LMFDB - Embedded newform 320.4.n.c.63.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(63,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.63"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			63.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	320.63
Dual form			320.4.n.c.127.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 11.0000i) q^{5} +27.0000i q^{9} +(37.0000 + 37.0000i) q^{13} +(99.0000 - 99.0000i) q^{17} +(-117.000 - 44.0000i) q^{25} -284.000i q^{29} +(91.0000 - 91.0000i) q^{37} +472.000 q^{41} +(297.000 + 54.0000i) q^{45} -343.000i q^{49} +(27.0000 + 27.0000i) q^{53} +468.000 q^{61} +(481.000 - 333.000i) q^{65} +(253.000 + 253.000i) q^{73} -729.000 q^{81} +(-891.000 - 1287.00i) q^{85} -176.000i q^{89} +(-611.000 + 611.000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} + 74 q^{13} + 198 q^{17} - 234 q^{25} + 182 q^{37} + 944 q^{41} + 594 q^{45} + 54 q^{53} + 936 q^{61} + 962 q^{65} + 506 q^{73} - 1458 q^{81} - 1782 q^{85} - 1222 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 74 * q^13 + 198 * q^17 - 234 * q^25 + 182 * q^37 + 944 * q^41 + 594 * q^45 + 54 * q^53 + 936 * q^61 + 962 * q^65 + 506 * q^73 - 1458 * q^81 - 1782 * q^85 - 1222 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	2.00000	−	11.0000i	0.178885	−	0.983870i
$5$	2.00000	−	11.0000i	0.178885	−	0.983870i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$			27.0000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	37.0000	+	37.0000i	0.789381	+	0.789381i	0.981393	−	0.192012i	$-0.0615011\pi$
$13$	37.0000	+	37.0000i	0.789381	+	0.789381i	−0.192012	+	0.981393i	$0.561501\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	99.0000	−	99.0000i	1.41241	−	1.41241i	0.670540	−	0.741874i	$-0.266063\pi$
$17$	99.0000	−	99.0000i	1.41241	−	1.41241i	0.741874	−	0.670540i	$-0.233937\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−117.000	−	44.0000i	−0.936000	−	0.352000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	284.000i		−	1.81853i	−0.416214	−	0.909267i	$-0.636643\pi$
$29$		−	284.000i		−	1.81853i	0.416214	−	0.909267i	$-0.363357\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	91.0000	−	91.0000i	0.404333	−	0.404333i	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	91.0000	−	91.0000i	0.404333	−	0.404333i	0.879757	+	0.475424i	$0.157705\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	472.000			1.79790			0.898951	−	0.438048i	$-0.144330\pi$
$41$	472.000			1.79790			0.898951	+	0.438048i	$0.144330\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	297.000	+	54.0000i	0.983870	+	0.178885i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	343.000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	27.0000	+	27.0000i	0.0699761	+	0.0699761i	0.741229	−	0.671253i	$-0.234243\pi$
$53$	27.0000	+	27.0000i	0.0699761	+	0.0699761i	−0.671253	+	0.741229i	$0.734243\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	468.000			0.982316			0.491158	−	0.871071i	$-0.336574\pi$
$61$	468.000			0.982316			0.491158	+	0.871071i	$0.336574\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	481.000	−	333.000i	0.917857	−	0.635439i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	253.000	+	253.000i	0.405636	+	0.405636i	0.880214	−	0.474578i	$-0.157399\pi$
$73$	253.000	+	253.000i	0.405636	+	0.405636i	−0.474578	+	0.880214i	$0.657399\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−729.000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−891.000	−	1287.00i	−1.13697	−	1.64229i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	176.000i		−	0.209618i	−0.994492	−	0.104809i	$-0.966577\pi$
$89$		−	176.000i		−	0.209618i	0.994492	−	0.104809i	$-0.0334231\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−611.000	+	611.000i	−0.639563	+	0.639563i	−0.950448	−	0.310884i	$-0.899375\pi$
$97$	−611.000	+	611.000i	−0.639563	+	0.639563i	0.310884	+	0.950448i	$0.399375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	598.000			0.589141			0.294570	−	0.955630i	$-0.404823\pi$
$101$	598.000			0.589141			0.294570	+	0.955630i	$0.404823\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$			1746.00i			1.53428i	0.641480	+	0.767140i	$0.278321\pi$
