LMFDB - Embedded newform 960.3.c.f.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(449,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 16x^{2} + 81 $ x^4 + 16*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.707107 + 2.91548i$ of defining polynomial
Character	$\chi$	$=$	960.449
Dual form			960.3.c.f.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 2.91548i) q^{3} +(2.82843 + 4.12311i) q^{5} -5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} +O(q^{10})$	$q+(0.707107 - 2.91548i) q^{3} +(2.82843 + 4.12311i) q^{5} -5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} -16.4924i q^{11} +(14.0208 - 5.33074i) q^{15} -11.3137 q^{17} -12.0000 q^{19} +(-17.0000 - 4.12311i) q^{21} +24.0416 q^{23} +(-9.00000 + 23.3238i) q^{25} +(-17.6777 + 20.4083i) q^{27} -32.0000 q^{31} +(-48.0833 - 11.6619i) q^{33} +(24.0416 - 16.4924i) q^{35} +23.3238i q^{37} -57.7235i q^{41} -40.8167i q^{43} +(-5.62742 - 44.6467i) q^{45} -35.3553 q^{47} +15.0000 q^{49} +(-8.00000 + 32.9848i) q^{51} -67.8823 q^{53} +(68.0000 - 46.6476i) q^{55} +(-8.48528 + 34.9857i) q^{57} -16.4924i q^{59} +16.0000 q^{61} +(-24.0416 + 46.6476i) q^{63} -5.83095i q^{67} +(17.0000 - 70.0928i) q^{69} -116.619i q^{73} +(61.6360 + 42.7317i) q^{75} -96.1665 q^{77} -72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} +43.8406 q^{83} +(-32.0000 - 46.6476i) q^{85} -65.9697i q^{89} +(-22.6274 + 93.2952i) q^{93} +(-33.9411 - 49.4773i) q^{95} +163.267i q^{97} +(-68.0000 + 131.939i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{9}+O(q^{10}) $ 4 * q - 32 * q^9	$ 4 q - 32 q^{9} + 8 q^{15} - 48 q^{19} - 68 q^{21} - 36 q^{25} - 128 q^{31} + 68 q^{45} + 60 q^{49} - 32 q^{51} + 272 q^{55} + 64 q^{61} + 68 q^{69} + 272 q^{75} - 288 q^{79} + 188 q^{81} - 128 q^{85} - 272 q^{99}+O(q^{100}) $ 4 * q - 32 * q^9 + 8 * q^15 - 48 * q^19 - 68 * q^21 - 36 * q^25 - 128 * q^31 + 68 * q^45 + 60 * q^49 - 32 * q^51 + 272 * q^55 + 64 * q^61 + 68 * q^69 + 272 * q^75 - 288 * q^79 + 188 * q^81 - 128 * q^85 - 272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0.707107	−	2.91548i	0.235702	−	0.971825i
$3$	0.707107	−	2.91548i	0.235702	−	0.971825i
$4$		0			0
$5$	2.82843	+	4.12311i	0.565685	+	0.824621i
$5$	2.82843	+	4.12311i	0.565685	+	0.824621i
$6$		0			0
$7$		−	5.83095i		−	0.832993i	−0.909137	−	0.416497i	$-0.863258\pi$
$7$		−	5.83095i		−	0.832993i	0.909137	−	0.416497i	$-0.136742\pi$
$8$		0			0
$9$	−8.00000	−	4.12311i	−0.888889	−	0.458123i
$10$		0			0
$11$		−	16.4924i		−	1.49931i	−0.661828	−	0.749656i	$-0.730219\pi$
$11$		−	16.4924i		−	1.49931i	0.661828	−	0.749656i	$-0.269781\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	14.0208	−	5.33074i	0.934721	−	0.355382i
$16$		0			0
$17$	−11.3137			−0.665512			−0.332756	−	0.943013i	$-0.607979\pi$
$17$	−11.3137			−0.665512			−0.332756	+	0.943013i	$0.607979\pi$
$18$		0			0
$19$	−12.0000			−0.631579			−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000			−0.631579			−0.315789	+	0.948829i	$0.602269\pi$
$20$		0			0
$21$	−17.0000	−	4.12311i	−0.809524	−	0.196338i
$22$		0			0
$23$	24.0416			1.04529			0.522644	−	0.852551i	$-0.324946\pi$
$23$	24.0416			1.04529			0.522644	+	0.852551i	$0.324946\pi$
$24$		0			0
$25$	−9.00000	+	23.3238i	−0.360000	+	0.932952i
$26$		0			0
$27$	−17.6777	+	20.4083i	−0.654729	+	0.755864i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−32.0000			−1.03226			−0.516129	−	0.856511i	$-0.672628\pi$
$31$	−32.0000			−1.03226			−0.516129	+	0.856511i	$0.672628\pi$
$32$		0			0
$33$	−48.0833	−	11.6619i	−1.45707	−	0.353391i
$34$		0			0
$35$	24.0416	−	16.4924i	0.686904	−	0.471212i
$36$		0			0
$37$			23.3238i			0.630373i	0.949030	+	0.315187i	$0.102067\pi$
$37$			23.3238i			0.630373i	−0.949030	+	0.315187i	$0.897933\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	57.7235i		−	1.40789i	−0.710255	−	0.703945i	$-0.751420\pi$
$41$		−	57.7235i		−	1.40789i	0.710255	−	0.703945i	$-0.248580\pi$
$42$		0			0
$43$		−	40.8167i		−	0.949225i	−0.880195	−	0.474612i	$-0.842588\pi$
$43$		−	40.8167i		−	0.949225i	0.880195	−	0.474612i	$-0.157412\pi$
$44$		0			0
$45$	−5.62742	−	44.6467i	−0.125054	−	0.992150i
$46$		0			0
$47$	−35.3553			−0.752241			−0.376121	−	0.926571i	$-0.622742\pi$
$47$	−35.3553			−0.752241			−0.376121	+	0.926571i	$0.622742\pi$
$48$		0			0
$49$	15.0000			0.306122
$50$		0			0
$51$	−8.00000	+	32.9848i	−0.156863	+	0.646762i
$52$		0			0
$53$	−67.8823			−1.28080			−0.640399	−	0.768043i	$-0.721231\pi$
$53$	−67.8823			−1.28080			−0.640399	+	0.768043i	$0.721231\pi$
$54$		0			0
$55$	68.0000	−	46.6476i	1.23636	−	0.848138i
$56$		0			0
$57$	−8.48528	+	34.9857i	−0.148865	+	0.613784i
$58$		0			0
$59$		−	16.4924i		−	0.279533i	−0.990185	−	0.139766i	$-0.955365\pi$
$59$		−	16.4924i		−	0.279533i	0.990185	−	0.139766i	$-0.0446351\pi$
$60$		0			0
$61$	16.0000			0.262295			0.131148	−	0.991363i	$-0.458134\pi$
$61$	16.0000			0.262295			0.131148	+	0.991363i	$0.458134\pi$
$62$		0			0
$63$	−24.0416	+	46.6476i	−0.381613	+	0.740438i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	5.83095i		−	0.0870291i	−0.999053	−	0.0435146i	$-0.986145\pi$
$67$		−	5.83095i		−	0.0870291i	0.999053	−	0.0435146i	$-0.0138555\pi$
$68$		0			0
$69$	17.0000	−	70.0928i	0.246377	−	1.01584i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	116.619i		−	1.59752i	−0.601649	−	0.798761i	$-0.705489\pi$
$73$		−	116.619i		−	1.59752i	0.601649	−	0.798761i	$-0.294511\pi$
$74$		0			0
$75$	61.6360	+	42.7317i	0.821814	+	0.569756i
$76$		0			0
$77$	−96.1665			−1.24892
$78$		0			0
$79$	−72.0000			−0.911392			−0.455696	−	0.890135i	$-0.650610\pi$
$79$	−72.0000			−0.911392			−0.455696	+	0.890135i	$0.650610\pi$
$80$		0			0
$81$	47.0000	+	65.9697i	0.580247	+	0.814441i
$82$		0			0
$83$	43.8406			0.528200			0.264100	−	0.964495i	$-0.414925\pi$
$83$	43.8406			0.528200			0.264100	+	0.964495i	$0.414925\pi$
$84$		0			0
