LMFDB - Embedded newform 160.7.p.b.33.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,7,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, 7, names="a")

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

chi := DirichletCharacter("160.33");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
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:	$36.8086533792$
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			33.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	160.33
Dual form			160.7.p.b.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(44.0000 + 117.000i) q^{5} -729.000i q^{9} +(1207.00 + 1207.00i) q^{13} +(5383.00 - 5383.00i) q^{17} +(-11753.0 + 10296.0i) q^{25} +36920.0i q^{29} +(14617.0 - 14617.0i) q^{37} +108560. q^{41} +(85293.0 - 32076.0i) q^{45} +117649. i q^{49} +(133433. + 133433. i) q^{53} +388440. q^{61} +(-88111.0 + 194327. i) q^{65} +(538793. + 538793. i) q^{73} -531441. q^{81} +(866663. + 392959. i) q^{85} -293920. i q^{89} +(-196903. + 196903. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 88 q^{5} + 2414 q^{13} + 10766 q^{17} - 23506 q^{25} + 29234 q^{37} + 217120 q^{41} + 170586 q^{45} + 266866 q^{53} + 776880 q^{61} - 176222 q^{65} + 1077586 q^{73} - 1062882 q^{81} + 1733326 q^{85}+ \cdots - 393806 q^{97}+O(q^{100}) $ 2 * q + 88 * q^5 + 2414 * q^13 + 10766 * q^17 - 23506 * q^25 + 29234 * q^37 + 217120 * q^41 + 170586 * q^45 + 266866 * q^53 + 776880 * q^61 - 176222 * q^65 + 1077586 * q^73 - 1062882 * q^81 + 1733326 * q^85 - 393806 * q^97

Character values

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

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	44.0000	+	117.000i	0.352000	+	0.936000i
$5$	44.0000	+	117.000i	0.352000	+	0.936000i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		−	729.000i		−	1.00000i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	1207.00	+	1207.00i	0.549386	+	0.549386i	0.926263	−	0.376878i	$-0.123002\pi$
$13$	1207.00	+	1207.00i	0.549386	+	0.549386i	−0.376878	+	0.926263i	$0.623002\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5383.00	−	5383.00i	1.09566	−	1.09566i	0.100753	−	0.994911i	$-0.467875\pi$
$17$	5383.00	−	5383.00i	1.09566	−	1.09566i	0.994911	−	0.100753i	$-0.0321252\pi$
$18$		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	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	−11753.0	+	10296.0i	−0.752192	+	0.658944i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			36920.0i			1.51380i	0.653532	+	0.756899i	$0.273286\pi$
$29$			36920.0i			1.51380i	−0.653532	+	0.756899i	$0.726714\pi$
$30$		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
$37$	14617.0	−	14617.0i	0.288571	−	0.288571i	−0.547944	−	0.836515i	$-0.684589\pi$
$37$	14617.0	−	14617.0i	0.288571	−	0.288571i	0.836515	+	0.547944i	$0.184589\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	108560.			1.57514			0.787568	−	0.616227i	$-0.211340\pi$
$41$	108560.			1.57514			0.787568	+	0.616227i	$0.211340\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	85293.0	−	32076.0i	0.936000	−	0.352000i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			117649.i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	133433.	+	133433.i	0.896263	+	0.896263i	0.995103	−	0.0988400i	$-0.0315132\pi$
$53$	133433.	+	133433.i	0.896263	+	0.896263i	−0.0988400	+	0.995103i	$0.531513\pi$
$54$		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
$61$	388440.			1.71133			0.855666	−	0.517528i	$-0.173148\pi$
$61$	388440.			1.71133			0.855666	+	0.517528i	$0.173148\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−88111.0	+	194327.i	−0.320841	+	0.707609i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		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
$73$	538793.	+	538793.i	1.38501	+	1.38501i	0.835460	+	0.549552i	$0.185202\pi$
