LMFDB - Embedded newform 1440.2.f.d.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(289,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, 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("1440.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.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:	$11.4984578911$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1440.289
Dual form			1440.2.f.d.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.00000i) q^{5} +4.00000i q^{7} +4.00000i q^{13} +8.00000 q^{19} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -6.00000 q^{29} -8.00000 q^{31} +(-8.00000 - 4.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} +4.00000i q^{43} -4.00000i q^{47} -9.00000 q^{49} +12.0000i q^{53} -6.00000 q^{61} +(-8.00000 - 4.00000i) q^{65} -12.0000i q^{67} +16.0000 q^{71} +8.00000 q^{79} +12.0000i q^{83} -10.0000 q^{89} -16.0000 q^{91} +(-8.00000 + 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 16 q^{19} - 6 q^{25} - 12 q^{29} - 16 q^{31} - 16 q^{35} - 12 q^{41} - 18 q^{49} - 12 q^{61} - 16 q^{65} + 32 q^{71} + 16 q^{79} - 20 q^{89} - 32 q^{91} - 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^5 + 16 * q^19 - 6 * q^25 - 12 * q^29 - 16 * q^31 - 16 * q^35 - 12 * q^41 - 18 * q^49 - 12 * q^61 - 16 * q^65 + 32 * q^71 + 16 * q^79 - 20 * q^89 - 32 * q^91 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.00000	−	4.00000i	−1.35225	−	0.676123i
$36$		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
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	−	4.00000i	−0.992278	−	0.496139i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		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
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−16.0000			−1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	+	16.0000i	−0.820783	+	1.64157i
$96$		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
$101$	−14.0000			−1.39305			−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000			−1.39305			−0.696526	+	0.717532i	$0.745272\pi$
$102$		0			0
$103$		−	12.0000i		−	1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$		−	12.0000i		−	1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$	8.00000	+	4.00000i	0.746004	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	16.0000			1.39793			0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000			1.39793			0.698963	+	0.715158i	$0.253645\pi$
$132$		0			0
$133$			32.0000i			2.77475i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	−	12.0000i	0.498273	−	0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	−	16.0000i	0.642575	−	1.28515i
$156$		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
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.00000i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$		−	4.00000i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4.00000i		−	0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$		−	4.00000i		−	0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.00000	−	4.00000i	−0.588172	−	0.294086i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	24.0000i		−	1.68447i
$204$		0			0
$205$	6.00000	−	12.0000i	0.419058	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	8.00000i		−	0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$		−	8.00000i		−	0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$		0			0
$235$	8.00000	+	4.00000i	0.521862	+	0.260931i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.00000	−	18.0000i	0.574989	−	1.14998i
$246$		0			0
$247$			32.0000i			2.03611i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.0000			1.00991			0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000			1.00991			0.504956	+	0.863145i	$0.331509\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.0000i		−	1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$		−	24.0000i		−	1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$		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	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$		0			0
$265$	−24.0000	−	12.0000i	−1.47431	−	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			4.00000i			0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$			4.00000i			0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$			20.0000i			1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$			20.0000i			1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	24.0000i		−	1.41668i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			20.0000i			1.16841i	0.811605	+	0.584206i	$0.198594\pi$
$293$			20.0000i			1.16841i	−0.811605	+	0.584206i	$0.801406\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.0000			0.925304
$300$		0			0
$301$	−16.0000			−0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.00000	−	12.0000i	0.343559	−	0.687118i
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	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
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	16.0000	−	12.0000i	0.887520	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000	+	12.0000i	1.31126	+	0.655630i
