LMFDB - Embedded newform 360.2.d.a.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,Mod(109,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("360.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	360.d (of order $2$, degree $1$, 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:	$2.87461447277$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.1
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.109
Dual form			360.2.d.a.109.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} +2.82843i q^{8} -3.16228 q^{10} -4.47214i q^{11} -6.32456 q^{13} +4.00000 q^{16} -2.82843i q^{17} +4.47214i q^{20} -6.32456 q^{22} +5.65685i q^{23} -5.00000 q^{25} +8.94427i q^{26} -4.47214i q^{29} +2.00000 q^{31} -5.65685i q^{32} -4.00000 q^{34} -6.32456 q^{37} +6.32456 q^{40} +12.6491 q^{43} +8.94427i q^{44} +8.00000 q^{46} -11.3137i q^{47} +7.00000 q^{49} +7.07107i q^{50} +12.6491 q^{52} -10.0000 q^{55} -6.32456 q^{58} -4.47214i q^{59} -2.82843i q^{62} -8.00000 q^{64} +14.1421i q^{65} +12.6491 q^{67} +5.65685i q^{68} +8.94427i q^{74} +14.0000 q^{79} -8.94427i q^{80} -6.32456 q^{85} -17.8885i q^{86} +12.6491 q^{88} -11.3137i q^{92} -16.0000 q^{94} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{31} - 16 q^{34} + 32 q^{46} + 28 q^{49} - 40 q^{55} - 32 q^{64} + 56 q^{79} - 64 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 8 * q^31 - 16 * q^34 + 32 * q^46 + 28 * q^49 - 40 * q^55 - 32 * q^64 + 56 * q^79 - 64 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	−3.16228	−1.00000
$11$	− 4.47214i	− 1.34840i	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	− 4.47214i	− 1.34840i	0.738549	−	0.674200i	$-0.235511\pi$
$12$	0	0
$13$	−6.32456	−1.75412	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	−6.32456	−1.75412	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 2.82843i	− 0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	− 2.82843i	− 0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	4.47214i	1.00000i
$21$	0	0
$22$	−6.32456	−1.34840
$23$	5.65685i	1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$23$	5.65685i	1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	8.94427i	1.75412i
$27$	0	0
$28$	0	0
$29$	− 4.47214i	− 0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$	− 4.47214i	− 0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	−6.32456	−1.03975	−0.519875	−	0.854242i	$-0.674022\pi$
$37$	−6.32456	−1.03975	−0.519875	+	0.854242i	$0.674022\pi$
$38$	0	0
$39$	0	0
$40$	6.32456	1.00000
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	12.6491	1.92897	0.964486	−	0.264135i	$-0.0850865\pi$
$43$	12.6491	1.92897	0.964486	+	0.264135i	$0.0850865\pi$
$44$	8.94427i	1.34840i
$45$	0	0
$46$	8.00000	1.17954
$47$	− 11.3137i	− 1.65027i	−0.564933	−	0.825137i	$-0.691098\pi$
$47$	− 11.3137i	− 1.65027i	0.564933	−	0.825137i	$-0.308902\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	7.07107i	1.00000i
$51$	0	0
$52$	12.6491	1.75412
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−10.0000	−1.34840
$56$	0	0
$57$	0	0
$58$	−6.32456	−0.830455
$59$	− 4.47214i	− 0.582223i	−0.956689	−	0.291111i	$-0.905975\pi$
$59$	− 4.47214i	− 0.582223i	0.956689	−	0.291111i	$-0.0940250\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	− 2.82843i	− 0.359211i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	14.1421i	1.75412i
