LMFDB - Embedded newform 360.4.m.b.179.2

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			179.2
Root			$1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.179
Dual form			360.4.m.b.179.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +11.1803i q^{5} +22.8377 q^{7} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} +11.1803i q^{5} +22.8377 q^{7} +22.6274i q^{8} +31.6228 q^{10} +9.70724i q^{11} -88.0833 q^{13} -64.5948i q^{14} +64.0000 q^{16} -107.517 q^{19} -89.4427i q^{20} +27.4562 q^{22} -219.135i q^{23} -125.000 q^{25} +249.137i q^{26} -182.702 q^{28} -181.019i q^{32} +255.333i q^{35} -423.767 q^{37} +304.105i q^{38} -252.982 q^{40} +499.229i q^{41} -77.6579i q^{44} -619.806 q^{46} +376.181i q^{47} +178.562 q^{49} +353.553i q^{50} +704.666 q^{52} -718.420i q^{53} -108.530 q^{55} +516.759i q^{56} +424.592i q^{59} -512.000 q^{64} -984.801i q^{65} +722.192 q^{70} +1198.59i q^{74} +860.140 q^{76} +221.691i q^{77} +715.542i q^{80} +1412.03 q^{82} -219.650 q^{88} -254.173i q^{89} -2011.62 q^{91} +1753.08i q^{92} +1064.00 q^{94} -1202.08i q^{95} -505.048i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	22.8377	1.23312	0.616561	−	0.787307i	$-0.288526\pi$
$7$	22.8377	1.23312	0.616561	+	0.787307i	$0.288526\pi$
$8$	22.6274i	1.00000i
$9$	0	0
$10$	31.6228	1.00000
$11$	9.70724i	0.266077i	0.991111	+	0.133038i	$0.0424733\pi$
$11$	9.70724i	0.266077i	−0.991111	+	0.133038i	$0.957527\pi$
$12$	0	0
$13$	−88.0833	−1.87922	−0.939612	−	0.342242i	$-0.888814\pi$
$13$	−88.0833	−1.87922	−0.939612	+	0.342242i	$0.888814\pi$
$14$	− 64.5948i	− 1.23312i
$15$	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	$-0.724859\pi$
$19$	−107.517	−1.29822	−0.649110	+	0.760694i	$0.724859\pi$
$20$	− 89.4427i	− 1.00000i
$21$	0	0
$22$	27.4562	0.266077
$23$	− 219.135i	− 1.98664i	−0.115389	−	0.993320i	$-0.536812\pi$
$23$	− 219.135i	− 1.98664i	0.115389	−	0.993320i	$-0.463188\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	249.137i	1.87922i
$27$	0	0
$28$	−182.702	−1.23312
$29$	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
$35$	255.333i	1.23312i
$36$	0	0
$37$	−423.767	−1.88289	−0.941445	−	0.337167i	$-0.890531\pi$
$37$	−423.767	−1.88289	−0.941445	+	0.337167i	$0.890531\pi$
$38$	304.105i	1.29822i
$39$	0	0
$40$	−252.982	−1.00000
$41$	499.229i	1.90162i	0.309770	+	0.950812i	$0.399748\pi$
$41$	499.229i	1.90162i	−0.309770	+	0.950812i	$0.600252\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 77.6579i	− 0.266077i
$45$	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
$49$	178.562	0.520588
$50$	353.553i	1.00000i
$51$	0	0
$52$	704.666	1.87922
$53$	− 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
$55$	−108.530	−0.266077
$56$	516.759i	1.23312i
$57$	0	0
$58$	0	0
$59$	424.592i	0.936902i	0.883490	+	0.468451i	$0.155188\pi$
$59$	424.592i	0.936902i	−0.883490	+	0.468451i	$0.844812\pi$
$60$	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
$65$	− 984.801i	− 1.87922i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	722.192	1.23312
$71$	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			$\pi$
$74$	1198.59i	1.88289i
$75$	0	0
$76$	860.140	1.29822
$77$	221.691i	0.328105i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	715.542i	1.00000i
$81$	0	0
$82$	1412.03	1.90162
$83$	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
$88$	−219.650	−0.266077
$89$	− 254.173i	− 0.302722i	−0.988479	−	0.151361i	$-0.951634\pi$
