LMFDB - Embedded newform 360.2.m.a.179.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,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, 2, names="a")

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

chi := DirichletCharacter("360.179");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
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:	$2.87461447277$
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			179.2
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.179
Dual form			360.2.m.a.179.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +1.16228 q^{7} +2.82843i q^{8} +3.16228 q^{10} -5.88635i q^{11} +7.16228 q^{13} -1.64371i q^{14} +4.00000 q^{16} +6.32456 q^{19} -4.47214i q^{20} -8.32456 q^{22} +4.47214i q^{23} -5.00000 q^{25} -10.1290i q^{26} -2.32456 q^{28} -5.65685i q^{32} +2.59893i q^{35} -4.83772 q^{37} -8.94427i q^{38} -6.32456 q^{40} +7.53006i q^{41} +11.7727i q^{44} +6.32456 q^{46} -2.82843i q^{47} -5.64911 q^{49} +7.07107i q^{50} -14.3246 q^{52} +5.65685i q^{53} +13.1623 q^{55} +3.28742i q^{56} -14.3716i q^{59} -8.00000 q^{64} +16.0153i q^{65} +3.67544 q^{70} +6.84157i q^{74} -12.6491 q^{76} -6.84157i q^{77} +8.94427i q^{80} +10.6491 q^{82} +16.6491 q^{88} -0.955223i q^{89} +8.32456 q^{91} -8.94427i q^{92} -4.00000 q^{94} +14.1421i q^{95} +7.98905i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 8 q^{7} + 16 q^{13} + 16 q^{16} - 8 q^{22} - 20 q^{25} + 16 q^{28} - 32 q^{37} + 28 q^{49} - 32 q^{52} + 40 q^{55} - 32 q^{64} + 40 q^{70} - 8 q^{82} + 16 q^{88} + 8 q^{91} - 16 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 8 * q^7 + 16 * q^13 + 16 * q^16 - 8 * q^22 - 20 * q^25 + 16 * q^28 - 32 * q^37 + 28 * q^49 - 32 * q^52 + 40 * q^55 - 32 * q^64 + 40 * q^70 - 8 * q^82 + 16 * q^88 + 8 * q^91 - 16 * q^94

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
$7$	1.16228	0.439300	0.219650	−	0.975579i	$-0.429509\pi$
$7$	1.16228	0.439300	0.219650	+	0.975579i	$0.429509\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	3.16228	1.00000
$11$	− 5.88635i	− 1.77480i	−0.460999	−	0.887401i	$-0.652509\pi$
$11$	− 5.88635i	− 1.77480i	0.460999	−	0.887401i	$-0.347491\pi$
$12$	0	0
$13$	7.16228	1.98646	0.993229	−	0.116171i	$-0.0370621\pi$
$13$	7.16228	1.98646	0.993229	+	0.116171i	$0.0370621\pi$
$14$	− 1.64371i	− 0.439300i
$15$	0	0
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	6.32456	1.45095	0.725476	−	0.688247i	$-0.241620\pi$
$19$	6.32456	1.45095	0.725476	+	0.688247i	$0.241620\pi$
$20$	− 4.47214i	− 1.00000i
$21$	0	0
$22$	−8.32456	−1.77480
$23$	4.47214i	0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$	4.47214i	0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	− 10.1290i	− 1.98646i
$27$	0	0
$28$	−2.32456	−0.439300
$29$	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			$\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	0	0
$35$	2.59893i	0.439300i
$36$	0	0
$37$	−4.83772	−0.795317	−0.397658	−	0.917534i	$-0.630177\pi$
$37$	−4.83772	−0.795317	−0.397658	+	0.917534i	$0.630177\pi$
$38$	− 8.94427i	− 1.45095i
$39$	0	0
$40$	−6.32456	−1.00000
$41$	7.53006i	1.17600i	0.808862	+	0.587999i	$0.200084\pi$
$41$	7.53006i	1.17600i	−0.808862	+	0.587999i	$0.799916\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	11.7727i	1.77480i
$45$	0	0
$46$	6.32456	0.932505
$47$	− 2.82843i	− 0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$	− 2.82843i	− 0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$	0	0
$49$	−5.64911	−0.807016
$50$	7.07107i	1.00000i
$51$	0	0
$52$	−14.3246	−1.98646
$53$	5.65685i	0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$	5.65685i	0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$	0	0
