LMFDB - Embedded newform 360.4.m.b.179.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,4,Mod(179,360)]

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([1, 1, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("360.179");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	360.m (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:	$21.2406876021$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			179.1
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.179
Dual form			360.4.m.b.179.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +29.1623 q^{7} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +29.1623 q^{7} +22.6274i q^{8} -31.6228 q^{10} +72.3171i q^{11} +32.0833 q^{13} -82.4834i q^{14} +64.0000 q^{16} +107.517 q^{19} +89.4427i q^{20} +204.544 q^{22} +219.135i q^{23} -125.000 q^{25} -90.7452i q^{26} -233.298 q^{28} -181.019i q^{32} -326.044i q^{35} +151.767 q^{37} -304.105i q^{38} +252.982 q^{40} -162.647i q^{41} -578.537i q^{44} +619.806 q^{46} +376.181i q^{47} +507.438 q^{49} +353.553i q^{50} -256.666 q^{52} -718.420i q^{53} +808.530 q^{55} +659.867i q^{56} -800.773i q^{59} -512.000 q^{64} -358.702i q^{65} -922.192 q^{70} -429.263i q^{74} -860.140 q^{76} +2108.93i q^{77} -715.542i q^{80} -460.034 q^{82} -1636.35 q^{88} +1659.90i q^{89} +935.621 q^{91} -1753.08i q^{92} +1064.00 q^{94} -1202.08i q^{95} -1435.25i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 104 q^{7}+O(q^{10}) $ 4 * q - 32 * q^4 + 104 * q^7	$ 4 q - 32 q^{4} + 104 q^{7} - 112 q^{13} + 256 q^{16} + 464 q^{22} - 500 q^{25} - 832 q^{28} - 544 q^{37} + 1372 q^{49} + 896 q^{52} + 1400 q^{55} - 2048 q^{64} - 400 q^{70} + 1904 q^{82} - 3712 q^{88} - 2152 q^{91} + 4256 q^{94}+O(q^{100}) $ 4 * q - 32 * q^4 + 104 * q^7 - 112 * q^13 + 256 * q^16 + 464 * q^22 - 500 * q^25 - 832 * q^28 - 544 * q^37 + 1372 * q^49 + 896 * q^52 + 1400 * q^55 - 2048 * q^64 - 400 * q^70 + 1904 * q^82 - 3712 * q^88 - 2152 * q^91 + 4256 * 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 $

$2$	− 2.82843i	− 1.00000i
$2$	− 2.82843i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−8.00000	−1.00000
$5$	− 11.1803i	− 1.00000i
$5$	− 11.1803i	− 1.00000i
$6$	0	0
$7$	29.1623	1.57461	0.787307	−	0.616561i	$-0.211474\pi$
$7$	29.1623	1.57461	0.787307	+	0.616561i	$0.211474\pi$
$8$	22.6274i	1.00000i
$9$	0	0
$10$	−31.6228	−1.00000
$11$	72.3171i	1.98222i	0.133038	+	0.991111i	$0.457527\pi$
$11$	72.3171i	1.98222i	−0.133038	+	0.991111i	$0.542473\pi$
$12$	0	0
$13$	32.0833	0.684485	0.342242	−	0.939612i	$-0.388814\pi$
$13$	32.0833	0.684485	0.342242	+	0.939612i	$0.388814\pi$
$14$	− 82.4834i	− 1.57461i
$15$	0	0
$16$	64.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	107.517	1.29822	0.649110	−	0.760694i	$-0.275141\pi$
$19$	107.517	1.29822	0.649110	+	0.760694i	$0.275141\pi$
$20$	89.4427i	1.00000i
$21$	0	0
$22$	204.544	1.98222
$23$	219.135i	1.98664i	0.115389	+	0.993320i	$0.463188\pi$
$23$	219.135i	1.98664i	−0.115389	+	0.993320i	$0.536812\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	− 90.7452i	− 0.684485i
$27$	0	0
$28$	−233.298	−1.57461
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 181.019i	− 1.00000i
$33$	0	0
$34$	0	0
$35$	− 326.044i	− 1.57461i
$36$	0	0
$37$	151.767	0.674335	0.337167	−	0.941445i	$-0.390531\pi$
$37$	151.767	0.674335	0.337167	+	0.941445i	$0.390531\pi$
$38$	− 304.105i	− 1.29822i
$39$	0	0
$40$	252.982	1.00000
$41$	− 162.647i	− 0.619540i	−0.950812	−	0.309770i	$-0.899748\pi$
$41$	− 162.647i	− 0.619540i	0.950812	−	0.309770i	$-0.100252\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 578.537i	− 1.98222i
$45$	0	0
$46$	619.806	1.98664
