LMFDB - Embedded newform 1600.2.s.e.207.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.7
Character	$\chi$	$=$	1600.207
Dual form			1600.2.s.e.943.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.619018 q^{3} +(-1.82373 - 1.82373i) q^{7} -2.61682 q^{9} +O(q^{10})$	$q+0.619018 q^{3} +(-1.82373 - 1.82373i) q^{7} -2.61682 q^{9} +(0.567849 - 0.567849i) q^{11} +2.78771i q^{13} +(3.65193 + 3.65193i) q^{17} +(-4.51065 + 4.51065i) q^{19} +(-1.12892 - 1.12892i) q^{21} +(2.15520 - 2.15520i) q^{23} -3.47691 q^{27} +(-3.20259 - 3.20259i) q^{29} +3.54087i q^{31} +(0.351509 - 0.351509i) q^{33} +5.22371i q^{37} +1.72564i q^{39} +8.76287i q^{41} +10.8604i q^{43} +(-3.22050 + 3.22050i) q^{47} -0.348024i q^{49} +(2.26061 + 2.26061i) q^{51} +12.8658 q^{53} +(-2.79218 + 2.79218i) q^{57} +(3.79319 + 3.79319i) q^{59} +(6.63395 - 6.63395i) q^{61} +(4.77236 + 4.77236i) q^{63} +7.78732i q^{67} +(1.33411 - 1.33411i) q^{69} -13.6650 q^{71} +(1.34382 + 1.34382i) q^{73} -2.07120 q^{77} -16.3528 q^{79} +5.69818 q^{81} +0.391056 q^{83} +(-1.98246 - 1.98246i) q^{87} -18.0317 q^{89} +(5.08402 - 5.08402i) q^{91} +2.19186i q^{93} +(-6.43517 - 6.43517i) q^{97} +(-1.48596 + 1.48596i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 40 q^{9}+O(q^{10}) $ 24 * q + 40 * q^9	$ 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) $ 24 * q + 40 * q^9 + 20 * q^11 - 12 * q^19 - 8 * q^29 - 20 * q^51 + 8 * q^59 - 48 * q^61 + 64 * q^69 + 16 * q^71 - 104 * q^79 + 48 * q^81 - 96 * q^89 - 64 * q^91 + 128 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.619018			0.357390			0.178695	−	0.983904i	$-0.442812\pi$
$3$	0.619018			0.357390			0.178695	+	0.983904i	$0.442812\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.82373	−	1.82373i	−0.689305	−	0.689305i	0.272773	−	0.962078i	$-0.412059\pi$
$7$	−1.82373	−	1.82373i	−0.689305	−	0.689305i	−0.962078	+	0.272773i	$0.912059\pi$
$8$		0			0
$9$	−2.61682			−0.872272
$10$		0			0
$11$	0.567849	−	0.567849i	0.171213	−	0.171213i	−0.616299	−	0.787512i	$-0.711369\pi$
$11$	0.567849	−	0.567849i	0.171213	−	0.171213i	0.787512	+	0.616299i	$0.211369\pi$
$12$		0			0
$13$			2.78771i			0.773171i	0.922254	+	0.386585i	$0.126346\pi$
$13$			2.78771i			0.773171i	−0.922254	+	0.386585i	$0.873654\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.65193	+	3.65193i	0.885723	+	0.885723i	0.994109	−	0.108386i	$-0.0345683\pi$
$17$	3.65193	+	3.65193i	0.885723	+	0.885723i	−0.108386	+	0.994109i	$0.534568\pi$
$18$		0			0
$19$	−4.51065	+	4.51065i	−1.03481	+	1.03481i	−0.0354432	+	0.999372i	$0.511284\pi$
$19$	−4.51065	+	4.51065i	−1.03481	+	1.03481i	−0.999372	+	0.0354432i	$0.988716\pi$
$20$		0			0
$21$	−1.12892	−	1.12892i	−0.246351	−	0.246351i
$22$		0			0
$23$	2.15520	−	2.15520i	0.449391	−	0.449391i	−0.445761	−	0.895152i	$-0.647067\pi$
$23$	2.15520	−	2.15520i	0.449391	−	0.449391i	0.895152	+	0.445761i	$0.147067\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.47691			−0.669132
$28$		0			0
$29$	−3.20259	−	3.20259i	−0.594705	−	0.594705i	0.344193	−	0.938899i	$-0.388152\pi$
$29$	−3.20259	−	3.20259i	−0.594705	−	0.594705i	−0.938899	+	0.344193i	$0.888152\pi$
$30$		0			0
$31$			3.54087i			0.635959i	0.948098	+	0.317980i	$0.103004\pi$
$31$			3.54087i			0.635959i	−0.948098	+	0.317980i	$0.896996\pi$
$32$		0			0
$33$	0.351509	−	0.351509i	0.0611898	−	0.0611898i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.22371i			0.858773i	0.903121	+	0.429386i	$0.141270\pi$
$37$			5.22371i			0.858773i	−0.903121	+	0.429386i	$0.858730\pi$
$38$		0			0
$39$			1.72564i			0.276324i
$40$		0			0
$41$			8.76287i			1.36853i	0.729233	+	0.684265i	$0.239877\pi$
$41$			8.76287i			1.36853i	−0.729233	+	0.684265i	$0.760123\pi$
$42$		0			0
$43$			10.8604i			1.65619i	0.560587	+	0.828096i	$0.310576\pi$
$43$			10.8604i			1.65619i	−0.560587	+	0.828096i	$0.689424\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.22050	+	3.22050i	−0.469758	+	0.469758i	−0.901836	−	0.432078i	$-0.857780\pi$
$47$	−3.22050	+	3.22050i	−0.469758	+	0.469758i	0.432078	+	0.901836i	$0.357780\pi$
$48$		0			0
$49$		−	0.348024i		−	0.0497176i
$50$		0			0
$51$	2.26061	+	2.26061i	0.316549	+	0.316549i
$52$		0			0
$53$	12.8658			1.76725			0.883626	−	0.468194i	$-0.155095\pi$
$53$	12.8658			1.76725			0.883626	+	0.468194i	$0.155095\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.79218	+	2.79218i	−0.369833	+	0.369833i
$58$		0			0
$59$	3.79319	+	3.79319i	0.493832	+	0.493832i	0.909511	−	0.415679i	$-0.136456\pi$
$59$	3.79319	+	3.79319i	0.493832	+	0.493832i	−0.415679	+	0.909511i	$0.636456\pi$
$60$		0			0
$61$	6.63395	−	6.63395i	0.849390	−	0.849390i	−0.140667	−	0.990057i	$-0.544925\pi$
$61$	6.63395	−	6.63395i	0.849390	−	0.849390i	0.990057	+	0.140667i	$0.0449246\pi$
$62$		0			0
$63$	4.77236	+	4.77236i	0.601261	+	0.601261i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			7.78732i			0.951373i	0.879615	+	0.475686i	$0.157800\pi$
$67$			7.78732i			0.951373i	−0.879615	+	0.475686i	$0.842200\pi$
$68$		0			0
$69$	1.33411	−	1.33411i	0.160608	−	0.160608i
$70$		0			0
$71$	−13.6650			−1.62174			−0.810868	−	0.585229i	$-0.801005\pi$
$71$	−13.6650			−1.62174			−0.810868	+	0.585229i	$0.801005\pi$
$72$		0			0
$73$	1.34382	+	1.34382i	0.157282	+	0.157282i	0.781361	−	0.624079i	$-0.214526\pi$
$73$	1.34382	+	1.34382i	0.157282	+	0.157282i	−0.624079	+	0.781361i	$0.714526\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.07120			−0.236036
$78$		0			0
$79$	−16.3528			−1.83984			−0.919918	−	0.392111i	$-0.871745\pi$
$79$	−16.3528			−1.83984			−0.919918	+	0.392111i	$0.871745\pi$
$80$		0			0
$81$	5.69818			0.633131
$82$		0			0
$83$	0.391056			0.0429240			0.0214620	−	0.999770i	$-0.493168\pi$
$83$	0.391056			0.0429240			0.0214620	+	0.999770i	$0.493168\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.98246	−	1.98246i	−0.212542	−	0.212542i
