LMFDB - Embedded newform 360.2.d.c.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("360.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.87461447277$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.4
Root			$0.809017 + 1.40126i$ of defining polynomial
Character	$\chi$	$=$	360.109
Dual form			360.2.d.c.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11803 + 0.866025i) q^{2} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(-1.11803 + 2.59808i) q^{8} +O(q^{10})$	$q+(1.11803 + 0.866025i) q^{2} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(-1.11803 + 2.59808i) q^{8} +(2.50000 + 1.93649i) q^{10} +(-3.50000 + 1.93649i) q^{16} -6.92820i q^{17} +7.74597i q^{19} +(1.11803 + 4.33013i) q^{20} -3.46410i q^{23} +5.00000 q^{25} -8.00000 q^{31} +(-5.59017 - 0.866025i) q^{32} +(6.00000 - 7.74597i) q^{34} +(-6.70820 + 8.66025i) q^{38} +(-2.50000 + 5.80948i) q^{40} +(3.00000 - 3.87298i) q^{46} -10.3923i q^{47} +7.00000 q^{49} +(5.59017 + 4.33013i) q^{50} -4.47214 q^{53} -15.4919i q^{61} +(-8.94427 - 6.92820i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(13.4164 - 3.46410i) q^{68} +(-15.0000 + 3.87298i) q^{76} -16.0000 q^{79} +(-7.82624 + 4.33013i) q^{80} +17.8885 q^{83} -15.4919i q^{85} +(6.70820 - 1.73205i) q^{92} +(9.00000 - 11.6190i) q^{94} +17.3205i q^{95} +(7.82624 + 6.06218i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} + 10 q^{10} - 14 q^{16} + 20 q^{25} - 32 q^{31} + 24 q^{34} - 10 q^{40} + 12 q^{46} + 28 q^{49} - 22 q^{64} - 60 q^{76} - 64 q^{79} + 36 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 + 10 * q^10 - 14 * q^16 + 20 * q^25 - 32 * q^31 + 24 * q^34 - 10 * q^40 + 12 * q^46 + 28 * q^49 - 22 * q^64 - 60 * q^76 - 64 * q^79 + 36 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.11803	+	0.866025i	0.790569	+	0.612372i
$2$	1.11803	+	0.866025i	0.790569	+	0.612372i
$3$		0			0
$3$		0			0
$4$	0.500000	+	1.93649i	0.250000	+	0.968246i
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−1.11803	+	2.59808i	−0.395285	+	0.918559i
$9$		0			0
$10$	2.50000	+	1.93649i	0.790569	+	0.612372i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$		−	6.92820i		−	1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$		−	6.92820i		−	1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$		0			0
$19$			7.74597i			1.77705i	0.458831	+	0.888523i	$0.348268\pi$
$19$			7.74597i			1.77705i	−0.458831	+	0.888523i	$0.651732\pi$
$20$	1.11803	+	4.33013i	0.250000	+	0.968246i
$21$		0			0
$22$		0			0
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.59017	−	0.866025i	−0.988212	−	0.153093i
$33$		0			0
$34$	6.00000	−	7.74597i	1.02899	−	1.32842i
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−6.70820	+	8.66025i	−1.08821	+	1.40488i
$39$		0			0
$40$	−2.50000	+	5.80948i	−0.395285	+	0.918559i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	3.00000	−	3.87298i	0.442326	−	0.571040i
$47$		−	10.3923i		−	1.51587i	−0.652328	−	0.757937i	$-0.726208\pi$
$47$		−	10.3923i		−	1.51587i	0.652328	−	0.757937i	$-0.273792\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	5.59017	+	4.33013i	0.790569	+	0.612372i
$51$		0			0
$52$		0			0
$53$	−4.47214			−0.614295			−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214			−0.614295			−0.307148	+	0.951662i	$0.599375\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	15.4919i		−	1.98354i	−0.128037	−	0.991769i	$-0.540868\pi$
$61$		−	15.4919i		−	1.98354i	0.128037	−	0.991769i	$-0.459132\pi$
$62$	−8.94427	−	6.92820i	−1.13592	−	0.879883i
$63$		0			0
$64$	−5.50000	−	5.80948i	−0.687500	−	0.726184i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	13.4164	−	3.46410i	1.62698	−	0.420084i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$	−15.0000	+	3.87298i	−1.72062	+	0.444262i
$77$		0			0
