LMFDB - Embedded newform 1600.2.j.e.143.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(143,1600)]

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, 3]))

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

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

chi := DirichletCharacter("1600.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.j (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			143.7
Character	$\chi$	$=$	1600.143
Dual form			1600.2.j.e.1007.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.619018i q^{3} +(-1.82373 + 1.82373i) q^{7} +2.61682 q^{9} +O(q^{10})$	$q+0.619018i q^{3} +(-1.82373 + 1.82373i) q^{7} +2.61682 q^{9} +(0.567849 - 0.567849i) q^{11} -2.78771 q^{13} +(3.65193 - 3.65193i) q^{17} +(4.51065 - 4.51065i) q^{19} +(-1.12892 - 1.12892i) q^{21} +(2.15520 + 2.15520i) q^{23} +3.47691i q^{27} +(3.20259 + 3.20259i) q^{29} +3.54087i q^{31} +(0.351509 + 0.351509i) q^{33} +5.22371 q^{37} -1.72564i q^{39} +8.76287i q^{41} -10.8604 q^{43} +(3.22050 + 3.22050i) q^{47} +0.348024i q^{49} +(2.26061 + 2.26061i) q^{51} +12.8658i q^{53} +(2.79218 + 2.79218i) q^{57} +(-3.79319 - 3.79319i) q^{59} +(6.63395 - 6.63395i) q^{61} +(-4.77236 + 4.77236i) q^{63} +7.78732 q^{67} +(-1.33411 + 1.33411i) q^{69} -13.6650 q^{71} +(-1.34382 + 1.34382i) q^{73} +2.07120i q^{77} +16.3528 q^{79} +5.69818 q^{81} +0.391056i q^{83} +(-1.98246 + 1.98246i) q^{87} +18.0317 q^{89} +(5.08402 - 5.08402i) q^{91} -2.19186 q^{93} +(-6.43517 + 6.43517i) q^{97} +(1.48596 - 1.48596i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

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

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
$3$			0.619018i			0.357390i	0.983904	+	0.178695i	$0.0571876\pi$
$3$			0.619018i			0.357390i	−0.983904	+	0.178695i	$0.942812\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.82373	+	1.82373i	−0.689305	+	0.689305i	−0.962078	−	0.272773i	$-0.912059\pi$
$7$	−1.82373	+	1.82373i	−0.689305	+	0.689305i	0.272773	+	0.962078i	$0.412059\pi$
$8$		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
$13$	−2.78771			−0.773171			−0.386585	−	0.922254i	$-0.626346\pi$
$13$	−2.78771			−0.773171			−0.386585	+	0.922254i	$0.626346\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.65193	−	3.65193i	0.885723	−	0.885723i	−0.108386	−	0.994109i	$-0.534568\pi$
$17$	3.65193	−	3.65193i	0.885723	−	0.885723i	0.994109	+	0.108386i	$0.0345683\pi$
$18$		0			0
$19$	4.51065	−	4.51065i	1.03481	−	1.03481i	0.0354432	−	0.999372i	$-0.488716\pi$
$19$	4.51065	−	4.51065i	1.03481	−	1.03481i	0.999372	−	0.0354432i	$-0.0112843\pi$
$20$		0			0
$21$	−1.12892	−	1.12892i	−0.246351	−	0.246351i
$22$		0			0
$23$	2.15520	+	2.15520i	0.449391	+	0.449391i	0.895152	−	0.445761i	$-0.147067\pi$
$23$	2.15520	+	2.15520i	0.449391	+	0.449391i	−0.445761	+	0.895152i	$0.647067\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			3.47691i			0.669132i
$28$		0			0
$29$	3.20259	+	3.20259i	0.594705	+	0.594705i	0.938899	−	0.344193i	$-0.111848\pi$
$29$	3.20259	+	3.20259i	0.594705	+	0.594705i	−0.344193	+	0.938899i	$0.611848\pi$
$30$		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
$37$	5.22371			0.858773			0.429386	−	0.903121i	$-0.358730\pi$
$37$	5.22371			0.858773			0.429386	+	0.903121i	$0.358730\pi$
$38$		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
$43$	−10.8604			−1.65619			−0.828096	−	0.560587i	$-0.810576\pi$
$43$	−10.8604			−1.65619			−0.828096	+	0.560587i	$0.810576\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.22050	+	3.22050i	0.469758	+	0.469758i	0.901836	−	0.432078i	$-0.142220\pi$
$47$	3.22050	+	3.22050i	0.469758	+	0.469758i	−0.432078	+	0.901836i	$0.642220\pi$
$48$		0			0
$49$			0.348024i			0.0497176i
$50$		0			0
$51$	2.26061	+	2.26061i	0.316549	+	0.316549i
$52$		0			0
$53$			12.8658i			1.76725i	0.468194	+	0.883626i	$0.344905\pi$
$53$			12.8658i			1.76725i	−0.468194	+	0.883626i	$0.655095\pi$