$85$	−32.0000	−	46.6476i	−0.376471	−	0.548795i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	65.9697i		−	0.741232i	−0.928786	−	0.370616i	$-0.879147\pi$
$89$		−	65.9697i		−	0.741232i	0.928786	−	0.370616i	$-0.120853\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−22.6274	+	93.2952i	−0.243306	+	1.00317i
$94$		0			0
$95$	−33.9411	−	49.4773i	−0.357275	−	0.520813i
$96$		0			0
$97$			163.267i			1.68316i	0.540131	+	0.841581i	$0.318375\pi$
$97$			163.267i			1.68316i	−0.540131	+	0.841581i	$0.681625\pi$
$98$		0			0
$99$	−68.0000	+	131.939i	−0.686869	+	1.33272i
$100$		0			0
$101$		−	131.939i		−	1.30633i	−0.757215	−	0.653165i	$-0.773441\pi$
$101$		−	131.939i		−	1.30633i	0.757215	−	0.653165i	$-0.226559\pi$
$102$		0			0
$103$			99.1262i			0.962390i	0.876614	+	0.481195i	$0.159797\pi$
$103$			99.1262i			0.962390i	−0.876614	+	0.481195i	$0.840203\pi$
$104$		0			0
$105$	−31.0833	−	81.7547i	−0.296031	−	0.778616i
$106$		0			0
$107$	55.1543			0.515461			0.257731	−	0.966217i	$-0.417025\pi$
$107$	55.1543			0.515461			0.257731	+	0.966217i	$0.417025\pi$
$108$		0			0
$109$	−80.0000			−0.733945			−0.366972	−	0.930232i	$-0.619606\pi$
$109$	−80.0000			−0.733945			−0.366972	+	0.930232i	$0.619606\pi$
$110$		0			0
$111$	68.0000	+	16.4924i	0.612613	+	0.148580i
$112$		0			0
$113$	−152.735			−1.35164			−0.675819	−	0.737068i	$-0.736210\pi$
$113$	−152.735			−1.35164			−0.675819	+	0.737068i	$0.736210\pi$
$114$		0			0
$115$	68.0000	+	99.1262i	0.591304	+	0.861967i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			65.9697i			0.554367i
$120$		0			0
$121$	−151.000			−1.24793
$122$		0			0
$123$	−168.291	−	40.8167i	−1.36822	−	0.331843i
$124$		0			0
$125$	−121.622	+	28.8617i	−0.972979	+	0.230894i
$126$		0			0
$127$		−	40.8167i		−	0.321391i	−0.987004	−	0.160696i	$-0.948626\pi$
$127$		−	40.8167i		−	0.321391i	0.987004	−	0.160696i	$-0.0513737\pi$
$128$		0			0
$129$	−119.000	−	28.8617i	−0.922481	−	0.223734i
$130$		0			0
$131$		−	49.4773i		−	0.377689i	−0.982007	−	0.188845i	$-0.939526\pi$
$131$		−	49.4773i		−	0.377689i	0.982007	−	0.188845i	$-0.0604742\pi$
$132$		0			0
$133$			69.9714i			0.526101i
$134$		0			0
$135$	−134.146	−	15.1634i	−0.993672	−	0.112322i
$136$		0			0
$137$	−50.9117			−0.371618			−0.185809	−	0.982586i	$-0.559491\pi$
$137$	−50.9117			−0.371618			−0.185809	+	0.982586i	$0.559491\pi$
$138$		0			0
$139$	−44.0000			−0.316547			−0.158273	−	0.987395i	$-0.550593\pi$
$139$	−44.0000			−0.316547			−0.158273	+	0.987395i	$0.550593\pi$
$140$		0			0
$141$	−25.0000	+	103.078i	−0.177305	+	0.731047i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.6066	−	43.7321i	0.0721538	−	0.297498i
$148$		0			0
$149$			8.24621i			0.0553437i	0.999617	+	0.0276718i	$0.00880935\pi$
$149$			8.24621i			0.0553437i	−0.999617	+	0.0276718i	$0.991191\pi$
$150$		0			0
$151$	136.000			0.900662			0.450331	−	0.892862i	$-0.351306\pi$
$151$	136.000			0.900662			0.450331	+	0.892862i	$0.351306\pi$
$152$		0			0
$153$	90.5097	+	46.6476i	0.591566	+	0.304886i
$154$		0			0
$155$	−90.5097	−	131.939i	−0.583933	−	0.851222i
$156$		0			0
$157$			116.619i			0.742796i	0.928474	+	0.371398i	$0.121122\pi$
$157$			116.619i			0.742796i	−0.928474	+	0.371398i	$0.878878\pi$
$158$		0			0
$159$	−48.0000	+	197.909i	−0.301887	+	1.24471i
$160$		0			0
$161$		−	140.186i		−	0.870718i
$162$		0			0
$163$		−	99.1262i		−	0.608136i	−0.952650	−	0.304068i	$-0.901655\pi$
$163$		−	99.1262i		−	0.608136i	0.952650	−	0.304068i	$-0.0983450\pi$
$164$		0			0
$165$	−87.9167	−	231.237i	−0.532829	−	1.40144i
$166$		0			0
$167$	292.742			1.75295			0.876474	−	0.481450i	$-0.159890\pi$
$167$	292.742			1.75295			0.876474	+	0.481450i	$0.159890\pi$
$168$		0			0
$169$	169.000			1.00000
$170$		0			0
$171$	96.0000	+	49.4773i	0.561404	+	0.289341i
$172$		0			0
$173$	164.049			0.948259			0.474129	−	0.880455i	$-0.342763\pi$
$173$	164.049			0.948259			0.474129	+	0.880455i	$0.342763\pi$
$174$		0			0
$175$	136.000	+	52.4786i	0.777143	+	0.299878i
$176$		0			0
$177$	−48.0833	−	11.6619i	−0.271657	−	0.0658865i
$178$		0			0
$179$		−	16.4924i		−	0.0921364i	−0.998938	−	0.0460682i	$-0.985331\pi$
$179$		−	16.4924i		−	0.0921364i	0.998938	−	0.0460682i	$-0.0146692\pi$
$180$		0			0
$181$	82.0000			0.453039			0.226519	−	0.974007i	$-0.427265\pi$
$181$	82.0000			0.453039			0.226519	+	0.974007i	$0.427265\pi$
$182$		0			0
$183$	11.3137	−	46.6476i	0.0618235	−	0.254905i
$184$		0			0
$185$	−96.1665	+	65.9697i	−0.519819	+	0.356593i
$186$		0			0
$187$			186.590i			0.997810i
$188$		0			0
$189$	119.000	+	103.078i	0.629630	+	0.545384i
$190$		0			0
$191$		−	296.864i		−	1.55426i	−0.629340	−	0.777130i	$-0.716675\pi$
$191$		−	296.864i		−	1.55426i	0.629340	−	0.777130i	$-0.283325\pi$
$192$		0			0
$193$		−	116.619i		−	0.604244i	−0.953269	−	0.302122i	$-0.902305\pi$
$193$		−	116.619i		−	0.604244i	0.953269	−	0.302122i	$-0.0976950\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	192.333			0.976310			0.488155	−	0.872757i	$-0.337670\pi$
$197$	192.333			0.976310			0.488155	+	0.872757i	$0.337670\pi$
$198$		0			0
$199$	−312.000			−1.56784			−0.783920	−	0.620862i	$-0.786783\pi$
$199$	−312.000			−1.56784			−0.783920	+	0.620862i	$0.786783\pi$
$200$		0			0
$201$	−17.0000	−	4.12311i	−0.0845771	−	0.0205130i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	238.000	−	163.267i	1.16098	−	0.796423i
$206$		0			0
$207$	−192.333	−	99.1262i	−0.929145	−	0.478870i
$208$		0			0
$209$			197.909i			0.946933i
$210$		0			0
$211$	12.0000			0.0568720			0.0284360	−	0.999596i	$-0.490947\pi$
$211$	12.0000			0.0568720			0.0284360	+	0.999596i	$0.490947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	168.291	−	115.447i	0.782751	−	0.536963i
$216$		0			0
$217$			186.590i			0.859864i
$218$		0			0
$219$	−340.000	−	82.4621i	−1.55251	−	0.376539i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			40.8167i			0.183034i	0.995803	+	0.0915172i	$0.0291716\pi$