$89$	− 254.173i	− 0.302722i	0.988479	−	0.151361i	$-0.0483656\pi$
$90$	0	0
$91$	−2011.62	−2.31731
$92$	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			$\pi$
$98$	− 505.048i	− 0.520588i
$99$	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
$103$	−2075.90	−1.98587	−0.992934	−	0.118671i	$-0.962137\pi$
$103$	−2075.90	−1.98587	−0.992934	+	0.118671i	$0.962137\pi$
$104$	− 1993.10i	− 1.87922i
$105$	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			$\pi$
$110$	306.970i	0.266077i
$111$	0	0
$112$	1461.61	1.23312
$113$	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
$118$	1200.93	0.936902
$119$	0	0
$120$	0	0
$121$	1236.77	0.929203
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1397.54i	− 1.00000i
$126$	0	0
$127$	−1135.56	−0.793419	−0.396710	−	0.917944i	$-0.629848\pi$
$127$	−1135.56	−0.793419	−0.396710	+	0.917944i	$0.629848\pi$
$128$	1448.15i	1.00000i
$129$	0	0
$130$	−2785.44	−1.87922
$131$	2994.29i	1.99704i	0.0543685	+	0.998521i	$0.482685\pi$
$131$	2994.29i	1.99704i	−0.0543685	+	0.998521i	$0.517315\pi$
$132$	0	0
$133$	−2455.45	−1.60086
$134$	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	$0.892962\pi$
$140$	− 2042.67i	− 1.23312i
$141$	0	0
$142$	0	0
$143$	− 855.046i	− 0.500018i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	3390.14	1.88289
$149$	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
$154$	627.038	0.328105
$155$	0	0
$156$	0	0
$157$	2285.47	1.16179	0.580893	−	0.813980i	$-0.302704\pi$
$157$	2285.47	1.16179	0.580893	+	0.813980i	$0.302704\pi$
$158$	0	0
$159$	0	0
$160$	2023.86	1.00000
$161$	− 5004.54i	− 2.44977i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 3993.84i	− 1.90162i
$165$	0	0
$166$	0	0
$167$	− 2151.10i	− 0.996748i	−0.866962	−	0.498374i	$-0.833931\pi$
$167$	− 2151.10i	− 0.996748i	0.866962	−	0.498374i	$-0.166069\pi$
$168$	0	0
$169$	5561.66	2.53148
$170$	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
$175$	−2854.72	−1.23312
$176$	621.263i	0.266077i
$177$	0	0
$178$	−718.910	−0.302722
$179$	− 332.832i	− 0.138978i	−0.997583	−	0.0694890i	$-0.977863\pi$
$179$	− 332.832i	− 0.138978i	0.997583	−	0.0694890i	$-0.0221369\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	5689.72i	2.31731i
$183$	0	0
$184$	4958.45	1.98664
$185$	− 4737.86i	− 1.88289i
$186$	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
$196$	−1428.49	−0.520588
$197$	2034.82i	0.735914i	0.929843	+	0.367957i	$0.119943\pi$
$197$	2034.82i	0.735914i	−0.929843	+	0.367957i	$0.880057\pi$
$198$	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
$205$	−5581.56	−1.90162
$206$	5871.53i	1.98587i
$207$	0	0
$208$	−5637.33	−1.87922
$209$	− 1043.70i	− 0.345426i
$210$	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
$220$	868.242	0.266077
$221$	0	0
$222$	0	0
$223$	6550.12	1.96695	0.983473	−	0.181057i	$-0.0579518\pi$
$223$	6550.12	1.96695	0.983473	+	0.181057i	$0.0579518\pi$
$224$	− 4134.07i	− 1.23312i
$225$	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
$236$	− 3396.74i	− 0.936902i
$237$	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	$-0.590539\pi$
$241$	−2099.75	−0.561232	−0.280616	+	0.959820i	$0.590539\pi$
$242$	− 3498.11i	− 0.929203i
$243$	0	0
$244$	0	0
$245$	1996.38i	0.520588i
$246$	0	0
$247$	9470.49	2.43965
$248$	0	0
$249$	0	0
$250$	−3952.85	−1.00000
$251$	6582.33i	1.65527i	0.561265	+	0.827636i	$0.310315\pi$
$251$	6582.33i	1.65527i	−0.561265	+	0.827636i	$0.689685\pi$