$54$		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	0.415679	−	0.909511i	$-0.363544\pi$
$59$	−3.79319	−	3.79319i	−0.493832	−	0.493832i	−0.909511	+	0.415679i	$0.863544\pi$
$60$		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
$67$	7.78732			0.951373			0.475686	−	0.879615i	$-0.342200\pi$
$67$	7.78732			0.951373			0.475686	+	0.879615i	$0.342200\pi$
$68$		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	$-0.785474\pi$
$73$	−1.34382	+	1.34382i	−0.157282	+	0.157282i	0.624079	+	0.781361i	$0.285474\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.07120i			0.236036i
$78$		0			0
$79$	16.3528			1.83984			0.919918	−	0.392111i	$-0.128255\pi$
$79$	16.3528			1.83984			0.919918	+	0.392111i	$0.128255\pi$
$80$		0			0
$81$	5.69818			0.633131
$82$		0			0
$83$			0.391056i			0.0429240i	0.999770	+	0.0214620i	$0.00683209\pi$
$83$			0.391056i			0.0429240i	−0.999770	+	0.0214620i	$0.993168\pi$
$84$		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	$-0.0951263\pi$
$89$	18.0317			1.91135			0.955676	+	0.294419i	$0.0951263\pi$
$90$		0			0
$91$	5.08402	−	5.08402i	0.532950	−	0.532950i
$92$		0			0
$93$	−2.19186			−0.227286
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.43517	+	6.43517i	−0.653392	+	0.653392i	−0.953808	−	0.300416i	$-0.902874\pi$
$97$	−6.43517	+	6.43517i	−0.653392	+	0.653392i	0.300416	+	0.953808i	$0.402874\pi$
$98$		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	0.988546	+	0.150919i	$0.0482231\pi$
$103$	11.5643	+	11.5643i	1.13946	+	1.13946i	0.150919	+	0.988546i	$0.451777\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.78732i		−	0.752829i	−0.926451	−	0.376414i	$-0.877157\pi$
$107$		−	7.78732i		−	0.752829i	0.926451	−	0.376414i	$-0.122843\pi$
$108$		0			0
$109$	−4.39771	−	4.39771i	−0.421225	−	0.421225i	0.464401	−	0.885625i	$-0.346270\pi$
$109$	−4.39771	−	4.39771i	−0.421225	−	0.421225i	−0.885625	+	0.464401i	$0.846270\pi$
$110$		0			0
$111$			3.23357i			0.306917i
$112$		0			0
$113$	−3.76206	−	3.76206i	−0.353905	−	0.353905i	0.507655	−	0.861560i	$-0.330512\pi$
$113$	−3.76206	−	3.76206i	−0.353905	−	0.353905i	−0.861560	+	0.507655i	$0.830512\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.29492			−0.674415
$118$		0			0
$119$			13.3203i			1.22107i
$120$		0			0
$121$			10.3551i			0.941372i
$122$		0			0
$123$	−5.42438			−0.489100
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.158731	−	0.158731i	−0.0140851	−	0.0140851i	0.700029	−	0.714114i	$-0.253170\pi$
$127$	−0.158731	−	0.158731i	−0.0140851	−	0.0140851i	−0.714114	+	0.700029i	$0.753170\pi$
$128$		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
$133$			16.4524i			1.42661i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.84792	+	6.84792i	0.585057	+	0.585057i	0.936289	−	0.351231i	$-0.114237\pi$
$137$	6.84792	+	6.84792i	0.585057	+	0.585057i	−0.351231	+	0.936289i	$0.614237\pi$
$138$		0			0
$139$	1.88900	+	1.88900i	0.160223	+	0.160223i	0.782665	−	0.622443i	$-0.213860\pi$
$139$	1.88900	+	1.88900i	0.160223	+	0.160223i	−0.622443	+	0.782665i	$0.713860\pi$
$140$		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
$147$	−0.215433			−0.0177686
$148$		0			0
$149$	3.47398	−	3.47398i	0.284600	−	0.284600i	−0.550341	−	0.834940i	$-0.685502\pi$
$149$	3.47398	−	3.47398i	0.284600	−	0.284600i	0.834940	+	0.550341i	$0.185502\pi$
$150$		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
$157$		−	1.52142i		−	0.121422i	−0.998155	−	0.0607112i	$-0.980663\pi$
$157$		−	1.52142i		−	0.121422i	0.998155	−	0.0607112i	$-0.0193369\pi$
$158$		0			0
$159$	−7.96416			−0.631599
$160$		0			0
$161$	−7.86102			−0.619535
$162$		0			0
$163$		−	8.50837i		−	0.666427i	−0.942851	−	0.333213i	$-0.891867\pi$
$163$		−	8.50837i		−	0.666427i	0.942851	−	0.333213i	$-0.108133\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.9639	−	10.9639i	0.848411	−	0.848411i	−0.141524	−	0.989935i	$-0.545200\pi$
$167$	10.9639	−	10.9639i	0.848411	−	0.848411i	0.989935	+	0.141524i	$0.0452002\pi$
$168$		0			0
$169$	−5.22869			−0.402207
$170$		0			0
