LMFDB - Embedded newform 160.3.p.a.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,3,Mod(33,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("160.33");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	160.p (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.35968422976$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			97.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	160.97
Dual form			160.3.p.a.33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.00000 + 3.00000i) q^{5} +9.00000i q^{9} +O(q^{10})$	$q+(-4.00000 + 3.00000i) q^{5} +9.00000i q^{9} +(-17.0000 + 17.0000i) q^{13} +(7.00000 + 7.00000i) q^{17} +(7.00000 - 24.0000i) q^{25} +40.0000i q^{29} +(-47.0000 - 47.0000i) q^{37} +80.0000 q^{41} +(-27.0000 - 36.0000i) q^{45} -49.0000i q^{49} +(17.0000 - 17.0000i) q^{53} +120.000 q^{61} +(17.0000 - 119.000i) q^{65} +(-103.000 + 103.000i) q^{73} -81.0000 q^{81} +(-49.0000 - 7.00000i) q^{85} +160.000i q^{89} +(-7.00000 - 7.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5}+O(q^{10}) $ 2 * q - 8 * q^5	$ 2 q - 8 q^{5} - 34 q^{13} + 14 q^{17} + 14 q^{25} - 94 q^{37} + 160 q^{41} - 54 q^{45} + 34 q^{53} + 240 q^{61} + 34 q^{65} - 206 q^{73} - 162 q^{81} - 98 q^{85} - 14 q^{97}+O(q^{100}) $ 2 * q - 8 * q^5 - 34 * q^13 + 14 * q^17 + 14 * q^25 - 94 * q^37 + 160 * q^41 - 54 * q^45 + 34 * q^53 + 240 * q^61 + 34 * q^65 - 206 * q^73 - 162 * q^81 - 98 * q^85 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$5$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$6$		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
$9$			9.00000i			1.00000i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−17.0000	+	17.0000i	−1.30769	+	1.30769i	−0.384615	+	0.923077i	$0.625666\pi$
$13$	−17.0000	+	17.0000i	−1.30769	+	1.30769i	−0.923077	+	0.384615i	$0.874334\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.00000	+	7.00000i	0.411765	+	0.411765i	0.882353	−	0.470588i	$-0.155958\pi$
$17$	7.00000	+	7.00000i	0.411765	+	0.411765i	−0.470588	+	0.882353i	$0.655958\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		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
$25$	7.00000	−	24.0000i	0.280000	−	0.960000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			40.0000i			1.37931i	0.724138	+	0.689655i	$0.242238\pi$
$29$			40.0000i			1.37931i	−0.724138	+	0.689655i	$0.757762\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−47.0000	−	47.0000i	−1.27027	−	1.27027i	−0.945946	−	0.324324i	$-0.894863\pi$
$37$	−47.0000	−	47.0000i	−1.27027	−	1.27027i	−0.324324	−	0.945946i	$-0.605137\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	80.0000			1.95122			0.975610	−	0.219512i	$-0.0704466\pi$
$41$	80.0000			1.95122			0.975610	+	0.219512i	$0.0704466\pi$
$42$		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
$45$	−27.0000	−	36.0000i	−0.600000	−	0.800000i
$46$		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
$49$		−	49.0000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	17.0000	−	17.0000i	0.320755	−	0.320755i	−0.528302	−	0.849057i	$-0.677171\pi$
$53$	17.0000	−	17.0000i	0.320755	−	0.320755i	0.849057	+	0.528302i	$0.177171\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	120.000			1.96721			0.983607	−	0.180328i	$-0.0577159\pi$
$61$	120.000			1.96721			0.983607	+	0.180328i	$0.0577159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17.0000	−	119.000i	0.261538	−	1.83077i
$66$		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			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−103.000	+	103.000i	−1.41096	+	1.41096i	−0.657534	+	0.753425i	$0.728401\pi$
$73$	−103.000	+	103.000i	−1.41096	+	1.41096i	−0.753425	+	0.657534i	$0.771599\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−81.0000			−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
