812$	3.30512i	0.115987i
$813$	22.8439i	0.801172i
$814$	5.82724	0.204245
$815$	0	0
$816$	−0.958413	−0.0335512
$817$	2.41858i	0.0846153i
$818$	22.2568i	0.778190i
$819$	3.50758	0.122565
$820$	0	0
$821$	33.2221	1.15946	0.579730	−	0.814809i	$-0.303158\pi$
$821$	33.2221	1.15946	0.579730	+	0.814809i	$0.303158\pi$
$822$	12.3361i	0.430270i
$823$	32.7898i	1.14298i	0.820609	+	0.571490i	$0.193634\pi$
$823$	32.7898i	1.14298i	−0.820609	+	0.571490i	$0.806366\pi$
$824$	−7.99871	−0.278648
$825$	0	0
$826$	−4.01479	−0.139692
$827$	5.18915i	0.180444i	0.995922	+	0.0902222i	$0.0287577\pi$
$827$	5.18915i	0.180444i	−0.995922	+	0.0902222i	$0.971242\pi$
$828$	3.76401i	0.130808i
$829$	5.55820	0.193044	0.0965221	−	0.995331i	$-0.469228\pi$
$829$	5.55820	0.193044	0.0965221	+	0.995331i	$0.469228\pi$
$830$	0	0
$831$	−31.4592	−1.09131
$832$	− 6.57392i	− 0.227910i
$833$	6.43605i	0.222996i
$834$	−4.53918	−0.157179
$835$	0	0
$836$	−0.305337	−0.0105603
$837$	2.33773i	0.0808037i
$838$	− 23.1294i	− 0.798991i
$839$	−43.4383	−1.49966	−0.749828	−	0.661633i	$-0.769864\pi$
$839$	−43.4383	−1.49966	−0.749828	+	0.661633i	$0.769864\pi$
$840$	0	0
$841$	9.37160	0.323159
$842$	− 9.76139i	− 0.336400i
$843$	− 10.7785i	− 0.371232i
$844$	−5.24920	−0.180685
$845$	0	0
$846$	10.1489	0.348927
$847$	− 4.77158i	− 0.163953i
$848$	− 3.23354i	− 0.111040i
$849$	4.16428	0.142918
$850$	0	0
$851$	15.2928	0.524231
$852$	− 10.1247i	− 0.346865i
$853$	− 3.34342i	− 0.114477i	−0.998361	−	0.0572383i	$-0.981771\pi$
$853$	− 3.34342i	− 0.114477i	0.998361	−	0.0572383i	$-0.0182295\pi$
$854$	6.70794	0.229541
$855$	0	0
$856$	18.9260	0.646877
$857$	− 34.5415i	− 1.17991i	−0.807434	−	0.589957i	$-0.799145\pi$
$857$	− 34.5415i	− 1.17991i	0.807434	−	0.589957i	$-0.200855\pi$
$858$	− 9.42867i	− 0.321889i
$859$	23.8513	0.813796	0.406898	−	0.913474i	$-0.366610\pi$
$859$	23.8513	0.813796	0.406898	+	0.913474i	$0.366610\pi$
$860$	0	0
$861$	4.23729	0.144407
$862$	− 8.49298i	− 0.289272i
$863$	− 5.73061i	− 0.195072i	−0.995232	−	0.0975361i	$-0.968904\pi$
$863$	− 5.73061i	− 0.195072i	0.995232	−	0.0975361i	$-0.0310962\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	22.0042	0.747733
$867$	16.0814i	0.546154i
$868$	1.24732i	0.0423367i
$869$	22.2710	0.755492
$870$	0	0
$871$	−45.4445	−1.53983
$872$	2.65524i	0.0899178i
$873$	− 5.56558i	− 0.188366i
$874$	−0.801317	−0.0271049
$875$	0	0
$876$	−13.9771	−0.472242
$877$	49.8482i	1.68325i	0.540059	+	0.841627i	$0.318402\pi$
$877$	49.8482i	1.68325i	−0.540059	+	0.841627i	$0.681598\pi$
$878$	− 38.3582i	− 1.29453i
$879$	17.9603	0.605786
$880$	0	0
$881$	−7.65609	−0.257940	−0.128970	−	0.991648i	$-0.541167\pi$
$881$	−7.65609	−0.257940	−0.128970	+	0.991648i	$0.541167\pi$
$882$	6.71531i	0.226116i
$883$	− 38.9959i	− 1.31232i	−0.754623	−	0.656158i	$-0.772180\pi$
$883$	− 38.9959i	− 1.31232i	0.754623	−	0.656158i	$-0.227820\pi$
$884$	6.30054	0.211910
$885$	0	0
$886$	12.7478	0.428270
$887$	− 32.5338i	− 1.09238i	−0.837662	−	0.546189i	$-0.816078\pi$
$887$	− 32.5338i	− 1.09238i	0.837662	−	0.546189i	$-0.183922\pi$
$888$	4.06291i	0.136342i
$889$	−3.45884	−0.116006
$890$	0	0
$891$	−1.43425	−0.0480493
$892$	19.2782i	0.645483i
$893$	2.16059i	0.0723015i
$894$	−11.0750	−0.370405
$895$	0	0
$896$	−0.533559	−0.0178250