$109$			1746.00i			1.53428i	−0.641480	+	0.767140i	$0.721679\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−337.000	−	337.000i	−0.280551	−	0.280551i	0.552778	−	0.833329i	$-0.313568\pi$
$113$	−337.000	−	337.000i	−0.280551	−	0.280551i	−0.833329	+	0.552778i	$0.813568\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−999.000	+	999.000i	−0.789381	+	0.789381i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−718.000	+	1199.00i	−0.513759	+	0.857935i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2191.00	+	2191.00i	−1.36635	+	1.36635i	−0.500766	+	0.865583i	$0.666948\pi$
$137$	−2191.00	+	2191.00i	−1.36635	+	1.36635i	−0.865583	+	0.500766i	$0.833052\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−3124.00	−	568.000i	−1.78920	−	0.325309i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	3514.00i		−	1.93207i	−0.258415	−	0.966034i	$-0.583200\pi$
$149$		−	3514.00i		−	1.93207i	0.258415	−	0.966034i	$-0.416800\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	2673.00	+	2673.00i	1.41241	+	1.41241i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1819.00	+	1819.00i	−0.924662	+	0.924662i	−0.997354	−	0.0726920i	$-0.976841\pi$
$157$	−1819.00	+	1819.00i	−0.924662	+	0.924662i	0.0726920	+	0.997354i	$0.476841\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			541.000i			0.246245i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3047.00	+	3047.00i	1.33907	+	1.33907i	0.896962	+	0.442108i	$0.145769\pi$
$173$	3047.00	+	3047.00i	1.33907	+	1.33907i	0.442108	+	0.896962i	$0.354231\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−3942.00			−1.61882			−0.809410	−	0.587243i	$-0.800213\pi$
$181$	−3942.00			−1.61882			−0.809410	+	0.587243i	$0.800213\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−819.000	−	1183.00i	−0.325482	−	0.470140i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−2717.00	−	2717.00i	−1.01334	−	1.01334i	−0.999910	−	0.0134266i	$-0.995726\pi$
$193$	−2717.00	−	2717.00i	−1.01334	−	1.01334i	−0.0134266	−	0.999910i	$-0.504274\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3289.00	+	3289.00i	−1.18950	+	1.18950i	−0.212295	+	0.977206i	$0.568094\pi$
$197$	−3289.00	+	3289.00i	−1.18950	+	1.18950i	−0.977206	+	0.212295i	$0.931906\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	944.000	−	5192.00i	0.321619	−	1.76890i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7326.00			2.22986
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	1188.00	−	3159.00i	0.352000	−	0.936000i
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	2684.00i		−	0.774514i	−0.921972	−	0.387257i	$-0.873423\pi$
$229$		−	2684.00i		−	0.774514i	0.921972	−	0.387257i	$-0.126577\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3843.00	+	3843.00i	1.08053	+	1.08053i	0.996460	+	0.0840693i	$0.0267917\pi$
$233$	3843.00	+	3843.00i	1.08053	+	1.08053i	0.0840693	+	0.996460i	$0.473208\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	5272.00			1.40913			0.704563	−	0.709641i	$-0.251143\pi$
$241$	5272.00			1.40913			0.704563	+	0.709641i	$0.251143\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3773.00	−	686.000i	−0.983870	−	0.178885i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3281.00	+	3281.00i	−0.796355	+	0.796355i	−0.982519	−	0.186164i	$-0.940394\pi$
$257$	−3281.00	+	3281.00i	−0.796355	+	0.796355i	0.186164	+	0.982519i	$0.440394\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7668.00			1.81853
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	351.000	−	243.000i	0.0813651	−	0.0563297i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3406.00i			0.771998i	0.922499	+	0.385999i	$0.126143\pi$
$269$			3406.00i			0.771998i	−0.922499	+	0.385999i	$0.873857\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5221.00	−	5221.00i	1.13249	−	1.13249i	0.142727	−	0.989762i	$-0.454413\pi$
$277$	5221.00	−	5221.00i	1.13249	−	1.13249i	0.989762	−	0.142727i	$-0.0455871\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5792.00			1.22961			0.614807	−	0.788677i	$-0.289234\pi$
