LMFDB - Embedded newform 5040.2.t.d.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 420)
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.d.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +O(q^{10})$	$q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +4.00000 q^{11} -2.00000i q^{13} -2.00000i q^{17} -2.00000 q^{19} +6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{29} -6.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -4.00000i q^{37} -4.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -2.00000i q^{53} +(-8.00000 + 4.00000i) q^{55} -4.00000 q^{59} -2.00000 q^{61} +(2.00000 + 4.00000i) q^{65} +12.0000i q^{67} -8.00000 q^{71} -14.0000i q^{73} -4.00000i q^{77} +16.0000 q^{79} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +16.0000 q^{89} -2.00000 q^{91} +(4.00000 - 2.00000i) q^{95} -14.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} + 8 q^{11} - 4 q^{19} + 6 q^{25} + 12 q^{29} - 12 q^{31} + 2 q^{35} - 2 q^{49} - 16 q^{55} - 8 q^{59} - 4 q^{61} + 4 q^{65} - 16 q^{71} + 32 q^{79} + 4 q^{85} + 32 q^{89} - 4 q^{91} + 8 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 + 8 * q^11 - 4 * q^19 + 6 * q^25 + 12 * q^29 - 12 * q^31 + 2 * q^35 - 2 * q^49 - 16 * q^55 - 8 * q^59 - 4 * q^61 + 4 * q^65 - 16 * q^71 + 32 * q^79 + 4 * q^85 + 32 * q^89 - 4 * q^91 + 8 * 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
$5$	−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
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		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
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		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	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		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
$35$	1.00000	+	2.00000i	0.169031	+	0.338062i
$36$		0			0
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		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	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$	−8.00000	+	4.00000i	−1.07872	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	4.00000i	0.248069	+	0.496139i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	2.00000	+	4.00000i	0.216930	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.0000			1.69600			0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000			1.69600			0.847998	+	0.529999i	$0.177808\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	2.00000i	0.410391	−	0.205196i
$96$		0			0
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18.0000i			1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$			18.0000i			1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	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
$115$	−6.00000	−	12.0000i	−0.559503	−	1.11901i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000			−0.183340
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$			2.00000i			0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.0000i		−	0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$		−	10.0000i		−	0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	8.00000i		−	0.668994i
$144$		0			0
$145$	−12.0000	+	6.00000i	−0.996546	+	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		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
$155$	12.0000	−	6.00000i	0.963863	−	0.481932i
$156$		0			0
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$		−	24.0000i		−	1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$		−	24.0000i		−	1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		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
$185$	4.00000	+	8.00000i	0.294086	+	0.588172i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	24.0000i		−	1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$		−	24.0000i		−	1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000			−0.553372
$210$		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
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4.00000i			0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$			4.00000i			0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$		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
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$	−4.00000	−	8.00000i	−0.260931	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.00000	−	1.00000i	0.127775	−	0.0638877i
$246$		0			0
$247$			4.00000i			0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			24.0000i			1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.0000i		−	1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$		−	22.0000i		−	1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	30.0000i		−	1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$		−	30.0000i		−	1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$		0			0
$265$	2.00000	+	4.00000i	0.122859	+	0.245718i
$266$		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
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.0000	−	16.0000i	0.723627	−	0.964836i
$276$		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
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	22.0000i		−	1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$		−	22.0000i		−	1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$		0			0
$295$	8.00000	−	4.00000i	0.465778	−	0.232889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000			0.693978
$300$		0			0
