LMFDB - Embedded newform 1600.2.q.f.849.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(49,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

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

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

chi := DirichletCharacter("1600.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.q (of order $4$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.4767670494822400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 $ x^12 - 4x^11 + 7x^10 - 4x^9 - 8x^8 + 24x^7 - 38x^6 + 48x^5 - 32x^4 - 32x^3 + 112x^2 - 128*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			849.3
Root			$-0.507829 - 1.31989i$ of defining polynomial
Character	$\chi$	$=$	1600.849
Dual form			1600.2.q.f.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0623209 - 0.0623209i) q^{3} -0.375877 q^{7} +2.99223i q^{9} +O(q^{10})$	$q+(0.0623209 - 0.0623209i) q^{3} -0.375877 q^{7} +2.99223i q^{9} +(-2.36756 + 2.36756i) q^{11} +(-1.76442 + 1.76442i) q^{13} -4.64955i q^{17} +(-2.34965 - 2.34965i) q^{19} +(-0.0234250 + 0.0234250i) q^{21} +2.07779 q^{23} +(0.373441 + 0.373441i) q^{27} +(-2.55422 - 2.55422i) q^{29} -8.51714 q^{31} +0.295096i q^{33} +(-7.62613 - 7.62613i) q^{37} +0.219921i q^{39} -3.77709i q^{41} +(6.21191 + 6.21191i) q^{43} -9.71696i q^{47} -6.85872 q^{49} +(-0.289764 - 0.289764i) q^{51} +(-3.03609 - 3.03609i) q^{53} -0.292864 q^{57} +(-8.11663 + 8.11663i) q^{59} +(0.728329 + 0.728329i) q^{61} -1.12471i q^{63} +(-0.969239 + 0.969239i) q^{67} +(0.129490 - 0.129490i) q^{69} +9.14230i q^{71} +7.56793 q^{73} +(0.889909 - 0.889909i) q^{77} +11.8065 q^{79} -8.93015 q^{81} +(-10.6393 + 10.6393i) q^{83} -0.318363 q^{87} +15.7111i q^{89} +(0.663205 - 0.663205i) q^{91} +(-0.530796 + 0.530796i) q^{93} -3.86020i q^{97} +(-7.08428 - 7.08428i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} - 12 q^{7}+O(q^{10}) $ 12 * q + 2 * q^3 - 12 * q^7	$ 12 q + 2 q^{3} - 12 q^{7} + 2 q^{11} + 4 q^{13} - 14 q^{19} - 20 q^{21} - 12 q^{23} - 10 q^{27} + 4 q^{31} - 8 q^{37} - 4 q^{49} - 10 q^{51} - 16 q^{53} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 50 q^{67} + 40 q^{73} - 8 q^{77} + 12 q^{79} - 8 q^{81} - 2 q^{83} + 64 q^{87} + 44 q^{93} + 12 q^{99}+O(q^{100}) $ 12 * q + 2 * q^3 - 12 * q^7 + 2 * q^11 + 4 * q^13 - 14 * q^19 - 20 * q^21 - 12 * q^23 - 10 * q^27 + 4 * q^31 - 8 * q^37 - 4 * q^49 - 10 * q^51 - 16 * q^53 - 16 * q^57 + 20 * q^59 + 4 * q^61 - 50 * q^67 + 40 * q^73 - 8 * q^77 + 12 * q^79 - 8 * q^81 - 2 * q^83 + 64 * q^87 + 44 * q^93 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	0.0623209	−	0.0623209i	0.0359810	−	0.0359810i	−0.688887	−	0.724868i	$-0.741901\pi$
$3$	0.0623209	−	0.0623209i	0.0359810	−	0.0359810i	0.724868	+	0.688887i	$0.241901\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.375877			−0.142068			−0.0710340	−	0.997474i	$-0.522630\pi$
$7$	−0.375877			−0.142068			−0.0710340	+	0.997474i	$0.522630\pi$
$8$		0			0
$9$			2.99223i			0.997411i
$10$		0			0
$11$	−2.36756	+	2.36756i	−0.713845	+	0.713845i	−0.967337	−	0.253492i	$-0.918421\pi$
$11$	−2.36756	+	2.36756i	−0.713845	+	0.713845i	0.253492	+	0.967337i	$0.418421\pi$
$12$		0			0
$13$	−1.76442	+	1.76442i	−0.489363	+	0.489363i	−0.908105	−	0.418742i	$-0.862471\pi$
$13$	−1.76442	+	1.76442i	−0.489363	+	0.489363i	0.418742	+	0.908105i	$0.362471\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.64955i		−	1.12768i	−0.825883	−	0.563841i	$-0.809323\pi$
$17$		−	4.64955i		−	1.12768i	0.825883	−	0.563841i	$-0.190677\pi$
$18$		0			0
$19$	−2.34965	−	2.34965i	−0.539047	−	0.539047i	0.384202	−	0.923249i	$-0.374476\pi$
$19$	−2.34965	−	2.34965i	−0.539047	−	0.539047i	−0.923249	+	0.384202i	$0.874476\pi$
$20$		0			0
$21$	−0.0234250	+	0.0234250i	−0.00511175	+	0.00511175i
$22$		0			0
$23$	2.07779			0.433250			0.216625	−	0.976255i	$-0.430495\pi$
$23$	2.07779			0.433250			0.216625	+	0.976255i	$0.430495\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.373441	+	0.373441i	0.0718688	+	0.0718688i
$28$		0			0
$29$	−2.55422	−	2.55422i	−0.474307	−	0.474307i	0.428998	−	0.903305i	$-0.358867\pi$
$29$	−2.55422	−	2.55422i	−0.474307	−	0.474307i	−0.903305	+	0.428998i	$0.858867\pi$
$30$		0			0
$31$	−8.51714			−1.52972			−0.764862	−	0.644194i	$-0.777193\pi$
$31$	−8.51714			−1.52972			−0.764862	+	0.644194i	$0.777193\pi$
$32$		0			0
$33$			0.295096i			0.0513697i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.62613	−	7.62613i	−1.25373	−	1.25373i	−0.954036	−	0.299691i	$-0.903116\pi$
$37$	−7.62613	−	7.62613i	−1.25373	−	1.25373i	−0.299691	−	0.954036i	$-0.596884\pi$
$38$		0			0
$39$			0.219921i			0.0352155i
$40$		0			0
$41$		−	3.77709i		−	0.589882i	−0.955515	−	0.294941i	$-0.904700\pi$
$41$		−	3.77709i		−	0.589882i	0.955515	−	0.294941i	$-0.0953001\pi$
$42$		0			0
$43$	6.21191	+	6.21191i	0.947307	+	0.947307i	0.998680	−	0.0513725i	$-0.0163596\pi$
$43$	6.21191	+	6.21191i	0.947307	+	0.947307i	−0.0513725	+	0.998680i	$0.516360\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.71696i		−	1.41736i	−0.705528	−	0.708682i	$-0.749290\pi$
$47$		−	9.71696i		−	1.41736i	0.705528	−	0.708682i	$-0.250710\pi$
$48$		0			0
$49$	−6.85872			−0.979817
$50$		0			0
$51$	−0.289764	−	0.289764i	−0.0405751	−	0.0405751i
$52$		0			0
$53$	−3.03609	−	3.03609i	−0.417040	−	0.417040i	0.467143	−	0.884182i	$-0.345283\pi$
$53$	−3.03609	−	3.03609i	−0.417040	−	0.417040i	−0.884182	+	0.467143i	$0.845283\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.292864			−0.0387908
$58$		0			0
$59$	−8.11663	+	8.11663i	−1.05670	+	1.05670i	−0.0584019	+	0.998293i	$0.518600\pi$
$59$	−8.11663	+	8.11663i	−1.05670	+	1.05670i	−0.998293	+	0.0584019i	$0.981400\pi$
$60$		0			0
$61$	0.728329	+	0.728329i	0.0932529	+	0.0932529i	0.752194	−	0.658941i	$-0.228995\pi$
$61$	0.728329	+	0.728329i	0.0932529	+	0.0932529i	−0.658941	+	0.752194i	$0.728995\pi$
$62$		0			0
$63$		−	1.12471i		−	0.141700i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.969239	+	0.969239i	−0.118411	+	0.118411i	−0.763829	−	0.645418i	$-0.776683\pi$