$66$	0	0
$67$	12.6491	1.54533	0.772667	−	0.634811i	$-0.218922\pi$
$67$	12.6491	1.54533	0.772667	+	0.634811i	$0.218922\pi$
$68$	5.65685i	0.685994i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	8.94427i	1.03975i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	− 8.94427i	− 1.00000i
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.32456	−0.685994
$86$	− 17.8885i	− 1.92897i
$87$	0	0
$88$	12.6491	1.34840
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	− 11.3137i	− 1.17954i
$93$	0	0
$94$	−16.0000	−1.65027
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 9.89949i	− 1.00000i
$99$	0	0
$100$	10.0000	1.00000
$101$	− 4.47214i	− 0.444994i	−0.974933	−	0.222497i	$-0.928579\pi$
$101$	− 4.47214i	− 0.444994i	0.974933	−	0.222497i	$-0.0714208\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	− 17.8885i	− 1.75412i
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	14.1421i	1.34840i
$111$	0	0
$112$	0	0
$113$	− 19.7990i	− 1.86253i	−0.364340	−	0.931266i	$-0.618705\pi$
$113$	− 19.7990i	− 1.86253i	0.364340	−	0.931266i	$-0.381295\pi$
$114$	0	0
$115$	12.6491	1.17954
$116$	8.94427i	0.830455i
$117$	0	0
$118$	−6.32456	−0.582223
$119$	0	0
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	−4.00000	−0.359211
$125$	11.1803i	1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	20.0000	1.75412
$131$	22.3607i	1.95366i	0.214013	+	0.976831i	$0.431347\pi$
$131$	22.3607i	1.95366i	−0.214013	+	0.976831i	$0.568653\pi$
$132$	0	0
$133$	0	0
$134$	− 17.8885i	− 1.54533i
$135$	0	0
$136$	8.00000	0.685994
$137$	− 2.82843i	− 0.241649i	−0.992674	−	0.120824i	$-0.961446\pi$
$137$	− 2.82843i	− 0.241649i	0.992674	−	0.120824i	$-0.0385538\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	28.2843i	2.36525i
$144$	0	0
$145$	−10.0000	−0.830455
$146$	0	0
$147$	0	0
$148$	12.6491	1.03975
$149$	22.3607i	1.83186i	0.401340	+	0.915929i	$0.368545\pi$
$149$	22.3607i	1.83186i	−0.401340	+	0.915929i	$0.631455\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.47214i	− 0.359211i
$156$	0	0
$157$	−6.32456	−0.504754	−0.252377	−	0.967629i	$-0.581212\pi$
$157$	−6.32456	−0.504754	−0.252377	+	0.967629i	$0.581212\pi$
$158$	− 19.7990i	− 1.57512i
$159$	0	0
$160$	−12.6491	−1.00000
$161$	0	0
$162$	0	0
$163$	−25.2982	−1.98151	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	−25.2982	−1.98151	−0.990755	+	0.135665i	$0.956683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.3137i	− 0.875481i	−0.899101	−	0.437741i	$-0.855779\pi$
$167$	− 11.3137i	− 0.875481i	0.899101	−	0.437741i	$-0.144221\pi$
$168$	0	0
$169$	27.0000	2.07692
$170$	8.94427i	0.685994i
$171$	0	0
$172$	−25.2982	−1.92897
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 17.8885i	− 1.34840i
$177$	0	0
$178$	0	0
$179$	22.3607i	1.67132i	0.549250	+	0.835658i	$0.314913\pi$
$179$	22.3607i	1.67132i	−0.549250	+	0.835658i	$0.685087\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−16.0000	−1.17954
$185$	14.1421i	1.03975i
$186$	0	0
$187$	−12.6491	−0.924995
$188$	22.6274i	1.65027i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−14.0000	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	− 14.1421i	− 1.00000i
$201$	0	0
$202$	−6.32456	−0.444994
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−25.2982	−1.75412
