LMFDB - Embedded newform 1440.2.b.d.431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(431,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("1440.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.b (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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2580992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + x^{4} + 2x^{2} - 8x + 8 $ x^6 - 2x^5 + x^4 + 2x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.5
Root			$1.38078 + 0.305697i$ of defining polynomial
Character	$\chi$	$=$	1440.431
Dual form			1440.2.b.d.431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.41421i q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.41421i q^{7} +0.191427i q^{11} -2.63700i q^{13} +6.20522i q^{17} +1.52311 q^{19} +5.25240 q^{23} +1.00000 q^{25} -0.270718 q^{29} +6.20522i q^{31} +1.41421i q^{35} -7.61944i q^{37} +9.22508i q^{41} +12.7755 q^{43} -3.79383 q^{47} +5.00000 q^{49} -8.77551 q^{53} +0.191427i q^{55} +10.4479i q^{59} +0.382853i q^{61} -2.63700i q^{65} -1.72928 q^{67} +9.72928 q^{71} -5.45856 q^{73} -0.270718 q^{77} -14.3077i q^{79} +15.2389i q^{83} +6.20522i q^{85} +3.56822i q^{89} +3.72928 q^{91} +1.52311 q^{95} +7.31695 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5}+O(q^{10}) $ 6 * q + 6 * q^5	$ 6 q + 6 q^{5} - 16 q^{19} - 4 q^{23} + 6 q^{25} - 12 q^{29} + 16 q^{43} - 8 q^{47} + 30 q^{49} + 8 q^{53} + 48 q^{71} - 12 q^{73} - 12 q^{77} + 12 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100}) $ 6 * q + 6 * q^5 - 16 * q^19 - 4 * q^23 + 6 * q^25 - 12 * q^29 + 16 * q^43 - 8 * q^47 + 30 * q^49 + 8 * q^53 + 48 * q^71 - 12 * q^73 - 12 * q^77 + 12 * q^91 - 16 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.41421i	0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$	1.41421i	0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.191427i	0.0577173i	0.999584	+	0.0288587i	$0.00918727\pi$
$11$	0.191427i	0.0577173i	−0.999584	+	0.0288587i	$0.990813\pi$
$12$	0	0
$13$	− 2.63700i	− 0.731372i	−0.930738	−	0.365686i	$-0.880834\pi$
$13$	− 2.63700i	− 0.731372i	0.930738	−	0.365686i	$-0.119166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.20522i	1.50499i	0.658599	+	0.752494i	$0.271149\pi$
$17$	6.20522i	1.50499i	−0.658599	+	0.752494i	$0.728851\pi$
$18$	0	0
$19$	1.52311	0.349426	0.174713	−	0.984619i	$-0.444100\pi$
$19$	1.52311	0.349426	0.174713	+	0.984619i	$0.444100\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.25240	1.09520	0.547600	−	0.836740i	$-0.315542\pi$
$23$	5.25240	1.09520	0.547600	+	0.836740i	$0.315542\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.270718	−0.0502711	−0.0251356	−	0.999684i	$-0.508002\pi$
$29$	−0.270718	−0.0502711	−0.0251356	+	0.999684i	$0.508002\pi$
$30$	0	0
$31$	6.20522i	1.11449i	0.830348	+	0.557245i	$0.188142\pi$
$31$	6.20522i	1.11449i	−0.830348	+	0.557245i	$0.811858\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.41421i	0.239046i
$36$	0	0
$37$	− 7.61944i	− 1.25263i	−0.779571	−	0.626314i	$-0.784563\pi$
$37$	− 7.61944i	− 1.25263i	0.779571	−	0.626314i	$-0.215437\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.22508i	1.44071i	0.693603	+	0.720357i	$0.256022\pi$
$41$	9.22508i	1.44071i	−0.693603	+	0.720357i	$0.743978\pi$
$42$	0	0
$43$	12.7755	1.94825	0.974124	−	0.226016i	$-0.0725702\pi$
$43$	12.7755	1.94825	0.974124	+	0.226016i	$0.0725702\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.79383	−0.553387	−0.276694	−	0.960958i	$-0.589239\pi$
$47$	−3.79383	−0.553387	−0.276694	+	0.960958i	$0.589239\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.77551	−1.20541	−0.602705	−	0.797964i	$-0.705910\pi$
$53$	−8.77551	−1.20541	−0.602705	+	0.797964i	$0.705910\pi$
$54$	0	0
$55$	0.191427i	0.0258120i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.4479i	1.36020i	0.733121	+	0.680098i	$0.238063\pi$
$59$	10.4479i	1.36020i	−0.733121	+	0.680098i	$0.761937\pi$
$60$	0	0
$61$	0.382853i	0.0490194i	0.999700	+	0.0245097i	$0.00780245\pi$
$61$	0.382853i	0.0490194i	−0.999700	+	0.0245097i	$0.992198\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.63700i	− 0.327080i
$66$	0	0
$67$	−1.72928	−0.211265	−0.105633	−	0.994405i	$-0.533687\pi$
$67$	−1.72928	−0.211265	−0.105633	+	0.994405i	$0.533687\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.72928	1.15465	0.577327	−	0.816513i	$-0.304096\pi$
$71$	9.72928	1.15465	0.577327	+	0.816513i	$0.304096\pi$
$72$	0	0
$73$	−5.45856	−0.638877	−0.319438	−	0.947607i	$-0.603494\pi$
$73$	−5.45856	−0.638877	−0.319438	+	0.947607i	$0.603494\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.270718	−0.0308512
$78$	0	0
$79$	− 14.3077i	− 1.60974i	−0.593454	−	0.804868i	$-0.702236\pi$
$79$	− 14.3077i	− 1.60974i	0.593454	−	0.804868i	$-0.297764\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.2389i	1.67268i	0.548208	+	0.836342i	$0.315310\pi$
