LMFDB - Embedded newform 1800.3.v.t.1657.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,3,Mod(793,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1800.793");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 $ x^8 - 4x^7 + 8x^6 + 269x^4 - 1116x^3 + 2312x^2 + 680x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1657.2
Root			$-2.88489 - 2.88489i$ of defining polynomial
Character	$\chi$	$=$	1800.1657
Dual form			1800.3.v.t.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.02356 - 6.02356i) q^{7} +O(q^{10})$	$q+(-6.02356 - 6.02356i) q^{7} -17.0038 q^{11} +(-0.124585 + 0.124585i) q^{13} +(10.7744 + 10.7744i) q^{17} +4.04713i q^{19} +(1.85568 - 1.85568i) q^{23} +54.4448i q^{29} -53.1995 q^{31} +(-21.2693 - 21.2693i) q^{37} +52.1414 q^{41} +(50.4554 - 50.4554i) q^{43} +(13.4045 + 13.4045i) q^{47} +23.5667i q^{49} +(12.4315 - 12.4315i) q^{53} -8.79413i q^{59} +105.860 q^{61} +(46.8595 + 46.8595i) q^{67} +56.8955 q^{71} +(52.7296 - 52.7296i) q^{73} +(102.424 + 102.424i) q^{77} +15.2920i q^{79} +(46.1914 - 46.1914i) q^{83} +0.948538i q^{89} +1.50089 q^{91} +(-77.6693 - 77.6693i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} - 32 q^{11} + 4 q^{13} + 52 q^{17} - 40 q^{23} + 96 q^{31} + 60 q^{37} + 152 q^{41} + 88 q^{43} - 16 q^{47} + 108 q^{53} + 264 q^{61} + 216 q^{67} + 240 q^{71} + 208 q^{73} + 168 q^{77} + 336 q^{83} + 592 q^{91} + 208 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 32 * q^11 + 4 * q^13 + 52 * q^17 - 40 * q^23 + 96 * q^31 + 60 * q^37 + 152 * q^41 + 88 * q^43 - 16 * q^47 + 108 * q^53 + 264 * q^61 + 216 * q^67 + 240 * q^71 + 208 * q^73 + 168 * q^77 + 336 * q^83 + 592 * q^91 + 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.02356	−	6.02356i	−0.860509	−	0.860509i	0.130888	−	0.991397i	$-0.458217\pi$
$7$	−6.02356	−	6.02356i	−0.860509	−	0.860509i	−0.991397	+	0.130888i	$0.958217\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.0038			−1.54580			−0.772901	−	0.634526i	$-0.781195\pi$
$11$	−17.0038			−1.54580			−0.772901	+	0.634526i	$0.781195\pi$
$12$		0			0
$13$	−0.124585	+	0.124585i	−0.00958344	+	0.00958344i	−0.711882	−	0.702299i	$-0.752157\pi$
$13$	−0.124585	+	0.124585i	−0.00958344	+	0.00958344i	0.702299	+	0.711882i	$0.252157\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	10.7744	+	10.7744i	0.633788	+	0.633788i	0.949016	−	0.315228i	$-0.102081\pi$
$17$	10.7744	+	10.7744i	0.633788	+	0.633788i	−0.315228	+	0.949016i	$0.602081\pi$
$18$		0			0
$19$			4.04713i			0.213007i	0.994312	+	0.106503i	$0.0339655\pi$
$19$			4.04713i			0.213007i	−0.994312	+	0.106503i	$0.966035\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.85568	−	1.85568i	0.0806817	−	0.0806817i	−0.665614	−	0.746296i	$-0.731830\pi$
$23$	1.85568	−	1.85568i	0.0806817	−	0.0806817i	0.746296	+	0.665614i	$0.231830\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			54.4448i			1.87741i	0.344724	+	0.938704i	$0.387972\pi$
$29$			54.4448i			1.87741i	−0.344724	+	0.938704i	$0.612028\pi$
$30$		0			0
$31$	−53.1995			−1.71611			−0.858056	−	0.513556i	$-0.828328\pi$
$31$	−53.1995			−1.71611			−0.858056	+	0.513556i	$0.828328\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−21.2693	−	21.2693i	−0.574846	−	0.574846i	0.358633	−	0.933479i	$-0.383243\pi$
$37$	−21.2693	−	21.2693i	−0.574846	−	0.574846i	−0.933479	+	0.358633i	$0.883243\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	52.1414			1.27174			0.635871	−	0.771796i	$-0.280641\pi$
$41$	52.1414			1.27174			0.635871	+	0.771796i	$0.280641\pi$
$42$		0			0
$43$	50.4554	−	50.4554i	1.17338	−	1.17338i	0.191984	−	0.981398i	$-0.438508\pi$
$43$	50.4554	−	50.4554i	1.17338	−	1.17338i	0.981398	−	0.191984i	$-0.0614921\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13.4045	+	13.4045i	0.285201	+	0.285201i	0.835179	−	0.549978i	$-0.185364\pi$
$47$	13.4045	+	13.4045i	0.285201	+	0.285201i	−0.549978	+	0.835179i	$0.685364\pi$
$48$		0			0
$49$			23.5667i			0.480952i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.4315	−	12.4315i	0.234556	−	0.234556i	−0.580035	−	0.814591i	$-0.696961\pi$
$53$	12.4315	−	12.4315i	0.234556	−	0.234556i	0.814591	+	0.580035i	$0.196961\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.79413i		−	0.149053i	−0.997219	−	0.0745265i	$-0.976255\pi$
$59$		−	8.79413i		−	0.149053i	0.997219	−	0.0745265i	$-0.0237445\pi$
$60$		0			0
$61$	105.860			1.73542			0.867708	−	0.497074i	$-0.165592\pi$
$61$	105.860			1.73542			0.867708	+	0.497074i	$0.165592\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	46.8595	+	46.8595i	0.699396	+	0.699396i	0.964280	−	0.264885i	$-0.0853338\pi$
$67$	46.8595	+	46.8595i	0.699396	+	0.699396i	−0.264885	+	0.964280i	$0.585334\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	56.8955			0.801346			0.400673	−	0.916221i	$-0.368776\pi$
$71$	56.8955			0.801346			0.400673	+	0.916221i	$0.368776\pi$
$72$		0			0
$73$	52.7296	−	52.7296i	0.722324	−	0.722324i	−0.246754	−	0.969078i	$-0.579364\pi$
$73$	52.7296	−	52.7296i	0.722324	−	0.722324i	0.969078	+	0.246754i	$0.0793640\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	102.424	+	102.424i	1.33018	+	1.33018i
$78$		0			0
$79$			15.2920i			0.193569i	0.995305	+	0.0967846i	$0.0308558\pi$