$171$	11.8036	−	11.8036i	0.902640	−	0.902640i
$172$		0			0
$173$	−15.9782			−1.21480			−0.607401	−	0.794395i	$-0.707788\pi$
$173$	−15.9782			−1.21480			−0.607401	+	0.794395i	$0.707788\pi$
$174$		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	−0.615161	−	0.788402i	$-0.710909\pi$
$179$	2.31781	−	2.31781i	0.173241	−	0.173241i	0.788402	+	0.615161i	$0.210909\pi$
$180$		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
$187$		−	4.14749i		−	0.303294i
$188$		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	0.632823	−	0.774297i	$-0.281896\pi$
$193$	−1.96542	−	1.96542i	−0.141474	−	0.141474i	−0.774297	+	0.632823i	$0.781896\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.27799			−0.162300			−0.0811500	−	0.996702i	$-0.525859\pi$
$197$	−2.27799			−0.162300			−0.0811500	+	0.996702i	$0.525859\pi$
$198$		0			0
$199$			10.0426i			0.711902i	0.934505	+	0.355951i	$0.115843\pi$
$199$			10.0426i			0.711902i	−0.934505	+	0.355951i	$0.884157\pi$
$200$		0			0
$201$			4.82049i			0.340011i
$202$		0			0
$203$	−11.6813			−0.819867
$204$		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
$213$		−	8.45888i		−	0.579593i
$214$		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	0.302246	−	0.953230i	$-0.402264\pi$
$223$	18.7483	−	18.7483i	1.25548	−	1.25548i	0.953230	−	0.302246i	$-0.0977365\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.29139			−0.616691			−0.308346	−	0.951274i	$-0.599775\pi$
$227$	−9.29139			−0.616691			−0.308346	+	0.951274i	$0.599775\pi$
$228$		0			0
$229$	−12.9965	+	12.9965i	−0.858833	+	0.858833i	−0.991201	−	0.132368i	$-0.957742\pi$
$229$	−12.9965	+	12.9965i	−0.858833	+	0.858833i	0.132368	+	0.991201i	$0.457742\pi$
$230$		0			0
$231$	−1.28211			−0.0843569
$232$		0			0
$233$	12.0407	−	12.0407i	0.788812	−	0.788812i	−0.192487	−	0.981299i	$-0.561655\pi$
$233$	12.0407	−	12.0407i	0.788812	−	0.788812i	0.981299	+	0.192487i	$0.0616554\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			10.1227i			0.657539i
$238$		0			0
$239$	−11.3495			−0.734140			−0.367070	−	0.930193i	$-0.619639\pi$
$239$	−11.3495			−0.734140			−0.367070	+	0.930193i	$0.619639\pi$
$240$		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
$243$			13.9580i			0.895407i
$244$		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
$253$	2.44766			0.153883
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.35316	−	6.35316i	0.396299	−	0.396299i	−0.480626	−	0.876926i	$-0.659591\pi$
$257$	6.35316	−	6.35316i	0.396299	−	0.396299i	0.876926	+	0.480626i	$0.159591\pi$
$258$		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	0.651933	−	0.758277i	$-0.273958\pi$
$263$	−1.72461	−	1.72461i	−0.106344	−	0.106344i	−0.758277	+	0.651933i	$0.773958\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			11.1619i			0.683099i
$268$		0			0
$269$	−8.87466	−	8.87466i	−0.541098	−	0.541098i	0.382753	−	0.923851i	$-0.374976\pi$
$269$	−8.87466	−	8.87466i	−0.541098	−	0.541098i	−0.923851	+	0.382753i	$0.874976\pi$
$270$		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
$277$	−8.53878			−0.513046			−0.256523	−	0.966538i	$-0.582577\pi$
$277$	−8.53878			−0.513046			−0.256523	+	0.966538i	$0.582577\pi$
$278$		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
$283$	18.5598			1.10327			0.551634	−	0.834086i	$-0.314005\pi$
$283$	18.5598			1.10327			0.551634	+	0.834086i	$0.314005\pi$
$284$		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
$293$		−	19.9056i		−	1.16290i	−0.813584	−	0.581448i	$-0.802486\pi$
$293$		−	19.9056i		−	1.16290i	0.813584	−	0.581448i	$-0.197514\pi$
$294$		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
$307$	7.92742			0.452442			0.226221	−	0.974076i	$-0.427363\pi$
$307$	7.92742			0.452442			0.226221	+	0.974076i	$0.427363\pi$
$308$		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	$-0.644949\pi$
$313$	8.10819	−	8.10819i	0.458302	−	0.458302i	0.898098	+	0.439796i	$0.144949\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.4721i		−	1.54298i	−0.636239	−	0.771492i	$-0.719511\pi$