$83$	15.2389i	1.67268i	−0.548208	+	0.836342i	$0.684690\pi$
$84$	0	0
$85$	6.20522i	0.673051i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.56822i	0.378231i	0.981955	+	0.189115i	$0.0605620\pi$
$89$	3.56822i	0.378231i	−0.981955	+	0.189115i	$0.939438\pi$
$90$	0	0
$91$	3.72928	0.390935
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.52311	0.156268
$96$	0	0
$97$	7.31695	0.742923	0.371462	−	0.928448i	$-0.378857\pi$
$97$	7.31695	0.742923	0.371462	+	0.928448i	$0.378857\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5510	1.54738	0.773692	−	0.633562i	$-0.218408\pi$
$101$	15.5510	1.54738	0.773692	+	0.633562i	$0.218408\pi$
$102$	0	0
$103$	2.08863i	0.205799i	0.994692	+	0.102899i	$0.0328120\pi$
$103$	2.08863i	0.205799i	−0.994692	+	0.102899i	$0.967188\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.82843i	0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$	2.82843i	0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$	0	0
$109$	5.27400i	0.505158i	0.967576	+	0.252579i	$0.0812787\pi$
$109$	5.27400i	0.505158i	−0.967576	+	0.252579i	$0.918721\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.82237i	− 0.547722i	−0.961769	−	0.273861i	$-0.911699\pi$
$113$	− 5.82237i	− 0.547722i	0.961769	−	0.273861i	$-0.0883009\pi$
$114$	0	0
$115$	5.25240	0.489788
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.77551	−0.804450
$120$	0	0
$121$	10.9634	0.996669
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	− 20.5388i	− 1.82252i	−0.411829	−	0.911261i	$-0.635110\pi$
$127$	− 20.5388i	− 1.82252i	0.411829	−	0.911261i	$-0.364890\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.5679i	− 1.18543i	−0.805413	−	0.592715i	$-0.798056\pi$
$131$	− 13.5679i	− 1.18543i	0.805413	−	0.592715i	$-0.201944\pi$
$132$	0	0
$133$	2.15401i	0.186776i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.72563i	0.403738i	0.979413	+	0.201869i	$0.0647015\pi$
$137$	4.72563i	0.403738i	−0.979413	+	0.201869i	$0.935298\pi$
$138$	0	0
$139$	−15.7572	−1.33651	−0.668254	−	0.743934i	$-0.732958\pi$
$139$	−15.7572	−1.33651	−0.668254	+	0.743934i	$0.732958\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.504792	0.0422129
$144$	0	0
$145$	−0.270718	−0.0224819
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.72928	0.633207	0.316604	−	0.948558i	$-0.397457\pi$
$149$	7.72928	0.633207	0.316604	+	0.948558i	$0.397457\pi$
$150$	0	0
$151$	22.2098i	1.80741i	0.428158	+	0.903704i	$0.359163\pi$
$151$	22.2098i	1.80741i	−0.428158	+	0.903704i	$0.640837\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.20522i	0.498416i
$156$	0	0
$157$	− 8.00229i	− 0.638652i	−0.947645	−	0.319326i	$-0.896543\pi$
$157$	− 8.00229i	− 0.638652i	0.947645	−	0.319326i	$-0.103457\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.42801i	0.585409i
$162$	0	0
$163$	0.953771	0.0747051	0.0373526	−	0.999302i	$-0.488108\pi$
$163$	0.953771	0.0747051	0.0373526	+	0.999302i	$0.488108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.7572	−0.909799	−0.454899	−	0.890543i	$-0.650325\pi$
$167$	−11.7572	−0.909799	−0.454899	+	0.890543i	$0.650325\pi$
$168$	0	0
$169$	6.04623	0.465095
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.22449	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$173$	3.22449	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$174$	0	0
$175$	1.41421i	0.106904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 12.6019i	− 0.941908i	−0.882158	−	0.470954i	$-0.843910\pi$
$179$	− 12.6019i	− 0.941908i	0.882158	−	0.470954i	$-0.156090\pi$
$180$	0	0
$181$	24.1070i	1.79186i	0.444195	+	0.895930i	$0.353490\pi$
$181$	24.1070i	1.79186i	−0.444195	+	0.895930i	$0.646510\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7.61944i	− 0.560192i
$186$	0	0
$187$	−1.18785	−0.0868639
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.9171	−0.789936	−0.394968	−	0.918695i	$-0.629244\pi$
$191$	−10.9171	−0.789936	−0.394968	+	0.918695i	$0.629244\pi$
$192$	0	0
$193$	−15.3169	−1.10254	−0.551269	−	0.834328i	$-0.685856\pi$
$193$	−15.3169	−1.10254	−0.551269	+	0.834328i	$0.685856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.0462	−1.21449	−0.607247	−	0.794513i	$-0.707726\pi$
$197$	−17.0462	−1.21449	−0.607247	+	0.794513i	$0.707726\pi$
$198$	0	0
$199$	4.72563i	0.334991i	0.985873	+	0.167496i	$0.0535680\pi$
$199$	4.72563i	0.334991i	−0.985873	+	0.167496i	$0.946432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.382853i	− 0.0268710i
$204$	0	0
$205$	9.22508i	0.644307i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.291565i	0.0201680i
$210$	0	0
$211$	−16.8401	−1.15932	−0.579659	−	0.814859i	$-0.696814\pi$
$211$	−16.8401	−1.15932	−0.579659	+	0.814859i	$0.696814\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.7755	0.871283
$216$	0	0
$217$	−8.77551	−0.595720