$209$	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
$215$	− 28.2843i	− 1.92897i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	20.0000	1.34840
$221$	17.8885i	1.20331i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	−28.0000	−1.86253
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	− 17.8885i	− 1.17954i
$231$	0	0
$232$	12.6491	0.830455
$233$	− 19.7990i	− 1.29707i	−0.761183	−	0.648537i	$-0.775381\pi$
$233$	− 19.7990i	− 1.29707i	0.761183	−	0.648537i	$-0.224619\pi$
$234$	0	0
$235$	−25.2982	−1.65027
$236$	8.94427i	0.582223i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	12.7279i	0.818182i
$243$	0	0
$244$	0	0
$245$	− 15.6525i	− 1.00000i
$246$	0	0
$247$	0	0
$248$	5.65685i	0.359211i
$249$	0	0
$250$	15.8114	1.00000
$251$	− 31.3050i	− 1.97595i	−0.154610	−	0.987976i	$-0.549412\pi$
$251$	− 31.3050i	− 1.97595i	0.154610	−	0.987976i	$-0.450588\pi$
$252$	0	0
$253$	25.2982	1.59049
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	31.1127i	1.94076i	0.241590	+	0.970378i	$0.422331\pi$
$257$	31.1127i	1.94076i	−0.241590	+	0.970378i	$0.577669\pi$
$258$	0	0
$259$	0	0
$260$	− 28.2843i	− 1.75412i
$261$	0	0
$262$	31.6228	1.95366
$263$	22.6274i	1.39527i	0.716455	+	0.697633i	$0.245763\pi$
$263$	22.6274i	1.39527i	−0.716455	+	0.697633i	$0.754237\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−25.2982	−1.54533
$269$	− 31.3050i	− 1.90870i	−0.298696	−	0.954348i	$-0.596552\pi$
$269$	− 31.3050i	− 1.90870i	0.298696	−	0.954348i	$-0.403448\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 11.3137i	− 0.685994i
$273$	0	0
$274$	−4.00000	−0.241649
$275$	22.3607i	1.34840i
$276$	0	0
$277$	31.6228	1.90003	0.950014	−	0.312207i	$-0.101068\pi$
$277$	31.6228	1.90003	0.950014	+	0.312207i	$0.101068\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	12.6491	0.751912	0.375956	−	0.926638i	$-0.377314\pi$
$283$	12.6491	0.751912	0.375956	+	0.926638i	$0.377314\pi$
$284$	0	0
$285$	0	0
$286$	40.0000	2.36525
$287$	0	0
$288$	0	0
$289$	9.00000	0.529412
$290$	14.1421i	0.830455i
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	− 17.8885i	− 1.03975i
$297$	0	0
$298$	31.6228	1.83186
$299$	− 35.7771i	− 2.06904i
$300$	0	0
$301$	0	0
$302$	31.1127i	1.79033i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.2982	−1.44385	−0.721923	−	0.691974i	$-0.756741\pi$
$307$	−25.2982	−1.44385	−0.721923	+	0.691974i	$0.756741\pi$
$308$	0	0
$309$	0	0
$310$	−6.32456	−0.359211
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	8.94427i	0.504754i
$315$	0	0
$316$	−28.0000	−1.57512
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	17.8885i	1.00000i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	31.6228	1.75412
$326$	35.7771i	1.98151i
$327$	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
$333$	0	0
$334$	−16.0000	−0.875481
$335$	− 28.2843i	− 1.54533i
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 38.1838i	− 2.07692i
$339$	0	0
$340$	12.6491	0.685994
$341$	− 8.94427i	− 0.484359i
$342$	0	0
$343$	0	0
$344$	35.7771i	1.92897i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−25.2982	−1.34840
$353$	− 36.7696i	− 1.95705i	−0.206138	−	0.978523i	$-0.566090\pi$
$353$	− 36.7696i	− 1.95705i	0.206138	−	0.978523i	$-0.433910\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	31.6228	1.67132