$301$	−4.00000			−0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000	−	2.00000i	0.229039	−	0.114520i
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$		−	22.0000i		−	1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$		−	22.0000i		−	1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			26.0000i			1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$			26.0000i			1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.00000i			0.222566i
$324$		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
$331$	24.0000			1.31916			0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000			1.31916			0.659580	+	0.751635i	$0.270734\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000	−	24.0000i	−0.655630	−	1.31126i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.0000			−1.29967
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.0000i			0.536828i	0.963304	+	0.268414i	$0.0864995\pi$
$347$			10.0000i			0.536828i	−0.963304	+	0.268414i	$0.913500\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			30.0000i			1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$			30.0000i			1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$		0			0
$355$	16.0000	−	8.00000i	0.849192	−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0000	+	28.0000i	0.732793	+	1.46559i
$366$		0			0
$367$		−	16.0000i		−	0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$		−	16.0000i		−	0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.00000			−0.103835
$372$		0			0
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$		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
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		0			0
$385$	4.00000	+	8.00000i	0.203859	+	0.407718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−32.0000	+	16.0000i	−1.61009	+	0.805047i
$396$		0			0
$397$		−	34.0000i		−	1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$		−	34.0000i		−	1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.00000i			0.196827i
$414$		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	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$			2.00000i			0.0967868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	12.0000i		−	0.574038i
$438$		0			0
$439$	−6.00000			−0.286364			−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000			−0.286364			−0.143182	+	0.989696i	$0.545733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			6.00000i			0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$			6.00000i			0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$		0			0
$445$	−32.0000	+	16.0000i	−1.51695	+	0.758473i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.00000	−	2.00000i	0.187523	−	0.0937614i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	24.0000			1.11779			0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000			1.11779			0.558896	+	0.829238i	$0.311225\pi$
$462$		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
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		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
$479$	4.00000			0.182765			0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000			0.182765			0.0913823	+	0.995816i	$0.470871\pi$
$480$		0			0
$481$	−8.00000			−0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.0000	+	28.0000i	0.635707	+	1.27141i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$		−	12.0000i		−	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$		0			0
$505$	16.0000	−	8.00000i	0.711991	−	0.355995i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.0000			−1.24108			−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000			−1.24108			−0.620539	+	0.784176i	$0.713086\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		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
$527$			12.0000i			0.522728i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−18.0000	−	36.0000i	−0.778208	−	1.55642i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.00000			−0.172292
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	36.0000	−	18.0000i	1.54207	−	0.771035i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000			−0.511217
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	6.00000	+	12.0000i	0.252422	+	0.504844i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	8.00000			0.334790			0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000			0.334790			0.167395	+	0.985890i	$0.446465\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000	+	18.0000i	1.00087	+	0.750652i
$576$		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
$583$		−	8.00000i		−	0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			16.0000i			0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$			16.0000i			0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$	4.00000	−	2.00000i	0.163984	−	0.0819920i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.00000			−0.326871			−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000			−0.326871			−0.163436	+	0.986554i	$0.552258\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	5.00000i	−0.406558	+	0.203279i
$606$		0			0
$607$		−	8.00000i		−	0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$		−	8.00000i		−	0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000			0.323645