$73$	538793.	+	538793.i	1.38501	+	1.38501i	0.549552	+	0.835460i	$0.314798\pi$
$74$		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
$81$	−531441.			−1.00000
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	866663.	+	392959.i	1.41122	+	0.639868i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	293920.i		−	0.416926i	−0.978030	−	0.208463i	$-0.933154\pi$
$89$		−	293920.i		−	0.416926i	0.978030	−	0.208463i	$-0.0668461\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−196903.	+	196903.i	−0.215743	+	0.215743i	−0.806702	−	0.590959i	$-0.798750\pi$
$97$	−196903.	+	196903.i	−0.215743	+	0.215743i	0.590959	+	0.806702i	$0.298750\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.70300e6			1.65291			0.826457	−	0.563001i	$-0.190353\pi$
$101$	1.70300e6			1.65291			0.826457	+	0.563001i	$0.190353\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$			458458.i			0.354014i	0.984210	+	0.177007i	$0.0566415\pi$
$109$			458458.i			0.354014i	−0.984210	+	0.177007i	$0.943359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.89043e6	−	1.89043e6i	−1.31016	−	1.31016i	−0.921291	−	0.388874i	$-0.872864\pi$
$113$	−1.89043e6	−	1.89043e6i	−1.31016	−	1.31016i	−0.388874	−	0.921291i	$-0.627136\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	879903.	−	879903.i	0.549386	−	0.549386i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.77156e6			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.72176e6	−	922077.i	−0.881543	−	0.472103i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		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
$137$	−3.51086e6	+	3.51086e6i	−1.36538	+	1.36538i	−0.498467	+	0.866909i	$0.666104\pi$
$137$	−3.51086e6	+	3.51086e6i	−1.36538	+	1.36538i	−0.866909	+	0.498467i	$0.833896\pi$
$138$		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
$145$	−4.31964e6	+	1.62448e6i	−1.41691	+	0.532857i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	5.73230e6i		−	1.73289i	−0.499276	−	0.866443i	$-0.666401\pi$
$149$		−	5.73230e6i		−	1.73289i	0.499276	−	0.866443i	$-0.333599\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−3.92421e6	−	3.92421e6i	−1.09566	−	1.09566i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.26786e6	−	3.26786e6i	0.844432	−	0.844432i	−0.144999	−	0.989432i	$-0.546318\pi$
$157$	3.26786e6	−	3.26786e6i	0.844432	−	0.844432i	0.989432	+	0.144999i	$0.0463180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	1.91311e6i		−	0.396351i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−952847.	−	952847.i	−0.184028	−	0.184028i	0.609080	−	0.793109i	$-0.291539\pi$
$173$	−952847.	−	952847.i	−0.184028	−	0.184028i	−0.793109	+	0.609080i	$0.791539\pi$
$174$		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
$181$	3.67988e6			0.620581			0.310290	−	0.950642i	$-0.399574\pi$
$181$	3.67988e6			0.620581			0.310290	+	0.950642i	$0.399574\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.35334e6	+	1.06704e6i	0.371680	+	0.168526i
$186$		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
$193$	−6.99343e6	−	6.99343e6i	−0.972789	−	0.972789i	0.0268508	−	0.999639i	$-0.491452\pi$
$193$	−6.99343e6	−	6.99343e6i	−0.972789	−	0.972789i	−0.999639	+	0.0268508i	$0.991452\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.01284e7	+	1.01284e7i	−1.32477	+	1.32477i	−0.414911	+	0.909862i	$0.636187\pi$
$197$	−1.01284e7	+	1.01284e7i	−1.32477	+	1.32477i	−0.909862	+	0.414911i	$0.863813\pi$
$198$		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
$205$	4.77664e6	+	1.27015e7i	0.554448	+	1.47433i
$206$		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
$221$	1.29946e7			1.20388
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	7.50578e6	+	8.56794e6i	0.658944	+	0.752192i
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		−	1.71508e7i		−	1.42816i	−0.700064	−	0.714080i	$-0.746845\pi$
