LMFDB - Embedded newform 1900.2.s.b.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(49,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.s (of order $6$, degree $2$, 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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			349.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1900.349
Dual form			1900.2.s.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{3} +4.00000i q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(1.73205 - 1.00000i) q^{3} +4.00000i q^{7} +(0.500000 - 0.866025i) q^{9} -3.00000 q^{11} +(-5.19615 - 3.00000i) q^{13} +(-1.73205 + 1.00000i) q^{17} +(-3.50000 + 2.59808i) q^{19} +(4.00000 + 6.92820i) q^{21} +(-3.46410 - 2.00000i) q^{23} +4.00000i q^{27} +(0.500000 - 0.866025i) q^{29} -5.00000 q^{31} +(-5.19615 + 3.00000i) q^{33} +4.00000i q^{37} -12.0000 q^{39} +(-1.00000 - 1.73205i) q^{41} +(5.19615 + 3.00000i) q^{47} -9.00000 q^{49} +(-2.00000 + 3.46410i) q^{51} +(5.19615 + 3.00000i) q^{53} +(-3.46410 + 8.00000i) q^{57} +(-0.500000 - 0.866025i) q^{59} +(3.50000 - 6.06218i) q^{61} +(3.46410 + 2.00000i) q^{63} +(-12.1244 - 7.00000i) q^{67} -8.00000 q^{69} +(-7.50000 - 12.9904i) q^{71} +(10.3923 - 6.00000i) q^{73} -12.0000i q^{77} +(-0.500000 - 0.866025i) q^{79} +(5.50000 + 9.52628i) q^{81} +16.0000i q^{83} -2.00000i q^{87} +(8.50000 - 14.7224i) q^{89} +(12.0000 - 20.7846i) q^{91} +(-8.66025 + 5.00000i) q^{93} +(-10.3923 + 6.00000i) q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} - 12 q^{11} - 14 q^{19} + 16 q^{21} + 2 q^{29} - 20 q^{31} - 48 q^{39} - 4 q^{41} - 36 q^{49} - 8 q^{51} - 2 q^{59} + 14 q^{61} - 32 q^{69} - 30 q^{71} - 2 q^{79} + 22 q^{81} + 34 q^{89} + 48 q^{91} - 6 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 - 12 * q^11 - 14 * q^19 + 16 * q^21 + 2 * q^29 - 20 * q^31 - 48 * q^39 - 4 * q^41 - 36 * q^49 - 8 * q^51 - 2 * q^59 + 14 * q^61 - 32 * q^69 - 30 * q^71 - 2 * q^79 + 22 * q^81 + 34 * q^89 + 48 * q^91 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.73205	−	1.00000i	1.00000	−	0.577350i	0.0917517	−	0.995782i	$-0.470753\pi$
$3$	1.73205	−	1.00000i	1.00000	−	0.577350i	0.908248	+	0.418432i	$0.137420\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	0.500000	−	0.866025i	0.166667	−	0.288675i
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−5.19615	−	3.00000i	−1.44115	−	0.832050i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	−5.19615	−	3.00000i	−1.44115	−	0.832050i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	0.275029	+	0.961436i	$0.411312\pi$
$18$		0			0
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$19$	−3.50000	+	2.59808i	−0.802955	+	0.596040i
$20$		0			0
$21$	4.00000	+	6.92820i	0.872872	+	1.51186i
$22$		0			0
$23$	−3.46410	−	2.00000i	−0.722315	−	0.417029i	0.0932891	−	0.995639i	$-0.470262\pi$
$23$	−3.46410	−	2.00000i	−0.722315	−	0.417029i	−0.815604	+	0.578610i	$0.803595\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	0.500000	−	0.866025i	0.0928477	−	0.160817i	−0.815861	−	0.578249i	$-0.803736\pi$
$29$	0.500000	−	0.866025i	0.0928477	−	0.160817i	0.908708	+	0.417432i	$0.137070\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$	−5.19615	+	3.00000i	−0.904534	+	0.522233i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	−12.0000			−1.92154
$40$		0			0
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	0.777312	−	0.629115i	$-0.216583\pi$
$41$	−1.00000	−	1.73205i	−0.156174	−	0.270501i	−0.933486	+	0.358614i	$0.883249\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.19615	+	3.00000i	0.757937	+	0.437595i	0.828554	−	0.559908i	$-0.189164\pi$
$47$	5.19615	+	3.00000i	0.757937	+	0.437595i	−0.0706177	+	0.997503i	$0.522497\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−2.00000	+	3.46410i	−0.280056	+	0.485071i
$52$		0			0
$53$	5.19615	+	3.00000i	0.713746	+	0.412082i	0.812447	−	0.583036i	$-0.198135\pi$
$53$	5.19615	+	3.00000i	0.713746	+	0.412082i	−0.0987002	+	0.995117i	$0.531468\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.46410	+	8.00000i	−0.458831	+	1.05963i
$58$		0			0
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	0.831641	−	0.555313i	$-0.187402\pi$
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	−0.896736	+	0.442566i	$0.854068\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$	3.46410	+	2.00000i	0.436436	+	0.251976i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.1244	−	7.00000i	−1.48123	−	0.855186i	−0.481452	−	0.876472i	$-0.659891\pi$
$67$	−12.1244	−	7.00000i	−1.48123	−	0.855186i	−0.999773	+	0.0212861i	$0.993224\pi$
$68$		0			0
$69$	−8.00000			−0.963087
$70$		0			0
$71$	−7.50000	−	12.9904i	−0.890086	−	1.54167i	−0.839771	−	0.542941i	$-0.817311\pi$
$71$	−7.50000	−	12.9904i	−0.890086	−	1.54167i	−0.0503155	−	0.998733i	$-0.516023\pi$
$72$		0			0
$73$	10.3923	−	6.00000i	1.21633	−	0.702247i	0.252197	−	0.967676i	$-0.418847\pi$
$73$	10.3923	−	6.00000i	1.21633	−	0.702247i	0.964130	+	0.265429i	$0.0855136\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	12.0000i		−	1.36753i
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	8.50000	−	14.7224i	0.900998	−	1.56057i	0.0747975	−	0.997199i	$-0.476169\pi$
$89$	8.50000	−	14.7224i	0.900998	−	1.56057i	0.826201	−	0.563376i	$-0.190498\pi$
$90$		0			0
$91$	12.0000	−	20.7846i	1.25794	−	2.17882i
$92$		0			0
