LMFDB - Embedded newform 2016.2.bu.c.1871.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(431,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))

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

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

chi := DirichletCharacter("2016.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.bu (of order $6$, degree $2$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1871.15
Character	$\chi$	$=$	2016.1871
Dual form			2016.2.bu.c.431.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.317795 - 0.550436i) q^{5} +(-2.11900 - 1.58425i) q^{7} +O(q^{10})$	$q+(0.317795 - 0.550436i) q^{5} +(-2.11900 - 1.58425i) q^{7} +(-3.16457 + 1.82707i) q^{11} +4.15869i q^{13} +(3.01908 - 1.74306i) q^{17} +(1.99238 - 3.45090i) q^{19} +(1.47015 - 2.54638i) q^{23} +(2.29801 + 3.98028i) q^{25} -6.35746 q^{29} +(-5.20467 + 3.00492i) q^{31} +(-1.54543 + 0.662908i) q^{35} +(1.59870 + 0.923007i) q^{37} +10.4931i q^{41} +2.83895 q^{43} +(-4.61494 + 7.99332i) q^{47} +(1.98031 + 6.71404i) q^{49} +(-2.99666 - 5.19038i) q^{53} +2.32253i q^{55} +(-9.10070 + 5.25429i) q^{59} +(1.72447 + 0.995622i) q^{61} +(2.28909 + 1.32161i) q^{65} +(8.01122 + 13.8758i) q^{67} +0.737952 q^{71} +(2.13696 + 3.70132i) q^{73} +(9.60025 + 1.14192i) q^{77} +(7.74900 + 4.47389i) q^{79} +3.67740i q^{83} -2.21574i q^{85} +(-4.63483 - 2.67592i) q^{89} +(6.58841 - 8.81226i) q^{91} +(-1.26633 - 2.19335i) q^{95} +17.6696 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 32 q^{19} + 160 q^{43} + 56 q^{49} + 16 q^{73} + 32 q^{91} + 240 q^{97}+O(q^{100}) $ 48 * q + 32 * q^19 + 160 * q^43 + 56 * q^49 + 16 * q^73 + 32 * q^91 + 240 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$

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
$5$	0.317795	−	0.550436i	0.142122	−	0.246163i	−0.786173	−	0.618006i	$-0.787941\pi$
$5$	0.317795	−	0.550436i	0.142122	−	0.246163i	0.928296	+	0.371843i	$0.121274\pi$
$6$		0			0
$7$	−2.11900	−	1.58425i	−0.800906	−	0.598790i
$7$	−2.11900	−	1.58425i	−0.800906	−	0.598790i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.16457	+	1.82707i	−0.954155	+	0.550881i	−0.894369	−	0.447330i	$-0.852375\pi$
$11$	−3.16457	+	1.82707i	−0.954155	+	0.550881i	−0.0597855	+	0.998211i	$0.519042\pi$
$12$		0			0
$13$			4.15869i			1.15341i	0.816951	+	0.576707i	$0.195662\pi$
$13$			4.15869i			1.15341i	−0.816951	+	0.576707i	$0.804338\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.01908	−	1.74306i	0.732233	−	0.422755i	−0.0870053	−	0.996208i	$-0.527730\pi$
$17$	3.01908	−	1.74306i	0.732233	−	0.422755i	0.819239	+	0.573453i	$0.194396\pi$
$18$		0			0
$19$	1.99238	−	3.45090i	0.457083	−	0.791691i	−0.541722	−	0.840558i	$-0.682228\pi$
$19$	1.99238	−	3.45090i	0.457083	−	0.791691i	0.998805	+	0.0488665i	$0.0155609\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.47015	−	2.54638i	0.306548	−	0.530957i	−0.671056	−	0.741406i	$-0.734159\pi$
$23$	1.47015	−	2.54638i	0.306548	−	0.530957i	0.977605	+	0.210449i	$0.0674925\pi$
$24$		0			0
$25$	2.29801	+	3.98028i	0.459603	+	0.796055i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.35746			−1.18055			−0.590275	−	0.807202i	$-0.700981\pi$
$29$	−6.35746			−1.18055			−0.590275	+	0.807202i	$0.700981\pi$
$30$		0			0
$31$	−5.20467	+	3.00492i	−0.934787	+	0.539699i	−0.888322	−	0.459221i	$-0.848129\pi$
$31$	−5.20467	+	3.00492i	−0.934787	+	0.539699i	−0.0464645	+	0.998920i	$0.514795\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.54543	+	0.662908i	−0.261226	+	0.112052i
$36$		0			0
$37$	1.59870	+	0.923007i	0.262824	+	0.151741i	0.625622	−	0.780126i	$-0.284845\pi$
$37$	1.59870	+	0.923007i	0.262824	+	0.151741i	−0.362798	+	0.931868i	$0.618179\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.4931i			1.63874i	0.573262	+	0.819372i	$0.305678\pi$
$41$			10.4931i			1.63874i	−0.573262	+	0.819372i	$0.694322\pi$
$42$		0			0
$43$	2.83895			0.432935			0.216468	−	0.976290i	$-0.430546\pi$
$43$	2.83895			0.432935			0.216468	+	0.976290i	$0.430546\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.61494	+	7.99332i	−0.673159	+	1.16595i	0.303845	+	0.952722i	$0.401730\pi$
$47$	−4.61494	+	7.99332i	−0.673159	+	1.16595i	−0.977003	+	0.213224i	$0.931604\pi$
$48$		0			0
$49$	1.98031	+	6.71404i	0.282901	+	0.959149i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.99666	−	5.19038i	−0.411624	−	0.712953i	0.583444	−	0.812153i	$-0.301705\pi$
$53$	−2.99666	−	5.19038i	−0.411624	−	0.712953i	−0.995067	+	0.0992005i	$0.968371\pi$
$54$		0			0
$55$			2.32253i			0.313170i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.10070	+	5.25429i	−1.18481	+	0.684050i	−0.957122	−	0.289684i	$-0.906450\pi$
$59$	−9.10070	+	5.25429i	−1.18481	+	0.684050i	−0.227688	+	0.973734i	$0.573117\pi$
$60$		0			0
$61$	1.72447	+	0.995622i	0.220795	+	0.127476i	0.606319	−	0.795222i	$-0.292646\pi$
$61$	1.72447	+	0.995622i	0.220795	+	0.127476i	−0.385523	+	0.922698i	$0.625979\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.28909	+	1.32161i	0.283927	+	0.163925i
$66$		0			0
$67$	8.01122	+	13.8758i	0.978726	+	1.69520i	0.667048	+	0.745015i	$0.267558\pi$
$67$	8.01122	+	13.8758i	0.978726	+	1.69520i	0.311678	+	0.950188i	$0.399109\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.737952			0.0875788			0.0437894	−	0.999041i	$-0.486057\pi$
$71$	0.737952			0.0875788			0.0437894	+	0.999041i	$0.486057\pi$
$72$		0			0
$73$	2.13696	+	3.70132i	0.250112	+	0.433207i	0.963557	−	0.267505i	$-0.0861991\pi$
$73$	2.13696	+	3.70132i	0.250112	+	0.433207i	−0.713444	+	0.700712i	$0.752866\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.60025	+	1.14192i	1.09405	+	0.130134i
$78$		0			0
$79$	7.74900	+	4.47389i	0.871830	+	0.503351i	0.867956	−	0.496641i	$-0.165433\pi$
$79$	7.74900	+	4.47389i	0.871830	+	0.503351i	0.00387425	+	0.999992i	$0.498767\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.67740i			0.403647i	0.979422	+	0.201823i	$0.0646867\pi$