$55$	13.1623	1.77480
$56$	3.28742i	0.439300i
$57$	0	0
$58$	0	0
$59$	− 14.3716i	− 1.87103i	−0.353291	−	0.935513i	$-0.614937\pi$
$59$	− 14.3716i	− 1.87103i	0.353291	−	0.935513i	$-0.385063\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	16.0153i	1.98646i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	3.67544	0.439300
$71$	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			$\pi$
$74$	6.84157i	0.795317i
$75$	0	0
$76$	−12.6491	−1.45095
$77$	− 6.84157i	− 0.779670i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	8.94427i	1.00000i
$81$	0	0
$82$	10.6491	1.17600
$83$	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
$88$	16.6491	1.77480
$89$	− 0.955223i	− 0.101253i	−0.998718	−	0.0506267i	$-0.983878\pi$
$89$	− 0.955223i	− 0.101253i	0.998718	−	0.0506267i	$-0.0161219\pi$
$90$	0	0
$91$	8.32456	0.872651
$92$	− 8.94427i	− 0.932505i
$93$	0	0
$94$	−4.00000	−0.412568
$95$	14.1421i	1.45095i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	7.98905i	0.807016i
$99$	0	0
$100$	10.0000	1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−10.8377	−1.06787	−0.533936	−	0.845525i	$-0.679288\pi$
$103$	−10.8377	−1.06787	−0.533936	+	0.845525i	$0.679288\pi$
$104$	20.2580i	1.98646i
$105$	0	0
$106$	8.00000	0.777029
$107$	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			$\pi$
$110$	− 18.6143i	− 1.77480i
$111$	0	0
$112$	4.64911	0.439300
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−10.0000	−0.932505
$116$	0	0
$117$	0	0
$118$	−20.3246	−1.87103
$119$	0	0
$120$	0	0
$121$	−23.6491	−2.14992
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	−17.8114	−1.58051	−0.790253	−	0.612781i	$-0.790051\pi$
$127$	−17.8114	−1.58051	−0.790253	+	0.612781i	$0.790051\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	22.6491	1.98646
$131$	11.0842i	0.968432i	0.874948	+	0.484216i	$0.160895\pi$
$131$	11.0842i	0.968432i	−0.874948	+	0.484216i	$0.839105\pi$
$132$	0	0
$133$	7.35089	0.637403
$134$	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
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	− 5.19786i	− 0.439300i
$141$	0	0
$142$	0	0
$143$	− 42.1597i	− 3.52557i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	9.67544	0.795317
$149$	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			$\pi$
$152$	17.8885i	1.45095i
$153$	0	0
$154$	−9.67544	−0.779670
$155$	0	0
$156$	0	0
$157$	−23.8114	−1.90036	−0.950178	−	0.311708i	$-0.899099\pi$
$157$	−23.8114	−1.90036	−0.950178	+	0.311708i	$0.899099\pi$
$158$	0	0
$159$	0	0
$160$	12.6491	1.00000
$161$	5.19786i	0.409649i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 15.0601i	− 1.17600i
$165$	0	0
$166$	0	0
$167$	4.47214i	0.346064i	0.984916	+	0.173032i	$0.0553564\pi$
$167$	4.47214i	0.346064i	−0.984916	+	0.173032i	$0.944644\pi$
$168$	0	0
$169$	38.2982	2.94602
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6274i	1.72033i	0.510015	+	0.860165i	$0.329640\pi$
$173$	22.6274i	1.72033i	−0.510015	+	0.860165i	$0.670360\pi$
$174$	0	0
$175$	−5.81139	−0.439300
$176$	− 23.5454i	− 1.77480i
$177$	0	0
$178$	−1.35089	−0.101253
$179$	− 22.8569i	− 1.70841i	−0.519940	−	0.854203i	$-0.674046\pi$
$179$	− 22.8569i	− 1.70841i	0.519940	−	0.854203i	$-0.325954\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	− 11.7727i	− 0.872651i
$183$	0	0
$184$	−12.6491	−0.932505
$185$	− 10.8175i	− 0.795317i
$186$	0	0
$187$	0	0
$188$	5.65685i	0.412568i
$189$	0	0
$190$	20.0000	1.45095
$191$	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
$196$	11.2982	0.807016
$197$	− 22.3607i	− 1.59313i	−0.604551	−	0.796566i	$-0.706648\pi$