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.3632	1.10071
$222$	0	0
$223$	− 22.3099i	− 1.49398i	−0.664833	−	0.746992i	$-0.731497\pi$
$223$	− 22.3099i	− 1.49398i	0.664833	−	0.746992i	$-0.268503\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.86813i	− 0.588599i	−0.955713	−	0.294299i	$-0.904914\pi$
$227$	− 8.86813i	− 0.588599i	0.955713	−	0.294299i	$-0.0950863\pi$
$228$	0	0
$229$	− 21.2786i	− 1.40613i	−0.711126	−	0.703064i	$-0.751815\pi$
$229$	− 21.2786i	− 1.40613i	0.711126	−	0.703064i	$-0.248185\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.0734i	− 0.987488i	−0.869607	−	0.493744i	$-0.835628\pi$
$233$	− 15.0734i	− 0.987488i	0.869607	−	0.493744i	$-0.164372\pi$
$234$	0	0
$235$	−3.79383	−0.247482
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.0462	−1.23200	−0.615999	−	0.787747i	$-0.711247\pi$
$239$	−19.0462	−1.23200	−0.615999	+	0.787747i	$0.711247\pi$
$240$	0	0
$241$	18.5048	1.19200	0.595999	−	0.802985i	$-0.296756\pi$
$241$	18.5048	1.19200	0.595999	+	0.802985i	$0.296756\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	− 4.01645i	− 0.255561i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16.8186i	− 1.06158i	−0.847503	−	0.530790i	$-0.821895\pi$
$251$	− 16.8186i	− 1.06158i	0.847503	−	0.530790i	$-0.178105\pi$
$252$	0	0
$253$	1.00545i	0.0632120i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.79936i	− 0.611267i	−0.952149	−	0.305634i	$-0.901132\pi$
$257$	− 9.79936i	− 0.611267i	0.952149	−	0.305634i	$-0.0988682\pi$
$258$	0	0
$259$	10.7755	0.669558
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.79383	0.357263	0.178632	−	0.983916i	$-0.442833\pi$
$263$	5.79383	0.357263	0.178632	+	0.983916i	$0.442833\pi$
$264$	0	0
$265$	−8.77551	−0.539075
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.1878	−0.926019	−0.463010	−	0.886353i	$-0.653230\pi$
$269$	−15.1878	−0.926019	−0.463010	+	0.886353i	$0.653230\pi$
$270$	0	0
$271$	10.1304i	0.615377i	0.951487	+	0.307689i	$0.0995555\pi$
$271$	10.1304i	0.615377i	−0.951487	+	0.307689i	$0.900444\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.191427i	0.0115435i
$276$	0	0
$277$	− 14.7559i	− 0.886595i	−0.896375	−	0.443298i	$-0.853809\pi$
$277$	− 14.7559i	− 0.886595i	0.896375	−	0.443298i	$-0.146191\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 21.5442i	− 1.28522i	−0.766193	−	0.642611i	$-0.777851\pi$
$281$	− 21.5442i	− 1.28522i	0.766193	−	0.642611i	$-0.222149\pi$
$282$	0	0
$283$	−18.2707	−1.08608	−0.543041	−	0.839706i	$-0.682727\pi$
$283$	−18.2707	−1.08608	−0.543041	+	0.839706i	$0.682727\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.0462	−0.770095
$288$	0	0
$289$	−21.5048	−1.26499
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	10.4479i	0.608298i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 13.8506i	− 0.800999i
$300$	0	0
$301$	18.0673i	1.04138i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.382853i	0.0219221i
$306$	0	0
$307$	−22.5048	−1.28442	−0.642208	−	0.766530i	$-0.721982\pi$
$307$	−22.5048	−1.28442	−0.642208	+	0.766530i	$0.721982\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.0462	1.76047	0.880235	−	0.474538i	$-0.157385\pi$
$311$	31.0462	1.76047	0.880235	+	0.474538i	$0.157385\pi$
$312$	0	0
$313$	−2.23407	−0.126277	−0.0631387	−	0.998005i	$-0.520111\pi$
$313$	−2.23407	−0.126277	−0.0631387	+	0.998005i	$0.520111\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.5048	−0.927001	−0.463501	−	0.886097i	$-0.653407\pi$
$317$	−16.5048	−0.927001	−0.463501	+	0.886097i	$0.653407\pi$
$318$	0	0
$319$	− 0.0518227i	− 0.00290151i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.45126i	0.525882i
$324$	0	0
$325$	− 2.63700i	− 0.146274i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.36529i	− 0.295798i
$330$	0	0
$331$	12.9817	0.713538	0.356769	−	0.934193i	$-0.383878\pi$
$331$	12.9817	0.713538	0.356769	+	0.934193i	$0.383878\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.72928	−0.0944808
$336$	0	0
$337$	−27.0096	−1.47131	−0.735653	−	0.677359i	$-0.763125\pi$
$337$	−27.0096	−1.47131	−0.735653	+	0.677359i	$0.763125\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.18785	−0.0643254
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.07372i	− 0.272372i	−0.990683	−	0.136186i	$-0.956516\pi$
$347$	− 5.07372i	− 0.272372i	0.990683	−	0.136186i	$-0.0434844\pi$
$348$	0	0
$349$	− 20.1300i	− 1.07754i	−0.842454	−	0.538768i	$-0.818890\pi$
$349$	− 20.1300i	− 1.07754i	0.842454	−	0.538768i	$-0.181110\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 14.3077i	− 0.761519i	−0.924674	−	0.380760i	$-0.875663\pi$
$353$	− 14.3077i	− 0.761519i	0.924674	−	0.380760i	$-0.124337\pi$
$354$	0	0
$355$	9.72928	0.516377
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.8217	1.67949	0.839744	−	0.542983i	$-0.182705\pi$
$359$	31.8217	1.67949	0.839744	+	0.542983i	$0.182705\pi$