$281$	5792.00			1.22961			0.614807	+	0.788677i	$0.289234\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	14689.0i		−	2.98982i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2983.00	−	2983.00i	−0.594774	−	0.594774i	0.344143	−	0.938917i	$-0.388169\pi$
$293$	−2983.00	−	2983.00i	−0.594774	−	0.594774i	−0.938917	+	0.344143i	$0.888169\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	936.000	−	5148.00i	0.175722	−	0.966471i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−937.000	−	937.000i	−0.169209	−	0.169209i	0.617423	−	0.786632i	$-0.288177\pi$
$313$	−937.000	−	937.000i	−0.169209	−	0.169209i	−0.786632	+	0.617423i	$0.788177\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2799.00	+	2799.00i	−0.495923	+	0.495923i	−0.910166	−	0.414243i	$-0.864046\pi$
$317$	−2799.00	+	2799.00i	−0.495923	+	0.495923i	0.414243	+	0.910166i	$0.364046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2701.00	−	5957.00i	−0.460999	−	1.01672i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	2457.00	+	2457.00i	0.404333	+	0.404333i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6391.00	+	6391.00i	−1.03306	+	1.03306i	−0.0336216	+	0.999435i	$0.510704\pi$
$337$	−6391.00	+	6391.00i	−1.03306	+	1.03306i	−0.999435	+	0.0336216i	$0.989296\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	8964.00i		−	1.37488i	−0.726243	−	0.687438i	$-0.758735\pi$
$349$		−	8964.00i		−	1.37488i	0.726243	−	0.687438i	$-0.241265\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8073.00	+	8073.00i	1.21723	+	1.21723i	0.968598	+	0.248633i	$0.0799813\pi$
$353$	8073.00	+	8073.00i	1.21723	+	1.21723i	0.248633	+	0.968598i	$0.420019\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−6859.00			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3289.00	−	2277.00i	0.471655	−	0.326530i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			12744.0i			1.79790i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	9647.00	+	9647.00i	1.33915	+	1.33915i	0.896884	+	0.442265i	$0.145825\pi$
$373$	9647.00	+	9647.00i	1.33915	+	1.33915i	0.442265	+	0.896884i	$0.354175\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10508.0	−	10508.0i	1.43552	−	1.43552i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	374.000i		−	0.0487469i	−0.999703	−	0.0243735i	$-0.992241\pi$
$389$		−	374.000i		−	0.0487469i	0.999703	−	0.0243735i	$-0.00775908\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−11089.0	+	11089.0i	−1.40187	+	1.40187i	−0.607699	+	0.794168i	$0.707907\pi$
$397$	−11089.0	+	11089.0i	−1.40187	+	1.40187i	−0.794168	+	0.607699i	$0.792093\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2398.00			−0.298629			−0.149315	−	0.988790i	$-0.547707\pi$
$401$	−2398.00			−0.298629			−0.149315	+	0.988790i	$0.547707\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1458.00	+	8019.00i	−0.178885	+	0.983870i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	7146.00i		−	0.863929i	−0.901891	−	0.431964i	$-0.857821\pi$
$409$		−	7146.00i		−	0.863929i	0.901891	−	0.431964i	$-0.142179\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−13412.0			−1.55264			−0.776319	−	0.630340i	$-0.782916\pi$
$421$	−13412.0			−1.55264			−0.776319	+	0.630340i	$0.782916\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15939.0	+	7227.00i	−1.81919	+	0.824849i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−11107.0	−	11107.0i	−1.23272	−	1.23272i	−0.962914	−	0.269807i	$-0.913040\pi$
$433$	−11107.0	−	11107.0i	−1.23272	−	1.23272i	−0.269807	−	0.962914i	$-0.586960\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	9261.00			1.00000
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−1936.00	−	352.000i	−0.206236	−	0.0374975i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16114.0i			1.69369i	0.531840	+	0.846845i	$0.321501\pi$
$449$			16114.0i			1.69369i	−0.531840	+	0.846845i	$0.678499\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13481.0	+	13481.0i	−1.37990	+	1.37990i	−0.535132	+	0.844768i	$0.679738\pi$
$457$	−13481.0	+	13481.0i	−1.37990	+	1.37990i	−0.844768	+	0.535132i	$0.820262\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2318.00			0.234187			0.117093	−	0.993121i	$-0.462642\pi$