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$	−30.0000			−1.20580			−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000			−1.20580			−0.602901	+	0.797816i	$0.705989\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	16.0000i		−	0.641026i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	−24.0000			−0.955425			−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000			−0.955425			−0.477712	+	0.878516i	$0.658534\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			2.00000i			0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			40.0000i			1.57256i	0.617869	+	0.786281i	$0.287996\pi$
$647$			40.0000i			1.57256i	−0.617869	+	0.786281i	$0.712004\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$	−8.00000	+	4.00000i	−0.312586	+	0.156293i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.00000	−	4.00000i	−0.0775567	−	0.155113i
$666$		0			0
$667$			36.0000i			1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			42.0000i			1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$			42.0000i			1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$		0			0
$679$	−14.0000			−0.537271
$680$		0			0
$681$		0			0
$682$		0			0
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$	10.0000	+	20.0000i	0.382080	+	0.764161i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00000			−0.152388
$690$		0			0
$691$	−22.0000			−0.836919			−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000			−0.836919			−0.418460	+	0.908235i	$0.637430\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.0000	−	14.0000i	1.06210	−	0.531050i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$			8.00000i			0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			8.00000i			0.300871i
$708$		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
$713$		−	36.0000i		−	1.34821i
$714$		0			0
$715$	8.00000	+	16.0000i	0.299183	+	0.598366i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.0000			−1.04422			−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000			−1.04422			−0.522112	+	0.852877i	$0.674856\pi$
$720$		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	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000			−0.295891
$732$		0			0
$733$		−	42.0000i		−	1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$		−	42.0000i		−	1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.0000i			1.76810i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	34.0000i		−	1.24734i	−0.781688	−	0.623670i	$-0.785641\pi$
$743$		−	34.0000i		−	1.24734i	0.781688	−	0.623670i	$-0.214359\pi$
$744$		0			0
$745$	−12.0000	+	6.00000i	−0.439646	+	0.219823i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000			0.657706
$750$		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
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$			18.0000i			0.651644i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8.00000i			0.288863i
$768$		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
$773$			10.0000i			0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$			10.0000i			0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$		0			0
$775$	−18.0000	+	24.0000i	−0.646579	+	0.862105i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.00000	−	4.00000i	−0.0713831	−	0.142766i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$			4.00000i			0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	56.0000i		−	1.97620i
$804$		0			0
$805$	−12.0000	+	6.00000i	−0.422944	+	0.211472i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$	22.0000			0.772524			0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000			0.772524			0.386262	+	0.922389i	$0.373766\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.0000	+	48.0000i	0.840683	+	1.68137i
$816$		0			0
$817$			8.00000i			0.279885i
$818$		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
$823$			8.00000i			0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$			8.00000i			0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	30.0000i		−	1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$		−	30.0000i		−	1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			2.00000i			0.0692959i
$834$		0			0
$835$	8.00000	+	16.0000i	0.276851	+	0.553703i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.0000			0.552381			0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000			0.552381			0.276191	+	0.961103i	$0.410928\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.0000	+	9.00000i	−0.619219	+	0.309609i
$846$		0			0
$847$		−	5.00000i		−	0.171802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			10.0000i			0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$			10.0000i			0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$		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
$863$			54.0000i			1.83818i	0.394046	+	0.919091i	$0.371075\pi$
$863$			54.0000i			1.83818i	−0.394046	+	0.919091i	$0.628925\pi$
$864$		0			0
$865$	−14.0000	−	28.0000i	−0.476014	−	0.952029i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	64.0000			2.17105
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.0000	+	2.00000i	0.371868	+	0.0676123i
$876$		0			0
$877$			36.0000i			1.21563i	0.794077	+	0.607817i	$0.207955\pi$
$877$			36.0000i			1.21563i	−0.794077	+	0.607817i	$0.792045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$		−	56.0000i		−	1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$		−	56.0000i		−	1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$		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