$360$	0	0
$361$	−16.6801	−0.877901
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.45856	−0.285714
$366$	0	0
$367$	− 22.9844i	− 1.19977i	−0.800085	−	0.599887i	$-0.795212\pi$
$367$	− 22.9844i	− 1.19977i	0.800085	−	0.599887i	$-0.204788\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 12.4104i	− 0.644318i
$372$	0	0
$373$	14.2423i	0.737437i	0.929541	+	0.368718i	$0.120203\pi$
$373$	14.2423i	0.737437i	−0.929541	+	0.368718i	$0.879797\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.713884i	0.0367669i
$378$	0	0
$379$	8.71096	0.447452	0.223726	−	0.974652i	$-0.428178\pi$
$379$	8.71096	0.447452	0.223726	+	0.974652i	$0.428178\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.7110	−0.956085	−0.478043	−	0.878337i	$-0.658654\pi$
$383$	−18.7110	−0.956085	−0.478043	+	0.878337i	$0.658654\pi$
$384$	0	0
$385$	−0.270718	−0.0137971
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−27.0096	−1.36944	−0.684720	−	0.728806i	$-0.740076\pi$
$389$	−27.0096	−1.36944	−0.684720	+	0.728806i	$0.740076\pi$
$390$	0	0
$391$	32.5923i	1.64826i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 14.3077i	− 0.719896i
$396$	0	0
$397$	− 25.3952i	− 1.27455i	−0.770638	−	0.637274i	$-0.780062\pi$
$397$	− 25.3952i	− 1.27455i	0.770638	−	0.637274i	$-0.219938\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6.01380i	− 0.300315i	−0.988662	−	0.150157i	$-0.952022\pi$
$401$	− 6.01380i	− 0.300315i	0.988662	−	0.150157i	$-0.0479780\pi$
$402$	0	0
$403$	16.3632	0.815108
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.45856	0.0722983
$408$	0	0
$409$	−19.5510	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$409$	−19.5510	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.7755	−0.727055
$414$	0	0
$415$	15.2389i	0.748047i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.96258i	0.0958784i	0.998850	+	0.0479392i	$0.0152654\pi$
$419$	1.96258i	0.0958784i	−0.998850	+	0.0479392i	$0.984735\pi$
$420$	0	0
$421$	4.30802i	0.209960i	0.994474	+	0.104980i	$0.0334778\pi$
$421$	4.30802i	0.209960i	−0.994474	+	0.104980i	$0.966522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.20522i	0.300998i
$426$	0	0
$427$	−0.541436	−0.0262019
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.8217	0.569433	0.284717	−	0.958612i	$-0.408101\pi$
$431$	11.8217	0.569433	0.284717	+	0.958612i	$0.408101\pi$
$432$	0	0
$433$	9.45856	0.454550	0.227275	−	0.973831i	$-0.427018\pi$
$433$	9.45856	0.454550	0.227275	+	0.973831i	$0.427018\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	16.0393i	0.765516i	0.923849	+	0.382758i	$0.125026\pi$
$439$	16.0393i	0.765516i	−0.923849	+	0.382758i	$0.874974\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 25.0730i	− 1.19125i	−0.803261	−	0.595627i	$-0.796904\pi$
$443$	− 25.0730i	− 1.19125i	0.803261	−	0.595627i	$-0.203096\pi$
$444$	0	0
$445$	3.56822i	0.169150i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.24264i	− 0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$	− 4.24264i	− 0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$	0	0
$451$	−1.76593	−0.0831542
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.72928	0.174831
$456$	0	0
$457$	28.8680	1.35039	0.675193	−	0.737641i	$-0.264060\pi$
$457$	28.8680	1.35039	0.675193	+	0.737641i	$0.264060\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	36.7389	1.71110	0.855550	−	0.517721i	$-0.173219\pi$
$461$	36.7389	1.71110	0.855550	+	0.517721i	$0.173219\pi$
$462$	0	0
$463$	− 16.6136i	− 0.772100i	−0.922478	−	0.386050i	$-0.873839\pi$
$463$	− 16.6136i	− 0.772100i	0.922478	−	0.386050i	$-0.126161\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.47959i	− 0.0684673i	−0.999414	−	0.0342336i	$-0.989101\pi$
$467$	− 1.47959i	− 0.0684673i	0.999414	−	0.0342336i	$-0.0108990\pi$
$468$	0	0
$469$	− 2.44557i	− 0.112926i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.44557i	0.112448i
$474$	0	0
$475$	1.52311	0.0698853
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.81215	−0.128491	−0.0642453	−	0.997934i	$-0.520464\pi$
$479$	−2.81215	−0.128491	−0.0642453	+	0.997934i	$0.520464\pi$
$480$	0	0
$481$	−20.0925	−0.916137
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.31695	0.332245
$486$	0	0
$487$	16.9447i	0.767835i	0.923367	+	0.383918i	$0.125425\pi$
$487$	16.9447i	0.767835i	−0.923367	+	0.383918i	$0.874575\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.0696i	1.17650i	0.808678	+	0.588252i	$0.200184\pi$
$491$	26.0696i	1.17650i	−0.808678	+	0.588252i	$0.799816\pi$
$492$	0	0
$493$	− 1.67987i	− 0.0756574i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.7593i	0.617188i
$498$	0	0
$499$	−22.5327	−1.00870	−0.504351	−	0.863499i	$-0.668268\pi$
$499$	−22.5327	−1.00870	−0.504351	+	0.863499i	$0.668268\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8.80342	−0.392525	−0.196262	−	0.980551i	$-0.562880\pi$