$83$			3.67740i			0.403647i	−0.979422	+	0.201823i	$0.935313\pi$
$84$		0			0
$85$		−	2.21574i		−	0.240331i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.63483	−	2.67592i	−0.491291	−	0.283647i	0.233819	−	0.972280i	$-0.424878\pi$
$89$	−4.63483	−	2.67592i	−0.491291	−	0.283647i	−0.725110	+	0.688633i	$0.758211\pi$
$90$		0			0
$91$	6.58841	−	8.81226i	0.690653	−	0.923776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.26633	−	2.19335i	−0.129923	−	0.225033i
$96$		0			0
$97$	17.6696			1.79407			0.897036	−	0.441958i	$-0.145716\pi$
$97$	17.6696			1.79407			0.897036	+	0.441958i	$0.145716\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.12560	−	10.6099i	−0.609520	−	1.05572i	−0.991320	−	0.131475i	$-0.958029\pi$
$101$	−6.12560	−	10.6099i	−0.609520	−	1.05572i	0.381799	−	0.924245i	$-0.375305\pi$
$102$		0			0
$103$	−2.71030	−	1.56479i	−0.267054	−	0.154184i	0.360494	−	0.932761i	$-0.382608\pi$
$103$	−2.71030	−	1.56479i	−0.267054	−	0.154184i	−0.627548	+	0.778578i	$0.715941\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.64238	−	3.25763i	−0.545470	−	0.314927i	0.201823	−	0.979422i	$-0.435313\pi$
$107$	−5.64238	−	3.25763i	−0.545470	−	0.314927i	−0.747293	+	0.664495i	$0.768647\pi$
$108$		0			0
$109$	−1.23586	+	0.713527i	−0.118374	+	0.0683435i	−0.558018	−	0.829829i	$-0.688438\pi$
$109$	−1.23586	+	0.713527i	−0.118374	+	0.0683435i	0.439644	+	0.898172i	$0.355105\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.27992i			0.308549i	0.988028	+	0.154275i	$0.0493040\pi$
$113$			3.27992i			0.308549i	−0.988028	+	0.154275i	$0.950696\pi$
$114$		0			0
$115$	−0.934414	−	1.61845i	−0.0871346	−	0.150922i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.15886	−	1.08942i	−0.839592	−	0.0998669i
$120$		0			0
$121$	1.17635	−	2.03749i	0.106941	−	0.185226i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.09913			0.545523
$126$		0			0
$127$			19.0629i			1.69156i	0.533535	+	0.845778i	$0.320863\pi$
$127$			19.0629i			1.69156i	−0.533535	+	0.845778i	$0.679137\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	11.2577	+	6.49965i	0.983591	+	0.567877i	0.903352	−	0.428899i	$-0.141098\pi$
$131$	11.2577	+	6.49965i	0.983591	+	0.567877i	0.0802388	+	0.996776i	$0.474432\pi$
$132$		0			0
$133$	−9.68893	+	4.15603i	−0.840137	+	0.360373i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.17651	+	0.679257i	−0.100516	+	0.0580329i	−0.549415	−	0.835549i	$-0.685149\pi$
$137$	−1.17651	+	0.679257i	−0.100516	+	0.0580329i	0.448899	+	0.893582i	$0.351816\pi$
$138$		0			0
$139$	3.06037			0.259577			0.129788	−	0.991542i	$-0.458570\pi$
$139$	3.06037			0.259577			0.129788	+	0.991542i	$0.458570\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.59821	−	13.1605i	−0.635394	−	1.10053i
$144$		0			0
$145$	−2.02037	+	3.49938i	−0.167782	+	0.290607i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.15986	+	14.1333i	−0.668481	+	1.15784i	0.309847	+	0.950786i	$0.399722\pi$
$149$	−8.15986	+	14.1333i	−0.668481	+	1.15784i	−0.978329	+	0.207057i	$0.933611\pi$
$150$		0			0
$151$	−7.81057	+	4.50943i	−0.635615	+	0.366972i	−0.782923	−	0.622118i	$-0.786272\pi$
$151$	−7.81057	+	4.50943i	−0.635615	+	0.366972i	0.147309	+	0.989091i	$0.452939\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.81979i			0.306813i
$156$		0			0
$157$	−20.0104	+	11.5530i	−1.59700	+	0.922031i	−0.604944	+	0.796268i	$0.706804\pi$
$157$	−20.0104	+	11.5530i	−1.59700	+	0.922031i	−0.992060	+	0.125763i	$0.959862\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.14936	+	3.06669i	−0.563449	+	0.241689i
$162$		0			0
$163$	−6.10214	+	10.5692i	−0.477956	+	0.827845i	−0.999681	−	0.0252693i	$-0.991956\pi$
$163$	−6.10214	+	10.5692i	−0.477956	+	0.827845i	0.521724	+	0.853114i	$0.325289\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.33534			0.258096			0.129048	−	0.991638i	$-0.458808\pi$
$167$	3.33534			0.258096			0.129048	+	0.991638i	$0.458808\pi$
$168$		0			0
$169$	−4.29472			−0.330363
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.79217	−	8.30028i	0.364342	−	0.631058i	−0.624329	−	0.781162i	$-0.714627\pi$
$173$	4.79217	−	8.30028i	0.364342	−	0.631058i	0.988670	+	0.150104i	$0.0479607\pi$
$174$		0			0
$175$	1.43626	−	12.0748i	0.108571	−	0.912771i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.0283182	−	0.0163495i	0.00211660	−	0.00122202i	−0.498941	−	0.866636i	$-0.666278\pi$
$179$	0.0283182	−	0.0163495i	0.00211660	−	0.00122202i	0.501058	+	0.865414i	$0.332944\pi$
$180$		0			0
$181$		−	19.3654i		−	1.43942i	−0.694277	−	0.719708i	$-0.744276\pi$
$181$		−	19.3654i		−	1.43942i	0.694277	−	0.719708i	$-0.255724\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.01611	−	0.586653i	0.0747061	−	0.0431316i
$186$		0			0
$187$	−6.36939	+	11.0321i	−0.465776	+	0.806747i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.50826	+	2.61239i	−0.109134	+	0.189026i	−0.915420	−	0.402501i	$-0.868141\pi$
$191$	−1.50826	+	2.61239i	−0.109134	+	0.189026i	0.806286	+	0.591526i	$0.201474\pi$
$192$		0			0
$193$	−10.9565	−	18.9772i	−0.788666	−	1.36601i	−0.926785	−	0.375593i	$-0.877439\pi$
$193$	−10.9565	−	18.9772i	−0.788666	−	1.36601i	0.138119	−	0.990416i	$-0.455894\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.2950			−1.08972			−0.544860	−	0.838527i	$-0.683417\pi$
$197$	−15.2950			−1.08972			−0.544860	+	0.838527i	$0.683417\pi$
$198$		0			0
$199$	2.77706	−	1.60334i	0.196861	−	0.113658i	−0.398330	−	0.917242i	$-0.630410\pi$
$199$	2.77706	−	1.60334i	0.196861	−	0.113658i	0.595190	+	0.803585i	$0.297077\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.4715	+	10.0718i	0.945510	+	0.706902i
$204$		0			0
$205$	5.77577	+	3.33464i	0.403398	+	0.232902i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			14.5608i			1.00719i
$210$		0			0
$211$	16.7059			1.15008			0.575040	−	0.818125i	$-0.304986\pi$
