LMFDB - Embedded newform 5040.2.t.c.1009.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1009,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5040.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.t (of order $2$, degree $1$, not 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:	$40.2446026187$
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:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.c.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +4.00000i q^{13} +2.00000i q^{17} +(3.00000 - 4.00000i) q^{25} +8.00000 q^{29} +4.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -8.00000i q^{37} -4.00000 q^{41} -8.00000i q^{43} +12.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} -8.00000 q^{59} +10.0000 q^{61} +(-4.00000 - 8.00000i) q^{65} +8.00000i q^{67} -16.0000 q^{71} +12.0000i q^{73} -8.00000 q^{79} +16.0000i q^{83} +(-2.00000 - 4.00000i) q^{85} -12.0000 q^{89} +4.00000 q^{91} -4.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 6 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 8 q^{41} - 2 q^{49} - 16 q^{59} + 20 q^{61} - 8 q^{65} - 32 q^{71} - 16 q^{79} - 4 q^{85} - 24 q^{89} + 8 q^{91}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^25 + 16 * q^29 + 8 * q^31 + 2 * q^35 - 8 * q^41 - 2 * q^49 - 16 * q^59 + 20 * q^61 - 8 * q^65 - 32 * q^71 - 16 * q^79 - 4 * q^85 - 24 * q^89 + 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		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
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		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
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		0			0
$85$	−2.00000	−	4.00000i	−0.216930	−	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	4.00000i		−	0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$		−	4.00000i		−	0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$		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
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000			0.183340
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−16.0000	+	8.00000i	−1.32873	+	0.664364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.0000			−1.31077			−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000			−1.31077			−0.655386	+	0.755295i	$0.727494\pi$
$150$		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
$155$	−8.00000	+	4.00000i	−0.642575	+	0.321288i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$		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
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.0000			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.00000	+	16.0000i	0.588172	+	1.17634i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$	8.00000	−	4.00000i	0.558744	−	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		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
$215$	8.00000	+	16.0000i	0.545595	+	1.09119i
$216$		0			0
$217$		−	4.00000i		−	0.271538i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.00000			−0.538138
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		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
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.0000i			1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$			22.0000i			1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$		0			0
$235$	−12.0000	−	24.0000i	−0.782794	−	1.56559i
$236$		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
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	−	1.00000i	0.127775	−	0.0638877i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.0000i			0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$			14.0000i			0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$		0			0
$261$		0			0
$262$		0			0
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.00000			−0.243884			−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000			−0.243884			−0.121942	+	0.992537i	$0.538912\pi$
$270$		0			0
$271$	−28.0000			−1.70088			−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000			−1.70088			−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			16.0000i			0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$			16.0000i			0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	18.0000i		−	1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$		−	18.0000i		−	1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$		0			0
$295$	16.0000	−	8.00000i	0.931556	−	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.0000	+	10.0000i	−1.14520	+	0.572598i
$306$		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
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	12.0000i		−	0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$		−	12.0000i		−	0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		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
$329$	12.0000			0.661581
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	−	16.0000i	−0.437087	−	0.874173i
$336$		0			0
$337$		−	24.0000i		−	1.30736i	−0.756770	−	0.653682i	$-0.773224\pi$
$337$		−	24.0000i		−	1.30736i	0.756770	−	0.653682i	$-0.226776\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		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
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		0			0
$355$	32.0000	−	16.0000i	1.69838	−	0.849192i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.0000	−	24.0000i	−0.628109	−	1.25622i
$366$		0			0
$367$		−	24.0000i		−	1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$		−	24.0000i		−	1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000			−0.311504
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			32.0000i			1.64808i
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.00000			−0.405616			−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000			−0.405616			−0.202808	+	0.979219i	$0.565007\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000	−	8.00000i	0.805047	−	0.402524i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$			16.0000i			0.797017i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	−16.0000	−	32.0000i	−0.785409	−	1.57082i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		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
