LMFDB - Embedded newform 5040.2.t.k.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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
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.k.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +O(q^{10})$	$q+(1.00000 + 2.00000i) q^{5} +1.00000i q^{7} -2.00000 q^{11} +2.00000i q^{13} +8.00000i q^{17} -2.00000 q^{19} +(-3.00000 + 4.00000i) q^{25} -6.00000 q^{29} -6.00000 q^{31} +(-2.00000 + 1.00000i) q^{35} -8.00000i q^{37} -6.00000 q^{41} -8.00000i q^{43} -4.00000i q^{47} -1.00000 q^{49} +2.00000i q^{53} +(-2.00000 - 4.00000i) q^{55} +8.00000 q^{59} +10.0000 q^{61} +(-4.00000 + 2.00000i) q^{65} -12.0000i q^{67} -14.0000 q^{71} -10.0000i q^{73} -2.00000i q^{77} +4.00000 q^{79} +16.0000i q^{83} +(-16.0000 + 8.00000i) q^{85} +10.0000 q^{89} -2.00000 q^{91} +(-2.00000 - 4.00000i) q^{95} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 6 q^{25} - 12 q^{29} - 12 q^{31} - 4 q^{35} - 12 q^{41} - 2 q^{49} - 4 q^{55} + 16 q^{59} + 20 q^{61} - 8 q^{65} - 28 q^{71} + 8 q^{79} - 32 q^{85} + 20 q^{89} - 4 q^{91} - 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^11 - 4 * q^19 - 6 * q^25 - 12 * q^29 - 12 * q^31 - 4 * q^35 - 12 * q^41 - 2 * q^49 - 4 * q^55 + 16 * q^59 + 20 * q^61 - 8 * q^65 - 28 * q^71 + 8 * q^79 - 32 * q^85 + 20 * q^89 - 4 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

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
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			8.00000i			1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$			8.00000i			1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		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
$29$	−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
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	+	1.00000i	−0.338062	+	0.169031i
$36$		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
$41$	−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
$43$		−	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
$47$		−	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
$49$	−1.00000			−0.142857
$50$		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
$55$	−2.00000	−	4.00000i	−0.269680	−	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		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	+	2.00000i	−0.496139	+	0.248069i
$66$		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
$71$	−14.0000			−1.66149			−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000			−1.66149			−0.830747	+	0.556650i	$0.812086\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.00000i		−	0.227921i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		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
$85$	−16.0000	+	8.00000i	−1.73544	+	0.867722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	−	4.00000i	−0.205196	−	0.410391i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		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
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.00000			−0.733359
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		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
$131$	−20.0000			−1.74741			−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000			−1.74741			−0.873704	+	0.486458i	$0.838289\pi$
$132$		0			0
$133$		−	2.00000i		−	0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.00000i		−	0.334497i
$144$		0			0
$145$	−6.00000	−	12.0000i	−0.498273	−	0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		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
$155$	−6.00000	−	12.0000i	−0.481932	−	0.963863i
$156$		0			0
$157$			10.0000i			0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$			10.0000i			0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$			16.0000i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$			16.0000i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$		0			0
$187$		−	16.0000i		−	1.17004i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.0000			1.01300			0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000			1.01300			0.506502	+	0.862239i	$0.330938\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	−26.0000			−1.84309			−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000			−1.84309			−0.921546	+	0.388270i	$0.873073\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$	−6.00000	−	12.0000i	−0.419058	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000			0.276686
$210$		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
$215$	16.0000	−	8.00000i	1.09119	−	0.545595i
$216$		0			0
$217$		−	6.00000i		−	0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.0000			−1.07628
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		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
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	8.00000	−	4.00000i	0.521862	−	0.260931i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.0000			1.42306			0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000			1.42306			0.711531	+	0.702655i	$0.248002\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.00000	−	2.00000i	−0.0638877	−	0.127775i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.0000i		−	1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$		−	20.0000i		−	1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$		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	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−4.00000	+	2.00000i	−0.245718	+	0.122859i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	−	8.00000i	0.361814	−	0.482418i