$211$	16.7059			1.15008			0.575040	+	0.818125i	$0.304986\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.902202	−	1.56266i	0.0615297	−	0.106573i
$216$		0			0
$217$	15.7892	+	1.87808i	1.07184	+	0.127492i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.24887	+	12.5554i	0.487611	+	0.844568i
$222$		0			0
$223$		−	19.1547i		−	1.28270i	−0.767250	−	0.641348i	$-0.778375\pi$
$223$		−	19.1547i		−	1.28270i	0.767250	−	0.641348i	$-0.221625\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.4962	−	8.36938i	0.962147	−	0.555496i	0.0653135	−	0.997865i	$-0.479195\pi$
$227$	14.4962	−	8.36938i	0.962147	−	0.555496i	0.896833	+	0.442369i	$0.145862\pi$
$228$		0			0
$229$	17.5029	+	10.1053i	1.15662	+	0.667777i	0.950492	−	0.310747i	$-0.100579\pi$
$229$	17.5029	+	10.1053i	1.15662	+	0.667777i	0.206131	+	0.978524i	$0.433913\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.3998	−	10.0458i	−1.13990	−	0.658122i	−0.193495	−	0.981101i	$-0.561982\pi$
$233$	−17.3998	−	10.0458i	−1.13990	−	0.658122i	−0.946406	+	0.322979i	$0.895316\pi$
$234$		0			0
$235$	2.93321	+	5.08047i	0.191341	+	0.331413i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.2312			0.855855			0.427928	−	0.903813i	$-0.359244\pi$
$239$	13.2312			0.855855			0.427928	+	0.903813i	$0.359244\pi$
$240$		0			0
$241$	−5.90444	−	10.2268i	−0.380338	−	0.658765i	0.610772	−	0.791806i	$-0.290859\pi$
$241$	−5.90444	−	10.2268i	−0.380338	−	0.658765i	−0.991111	+	0.133041i	$0.957526\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.32498	+	1.04365i	0.276313	+	0.0666766i
$246$		0			0
$247$	14.3512	+	8.28569i	0.913147	+	0.527206i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	18.9939i		−	1.19888i	−0.800419	−	0.599441i	$-0.795390\pi$
$251$		−	18.9939i		−	1.19888i	0.800419	−	0.599441i	$-0.204610\pi$
$252$		0			0
$253$			10.7443i			0.675487i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.5458	−	13.0168i	−1.40637	−	0.811966i	−0.411331	−	0.911486i	$-0.634936\pi$
$257$	−22.5458	−	13.0168i	−1.40637	−	0.811966i	−0.995036	+	0.0995204i	$0.968269\pi$
$258$		0			0
$259$	−1.92536	−	4.48858i	−0.119636	−	0.278907i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.3095	+	23.0527i	0.820698	+	1.42149i	0.905163	+	0.425064i	$0.139749\pi$
$263$	13.3095	+	23.0527i	0.820698	+	1.42149i	−0.0844650	+	0.996426i	$0.526918\pi$
$264$		0			0
$265$	−3.80929			−0.234003
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.6453	+	21.9022i	0.770995	+	1.33540i	0.937018	+	0.349281i	$0.113574\pi$
$269$	12.6453	+	21.9022i	0.770995	+	1.33540i	−0.166022	+	0.986122i	$0.553092\pi$
$270$		0			0
$271$	−14.4561	−	8.34622i	−0.878145	−	0.506997i	−0.00809834	−	0.999967i	$-0.502578\pi$
$271$	−14.4561	−	8.34622i	−0.878145	−	0.506997i	−0.870046	+	0.492970i	$0.835911\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.5445	−	8.39725i	−0.877064	−	0.506373i
$276$		0			0
$277$	23.0279	−	13.2952i	1.38361	−	0.798830i	0.391029	−	0.920379i	$-0.372119\pi$
$277$	23.0279	−	13.2952i	1.38361	−	0.798830i	0.992585	+	0.121549i	$0.0387860\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8.11615i			0.484169i	0.970255	+	0.242085i	$0.0778311\pi$
$281$			8.11615i			0.484169i	−0.970255	+	0.242085i	$0.922169\pi$
$282$		0			0
$283$	−4.12929	−	7.15214i	−0.245461	−	0.425150i	0.716800	−	0.697278i	$-0.245606\pi$
$283$	−4.12929	−	7.15214i	−0.245461	−	0.425150i	−0.962261	+	0.272128i	$0.912273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.6237	−	22.2348i	0.981264	−	1.31248i
$288$		0			0
$289$	−2.42346	+	4.19755i	−0.142556	+	0.246915i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−20.9952			−1.22655			−0.613277	−	0.789868i	$-0.710149\pi$
$293$	−20.9952			−1.22655			−0.613277	+	0.789868i	$0.710149\pi$
$294$		0			0
$295$			6.67914i			0.388875i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.5896	+	6.11392i	0.612414	+	0.353577i
$300$		0			0
$301$	−6.01572	−	4.49760i	−0.346741	−	0.259237i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.09605	−	0.632806i	0.0627598	−	0.0362344i
$306$		0			0
$307$	−18.1231			−1.03434			−0.517170	−	0.855882i	$-0.673015\pi$
$307$	−18.1231			−1.03434			−0.517170	+	0.855882i	$0.673015\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.4208	−	26.7096i	−0.874433	−	1.51456i	−0.857366	−	0.514707i	$-0.827901\pi$
$311$	−15.4208	−	26.7096i	−0.874433	−	1.51456i	−0.0170667	−	0.999854i	$-0.505433\pi$
$312$		0			0
$313$	0.521916	−	0.903984i	0.0295004	−	0.0510962i	−0.850898	−	0.525330i	$-0.823942\pi$
$313$	0.521916	−	0.903984i	0.0295004	−	0.0510962i	0.880399	+	0.474234i	$0.157275\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.12901	+	8.88370i	−0.288074	+	0.498958i	−0.973350	−	0.229325i	$-0.926348\pi$
$317$	−5.12901	+	8.88370i	−0.288074	+	0.498958i	0.685276	+	0.728283i	$0.259681\pi$
$318$		0			0
$319$	20.1186	−	11.6155i	1.12643	−	0.650344i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13.8914i		−	0.772937i
$324$		0			0
$325$	−16.5527	+	9.55673i	−0.918181	+	0.530112i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.4425	−	9.62660i	1.23729	−	0.530732i
$330$		0			0
$331$	−2.31756	+	4.01414i	−0.127385	+	0.220637i	−0.922663	−	0.385608i	$-0.873992\pi$
$331$	−2.31756	+	4.01414i	−0.127385	+	0.220637i	0.795278	+	0.606245i	$0.207325\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.1837			0.556394
$336$		0			0
$337$	−33.0263			−1.79906			−0.899528	−	0.436863i	$-0.856089\pi$
$337$	−33.0263			−1.79906			−0.899528	+	0.436863i	$0.856089\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.9804	−	19.0186i	0.594621	−	1.02991i
$342$		0			0
$343$	6.44045	−	17.3643i	0.347752	−	0.937587i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.47551	−	2.00659i	0.186575	−	0.107719i	−0.403803	−	0.914846i	$-0.632312\pi$
$347$	3.47551	−	2.00659i	0.186575	−	0.107719i	0.590378	+	0.807127i	$0.298979\pi$
$348$		0			0
$349$		−	10.4671i		−	0.560291i	−0.959958	−	0.280145i	$-0.909617\pi$