$79$			15.2920i			0.193569i	−0.995305	+	0.0967846i	$0.969144\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	46.1914	−	46.1914i	0.556523	−	0.556523i	−0.371792	−	0.928316i	$-0.621257\pi$
$83$	46.1914	−	46.1914i	0.556523	−	0.556523i	0.928316	+	0.371792i	$0.121257\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			0.948538i			0.0106577i	0.999986	+	0.00532886i	$0.00169624\pi$
$89$			0.948538i			0.0106577i	−0.999986	+	0.00532886i	$0.998304\pi$
$90$		0			0
$91$	1.50089			0.0164933
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−77.6693	−	77.6693i	−0.800715	−	0.800715i	0.182492	−	0.983207i	$-0.441583\pi$
$97$	−77.6693	−	77.6693i	−0.800715	−	0.800715i	−0.983207	+	0.182492i	$0.941583\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	49.6243			0.491329			0.245665	−	0.969355i	$-0.420994\pi$
$101$	49.6243			0.491329			0.245665	+	0.969355i	$0.420994\pi$
$102$		0			0
$103$	−18.7930	+	18.7930i	−0.182457	+	0.182457i	−0.792425	−	0.609969i	$-0.791182\pi$
$103$	−18.7930	+	18.7930i	−0.182457	+	0.182457i	0.609969	+	0.792425i	$0.291182\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−69.0004	−	69.0004i	−0.644863	−	0.644863i	0.306884	−	0.951747i	$-0.400714\pi$
$107$	−69.0004	−	69.0004i	−0.644863	−	0.644863i	−0.951747	+	0.306884i	$0.900714\pi$
$108$		0			0
$109$		−	130.003i		−	1.19268i	−0.802730	−	0.596342i	$-0.796620\pi$
$109$		−	130.003i		−	1.19268i	0.802730	−	0.596342i	$-0.203380\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−157.585	+	157.585i	−1.39456	+	1.39456i	−0.579799	+	0.814760i	$0.696869\pi$
$113$	−157.585	+	157.585i	−1.39456	+	1.39456i	−0.814760	+	0.579799i	$0.803131\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	129.801i		−	1.09076i
$120$		0			0
$121$	168.130			1.38951
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	106.268	+	106.268i	0.836753	+	0.836753i	0.988430	−	0.151677i	$-0.0484674\pi$
$127$	106.268	+	106.268i	0.836753	+	0.836753i	−0.151677	+	0.988430i	$0.548467\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	102.890			0.785421			0.392711	−	0.919662i	$-0.371537\pi$
$131$	102.890			0.785421			0.392711	+	0.919662i	$0.371537\pi$
$132$		0			0
$133$	24.3781	−	24.3781i	0.183294	−	0.183294i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−84.7079	−	84.7079i	−0.618306	−	0.618306i	0.326791	−	0.945097i	$-0.394033\pi$
$137$	−84.7079	−	84.7079i	−0.618306	−	0.618306i	−0.945097	+	0.326791i	$0.894033\pi$
$138$		0			0
$139$			244.050i			1.75575i	0.478887	+	0.877877i	$0.341040\pi$
$139$			244.050i			1.75575i	−0.478887	+	0.877877i	$0.658960\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.11842	−	2.11842i	0.0148141	−	0.0148141i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			220.759i			1.48161i	0.671722	+	0.740803i	$0.265555\pi$
$149$			220.759i			1.48161i	−0.671722	+	0.740803i	$0.734445\pi$
$150$		0			0
$151$	217.984			1.44360			0.721802	−	0.692099i	$-0.243314\pi$
$151$	217.984			1.44360			0.721802	+	0.692099i	$0.243314\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	185.043	+	185.043i	1.17862	+	1.17862i	0.980096	+	0.198522i	$0.0636141\pi$
$157$	185.043	+	185.043i	1.17862	+	1.17862i	0.198522	+	0.980096i	$0.436386\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.3556			−0.138855
$162$		0			0
$163$	15.5263	−	15.5263i	0.0952533	−	0.0952533i	−0.657874	−	0.753128i	$-0.728544\pi$
$163$	15.5263	−	15.5263i	0.0952533	−	0.0952533i	0.753128	+	0.657874i	$0.228544\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.5448	−	21.5448i	−0.129011	−	0.129011i	0.639653	−	0.768664i	$-0.279078\pi$
$167$	−21.5448	−	21.5448i	−0.129011	−	0.129011i	−0.768664	+	0.639653i	$0.779078\pi$
$168$		0			0
$169$			168.969i			0.999816i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	86.5808	−	86.5808i	0.500467	−	0.500467i	−0.411116	−	0.911583i	$-0.634861\pi$
$173$	86.5808	−	86.5808i	0.500467	−	0.500467i	0.911583	+	0.411116i	$0.134861\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	45.4647i		−	0.253993i	−0.991903	−	0.126996i	$-0.959466\pi$
$179$		−	45.4647i		−	0.253993i	0.991903	−	0.126996i	$-0.0405336\pi$
$180$		0			0
$181$	13.4991			0.0745807			0.0372903	−	0.999304i	$-0.488127\pi$
$181$	13.4991			0.0745807			0.0372903	+	0.999304i	$0.488127\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−183.206	−	183.206i	−0.979711	−	0.979711i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−67.6690			−0.354288			−0.177144	−	0.984185i	$-0.556686\pi$
$191$	−67.6690			−0.354288			−0.177144	+	0.984185i	$0.556686\pi$
$192$		0			0
$193$	−31.9967	+	31.9967i	−0.165786	+	0.165786i	−0.785124	−	0.619338i	$-0.787401\pi$
$193$	−31.9967	+	31.9967i	−0.165786	+	0.165786i	0.619338	+	0.785124i	$0.287401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	122.056	+	122.056i	0.619575	+	0.619575i	0.945422	−	0.325848i	$-0.105650\pi$
$197$	122.056	+	122.056i	0.619575	+	0.619575i	−0.325848	+	0.945422i	$0.605650\pi$
$198$		0			0
$199$			4.36392i			0.0219292i	0.999940	+	0.0109646i	$0.00349022\pi$
$199$			4.36392i			0.0219292i	−0.999940	+	0.0109646i	$0.996510\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	327.952	−	327.952i	1.61553	−	1.61553i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	68.8167i		−	0.329266i
$210$		0			0
$211$	183.096			0.867756			0.433878	−	0.900972i	$-0.357145\pi$