$88$		0			0
$89$	−18.0317			−1.91135			−0.955676	−	0.294419i	$-0.904874\pi$
$89$	−18.0317			−1.91135			−0.955676	+	0.294419i	$0.904874\pi$
$90$		0			0
$91$	5.08402	−	5.08402i	0.532950	−	0.532950i
$92$		0			0
$93$			2.19186i			0.227286i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.43517	−	6.43517i	−0.653392	−	0.653392i	0.300416	−	0.953808i	$-0.402874\pi$
$97$	−6.43517	−	6.43517i	−0.653392	−	0.653392i	−0.953808	+	0.300416i	$0.902874\pi$
$98$		0			0
$99$	−1.48596	+	1.48596i	−0.149344	+	0.149344i
$100$		0			0
$101$	−5.36516	−	5.36516i	−0.533853	−	0.533853i	0.387864	−	0.921717i	$-0.373213\pi$
$101$	−5.36516	−	5.36516i	−0.533853	−	0.533853i	−0.921717	+	0.387864i	$0.873213\pi$
$102$		0			0
$103$	11.5643	−	11.5643i	1.13946	−	1.13946i	0.150919	−	0.988546i	$-0.451777\pi$
$103$	11.5643	−	11.5643i	1.13946	−	1.13946i	0.988546	−	0.150919i	$-0.0482231\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.78732			0.752829			0.376414	−	0.926451i	$-0.377157\pi$
$107$	7.78732			0.752829			0.376414	+	0.926451i	$0.377157\pi$
$108$		0			0
$109$	4.39771	+	4.39771i	0.421225	+	0.421225i	0.885625	−	0.464401i	$-0.153730\pi$
$109$	4.39771	+	4.39771i	0.421225	+	0.421225i	−0.464401	+	0.885625i	$0.653730\pi$
$110$		0			0
$111$			3.23357i			0.306917i
$112$		0			0
$113$	−3.76206	+	3.76206i	−0.353905	+	0.353905i	−0.861560	−	0.507655i	$-0.830512\pi$
$113$	−3.76206	+	3.76206i	−0.353905	+	0.353905i	0.507655	+	0.861560i	$0.330512\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	7.29492i		−	0.674415i
$118$		0			0
$119$		−	13.3203i		−	1.22107i
$120$		0			0
$121$			10.3551i			0.941372i
$122$		0			0
$123$			5.42438i			0.489100i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.158731	−	0.158731i	0.0140851	−	0.0140851i	−0.700029	−	0.714114i	$-0.746830\pi$
$127$	0.158731	−	0.158731i	0.0140851	−	0.0140851i	0.714114	+	0.700029i	$0.246830\pi$
$128$		0			0
$129$			6.72277i			0.591907i
$130$		0			0
$131$	0.00483713	+	0.00483713i	0.000422622	+	0.000422622i	0.707318	−	0.706895i	$-0.249905\pi$
$131$	0.00483713	+	0.00483713i	0.000422622	+	0.000422622i	−0.706895	+	0.707318i	$0.749905\pi$
$132$		0			0
$133$	16.4524			1.42661
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.84792	+	6.84792i	−0.585057	+	0.585057i	−0.936289	−	0.351231i	$-0.885763\pi$
$137$	−6.84792	+	6.84792i	−0.585057	+	0.585057i	0.351231	+	0.936289i	$0.385763\pi$
$138$		0			0
$139$	−1.88900	−	1.88900i	−0.160223	−	0.160223i	0.622443	−	0.782665i	$-0.286140\pi$
$139$	−1.88900	−	1.88900i	−0.160223	−	0.160223i	−0.782665	+	0.622443i	$0.786140\pi$
$140$		0			0
$141$	−1.99355	+	1.99355i	−0.167887	+	0.167887i
$142$		0			0
$143$	1.58300	+	1.58300i	0.132377	+	0.132377i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	0.215433i		−	0.0177686i
$148$		0			0
$149$	−3.47398	+	3.47398i	−0.284600	+	0.284600i	−0.834940	−	0.550341i	$-0.814498\pi$
$149$	−3.47398	+	3.47398i	−0.284600	+	0.284600i	0.550341	+	0.834940i	$0.314498\pi$
$150$		0			0
$151$	−1.93929			−0.157818			−0.0789088	−	0.996882i	$-0.525144\pi$
$151$	−1.93929			−0.157818			−0.0789088	+	0.996882i	$0.525144\pi$
$152$		0			0
$153$	−9.55643	−	9.55643i	−0.772591	−	0.772591i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.52142			0.121422			0.0607112	−	0.998155i	$-0.480663\pi$
$157$	1.52142			0.121422			0.0607112	+	0.998155i	$0.480663\pi$
$158$		0			0
$159$	7.96416			0.631599
$160$		0			0
$161$	−7.86102			−0.619535
$162$		0			0
$163$	−8.50837			−0.666427			−0.333213	−	0.942851i	$-0.608133\pi$
$163$	−8.50837			−0.666427			−0.333213	+	0.942851i	$0.608133\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.9639	+	10.9639i	0.848411	+	0.848411i	0.989935	−	0.141524i	$-0.0452002\pi$
$167$	10.9639	+	10.9639i	0.848411	+	0.848411i	−0.141524	+	0.989935i	$0.545200\pi$
$168$		0			0
$169$	5.22869			0.402207
$170$		0			0
$171$	11.8036	−	11.8036i	0.902640	−	0.902640i
$172$		0			0
$173$			15.9782i			1.21480i	0.794395	+	0.607401i	$0.207788\pi$
$173$			15.9782i			1.21480i	−0.794395	+	0.607401i	$0.792212\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.34806	+	2.34806i	0.176491	+	0.176491i
$178$		0			0
$179$	−2.31781	+	2.31781i	−0.173241	+	0.173241i	−0.788402	−	0.615161i	$-0.789091\pi$
$179$	−2.31781	+	2.31781i	−0.173241	+	0.173241i	0.615161	+	0.788402i	$0.289091\pi$
$180$		0			0
$181$	−16.9288	−	16.9288i	−1.25831	−	1.25831i	−0.951899	−	0.306412i	$-0.900871\pi$
$181$	−16.9288	−	16.9288i	−1.25831	−	1.25831i	−0.306412	−	0.951899i	$-0.599129\pi$
$182$		0			0
$183$	4.10654	−	4.10654i	0.303564	−	0.303564i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.14749			0.303294
$188$		0			0
$189$	6.34095	+	6.34095i	0.461236	+	0.461236i
$190$		0			0
$191$			1.77412i			0.128371i	0.997938	+	0.0641855i	$0.0204449\pi$
$191$			1.77412i			0.128371i	−0.997938	+	0.0641855i	$0.979555\pi$
$192$		0			0
$193$	−1.96542	+	1.96542i	−0.141474	+	0.141474i	−0.774297	−	0.632823i	$-0.781896\pi$
$193$	−1.96542	+	1.96542i	−0.141474	+	0.141474i	0.632823	+	0.774297i	$0.281896\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.27799i		−	0.162300i	−0.996702	−	0.0811500i	$-0.974141\pi$
$197$		−	2.27799i		−	0.162300i	0.996702	−	0.0811500i	$-0.0258593\pi$
$198$		0			0
$199$		−	10.0426i		−	0.711902i	−0.934505	−	0.355951i	$-0.884157\pi$
$199$		−	10.0426i		−	0.711902i	0.934505	−	0.355951i	$-0.115843\pi$
$200$		0			0
$201$			4.82049i			0.340011i
$202$		0			0
$203$			11.6813i			0.819867i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−5.63977	+	5.63977i	−0.391991	+	0.391991i
$208$		0			0
$209$			5.12274i			0.354347i
$210$		0			0
$211$	18.4884	+	18.4884i	1.27279	+	1.27279i	0.944616	+	0.328179i	$0.106435\pi$
$211$	18.4884	+	18.4884i	1.27279	+	1.27279i	0.328179	+	0.944616i	$0.393565\pi$
$212$		0			0
$213$	−8.45888			−0.579593