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	22.6274i	1.17954i
$369$	0	0
$370$	20.0000	1.03975
$371$	0	0
$372$	0	0
$373$	−6.32456	−0.327473	−0.163737	−	0.986504i	$-0.552355\pi$
$373$	−6.32456	−0.327473	−0.163737	+	0.986504i	$0.552355\pi$
$374$	17.8885i	0.924995i
$375$	0	0
$376$	32.0000	1.65027
$377$	28.2843i	1.45671i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.65685i	0.289052i	0.989501	+	0.144526i	$0.0461657\pi$
$383$	5.65685i	0.289052i	−0.989501	+	0.144526i	$0.953834\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.47214i	− 0.226746i	−0.993552	−	0.113373i	$-0.963834\pi$
$389$	− 4.47214i	− 0.226746i	0.993552	−	0.113373i	$-0.0361656\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	19.7990i	1.00000i
$393$	0	0
$394$	0	0
$395$	− 31.3050i	− 1.57512i
$396$	0	0
$397$	31.6228	1.58710	0.793551	−	0.608504i	$-0.208230\pi$
$397$	31.6228	1.58710	0.793551	+	0.608504i	$0.208230\pi$
$398$	− 36.7696i	− 1.84309i
$399$	0	0
$400$	−20.0000	−1.00000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−12.6491	−0.630097
$404$	8.94427i	0.444994i
$405$	0	0
$406$	0	0
$407$	28.2843i	1.40200i
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	35.7771i	1.75412i
$417$	0	0
$418$	0	0
$419$	22.3607i	1.09239i	0.837658	+	0.546195i	$0.183924\pi$
$419$	22.3607i	1.09239i	−0.837658	+	0.546195i	$0.816076\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.1421i	0.685994i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−40.0000	−1.92897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	− 28.2843i	− 1.34840i
$441$	0	0
$442$	25.2982	1.20331
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	39.5980i	1.86253i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−25.2982	−1.17954
$461$	− 31.3050i	− 1.45802i	−0.684505	−	0.729008i	$-0.739981\pi$
$461$	− 31.3050i	− 1.45802i	0.684505	−	0.729008i	$-0.260019\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	− 17.8885i	− 0.830455i
$465$	0	0
$466$	−28.0000	−1.29707
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	35.7771i	1.65027i
$471$	0	0
$472$	12.6491	0.582223
$473$	− 56.5685i	− 2.60102i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	31.1127i	1.41714i
$483$	0	0
$484$	18.0000	0.818182
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−22.1359	−1.00000
$491$	− 4.47214i	− 0.201825i	−0.994895	−	0.100912i	$-0.967824\pi$
$491$	− 4.47214i	− 0.201825i	0.994895	−	0.100912i	$-0.0321762\pi$
$492$	0	0
$493$	−12.6491	−0.569687
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 22.3607i	− 1.00000i
$501$	0	0
$502$	−44.2719	−1.97595
$503$	22.6274i	1.00891i	0.863439	+	0.504453i	$0.168306\pi$
$503$	22.6274i	1.00891i	−0.863439	+	0.504453i	$0.831694\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	− 35.7771i	− 1.59049i
$507$	0	0
$508$	0	0
$509$	22.3607i	0.991120i	0.868574	+	0.495560i	$0.165037\pi$
$509$	22.3607i	0.991120i	−0.868574	+	0.495560i	$0.834963\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	44.0000	1.94076
$515$	0	0
$516$	0	0
$517$	−50.5964	−2.22523
$518$	0	0
$519$	0	0
$520$	−40.0000	−1.75412
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−25.2982	−1.10621	−0.553107	−	0.833110i	$-0.686558\pi$
$523$	−25.2982	−1.10621	−0.553107	+	0.833110i	$0.686558\pi$
$524$	− 44.7214i	− 1.95366i
$525$	0	0