$211$	183.096			0.867756			0.433878	+	0.900972i	$0.357145\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	320.451	+	320.451i	1.47673	+	1.47673i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.68465			−0.0121477
$222$		0			0
$223$	5.03964	−	5.03964i	0.0225993	−	0.0225993i	−0.695717	−	0.718316i	$-0.744913\pi$
$223$	5.03964	−	5.03964i	0.0225993	−	0.0225993i	0.718316	+	0.695717i	$0.244913\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−226.475	−	226.475i	−0.997689	−	0.997689i	0.00230792	−	0.999997i	$-0.499265\pi$
$227$	−226.475	−	226.475i	−0.997689	−	0.997689i	−0.999997	+	0.00230792i	$0.999265\pi$
$228$		0			0
$229$			276.917i			1.20924i	0.796513	+	0.604621i	$0.206676\pi$
$229$			276.917i			1.20924i	−0.796513	+	0.604621i	$0.793324\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	198.106	−	198.106i	0.850242	−	0.850242i	−0.139921	−	0.990163i	$-0.544685\pi$
$233$	198.106	−	198.106i	0.850242	−	0.850242i	0.990163	+	0.139921i	$0.0446847\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			230.183i			0.963111i	0.876416	+	0.481555i	$0.159928\pi$
$239$			230.183i			0.963111i	−0.876416	+	0.481555i	$0.840072\pi$
$240$		0			0
$241$	283.542			1.17652			0.588261	−	0.808671i	$-0.299813\pi$
$241$	283.542			1.17652			0.588261	+	0.808671i	$0.299813\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.504210	−	0.504210i	−0.00204134	−	0.00204134i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−202.731			−0.807694			−0.403847	−	0.914826i	$-0.632327\pi$
$251$	−202.731			−0.807694			−0.403847	+	0.914826i	$0.632327\pi$
$252$		0			0
$253$	−31.5536	+	31.5536i	−0.124718	+	0.124718i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	202.151	+	202.151i	0.786580	+	0.786580i	0.980932	−	0.194352i	$-0.0622604\pi$
$257$	202.151	+	202.151i	0.786580	+	0.786580i	−0.194352	+	0.980932i	$0.562260\pi$
$258$		0			0
$259$			256.234i			0.989320i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−34.3895	+	34.3895i	−0.130759	+	0.130759i	−0.769457	−	0.638699i	$-0.779473\pi$
$263$	−34.3895	+	34.3895i	−0.130759	+	0.130759i	0.638699	+	0.769457i	$0.279473\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	32.5471i		−	0.120993i	−0.998168	−	0.0604964i	$-0.980732\pi$
$269$		−	32.5471i		−	0.120993i	0.998168	−	0.0604964i	$-0.0192684\pi$
$270$		0			0
$271$	−91.9490			−0.339295			−0.169648	−	0.985505i	$-0.554263\pi$
$271$	−91.9490			−0.339295			−0.169648	+	0.985505i	$0.554263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−27.6712	−	27.6712i	−0.0998959	−	0.0998959i	0.655393	−	0.755288i	$-0.272503\pi$
$277$	−27.6712	−	27.6712i	−0.0998959	−	0.0998959i	−0.755288	+	0.655393i	$0.772503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−183.408			−0.652697			−0.326348	−	0.945250i	$-0.605818\pi$
$281$	−183.408			−0.652697			−0.326348	+	0.945250i	$0.605818\pi$
$282$		0			0
$283$	333.158	−	333.158i	1.17724	−	1.17724i	0.196793	−	0.980445i	$-0.436947\pi$
$283$	333.158	−	333.158i	1.17724	−	1.17724i	0.980445	−	0.196793i	$-0.0630527\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−314.077	−	314.077i	−1.09434	−	1.09434i
$288$		0			0
$289$		−	56.8248i		−	0.196626i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−185.884	+	185.884i	−0.634417	+	0.634417i	−0.949173	−	0.314756i	$-0.898077\pi$
$293$	−185.884	+	185.884i	−0.634417	+	0.634417i	0.314756	+	0.949173i	$0.398077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			0.462378i			0.00154642i
$300$		0			0
$301$	−607.843			−2.01941
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	184.175	+	184.175i	0.599920	+	0.599920i	0.940291	−	0.340371i	$-0.110553\pi$
$307$	184.175	+	184.175i	0.599920	+	0.599920i	−0.340371	+	0.940291i	$0.610553\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−191.522			−0.615826			−0.307913	−	0.951415i	$-0.599631\pi$
$311$	−191.522			−0.615826			−0.307913	+	0.951415i	$0.599631\pi$
$312$		0			0
$313$	257.541	−	257.541i	0.822815	−	0.822815i	−0.163696	−	0.986511i	$-0.552342\pi$
$313$	257.541	−	257.541i	0.822815	−	0.822815i	0.986511	+	0.163696i	$0.0523415\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	162.518	+	162.518i	0.512674	+	0.512674i	0.915345	−	0.402671i	$-0.131918\pi$
$317$	162.518	+	162.518i	0.512674	+	0.512674i	−0.402671	+	0.915345i	$0.631918\pi$
$318$		0			0
$319$		−	925.771i		−	2.90210i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−43.6054	+	43.6054i	−0.135001	+	0.135001i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	161.485i		−	0.490837i
$330$		0			0
$331$	49.0405			0.148159			0.0740793	−	0.997252i	$-0.476398\pi$
$331$	49.0405			0.148159			0.0740793	+	0.997252i	$0.476398\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	17.6580	+	17.6580i	0.0523977	+	0.0523977i	0.732820	−	0.680422i	$-0.238204\pi$
$337$	17.6580	+	17.6580i	0.0523977	+	0.0523977i	−0.680422	+	0.732820i	$0.738204\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	904.595			2.65277
$342$		0			0
$343$	−153.199	+	153.199i	−0.446646	+	0.446646i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	296.136	+	296.136i	0.853417	+	0.853417i	0.990552	−	0.137135i	$-0.0437894\pi$
$347$	296.136	+	296.136i	0.853417	+	0.853417i	−0.137135	+	0.990552i	$0.543789\pi$
$348$		0			0