$85$	−49.0000	−	7.00000i	−0.576471	−	0.0823529i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			160.000i			1.79775i	0.438202	+	0.898876i	$0.355615\pi$
$89$			160.000i			1.79775i	−0.438202	+	0.898876i	$0.644385\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.00000	−	7.00000i	−0.0721649	−	0.0721649i	0.670103	−	0.742268i	$-0.266250\pi$
$97$	−7.00000	−	7.00000i	−0.0721649	−	0.0721649i	−0.742268	+	0.670103i	$0.766250\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	198.000			1.96040			0.980198	−	0.198020i	$-0.0634510\pi$
$101$	198.000			1.96040			0.980198	+	0.198020i	$0.0634510\pi$
$102$		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
$109$			182.000i			1.66972i	0.550459	+	0.834862i	$0.314453\pi$
$109$			182.000i			1.66972i	−0.550459	+	0.834862i	$0.685547\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−97.0000	+	97.0000i	−0.858407	+	0.858407i	−0.991150	−	0.132743i	$-0.957621\pi$
$113$	−97.0000	+	97.0000i	−0.858407	+	0.858407i	0.132743	+	0.991150i	$0.457621\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−153.000	−	153.000i	−1.30769	−	1.30769i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	44.0000	+	117.000i	0.352000	+	0.936000i
$126$		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			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	193.000	+	193.000i	1.40876	+	1.40876i	0.766423	+	0.642336i	$0.222035\pi$
$137$	193.000	+	193.000i	1.40876	+	1.40876i	0.642336	+	0.766423i	$0.277965\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−120.000	−	160.000i	−0.827586	−	1.10345i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	102.000i		−	0.684564i	−0.939597	−	0.342282i	$-0.888800\pi$
$149$		−	102.000i		−	0.684564i	0.939597	−	0.342282i	$-0.111200\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−63.0000	+	63.0000i	−0.411765	+	0.411765i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	47.0000	+	47.0000i	0.299363	+	0.299363i	0.840764	−	0.541401i	$-0.182106\pi$
$157$	47.0000	+	47.0000i	0.299363	+	0.299363i	−0.541401	+	0.840764i	$0.682106\pi$
$158$		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
$169$		−	409.000i		−	2.42012i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	217.000	−	217.000i	1.25434	−	1.25434i	0.300578	−	0.953757i	$-0.402820\pi$
$173$	217.000	−	217.000i	1.25434	−	1.25434i	0.953757	−	0.300578i	$-0.0971796\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−38.0000			−0.209945			−0.104972	−	0.994475i	$-0.533475\pi$
$181$	−38.0000			−0.209945			−0.104972	+	0.994475i	$0.533475\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	329.000	+	47.0000i	1.77838	+	0.254054i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	263.000	−	263.000i	1.36269	−	1.36269i	0.492228	−	0.870466i	$-0.336183\pi$
$193$	263.000	−	263.000i	1.36269	−	1.36269i	0.870466	−	0.492228i	$-0.163817\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−167.000	−	167.000i	−0.847716	−	0.847716i	0.142132	−	0.989848i	$-0.454604\pi$
$197$	−167.000	−	167.000i	−0.847716	−	0.847716i	−0.989848	+	0.142132i	$0.954604\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−320.000	+	240.000i	−1.56098	+	1.17073i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		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
$221$	−238.000			−1.07692
$222$		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
$225$	216.000	+	63.0000i	0.960000	+	0.280000i
$226$		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
$229$		−	120.000i		−	0.524017i	−0.965066	−	0.262009i	$-0.915615\pi$
$229$		−	120.000i		−	0.524017i	0.965066	−	0.262009i	$-0.0843849\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	103.000	−	103.000i	0.442060	−	0.442060i	−0.450644	−	0.892704i	$-0.648806\pi$