$526$	32.0000	1.39527
$527$	− 5.65685i	− 0.246416i
$528$	0	0
$529$	−9.00000	−0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	35.7771i	1.54533i
$537$	0	0
$538$	−44.2719	−1.90870
$539$	− 31.3050i	− 1.34840i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	− 2.82843i	− 0.121491i
$543$	0	0
$544$	−16.0000	−0.685994
$545$	0	0
$546$	0	0
$547$	12.6491	0.540837	0.270418	−	0.962743i	$-0.412838\pi$
$547$	12.6491	0.540837	0.270418	+	0.962743i	$0.412838\pi$
$548$	5.65685i	0.241649i
$549$	0	0
$550$	31.6228	1.34840
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 44.7214i	− 1.90003i
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−80.0000	−3.38364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−44.2719	−1.86253
$566$	− 17.8885i	− 0.751912i
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	− 56.5685i	− 2.36525i
$573$	0	0
$574$	0	0
$575$	− 28.2843i	− 1.17954i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 12.7279i	− 0.529412i
$579$	0	0
$580$	20.0000	0.830455
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	14.1421i	0.582223i
$591$	0	0
$592$	−25.2982	−1.03975
$593$	48.0833i	1.97454i	0.159044	+	0.987271i	$0.449159\pi$
$593$	48.0833i	1.97454i	−0.159044	+	0.987271i	$0.550841\pi$
$594$	0	0
$595$	0	0
$596$	− 44.7214i	− 1.83186i
$597$	0	0
$598$	−50.5964	−2.06904
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	44.0000	1.79033
$605$	20.1246i	0.818182i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	71.5542i	2.89477i
$612$	0	0
$613$	31.6228	1.27723	0.638616	−	0.769526i	$-0.279507\pi$
$613$	31.6228	1.27723	0.638616	+	0.769526i	$0.279507\pi$
$614$	35.7771i	1.44385i
$615$	0	0
$616$	0	0
$617$	31.1127i	1.25255i	0.779602	+	0.626275i	$0.215421\pi$
$617$	31.1127i	1.25255i	−0.779602	+	0.626275i	$0.784579\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	8.94427i	0.359211i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	12.6491	0.504754
$629$	17.8885i	0.713263i
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	39.5980i	1.57512i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−44.2719	−1.75412
$638$	28.2843i	1.11979i
$639$	0	0
$640$	25.2982	1.00000
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	50.5964	1.99533	0.997664	−	0.0683054i	$-0.0217592\pi$
$643$	50.5964	1.99533	0.997664	+	0.0683054i	$0.0217592\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 45.2548i	− 1.77915i	−0.456788	−	0.889576i	$-0.651000\pi$
$647$	− 45.2548i	− 1.77915i	0.456788	−	0.889576i	$-0.349000\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	− 44.7214i	− 1.75412i
$651$	0	0
$652$	50.5964	1.98151
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	50.0000	1.95366
$656$	0	0
$657$	0	0
$658$	0	0
$659$	49.1935i	1.91631i	0.286256	+	0.958153i	$0.407589\pi$
$659$	49.1935i	1.91631i	−0.286256	+	0.958153i	$0.592411\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.2982	0.979551
$668$	22.6274i	0.875481i
$669$	0	0
$670$	−40.0000	−1.54533
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−54.0000	−2.07692
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	− 17.8885i	− 0.685994i
$681$	0	0
$682$	−12.6491	−0.484359
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−6.32456	−0.241649
$686$	0	0
$687$	0	0
$688$	50.5964	1.92897
$689$	0	0