$349$			217.733i			0.623877i	0.950102	+	0.311939i	$0.100978\pi$
$349$			217.733i			0.623877i	−0.950102	+	0.311939i	$0.899022\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	70.0501	−	70.0501i	0.198442	−	0.198442i	−0.600890	−	0.799332i	$-0.705187\pi$
$353$	70.0501	−	70.0501i	0.198442	−	0.198442i	0.799332	+	0.600890i	$0.205187\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	265.093i		−	0.738422i	−0.929346	−	0.369211i	$-0.879628\pi$
$359$		−	265.093i		−	0.738422i	0.929346	−	0.369211i	$-0.120372\pi$
$360$		0			0
$361$	344.621			0.954628
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−473.377	−	473.377i	−1.28986	−	1.28986i	−0.934872	−	0.354984i	$-0.884486\pi$
$367$	−473.377	−	473.377i	−1.28986	−	1.28986i	−0.354984	−	0.934872i	$-0.615514\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−149.764			−0.403675
$372$		0			0
$373$	−201.623	+	201.623i	−0.540544	+	0.540544i	−0.923688	−	0.383144i	$-0.874841\pi$
$373$	−201.623	+	201.623i	−0.540544	+	0.540544i	0.383144	+	0.923688i	$0.374841\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.78299	−	6.78299i	−0.0179920	−	0.0179920i
$378$		0			0
$379$		−	230.312i		−	0.607684i	−0.952722	−	0.303842i	$-0.901730\pi$
$379$		−	230.312i		−	0.607684i	0.952722	−	0.303842i	$-0.0982695\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−312.599	+	312.599i	−0.816185	+	0.816185i	−0.985553	−	0.169368i	$-0.945827\pi$
$383$	−312.599	+	312.599i	−0.816185	+	0.816185i	0.169368	+	0.985553i	$0.445827\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			555.109i			1.42702i	0.700647	+	0.713508i	$0.252895\pi$
$389$			555.109i			1.42702i	−0.700647	+	0.713508i	$0.747105\pi$
$390$		0			0
$391$	39.9876			0.102270
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−221.888	−	221.888i	−0.558911	−	0.558911i	0.370086	−	0.928997i	$-0.379328\pi$
$397$	−221.888	−	221.888i	−0.558911	−	0.558911i	−0.928997	+	0.370086i	$0.879328\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	319.629			0.797079			0.398540	−	0.917151i	$-0.369517\pi$
$401$	319.629			0.797079			0.398540	+	0.917151i	$0.369517\pi$
$402$		0			0
$403$	6.62784	−	6.62784i	0.0164463	−	0.0164463i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	361.659	+	361.659i	0.888598	+	0.888598i
$408$		0			0
$409$		−	40.1118i		−	0.0980730i	−0.998797	−	0.0490365i	$-0.984385\pi$
$409$		−	40.1118i		−	0.0980730i	0.998797	−	0.0490365i	$-0.0156151\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−52.9720	+	52.9720i	−0.128262	+	0.128262i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$			345.933i			0.825616i	0.910818	+	0.412808i	$0.135452\pi$
$419$			345.933i			0.825616i	−0.910818	+	0.412808i	$0.864548\pi$
$420$		0			0
$421$	−105.186			−0.249849			−0.124925	−	0.992166i	$-0.539869\pi$
$421$	−105.186			−0.249849			−0.124925	+	0.992166i	$0.539869\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−637.657	−	637.657i	−1.49334	−	1.49334i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−147.504			−0.342236			−0.171118	−	0.985251i	$-0.554738\pi$
$431$	−147.504			−0.342236			−0.171118	+	0.985251i	$0.554738\pi$
$432$		0			0
$433$	−202.619	+	202.619i	−0.467942	+	0.467942i	−0.901247	−	0.433305i	$-0.857347\pi$
$433$	−202.619	+	202.619i	−0.467942	+	0.467942i	0.433305	+	0.901247i	$0.357347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.51017	+	7.51017i	0.0171857	+	0.0171857i
$438$		0			0
$439$			513.562i			1.16985i	0.811089	+	0.584923i	$0.198875\pi$
$439$			513.562i			1.16985i	−0.811089	+	0.584923i	$0.801125\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−200.016	+	200.016i	−0.451503	+	0.451503i	−0.895853	−	0.444350i	$-0.853435\pi$
$443$	−200.016	+	200.016i	−0.451503	+	0.451503i	0.444350	+	0.895853i	$0.353435\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	387.201i		−	0.862363i	−0.902265	−	0.431182i	$-0.858097\pi$
$449$		−	387.201i		−	0.862363i	0.902265	−	0.431182i	$-0.141903\pi$
$450$		0			0
$451$	−886.603			−1.96586
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	124.992	+	124.992i	0.273505	+	0.273505i	0.830509	−	0.557005i	$-0.188049\pi$
$457$	124.992	+	124.992i	0.273505	+	0.273505i	−0.557005	+	0.830509i	$0.688049\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	420.024			0.911114			0.455557	−	0.890207i	$-0.349440\pi$
$461$	420.024			0.911114			0.455557	+	0.890207i	$0.349440\pi$
$462$		0			0
$463$	−228.184	+	228.184i	−0.492839	+	0.492839i	−0.909200	−	0.416361i	$-0.863305\pi$
$463$	−228.184	+	228.184i	−0.492839	+	0.492839i	0.416361	+	0.909200i	$0.363305\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.4129	+	17.4129i	0.0372867	+	0.0372867i	0.725504	−	0.688218i	$-0.241607\pi$
$467$	17.4129	+	17.4129i	0.0372867	+	0.0372867i	−0.688218	+	0.725504i	$0.741607\pi$
$468$		0			0
$469$		−	564.522i		−	1.20367i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−857.935	+	857.935i	−1.81382	+	1.81382i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	319.260i		−	0.666514i	−0.942836	−	0.333257i	$-0.891852\pi$
$479$		−	319.260i		−	0.666514i	0.942836	−	0.333257i	$-0.108148\pi$
$480$		0			0
$481$	5.29966			0.0110180
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	276.414	+	276.414i	0.567586	+	0.567586i	0.931452	−	0.363865i	$-0.118543\pi$