$47$	376.181i	1.16748i	0.811940	+	0.583741i	$0.198411\pi$
$47$	376.181i	1.16748i	−0.811940	+	0.583741i	$0.801589\pi$
$48$	0	0
$49$	507.438	1.47941
$50$	353.553i	1.00000i
$51$	0	0
$52$	−256.666	−0.684485
$53$	− 718.420i	− 1.86194i	−0.365100	−	0.930968i	$-0.618965\pi$
$53$	− 718.420i	− 1.86194i	0.365100	−	0.930968i	$-0.381035\pi$
$54$	0	0
$55$	808.530	1.98222
$56$	659.867i	1.57461i
$57$	0	0
$58$	0	0
$59$	− 800.773i	− 1.76698i	−0.468451	−	0.883490i	$-0.655188\pi$
$59$	− 800.773i	− 1.76698i	0.468451	−	0.883490i	$-0.344812\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−512.000	−1.00000
$65$	− 358.702i	− 0.684485i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−922.192	−1.57461
$71$	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			$\pi$
$74$	− 429.263i	− 0.674335i
$75$	0	0
$76$	−860.140	−1.29822
$77$	2108.93i	3.12124i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 715.542i	− 1.00000i
$81$	0	0
$82$	−460.034	−0.619540
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−1636.35	−1.98222
$89$	1659.90i	1.97696i	0.151361	+	0.988479i	$0.451634\pi$
$89$	1659.90i	1.97696i	−0.151361	+	0.988479i	$0.548366\pi$
$90$	0	0
$91$	935.621	1.07780
$92$	− 1753.08i	− 1.98664i
$93$	0	0
$94$	1064.00	1.16748
$95$	− 1202.08i	− 1.29822i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 1435.25i	− 1.47941i
$99$	0	0
$100$	1000.00	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−248.102	−0.237342	−0.118671	−	0.992934i	$-0.537863\pi$
$103$	−248.102	−0.237342	−0.118671	+	0.992934i	$0.537863\pi$
$104$	725.962i	0.684485i
$105$	0	0
$106$	−2032.00	−1.86194
$107$	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			$\pi$
$110$	− 2286.87i	− 1.98222i
$111$	0	0
$112$	1866.39	1.57461
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	2450.00	1.98664
$116$	0	0
$117$	0	0
$118$	−2264.93	−1.76698
$119$	0	0
$120$	0	0
$121$	−3898.77	−2.92920
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1397.54i	1.00000i
$126$	0	0
$127$	2627.56	1.83589	0.917944	−	0.396710i	$-0.129848\pi$
$127$	2627.56	1.83589	0.917944	+	0.396710i	$0.129848\pi$
$128$	1448.15i	1.00000i
$129$	0	0
$130$	−1014.56	−0.684485
$131$	− 163.036i	− 0.108737i	−0.998521	−	0.0543685i	$-0.982685\pi$
$131$	− 163.036i	− 0.108737i	0.998521	−	0.0543685i	$-0.0173146\pi$
$132$	0	0
$133$	3135.45	2.04420
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−3094.00	−1.88798	−0.943992	−	0.329970i	$-0.892962\pi$
$139$	−3094.00	−1.88798	−0.943992	+	0.329970i	$0.892962\pi$
$140$	2608.35i	1.57461i
$141$	0	0
$142$	0	0
$143$	2320.17i	1.35680i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−1214.14	−0.674335
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	2432.84i	1.29822i
$153$	0	0
$154$	5964.96	3.12124
$155$	0	0
$156$	0	0
$157$	3202.53	1.62796	0.813980	−	0.580893i	$-0.197296\pi$
$157$	3202.53	1.62796	0.813980	+	0.580893i	$0.197296\pi$
$158$	0	0
$159$	0	0
$160$	−2023.86	−1.00000
$161$	6390.47i	3.12819i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	1301.17i	0.619540i
$165$	0	0
$166$	0	0
$167$	2151.10i	0.996748i	0.866962	+	0.498374i	$0.166069\pi$
$167$	2151.10i	0.996748i	−0.866962	+	0.498374i	$0.833931\pi$
$168$	0	0
$169$	−1167.66	−0.531481
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 158.392i	− 0.0696088i	−0.999394	−	0.0348044i	$-0.988919\pi$
$173$	− 158.392i	− 0.0696088i	0.999394	−	0.0348044i	$-0.0110808\pi$
$174$	0	0
$175$	−3645.28	−1.57461
$176$	4628.30i	1.98222i
$177$	0	0
$178$	4694.91	1.97696
$179$	− 4778.14i	− 1.99517i	−0.0694890	−	0.997583i	$-0.522137\pi$
$179$	− 4778.14i	− 1.99517i	0.0694890	−	0.997583i	$-0.477863\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	− 2646.34i	− 1.07780i