$690$	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
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	49.1935i	1.85801i	0.370064	+	0.929006i	$0.379336\pi$
$701$	49.1935i	1.85801i	−0.370064	+	0.929006i	$0.620664\pi$
$702$	0	0
$703$	0	0
$704$	35.7771i	1.34840i
$705$	0	0
$706$	−52.0000	−1.95705
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.3137i	0.423702i
$714$	0	0
$715$	63.2456	2.36525
$716$	− 44.7214i	− 1.67132i
$717$	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
$722$	− 26.8701i	− 1.00000i
$723$	0	0
$724$	0	0
$725$	22.3607i	0.830455i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 35.7771i	− 1.32326i
$732$	0	0
$733$	−44.2719	−1.63522	−0.817610	−	0.575773i	$-0.804701\pi$
$733$	−44.2719	−1.63522	−0.817610	+	0.575773i	$0.804701\pi$
$734$	0	0
$735$	0	0
$736$	32.0000	1.17954
$737$	− 56.5685i	− 2.08373i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 28.2843i	− 1.03975i
$741$	0	0
$742$	0	0
$743$	5.65685i	0.207530i	0.994602	+	0.103765i	$0.0330890\pi$
$743$	5.65685i	0.207530i	−0.994602	+	0.103765i	$0.966911\pi$
$744$	0	0
$745$	50.0000	1.83186
$746$	8.94427i	0.327473i
$747$	0	0
$748$	25.2982	0.924995
$749$	0	0
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	− 45.2548i	− 1.65027i
$753$	0	0
$754$	40.0000	1.45671
$755$	49.1935i	1.79033i
$756$	0	0
$757$	31.6228	1.14935	0.574675	−	0.818382i	$-0.305129\pi$
$757$	31.6228	1.14935	0.574675	+	0.818382i	$0.305129\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	8.00000	0.289052
$767$	28.2843i	1.02129i
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	0	0
$778$	−6.32456	−0.226746
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 22.6274i	− 0.809155i
$783$	0	0
$784$	28.0000	1.00000
$785$	14.1421i	0.504754i
$786$	0	0
$787$	50.5964	1.80357	0.901784	−	0.432187i	$-0.142258\pi$
$787$	50.5964	1.80357	0.901784	+	0.432187i	$0.142258\pi$
$788$	0	0
$789$	0	0
$790$	−44.2719	−1.57512
$791$	0	0
$792$	0	0
$793$	0	0
$794$	− 44.7214i	− 1.58710i
$795$	0	0
$796$	−52.0000	−1.84309
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	28.2843i	1.00000i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	17.8885i	0.630097i
$807$	0	0
$808$	12.6491	0.444994
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	40.0000	1.40200
$815$	56.5685i	1.98151i
$816$	0	0
$817$	0	0
$818$	48.0833i	1.68119i
$819$	0	0
$820$	0	0
$821$	49.1935i	1.71686i	0.512927	+	0.858432i	$0.328561\pi$
$821$	49.1935i	1.71686i	−0.512927	+	0.858432i	$0.671439\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	50.5964	1.75412
$833$	− 19.7990i	− 0.685994i
$834$	0	0
$835$	−25.2982	−0.875481
$836$	0	0
$837$	0	0
$838$	31.6228	1.09239
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	9.00000	0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 60.3738i	− 2.07692i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	20.0000	0.685994
$851$	− 35.7771i	− 1.22642i
$852$	0	0
$853$	−44.2719	−1.51584	−0.757920	−	0.652347i	$-0.773784\pi$
$853$	−44.2719	−1.51584	−0.757920	+	0.652347i	$0.773784\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 53.7401i	− 1.83573i	−0.396896	−	0.917864i	$-0.629913\pi$
$857$	− 53.7401i	− 1.83573i	0.396896	−	0.917864i	$-0.370087\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	56.5685i	1.92897i
$861$	0	0
$862$	0	0