$487$	276.414	+	276.414i	0.567586	+	0.567586i	−0.363865	+	0.931452i	$0.618543\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−42.5613			−0.0866829			−0.0433414	−	0.999060i	$-0.513800\pi$
$491$	−42.5613			−0.0866829			−0.0433414	+	0.999060i	$0.513800\pi$
$492$		0			0
$493$	−586.610	+	586.610i	−1.18988	+	1.18988i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−342.714	−	342.714i	−0.689565	−	0.689565i
$498$		0			0
$499$		−	669.280i		−	1.34124i	−0.741800	−	0.670621i	$-0.766028\pi$
$499$		−	669.280i		−	1.34124i	0.741800	−	0.670621i	$-0.233972\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.01493	+	3.01493i	−0.00599389	+	0.00599389i	−0.710097	−	0.704103i	$-0.751349\pi$
$503$	−3.01493	+	3.01493i	−0.00599389	+	0.00599389i	0.704103	+	0.710097i	$0.251349\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	506.628i		−	0.995339i	−0.867367	−	0.497670i	$-0.834189\pi$
$509$		−	506.628i		−	0.995339i	0.867367	−	0.497670i	$-0.165811\pi$
$510$		0			0
$511$	−635.241			−1.24313
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−227.927	−	227.927i	−0.440865	−	0.440865i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−399.589			−0.766966			−0.383483	−	0.923548i	$-0.625275\pi$
$521$	−399.589			−0.766966			−0.383483	+	0.923548i	$0.625275\pi$
$522$		0			0
$523$	404.156	−	404.156i	0.772764	−	0.772764i	−0.205824	−	0.978589i	$-0.565988\pi$
$523$	404.156	−	404.156i	0.772764	−	0.772764i	0.978589	+	0.205824i	$0.0659876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−573.192	−	573.192i	−1.08765	−	1.08765i
$528$		0			0
$529$			522.113i			0.986981i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.49602	+	6.49602i	−0.0121877	+	0.0121877i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	400.723i		−	0.743457i
$540$		0			0
$541$	−766.639			−1.41708			−0.708538	−	0.705672i	$-0.750645\pi$
$541$	−766.639			−1.41708			−0.708538	+	0.705672i	$0.750645\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	192.460	+	192.460i	0.351846	+	0.351846i	0.860796	−	0.508950i	$-0.169966\pi$
$547$	192.460	+	192.460i	0.351846	+	0.351846i	−0.508950	+	0.860796i	$0.669966\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−220.345			−0.399901
$552$		0			0
$553$	92.1121	−	92.1121i	0.166568	−	0.166568i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−162.407	−	162.407i	−0.291574	−	0.291574i	0.546128	−	0.837702i	$-0.316101\pi$
$557$	−162.407	−	162.407i	−0.291574	−	0.291574i	−0.837702	+	0.546128i	$0.816101\pi$
$558$		0			0
$559$			12.5719i			0.0224901i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	687.682	−	687.682i	1.22146	−	1.22146i	0.254346	−	0.967113i	$-0.418140\pi$
$563$	687.682	−	687.682i	1.22146	−	1.22146i	0.967113	−	0.254346i	$-0.0818603\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			452.621i			0.795468i	0.917501	+	0.397734i	$0.130203\pi$
$569$			452.621i			0.795468i	−0.917501	+	0.397734i	$0.869797\pi$
$570$		0			0
$571$	8.52498			0.0149299			0.00746496	−	0.999972i	$-0.497624\pi$
$571$	8.52498			0.0149299			0.00746496	+	0.999972i	$0.497624\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−271.812	−	271.812i	−0.471078	−	0.471078i	0.431186	−	0.902263i	$-0.358095\pi$
$577$	−271.812	−	271.812i	−0.471078	−	0.471078i	−0.902263	+	0.431186i	$0.858095\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−556.474			−0.957787
$582$		0			0
$583$	−211.383	+	211.383i	−0.362577	+	0.362577i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	160.042	+	160.042i	0.272644	+	0.272644i	0.830164	−	0.557519i	$-0.188247\pi$
$587$	160.042	+	160.042i	0.272644	+	0.272644i	−0.557519	+	0.830164i	$0.688247\pi$
$588$		0			0
$589$		−	215.305i		−	0.365544i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−25.3238	+	25.3238i	−0.0427045	+	0.0427045i	−0.728137	−	0.685432i	$-0.759613\pi$
$593$	−25.3238	+	25.3238i	−0.0427045	+	0.0427045i	0.685432	+	0.728137i	$0.259613\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1017.30i			1.69832i	0.528132	+	0.849162i	$0.322893\pi$
$599$			1017.30i			1.69832i	−0.528132	+	0.849162i	$0.677107\pi$
$600$		0			0
$601$	−322.485			−0.536580			−0.268290	−	0.963338i	$-0.586459\pi$
$601$	−322.485			−0.536580			−0.268290	+	0.963338i	$0.586459\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	151.189	+	151.189i	0.249076	+	0.249076i	0.820591	−	0.571515i	$-0.193644\pi$
$607$	151.189	+	151.189i	0.249076	+	0.249076i	−0.571515	+	0.820591i	$0.693644\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.33998			−0.00546642
$612$		0			0
$613$	−397.727	+	397.727i	−0.648821	+	0.648821i	−0.952708	−	0.303887i	$-0.901715\pi$
$613$	−397.727	+	397.727i	−0.648821	+	0.648821i	0.303887	+	0.952708i	$0.401715\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	657.460	+	657.460i	1.06557	+	1.06557i	0.997693	+	0.0678812i	$0.0216239\pi$
$617$	657.460	+	657.460i	1.06557	+	1.06557i	0.0678812	+	0.997693i	$0.478376\pi$
$618$		0			0
$619$		−	984.228i		−	1.59003i	−0.606590	−	0.795015i	$-0.707463\pi$
$619$		−	984.228i		−	1.59003i	0.606590	−	0.795015i	$-0.292537\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.71358	−	5.71358i	0.00917107	−	0.00917107i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	458.328i		−	0.728661i
$630$		0			0