$223$			40.8167i			0.183034i	−0.995803	+	0.0915172i	$0.970828\pi$
$224$		0			0
$225$	168.167	−	149.483i	0.747407	−	0.664367i
$226$		0			0
$227$	159.806			0.703992			0.351996	−	0.936002i	$-0.385503\pi$
$227$	159.806			0.703992			0.351996	+	0.936002i	$0.385503\pi$
$228$		0			0
$229$	−82.0000			−0.358079			−0.179039	−	0.983842i	$-0.557299\pi$
$229$	−82.0000			−0.358079			−0.179039	+	0.983842i	$0.557299\pi$
$230$		0			0
$231$	−68.0000	+	280.371i	−0.294372	+	1.21373i
$232$		0			0
$233$	192.333			0.825464			0.412732	−	0.910853i	$-0.364575\pi$
$233$	192.333			0.825464			0.412732	+	0.910853i	$0.364575\pi$
$234$		0			0
$235$	−100.000	−	145.774i	−0.425532	−	0.620314i
$236$		0			0
$237$	−50.9117	+	209.914i	−0.214817	+	0.885714i
$238$		0			0
$239$		−	461.788i		−	1.93217i	−0.258231	−	0.966083i	$-0.583139\pi$
$239$		−	461.788i		−	1.93217i	0.258231	−	0.966083i	$-0.416861\pi$
$240$		0			0
$241$	304.000			1.26141			0.630705	−	0.776022i	$-0.282766\pi$
$241$	304.000			1.26141			0.630705	+	0.776022i	$0.282766\pi$
$242$		0			0
$243$	225.567	−	90.3798i	0.928260	−	0.371933i
$244$		0			0
$245$	42.4264	+	61.8466i	0.173169	+	0.252435i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	31.0000	−	127.816i	0.124498	−	0.513318i
$250$		0			0
$251$		−	346.341i		−	1.37984i	−0.723884	−	0.689922i	$-0.757645\pi$
$251$		−	346.341i		−	1.37984i	0.723884	−	0.689922i	$-0.242355\pi$
$252$		0			0
$253$		−	396.505i		−	1.56721i
$254$		0			0
$255$	−158.627	+	60.3104i	−0.622068	+	0.236511i
$256$		0			0
$257$	390.323			1.51877			0.759383	−	0.650644i	$-0.225501\pi$
$257$	390.323			1.51877			0.759383	+	0.650644i	$0.225501\pi$
$258$		0			0
$259$	136.000			0.525097
$260$		0			0
$261$		0			0
$262$		0			0
$263$	295.571			1.12384			0.561921	−	0.827191i	$-0.310062\pi$
$263$	295.571			1.12384			0.561921	+	0.827191i	$0.310062\pi$
$264$		0			0
$265$	−192.000	−	279.886i	−0.724528	−	1.05617i
$266$		0			0
$267$	−192.333	−	46.6476i	−0.720348	−	0.174710i
$268$		0			0
$269$		−	74.2159i		−	0.275896i	−0.990440	−	0.137948i	$-0.955949\pi$
$269$		−	74.2159i		−	0.275896i	0.990440	−	0.137948i	$-0.0440506\pi$
$270$		0			0
$271$	40.0000			0.147601			0.0738007	−	0.997273i	$-0.476487\pi$
$271$	40.0000			0.147601			0.0738007	+	0.997273i	$0.476487\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	384.666	+	148.432i	1.39879	+	0.539752i
$276$		0			0
$277$			443.152i			1.59983i	0.600115	+	0.799914i	$0.295122\pi$
$277$			443.152i			1.59983i	−0.600115	+	0.799914i	$0.704878\pi$
$278$		0			0
$279$	256.000	+	131.939i	0.917563	+	0.472901i
$280$		0			0
$281$			519.511i			1.84879i	0.381431	+	0.924397i	$0.375431\pi$
$281$			519.511i			1.84879i	−0.381431	+	0.924397i	$0.624569\pi$
$282$		0			0
$283$			320.702i			1.13322i	0.823985	+	0.566612i	$0.191746\pi$
$283$			320.702i			1.13322i	−0.823985	+	0.566612i	$0.808254\pi$
$284$		0			0
$285$	−168.250	+	63.9688i	−0.590350	+	0.224452i
$286$		0			0
$287$	−336.583			−1.17276
$288$		0			0
$289$	−161.000			−0.557093
$290$		0			0
$291$	476.000	+	115.447i	1.63574	+	0.396725i
$292$		0			0
$293$	−84.8528			−0.289600			−0.144800	−	0.989461i	$-0.546254\pi$
$293$	−84.8528			−0.289600			−0.144800	+	0.989461i	$0.546254\pi$
$294$		0			0
$295$	68.0000	−	46.6476i	0.230508	−	0.158128i
$296$		0			0
$297$	336.583	+	291.548i	1.13328	+	0.981642i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−238.000			−0.790698
$302$		0			0
$303$	−384.666	−	93.2952i	−1.26953	−	0.307905i
$304$		0			0
$305$	45.2548	+	65.9697i	0.148377	+	0.216294i
$306$		0			0
$307$		−	367.350i		−	1.19658i	−0.801280	−	0.598290i	$-0.795847\pi$
$307$		−	367.350i		−	1.19658i	0.801280	−	0.598290i	$-0.204153\pi$
$308$		0			0
$309$	289.000	+	70.0928i	0.935275	+	0.226838i
$310$		0			0
$311$		−	98.9545i		−	0.318182i	−0.987264	−	0.159091i	$-0.949144\pi$
$311$		−	98.9545i		−	0.318182i	0.987264	−	0.159091i	$-0.0508563\pi$
$312$		0			0
$313$		−	186.590i		−	0.596136i	−0.954545	−	0.298068i	$-0.903658\pi$
$313$		−	186.590i		−	0.596136i	0.954545	−	0.298068i	$-0.0963422\pi$
$314$		0			0
$315$	−260.333	+	32.8132i	−0.826454	+	0.104169i
$316$		0			0
$317$	520.431			1.64174			0.820868	−	0.571117i	$-0.193490\pi$
$317$	520.431			1.64174			0.820868	+	0.571117i	$0.193490\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	39.0000	−	160.801i	0.121495	−	0.500938i
$322$		0			0
$323$	135.765			0.420324
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−56.5685	+	233.238i	−0.172992	+	0.713266i
$328$		0			0
$329$			206.155i			0.626612i
$330$		0			0
$331$	−292.000			−0.882175			−0.441088	−	0.897464i	$-0.645407\pi$
$331$	−292.000			−0.882175			−0.441088	+	0.897464i	$0.645407\pi$
$332$		0			0
$333$	96.1665	−	186.590i	0.288788	−	0.560332i
$334$		0			0
$335$	24.0416	−	16.4924i	0.0717661	−	0.0492311i
$336$		0			0
$337$			326.533i			0.968942i	0.874807	+	0.484471i	$0.160988\pi$
$337$			326.533i			0.968942i	−0.874807	+	0.484471i	$0.839012\pi$
$338$		0			0
$339$	−108.000	+	445.295i	−0.318584	+	1.31356i
$340$		0			0
$341$			527.758i			1.54768i
$342$		0			0
$343$		−	373.181i		−	1.08799i
$344$		0			0
$345$	337.083	−	128.160i	0.977053	−	0.371477i
$346$		0			0
$347$	−394.566			−1.13708			−0.568538	−	0.822657i	$-0.692491\pi$
$347$	−394.566			−1.13708			−0.568538	+	0.822657i	$0.692491\pi$
$348$		0			0
$349$	−254.000			−0.727794			−0.363897	−	0.931439i	$-0.618554\pi$
$349$	−254.000			−0.727794			−0.363897	+	0.931439i	$0.618554\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	345.068			0.977530			0.488765	−	0.872415i	$-0.337448\pi$
$353$	345.068			0.977530			0.488765	+	0.872415i	$0.337448\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	192.333	+	46.6476i	0.538748	+	0.130666i
$358$		0			0
$359$			395.818i			1.10256i	0.834321	+	0.551279i	$0.185860\pi$
$359$			395.818i			1.10256i	−0.834321	+	0.551279i	$0.814140\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$		0			0