$503$	−8.80342	−0.392525	−0.196262	+	0.980551i	$0.562880\pi$
$504$	0	0
$505$	15.5510	0.692011
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.6339	−0.914581	−0.457291	−	0.889317i	$-0.651180\pi$
$509$	−20.6339	−0.914581	−0.457291	+	0.889317i	$0.651180\pi$
$510$	0	0
$511$	− 7.71957i	− 0.341494i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.08863i	0.0920361i
$516$	0	0
$517$	− 0.726241i	− 0.0319400i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.55066i	0.374611i	0.982302	+	0.187306i	$0.0599755\pi$
$521$	8.55066i	0.374611i	−0.982302	+	0.187306i	$0.940025\pi$
$522$	0	0
$523$	−33.0096	−1.44341	−0.721704	−	0.692202i	$-0.756641\pi$
$523$	−33.0096	−1.44341	−0.721704	+	0.692202i	$0.756641\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.5048	−1.67730
$528$	0	0
$529$	4.58767	0.199464
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.3265	1.05370
$534$	0	0
$535$	2.82843i	0.122284i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.957133i	0.0412267i
$540$	0	0
$541$	27.6493i	1.18874i	0.804193	+	0.594369i	$0.202598\pi$
$541$	27.6493i	1.18874i	−0.804193	+	0.594369i	$0.797402\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.27400i	0.225913i
$546$	0	0
$547$	9.85838	0.421514	0.210757	−	0.977538i	$-0.432407\pi$
$547$	9.85838	0.421514	0.210757	+	0.977538i	$0.432407\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.412335	−0.0175661
$552$	0	0
$553$	20.2341	0.860440
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.68305	0.113685	0.0568423	−	0.998383i	$-0.481897\pi$
$557$	2.68305	0.113685	0.0568423	+	0.998383i	$0.481897\pi$
$558$	0	0
$559$	− 33.6890i	− 1.42489i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0794i	0.509087i	0.967061	+	0.254543i	$0.0819251\pi$
$563$	12.0794i	0.509087i	−0.967061	+	0.254543i	$0.918075\pi$
$564$	0	0
$565$	− 5.82237i	− 0.244949i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.8479i	0.664379i	0.943213	+	0.332190i	$0.107787\pi$
$569$	15.8479i	0.664379i	−0.943213	+	0.332190i	$0.892213\pi$
$570$	0	0
$571$	−1.11078	−0.0464847	−0.0232423	−	0.999730i	$-0.507399\pi$
$571$	−1.11078	−0.0464847	−0.0232423	+	0.999730i	$0.507399\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.25240	0.219040
$576$	0	0
$577$	14.7755	0.615113	0.307556	−	0.951530i	$-0.400489\pi$
$577$	14.7755	0.615113	0.307556	+	0.951530i	$0.400489\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−21.5510	−0.894087
$582$	0	0
$583$	− 1.67987i	− 0.0695730i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.5590i	0.559640i	0.960052	+	0.279820i	$0.0902748\pi$
$587$	13.5590i	0.559640i	−0.960052	+	0.279820i	$0.909725\pi$
$588$	0	0
$589$	9.45126i	0.389433i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.9921i	1.31376i	0.753996	+	0.656879i	$0.228124\pi$
$593$	31.9921i	1.31376i	−0.753996	+	0.656879i	$0.771876\pi$
$594$	0	0
$595$	−8.77551	−0.359761
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.82174	−0.156152	−0.0780760	−	0.996947i	$-0.524878\pi$
$599$	−3.82174	−0.156152	−0.0780760	+	0.996947i	$0.524878\pi$
$600$	0	0
$601$	31.6435	1.29076	0.645382	−	0.763860i	$-0.276698\pi$
$601$	31.6435	1.29076	0.645382	+	0.763860i	$0.276698\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.9634	0.445724
$606$	0	0
$607$	− 34.7204i	− 1.40926i	−0.709576	−	0.704629i	$-0.751114\pi$
$607$	− 34.7204i	− 1.40926i	0.709576	−	0.704629i	$-0.248886\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.0043i	0.404732i
$612$	0	0
$613$	− 26.3612i	− 1.06472i	−0.846519	−	0.532359i	$-0.821306\pi$
$613$	− 26.3612i	− 1.06472i	0.846519	−	0.532359i	$-0.178694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.88509i	− 0.317442i	−0.987323	−	0.158721i	$-0.949263\pi$
$617$	− 7.88509i	− 0.317442i	0.987323	−	0.158721i	$-0.0507370\pi$
$618$	0	0
$619$	38.9325	1.56483	0.782415	−	0.622757i	$-0.213988\pi$
$619$	38.9325	1.56483	0.782415	+	0.622757i	$0.213988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.04623	−0.202173
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47.2803	1.88519
$630$	0	0
$631$	0.800468i	0.0318661i	0.999873	+	0.0159331i	$0.00507186\pi$
$631$	0.800468i	0.0318661i	−0.999873	+	0.0159331i	$0.994928\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 20.5388i	− 0.815057i
$636$	0	0
$637$	− 13.1850i	− 0.522409i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 22.8931i	− 0.904222i	−0.891962	−	0.452111i	$-0.850671\pi$
$641$	− 22.8931i	− 0.904222i	0.891962	−	0.452111i	$-0.149329\pi$
$642$	0	0
$643$	12.9538	0.510847	0.255423	−	0.966829i	$-0.417785\pi$
$643$	12.9538	0.510847	0.255423	+	0.966829i	$0.417785\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6647	0.615844	0.307922	−	0.951412i	$-0.400366\pi$
$647$	15.6647	0.615844	0.307922	+	0.951412i	$0.400366\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.5144	1.62458	0.812292	−	0.583252i	$-0.198220\pi$