$93$	−8.66025	+	5.00000i	−0.898027	+	0.518476i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.3923	+	6.00000i	−1.05518	+	0.609208i	−0.924095	−	0.382164i	$-0.875179\pi$
$97$	−10.3923	+	6.00000i	−1.05518	+	0.609208i	−0.131084	+	0.991371i	$0.541846\pi$
$98$		0			0
$99$	−1.50000	+	2.59808i	−0.150756	+	0.261116i
$100$		0			0
$101$	−5.50000	+	9.52628i	−0.547270	+	0.947900i	0.451190	+	0.892428i	$0.351000\pi$
$101$	−5.50000	+	9.52628i	−0.547270	+	0.947900i	−0.998460	+	0.0554722i	$0.982334\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	3.50000	+	6.06218i	0.335239	+	0.580651i	0.983531	−	0.180741i	$-0.0578495\pi$
$109$	3.50000	+	6.06218i	0.335239	+	0.580651i	−0.648292	+	0.761392i	$0.724516\pi$
$110$		0			0
$111$	4.00000	+	6.92820i	0.379663	+	0.657596i
$112$		0			0
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.19615	+	3.00000i	−0.480384	+	0.277350i
$118$		0			0
$119$	−4.00000	−	6.92820i	−0.366679	−	0.635107i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	−3.46410	−	2.00000i	−0.312348	−	0.180334i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	15.5885	+	9.00000i	1.38325	+	0.798621i	0.992543	−	0.121894i	$-0.0388966\pi$
$127$	15.5885	+	9.00000i	1.38325	+	0.798621i	0.390709	+	0.920514i	$0.372230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.0000	+	17.3205i	0.873704	+	1.51330i	0.858137	+	0.513421i	$0.171622\pi$
$131$	10.0000	+	17.3205i	0.873704	+	1.51330i	0.0155672	+	0.999879i	$0.495045\pi$
$132$		0			0
$133$	−10.3923	−	14.0000i	−0.901127	−	1.21395i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3923	+	6.00000i	0.887875	+	0.512615i	0.873247	−	0.487278i	$-0.162010\pi$
$137$	10.3923	+	6.00000i	0.887875	+	0.512615i	0.0146279	+	0.999893i	$0.495344\pi$
$138$		0			0
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$	15.5885	+	9.00000i	1.30357	+	0.752618i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−15.5885	+	9.00000i	−1.28571	+	0.742307i
$148$		0			0
$149$	−8.50000	−	14.7224i	−0.696347	−	1.20611i	−0.969724	−	0.244202i	$-0.921474\pi$
$149$	−8.50000	−	14.7224i	−0.696347	−	1.20611i	0.273377	−	0.961907i	$-0.411859\pi$
$150$		0			0
$151$	3.00000			0.244137			0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000			0.244137			0.122068	+	0.992522i	$0.461047\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−15.5885	+	9.00000i	−1.24409	+	0.718278i	−0.969925	−	0.243403i	$-0.921736\pi$
$157$	−15.5885	+	9.00000i	−1.24409	+	0.718278i	−0.274169	+	0.961681i	$0.588403\pi$
$158$		0			0
$159$	12.0000			0.951662
$160$		0			0
$161$	8.00000	−	13.8564i	0.630488	−	1.09204i
$162$		0			0
$163$			18.0000i			1.40987i	0.709273	+	0.704934i	$0.249024\pi$
$163$			18.0000i			1.40987i	−0.709273	+	0.704934i	$0.750976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.92820	+	4.00000i	0.536120	+	0.309529i	0.743505	−	0.668730i	$-0.233162\pi$
$167$	6.92820	+	4.00000i	0.536120	+	0.309529i	−0.207385	+	0.978259i	$0.566495\pi$
$168$		0			0
$169$	11.5000	+	19.9186i	0.884615	+	1.53220i
$170$		0			0
$171$	0.500000	+	4.33013i	0.0382360	+	0.331133i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.73205	−	1.00000i	−0.130189	−	0.0751646i
$178$		0			0
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	−1.00000	+	1.73205i	−0.0743294	+	0.128742i	−0.900794	−	0.434246i	$-0.857015\pi$
$181$	−1.00000	+	1.73205i	−0.0743294	+	0.128742i	0.826465	+	0.562988i	$0.190348\pi$
$182$		0			0
$183$		−	14.0000i		−	1.03491i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	5.19615	−	3.00000i	0.379980	−	0.219382i
$188$		0			0
$189$	−16.0000			−1.16383
$190$		0			0
$191$	−13.0000			−0.940647			−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000			−0.940647			−0.470323	+	0.882494i	$0.655863\pi$
$192$		0			0
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.0899262	−	0.995948i	$-0.471337\pi$
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.907480	+	0.420096i	$0.138004\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	11.5000	−	19.9186i	0.815213	−	1.41199i	−0.0939612	−	0.995576i	$-0.529953\pi$
$199$	11.5000	−	19.9186i	0.815213	−	1.41199i	0.909175	−	0.416415i	$-0.136714\pi$
$200$		0			0
$201$	−28.0000			−1.97497
$202$		0			0
$203$	3.46410	+	2.00000i	0.243132	+	0.140372i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.46410	+	2.00000i	−0.240772	+	0.139010i
$208$		0			0
$209$	10.5000	−	7.79423i	0.726300	−	0.539138i
$210$		0			0
$211$	0.500000	+	0.866025i	0.0344214	+	0.0596196i	0.882723	−	0.469894i	$-0.155708\pi$
$211$	0.500000	+	0.866025i	0.0344214	+	0.0596196i	−0.848301	+	0.529514i	$0.822374\pi$
$212$		0			0
$213$	−25.9808	−	15.0000i	−1.78017	−	1.02778i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	20.0000i		−	1.35769i
$218$		0			0
$219$	12.0000	−	20.7846i	0.810885	−	1.40449i
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$	6.92820	−	4.00000i	0.463947	−	0.267860i	−0.249756	−	0.968309i	$-0.580350\pi$
$223$	6.92820	−	4.00000i	0.463947	−	0.267860i	0.713702	+	0.700449i	$0.247017\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4.00000i			0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$			4.00000i			0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$		0			0
$229$	−21.0000			−1.38772			−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000			−1.38772			−0.693860	+	0.720110i	$0.744091\pi$
$230$		0			0