$863$	22.6274i	0.770246i	0.922865	+	0.385123i	$0.125841\pi$
$863$	22.6274i	0.770246i	−0.922865	+	0.385123i	$0.874159\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 62.6099i	− 2.12390i
$870$	0	0
$871$	−80.0000	−2.71070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.32456	−0.213565	−0.106783	−	0.994282i	$-0.534055\pi$
$877$	−6.32456	−0.213565	−0.106783	+	0.994282i	$0.534055\pi$
$878$	− 36.7696i	− 1.24091i
$879$	0	0
$880$	−40.0000	−1.34840
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	50.5964	1.70271	0.851353	−	0.524593i	$-0.175783\pi$
$883$	50.5964	1.70271	0.851353	+	0.524593i	$0.175783\pi$
$884$	− 35.7771i	− 1.20331i
$885$	0	0
$886$	0	0
$887$	− 45.2548i	− 1.51951i	−0.650210	−	0.759754i	$-0.725319\pi$
$887$	− 45.2548i	− 1.51951i	0.650210	−	0.759754i	$-0.274681\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	50.0000	1.67132
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 8.94427i	− 0.298308i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	56.0000	1.86253
$905$	0	0
$906$	0	0
$907$	12.6491	0.420007	0.210003	−	0.977701i	$-0.432652\pi$
$907$	12.6491	0.420007	0.210003	+	0.977701i	$0.432652\pi$
$908$	0	0
$909$	0	0
$910$	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
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	35.7771i	1.17954i
$921$	0	0
$922$	−44.2719	−1.45802
$923$	0	0
$924$	0	0
$925$	31.6228	1.03975
$926$	0	0
$927$	0	0
$928$	−25.2982	−0.830455
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	39.5980i	1.29707i
$933$	0	0
$934$	0	0
$935$	28.2843i	0.924995i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	50.5964	1.65027
$941$	− 58.1378i	− 1.89524i	−0.319404	−	0.947619i	$-0.603483\pi$
$941$	− 58.1378i	− 1.89524i	0.319404	−	0.947619i	$-0.396517\pi$
$942$	0	0
$943$	0	0
$944$	− 17.8885i	− 0.582223i
$945$	0	0
$946$	−80.0000	−2.60102
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.0833i	1.55757i	0.627291	+	0.778785i	$0.284164\pi$
$953$	48.0833i	1.55757i	−0.627291	+	0.778785i	$0.715836\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 56.5685i	− 1.82384i
$963$	0	0
$964$	44.0000	1.41714
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 25.4558i	− 0.818182i
$969$	0	0
$970$	0	0
$971$	− 31.3050i	− 1.00462i	−0.864687	−	0.502312i	$-0.832483\pi$
$971$	− 31.3050i	− 1.00462i	0.864687	−	0.502312i	$-0.167517\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1127i	0.995383i	0.867354	+	0.497692i	$0.165819\pi$
$977$	31.1127i	0.995383i	−0.867354	+	0.497692i	$0.834181\pi$
$978$	0	0
$979$	0	0
$980$	31.3050i	1.00000i
$981$	0	0
$982$	−6.32456	−0.201825
$983$	− 62.2254i	− 1.98468i	−0.123529	−	0.992341i	$-0.539421\pi$
$983$	− 62.2254i	− 1.98468i	0.123529	−	0.992341i	$-0.460579\pi$
$984$	0	0
$985$	0	0
$986$	17.8885i	0.569687i
$987$	0	0
$988$	0	0
$989$	71.5542i	2.27529i
$990$	0	0
$991$	62.0000	1.96949	0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000	1.96949	0.984747	+	0.173990i	$0.0556660\pi$
$992$	− 11.3137i	− 0.359211i
$993$	0	0
$994$	0	0
$995$	− 58.1378i	− 1.84309i
$996$	0	0
$997$	−44.2719	−1.40210	−0.701052	−	0.713110i	$-0.747286\pi$
$997$	−44.2719	−1.40210	−0.701052	+	0.713110i	$0.747286\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.d.a.109.1	✓	4
3.2	odd	2	inner	360.2.d.a.109.4	yes	4
4.3	odd	2		1440.2.d.a.1009.1		4
5.2	odd	4		1800.2.k.o.901.3		4