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.45759	−	6.45759i	0.438370	−	0.438370i
$218$		0			0
$219$	0.831850	+	0.831850i	0.0562112	+	0.0562112i
$220$		0			0
$221$	−10.1805	+	10.1805i	−0.684815	+	0.684815i
$222$		0			0
$223$	−18.7483	−	18.7483i	−1.25548	−	1.25548i	−0.953230	−	0.302246i	$-0.902264\pi$
$223$	−18.7483	−	18.7483i	−1.25548	−	1.25548i	−0.302246	−	0.953230i	$-0.597736\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	9.29139i		−	0.616691i	−0.951274	−	0.308346i	$-0.900225\pi$
$227$		−	9.29139i		−	0.616691i	0.951274	−	0.308346i	$-0.0997753\pi$
$228$		0			0
$229$	12.9965	−	12.9965i	0.858833	−	0.858833i	−0.132368	−	0.991201i	$-0.542258\pi$
$229$	12.9965	−	12.9965i	0.858833	−	0.858833i	0.991201	+	0.132368i	$0.0422579\pi$
$230$		0			0
$231$	−1.28211			−0.0843569
$232$		0			0
$233$	−12.0407	−	12.0407i	−0.788812	−	0.788812i	0.192487	−	0.981299i	$-0.438345\pi$
$233$	−12.0407	−	12.0407i	−0.788812	−	0.788812i	−0.981299	+	0.192487i	$0.938345\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−10.1227			−0.657539
$238$		0			0
$239$	11.3495			0.734140			0.367070	−	0.930193i	$-0.380361\pi$
$239$	11.3495			0.734140			0.367070	+	0.930193i	$0.380361\pi$
$240$		0			0
$241$	7.83447			0.504662			0.252331	−	0.967641i	$-0.418803\pi$
$241$	7.83447			0.504662			0.252331	+	0.967641i	$0.418803\pi$
$242$		0			0
$243$	13.9580			0.895407
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.5744	−	12.5744i	−0.800089	−	0.800089i
$248$		0			0
$249$	0.242071			0.0153406
$250$		0			0
$251$	11.2522	−	11.2522i	0.710234	−	0.710234i	−0.256350	−	0.966584i	$-0.582520\pi$
$251$	11.2522	−	11.2522i	0.710234	−	0.710234i	0.966584	+	0.256350i	$0.0825201\pi$
$252$		0			0
$253$		−	2.44766i		−	0.153883i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.35316	+	6.35316i	0.396299	+	0.396299i	0.876926	−	0.480626i	$-0.159591\pi$
$257$	6.35316	+	6.35316i	0.396299	+	0.396299i	−0.480626	+	0.876926i	$0.659591\pi$
$258$		0			0
$259$	9.52664	−	9.52664i	0.591956	−	0.591956i
$260$		0			0
$261$	8.38058	+	8.38058i	0.518745	+	0.518745i
$262$		0			0
$263$	−1.72461	+	1.72461i	−0.106344	+	0.106344i	−0.758277	−	0.651933i	$-0.773958\pi$
$263$	−1.72461	+	1.72461i	−0.106344	+	0.106344i	0.651933	+	0.758277i	$0.273958\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−11.1619			−0.683099
$268$		0			0
$269$	8.87466	+	8.87466i	0.541098	+	0.541098i	0.923851	−	0.382753i	$-0.125024\pi$
$269$	8.87466	+	8.87466i	0.541098	+	0.541098i	−0.382753	+	0.923851i	$0.625024\pi$
$270$		0			0
$271$		−	2.61613i		−	0.158919i	−0.996838	−	0.0794594i	$-0.974681\pi$
$271$		−	2.61613i		−	0.158919i	0.996838	−	0.0794594i	$-0.0253194\pi$
$272$		0			0
$273$	3.14710	−	3.14710i	0.190471	−	0.190471i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	8.53878i		−	0.513046i	−0.966538	−	0.256523i	$-0.917423\pi$
$277$		−	8.53878i		−	0.513046i	0.966538	−	0.256523i	$-0.0825769\pi$
$278$		0			0
$279$		−	9.26581i		−	0.554729i
$280$		0			0
$281$			3.31230i			0.197595i	0.995108	+	0.0987976i	$0.0314996\pi$
$281$			3.31230i			0.197595i	−0.995108	+	0.0987976i	$0.968500\pi$
$282$		0			0
$283$		−	18.5598i		−	1.10327i	−0.834086	−	0.551634i	$-0.814005\pi$
$283$		−	18.5598i		−	1.10327i	0.834086	−	0.551634i	$-0.185995\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.9811	−	15.9811i	0.943335	−	0.943335i
$288$		0			0
$289$			9.67316i			0.569010i
$290$		0			0
$291$	−3.98348	−	3.98348i	−0.233516	−	0.233516i
$292$		0			0
$293$	−19.9056			−1.16290			−0.581448	−	0.813584i	$-0.697514\pi$
$293$	−19.9056			−1.16290			−0.581448	+	0.813584i	$0.697514\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.97436	+	1.97436i	−0.114564	+	0.114564i
$298$		0			0
$299$	6.00808	+	6.00808i	0.347456	+	0.347456i
$300$		0			0
$301$	19.8064	−	19.8064i	1.14162	−	1.14162i
$302$		0			0
$303$	−3.32113	−	3.32113i	−0.190794	−	0.190794i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			7.92742i			0.452442i	0.974076	+	0.226221i	$0.0726371\pi$
$307$			7.92742i			0.452442i	−0.974076	+	0.226221i	$0.927363\pi$
$308$		0			0
$309$	7.15852	−	7.15852i	0.407234	−	0.407234i
$310$		0			0
$311$	−20.6190			−1.16920			−0.584598	−	0.811323i	$-0.698748\pi$
$311$	−20.6190			−1.16920			−0.584598	+	0.811323i	$0.698748\pi$
$312$		0			0
$313$	−8.10819	−	8.10819i	−0.458302	−	0.458302i	0.439796	−	0.898098i	$-0.355051\pi$
$313$	−8.10819	−	8.10819i	−0.458302	−	0.458302i	−0.898098	+	0.439796i	$0.855051\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	27.4721			1.54298			0.771492	−	0.636239i	$-0.219511\pi$
$317$	27.4721			1.54298			0.771492	+	0.636239i	$0.219511\pi$
$318$		0			0
$319$	−3.63717			−0.203642
$320$		0			0
$321$	4.82049			0.269054
$322$		0			0
$323$	−32.9452			−1.83312
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.72227	+	2.72227i	0.150542	+	0.150542i
$328$		0			0
$329$	11.7466			0.647612
$330$		0			0
$331$	−18.5236	+	18.5236i	−1.01815	+	1.01815i	−0.0183151	+	0.999832i	$0.505830\pi$
$331$	−18.5236	+	18.5236i	−1.01815	+	1.01815i	−0.999832	+	0.0183151i	$0.994170\pi$
$332$		0			0
$333$		−	13.6695i		−	0.749084i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.6567	−	14.6567i	−0.798402	−	0.798402i	0.184442	−	0.982843i	$-0.440952\pi$
$337$	−14.6567	−	14.6567i	−0.798402	−	0.798402i	−0.982843	+	0.184442i	$0.940952\pi$
$338$		0			0
$339$	−2.32879	+	2.32879i	−0.126482	+	0.126482i
$340$		0			0
$341$	2.01068	+	2.01068i	0.108884	+	0.108884i
$342$		0			0
$343$	−13.4008	+	13.4008i	−0.723575	+	0.723575i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25.0440			1.34443			0.672217	−	0.740354i	$-0.265342\pi$
$347$	25.0440			1.34443			0.672217	+	0.740354i	$0.265342\pi$
$348$		0			0
$349$	−17.5767	−	17.5767i	−0.940860	−	0.940860i	0.0574866	−	0.998346i	$-0.481691\pi$
$349$	−17.5767	−	17.5767i	−0.940860	−	0.940860i	−0.998346	+	0.0574866i	$0.981691\pi$
$350$		0			0