$67$	−0.969239	+	0.969239i	−0.118411	+	0.118411i	0.645418	+	0.763829i	$0.276683\pi$
$68$		0			0
$69$	0.129490	−	0.129490i	0.0155887	−	0.0155887i
$70$		0			0
$71$			9.14230i			1.08499i	0.840058	+	0.542496i	$0.182521\pi$
$71$			9.14230i			1.08499i	−0.840058	+	0.542496i	$0.817479\pi$
$72$		0			0
$73$	7.56793			0.885759			0.442879	−	0.896581i	$-0.353957\pi$
$73$	7.56793			0.885759			0.442879	+	0.896581i	$0.353957\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.889909	−	0.889909i	0.101415	−	0.101415i
$78$		0			0
$79$	11.8065			1.32834			0.664169	−	0.747583i	$-0.268786\pi$
$79$	11.8065			1.32834			0.664169	+	0.747583i	$0.268786\pi$
$80$		0			0
$81$	−8.93015			−0.992239
$82$		0			0
$83$	−10.6393	+	10.6393i	−1.16782	+	1.16782i	−0.185101	+	0.982720i	$0.559261\pi$
$83$	−10.6393	+	10.6393i	−1.16782	+	1.16782i	−0.982720	+	0.185101i	$0.940739\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.318363			−0.0341320
$88$		0			0
$89$			15.7111i			1.66538i	0.553741	+	0.832689i	$0.313200\pi$
$89$			15.7111i			1.66538i	−0.553741	+	0.832689i	$0.686800\pi$
$90$		0			0
$91$	0.663205	−	0.663205i	0.0695228	−	0.0695228i
$92$		0			0
$93$	−0.530796	+	0.530796i	−0.0550410	+	0.0550410i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	3.86020i		−	0.391943i	−0.980610	−	0.195972i	$-0.937214\pi$
$97$		−	3.86020i		−	0.391943i	0.980610	−	0.195972i	$-0.0627861\pi$
$98$		0			0
$99$	−7.08428	−	7.08428i	−0.711997	−	0.711997i
$100$		0			0
$101$	−6.87437	+	6.87437i	−0.684026	+	0.684026i	−0.960905	−	0.276879i	$-0.910700\pi$
$101$	−6.87437	+	6.87437i	−0.684026	+	0.684026i	0.276879	+	0.960905i	$0.410700\pi$
$102$		0			0
$103$	−1.15407			−0.113714			−0.0568571	−	0.998382i	$-0.518108\pi$
$103$	−1.15407			−0.113714			−0.0568571	+	0.998382i	$0.518108\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.70435	−	5.70435i	−0.551460	−	0.551460i	0.375402	−	0.926862i	$-0.377505\pi$
$107$	−5.70435	−	5.70435i	−0.551460	−	0.551460i	−0.926862	+	0.375402i	$0.877505\pi$
$108$		0			0
$109$	−11.1863	−	11.1863i	−1.07145	−	1.07145i	−0.997243	−	0.0742092i	$-0.976357\pi$
$109$	−11.1863	−	11.1863i	−1.07145	−	1.07145i	−0.0742092	−	0.997243i	$-0.523643\pi$
$110$		0			0
$111$	−0.950534			−0.0902207
$112$		0			0
$113$			4.08163i			0.383967i	0.981398	+	0.191984i	$0.0614920\pi$
$113$			4.08163i			0.383967i	−0.981398	+	0.191984i	$0.938508\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.27956	−	5.27956i	−0.488096	−	0.488096i
$118$		0			0
$119$			1.74766i			0.160208i
$120$		0			0
$121$		−	0.210643i		−	0.0191493i
$122$		0			0
$123$	−0.235392	−	0.235392i	−0.0212245	−	0.0212245i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	17.0918i		−	1.51665i	−0.651876	−	0.758326i	$-0.726018\pi$
$127$		−	17.0918i		−	1.51665i	0.651876	−	0.758326i	$-0.273982\pi$
$128$		0			0
$129$	0.774263			0.0681701
$130$		0			0
$131$	3.56424	+	3.56424i	0.311409	+	0.311409i	0.845455	−	0.534046i	$-0.179329\pi$
$131$	3.56424	+	3.56424i	0.311409	+	0.311409i	−0.534046	+	0.845455i	$0.679329\pi$
$132$		0			0
$133$	0.883179	+	0.883179i	0.0765813	+	0.0765813i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.6995			1.42673			0.713366	−	0.700792i	$-0.247170\pi$
$137$	16.6995			1.42673			0.713366	+	0.700792i	$0.247170\pi$
$138$		0			0
$139$	−7.56455	+	7.56455i	−0.641616	+	0.641616i	−0.950953	−	0.309336i	$-0.899893\pi$
$139$	−7.56455	+	7.56455i	−0.641616	+	0.641616i	0.309336	+	0.950953i	$0.399893\pi$
$140$		0			0
$141$	−0.605569	−	0.605569i	−0.0509982	−	0.0509982i
$142$		0			0
$143$		−	8.35474i		−	0.698658i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.427441	+	0.427441i	−0.0352548	+	0.0352548i
$148$		0			0
$149$	10.2542	−	10.2542i	0.840056	−	0.840056i	−0.148810	−	0.988866i	$-0.547544\pi$
$149$	10.2542	−	10.2542i	0.840056	−	0.840056i	0.988866	+	0.148810i	$0.0475444\pi$
$150$		0			0
$151$		−	19.0430i		−	1.54970i	−0.632147	−	0.774849i	$-0.717826\pi$
$151$		−	19.0430i		−	1.54970i	0.632147	−	0.774849i	$-0.282174\pi$
$152$		0			0
$153$	13.9125			1.12476
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.1335	+	10.1335i	−0.808741	+	0.808741i	−0.984443	−	0.175702i	$-0.943780\pi$
$157$	−10.1335	+	10.1335i	−0.808741	+	0.808741i	0.175702	+	0.984443i	$0.443780\pi$
$158$		0			0
$159$	−0.378424			−0.0300110
$160$		0			0
$161$	−0.780994			−0.0615509
$162$		0			0
$163$	7.35501	−	7.35501i	0.576089	−	0.576089i	−0.357735	−	0.933823i	$-0.616451\pi$
$163$	7.35501	−	7.35501i	0.576089	−	0.576089i	0.933823	+	0.357735i	$0.116451\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.02936			−0.621331			−0.310665	−	0.950519i	$-0.600552\pi$
$167$	−8.02936			−0.621331			−0.310665	+	0.950519i	$0.600552\pi$
$168$		0			0
$169$			6.77363i			0.521048i
$170$		0			0
$171$	7.03070	−	7.03070i	0.537651	−	0.537651i
$172$		0			0
$173$	−10.4326	+	10.4326i	−0.793177	+	0.793177i	−0.982009	−	0.188832i	$-0.939530\pi$
$173$	−10.4326	+	10.4326i	−0.793177	+	0.793177i	0.188832	+	0.982009i	$0.439530\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			1.01167i			0.0760418i
$178$		0			0
$179$	−8.30280	−	8.30280i	−0.620580	−	0.620580i	0.325099	−	0.945680i	$-0.394602\pi$
$179$	−8.30280	−	8.30280i	−0.620580	−	0.620580i	−0.945680	+	0.325099i	$0.894602\pi$
$180$		0			0
$181$	−10.4772	+	10.4772i	−0.778765	+	0.778765i	−0.979621	−	0.200856i	$-0.935628\pi$
$181$	−10.4772	+	10.4772i	−0.778765	+	0.778765i	0.200856	+	0.979621i	$0.435628\pi$
$182$		0			0
$183$	0.0907802			0.00671066
$184$		0			0
$185$		0			0
$186$		0			0
$187$	11.0081	+	11.0081i	0.804990	+	0.804990i
$188$		0			0
$189$	−0.140368	−	0.140368i	−0.0102103	−	0.0102103i
$190$		0			0
$191$	−1.68079			−0.121618			−0.0608089	−	0.998149i	$-0.519368\pi$
$191$	−1.68079			−0.121618			−0.0608089	+	0.998149i	$0.519368\pi$
$192$		0			0
$193$			1.61403i			0.116181i	0.998311	+	0.0580903i	$0.0185011\pi$
$193$			1.61403i			0.116181i	−0.998311	+	0.0580903i	$0.981499\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.10322	+	5.10322i	0.363589	+	0.363589i	0.865133	−	0.501543i	$-0.167234\pi$