5.3	odd	4		1800.2.k.o.901.1		4
5.4	even	2	inner	360.2.d.a.109.3	yes	4
8.3	odd	2		1440.2.d.a.1009.4		4
8.5	even	2	inner	360.2.d.a.109.2	yes	4
12.11	even	2		1440.2.d.a.1009.3		4
15.2	even	4		1800.2.k.o.901.2		4
15.8	even	4		1800.2.k.o.901.4		4
15.14	odd	2	inner	360.2.d.a.109.2	yes	4
20.3	even	4		7200.2.k.m.3601.3		4
20.7	even	4		7200.2.k.m.3601.4		4
20.19	odd	2		1440.2.d.a.1009.2		4
24.5	odd	2	inner	360.2.d.a.109.3	yes	4
24.11	even	2		1440.2.d.a.1009.2		4
40.3	even	4		7200.2.k.m.3601.2		4
40.13	odd	4		1800.2.k.o.901.2		4
40.19	odd	2		1440.2.d.a.1009.3		4
40.27	even	4		7200.2.k.m.3601.1		4
40.29	even	2	inner	360.2.d.a.109.4	yes	4
40.37	odd	4		1800.2.k.o.901.4		4
60.23	odd	4		7200.2.k.m.3601.1		4
60.47	odd	4		7200.2.k.m.3601.2		4
60.59	even	2		1440.2.d.a.1009.4		4
120.29	odd	2	CM	360.2.d.a.109.1	✓	4
120.53	even	4		1800.2.k.o.901.3		4
120.59	even	2		1440.2.d.a.1009.1		4
120.77	even	4		1800.2.k.o.901.1		4
120.83	odd	4		7200.2.k.m.3601.4		4
120.107	odd	4		7200.2.k.m.3601.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.d.a.109.1	✓	4	1.1	even	1	trivial
360.2.d.a.109.1	✓	4	120.29	odd	2	CM
360.2.d.a.109.2	yes	4	8.5	even	2	inner
360.2.d.a.109.2	yes	4	15.14	odd	2	inner
360.2.d.a.109.3	yes	4	5.4	even	2	inner
360.2.d.a.109.3	yes	4	24.5	odd	2	inner
360.2.d.a.109.4	yes	4	3.2	odd	2	inner
360.2.d.a.109.4	yes	4	40.29	even	2	inner
1440.2.d.a.1009.1		4	4.3	odd	2
1440.2.d.a.1009.1		4	120.59	even	2
1440.2.d.a.1009.2		4	20.19	odd	2
1440.2.d.a.1009.2		4	24.11	even	2
1440.2.d.a.1009.3		4	12.11	even	2
1440.2.d.a.1009.3		4	40.19	odd	2
1440.2.d.a.1009.4		4	8.3	odd	2
1440.2.d.a.1009.4		4	60.59	even	2
1800.2.k.o.901.1		4	5.3	odd	4
1800.2.k.o.901.1		4	120.77	even	4
1800.2.k.o.901.2		4	15.2	even	4
1800.2.k.o.901.2		4	40.13	odd	4
1800.2.k.o.901.3		4	5.2	odd	4
1800.2.k.o.901.3		4	120.53	even	4
1800.2.k.o.901.4		4	15.8	even	4
1800.2.k.o.901.4		4	40.37	odd	4
7200.2.k.m.3601.1		4	40.27	even	4
7200.2.k.m.3601.1		4	60.23	odd	4
7200.2.k.m.3601.2		4	40.3	even	4
7200.2.k.m.3601.2		4	60.47	odd	4
7200.2.k.m.3601.3		4	20.3	even	4
7200.2.k.m.3601.3		4	120.107	odd	4
7200.2.k.m.3601.4		4	20.7	even	4
7200.2.k.m.3601.4		4	120.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} +2.82843i q^{8} -3.16228 q^{10} -4.47214i q^{11} -6.32456 q^{13} +4.00000 q^{16} -2.82843i q^{17} +4.47214i q^{20} -6.32456 q^{22} +5.65685i q^{23} -5.00000 q^{25} +8.94427i q^{26} -4.47214i q^{29} +2.00000 q^{31} -5.65685i q^{32} -4.00000 q^{34} -6.32456 q^{37} +6.32456 q^{40} +12.6491 q^{43} +8.94427i q^{44} +8.00000 q^{46} -11.3137i q^{47} +7.00000 q^{49} +7.07107i q^{50} +12.6491 q^{52} -10.0000 q^{55} -6.32456 q^{58} -4.47214i q^{59} -2.82843i q^{62} -8.00000 q^{64} +14.1421i q^{65} +12.6491 q^{67} +5.65685i q^{68} +8.94427i q^{74} +14.0000 q^{79} -8.94427i q^{80} -6.32456 q^{85} -17.8885i q^{86} +12.6491 q^{88} -11.3137i q^{92} -16.0000 q^{94} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{31} - 16 q^{34} + 32 q^{46} + 28 q^{49} - 40 q^{55} - 32 q^{64} + 56 q^{79} - 64 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 - 20 * q^25 + 8 * q^31 - 16 * q^34 + 32 * q^46 + 28 * q^49 - 40 * q^55 - 32 * q^64 + 56 * q^79 - 64 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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