$252$	0	0
$253$	2127.19	0.528599
$254$	3211.84i	0.793419i
$255$	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
$259$	−9677.88	−2.32183
$260$	7878.41i	1.87922i
$261$	0	0
$262$	8469.14	1.99704
$263$	5979.25i	1.40189i	0.713217	+	0.700943i	$0.247237\pi$
$263$	5979.25i	1.40189i	−0.713217	+	0.700943i	$0.752763\pi$
$264$	0	0
$265$	8032.19	1.86194
$266$	6945.07i	1.60086i
$267$	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
$275$	− 1213.41i	− 0.266077i
$276$	0	0
$277$	8922.26	1.93533	0.967665	−	0.252239i	$-0.0811669\pi$
$277$	8922.26	1.93533	0.967665	+	0.252239i	$0.0811669\pi$
$278$	8751.15i	1.88798i
$279$	0	0
$280$	−5777.54	−1.23312
$281$	− 1953.35i	− 0.414688i	−0.978268	−	0.207344i	$-0.933518\pi$
$281$	− 1953.35i	− 0.414688i	0.978268	−	0.207344i	$-0.0664820\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−2418.43	−0.500018
$287$	11401.3i	2.34493i
$288$	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	$-0.874898\pi$
$293$	− 3841.56i	− 0.765961i	0.923756	−	0.382981i	$-0.125102\pi$
$294$	0	0
$295$	−4747.09	−0.936902
$296$	− 9588.76i	− 1.88289i
$297$	0	0
$298$	0	0
$299$	19302.1i	3.73334i
$300$	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			$\pi$
$308$	− 1773.53i	− 0.328105i
$309$	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			$\pi$
$314$	− 6464.28i	− 1.16179i
$315$	0	0
$316$	0	0
$317$	907.844i	0.160850i	0.996761	+	0.0804252i	$0.0256278\pi$
$317$	907.844i	0.160850i	−0.996761	+	0.0804252i	$0.974372\pi$
$318$	0	0
$319$	0	0
$320$	− 5724.33i	− 1.00000i
$321$	0	0
$322$	−14155.0	−2.44977
$323$	0	0
$324$	0	0
$325$	11010.4	1.87922
$326$	0	0
$327$	0	0
$328$	−11296.3	−1.90162
$329$	8591.11i	1.43965i
$330$	0	0
$331$	−221.359	−0.0367584	−0.0183792	−	0.999831i	$-0.505851\pi$
$331$	−221.359	−0.0367584	−0.0183792	+	0.999831i	$0.505851\pi$
$332$	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			$\pi$
$338$	− 15730.8i	− 2.53148i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3755.40	−0.591173
$344$	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			$\pi$
$350$	8074.35i	1.23312i
$351$	0	0
$352$	1757.20	0.266077
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	2033.38i	0.302722i
$357$	0	0
$358$	−941.392	−0.138978
$359$	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
$364$	16093.0	2.31731
$365$	0	0
$366$	0	0
$367$	6576.64	0.935417	0.467709	−	0.883883i	$-0.345080\pi$
$367$	6576.64	0.935417	0.467709	+	0.883883i	$0.345080\pi$
$368$	− 14024.6i	− 1.98664i
$369$	0	0
$370$	−13400.7	−1.88289
$371$	− 16407.1i	− 2.29599i
$372$	0	0
$373$	−12788.9	−1.77529	−0.887646	−	0.460526i	$-0.847661\pi$
$373$	−12788.9	−1.77529	−0.887646	+	0.460526i	$0.847661\pi$
$374$	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
$385$	−2478.58	−0.328105
$386$	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
$392$	4040.39i	0.520588i
$393$	0	0
$394$	5755.35	0.735914
$395$	0	0
$396$	0	0
$397$	6852.97	0.866350	0.433175	−	0.901310i	$-0.357393\pi$
$397$	6852.97	0.866350	0.433175	+	0.901310i	$0.357393\pi$
$398$	0	0
$399$	0	0
$400$	−8000.00	−1.00000
$401$	− 15804.6i	− 1.96818i	−0.177658	−	0.984092i	$-0.556852\pi$
$401$	− 15804.6i	− 1.96818i	0.177658	−	0.984092i	$-0.443148\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4113.61i	− 0.500993i
$408$	0	0
$409$	−14434.0	−1.74503	−0.872513	−	0.488591i	$-0.837511\pi$
$409$	−14434.0	−1.74503	−0.872513	+	0.488591i	$0.837511\pi$
$410$	15787.0i	1.90162i
$411$	0	0