$233$	103.000	−	103.000i	0.442060	−	0.442060i	0.892704	+	0.450644i	$0.148806\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−240.000			−0.995851			−0.497925	−	0.867220i	$-0.665905\pi$
$241$	−240.000			−0.995851			−0.497925	+	0.867220i	$0.665905\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	147.000	+	196.000i	0.600000	+	0.800000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	287.000	+	287.000i	1.11673	+	1.11673i	0.992218	+	0.124514i	$0.0397370\pi$
$257$	287.000	+	287.000i	1.11673	+	1.11673i	0.124514	+	0.992218i	$0.460263\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−360.000			−1.37931
$262$		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
$265$	−17.0000	+	119.000i	−0.0641509	+	0.449057i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			138.000i			0.513011i	0.966543	+	0.256506i	$0.0825712\pi$
$269$			138.000i			0.513011i	−0.966543	+	0.256506i	$0.917429\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	367.000	+	367.000i	1.32491	+	1.32491i	0.909747	+	0.415162i	$0.136275\pi$
$277$	367.000	+	367.000i	1.32491	+	1.32491i	0.415162	+	0.909747i	$0.363725\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−320.000			−1.13879			−0.569395	−	0.822064i	$-0.692822\pi$
$281$	−320.000			−1.13879			−0.569395	+	0.822064i	$0.692822\pi$
$282$		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
$289$		−	191.000i		−	0.660900i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−217.000	+	217.000i	−0.740614	+	0.740614i	−0.972696	−	0.232082i	$-0.925446\pi$
$293$	−217.000	+	217.000i	−0.740614	+	0.740614i	0.232082	+	0.972696i	$0.425446\pi$
$294$		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
$305$	−480.000	+	360.000i	−1.57377	+	1.18033i
$306$		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			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−337.000	+	337.000i	−1.07668	+	1.07668i	−0.0798722	+	0.996805i	$0.525451\pi$
$313$	−337.000	+	337.000i	−1.07668	+	1.07668i	−0.996805	+	0.0798722i	$0.974549\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−233.000	−	233.000i	−0.735016	−	0.735016i	0.236593	−	0.971609i	$-0.423969\pi$
$317$	−233.000	−	233.000i	−0.735016	−	0.735016i	−0.971609	+	0.236593i	$0.923969\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	289.000	+	527.000i	0.889231	+	1.62154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	423.000	−	423.000i	1.27027	−	1.27027i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	113.000	+	113.000i	0.335312	+	0.335312i	0.854599	−	0.519288i	$-0.173803\pi$
$337$	113.000	+	113.000i	0.335312	+	0.335312i	−0.519288	+	0.854599i	$0.673803\pi$
$338$		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
$349$		−	360.000i		−	1.03152i	−0.856734	−	0.515759i	$-0.827510\pi$
$349$		−	360.000i		−	1.03152i	0.856734	−	0.515759i	$-0.172490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	497.000	−	497.000i	1.40793	−	1.40793i	0.637394	−	0.770538i	$-0.280012\pi$
$353$	497.000	−	497.000i	1.40793	−	1.40793i	0.770538	−	0.637394i	$-0.219988\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	103.000	−	721.000i	0.282192	−	1.97534i
$366$		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
$369$			720.000i			1.95122i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−23.0000	+	23.0000i	−0.0616622	+	0.0616622i	−0.737265	−	0.675603i	$-0.763883\pi$
$373$	−23.0000	+	23.0000i	−0.0616622	+	0.0616622i	0.675603	+	0.737265i	$0.263883\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−680.000	−	680.000i	−1.80371	−	1.80371i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		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
$389$		−	378.000i		−	0.971722i	−0.874036	−	0.485861i	$-0.838506\pi$