$461$	2318.00			0.234187			0.117093	+	0.993121i	$0.462642\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−729.000	+	729.000i	−0.0699761	+	0.0699761i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	6734.00			0.638345
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5499.00	+	7943.00i	0.514839	+	0.743656i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−28116.0	−	28116.0i	−2.56852	−	2.56852i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	1196.00	−	6578.00i	0.105389	−	0.579638i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			17996.0i			1.56711i	0.621323	+	0.783555i	$0.286596\pi$
$509$			17996.0i			1.56711i	−0.621323	+	0.783555i	$0.713404\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−23738.0			−1.99612			−0.998062	−	0.0622265i	$-0.980180\pi$
$521$	−23738.0			−1.99612			−0.998062	+	0.0622265i	$0.980180\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			12167.0i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17464.0	+	17464.0i	1.41923	+	1.41923i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5922.00			−0.470622			−0.235311	−	0.971920i	$-0.575611\pi$
$541$	−5922.00			−0.470622			−0.235311	+	0.971920i	$0.575611\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19206.0	+	3492.00i	1.50953	+	0.274460i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$			12636.0i			0.982316i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16731.0	−	16731.0i	1.27274	−	1.27274i	0.328093	−	0.944646i	$-0.393594\pi$
$557$	16731.0	−	16731.0i	1.27274	−	1.27274i	0.944646	−	0.328093i	$-0.106406\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−4381.00	+	3033.00i	−0.326212	+	0.225839i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26806.0i		−	1.97498i	−0.157669	−	0.987492i	$-0.550398\pi$
$569$		−	26806.0i		−	1.97498i	0.157669	−	0.987492i	$-0.449602\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	15479.0	−	15479.0i	1.11681	−	1.11681i	0.124603	−	0.992207i	$-0.460234\pi$
$577$	15479.0	−	15479.0i	1.11681	−	1.11681i	0.992207	−	0.124603i	$-0.0397657\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	8991.00	+	12987.0i	0.635439	+	0.917857i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4433.00	+	4433.00i	0.306984	+	0.306984i	0.843738	−	0.536755i	$-0.180350\pi$
$593$	4433.00	+	4433.00i	0.306984	+	0.306984i	−0.536755	+	0.843738i	$0.680350\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−24048.0			−1.63218			−0.816089	−	0.577927i	$-0.803862\pi$
$601$	−24048.0			−1.63218			−0.816089	+	0.577927i	$0.803862\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2662.00	−	14641.0i	0.178885	−	0.983870i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1837.00	+	1837.00i	0.121037	+	0.121037i	0.765031	−	0.643994i	$-0.222724\pi$
$613$	1837.00	+	1837.00i	0.121037	+	0.121037i	−0.643994	+	0.765031i	$0.722724\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5499.00	−	5499.00i	0.358803	−	0.358803i	−0.504569	−	0.863372i	$-0.668348\pi$
$617$	5499.00	−	5499.00i	0.358803	−	0.358803i	0.863372	+	0.504569i	$0.168348\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11753.0	+	10296.0i	0.752192	+	0.658944i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	18018.0i		−	1.14217i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12691.0	−	12691.0i	0.789381	−	0.789381i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14872.0			0.916394			0.458197	−	0.888851i	$-0.348495\pi$
$641$	14872.0			0.916394			0.458197	+	0.888851i	$0.348495\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16173.0	−	16173.0i	−0.969217	−	0.969217i	0.0303236	−	0.999540i	$-0.490346\pi$
$653$	−16173.0	−	16173.0i	−0.969217	−	0.969217i	−0.999540	+	0.0303236i	$0.990346\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6831.00	+	6831.00i	−0.405636	+	0.405636i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	22068.0			1.29856			0.649278	−	0.760551i	$-0.275071\pi$
$661$	22068.0			1.29856			0.649278	+	0.760551i	$0.275071\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−19547.0	−	19547.0i	−1.11959	−	1.11959i	−0.991802	−	0.127784i	$-0.959214\pi$
$673$	−19547.0	−	19547.0i	−1.11959	−	1.11959i	−0.127784	−	0.991802i	$-0.540786\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15471.0	−	15471.0i	0.878285	−	0.878285i	−0.115072	−	0.993357i	$-0.536710\pi$