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$	−7.82624	+	4.33013i	−0.875000	+	0.484123i
$81$		0			0
$82$		0			0
$83$	17.8885			1.96352			0.981761	−	0.190117i	$-0.0608868\pi$
$83$	17.8885			1.96352			0.981761	+	0.190117i	$0.0608868\pi$
$84$		0			0
$85$		−	15.4919i		−	1.68034i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	6.70820	−	1.73205i	0.699379	−	0.180579i
$93$		0			0
$94$	9.00000	−	11.6190i	0.928279	−	1.19840i
$95$			17.3205i			1.77705i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	7.82624	+	6.06218i	0.790569	+	0.612372i
$99$		0			0
$100$	2.50000	+	9.68246i	0.250000	+	0.968246i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−5.00000	−	3.87298i	−0.485643	−	0.376177i
$107$	−17.8885			−1.72935			−0.864675	−	0.502331i	$-0.832476\pi$
$107$	−17.8885			−1.72935			−0.864675	+	0.502331i	$0.832476\pi$
$108$		0			0
$109$			15.4919i			1.48386i	0.670478	+	0.741929i	$0.266089\pi$
$109$			15.4919i			1.48386i	−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			20.7846i			1.95525i	0.210352	+	0.977626i	$0.432539\pi$
$113$			20.7846i			1.95525i	−0.210352	+	0.977626i	$0.567461\pi$
$114$		0			0
$115$		−	7.74597i		−	0.722315i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	13.4164	−	17.3205i	1.21466	−	1.56813i
$123$		0			0
$124$	−4.00000	−	15.4919i	−0.359211	−	1.39122i
$125$	11.1803			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−1.11803	−	11.2583i	−0.0988212	−	0.995105i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	18.0000	+	7.74597i	1.54349	+	0.664211i
$137$		−	6.92820i		−	0.591916i	−0.955201	−	0.295958i	$-0.904361\pi$
$137$		−	6.92820i		−	0.591916i	0.955201	−	0.295958i	$-0.0956389\pi$
$138$		0			0
$139$			23.2379i			1.97101i	0.169638	+	0.985506i	$0.445740\pi$
$139$			23.2379i			1.97101i	−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$	−20.1246	−	8.66025i	−1.63232	−	0.702439i
$153$		0			0
$154$		0			0
$155$	−17.8885			−1.43684
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−17.8885	−	13.8564i	−1.42314	−	1.10236i
$159$		0			0
$160$	−12.5000	−	1.93649i	−0.988212	−	0.153093i
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	20.0000	+	15.4919i	1.55230	+	1.20241i
$167$			24.2487i			1.87642i	0.346064	+	0.938211i	$0.387518\pi$
$167$			24.2487i			1.87642i	−0.346064	+	0.938211i	$0.612482\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$	13.4164	−	17.3205i	1.02899	−	1.32842i
$171$		0			0
$172$		0			0
$173$	−22.3607			−1.70005			−0.850026	−	0.526742i	$-0.823414\pi$
$173$	−22.3607			−1.70005			−0.850026	+	0.526742i	$0.823414\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			15.4919i			1.15151i	0.817624	+	0.575753i	$0.195291\pi$
$181$			15.4919i			1.15151i	−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		0			0
$184$	9.00000	+	3.87298i	0.663489	+	0.285520i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	20.1246	−	5.19615i	1.46774	−	0.378968i
$189$		0			0
$190$	−15.0000	+	19.3649i	−1.08821	+	1.40488i
$191$		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
$196$	3.50000	+	13.5554i	0.250000	+	0.968246i
$197$	4.47214			0.318626			0.159313	−	0.987228i	$-0.449072\pi$
$197$	4.47214			0.318626			0.159313	+	0.987228i	$0.449072\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$	−5.59017	+	12.9904i	−0.395285	+	0.918559i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$	−2.23607	−	8.66025i	−0.153574	−	0.594789i
$213$		0			0
$214$	−20.0000	−	15.4919i	−1.36717	−	1.05901i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−13.4164	+	17.3205i	−0.908674	+	1.17309i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−18.0000	+	23.2379i	−1.19734	+	1.54576i
$227$	17.8885			1.18730			0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885			1.18730			0.593652	+	0.804722i	$0.297686\pi$
$228$		0			0
$229$			15.4919i			1.02374i	0.859064	+	0.511868i	$0.171046\pi$