$231$	−12.0000	−	20.7846i	−0.789542	−	1.36753i
$232$		0			0
$233$	−8.66025	+	5.00000i	−0.567352	+	0.327561i	−0.756091	−	0.654466i	$-0.772893\pi$
$233$	−8.66025	+	5.00000i	−0.567352	+	0.327561i	0.188739	+	0.982027i	$0.439560\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.73205	−	1.00000i	−0.112509	−	0.0649570i
$238$		0			0
$239$	−15.0000			−0.970269			−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000			−0.970269			−0.485135	+	0.874439i	$0.661229\pi$
$240$		0			0
$241$	−2.50000	+	4.33013i	−0.161039	+	0.278928i	−0.935242	−	0.354010i	$-0.884818\pi$
$241$	−2.50000	+	4.33013i	−0.161039	+	0.278928i	0.774202	+	0.632938i	$0.218151\pi$
$242$		0			0
$243$	8.66025	+	5.00000i	0.555556	+	0.320750i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	25.9808	−	3.00000i	1.65312	−	0.190885i
$248$		0			0
$249$	16.0000	+	27.7128i	1.01396	+	1.75623i
$250$		0			0
$251$	−15.5000	+	26.8468i	−0.978351	+	1.69455i	−0.309951	+	0.950753i	$0.600313\pi$
$251$	−15.5000	+	26.8468i	−0.978351	+	1.69455i	−0.668400	+	0.743802i	$0.733021\pi$
$252$		0			0
$253$	10.3923	+	6.00000i	0.653359	+	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	15.5885	+	9.00000i	0.972381	+	0.561405i	0.899961	−	0.435970i	$-0.143595\pi$
$257$	15.5885	+	9.00000i	0.972381	+	0.561405i	0.0724199	+	0.997374i	$0.476928\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$	−0.500000	−	0.866025i	−0.0309492	−	0.0536056i
$262$		0			0
$263$	8.66025	−	5.00000i	0.534014	−	0.308313i	−0.208635	−	0.977993i	$-0.566902\pi$
$263$	8.66025	−	5.00000i	0.534014	−	0.308313i	0.742650	+	0.669680i	$0.233569\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	34.0000i		−	2.08077i
$268$		0			0
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	0.850380	−	0.526169i	$-0.176372\pi$
$269$	−0.500000	−	0.866025i	−0.0304855	−	0.0528025i	−0.880866	+	0.473366i	$0.843039\pi$
$270$		0			0
$271$	−8.50000	−	14.7224i	−0.516338	−	0.894324i	−0.999820	−	0.0189696i	$-0.993961\pi$
$271$	−8.50000	−	14.7224i	−0.516338	−	0.894324i	0.483482	−	0.875354i	$-0.339372\pi$
$272$		0			0
$273$		−	48.0000i		−	2.90509i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	16.0000i		−	0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$		−	16.0000i		−	0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$		0			0
$279$	−2.50000	+	4.33013i	−0.149671	+	0.259238i
$280$		0			0
$281$	13.0000	−	22.5167i	0.775515	−	1.34323i	−0.158990	−	0.987280i	$-0.550824\pi$
$281$	13.0000	−	22.5167i	0.775515	−	1.34323i	0.934505	−	0.355951i	$-0.115843\pi$
$282$		0			0
$283$	−22.5167	+	13.0000i	−1.33848	+	0.772770i	−0.986581	−	0.163270i	$-0.947796\pi$
$283$	−22.5167	+	13.0000i	−1.33848	+	0.772770i	−0.351895	+	0.936039i	$0.614463\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.92820	−	4.00000i	0.408959	−	0.236113i
$288$		0			0
$289$	−6.50000	+	11.2583i	−0.382353	+	0.662255i
$290$		0			0
$291$	−12.0000	+	20.7846i	−0.703452	+	1.21842i
$292$		0			0
$293$			16.0000i			0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$			16.0000i			0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	12.0000i		−	0.696311i
$298$		0			0
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$			22.0000i			1.26387i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.1244	−	7.00000i	0.691974	−	0.399511i	−0.112377	−	0.993666i	$-0.535847\pi$
$307$	12.1244	−	7.00000i	0.691974	−	0.399511i	0.804351	+	0.594154i	$0.202513\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−29.4449	−	17.0000i	−1.66432	−	0.960897i	−0.970614	−	0.240640i	$-0.922643\pi$
$313$	−29.4449	−	17.0000i	−1.66432	−	0.960897i	−0.693708	−	0.720257i	$-0.744024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.3923	−	6.00000i	−0.583690	−	0.336994i	0.178908	−	0.983866i	$-0.442743\pi$
$317$	−10.3923	−	6.00000i	−0.583690	−	0.336994i	−0.762598	+	0.646872i	$0.776077\pi$
$318$		0			0
$319$	−1.50000	+	2.59808i	−0.0839839	+	0.145464i
$320$		0			0
$321$	12.0000	+	20.7846i	0.669775	+	1.16008i
$322$		0			0
$323$	3.46410	−	8.00000i	0.192748	−	0.445132i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	12.1244	+	7.00000i	0.670478	+	0.387101i
$328$		0			0
$329$	−12.0000	+	20.7846i	−0.661581	+	1.14589i
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$	3.46410	+	2.00000i	0.189832	+	0.109599i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.46410	+	2.00000i	−0.188702	+	0.108947i	−0.591375	−	0.806397i	$-0.701415\pi$
$337$	−3.46410	+	2.00000i	−0.188702	+	0.108947i	0.402673	+	0.915344i	$0.368081\pi$
$338$		0			0
$339$	14.0000	+	24.2487i	0.760376	+	1.31701i
$340$		0			0
$341$	15.0000			0.812296
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	−0.752297	−	0.658824i	$-0.771054\pi$
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	0.194409	+	0.980921i	$0.437721\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	12.0000	−	20.7846i	0.640513	−	1.10940i
$352$		0			0
$353$		−	36.0000i		−	1.91609i	−0.286623	−	0.958043i	$-0.592533\pi$
$353$		−	36.0000i		−	1.91609i	0.286623	−	0.958043i	$-0.407467\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−13.8564	−	8.00000i	−0.733359	−	0.423405i
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		0			0
$363$	−3.46410	+	2.00000i	−0.181818	+	0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−24.2487	−	14.0000i	−1.26577	−	0.730794i	−0.291587	−	0.956544i	$-0.594183\pi$