$677$	15471.0	−	15471.0i	0.878285	−	0.878285i	0.993357	+	0.115072i	$0.0367100\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	19719.0	+	28483.0i	1.09989	+	1.58873i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1998.00i			0.110476i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	46728.0	−	46728.0i	2.53938	−	2.53938i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31252.0			−1.68384			−0.841920	−	0.539602i	$-0.818575\pi$
$701$	−31252.0			−1.68384			−0.841920	+	0.539602i	$0.818575\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	8404.00i		−	0.445161i	−0.974914	−	0.222580i	$-0.928552\pi$
$709$		−	8404.00i		−	0.445161i	0.974914	−	0.222580i	$-0.0714479\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12496.0	+	33228.0i	−0.640124	+	1.70215i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	19683.0i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−14993.0	−	14993.0i	−0.755497	−	0.755497i	0.220003	−	0.975499i	$-0.429393\pi$
$733$	−14993.0	−	14993.0i	−0.755497	−	0.755497i	−0.975499	+	0.220003i	$0.929393\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	−38654.0	−	7028.00i	−1.90090	−	0.345619i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	28781.0	−	28781.0i	1.38185	−	1.38185i	0.540527	−	0.841327i	$-0.318225\pi$
$757$	28781.0	−	28781.0i	1.38185	−	1.38185i	0.841327	−	0.540527i	$-0.181775\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31882.0			1.51869			0.759344	−	0.650689i	$-0.225520\pi$
$761$	31882.0			1.51869			0.759344	+	0.650689i	$0.225520\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	34749.0	−	24057.0i	1.64229	−	1.13697i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			41544.0i			1.94813i	0.226260	+	0.974067i	$0.427350\pi$
$769$			41544.0i			1.94813i	−0.226260	+	0.974067i	$0.572650\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28197.0	+	28197.0i	1.31200	+	1.31200i	0.919940	+	0.392060i	$0.128237\pi$
$773$	28197.0	+	28197.0i	1.31200	+	1.31200i	0.392060	+	0.919940i	$0.371763\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16371.0	+	23647.0i	0.744339	+	1.07516i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	17316.0	+	17316.0i	0.775421	+	0.775421i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12839.0	+	12839.0i	−0.570616	+	0.570616i	−0.932300	−	0.361685i	$-0.882202\pi$
$797$	−12839.0	+	12839.0i	−0.570616	+	0.570616i	0.361685	+	0.932300i	$0.382202\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	4752.00			0.209618
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			39704.0i			1.72549i	0.505643	+	0.862743i	$0.331255\pi$
$809$			39704.0i			1.72549i	−0.505643	+	0.862743i	$0.668745\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−47012.0			−1.99845			−0.999227	−	0.0393212i	$-0.987480\pi$
$821$	−47012.0			−1.99845			−0.999227	+	0.0393212i	$0.987480\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			23166.0i			0.970553i	0.874361	+	0.485276i	$0.161281\pi$
$829$			23166.0i			0.970553i	−0.874361	+	0.485276i	$0.838719\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−33957.0	−	33957.0i	−1.41241	−	1.41241i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−56267.0			−2.30706
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5951.00	+	1082.00i	0.242273	+	0.0440496i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−32923.0	−	32923.0i	−1.32153	−	1.32153i	−0.912541	−	0.408986i	$-0.865883\pi$
$853$	−32923.0	−	32923.0i	−1.32153	−	1.32153i	−0.408986	−	0.912541i	$-0.634117\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12969.0	−	12969.0i	0.516934	−	0.516934i	−0.399708	−	0.916642i	$-0.630889\pi$
$857$	12969.0	−	12969.0i	0.516934	−	0.516934i	0.916642	+	0.399708i	$0.130889\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	39611.0	−	27423.0i	1.55701	−	1.07793i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−16497.0	−	16497.0i	−0.639563	−	0.639563i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36179.0	+	36179.0i	−1.39302	+	1.39302i	−0.574550	+	0.818470i	$0.694823\pi$