$229$			15.4919i			1.02374i	−0.859064	+	0.511868i	$0.828954\pi$
$230$	6.70820	−	8.66025i	0.442326	−	0.571040i
$231$		0			0
$232$		0			0
$233$			20.7846i			1.36165i	0.732448	+	0.680823i	$0.238378\pi$
$233$			20.7846i			1.36165i	−0.732448	+	0.680823i	$0.761622\pi$
$234$		0			0
$235$		−	23.2379i		−	1.51587i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$	12.2984	+	9.52628i	0.790569	+	0.612372i
$243$		0			0
$244$	30.0000	−	7.74597i	1.92055	−	0.495885i
$245$	15.6525			1.00000
$246$		0			0
$247$		0			0
$248$	8.94427	−	20.7846i	0.567962	−	1.31982i
$249$		0			0
$250$	12.5000	+	9.68246i	0.790569	+	0.612372i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$			6.92820i			0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$			6.92820i			0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	31.1769i		−	1.92245i	−0.275764	−	0.961225i	$-0.588931\pi$
$263$		−	31.1769i		−	1.92245i	0.275764	−	0.961225i	$-0.411069\pi$
$264$		0			0
$265$	−10.0000			−0.614295
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	13.4164	+	24.2487i	0.813489	+	1.47029i
$273$		0			0
$274$	6.00000	−	7.74597i	0.362473	−	0.467951i
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−20.1246	+	25.9808i	−1.20699	+	1.55822i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−31.0000			−1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−31.3050			−1.82885			−0.914427	−	0.404750i	$-0.867359\pi$
$293$	−31.3050			−1.82885			−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	8.94427	+	6.92820i	0.514685	+	0.398673i
$303$		0			0
$304$	−15.0000	−	27.1109i	−0.860309	−	1.55492i
$305$		−	34.6410i		−	1.98354i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	−20.0000	−	15.4919i	−1.13592	−	0.879883i
$311$		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			$\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000	−	30.9839i	−0.450035	−	1.74298i
$317$	22.3607			1.25590			0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607			1.25590			0.627950	+	0.778253i	$0.283894\pi$
$318$		0			0
$319$		0			0
$320$	−12.2984	−	12.9904i	−0.687500	−	0.726184i
$321$		0			0
$322$		0			0
$323$	53.6656			2.98604
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	23.2379i		−	1.27727i	−0.769510	−	0.638635i	$-0.779499\pi$
$331$		−	23.2379i		−	1.27727i	0.769510	−	0.638635i	$-0.220501\pi$
$332$	8.94427	+	34.6410i	0.490881	+	1.90117i
$333$		0			0
$334$	−21.0000	+	27.1109i	−1.14907	+	1.48344i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−14.5344	−	11.2583i	−0.790569	−	0.612372i
$339$		0			0
$340$	30.0000	−	7.74597i	1.62698	−	0.420084i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−25.0000	−	19.3649i	−1.34401	−	1.04106i
$347$	−35.7771			−1.92061			−0.960307	−	0.278944i	$-0.910016\pi$
$347$	−35.7771			−1.92061			−0.960307	+	0.278944i	$0.910016\pi$
$348$		0			0
$349$			15.4919i			0.829264i	0.909989	+	0.414632i	$0.136090\pi$
$349$			15.4919i			0.829264i	−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.7846i		−	1.10625i	−0.833097	−	0.553127i	$-0.813435\pi$
$353$		−	20.7846i		−	1.10625i	0.833097	−	0.553127i	$-0.186565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−41.0000			−2.15789
$362$	−13.4164	+	17.3205i	−0.705151	+	0.910346i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	6.70820	+	12.1244i	0.349689	+	0.632026i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	27.0000	+	11.6190i	1.39242	+	0.599202i
$377$		0			0
$378$		0			0
$379$		−	38.7298i		−	1.98942i	−0.102733	−	0.994709i	$-0.532759\pi$
$379$		−	38.7298i		−	1.98942i	0.102733	−	0.994709i	$-0.467241\pi$
$380$	−33.5410	+	8.66025i	−1.72062	+	0.444262i
$381$		0			0
$382$		0			0
$383$		−	38.1051i		−	1.94708i	−0.228515	−	0.973540i	$-0.573387\pi$