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6.00000i		−	0.354169i
$288$		0			0
$289$	−47.0000			−2.76471
$290$		0			0
$291$		0			0
$292$		0			0
$293$			16.0000i			0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$			16.0000i			0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$		0			0
$295$	8.00000	+	16.0000i	0.465778	+	0.931556i
$296$		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
$305$	10.0000	+	20.0000i	0.572598	+	1.14520i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000			0.220527
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000	−	12.0000i	1.31126	−	0.655630i
$336$		0			0
$337$		−	32.0000i		−	1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$		−	32.0000i		−	1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.0000			0.649836
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−14.0000	−	28.0000i	−0.743043	−	1.48609i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.0000			−0.738892			−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000			−0.738892			−0.369446	+	0.929252i	$0.620452\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	20.0000	−	10.0000i	1.04685	−	0.523424i
$366$		0			0
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.00000			−0.103835
$372$		0			0
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	4.00000	−	2.00000i	0.203859	−	0.101929i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000	+	8.00000i	0.201262	+	0.402524i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		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	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	−32.0000	+	16.0000i	−1.57082	+	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
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−32.0000	−	24.0000i	−1.55223	−	1.16417i
$426$		0			0
$427$			10.0000i			0.483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$		0			0
$433$			26.0000i			1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$			26.0000i			1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	6.00000			0.286364			0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000			0.286364			0.143182	+	0.989696i	$0.454267\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		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
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.00000	−	4.00000i	−0.0937614	−	0.187523i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$			12.0000i			0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$			12.0000i			0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	16.0000i		−	0.740392i	−0.928954	−	0.370196i	$-0.879291\pi$
$467$		−	16.0000i		−	0.740392i	0.928954	−	0.370196i	$-0.120709\pi$
$468$		0			0
$469$	12.0000			0.554109
$470$		0			0
$471$		0			0
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$	6.00000	−	8.00000i	0.275299	−	0.367065i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.0000	−	10.0000i	0.908153	−	0.454077i
$486$		0			0
$487$			36.0000i			1.63132i	0.578535	+	0.815658i	$0.303625\pi$
$487$			36.0000i			1.63132i	−0.578535	+	0.815658i	$0.696375\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−42.0000			−1.89543			−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000			−1.89543			−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$		−	48.0000i		−	2.16181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	14.0000i		−	0.627986i
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	10.0000	+	20.0000i	0.444994	+	0.889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.00000			0.0886484			0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000			0.0886484			0.0443242	+	0.999017i	$0.485887\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			8.00000i			0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			12.0000i			0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$			12.0000i			0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	48.0000i		−	2.09091i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−24.0000	+	12.0000i	−1.03761	+	0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.00000	+	12.0000i	0.257012	+	0.514024i
$546$		0			0
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.0000			0.511217
$552$		0			0
$553$			4.00000i			0.170097i
$554$		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
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	12.0000	−	6.00000i	0.504844	−	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	18.0000i		−	0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$		−	18.0000i		−	0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.0000			−0.663792
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	4.00000i		−	0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$		−	4.00000i		−	0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$			32.0000i			1.31408i	0.753855	+	0.657041i	$0.228192\pi$
$593$			32.0000i			1.31408i	−0.753855	+	0.657041i	$0.771808\pi$
$594$		0			0
$595$	−8.00000	−	16.0000i	−0.327968	−	0.655936i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14.0000			−0.572024			−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000			−0.572024			−0.286012	+	0.958226i	$0.592330\pi$