$349$		−	10.4671i		−	0.560291i	0.959958	−	0.280145i	$-0.0903826\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.3486	−	8.86154i	0.816925	−	0.471652i	−0.0324298	−	0.999474i	$-0.510325\pi$
$353$	15.3486	−	8.86154i	0.816925	−	0.471652i	0.849355	+	0.527822i	$0.176991\pi$
$354$		0			0
$355$	0.234517	−	0.406196i	0.0124469	−	0.0215586i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.52933	+	11.3091i	−0.344605	+	0.596873i	−0.985282	−	0.170937i	$-0.945320\pi$
$359$	−6.52933	+	11.3091i	−0.344605	+	0.596873i	0.640677	+	0.767811i	$0.278654\pi$
$360$		0			0
$361$	1.56085	+	2.70348i	0.0821503	+	0.142288i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.71646			0.142186
$366$		0			0
$367$	−19.3899	+	11.1948i	−1.01215	+	0.584363i	−0.911819	−	0.410592i	$-0.865322\pi$
$367$	−19.3899	+	11.1948i	−1.01215	+	0.584363i	−0.100327	+	0.994955i	$0.531989\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.87292	+	15.7459i	−0.0972373	+	0.817484i
$372$		0			0
$373$	3.83798	+	2.21586i	0.198723	+	0.114733i	0.596060	−	0.802940i	$-0.296732\pi$
$373$	3.83798	+	2.21586i	0.198723	+	0.114733i	−0.397337	+	0.917673i	$0.630065\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	26.4387i		−	1.36166i
$378$		0			0
$379$	−28.8901			−1.48399			−0.741993	−	0.670407i	$-0.766119\pi$
$379$	−28.8901			−1.48399			−0.741993	+	0.670407i	$0.766119\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.3811	+	28.3729i	−0.837037	+	1.44979i	0.0553247	+	0.998468i	$0.482381\pi$
$383$	−16.3811	+	28.3729i	−0.837037	+	1.44979i	−0.892361	+	0.451322i	$0.850953\pi$
$384$		0			0
$385$	3.67946	−	4.92143i	0.187523	−	0.250819i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.24881	−	10.8233i	−0.316827	−	0.548761i	0.662997	−	0.748622i	$-0.269284\pi$
$389$	−6.24881	−	10.8233i	−0.316827	−	0.548761i	−0.979824	+	0.199861i	$0.935951\pi$
$390$		0			0
$391$		−	10.2503i		−	0.518380i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.92518	−	2.84355i	0.247813	−	0.143075i
$396$		0			0
$397$	6.09678	+	3.51998i	0.305989	+	0.176663i	0.645130	−	0.764073i	$-0.276803\pi$
$397$	6.09678	+	3.51998i	0.305989	+	0.176663i	−0.339141	+	0.940735i	$0.610137\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	25.7480	+	14.8656i	1.28579	+	0.742353i	0.977901	−	0.209068i	$-0.0670429\pi$
$401$	25.7480	+	14.8656i	1.28579	+	0.742353i	0.307893	+	0.951421i	$0.400376\pi$
$402$		0			0
$403$	−12.4965	−	21.6446i	−0.622497	−	1.07820i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.74558			−0.334366
$408$		0			0
$409$	−16.6222	−	28.7905i	−0.821914	−	1.42360i	−0.904255	−	0.426993i	$-0.859573\pi$
$409$	−16.6222	−	28.7905i	−0.821914	−	1.42360i	0.0823411	−	0.996604i	$-0.473760\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.6085	+	3.28394i	1.35852	+	0.161592i
$414$		0			0
$415$	2.02417	+	1.16866i	0.0993627	+	0.0573671i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.95024i			0.437248i	0.975809	+	0.218624i	$0.0701568\pi$
$419$			8.95024i			0.437248i	−0.975809	+	0.218624i	$0.929843\pi$
$420$		0			0
$421$		−	26.3473i		−	1.28409i	−0.766667	−	0.642044i	$-0.778086\pi$
$421$		−	26.3473i		−	1.28409i	0.766667	−	0.642044i	$-0.221914\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.8758	+	8.01117i	0.673073	+	0.388599i
$426$		0			0
$427$	−2.07683	−	4.84171i	−0.100505	−	0.234307i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.97354	+	13.8106i	0.384072	+	0.665232i	0.991640	−	0.129036i	$-0.0411882\pi$
$431$	7.97354	+	13.8106i	0.384072	+	0.665232i	−0.607568	+	0.794268i	$0.707855\pi$
$432$		0			0
$433$	12.5058			0.600991			0.300496	−	0.953783i	$-0.402848\pi$
$433$	12.5058			0.600991			0.300496	+	0.953783i	$0.402848\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.85821	−	10.1467i	−0.280236	−	0.485383i
$438$		0			0
$439$	14.0552	+	8.11479i	0.670820	+	0.387298i	0.796387	−	0.604787i	$-0.206742\pi$
$439$	14.0552	+	8.11479i	0.670820	+	0.387298i	−0.125568	+	0.992085i	$0.540075\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.80303	+	3.35038i	0.275710	+	0.159181i	0.631480	−	0.775392i	$-0.282448\pi$
$443$	5.80303	+	3.35038i	0.275710	+	0.159181i	−0.355769	+	0.934574i	$0.615781\pi$
$444$		0			0
$445$	−2.94584	+	1.70078i	−0.139646	+	0.0806249i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	26.1155i		−	1.23246i	−0.787565	−	0.616232i	$-0.788658\pi$
$449$		−	26.1155i		−	1.23246i	0.787565	−	0.616232i	$-0.211342\pi$
$450$		0			0
$451$	−19.1716	−	33.2061i	−0.902754	−	1.56362i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.75683	−	6.42699i	−0.129242	−	0.301302i
$456$		0			0
$457$	−16.0328	+	27.7697i	−0.749984	+	1.29901i	0.197846	+	0.980233i	$0.436606\pi$
$457$	−16.0328	+	27.7697i	−0.749984	+	1.29901i	−0.947830	+	0.318777i	$0.896728\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	39.2781			1.82936			0.914682	−	0.404175i	$-0.132441\pi$
$461$	39.2781			1.82936			0.914682	+	0.404175i	$0.132441\pi$
$462$		0			0
$463$		−	2.26863i		−	0.105432i	−0.998610	−	0.0527161i	$-0.983212\pi$
$463$		−	2.26863i		−	0.105432i	0.998610	−	0.0527161i	$-0.0167878\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.8867	−	12.0590i	−0.966522	−	0.558022i	−0.0683480	−	0.997662i	$-0.521773\pi$
$467$	−20.8867	−	12.0590i	−0.966522	−	0.558022i	−0.898174	+	0.439640i	$0.855106\pi$
$468$		0			0
$469$	5.00703	−	42.0946i	0.231203	−	1.94375i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.98406	+	5.18695i	−0.413087	+	0.238496i
$474$		0			0
$475$	18.3141			0.840306
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.19574	−	10.7313i	−0.283091	−	0.490328i	0.689054	−	0.724710i	$-0.258026\pi$
$479$	−6.19574	−	10.7313i	−0.283091	−	0.490328i	−0.972144	+	0.234383i	$0.924693\pi$
$480$		0			0
$481$	−3.83850	+	6.64848i	−0.175021	+	0.303145i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.61529	−	9.72596i	0.254977	−	0.441633i
$486$		0			0