$183$	0	0
$184$	−4958.45	−1.98664
$185$	− 1696.81i	− 0.674335i
$186$	0	0
$187$	0	0
$188$	− 3009.45i	− 1.16748i
$189$	0	0
$190$	−3400.00	−1.29822
$191$	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
$196$	−4059.51	−1.47941
$197$	− 2034.82i	− 0.735914i	−0.929843	−	0.367957i	$-0.880057\pi$
$197$	− 2034.82i	− 0.735914i	0.929843	−	0.367957i	$-0.119943\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 2828.43i	− 1.00000i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1818.44	−0.619540
$206$	701.738i	0.237342i
$207$	0	0
$208$	2053.33	0.684485
$209$	7775.35i	2.57336i
$210$	0	0
$211$	3278.00	1.06951	0.534756	−	0.845007i	$-0.320404\pi$
$211$	3278.00	1.06951	0.534756	+	0.845007i	$0.320404\pi$
$212$	5747.36i	1.86194i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−6468.24	−1.98222
$221$	0	0
$222$	0	0
$223$	1205.88	0.362114	0.181057	−	0.983473i	$-0.442048\pi$
$223$	1205.88	0.362114	0.181057	+	0.983473i	$0.442048\pi$
$224$	− 5278.94i	− 1.57461i
$225$	0	0
$226$	0	0
$227$	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			$\pi$
$230$	− 6929.65i	− 1.98664i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	4205.83	1.16748
$236$	6406.18i	1.76698i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2099.75	0.561232	0.280616	−	0.959820i	$-0.409461\pi$
$241$	2099.75	0.561232	0.280616	+	0.959820i	$0.409461\pi$
$242$	11027.4i	2.92920i
$243$	0	0
$244$	0	0
$245$	− 5673.33i	− 1.47941i
$246$	0	0
$247$	3449.51	0.888612
$248$	0	0
$249$	0	0
$250$	3952.85	1.00000
$251$	− 4463.84i	− 1.12253i	−0.827636	−	0.561265i	$-0.810315\pi$
$251$	− 4463.84i	− 1.12253i	0.827636	−	0.561265i	$-0.189685\pi$
$252$	0	0
$253$	−15847.2	−3.93796
$254$	− 7431.85i	− 1.83589i
$255$	0	0
$256$	4096.00	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	4425.88	1.06182
$260$	2869.62i	0.684485i
$261$	0	0
$262$	−461.136	−0.108737
$263$	− 5979.25i	− 1.40189i	−0.713217	−	0.700943i	$-0.752763\pi$
$263$	− 5979.25i	− 1.40189i	0.713217	−	0.700943i	$-0.247237\pi$
$264$	0	0
$265$	−8032.19	−1.86194
$266$	− 8868.40i	− 2.04420i
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9039.64i	− 1.98222i
$276$	0	0
$277$	2325.74	0.504478	0.252239	−	0.967665i	$-0.418833\pi$
$277$	2325.74	0.504478	0.252239	+	0.967665i	$0.418833\pi$
$278$	8751.15i	1.88798i
$279$	0	0
$280$	7377.54	1.57461
$281$	− 9216.10i	− 1.95654i	−0.207344	−	0.978268i	$-0.566482\pi$
$281$	− 9216.10i	− 1.95654i	0.207344	−	0.978268i	$-0.433518\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	6562.43	1.35680
$287$	− 4743.15i	− 0.975537i
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3841.56i	0.765961i	0.923756	+	0.382981i	$0.125102\pi$
$293$	3841.56i	0.765961i	−0.923756	+	0.382981i	$0.874898\pi$
$294$	0	0
$295$	−8952.91	−1.76698
$296$	3434.10i	0.674335i
$297$	0	0
$298$	0	0
$299$	7030.56i	1.35982i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	6881.12	1.29822
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 16871.5i	− 3.12124i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	− 9058.12i	− 1.62796i
$315$	0	0
$316$	0	0
$317$	− 907.844i	− 0.160850i	−0.996761	−	0.0804252i	$-0.974372\pi$
$317$	− 907.844i	− 0.160850i	0.996761	−	0.0804252i	$-0.0256278\pi$
$318$	0	0
$319$	0	0
$320$	5724.33i	1.00000i
$321$	0	0
$322$	18075.0	3.12819
$323$	0	0
$324$	0	0
$325$	−4010.41	−0.684485
$326$	0	0
$327$	0	0
$328$	3680.27	0.619540
$329$	10970.3i	1.83833i
$330$	0	0
$331$	221.359	0.0367584	0.0183792	−	0.999831i	$-0.494149\pi$
$331$	221.359	0.0367584	0.0183792	+	0.999831i	$0.494149\pi$
$332$	0	0
$333$	0	0
$334$	6084.22	0.996748