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		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
$347$			28.0000i			1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$			28.0000i			1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$	−16.0000	+	32.0000i	−0.849192	+	1.69838i
$356$		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
$361$	45.0000			2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−48.0000			−2.49204
$372$		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
$377$		−	24.0000i		−	1.23606i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			12.0000i			0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$			12.0000i			0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.00000	+	16.0000i	−0.402524	+	0.805047i
$396$		0			0
$397$		−	28.0000i		−	1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$		−	28.0000i		−	1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		−	32.0000i		−	1.59403i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−24.0000	−	12.0000i	−1.17811	−	0.589057i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	24.0000i		−	1.16144i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$			40.0000i			1.92228i	0.276066	+	0.961139i	$0.410969\pi$
$433$			40.0000i			1.92228i	−0.276066	+	0.961139i	$0.589031\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	32.0000i		−	1.53077i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16.0000	−	32.0000i	0.750092	−	1.50018i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.0000			−1.02464			−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000			−1.02464			−0.512321	+	0.858794i	$0.671214\pi$
$462$		0			0
$463$			20.0000i			0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$			20.0000i			0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4.00000i		−	0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$		−	4.00000i		−	0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$		0			0
$469$	48.0000			2.21643
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.0000	−	32.0000i	−1.10120	−	1.46826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	+	8.00000i	0.726523	+	0.363261i
$486$		0			0
$487$			4.00000i			0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$			4.00000i			0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−32.0000			−1.44414			−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000			−1.44414			−0.722070	+	0.691820i	$0.756809\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			64.0000i			2.87079i
$498$		0			0
$499$	24.0000			1.07439			0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000			1.07439			0.537194	+	0.843459i	$0.319484\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			28.0000i			1.24846i	0.781241	+	0.624229i	$0.214587\pi$
$503$			28.0000i			1.24846i	−0.781241	+	0.624229i	$0.785413\pi$
$504$		0			0
$505$	14.0000	−	28.0000i	0.622992	−	1.24598i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0000	+	12.0000i	1.05757	+	0.528783i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$		−	12.0000i		−	0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$		−	12.0000i		−	0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$		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
$533$		−	24.0000i		−	1.03956i
$534$		0			0
$535$	−24.0000	−	12.0000i	−1.03761	−	0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.0000	+	20.0000i	−0.428353	+	0.856706i
$546$		0			0
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−48.0000			−2.04487
$552$		0			0
$553$			32.0000i			1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4.00000i		−	0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$		−	4.00000i		−	0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$			28.0000i			1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$			28.0000i			1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$		0			0
$565$	−16.0000	−	8.00000i	−0.673125	−	0.336563i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		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	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−48.0000			−1.99138
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$		0			0
$589$	−64.0000			−2.63707
$590$		0			0
$591$		0			0
$592$		0			0
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	−	22.0000i	0.447214	−	0.894427i
$606$		0			0
$607$			20.0000i			0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$			20.0000i			0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$		−	20.0000i		−	0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$		−	20.0000i		−	0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		0			0
$619$	24.0000			0.964641			0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000			0.964641			0.482321	+	0.875995i	$0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	40.0000i		−	1.60257i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		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
$635$	−8.00000	−	4.00000i	−0.317470	−	0.158735i
$636$		0			0