$653$	41.5144	1.62458	0.812292	+	0.583252i	$0.198220\pi$
$654$	0	0
$655$	− 13.5679i	− 0.530140i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.8859i	1.35896i	0.733693	+	0.679481i	$0.237795\pi$
$659$	34.8859i	1.35896i	−0.733693	+	0.679481i	$0.762205\pi$
$660$	0	0
$661$	19.7990i	0.770091i	0.922897	+	0.385046i	$0.125814\pi$
$661$	19.7990i	0.770091i	−0.922897	+	0.385046i	$0.874186\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.15401i	0.0835289i
$666$	0	0
$667$	−1.42192	−0.0550569
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.0732884	−0.00282927
$672$	0	0
$673$	34.1849	1.31773	0.658866	−	0.752260i	$-0.271036\pi$
$673$	34.1849	1.31773	0.658866	+	0.752260i	$0.271036\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.0462	−0.655140	−0.327570	−	0.944827i	$-0.606230\pi$
$677$	−17.0462	−0.655140	−0.327570	+	0.944827i	$0.606230\pi$
$678$	0	0
$679$	10.3477i	0.397109i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 28.6671i	− 1.09692i	−0.836178	−	0.548459i	$-0.815215\pi$
$683$	− 28.6671i	− 1.09692i	0.836178	−	0.548459i	$-0.184785\pi$
$684$	0	0
$685$	4.72563i	0.180557i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.1410i	0.881603i
$690$	0	0
$691$	27.0741	1.02995	0.514974	−	0.857206i	$-0.327801\pi$
$691$	27.0741	1.02995	0.514974	+	0.857206i	$0.327801\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.7572	−0.597704
$696$	0	0
$697$	−57.2437	−2.16826
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.5510	−0.889510	−0.444755	−	0.895652i	$-0.646709\pi$
$701$	−23.5510	−0.889510	−0.444755	+	0.895652i	$0.646709\pi$
$702$	0	0
$703$	− 11.6053i	− 0.437701i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.9925i	0.827112i
$708$	0	0
$709$	7.85033i	0.294825i	0.989075	+	0.147413i	$0.0470945\pi$
$709$	7.85033i	0.294825i	−0.989075	+	0.147413i	$0.952905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.5923i	1.22059i
$714$	0	0
$715$	0.504792	0.0188782
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.1416	−0.378218	−0.189109	−	0.981956i	$-0.560560\pi$
$719$	−10.1416	−0.378218	−0.189109	+	0.981956i	$0.560560\pi$
$720$	0	0
$721$	−2.95377	−0.110004
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.270718	−0.0100542
$726$	0	0
$727$	− 4.86524i	− 0.180442i	−0.995922	−	0.0902208i	$-0.971243\pi$
$727$	− 4.86524i	− 0.180442i	0.995922	−	0.0902208i	$-0.0287573\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	79.2749i	2.93209i
$732$	0	0
$733$	− 9.09903i	− 0.336080i	−0.985780	−	0.168040i	$-0.946256\pi$
$733$	− 9.09903i	− 0.336080i	0.985780	−	0.168040i	$-0.0537438\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.331031i	− 0.0121937i
$738$	0	0
$739$	−12.5693	−0.462371	−0.231185	−	0.972910i	$-0.574260\pi$
$739$	−12.5693	−0.462371	−0.231185	+	0.972910i	$0.574260\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.6647	0.868175	0.434087	−	0.900871i	$-0.357071\pi$
$743$	23.6647	0.868175	0.434087	+	0.900871i	$0.357071\pi$
$744$	0	0
$745$	7.72928	0.283179
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	− 31.2782i	− 1.14136i	−0.821173	−	0.570679i	$-0.806680\pi$
$751$	− 31.2782i	− 1.14136i	0.821173	−	0.570679i	$-0.193320\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.2098i	0.808297i
$756$	0	0
$757$	39.1150i	1.42166i	0.703365	+	0.710829i	$0.251680\pi$
$757$	39.1150i	1.42166i	−0.703365	+	0.710829i	$0.748320\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.76305i	0.100160i	0.998745	+	0.0500802i	$0.0159477\pi$
$761$	2.76305i	0.100160i	−0.998745	+	0.0500802i	$0.984052\pi$
$762$	0	0
$763$	−7.45856	−0.270018
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.5510	0.994810
$768$	0	0
$769$	46.5606	1.67902	0.839509	−	0.543345i	$-0.182843\pi$
$769$	46.5606	1.67902	0.839509	+	0.543345i	$0.182843\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29.5877	−1.06419	−0.532097	−	0.846683i	$-0.678596\pi$
$773$	−29.5877	−1.06419	−0.532097	+	0.846683i	$0.678596\pi$
$774$	0	0
$775$	6.20522i	0.222898i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.0508i	0.503424i
$780$	0	0
$781$	1.86244i	0.0666435i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 8.00229i	− 0.285614i
$786$	0	0
$787$	35.9267	1.28065	0.640324	−	0.768105i	$-0.278800\pi$
$787$	35.9267	1.28065	0.640324	+	0.768105i	$0.278800\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.23407	0.292770
$792$	0	0
$793$	1.00958	0.0358514
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.31695	−0.330023	−0.165012	−	0.986292i	$-0.552766\pi$
$797$	−9.31695	−0.330023	−0.165012	+	0.986292i	$0.552766\pi$
$798$	0	0
$799$	− 23.5416i	− 0.832841i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.04491i	− 0.0368742i
$804$	0	0
$805$	7.42801i	0.261803i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 34.9207i	− 1.22775i	−0.789405	−	0.613873i	$-0.789611\pi$