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	3302.65i	0.531481i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4795.40	0.754890
$344$	0	0
$345$	0	0
$346$	−448.000	−0.0696088
$347$	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			$\pi$
$350$	10310.4i	1.57461i
$351$	0	0
$352$	13090.8	1.98222
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	− 13279.2i	− 1.97696i
$357$	0	0
$358$	−13514.6	−1.99517
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	4701.00	0.685377
$362$	0	0
$363$	0	0
$364$	−7484.97	−1.07780
$365$	0	0
$366$	0	0
$367$	−12428.6	−1.76777	−0.883883	−	0.467709i	$-0.845080\pi$
$367$	−12428.6	−1.76777	−0.883883	+	0.467709i	$0.845080\pi$
$368$	14024.6i	1.98664i
$369$	0	0
$370$	−4799.30	−0.674335
$371$	− 20950.8i	− 2.93183i
$372$	0	0
$373$	−6635.10	−0.921053	−0.460526	−	0.887646i	$-0.652339\pi$
$373$	−6635.10	−0.921053	−0.460526	+	0.887646i	$0.652339\pi$
$374$	0	0
$375$	0	0
$376$	−8512.00	−1.16748
$377$	0	0
$378$	0	0
$379$	−646.000	−0.0875535	−0.0437768	−	0.999041i	$-0.513939\pi$
$379$	−646.000	−0.0875535	−0.0437768	+	0.999041i	$0.513939\pi$
$380$	9616.65i	1.29822i
$381$	0	0
$382$	0	0
$383$	− 7464.22i	− 0.995832i	−0.867225	−	0.497916i	$-0.834099\pi$
$383$	− 7464.22i	− 0.995832i	0.867225	−	0.497916i	$-0.165901\pi$
$384$	0	0
$385$	23578.6	3.12124
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	11482.0i	1.47941i
$393$	0	0
$394$	−5755.35	−0.735914
$395$	0	0
$396$	0	0
$397$	14259.0	1.80262	0.901310	−	0.433175i	$-0.142607\pi$
$397$	14259.0	1.80262	0.901310	+	0.433175i	$0.142607\pi$
$398$	0	0
$399$	0	0
$400$	−8000.00	−1.00000
$401$	2853.19i	0.355316i	0.984092	+	0.177658i	$0.0568520\pi$
$401$	2853.19i	0.355316i	−0.984092	+	0.177658i	$0.943148\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10975.4i	1.33668i
$408$	0	0
$409$	−14434.0	−1.74503	−0.872513	−	0.488591i	$-0.837511\pi$
$409$	−14434.0	−1.74503	−0.872513	+	0.488591i	$0.837511\pi$
$410$	5143.34i	0.619540i
$411$	0	0
$412$	1984.81	0.237342
$413$	− 23352.4i	− 2.78231i
$414$	0	0
$415$	0	0
$416$	− 5807.69i	− 0.684485i
$417$	0	0
$418$	21992.0	2.57336
$419$	16428.2i	1.91545i	0.287689	+	0.957724i	$0.407113\pi$
$419$	16428.2i	1.91545i	−0.287689	+	0.957724i	$0.592887\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 9271.58i	− 1.06951i
$423$	0	0
$424$	16256.0	1.86194
$425$	0	0
$426$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23560.8i	2.57910i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	18295.0i	1.98222i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	18558.3	1.97696
$446$	− 3410.73i	− 0.362114i
$447$	0	0
$448$	−14931.1	−1.57461
$449$	− 17239.2i	− 1.81195i	−0.423330	−	0.905975i	$-0.639139\pi$
$449$	− 17239.2i	− 1.81195i	0.423330	−	0.905975i	$-0.360861\pi$
$450$	0	0
$451$	11762.1	1.22807
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 10460.6i	− 1.07780i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	−19600.0	−1.98664
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−13094.3	−1.31435	−0.657176	−	0.753737i	$-0.728249\pi$
$463$	−13094.3	−1.31435	−0.657176	+	0.753737i	$0.728249\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	− 11895.9i	− 1.16748i
$471$	0	0
$472$	18119.4	1.76698
$473$	0	0
$474$	0	0
$475$	−13439.7	−1.29822
$476$	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
$481$	4869.19	0.461572
$482$	− 5939.00i	− 0.561232i
$483$	0	0
$484$	31190.2	2.92920
$485$	0	0
$486$	0	0
$487$	−198.611	−0.0184803	−0.00924017	−	0.999957i	$-0.502941\pi$
$487$	−198.611	−0.0184803	−0.00924017	+	0.999957i	$0.502941\pi$
$488$	0	0
$489$	0	0
$490$	−16046.6	−1.47941
$491$	17323.0i	1.59221i	0.605158	+	0.796106i	$0.293110\pi$