$389$		−	378.000i		−	0.971722i	0.874036	−	0.485861i	$-0.161494\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	553.000	+	553.000i	1.39295	+	1.39295i	0.818640	+	0.574307i	$0.194729\pi$
$397$	553.000	+	553.000i	1.39295	+	1.39295i	0.574307	+	0.818640i	$0.305271\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−798.000			−1.99002			−0.995012	−	0.0997506i	$-0.968195\pi$
$401$	−798.000			−1.99002			−0.995012	+	0.0997506i	$0.968195\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	324.000	−	243.000i	0.800000	−	0.600000i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	782.000i		−	1.91198i	−0.293399	−	0.955990i	$-0.594786\pi$
$409$		−	782.000i		−	1.91198i	0.293399	−	0.955990i	$-0.405214\pi$
$410$		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		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	840.000			1.99525			0.997625	−	0.0688836i	$-0.0219437\pi$
$421$	840.000			1.99525			0.997625	+	0.0688836i	$0.0219437\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	217.000	−	119.000i	0.510588	−	0.280000i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−263.000	+	263.000i	−0.607390	+	0.607390i	−0.942263	−	0.334873i	$-0.891307\pi$
$433$	−263.000	+	263.000i	−0.607390	+	0.607390i	0.334873	+	0.942263i	$0.391307\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	441.000			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
$445$	−480.000	−	640.000i	−1.07865	−	1.43820i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			702.000i			1.56347i	0.623608	+	0.781737i	$0.285666\pi$
$449$			702.000i			1.56347i	−0.623608	+	0.781737i	$0.714334\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−593.000	−	593.000i	−1.29759	−	1.29759i	−0.929978	−	0.367615i	$-0.880174\pi$
$457$	−593.000	−	593.000i	−1.29759	−	1.29759i	−0.367615	−	0.929978i	$-0.619826\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−522.000			−1.13232			−0.566161	−	0.824295i	$-0.691572\pi$
$461$	−522.000			−1.13232			−0.566161	+	0.824295i	$0.691572\pi$
$462$		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
$477$	153.000	+	153.000i	0.320755	+	0.320755i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	1598.00			3.32225
$482$		0			0
$483$		0			0
$484$		0			0
$485$	49.0000	+	7.00000i	0.101031	+	0.0144330i
$486$		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			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−280.000	+	280.000i	−0.567951	+	0.567951i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		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
$505$	−792.000	+	594.000i	−1.56832	+	1.17624i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			440.000i			0.864440i	0.901768	+	0.432220i	$0.142270\pi$
$509$			440.000i			0.864440i	−0.901768	+	0.432220i	$0.857730\pi$
$510$		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
$521$	558.000			1.07102			0.535509	−	0.844530i	$-0.320120\pi$
$521$	558.000			1.07102			0.535509	+	0.844530i	$0.320120\pi$
$522$		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
$529$			529.000i			1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1360.00	+	1360.00i	−2.55159	+	2.55159i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−682.000			−1.26063			−0.630314	−	0.776340i	$-0.717074\pi$
$541$	−682.000			−1.26063			−0.630314	+	0.776340i	$0.717074\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−546.000	−	728.000i	−1.00183	−	1.33578i
$546$		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
$549$			1080.00i			1.96721i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−367.000	−	367.000i	−0.658887	−	0.658887i	0.296230	−	0.955117i	$-0.404271\pi$
$557$	−367.000	−	367.000i	−0.658887	−	0.658887i	−0.955117	+	0.296230i	$0.904271\pi$
$558$		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
$565$	97.0000	−	679.000i	0.171681	−	1.20177i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			462.000i			0.811951i	0.913884	+	0.405975i	$0.133068\pi$