$363$	−106.773	+	440.237i	−0.294141	+	1.21277i
$364$		0			0
$365$	480.833	−	329.848i	1.31735	−	0.903694i
$366$		0			0
$367$			413.998i			1.12806i	0.825755	+	0.564029i	$0.190750\pi$
$367$			413.998i			1.12806i	−0.825755	+	0.564029i	$0.809250\pi$
$368$		0			0
$369$	−238.000	+	461.788i	−0.644986	+	1.25146i
$370$		0			0
$371$			395.818i			1.06690i
$372$		0			0
$373$		−	629.743i		−	1.68832i	−0.536092	−	0.844159i	$-0.680100\pi$
$373$		−	629.743i		−	1.68832i	0.536092	−	0.844159i	$-0.319900\pi$
$374$		0			0
$375$	−1.85429	+	374.995i	−0.00494478	+	0.999988i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	572.000			1.50923			0.754617	−	0.656165i	$-0.227822\pi$
$379$	572.000			1.50923			0.754617	+	0.656165i	$0.227822\pi$
$380$		0			0
$381$	−119.000	−	28.8617i	−0.312336	−	0.0757526i
$382$		0			0
$383$	−193.747			−0.505868			−0.252934	−	0.967484i	$-0.581395\pi$
$383$	−193.747			−0.505868			−0.252934	+	0.967484i	$0.581395\pi$
$384$		0			0
$385$	−272.000	−	396.505i	−0.706494	−	1.02988i
$386$		0			0
$387$	−168.291	+	326.533i	−0.434862	+	0.843755i
$388$		0			0
$389$			387.572i			0.996329i	0.867083	+	0.498164i	$0.165992\pi$
$389$			387.572i			0.996329i	−0.867083	+	0.498164i	$0.834008\pi$
$390$		0			0
$391$	−272.000			−0.695652
$392$		0			0
$393$	−144.250	−	34.9857i	−0.367048	−	0.0890222i
$394$		0			0
$395$	−203.647	−	296.864i	−0.515561	−	0.751553i
$396$		0			0
$397$			513.124i			1.29250i	0.763124	+	0.646252i	$0.223664\pi$
$397$			513.124i			1.29250i	−0.763124	+	0.646252i	$0.776336\pi$
$398$		0			0
$399$	204.000	+	49.4773i	0.511278	+	0.124003i
$400$		0			0
$401$		−	65.9697i		−	0.164513i	−0.996611	−	0.0822565i	$-0.973787\pi$
$401$		−	65.9697i		−	0.164513i	0.996611	−	0.0822565i	$-0.0262127\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−139.064	+	380.376i	−0.343368	+	0.939201i
$406$		0			0
$407$	384.666			0.945126
$408$		0			0
$409$	640.000			1.56479			0.782396	−	0.622781i	$-0.213997\pi$
$409$	640.000			1.56479			0.782396	+	0.622781i	$0.213997\pi$
$410$		0			0
$411$	−36.0000	+	148.432i	−0.0875912	+	0.361148i
$412$		0			0
$413$	−96.1665			−0.232849
$414$		0			0
$415$	124.000	+	180.760i	0.298795	+	0.435565i
$416$		0			0
$417$	−31.1127	+	128.281i	−0.0746108	+	0.307628i
$418$		0			0
$419$		−	577.235i		−	1.37765i	−0.724928	−	0.688824i	$-0.758127\pi$
$419$		−	577.235i		−	1.37765i	0.724928	−	0.688824i	$-0.241873\pi$
$420$		0			0
$421$	656.000			1.55819			0.779097	−	0.626903i	$-0.215678\pi$
$421$	656.000			1.55819			0.779097	+	0.626903i	$0.215678\pi$
$422$		0			0
$423$	282.843	+	145.774i	0.668659	+	0.344619i
$424$		0			0
$425$	101.823	−	263.879i	0.239584	−	0.620891i
$426$		0			0
$427$		−	93.2952i		−	0.218490i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			362.833i			0.841841i	0.907098	+	0.420920i	$0.138293\pi$
$431$			362.833i			0.841841i	−0.907098	+	0.420920i	$0.861707\pi$
$432$		0			0
$433$			163.267i			0.377059i	0.982068	+	0.188530i	$0.0603722\pi$
$433$			163.267i			0.377059i	−0.982068	+	0.188530i	$0.939628\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−288.500			−0.660182
$438$		0			0
$439$	432.000			0.984055			0.492027	−	0.870580i	$-0.336256\pi$
$439$	432.000			0.984055			0.492027	+	0.870580i	$0.336256\pi$
$440$		0			0
$441$	−120.000	−	61.8466i	−0.272109	−	0.140242i
$442$		0			0
$443$	−123.037			−0.277735			−0.138867	−	0.990311i	$-0.544346\pi$
$443$	−123.037			−0.277735			−0.138867	+	0.990311i	$0.544346\pi$
$444$		0			0
$445$	272.000	−	186.590i	0.611236	−	0.419304i
$446$		0			0
$447$	24.0416	+	5.83095i	0.0537844	+	0.0130446i
$448$		0			0
$449$		−	865.852i		−	1.92840i	−0.265174	−	0.964201i	$-0.585429\pi$
$449$		−	865.852i		−	1.92840i	0.265174	−	0.964201i	$-0.414571\pi$
$450$		0			0
$451$	−952.000			−2.11086
$452$		0			0
$453$	96.1665	−	396.505i	0.212288	−	0.875286i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			466.476i			1.02074i	0.859956	+	0.510368i	$0.170491\pi$
$457$			466.476i			1.02074i	−0.859956	+	0.510368i	$0.829509\pi$
$458$		0			0
$459$	200.000	−	230.894i	0.435730	−	0.503037i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		−	612.250i		−	1.32235i	−0.750230	−	0.661177i	$-0.770057\pi$
$463$		−	612.250i		−	1.32235i	0.750230	−	0.661177i	$-0.229943\pi$
$464$		0			0
$465$	−448.666	+	170.584i	−0.964873	+	0.366846i
$466$		0			0
$467$	767.918			1.64436			0.822182	−	0.569225i	$-0.192757\pi$
$467$	767.918			1.64436			0.822182	+	0.569225i	$0.192757\pi$
$468$		0			0
$469$	−34.0000			−0.0724947
$470$		0			0
$471$	340.000	+	82.4621i	0.721868	+	0.175079i
$472$		0			0
$473$	−673.166			−1.42318
$474$		0			0
$475$	108.000	−	279.886i	0.227368	−	0.589233i
$476$		0			0
$477$	543.058	+	279.886i	1.13849	+	0.586762i
$478$		0			0
$479$		−	560.742i		−	1.17065i	−0.810798	−	0.585326i	$-0.800967\pi$
$479$		−	560.742i		−	1.17065i	0.810798	−	0.585326i	$-0.199033\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−408.708	−	99.1262i	−0.846186	−	0.205230i
$484$		0			0
$485$	−673.166	+	461.788i	−1.38797	+	0.952140i
$486$		0			0
$487$		−	647.236i		−	1.32903i	−0.747277	−	0.664513i	$-0.768639\pi$
$487$		−	647.236i		−	1.32903i	0.747277	−	0.664513i	$-0.231361\pi$
$488$		0			0
$489$	−289.000	−	70.0928i	−0.591002	−	0.143339i
$490$		0			0
$491$		−	346.341i		−	0.705379i	−0.935740	−	0.352689i	$-0.885267\pi$
$491$		−	346.341i		−	0.705379i	0.935740	−	0.352689i	$-0.114733\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−736.333	+	92.8097i	−1.48754	+	0.187494i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	660.000			1.32265			0.661323	−	0.750102i	$-0.269995\pi$
$499$	660.000			1.32265			0.661323	+	0.750102i	$0.269995\pi$
$500$		0			0
$501$	207.000	−	853.483i	0.413174	−	1.70356i
$502$		0			0
$503$	−182.434			−0.362691			−0.181345	−	0.983419i	$-0.558045\pi$
$503$	−182.434			−0.362691			−0.181345	+	0.983419i	$0.558045\pi$
$504$		0			0
$505$	544.000	−	373.181i	1.07723	−	0.738972i