$487$	−10.1179	+	5.84159i	−0.458487	+	0.264708i	−0.711408	−	0.702779i	$-0.751942\pi$
$487$	−10.1179	+	5.84159i	−0.458487	+	0.264708i	0.252921	+	0.967487i	$0.418609\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	42.8926i		−	1.93572i	−0.251498	−	0.967858i	$-0.580923\pi$
$491$		−	42.8926i		−	1.93572i	0.251498	−	0.967858i	$-0.419077\pi$
$492$		0			0
$493$	−19.1937	+	11.0815i	−0.864439	+	0.499084i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.56372	−	1.16910i	−0.0701424	−	0.0524413i
$498$		0			0
$499$	4.49173	−	7.77991i	0.201078	−	0.348277i	−0.747798	−	0.663926i	$-0.768889\pi$
$499$	4.49173	−	7.77991i	0.201078	−	0.348277i	0.948876	+	0.315649i	$0.102222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	43.5055			1.93981			0.969907	−	0.243476i	$-0.0782878\pi$
$503$	43.5055			1.93981			0.969907	+	0.243476i	$0.0782878\pi$
$504$		0			0
$505$	−7.78673			−0.346505
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.45911	−	9.45546i	0.241971	−	0.419106i	−0.719305	−	0.694695i	$-0.755540\pi$
$509$	5.45911	−	9.45546i	0.241971	−	0.419106i	0.961276	+	0.275589i	$0.0888728\pi$
$510$		0			0
$511$	1.33561	−	11.2286i	0.0590837	−	0.496723i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.72264	+	0.994565i	−0.0759085	+	0.0438258i
$516$		0			0
$517$		−	33.7272i		−	1.48332i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.4185	−	11.2113i	0.850739	−	0.491174i	−0.0101612	−	0.999948i	$-0.503234\pi$
$521$	19.4185	−	11.2113i	0.850739	−	0.491174i	0.860900	+	0.508774i	$0.169901\pi$
$522$		0			0
$523$	0.483187	−	0.836904i	0.0211283	−	0.0365953i	−0.855268	−	0.518186i	$-0.826607\pi$
$523$	0.483187	−	0.836904i	0.0211283	−	0.0365953i	0.876396	+	0.481591i	$0.159941\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.4755	+	18.1442i	−0.456321	+	0.790372i
$528$		0			0
$529$	7.17729	+	12.4314i	0.312056	+	0.540497i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−43.6375			−1.89015
$534$		0			0
$535$	−3.58624	+	2.07051i	−0.155046	+	0.0895161i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.5338	−	17.6289i	−0.798309	−	0.759332i
$540$		0			0
$541$	32.5318	+	18.7823i	1.39865	+	0.807512i	0.994251	−	0.107070i	$-0.0341469\pi$
$541$	32.5318	+	18.7823i	1.39865	+	0.807512i	0.404400	+	0.914582i	$0.367480\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			0.907020i			0.0388525i
$546$		0			0
$547$	−13.8253			−0.591126			−0.295563	−	0.955323i	$-0.595507\pi$
$547$	−13.8253			−0.591126			−0.295563	+	0.955323i	$0.595507\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.6665	+	21.9390i	−0.539610	+	0.934632i
$552$		0			0
$553$	−9.33236	−	21.7565i	−0.396852	−	0.925180i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.15732	+	3.73659i	0.0914087	+	0.158325i	0.908104	−	0.418744i	$-0.137530\pi$
$557$	2.15732	+	3.73659i	0.0914087	+	0.158325i	−0.816695	+	0.577069i	$0.804196\pi$
$558$		0			0
$559$			11.8063i			0.499354i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.35371	−	3.09097i	0.225632	−	0.130269i	−0.382923	−	0.923780i	$-0.625083\pi$
$563$	5.35371	−	3.09097i	0.225632	−	0.130269i	0.608555	+	0.793511i	$0.291749\pi$
$564$		0			0
$565$	1.80539	+	1.04234i	0.0759532	+	0.0438516i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.65848	+	3.26692i	0.237216	+	0.136957i	0.613896	−	0.789387i	$-0.289601\pi$
$569$	5.65848	+	3.26692i	0.237216	+	0.136957i	−0.376681	+	0.926343i	$0.622935\pi$
$570$		0			0
$571$	6.00146	+	10.3948i	0.251153	+	0.435011i	0.963844	−	0.266468i	$-0.0858568\pi$
$571$	6.00146	+	10.3948i	0.251153	+	0.435011i	−0.712690	+	0.701479i	$0.752523\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.5137			0.563562
$576$		0			0
$577$	1.42990	+	2.47666i	0.0595276	+	0.103105i	0.894253	−	0.447561i	$-0.147707\pi$
$577$	1.42990	+	2.47666i	0.0595276	+	0.103105i	−0.834726	+	0.550666i	$0.814374\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.82591	−	7.79240i	0.241700	−	0.323283i
$582$		0			0
$583$	18.9663	+	10.9502i	0.785505	+	0.453512i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			30.3998i			1.25473i	0.778723	+	0.627367i	$0.215868\pi$
$587$			30.3998i			1.25473i	−0.778723	+	0.627367i	$0.784132\pi$
$588$		0			0
$589$			23.9477i			0.986750i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.46791	+	0.847496i	0.0602797	+	0.0348025i	0.529837	−	0.848100i	$-0.322253\pi$
$593$	1.46791	+	0.847496i	0.0602797	+	0.0348025i	−0.469557	+	0.882902i	$0.655586\pi$
$594$		0			0
$595$	−3.51029	+	4.69516i	−0.143908	+	0.192483i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.00010	+	3.46428i	0.0817219	+	0.141547i	0.903990	−	0.427554i	$-0.140625\pi$
$599$	2.00010	+	3.46428i	0.0817219	+	0.141547i	−0.822268	+	0.569101i	$0.807291\pi$
$600$		0			0
$601$	−33.9144			−1.38340			−0.691699	−	0.722186i	$-0.743138\pi$
$601$	−33.9144			−1.38340			−0.691699	+	0.722186i	$0.743138\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.747673	−	1.29501i	−0.0303972	−	0.0526495i
$606$		0			0
$607$	−2.15442	−	1.24386i	−0.0874452	−	0.0504865i	0.455640	−	0.890164i	$-0.349411\pi$
$607$	−2.15442	−	1.24386i	−0.0874452	−	0.0504865i	−0.543085	+	0.839678i	$0.682744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−33.2417	−	19.1921i	−1.34482	−	0.776431i
$612$		0			0
$613$	26.9572	−	15.5638i	1.08879	−	0.628614i	0.155537	−	0.987830i	$-0.450289\pi$
$613$	26.9572	−	15.5638i	1.08879	−	0.628614i	0.933255	+	0.359216i	$0.116956\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			8.23255i			0.331430i	0.986174	+	0.165715i	$0.0529932\pi$
$617$			8.23255i			0.331430i	−0.986174	+	0.165715i	$0.947007\pi$
$618$		0			0
$619$	−18.6796	−	32.3540i	−0.750796	−	1.30042i	−0.947437	−	0.319941i	$-0.896337\pi$
$619$	−18.6796	−	32.3540i	−0.750796	−	1.30042i	0.196641	−	0.980475i	$-0.436997\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.58187	+	13.0130i	0.223633	+	0.521354i
$624$		0			0
