LMFDB - Embedded newform 360.2.f.b.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,Mod(289,360)]

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, 0, 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("360.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	360.f (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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.2.f.b.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +O(q^{10})$	$q+(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +4.00000 q^{11} -4.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{29} +(4.00000 + 8.00000i) q^{35} +4.00000i q^{37} -8.00000 q^{41} -12.0000i q^{47} -9.00000 q^{49} +2.00000i q^{53} +(-8.00000 + 4.00000i) q^{55} +12.0000 q^{59} +2.00000 q^{61} +(4.00000 + 8.00000i) q^{65} +8.00000i q^{67} -8.00000 q^{71} +16.0000i q^{73} -16.0000i q^{77} +8.00000 q^{79} +8.00000i q^{83} +(6.00000 + 12.0000i) q^{85} -16.0000 q^{91} +(-8.00000 + 4.00000i) q^{95} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} + 8 q^{11} + 8 q^{19} + 6 q^{25} - 8 q^{29} + 8 q^{35} - 16 q^{41} - 18 q^{49} - 16 q^{55} + 24 q^{59} + 4 q^{61} + 8 q^{65} - 16 q^{71} + 16 q^{79} + 12 q^{85} - 32 q^{91} - 16 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 + 8 * q^11 + 8 * q^19 + 6 * q^25 - 8 * q^29 + 8 * q^35 - 16 * q^41 - 18 * q^49 - 16 * q^55 + 24 * q^59 + 4 * q^61 + 8 * q^65 - 16 * q^71 + 16 * q^79 + 12 * q^85 - 32 * q^91 - 16 * q^95

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000	+	8.00000i	0.676123	+	1.35225i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$	−8.00000	+	4.00000i	−1.07872	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	16.0000i		−	1.82337i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$	6.00000	+	12.0000i	0.650791	+	1.30158i
$86$		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
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	+	4.00000i	−0.820783	+	0.410391i
$96$		0			0
$97$		−	8.00000i		−	0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$		−	8.00000i		−	0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$	−4.00000	−	8.00000i	−0.373002	−	0.746004i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−24.0000			−2.20008
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$		−	16.0000i		−	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	16.0000i		−	1.33799i
$144$		0			0
$145$	8.00000	−	4.00000i	0.664364	−	0.332182i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.0000			0.983078			0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000			0.983078			0.491539	+	0.870855i	$0.336434\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	8.00000i	−0.294086	−	0.588172i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	8.00000i		−	0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			16.0000i			1.12298i
$204$		0			0
$205$	16.0000	−	8.00000i	1.11749	−	0.558744i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.0000			1.10674
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.0000			−1.61441
$222$		0			0
$223$		−	12.0000i		−	0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$		−	12.0000i		−	0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$	12.0000	+	24.0000i	0.782794	+	1.56559i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	12.0000i		−	0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$		−	12.0000i		−	0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$		0			0
$265$	−2.00000	−	4.00000i	−0.122859	−	0.245718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0000			−0.731653			−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			32.0000i			1.88890i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$		0			0
$295$	−24.0000	+	12.0000i	−1.39733	+	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.0000			0.925304
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000	+	2.00000i	−0.229039	+	0.114520i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$	−16.0000			−0.895828
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−48.0000			−2.64633
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	−	16.0000i	−0.437087	−	0.874173i
$336$		0			0
$337$			32.0000i			1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$			32.0000i			1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			32.0000i			1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$			32.0000i			1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$		0			0
$355$	16.0000	−	8.00000i	0.849192	−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.0000	−	32.0000i	−0.837478	−	1.67496i
$366$		0			0
$367$			28.0000i			1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$			28.0000i			1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$			20.0000i			1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$			20.0000i			1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			36.0000i			1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$			36.0000i			1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$		0			0