$491$	17323.0i	1.59221i	−0.605158	+	0.796106i	$0.706890\pi$
$492$	0	0
$493$	0	0
$494$	− 9756.69i	− 0.888612i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20497.9	−1.83890	−0.919450	−	0.393206i	$-0.871366\pi$
$499$	−20497.9	−1.83890	−0.919450	+	0.393206i	$0.871366\pi$
$500$	− 11180.3i	− 1.00000i
$501$	0	0
$502$	−12625.7	−1.12253
$503$	22115.5i	1.96040i	0.198015	+	0.980199i	$0.436550\pi$
$503$	22115.5i	1.96040i	−0.198015	+	0.980199i	$0.563450\pi$
$504$	0	0
$505$	0	0
$506$	44822.6i	3.93796i
$507$	0	0
$508$	−21020.4	−1.83589
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 1.00000i
$513$	0	0
$514$	0	0
$515$	2773.86i	0.237342i
$516$	0	0
$517$	−27204.3	−2.31421
$518$	− 12518.3i	− 1.06182i
$519$	0	0
$520$	8116.50	0.684485
$521$	− 13313.6i	− 1.11954i	−0.828649	−	0.559769i	$-0.810890\pi$
$521$	− 13313.6i	− 1.11954i	0.828649	−	0.559769i	$-0.189110\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1304.29i	0.108737i
$525$	0	0
$526$	−16911.9	−1.40189
$527$	0	0
$528$	0	0
$529$	−35853.0	−2.94674
$530$	22718.5i	1.86194i
$531$	0	0
$532$	−25083.6	−2.04420
$533$	− 5218.24i	− 0.424066i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36696.5i	2.93252i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−25568.0	−1.98222
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 6578.20i	− 0.504478i
$555$	0	0
$556$	24752.0	1.88798
$557$	− 17061.1i	− 1.29785i	−0.760854	−	0.648924i	$-0.775219\pi$
$557$	− 17061.1i	− 1.29785i	0.760854	−	0.648924i	$-0.224781\pi$
$558$	0	0
$559$	0	0
$560$	− 20866.8i	− 1.57461i
$561$	0	0
$562$	−26067.1	−1.95654
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27140.3i	1.99961i	0.0196453	+	0.999807i	$0.493746\pi$
$569$	27140.3i	1.99961i	−0.0196453	+	0.999807i	$0.506254\pi$
$570$	0	0
$571$	−10935.2	−0.801440	−0.400720	−	0.916201i	$-0.631240\pi$
$571$	−10935.2	−0.801440	−0.400720	+	0.916201i	$0.631240\pi$
$572$	− 18561.4i	− 1.35680i
$573$	0	0
$574$	−13415.6	−0.975537
$575$	− 27391.8i	− 1.98664i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	13896.1i	1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	51954.1	3.69077
$584$	0	0
$585$	0	0
$586$	10865.6	0.765961
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	25322.7i	1.76698i
$591$	0	0
$592$	9713.11	0.674335
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	19885.4	1.35982
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	29421.8	1.99691	0.998454	−	0.0555869i	$-0.0177030\pi$
$601$	29421.8	1.99691	0.998454	+	0.0555869i	$0.0177030\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	43589.6i	2.92920i
$606$	0	0
$607$	24462.2	1.63574	0.817868	−	0.575406i	$-0.195156\pi$
$607$	24462.2	1.63574	0.817868	+	0.575406i	$0.195156\pi$
$608$	− 19462.7i	− 1.29822i
$609$	0	0
$610$	0	0
$611$	12069.1i	0.799123i
$612$	0	0
$613$	12982.1	0.855372	0.427686	−	0.903927i	$-0.359329\pi$
$613$	12982.1	0.855372	0.427686	+	0.903927i	$0.359329\pi$
$614$	0	0
$615$	0	0
$616$	−47719.7	−3.12124
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−11914.0	−0.773609	−0.386805	−	0.922162i	$-0.626421\pi$
$619$	−11914.0	−0.773609	−0.386805	+	0.922162i	$0.626421\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48406.5i	3.11295i
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	−25620.2	−1.62796
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−2567.77	−0.160850
$635$	− 29377.0i	− 1.83589i
$636$	0	0
$637$	16280.3	1.01263
$638$	0	0
$639$	0	0
$640$	16190.9	1.00000
$641$	− 20076.1i	− 1.23707i	−0.785759	−	0.618533i	$-0.787727\pi$
$641$	− 20076.1i	− 1.23707i	0.785759	−	0.618533i	$-0.212273\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 51123.7i	− 3.12819i