$317$		−	27.4721i		−	1.54298i	0.636239	−	0.771492i	$-0.280489\pi$
$318$		0			0
$319$	3.63717			0.203642
$320$		0			0
$321$	4.82049			0.269054
$322$		0			0
$323$		−	32.9452i		−	1.83312i
$324$		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
$333$	13.6695			0.749084
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.6567	+	14.6567i	−0.798402	+	0.798402i	−0.982843	−	0.184442i	$-0.940952\pi$
$337$	−14.6567	+	14.6567i	−0.798402	+	0.798402i	0.184442	+	0.982843i	$0.440952\pi$
$338$		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
$347$		−	25.0440i		−	1.34443i	−0.740354	−	0.672217i	$-0.765342\pi$
$347$		−	25.0440i		−	1.34443i	0.740354	−	0.672217i	$-0.234658\pi$
$348$		0			0
$349$	17.5767	+	17.5767i	0.940860	+	0.940860i	0.998346	−	0.0574866i	$-0.0183087\pi$
$349$	17.5767	+	17.5767i	0.940860	+	0.940860i	−0.0574866	+	0.998346i	$0.518309\pi$
$350$		0			0
$351$		−	9.69261i		−	0.517353i
$352$		0			0
$353$	−6.30855	−	6.30855i	−0.335770	−	0.335770i	0.519002	−	0.854773i	$-0.326304\pi$
$353$	−6.30855	−	6.30855i	−0.335770	−	0.335770i	−0.854773	+	0.519002i	$0.826304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.24548			−0.436397
$358$		0			0
$359$		−	27.8527i		−	1.47001i	−0.678061	−	0.735005i	$-0.737180\pi$
$359$		−	27.8527i		−	1.47001i	0.678061	−	0.735005i	$-0.262820\pi$
$360$		0			0
$361$		−	21.6920i		−	1.14168i
$362$		0			0
$363$	−6.40999			−0.336437
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−3.29618	−	3.29618i	−0.172059	−	0.172059i	0.615824	−	0.787883i	$-0.288823\pi$
$367$	−3.29618	−	3.29618i	−0.172059	−	0.172059i	−0.787883	+	0.615824i	$0.788823\pi$
$368$		0			0
$369$			22.9308i			1.19373i
$370$		0			0
$371$	−23.4637	−	23.4637i	−1.21818	−	1.21818i
$372$		0			0
$373$			5.93152i			0.307122i	0.988139	+	0.153561i	$0.0490742\pi$
$373$			5.93152i			0.307122i	−0.988139	+	0.153561i	$0.950926\pi$
$374$		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	$-0.980287\pi$
$379$	−20.6355	−	20.6355i	−1.05997	−	1.05997i	−0.0618894	−	0.998083i	$-0.519713\pi$
$380$		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	$-0.672713\pi$
$383$	6.65419	−	6.65419i	0.340013	−	0.340013i	0.856372	+	0.516359i	$0.172713\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−28.4196			−1.44465
$388$		0			0
$389$	2.50799	−	2.50799i	0.127160	−	0.127160i	−0.640662	−	0.767823i	$-0.721340\pi$
$389$	2.50799	−	2.50799i	0.127160	−	0.127160i	0.767823	+	0.640662i	$0.221340\pi$
$390$		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
$397$		−	17.9719i		−	0.901986i	−0.892528	−	0.450993i	$-0.851070\pi$
$397$		−	17.9719i		−	0.901986i	0.892528	−	0.450993i	$-0.148930\pi$
$398$		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
$403$		−	9.87091i		−	0.491705i
$404$		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	$-0.653808\pi$
$409$	−18.7926			−0.929233			−0.464617	+	0.885512i	$0.653808\pi$
$410$		0			0
$411$	−4.23899	+	4.23899i	−0.209094	+	0.209094i
$412$		0			0
$413$	13.8355			0.680801
$414$		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	−0.897036	−	0.441958i	$-0.854284\pi$
$419$	−9.31520	+	9.31520i	−0.455077	+	0.455077i	0.441958	+	0.897036i	$0.354284\pi$
$420$		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
$427$			24.1971i			1.17098i
$428$		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	0.681239	−	0.732061i	$-0.261441\pi$
$433$	−1.05752	−	1.05752i	−0.0508212	−	0.0508212i	−0.732061	+	0.681239i	$0.761441\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	19.4428			0.930073
$438$		0			0
$439$		−	12.3556i		−	0.589702i	−0.955543	−	0.294851i	$-0.904730\pi$
$439$		−	12.3556i		−	0.589702i	0.955543	−	0.294851i	$-0.0952700\pi$
$440$		0			0
$441$			0.910714i			0.0433673i
$442$		0			0
$443$	10.9254			0.519082			0.259541	−	0.965732i	$-0.416429\pi$
$443$	10.9254			0.519082			0.259541	+	0.965732i	$0.416429\pi$
$444$		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	$0.352066\pi$