$383$		−	38.1051i		−	1.94708i	0.228515	−	0.973540i	$-0.426613\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$	−7.82624	+	18.1865i	−0.395285	+	0.918559i
$393$		0			0
$394$	5.00000	+	3.87298i	0.251896	+	0.195118i
$395$	−35.7771			−1.80014
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	17.8885	+	13.8564i	0.896672	+	0.694559i
$399$		0			0
$400$	−17.5000	+	9.68246i	−0.875000	+	0.484123i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	40.0000			1.96352
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	15.4919i		−	0.755031i	−0.926003	−	0.377515i	$-0.876779\pi$
$421$		−	15.4919i		−	0.755031i	0.926003	−	0.377515i	$-0.123221\pi$
$422$	6.70820	−	8.66025i	0.326550	−	0.421575i
$423$		0			0
$424$	5.00000	−	11.6190i	0.242821	−	0.564266i
$425$		−	34.6410i		−	1.68034i
$426$		0			0
$427$		0			0
$428$	−8.94427	−	34.6410i	−0.432338	−	1.67444i
$429$		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
$436$	−30.0000	+	7.74597i	−1.43674	+	0.370965i
$437$	26.8328			1.28359
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.8885			−0.849910			−0.424955	−	0.905214i	$-0.639710\pi$
$443$	−17.8885			−0.849910			−0.424955	+	0.905214i	$0.639710\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−40.2492	+	10.3923i	−1.89316	+	0.488813i
$453$		0			0
$454$	20.0000	+	15.4919i	0.938647	+	0.727072i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−13.4164	+	17.3205i	−0.626908	+	0.809334i
$459$		0			0
$460$	15.0000	−	3.87298i	0.699379	−	0.180579i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−18.0000	+	23.2379i	−0.833834	+	1.07647i
$467$	35.7771			1.65557			0.827783	−	0.561048i	$-0.189602\pi$
$467$	35.7771			1.65557			0.827783	+	0.561048i	$0.189602\pi$
$468$		0			0
$469$		0			0
$470$	20.1246	−	25.9808i	0.928279	−	1.19840i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			38.7298i			1.77705i
$476$		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
$481$		0			0
$482$	−2.23607	−	1.73205i	−0.101850	−	0.0788928i
$483$		0			0
$484$	5.50000	+	21.3014i	0.250000	+	0.968246i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	40.2492	+	17.3205i	1.82200	+	0.784063i
$489$		0			0
$490$	17.5000	+	13.5554i	0.790569	+	0.612372i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	28.0000	−	15.4919i	1.25724	−	0.695608i
$497$		0			0
$498$		0			0
$499$			7.74597i			0.346757i	0.984855	+	0.173379i	$0.0554684\pi$
$499$			7.74597i			0.346757i	−0.984855	+	0.173379i	$0.944532\pi$
$500$	5.59017	+	21.6506i	0.250000	+	0.968246i
$501$		0			0
$502$		0			0
$503$			3.46410i			0.154457i	0.997013	+	0.0772283i	$0.0246070\pi$
$503$			3.46410i			0.154457i	−0.997013	+	0.0772283i	$0.975393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	21.2426	−	7.79423i	0.938801	−	0.344459i
$513$		0			0
$514$	−6.00000	+	7.74597i	−0.264649	+	0.341660i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	27.0000	−	34.8569i	1.17726	−	1.51983i
$527$			55.4256i			2.41438i
$528$		0			0
$529$	11.0000			0.478261
$530$	−11.1803	−	8.66025i	−0.485643	−	0.376177i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−40.0000			−1.72935
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$			46.4758i			1.99815i	0.0429934	+	0.999075i	$0.486311\pi$
$541$			46.4758i			1.99815i	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	35.7771	+	27.7128i	1.53676	+	1.19037i
$543$		0			0
$544$	−6.00000	+	38.7298i	−0.257248	+	1.66053i
$545$			34.6410i			1.48386i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	13.4164	−	3.46410i	0.573121	−	0.147979i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−45.0000	+	11.6190i	−1.90843	+	0.492753i
$557$	−22.3607			−0.947452			−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607			−0.947452			−0.473726	+	0.880672i	$0.657091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	35.7771			1.50782			0.753912	−	0.656975i	$-0.228164\pi$