$897$	− 24.7443i	− 0.826189i
$898$	18.0358i	0.601862i
$899$	14.4810	0.482969
$900$	0	0
$901$	3.09907	0.103245
$902$	− 11.3902i	− 0.379253i
$903$	6.06163i	0.201718i
$904$	−3.06283	−0.101868
$905$	0	0
$906$	−1.63387	−0.0542817
$907$	16.5820i	0.550595i	0.961359	+	0.275297i	$0.0887763\pi$
$907$	16.5820i	0.550595i	−0.961359	+	0.275297i	$0.911224\pi$
$908$	8.43957i	0.280077i
$909$	9.42708	0.312676
$910$	0	0
$911$	−28.5720	−0.946633	−0.473316	−	0.880893i	$-0.656943\pi$
$911$	−28.5720	−0.946633	−0.473316	+	0.880893i	$0.656943\pi$
$912$	− 0.212889i	− 0.00704946i
$913$	− 23.4225i	− 0.775173i
$914$	21.1495	0.699565
$915$	0	0
$916$	5.95568	0.196781
$917$	− 5.04637i	− 0.166646i
$918$	− 0.958413i	− 0.0316324i
$919$	0.382861	0.0126294	0.00631471	−	0.999980i	$-0.497990\pi$
$919$	0.382861	0.0126294	0.00631471	+	0.999980i	$0.497990\pi$
$920$	0	0
$921$	−20.8174	−0.685957
$922$	− 32.0085i	− 1.05414i
$923$	66.5588i	2.19081i
$924$	−0.765259	−0.0251752
$925$	0	0
$926$	28.7776	0.945689
$927$	− 7.99871i	− 0.262712i
$928$	6.19448i	0.203344i
$929$	6.25711	0.205289	0.102645	−	0.994718i	$-0.467270\pi$
$929$	6.25711	0.205289	0.102645	+	0.994718i	$0.467270\pi$
$930$	0	0
$931$	−1.42962	−0.0468538
$932$	2.09293i	0.0685561i
$933$	− 19.4526i	− 0.636849i
$934$	−5.20326	−0.170256
$935$	0	0
$936$	6.57392	0.214875
$937$	− 4.93214i	− 0.161126i	−0.996750	−	0.0805630i	$-0.974328\pi$
$937$	− 4.93214i	− 0.161126i	0.996750	−	0.0805630i	$-0.0256718\pi$
$938$	3.68841i	0.120431i
$939$	−5.88310	−0.191988
$940$	0	0
$941$	29.8107	0.971801	0.485901	−	0.874014i	$-0.338492\pi$
$941$	29.8107	0.971801	0.485901	+	0.874014i	$0.338492\pi$
$942$	− 6.64544i	− 0.216520i
$943$	− 29.8921i	− 0.973422i
$944$	−7.52455	−0.244903
$945$	0	0
$946$	16.2942	0.529769
$947$	− 26.9125i	− 0.874539i	−0.899330	−	0.437270i	$-0.855946\pi$
$947$	− 26.9125i	− 0.874539i	0.899330	−	0.437270i	$-0.144054\pi$
$948$	15.5279i	0.504324i
$949$	91.8843	2.98269
$950$	0	0
$951$	−10.7610	−0.348950
$952$	− 0.511370i	− 0.0165736i
$953$	− 37.2337i	− 1.20612i	−0.797697	−	0.603059i	$-0.793949\pi$
$953$	− 37.2337i	− 1.20612i	0.797697	−	0.603059i	$-0.206051\pi$
$954$	3.23354	0.104690
$955$	0	0
$956$	9.89840	0.320137
$957$	8.88445i	0.287194i
$958$	− 39.6385i	− 1.28066i
$959$	−6.58203	−0.212545
$960$	0	0
$961$	−25.5350	−0.823710
$962$	− 26.7093i	− 0.861141i
$963$	18.9260i	0.609882i
$964$	21.4567	0.691074
$965$	0	0
$966$	−2.00832	−0.0646167
$967$	16.0845i	0.517244i	0.965979	+	0.258622i	$0.0832683\pi$
$967$	16.0845i	0.517244i	−0.965979	+	0.258622i	$0.916732\pi$
$968$	− 8.94292i	− 0.287436i
$969$	0.204036	0.00655457
$970$	0	0
$971$	12.2569	0.393342	0.196671	−	0.980470i	$-0.436987\pi$
$971$	12.2569	0.393342	0.196671	+	0.980470i	$0.436987\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 2.42192i	− 0.0776431i
$974$	5.44308	0.174407
$975$	0	0
$976$	12.5721	0.402422
$977$	18.0218i	0.576569i	0.957545	+	0.288285i	$0.0930849\pi$
$977$	18.0218i	0.576569i	−0.957545	+	0.288285i	$0.906915\pi$
$978$	− 10.1134i	− 0.323391i
$979$	8.06365	0.257715
$980$	0	0
$981$	−2.65524	−0.0847753
$982$	− 0.873965i	− 0.0278893i
$983$	− 49.4029i	− 1.57571i	−0.615862	−	0.787854i	$-0.711192\pi$
$983$	− 49.4029i	− 1.57571i	0.615862	−	0.787854i	$-0.288808\pi$