$367$	−24.2487	−	14.0000i	−1.26577	−	0.730794i	−0.974185	+	0.225750i	$0.927517\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$	−12.0000	+	20.7846i	−0.623009	+	1.07908i
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.19615	+	3.00000i	−0.267615	+	0.154508i
$378$		0			0
$379$	19.0000			0.975964			0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000			0.975964			0.487982	+	0.872854i	$0.337733\pi$
$380$		0			0
$381$	36.0000			1.84434
$382$		0			0
$383$	−15.5885	+	9.00000i	−0.796533	+	0.459879i	−0.842257	−	0.539076i	$-0.818774\pi$
$383$	−15.5885	+	9.00000i	−0.796533	+	0.459879i	0.0457244	+	0.998954i	$0.485440\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	0.466915	+	0.884302i	$0.345366\pi$
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	−0.999286	+	0.0377914i	$0.987968\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$	34.6410	+	20.0000i	1.74741	+	1.00887i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.92820	−	4.00000i	0.347717	−	0.200754i	−0.315963	−	0.948772i	$-0.602327\pi$
$397$	6.92820	−	4.00000i	0.347717	−	0.200754i	0.663679	+	0.748017i	$0.268994\pi$
$398$		0			0
$399$	−32.0000	−	13.8564i	−1.60200	−	0.693688i
$400$		0			0
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	0.615725	−	0.787961i	$-0.288863\pi$
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	−0.990257	+	0.139253i	$0.955530\pi$
$402$		0			0
$403$	25.9808	+	15.0000i	1.29419	+	0.747203i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	12.0000i		−	0.594818i
$408$		0			0
$409$	−0.500000	+	0.866025i	−0.0247234	+	0.0428222i	−0.878122	−	0.478436i	$-0.841204\pi$
$409$	−0.500000	+	0.866025i	−0.0247234	+	0.0428222i	0.853399	+	0.521258i	$0.174537\pi$
$410$		0			0
$411$	24.0000			1.18383
$412$		0			0
$413$	3.46410	−	2.00000i	0.170457	−	0.0984136i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.00000			0.244266			0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000			0.244266			0.122133	+	0.992514i	$0.461027\pi$
$420$		0			0
$421$	−12.5000	−	21.6506i	−0.609213	−	1.05519i	−0.991370	−	0.131090i	$-0.958152\pi$
$421$	−12.5000	−	21.6506i	−0.609213	−	1.05519i	0.382158	−	0.924097i	$-0.375181\pi$
$422$		0			0
$423$	5.19615	−	3.00000i	0.252646	−	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	24.2487	+	14.0000i	1.17348	+	0.677507i
$428$		0			0
$429$	36.0000			1.73810
$430$		0			0
$431$	−13.5000	+	23.3827i	−0.650272	+	1.12630i	0.332785	+	0.943003i	$0.392012\pi$
$431$	−13.5000	+	23.3827i	−0.650272	+	1.12630i	−0.983057	+	0.183301i	$0.941322\pi$
$432$		0			0
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	17.3205	−	2.00000i	0.828552	−	0.0956730i
$438$		0			0
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	0.877711	−	0.479191i	$-0.159070\pi$
$439$	0.500000	+	0.866025i	0.0238637	+	0.0413331i	−0.853847	+	0.520524i	$0.825737\pi$
$440$		0			0
$441$	−4.50000	+	7.79423i	−0.214286	+	0.371154i
$442$		0			0
$443$	10.3923	+	6.00000i	0.493753	+	0.285069i	0.726130	−	0.687557i	$-0.241317\pi$
$443$	10.3923	+	6.00000i	0.493753	+	0.285069i	−0.232377	+	0.972626i	$0.574650\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−29.4449	−	17.0000i	−1.39269	−	0.804072i
$448$		0			0
$449$	1.00000			0.0471929			0.0235965	−	0.999722i	$-0.492488\pi$
$449$	1.00000			0.0471929			0.0235965	+	0.999722i	$0.492488\pi$
$450$		0			0
$451$	3.00000	+	5.19615i	0.141264	+	0.244677i
$452$		0			0
$453$	5.19615	−	3.00000i	0.244137	−	0.140952i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			2.00000i			0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$			2.00000i			0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$		0			0
$459$	−4.00000	−	6.92820i	−0.186704	−	0.323381i
$460$		0			0
$461$	−17.5000	−	30.3109i	−0.815056	−	1.41172i	−0.909288	−	0.416169i	$-0.863373\pi$
$461$	−17.5000	−	30.3109i	−0.815056	−	1.41172i	0.0942312	−	0.995550i	$-0.469961\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			20.0000i			0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$			20.0000i			0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$		0			0
$469$	28.0000	−	48.4974i	1.29292	−	2.23940i
$470$		0			0
$471$	−18.0000	+	31.1769i	−0.829396	+	1.43656i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.19615	−	3.00000i	0.237915	−	0.137361i
$478$		0			0
$479$	−1.50000	+	2.59808i	−0.0685367	+	0.118709i	−0.898257	−	0.439470i	$-0.855166\pi$
$479$	−1.50000	+	2.59808i	−0.0685367	+	0.118709i	0.829721	+	0.558179i	$0.188500\pi$
$480$		0			0
$481$	12.0000	−	20.7846i	0.547153	−	0.947697i
$482$		0			0
$483$		−	32.0000i		−	1.45605i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	34.0000i		−	1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$		−	34.0000i		−	1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$		0			0
$489$	18.0000	+	31.1769i	0.813988	+	1.40987i
$490$		0			0
$491$	12.5000	+	21.6506i	0.564117	+	0.977079i	0.997131	+	0.0756923i	$0.0241167\pi$
$491$	12.5000	+	21.6506i	0.564117	+	0.977079i	−0.433014	+	0.901387i	$0.642550\pi$
$492$		0			0
$493$			2.00000i			0.0900755i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	51.9615	−	30.0000i	2.33079	−	1.34568i
$498$		0			0
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	0.915756	+	0.401735i	$0.131593\pi$
$499$	18.0000	+	31.1769i	0.805791	+	1.39567i	−0.109965	+	0.993935i	$0.535074\pi$
$500$		0			0