$563$	35.7771			1.50782			0.753912	+	0.656975i	$0.228164\pi$
$564$		0			0
$565$			46.4758i			1.95525i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			38.7298i			1.62079i	0.585882	+	0.810397i	$0.300748\pi$
$571$			38.7298i			1.62079i	−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17.3205i		−	0.722315i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−34.6591	−	26.8468i	−1.44163	−	1.11668i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−35.0000	−	27.1109i	−1.44584	−	1.11994i
$587$	−17.8885			−0.738339			−0.369170	−	0.929362i	$-0.620358\pi$
$587$	−17.8885			−0.738339			−0.369170	+	0.929362i	$0.620358\pi$
$588$		0			0
$589$		−	61.9677i		−	2.55334i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			48.4974i			1.99155i	0.0918243	+	0.995775i	$0.470730\pi$
$593$			48.4974i			1.99155i	−0.0918243	+	0.995775i	$0.529270\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$	4.00000	+	15.4919i	0.162758	+	0.630358i
$605$	24.5967			1.00000
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	6.70820	−	43.3013i	0.272054	−	1.75610i
$609$		0			0
$610$	30.0000	−	38.7298i	1.21466	−	1.56813i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.92820i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$617$			6.92820i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$618$		0			0
$619$			23.2379i			0.934010i	0.884255	+	0.467005i	$0.154667\pi$
$619$			23.2379i			0.934010i	−0.884255	+	0.467005i	$0.845333\pi$
$620$	−8.94427	−	34.6410i	−0.359211	−	1.39122i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$	17.8885	−	41.5692i	0.711568	−	1.65353i
$633$		0			0
$634$	25.0000	+	19.3649i	0.992877	+	0.769079i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−2.50000	−	25.1744i	−0.0988212	−	0.995105i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	60.0000	+	46.4758i	2.36067	+	1.82857i
$647$		−	24.2487i		−	0.953315i	−0.879089	−	0.476658i	$-0.841848\pi$
$647$		−	24.2487i		−	0.953315i	0.879089	−	0.476658i	$-0.158152\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−49.1935			−1.92509			−0.962545	−	0.271122i	$-0.912605\pi$
$653$	−49.1935			−1.92509			−0.962545	+	0.271122i	$0.912605\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	46.4758i		−	1.80770i	−0.427850	−	0.903850i	$-0.640729\pi$
$661$		−	46.4758i		−	1.80770i	0.427850	−	0.903850i	$-0.359271\pi$
$662$	20.1246	−	25.9808i	0.782165	−	1.00977i
$663$		0			0
$664$	−20.0000	+	46.4758i	−0.776151	+	1.80361i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−46.9574	+	12.1244i	−1.81684	+	0.469105i
$669$		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
$676$	−6.50000	−	25.1744i	−0.250000	−	0.968246i
$677$	−31.3050			−1.20315			−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050			−1.20315			−0.601574	+	0.798817i	$0.705459\pi$
$678$		0			0
$679$		0			0
$680$	40.2492	+	17.3205i	1.54349	+	0.664211i
$681$		0			0
$682$		0			0
$683$	−35.7771			−1.36897			−0.684486	−	0.729026i	$-0.739973\pi$
$683$	−35.7771			−1.36897			−0.684486	+	0.729026i	$0.739973\pi$
$684$		0			0
$685$		−	15.4919i		−	0.591916i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			7.74597i			0.294670i	0.989087	+	0.147335i	$0.0470696\pi$
$691$			7.74597i			0.294670i	−0.989087	+	0.147335i	$0.952930\pi$
$692$	−11.1803	−	43.3013i	−0.425013	−	1.64607i
$693$		0			0
$694$	−40.0000	−	30.9839i	−1.51838	−	1.17613i
$695$			51.9615i			1.97101i
$696$		0			0
$697$		0			0
$698$	−13.4164	+	17.3205i	−0.507819	+	0.655591i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	18.0000	−	23.2379i	0.677439	−	0.874570i
$707$		0			0
$708$		0			0
$709$		−	46.4758i		−	1.74544i	−0.488225	−	0.872718i	$-0.662356\pi$
$709$		−	46.4758i		−	1.74544i	0.488225	−	0.872718i	$-0.337644\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			27.7128i			1.03785i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−45.8394	−	35.5070i	−1.70597	−	1.32144i