$197$	5.10322	+	5.10322i	0.363589	+	0.363589i	−0.501543	+	0.865133i	$0.667234\pi$
$198$		0			0
$199$			11.1545i			0.790725i	0.918525	+	0.395362i	$0.129381\pi$
$199$			11.1545i			0.790725i	−0.918525	+	0.395362i	$0.870619\pi$
$200$		0			0
$201$			0.120808i			0.00852111i
$202$		0			0
$203$	0.960072	+	0.960072i	0.0673839	+	0.0673839i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			6.21724i			0.432128i
$208$		0			0
$209$	11.1259			0.769591
$210$		0			0
$211$	2.48377	+	2.48377i	0.170989	+	0.170989i	0.787414	−	0.616425i	$-0.211419\pi$
$211$	2.48377	+	2.48377i	0.170989	+	0.170989i	−0.616425	+	0.787414i	$0.711419\pi$
$212$		0			0
$213$	0.569756	+	0.569756i	0.0390391	+	0.0390391i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.20140			0.217325
$218$		0			0
$219$	0.471640	−	0.471640i	0.0318705	−	0.0318705i
$220$		0			0
$221$	8.20377	+	8.20377i	0.551846	+	0.551846i
$222$		0			0
$223$		−	21.1384i		−	1.41553i	−0.706448	−	0.707765i	$-0.749703\pi$
$223$		−	21.1384i		−	1.41553i	0.706448	−	0.707765i	$-0.250297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.4885	+	14.4885i	−0.961634	+	0.961634i	−0.999291	−	0.0376566i	$-0.988011\pi$
$227$	−14.4885	+	14.4885i	−0.961634	+	0.961634i	0.0376566	+	0.999291i	$0.488011\pi$
$228$		0			0
$229$	10.0956	−	10.0956i	0.667138	−	0.667138i	−0.289914	−	0.957053i	$-0.593627\pi$
$229$	10.0956	−	10.0956i	0.667138	−	0.667138i	0.957053	+	0.289914i	$0.0936268\pi$
$230$		0			0
$231$		−	0.110920i		−	0.00729799i
$232$		0			0
$233$	−3.44995			−0.226014			−0.113007	−	0.993594i	$-0.536048\pi$
$233$	−3.44995			−0.226014			−0.113007	+	0.993594i	$0.536048\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.735793	−	0.735793i	0.0477949	−	0.0477949i
$238$		0			0
$239$	18.0060			1.16471			0.582354	−	0.812935i	$-0.302132\pi$
$239$	18.0060			1.16471			0.582354	+	0.812935i	$0.302132\pi$
$240$		0			0
$241$	12.6235			0.813154			0.406577	−	0.913617i	$-0.366722\pi$
$241$	12.6235			0.813154			0.406577	+	0.913617i	$0.366722\pi$
$242$		0			0
$243$	−1.67686	+	1.67686i	−0.107571	+	0.107571i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.29155			0.527578
$248$		0			0
$249$			1.32611i			0.0840386i
$250$		0			0
$251$	9.17919	−	9.17919i	0.579386	−	0.579386i	−0.355348	−	0.934734i	$-0.615638\pi$
$251$	9.17919	−	9.17919i	0.579386	−	0.579386i	0.934734	+	0.355348i	$0.115638\pi$
$252$		0			0
$253$	−4.91929	+	4.91929i	−0.309273	+	0.309273i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			16.2897i			1.01612i	0.861321	+	0.508061i	$0.169637\pi$
$257$			16.2897i			1.01612i	−0.861321	+	0.508061i	$0.830363\pi$
$258$		0			0
$259$	2.86648	+	2.86648i	0.178115	+	0.178115i
$260$		0			0
$261$	7.64282	−	7.64282i	0.473079	−	0.473079i
$262$		0			0
$263$	−10.4898			−0.646831			−0.323416	−	0.946257i	$-0.604831\pi$
$263$	−10.4898			−0.646831			−0.323416	+	0.946257i	$0.604831\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.979132	+	0.979132i	0.0599219	+	0.0599219i
$268$		0			0
$269$	8.46636	+	8.46636i	0.516203	+	0.516203i	0.916420	−	0.400217i	$-0.131065\pi$
$269$	8.46636	+	8.46636i	0.516203	+	0.516203i	−0.400217	+	0.916420i	$0.631065\pi$
$270$		0			0
$271$	8.92117			0.541923			0.270961	−	0.962590i	$-0.412658\pi$
$271$	8.92117			0.541923			0.270961	+	0.962590i	$0.412658\pi$
$272$		0			0
$273$		−	0.0826631i		−	0.00500300i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	9.36430	+	9.36430i	0.562646	+	0.562646i	0.930058	−	0.367412i	$-0.119756\pi$
$277$	9.36430	+	9.36430i	0.562646	+	0.562646i	−0.367412	+	0.930058i	$0.619756\pi$
$278$		0			0
$279$		−	25.4853i		−	1.52576i
$280$		0			0
$281$		−	3.12921i		−	0.186673i	−0.995635	−	0.0933365i	$-0.970247\pi$
$281$		−	3.12921i		−	0.186673i	0.995635	−	0.0933365i	$-0.0297532\pi$
$282$		0			0
$283$	−2.07308	−	2.07308i	−0.123232	−	0.123232i	0.642801	−	0.766033i	$-0.277772\pi$
$283$	−2.07308	−	2.07308i	−0.123232	−	0.123232i	−0.766033	+	0.642801i	$0.777772\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.41972i			0.0838035i
$288$		0			0
$289$	−4.61834			−0.271667
$290$		0			0
$291$	−0.240571	−	0.240571i	−0.0141025	−	0.0141025i
$292$		0			0
$293$	−12.3528	−	12.3528i	−0.721659	−	0.721659i	0.247284	−	0.968943i	$-0.420462\pi$
$293$	−12.3528	−	12.3528i	−0.721659	−	0.721659i	−0.968943	+	0.247284i	$0.920462\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.76829			−0.102606
$298$		0			0
$299$	−3.66610	+	3.66610i	−0.212016	+	0.212016i
$300$		0			0
$301$	−2.33491	−	2.33491i	−0.134582	−	0.134582i
$302$		0			0
$303$			0.856834i			0.0492238i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.5938	−	10.5938i	0.604619	−	0.604619i	−0.336916	−	0.941535i	$-0.609384\pi$
$307$	10.5938	−	10.5938i	0.604619	−	0.604619i	0.941535	+	0.336916i	$0.109384\pi$
$308$		0			0
$309$	−0.0719229	+	0.0719229i	−0.00409155	+	0.00409155i
$310$		0			0
$311$			19.4153i			1.10094i	0.834854	+	0.550471i	$0.185552\pi$
$311$			19.4153i			1.10094i	−0.834854	+	0.550471i	$0.814448\pi$
$312$		0			0
$313$	−2.56569			−0.145022			−0.0725108	−	0.997368i	$-0.523101\pi$
$313$	−2.56569			−0.145022			−0.0725108	+	0.997368i	$0.523101\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.32418	−	7.32418i	0.411367	−	0.411367i	−0.470848	−	0.882215i	$-0.656052\pi$
$317$	7.32418	−	7.32418i	0.411367	−	0.411367i	0.882215	+	0.470848i	$0.156052\pi$
$318$		0			0
$319$	12.0945			0.677163
$320$		0			0
$321$	−0.711000			−0.0396842
$322$		0			0
$323$	−10.9248	+	10.9248i	−0.607873	+	0.607873i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.39428			−0.0771038
$328$		0			0
$329$			3.65238i			0.201362i
$330$		0			0
$331$	−4.17652	+	4.17652i	−0.229562	+	0.229562i	−0.812510	−	0.582948i	$-0.801899\pi$
$331$	−4.17652	+	4.17652i	−0.229562	+	0.229562i	0.582948	+	0.812510i	$0.301899\pi$
$332$		0			0
$333$	22.8191	−	22.8191i	1.25048	−	1.25048i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			12.4540i			0.678410i	0.940712	+	0.339205i	$0.110158\pi$
$337$			12.4540i			0.678410i	−0.940712	+	0.339205i	$0.889842\pi$
$338$		0			0
$339$	0.254371	+	0.254371i	0.0138155	+	0.0138155i
$340$		0			0