$569$			462.000i			0.811951i	−0.913884	+	0.405975i	$0.866932\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	527.000	+	527.000i	0.913345	+	0.913345i	0.996534	−	0.0831889i	$-0.0265105\pi$
$577$	527.000	+	527.000i	0.913345	+	0.913345i	−0.0831889	+	0.996534i	$0.526510\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1071.00	+	153.000i	1.83077	+	0.261538i
$586$		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
$593$	97.0000	−	97.0000i	0.163575	−	0.163575i	−0.620573	−	0.784148i	$-0.713100\pi$
$593$	97.0000	−	97.0000i	0.163575	−	0.163575i	0.784148	+	0.620573i	$0.213100\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	480.000			0.798669			0.399334	−	0.916805i	$-0.369241\pi$
$601$	480.000			0.798669			0.399334	+	0.916805i	$0.369241\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	484.000	−	363.000i	0.800000	−	0.600000i
$606$		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
$613$	−577.000	+	577.000i	−0.941272	+	0.941272i	−0.998369	−	0.0570962i	$-0.981816\pi$
$613$	−577.000	+	577.000i	−0.941272	+	0.941272i	0.0570962	+	0.998369i	$0.481816\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−713.000	−	713.000i	−1.15559	−	1.15559i	−0.985413	−	0.170178i	$-0.945566\pi$
$617$	−713.000	−	713.000i	−1.15559	−	1.15559i	−0.170178	−	0.985413i	$-0.554434\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−527.000	−	336.000i	−0.843200	−	0.537600i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	658.000i		−	1.04610i
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	833.000	+	833.000i	1.30769	+	1.30769i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	400.000			0.624025			0.312012	−	0.950078i	$-0.398997\pi$
$641$	400.000			0.624025			0.312012	+	0.950078i	$0.398997\pi$
$642$		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
$653$	257.000	−	257.000i	0.393568	−	0.393568i	−0.482389	−	0.875957i	$-0.660231\pi$
$653$	257.000	−	257.000i	0.393568	−	0.393568i	0.875957	+	0.482389i	$0.160231\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−927.000	−	927.000i	−1.41096	−	1.41096i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	600.000			0.907716			0.453858	−	0.891074i	$-0.350047\pi$
$661$	600.000			0.907716			0.453858	+	0.891074i	$0.350047\pi$
$662$		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
$673$	937.000	−	937.000i	1.39227	−	1.39227i	0.572065	−	0.820208i	$-0.306142\pi$
$673$	937.000	−	937.000i	1.39227	−	1.39227i	0.820208	−	0.572065i	$-0.193858\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−727.000	−	727.000i	−1.07386	−	1.07386i	−0.997046	−	0.0768095i	$-0.975527\pi$
$677$	−727.000	−	727.000i	−1.07386	−	1.07386i	−0.0768095	−	0.997046i	$-0.524473\pi$
$678$		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
$685$	−1351.00	−	193.000i	−1.97226	−	0.281752i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			578.000i			0.838897i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	560.000	+	560.000i	0.803443	+	0.803443i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	520.000			0.741797			0.370899	−	0.928673i	$-0.379050\pi$
$701$	520.000			0.741797			0.370899	+	0.928673i	$0.379050\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1320.00i		−	1.86178i	−0.365303	−	0.930889i	$-0.619035\pi$
$709$		−	1320.00i		−	1.86178i	0.365303	−	0.930889i	$-0.380965\pi$
$710$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	960.000	+	280.000i	1.32414	+	0.386207i
$726$		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
$729$		−	729.000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	617.000	−	617.000i	0.841746	−	0.841746i	−0.147340	−	0.989086i	$-0.547071\pi$