$351$		−	9.69261i		−	0.517353i
$352$		0			0
$353$	−6.30855	+	6.30855i	−0.335770	+	0.335770i	−0.854773	−	0.519002i	$-0.826304\pi$
$353$	−6.30855	+	6.30855i	−0.335770	+	0.335770i	0.519002	+	0.854773i	$0.326304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	8.24548i		−	0.436397i
$358$		0			0
$359$			27.8527i			1.47001i	0.678061	+	0.735005i	$0.262820\pi$
$359$			27.8527i			1.47001i	−0.678061	+	0.735005i	$0.737180\pi$
$360$		0			0
$361$		−	21.6920i		−	1.14168i
$362$		0			0
$363$			6.40999i			0.336437i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.29618	−	3.29618i	0.172059	−	0.172059i	−0.615824	−	0.787883i	$-0.711177\pi$
$367$	3.29618	−	3.29618i	0.172059	−	0.172059i	0.787883	+	0.615824i	$0.211177\pi$
$368$		0			0
$369$		−	22.9308i		−	1.19373i
$370$		0			0
$371$	−23.4637	−	23.4637i	−1.21818	−	1.21818i
$372$		0			0
$373$	5.93152			0.307122			0.153561	−	0.988139i	$-0.450926\pi$
$373$	5.93152			0.307122			0.153561	+	0.988139i	$0.450926\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.92787	−	8.92787i	0.459809	−	0.459809i
$378$		0			0
$379$	20.6355	+	20.6355i	1.05997	+	1.05997i	0.998083	+	0.0618894i	$0.0197126\pi$
$379$	20.6355	+	20.6355i	1.05997	+	1.05997i	0.0618894	+	0.998083i	$0.480287\pi$
$380$		0			0
$381$	0.0982571	−	0.0982571i	0.00503386	−	0.00503386i
$382$		0			0
$383$	−6.65419	−	6.65419i	−0.340013	−	0.340013i	0.516359	−	0.856372i	$-0.327287\pi$
$383$	−6.65419	−	6.65419i	−0.340013	−	0.340013i	−0.856372	+	0.516359i	$0.827287\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	28.4196i		−	1.44465i
$388$		0			0
$389$	−2.50799	+	2.50799i	−0.127160	+	0.127160i	−0.767823	−	0.640662i	$-0.778660\pi$
$389$	−2.50799	+	2.50799i	−0.127160	+	0.127160i	0.640662	+	0.767823i	$0.278660\pi$
$390$		0			0
$391$	15.7413			0.796072
$392$		0			0
$393$	0.00299427	+	0.00299427i	0.000151041	+	0.000151041i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	17.9719			0.901986			0.450993	−	0.892528i	$-0.351070\pi$
$397$	17.9719			0.901986			0.450993	+	0.892528i	$0.351070\pi$
$398$		0			0
$399$	10.1843			0.509855
$400$		0			0
$401$	4.90722			0.245055			0.122527	−	0.992465i	$-0.460900\pi$
$401$	4.90722			0.245055			0.122527	+	0.992465i	$0.460900\pi$
$402$		0			0
$403$	−9.87091			−0.491705
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.96628	+	2.96628i	0.147033	+	0.147033i
$408$		0			0
$409$	18.7926			0.929233			0.464617	−	0.885512i	$-0.346192\pi$
$409$	18.7926			0.929233			0.464617	+	0.885512i	$0.346192\pi$
$410$		0			0
$411$	−4.23899	+	4.23899i	−0.209094	+	0.209094i
$412$		0			0
$413$		−	13.8355i		−	0.680801i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.16933	−	1.16933i	−0.0572621	−	0.0572621i
$418$		0			0
$419$	9.31520	−	9.31520i	0.455077	−	0.455077i	−0.441958	−	0.897036i	$-0.645716\pi$
$419$	9.31520	−	9.31520i	0.455077	−	0.455077i	0.897036	+	0.441958i	$0.145716\pi$
$420$		0			0
$421$	7.06682	+	7.06682i	0.344416	+	0.344416i	0.858025	−	0.513609i	$-0.171692\pi$
$421$	7.06682	+	7.06682i	0.344416	+	0.344416i	−0.513609	+	0.858025i	$0.671692\pi$
$422$		0			0
$423$	8.42745	−	8.42745i	0.409757	−	0.409757i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−24.1971			−1.17098
$428$		0			0
$429$	0.979903	+	0.979903i	0.0473102	+	0.0473102i
$430$		0			0
$431$			6.87936i			0.331367i	0.986179	+	0.165684i	$0.0529830\pi$
$431$			6.87936i			0.331367i	−0.986179	+	0.165684i	$0.947017\pi$
$432$		0			0
$433$	−1.05752	+	1.05752i	−0.0508212	+	0.0508212i	−0.732061	−	0.681239i	$-0.761441\pi$
$433$	−1.05752	+	1.05752i	−0.0508212	+	0.0508212i	0.681239	+	0.732061i	$0.261441\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			19.4428i			0.930073i
$438$		0			0
$439$			12.3556i			0.589702i	0.955543	+	0.294851i	$0.0952700\pi$
$439$			12.3556i			0.589702i	−0.955543	+	0.294851i	$0.904730\pi$
$440$		0			0
$441$			0.910714i			0.0433673i
$442$		0			0
$443$		−	10.9254i		−	0.519082i	−0.965732	−	0.259541i	$-0.916429\pi$
$443$		−	10.9254i		−	0.519082i	0.965732	−	0.259541i	$-0.0835713\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.15046	+	2.15046i	−0.101713	+	0.101713i
$448$		0			0
$449$		−	37.8842i		−	1.78787i	−0.448199	−	0.893934i	$-0.647934\pi$
$449$		−	37.8842i		−	1.78787i	0.448199	−	0.893934i	$-0.352066\pi$
$450$		0			0
$451$	4.97599	+	4.97599i	0.234310	+	0.234310i
$452$		0			0
$453$	−1.20046			−0.0564025
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.82262	−	6.82262i	0.319149	−	0.319149i	−0.529291	−	0.848440i	$-0.677542\pi$
$457$	6.82262	−	6.82262i	0.319149	−	0.319149i	0.848440	+	0.529291i	$0.177542\pi$
$458$		0			0
$459$	−12.6974	−	12.6974i	−0.592665	−	0.592665i
$460$		0			0
$461$	−12.5451	+	12.5451i	−0.584285	+	0.584285i	−0.936078	−	0.351793i	$-0.885572\pi$
$461$	−12.5451	+	12.5451i	−0.584285	+	0.584285i	0.351793	+	0.936078i	$0.385572\pi$
$462$		0			0
$463$	17.9408	+	17.9408i	0.833780	+	0.833780i	0.988032	−	0.154251i	$-0.0492966\pi$
$463$	17.9408	+	17.9408i	0.833780	+	0.833780i	−0.154251	+	0.988032i	$0.549297\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			11.2034i			0.518430i	0.965820	+	0.259215i	$0.0834639\pi$
$467$			11.2034i			0.518430i	−0.965820	+	0.259215i	$0.916536\pi$
$468$		0			0
$469$	14.2020	−	14.2020i	0.655786	−	0.655786i
$470$		0			0
$471$	0.941786			0.0433952
$472$		0			0
$473$	6.16705	+	6.16705i	0.283561	+	0.283561i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−33.6674			−1.54152
$478$		0			0
$479$	25.3283			1.15728			0.578641	−	0.815583i	$-0.303583\pi$
$479$	25.3283			1.15728			0.578641	+	0.815583i	$0.303583\pi$
$480$		0			0
$481$	−14.5622			−0.663978
$482$		0			0
$483$	−4.86611			−0.221416
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.6181	−	11.6181i	−0.526466	−	0.526466i	0.393051	−	0.919517i	$-0.371420\pi$
$487$	−11.6181	−	11.6181i	−0.526466	−	0.526466i	−0.919517	+	0.393051i	$0.871420\pi$