$341$	20.1648	−	20.1648i	1.09199	−	1.09199i
$342$		0			0
$343$	5.20917			0.281269
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.5107	+	17.5107i	0.940024	+	0.940024i	0.998300	−	0.0582766i	$-0.0185605\pi$
$347$	17.5107	+	17.5107i	0.940024	+	0.940024i	−0.0582766	+	0.998300i	$0.518561\pi$
$348$		0			0
$349$	−8.42042	−	8.42042i	−0.450735	−	0.450735i	0.444863	−	0.895598i	$-0.353252\pi$
$349$	−8.42042	−	8.42042i	−0.450735	−	0.450735i	−0.895598	+	0.444863i	$0.853252\pi$
$350$		0			0
$351$	−1.31782			−0.0703398
$352$		0			0
$353$		−	9.71293i		−	0.516967i	−0.966016	−	0.258484i	$-0.916777\pi$
$353$		−	9.71293i		−	0.516967i	0.966016	−	0.258484i	$-0.0832228\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	0.108916	+	0.108916i	0.00576443	+	0.00576443i
$358$		0			0
$359$			6.77551i			0.357598i	0.983886	+	0.178799i	$0.0572212\pi$
$359$			6.77551i			0.357598i	−0.983886	+	0.178799i	$0.942779\pi$
$360$		0			0
$361$		−	7.95830i		−	0.418858i
$362$		0			0
$363$	−0.0131274	−	0.0131274i	−0.000689012	−	0.000689012i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			34.4591i			1.79875i	0.437176	+	0.899376i	$0.355979\pi$
$367$			34.4591i			1.79875i	−0.437176	+	0.899376i	$0.644021\pi$
$368$		0			0
$369$	11.3019			0.588355
$370$		0			0
$371$	1.14120	+	1.14120i	0.0592480	+	0.0592480i
$372$		0			0
$373$	−3.55187	−	3.55187i	−0.183909	−	0.183909i	0.609148	−	0.793057i	$-0.291512\pi$
$373$	−3.55187	−	3.55187i	−0.183909	−	0.183909i	−0.793057	+	0.609148i	$0.791512\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.01345			0.464216
$378$		0			0
$379$	26.4464	−	26.4464i	1.35846	−	1.35846i	0.482644	−	0.875817i	$-0.339677\pi$
$379$	26.4464	−	26.4464i	1.35846	−	1.35846i	0.875817	−	0.482644i	$-0.160323\pi$
$380$		0			0
$381$	−1.06518	−	1.06518i	−0.0545706	−	0.0545706i
$382$		0			0
$383$			30.8614i			1.57695i	0.615069	+	0.788473i	$0.289128\pi$
$383$			30.8614i			1.57695i	−0.615069	+	0.788473i	$0.710872\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−18.5875	+	18.5875i	−0.944854	+	0.944854i
$388$		0			0
$389$	−9.50959	+	9.50959i	−0.482155	+	0.482155i	−0.905819	−	0.423664i	$-0.860744\pi$
$389$	−9.50959	+	9.50959i	−0.482155	+	0.482155i	0.423664	+	0.905819i	$0.360744\pi$
$390$		0			0
$391$		−	9.66080i		−	0.488568i
$392$		0			0
$393$	0.444253			0.0224096
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−24.8540	+	24.8540i	−1.24739	+	1.24739i	−0.290518	+	0.956870i	$0.593828\pi$
$397$	−24.8540	+	24.8540i	−1.24739	+	1.24739i	−0.956870	+	0.290518i	$0.906172\pi$
$398$		0			0
$399$	0.110081			0.00551094
$400$		0			0
$401$	4.69303			0.234359			0.117179	−	0.993111i	$-0.462615\pi$
$401$	4.69303			0.234359			0.117179	+	0.993111i	$0.462615\pi$
$402$		0			0
$403$	15.0278	−	15.0278i	0.748590	−	0.748590i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	36.1106			1.78993
$408$		0			0
$409$		−	28.2641i		−	1.39757i	−0.715331	−	0.698786i	$-0.753724\pi$
$409$		−	28.2641i		−	1.39757i	0.715331	−	0.698786i	$-0.246276\pi$
$410$		0			0
$411$	1.04073	−	1.04073i	0.0513352	−	0.0513352i
$412$		0			0
$413$	3.05085	−	3.05085i	0.150123	−	0.150123i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			0.942858i			0.0461720i
$418$		0			0
$419$	23.0355	+	23.0355i	1.12536	+	1.12536i	0.990923	+	0.134433i	$0.0429213\pi$
$419$	23.0355	+	23.0355i	1.12536	+	1.12536i	0.134433	+	0.990923i	$0.457079\pi$
$420$		0			0
$421$	5.40760	−	5.40760i	0.263550	−	0.263550i	−0.562945	−	0.826495i	$-0.690332\pi$
$421$	5.40760	−	5.40760i	0.263550	−	0.263550i	0.826495	+	0.562945i	$0.190332\pi$
$422$		0			0
$423$	29.0754			1.41369
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.273762	−	0.273762i	−0.0132483	−	0.0132483i
$428$		0			0
$429$	−0.520675	−	0.520675i	−0.0251384	−	0.0251384i
$430$		0			0
$431$	12.6839			0.610961			0.305481	−	0.952198i	$-0.401183\pi$
$431$	12.6839			0.610961			0.305481	+	0.952198i	$0.401183\pi$
$432$		0			0
$433$		−	23.8511i		−	1.14621i	−0.819482	−	0.573104i	$-0.805739\pi$
$433$		−	23.8511i		−	1.14621i	0.819482	−	0.573104i	$-0.194261\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.88208	−	4.88208i	−0.233542	−	0.233542i
$438$		0			0
$439$			4.65878i			0.222352i	0.993801	+	0.111176i	$0.0354617\pi$
$439$			4.65878i			0.222352i	−0.993801	+	0.111176i	$0.964538\pi$
$440$		0			0
$441$		−	20.5229i		−	0.977280i
$442$		0			0
$443$	−8.74048	−	8.74048i	−0.415273	−	0.415273i	0.468298	−	0.883571i	$-0.344867\pi$
$443$	−8.74048	−	8.74048i	−0.415273	−	0.415273i	−0.883571	+	0.468298i	$0.844867\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	1.27810i		−	0.0604520i
$448$		0			0
$449$	7.28525			0.343812			0.171906	−	0.985113i	$-0.445007\pi$
$449$	7.28525			0.343812			0.171906	+	0.985113i	$0.445007\pi$
$450$		0			0
$451$	8.94247	+	8.94247i	0.421085	+	0.421085i
$452$		0			0
$453$	−1.18678	−	1.18678i	−0.0557596	−	0.0557596i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.2194			−1.17971			−0.589857	−	0.807508i	$-0.700816\pi$
$457$	−25.2194			−1.17971			−0.589857	+	0.807508i	$0.700816\pi$
$458$		0			0
$459$	1.73633	−	1.73633i	0.0810451	−	0.0810451i
$460$		0			0
$461$	13.6698	+	13.6698i	0.636667	+	0.636667i	0.949732	−	0.313064i	$-0.101356\pi$
$461$	13.6698	+	13.6698i	0.636667	+	0.636667i	−0.313064	+	0.949732i	$0.601356\pi$
$462$		0			0
$463$		−	2.77045i		−	0.128754i	−0.997926	−	0.0643768i	$-0.979494\pi$
$463$		−	2.77045i		−	0.128754i	0.997926	−	0.0643768i	$-0.0205060\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.9587	−	17.9587i	0.831031	−	0.831031i	−0.156627	−	0.987658i	$-0.550062\pi$
$467$	17.9587	−	17.9587i	0.831031	−	0.831031i	0.987658	+	0.156627i	$0.0500621\pi$
$468$		0			0
$469$	0.364314	−	0.364314i	0.0168225	−	0.0168225i
$470$		0			0
$471$			1.26306i			0.0581986i
$472$		0			0
$473$	−29.4141			−1.35246
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.08470	−	9.08470i	0.415960	−	0.415960i
$478$		0			0
$479$	−22.4540			−1.02595			−0.512975	−	0.858403i	$-0.671457\pi$
$479$	−22.4540			−1.02595			−0.512975	+	0.858403i	$0.671457\pi$
$480$		0			0