$501$	16.0000			0.714827
$502$		0			0
$503$	−19.0526	−	11.0000i	−0.849512	−	0.490466i	0.0109744	−	0.999940i	$-0.496507\pi$
$503$	−19.0526	−	11.0000i	−0.849512	−	0.490466i	−0.860486	+	0.509474i	$0.829840\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	39.8372	+	23.0000i	1.76923	+	1.02147i
$508$		0			0
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	24.0000	+	41.5692i	1.06170	+	1.83891i
$512$		0			0
$513$	−10.3923	−	14.0000i	−0.458831	−	0.618115i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−15.5885	−	9.00000i	−0.685580	−	0.395820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−45.0000			−1.97149			−0.985743	−	0.168259i	$-0.946186\pi$
$521$	−45.0000			−1.97149			−0.985743	+	0.168259i	$0.946186\pi$
$522$		0			0
$523$	8.66025	+	5.00000i	0.378686	+	0.218635i	0.677247	−	0.735756i	$-0.263173\pi$
$523$	8.66025	+	5.00000i	0.378686	+	0.218635i	−0.298560	+	0.954391i	$0.596506\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.66025	−	5.00000i	0.377247	−	0.217803i
$528$		0			0
$529$	−3.50000	−	6.06218i	−0.152174	−	0.263573i
$530$		0			0
$531$	−1.00000			−0.0433963
$532$		0			0
$533$			12.0000i			0.519778i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	25.9808	−	15.0000i	1.12115	−	0.647298i
$538$		0			0
$539$	27.0000			1.16297
$540$		0			0
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	−0.461625	−	0.887075i	$-0.652733\pi$
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	0.999042	−	0.0437584i	$-0.0139332\pi$
$542$		0			0
$543$			4.00000i			0.171656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.92820	−	4.00000i	−0.296229	−	0.171028i	0.344519	−	0.938779i	$-0.388042\pi$
$547$	−6.92820	−	4.00000i	−0.296229	−	0.171028i	−0.640747	+	0.767752i	$0.721375\pi$
$548$		0			0
$549$	−3.50000	−	6.06218i	−0.149376	−	0.258727i
$550$		0			0
$551$	0.500000	+	4.33013i	0.0213007	+	0.184470i
$552$		0			0
$553$	3.46410	−	2.00000i	0.147309	−	0.0850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−34.6410	−	20.0000i	−1.46779	−	0.847427i	−0.468438	−	0.883497i	$-0.655183\pi$
$557$	−34.6410	−	20.0000i	−1.46779	−	0.847427i	−0.999349	+	0.0360693i	$0.988516\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	6.00000	−	10.3923i	0.253320	−	0.438763i
$562$		0			0
$563$			10.0000i			0.421450i	0.977545	+	0.210725i	$0.0675824\pi$
$563$			10.0000i			0.421450i	−0.977545	+	0.210725i	$0.932418\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−38.1051	+	22.0000i	−1.60026	+	0.923913i
$568$		0			0
$569$	17.0000			0.712677			0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000			0.712677			0.356339	+	0.934357i	$0.384025\pi$
$570$		0			0
$571$	−17.0000			−0.711428			−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000			−0.711428			−0.355714	+	0.934595i	$0.615762\pi$
$572$		0			0
$573$	−22.5167	+	13.0000i	−0.940647	+	0.543083i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			20.0000i			0.832611i	0.909225	+	0.416305i	$0.136675\pi$
$577$			20.0000i			0.832611i	−0.909225	+	0.416305i	$0.863325\pi$
$578$		0			0
$579$	16.0000	−	27.7128i	0.664937	−	1.15171i
$580$		0			0
$581$	−64.0000			−2.65517
$582$		0			0
$583$	−15.5885	−	9.00000i	−0.645608	−	0.372742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.3923	−	6.00000i	0.428936	−	0.247647i	−0.269957	−	0.962872i	$-0.587010\pi$
$587$	10.3923	−	6.00000i	0.428936	−	0.247647i	0.698893	+	0.715226i	$0.253676\pi$
$588$		0			0
$589$	17.5000	−	12.9904i	0.721075	−	0.535259i
$590$		0			0
$591$	−2.00000	−	3.46410i	−0.0822690	−	0.142494i
$592$		0			0
$593$	3.46410	+	2.00000i	0.142254	+	0.0821302i	0.569438	−	0.822035i	$-0.307161\pi$
$593$	3.46410	+	2.00000i	0.142254	+	0.0821302i	−0.427184	+	0.904165i	$0.640494\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	46.0000i		−	1.88265i
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$		0			0
$603$	−12.1244	+	7.00000i	−0.493742	+	0.285062i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$		0			0
$609$	8.00000			0.324176
$610$		0			0
$611$	−18.0000	−	31.1769i	−0.728202	−	1.26128i
$612$		0			0
$613$	−19.0526	+	11.0000i	−0.769526	+	0.444286i	−0.832705	−	0.553716i	$-0.813209\pi$
$613$	−19.0526	+	11.0000i	−0.769526	+	0.444286i	0.0631797	+	0.998002i	$0.479876\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.92820	−	4.00000i	−0.278919	−	0.161034i	0.354015	−	0.935240i	$-0.384816\pi$
$617$	−6.92820	−	4.00000i	−0.278919	−	0.161034i	−0.632934	+	0.774206i	$0.718150\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	8.00000	−	13.8564i	0.321029	−	0.556038i
$622$		0			0
$623$	58.8897	+	34.0000i	2.35937	+	1.36218i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	10.3923	−	24.0000i	0.415029	−	0.958468i
$628$		0			0
$629$	−4.00000	−	6.92820i	−0.159490	−	0.276246i
$630$		0			0
$631$	16.5000	−	28.5788i	0.656855	−	1.13771i	−0.324571	−	0.945861i	$-0.605220\pi$
$631$	16.5000	−	28.5788i	0.656855	−	1.13771i	0.981425	−	0.191844i	$-0.0614468\pi$
$632$		0			0
$633$	1.73205	+	1.00000i	0.0688428	+	0.0397464i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	46.7654	+	27.0000i	1.85291	+	1.06978i
$638$		0			0
$639$	−15.0000			−0.593391
$640$		0			0
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	0.763367	−	0.645966i	$-0.223545\pi$
$641$	−4.50000	−	7.79423i	−0.177739	−	0.307854i	−0.941106	+	0.338112i	$0.890212\pi$
$642$		0			0