$625$	−9.55180	+	16.5442i	−0.382072	+	0.661768i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.43544			0.256598
$630$		0			0
$631$			1.68550i			0.0670986i	0.999437	+	0.0335493i	$0.0106811\pi$
$631$			1.68550i			0.0670986i	−0.999437	+	0.0335493i	$0.989319\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.4929	+	6.05808i	0.416398	+	0.240407i
$636$		0			0
$637$	−27.9216	+	8.23549i	−1.10630	+	0.326302i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.16827	−	5.29330i	0.362125	−	0.209073i	−0.307888	−	0.951423i	$-0.599622\pi$
$641$	9.16827	−	5.29330i	0.362125	−	0.209073i	0.670012	+	0.742350i	$0.266289\pi$
$642$		0			0
$643$	27.0154			1.06538			0.532692	−	0.846309i	$-0.321180\pi$
$643$	27.0154			1.06538			0.532692	+	0.846309i	$0.321180\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.31353	−	2.27510i	−0.0516402	−	0.0894434i	0.839050	−	0.544055i	$-0.183112\pi$
$647$	−1.31353	−	2.27510i	−0.0516402	−	0.0894434i	−0.890690	+	0.454611i	$0.849778\pi$
$648$		0			0
$649$	19.1999	−	33.2552i	0.753661	−	1.30538i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.6027	+	28.7567i	−0.649712	+	1.12533i	0.333479	+	0.942757i	$0.391777\pi$
$653$	−16.6027	+	28.7567i	−0.649712	+	1.12533i	−0.983191	+	0.182577i	$0.941556\pi$
$654$		0			0
$655$	7.15528	−	4.13110i	0.279580	−	0.161416i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.108371i		−	0.00422154i	−0.999998	−	0.00211077i	$-0.999328\pi$
$659$		−	0.108371i		−	0.00422154i	0.999998	−	0.00211077i	$-0.000671879\pi$
$660$		0			0
$661$	−2.37028	+	1.36848i	−0.0921932	+	0.0532278i	−0.545388	−	0.838184i	$-0.683618\pi$
$661$	−2.37028	+	1.36848i	−0.0921932	+	0.0532278i	0.453195	+	0.891412i	$0.350284\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.791462	+	6.65390i	−0.0306916	+	0.258027i
$666$		0			0
$667$	−9.34645	+	16.1885i	−0.361896	+	0.626822i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.27627			−0.280897
$672$		0			0
$673$	15.7488			0.607071			0.303536	−	0.952820i	$-0.401833\pi$
$673$	15.7488			0.607071			0.303536	+	0.952820i	$0.401833\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.1246	+	33.1248i	−0.735019	+	1.27309i	0.219696	+	0.975568i	$0.429493\pi$
$677$	−19.1246	+	33.1248i	−0.735019	+	1.27309i	−0.954715	+	0.297522i	$0.903840\pi$
$678$		0			0
$679$	−37.4418	−	27.9930i	−1.43688	−	1.07427i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.06255	−	1.76816i	0.117185	−	0.0676569i	−0.440262	−	0.897869i	$-0.645114\pi$
$683$	3.06255	−	1.76816i	0.117185	−	0.0676569i	0.557447	+	0.830213i	$0.311781\pi$
$684$		0			0
$685$			0.863457i			0.0329910i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	21.5852	−	12.4622i	0.822330	−	0.474772i
$690$		0			0
$691$	−7.99597	+	13.8494i	−0.304181	+	0.526857i	−0.977079	−	0.212879i	$-0.931716\pi$
$691$	−7.99597	+	13.8494i	−0.304181	+	0.526857i	0.672898	+	0.739736i	$0.265049\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.972568	−	1.68454i	0.0368916	−	0.0638981i
$696$		0			0
$697$	18.2901	+	31.6794i	0.692788	+	1.19994i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	42.7818			1.61585			0.807924	−	0.589287i	$-0.200591\pi$
$701$	42.7818			1.61585			0.807924	+	0.589287i	$0.200591\pi$
$702$		0			0
$703$	6.37041	−	3.67796i	0.240265	−	0.138717i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.82852	+	32.1867i	−0.143986	+	1.21051i
$708$		0			0
$709$	17.0615	+	9.85048i	0.640759	+	0.369943i	0.784907	−	0.619614i	$-0.212711\pi$
$709$	17.0615	+	9.85048i	0.640759	+	0.369943i	−0.144148	+	0.989556i	$0.546044\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			17.6708i			0.661776i
$714$		0			0
$715$	−9.65868			−0.361214
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.14423	−	8.91007i	0.191847	−	0.332290i	−0.754015	−	0.656857i	$-0.771885\pi$
$719$	5.14423	−	8.91007i	0.191847	−	0.332290i	0.945863	+	0.324568i	$0.105219\pi$
$720$		0			0
$721$	3.26410	+	7.60958i	0.121561	+	0.283396i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.6095	−	25.3045i	−0.542584	−	0.939784i
$726$		0			0
$727$		−	40.0773i		−	1.48638i	−0.669078	−	0.743192i	$-0.733311\pi$
$727$		−	40.0773i		−	1.48638i	0.669078	−	0.743192i	$-0.266689\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.57100	−	4.94847i	0.317010	−	0.183026i
$732$		0			0
$733$	17.9716	+	10.3759i	0.663795	+	0.383242i	0.793721	−	0.608281i	$-0.208141\pi$
$733$	17.9716	+	10.3759i	0.663795	+	0.383242i	−0.129926	+	0.991524i	$0.541474\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−50.7041	−	29.2741i	−1.86771	−	1.07832i
$738$		0			0
$739$	−8.47770	−	14.6838i	−0.311857	−	0.540152i	0.666907	−	0.745141i	$-0.267618\pi$
$739$	−8.47770	−	14.6838i	−0.311857	−	0.540152i	−0.978764	+	0.204988i	$0.934284\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.45125			−0.199987			−0.0999935	−	0.994988i	$-0.531882\pi$
$743$	−5.45125			−0.199987			−0.0999935	+	0.994988i	$0.531882\pi$
$744$		0			0
$745$	5.18631	+	8.98296i	0.190012	+	0.329110i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.79530	+	15.8419i	0.248295	+	0.578849i
$750$		0			0
$751$	13.5551	+	7.82606i	0.494634	+	0.285577i	0.726495	−	0.687172i	$-0.241148\pi$
$751$	13.5551	+	7.82606i	0.494634	+	0.285577i	−0.231861	+	0.972749i	$0.574481\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			5.73229i			0.208619i
$756$		0			0
$757$			9.16577i			0.333136i	0.986030	+	0.166568i	$0.0532685\pi$
$757$			9.16577i			0.333136i	−0.986030	+	0.166568i	$0.946732\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.73152	−	0.999696i	−0.0627677	−	0.0362389i	0.468288	−	0.883576i	$-0.344871\pi$
$761$	−1.73152	−	0.999696i	−0.0627677	−	0.0362389i	−0.531055	+	0.847337i	$0.678204\pi$
$762$		0			0
$763$	3.74920	+	0.445956i	0.135730	+	0.0161447i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.8510	−	37.8470i	−0.788993	−	1.36658i
$768$		0			0
$769$	2.48446			0.0895920			0.0447960	−	0.998996i	$-0.485736\pi$