$481$	26.9114			1.22705
$482$		0			0
$483$	−0.0486722	+	0.0486722i	−0.00221466	+	0.00221466i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−27.7615			−1.25799			−0.628997	−	0.777408i	$-0.716534\pi$
$487$	−27.7615			−1.25799			−0.628997	+	0.777408i	$0.716534\pi$
$488$		0			0
$489$		−	0.916741i		−	0.0414565i
$490$		0			0
$491$	−16.8993	+	16.8993i	−0.762656	+	0.762656i	−0.976802	−	0.214146i	$-0.931303\pi$
$491$	−16.8993	+	16.8993i	−0.762656	+	0.762656i	0.214146	+	0.976802i	$0.431303\pi$
$492$		0			0
$493$	−11.8760	+	11.8760i	−0.534867	+	0.534867i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3.43638i		−	0.154143i
$498$		0			0
$499$	−1.81950	−	1.81950i	−0.0814520	−	0.0814520i	0.665207	−	0.746659i	$-0.268343\pi$
$499$	−1.81950	−	1.81950i	−0.0814520	−	0.0814520i	−0.746659	+	0.665207i	$0.768343\pi$
$500$		0			0
$501$	−0.500397	+	0.500397i	−0.0223561	+	0.0223561i
$502$		0			0
$503$	−42.2076			−1.88195			−0.940973	−	0.338482i	$-0.890087\pi$
$503$	−42.2076			−1.88195			−0.940973	+	0.338482i	$0.890087\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.422139	+	0.422139i	0.0187478	+	0.0187478i
$508$		0			0
$509$	21.9831	+	21.9831i	0.974382	+	0.974382i	0.999680	−	0.0252980i	$-0.00805345\pi$
$509$	21.9831	+	21.9831i	0.974382	+	0.974382i	−0.0252980	+	0.999680i	$0.508053\pi$
$510$		0			0
$511$	−2.84461			−0.125838
$512$		0			0
$513$		−	1.75491i		−	0.0774812i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	23.0054	+	23.0054i	1.01178	+	1.01178i
$518$		0			0
$519$			1.30034i			0.0570786i
$520$		0			0
$521$			28.1418i			1.23291i	0.787388	+	0.616457i	$0.211433\pi$
$521$			28.1418i			1.23291i	−0.787388	+	0.616457i	$0.788567\pi$
$522$		0			0
$523$	9.58093	+	9.58093i	0.418945	+	0.418945i	0.884840	−	0.465895i	$-0.154268\pi$
$523$	9.58093	+	9.58093i	0.418945	+	0.418945i	−0.465895	+	0.884840i	$0.654268\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			39.6009i			1.72504i
$528$		0			0
$529$	−18.6828			−0.812295
$530$		0			0
$531$	−24.2868	−	24.2868i	−1.05396	−	1.05396i
$532$		0			0
$533$	6.66438	+	6.66438i	0.288666	+	0.288666i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.03488			−0.0446582
$538$		0			0
$539$	16.2384	−	16.2384i	0.699437	−	0.699437i
$540$		0			0
$541$	−26.9128	−	26.9128i	−1.15707	−	1.15707i	−0.985102	−	0.171972i	$-0.944986\pi$
$541$	−26.9128	−	26.9128i	−1.15707	−	1.15707i	−0.171972	−	0.985102i	$-0.555014\pi$
$542$		0			0
$543$			1.30590i			0.0560415i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.6627	−	10.6627i	0.455902	−	0.455902i	−0.441406	−	0.897308i	$-0.645520\pi$
$547$	10.6627	−	10.6627i	0.455902	−	0.455902i	0.897308	+	0.441406i	$0.145520\pi$
$548$		0			0
$549$	−2.17933	+	2.17933i	−0.0930115	+	0.0930115i
$550$		0			0
$551$			12.0030i			0.511347i
$552$		0			0
$553$	−4.43780			−0.188714
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.8060	−	22.8060i	0.966320	−	0.966320i	−0.0331307	−	0.999451i	$-0.510548\pi$
$557$	22.8060	−	22.8060i	0.966320	−	0.966320i	0.999451	+	0.0331307i	$0.0105477\pi$
$558$		0			0
$559$	−21.9209			−0.927153
$560$		0			0
$561$	1.37207			0.0579287
$562$		0			0
$563$	−0.472513	+	0.472513i	−0.0199140	+	0.0199140i	−0.716994	−	0.697080i	$-0.754482\pi$
$563$	−0.472513	+	0.472513i	−0.0199140	+	0.0199140i	0.697080	+	0.716994i	$0.254482\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.35664			0.140965
$568$		0			0
$569$		−	3.14792i		−	0.131967i	−0.997821	−	0.0659837i	$-0.978981\pi$
$569$		−	3.14792i		−	0.131967i	0.997821	−	0.0659837i	$-0.0210185\pi$
$570$		0			0
$571$	−5.78162	+	5.78162i	−0.241953	+	0.241953i	−0.817658	−	0.575704i	$-0.804728\pi$
$571$	−5.78162	+	5.78162i	−0.241953	+	0.241953i	0.575704	+	0.817658i	$0.304728\pi$
$572$		0			0
$573$	−0.104748	+	0.104748i	−0.00437593	+	0.00437593i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	29.0110i		−	1.20774i	−0.797081	−	0.603872i	$-0.793624\pi$
$577$		−	29.0110i		−	1.20774i	0.797081	−	0.603872i	$-0.206376\pi$
$578$		0			0
$579$	0.100588	+	0.100588i	0.00418029	+	0.00418029i
$580$		0			0
$581$	3.99908	−	3.99908i	0.165910	−	0.165910i
$582$		0			0
$583$	14.3762			0.595403
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.4099	−	20.4099i	−0.842408	−	0.842408i	0.146764	−	0.989172i	$-0.453114\pi$
$587$	−20.4099	−	20.4099i	−0.842408	−	0.842408i	−0.989172	+	0.146764i	$0.953114\pi$
$588$		0			0
$589$	20.0123	+	20.0123i	0.824592	+	0.824592i
$590$		0			0
$591$	0.636074			0.0261646
$592$		0			0
$593$			28.9098i			1.18718i	0.804766	+	0.593592i	$0.202291\pi$
$593$			28.9098i			1.18718i	−0.804766	+	0.593592i	$0.797709\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.695161	+	0.695161i	0.0284511	+	0.0284511i
$598$		0			0
$599$		−	11.7893i		−	0.481696i	−0.970563	−	0.240848i	$-0.922574\pi$
$599$		−	11.7893i		−	0.481696i	0.970563	−	0.240848i	$-0.0774256\pi$
$600$		0			0
$601$			17.7398i			0.723621i	0.932252	+	0.361810i	$0.117841\pi$
$601$			17.7398i			0.723621i	−0.932252	+	0.361810i	$0.882159\pi$
$602$		0			0
$603$	−2.90019	−	2.90019i	−0.118105	−	0.118105i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	25.8393i		−	1.04878i	−0.851477	−	0.524392i	$-0.824293\pi$
$607$		−	25.8393i		−	1.04878i	0.851477	−	0.524392i	$-0.175707\pi$
$608$		0			0
$609$	0.119665			0.00484907
$610$		0			0
$611$	17.1448	+	17.1448i	0.693605	+	0.693605i
$612$		0			0
$613$	31.5411	+	31.5411i	1.27393	+	1.27393i	0.944006	+	0.329929i	$0.107025\pi$
$613$	31.5411	+	31.5411i	1.27393	+	1.27393i	0.329929	+	0.944006i	$0.392975\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0637			−0.606440			−0.303220	−	0.952921i	$-0.598062\pi$
$617$	−15.0637			−0.606440			−0.303220	+	0.952921i	$0.598062\pi$
$618$		0			0
$619$	−10.4975	+	10.4975i	−0.421929	+	0.421929i	−0.885868	−	0.463938i	$-0.846436\pi$
$619$	−10.4975	+	10.4975i	−0.421929	+	0.421929i	0.463938	+	0.885868i	$0.346436\pi$
$620$		0			0
$621$	0.775933	+	0.775933i	0.0311371	+	0.0311371i
$622$		0			0
$623$		−	5.90545i		−	0.236597i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	0.693373	−	0.693373i	0.0276906	−	0.0276906i