$433$			20.0000i			0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$			20.0000i			0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	12.0000			0.572729			0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000			0.572729			0.286364	+	0.958121i	$0.407553\pi$
$440$		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
$445$	24.0000	−	12.0000i	1.13771	−	0.568855i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.0000			−0.755087			−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000			−0.755087			−0.377543	+	0.925992i	$0.623231\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		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
$481$	32.0000			1.45907
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	8.00000i	0.181631	+	0.363261i
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	32.0000			1.44414			0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000			1.44414			0.722070	+	0.691820i	$0.243191\pi$
$492$		0			0
$493$			16.0000i			0.720604i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			16.0000i			0.717698i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$		0			0
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	36.0000			1.59567			0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000			1.59567			0.797836	+	0.602875i	$0.205978\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	+	16.0000i	0.352522	+	0.705044i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.00000			−0.175243			−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000			−0.175243			−0.0876216	+	0.996154i	$0.527927\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	16.0000i		−	0.693037i
$534$		0			0
$535$	−4.00000	−	8.00000i	−0.172935	−	0.345870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.00000	−	2.00000i	0.171341	−	0.0856706i
$546$		0			0
$547$		−	40.0000i		−	1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$		−	40.0000i		−	1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			8.00000i			0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		0			0
$559$	32.0000			1.35346
$560$		0			0
$561$		0			0
$562$		0			0
$563$			24.0000i			1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$			24.0000i			1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$		0			0
$565$	−18.0000	−	36.0000i	−0.757266	−	1.51453i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	20.0000i		−	0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$		−	20.0000i		−	0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0000			0.663792
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.00000i			0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$			2.00000i			0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$		0			0
$595$	−4.00000	+	2.00000i	−0.163984	+	0.0819920i
$596$		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
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	22.0000	−	11.0000i	0.894427	−	0.447214i
$606$		0			0
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$		−	8.00000i		−	0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$		−	8.00000i		−	0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		0			0
$619$	40.0000			1.60774			0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000			1.60774			0.803868	+	0.594808i	$0.202772\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.0000			0.637962
$630$		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	−	16.0000i	−0.317470	−	0.634941i
$636$		0			0
$637$		−	4.00000i		−	0.158486i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$		−	16.0000i		−	0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$		−	16.0000i		−	0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$		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
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	46.0000i		−	1.80012i	−0.435767	−	0.900060i	$-0.643523\pi$
$653$		−	46.0000i		−	1.80012i	0.435767	−	0.900060i	$-0.356477\pi$
$654$		0			0
$655$	−16.0000	+	8.00000i	−0.625172	+	0.312586i
$656$		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
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			24.0000i			0.925132i	0.886585	+	0.462566i	$0.153071\pi$
$673$			24.0000i			0.925132i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	50.0000i		−	1.92166i	−0.277145	−	0.960828i	$-0.589388\pi$
$677$		−	50.0000i		−	1.92166i	0.277145	−	0.960828i	$-0.410612\pi$
$678$		0			0
$679$	−4.00000			−0.153506
$680$		0			0
$681$		0			0
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−6.00000	−	12.0000i	−0.229248	−	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.0000			0.914327
$690$		0			0
$691$	−40.0000			−1.52167			−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000			−1.52167			−0.760836	+	0.648944i	$0.775211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.0000	+	8.00000i	−0.606915	+	0.303457i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.00000			0.302156			0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000			0.302156			0.151078	+	0.988522i	$0.451726\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.0000	−	32.0000i	0.891338	−	1.18845i
$726$		0			0
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.0000			0.591781
$732$		0			0
$733$			52.0000i			1.92066i	0.278859	+	0.960332i	$0.410044\pi$
$733$			52.0000i			1.92066i	−0.278859	+	0.960332i	$0.589956\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$	32.0000	−	16.0000i	1.17239	−	0.586195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.00000			0.146157
$750$		0			0
$751$	24.0000			0.875772			0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000			0.875772			0.437886	+	0.899030i	$0.355727\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.0000			0.724999			0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000			0.724999			0.362500	+	0.931984i	$0.381923\pi$
$762$		0			0
$763$			2.00000i			0.0724049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	32.0000i		−	1.15545i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	12.0000	−	16.0000i	0.431053	−	0.574737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	12.0000	+	24.0000i	0.428298	+	0.856597i
$786$		0			0
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	18.0000			0.640006
$792$		0			0