$229$		−	1.71508e7i		−	1.42816i	0.700064	−	0.714080i	$-0.253155\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.45898e7	+	1.45898e7i	1.15341	+	1.15341i	0.985865	+	0.167543i	$0.0535834\pi$
$233$	1.45898e7	+	1.45898e7i	1.15341	+	1.15341i	0.167543	+	0.985865i	$0.446417\pi$
$234$		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
$241$	2.79943e7			1.99995			0.999974	−	0.00718191i	$-0.00228609\pi$
$241$	2.79943e7			1.99995			0.999974	+	0.00718191i	$0.00228609\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.37649e7	+	5.17656e6i	−0.936000	+	0.352000i
$246$		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
$257$	9.58838e6	−	9.58838e6i	0.564867	−	0.564867i	−0.365819	−	0.930686i	$-0.619211\pi$
$257$	9.58838e6	−	9.58838e6i	0.564867	−	0.564867i	0.930686	+	0.365819i	$0.119211\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.69147e7			1.51380
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	−9.74061e6	+	2.14827e7i	−0.523418	+	1.15439i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2.73294e7i			1.40402i	0.712168	+	0.702009i	$0.247714\pi$
$269$			2.73294e7i			1.40402i	−0.712168	+	0.702009i	$0.752286\pi$
$270$		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
$277$	−1.43831e7	+	1.43831e7i	−0.676726	+	0.676726i	−0.959258	−	0.282532i	$-0.908826\pi$
$277$	−1.43831e7	+	1.43831e7i	−0.676726	+	0.676726i	0.282532	+	0.959258i	$0.408826\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.30346e7			1.93954			0.969769	−	0.244024i	$-0.0784675\pi$
$281$	4.30346e7			1.93954			0.969769	+	0.244024i	$0.0784675\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	3.38158e7i		−	1.40096i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.54511e7	−	3.54511e7i	−1.40937	−	1.40937i	−0.763131	−	0.646244i	$-0.776339\pi$
$293$	−3.54511e7	−	3.54511e7i	−1.40937	−	1.40937i	−0.646244	−	0.763131i	$-0.723661\pi$
$294$		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
$305$	1.70914e7	+	4.54475e7i	0.602389	+	1.60181i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		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
$313$	−2.25012e7	−	2.25012e7i	−0.733790	−	0.733790i	0.237578	−	0.971368i	$-0.423646\pi$
$313$	−2.25012e7	−	2.25012e7i	−0.733790	−	0.733790i	−0.971368	+	0.237578i	$0.923646\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.49431e7	+	4.49431e7i	−1.41087	+	1.41087i	−0.656805	+	0.754061i	$0.728092\pi$
$317$	−4.49431e7	+	4.49431e7i	−1.41087	+	1.41087i	−0.754061	+	0.656805i	$0.771908\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.66131e7	−	1.75860e6i	−0.775258	−	0.0512291i
$326$		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
$333$	−1.06558e7	−	1.06558e7i	−0.288571	−	0.288571i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.56141e7	−	3.56141e7i	0.930534	−	0.930534i	−0.0672052	−	0.997739i	$-0.521408\pi$
$337$	3.56141e7	−	3.56141e7i	0.930534	−	0.930534i	0.997739	+	0.0672052i	$0.0214082\pi$
$338$		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	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	8.48891e7i		−	1.99699i	−0.0548572	−	0.998494i	$-0.517470\pi$
$349$		−	8.48891e7i		−	1.99699i	0.0548572	−	0.998494i	$-0.482530\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.97349e7	−	5.97349e7i	−1.35801	−	1.35801i	−0.876363	−	0.481651i	$-0.840037\pi$
$353$	−5.97349e7	−	5.97349e7i	−1.35801	−	1.35801i	−0.481651	−	0.876363i	$-0.659963\pi$
$354$		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
$361$	4.70459e7			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.93319e7	+	8.67457e7i	−0.808847	+	1.78389i
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		−	7.91402e7i		−	1.57514i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−9.57557e6	−	9.57557e6i	−0.184518	−	0.184518i	0.608803	−	0.793321i	$-0.291650\pi$