$631$	−242.533			−0.384362			−0.192181	−	0.981359i	$-0.561556\pi$
$631$	−242.533			−0.384362			−0.192181	+	0.981359i	$0.561556\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.93604	−	2.93604i	−0.00460917	−	0.00460917i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−761.727			−1.18834			−0.594171	−	0.804339i	$-0.702520\pi$
$641$	−761.727			−1.18834			−0.594171	+	0.804339i	$0.702520\pi$
$642$		0			0
$643$	−11.3113	+	11.3113i	−0.0175914	+	0.0175914i	−0.715848	−	0.698256i	$-0.753960\pi$
$643$	−11.3113	+	11.3113i	−0.0175914	+	0.0175914i	0.698256	+	0.715848i	$0.253960\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−290.292	−	290.292i	−0.448674	−	0.448674i	0.446240	−	0.894914i	$-0.352763\pi$
$647$	−290.292	−	290.292i	−0.448674	−	0.448674i	−0.894914	+	0.446240i	$0.852763\pi$
$648$		0			0
$649$			149.534i			0.230407i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	791.306	−	791.306i	1.21180	−	1.21180i	0.241368	−	0.970434i	$-0.422404\pi$
$653$	791.306	−	791.306i	1.21180	−	1.21180i	0.970434	−	0.241368i	$-0.0775961\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1153.66i		−	1.75062i	−0.483559	−	0.875312i	$-0.660656\pi$
$659$		−	1153.66i		−	1.75062i	0.483559	−	0.875312i	$-0.339344\pi$
$660$		0			0
$661$	1250.76			1.89222			0.946111	−	0.323842i	$-0.104975\pi$
$661$	1250.76			1.89222			0.946111	+	0.323842i	$0.104975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	101.032	+	101.032i	0.151472	+	0.151472i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1800.03			−2.68261
$672$		0			0
$673$	320.198	−	320.198i	0.475778	−	0.475778i	−0.428001	−	0.903778i	$-0.640782\pi$
$673$	320.198	−	320.198i	0.475778	−	0.475778i	0.903778	+	0.428001i	$0.140782\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	105.922	+	105.922i	0.156457	+	0.156457i	0.780995	−	0.624538i	$-0.214713\pi$
$677$	105.922	+	105.922i	0.156457	+	0.156457i	−0.624538	+	0.780995i	$0.714713\pi$
$678$		0			0
$679$			935.692i			1.37804i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	467.606	−	467.606i	0.684636	−	0.684636i	−0.276405	−	0.961041i	$-0.589143\pi$
$683$	467.606	−	467.606i	0.684636	−	0.684636i	0.961041	+	0.276405i	$0.0891432\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.09754i			0.00449571i
$690$		0			0
$691$	47.2980			0.0684487			0.0342243	−	0.999414i	$-0.489104\pi$
$691$	47.2980			0.0684487			0.0342243	+	0.999414i	$0.489104\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	561.792	+	561.792i	0.806014	+	0.806014i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−444.721			−0.634409			−0.317204	−	0.948357i	$-0.602744\pi$
$701$	−444.721			−0.634409			−0.317204	+	0.948357i	$0.602744\pi$
$702$		0			0
$703$	86.0796	−	86.0796i	0.122446	−	0.122446i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−298.915	−	298.915i	−0.422793	−	0.422793i
$708$		0			0
$709$		−	996.549i		−	1.40557i	−0.711403	−	0.702785i	$-0.751940\pi$
$709$		−	996.549i		−	1.40557i	0.711403	−	0.702785i	$-0.248060\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−98.7212	+	98.7212i	−0.138459	+	0.138459i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			319.778i			0.444753i	0.974961	+	0.222377i	$0.0713814\pi$
$719$			319.778i			0.444753i	−0.974961	+	0.222377i	$0.928619\pi$
$720$		0			0
$721$	226.402			0.314011
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	116.716	+	116.716i	0.160545	+	0.160545i	0.782808	−	0.622263i	$-0.213787\pi$
$727$	116.716	+	116.716i	0.160545	+	0.160545i	−0.622263	+	0.782808i	$0.713787\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1087.25			1.48735
$732$		0			0
$733$	121.699	−	121.699i	0.166028	−	0.166028i	−0.619203	−	0.785231i	$-0.712544\pi$
$733$	121.699	−	121.699i	0.166028	−	0.166028i	0.785231	+	0.619203i	$0.212544\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−796.791	−	796.791i	−1.08113	−	1.08113i
$738$		0			0
$739$			498.476i			0.674528i	0.941410	+	0.337264i	$0.109501\pi$
$739$			498.476i			0.674528i	−0.941410	+	0.337264i	$0.890499\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.9317	−	13.9317i	0.0187507	−	0.0187507i	−0.697669	−	0.716420i	$-0.745779\pi$
$743$	13.9317	−	13.9317i	0.0187507	−	0.0187507i	0.716420	+	0.697669i	$0.245779\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			831.257i			1.10982i
$750$		0			0
$751$	−148.692			−0.197992			−0.0989962	−	0.995088i	$-0.531563\pi$
$751$	−148.692			−0.197992			−0.0989962	+	0.995088i	$0.531563\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−243.978	−	243.978i	−0.322296	−	0.322296i	0.527352	−	0.849647i	$-0.323185\pi$
$757$	−243.978	−	243.978i	−0.322296	−	0.322296i	−0.849647	+	0.527352i	$0.823185\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−838.059			−1.10126			−0.550630	−	0.834749i	$-0.685613\pi$
$761$	−838.059			−1.10126			−0.550630	+	0.834749i	$0.685613\pi$
$762$		0			0
$763$	−783.079	+	783.079i	−1.02632	+	1.02632i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.09561	+	1.09561i	0.00142844	+	0.00142844i
$768$		0			0
$769$			48.7943i			0.0634516i	0.999497	+	0.0317258i	$0.0101003\pi$
$769$			48.7943i			0.0634516i	−0.999497	+	0.0317258i	$0.989900\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	349.857	−	349.857i	0.452596	−	0.452596i	−0.443619	−	0.896215i	$-0.646306\pi$