$793$			40.0000i			1.42044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	38.0000i		−	1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$		−	38.0000i		−	1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		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
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		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
$821$	−32.0000			−1.11681			−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000			−1.11681			−0.558404	+	0.829569i	$0.688586\pi$
$822$		0			0
$823$		−	8.00000i		−	0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$		−	8.00000i		−	0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$		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
$829$	6.00000			0.208389			0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000			0.208389			0.104194	+	0.994557i	$0.466774\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.00000i		−	0.0692959i
$834$		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
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.00000	−	3.00000i	0.206406	−	0.103203i
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			36.0000i			1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$			36.0000i			1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	−8.00000			−0.272956			−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000			−0.272956			−0.136478	+	0.990643i	$0.543578\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−6.00000	−	12.0000i	−0.204006	−	0.408012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−32.0000			−1.08428
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$			40.0000i			1.35070i	0.737496	+	0.675352i	$0.236008\pi$
$877$			40.0000i			1.35070i	−0.737496	+	0.675352i	$0.763992\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−28.0000			−0.943344			−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000			−0.943344			−0.471672	+	0.881774i	$0.656349\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	4.00000i		−	0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$		−	4.00000i		−	0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−32.0000	+	16.0000i	−1.06964	+	0.534821i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.0000			1.06726
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.0000	+	18.0000i	−1.19668	+	0.598340i
$906$		0			0
$907$		−	8.00000i		−	0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$		−	8.00000i		−	0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	8.00000i		−	0.264183i
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	64.0000i		−	2.10659i
$924$		0			0
$925$	−32.0000	−	24.0000i	−1.05215	−	0.789115i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.00000			−0.131236			−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000			−0.131236			−0.0656179	+	0.997845i	$0.520902\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	44.0000i		−	1.43742i	−0.695311	−	0.718709i	$-0.744734\pi$
$937$		−	44.0000i		−	1.43742i	0.695311	−	0.718709i	$-0.255266\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	36.0000			1.17357			0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000			1.17357			0.586783	+	0.809744i	$0.300394\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.0000i		−	1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$		−	36.0000i		−	1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$		0			0
$949$	−48.0000			−1.55815
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.0000	−	32.0000i	−0.515058	−	1.03012i
$966$		0			0
$967$		−	8.00000i		−	0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$		−	8.00000i		−	0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	56.0000			1.79713			0.898563	−	0.438845i	$-0.144612\pi$
$971$	56.0000			1.79713			0.898563	+	0.438845i	$0.144612\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$	−10.0000	−	20.0000i	−0.318626	−	0.637253i
$986$		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	$0.330290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.00000	−	4.00000i	0.253617	−	0.126809i
$996$		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	5040.2.t.c.1009.2		2
3.2	odd	2		5040.2.t.r.1009.1		2
4.3	odd	2		315.2.d.b.64.2	yes	2
5.4	even	2	inner	5040.2.t.c.1009.1		2
12.11	even	2		315.2.d.d.64.1	yes	2
15.14	odd	2		5040.2.t.r.1009.2		2
20.3	even	4		1575.2.a.j.1.1		1
20.7	even	4		1575.2.a.b.1.1		1
20.19	odd	2		315.2.d.b.64.1	✓	2
28.27	even	2		2205.2.d.g.1324.2		2
60.23	odd	4		1575.2.a.d.1.1		1
60.47	odd	4		1575.2.a.g.1.1		1
60.59	even	2		315.2.d.d.64.2	yes	2
84.83	odd	2		2205.2.d.c.1324.1		2
140.139	even	2		2205.2.d.g.1324.1		2
420.419	odd	2		2205.2.d.c.1324.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.d.b.64.1	✓	2	20.19	odd	2
315.2.d.b.64.2	yes	2	4.3	odd	2
315.2.d.d.64.1	yes	2	12.11	even	2
315.2.d.d.64.2	yes	2	60.59	even	2
1575.2.a.b.1.1		1	20.7	even	4
1575.2.a.d.1.1		1	60.23	odd	4
1575.2.a.g.1.1		1	60.47	odd	4
1575.2.a.j.1.1		1	20.3	even	4
2205.2.d.c.1324.1		2	84.83	odd	2
2205.2.d.c.1324.2		2	420.419	odd	2
2205.2.d.g.1324.1		2	140.139	even	2
2205.2.d.g.1324.2		2	28.27	even	2
5040.2.t.c.1009.1		2	5.4	even	2	inner
5040.2.t.c.1009.2		2	1.1	even	1	trivial
5040.2.t.r.1009.1		2	3.2	odd	2
5040.2.t.r.1009.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +4.00000i q^{13} +2.00000i q^{17} +(3.00000 - 4.00000i) q^{25} +8.00000 q^{29} +4.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -8.00000i q^{37} -4.00000 q^{41} -8.00000i q^{43} +12.0000i q^{47} -1.00000 q^{49} -6.00000i q^{53} -8.00000 q^{59} +10.0000 q^{61} +(-4.00000 - 8.00000i) q^{65} +8.00000i q^{67} -16.0000 q^{71} +12.0000i q^{73} -8.00000 q^{79} +16.0000i q^{83} +(-2.00000 - 4.00000i) q^{85} -12.0000 q^{89} +4.00000 q^{91} -4.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 6 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 8 q^{41} - 2 q^{49} - 16 q^{59} + 20 q^{61} - 8 q^{65} - 32 q^{71} - 16 q^{79} - 4 q^{85} - 24 q^{89} + 8 q^{91}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^25 + 16 * q^29 + 8 * q^31 + 2 * q^35 - 8 * q^41 - 2 * q^49 - 16 * q^59 + 20 * q^61 - 8 * q^65 - 32 * q^71 - 16 * q^79 - 4 * q^85 - 24 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

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