$385$	16.0000	+	32.0000i	0.815436	+	1.63087i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	8.00000			0.399501			0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000			0.399501			0.199750	+	0.979847i	$0.435987\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			16.0000i			0.793091i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	48.0000i		−	2.36193i
$414$		0			0
$415$	−8.00000	−	16.0000i	−0.392705	−	0.785409i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.0000	−	18.0000i	−1.16417	−	0.873128i
$426$		0			0
$427$		−	8.00000i		−	0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$		−	24.0000i		−	1.15337i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$		−	24.0000i		−	1.15337i	0.816968	−	0.576683i	$-0.195653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			16.0000i			0.765384i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$		0			0
$453$		0			0
$454$		0			0
$455$	32.0000	−	16.0000i	1.50018	−	0.750092i
$456$		0			0
$457$		−	24.0000i		−	1.12267i	−0.827588	−	0.561336i	$-0.810287\pi$
$457$		−	24.0000i		−	1.12267i	0.827588	−	0.561336i	$-0.189713\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	32.0000			1.47762
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.00000			0.365529			0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000			0.365529			0.182765	+	0.983157i	$0.441495\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000	+	16.0000i	0.363261	+	0.726523i
$486$		0			0
$487$		−	28.0000i		−	1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$		−	28.0000i		−	1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$			24.0000i			1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			32.0000i			1.43540i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.0000			−0.886484			−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000			−0.886484			−0.443242	+	0.896402i	$0.646172\pi$
$510$		0			0
$511$	64.0000			2.83119
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	−	8.00000i	−0.176261	−	0.352522i
$516$		0			0
$517$		−	48.0000i		−	2.11104i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.0000			−1.75243			−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000			−1.75243			−0.876216	+	0.481919i	$0.839940\pi$
$522$		0			0
$523$		−	40.0000i		−	1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$		−	40.0000i		−	1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$			32.0000i			1.38607i
$534$		0			0
$535$	8.00000	+	16.0000i	0.345870	+	0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−36.0000			−1.55063
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	36.0000	−	18.0000i	1.54207	−	0.771035i
$546$		0			0
$547$			32.0000i			1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$			32.0000i			1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.0000			−0.681623
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	−10.0000	−	20.0000i	−0.420703	−	0.841406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$		0			0
$577$		−	16.0000i		−	0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$		−	16.0000i		−	0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	32.0000			1.32758
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	8.00000i		−	0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$		−	8.00000i		−	0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$	48.0000	−	24.0000i	1.96781	−	0.983904i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	16.0000			0.653742			0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000			0.653742			0.326871	+	0.945069i	$0.394006\pi$
$600$		0			0
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		0			0
$607$			4.00000i			0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$			4.00000i			0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$		−	4.00000i		−	0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$		−	4.00000i		−	0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	46.0000i		−	1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$		−	46.0000i		−	1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000	−	24.0000i	−0.476205	−	0.952411i
$636$		0			0
$637$			36.0000i			1.42637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	24.0000			0.947943			0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000			0.947943			0.473972	+	0.880540i	$0.342820\pi$
$642$		0			0
$643$		−	24.0000i		−	0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$		−	24.0000i		−	0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.0000i			1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$			28.0000i			1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$		0			0
$649$	48.0000			1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$			42.0000i			1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$			42.0000i			1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$		0			0