$197$	− 22.3607i	− 1.59313i	0.604551	−	0.796566i	$-0.293352\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 14.1421i	− 1.00000i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−16.8377	−1.17600
$206$	15.3269i	1.06787i
$207$	0	0
$208$	28.6491	1.98646
$209$	− 37.2285i	− 2.57515i
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	− 11.3137i	− 0.777029i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−26.3246	−1.77480
$221$	0	0
$222$	0	0
$223$	−29.8114	−1.99632	−0.998159	−	0.0606498i	$-0.980683\pi$
$223$	−29.8114	−1.99632	−0.998159	+	0.0606498i	$0.980683\pi$
$224$	− 6.57484i	− 0.439300i
$225$	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			$\pi$
$230$	14.1421i	0.932505i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	6.32456	0.412568
$236$	28.7433i	1.87103i
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	25.2982	1.62960	0.814801	−	0.579741i	$-0.196846\pi$
$241$	25.2982	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$242$	33.4449i	2.14992i
$243$	0	0
$244$	0	0
$245$	− 12.6318i	− 0.807016i
$246$	0	0
$247$	45.2982	2.88226
$248$	0	0
$249$	0	0
$250$	−15.8114	−1.00000
$251$	20.9465i	1.32213i	0.750329	+	0.661065i	$0.229895\pi$
$251$	20.9465i	1.32213i	−0.750329	+	0.661065i	$0.770105\pi$
$252$	0	0
$253$	26.3246	1.65501
$254$	25.1891i	1.58051i
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−5.62278	−0.349382
$260$	− 32.0307i	− 1.98646i
$261$	0	0
$262$	15.6754	0.968432
$263$	31.3050i	1.93035i	0.261612	+	0.965173i	$0.415746\pi$
$263$	31.3050i	1.93035i	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−12.6491	−0.777029
$266$	− 10.3957i	− 0.637403i
$267$	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
$275$	29.4317i	1.77480i
$276$	0	0
$277$	14.1359	0.849347	0.424673	−	0.905347i	$-0.360389\pi$
$277$	14.1359	0.849347	0.424673	+	0.905347i	$0.360389\pi$
$278$	− 19.7990i	− 1.18746i
$279$	0	0
$280$	−7.35089	−0.439300
$281$	− 2.33219i	− 0.139127i	−0.997578	−	0.0695635i	$-0.977839\pi$
$281$	− 2.33219i	− 0.139127i	0.997578	−	0.0695635i	$-0.0221607\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	−59.6228	−3.52557
$287$	8.75202i	0.516615i
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.47214i	0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$293$	4.47214i	0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$294$	0	0
$295$	32.1359	1.87103
$296$	− 13.6831i	− 0.795317i
$297$	0	0
$298$	0	0
$299$	32.0307i	1.85238i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	25.2982	1.45095
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	13.6831i	0.779670i
$309$	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			$\pi$
$314$	33.6744i	1.90036i
$315$	0	0
$316$	0	0
$317$	31.3050i	1.75826i	0.476581	+	0.879131i	$0.341876\pi$
$317$	31.3050i	1.75826i	−0.476581	+	0.879131i	$0.658124\pi$
$318$	0	0
$319$	0	0
$320$	− 17.8885i	− 1.00000i
$321$	0	0
$322$	7.35089	0.409649
$323$	0	0
$324$	0	0
$325$	−35.8114	−1.98646
$326$	0	0
$327$	0	0
$328$	−21.2982	−1.17600
$329$	− 3.28742i	− 0.181241i
$330$	0	0
$331$	−31.6228	−1.73814	−0.869072	−	0.494685i	$-0.835284\pi$
$331$	−31.6228	−1.73814	−0.869072	+	0.494685i	$0.835284\pi$
$332$	0	0
$333$	0	0
$334$	6.32456	0.346064
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	− 54.1619i	− 2.94602i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.7018	−0.793821
$344$	0	0
$345$	0	0
$346$	32.0000	1.72033
$347$	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			$\pi$
$350$	8.21854i	0.439300i
$351$	0	0
$352$	−33.2982	−1.77480
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	1.91045i	0.101253i
$357$	0	0
$358$	−32.3246	−1.70841