$628$		0			0
$629$	−35.4581	+	35.4581i	−1.41381	+	1.41381i
$630$		0			0
$631$		−	43.6349i		−	1.73708i	−0.495621	−	0.868539i	$-0.665060\pi$
$631$		−	43.6349i		−	1.73708i	0.495621	−	0.868539i	$-0.334940\pi$
$632$		0			0
$633$	0.309581			0.0123047
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12.1017	−	12.1017i	0.479486	−	0.479486i
$638$		0			0
$639$	−27.3559			−1.08218
$640$		0			0
$641$	−34.2710			−1.35362			−0.676812	−	0.736156i	$-0.736639\pi$
$641$	−34.2710			−1.35362			−0.676812	+	0.736156i	$0.736639\pi$
$642$		0			0
$643$	30.1937	−	30.1937i	1.19072	−	1.19072i	0.213857	−	0.976865i	$-0.431397\pi$
$643$	30.1937	−	30.1937i	1.19072	−	1.19072i	0.976865	−	0.213857i	$-0.0686027\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.7474			−0.619096			−0.309548	−	0.950884i	$-0.600178\pi$
$647$	−15.7474			−0.619096			−0.309548	+	0.950884i	$0.600178\pi$
$648$		0			0
$649$		−	38.4331i		−	1.50863i
$650$		0			0
$651$	0.199514	−	0.199514i	0.00781956	−	0.00781956i
$652$		0			0
$653$	5.80619	−	5.80619i	0.227214	−	0.227214i	−0.584314	−	0.811528i	$-0.698636\pi$
$653$	5.80619	−	5.80619i	0.227214	−	0.227214i	0.811528	+	0.584314i	$0.198636\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			22.6450i			0.883465i
$658$		0			0
$659$	1.76782	+	1.76782i	0.0688647	+	0.0688647i	0.740700	−	0.671836i	$-0.234494\pi$
$659$	1.76782	+	1.76782i	0.0688647	+	0.0688647i	−0.671836	+	0.740700i	$0.734494\pi$
$660$		0			0
$661$	−12.4824	+	12.4824i	−0.485509	+	0.485509i	−0.906886	−	0.421377i	$-0.861547\pi$
$661$	−12.4824	+	12.4824i	−0.485509	+	0.485509i	0.421377	+	0.906886i	$0.361547\pi$
$662$		0			0
$663$	1.02253			0.0397119
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.30714	−	5.30714i	−0.205493	−	0.205493i
$668$		0			0
$669$	−1.31736	−	1.31736i	−0.0509322	−	0.0509322i
$670$		0			0
$671$	−3.44872			−0.133136
$672$		0			0
$673$			14.1113i			0.543950i	0.962304	+	0.271975i	$0.0876768\pi$
$673$			14.1113i			0.543950i	−0.962304	+	0.271975i	$0.912323\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.31191	−	8.31191i	−0.319453	−	0.319453i	0.529104	−	0.848557i	$-0.322528\pi$
$677$	−8.31191	−	8.31191i	−0.319453	−	0.319453i	−0.848557	+	0.529104i	$0.822528\pi$
$678$		0			0
$679$			1.45096i			0.0556827i
$680$		0			0
$681$			1.80587i			0.0692011i
$682$		0			0
$683$	−30.0811	−	30.0811i	−1.15102	−	1.15102i	−0.986348	−	0.164673i	$-0.947343\pi$
$683$	−30.0811	−	30.0811i	−1.15102	−	1.15102i	−0.164673	−	0.986348i	$-0.552657\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	1.25834i		−	0.0480086i
$688$		0			0
$689$	10.7139			0.408167
$690$		0			0
$691$	−24.0212	−	24.0212i	−0.913810	−	0.913810i	0.0827600	−	0.996570i	$-0.473627\pi$
$691$	−24.0212	−	24.0212i	−0.913810	−	0.913810i	−0.996570	+	0.0827600i	$0.973627\pi$
$692$		0			0
$693$	2.66282	+	2.66282i	0.101152	+	0.101152i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17.5618			−0.665200
$698$		0			0
$699$	−0.215004	+	0.215004i	−0.00813219	+	0.00813219i
$700$		0			0
$701$	10.0971	+	10.0971i	0.381363	+	0.381363i	0.871593	−	0.490230i	$-0.163087\pi$
$701$	10.0971	+	10.0971i	0.381363	+	0.381363i	−0.490230	+	0.871593i	$0.663087\pi$
$702$		0			0
$703$			35.8375i			1.35164i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.58392	−	2.58392i	0.0971782	−	0.0971782i
$708$		0			0
$709$	4.67310	−	4.67310i	0.175502	−	0.175502i	−0.613890	−	0.789392i	$-0.710396\pi$
$709$	4.67310	−	4.67310i	0.175502	−	0.175502i	0.789392	+	0.613890i	$0.210396\pi$
$710$		0			0
$711$			35.3279i			1.32490i
$712$		0			0
$713$	−17.6968			−0.662752
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.12215	−	1.12215i	0.0419074	−	0.0419074i
$718$		0			0
$719$	−23.5339			−0.877667			−0.438833	−	0.898568i	$-0.644608\pi$
$719$	−23.5339			−0.877667			−0.438833	+	0.898568i	$0.644608\pi$
$720$		0			0
$721$	0.433789			0.0161552
$722$		0			0
$723$	0.786710	−	0.786710i	0.0292581	−	0.0292581i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−16.6692			−0.618226			−0.309113	−	0.951025i	$-0.600032\pi$
$727$	−16.6692			−0.618226			−0.309113	+	0.951025i	$0.600032\pi$
$728$		0			0
$729$		−	26.5814i		−	0.984498i
$730$		0			0
$731$	28.8826	−	28.8826i	1.06826	−	1.06826i
$732$		0			0
$733$	27.4684	−	27.4684i	1.01457	−	1.01457i	0.0146760	−	0.999892i	$-0.495328\pi$
$733$	27.4684	−	27.4684i	1.01457	−	1.01457i	0.999892	−	0.0146760i	$-0.00467168\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.58945i		−	0.169055i
$738$		0			0
$739$	22.7939	+	22.7939i	0.838486	+	0.838486i	0.988660	−	0.150174i	$-0.0479832\pi$
$739$	22.7939	+	22.7939i	0.838486	+	0.838486i	−0.150174	+	0.988660i	$0.547983\pi$
$740$		0			0
$741$	0.516736	−	0.516736i	0.0189828	−	0.0189828i
$742$		0			0
$743$	16.4964			0.605196			0.302598	−	0.953118i	$-0.402146\pi$
$743$	16.4964			0.605196			0.302598	+	0.953118i	$0.402146\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−31.8354	−	31.8354i	−1.16480	−	1.16480i
$748$		0			0
$749$	2.14413	+	2.14413i	0.0783449	+	0.0783449i
$750$		0			0
$751$	21.6997			0.791833			0.395917	−	0.918286i	$-0.370427\pi$
$751$	21.6997			0.791833			0.395917	+	0.918286i	$0.370427\pi$
$752$		0			0
$753$		−	1.14411i		−	0.0416937i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.73819	−	1.73819i	−0.0631757	−	0.0631757i	0.674813	−	0.737989i	$-0.264224\pi$
$757$	−1.73819	−	1.73819i	−0.0631757	−	0.0631757i	−0.737989	+	0.674813i	$0.764224\pi$
$758$		0			0
$759$			0.613149i			0.0222559i
$760$		0			0
$761$			46.5311i			1.68675i	0.537323	+	0.843376i	$0.319435\pi$
$761$			46.5311i			1.68675i	−0.537323	+	0.843376i	$0.680565\pi$
$762$		0			0
$763$	4.20467	+	4.20467i	0.152219	+	0.152219i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28.6423i		−	1.03421i
$768$		0			0
$769$	−15.4731			−0.557976			−0.278988	−	0.960295i	$-0.589999\pi$
$769$	−15.4731			−0.557976			−0.278988	+	0.960295i	$0.589999\pi$
$770$		0			0
$771$	1.01519	+	1.01519i	0.0365610	+	0.0365610i
$772$		0			0
$773$	−5.69848	−	5.69848i	−0.204960	−	0.204960i	0.597161	−	0.802121i	$-0.296295\pi$