$984$	7.94156	0.253168
$985$	0	0
$986$	−5.93687	−0.189069
$987$	5.41505i	0.172363i
$988$	1.39952i	0.0445246i
$989$	42.7619	1.35975
$990$	0	0
$991$	−30.5664	−0.970972	−0.485486	−	0.874244i	$-0.661357\pi$
$991$	−30.5664	−0.970972	−0.485486	+	0.874244i	$0.661357\pi$
$992$	2.33773i	0.0742230i
$993$	− 0.617092i	− 0.0195828i
$994$	5.40211	0.171345
$995$	0	0
$996$	16.3308	0.517462
$997$	6.81050i	0.215691i	0.994168	+	0.107845i	$0.0343952\pi$
$997$	6.81050i	0.215691i	−0.994168	+	0.107845i	$0.965605\pi$
$998$	− 34.9604i	− 1.10665i
$999$	−4.06291	−0.128545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.k.1249.6		16
5.2	odd	4		3750.2.a.v.1.3		8
5.3	odd	4		3750.2.a.u.1.6		8
5.4	even	2	inner	3750.2.c.k.1249.11		16
25.3	odd	20		750.2.g.g.451.3		16
25.4	even	10		150.2.h.b.109.3	✓	16
25.6	even	5		150.2.h.b.139.3	yes	16
25.8	odd	20		750.2.g.g.301.3		16
25.17	odd	20		750.2.g.f.301.2		16
25.19	even	10		750.2.h.d.199.1		16
25.21	even	5		750.2.h.d.49.2		16
25.22	odd	20		750.2.g.f.451.2		16
75.29	odd	10		450.2.l.c.109.2		16
75.56	odd	10		450.2.l.c.289.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.b.109.3	✓	16	25.4	even	10
150.2.h.b.139.3	yes	16	25.6	even	5
450.2.l.c.109.2		16	75.29	odd	10
450.2.l.c.289.2		16	75.56	odd	10
750.2.g.f.301.2		16	25.17	odd	20
750.2.g.f.451.2		16	25.22	odd	20
750.2.g.g.301.3		16	25.8	odd	20
750.2.g.g.451.3		16	25.3	odd	20
750.2.h.d.49.2		16	25.21	even	5
750.2.h.d.199.1		16	25.19	even	10
3750.2.a.u.1.6		8	5.3	odd	4
3750.2.a.v.1.3		8	5.2	odd	4
3750.2.c.k.1249.6		16	1.1	even	1	trivial
3750.2.c.k.1249.11		16	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.533559i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +0.533559i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.43425 q^{11} -1.00000i q^{12} +6.57392i q^{13} +0.533559 q^{14} +1.00000 q^{16} +0.958413i q^{17} +1.00000i q^{18} -0.212889 q^{19} -0.533559 q^{21} +1.43425i q^{22} +3.76401i q^{23} -1.00000 q^{24} +6.57392 q^{26} -1.00000i q^{27} -0.533559i q^{28} -6.19448 q^{29} -2.33773 q^{31} -1.00000i q^{32} -1.43425i q^{33} +0.958413 q^{34} +1.00000 q^{36} -4.06291i q^{37} +0.212889i q^{38} -6.57392 q^{39} -7.94156 q^{41} +0.533559i q^{42} -11.3607i q^{43} +1.43425 q^{44} +3.76401 q^{46} -10.1489i q^{47} +1.00000i q^{48} +6.71531 q^{49} -0.958413 q^{51} -6.57392i q^{52} -3.23354i q^{53} -1.00000 q^{54} -0.533559 q^{56} -0.212889i q^{57} +6.19448i q^{58} -7.52455 q^{59} +12.5721 q^{61} +2.33773i q^{62} -0.533559i q^{63} -1.00000 q^{64} -1.43425 q^{66} +6.91285i q^{67} -0.958413i q^{68} -3.76401 q^{69} +10.1247 q^{71} -1.00000i q^{72} -13.9771i q^{73} -4.06291 q^{74} +0.212889 q^{76} -0.765259i q^{77} +6.57392i q^{78} -15.5279 q^{79} +1.00000 q^{81} +7.94156i q^{82} +16.3308i q^{83} +0.533559 q^{84} -11.3607 q^{86} -6.19448i q^{87} -1.43425i q^{88} -5.62220 q^{89} -3.50758 q^{91} -3.76401i q^{92} -2.33773i q^{93} -10.1489 q^{94} +1.00000 q^{96} +5.56558i q^{97} -6.71531i q^{98} +1.43425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) \) 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

Varieties

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Representations

Groups

Database

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