$645$	0	0
$646$	0	0
$647$	− 30272.7i	− 1.83948i	−0.392533	−	0.919738i	$-0.628401\pi$
$647$	− 30272.7i	− 1.83948i	0.392533	−	0.919738i	$-0.371599\pi$
$648$	0	0
$649$	57909.6	3.50254
$650$	11343.2i	0.684485i
$651$	0	0
$652$	0	0
$653$	8671.47i	0.519664i	0.965654	+	0.259832i	$0.0836673\pi$
$653$	8671.47i	0.519664i	−0.965654	+	0.259832i	$0.916333\pi$
$654$	0	0
$655$	−1822.80	−0.108737
$656$	− 10409.4i	− 0.619540i
$657$	0	0
$658$	31028.7	1.83833
$659$	23386.4i	1.38241i	0.722661	+	0.691203i	$0.242919\pi$
$659$	23386.4i	1.38241i	−0.722661	+	0.691203i	$0.757081\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 626.099i	− 0.0367584i
$663$	0	0
$664$	0	0
$665$	− 35055.4i	− 2.04420i
$666$	0	0
$667$	0	0
$668$	− 17208.8i	− 0.996748i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	9341.31	0.531481
$677$	19136.3i	1.08636i	0.839616	+	0.543181i	$0.182780\pi$
$677$	19136.3i	1.08636i	−0.839616	+	0.543181i	$0.817220\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	− 13563.4i	− 0.754890i
$687$	0	0
$688$	0	0
$689$	− 23049.3i	− 1.27447i
$690$	0	0
$691$	−33931.2	−1.86802	−0.934012	−	0.357241i	$-0.883718\pi$
$691$	−33931.2	−1.86802	−0.934012	+	0.357241i	$0.883718\pi$
$692$	1267.14i	0.0696088i
$693$	0	0
$694$	0	0
$695$	34592.0i	1.88798i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	29162.3	1.57461
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	16317.6	0.875435
$704$	− 37026.4i	− 1.98222i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−37559.3	−1.97696
$713$	0	0
$714$	0	0
$715$	25940.3	1.35680
$716$	38225.1i	1.99517i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−7235.21	−0.373722
$722$	− 13296.4i	− 0.685377i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−20032.3	−1.02195	−0.510975	−	0.859596i	$-0.670716\pi$
$727$	−20032.3	−1.02195	−0.510975	+	0.859596i	$0.670716\pi$
$728$	21170.7i	1.07780i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−19304.5	−0.972753	−0.486377	−	0.873749i	$-0.661682\pi$
$733$	−19304.5	−0.972753	−0.486377	+	0.873749i	$0.661682\pi$
$734$	35153.5i	1.76777i
$735$	0	0
$736$	39667.6	1.98664
$737$	0	0
$738$	0	0
$739$	13768.6	0.685365	0.342682	−	0.939451i	$-0.388665\pi$
$739$	13768.6	0.685365	0.342682	+	0.939451i	$0.388665\pi$
$740$	13574.5i	0.674335i
$741$	0	0
$742$	−59257.7	−2.93183
$743$	32246.9i	1.59223i	0.605148	+	0.796113i	$0.293114\pi$
$743$	32246.9i	1.59223i	−0.605148	+	0.796113i	$0.706886\pi$
$744$	0	0
$745$	0	0
$746$	18766.9i	0.921053i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	24075.6i	1.16748i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35713.5	1.71470	0.857351	−	0.514732i	$-0.172109\pi$
$757$	35713.5	1.71470	0.857351	+	0.514732i	$0.172109\pi$
$758$	1827.16i	0.0875535i
$759$	0	0
$760$	27200.0	1.29822
$761$	− 37896.1i	− 1.80517i	−0.430513	−	0.902584i	$-0.641667\pi$
$761$	− 37896.1i	− 1.80517i	0.430513	−	0.902584i	$-0.358333\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−21112.0	−0.995832
$767$	− 25691.4i	− 1.20947i
$768$	0	0
$769$	−27157.6	−1.27351	−0.636755	−	0.771066i	$-0.719724\pi$
$769$	−27157.6	−1.27351	−0.636755	+	0.771066i	$0.719724\pi$
$770$	− 66690.3i	− 3.12124i
$771$	0	0
$772$	0	0
$773$	− 40674.1i	− 1.89255i	−0.323356	−	0.946277i	$-0.604811\pi$
$773$	− 40674.1i	− 1.89255i	0.323356	−	0.946277i	$-0.395189\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 17487.4i	− 0.804300i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	32476.1	1.47941
$785$	− 35805.4i	− 1.62796i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	16278.6i	0.735914i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	− 40330.6i	− 1.80262i
$795$	0	0
$796$	0	0