$373$	−9.57557e6	−	9.57557e6i	−0.184518	−	0.184518i	−0.793321	+	0.608803i	$0.791650\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.45624e7	+	4.45624e7i	−0.831658	+	0.831658i
$378$		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	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	1.17588e8i		−	1.99762i	−0.0487324	−	0.998812i	$-0.515518\pi$
$389$		−	1.17588e8i		−	1.99762i	0.0487324	−	0.998812i	$-0.484482\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.67514e7	+	7.67514e7i	−1.22663	+	1.22663i	−0.261404	+	0.965229i	$0.584186\pi$
$397$	−7.67514e7	+	7.67514e7i	−1.22663	+	1.22663i	−0.965229	+	0.261404i	$0.915814\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.23212e8			−1.91082			−0.955410	−	0.295282i	$-0.904586\pi$
$401$	−1.23212e8			−1.91082			−0.955410	+	0.295282i	$0.904586\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.33834e7	−	6.21786e7i	−0.352000	−	0.936000i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			8.57705e7i			1.25363i	0.779169	+	0.626813i	$0.215641\pi$
$409$			8.57705e7i			1.25363i	−0.779169	+	0.626813i	$0.784359\pi$
$410$		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
$421$	1.46057e8			1.95738			0.978690	−	0.205343i	$-0.0658311\pi$
$421$	1.46057e8			1.95738			0.978690	+	0.205343i	$0.0658311\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.84303e6	+	1.18690e8i	−0.102169	+	1.54613i
$426$		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
$433$	−1.11546e8	−	1.11546e8i	−1.37401	−	1.37401i	−0.854408	−	0.519602i	$-0.826080\pi$
$433$	−1.11546e8	−	1.11546e8i	−1.37401	−	1.37401i	−0.519602	−	0.854408i	$-0.673920\pi$
$434$		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
$441$	8.57661e7			1.00000
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	3.43886e7	−	1.29325e7i	0.390243	−	0.146758i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.86233e7i			0.868585i	0.900772	+	0.434292i	$0.143002\pi$
$449$			7.86233e7i			0.868585i	−0.900772	+	0.434292i	$0.856998\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.55133e7	−	4.55133e7i	0.476859	−	0.476859i	−0.427267	−	0.904126i	$-0.640523\pi$
$457$	4.55133e7	−	4.55133e7i	0.476859	−	0.476859i	0.904126	+	0.427267i	$0.140523\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.90571e8			1.94516			0.972578	−	0.232576i	$-0.0747153\pi$
$461$	1.90571e8			1.94516			0.972578	+	0.232576i	$0.0747153\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		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
$477$	9.72727e7	−	9.72727e7i	0.896263	−	0.896263i
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	3.52854e7			0.317074
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.17014e7	−	1.43739e7i	−0.277877	−	0.125994i
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		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
$493$	1.98740e8	+	1.98740e8i	1.65861	+	1.65861i
$494$		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	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	7.49319e7	+	1.99251e8i	0.581825	+	1.54713i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			2.56803e8i			1.94736i	0.227916	+	0.973681i	$0.426809\pi$
$509$			2.56803e8i			1.94736i	−0.227916	+	0.973681i	$0.573191\pi$
$510$		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
$521$	−2.80651e8			−1.98451			−0.992256	−	0.124212i	$-0.960360\pi$
$521$	−2.80651e8			−1.98451			−0.992256	+	0.124212i	$0.960360\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	1.48036e8i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.31032e8	+	1.31032e8i	0.865357	+	0.865357i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.81611e8			1.77851			0.889257	−	0.457407i	$-0.151222\pi$