$773$	−5.69848	−	5.69848i	−0.204960	−	0.204960i	−0.802121	+	0.597161i	$0.796295\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.357284			0.0128175
$778$		0			0
$779$	−8.87484	+	8.87484i	−0.317974	+	0.317974i
$780$		0			0
$781$	−21.6449	−	21.6449i	−0.774516	−	0.774516i
$782$		0			0
$783$		−	1.90770i		−	0.0681757i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	18.1351	−	18.1351i	0.646448	−	0.646448i	−0.305685	−	0.952133i	$-0.598885\pi$
$787$	18.1351	−	18.1351i	0.646448	−	0.646448i	0.952133	+	0.305685i	$0.0988855\pi$
$788$		0			0
$789$	−0.653736	+	0.653736i	−0.0232736	+	0.0232736i
$790$		0			0
$791$		−	1.53419i		−	0.0545495i
$792$		0			0
$793$	−2.57016			−0.0912690
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.51575	+	4.51575i	−0.159956	+	0.159956i	−0.782547	−	0.622591i	$-0.786080\pi$
$797$	−4.51575	+	4.51575i	−0.159956	+	0.159956i	0.622591	+	0.782547i	$0.286080\pi$
$798$		0			0
$799$	−45.1795			−1.59834
$800$		0			0
$801$	−47.0114			−1.66107
$802$		0			0
$803$	−17.9175	+	17.9175i	−0.632294	+	0.632294i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.05526			0.0371470
$808$		0			0
$809$		−	3.33368i		−	0.117206i	−0.998281	−	0.0586030i	$-0.981335\pi$
$809$		−	3.33368i		−	0.117206i	0.998281	−	0.0586030i	$-0.0186646\pi$
$810$		0			0
$811$	−37.1948	+	37.1948i	−1.30609	+	1.30609i	−0.381873	+	0.924215i	$0.624721\pi$
$811$	−37.1948	+	37.1948i	−1.30609	+	1.30609i	−0.924215	+	0.381873i	$0.875279\pi$
$812$		0			0
$813$	0.555975	−	0.555975i	0.0194989	−	0.0194989i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	29.1916i		−	1.02129i
$818$		0			0
$819$	1.98446	+	1.98446i	0.0693428	+	0.0693428i
$820$		0			0
$821$	−25.5278	+	25.5278i	−0.890926	+	0.890926i	−0.994610	−	0.103684i	$-0.966937\pi$
$821$	−25.5278	+	25.5278i	−0.890926	+	0.890926i	0.103684	+	0.994610i	$0.466937\pi$
$822$		0			0
$823$	−16.8858			−0.588603			−0.294301	−	0.955713i	$-0.595087\pi$
$823$	−16.8858			−0.588603			−0.294301	+	0.955713i	$0.595087\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.5765	−	11.5765i	−0.402556	−	0.402556i	0.476577	−	0.879133i	$-0.341877\pi$
$827$	−11.5765	−	11.5765i	−0.402556	−	0.402556i	−0.879133	+	0.476577i	$0.841877\pi$
$828$		0			0
$829$	−5.97296	−	5.97296i	−0.207449	−	0.207449i	0.595733	−	0.803182i	$-0.296862\pi$
$829$	−5.97296	−	5.97296i	−0.207449	−	0.207449i	−0.803182	+	0.595733i	$0.796862\pi$
$830$		0			0
$831$	1.16718			0.0404891
$832$		0			0
$833$			31.8900i			1.10492i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.18065	−	3.18065i	−0.109939	−	0.109939i
$838$		0			0
$839$			27.2960i			0.942362i	0.882037	+	0.471181i	$0.156172\pi$
$839$			27.2960i			0.942362i	−0.882037	+	0.471181i	$0.843828\pi$
$840$		0			0
$841$		−	15.9519i		−	0.550066i
$842$		0			0
$843$	−0.195015	−	0.195015i	−0.00671668	−	0.00671668i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			0.0791757i			0.00272051i
$848$		0			0
$849$	−0.258392			−0.00886799
$850$		0			0
$851$	−15.8455	−	15.8455i	−0.543177	−	0.543177i
$852$		0			0
$853$	29.1167	+	29.1167i	0.996938	+	0.996938i	0.999995	−	0.00305738i	$-0.000973195\pi$
$853$	29.1167	+	29.1167i	0.996938	+	0.996938i	−0.00305738	+	0.999995i	$0.500973\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.0170			−0.752087			−0.376044	−	0.926602i	$-0.622716\pi$
$857$	−22.0170			−0.752087			−0.376044	+	0.926602i	$0.622716\pi$
$858$		0			0
$859$	16.8910	−	16.8910i	0.576312	−	0.576312i	−0.357573	−	0.933885i	$-0.616396\pi$
$859$	16.8910	−	16.8910i	0.576312	−	0.576312i	0.933885	+	0.357573i	$0.116396\pi$
$860$		0			0
$861$	0.0884782	+	0.0884782i	0.00301533	+	0.00301533i
$862$		0			0
$863$			46.2073i			1.57292i	0.617644	+	0.786458i	$0.288087\pi$
$863$			46.2073i			1.57292i	−0.617644	+	0.786458i	$0.711913\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.287819	+	0.287819i	−0.00977484	+	0.00977484i
$868$		0			0
$869$	−27.9526	+	27.9526i	−0.948227	+	0.948227i
$870$		0			0
$871$		−	3.42029i		−	0.115892i
$872$		0			0
$873$	11.5506			0.390929
$874$		0			0
$875$		0			0
$876$		0			0
$877$	21.6380	−	21.6380i	0.730664	−	0.730664i	−0.240087	−	0.970751i	$-0.577176\pi$
$877$	21.6380	−	21.6380i	0.730664	−	0.730664i	0.970751	+	0.240087i	$0.0771760\pi$
$878$		0			0
$879$	−1.53968			−0.0519320
$880$		0			0
$881$	−35.6649			−1.20158			−0.600790	−	0.799407i	$-0.705147\pi$
$881$	−35.6649			−1.20158			−0.600790	+	0.799407i	$0.705147\pi$
$882$		0			0
$883$	−17.5681	+	17.5681i	−0.591213	+	0.591213i	−0.937959	−	0.346746i	$-0.887287\pi$
$883$	−17.5681	+	17.5681i	−0.591213	+	0.591213i	0.346746	+	0.937959i	$0.387287\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	51.2454			1.72065			0.860326	−	0.509744i	$-0.170260\pi$
$887$	51.2454			1.72065			0.860326	+	0.509744i	$0.170260\pi$
$888$		0			0
$889$			6.42441i			0.215468i
$890$		0			0
$891$	21.1426	−	21.1426i	0.708305	−	0.708305i
$892$		0			0
$893$	−22.8315	+	22.8315i	−0.764025	+	0.764025i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			0.456949i			0.0152571i
$898$		0			0
$899$	21.7547	+	21.7547i	0.725559	+	0.725559i
$900$		0			0
$901$	−14.1165	+	14.1165i	−0.470288	+	0.470288i
$902$		0			0
$903$	−0.291028			−0.00968479
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−29.6745	−	29.6745i	−0.985324	−	0.985324i	0.0145695	−	0.999894i	$-0.495362\pi$
$907$	−29.6745	−	29.6745i	−0.985324	−	0.985324i	−0.999894	+	0.0145695i	$0.995362\pi$
$908$		0			0
$909$	−20.5697	−	20.5697i	−0.682255	−	0.682255i
$910$		0			0
$911$	24.9064			0.825187			0.412593	−	0.910915i	$-0.364623\pi$
$911$	24.9064			0.825187			0.412593	+	0.910915i	$0.364623\pi$
$912$		0			0
$913$		−	50.3785i		−	1.66729i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.33971	−	1.33971i	−0.0442413	−	0.0442413i
$918$		0			0
$919$			15.6940i			0.517696i	0.965918	+	0.258848i	$0.0833429\pi$
$919$			15.6940i			0.517696i	−0.965918	+	0.258848i	$0.916657\pi$
$920$		0			0
$921$		−	1.32043i		−	0.0435096i
$922$		0			0
$923$	−16.1309	−	16.1309i	−0.530954	−	0.530954i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	3.45326i		−	0.113420i