$797$	− 32589.1i	− 1.44839i	−0.689595	−	0.724195i	$-0.742212\pi$
$797$	− 32589.1i	− 1.44839i	0.689595	−	0.724195i	$-0.257788\pi$
$798$	0	0
$799$	0	0
$800$	22627.4i	1.00000i
$801$	0	0
$802$	8070.04	0.355316
$803$	0	0
$804$	0	0
$805$	71447.6	3.12819
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27433.4i	1.19222i	0.802902	+	0.596110i	$0.203288\pi$
$809$	27433.4i	1.19222i	−0.802902	+	0.596110i	$0.796712\pi$
$810$	0	0
$811$	−4858.00	−0.210342	−0.105171	−	0.994454i	$-0.533539\pi$
$811$	−4858.00	−0.210342	−0.105171	+	0.994454i	$0.533539\pi$
$812$	0	0
$813$	0	0
$814$	31043.0	1.33668
$815$	0	0
$816$	0	0
$817$	0	0
$818$	40825.5i	1.74503i
$819$	0	0
$820$	14547.6	0.619540
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	27622.4	1.16994	0.584968	−	0.811057i	$-0.301107\pi$
$823$	27622.4	1.16994	0.584968	+	0.811057i	$0.301107\pi$
$824$	− 5613.90i	− 0.237342i
$825$	0	0
$826$	−66050.5	−2.78231
$827$	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
$832$	−16426.6	−0.684485
$833$	0	0
$834$	0	0
$835$	24050.0	0.996748
$836$	− 62202.8i	− 2.57336i
$837$	0	0
$838$	46466.1	1.91545
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	−26224.0	−1.06951
$845$	13054.9i	0.531481i
$846$	0	0
$847$	−113697.	−4.61237
$848$	− 45978.9i	− 1.86194i
$849$	0	0
$850$	0	0
$851$	33257.5i	1.33966i
$852$	0	0
$853$	−43859.0	−1.76050	−0.880248	−	0.474514i	$-0.842624\pi$
$853$	−43859.0	−1.76050	−0.880248	+	0.474514i	$0.842624\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−33334.0	−1.32403	−0.662015	−	0.749491i	$-0.730298\pi$
$859$	−33334.0	−1.32403	−0.662015	+	0.749491i	$0.730298\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 45987.0i	− 1.81392i	−0.421214	−	0.906961i	$-0.638396\pi$
$863$	− 45987.0i	− 1.81392i	0.421214	−	0.906961i	$-0.361604\pi$
$864$	0	0
$865$	−1770.88	−0.0696088
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	66640.0	2.57910
$875$	40755.5i	1.57461i
$876$	0	0
$877$	−10767.1	−0.414571	−0.207286	−	0.978280i	$-0.566463\pi$
$877$	−10767.1	−0.414571	−0.207286	+	0.978280i	$0.566463\pi$
$878$	0	0
$879$	0	0
$880$	51745.9	1.98222
$881$	51778.4i	1.98009i	0.140751	+	0.990045i	$0.455048\pi$
$881$	51778.4i	1.98009i	−0.140751	+	0.990045i	$0.544952\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11810.9i	0.447093i	0.974693	+	0.223547i	$0.0717635\pi$
$887$	11810.9i	0.447093i	−0.974693	+	0.223547i	$0.928237\pi$
$888$	0	0
$889$	76625.5	2.89082
$890$	− 52490.7i	− 1.97696i
$891$	0	0
$892$	−9647.00	−0.362114
$893$	40446.0i	1.51565i
$894$	0	0
$895$	−53421.2	−1.99517
$896$	42231.5i	1.57461i
$897$	0	0
$898$	−48759.7	−1.81195
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 33268.4i	− 1.22807i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	−29586.9	−1.07780
$911$	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
$917$	− 4754.51i	− 0.171219i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	55437.2i	1.98664i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−18970.9	−0.674335
$926$	37036.3i	1.31435i
$927$	0	0
$928$	0	0
$929$	− 31090.5i	− 1.09800i	−0.835821	−	0.549002i	$-0.815008\pi$
$929$	− 31090.5i	− 1.09800i	0.835821	−	0.549002i	$-0.184992\pi$
$930$	0	0
$931$	54558.5	1.92060
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	−33646.6	−1.16748
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	35641.5	1.23080
$944$	− 51249.5i	− 1.76698i
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	38013.2i	1.29822i
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29791.0	1.00000
$962$	− 13772.2i	− 0.461572i
$963$	0	0
$964$	−16798.0	−0.561232
$965$	0	0
$966$	0	0
$967$	43017.1	1.43055	0.715273	−	0.698845i	$-0.246302\pi$