$877$	−36179.0	+	36179.0i	−1.39302	+	1.39302i	−0.818470	+	0.574550i	$0.805177\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51808.0			−1.98122			−0.990611	−	0.136714i	$-0.956346\pi$
$881$	−51808.0			−1.98122			−0.990611	+	0.136714i	$0.956346\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	5346.00			0.197670
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7884.00	+	43362.0i	−0.289583	+	1.59271i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			16146.0i			0.589141i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−14651.0	+	6643.00i	−0.520780	+	0.236130i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	30866.0i		−	1.09008i	−0.838411	−	0.545038i	$-0.816515\pi$
$929$		−	30866.0i		−	1.09008i	0.838411	−	0.545038i	$-0.183485\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38141.0	+	38141.0i	−1.32979	+	1.32979i	−0.424238	+	0.905551i	$0.639458\pi$
$937$	−38141.0	+	38141.0i	−1.32979	+	1.32979i	−0.905551	+	0.424238i	$0.860542\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	31378.0			1.08703			0.543514	−	0.839400i	$-0.317093\pi$
$941$	31378.0			1.08703			0.543514	+	0.839400i	$0.317093\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$			18722.0i			0.640402i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20623.0	+	20623.0i	0.700991	+	0.700991i	0.964623	−	0.263632i	$-0.0849205\pi$
$953$	20623.0	+	20623.0i	0.700991	+	0.700991i	−0.263632	+	0.964623i	$0.584921\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−35321.0	+	24453.0i	−1.17826	+	0.815720i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−39771.0	+	39771.0i	−1.30234	+	1.30234i	−0.375531	+	0.926810i	$0.622540\pi$
$977$	−39771.0	+	39771.0i	−1.30234	+	1.30234i	−0.926810	+	0.375531i	$0.877460\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−47142.0			−1.53428
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	29601.0	+	42757.0i	0.957529	+	1.38310i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9361.00	−	9361.00i	0.297358	−	0.297358i	−0.542620	−	0.839978i	$-0.682568\pi$
$997$	9361.00	−	9361.00i	0.297358	−	0.297358i	0.839978	+	0.542620i	$0.182568\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.n.c.63.1		2
4.3	odd	2	CM	320.4.n.c.63.1		2
5.2	odd	4	inner	320.4.n.c.127.1		2
8.3	odd	2		20.4.e.a.3.1	✓	2
8.5	even	2		20.4.e.a.3.1	✓	2
20.7	even	4	inner	320.4.n.c.127.1		2
24.5	odd	2		180.4.k.a.163.1		2
24.11	even	2		180.4.k.a.163.1		2
40.3	even	4		100.4.e.a.7.1		2
40.13	odd	4		100.4.e.a.7.1		2
40.19	odd	2		100.4.e.a.43.1		2
40.27	even	4		20.4.e.a.7.1	yes	2
40.29	even	2		100.4.e.a.43.1		2
40.37	odd	4		20.4.e.a.7.1	yes	2
120.77	even	4		180.4.k.a.127.1		2
120.107	odd	4		180.4.k.a.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.a.3.1	✓	2	8.3	odd	2
20.4.e.a.3.1	✓	2	8.5	even	2
20.4.e.a.7.1	yes	2	40.27	even	4
20.4.e.a.7.1	yes	2	40.37	odd	4
100.4.e.a.7.1		2	40.3	even	4
100.4.e.a.7.1		2	40.13	odd	4
100.4.e.a.43.1		2	40.19	odd	2
100.4.e.a.43.1		2	40.29	even	2
180.4.k.a.127.1		2	120.77	even	4
180.4.k.a.127.1		2	120.107	odd	4
180.4.k.a.163.1		2	24.5	odd	2
180.4.k.a.163.1		2	24.11	even	2
320.4.n.c.63.1		2	1.1	even	1	trivial
320.4.n.c.63.1		2	4.3	odd	2	CM
320.4.n.c.127.1		2	5.2	odd	4	inner
320.4.n.c.127.1		2	20.7	even	4	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 11.0000i) q^{5} +27.0000i q^{9} +(37.0000 + 37.0000i) q^{13} +(99.0000 - 99.0000i) q^{17} +(-117.000 - 44.0000i) q^{25} -284.000i q^{29} +(91.0000 - 91.0000i) q^{37} +472.000 q^{41} +(297.000 + 54.0000i) q^{45} -343.000i q^{49} +(27.0000 + 27.0000i) q^{53} +468.000 q^{61} +(481.000 - 333.000i) q^{65} +(253.000 + 253.000i) q^{73} -729.000 q^{81} +(-891.000 - 1287.00i) q^{85} -176.000i q^{89} +(-611.000 + 611.000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} + 74 q^{13} + 198 q^{17} - 234 q^{25} + 182 q^{37} + 944 q^{41} + 594 q^{45} + 54 q^{53} + 936 q^{61} + 962 q^{65} + 506 q^{73} - 1458 q^{81} - 1782 q^{85} - 1222 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 74 * q^13 + 198 * q^17 - 234 * q^25 + 182 * q^37 + 944 * q^41 + 594 * q^45 + 54 * q^53 + 936 * q^61 + 962 * q^65 + 506 * q^73 - 1458 * q^81 - 1782 * q^85 - 1222 * q^97

Introduction

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