$769$	2.48446			0.0895920			0.0447960	+	0.998996i	$0.485736\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.3968	−	45.7206i	−0.949427	−	1.64446i	−0.746635	−	0.665234i	$-0.768332\pi$
$773$	−26.3968	−	45.7206i	−0.949427	−	1.64446i	−0.202792	−	0.979222i	$-0.565002\pi$
$774$		0			0
$775$	−23.9208	−	13.8107i	−0.859261	−	0.496095i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36.2106	+	20.9062i	1.29738	+	0.749042i
$780$		0			0
$781$	−2.33530	+	1.34829i	−0.0835637	+	0.0482455i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			14.6859i			0.524164i
$786$		0			0
$787$	20.0243	+	34.6831i	0.713789	+	1.23632i	0.963425	+	0.267979i	$0.0863559\pi$
$787$	20.0243	+	34.6831i	0.713789	+	1.23632i	−0.249635	+	0.968340i	$0.580311\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.19621	−	6.95015i	0.184756	−	0.247119i
$792$		0			0
$793$	−4.14048	+	7.17153i	−0.147033	+	0.254668i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.9157			0.847139			0.423570	−	0.905864i	$-0.360777\pi$
$797$	23.9157			0.847139			0.423570	+	0.905864i	$0.360777\pi$
$798$		0			0
$799$			32.1766i			1.13833i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−13.5251	−	7.80874i	−0.477292	−	0.275564i
$804$		0			0
$805$	−0.584011	+	4.90984i	−0.0205837	+	0.173049i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.2044	+	13.3971i	−0.815824	+	0.471016i	−0.848974	−	0.528434i	$-0.822779\pi$
$809$	−23.2044	+	13.3971i	−0.815824	+	0.471016i	0.0331504	+	0.999450i	$0.489446\pi$
$810$		0			0
$811$	43.7619			1.53669			0.768345	−	0.640036i	$-0.221081\pi$
$811$	43.7619			1.53669			0.768345	+	0.640036i	$0.221081\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.87845	+	6.71768i	0.135856	+	0.235310i
$816$		0			0
$817$	5.65626	−	9.79693i	0.197887	−	0.342751i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−18.9005	+	32.7366i	−0.659632	+	1.14252i	0.321080	+	0.947052i	$0.395954\pi$
$821$	−18.9005	+	32.7366i	−0.659632	+	1.14252i	−0.980711	+	0.195463i	$0.937379\pi$
$822$		0			0
$823$	19.9129	−	11.4967i	0.694121	−	0.400751i	−0.111033	−	0.993817i	$-0.535416\pi$
$823$	19.9129	−	11.4967i	0.694121	−	0.400751i	0.805154	+	0.593066i	$0.202083\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			31.3424i			1.08988i	0.838474	+	0.544941i	$0.183448\pi$
$827$			31.3424i			1.08988i	−0.838474	+	0.544941i	$0.816552\pi$
$828$		0			0
$829$	−35.5914	+	20.5487i	−1.23614	+	0.713686i	−0.968303	−	0.249779i	$-0.919642\pi$
$829$	−35.5914	+	20.5487i	−1.23614	+	0.713686i	−0.267837	+	0.963464i	$0.586309\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.6817	+	16.8184i	0.612635	+	0.582723i
$834$		0			0
$835$	1.05995	−	1.83589i	0.0366812	−	0.0635337i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.17046			0.178504			0.0892520	−	0.996009i	$-0.471552\pi$
$839$	5.17046			0.178504			0.0892520	+	0.996009i	$0.471552\pi$
$840$		0			0
$841$	11.4173			0.393701
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.36484	+	2.36397i	−0.0469518	+	0.0813230i
$846$		0			0
$847$	−5.72057	+	2.45381i	−0.196561	+	0.0843141i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.70066	−	2.71393i	0.161136	−	0.0930322i
$852$		0			0
$853$			23.8844i			0.817786i	0.912582	+	0.408893i	$0.134085\pi$
$853$			23.8844i			0.817786i	−0.912582	+	0.408893i	$0.865915\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−16.7053	+	9.64484i	−0.570644	+	0.329461i	−0.757406	−	0.652944i	$-0.773534\pi$
$857$	−16.7053	+	9.64484i	−0.570644	+	0.329461i	0.186763	+	0.982405i	$0.440200\pi$
$858$		0			0
$859$	−4.54550	+	7.87303i	−0.155090	+	0.268624i	−0.933092	−	0.359638i	$-0.882900\pi$
$859$	−4.54550	+	7.87303i	−0.155090	+	0.268624i	0.778002	+	0.628262i	$0.216234\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.0104	−	19.0706i	0.374799	−	0.649170i	−0.615498	−	0.788138i	$-0.711045\pi$
$863$	11.0104	−	19.0706i	0.374799	−	0.649170i	0.990297	+	0.138968i	$0.0443785\pi$
$864$		0			0
$865$	−3.04585	−	5.27556i	−0.103562	−	0.179375i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−32.6964			−1.10915
$870$		0			0
$871$	−57.7053	+	33.3162i	−1.95527	+	1.12888i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.9240	−	9.66254i	−0.436912	−	0.326654i
$876$		0			0
$877$	−23.0512	−	13.3086i	−0.778385	−	0.449401i	0.0574726	−	0.998347i	$-0.481696\pi$
$877$	−23.0512	−	13.3086i	−0.778385	−	0.449401i	−0.835858	+	0.548946i	$0.815029\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.95975i		−	0.200789i	−0.994948	−	0.100395i	$-0.967990\pi$
$881$		−	5.95975i		−	0.200789i	0.994948	−	0.100395i	$-0.0320105\pi$
$882$		0			0
$883$	8.91564			0.300035			0.150018	−	0.988683i	$-0.452067\pi$
$883$	8.91564			0.300035			0.150018	+	0.988683i	$0.452067\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.1142	−	45.2311i	0.876829	−	1.51871i	0.0220275	−	0.999757i	$-0.492988\pi$
$887$	26.1142	−	45.2311i	0.876829	−	1.51871i	0.854802	−	0.518955i	$-0.173679\pi$
$888$		0			0
$889$	30.2003	−	40.3942i	1.01289	−	1.35478i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.3894	+	31.8514i	0.615379	+	1.06587i
$894$		0			0
$895$		−	0.0207832i		−	0.000694704i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	33.0885	−	19.1037i	1.10356	−	0.637143i
$900$		0			0
$901$	−18.0943	−	10.4468i	−0.602809	−	0.348032i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.6594	−	6.15421i	−0.354330	−	0.204573i
$906$		0			0
$907$	8.71744	+	15.0990i	0.289458	+	0.501355i	0.973680	−	0.227918i	$-0.0731917\pi$
$907$	8.71744	+	15.0990i	0.289458	+	0.501355i	−0.684223	+	0.729273i	$0.739858\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−39.1655			−1.29761			−0.648805	−	0.760954i	$-0.724731\pi$
$911$	−39.1655			−1.29761			−0.648805	+	0.760954i	$0.724731\pi$
$912$		0			0
$913$	−6.71885	−	11.6374i	−0.222361	−	0.385141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.5580	−	31.6078i	−0.447725	−	1.04378i
$918$		0			0