$506$		0			0
$507$	119.501	−	492.715i	0.235702	−	0.971825i
$508$		0			0
$509$		−	395.818i		−	0.777639i	−0.921314	−	0.388819i	$-0.872883\pi$
$509$		−	395.818i		−	0.777639i	0.921314	−	0.388819i	$-0.127117\pi$
$510$		0			0
$511$	−680.000			−1.33072
$512$		0			0
$513$	212.132	−	244.900i	0.413513	−	0.477388i
$514$		0			0
$515$	−408.708	+	280.371i	−0.793607	+	0.544410i
$516$		0			0
$517$			583.095i			1.12784i
$518$		0			0
$519$	116.000	−	478.280i	0.223507	−	0.921542i
$520$		0			0
$521$			131.939i			0.253243i	0.991951	+	0.126621i	$0.0404133\pi$
$521$			131.939i			0.253243i	−0.991951	+	0.126621i	$0.959587\pi$
$522$		0			0
$523$			145.774i			0.278726i	0.990241	+	0.139363i	$0.0445055\pi$
$523$			145.774i			0.278726i	−0.990241	+	0.139363i	$0.955494\pi$
$524$		0			0
$525$	249.167	−	359.397i	0.474603	−	0.684565i
$526$		0			0
$527$	362.039			0.686980
$528$		0			0
$529$	49.0000			0.0926276
$530$		0			0
$531$	−68.0000	+	131.939i	−0.128060	+	0.248473i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	156.000	+	227.407i	0.291589	+	0.425060i
$536$		0			0
$537$	−48.0833	−	11.6619i	−0.0895405	−	0.0217168i
$538$		0			0
$539$		−	247.386i		−	0.458973i
$540$		0			0
$541$	−418.000			−0.772643			−0.386322	−	0.922364i	$-0.626255\pi$
$541$	−418.000			−0.772643			−0.386322	+	0.922364i	$0.626255\pi$
$542$		0			0
$543$	57.9828	−	239.069i	0.106782	−	0.440274i
$544$		0			0
$545$	−226.274	−	329.848i	−0.415182	−	0.605227i
$546$		0			0
$547$			285.717i			0.522334i	0.965294	+	0.261167i	$0.0841073\pi$
$547$			285.717i			0.522334i	−0.965294	+	0.261167i	$0.915893\pi$
$548$		0			0
$549$	−128.000	−	65.9697i	−0.233151	−	0.120163i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			419.829i			0.759184i
$554$		0			0
$555$	124.333	+	327.019i	0.224024	+	0.589223i
$556$		0			0
$557$	−424.264			−0.761695			−0.380847	−	0.924638i	$-0.624368\pi$
$557$	−424.264			−0.761695			−0.380847	+	0.924638i	$0.624368\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	544.000	+	131.939i	0.969697	+	0.235186i
$562$		0			0
$563$	813.173			1.44436			0.722178	−	0.691707i	$-0.243141\pi$
$563$	813.173			1.44436			0.722178	+	0.691707i	$0.243141\pi$
$564$		0			0
$565$	−432.000	−	629.743i	−0.764602	−	1.11459i
$566$		0			0
$567$	384.666	−	274.055i	0.678423	−	0.483342i
$568$		0			0
$569$			453.542i			0.797085i	0.917150	+	0.398543i	$0.130484\pi$
$569$			453.542i			0.797085i	−0.917150	+	0.398543i	$0.869516\pi$
$570$		0			0
$571$	−220.000			−0.385289			−0.192644	−	0.981269i	$-0.561706\pi$
$571$	−220.000			−0.385289			−0.192644	+	0.981269i	$0.561706\pi$
$572$		0			0
$573$	−865.499	−	209.914i	−1.51047	−	0.366343i
$574$		0			0
$575$	−216.375	+	560.742i	−0.376304	+	0.975204i
$576$		0			0
$577$			46.6476i			0.0808451i	0.999183	+	0.0404225i	$0.0128704\pi$
$577$			46.6476i			0.0808451i	−0.999183	+	0.0404225i	$0.987130\pi$
$578$		0			0
$579$	−340.000	−	82.4621i	−0.587219	−	0.142422i
$580$		0			0
$581$		−	255.633i		−	0.439987i
$582$		0			0
$583$			1119.54i			1.92031i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−55.1543			−0.0939597			−0.0469798	−	0.998896i	$-0.514960\pi$
$587$	−55.1543			−0.0939597			−0.0469798	+	0.998896i	$0.514960\pi$
$588$		0			0
$589$	384.000			0.651952
$590$		0			0
$591$	136.000	−	560.742i	0.230118	−	0.948803i
$592$		0			0
$593$	390.323			0.658217			0.329109	−	0.944292i	$-0.393252\pi$
$593$	390.323			0.658217			0.329109	+	0.944292i	$0.393252\pi$
$594$		0			0
$595$	−272.000	+	186.590i	−0.457143	+	0.313597i
$596$		0			0
$597$	−220.617	+	909.628i	−0.369543	+	1.52367i
$598$		0			0
$599$		−	98.9545i		−	0.165200i	−0.996583	−	0.0825998i	$-0.973678\pi$
$599$		−	98.9545i		−	0.165200i	0.996583	−	0.0825998i	$-0.0263223\pi$
$600$		0			0
$601$	−880.000			−1.46423			−0.732113	−	0.681183i	$-0.761466\pi$
$601$	−880.000			−1.46423			−0.732113	+	0.681183i	$0.761466\pi$
$602$		0			0
$603$	−24.0416	+	46.6476i	−0.0398700	+	0.0773592i
$604$		0			0
$605$	−427.092	−	622.589i	−0.705938	−	1.02907i
$606$		0			0
$607$			425.659i			0.701251i	0.936516	+	0.350626i	$0.114031\pi$
$607$			425.659i			0.701251i	−0.936516	+	0.350626i	$0.885969\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			606.419i			0.989264i	0.869102	+	0.494632i	$0.164697\pi$
$613$			606.419i			0.989264i	−0.869102	+	0.494632i	$0.835303\pi$
$614$		0			0
$615$	−307.709	−	809.330i	−0.500339	−	1.31598i
$616$		0			0
$617$	−113.137			−0.183366			−0.0916832	−	0.995788i	$-0.529225\pi$
$617$	−113.137			−0.183366			−0.0916832	+	0.995788i	$0.529225\pi$
$618$		0			0
$619$	−52.0000			−0.0840065			−0.0420032	−	0.999117i	$-0.513374\pi$
$619$	−52.0000			−0.0840065			−0.0420032	+	0.999117i	$0.513374\pi$
$620$		0			0
$621$	−425.000	+	490.650i	−0.684380	+	0.790096i
$622$		0			0
$623$	−384.666			−0.617442
$624$		0			0
$625$	−463.000	−	419.829i	−0.740800	−	0.671726i
$626$		0			0
$627$	576.999	+	139.943i	0.920254	+	0.223194i
$628$		0			0
$629$		−	263.879i		−	0.419521i
$630$		0			0
$631$	−544.000			−0.862124			−0.431062	−	0.902322i	$-0.641861\pi$
$631$	−544.000			−0.862124			−0.431062	+	0.902322i	$0.641861\pi$
$632$		0			0
$633$	8.48528	−	34.9857i	0.0134049	−	0.0552697i
$634$		0			0
$635$	168.291	−	115.447i	0.265026	−	0.181806i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	849.360i		−	1.32505i	−0.749038	−	0.662527i	$-0.769484\pi$
$641$		−	849.360i		−	1.32505i	0.749038	−	0.662527i	$-0.230516\pi$
$642$		0			0
$643$		−	367.350i		−	0.571306i	−0.958333	−	0.285653i	$-0.907789\pi$
$643$		−	367.350i		−	0.571306i	0.958333	−	0.285653i	$-0.0922105\pi$
$644$		0			0
$645$	−217.583	−	572.283i	−0.337338	−	0.887260i
$646$		0			0
$647$	−971.565			−1.50165			−0.750823	−	0.660504i	$-0.770343\pi$
$647$	−971.565			−1.50165			−0.750823	+	0.660504i	$0.770343\pi$
$648$		0			0
$649$	−272.000			−0.419106
$650$		0			0
$651$	544.000	+	131.939i	0.835637	+	0.202672i
$652$		0			0