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	21.0000	1.10526
$362$	0	0
$363$	0	0
$364$	−16.6491	−0.872651
$365$	0	0
$366$	0	0
$367$	6.18861	0.323043	0.161521	−	0.986869i	$-0.448360\pi$
$367$	6.18861	0.323043	0.161521	+	0.986869i	$0.448360\pi$
$368$	17.8885i	0.932505i
$369$	0	0
$370$	−15.2982	−0.795317
$371$	6.57484i	0.341348i
$372$	0	0
$373$	38.1359	1.97460	0.987302	−	0.158854i	$-0.0507798\pi$
$373$	38.1359	1.97460	0.987302	+	0.158854i	$0.0507798\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	− 28.2843i	− 1.45095i
$381$	0	0
$382$	0	0
$383$	− 36.7696i	− 1.87884i	−0.342773	−	0.939418i	$-0.611366\pi$
$383$	− 36.7696i	− 1.87884i	0.342773	−	0.939418i	$-0.388634\pi$
$384$	0	0
$385$	15.2982	0.779670
$386$	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
$392$	− 15.9781i	− 0.807016i
$393$	0	0
$394$	−31.6228	−1.59313
$395$	0	0
$396$	0	0
$397$	31.1623	1.56399	0.781995	−	0.623285i	$-0.214202\pi$
$397$	31.1623	1.56399	0.781995	+	0.623285i	$0.214202\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	− 17.9258i	− 0.895171i	−0.894241	−	0.447586i	$-0.852284\pi$
$401$	− 17.9258i	− 0.895171i	0.894241	−	0.447586i	$-0.147716\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.4765i	1.41153i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	23.8121i	1.17600i
$411$	0	0
$412$	21.6754	1.06787
$413$	− 16.7038i	− 0.821942i
$414$	0	0
$415$	0	0
$416$	− 40.5160i	− 1.98646i
$417$	0	0
$418$	−52.6491	−2.57515
$419$	3.97590i	0.194236i	0.995273	+	0.0971178i	$0.0309624\pi$
$419$	3.97590i	0.194236i	−0.995273	+	0.0971178i	$0.969038\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	31.1127i	1.51454i
$423$	0	0
$424$	−16.0000	−0.777029
$425$	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
$437$	28.2843i	1.35302i
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	37.2285i	1.77480i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	2.13594	0.101253
$446$	42.1597i	1.99632i
$447$	0	0
$448$	−9.29822	−0.439300
$449$	6.15309i	0.290382i	0.989404	+	0.145191i	$0.0463797\pi$
$449$	6.15309i	0.290382i	−0.989404	+	0.145191i	$0.953620\pi$
$450$	0	0
$451$	44.3246	2.08716
$452$	0	0
$453$	0	0
$454$	0	0
$455$	18.6143i	0.872651i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	20.0000	0.932505
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−41.8114	−1.94314	−0.971570	−	0.236752i	$-0.923917\pi$
$463$	−41.8114	−1.94314	−0.971570	+	0.236752i	$0.923917\pi$
$464$	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
$470$	− 8.94427i	− 0.412568i
$471$	0	0
$472$	40.6491	1.87103
$473$	0	0
$474$	0	0
$475$	−31.6228	−1.45095
$476$	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
$481$	−34.6491	−1.57986
$482$	− 35.7771i	− 1.62960i
$483$	0	0
$484$	47.2982	2.14992
$485$	0	0
$486$	0	0
$487$	44.1359	1.99999	0.999995	−	0.00308010i	$-0.000980427\pi$
$487$	44.1359	1.99999	0.999995	+	0.00308010i	$0.000980427\pi$
$488$	0	0
$489$	0	0
$490$	−17.8641	−0.807016
$491$	− 32.7192i	− 1.47660i	−0.674475	−	0.738298i	$-0.735630\pi$
$491$	− 32.7192i	− 1.47660i	0.674475	−	0.738298i	$-0.264370\pi$
$492$	0	0
$493$	0	0
$494$	− 64.0614i	− 2.88226i
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	44.2719	1.98188	0.990941	−	0.134298i	$-0.0428781\pi$
$499$	44.2719	1.98188	0.990941	+	0.134298i	$0.0428781\pi$
$500$	22.3607i	1.00000i
$501$	0	0
$502$	29.6228	1.32213
$503$	− 19.7990i	− 0.882793i	−0.897312	−	0.441397i	$-0.854483\pi$
$503$	− 19.7990i	− 0.882793i	0.897312	−	0.441397i	$-0.145517\pi$
$504$	0	0
$505$	0	0
$506$	− 37.2285i	− 1.65501i
$507$	0	0
$508$	35.6228	1.58051
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	0	0
$515$	− 24.2339i	− 1.06787i
$516$	0	0