$488$		0			0
$489$	−5.26683			−0.238174
$490$		0			0
$491$	−11.4614	+	11.4614i	−0.517244	+	0.517244i	−0.916737	−	0.399492i	$-0.869186\pi$
$491$	−11.4614	+	11.4614i	−0.517244	+	0.517244i	0.399492	+	0.916737i	$0.369186\pi$
$492$		0			0
$493$		−	23.3912i		−	1.05349i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.9213	+	24.9213i	1.11787	+	1.11787i
$498$		0			0
$499$	9.37358	−	9.37358i	0.419619	−	0.419619i	−0.465453	−	0.885072i	$-0.654109\pi$
$499$	9.37358	−	9.37358i	0.419619	−	0.419619i	0.885072	+	0.465453i	$0.154109\pi$
$500$		0			0
$501$	6.78685	+	6.78685i	0.303214	+	0.303214i
$502$		0			0
$503$	−24.6142	+	24.6142i	−1.09749	+	1.09749i	−0.102788	+	0.994703i	$0.532776\pi$
$503$	−24.6142	+	24.6142i	−1.09749	+	1.09749i	−0.994703	+	0.102788i	$0.967224\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.23666			0.143745
$508$		0			0
$509$	17.7086	+	17.7086i	0.784920	+	0.784920i	0.980656	−	0.195737i	$-0.0627098\pi$
$509$	17.7086	+	17.7086i	0.784920	+	0.784920i	−0.195737	+	0.980656i	$0.562710\pi$
$510$		0			0
$511$		−	4.90153i		−	0.216831i
$512$		0			0
$513$	15.6831	−	15.6831i	0.692428	−	0.692428i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			3.65751i			0.160857i
$518$		0			0
$519$			9.89081i			0.434159i
$520$		0			0
$521$		−	8.92968i		−	0.391216i	−0.980682	−	0.195608i	$-0.937332\pi$
$521$		−	8.92968i		−	0.391216i	0.980682	−	0.195608i	$-0.0626681\pi$
$522$		0			0
$523$		−	30.6587i		−	1.34061i	−0.742085	−	0.670306i	$-0.766163\pi$
$523$		−	30.6587i		−	1.34061i	0.742085	−	0.670306i	$-0.233837\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.9310	+	12.9310i	−0.563284	+	0.563284i
$528$		0			0
$529$			13.7102i			0.596095i
$530$		0			0
$531$	−9.92609	−	9.92609i	−0.430756	−	0.430756i
$532$		0			0
$533$	−24.4283			−1.05811
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.43476	+	1.43476i	−0.0619146	+	0.0619146i
$538$		0			0
$539$	−0.197625	−	0.197625i	−0.00851230	−	0.00851230i
$540$		0			0
$541$	8.72277	−	8.72277i	0.375021	−	0.375021i	−0.494281	−	0.869302i	$-0.664569\pi$
$541$	8.72277	−	8.72277i	0.375021	−	0.375021i	0.869302	+	0.494281i	$0.164569\pi$
$542$		0			0
$543$	−10.4793	−	10.4793i	−0.449708	−	0.449708i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	12.3389i		−	0.527575i	−0.964581	−	0.263788i	$-0.915028\pi$
$547$		−	12.3389i		−	0.527575i	0.964581	−	0.263788i	$-0.0849718\pi$
$548$		0			0
$549$	−17.3598	+	17.3598i	−0.740899	+	0.740899i
$550$		0			0
$551$	28.8915			1.23082
$552$		0			0
$553$	29.8231	+	29.8231i	1.26821	+	1.26821i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.93717			−0.124452			−0.0622259	−	0.998062i	$-0.519820\pi$
$557$	−2.93717			−0.124452			−0.0622259	+	0.998062i	$0.519820\pi$
$558$		0			0
$559$	−30.2755			−1.28052
$560$		0			0
$561$	2.56737			0.108394
$562$		0			0
$563$	41.2139			1.73696			0.868480	−	0.495725i	$-0.165097\pi$
$563$	41.2139			1.73696			0.868480	+	0.495725i	$0.165097\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−10.3919	−	10.3919i	−0.436420	−	0.436420i
$568$		0			0
$569$	17.7646			0.744730			0.372365	−	0.928086i	$-0.378547\pi$
$569$	17.7646			0.744730			0.372365	+	0.928086i	$0.378547\pi$
$570$		0			0
$571$	−4.58877	+	4.58877i	−0.192034	+	0.192034i	−0.796575	−	0.604540i	$-0.793357\pi$
$571$	−4.58877	+	4.58877i	−0.192034	+	0.192034i	0.604540	+	0.796575i	$0.293357\pi$
$572$		0			0
$573$			1.09821i			0.0458786i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.54451	−	3.54451i	−0.147560	−	0.147560i	0.629467	−	0.777027i	$-0.283273\pi$
$577$	−3.54451	−	3.54451i	−0.147560	−	0.147560i	−0.777027	+	0.629467i	$0.783273\pi$
$578$		0			0
$579$	−1.21663	+	1.21663i	−0.0505615	+	0.0505615i
$580$		0			0
$581$	−0.713180	−	0.713180i	−0.0295877	−	0.0295877i
$582$		0			0
$583$	7.30582	−	7.30582i	0.302576	−	0.302576i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.69644			−0.111294			−0.0556469	−	0.998451i	$-0.517722\pi$
$587$	−2.69644			−0.111294			−0.0556469	+	0.998451i	$0.517722\pi$
$588$		0			0
$589$	−15.9716	−	15.9716i	−0.658100	−	0.658100i
$590$		0			0
$591$		−	1.41012i		−	0.0580044i
$592$		0			0
$593$	−14.8282	+	14.8282i	−0.608922	+	0.608922i	−0.942664	−	0.333743i	$-0.891688\pi$
$593$	−14.8282	+	14.8282i	−0.608922	+	0.608922i	0.333743	+	0.942664i	$0.391688\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	6.21656i		−	0.254427i
$598$		0			0
$599$		−	29.5933i		−	1.20915i	−0.796548	−	0.604576i	$-0.793343\pi$
$599$		−	29.5933i		−	1.20915i	0.796548	−	0.604576i	$-0.206657\pi$
$600$		0			0
$601$		−	22.7968i		−	0.929900i	−0.885337	−	0.464950i	$-0.846072\pi$
$601$		−	22.7968i		−	0.929900i	0.885337	−	0.464950i	$-0.153928\pi$
$602$		0			0
$603$		−	20.3780i		−	0.829856i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	9.33600	−	9.33600i	0.378937	−	0.378937i	−0.491782	−	0.870718i	$-0.663654\pi$
$607$	9.33600	−	9.33600i	0.378937	−	0.378937i	0.870718	+	0.491782i	$0.163654\pi$
$608$		0			0
$609$			7.23094i			0.293012i
$610$		0			0
$611$	−8.97780	−	8.97780i	−0.363203	−	0.363203i
$612$		0			0
$613$	−30.5843			−1.23529			−0.617643	−	0.786458i	$-0.711912\pi$
$613$	−30.5843			−1.23529			−0.617643	+	0.786458i	$0.711912\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.7311	−	21.7311i	0.874862	−	0.874862i	−0.118135	−	0.992998i	$-0.537692\pi$
$617$	21.7311	−	21.7311i	0.874862	−	0.874862i	0.992998	+	0.118135i	$0.0376916\pi$
$618$		0			0
$619$	−28.6904	−	28.6904i	−1.15317	−	1.15317i	−0.985914	−	0.167252i	$-0.946511\pi$
$619$	−28.6904	−	28.6904i	−1.15317	−	1.15317i	−0.167252	−	0.985914i	$-0.553489\pi$
$620$		0			0
$621$	−7.49345	+	7.49345i	−0.300702	+	0.300702i
$622$		0			0
$623$	32.8849	+	32.8849i	1.31750	+	1.31750i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			3.17107i			0.126640i
$628$		0			0
$629$	−19.0766	+	19.0766i	−0.760635	+	0.760635i