$733$	617.000	−	617.000i	0.841746	−	0.841746i	0.989086	+	0.147340i	$0.0470711\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		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
$745$	306.000	+	408.000i	0.410738	+	0.547651i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	127.000	+	127.000i	0.167768	+	0.167768i	0.785997	−	0.618230i	$-0.212150\pi$
$757$	127.000	+	127.000i	0.167768	+	0.167768i	−0.618230	+	0.785997i	$0.712150\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−78.0000			−0.102497			−0.0512484	−	0.998686i	$-0.516320\pi$
$761$	−78.0000			−0.102497			−0.0512484	+	0.998686i	$0.516320\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	63.0000	−	441.000i	0.0823529	−	0.576471i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	1200.00i		−	1.56047i	−0.625488	−	0.780234i	$-0.715100\pi$
$769$		−	1200.00i		−	1.56047i	0.625488	−	0.780234i	$-0.284900\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	943.000	−	943.000i	1.21992	−	1.21992i	0.252264	−	0.967658i	$-0.418825\pi$
$773$	943.000	−	943.000i	1.21992	−	1.21992i	0.967658	−	0.252264i	$-0.0811751\pi$
$774$		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
$785$	−329.000	−	47.0000i	−0.419108	−	0.0598726i
$786$		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
$793$	−2040.00	+	2040.00i	−2.57251	+	2.57251i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1127.00	+	1127.00i	1.41405	+	1.41405i	0.717691	+	0.696361i	$0.245199\pi$
$797$	1127.00	+	1127.00i	1.41405	+	1.41405i	0.696361	+	0.717691i	$0.254801\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1440.00			−1.79775
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			560.000i			0.692213i	0.938195	+	0.346106i	$0.112496\pi$
$809$			560.000i			0.692213i	−0.938195	+	0.346106i	$0.887504\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		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
$821$	−1400.00			−1.70524			−0.852619	−	0.522533i	$-0.824987\pi$
$821$	−1400.00			−1.70524			−0.852619	+	0.522533i	$0.824987\pi$
$822$		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
$829$			1258.00i			1.51749i	0.651387	+	0.758745i	$0.274187\pi$
$829$			1258.00i			1.51749i	−0.651387	+	0.758745i	$0.725813\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	343.000	−	343.000i	0.411765	−	0.411765i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−759.000			−0.902497
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1227.00	+	1636.00i	1.45207	+	1.93609i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	623.000	−	623.000i	0.730363	−	0.730363i	−0.240328	−	0.970692i	$-0.577255\pi$
$853$	623.000	−	623.000i	0.730363	−	0.730363i	0.970692	+	0.240328i	$0.0772551\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	593.000	+	593.000i	0.691949	+	0.691949i	0.962660	−	0.270712i	$-0.0872590\pi$
$857$	593.000	+	593.000i	0.691949	+	0.691949i	−0.270712	+	0.962660i	$0.587259\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		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
$865$	−217.000	+	1519.00i	−0.250867	+	1.75607i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	63.0000	−	63.0000i	0.0721649	−	0.0721649i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1153.00	−	1153.00i	−1.31471	−	1.31471i	−0.917902	−	0.396807i	$-0.870118\pi$
$877$	−1153.00	−	1153.00i	−1.31471	−	1.31471i	−0.396807	−	0.917902i	$-0.629882\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1600.00			−1.81612			−0.908059	−	0.418842i	$-0.862436\pi$
$881$	−1600.00			−1.81612			−0.908059	+	0.418842i	$0.862436\pi$
$882$		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
$901$	238.000			0.264151
$902$		0			0
$903$		0			0
$904$		0			0
$905$	152.000	−	114.000i	0.167956	−	0.125967i
$906$		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