$967$	43017.1	1.43055	0.715273	+	0.698845i	$0.246302\pi$
$968$	− 88219.1i	− 2.92920i
$969$	0	0
$970$	0	0
$971$	− 53664.1i	− 1.77360i	−0.462157	−	0.886798i	$-0.652924\pi$
$971$	− 53664.1i	− 1.77360i	0.462157	−	0.886798i	$-0.347076\pi$
$972$	0	0
$973$	−90228.1	−2.97285
$974$	561.757i	0.0184803i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−120039.	−3.91877
$980$	45386.7i	1.47941i
$981$	0	0
$982$	48996.8	1.59221
$983$	− 30629.0i	− 0.993809i	−0.867805	−	0.496905i	$-0.834470\pi$
$983$	− 30629.0i	− 0.993809i	0.867805	−	0.496905i	$-0.165530\pi$
$984$	0	0
$985$	−22750.0	−0.735914
$986$	0	0
$987$	0	0
$988$	−27596.1	−0.888612
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−59706.3	−1.89661	−0.948303	−	0.317366i	$-0.897202\pi$
$997$	−59706.3	−1.89661	−0.948303	+	0.317366i	$0.897202\pi$
$998$	57976.8i	1.83890i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.4.m.b.179.1	yes	4
3.2	odd	2	inner	360.4.m.b.179.4	yes	4
4.3	odd	2		1440.4.m.a.719.1		4
5.4	even	2		360.4.m.a.179.4	yes	4
8.3	odd	2		360.4.m.a.179.4	yes	4
8.5	even	2		1440.4.m.b.719.4		4
12.11	even	2		1440.4.m.a.719.3		4
15.14	odd	2		360.4.m.a.179.1	✓	4
20.19	odd	2		1440.4.m.b.719.4		4
24.5	odd	2		1440.4.m.b.719.2		4
24.11	even	2		360.4.m.a.179.1	✓	4
40.19	odd	2	CM	360.4.m.b.179.1	yes	4
40.29	even	2		1440.4.m.a.719.1		4
60.59	even	2		1440.4.m.b.719.2		4
120.29	odd	2		1440.4.m.a.719.3		4
120.59	even	2	inner	360.4.m.b.179.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.m.a.179.1	✓	4	15.14	odd	2
360.4.m.a.179.1	✓	4	24.11	even	2
360.4.m.a.179.4	yes	4	5.4	even	2
360.4.m.a.179.4	yes	4	8.3	odd	2
360.4.m.b.179.1	yes	4	1.1	even	1	trivial
360.4.m.b.179.1	yes	4	40.19	odd	2	CM
360.4.m.b.179.4	yes	4	3.2	odd	2	inner
360.4.m.b.179.4	yes	4	120.59	even	2	inner
1440.4.m.a.719.1		4	4.3	odd	2
1440.4.m.a.719.1		4	40.29	even	2
1440.4.m.a.719.3		4	12.11	even	2
1440.4.m.a.719.3		4	120.29	odd	2
1440.4.m.b.719.2		4	24.5	odd	2
1440.4.m.b.719.2		4	60.59	even	2
1440.4.m.b.719.4		4	8.5	even	2
1440.4.m.b.719.4		4	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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +29.1623 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +29.1623 q^{7} +22.6274i q^{8} -31.6228 q^{10} +72.3171i q^{11} +32.0833 q^{13} -82.4834i q^{14} +64.0000 q^{16} +107.517 q^{19} +89.4427i q^{20} +204.544 q^{22} +219.135i q^{23} -125.000 q^{25} -90.7452i q^{26} -233.298 q^{28} -181.019i q^{32} -326.044i q^{35} +151.767 q^{37} -304.105i q^{38} +252.982 q^{40} -162.647i q^{41} -578.537i q^{44} +619.806 q^{46} +376.181i q^{47} +507.438 q^{49} +353.553i q^{50} -256.666 q^{52} -718.420i q^{53} +808.530 q^{55} +659.867i q^{56} -800.773i q^{59} -512.000 q^{64} -358.702i q^{65} -922.192 q^{70} -429.263i q^{74} -860.140 q^{76} +2108.93i q^{77} -715.542i q^{80} -460.034 q^{82} -1636.35 q^{88} +1659.90i q^{89} +935.621 q^{91} -1753.08i q^{92} +1064.00 q^{94} -1202.08i q^{95} -1435.25i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 104 q^{7}+O(q^{10}) \) 4 * q - 32 * q^4 + 104 * q^7	\( 4 q - 32 q^{4} + 104 q^{7} - 112 q^{13} + 256 q^{16} + 464 q^{22} - 500 q^{25} - 832 q^{28} - 544 q^{37} + 1372 q^{49} + 896 q^{52} + 1400 q^{55} - 2048 q^{64} - 400 q^{70} + 1904 q^{82} - 3712 q^{88} - 2152 q^{91} + 4256 q^{94}+O(q^{100}) \) 4 * q - 32 * q^4 + 104 * q^7 - 112 * q^13 + 256 * q^16 + 464 * q^22 - 500 * q^25 - 832 * q^28 - 544 * q^37 + 1372 * q^49 + 896 * q^52 + 1400 * q^55 - 2048 * q^64 - 400 * q^70 + 1904 * q^82 - 3712 * q^88 - 2152 * q^91 + 4256 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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