$630$		0			0
$631$	20.4625			0.814597			0.407299	−	0.913295i	$-0.366471\pi$
$631$	20.4625			0.814597			0.407299	+	0.913295i	$0.366471\pi$
$632$		0			0
$633$	11.4447	+	11.4447i	0.454884	+	0.454884i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.970188			0.0384402
$638$		0			0
$639$	35.7588			1.41460
$640$		0			0
$641$	−19.4539			−0.768382			−0.384191	−	0.923254i	$-0.625520\pi$
$641$	−19.4539			−0.768382			−0.384191	+	0.923254i	$0.625520\pi$
$642$		0			0
$643$	23.2479			0.916807			0.458403	−	0.888744i	$-0.348422\pi$
$643$	23.2479			0.916807			0.458403	+	0.888744i	$0.348422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.1518	+	19.1518i	0.752937	+	0.752937i	0.975026	−	0.222090i	$-0.0712878\pi$
$647$	19.1518	+	19.1518i	0.752937	+	0.752937i	−0.222090	+	0.975026i	$0.571288\pi$
$648$		0			0
$649$	4.30792			0.169101
$650$		0			0
$651$	3.99737	−	3.99737i	0.156669	−	0.156669i
$652$		0			0
$653$			26.8221i			1.04963i	0.851216	+	0.524816i	$0.175866\pi$
$653$			26.8221i			1.04963i	−0.851216	+	0.524816i	$0.824134\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.51653	−	3.51653i	−0.137193	−	0.137193i
$658$		0			0
$659$	−35.2877	+	35.2877i	−1.37461	+	1.37461i	−0.521148	+	0.853467i	$0.674496\pi$
$659$	−35.2877	+	35.2877i	−1.37461	+	1.37461i	−0.853467	+	0.521148i	$0.825504\pi$
$660$		0			0
$661$	24.0302	+	24.0302i	0.934665	+	0.934665i	0.997993	−	0.0633282i	$-0.0201715\pi$
$661$	24.0302	+	24.0302i	0.934665	+	0.934665i	−0.0633282	+	0.997993i	$0.520171\pi$
$662$		0			0
$663$	−6.30192	+	6.30192i	−0.244746	+	0.244746i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−13.8045			−0.534511
$668$		0			0
$669$	−11.6055	−	11.6055i	−0.448695	−	0.448695i
$670$		0			0
$671$		−	7.53416i		−	0.290853i
$672$		0			0
$673$	18.6258	−	18.6258i	0.717972	−	0.717972i	−0.250218	−	0.968190i	$-0.580502\pi$
$673$	18.6258	−	18.6258i	0.717972	−	0.717972i	0.968190	+	0.250218i	$0.0805022\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15.8471i			0.609054i	0.952504	+	0.304527i	$0.0984984\pi$
$677$			15.8471i			0.609054i	−0.952504	+	0.304527i	$0.901502\pi$
$678$		0			0
$679$			23.4720i			0.900773i
$680$		0			0
$681$		−	5.75154i		−	0.220400i
$682$		0			0
$683$			34.7894i			1.33118i	0.746318	+	0.665590i	$0.231820\pi$
$683$			34.7894i			1.33118i	−0.746318	+	0.665590i	$0.768180\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	8.04507	−	8.04507i	0.306939	−	0.306939i
$688$		0			0
$689$			35.8660i			1.36639i
$690$		0			0
$691$	4.03081	+	4.03081i	0.153339	+	0.153339i	0.779608	−	0.626268i	$-0.215419\pi$
$691$	4.03081	+	4.03081i	0.153339	+	0.153339i	−0.626268	+	0.779608i	$0.715419\pi$
$692$		0			0
$693$	5.41996			0.205887
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−32.0014	+	32.0014i	−1.21214	+	1.21214i
$698$		0			0
$699$	−7.45341	−	7.45341i	−0.281914	−	0.281914i
$700$		0			0
$701$	−29.1089	+	29.1089i	−1.09943	+	1.09943i	−0.104951	+	0.994477i	$0.533469\pi$
$701$	−29.1089	+	29.1089i	−1.09943	+	1.09943i	−0.994477	+	0.104951i	$0.966531\pi$
$702$		0			0
$703$	−23.5624	−	23.5624i	−0.888671	−	0.888671i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			19.5692i			0.735975i
$708$		0			0
$709$	−28.5783	+	28.5783i	−1.07328	+	1.07328i	−0.0761883	+	0.997093i	$0.524275\pi$
$709$	−28.5783	+	28.5783i	−1.07328	+	1.07328i	−0.997093	+	0.0761883i	$0.975725\pi$
$710$		0			0
$711$	42.7923			1.60484
$712$		0			0
$713$	7.63130	+	7.63130i	0.285794	+	0.285794i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.02557			0.262375
$718$		0			0
$719$	−37.8803			−1.41270			−0.706348	−	0.707865i	$-0.749658\pi$
$719$	−37.8803			−1.41270			−0.706348	+	0.707865i	$0.749658\pi$
$720$		0			0
$721$	−42.1803			−1.57088
$722$		0			0
$723$	4.84968			0.180361
$724$		0			0
$725$		0			0
$726$		0			0
$727$	12.8109	+	12.8109i	0.475130	+	0.475130i	0.903570	−	0.428440i	$-0.140937\pi$
$727$	12.8109	+	12.8109i	0.475130	+	0.475130i	−0.428440	+	0.903570i	$0.640937\pi$
$728$		0			0
$729$	−8.45427			−0.313121
$730$		0			0
$731$	−39.6613	+	39.6613i	−1.46693	+	1.46693i
$732$		0			0
$733$			8.70612i			0.321568i	0.986990	+	0.160784i	$0.0514023\pi$
$733$			8.70612i			0.321568i	−0.986990	+	0.160784i	$0.948598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.42202	+	4.42202i	0.162887	+	0.162887i
$738$		0			0
$739$	33.1477	−	33.1477i	1.21936	−	1.21936i	0.251502	−	0.967857i	$-0.419076\pi$
$739$	33.1477	−	33.1477i	1.21936	−	1.21936i	0.967857	−	0.251502i	$-0.0809244\pi$
$740$		0			0
$741$	−7.78377	−	7.78377i	−0.285944	−	0.285944i
$742$		0			0
$743$	−6.97405	+	6.97405i	−0.255853	+	0.255853i	−0.823365	−	0.567512i	$-0.807906\pi$
$743$	−6.97405	+	6.97405i	−0.255853	+	0.255853i	0.567512	+	0.823365i	$0.307906\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.02332			−0.0374414
$748$		0			0
$749$	−14.2020	−	14.2020i	−0.518928	−	0.518928i
$750$		0			0
$751$		−	4.95216i		−	0.180707i	−0.995910	−	0.0903535i	$-0.971200\pi$
$751$		−	4.95216i		−	0.180707i	0.995910	−	0.0903535i	$-0.0287997\pi$
$752$		0			0
$753$	6.96533	−	6.96533i	0.253831	−	0.253831i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			8.40963i			0.305653i	0.988253	+	0.152827i	$0.0488376\pi$
$757$			8.40963i			0.305653i	−0.988253	+	0.152827i	$0.951162\pi$
$758$		0			0
$759$		−	1.51515i		−	0.0549963i
$760$		0			0
$761$		−	13.9357i		−	0.505170i	−0.967575	−	0.252585i	$-0.918719\pi$
$761$		−	13.9357i		−	0.505170i	0.967575	−	0.252585i	$-0.0812808\pi$
$762$		0			0
$763$		−	16.0405i		−	0.580704i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.5743	+	10.5743i	−0.381816	+	0.381816i
$768$		0			0
$769$			19.0307i			0.686264i	0.939287	+	0.343132i	$0.111488\pi$
$769$			19.0307i			0.686264i	−0.939287	+	0.343132i	$0.888512\pi$
$770$		0			0
$771$	3.93272	+	3.93272i	0.141634	+	0.141634i
$772$		0			0