$928$		0			0
$929$	−6.00598			−0.197050			−0.0985249	−	0.995135i	$-0.531412\pi$
$929$	−6.00598			−0.197050			−0.0985249	+	0.995135i	$0.531412\pi$
$930$		0			0
$931$	16.1156	+	16.1156i	0.528167	+	0.528167i
$932$		0			0
$933$	1.20998	+	1.20998i	0.0396130	+	0.0396130i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.83037			0.157801			0.0789006	−	0.996882i	$-0.474859\pi$
$937$	4.83037			0.157801			0.0789006	+	0.996882i	$0.474859\pi$
$938$		0			0
$939$	−0.159896	+	0.159896i	−0.00521802	+	0.00521802i
$940$		0			0
$941$	−3.63878	−	3.63878i	−0.118621	−	0.118621i	0.645305	−	0.763925i	$-0.276730\pi$
$941$	−3.63878	−	3.63878i	−0.118621	−	0.118621i	−0.763925	+	0.645305i	$0.776730\pi$
$942$		0			0
$943$		−	7.84801i		−	0.255566i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.7720	−	34.7720i	1.12994	−	1.12994i	0.139751	−	0.990187i	$-0.455370\pi$
$947$	34.7720	−	34.7720i	1.12994	−	1.12994i	0.990187	−	0.139751i	$-0.0446302\pi$
$948$		0			0
$949$	−13.3530	+	13.3530i	−0.433457	+	0.433457i
$950$		0			0
$951$		−	0.912899i		−	0.0296028i
$952$		0			0
$953$	1.89827			0.0614910			0.0307455	−	0.999527i	$-0.490212\pi$
$953$	1.89827			0.0614910			0.0307455	+	0.999527i	$0.490212\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	0.753741	−	0.753741i	0.0243650	−	0.0243650i
$958$		0			0
$959$	−6.27694			−0.202693
$960$		0			0
$961$	41.5417			1.34006
$962$		0			0
$963$	17.0687	−	17.0687i	0.550033	−	0.550033i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−61.5753			−1.98013			−0.990064	−	0.140616i	$-0.955092\pi$
$967$	−61.5753			−1.98013			−0.990064	+	0.140616i	$0.955092\pi$
$968$		0			0
$969$			1.36169i			0.0437437i
$970$		0			0
$971$	−22.6595	+	22.6595i	−0.727179	+	0.727179i	−0.970057	−	0.242878i	$-0.921909\pi$
$971$	−22.6595	+	22.6595i	−0.727179	+	0.727179i	0.242878	+	0.970057i	$0.421909\pi$
$972$		0			0
$973$	2.84334	−	2.84334i	0.0911532	−	0.0911532i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.3240i		−	0.810186i	−0.914276	−	0.405093i	$-0.867239\pi$
$977$		−	25.3240i		−	0.810186i	0.914276	−	0.405093i	$-0.132761\pi$
$978$		0			0
$979$	−37.1970	−	37.1970i	−1.18882	−	1.18882i
$980$		0			0
$981$	33.4720	−	33.4720i	1.06868	−	1.06868i
$982$		0			0
$983$	−5.01686			−0.160013			−0.0800065	−	0.996794i	$-0.525494\pi$
$983$	−5.01686			−0.160013			−0.0800065	+	0.996794i	$0.525494\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.227620	+	0.227620i	0.00724521	+	0.00724521i
$988$		0			0
$989$	12.9071	+	12.9071i	0.410420	+	0.410420i
$990$		0			0
$991$	−41.2998			−1.31193			−0.655965	−	0.754791i	$-0.727738\pi$
$991$	−41.2998			−1.31193			−0.655965	+	0.754791i	$0.727738\pi$
$992$		0			0
$993$			0.520569i			0.0165198i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.23850	+	1.23850i	0.0392235	+	0.0392235i	0.726447	−	0.687223i	$-0.241170\pi$
$997$	1.23850	+	1.23850i	0.0392235	+	0.0392235i	−0.687223	+	0.726447i	$0.741170\pi$
$998$		0			0
$999$		−	5.69582i		−	0.180208i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.q.f.849.3		12
4.3	odd	2		400.2.q.f.349.6		12
5.2	odd	4		1600.2.l.f.401.4		12
5.3	odd	4		1600.2.l.g.401.3		12
5.4	even	2		1600.2.q.e.849.4		12
16.5	even	4		1600.2.q.e.49.4		12
16.11	odd	4		400.2.q.e.149.1		12
20.3	even	4		400.2.l.f.301.5	yes	12
20.7	even	4		400.2.l.g.301.2	yes	12
20.19	odd	2		400.2.q.e.349.1		12
80.27	even	4		400.2.l.g.101.2	yes	12
80.37	odd	4		1600.2.l.f.1201.4		12
80.43	even	4		400.2.l.f.101.5	✓	12
80.53	odd	4		1600.2.l.g.1201.3		12
80.59	odd	4		400.2.q.f.149.6		12
80.69	even	4	inner	1600.2.q.f.49.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.l.f.101.5	✓	12	80.43	even	4
400.2.l.f.301.5	yes	12	20.3	even	4
400.2.l.g.101.2	yes	12	80.27	even	4
400.2.l.g.301.2	yes	12	20.7	even	4
400.2.q.e.149.1		12	16.11	odd	4
400.2.q.e.349.1		12	20.19	odd	2
400.2.q.f.149.6		12	80.59	odd	4
400.2.q.f.349.6		12	4.3	odd	2
1600.2.l.f.401.4		12	5.2	odd	4
1600.2.l.f.1201.4		12	80.37	odd	4
1600.2.l.g.401.3		12	5.3	odd	4
1600.2.l.g.1201.3		12	80.53	odd	4
1600.2.q.e.49.4		12	16.5	even	4
1600.2.q.e.849.4		12	5.4	even	2
1600.2.q.f.49.3		12	80.69	even	4	inner
1600.2.q.f.849.3		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.q (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0623209 - 0.0623209i) q^{3} -0.375877 q^{7} +2.99223i q^{9} +O(q^{10})\)	\(q+(0.0623209 - 0.0623209i) q^{3} -0.375877 q^{7} +2.99223i q^{9} +(-2.36756 + 2.36756i) q^{11} +(-1.76442 + 1.76442i) q^{13} -4.64955i q^{17} +(-2.34965 - 2.34965i) q^{19} +(-0.0234250 + 0.0234250i) q^{21} +2.07779 q^{23} +(0.373441 + 0.373441i) q^{27} +(-2.55422 - 2.55422i) q^{29} -8.51714 q^{31} +0.295096i q^{33} +(-7.62613 - 7.62613i) q^{37} +0.219921i q^{39} -3.77709i q^{41} +(6.21191 + 6.21191i) q^{43} -9.71696i q^{47} -6.85872 q^{49} +(-0.289764 - 0.289764i) q^{51} +(-3.03609 - 3.03609i) q^{53} -0.292864 q^{57} +(-8.11663 + 8.11663i) q^{59} +(0.728329 + 0.728329i) q^{61} -1.12471i q^{63} +(-0.969239 + 0.969239i) q^{67} +(0.129490 - 0.129490i) q^{69} +9.14230i q^{71} +7.56793 q^{73} +(0.889909 - 0.889909i) q^{77} +11.8065 q^{79} -8.93015 q^{81} +(-10.6393 + 10.6393i) q^{83} -0.318363 q^{87} +15.7111i q^{89} +(0.663205 - 0.663205i) q^{91} +(-0.530796 + 0.530796i) q^{93} -3.86020i q^{97} +(-7.08428 - 7.08428i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} - 12 q^{7}+O(q^{10}) \) 12 * q + 2 * q^3 - 12 * q^7	\( 12 q + 2 q^{3} - 12 q^{7} + 2 q^{11} + 4 q^{13} - 14 q^{19} - 20 q^{21} - 12 q^{23} - 10 q^{27} + 4 q^{31} - 8 q^{37} - 4 q^{49} - 10 q^{51} - 16 q^{53} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 50 q^{67} + 40 q^{73} - 8 q^{77} + 12 q^{79} - 8 q^{81} - 2 q^{83} + 64 q^{87} + 44 q^{93} + 12 q^{99}+O(q^{100}) \) 12 * q + 2 * q^3 - 12 * q^7 + 2 * q^11 + 4 * q^13 - 14 * q^19 - 20 * q^21 - 12 * q^23 - 10 * q^27 + 4 * q^31 - 8 * q^37 - 4 * q^49 - 10 * q^51 - 16 * q^53 - 16 * q^57 + 20 * q^59 + 4 * q^61 - 50 * q^67 + 40 * q^73 - 8 * q^77 + 12 * q^79 - 8 * q^81 - 2 * q^83 + 64 * q^87 + 44 * q^93 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more