$600$		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
$605$	−7.00000	−	14.0000i	−0.284590	−	0.569181i
$606$		0			0
$607$		−	16.0000i		−	0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$		−	16.0000i		−	0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000			0.323645
$612$		0			0
$613$			32.0000i			1.29247i	0.763139	+	0.646234i	$0.223657\pi$
$613$			32.0000i			1.29247i	−0.763139	+	0.646234i	$0.776343\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$	−18.0000			−0.723481			−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000			−0.723481			−0.361741	+	0.932279i	$0.617817\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.0000i			0.400642i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	64.0000			2.55185
$630$		0			0
$631$	−12.0000			−0.477712			−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000			−0.477712			−0.238856	+	0.971055i	$0.576772\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000	+	12.0000i	−0.952411	+	0.476205i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16.0000i		−	0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$		−	16.0000i		−	0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$			14.0000i			0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$			14.0000i			0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$		0			0
$655$	−20.0000	−	40.0000i	−0.781465	−	1.56293i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.00000	−	2.00000i	0.155113	−	0.0775567i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−20.0000			−0.772091
$672$		0			0
$673$			12.0000i			0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$			12.0000i			0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$		0			0
$681$		0			0
$682$		0			0
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$		0			0
$685$	28.0000	−	14.0000i	1.06983	−	0.534913i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00000			−0.152388
$690$		0			0
$691$	14.0000			0.532585			0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000			0.532585			0.266293	+	0.963892i	$0.414201\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.00000	−	4.00000i	−0.0758643	−	0.151729i
$696$		0			0
$697$		−	48.0000i		−	1.81813i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			10.0000i			0.376089i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	8.00000	−	4.00000i	0.299183	−	0.149592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000	−	24.0000i	0.668503	−	0.891338i
$726$		0			0
$727$			16.0000i			0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$			16.0000i			0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	64.0000			2.36713
$732$		0			0
$733$		−	30.0000i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$		−	30.0000i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			24.0000i			0.884051i
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−18.0000	−	36.0000i	−0.659469	−	1.31894i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$			4.00000i			0.145382i	0.997354	+	0.0726912i	$0.0231588\pi$
$757$			4.00000i			0.145382i	−0.997354	+	0.0726912i	$0.976841\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$			6.00000i			0.217215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			16.0000i			0.577727i
$768$		0			0
$769$	42.0000			1.51456			0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000			1.51456			0.757279	+	0.653091i	$0.226528\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			32.0000i			1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$			32.0000i			1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$		0			0
$775$	18.0000	−	24.0000i	0.646579	−	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−20.0000	+	10.0000i	−0.713831	+	0.356915i
$786$		0			0
$787$		−	44.0000i		−	1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$		−	44.0000i		−	1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$			20.0000i			0.710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$		0			0
$802$		0			0
$803$			20.0000i			0.705785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	−14.0000			−0.491606			−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000			−0.491606			−0.245803	+	0.969320i	$0.579052\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$			16.0000i			0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$			4.00000i			0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$			4.00000i			0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	8.00000i		−	0.277184i
$834$		0			0
$835$	−16.0000	+	8.00000i	−0.553703	+	0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−44.0000			−1.51905			−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000			−1.51905			−0.759524	+	0.650479i	$0.774568\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.00000	+	18.0000i	0.309609	+	0.619219i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			10.0000i			0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$			10.0000i			0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			20.0000i			0.683187i	0.939848	+	0.341593i	$0.110967\pi$
$857$			20.0000i			0.683187i	−0.939848	+	0.341593i	$0.889033\pi$
$858$		0			0
$859$	−6.00000			−0.204717			−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000			−0.204717			−0.102359	+	0.994748i	$0.532639\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$	−32.0000	+	16.0000i	−1.08803	+	0.544016i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.00000	−	11.0000i	0.0676123	−	0.371868i