$541$	2.81611e8			1.77851			0.889257	+	0.457407i	$0.151222\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.36396e7	+	2.01722e7i	−0.331357	+	0.124613i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		−	2.83173e8i		−	1.71133i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.42722e8	+	2.42722e8i	−1.40457	+	1.40457i	−0.619863	+	0.784710i	$0.712812\pi$
$557$	−2.42722e8	+	2.42722e8i	−1.40457	+	1.40457i	−0.784710	+	0.619863i	$0.787188\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	1.38002e8	−	3.04360e8i	0.765136	−	1.68749i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.50122e8i			1.90056i	0.311393	+	0.950281i	$0.399204\pi$
$569$			3.50122e8i			1.90056i	−0.311393	+	0.950281i	$0.600796\pi$
$570$		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
$577$	2.33634e8	−	2.33634e8i	1.21621	−	1.21621i	0.247264	−	0.968948i	$-0.420469\pi$
$577$	2.33634e8	−	2.33634e8i	1.21621	−	1.21621i	0.968948	−	0.247264i	$-0.0795315\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1.41664e8	+	6.42329e7i	0.707609	+	0.320841i
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.00505e8	+	1.00505e8i	0.481972	+	0.481972i	0.905761	−	0.423789i	$-0.139300\pi$
$593$	1.00505e8	+	1.00505e8i	0.481972	+	0.481972i	−0.423789	+	0.905761i	$0.639300\pi$
$594$		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
$601$	−4.09537e8			−1.88656			−0.943279	−	0.332001i	$-0.892276\pi$
$601$	−4.09537e8			−1.88656			−0.943279	+	0.332001i	$0.892276\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.79487e7	−	2.07273e8i	−0.352000	−	0.936000i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.66256e8	−	2.66256e8i	−1.15589	−	1.15589i	−0.985350	−	0.170544i	$-0.945447\pi$
$613$	−2.66256e8	−	2.66256e8i	−1.15589	−	1.15589i	−0.170544	−	0.985350i	$-0.554553\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.93596e7	+	8.93596e7i	−0.380440	+	0.380440i	−0.871260	−	0.490821i	$-0.836697\pi$
$617$	−8.93596e7	+	8.93596e7i	−0.380440	+	0.380440i	0.490821	+	0.871260i	$0.336697\pi$
$618$		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
$625$	3.21254e7	−	2.42018e8i	0.131586	−	0.991305i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.57367e8i		−	0.632355i
$630$		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
$637$	−1.42002e8	+	1.42002e8i	−0.549386	+	0.549386i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.29057e8			−1.62908			−0.814538	−	0.580111i	$-0.803009\pi$
$641$	−4.29057e8			−1.62908			−0.814538	+	0.580111i	$0.803009\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.94812e8	+	2.94812e8i	1.05878	+	1.05878i	0.998161	+	0.0606193i	$0.0193076\pi$
$653$	2.94812e8	+	2.94812e8i	1.05878	+	1.05878i	0.0606193	+	0.998161i	$0.480692\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.92780e8	−	3.92780e8i	1.38501	−	1.38501i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−5.70458e8			−1.97524			−0.987618	−	0.156876i	$-0.949858\pi$
$661$	−5.70458e8			−1.97524			−0.987618	+	0.156876i	$0.949858\pi$
$662$		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
$673$	−3.72130e8	−	3.72130e8i	−1.22082	−	1.22082i	−0.967342	−	0.253473i	$-0.918427\pi$
$673$	−3.72130e8	−	3.72130e8i	−1.22082	−	1.22082i	−0.253473	−	0.967342i	$-0.581573\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.31134e8	+	2.31134e8i	−0.744901	+	0.744901i	−0.973517	−	0.228616i	$-0.926580\pi$
$677$	−2.31134e8	+	2.31134e8i	−0.744901	+	0.744901i	0.228616	+	0.973517i	$0.426580\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−5.65249e8	−	2.56293e8i	−1.75860	−	0.797379i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.22107e8i			0.984788i
$690$		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
$697$	5.84378e8	−	5.84378e8i	1.72582	−	1.72582i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.25978e8			−1.81721			−0.908604	−	0.417658i	$-0.862851\pi$
$701$	−6.25978e8			−1.81721			−0.908604	+	0.417658i	$0.862851\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			3.09351e8i			0.867987i	0.900916	+	0.433993i	$0.142896\pi$