$773$	3.20152			0.115151			0.0575753	−	0.998341i	$-0.481663\pi$
$773$	3.20152			0.115151			0.0575753	+	0.998341i	$0.481663\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.89716	−	5.89716i	0.211560	−	0.211560i
$778$		0			0
$779$	−39.5263	−	39.5263i	−1.41618	−	1.41618i
$780$		0			0
$781$	−7.75965	+	7.75965i	−0.277662	+	0.277662i
$782$		0			0
$783$	11.1351	+	11.1351i	0.397936	+	0.397936i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			52.2753i			1.86341i	0.363213	+	0.931706i	$0.381680\pi$
$787$			52.2753i			1.86341i	−0.363213	+	0.931706i	$0.618320\pi$
$788$		0			0
$789$	−1.06757	+	1.06757i	−0.0380064	+	0.0380064i
$790$		0			0
$791$	13.7220			0.487897
$792$		0			0
$793$	18.4935	+	18.4935i	0.656724	+	0.656724i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−49.3883			−1.74942			−0.874712	−	0.484643i	$-0.838950\pi$
$797$	−49.3883			−1.74942			−0.874712	+	0.484643i	$0.838950\pi$
$798$		0			0
$799$	−23.5220			−0.832150
$800$		0			0
$801$	47.1856			1.66722
$802$		0			0
$803$	1.52617			0.0538575
$804$		0			0
$805$		0			0
$806$		0			0
$807$	5.49358	+	5.49358i	0.193383	+	0.193383i
$808$		0			0
$809$	−8.22627			−0.289220			−0.144610	−	0.989489i	$-0.546193\pi$
$809$	−8.22627			−0.289220			−0.144610	+	0.989489i	$0.546193\pi$
$810$		0			0
$811$	14.7637	−	14.7637i	0.518424	−	0.518424i	−0.398670	−	0.917094i	$-0.630528\pi$
$811$	14.7637	−	14.7637i	0.518424	−	0.518424i	0.917094	+	0.398670i	$0.130528\pi$
$812$		0			0
$813$		−	1.61943i		−	0.0567960i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−48.9874	−	48.9874i	−1.71385	−	1.71385i
$818$		0			0
$819$	−13.3040	+	13.3040i	−0.464878	+	0.464878i
$820$		0			0
$821$	13.7757	+	13.7757i	0.480775	+	0.480775i	0.905379	−	0.424604i	$-0.139587\pi$
$821$	13.7757	+	13.7757i	0.480775	+	0.480775i	−0.424604	+	0.905379i	$0.639587\pi$
$822$		0			0
$823$	−29.0916	+	29.0916i	−1.01407	+	1.01407i	−0.0141712	+	0.999900i	$0.504511\pi$
$823$	−29.0916	+	29.0916i	−1.01407	+	1.01407i	−0.999900	+	0.0141712i	$0.995489\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0413			1.25328			0.626639	−	0.779310i	$-0.284430\pi$
$827$	36.0413			1.25328			0.626639	+	0.779310i	$0.284430\pi$
$828$		0			0
$829$	−3.76098	−	3.76098i	−0.130624	−	0.130624i	0.638772	−	0.769396i	$-0.279443\pi$
$829$	−3.76098	−	3.76098i	−0.130624	−	0.130624i	−0.769396	+	0.638772i	$0.779443\pi$
$830$		0			0
$831$		−	5.28566i		−	0.183358i
$832$		0			0
$833$	1.27096	−	1.27096i	0.0440361	−	0.0440361i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	12.3113i		−	0.425541i
$838$		0			0
$839$			44.0039i			1.51918i	0.650401	+	0.759591i	$0.274601\pi$
$839$			44.0039i			1.51918i	−0.650401	+	0.759591i	$0.725399\pi$
$840$		0			0
$841$		−	8.48688i		−	0.292651i
$842$		0			0
$843$			2.05037i			0.0706186i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	18.8849	−	18.8849i	0.648893	−	0.648893i
$848$		0			0
$849$		−	11.4889i		−	0.394297i
$850$		0			0
$851$	11.2582	+	11.2582i	0.385925	+	0.385925i
$852$		0			0
$853$	31.3292			1.07269			0.536345	−	0.843999i	$-0.319805\pi$
$853$	31.3292			1.07269			0.536345	+	0.843999i	$0.319805\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.52558	+	3.52558i	−0.120432	+	0.120432i	−0.764754	−	0.644322i	$-0.777139\pi$
$857$	−3.52558	+	3.52558i	−0.120432	+	0.120432i	0.644322	+	0.764754i	$0.277139\pi$
$858$		0			0
$859$	24.7943	+	24.7943i	0.845972	+	0.845972i	0.989628	−	0.143656i	$-0.0458858\pi$
$859$	24.7943	+	24.7943i	0.845972	+	0.845972i	−0.143656	+	0.989628i	$0.545886\pi$
$860$		0			0
$861$	9.89260	−	9.89260i	0.337139	−	0.337139i
$862$		0			0
$863$	29.1489	+	29.1489i	0.992240	+	0.992240i	0.999970	−	0.00772992i	$-0.00246054\pi$
$863$	29.1489	+	29.1489i	0.992240	+	0.992240i	−0.00772992	+	0.999970i	$0.502461\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			5.98787i			0.203359i
$868$		0			0
$869$	−9.28593	+	9.28593i	−0.315003	+	0.315003i
$870$		0			0
$871$	−21.7088			−0.735573
$872$		0			0
$873$	16.8396	+	16.8396i	0.569936	+	0.569936i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.88742			−0.333874			−0.166937	−	0.985968i	$-0.553388\pi$
$877$	−9.88742			−0.333874			−0.166937	+	0.985968i	$0.553388\pi$
$878$		0			0
$879$	−12.3219			−0.415608
$880$		0			0
$881$	−15.9218			−0.536418			−0.268209	−	0.963361i	$-0.586432\pi$
$881$	−15.9218			−0.536418			−0.268209	+	0.963361i	$0.586432\pi$
$882$		0			0
$883$	45.1080			1.51800			0.759002	−	0.651088i	$-0.225687\pi$
$883$	45.1080			1.51800			0.759002	+	0.651088i	$0.225687\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.5079	+	16.5079i	0.554280	+	0.554280i	0.927673	−	0.373393i	$-0.121806\pi$
$887$	16.5079	+	16.5079i	0.554280	+	0.554280i	−0.373393	+	0.927673i	$0.621806\pi$
$888$		0			0
$889$	−0.578963			−0.0194178
$890$		0			0
$891$	3.23570	−	3.23570i	0.108400	−	0.108400i
$892$		0			0
$893$		−	29.0531i		−	0.972224i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	3.71911	+	3.71911i	0.124177	+	0.124177i
$898$		0			0
$899$	11.3399	−	11.3399i	0.378208	−	0.378208i
$900$		0			0
$901$	46.9849	+	46.9849i	1.56529	+	1.56529i
$902$		0			0
$903$	12.2605	−	12.2605i	0.408004	−	0.408004i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−38.9131			−1.29209			−0.646044	−	0.763300i	$-0.723578\pi$
$907$	−38.9131			−1.29209			−0.646044	+	0.763300i	$0.723578\pi$
$908$		0			0
$909$	14.0396	+	14.0396i	0.465665	+	0.465665i
$910$		0			0
$911$			17.5963i			0.582991i	0.956572	+	0.291495i	$0.0941528\pi$
$911$			17.5963i			0.582991i	−0.956572	+	0.291495i	$0.905847\pi$
$912$		0			0
$913$	0.222061	−	0.222061i	0.00734914	−	0.00734914i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	0.0176432i		−	0.000582631i
$918$		0			0
$919$		−	17.2370i		−	0.568596i	−0.958736	−	0.284298i	$-0.908240\pi$
$919$		−	17.2370i		−	0.568596i	0.958736	−	0.284298i	$-0.0917605\pi$