$643$	−1.73205	+	1.00000i	−0.0683054	+	0.0394362i	−0.533764	−	0.845634i	$-0.679223\pi$
$643$	−1.73205	+	1.00000i	−0.0683054	+	0.0394362i	0.465458	+	0.885070i	$0.345890\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.0000i			1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$			30.0000i			1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$		0			0
$649$	1.50000	+	2.59808i	0.0588802	+	0.101983i
$650$		0			0
$651$	−20.0000	−	34.6410i	−0.783862	−	1.35769i
$652$		0			0
$653$			12.0000i			0.469596i	0.972044	+	0.234798i	$0.0754429\pi$
$653$			12.0000i			0.469596i	−0.972044	+	0.234798i	$0.924557\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	0.266872	+	0.963732i	$0.414010\pi$
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	−0.968052	+	0.250748i	$0.919323\pi$
$660$		0			0
$661$	−20.5000	+	35.5070i	−0.797358	+	1.38106i	0.123974	+	0.992286i	$0.460436\pi$
$661$	−20.5000	+	35.5070i	−0.797358	+	1.38106i	−0.921331	+	0.388778i	$0.872897\pi$
$662$		0			0
$663$	20.7846	−	12.0000i	0.807207	−	0.466041i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.46410	+	2.00000i	−0.134131	+	0.0774403i
$668$		0			0
$669$	8.00000	−	13.8564i	0.309298	−	0.535720i
$670$		0			0
$671$	−10.5000	+	18.1865i	−0.405348	+	0.702083i
$672$		0			0
$673$		−	28.0000i		−	1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$		−	28.0000i		−	1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.0000i			1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$			32.0000i			1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$		0			0
$679$	−24.0000	−	41.5692i	−0.921035	−	1.59528i
$680$		0			0
$681$	4.00000	+	6.92820i	0.153280	+	0.265489i
$682$		0			0
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−36.3731	+	21.0000i	−1.38772	+	0.801200i
$688$		0			0
$689$	−18.0000	−	31.1769i	−0.685745	−	1.18775i
$690$		0			0
$691$	37.0000			1.40755			0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000			1.40755			0.703773	+	0.710425i	$0.251497\pi$
$692$		0			0
$693$	−10.3923	−	6.00000i	−0.394771	−	0.227921i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	3.46410	+	2.00000i	0.131212	+	0.0757554i
$698$		0			0
$699$	−10.0000	+	17.3205i	−0.378235	+	0.655122i
$700$		0			0
$701$	21.0000	+	36.3731i	0.793159	+	1.37379i	0.924002	+	0.382389i	$0.124898\pi$
$701$	21.0000	+	36.3731i	0.793159	+	1.37379i	−0.130843	+	0.991403i	$0.541768\pi$
$702$		0			0
$703$	−10.3923	−	14.0000i	−0.391953	−	0.528020i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−38.1051	−	22.0000i	−1.43309	−	0.827395i
$708$		0			0
$709$	2.50000	−	4.33013i	0.0938895	−	0.162621i	−0.815255	−	0.579102i	$-0.803403\pi$
$709$	2.50000	−	4.33013i	0.0938895	−	0.162621i	0.909145	+	0.416481i	$0.136737\pi$
$710$		0			0
$711$	−1.00000			−0.0375029
$712$		0			0
$713$	17.3205	+	10.0000i	0.648658	+	0.374503i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−25.9808	+	15.0000i	−0.970269	+	0.560185i
$718$		0			0
$719$	14.5000	+	25.1147i	0.540759	+	0.936622i	0.998861	+	0.0477220i	$0.0151961\pi$
$719$	14.5000	+	25.1147i	0.540759	+	0.936622i	−0.458102	+	0.888900i	$0.651471\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			10.0000i			0.371904i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−22.5167	+	13.0000i	−0.835097	+	0.482143i	−0.855595	−	0.517647i	$-0.826808\pi$
$727$	−22.5167	+	13.0000i	−0.835097	+	0.482143i	0.0204978	+	0.999790i	$0.493475\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	12.0000i		−	0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$		−	12.0000i		−	0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36.3731	+	21.0000i	1.33982	+	0.773545i
$738$		0			0
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	0.636691	−	0.771119i	$-0.280303\pi$
$739$	−9.50000	−	16.4545i	−0.349463	−	0.605288i	−0.986154	+	0.165831i	$0.946969\pi$
$740$		0			0
$741$	42.0000	−	31.1769i	1.54291	−	1.14531i
$742$		0			0
$743$	45.0333	−	26.0000i	1.65211	−	0.953847i	0.675910	−	0.736984i	$-0.263751\pi$
$743$	45.0333	−	26.0000i	1.65211	−	0.953847i	0.976202	−	0.216864i	$-0.0695827\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	13.8564	+	8.00000i	0.506979	+	0.292705i
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	−0.985654	−	0.168779i	$-0.946018\pi$
$751$	−9.50000	+	16.4545i	−0.346660	+	0.600433i	0.638994	+	0.769212i	$0.279351\pi$
$752$		0			0
$753$			62.0000i			2.25941i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	12.1244	−	7.00000i	0.440667	−	0.254419i	−0.263213	−	0.964738i	$-0.584782\pi$
$757$	12.1244	−	7.00000i	0.440667	−	0.254419i	0.703881	+	0.710318i	$0.251449\pi$
$758$		0			0
$759$	24.0000			0.871145
$760$		0			0
$761$	54.0000			1.95750			0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000			1.95750			0.978749	+	0.205061i	$0.0657392\pi$
$762$		0			0
$763$	−24.2487	+	14.0000i	−0.877862	+	0.506834i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			6.00000i			0.216647i
$768$		0			0
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	−0.724701	−	0.689063i	$-0.758022\pi$
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	0.959097	+	0.283078i	$0.0913554\pi$
$770$		0			0
$771$	36.0000			1.29651
$772$		0			0
$773$	46.7654	+	27.0000i	1.68203	+	0.971123i	0.960307	+	0.278944i	$0.0899843\pi$
$773$	46.7654	+	27.0000i	1.68203	+	0.971123i	0.721726	+	0.692179i	$0.243349\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−27.7128	+	16.0000i	−0.994192	+	0.573997i