$773$	349.857	−	349.857i	0.452596	−	0.452596i	0.896215	+	0.443619i	$0.146306\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			211.023i			0.270889i
$780$		0			0
$781$	−967.442			−1.23872
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	504.294	+	504.294i	0.640780	+	0.640780i	0.950747	−	0.309967i	$-0.100318\pi$
$787$	504.294	+	504.294i	0.640780	+	0.640780i	−0.309967	+	0.950747i	$0.600318\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1898.45			2.40006
$792$		0			0
$793$	−13.1886	+	13.1886i	−0.0166313	+	0.0166313i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−275.451	−	275.451i	−0.345610	−	0.345610i	0.512862	−	0.858471i	$-0.328585\pi$
$797$	−275.451	−	275.451i	−0.345610	−	0.345610i	−0.858471	+	0.512862i	$0.828585\pi$
$798$		0			0
$799$			288.850i			0.361514i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−896.606	+	896.606i	−1.11657	+	1.11657i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	472.171i		−	0.583648i	−0.956472	−	0.291824i	$-0.905738\pi$
$809$		−	472.171i		−	0.583648i	0.956472	−	0.291824i	$-0.0942621\pi$
$810$		0			0
$811$	1383.85			1.70636			0.853178	−	0.521620i	$-0.174672\pi$
$811$	1383.85			1.70636			0.853178	+	0.521620i	$0.174672\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	204.200	+	204.200i	0.249938	+	0.249938i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	280.480			0.341632			0.170816	−	0.985303i	$-0.445360\pi$
$821$	280.480			0.341632			0.170816	+	0.985303i	$0.445360\pi$
$822$		0			0
$823$	−256.396	+	256.396i	−0.311539	+	0.311539i	−0.845505	−	0.533967i	$-0.820701\pi$
$823$	−256.396	+	256.396i	−0.311539	+	0.311539i	0.533967	+	0.845505i	$0.320701\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	325.679	+	325.679i	0.393808	+	0.393808i	0.876042	−	0.482234i	$-0.160175\pi$
$827$	325.679	+	325.679i	0.393808	+	0.393808i	−0.482234	+	0.876042i	$0.660175\pi$
$828$		0			0
$829$		−	5.87954i		−	0.00709232i	−0.999994	−	0.00354616i	$-0.998871\pi$
$829$		−	5.87954i		−	0.00709232i	0.999994	−	0.00354616i	$-0.00112878\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−253.916	+	253.916i	−0.304822	+	0.304822i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	141.023i		−	0.168085i	−0.996462	−	0.0840424i	$-0.973217\pi$
$839$		−	141.023i		−	0.168085i	0.996462	−	0.0840424i	$-0.0267831\pi$
$840$		0			0
$841$	−2123.24			−2.52466
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1012.74	−	1012.74i	−1.19568	−	1.19568i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−78.9380			−0.0927591
$852$		0			0
$853$	322.940	−	322.940i	0.378593	−	0.378593i	−0.492002	−	0.870594i	$-0.663735\pi$
$853$	322.940	−	322.940i	0.378593	−	0.378593i	0.870594	+	0.492002i	$0.163735\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	277.382	+	277.382i	0.323667	+	0.323667i	0.850172	−	0.526505i	$-0.176498\pi$
$857$	277.382	+	277.382i	0.323667	+	0.323667i	−0.526505	+	0.850172i	$0.676498\pi$
$858$		0			0
$859$			530.257i			0.617295i	0.951176	+	0.308648i	$0.0998764\pi$
$859$			530.257i			0.617295i	−0.951176	+	0.308648i	$0.900124\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−725.484	+	725.484i	−0.840653	+	0.840653i	−0.988944	−	0.148291i	$-0.952623\pi$
$863$	−725.484	+	725.484i	−0.840653	+	0.840653i	0.148291	+	0.988944i	$0.452623\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	260.022i		−	0.299220i
$870$		0			0
$871$	−11.6760			−0.0134052
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−114.761	−	114.761i	−0.130856	−	0.130856i	0.638645	−	0.769501i	$-0.279495\pi$
$877$	−114.761	−	114.761i	−0.130856	−	0.130856i	−0.769501	+	0.638645i	$0.779495\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	582.431			0.661102			0.330551	−	0.943788i	$-0.392765\pi$
$881$	582.431			0.661102			0.330551	+	0.943788i	$0.392765\pi$
$882$		0			0
$883$	−1140.67	+	1140.67i	−1.29182	+	1.29182i	−0.358153	+	0.933663i	$0.616594\pi$
$883$	−1140.67	+	1140.67i	−1.29182	+	1.29182i	−0.933663	+	0.358153i	$0.883406\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−910.472	−	910.472i	−1.02646	−	1.02646i	−0.999640	−	0.0268221i	$-0.991461\pi$
$887$	−910.472	−	910.472i	−1.02646	−	1.02646i	−0.0268221	−	0.999640i	$-0.508539\pi$
$888$		0			0
$889$		−	1280.22i		−	1.44007i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−54.2496	+	54.2496i	−0.0607498	+	0.0607498i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2896.44i		−	3.22184i
$900$		0			0
$901$	267.883			0.297318
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	928.527	+	928.527i	1.02373	+	1.02373i	0.999711	+	0.0240232i	$0.00764757\pi$
$907$	928.527	+	928.527i	1.02373	+	1.02373i	0.0240232	+	0.999711i	$0.492352\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−751.939			−0.825400			−0.412700	−	0.910867i	$-0.635414\pi$
$911$	−751.939			−0.825400			−0.412700	+	0.910867i	$0.635414\pi$
$912$		0			0
$913$	−785.431	+	785.431i	−0.860275	+	0.860275i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−619.766	−	619.766i	−0.675862	−	0.675862i
$918$		0			0
$919$			612.491i			0.666476i	0.942843	+	0.333238i	$0.108141\pi$
$919$			612.491i			0.666476i	−0.942843	+	0.333238i	$0.891859\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.08831	+	7.08831i	−0.00767965	+	0.00767965i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			487.299i			0.524541i	0.964994	+	0.262271i	$0.0844713\pi$