$412$	16607.2	1.98587
$413$	9696.72i	1.15531i
$414$	0	0
$415$	0	0
$416$	15944.8i	1.87922i
$417$	0	0
$418$	−2952.02	−0.345426
$419$	4934.86i	0.575379i	0.957724	+	0.287689i	$0.0928871\pi$
$419$	4934.86i	0.575379i	−0.957724	+	0.287689i	$0.907113\pi$
$420$	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			$\pi$
$440$	− 2455.76i	− 0.266077i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	2841.74	0.302722
$446$	− 18526.6i	− 1.96695i
$447$	0	0
$448$	−11692.9	−1.23312
$449$	8055.25i	0.846661i	0.905975	+	0.423330i	$0.139139\pi$
$449$	8055.25i	0.846661i	−0.905975	+	0.423330i	$0.860861\pi$
$450$	0	0
$451$	−4846.14	−0.505978
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 22490.6i	− 2.31731i
$456$	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
$463$	15018.3	1.50747	0.753737	−	0.657176i	$-0.228249\pi$
$463$	15018.3	1.50747	0.753737	+	0.657176i	$0.228249\pi$
$464$	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
$472$	−9607.43	−0.936902
$473$	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
$481$	37326.8	3.53837
$482$	5939.00i	0.561232i
$483$	0	0
$484$	−9894.16	−0.929203
$485$	0	0
$486$	0	0
$487$	−21493.4	−1.99991	−0.999957	−	0.00924017i	$-0.997059\pi$
$487$	−21493.4	−1.99991	−0.999957	+	0.00924017i	$0.997059\pi$
$488$	0	0
$489$	0	0
$490$	5646.61	0.520588
$491$	− 13168.0i	− 1.21032i	−0.796106	−	0.605158i	$-0.793110\pi$
$491$	− 13168.0i	− 1.21032i	0.796106	−	0.605158i	$-0.206890\pi$
$492$	0	0
$493$	0	0
$494$	− 26786.6i	− 2.43965i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20497.9	1.83890	0.919450	−	0.393206i	$-0.128634\pi$
$499$	20497.9	1.83890	0.919450	+	0.393206i	$0.128634\pi$
$500$	11180.3i	1.00000i
$501$	0	0
$502$	18617.7	1.65527
$503$	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
$506$	− 6016.61i	− 0.528599i
$507$	0	0
$508$	9084.44	0.793419
$509$	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
$515$	− 23209.2i	− 1.98587i
$516$	0	0
$517$	−3651.68	−0.310639
$518$	27373.2i	2.32183i
$519$	0	0
$520$	22283.5	1.87922
$521$	19708.7i	1.65730i	0.559769	+	0.828649i	$0.310890\pi$
$521$	19708.7i	1.65730i	−0.559769	+	0.828649i	$0.689110\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	− 23954.3i	− 1.99704i
$525$	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
$532$	19643.6	1.60086
$533$	− 43973.8i	− 3.57357i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1733.34i	0.138516i
$540$	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
$550$	−3432.03	−0.266077
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 25235.9i	− 1.93533i
$555$	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
$560$	16341.3i	1.23312i
$561$	0	0
$562$	−5524.92	−0.414688
$563$	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
$569$	− 533.282i	− 0.0392906i	−0.999807	−	0.0196453i	$-0.993746\pi$
$569$	− 533.282i	− 0.0392906i	0.999807	−	0.0196453i	$-0.00625369\pi$
$570$	0	0
$571$	10935.2	0.801440	0.400720	−	0.916201i	$-0.368760\pi$
$571$	10935.2	0.801440	0.400720	+	0.916201i	$0.368760\pi$
$572$	6840.37i	0.500018i
$573$	0	0
$574$	32247.6	2.34493
$575$	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
$583$	6973.88	0.495418
$584$	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
$590$	13426.8i	0.936902i
$591$	0	0
$592$	−27121.1	−1.88289
$593$	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
$598$	54594.6	3.73334
$599$	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	$-0.982297\pi$