$723$		0			0
$724$	−30.0000	+	7.74597i	−1.11494	+	0.287877i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	−3.00000	+	19.3649i	−0.110581	+	0.713800i
$737$		0			0
$738$		0			0
$739$		−	54.2218i		−	1.99458i	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$		−	54.2218i		−	1.99458i	0.0735712	−	0.997290i	$-0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.1769i			1.14377i	0.820334	+	0.571885i	$0.193788\pi$
$743$			31.1769i			1.14377i	−0.820334	+	0.571885i	$0.806212\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$	20.1246	+	36.3731i	0.733869	+	1.32639i
$753$		0			0
$754$		0			0
$755$	17.8885			0.651031
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	33.5410	−	43.3013i	1.21826	−	1.57277i
$759$		0			0
$760$	−45.0000	−	19.3649i	−1.63232	−	0.702439i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	33.0000	−	42.6028i	1.19234	−	1.53930i
$767$		0			0
$768$		0			0
$769$	46.0000			1.65880			0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000			1.65880			0.829401	+	0.558653i	$0.188682\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.47214			0.160852			0.0804258	−	0.996761i	$-0.474372\pi$
$773$	4.47214			0.160852			0.0804258	+	0.996761i	$0.474372\pi$
$774$		0			0
$775$	−40.0000			−1.43684
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−26.8328	−	20.7846i	−0.959540	−	0.743256i
$783$		0			0
$784$	−24.5000	+	13.5554i	−0.875000	+	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	2.23607	+	8.66025i	0.0796566	+	0.308509i
$789$		0			0
$790$	−40.0000	−	30.9839i	−1.42314	−	1.10236i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	8.00000	+	30.9839i	0.283552	+	1.09819i
$797$	49.1935			1.74252			0.871262	−	0.490819i	$-0.163302\pi$
$797$	49.1935			1.74252			0.871262	+	0.490819i	$0.163302\pi$
$798$		0			0
$799$	−72.0000			−2.54718
$800$	−27.9508	−	4.33013i	−0.988212	−	0.153093i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	29.0689	+	22.5167i	1.01637	+	0.787277i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.7771			−1.24409			−0.622046	−	0.782981i	$-0.713698\pi$
$827$	−35.7771			−1.24409			−0.622046	+	0.782981i	$0.713698\pi$
$828$		0			0
$829$		−	46.4758i		−	1.61417i	−0.590434	−	0.807086i	$-0.701044\pi$
$829$		−	46.4758i		−	1.61417i	0.590434	−	0.807086i	$-0.298956\pi$
$830$	44.7214	+	34.6410i	1.55230	+	1.20241i
$831$		0			0
$832$		0			0
$833$		−	48.4974i		−	1.68034i
$834$		0			0
$835$			54.2218i			1.87642i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	13.4164	−	17.3205i	0.462360	−	0.596904i
$843$		0			0
$844$	15.0000	−	3.87298i	0.516321	−	0.133314i
$845$	−29.0689			−1.00000
$846$		0			0
$847$		0			0
$848$	15.6525	−	8.66025i	0.537508	−	0.297394i
$849$		0			0
$850$	30.0000	−	38.7298i	1.02899	−	1.32842i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	20.0000	−	46.4758i	0.683586	−	1.58851i
$857$			6.92820i			0.236663i	0.992974	+	0.118331i	$0.0377545\pi$
$857$			6.92820i			0.236663i	−0.992974	+	0.118331i	$0.962245\pi$
$858$		0			0
$859$		−	38.7298i		−	1.32144i	−0.750630	−	0.660722i	$-0.770250\pi$
$859$		−	38.7298i		−	1.32144i	0.750630	−	0.660722i	$-0.229750\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.1051i			1.29711i	0.761166	+	0.648557i	$0.224627\pi$
$863$			38.1051i			1.29711i	−0.761166	+	0.648557i	$0.775373\pi$
$864$		0			0
$865$	−50.0000			−1.70005
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−40.2492	−	17.3205i	−1.36301	−	0.586546i
$873$		0			0
$874$	30.0000	+	23.2379i	1.01477	+	0.786034i
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	17.8885	+	13.8564i	0.603709	+	0.467631i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−20.0000	−	15.4919i	−0.671913	−	0.520462i
$887$		−	58.8897i		−	1.97732i	−0.150160	−	0.988662i	$-0.547979\pi$