$517$	−16.6491	−0.732227
$518$	7.95181i	0.349382i
$519$	0	0
$520$	−45.2982	−1.98646
$521$	− 44.7586i	− 1.96091i	−0.196744	−	0.980455i	$-0.563037\pi$
$521$	− 44.7586i	− 1.96091i	0.196744	−	0.980455i	$-0.436963\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	− 22.1684i	− 0.968432i
$525$	0	0
$526$	44.2719	1.93035
$527$	0	0
$528$	0	0
$529$	3.00000	0.130435
$530$	17.8885i	0.777029i
$531$	0	0
$532$	−14.7018	−0.637403
$533$	53.9324i	2.33607i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	33.2526i	1.43229i
$540$	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
$550$	41.6228	1.77480
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 19.9912i	− 0.849347i
$555$	0	0
$556$	−28.0000	−1.18746
$557$	− 45.2548i	− 1.91751i	−0.284236	−	0.958754i	$-0.591740\pi$
$557$	− 45.2548i	− 1.91751i	0.284236	−	0.958754i	$-0.408260\pi$
$558$	0	0
$559$	0	0
$560$	10.3957i	0.439300i
$561$	0	0
$562$	−3.29822	−0.139127
$563$	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
$569$	41.4712i	1.73856i	0.494318	+	0.869281i	$0.335418\pi$
$569$	41.4712i	1.73856i	−0.494318	+	0.869281i	$0.664582\pi$
$570$	0	0
$571$	44.2719	1.85272	0.926360	−	0.376638i	$-0.122920\pi$
$571$	44.2719	1.85272	0.926360	+	0.376638i	$0.122920\pi$
$572$	84.3193i	3.52557i
$573$	0	0
$574$	12.3772	0.516615
$575$	− 22.3607i	− 0.932505i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	24.0416i	1.00000i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	33.2982	1.37907
$584$	0	0
$585$	0	0
$586$	6.32456	0.261265
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	− 45.4471i	− 1.87103i
$591$	0	0
$592$	−19.3509	−0.795317
$593$	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
$598$	45.2982	1.85238
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	25.2982	1.03194	0.515968	−	0.856608i	$-0.327432\pi$
$601$	25.2982	1.03194	0.515968	+	0.856608i	$0.327432\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 52.8810i	− 2.14992i
$606$	0	0
$607$	−36.7851	−1.49306	−0.746530	−	0.665352i	$-0.768281\pi$
$607$	−36.7851	−1.49306	−0.746530	+	0.665352i	$0.768281\pi$
$608$	− 35.7771i	− 1.45095i
$609$	0	0
$610$	0	0
$611$	− 20.2580i	− 0.819550i
$612$	0	0
$613$	−30.7851	−1.24340	−0.621698	−	0.783257i	$-0.713557\pi$
$613$	−30.7851	−1.24340	−0.621698	+	0.783257i	$0.713557\pi$
$614$	0	0
$615$	0	0
$616$	19.3509	0.779670
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.11023i	− 0.0444806i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	47.6228	1.90036
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	44.2719	1.75826
$635$	− 39.8275i	− 1.58051i
$636$	0	0
$637$	−40.4605	−1.60310
$638$	0	0
$639$	0	0
$640$	−25.2982	−1.00000
$641$	34.3629i	1.35725i	0.734484	+	0.678626i	$0.237424\pi$
$641$	34.3629i	1.35725i	−0.734484	+	0.678626i	$0.762576\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 10.3957i	− 0.409649i
$645$	0	0
$646$	0	0
$647$	31.1127i	1.22317i	0.791180	+	0.611583i	$0.209467\pi$
$647$	31.1127i	1.22317i	−0.791180	+	0.611583i	$0.790533\pi$
$648$	0	0
$649$	−84.5964	−3.32070
$650$	50.6450i	1.98646i
$651$	0	0
$652$	0	0
$653$	4.47214i	0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$	4.47214i	0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$	0	0
$655$	−24.7851	−0.968432
$656$	30.1202i	1.17600i
$657$	0	0
$658$	−4.64911	−0.181241
$659$	− 49.6897i	− 1.93564i	−0.251649	−	0.967818i	$-0.580973\pi$
$659$	− 49.6897i	− 1.93564i	0.251649	−	0.967818i	$-0.419027\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	44.7214i	1.73814i
$663$	0	0
$664$	0	0
$665$	16.4371i	0.637403i
$666$	0	0
$667$	0	0
$668$	− 8.94427i	− 0.346064i
$669$	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