$449$			37.8842i			1.78787i	−0.448199	+	0.893934i	$0.647934\pi$
$450$		0			0
$451$	4.97599	+	4.97599i	0.234310	+	0.234310i
$452$		0			0
$453$		−	1.20046i		−	0.0564025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.82262	−	6.82262i	−0.319149	−	0.319149i	0.529291	−	0.848440i	$-0.322458\pi$
$457$	−6.82262	−	6.82262i	−0.319149	−	0.319149i	−0.848440	+	0.529291i	$0.822458\pi$
$458$		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	$-0.950703\pi$
$463$	−17.9408	+	17.9408i	−0.833780	+	0.833780i	0.154251	+	0.988032i	$0.450703\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.2034			0.518430			0.259215	−	0.965820i	$-0.416536\pi$
$467$	11.2034			0.518430			0.259215	+	0.965820i	$0.416536\pi$
$468$		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
$477$			33.6674i			1.54152i
$478$		0			0
$479$	−25.3283			−1.15728			−0.578641	−	0.815583i	$-0.696417\pi$
$479$	−25.3283			−1.15728			−0.578641	+	0.815583i	$0.696417\pi$
$480$		0			0
$481$	−14.5622			−0.663978
$482$		0			0
$483$		−	4.86611i		−	0.221416i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.6181	+	11.6181i	−0.526466	+	0.526466i	−0.919517	−	0.393051i	$-0.871420\pi$
$487$	−11.6181	+	11.6181i	−0.526466	+	0.526466i	0.393051	+	0.919517i	$0.371420\pi$
$488$		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
$493$	23.3912			1.05349
$494$		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	−0.885072	−	0.465453i	$-0.845891\pi$
$499$	−9.37358	+	9.37358i	−0.419619	+	0.419619i	0.465453	+	0.885072i	$0.345891\pi$
$500$		0			0
$501$	6.78685	+	6.78685i	0.303214	+	0.303214i
$502$		0			0
$503$	−24.6142	−	24.6142i	−1.09749	−	1.09749i	−0.994703	−	0.102788i	$-0.967224\pi$
$503$	−24.6142	−	24.6142i	−1.09749	−	1.09749i	−0.102788	−	0.994703i	$-0.532776\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	3.23666i		−	0.143745i
$508$		0			0
$509$	−17.7086	−	17.7086i	−0.784920	−	0.784920i	0.195737	−	0.980656i	$-0.437290\pi$
$509$	−17.7086	−	17.7086i	−0.784920	−	0.784920i	−0.980656	+	0.195737i	$0.937290\pi$
$510$		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
$517$	3.65751			0.160857
$518$		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
$523$	30.6587			1.34061			0.670306	−	0.742085i	$-0.266163\pi$
$523$	30.6587			1.34061			0.670306	+	0.742085i	$0.266163\pi$
$524$		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
$533$		−	24.4283i		−	1.05811i
$534$		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
$547$	−12.3389			−0.527575			−0.263788	−	0.964581i	$-0.584972\pi$
$547$	−12.3389			−0.527575			−0.263788	+	0.964581i	$0.584972\pi$
$548$		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
$557$			2.93717i			0.124452i	0.998062	+	0.0622259i	$0.0198199\pi$
$557$			2.93717i			0.124452i	−0.998062	+	0.0622259i	$0.980180\pi$
$558$		0			0
$559$	30.2755			1.28052
$560$		0			0
$561$	2.56737			0.108394
$562$		0			0
$563$			41.2139i			1.73696i	0.495725	+	0.868480i	$0.334903\pi$
$563$			41.2139i			1.73696i	−0.495725	+	0.868480i	$0.665097\pi$
$564$		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	$-0.621453\pi$
$569$	−17.7646			−0.744730			−0.372365	+	0.928086i	$0.621453\pi$
$570$		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
$573$	−1.09821			−0.0458786
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.54451	+	3.54451i	−0.147560	+	0.147560i	−0.777027	−	0.629467i	$-0.783273\pi$
$577$	−3.54451	+	3.54451i	−0.147560	+	0.147560i	0.629467	+	0.777027i	$0.283273\pi$
$578$		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
$587$			2.69644i			0.111294i	0.998451	+	0.0556469i	$0.0177221\pi$
$587$			2.69644i			0.111294i	−0.998451	+	0.0556469i	$0.982278\pi$
$588$		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	0.333743	−	0.942664i	$-0.391688\pi$
$593$	−14.8282	−	14.8282i	−0.608922	−	0.608922i	−0.942664	+	0.333743i	$0.891688\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−6.21656			−0.254427
$598$		0			0
$599$			29.5933i			1.20915i	0.796548	+	0.604576i	$0.206657\pi$