$637$		−	36.0000i		−	1.42637i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	34.0000			1.34292			0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000			1.34292			0.671460	+	0.741041i	$0.265668\pi$
$642$		0			0
$643$		−	12.0000i		−	0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$		−	12.0000i		−	0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.0000i			0.782660i	0.920250	+	0.391330i	$0.127985\pi$
$653$			20.0000i			0.782660i	−0.920250	+	0.391330i	$0.872015\pi$
$654$		0			0
$655$	−16.0000	+	32.0000i	−0.625172	+	1.25034i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−64.0000	−	32.0000i	−2.48181	−	1.24091i
$666$		0			0
$667$			24.0000i			0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			8.00000i			0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$			8.00000i			0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$		0			0
$679$	32.0000			1.22805
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	36.0000i		−	1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$		−	36.0000i		−	1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−48.0000			−1.82865
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	−	16.0000i	0.303457	−	0.606915i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$			32.0000i			1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	56.0000i		−	2.10610i
$708$		0			0
$709$	−2.00000			−0.0751116			−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000			−0.0751116			−0.0375558	+	0.999295i	$0.511957\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0000			−0.596699			−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000			−0.596699			−0.298350	+	0.954457i	$0.596436\pi$
$720$		0			0
$721$	48.0000			1.78761
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		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
$745$	−14.0000	+	28.0000i	−0.512920	+	1.02584i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	12.0000i		−	0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$		−	12.0000i		−	0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$			40.0000i			1.44810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			28.0000i			1.00709i	0.863969	+	0.503545i	$0.167971\pi$
$773$			28.0000i			1.00709i	−0.863969	+	0.503545i	$0.832029\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.00000	+	4.00000i	0.285532	+	0.142766i
$786$		0			0
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−32.0000			−1.13779
$792$		0			0
$793$		−	24.0000i		−	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−16.0000	+	32.0000i	−0.563926	+	1.12785i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22.0000			0.773479			0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000			0.773479			0.386739	+	0.922189i	$0.373601\pi$
$810$		0			0
$811$	8.00000			0.280918			0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000			0.280918			0.140459	+	0.990086i	$0.455142\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$			32.0000i			1.11954i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$		0			0
$823$		−	44.0000i		−	1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$		−	44.0000i		−	1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	26.0000			0.903017			0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000			0.903017			0.451509	+	0.892267i	$0.350886\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	8.00000	+	4.00000i	0.276851	+	0.138426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−32.0000			−1.10476			−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000			−1.10476			−0.552381	+	0.833592i	$0.686281\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000	−	6.00000i	0.103203	−	0.206406i
$846$		0			0
$847$		−	44.0000i		−	1.51186i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16.0000			0.548473
$852$		0			0
$853$			12.0000i			0.410872i	0.978671	+	0.205436i	$0.0658613\pi$
$853$			12.0000i			0.410872i	−0.978671	+	0.205436i	$0.934139\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$	−56.0000			−1.91070			−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000			−1.91070			−0.955348	+	0.295484i	$0.904519\pi$
$860$		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
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.00000	+	44.0000i	0.270449	+	1.48747i
$876$		0			0
$877$			28.0000i			0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$			28.0000i			0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$			4.00000i			0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$			4.00000i			0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$	16.0000	−	32.0000i	0.534821	−	1.06964i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	48.0000			1.60089
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.0000	+	28.0000i	−0.465376	+	0.930751i
$906$		0			0
$907$			20.0000i			0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$			20.0000i			0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			64.0000i			2.11347i
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			64.0000i			2.10659i
$924$		0			0