$778$		0			0
$779$	8.00000	+	3.46410i	0.286630	+	0.124114i
$780$		0			0
$781$	22.5000	+	38.9711i	0.805113	+	1.39450i
$782$		0			0
$783$	3.46410	+	2.00000i	0.123797	+	0.0714742i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$		0			0
$789$	10.0000	−	17.3205i	0.356009	−	0.616626i
$790$		0			0
$791$	−56.0000			−1.99113
$792$		0			0
$793$	−36.3731	+	21.0000i	−1.29165	+	0.745732i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			22.0000i			0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$			22.0000i			0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$		0			0
$801$	−8.50000	−	14.7224i	−0.300333	−	0.520192i
$802$		0			0
$803$	−31.1769	+	18.0000i	−1.10021	+	0.635206i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.73205	−	1.00000i	−0.0609711	−	0.0352017i
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	−0.729042	−	0.684468i	$-0.760034\pi$
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	0.957288	+	0.289135i	$0.0933677\pi$
$812$		0			0
$813$	−29.4449	−	17.0000i	−1.03268	−	0.596216i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−12.0000	−	20.7846i	−0.419314	−	0.726273i
$820$		0			0
$821$	−12.5000	+	21.6506i	−0.436253	+	0.755612i	−0.997397	−	0.0721058i	$-0.977028\pi$
$821$	−12.5000	+	21.6506i	−0.436253	+	0.755612i	0.561144	+	0.827718i	$0.310361\pi$
$822$		0			0
$823$	−22.5167	−	13.0000i	−0.784881	−	0.453152i	0.0532760	−	0.998580i	$-0.483034\pi$
$823$	−22.5167	−	13.0000i	−0.784881	−	0.453152i	−0.838157	+	0.545428i	$0.816367\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.5885	−	9.00000i	−0.542064	−	0.312961i	0.203851	−	0.979002i	$-0.434654\pi$
$827$	−15.5885	−	9.00000i	−0.542064	−	0.312961i	−0.745915	+	0.666041i	$0.767987\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$	−16.0000	−	27.7128i	−0.555034	−	0.961347i
$832$		0			0
$833$	15.5885	−	9.00000i	0.540108	−	0.311832i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	20.0000i		−	0.691301i
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	14.0000	+	24.2487i	0.482759	+	0.836162i
$842$		0			0
$843$		−	52.0000i		−	1.79098i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	8.00000i		−	0.274883i
$848$		0			0
$849$	−26.0000	+	45.0333i	−0.892318	+	1.54554i
$850$		0			0
$851$	8.00000	−	13.8564i	0.274236	−	0.474991i
$852$		0			0
$853$	32.9090	−	19.0000i	1.12678	−	0.650548i	0.183658	−	0.982990i	$-0.441206\pi$
$853$	32.9090	−	19.0000i	1.12678	−	0.650548i	0.943123	+	0.332443i	$0.107873\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−0.500000	+	0.866025i	−0.0170598	+	0.0295484i	−0.874429	−	0.485153i	$-0.838764\pi$
$859$	−0.500000	+	0.866025i	−0.0170598	+	0.0295484i	0.857369	+	0.514701i	$0.172097\pi$
$860$		0			0
$861$	8.00000	−	13.8564i	0.272639	−	0.472225i
$862$		0			0
$863$		−	18.0000i		−	0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$		−	18.0000i		−	0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			26.0000i			0.883006i
$868$		0			0
$869$	1.50000	+	2.59808i	0.0508840	+	0.0881337i
$870$		0			0
$871$	42.0000	+	72.7461i	1.42312	+	2.46491i
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−34.6410	+	20.0000i	−1.16974	+	0.675352i	−0.953620	−	0.301014i	$-0.902675\pi$
$877$	−34.6410	+	20.0000i	−1.16974	+	0.675352i	−0.216124	+	0.976366i	$0.569342\pi$
$878$		0			0
$879$	16.0000	+	27.7128i	0.539667	+	0.934730i
$880$		0			0
$881$	−7.00000			−0.235836			−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000			−0.235836			−0.117918	+	0.993023i	$0.537622\pi$
$882$		0			0
$883$	−6.92820	−	4.00000i	−0.233153	−	0.134611i	0.378873	−	0.925449i	$-0.376312\pi$
$883$	−6.92820	−	4.00000i	−0.233153	−	0.134611i	−0.612026	+	0.790838i	$0.709645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.2487	+	14.0000i	0.814192	+	0.470074i	0.848410	−	0.529340i	$-0.177561\pi$
$887$	24.2487	+	14.0000i	0.814192	+	0.470074i	−0.0342175	+	0.999414i	$0.510894\pi$
$888$		0			0
$889$	−36.0000	+	62.3538i	−1.20740	+	2.09128i
$890$		0			0
$891$	−16.5000	−	28.5788i	−0.552771	−	0.957427i
$892$		0			0
$893$	−25.9808	+	3.00000i	−0.869413	+	0.100391i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	41.5692	+	24.0000i	1.38796	+	0.801337i
$898$		0			0
$899$	−2.50000	+	4.33013i	−0.0833797	+	0.144418i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−27.7128	+	16.0000i	−0.920189	+	0.531271i	−0.883695	−	0.468063i	$-0.844952\pi$
$907$	−27.7128	+	16.0000i	−0.920189	+	0.531271i	−0.0364935	+	0.999334i	$0.511619\pi$
$908$		0			0
$909$	5.50000	+	9.52628i	0.182423	+	0.315967i
$910$		0			0
$911$	7.00000			0.231920			0.115960	−	0.993254i	$-0.463006\pi$
$911$	7.00000			0.231920			0.115960	+	0.993254i	$0.463006\pi$
$912$		0			0
$913$		−	48.0000i		−	1.58857i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−69.2820	+	40.0000i	−2.28789	+	1.32092i
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$		0			0
$921$	14.0000	−	24.2487i	0.461316	−	0.799022i
$922$		0			0
$923$			90.0000i			2.96239i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.5000	−	18.1865i	−0.344494	−	0.596681i	0.640768	−	0.767735i	$-0.278616\pi$
$929$	−10.5000	−	18.1865i	−0.344494	−	0.596681i	−0.985262	+	0.171054i	$0.945283\pi$
$930$		0			0
$931$	31.5000	−	23.3827i	1.03237	−	0.766337i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.73205	+	1.00000i	0.0565836	+	0.0326686i	0.528025	−	0.849229i	$-0.322933\pi$