$653$	−350.725			−0.537098			−0.268549	−	0.963266i	$-0.586544\pi$
$653$	−350.725			−0.537098			−0.268549	+	0.963266i	$0.586544\pi$
$654$		0			0
$655$	204.000	−	139.943i	0.311450	−	0.213653i
$656$		0			0
$657$	−480.833	+	932.952i	−0.731861	+	1.42002i
$658$		0			0
$659$			577.235i			0.875925i	0.898993	+	0.437963i	$0.144300\pi$
$659$			577.235i			0.875925i	−0.898993	+	0.437963i	$0.855700\pi$
$660$		0			0
$661$	−80.0000			−0.121029			−0.0605144	−	0.998167i	$-0.519274\pi$
$661$	−80.0000			−0.121029			−0.0605144	+	0.998167i	$0.519274\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−288.500	+	197.909i	−0.433834	+	0.297608i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	119.000	+	28.8617i	0.177877	+	0.0431416i
$670$		0			0
$671$		−	263.879i		−	0.393262i
$672$		0			0
$673$		−	489.800i		−	0.727786i	−0.931441	−	0.363893i	$-0.881447\pi$
$673$		−	489.800i		−	0.727786i	0.931441	−	0.363893i	$-0.118553\pi$
$674$		0			0
$675$	−316.901	−	595.986i	−0.469483	−	0.882942i
$676$		0			0
$677$	−192.333			−0.284096			−0.142048	−	0.989860i	$-0.545369\pi$
$677$	−192.333			−0.284096			−0.142048	+	0.989860i	$0.545369\pi$
$678$		0			0
$679$	952.000			1.40206
$680$		0			0
$681$	113.000	−	465.911i	0.165932	−	0.684157i
$682$		0			0
$683$	−236.174			−0.345789			−0.172894	−	0.984940i	$-0.555312\pi$
$683$	−236.174			−0.345789			−0.172894	+	0.984940i	$0.555312\pi$
$684$		0			0
$685$	−144.000	−	209.914i	−0.210219	−	0.306444i
$686$		0			0
$687$	−57.9828	+	239.069i	−0.0843999	+	0.347990i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	548.000			0.793054			0.396527	−	0.918023i	$-0.370215\pi$
$691$	548.000			0.793054			0.396527	+	0.918023i	$0.370215\pi$
$692$		0			0
$693$	769.332	+	396.505i	1.11015	+	0.572157i
$694$		0			0
$695$	−124.451	−	181.417i	−0.179066	−	0.261031i
$696$		0			0
$697$			653.067i			0.936968i
$698$		0			0
$699$	136.000	−	560.742i	0.194564	−	0.802207i
$700$		0			0
$701$			57.7235i			0.0823445i	0.999152	+	0.0411722i	$0.0131092\pi$
$701$			57.7235i			0.0823445i	−0.999152	+	0.0411722i	$0.986891\pi$
$702$		0			0
$703$		−	279.886i		−	0.398130i
$704$		0			0
$705$	−495.711	+	188.470i	−0.703136	+	0.267333i
$706$		0			0
$707$	−769.332			−1.08816
$708$		0			0
$709$	−1230.00			−1.73484			−0.867419	−	0.497579i	$-0.834223\pi$
$709$	−1230.00			−1.73484			−0.867419	+	0.497579i	$0.834223\pi$
$710$		0			0
$711$	576.000	+	296.864i	0.810127	+	0.417530i
$712$		0			0
$713$	−769.332			−1.07901
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−1346.33	−	326.533i	−1.87773	−	0.455416i
$718$		0			0
$719$			626.712i			0.871644i	0.900033	+	0.435822i	$0.143542\pi$
$719$			626.712i			0.871644i	−0.900033	+	0.435822i	$0.856458\pi$
$720$		0			0
$721$	578.000			0.801664
$722$		0			0
$723$	214.960	−	886.305i	0.297317	−	1.22587i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	367.350i		−	0.505296i	−0.967558	−	0.252648i	$-0.918699\pi$
$727$		−	367.350i		−	0.505296i	0.967558	−	0.252648i	$-0.0813014\pi$
$728$		0			0
$729$	−104.000	−	721.543i	−0.142661	−	0.989772i
$730$		0			0
$731$			461.788i			0.631721i
$732$		0			0
$733$			1002.92i			1.36825i	0.729367	+	0.684123i	$0.239815\pi$
$733$			1002.92i			1.36825i	−0.729367	+	0.684123i	$0.760185\pi$
$734$		0			0
$735$	210.312	−	79.9610i	0.286139	−	0.108791i
$736$		0			0
$737$	−96.1665			−0.130484
$738$		0			0
$739$	340.000			0.460081			0.230041	−	0.973181i	$-0.426114\pi$
$739$	340.000			0.460081			0.230041	+	0.973181i	$0.426114\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1243.09			1.67307			0.836537	−	0.547911i	$-0.184577\pi$
$743$	1243.09			1.67307			0.836537	+	0.547911i	$0.184577\pi$
$744$		0			0
$745$	−34.0000	+	23.3238i	−0.0456376	+	0.0313071i
$746$		0			0
$747$	−350.725	−	180.760i	−0.469511	−	0.241981i
$748$		0			0
$749$		−	321.602i		−	0.429375i
$750$		0			0
$751$	−520.000			−0.692410			−0.346205	−	0.938159i	$-0.612530\pi$
$751$	−520.000			−0.692410			−0.346205	+	0.938159i	$0.612530\pi$
$752$		0			0
$753$	−1009.75	−	244.900i	−1.34097	−	0.325232i
$754$		0			0
$755$	384.666	+	560.742i	0.509492	+	0.742705i
$756$		0			0
$757$		−	816.333i		−	1.07838i	−0.842184	−	0.539190i	$-0.818731\pi$
$757$		−	816.333i		−	1.07838i	0.842184	−	0.539190i	$-0.181269\pi$
$758$		0			0
$759$	−1156.00	−	280.371i	−1.52306	−	0.369395i
$760$		0			0
$761$			395.818i			0.520129i	0.965591	+	0.260064i	$0.0837438\pi$
$761$			395.818i			0.520129i	−0.965591	+	0.260064i	$0.916256\pi$
$762$		0			0
$763$			466.476i			0.611371i
$764$		0			0
$765$	63.6670	+	505.120i	0.0832248	+	0.660288i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−306.000			−0.397919			−0.198960	−	0.980008i	$-0.563756\pi$
$769$	−306.000			−0.397919			−0.198960	+	0.980008i	$0.563756\pi$
$770$		0			0
$771$	276.000	−	1137.98i	0.357977	−	1.47598i
$772$		0			0
$773$	−305.470			−0.395175			−0.197587	−	0.980285i	$-0.563311\pi$
$773$	−305.470			−0.395175			−0.197587	+	0.980285i	$0.563311\pi$
$774$		0			0
$775$	288.000	−	746.362i	0.371613	−	0.963048i
$776$		0			0
$777$	96.1665	−	396.505i	0.123766	−	0.510302i
$778$		0			0
$779$			692.682i			0.889194i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−480.833	+	329.848i	−0.612526	+	0.420189i
$786$		0			0
$787$			565.602i			0.718681i	0.933206	+	0.359341i	$0.116998\pi$
$787$			565.602i			0.718681i	−0.933206	+	0.359341i	$0.883002\pi$
$788$		0			0
$789$	209.000	−	861.729i	0.264892	−	1.09218i
$790$		0			0
$791$			890.591i			1.12590i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−951.765	+	361.862i	−1.19719	+	0.455173i
$796$		0			0
$797$	1063.49			1.33436			0.667182	−	0.744894i	$-0.267500\pi$
$797$	1063.49			1.33436			0.667182	+	0.744894i	$0.267500\pi$
$798$		0			0
$799$	400.000			0.500626
$800$		0			0
$801$	−272.000	+	527.758i	−0.339576	+	0.658873i
$802$		0			0
$803$	−1923.33			−2.39518
$804$		0			0
$805$	578.000	−	396.505i	0.718012	−	0.492552i
$806$		0			0