$919$	35.0892	+	20.2588i	1.15749	+	0.668276i	0.950701	−	0.310110i	$-0.100366\pi$
$919$	35.0892	+	20.2588i	1.15749	+	0.668276i	0.206787	+	0.978386i	$0.433699\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			3.06892i			0.101015i
$924$		0			0
$925$			8.48433i			0.278963i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0117	+	8.08966i	0.459709	+	0.265413i	0.711922	−	0.702259i	$-0.247825\pi$
$929$	14.0117	+	8.08966i	0.459709	+	0.265413i	−0.252213	+	0.967672i	$0.581158\pi$
$930$		0			0
$931$	27.1150	+	6.54307i	0.888659	+	0.214441i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.04831	+	7.01188i	0.132394	+	0.229313i
$936$		0			0
$937$	7.65486			0.250073			0.125037	−	0.992152i	$-0.460095\pi$
$937$	7.65486			0.250073			0.125037	+	0.992152i	$0.460095\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.8841	−	20.5839i	−0.387411	−	0.671016i	0.604689	−	0.796462i	$-0.293297\pi$
$941$	−11.8841	−	20.5839i	−0.387411	−	0.671016i	−0.992101	+	0.125445i	$0.959964\pi$
$942$		0			0
$943$	26.7194	+	15.4265i	0.870104	+	0.502355i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.2481	+	20.3505i	1.14541	+	0.661302i	0.947764	−	0.318972i	$-0.103338\pi$
$947$	35.2481	+	20.3505i	1.14541	+	0.661302i	0.197644	+	0.980274i	$0.436671\pi$
$948$		0			0
$949$	−15.3927	+	8.88696i	−0.499667	+	0.288483i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	19.5679i		−	0.633867i	−0.948448	−	0.316934i	$-0.897347\pi$
$953$		−	19.5679i		−	0.633867i	0.948448	−	0.316934i	$-0.102653\pi$
$954$		0			0
$955$	0.958636	+	1.66041i	0.0310207	+	0.0537295i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.56913	+	0.424537i	0.115253	+	0.0137090i
$960$		0			0
$961$	2.55908	−	4.43245i	0.0825509	−	0.142982i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.9277			−0.448347
$966$		0			0
$967$			24.6339i			0.792174i	0.918213	+	0.396087i	$0.129632\pi$
$967$			24.6339i			0.792174i	−0.918213	+	0.396087i	$0.870368\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41.0803	−	23.7177i	−1.31833	−	0.761137i	−0.334869	−	0.942265i	$-0.608692\pi$
$971$	−41.0803	−	23.7177i	−1.31833	−	0.761137i	−0.983460	+	0.181128i	$0.942025\pi$
$972$		0			0
$973$	−6.48491	−	4.84838i	−0.207897	−	0.155432i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−48.9364	+	28.2534i	−1.56561	+	0.903908i	−0.568943	+	0.822377i	$0.692648\pi$
$977$	−48.9364	+	28.2534i	−1.56561	+	0.903908i	−0.996671	+	0.0815309i	$0.974019\pi$
$978$		0			0
$979$	19.5563			0.625023
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0801	−	31.3156i	−0.576665	−	0.998814i	−0.995859	−	0.0909167i	$-0.971020\pi$
$983$	−18.0801	−	31.3156i	−0.576665	−	0.998814i	0.419193	−	0.907897i	$-0.362313\pi$
$984$		0			0
$985$	−4.86065	+	8.41890i	−0.154873	+	0.268248i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.17369	−	7.22905i	0.132716	−	0.229870i
$990$		0			0
$991$	8.72687	−	5.03846i	0.277218	−	0.160052i	−0.354945	−	0.934887i	$-0.615500\pi$
$991$	8.72687	−	5.03846i	0.277218	−	0.160052i	0.632163	+	0.774835i	$0.282167\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2.03813i		−	0.0646130i
$996$		0			0
$997$	14.5584	−	8.40532i	0.461070	−	0.266199i	−0.251424	−	0.967877i	$-0.580899\pi$
$997$	14.5584	−	8.40532i	0.461070	−	0.266199i	0.712494	+	0.701678i	$0.247565\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.bu.c.1871.15		48
3.2	odd	2	inner	2016.2.bu.c.1871.10		48
4.3	odd	2		504.2.bm.c.107.13	yes	48
7.4	even	3	inner	2016.2.bu.c.431.16		48
8.3	odd	2	inner	2016.2.bu.c.1871.9		48
8.5	even	2		504.2.bm.c.107.21	yes	48
12.11	even	2		504.2.bm.c.107.12	yes	48
21.11	odd	6	inner	2016.2.bu.c.431.9		48
24.5	odd	2		504.2.bm.c.107.4	✓	48
24.11	even	2	inner	2016.2.bu.c.1871.16		48
28.11	odd	6		504.2.bm.c.179.4	yes	48
56.11	odd	6	inner	2016.2.bu.c.431.10		48
56.53	even	6		504.2.bm.c.179.12	yes	48
84.11	even	6		504.2.bm.c.179.21	yes	48
168.11	even	6	inner	2016.2.bu.c.431.15		48
168.53	odd	6		504.2.bm.c.179.13	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bm.c.107.4	✓	48	24.5	odd	2
504.2.bm.c.107.12	yes	48	12.11	even	2
504.2.bm.c.107.13	yes	48	4.3	odd	2
504.2.bm.c.107.21	yes	48	8.5	even	2
504.2.bm.c.179.4	yes	48	28.11	odd	6
504.2.bm.c.179.12	yes	48	56.53	even	6
504.2.bm.c.179.13	yes	48	168.53	odd	6
504.2.bm.c.179.21	yes	48	84.11	even	6
2016.2.bu.c.431.9		48	21.11	odd	6	inner
2016.2.bu.c.431.10		48	56.11	odd	6	inner
2016.2.bu.c.431.15		48	168.11	even	6	inner
2016.2.bu.c.431.16		48	7.4	even	3	inner
2016.2.bu.c.1871.9		48	8.3	odd	2	inner
2016.2.bu.c.1871.10		48	3.2	odd	2	inner
2016.2.bu.c.1871.15		48	1.1	even	1	trivial
2016.2.bu.c.1871.16		48	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.bu (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.317795 - 0.550436i) q^{5} +(-2.11900 - 1.58425i) q^{7} +O(q^{10})\)	\(q+(0.317795 - 0.550436i) q^{5} +(-2.11900 - 1.58425i) q^{7} +(-3.16457 + 1.82707i) q^{11} +4.15869i q^{13} +(3.01908 - 1.74306i) q^{17} +(1.99238 - 3.45090i) q^{19} +(1.47015 - 2.54638i) q^{23} +(2.29801 + 3.98028i) q^{25} -6.35746 q^{29} +(-5.20467 + 3.00492i) q^{31} +(-1.54543 + 0.662908i) q^{35} +(1.59870 + 0.923007i) q^{37} +10.4931i q^{41} +2.83895 q^{43} +(-4.61494 + 7.99332i) q^{47} +(1.98031 + 6.71404i) q^{49} +(-2.99666 - 5.19038i) q^{53} +2.32253i q^{55} +(-9.10070 + 5.25429i) q^{59} +(1.72447 + 0.995622i) q^{61} +(2.28909 + 1.32161i) q^{65} +(8.01122 + 13.8758i) q^{67} +0.737952 q^{71} +(2.13696 + 3.70132i) q^{73} +(9.60025 + 1.14192i) q^{77} +(7.74900 + 4.47389i) q^{79} +3.67740i q^{83} -2.21574i q^{85} +(-4.63483 - 2.67592i) q^{89} +(6.58841 - 8.81226i) q^{91} +(-1.26633 - 2.19335i) q^{95} +17.6696 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 32 q^{19} + 160 q^{43} + 56 q^{49} + 16 q^{73} + 32 q^{91} + 240 q^{97}+O(q^{100}) \) 48 * q + 32 * q^19 + 160 * q^43 + 56 * q^49 + 16 * q^73 + 32 * q^91 + 240 * q^97

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