$876$		0			0
$877$			48.0000i			1.62084i	0.585846	+	0.810422i	$0.300762\pi$
$877$			48.0000i			1.62084i	−0.585846	+	0.810422i	$0.699238\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$		−	52.0000i		−	1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$		−	52.0000i		−	1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36.0000i		−	1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$		−	36.0000i		−	1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$		0			0
$892$		0			0
$893$			8.00000i			0.267710i
$894$		0			0
$895$	2.00000	+	4.00000i	0.0668526	+	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36.0000			1.20067
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.0000	+	44.0000i	0.731305	+	1.46261i
$906$		0			0
$907$		−	44.0000i		−	1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$		−	44.0000i		−	1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$		−	32.0000i		−	1.05905i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	20.0000i		−	0.660458i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	28.0000i		−	0.921631i
$924$		0			0
$925$	32.0000	+	24.0000i	1.05215	+	0.789115i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	2.00000			0.0655474
$932$		0			0
$933$		0			0
$934$		0			0
$935$	32.0000	−	16.0000i	1.04651	−	0.523256i
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	20.0000			0.649227
$950$		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
$955$	14.0000	+	28.0000i	0.453029	+	0.906059i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.0000			0.452084
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$		−	2.00000i		−	0.0641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	−20.0000			−0.639203
$980$		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
$985$	12.0000	−	6.00000i	0.382352	−	0.191176i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	60.0000			1.90596			0.952981	−	0.303029i	$-0.0979978\pi$
$991$	60.0000			1.90596			0.952981	+	0.303029i	$0.0979978\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−26.0000	−	52.0000i	−0.824255	−	1.64851i
$996$		0			0
$997$			38.0000i			1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$			38.0000i			1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.k.1009.2		2
3.2	odd	2		1680.2.t.d.1009.2		2
4.3	odd	2		630.2.g.d.379.2		2
5.4	even	2	inner	5040.2.t.k.1009.1		2
12.11	even	2		210.2.g.a.169.1	✓	2
15.2	even	4		8400.2.a.ca.1.1		1
15.8	even	4		8400.2.a.bd.1.1		1
15.14	odd	2		1680.2.t.d.1009.1		2
20.3	even	4		3150.2.a.be.1.1		1
20.7	even	4		3150.2.a.q.1.1		1
20.19	odd	2		630.2.g.d.379.1		2
60.23	odd	4		1050.2.a.g.1.1		1
60.47	odd	4		1050.2.a.m.1.1		1
60.59	even	2		210.2.g.a.169.2	yes	2
84.11	even	6		1470.2.n.g.79.2		4
84.23	even	6		1470.2.n.g.949.1		4
84.47	odd	6		1470.2.n.c.949.1		4
84.59	odd	6		1470.2.n.c.79.2		4
84.83	odd	2		1470.2.g.e.589.1		2
420.59	odd	6		1470.2.n.c.79.1		4
420.83	even	4		7350.2.a.g.1.1		1
420.167	even	4		7350.2.a.co.1.1		1
420.179	even	6		1470.2.n.g.79.1		4
420.299	odd	6		1470.2.n.c.949.2		4
420.359	even	6		1470.2.n.g.949.2		4
420.419	odd	2		1470.2.g.e.589.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.a.169.1	✓	2	12.11	even	2
210.2.g.a.169.2	yes	2	60.59	even	2
630.2.g.d.379.1		2	20.19	odd	2
630.2.g.d.379.2		2	4.3	odd	2
1050.2.a.g.1.1		1	60.23	odd	4
1050.2.a.m.1.1		1	60.47	odd	4
1470.2.g.e.589.1		2	84.83	odd	2
1470.2.g.e.589.2		2	420.419	odd	2
1470.2.n.c.79.1		4	420.59	odd	6
1470.2.n.c.79.2		4	84.59	odd	6
1470.2.n.c.949.1		4	84.47	odd	6
1470.2.n.c.949.2		4	420.299	odd	6
1470.2.n.g.79.1		4	420.179	even	6
1470.2.n.g.79.2		4	84.11	even	6
1470.2.n.g.949.1		4	84.23	even	6
1470.2.n.g.949.2		4	420.359	even	6
1680.2.t.d.1009.1		2	15.14	odd	2
1680.2.t.d.1009.2		2	3.2	odd	2
3150.2.a.q.1.1		1	20.7	even	4
3150.2.a.be.1.1		1	20.3	even	4
5040.2.t.k.1009.1		2	5.4	even	2	inner
5040.2.t.k.1009.2		2	1.1	even	1	trivial
7350.2.a.g.1.1		1	420.83	even	4
7350.2.a.co.1.1		1	420.167	even	4
8400.2.a.bd.1.1		1	15.8	even	4
8400.2.a.ca.1.1		1	15.2	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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(1.00000 + 2.00000i) q^{5} +1.00000i q^{7} -2.00000 q^{11} +2.00000i q^{13} +8.00000i q^{17} -2.00000 q^{19} +(-3.00000 + 4.00000i) q^{25} -6.00000 q^{29} -6.00000 q^{31} +(-2.00000 + 1.00000i) q^{35} -8.00000i q^{37} -6.00000 q^{41} -8.00000i q^{43} -4.00000i q^{47} -1.00000 q^{49} +2.00000i q^{53} +(-2.00000 - 4.00000i) q^{55} +8.00000 q^{59} +10.0000 q^{61} +(-4.00000 + 2.00000i) q^{65} -12.0000i q^{67} -14.0000 q^{71} -10.0000i q^{73} -2.00000i q^{77} +4.00000 q^{79} +16.0000i q^{83} +(-16.0000 + 8.00000i) q^{85} +10.0000 q^{89} -2.00000 q^{91} +(-2.00000 - 4.00000i) q^{95} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 4 q^{11} - 4 q^{19} - 6 q^{25} - 12 q^{29} - 12 q^{31} - 4 q^{35} - 12 q^{41} - 2 q^{49} - 4 q^{55} + 16 q^{59} + 20 q^{61} - 8 q^{65} - 28 q^{71} + 8 q^{79} - 32 q^{85} + 20 q^{89} - 4 q^{91} - 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^11 - 4 * q^19 - 6 * q^25 - 12 * q^29 - 12 * q^31 - 4 * q^35 - 12 * q^41 - 2 * q^49 - 4 * q^55 + 16 * q^59 + 20 * q^61 - 8 * q^65 - 28 * q^71 + 8 * q^79 - 32 * q^85 + 20 * q^89 - 4 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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