$937$	1.73205	+	1.00000i	0.0565836	+	0.0326686i	−0.471441	+	0.881897i	$0.656266\pi$
$938$		0			0
$939$	−68.0000			−2.21910
$940$		0			0
$941$	−17.5000	+	30.3109i	−0.570484	+	0.988107i	0.426033	+	0.904708i	$0.359911\pi$
$941$	−17.5000	+	30.3109i	−0.570484	+	0.988107i	−0.996516	+	0.0833989i	$0.973422\pi$
$942$		0			0
$943$			8.00000i			0.260516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.66025	−	5.00000i	0.281420	−	0.162478i	−0.352646	−	0.935757i	$-0.614718\pi$
$947$	8.66025	−	5.00000i	0.281420	−	0.162478i	0.634066	+	0.773279i	$0.281385\pi$
$948$		0			0
$949$	−72.0000			−2.33722
$950$		0			0
$951$	−24.0000			−0.778253
$952$		0			0
$953$	−24.2487	+	14.0000i	−0.785493	+	0.453504i	−0.838373	−	0.545096i	$-0.816493\pi$
$953$	−24.2487	+	14.0000i	−0.785493	+	0.453504i	0.0528806	+	0.998601i	$0.483160\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			6.00000i			0.193952i
$958$		0			0
$959$	−24.0000	+	41.5692i	−0.775000	+	1.34234i
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$	10.3923	+	6.00000i	0.334887	+	0.193347i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.3923	−	6.00000i	0.334194	−	0.192947i	−0.323508	−	0.946226i	$-0.604862\pi$
$967$	10.3923	−	6.00000i	0.334194	−	0.192947i	0.657702	+	0.753279i	$0.271529\pi$
$968$		0			0
$969$	−2.00000	−	17.3205i	−0.0642493	−	0.556415i
$970$		0			0
$971$	−30.0000	−	51.9615i	−0.962746	−	1.66752i	−0.715553	−	0.698558i	$-0.753825\pi$
$971$	−30.0000	−	51.9615i	−0.962746	−	1.66752i	−0.247193	−	0.968966i	$-0.579508\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	12.0000i		−	0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$		−	12.0000i		−	0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$		0			0
$979$	−25.5000	+	44.1673i	−0.814984	+	1.41159i
$980$		0			0
$981$	7.00000			0.223493
$982$		0			0
$983$	−31.1769	+	18.0000i	−0.994389	+	0.574111i	−0.906583	−	0.422027i	$-0.861319\pi$
$983$	−31.1769	+	18.0000i	−0.994389	+	0.574111i	−0.0878058	+	0.996138i	$0.527985\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			48.0000i			1.52786i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	−0.999954	−	0.00956046i	$-0.996957\pi$
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	0.491698	−	0.870766i	$-0.336377\pi$
$992$		0			0
$993$	−34.6410	+	20.0000i	−1.09930	+	0.634681i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−8.66025	−	5.00000i	−0.274273	−	0.158352i	0.356555	−	0.934274i	$-0.383951\pi$
$997$	−8.66025	−	5.00000i	−0.274273	−	0.158352i	−0.630828	+	0.775923i	$0.717285\pi$
$998$		0			0
$999$	−16.0000			−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.s.b.349.2		4
5.2	odd	4		380.2.i.a.121.1	✓	2
5.3	odd	4		1900.2.i.b.501.1		2
5.4	even	2	inner	1900.2.s.b.349.1		4
15.2	even	4		3420.2.t.b.1261.1		2
19.11	even	3	inner	1900.2.s.b.49.1		4
20.7	even	4		1520.2.q.g.881.1		2
95.7	odd	12		7220.2.a.e.1.1		1
95.12	even	12		7220.2.a.a.1.1		1
95.49	even	6	inner	1900.2.s.b.49.2		4
95.68	odd	12		1900.2.i.b.201.1		2
95.87	odd	12		380.2.i.a.201.1	yes	2
285.182	even	12		3420.2.t.b.3241.1		2
380.87	even	12		1520.2.q.g.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.a.121.1	✓	2	5.2	odd	4
380.2.i.a.201.1	yes	2	95.87	odd	12
1520.2.q.g.881.1		2	20.7	even	4
1520.2.q.g.961.1		2	380.87	even	12
1900.2.i.b.201.1		2	95.68	odd	12
1900.2.i.b.501.1		2	5.3	odd	4
1900.2.s.b.49.1		4	19.11	even	3	inner
1900.2.s.b.49.2		4	95.49	even	6	inner
1900.2.s.b.349.1		4	5.4	even	2	inner
1900.2.s.b.349.2		4	1.1	even	1	trivial
3420.2.t.b.1261.1		2	15.2	even	4
3420.2.t.b.3241.1		2	285.182	even	12
7220.2.a.a.1.1		1	95.12	even	12
7220.2.a.e.1.1		1	95.7	odd	12

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{3} +4.00000i q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{3} +4.00000i q^{7} +(0.500000 - 0.866025i) q^{9} -3.00000 q^{11} +(-5.19615 - 3.00000i) q^{13} +(-1.73205 + 1.00000i) q^{17} +(-3.50000 + 2.59808i) q^{19} +(4.00000 + 6.92820i) q^{21} +(-3.46410 - 2.00000i) q^{23} +4.00000i q^{27} +(0.500000 - 0.866025i) q^{29} -5.00000 q^{31} +(-5.19615 + 3.00000i) q^{33} +4.00000i q^{37} -12.0000 q^{39} +(-1.00000 - 1.73205i) q^{41} +(5.19615 + 3.00000i) q^{47} -9.00000 q^{49} +(-2.00000 + 3.46410i) q^{51} +(5.19615 + 3.00000i) q^{53} +(-3.46410 + 8.00000i) q^{57} +(-0.500000 - 0.866025i) q^{59} +(3.50000 - 6.06218i) q^{61} +(3.46410 + 2.00000i) q^{63} +(-12.1244 - 7.00000i) q^{67} -8.00000 q^{69} +(-7.50000 - 12.9904i) q^{71} +(10.3923 - 6.00000i) q^{73} -12.0000i q^{77} +(-0.500000 - 0.866025i) q^{79} +(5.50000 + 9.52628i) q^{81} +16.0000i q^{83} -2.00000i q^{87} +(8.50000 - 14.7224i) q^{89} +(12.0000 - 20.7846i) q^{91} +(-8.66025 + 5.00000i) q^{93} +(-10.3923 + 6.00000i) q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} - 12 q^{11} - 14 q^{19} + 16 q^{21} + 2 q^{29} - 20 q^{31} - 48 q^{39} - 4 q^{41} - 36 q^{49} - 8 q^{51} - 2 q^{59} + 14 q^{61} - 32 q^{69} - 30 q^{71} - 2 q^{79} + 22 q^{81} + 34 q^{89} + 48 q^{91} - 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 - 12 * q^11 - 14 * q^19 + 16 * q^21 + 2 * q^29 - 20 * q^31 - 48 * q^39 - 4 * q^41 - 36 * q^49 - 8 * q^51 - 2 * q^59 + 14 * q^61 - 32 * q^69 - 30 * q^71 - 2 * q^79 + 22 * q^81 + 34 * q^89 + 48 * q^91 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more