$887$		−	58.8897i		−	1.97732i	0.150160	−	0.988662i	$-0.452021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	80.4984			2.69378
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			30.9839i			1.03222i
$902$		0			0
$903$		0			0
$904$	−54.0000	−	23.2379i	−1.79601	−	0.772881i
$905$			34.6410i			1.15151i
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	8.94427	+	34.6410i	0.296826	+	1.14960i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−30.0000	+	7.74597i	−0.991228	+	0.255934i
$917$		0			0
$918$		0			0
$919$	56.0000			1.84727			0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000			1.84727			0.923635	+	0.383274i	$0.125203\pi$
$920$	20.1246	+	8.66025i	0.663489	+	0.285520i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$			54.2218i			1.77705i
$932$	−40.2492	+	10.3923i	−1.31841	+	0.340411i
$933$		0			0
$934$	40.0000	+	30.9839i	1.30884	+	1.01382i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	45.0000	−	11.6190i	1.46774	−	0.378968i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.8885			0.581300			0.290650	−	0.956830i	$-0.406129\pi$
$947$	17.8885			0.581300			0.290650	+	0.956830i	$0.406129\pi$
$948$		0			0
$949$		0			0
$950$	−33.5410	+	43.3013i	−1.08821	+	1.40488i
$951$		0			0
$952$		0			0
$953$		−	20.7846i		−	0.673280i	−0.941634	−	0.336640i	$-0.890710\pi$
$953$		−	20.7846i		−	0.673280i	0.941634	−	0.336640i	$-0.109290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		0			0
$964$	−1.00000	−	3.87298i	−0.0322078	−	0.124740i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−12.2984	+	28.5788i	−0.395285	+	0.918559i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	30.0000	+	54.2218i	0.960277	+	1.73560i
$977$		−	62.3538i		−	1.99488i	−0.0715382	−	0.997438i	$-0.522791\pi$
$977$		−	62.3538i		−	1.99488i	0.0715382	−	0.997438i	$-0.477209\pi$
$978$		0			0
$979$		0			0
$980$	7.82624	+	30.3109i	0.250000	+	0.968246i
$981$		0			0
$982$		0			0
$983$			3.46410i			0.110488i	0.998473	+	0.0552438i	$0.0175936\pi$
$983$			3.46410i			0.110488i	−0.998473	+	0.0552438i	$0.982406\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	44.7214	+	6.92820i	1.41990	+	0.219971i
$993$		0			0
$994$		0			0
$995$	35.7771			1.13421
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−6.70820	+	8.66025i	−0.212344	+	0.274136i
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.11803 + 0.866025i) q^{2} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(-1.11803 + 2.59808i) q^{8} +O(q^{10})\)	\(q+(1.11803 + 0.866025i) q^{2} +(0.500000 + 1.93649i) q^{4} +2.23607 q^{5} +(-1.11803 + 2.59808i) q^{8} +(2.50000 + 1.93649i) q^{10} +(-3.50000 + 1.93649i) q^{16} -6.92820i q^{17} +7.74597i q^{19} +(1.11803 + 4.33013i) q^{20} -3.46410i q^{23} +5.00000 q^{25} -8.00000 q^{31} +(-5.59017 - 0.866025i) q^{32} +(6.00000 - 7.74597i) q^{34} +(-6.70820 + 8.66025i) q^{38} +(-2.50000 + 5.80948i) q^{40} +(3.00000 - 3.87298i) q^{46} -10.3923i q^{47} +7.00000 q^{49} +(5.59017 + 4.33013i) q^{50} -4.47214 q^{53} -15.4919i q^{61} +(-8.94427 - 6.92820i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(13.4164 - 3.46410i) q^{68} +(-15.0000 + 3.87298i) q^{76} -16.0000 q^{79} +(-7.82624 + 4.33013i) q^{80} +17.8885 q^{83} -15.4919i q^{85} +(6.70820 - 1.73205i) q^{92} +(9.00000 - 11.6190i) q^{94} +17.3205i q^{95} +(7.82624 + 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} + 10 q^{10} - 14 q^{16} + 20 q^{25} - 32 q^{31} + 24 q^{34} - 10 q^{40} + 12 q^{46} + 28 q^{49} - 22 q^{64} - 60 q^{76} - 64 q^{79} + 36 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 + 10 * q^10 - 14 * q^16 + 20 * q^25 - 32 * q^31 + 24 * q^34 - 10 * q^40 + 12 * q^46 + 28 * q^49 - 22 * q^64 - 60 * q^76 - 64 * q^79 + 36 * q^94

Introduction

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