$709$			3.09351e8i			0.867987i	−0.900916	+	0.433993i	$0.857104\pi$
$710$		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
$725$	−3.80128e8	−	4.33921e8i	−0.997508	−	1.13867i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			3.87420e8i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.24752e8	+	5.24752e8i	1.33242	+	1.33242i	0.903198	+	0.429225i	$0.141213\pi$
$733$	5.24752e8	+	5.24752e8i	1.33242	+	1.33242i	0.429225	+	0.903198i	$0.358787\pi$
$734$		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	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	6.70679e8	−	2.52221e8i	1.62198	−	0.609976i
$746$		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
$757$	2.14235e8	−	2.14235e8i	0.493859	−	0.493859i	−0.415661	−	0.909520i	$-0.636450\pi$
$757$	2.14235e8	−	2.14235e8i	0.493859	−	0.493859i	0.909520	+	0.415661i	$0.136450\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.35040e8			0.306413			0.153207	−	0.988194i	$-0.451040\pi$
$761$	1.35040e8			0.306413			0.153207	+	0.988194i	$0.451040\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.86467e8	−	6.31797e8i	0.639868	−	1.41122i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	4.00900e8i		−	0.881570i	−0.897613	−	0.440785i	$-0.854700\pi$
$769$		−	4.00900e8i		−	0.881570i	0.897613	−	0.440785i	$-0.145300\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.32859e7	+	1.32859e7i	0.0287643	+	0.0287643i	0.721343	−	0.692578i	$-0.243525\pi$
$773$	1.32859e7	+	1.32859e7i	0.0287643	+	0.0287643i	−0.692578	+	0.721343i	$0.743525\pi$
$774$		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
$785$	5.26126e8	+	2.38554e8i	1.08763	+	0.493149i
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.68847e8	+	4.68847e8i	0.940181	+	0.940181i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.15229e8	+	7.15229e8i	−1.41277	+	1.41277i	−0.674398	+	0.738368i	$0.735597\pi$
$797$	−7.15229e8	+	7.15229e8i	−1.41277	+	1.41277i	−0.738368	+	0.674398i	$0.764403\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−2.14268e8			−0.416926
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			9.23912e8i			1.74496i	0.488651	+	0.872479i	$0.337489\pi$
$809$			9.23912e8i			1.74496i	−0.488651	+	0.872479i	$0.662511\pi$
$810$		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
$821$	8.69722e7			0.157163			0.0785816	−	0.996908i	$-0.474961\pi$
$821$	8.69722e7			0.157163			0.0785816	+	0.996908i	$0.474961\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$			6.02782e8i			1.05803i	0.848613	+	0.529013i	$0.177438\pi$
$829$			6.02782e8i			1.05803i	−0.848613	+	0.529013i	$0.822562\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.33305e8	+	6.33305e8i	1.09566	+	1.09566i
$834$		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
$841$	−7.68263e8			−1.29158
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.23834e8	−	8.41769e7i	0.370985	−	0.139516i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	8.76292e8	+	8.76292e8i	1.41189	+	1.41189i	0.746432	+	0.665462i	$0.231765\pi$
$853$	8.76292e8	+	8.76292e8i	1.41189	+	1.41189i	0.665462	+	0.746432i	$0.268235\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.89529e8	−	8.89529e8i	1.41325	−	1.41325i	0.680467	−	0.732779i	$-0.261777\pi$
$857$	8.89529e8	−	8.89529e8i	1.41325	−	1.41325i	0.732779	−	0.680467i	$-0.238223\pi$
$858$		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	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	6.95578e7	−	1.53408e8i	0.107473	−	0.237029i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.43542e8	+	1.43542e8i	0.215743	+	0.215743i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.05200e8	−	4.05200e8i	0.600718	−	0.600718i	−0.339785	−	0.940503i	$-0.610354\pi$
$877$	4.05200e8	−	4.05200e8i	0.600718	−	0.600718i	0.940503	+	0.339785i	$0.110354\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.70427e8			−0.541720			−0.270860	−	0.962619i	$-0.587308\pi$