$920$		0			0
$921$			4.90722i			0.161698i
$922$		0			0
$923$		−	38.0940i		−	1.25388i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−30.2617	+	30.2617i	−0.993923	+	0.993923i
$928$		0			0
$929$		−	30.2159i		−	0.991350i	−0.868508	−	0.495675i	$-0.834921\pi$
$929$		−	30.2159i		−	0.991350i	0.868508	−	0.495675i	$-0.165079\pi$
$930$		0			0
$931$	1.56981	+	1.56981i	0.0514486	+	0.0514486i
$932$		0			0
$933$	−12.7635			−0.417860
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−29.8348	+	29.8348i	−0.974661	+	0.974661i	−0.999687	−	0.0250259i	$-0.992033\pi$
$937$	−29.8348	+	29.8348i	−0.974661	+	0.974661i	0.0250259	+	0.999687i	$0.492033\pi$
$938$		0			0
$939$	−5.01912	−	5.01912i	−0.163793	−	0.163793i
$940$		0			0
$941$	11.5151	−	11.5151i	0.375381	−	0.375381i	−0.494052	−	0.869432i	$-0.664485\pi$
$941$	11.5151	−	11.5151i	0.375381	−	0.375381i	0.869432	+	0.494052i	$0.164485\pi$
$942$		0			0
$943$	18.8858	+	18.8858i	0.615005	+	0.615005i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	25.0570i		−	0.814243i	−0.913374	−	0.407122i	$-0.866533\pi$
$947$		−	25.0570i		−	0.814243i	0.913374	−	0.407122i	$-0.133467\pi$
$948$		0			0
$949$	−3.74618	+	3.74618i	−0.121606	+	0.121606i
$950$		0			0
$951$	17.0057			0.551448
$952$		0			0
$953$	24.0330	+	24.0330i	0.778505	+	0.778505i	0.979577	−	0.201072i	$-0.0644424\pi$
$953$	24.0330	+	24.0330i	0.778505	+	0.778505i	−0.201072	+	0.979577i	$0.564442\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2.25147			−0.0727798
$958$		0			0
$959$	24.9775			0.806566
$960$		0			0
$961$	18.4622			0.595556
$962$		0			0
$963$	−20.3780			−0.656671
$964$		0			0
$965$		0			0
$966$		0			0
$967$	11.0419	+	11.0419i	0.355082	+	0.355082i	0.861996	−	0.506914i	$-0.169214\pi$
$967$	11.0419	+	11.0419i	0.355082	+	0.355082i	−0.506914	+	0.861996i	$0.669214\pi$
$968$		0			0
$969$	−20.3937			−0.655139
$970$		0			0
$971$	−31.6056	+	31.6056i	−1.01427	+	1.01427i	−0.0143746	+	0.999897i	$0.504576\pi$
$971$	−31.6056	+	31.6056i	−1.01427	+	1.01427i	−0.999897	+	0.0143746i	$0.995424\pi$
$972$		0			0
$973$			6.89005i			0.220885i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.8941	+	24.8941i	0.796433	+	0.796433i	0.982531	−	0.186098i	$-0.0595842\pi$
$977$	24.8941	+	24.8941i	0.796433	+	0.796433i	−0.186098	+	0.982531i	$0.559584\pi$
$978$		0			0
$979$	−10.2393	+	10.2393i	−0.327248	+	0.327248i
$980$		0			0
$981$	−11.5080	−	11.5080i	−0.367423	−	0.367423i
$982$		0			0
$983$	−15.8011	+	15.8011i	−0.503976	+	0.503976i	−0.912671	−	0.408695i	$-0.865984\pi$
$983$	−15.8011	+	15.8011i	−0.503976	+	0.503976i	0.408695	+	0.912671i	$0.365984\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.27138			0.231450
$988$		0			0
$989$	23.4063	+	23.4063i	0.744278	+	0.744278i
$990$		0			0
$991$		−	29.3466i		−	0.932226i	−0.884725	−	0.466113i	$-0.845654\pi$
$991$		−	29.3466i		−	0.932226i	0.884725	−	0.466113i	$-0.154346\pi$
$992$		0			0
$993$	−11.4664	+	11.4664i	−0.363876	+	0.363876i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			26.7001i			0.845601i	0.906223	+	0.422800i	$0.138953\pi$
$997$			26.7001i			0.845601i	−0.906223	+	0.422800i	$0.861047\pi$
$998$		0			0
$999$		−	18.1624i		−	0.574633i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.s.e.207.7		24
4.3	odd	2		400.2.s.e.107.9	yes	24
5.2	odd	4		1600.2.j.e.143.6		24
5.3	odd	4		1600.2.j.e.143.7		24
5.4	even	2	inner	1600.2.s.e.207.6		24
16.3	odd	4		1600.2.j.e.1007.6		24
16.13	even	4		400.2.j.e.307.10	yes	24
20.3	even	4		400.2.j.e.43.10	yes	24
20.7	even	4		400.2.j.e.43.3	✓	24
20.19	odd	2		400.2.s.e.107.4	yes	24
80.3	even	4	inner	1600.2.s.e.943.7		24
80.13	odd	4		400.2.s.e.243.9	yes	24
80.19	odd	4		1600.2.j.e.1007.7		24
80.29	even	4		400.2.j.e.307.3	yes	24
80.67	even	4	inner	1600.2.s.e.943.6		24
80.77	odd	4		400.2.s.e.243.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.3	✓	24	20.7	even	4
400.2.j.e.43.10	yes	24	20.3	even	4
400.2.j.e.307.3	yes	24	80.29	even	4
400.2.j.e.307.10	yes	24	16.13	even	4
400.2.s.e.107.4	yes	24	20.19	odd	2
400.2.s.e.107.9	yes	24	4.3	odd	2
400.2.s.e.243.4	yes	24	80.77	odd	4
400.2.s.e.243.9	yes	24	80.13	odd	4
1600.2.j.e.143.6		24	5.2	odd	4
1600.2.j.e.143.7		24	5.3	odd	4
1600.2.j.e.1007.6		24	16.3	odd	4
1600.2.j.e.1007.7		24	80.19	odd	4
1600.2.s.e.207.6		24	5.4	even	2	inner
1600.2.s.e.207.7		24	1.1	even	1	trivial
1600.2.s.e.943.6		24	80.67	even	4	inner
1600.2.s.e.943.7		24	80.3	even	4	inner

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.619018 q^{3} +(-1.82373 - 1.82373i) q^{7} -2.61682 q^{9} +O(q^{10})\)	\(q+0.619018 q^{3} +(-1.82373 - 1.82373i) q^{7} -2.61682 q^{9} +(0.567849 - 0.567849i) q^{11} +2.78771i q^{13} +(3.65193 + 3.65193i) q^{17} +(-4.51065 + 4.51065i) q^{19} +(-1.12892 - 1.12892i) q^{21} +(2.15520 - 2.15520i) q^{23} -3.47691 q^{27} +(-3.20259 - 3.20259i) q^{29} +3.54087i q^{31} +(0.351509 - 0.351509i) q^{33} +5.22371i q^{37} +1.72564i q^{39} +8.76287i q^{41} +10.8604i q^{43} +(-3.22050 + 3.22050i) q^{47} -0.348024i q^{49} +(2.26061 + 2.26061i) q^{51} +12.8658 q^{53} +(-2.79218 + 2.79218i) q^{57} +(3.79319 + 3.79319i) q^{59} +(6.63395 - 6.63395i) q^{61} +(4.77236 + 4.77236i) q^{63} +7.78732i q^{67} +(1.33411 - 1.33411i) q^{69} -13.6650 q^{71} +(1.34382 + 1.34382i) q^{73} -2.07120 q^{77} -16.3528 q^{79} +5.69818 q^{81} +0.391056 q^{83} +(-1.98246 - 1.98246i) q^{87} -18.0317 q^{89} +(5.08402 - 5.08402i) q^{91} +2.19186i q^{93} +(-6.43517 - 6.43517i) q^{97} +(-1.48596 + 1.48596i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{9}+O(q^{10}) \) 24 * q + 40 * q^9	\( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \) 24 * q + 40 * q^9 + 20 * q^11 - 12 * q^19 - 8 * q^29 - 20 * q^51 + 8 * q^59 - 48 * q^61 + 64 * q^69 + 16 * q^71 - 104 * q^79 + 48 * q^81 - 96 * q^89 - 64 * q^91 + 128 * q^99

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