$807$	−216.375	−	52.4786i	−0.268122	−	0.0650292i
$808$		0			0
$809$		−	791.636i		−	0.978537i	−0.872133	−	0.489268i	$-0.837264\pi$
$809$		−	791.636i		−	0.978537i	0.872133	−	0.489268i	$-0.162736\pi$
$810$		0			0
$811$	−436.000			−0.537608			−0.268804	−	0.963195i	$-0.586628\pi$
$811$	−436.000			−0.537608			−0.268804	+	0.963195i	$0.586628\pi$
$812$		0			0
$813$	28.2843	−	116.619i	0.0347900	−	0.143443i
$814$		0			0
$815$	408.708	−	280.371i	0.501482	−	0.344014i
$816$		0			0
$817$			489.800i			0.599510i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	799.882i		−	0.974278i	−0.873324	−	0.487139i	$-0.838040\pi$
$821$		−	799.882i		−	0.974278i	0.873324	−	0.487139i	$-0.161960\pi$
$822$		0			0
$823$			1311.96i			1.59412i	0.603897	+	0.797062i	$0.293614\pi$
$823$			1311.96i			1.59412i	−0.603897	+	0.797062i	$0.706386\pi$
$824$		0			0
$825$	704.749	−	1016.53i	0.854242	−	1.23215i
$826$		0			0
$827$	−985.707			−1.19191			−0.595953	−	0.803019i	$-0.703226\pi$
$827$	−985.707			−1.19191			−0.595953	+	0.803019i	$0.703226\pi$
$828$		0			0
$829$	1488.00			1.79493			0.897467	−	0.441082i	$-0.145405\pi$
$829$	1488.00			1.79493			0.897467	+	0.441082i	$0.145405\pi$
$830$		0			0
$831$	1292.00	+	313.356i	1.55475	+	0.377083i
$832$		0			0
$833$	−169.706			−0.203728
$834$		0			0
$835$	828.000	+	1207.01i	0.991617	+	1.44552i
$836$		0			0
$837$	565.685	−	653.067i	0.675849	−	0.780247i
$838$		0			0
$839$			461.788i			0.550403i	0.961387	+	0.275201i	$0.0887445\pi$
$839$			461.788i			0.550403i	−0.961387	+	0.275201i	$0.911255\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	1514.62	+	367.350i	1.79671	+	0.435765i
$844$		0			0
$845$	478.004	+	696.805i	0.565685	+	0.824621i
$846$		0			0
$847$			880.474i			1.03952i
$848$		0			0
$849$	935.000	+	226.771i	1.10130	+	0.267103i
$850$		0			0
$851$			560.742i			0.658922i
$852$		0			0
$853$		−	326.533i		−	0.382806i	−0.981512	−	0.191403i	$-0.938696\pi$
$853$		−	326.533i		−	0.382806i	0.981512	−	0.191403i	$-0.0613037\pi$
$854$		0			0
$855$	67.5290	+	535.761i	0.0789813	+	0.626621i
$856$		0			0
$857$	−288.500			−0.336639			−0.168319	−	0.985732i	$-0.553834\pi$
$857$	−288.500			−0.336639			−0.168319	+	0.985732i	$0.553834\pi$
$858$		0			0
$859$	−796.000			−0.926659			−0.463329	−	0.886186i	$-0.653345\pi$
$859$	−796.000			−0.926659			−0.463329	+	0.886186i	$0.653345\pi$
$860$		0			0
$861$	−238.000	+	981.299i	−0.276423	+	1.13972i
$862$		0			0
$863$	−352.139			−0.408041			−0.204020	−	0.978967i	$-0.565401\pi$
$863$	−352.139			−0.408041			−0.204020	+	0.978967i	$0.565401\pi$
$864$		0			0
$865$	464.000	+	676.390i	0.536416	+	0.781954i
$866$		0			0
$867$	−113.844	+	469.392i	−0.131308	+	0.541397i
$868$		0			0
$869$			1187.45i			1.36646i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	673.166	−	1306.13i	0.771095	−	1.49614i
$874$		0			0
$875$	168.291	+	709.174i	0.192333	+	0.810485i
$876$		0			0
$877$			676.390i			0.771255i	0.922655	+	0.385627i	$0.126015\pi$
$877$			676.390i			0.771255i	−0.922655	+	0.385627i	$0.873985\pi$
$878$		0			0
$879$	−60.0000	+	247.386i	−0.0682594	+	0.281441i
$880$		0			0
$881$		−	519.511i		−	0.589684i	−0.955546	−	0.294842i	$-0.904733\pi$
$881$		−	519.511i		−	0.589684i	0.955546	−	0.294842i	$-0.0952670\pi$
$882$		0			0
$883$		−	1591.85i		−	1.80277i	−0.433014	−	0.901387i	$-0.642550\pi$
$883$		−	1591.85i		−	1.80277i	0.433014	−	0.901387i	$-0.357450\pi$
$884$		0			0
$885$	−87.9167	−	231.237i	−0.0993409	−	0.261285i
$886$		0			0
$887$	−767.918			−0.865747			−0.432874	−	0.901455i	$-0.642500\pi$
$887$	−767.918			−0.865747			−0.432874	+	0.901455i	$0.642500\pi$
$888$		0			0
$889$	−238.000			−0.267717
$890$		0			0
$891$	1088.00	−	775.144i	1.22110	−	0.869971i
$892$		0			0
$893$	424.264			0.475100
$894$		0			0
$895$	68.0000	−	46.6476i	0.0759777	−	0.0521202i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	768.000			0.852386
$902$		0			0
$903$	−168.291	+	693.883i	−0.186369	+	0.768420i
$904$		0			0
$905$	231.931	+	338.095i	0.256277	+	0.373585i
$906$		0			0
$907$		−	332.364i		−	0.366444i	−0.983072	−	0.183222i	$-0.941347\pi$
$907$		−	332.364i		−	0.366444i	0.983072	−	0.183222i	$-0.0586526\pi$
$908$		0			0
$909$	−544.000	+	1055.52i	−0.598460	+	1.16118i
$910$		0			0
$911$			692.682i			0.760353i	0.924914	+	0.380177i	$0.124137\pi$
$911$			692.682i			0.760353i	−0.924914	+	0.380177i	$0.875863\pi$
$912$		0			0
$913$		−	723.038i		−	0.791937i
$914$		0			0
$915$	224.333	−	85.2918i	0.245173	−	0.0932150i
$916$		0			0
$917$	−288.500			−0.314612
$918$		0			0
$919$	−912.000			−0.992383			−0.496192	−	0.868213i	$-0.665269\pi$
$919$	−912.000			−0.992383			−0.496192	+	0.868213i	$0.665269\pi$
$920$		0			0
$921$	−1071.00	−	259.756i	−1.16287	−	0.282037i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−544.000	−	209.914i	−0.588108	−	0.226934i
$926$		0			0
$927$	408.708	−	793.009i	0.440893	−	0.855458i
$928$		0			0
$929$			1261.67i			1.35810i	0.734094	+	0.679048i	$0.237607\pi$
$929$			1261.67i			1.35810i	−0.734094	+	0.679048i	$0.762393\pi$
$930$		0			0
$931$	−180.000			−0.193340
$932$		0			0
$933$	−288.500	−	69.9714i	−0.309217	−	0.0749962i
$934$		0			0
$935$	−769.332	+	527.758i	−0.822815	+	0.564447i
$936$		0			0
$937$		−	816.333i		−	0.871220i	−0.900135	−	0.435610i	$-0.856533\pi$
$937$		−	816.333i		−	0.871220i	0.900135	−	0.435610i	$-0.143467\pi$
$938$		0			0
$939$	−544.000	−	131.939i	−0.579340	−	0.140511i
$940$		0			0
$941$		−	1055.52i		−	1.12170i	−0.827919	−	0.560848i	$-0.810475\pi$
$941$		−	1055.52i		−	1.12170i	0.827919	−	0.560848i	$-0.189525\pi$
$942$		0			0
$943$		−	1387.77i		−	1.47165i
$944$		0			0
$945$	−88.4172	+	782.197i	−0.0935631	+	0.827722i
$946$		0			0
$947$	114.551			0.120962			0.0604812	−	0.998169i	$-0.480736\pi$
$947$	114.551			0.120962			0.0604812	+	0.998169i	$0.480736\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	368.000	−	1517.30i	0.386961	−	1.59548i
$952$		0			0