$676$	−76.5964	−2.94602
$677$	− 49.1935i	− 1.89066i	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	− 49.1935i	− 1.89066i	0.326116	−	0.945330i	$-0.394260\pi$
$678$	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
$686$	20.7915i	0.793821i
$687$	0	0
$688$	0	0
$689$	40.5160i	1.54354i
$690$	0	0
$691$	−31.6228	−1.20299	−0.601494	−	0.798878i	$-0.705427\pi$
$691$	−31.6228	−1.20299	−0.601494	+	0.798878i	$0.705427\pi$
$692$	− 45.2548i	− 1.72033i
$693$	0	0
$694$	0	0
$695$	31.3050i	1.18746i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	11.6228	0.439300
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−30.5964	−1.15397
$704$	47.0908i	1.77480i
$705$	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
$712$	2.70178	0.101253
$713$	0	0
$714$	0	0
$715$	94.2719	3.52557
$716$	45.7138i	1.70841i
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−12.5964	−0.469116
$722$	− 29.6985i	− 1.10526i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8377	−1.29206	−0.646030	−	0.763312i	$-0.723572\pi$
$727$	−34.8377	−1.29206	−0.646030	+	0.763312i	$0.723572\pi$
$728$	23.5454i	0.872651i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−18.7851	−0.693842	−0.346921	−	0.937894i	$-0.612773\pi$
$733$	−18.7851	−0.693842	−0.346921	+	0.937894i	$0.612773\pi$
$734$	− 8.75202i	− 0.323043i
$735$	0	0
$736$	25.2982	0.932505
$737$	0	0
$738$	0	0
$739$	6.32456	0.232653	0.116326	−	0.993211i	$-0.462888\pi$
$739$	6.32456	0.232653	0.116326	+	0.993211i	$0.462888\pi$
$740$	21.6350i	0.795317i
$741$	0	0
$742$	9.29822	0.341348
$743$	− 36.7696i	− 1.34894i	−0.738300	−	0.674472i	$-0.764371\pi$
$743$	− 36.7696i	− 1.34894i	0.738300	−	0.674472i	$-0.235629\pi$
$744$	0	0
$745$	0	0
$746$	− 53.9324i	− 1.97460i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 11.3137i	− 0.412568i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.86406	−0.358515	−0.179258	−	0.983802i	$-0.557370\pi$
$757$	−9.86406	−0.358515	−0.179258	+	0.983802i	$0.557370\pi$
$758$	48.0833i	1.74646i
$759$	0	0
$760$	−40.0000	−1.45095
$761$	51.3334i	1.86084i	0.366501	+	0.930418i	$0.380556\pi$
$761$	51.3334i	1.86084i	−0.366501	+	0.930418i	$0.619444\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−52.0000	−1.87884
$767$	− 102.934i	− 3.71672i
$768$	0	0
$769$	−12.6491	−0.456139	−0.228069	−	0.973645i	$-0.573241\pi$
$769$	−12.6491	−0.456139	−0.228069	+	0.973645i	$0.573241\pi$
$770$	− 21.6350i	− 0.779670i
$771$	0	0
$772$	0	0
$773$	− 22.3607i	− 0.804258i	−0.915583	−	0.402129i	$-0.868270\pi$
$773$	− 22.3607i	− 0.804258i	0.915583	−	0.402129i	$-0.131730\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47.6243i	1.70632i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−22.5964	−0.807016
$785$	− 53.2439i	− 1.90036i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	44.7214i	1.59313i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	− 44.0701i	− 1.56399i
$795$	0	0
$796$	0	0
$797$	39.5980i	1.40263i	0.712850	+	0.701316i	$0.247404\pi$
$797$	39.5980i	1.40263i	−0.712850	+	0.701316i	$0.752596\pi$
$798$	0	0
$799$	0	0
$800$	28.2843i	1.00000i
$801$	0	0
$802$	−25.3509	−0.895171
$803$	0	0
$804$	0	0
$805$	−11.6228	−0.409649
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.3923i	0.611481i	0.952115	+	0.305741i	$0.0989040\pi$
$809$	17.3923i	0.611481i	−0.952115	+	0.305741i	$0.901096\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	40.2719	1.41153
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 19.7990i	− 0.692255i
$819$	0	0
$820$	33.6754	1.17600
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	56.1359	1.95678	0.978388	−	0.206778i	$-0.0662976\pi$