$655$	−24.0000	+	12.0000i	−0.937758	+	0.468879i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000	+	32.0000i	0.620453	+	1.24091i
$666$		0			0
$667$		−	16.0000i		−	0.619522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.00000			0.308837
$672$		0			0
$673$			40.0000i			1.54189i	0.636904	+	0.770943i	$0.280215\pi$
$673$			40.0000i			1.54189i	−0.636904	+	0.770943i	$0.719785\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	32.0000i		−	1.22445i	−0.790685	−	0.612223i	$-0.790275\pi$
$683$		−	32.0000i		−	1.22445i	0.790685	−	0.612223i	$-0.209725\pi$
$684$		0			0
$685$	2.00000	+	4.00000i	0.0764161	+	0.152832i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.00000			0.304776
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−40.0000	+	20.0000i	−1.51729	+	0.758643i
$696$		0			0
$697$			48.0000i			1.81813i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	48.0000i		−	1.80523i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	16.0000	+	32.0000i	0.598366	+	1.19673i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.0000	+	16.0000i	−0.445669	+	0.594225i
$726$		0			0
$727$		−	12.0000i		−	0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$		−	12.0000i		−	0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	12.0000i		−	0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$		−	12.0000i		−	0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			32.0000i			1.17874i
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.0000i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$			12.0000i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$		0			0
$745$	−24.0000	+	12.0000i	−0.879292	+	0.439646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.0000			−1.16925
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$			28.0000i			1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$757$			28.0000i			1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.00000			0.290000			0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000			0.290000			0.145000	+	0.989432i	$0.453682\pi$
$762$		0			0
$763$			72.0000i			2.60658i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	48.0000i		−	1.73318i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	14.0000i		−	0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$		−	14.0000i		−	0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−32.0000			−1.14652
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.00000	+	8.00000i	0.142766	+	0.285532i
$786$		0			0
$787$		−	8.00000i		−	0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$		−	8.00000i		−	0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	40.0000			1.42224
$792$		0			0
$793$		−	8.00000i		−	0.284088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	6.00000i		−	0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$		−	6.00000i		−	0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$		0			0
$799$	−72.0000			−2.54718
$800$		0			0
$801$		0			0
$802$		0			0
$803$			64.0000i			2.25851i
$804$		0			0
$805$	−32.0000	+	16.0000i	−1.12785	+	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.00000	+	16.0000i	0.280228	+	0.560456i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	36.0000			1.25641			0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000			1.25641			0.628204	+	0.778048i	$0.283790\pi$
$822$		0			0
$823$		−	4.00000i		−	0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$		−	4.00000i		−	0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			54.0000i			1.87099i
$834$		0			0
$835$	12.0000	+	24.0000i	0.415277	+	0.830554i
$836$		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
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.00000	−	3.00000i	0.206406	−	0.103203i
$846$		0			0
$847$		−	20.0000i		−	0.687208i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.0000			−0.548473
$852$		0			0
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46.0000i		−	1.57133i	−0.618652	−	0.785665i	$-0.712321\pi$
$857$		−	46.0000i		−	1.57133i	0.618652	−	0.785665i	$-0.287679\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.0000			1.08553
$870$		0			0
$871$	32.0000			1.08428
$872$		0			0
$873$		0			0
$874$		0			0
$875$	44.0000	+	8.00000i	1.48747	+	0.270449i
$876$		0			0
$877$		−	4.00000i		−	0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$877$		−	4.00000i		−	0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16.0000			−0.539054			−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000			−0.539054			−0.269527	+	0.962993i	$0.586867\pi$
$882$		0			0
$883$			40.0000i			1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$			40.0000i			1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	48.0000			1.60987
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	48.0000i		−	1.60626i
$894$		0			0
$895$	8.00000	−	4.00000i	0.267411	−	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000	+	10.0000i	−0.664822	+	0.332411i
$906$		0			0