$909$			1782.00i			1.96040i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1457.00	+	799.000i	−1.57514	+	0.863784i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			258.000i			0.277718i	0.990312	+	0.138859i	$0.0443435\pi$
$929$			258.000i			0.277718i	−0.990312	+	0.138859i	$0.955657\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1127.00	+	1127.00i	1.20277	+	1.20277i	0.973319	+	0.229456i	$0.0736946\pi$
$937$	1127.00	+	1127.00i	1.20277	+	1.20277i	0.229456	+	0.973319i	$0.426305\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1482.00			1.57492			0.787460	−	0.616366i	$-0.211396\pi$
$941$	1482.00			1.57492			0.787460	+	0.616366i	$0.211396\pi$
$942$		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
$949$		−	3502.00i		−	3.69020i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1343.00	−	1343.00i	1.40923	−	1.40923i	0.645331	−	0.763903i	$-0.276720\pi$
$953$	1343.00	−	1343.00i	1.40923	−	1.40923i	0.763903	−	0.645331i	$-0.223280\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−961.000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−263.000	+	1841.00i	−0.272539	+	1.90777i
$966$		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			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1193.00	+	1193.00i	1.22108	+	1.22108i	0.967247	+	0.253838i	$0.0816931\pi$
$977$	1193.00	+	1193.00i	1.22108	+	1.22108i	0.253838	+	0.967247i	$0.418307\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1638.00			−1.66972
$982$		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
$985$	1169.00	+	167.000i	1.18680	+	0.169543i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−553.000	−	553.000i	−0.554664	−	0.554664i	0.373119	−	0.927783i	$-0.378288\pi$
$997$	−553.000	−	553.000i	−0.554664	−	0.554664i	−0.927783	+	0.373119i	$0.878288\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.3.p.a.97.1	yes	2
4.3	odd	2	CM	160.3.p.a.97.1	yes	2
5.2	odd	4		800.3.p.b.193.1		2
5.3	odd	4	inner	160.3.p.a.33.1	✓	2
5.4	even	2		800.3.p.b.257.1		2
8.3	odd	2		320.3.p.e.257.1		2
8.5	even	2		320.3.p.e.257.1		2
20.3	even	4	inner	160.3.p.a.33.1	✓	2
20.7	even	4		800.3.p.b.193.1		2
20.19	odd	2		800.3.p.b.257.1		2
40.3	even	4		320.3.p.e.193.1		2
40.13	odd	4		320.3.p.e.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.p.a.33.1	✓	2	5.3	odd	4	inner
160.3.p.a.33.1	✓	2	20.3	even	4	inner
160.3.p.a.97.1	yes	2	1.1	even	1	trivial
160.3.p.a.97.1	yes	2	4.3	odd	2	CM
320.3.p.e.193.1		2	40.3	even	4
320.3.p.e.193.1		2	40.13	odd	4
320.3.p.e.257.1		2	8.3	odd	2
320.3.p.e.257.1		2	8.5	even	2
800.3.p.b.193.1		2	5.2	odd	4
800.3.p.b.193.1		2	20.7	even	4
800.3.p.b.257.1		2	5.4	even	2
800.3.p.b.257.1		2	20.19	odd	2

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	160.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 + 3.00000i) q^{5} +9.00000i q^{9} +O(q^{10})\)	\(q+(-4.00000 + 3.00000i) q^{5} +9.00000i q^{9} +(-17.0000 + 17.0000i) q^{13} +(7.00000 + 7.00000i) q^{17} +(7.00000 - 24.0000i) q^{25} +40.0000i q^{29} +(-47.0000 - 47.0000i) q^{37} +80.0000 q^{41} +(-27.0000 - 36.0000i) q^{45} -49.0000i q^{49} +(17.0000 - 17.0000i) q^{53} +120.000 q^{61} +(17.0000 - 119.000i) q^{65} +(-103.000 + 103.000i) q^{73} -81.0000 q^{81} +(-49.0000 - 7.00000i) q^{85} +160.000i q^{89} +(-7.00000 - 7.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5}+O(q^{10}) \) 2 * q - 8 * q^5	\( 2 q - 8 q^{5} - 34 q^{13} + 14 q^{17} + 14 q^{25} - 94 q^{37} + 160 q^{41} - 54 q^{45} + 34 q^{53} + 240 q^{61} + 34 q^{65} - 206 q^{73} - 162 q^{81} - 98 q^{85} - 14 q^{97}+O(q^{100}) \) 2 * q - 8 * q^5 - 34 * q^13 + 14 * q^17 + 14 * q^25 - 94 * q^37 + 160 * q^41 - 54 * q^45 + 34 * q^53 + 240 * q^61 + 34 * q^65 - 206 * q^73 - 162 * q^81 - 98 * q^85 - 14 * q^97

Introduction

L-functions

Modular forms

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