$601$	−29421.8	−1.99691	−0.998454	+	0.0555869i	$0.982297\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13827.5i	0.929203i
$606$	0	0
$607$	−17210.2	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$607$	−17210.2	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$608$	19462.7i	1.29822i
$609$	0	0
$610$	0	0
$611$	− 33135.2i	− 2.19396i
$612$	0	0
$613$	−27438.1	−1.80785	−0.903927	−	0.427686i	$-0.859329\pi$
$613$	−27438.1	−1.80785	−0.903927	+	0.427686i	$0.859329\pi$
$614$	0	0
$615$	0	0
$616$	−5016.30	−0.328105
$617$	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
$623$	− 5804.73i	− 0.373293i
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	−18283.8	−1.16179
$629$	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
$635$	− 12695.9i	− 0.793419i
$636$	0	0
$637$	−15728.3	−0.978301
$638$	0	0
$639$	0	0
$640$	−16190.9	−1.00000
$641$	25503.9i	1.57152i	0.618533	+	0.785759i	$0.287727\pi$
$641$	25503.9i	1.57152i	−0.618533	+	0.785759i	$0.712273\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	40036.3i	2.44977i
$645$	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
$649$	−4121.62	−0.249288
$650$	− 31142.1i	− 1.87922i
$651$	0	0
$652$	0	0
$653$	− 8671.47i	− 0.519664i	−0.965654	−	0.259832i	$-0.916333\pi$
$653$	− 8671.47i	− 0.519664i	0.965654	−	0.259832i	$-0.0836673\pi$
$654$	0	0
$655$	−33477.2	−1.99704
$656$	31950.7i	1.90162i
$657$	0	0
$658$	24299.3	1.43965
$659$	24450.8i	1.44532i	0.691203	+	0.722661i	$0.257081\pi$
$659$	24450.8i	1.44532i	−0.691203	+	0.722661i	$0.742919\pi$
$660$	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
$665$	− 27452.8i	− 1.60086i
$666$	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
$676$	−44493.3	−2.53148
$677$	− 19136.3i	− 1.08636i	−0.839616	−	0.543181i	$-0.817220\pi$
$677$	− 19136.3i	− 1.08636i	0.839616	−	0.543181i	$-0.182780\pi$
$678$	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
$686$	10621.9i	0.591173i
$687$	0	0
$688$	0	0
$689$	63280.8i	3.49900i
$690$	0	0
$691$	33931.2	1.86802	0.934012	−	0.357241i	$-0.116282\pi$
$691$	33931.2	1.86802	0.934012	+	0.357241i	$0.116282\pi$
$692$	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
$700$	22837.7	1.23312
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	45562.4	2.44441
$704$	− 4970.11i	− 0.266077i
$705$	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
$712$	5751.28	0.302722
$713$	0	0
$714$	0	0
$715$	9559.70	0.500018
$716$	2662.66i	0.138978i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−47408.8	−2.44881
$722$	− 13296.4i	− 0.685377i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33699.7	−1.71919	−0.859596	−	0.510975i	$-0.829284\pi$
$727$	−33699.7	−1.71919	−0.859596	+	0.510975i	$0.829284\pi$
$728$	− 45517.8i	− 2.31731i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−34679.5	−1.74750	−0.873749	−	0.486377i	$-0.838318\pi$
$733$	−34679.5	−1.74750	−0.873749	+	0.486377i	$0.838318\pi$
$734$	− 18601.6i	− 0.935417i
$735$	0	0
$736$	−39667.6	−1.98664
$737$	0	0
$738$	0	0
$739$	−13768.6	−0.685365	−0.342682	−	0.939451i	$-0.611335\pi$
$739$	−13768.6	−0.685365	−0.342682	+	0.939451i	$0.611335\pi$
$740$	37902.9i	1.88289i
$741$	0	0
$742$	−46406.3	−2.29599
$743$	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
$746$	36172.5i	1.77529i
$747$	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
$757$	−21441.5	−1.02946	−0.514732	−	0.857351i	$-0.672109\pi$