$881$	−3.70427e8			−0.541720			−0.270860	+	0.962619i	$0.587308\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		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
$901$	1.43654e9			1.96401
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.61915e8	+	4.30546e8i	0.218444	+	0.580863i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	1.24149e9i		−	1.65291i
$910$		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
$925$	−2.12970e7	+	3.22290e8i	−0.0269087	+	0.407213i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			6.50820e8i			0.811734i	0.913932	+	0.405867i	$0.133030\pi$
$929$			6.50820e8i			0.811734i	−0.913932	+	0.405867i	$0.866970\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.05542e8	−	1.05542e8i	0.128295	−	0.128295i	−0.640044	−	0.768338i	$-0.721084\pi$
$937$	1.05542e8	−	1.05542e8i	0.128295	−	0.128295i	0.768338	+	0.640044i	$0.221084\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6.81896e8			−0.818370			−0.409185	−	0.912452i	$-0.634187\pi$
$941$	−6.81896e8			−0.818370			−0.409185	+	0.912452i	$0.634187\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$			1.30065e9i			1.52181i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.18543e9	−	1.18543e9i	−1.36961	−	1.36961i	−0.860996	−	0.508612i	$-0.830159\pi$
$953$	−1.18543e9	−	1.18543e9i	−1.36961	−	1.36961i	−0.508612	−	0.860996i	$-0.669841\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8.87504e8			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.10521e8	−	1.12594e9i	0.568109	−	1.25295i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		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
$977$	2.03872e7	−	2.03872e7i	0.0218612	−	0.0218612i	−0.696092	−	0.717953i	$-0.745079\pi$
$977$	2.03872e7	−	2.03872e7i	0.0218612	−	0.0218612i	0.717953	+	0.696092i	$0.245079\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	3.34216e8			0.354014
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−1.63067e9	−	7.39372e8i	−1.70631	−	0.773667i
$986$		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
$997$	−1.31084e9	+	1.31084e9i	−1.32270	+	1.32270i	−0.411127	+	0.911578i	$0.634865\pi$
$997$	−1.31084e9	+	1.31084e9i	−1.32270	+	1.32270i	−0.911578	+	0.411127i	$0.865135\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.7.p.b.33.1	✓	2
4.3	odd	2	CM	160.7.p.b.33.1	✓	2
5.2	odd	4	inner	160.7.p.b.97.1	yes	2
20.7	even	4	inner	160.7.p.b.97.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.7.p.b.33.1	✓	2	1.1	even	1	trivial
160.7.p.b.33.1	✓	2	4.3	odd	2	CM
160.7.p.b.97.1	yes	2	5.2	odd	4	inner
160.7.p.b.97.1	yes	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	160.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(44.0000 + 117.000i) q^{5} -729.000i q^{9} +(1207.00 + 1207.00i) q^{13} +(5383.00 - 5383.00i) q^{17} +(-11753.0 + 10296.0i) q^{25} +36920.0i q^{29} +(14617.0 - 14617.0i) q^{37} +108560. q^{41} +(85293.0 - 32076.0i) q^{45} +117649. i q^{49} +(133433. + 133433. i) q^{53} +388440. q^{61} +(-88111.0 + 194327. i) q^{65} +(538793. + 538793. i) q^{73} -531441. q^{81} +(866663. + 392959. i) q^{85} -293920. i q^{89} +(-196903. + 196903. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 88 q^{5} + 2414 q^{13} + 10766 q^{17} - 23506 q^{25} + 29234 q^{37} + 217120 q^{41} + 170586 q^{45} + 266866 q^{53} + 776880 q^{61} - 176222 q^{65} + 1077586 q^{73} - 1062882 q^{81} + 1733326 q^{85}+ \cdots - 393806 q^{97}+O(q^{100}) \) 2 * q + 88 * q^5 + 2414 * q^13 + 10766 * q^17 - 23506 * q^25 + 29234 * q^37 + 217120 * q^41 + 170586 * q^45 + 266866 * q^53 + 776880 * q^61 - 176222 * q^65 + 1077586 * q^73 - 1062882 * q^81 + 1733326 * q^85 - 393806 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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