$929$			487.299i			0.524541i	−0.964994	+	0.262271i	$0.915529\pi$
$930$		0			0
$931$	−95.3773			−0.102446
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−943.521	−	943.521i	−1.00696	−	1.00696i	−0.999976	−	0.00698356i	$-0.997777\pi$
$937$	−943.521	−	943.521i	−1.00696	−	1.00696i	−0.00698356	−	0.999976i	$-0.502223\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−243.515			−0.258784			−0.129392	−	0.991594i	$-0.541302\pi$
$941$	−243.515			−0.258784			−0.129392	+	0.991594i	$0.541302\pi$
$942$		0			0
$943$	96.7577	−	96.7577i	0.102606	−	0.102606i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	792.978	+	792.978i	0.837358	+	0.837358i	0.988510	−	0.151153i	$-0.0482985\pi$
$947$	792.978	+	792.978i	0.837358	+	0.837358i	−0.151153	+	0.988510i	$0.548299\pi$
$948$		0			0
$949$			13.1386i			0.0138447i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−77.6634	+	77.6634i	−0.0814936	+	0.0814936i	−0.746679	−	0.665185i	$-0.768353\pi$
$953$	−77.6634	+	77.6634i	−0.0814936	+	0.0814936i	0.665185	+	0.746679i	$0.268353\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1020.49i			1.06412i
$960$		0			0
$961$	1869.19			1.94504
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−294.078	−	294.078i	−0.304113	−	0.304113i	0.538507	−	0.842621i	$-0.318988\pi$
$967$	−294.078	−	294.078i	−0.304113	−	0.304113i	−0.842621	+	0.538507i	$0.818988\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	322.263			0.331888			0.165944	−	0.986135i	$-0.446933\pi$
$971$	322.263			0.331888			0.165944	+	0.986135i	$0.446933\pi$
$972$		0			0
$973$	1470.05	−	1470.05i	1.51084	−	1.51084i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	739.522	+	739.522i	0.756932	+	0.756932i	0.975763	−	0.218831i	$-0.0702243\pi$
$977$	739.522	+	739.522i	0.756932	+	0.756932i	−0.218831	+	0.975763i	$0.570224\pi$
$978$		0			0
$979$		−	16.1288i		−	0.0164747i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−767.218	+	767.218i	−0.780486	+	0.780486i	−0.979913	−	0.199427i	$-0.936092\pi$
$983$	−767.218	+	767.218i	−0.780486	+	0.780486i	0.199427	+	0.979913i	$0.436092\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	187.258i		−	0.189341i
$990$		0			0
$991$	908.418			0.916668			0.458334	−	0.888780i	$-0.348446\pi$
$991$	908.418			0.916668			0.458334	+	0.888780i	$0.348446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	397.062	+	397.062i	0.398257	+	0.398257i	0.877618	−	0.479361i	$-0.159132\pi$
$997$	397.062	+	397.062i	0.398257	+	0.398257i	−0.479361	+	0.877618i	$0.659132\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.t.1657.2		8
3.2	odd	2		600.3.u.h.457.3		8
5.2	odd	4		360.3.v.f.73.1		8
5.3	odd	4	inner	1800.3.v.t.793.2		8
5.4	even	2		360.3.v.f.217.1		8
12.11	even	2		1200.3.bg.q.1057.2		8
15.2	even	4		120.3.u.b.73.2	✓	8
15.8	even	4		600.3.u.h.193.3		8
15.14	odd	2		120.3.u.b.97.2	yes	8
20.7	even	4		720.3.bh.o.433.1		8
20.19	odd	2		720.3.bh.o.577.1		8
60.23	odd	4		1200.3.bg.q.193.2		8
60.47	odd	4		240.3.bg.e.193.4		8
60.59	even	2		240.3.bg.e.97.4		8
120.29	odd	2		960.3.bg.l.577.3		8
120.59	even	2		960.3.bg.k.577.1		8
120.77	even	4		960.3.bg.l.193.3		8
120.107	odd	4		960.3.bg.k.193.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.b.73.2	✓	8	15.2	even	4
120.3.u.b.97.2	yes	8	15.14	odd	2
240.3.bg.e.97.4		8	60.59	even	2
240.3.bg.e.193.4		8	60.47	odd	4
360.3.v.f.73.1		8	5.2	odd	4
360.3.v.f.217.1		8	5.4	even	2
600.3.u.h.193.3		8	15.8	even	4
600.3.u.h.457.3		8	3.2	odd	2
720.3.bh.o.433.1		8	20.7	even	4
720.3.bh.o.577.1		8	20.19	odd	2
960.3.bg.k.193.1		8	120.107	odd	4
960.3.bg.k.577.1		8	120.59	even	2
960.3.bg.l.193.3		8	120.77	even	4
960.3.bg.l.577.3		8	120.29	odd	2
1200.3.bg.q.193.2		8	60.23	odd	4
1200.3.bg.q.1057.2		8	12.11	even	2
1800.3.v.t.793.2		8	5.3	odd	4	inner
1800.3.v.t.1657.2		8	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-6.02356 - 6.02356i) q^{7} +O(q^{10})\)	\(q+(-6.02356 - 6.02356i) q^{7} -17.0038 q^{11} +(-0.124585 + 0.124585i) q^{13} +(10.7744 + 10.7744i) q^{17} +4.04713i q^{19} +(1.85568 - 1.85568i) q^{23} +54.4448i q^{29} -53.1995 q^{31} +(-21.2693 - 21.2693i) q^{37} +52.1414 q^{41} +(50.4554 - 50.4554i) q^{43} +(13.4045 + 13.4045i) q^{47} +23.5667i q^{49} +(12.4315 - 12.4315i) q^{53} -8.79413i q^{59} +105.860 q^{61} +(46.8595 + 46.8595i) q^{67} +56.8955 q^{71} +(52.7296 - 52.7296i) q^{73} +(102.424 + 102.424i) q^{77} +15.2920i q^{79} +(46.1914 - 46.1914i) q^{83} +0.948538i q^{89} +1.50089 q^{91} +(-77.6693 - 77.6693i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} - 32 q^{11} + 4 q^{13} + 52 q^{17} - 40 q^{23} + 96 q^{31} + 60 q^{37} + 152 q^{41} + 88 q^{43} - 16 q^{47} + 108 q^{53} + 264 q^{61} + 216 q^{67} + 240 q^{71} + 208 q^{73} + 168 q^{77} + 336 q^{83} + 592 q^{91} + 208 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 32 * q^11 + 4 * q^13 + 52 * q^17 - 40 * q^23 + 96 * q^31 + 60 * q^37 + 152 * q^41 + 88 * q^43 - 16 * q^47 + 108 * q^53 + 264 * q^61 + 216 * q^67 + 240 * q^71 + 208 * q^73 + 168 * q^77 + 336 * q^83 + 592 * q^91 + 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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