$757$	−21441.5	−1.02946	−0.514732	+	0.857351i	$0.672109\pi$
$758$	1827.16i	0.0875535i
$759$	0	0
$760$	27200.0	1.29822
$761$	− 18075.6i	− 0.861026i	−0.902584	−	0.430513i	$-0.858333\pi$
$761$	− 18075.6i	− 0.861026i	0.902584	−	0.430513i	$-0.141667\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−21112.0	−0.995832
$767$	− 37399.5i	− 1.76065i
$768$	0	0
$769$	27157.6	1.27351	0.636755	−	0.771066i	$-0.280276\pi$
$769$	27157.6	1.27351	0.636755	+	0.771066i	$0.280276\pi$
$770$	7010.49i	0.328105i
$771$	0	0
$772$	0	0
$773$	40674.1i	1.89255i	0.323356	+	0.946277i	$0.395189\pi$
$773$	40674.1i	1.89255i	−0.323356	+	0.946277i	$0.604811\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 53675.9i	− 2.46873i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	11427.9	0.520588
$785$	25552.3i	1.16179i
$786$	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
$794$	− 19383.1i	− 0.866350i
$795$	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
$802$	−44702.0	−1.96818
$803$	0	0
$804$	0	0
$805$	55952.4	2.44977
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36950.1i	1.60580i	0.596110	+	0.802902i	$0.296712\pi$
$809$	36950.1i	1.60580i	−0.596110	+	0.802902i	$0.703288\pi$
$810$	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
$814$	−11635.0	−0.500993
$815$	0	0
$816$	0	0
$817$	0	0
$818$	40825.5i	1.74503i
$819$	0	0
$820$	44652.4	1.90162
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−38298.4	−1.62211	−0.811057	−	0.584968i	$-0.801107\pi$
$823$	−38298.4	−1.62211	−0.811057	+	0.584968i	$0.801107\pi$
$824$	− 46972.2i	− 1.98587i
$825$	0	0
$826$	27426.5	1.15531
$827$	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
$832$	45098.6	1.87922
$833$	0	0
$834$	0	0
$835$	24050.0	0.996748
$836$	8349.58i	0.345426i
$837$	0	0
$838$	13957.9	0.575379
$839$	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
$845$	62181.3i	2.53148i
$846$	0	0
$847$	28245.0	1.14582
$848$	− 45978.9i	− 1.86194i
$849$	0	0
$850$	0	0
$851$	92862.1i	3.74062i
$852$	0	0
$853$	23643.0	0.949028	0.474514	−	0.880248i	$-0.342624\pi$
$853$	23643.0	0.949028	0.474514	+	0.880248i	$0.342624\pi$
$854$	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	$0.361604\pi$
$863$	45987.0i	1.81392i	−0.421214	+	0.906961i	$0.638396\pi$
$864$	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
$875$	− 31916.7i	− 1.23312i
$876$	0	0
$877$	50815.1	1.95656	0.978280	−	0.207286i	$-0.0664630\pi$
$877$	50815.1	1.95656	0.978280	+	0.207286i	$0.0664630\pi$
$878$	0	0
$879$	0	0
$880$	−6945.94	−0.266077
$881$	7361.16i	0.281503i	0.990045	+	0.140751i	$0.0449518\pi$
$881$	7361.16i	0.281503i	−0.990045	+	0.140751i	$0.955048\pi$
$882$	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	$-0.928237\pi$
$887$	− 11810.9i	− 0.447093i	0.974693	−	0.223547i	$-0.0717635\pi$
$888$	0	0
$889$	−25933.5	−0.978382
$890$	− 8037.65i	− 0.302722i
$891$	0	0
$892$	−52401.0	−1.96695
$893$	− 40446.0i	− 1.51565i
$894$	0	0
$895$	3721.18	0.138978
$896$	33072.6i	1.23312i
$897$	0	0
$898$	22783.7	0.846661
$899$	0	0
$900$	0	0
$901$	0	0
$902$	13707.0i	0.505978i
$903$	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
$910$	−63613.1	−2.31731
$911$	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
$917$	68382.8i	2.46259i
$918$	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
$925$	52970.9	1.88289
$926$	− 42478.2i	− 1.50747i
$927$	0	0
$928$	0	0
$929$	− 47333.3i	− 1.67164i	−0.549002	−	0.835821i	$-0.684992\pi$