$925$	16.0000	−	12.0000i	0.526077	−	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.00000			0.0656179			0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000			0.0656179			0.0328089	+	0.999462i	$0.489555\pi$
$930$		0			0
$931$	−72.0000			−2.35970
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		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
$941$	10.0000			0.325991			0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000			0.325991			0.162995	+	0.986627i	$0.447884\pi$
$942$		0			0
$943$			24.0000i			0.781548i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	48.0000i		−	1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$		−	48.0000i		−	1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$		0			0
$955$	16.0000	−	32.0000i	0.517748	−	1.03550i
$956$		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
$964$		0			0
$965$	−48.0000	−	24.0000i	−1.54517	−	0.772587i
$966$		0			0
$967$		−	12.0000i		−	0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$		−	12.0000i		−	0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.0000			0.513464			0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000			0.513464			0.256732	+	0.966483i	$0.417354\pi$
$972$		0			0
$973$		−	32.0000i		−	1.02587i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	48.0000i		−	1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$		−	48.0000i		−	1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	−24.0000	−	12.0000i	−0.764704	−	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.0000			0.508770
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.00000	−	16.0000i	0.253617	−	0.507234i
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.f.d.289.2		2
3.2	odd	2		480.2.f.d.289.2	yes	2
4.3	odd	2		1440.2.f.b.289.2		2
5.2	odd	4		7200.2.a.c.1.1		1
5.3	odd	4		7200.2.a.by.1.1		1
5.4	even	2	inner	1440.2.f.d.289.1		2
8.3	odd	2		2880.2.f.o.1729.1		2
8.5	even	2		2880.2.f.m.1729.1		2
12.11	even	2		480.2.f.c.289.1	✓	2
15.2	even	4		2400.2.a.t.1.1		1
15.8	even	4		2400.2.a.o.1.1		1
15.14	odd	2		480.2.f.d.289.1	yes	2
20.3	even	4		7200.2.a.a.1.1		1
20.7	even	4		7200.2.a.ca.1.1		1
20.19	odd	2		1440.2.f.b.289.1		2
24.5	odd	2		960.2.f.e.769.1		2
24.11	even	2		960.2.f.d.769.2		2
40.19	odd	2		2880.2.f.o.1729.2		2
40.29	even	2		2880.2.f.m.1729.2		2
48.5	odd	4		3840.2.d.bf.2689.2		2
48.11	even	4		3840.2.d.p.2689.2		2
48.29	odd	4		3840.2.d.a.2689.1		2
48.35	even	4		3840.2.d.q.2689.1		2
60.23	odd	4		2400.2.a.s.1.1		1
60.47	odd	4		2400.2.a.p.1.1		1
60.59	even	2		480.2.f.c.289.2	yes	2
120.29	odd	2		960.2.f.e.769.2		2
120.53	even	4		4800.2.a.cr.1.1		1
120.59	even	2		960.2.f.d.769.1		2
120.77	even	4		4800.2.a.c.1.1		1
120.83	odd	4		4800.2.a.e.1.1		1
120.107	odd	4		4800.2.a.cp.1.1		1
240.29	odd	4		3840.2.d.bf.2689.1		2
240.59	even	4		3840.2.d.q.2689.2		2
240.149	odd	4		3840.2.d.a.2689.2		2
240.179	even	4		3840.2.d.p.2689.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.f.c.289.1	✓	2	12.11	even	2
480.2.f.c.289.2	yes	2	60.59	even	2
480.2.f.d.289.1	yes	2	15.14	odd	2
480.2.f.d.289.2	yes	2	3.2	odd	2
960.2.f.d.769.1		2	120.59	even	2
960.2.f.d.769.2		2	24.11	even	2
960.2.f.e.769.1		2	24.5	odd	2
960.2.f.e.769.2		2	120.29	odd	2
1440.2.f.b.289.1		2	20.19	odd	2
1440.2.f.b.289.2		2	4.3	odd	2
1440.2.f.d.289.1		2	5.4	even	2	inner
1440.2.f.d.289.2		2	1.1	even	1	trivial
2400.2.a.o.1.1		1	15.8	even	4
2400.2.a.p.1.1		1	60.47	odd	4
2400.2.a.s.1.1		1	60.23	odd	4
2400.2.a.t.1.1		1	15.2	even	4
2880.2.f.m.1729.1		2	8.5	even	2
2880.2.f.m.1729.2		2	40.29	even	2
2880.2.f.o.1729.1		2	8.3	odd	2
2880.2.f.o.1729.2		2	40.19	odd	2
3840.2.d.a.2689.1		2	48.29	odd	4
3840.2.d.a.2689.2		2	240.149	odd	4
3840.2.d.p.2689.1		2	240.179	even	4
3840.2.d.p.2689.2		2	48.11	even	4
3840.2.d.q.2689.1		2	48.35	even	4
3840.2.d.q.2689.2		2	240.59	even	4
3840.2.d.bf.2689.1		2	240.29	odd	4
3840.2.d.bf.2689.2		2	48.5	odd	4
4800.2.a.c.1.1		1	120.77	even	4
4800.2.a.e.1.1		1	120.83	odd	4
4800.2.a.cp.1.1		1	120.107	odd	4
4800.2.a.cr.1.1		1	120.53	even	4
7200.2.a.a.1.1		1	20.3	even	4
7200.2.a.c.1.1		1	5.2	odd	4
7200.2.a.by.1.1		1	5.3	odd	4
7200.2.a.ca.1.1		1	20.7	even	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.00000i) q^{5} +4.00000i q^{7} +4.00000i q^{13} +8.00000 q^{19} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -6.00000 q^{29} -8.00000 q^{31} +(-8.00000 - 4.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} +4.00000i q^{43} -4.00000i q^{47} -9.00000 q^{49} +12.0000i q^{53} -6.00000 q^{61} +(-8.00000 - 4.00000i) q^{65} -12.0000i q^{67} +16.0000 q^{71} +8.00000 q^{79} +12.0000i q^{83} -10.0000 q^{89} -16.0000 q^{91} +(-8.00000 + 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 16 q^{19} - 6 q^{25} - 12 q^{29} - 16 q^{31} - 16 q^{35} - 12 q^{41} - 18 q^{49} - 12 q^{61} - 16 q^{65} + 32 q^{71} + 16 q^{79} - 20 q^{89} - 32 q^{91} - 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^5 + 16 * q^19 - 6 * q^25 - 12 * q^29 - 16 * q^31 - 16 * q^35 - 12 * q^41 - 18 * q^49 - 12 * q^61 - 16 * q^65 + 32 * q^71 + 16 * q^79 - 20 * q^89 - 32 * q^91 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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