$599$			29.5933i			1.20915i	−0.796548	+	0.604576i	$0.793343\pi$
$600$		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
$603$	20.3780			0.829856
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−9.33600	−	9.33600i	−0.378937	−	0.378937i	0.491782	−	0.870718i	$-0.336346\pi$
$607$	−9.33600	−	9.33600i	−0.378937	−	0.378937i	−0.870718	+	0.491782i	$0.836346\pi$
$608$		0			0
$609$		−	7.23094i		−	0.293012i
$610$		0			0
$611$	−8.97780	−	8.97780i	−0.363203	−	0.363203i
$612$		0			0
$613$		−	30.5843i		−	1.23529i	−0.786458	−	0.617643i	$-0.788088\pi$
$613$		−	30.5843i		−	1.23529i	0.786458	−	0.617643i	$-0.211912\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.7311	−	21.7311i	−0.874862	−	0.874862i	0.118135	−	0.992998i	$-0.462308\pi$
$617$	−21.7311	−	21.7311i	−0.874862	−	0.874862i	−0.992998	+	0.118135i	$0.962308\pi$
$618$		0			0
$619$	28.6904	+	28.6904i	1.15317	+	1.15317i	0.985914	+	0.167252i	$0.0534893\pi$
$619$	28.6904	+	28.6904i	1.15317	+	1.15317i	0.167252	+	0.985914i	$0.446511\pi$
$620$		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
$627$	3.17107			0.126640
$628$		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
$637$		−	0.970188i		−	0.0384402i
$638$		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
$643$			23.2479i			0.916807i	0.888744	+	0.458403i	$0.151578\pi$
$643$			23.2479i			0.916807i	−0.888744	+	0.458403i	$0.848422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.1518	−	19.1518i	0.752937	−	0.752937i	−0.222090	−	0.975026i	$-0.571288\pi$
$647$	19.1518	−	19.1518i	0.752937	−	0.752937i	0.975026	+	0.222090i	$0.0712878\pi$
$648$		0			0
$649$	−4.30792			−0.169101
$650$		0			0
$651$	3.99737	−	3.99737i	0.156669	−	0.156669i
$652$		0			0
$653$	−26.8221			−1.04963			−0.524816	−	0.851216i	$-0.675866\pi$
$653$	−26.8221			−1.04963			−0.524816	+	0.851216i	$0.675866\pi$
$654$		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	$-0.325504\pi$
$659$	35.2877	−	35.2877i	1.37461	−	1.37461i	0.853467	−	0.521148i	$-0.174496\pi$
$660$		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
$667$			13.8045i			0.534511i
$668$		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	0.968190	−	0.250218i	$-0.0805022\pi$
$673$	18.6258	+	18.6258i	0.717972	+	0.717972i	−0.250218	+	0.968190i	$0.580502\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.8471			0.609054			0.304527	−	0.952504i	$-0.401502\pi$
$677$	15.8471			0.609054			0.304527	+	0.952504i	$0.401502\pi$
$678$		0			0
$679$		−	23.4720i		−	0.900773i
$680$		0			0
$681$		−	5.75154i		−	0.220400i
$682$		0			0
$683$	−34.7894			−1.33118			−0.665590	−	0.746318i	$-0.731820\pi$
$683$	−34.7894			−1.33118			−0.665590	+	0.746318i	$0.731820\pi$
$684$		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
$693$			5.41996i			0.205887i
$694$		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
$707$	19.5692			0.735975
$708$		0			0
$709$	28.5783	−	28.5783i	1.07328	−	1.07328i	0.0761883	−	0.997093i	$-0.475725\pi$
$709$	28.5783	−	28.5783i	1.07328	−	1.07328i	0.997093	−	0.0761883i	$-0.0242750\pi$
$710$		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
$717$		−	7.02557i		−	0.262375i
$718$		0			0
$719$	37.8803			1.41270			0.706348	−	0.707865i	$-0.250342\pi$
$719$	37.8803			1.41270			0.706348	+	0.707865i	$0.250342\pi$
$720$		0			0
$721$	−42.1803			−1.57088
$722$		0			0
$723$			4.84968i			0.180361i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	12.8109	−	12.8109i	0.475130	−	0.475130i	−0.428440	−	0.903570i	$-0.640937\pi$
$727$	12.8109	−	12.8109i	0.475130	−	0.475130i	0.903570	+	0.428440i	$0.140937\pi$
$728$		0			0
$729$	8.45427			0.313121
$730$		0			0
$731$	−39.6613	+	39.6613i	−1.46693	+	1.46693i
$732$		0			0
$733$	−8.70612			−0.321568			−0.160784	−	0.986990i	$-0.551402\pi$
$733$	−8.70612			−0.321568			−0.160784	+	0.986990i	$0.551402\pi$
$734$		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	$0.580924\pi$
$739$	−33.1477	+	33.1477i	−1.21936	+	1.21936i	−0.967857	+	0.251502i	$0.919076\pi$
$740$		0			0