$929$	− 47333.3i	− 1.67164i	0.549002	−	0.835821i	$-0.315008\pi$
$930$	0	0
$931$	−19198.5	−0.675838
$932$	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
$943$	109398.	3.77784
$944$	27173.9i	0.936902i
$945$	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
$962$	− 105576.i	− 3.53837i
$963$	0	0
$964$	16798.0	0.561232
$965$	0	0
$966$	0	0
$967$	−42029.1	−1.39769	−0.698845	−	0.715273i	$-0.746302\pi$
$967$	−42029.1	−1.39769	−0.698845	+	0.715273i	$0.746302\pi$
$968$	27984.9i	0.929203i
$969$	0	0
$970$	0	0
$971$	− 27967.2i	− 0.924314i	−0.886798	−	0.462157i	$-0.847076\pi$
$971$	− 27967.2i	− 0.924314i	0.886798	−	0.462157i	$-0.152924\pi$
$972$	0	0
$973$	−70659.9	−2.32811
$974$	60792.5i	1.99991i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	2467.32	0.0805473
$980$	− 15971.0i	− 0.520588i
$981$	0	0
$982$	−37244.8	−1.21032
$983$	− 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
$988$	−75763.9	−2.43965
$989$	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
$997$	−19981.7	−0.634732	−0.317366	−	0.948303i	$-0.602798\pi$
$997$	−19981.7	−0.634732	−0.317366	+	0.948303i	$0.602798\pi$
$998$	− 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.179.2	yes	4
3.2	odd	2	inner	360.4.m.b.179.3	yes	4
4.3	odd	2		1440.4.m.a.719.4		4
5.4	even	2		360.4.m.a.179.3	yes	4
8.3	odd	2		360.4.m.a.179.3	yes	4
8.5	even	2		1440.4.m.b.719.1		4
12.11	even	2		1440.4.m.a.719.2		4
15.14	odd	2		360.4.m.a.179.2	✓	4
20.19	odd	2		1440.4.m.b.719.1		4
24.5	odd	2		1440.4.m.b.719.3		4
24.11	even	2		360.4.m.a.179.2	✓	4
40.19	odd	2	CM	360.4.m.b.179.2	yes	4
40.29	even	2		1440.4.m.a.719.4		4
60.59	even	2		1440.4.m.b.719.3		4
120.29	odd	2		1440.4.m.a.719.2		4
120.59	even	2	inner	360.4.m.b.179.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.m.a.179.2	✓	4	15.14	odd	2
360.4.m.a.179.2	✓	4	24.11	even	2
360.4.m.a.179.3	yes	4	5.4	even	2
360.4.m.a.179.3	yes	4	8.3	odd	2
360.4.m.b.179.2	yes	4	1.1	even	1	trivial
360.4.m.b.179.2	yes	4	40.19	odd	2	CM
360.4.m.b.179.3	yes	4	3.2	odd	2	inner
360.4.m.b.179.3	yes	4	120.59	even	2	inner
1440.4.m.a.719.2		4	12.11	even	2
1440.4.m.a.719.2		4	120.29	odd	2
1440.4.m.a.719.4		4	4.3	odd	2
1440.4.m.a.719.4		4	40.29	even	2
1440.4.m.b.719.1		4	8.5	even	2
1440.4.m.b.719.1		4	20.19	odd	2
1440.4.m.b.719.3		4	24.5	odd	2
1440.4.m.b.719.3		4	60.59	even	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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +11.1803i q^{5} +22.8377 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +11.1803i q^{5} +22.8377 q^{7} +22.6274i q^{8} +31.6228 q^{10} +9.70724i q^{11} -88.0833 q^{13} -64.5948i q^{14} +64.0000 q^{16} -107.517 q^{19} -89.4427i q^{20} +27.4562 q^{22} -219.135i q^{23} -125.000 q^{25} +249.137i q^{26} -182.702 q^{28} -181.019i q^{32} +255.333i q^{35} -423.767 q^{37} +304.105i q^{38} -252.982 q^{40} +499.229i q^{41} -77.6579i q^{44} -619.806 q^{46} +376.181i q^{47} +178.562 q^{49} +353.553i q^{50} +704.666 q^{52} -718.420i q^{53} -108.530 q^{55} +516.759i q^{56} +424.592i q^{59} -512.000 q^{64} -984.801i q^{65} +722.192 q^{70} +1198.59i q^{74} +860.140 q^{76} +221.691i q^{77} +715.542i q^{80} +1412.03 q^{82} -219.650 q^{88} -254.173i q^{89} -2011.62 q^{91} +1753.08i q^{92} +1064.00 q^{94} -1202.08i q^{95} -505.048i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 104 q^{7}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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