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	−40.0000	+	20.0000i	−1.33705	+	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.0000			−1.20067
$900$		0			0
$901$	−4.00000			−0.133259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−44.0000	+	22.0000i	−1.46261	+	0.731305i
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		−	64.0000i		−	2.11809i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.00000i		−	0.132092i
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			16.0000i			0.526646i
$924$		0			0
$925$	−16.0000	−	12.0000i	−0.526077	−	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.0000			1.57483			0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000			1.57483			0.787414	+	0.616424i	$0.211419\pi$
$930$		0			0
$931$	2.00000			0.0655474
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.00000	+	16.0000i	0.261628	+	0.523256i
$936$		0			0
$937$			10.0000i			0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$			10.0000i			0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	60.0000			1.95594			0.977972	−	0.208736i	$-0.0669349\pi$
$941$	60.0000			1.95594			0.977972	+	0.208736i	$0.0669349\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		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
$955$	−16.0000	+	8.00000i	−0.517748	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.0000			−0.322917
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.0000	+	48.0000i	0.772587	+	1.54517i
$966$		0			0
$967$			36.0000i			1.15768i	0.815440	+	0.578841i	$0.196495\pi$
$967$			36.0000i			1.15768i	−0.815440	+	0.578841i	$0.803505\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$			14.0000i			0.448819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			42.0000i			1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		0			0
$979$	64.0000			2.04545
$980$		0			0
$981$		0			0
$982$		0			0
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$		0			0
$985$	18.0000	+	36.0000i	0.573528	+	1.14706i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	−36.0000			−1.14358			−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000			−1.14358			−0.571789	+	0.820401i	$0.693750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.0000	+	10.0000i	−0.634043	+	0.317021i
$996$		0			0
$997$			22.0000i			0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$			22.0000i			0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.t.d.1009.2		2
3.2	odd	2		1680.2.t.g.1009.1		2
4.3	odd	2		1260.2.k.a.1009.2		2
5.4	even	2	inner	5040.2.t.d.1009.1		2
12.11	even	2		420.2.k.b.169.2	yes	2
15.2	even	4		8400.2.a.o.1.1		1
15.8	even	4		8400.2.a.bm.1.1		1
15.14	odd	2		1680.2.t.g.1009.2		2
20.3	even	4		6300.2.a.r.1.1		1
20.7	even	4		6300.2.a.b.1.1		1
20.19	odd	2		1260.2.k.a.1009.1		2
60.23	odd	4		2100.2.a.i.1.1		1
60.47	odd	4		2100.2.a.n.1.1		1
60.59	even	2		420.2.k.b.169.1	✓	2
84.11	even	6		2940.2.bb.a.1549.2		4
84.23	even	6		2940.2.bb.a.949.1		4
84.47	odd	6		2940.2.bb.f.949.2		4
84.59	odd	6		2940.2.bb.f.1549.1		4
84.83	odd	2		2940.2.k.b.589.1		2
420.59	odd	6		2940.2.bb.f.1549.2		4
420.179	even	6		2940.2.bb.a.1549.1		4
420.299	odd	6		2940.2.bb.f.949.1		4
420.359	even	6		2940.2.bb.a.949.2		4
420.419	odd	2		2940.2.k.b.589.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.k.b.169.1	✓	2	60.59	even	2
420.2.k.b.169.2	yes	2	12.11	even	2
1260.2.k.a.1009.1		2	20.19	odd	2
1260.2.k.a.1009.2		2	4.3	odd	2
1680.2.t.g.1009.1		2	3.2	odd	2
1680.2.t.g.1009.2		2	15.14	odd	2
2100.2.a.i.1.1		1	60.23	odd	4
2100.2.a.n.1.1		1	60.47	odd	4
2940.2.k.b.589.1		2	84.83	odd	2
2940.2.k.b.589.2		2	420.419	odd	2
2940.2.bb.a.949.1		4	84.23	even	6
2940.2.bb.a.949.2		4	420.359	even	6
2940.2.bb.a.1549.1		4	420.179	even	6
2940.2.bb.a.1549.2		4	84.11	even	6
2940.2.bb.f.949.1		4	420.299	odd	6
2940.2.bb.f.949.2		4	84.47	odd	6
2940.2.bb.f.1549.1		4	84.59	odd	6
2940.2.bb.f.1549.2		4	420.59	odd	6
5040.2.t.d.1009.1		2	5.4	even	2	inner
5040.2.t.d.1009.2		2	1.1	even	1	trivial
6300.2.a.b.1.1		1	20.7	even	4
6300.2.a.r.1.1		1	20.3	even	4
8400.2.a.o.1.1		1	15.2	even	4
8400.2.a.bm.1.1		1	15.8	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)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+(-2.00000 + 1.00000i) q^{5} -1.00000i q^{7} +4.00000 q^{11} -2.00000i q^{13} -2.00000i q^{17} -2.00000 q^{19} +6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{29} -6.00000 q^{31} +(1.00000 + 2.00000i) q^{35} -4.00000i q^{37} -4.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -2.00000i q^{53} +(-8.00000 + 4.00000i) q^{55} -4.00000 q^{59} -2.00000 q^{61} +(2.00000 + 4.00000i) q^{65} +12.0000i q^{67} -8.00000 q^{71} -14.0000i q^{73} -4.00000i q^{77} +16.0000 q^{79} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +16.0000 q^{89} -2.00000 q^{91} +(4.00000 - 2.00000i) q^{95} -14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} + 8 q^{11} - 4 q^{19} + 6 q^{25} + 12 q^{29} - 12 q^{31} + 2 q^{35} - 2 q^{49} - 16 q^{55} - 8 q^{59} - 4 q^{61} + 4 q^{65} - 16 q^{71} + 32 q^{79} + 4 q^{85} + 32 q^{89} - 4 q^{91} + 8 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 + 8 * q^11 - 4 * q^19 + 6 * q^25 + 12 * q^29 - 12 * q^31 + 2 * q^35 - 2 * q^49 - 16 * q^55 - 8 * q^59 - 4 * q^61 + 4 * q^65 - 16 * q^71 + 32 * q^79 + 4 * q^85 + 32 * q^89 - 4 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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