$953$	243.245			0.255241			0.127621	−	0.991823i	$-0.459266\pi$
$953$	243.245			0.255241			0.127621	+	0.991823i	$0.459266\pi$
$954$		0			0
$955$	1224.00	−	839.657i	1.28168	−	0.879222i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			296.864i			0.309555i
$960$		0			0
$961$	63.0000			0.0655567
$962$		0			0
$963$	−441.235	−	227.407i	−0.458188	−	0.236144i
$964$		0			0
$965$	480.833	−	329.848i	0.498272	−	0.341812i
$966$		0			0
$967$		−	938.783i		−	0.970820i	−0.874287	−	0.485410i	$-0.838670\pi$
$967$		−	938.783i		−	0.970820i	0.874287	−	0.485410i	$-0.161330\pi$
$968$		0			0
$969$	96.0000	−	395.818i	0.0990712	−	0.408481i
$970$		0			0
$971$			1335.89i			1.37578i	0.725813	+	0.687892i	$0.241464\pi$
$971$			1335.89i			1.37578i	−0.725813	+	0.687892i	$0.758536\pi$
$972$		0			0
$973$			256.562i			0.263681i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	305.470			0.312661			0.156331	−	0.987705i	$-0.450033\pi$
$977$	305.470			0.312661			0.156331	+	0.987705i	$0.450033\pi$
$978$		0			0
$979$	−1088.00			−1.11134
$980$		0			0
$981$	640.000	+	329.848i	0.652396	+	0.336237i
$982$		0			0
$983$	−1110.16			−1.12936			−0.564678	−	0.825311i	$-0.691000\pi$
$983$	−1110.16			−1.12936			−0.564678	+	0.825311i	$0.691000\pi$
$984$		0			0
$985$	544.000	+	793.009i	0.552284	+	0.805086i
$986$		0			0
$987$	601.041	+	145.774i	0.608957	+	0.147694i
$988$		0			0
$989$		−	981.299i		−	0.992213i
$990$		0			0
$991$	640.000			0.645812			0.322906	−	0.946431i	$-0.395340\pi$
$991$	640.000			0.645812			0.322906	+	0.946431i	$0.395340\pi$
$992$		0			0
$993$	−206.475	+	851.319i	−0.207931	+	0.857320i
$994$		0			0
$995$	−882.469	−	1286.41i	−0.886904	−	1.29287i
$996$		0			0
$997$			279.886i			0.280728i	0.990100	+	0.140364i	$0.0448273\pi$
$997$			279.886i			0.280728i	−0.990100	+	0.140364i	$0.955173\pi$
$998$		0			0
$999$	−476.000	−	412.311i	−0.476476	−	0.412723i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.c.f.449.3		4
3.2	odd	2	inner	960.3.c.f.449.1		4
4.3	odd	2		960.3.c.e.449.2		4
5.4	even	2	inner	960.3.c.f.449.2		4
8.3	odd	2		240.3.c.c.209.3		4
8.5	even	2		30.3.b.a.29.4	yes	4
12.11	even	2		960.3.c.e.449.4		4
15.14	odd	2	inner	960.3.c.f.449.4		4
20.19	odd	2		960.3.c.e.449.3		4
24.5	odd	2		30.3.b.a.29.2	yes	4
24.11	even	2		240.3.c.c.209.1		4
40.3	even	4		1200.3.l.t.401.4		4
40.13	odd	4		150.3.d.d.101.1		4
40.19	odd	2		240.3.c.c.209.2		4
40.27	even	4		1200.3.l.t.401.1		4
40.29	even	2		30.3.b.a.29.1	✓	4
40.37	odd	4		150.3.d.d.101.4		4
60.59	even	2		960.3.c.e.449.1		4
72.5	odd	6		810.3.j.c.269.4		8
72.13	even	6		810.3.j.c.269.1		8
72.29	odd	6		810.3.j.c.539.3		8
72.61	even	6		810.3.j.c.539.2		8
120.29	odd	2		30.3.b.a.29.3	yes	4
120.53	even	4		150.3.d.d.101.3		4
120.59	even	2		240.3.c.c.209.4		4
120.77	even	4		150.3.d.d.101.2		4
120.83	odd	4		1200.3.l.t.401.3		4
120.107	odd	4		1200.3.l.t.401.2		4
360.29	odd	6		810.3.j.c.539.1		8
360.149	odd	6		810.3.j.c.269.2		8
360.229	even	6		810.3.j.c.269.3		8
360.349	even	6		810.3.j.c.539.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.b.a.29.1	✓	4	40.29	even	2
30.3.b.a.29.2	yes	4	24.5	odd	2
30.3.b.a.29.3	yes	4	120.29	odd	2
30.3.b.a.29.4	yes	4	8.5	even	2
150.3.d.d.101.1		4	40.13	odd	4
150.3.d.d.101.2		4	120.77	even	4
150.3.d.d.101.3		4	120.53	even	4
150.3.d.d.101.4		4	40.37	odd	4
240.3.c.c.209.1		4	24.11	even	2
240.3.c.c.209.2		4	40.19	odd	2
240.3.c.c.209.3		4	8.3	odd	2
240.3.c.c.209.4		4	120.59	even	2
810.3.j.c.269.1		8	72.13	even	6
810.3.j.c.269.2		8	360.149	odd	6
810.3.j.c.269.3		8	360.229	even	6
810.3.j.c.269.4		8	72.5	odd	6
810.3.j.c.539.1		8	360.29	odd	6
810.3.j.c.539.2		8	72.61	even	6
810.3.j.c.539.3		8	72.29	odd	6
810.3.j.c.539.4		8	360.349	even	6
960.3.c.e.449.1		4	60.59	even	2
960.3.c.e.449.2		4	4.3	odd	2
960.3.c.e.449.3		4	20.19	odd	2
960.3.c.e.449.4		4	12.11	even	2
960.3.c.f.449.1		4	3.2	odd	2	inner
960.3.c.f.449.2		4	5.4	even	2	inner
960.3.c.f.449.3		4	1.1	even	1	trivial
960.3.c.f.449.4		4	15.14	odd	2	inner
1200.3.l.t.401.1		4	40.27	even	4
1200.3.l.t.401.2		4	120.107	odd	4
1200.3.l.t.401.3		4	120.83	odd	4
1200.3.l.t.401.4		4	40.3	even	4

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 2.91548i) q^{3} +(2.82843 + 4.12311i) q^{5} -5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} +O(q^{10})\)	\(q+(0.707107 - 2.91548i) q^{3} +(2.82843 + 4.12311i) q^{5} -5.83095i q^{7} +(-8.00000 - 4.12311i) q^{9} -16.4924i q^{11} +(14.0208 - 5.33074i) q^{15} -11.3137 q^{17} -12.0000 q^{19} +(-17.0000 - 4.12311i) q^{21} +24.0416 q^{23} +(-9.00000 + 23.3238i) q^{25} +(-17.6777 + 20.4083i) q^{27} -32.0000 q^{31} +(-48.0833 - 11.6619i) q^{33} +(24.0416 - 16.4924i) q^{35} +23.3238i q^{37} -57.7235i q^{41} -40.8167i q^{43} +(-5.62742 - 44.6467i) q^{45} -35.3553 q^{47} +15.0000 q^{49} +(-8.00000 + 32.9848i) q^{51} -67.8823 q^{53} +(68.0000 - 46.6476i) q^{55} +(-8.48528 + 34.9857i) q^{57} -16.4924i q^{59} +16.0000 q^{61} +(-24.0416 + 46.6476i) q^{63} -5.83095i q^{67} +(17.0000 - 70.0928i) q^{69} -116.619i q^{73} +(61.6360 + 42.7317i) q^{75} -96.1665 q^{77} -72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} +43.8406 q^{83} +(-32.0000 - 46.6476i) q^{85} -65.9697i q^{89} +(-22.6274 + 93.2952i) q^{93} +(-33.9411 - 49.4773i) q^{95} +163.267i q^{97} +(-68.0000 + 131.939i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{9}+O(q^{10}) \) 4 * q - 32 * q^9	\( 4 q - 32 q^{9} + 8 q^{15} - 48 q^{19} - 68 q^{21} - 36 q^{25} - 128 q^{31} + 68 q^{45} + 60 q^{49} - 32 q^{51} + 272 q^{55} + 64 q^{61} + 68 q^{69} + 272 q^{75} - 288 q^{79} + 188 q^{81} - 128 q^{85} - 272 q^{99}+O(q^{100}) \) 4 * q - 32 * q^9 + 8 * q^15 - 48 * q^19 - 68 * q^21 - 36 * q^25 - 128 * q^31 + 68 * q^45 + 60 * q^49 - 32 * q^51 + 272 * q^55 + 64 * q^61 + 68 * q^69 + 272 * q^75 - 288 * q^79 + 188 * q^81 - 128 * q^85 - 272 * q^99

Introduction

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