$823$	56.1359	1.95678	0.978388	+	0.206778i	$0.0662976\pi$
$824$	− 30.6537i	− 1.06787i
$825$	0	0
$826$	−23.6228	−0.821942
$827$	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
$832$	−57.2982	−1.98646
$833$	0	0
$834$	0	0
$835$	−10.0000	−0.346064
$836$	74.4571i	2.57515i
$837$	0	0
$838$	5.62278	0.194236
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	44.0000	1.51454
$845$	85.6374i	2.94602i
$846$	0	0
$847$	−27.4868	−0.944459
$848$	22.6274i	0.777029i
$849$	0	0
$850$	0	0
$851$	− 21.6350i	− 0.741637i
$852$	0	0
$853$	45.1096	1.54452	0.772262	−	0.635304i	$-0.219125\pi$
$853$	45.1096	1.54452	0.772262	+	0.635304i	$0.219125\pi$
$854$	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
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.1378i	1.97903i	0.144421	+	0.989516i	$0.453868\pi$
$863$	58.1378i	1.97903i	−0.144421	+	0.989516i	$0.546132\pi$
$864$	0	0
$865$	−50.5964	−1.72033
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	40.0000	1.35302
$875$	− 12.9947i	− 0.439300i
$876$	0	0
$877$	33.1096	1.11803	0.559016	−	0.829157i	$-0.311179\pi$
$877$	33.1096	1.11803	0.559016	+	0.829157i	$0.311179\pi$
$878$	0	0
$879$	0	0
$880$	52.6491	1.77480
$881$	49.9565i	1.68308i	0.540198	+	0.841538i	$0.318349\pi$
$881$	49.9565i	1.68308i	−0.540198	+	0.841538i	$0.681651\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.47214i	0.150160i	0.997178	+	0.0750798i	$0.0239212\pi$
$887$	4.47214i	0.150160i	−0.997178	+	0.0750798i	$0.976079\pi$
$888$	0	0
$889$	−20.7018	−0.694315
$890$	− 3.02068i	− 0.101253i
$891$	0	0
$892$	59.6228	1.99632
$893$	− 17.8885i	− 0.598617i
$894$	0	0
$895$	51.1096	1.70841
$896$	13.1497i	0.439300i
$897$	0	0
$898$	8.70178	0.290382
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 62.6844i	− 2.08716i
$903$	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
$910$	26.3246	0.872651
$911$	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
$917$	12.8829i	0.425432i
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 28.2843i	− 0.932505i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	24.1886	0.795317
$926$	59.1302i	1.94314i
$927$	0	0
$928$	0	0
$929$	59.8187i	1.96259i	0.192514	+	0.981294i	$0.438336\pi$
$929$	59.8187i	1.96259i	−0.192514	+	0.981294i	$0.561664\pi$
$930$	0	0
$931$	−35.7281	−1.17094
$932$	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
$940$	−12.6491	−0.412568
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−33.6754	−1.09662
$944$	− 57.4865i	− 1.87103i
$945$	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
$950$	44.7214i	1.45095i
$951$	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
$961$	31.0000	1.00000
$962$	49.0012i	1.57986i
$963$	0	0
$964$	−50.5964	−1.62960
$965$	0	0
$966$	0	0
$967$	−15.8641	−0.510154	−0.255077	−	0.966921i	$-0.582101\pi$
$967$	−15.8641	−0.510154	−0.255077	+	0.966921i	$0.582101\pi$
$968$	− 66.8898i	− 2.14992i
$969$	0	0
$970$	0	0
$971$	− 48.3128i	− 1.55043i	−0.631697	−	0.775215i	$-0.717641\pi$
$971$	− 48.3128i	− 1.55043i	0.631697	−	0.775215i	$-0.282359\pi$
$972$	0	0
$973$	16.2719	0.521653
$974$	− 62.4177i	− 1.99999i
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−5.62278	−0.179705
$980$	25.2636i	0.807016i
$981$	0	0
$982$	−46.2719	−1.47660
$983$	48.0833i	1.53362i	0.641875	+	0.766809i	$0.278157\pi$
$983$	48.0833i	1.53362i	−0.641875	+	0.766809i	$0.721843\pi$
$984$	0	0
$985$	50.0000	1.59313
$986$	0	0
$987$	0	0
$988$	−90.5964	−2.88226
$989$	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
$997$	−6.78505	−0.214885	−0.107442	−	0.994211i	$-0.534266\pi$
$997$	−6.78505	−0.214885	−0.107442	+	0.994211i	$0.534266\pi$