$809$	− 34.9207i	− 1.22775i	0.789405	−	0.613873i	$-0.210389\pi$
$810$	0	0
$811$	25.7938	0.905744	0.452872	−	0.891576i	$-0.350399\pi$
$811$	25.7938	0.905744	0.452872	+	0.891576i	$0.350399\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.953771	0.0334092
$816$	0	0
$817$	19.4586	0.680769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1878	1.08846	0.544232	−	0.838935i	$-0.316821\pi$
$821$	31.1878	1.08846	0.544232	+	0.838935i	$0.316821\pi$
$822$	0	0
$823$	33.5718i	1.17024i	0.810947	+	0.585120i	$0.198953\pi$
$823$	33.5718i	1.17024i	−0.810947	+	0.585120i	$0.801047\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.4362i	1.85816i	0.369881	+	0.929079i	$0.379399\pi$
$827$	53.4362i	1.85816i	−0.369881	+	0.929079i	$0.620601\pi$
$828$	0	0
$829$	− 0.634952i	− 0.0220528i	−0.999939	−	0.0110264i	$-0.996490\pi$
$829$	− 0.634952i	− 0.0220528i	0.999939	−	0.0110264i	$-0.00350988\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.0261i	1.07499i
$834$	0	0
$835$	−11.7572	−0.406874
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.9046	−0.859803	−0.429901	−	0.902876i	$-0.641452\pi$
$839$	−24.9046	−0.859803	−0.429901	+	0.902876i	$0.641452\pi$
$840$	0	0
$841$	−28.9267	−0.997473
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.04623	0.207997
$846$	0	0
$847$	15.5045i	0.532742i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 40.0203i	− 1.37188i
$852$	0	0
$853$	17.5448i	0.600724i	0.953825	+	0.300362i	$0.0971075\pi$
$853$	17.5448i	0.600724i	−0.953825	+	0.300362i	$0.902893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.30196i	0.249430i	0.992193	+	0.124715i	$0.0398017\pi$
$857$	7.30196i	0.249430i	−0.992193	+	0.124715i	$0.960198\pi$
$858$	0	0
$859$	−15.2890	−0.521655	−0.260828	−	0.965385i	$-0.583995\pi$
$859$	−15.2890	−0.521655	−0.260828	+	0.965385i	$0.583995\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.9787	−1.42897	−0.714487	−	0.699649i	$-0.753340\pi$
$863$	−41.9787	−1.42897	−0.714487	+	0.699649i	$0.753340\pi$
$864$	0	0
$865$	3.22449	0.109636
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.73887	0.0929097
$870$	0	0
$871$	4.56012i	0.154514i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.41421i	0.0478091i
$876$	0	0
$877$	38.4800i	1.29938i	0.760200	+	0.649689i	$0.225101\pi$
$877$	38.4800i	1.29938i	−0.760200	+	0.649689i	$0.774899\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.11053i	− 0.239560i	−0.992800	−	0.119780i	$-0.961781\pi$
$881$	− 7.11053i	− 0.239560i	0.992800	−	0.119780i	$-0.0382189\pi$
$882$	0	0
$883$	26.5048	0.891957	0.445979	−	0.895044i	$-0.352856\pi$
$883$	26.5048	0.891957	0.445979	+	0.895044i	$0.352856\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.3449	1.32107	0.660535	−	0.750795i	$-0.270329\pi$
$887$	39.3449	1.32107	0.660535	+	0.750795i	$0.270329\pi$
$888$	0	0
$889$	29.0462	0.974179
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.77844	−0.193368
$894$	0	0
$895$	− 12.6019i	− 0.421234i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.67987i	− 0.0560267i
$900$	0	0
$901$	− 54.4540i	− 1.81413i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.1070i	0.801344i
$906$	0	0
$907$	22.1974	0.737054	0.368527	−	0.929617i	$-0.379862\pi$
$907$	22.1974	0.737054	0.368527	+	0.929617i	$0.379862\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.9538	0.694229	0.347115	−	0.937823i	$-0.387161\pi$
$911$	20.9538	0.694229	0.347115	+	0.937823i	$0.387161\pi$
$912$	0	0
$913$	−2.91713	−0.0965428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.1878	0.633638
$918$	0	0
$919$	− 7.88509i	− 0.260105i	−0.991507	−	0.130053i	$-0.958485\pi$
$919$	− 7.88509i	− 0.260105i	0.991507	−	0.130053i	$-0.0415146\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 25.6561i	− 0.844481i
$924$	0	0
$925$	− 7.61944i	− 0.250526i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6.35718i	− 0.208572i	−0.994547	−	0.104286i	$-0.966744\pi$
$929$	− 6.35718i	− 0.208572i	0.994547	−	0.104286i	$-0.0332558\pi$
$930$	0	0
$931$	7.61557	0.249590
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.18785	−0.0388467
$936$	0	0
$937$	35.0096	1.14371	0.571857	−	0.820354i	$-0.306223\pi$
$937$	35.0096	1.14371	0.571857	+	0.820354i	$0.306223\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.5606	−0.931049	−0.465525	−	0.885035i	$-0.654134\pi$
$941$	−28.5606	−0.931049	−0.465525	+	0.885035i	$0.654134\pi$
$942$	0	0
$943$	48.4538i	1.57787i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.7650i	− 0.674771i	−0.941367	−	0.337385i	$-0.890457\pi$
$947$	− 20.7650i	− 0.674771i	0.941367	−	0.337385i	$-0.109543\pi$
$948$	0	0
$949$	14.3942i	0.467257i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.4147i	1.24437i	0.782869	+	0.622186i	$0.213755\pi$
$953$	38.4147i	1.24437i	−0.782869	+	0.622186i	$0.786245\pi$
$954$	0	0
$955$	−10.9171	−0.353270