$741$	−7.78377	−	7.78377i	−0.285944	−	0.285944i
$742$		0			0
$743$	−6.97405	−	6.97405i	−0.255853	−	0.255853i	0.567512	−	0.823365i	$-0.307906\pi$
$743$	−6.97405	−	6.97405i	−0.255853	−	0.255853i	−0.823365	+	0.567512i	$0.807906\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			1.02332i			0.0374414i
$748$		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
$757$	8.40963			0.305653			0.152827	−	0.988253i	$-0.451162\pi$
$757$	8.40963			0.305653			0.152827	+	0.988253i	$0.451162\pi$
$758$		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
$763$	16.0405			0.580704
$764$		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	$-0.888512\pi$
$769$		−	19.0307i		−	0.686264i	0.939287	−	0.343132i	$-0.111488\pi$
$770$		0			0
$771$	3.93272	+	3.93272i	0.141634	+	0.141634i
$772$		0			0
$773$			3.20152i			0.115151i	0.998341	+	0.0575753i	$0.0183369\pi$
$773$			3.20152i			0.115151i	−0.998341	+	0.0575753i	$0.981663\pi$
$774$		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
$787$	52.2753			1.86341			0.931706	−	0.363213i	$-0.118320\pi$
$787$	52.2753			1.86341			0.931706	+	0.363213i	$0.118320\pi$
$788$		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
$797$			49.3883i			1.74942i	0.484643	+	0.874712i	$0.338950\pi$
$797$			49.3883i			1.74942i	−0.484643	+	0.874712i	$0.661050\pi$
$798$		0			0
$799$	23.5220			0.832150
$800$		0			0
$801$	47.1856			1.66722
$802$		0			0
$803$			1.52617i			0.0538575i
$804$		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	$-0.453807\pi$
$809$	8.22627			0.289220			0.144610	+	0.989489i	$0.453807\pi$
$810$		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
$813$	1.61943			0.0567960
$814$		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	−0.999900	−	0.0141712i	$-0.995489\pi$
$823$	−29.0916	−	29.0916i	−1.01407	−	1.01407i	−0.0141712	−	0.999900i	$-0.504511\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	36.0413i		−	1.25328i	−0.779310	−	0.626639i	$-0.784430\pi$
$827$		−	36.0413i		−	1.25328i	0.779310	−	0.626639i	$-0.215570\pi$
$828$		0			0
$829$	3.76098	+	3.76098i	0.130624	+	0.130624i	0.769396	−	0.638772i	$-0.220557\pi$
$829$	3.76098	+	3.76098i	0.130624	+	0.130624i	−0.638772	+	0.769396i	$0.720557\pi$
$830$		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
$837$	−12.3113			−0.425541
$838$		0			0
$839$		−	44.0039i		−	1.51918i	−0.650401	−	0.759591i	$-0.725399\pi$
$839$		−	44.0039i		−	1.51918i	0.650401	−	0.759591i	$-0.274601\pi$
$840$		0			0
$841$		−	8.48688i		−	0.292651i
$842$		0			0
$843$	−2.05037			−0.0706186
$844$		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
$853$			31.3292i			1.07269i	0.843999	+	0.536345i	$0.180195\pi$
$853$			31.3292i			1.07269i	−0.843999	+	0.536345i	$0.819805\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.52558	+	3.52558i	0.120432	+	0.120432i	0.764754	−	0.644322i	$-0.222861\pi$
$857$	3.52558	+	3.52558i	0.120432	+	0.120432i	−0.644322	+	0.764754i	$0.722861\pi$
$858$		0			0
$859$	−24.7943	−	24.7943i	−0.845972	−	0.845972i	0.143656	−	0.989628i	$-0.454114\pi$
$859$	−24.7943	−	24.7943i	−0.845972	−	0.845972i	−0.989628	+	0.143656i	$0.954114\pi$
$860$		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	$-0.997539\pi$
$863$	−29.1489	+	29.1489i	−0.992240	+	0.992240i	0.00772992	+	0.999970i	$0.497539\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.98787			0.203359
$868$		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
$877$			9.88742i			0.333874i	0.985968	+	0.166937i	$0.0533877\pi$
$877$			9.88742i			0.333874i	−0.985968	+	0.166937i	$0.946612\pi$
$878$		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
$883$			45.1080i			1.51800i	0.651088	+	0.759002i	$0.274313\pi$
$883$			45.1080i			1.51800i	−0.651088	+	0.759002i	$0.725687\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.5079	−	16.5079i	0.554280	−	0.554280i	−0.373393	−	0.927673i	$-0.621806\pi$
$887$	16.5079	−	16.5079i	0.554280	−	0.554280i	0.927673	+	0.373393i	$0.121806\pi$
$888$		0			0
$889$	0.578963			0.0194178