$907$			16.0000i			0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$			16.0000i			0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$			32.0000i			1.05905i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	48.0000i		−	1.58510i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			32.0000i			1.05329i
$924$		0			0
$925$	16.0000	+	12.0000i	0.526077	+	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.00000			0.262471			0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000			0.262471			0.131236	+	0.991351i	$0.458106\pi$
$930$		0			0
$931$	−36.0000			−1.17985
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.0000	+	48.0000i	0.784884	+	1.56977i
$936$		0			0
$937$		−	16.0000i		−	0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$		−	16.0000i		−	0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−44.0000			−1.43436			−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000			−1.43436			−0.717180	+	0.696888i	$0.754567\pi$
$942$		0			0
$943$		−	32.0000i		−	1.04206i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			24.0000i			0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$			24.0000i			0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$		0			0
$949$	64.0000			2.07753
$950$		0			0
$951$		0			0
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$	48.0000	−	24.0000i	1.55324	−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000			−0.258333
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.00000	+	16.0000i	0.257529	+	0.515058i
$966$		0			0
$967$			44.0000i			1.41494i	0.706741	+	0.707472i	$0.250165\pi$
$967$			44.0000i			1.41494i	−0.706741	+	0.707472i	$0.749835\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$		−	80.0000i		−	2.56468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	38.0000i		−	1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$		−	38.0000i		−	1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$			28.0000i			0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$			28.0000i			0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$		0			0
$985$	−18.0000	−	36.0000i	−0.573528	−	1.14706i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000	+	8.00000i	−0.507234	+	0.253617i
$996$		0			0
$997$			12.0000i			0.380044i	0.981780	+	0.190022i	$0.0608559\pi$
$997$			12.0000i			0.380044i	−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.f.b.289.2	yes	2
3.2	odd	2		360.2.f.d.289.1	yes	2
4.3	odd	2		720.2.f.a.289.2		2
5.2	odd	4		1800.2.a.w.1.1		1
5.3	odd	4		1800.2.a.d.1.1		1
5.4	even	2	inner	360.2.f.b.289.1	✓	2
8.3	odd	2		2880.2.f.u.1729.1		2
8.5	even	2		2880.2.f.q.1729.1		2
12.11	even	2		720.2.f.g.289.1		2
15.2	even	4		1800.2.a.u.1.1		1
15.8	even	4		1800.2.a.b.1.1		1
15.14	odd	2		360.2.f.d.289.2	yes	2
20.3	even	4		3600.2.a.bn.1.1		1
20.7	even	4		3600.2.a.c.1.1		1
20.19	odd	2		720.2.f.a.289.1		2
24.5	odd	2		2880.2.f.f.1729.2		2
24.11	even	2		2880.2.f.b.1729.2		2
40.19	odd	2		2880.2.f.u.1729.2		2
40.29	even	2		2880.2.f.q.1729.2		2
60.23	odd	4		3600.2.a.bp.1.1		1
60.47	odd	4		3600.2.a.g.1.1		1
60.59	even	2		720.2.f.g.289.2		2
120.29	odd	2		2880.2.f.f.1729.1		2
120.59	even	2		2880.2.f.b.1729.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.f.b.289.1	✓	2	5.4	even	2	inner
360.2.f.b.289.2	yes	2	1.1	even	1	trivial
360.2.f.d.289.1	yes	2	3.2	odd	2
360.2.f.d.289.2	yes	2	15.14	odd	2
720.2.f.a.289.1		2	20.19	odd	2
720.2.f.a.289.2		2	4.3	odd	2
720.2.f.g.289.1		2	12.11	even	2
720.2.f.g.289.2		2	60.59	even	2
1800.2.a.b.1.1		1	15.8	even	4
1800.2.a.d.1.1		1	5.3	odd	4
1800.2.a.u.1.1		1	15.2	even	4
1800.2.a.w.1.1		1	5.2	odd	4
2880.2.f.b.1729.1		2	120.59	even	2
2880.2.f.b.1729.2		2	24.11	even	2
2880.2.f.f.1729.1		2	120.29	odd	2
2880.2.f.f.1729.2		2	24.5	odd	2
2880.2.f.q.1729.1		2	8.5	even	2
2880.2.f.q.1729.2		2	40.29	even	2
2880.2.f.u.1729.1		2	8.3	odd	2
2880.2.f.u.1729.2		2	40.19	odd	2
3600.2.a.c.1.1		1	20.7	even	4
3600.2.a.g.1.1		1	60.47	odd	4
3600.2.a.bn.1.1		1	20.3	even	4
3600.2.a.bp.1.1		1	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +O(q^{10})\)	\(q+(-2.00000 + 1.00000i) q^{5} -4.00000i q^{7} +4.00000 q^{11} -4.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{29} +(4.00000 + 8.00000i) q^{35} +4.00000i q^{37} -8.00000 q^{41} -12.0000i q^{47} -9.00000 q^{49} +2.00000i q^{53} +(-8.00000 + 4.00000i) q^{55} +12.0000 q^{59} +2.00000 q^{61} +(4.00000 + 8.00000i) q^{65} +8.00000i q^{67} -8.00000 q^{71} +16.0000i q^{73} -16.0000i q^{77} +8.00000 q^{79} +8.00000i q^{83} +(6.00000 + 12.0000i) q^{85} -16.0000 q^{91} +(-8.00000 + 4.00000i) q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} + 8 q^{11} + 8 q^{19} + 6 q^{25} - 8 q^{29} + 8 q^{35} - 16 q^{41} - 18 q^{49} - 16 q^{55} + 24 q^{59} + 4 q^{61} + 8 q^{65} - 16 q^{71} + 16 q^{79} + 12 q^{85} - 32 q^{91} - 16 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 + 8 * q^11 + 8 * q^19 + 6 * q^25 - 8 * q^29 + 8 * q^35 - 16 * q^41 - 18 * q^49 - 16 * q^55 + 24 * q^59 + 4 * q^61 + 8 * q^65 - 16 * q^71 + 16 * q^79 + 12 * q^85 - 32 * q^91 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more