$998$	− 62.6099i	− 1.98188i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.m.a.179.2	✓	4
3.2	odd	2	inner	360.2.m.a.179.3	yes	4
4.3	odd	2		1440.2.m.b.719.3		4
5.2	odd	4		1800.2.b.f.251.7		8
5.3	odd	4		1800.2.b.f.251.2		8
5.4	even	2		360.2.m.b.179.3	yes	4
8.3	odd	2		360.2.m.b.179.3	yes	4
8.5	even	2		1440.2.m.a.719.2		4
12.11	even	2		1440.2.m.b.719.1		4
15.2	even	4		1800.2.b.f.251.3		8
15.8	even	4		1800.2.b.f.251.6		8
15.14	odd	2		360.2.m.b.179.2	yes	4
20.3	even	4		7200.2.b.f.4751.6		8
20.7	even	4		7200.2.b.f.4751.4		8
20.19	odd	2		1440.2.m.a.719.2		4
24.5	odd	2		1440.2.m.a.719.4		4
24.11	even	2		360.2.m.b.179.2	yes	4
40.3	even	4		1800.2.b.f.251.7		8
40.13	odd	4		7200.2.b.f.4751.4		8
40.19	odd	2	CM	360.2.m.a.179.2	✓	4
40.27	even	4		1800.2.b.f.251.2		8
40.29	even	2		1440.2.m.b.719.3		4
40.37	odd	4		7200.2.b.f.4751.6		8
60.23	odd	4		7200.2.b.f.4751.5		8
60.47	odd	4		7200.2.b.f.4751.3		8
60.59	even	2		1440.2.m.a.719.4		4
120.29	odd	2		1440.2.m.b.719.1		4
120.53	even	4		7200.2.b.f.4751.3		8
120.59	even	2	inner	360.2.m.a.179.3	yes	4
120.77	even	4		7200.2.b.f.4751.5		8
120.83	odd	4		1800.2.b.f.251.3		8
120.107	odd	4		1800.2.b.f.251.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.a.179.2	✓	4	1.1	even	1	trivial
360.2.m.a.179.2	✓	4	40.19	odd	2	CM
360.2.m.a.179.3	yes	4	3.2	odd	2	inner
360.2.m.a.179.3	yes	4	120.59	even	2	inner
360.2.m.b.179.2	yes	4	15.14	odd	2
360.2.m.b.179.2	yes	4	24.11	even	2
360.2.m.b.179.3	yes	4	5.4	even	2
360.2.m.b.179.3	yes	4	8.3	odd	2
1440.2.m.a.719.2		4	8.5	even	2
1440.2.m.a.719.2		4	20.19	odd	2
1440.2.m.a.719.4		4	24.5	odd	2
1440.2.m.a.719.4		4	60.59	even	2
1440.2.m.b.719.1		4	12.11	even	2
1440.2.m.b.719.1		4	120.29	odd	2
1440.2.m.b.719.3		4	4.3	odd	2
1440.2.m.b.719.3		4	40.29	even	2
1800.2.b.f.251.2		8	5.3	odd	4
1800.2.b.f.251.2		8	40.27	even	4
1800.2.b.f.251.3		8	15.2	even	4
1800.2.b.f.251.3		8	120.83	odd	4
1800.2.b.f.251.6		8	15.8	even	4
1800.2.b.f.251.6		8	120.107	odd	4
1800.2.b.f.251.7		8	5.2	odd	4
1800.2.b.f.251.7		8	40.3	even	4
7200.2.b.f.4751.3		8	60.47	odd	4
7200.2.b.f.4751.3		8	120.53	even	4
7200.2.b.f.4751.4		8	20.7	even	4
7200.2.b.f.4751.4		8	40.13	odd	4
7200.2.b.f.4751.5		8	60.23	odd	4
7200.2.b.f.4751.5		8	120.77	even	4
7200.2.b.f.4751.6		8	20.3	even	4
7200.2.b.f.4751.6		8	40.37	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.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +1.16228 q^{7} +2.82843i q^{8} +3.16228 q^{10} -5.88635i q^{11} +7.16228 q^{13} -1.64371i q^{14} +4.00000 q^{16} +6.32456 q^{19} -4.47214i q^{20} -8.32456 q^{22} +4.47214i q^{23} -5.00000 q^{25} -10.1290i q^{26} -2.32456 q^{28} -5.65685i q^{32} +2.59893i q^{35} -4.83772 q^{37} -8.94427i q^{38} -6.32456 q^{40} +7.53006i q^{41} +11.7727i q^{44} +6.32456 q^{46} -2.82843i q^{47} -5.64911 q^{49} +7.07107i q^{50} -14.3246 q^{52} +5.65685i q^{53} +13.1623 q^{55} +3.28742i q^{56} -14.3716i q^{59} -8.00000 q^{64} +16.0153i q^{65} +3.67544 q^{70} +6.84157i q^{74} -12.6491 q^{76} -6.84157i q^{77} +8.94427i q^{80} +10.6491 q^{82} +16.6491 q^{88} -0.955223i q^{89} +8.32456 q^{91} -8.94427i q^{92} -4.00000 q^{94} +14.1421i q^{95} +7.98905i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 8 q^{7} + 16 q^{13} + 16 q^{16} - 8 q^{22} - 20 q^{25} + 16 q^{28} - 32 q^{37} + 28 q^{49} - 32 q^{52} + 40 q^{55} - 32 q^{64} + 40 q^{70} - 8 q^{82} + 16 q^{88} + 8 q^{91} - 16 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 8 * q^7 + 16 * q^13 + 16 * q^16 - 8 * q^22 - 20 * q^25 + 16 * q^28 - 32 * q^37 + 28 * q^49 - 32 * q^52 + 40 * q^55 - 32 * q^64 + 40 * q^70 - 8 * q^82 + 16 * q^88 + 8 * q^91 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more