$890$		0			0
$891$	3.23570	−	3.23570i	0.108400	−	0.108400i
$892$		0			0
$893$	29.0531			0.972224
$894$		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
$907$			38.9131i			1.29209i	0.763300	+	0.646044i	$0.223578\pi$
$907$			38.9131i			1.29209i	−0.763300	+	0.646044i	$0.776422\pi$
$908$		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
$917$	−0.0176432			−0.000582631
$918$		0			0
$919$			17.2370i			0.568596i	0.958736	+	0.284298i	$0.0917605\pi$
$919$			17.2370i			0.568596i	−0.958736	+	0.284298i	$0.908240\pi$
$920$		0			0
$921$			4.90722i			0.161698i
$922$		0			0
$923$	38.0940			1.25388
$924$		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	$0.165079\pi$
$929$			30.2159i			0.991350i	−0.868508	+	0.495675i	$0.834921\pi$
$930$		0			0
$931$	1.56981	+	1.56981i	0.0514486	+	0.0514486i
$932$		0			0
$933$		−	12.7635i		−	0.417860i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	29.8348	+	29.8348i	0.974661	+	0.974661i	0.999687	−	0.0250259i	$-0.00796681\pi$
$937$	29.8348	+	29.8348i	0.974661	+	0.974661i	−0.0250259	+	0.999687i	$0.507967\pi$
$938$		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
$947$	−25.0570			−0.814243			−0.407122	−	0.913374i	$-0.633467\pi$
$947$	−25.0570			−0.814243			−0.407122	+	0.913374i	$0.633467\pi$
$948$		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	$-0.935558\pi$
$953$	−24.0330	+	24.0330i	−0.778505	+	0.778505i	0.201072	+	0.979577i	$0.435558\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			2.25147i			0.0727798i
$958$		0			0
$959$	−24.9775			−0.806566
$960$		0			0
$961$	18.4622			0.595556
$962$		0			0
$963$		−	20.3780i		−	0.656671i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	11.0419	−	11.0419i	0.355082	−	0.355082i	−0.506914	−	0.861996i	$-0.669214\pi$
$967$	11.0419	−	11.0419i	0.355082	−	0.355082i	0.861996	+	0.506914i	$0.169214\pi$
$968$		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
$973$	−6.89005			−0.220885
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.8941	−	24.8941i	0.796433	−	0.796433i	−0.186098	−	0.982531i	$-0.559584\pi$
$977$	24.8941	−	24.8941i	0.796433	−	0.796433i	0.982531	+	0.186098i	$0.0595842\pi$
$978$		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	0.408695	−	0.912671i	$-0.365984\pi$
$983$	−15.8011	−	15.8011i	−0.503976	−	0.503976i	−0.912671	+	0.408695i	$0.865984\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	7.27138i		−	0.231450i
$988$		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
$997$	26.7001			0.845601			0.422800	−	0.906223i	$-0.361047\pi$
$997$	26.7001			0.845601			0.422800	+	0.906223i	$0.361047\pi$
$998$		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.j.e.143.7		24
4.3	odd	2		400.2.j.e.43.10	yes	24
5.2	odd	4		1600.2.s.e.207.7		24
5.3	odd	4		1600.2.s.e.207.6		24
5.4	even	2	inner	1600.2.j.e.143.6		24
16.3	odd	4		1600.2.s.e.943.7		24
16.13	even	4		400.2.s.e.243.9	yes	24
20.3	even	4		400.2.s.e.107.4	yes	24
20.7	even	4		400.2.s.e.107.9	yes	24
20.19	odd	2		400.2.j.e.43.3	✓	24
80.3	even	4	inner	1600.2.j.e.1007.7		24
80.13	odd	4		400.2.j.e.307.3	yes	24
80.19	odd	4		1600.2.s.e.943.6		24
80.29	even	4		400.2.s.e.243.4	yes	24
80.67	even	4	inner	1600.2.j.e.1007.6		24
80.77	odd	4		400.2.j.e.307.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.3	✓	24	20.19	odd	2
400.2.j.e.43.10	yes	24	4.3	odd	2
400.2.j.e.307.3	yes	24	80.13	odd	4
400.2.j.e.307.10	yes	24	80.77	odd	4
400.2.s.e.107.4	yes	24	20.3	even	4
400.2.s.e.107.9	yes	24	20.7	even	4
400.2.s.e.243.4	yes	24	80.29	even	4
400.2.s.e.243.9	yes	24	16.13	even	4
1600.2.j.e.143.6		24	5.4	even	2	inner
1600.2.j.e.143.7		24	1.1	even	1	trivial
1600.2.j.e.1007.6		24	80.67	even	4	inner
1600.2.j.e.1007.7		24	80.3	even	4	inner
1600.2.s.e.207.6		24	5.3	odd	4
1600.2.s.e.207.7		24	5.2	odd	4
1600.2.s.e.943.6		24	80.19	odd	4
1600.2.s.e.943.7		24	16.3	odd	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.j (of order \(4\), degree \(2\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more