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.68305	−0.215807
$960$	0	0
$961$	−7.50479	−0.242090
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.3169	−0.493070
$966$	0	0
$967$	− 16.6013i	− 0.533861i	−0.963716	−	0.266930i	$-0.913991\pi$
$967$	− 16.6013i	− 0.533861i	0.963716	−	0.266930i	$-0.0860094\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.3171i	0.331092i	0.986202	+	0.165546i	$0.0529386\pi$
$971$	10.3171i	0.331092i	−0.986202	+	0.165546i	$0.947061\pi$
$972$	0	0
$973$	− 22.2840i	− 0.714393i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.0439i	− 1.02518i	−0.858635	−	0.512588i	$-0.828687\pi$
$977$	− 32.0439i	− 1.02518i	0.858635	−	0.512588i	$-0.171313\pi$
$978$	0	0
$979$	−0.683053	−0.0218305
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.8034	0.791106	0.395553	−	0.918443i	$-0.370553\pi$
$983$	24.8034	0.791106	0.395553	+	0.918443i	$0.370553\pi$
$984$	0	0
$985$	−17.0462	−0.543138
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.1020	2.13372
$990$	0	0
$991$	− 15.7872i	− 0.501498i	−0.968052	−	0.250749i	$-0.919323\pi$
$991$	− 15.7872i	− 0.501498i	0.968052	−	0.250749i	$-0.0806769\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.72563i	0.149813i
$996$	0	0
$997$	− 51.3823i	− 1.62729i	−0.581359	−	0.813647i	$-0.697479\pi$
$997$	− 51.3823i	− 1.62729i	0.581359	−	0.813647i	$-0.302521\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.b.d.431.5		6
3.2	odd	2		1440.2.b.c.431.5		6
4.3	odd	2		360.2.b.d.251.6	yes	6
5.2	odd	4		7200.2.m.e.3599.4		12
5.3	odd	4		7200.2.m.e.3599.10		12
5.4	even	2		7200.2.b.e.4751.2		6
8.3	odd	2		1440.2.b.c.431.2		6
8.5	even	2		360.2.b.c.251.2	yes	6
12.11	even	2		360.2.b.c.251.1	✓	6
15.2	even	4		7200.2.m.d.3599.3		12
15.8	even	4		7200.2.m.d.3599.9		12
15.14	odd	2		7200.2.b.d.4751.2		6
20.3	even	4		1800.2.m.e.899.7		12
20.7	even	4		1800.2.m.e.899.6		12
20.19	odd	2		1800.2.b.d.251.1		6
24.5	odd	2		360.2.b.d.251.5	yes	6
24.11	even	2	inner	1440.2.b.d.431.2		6
40.3	even	4		7200.2.m.d.3599.4		12
40.13	odd	4		1800.2.m.d.899.8		12
40.19	odd	2		7200.2.b.d.4751.5		6
40.27	even	4		7200.2.m.d.3599.10		12
40.29	even	2		1800.2.b.e.251.5		6
40.37	odd	4		1800.2.m.d.899.5		12
60.23	odd	4		1800.2.m.d.899.6		12
60.47	odd	4		1800.2.m.d.899.7		12
60.59	even	2		1800.2.b.e.251.6		6
120.29	odd	2		1800.2.b.d.251.2		6
120.53	even	4		1800.2.m.e.899.5		12
120.59	even	2		7200.2.b.e.4751.5		6
120.77	even	4		1800.2.m.e.899.8		12
120.83	odd	4		7200.2.m.e.3599.3		12
120.107	odd	4		7200.2.m.e.3599.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.b.c.251.1	✓	6	12.11	even	2
360.2.b.c.251.2	yes	6	8.5	even	2
360.2.b.d.251.5	yes	6	24.5	odd	2
360.2.b.d.251.6	yes	6	4.3	odd	2
1440.2.b.c.431.2		6	8.3	odd	2
1440.2.b.c.431.5		6	3.2	odd	2
1440.2.b.d.431.2		6	24.11	even	2	inner
1440.2.b.d.431.5		6	1.1	even	1	trivial
1800.2.b.d.251.1		6	20.19	odd	2
1800.2.b.d.251.2		6	120.29	odd	2
1800.2.b.e.251.5		6	40.29	even	2
1800.2.b.e.251.6		6	60.59	even	2
1800.2.m.d.899.5		12	40.37	odd	4
1800.2.m.d.899.6		12	60.23	odd	4
1800.2.m.d.899.7		12	60.47	odd	4
1800.2.m.d.899.8		12	40.13	odd	4
1800.2.m.e.899.5		12	120.53	even	4
1800.2.m.e.899.6		12	20.7	even	4
1800.2.m.e.899.7		12	20.3	even	4
1800.2.m.e.899.8		12	120.77	even	4
7200.2.b.d.4751.2		6	15.14	odd	2
7200.2.b.d.4751.5		6	40.19	odd	2
7200.2.b.e.4751.2		6	5.4	even	2
7200.2.b.e.4751.5		6	120.59	even	2
7200.2.m.d.3599.3		12	15.2	even	4
7200.2.m.d.3599.4		12	40.3	even	4
7200.2.m.d.3599.9		12	15.8	even	4
7200.2.m.d.3599.10		12	40.27	even	4
7200.2.m.e.3599.3		12	120.83	odd	4
7200.2.m.e.3599.4		12	5.2	odd	4
7200.2.m.e.3599.9		12	120.107	odd	4
7200.2.m.e.3599.10		12	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.41421i q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.41421i q^{7} +0.191427i q^{11} -2.63700i q^{13} +6.20522i q^{17} +1.52311 q^{19} +5.25240 q^{23} +1.00000 q^{25} -0.270718 q^{29} +6.20522i q^{31} +1.41421i q^{35} -7.61944i q^{37} +9.22508i q^{41} +12.7755 q^{43} -3.79383 q^{47} +5.00000 q^{49} -8.77551 q^{53} +0.191427i q^{55} +10.4479i q^{59} +0.382853i q^{61} -2.63700i q^{65} -1.72928 q^{67} +9.72928 q^{71} -5.45856 q^{73} -0.270718 q^{77} -14.3077i q^{79} +15.2389i q^{83} +6.20522i q^{85} +3.56822i q^{89} +3.72928 q^{91} +1.52311 q^{95} +7.31695 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5}+O(q^{10}) \) 6 * q + 6 * q^5	\( 6 q + 6 q^{5} - 16 q^{19} - 4 q^{23} + 6 q^{25} - 12 q^{29} + 16 q^{43} - 8 q^{47} + 30 q^{49} + 8 q^{53} + 48 q^{71} - 12 q^{73} - 12 q^{77} + 12 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100}) \) 6 * q + 6 * q^5 - 16 * q^19 - 4 * q^23 + 6 * q^25 - 12 * q^29 + 16 * q^43 - 8 * q^47 + 30 * q^49 + 8 * q^53 + 48 * q^71 - 12 * q^73 - 12 * q^77 + 12 * q^91 - 16 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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