LMFDB - Embedded newform 2016.2.cp.b.17.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(17,2016)]

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([0, 3, 3, 1]))

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

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

chi := DirichletCharacter("2016.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.cp (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:	$56$
Relative dimension:	$28$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.23
Character	$\chi$	$=$	2016.17
Dual form			2016.2.cp.b.593.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87230 - 1.08097i) q^{5} +(-2.55958 - 0.669737i) q^{7} +O(q^{10})$	$q+(1.87230 - 1.08097i) q^{5} +(-2.55958 - 0.669737i) q^{7} +(-1.67157 + 2.89525i) q^{11} +1.74247 q^{13} +(-0.283937 + 0.491793i) q^{17} +(0.270105 + 0.467836i) q^{19} +(5.21616 - 3.01155i) q^{23} +(-0.162997 + 0.282319i) q^{25} +1.77912 q^{29} +(6.56726 + 3.79161i) q^{31} +(-5.51627 + 1.51289i) q^{35} +(9.60029 - 5.54273i) q^{37} +7.77201 q^{41} -1.80152i q^{43} +(-0.679499 - 1.17693i) q^{47} +(6.10291 + 3.42849i) q^{49} +(1.46832 - 2.54321i) q^{53} +7.22769i q^{55} +(-9.84763 - 5.68553i) q^{59} +(-5.60858 - 9.71434i) q^{61} +(3.26242 - 1.88356i) q^{65} +(10.7563 + 6.21014i) q^{67} -7.79753i q^{71} +(7.56066 + 4.36515i) q^{73} +(6.21757 - 6.29110i) q^{77} +(4.77913 + 8.27770i) q^{79} -15.9958i q^{83} +1.22771i q^{85} +(-6.30930 - 10.9280i) q^{89} +(-4.45999 - 1.16699i) q^{91} +(1.01144 + 0.583953i) q^{95} +16.4013i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 20 q^{7}+O(q^{10}) $ 56 * q + 20 * q^7	$ 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) $ 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

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
$5$	1.87230	−	1.08097i	0.837318	−	0.483426i	−0.0190339	−	0.999819i	$-0.506059\pi$
$5$	1.87230	−	1.08097i	0.837318	−	0.483426i	0.856352	+	0.516393i	$0.172726\pi$
$6$		0			0
$7$	−2.55958	−	0.669737i	−0.967431	−	0.253137i
$7$	−2.55958	−	0.669737i	−0.967431	−	0.253137i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.67157	+	2.89525i	−0.503998	+	0.872949i	0.495992	+	0.868327i	$0.334805\pi$
$11$	−1.67157	+	2.89525i	−0.503998	+	0.872949i	−0.999989	+	0.00462217i	$0.998529\pi$
$12$		0			0
$13$	1.74247			0.483274			0.241637	−	0.970367i	$-0.422316\pi$
$13$	1.74247			0.483274			0.241637	+	0.970367i	$0.422316\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.283937	+	0.491793i	−0.0688648	+	0.119277i	−0.898402	−	0.439174i	$-0.855271\pi$
$17$	−0.283937	+	0.491793i	−0.0688648	+	0.119277i	0.829537	+	0.558452i	$0.188604\pi$
$18$		0			0
$19$	0.270105	+	0.467836i	0.0619664	+	0.107329i	0.895344	−	0.445375i	$-0.146929\pi$
$19$	0.270105	+	0.467836i	0.0619664	+	0.107329i	−0.833378	+	0.552704i	$0.813596\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.21616	−	3.01155i	1.08764	−	0.627952i	0.154696	−	0.987962i	$-0.450560\pi$
$23$	5.21616	−	3.01155i	1.08764	−	0.627952i	0.932948	+	0.360010i	$0.117227\pi$
$24$		0			0
$25$	−0.162997	+	0.282319i	−0.0325994	+	0.0564638i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.77912			0.330374			0.165187	−	0.986262i	$-0.447177\pi$
$29$	1.77912			0.330374			0.165187	+	0.986262i	$0.447177\pi$
$30$		0			0
$31$	6.56726	+	3.79161i	1.17952	+	0.680993i	0.955902	−	0.293685i	$-0.0948815\pi$
$31$	6.56726	+	3.79161i	1.17952	+	0.680993i	0.223613	+	0.974678i	$0.428215\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.51627	+	1.51289i	−0.932419	+	0.255725i
$36$		0			0
$37$	9.60029	−	5.54273i	1.57828	−	0.911219i	0.583177	−	0.812345i	$-0.301809\pi$
$37$	9.60029	−	5.54273i	1.57828	−	0.911219i	0.995100	−	0.0988741i	$-0.0315241\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.77201			1.21378			0.606892	−	0.794785i	$-0.292416\pi$
$41$	7.77201			1.21378			0.606892	+	0.794785i	$0.292416\pi$
$42$		0			0
$43$		−	1.80152i		−	0.274730i	−0.990521	−	0.137365i	$-0.956137\pi$
$43$		−	1.80152i		−	0.274730i	0.990521	−	0.137365i	$-0.0438633\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.679499	−	1.17693i	−0.0991151	−	0.171672i	0.812204	−	0.583374i	$-0.198268\pi$
$47$	−0.679499	−	1.17693i	−0.0991151	−	0.171672i	−0.911319	+	0.411702i	$0.864935\pi$
$48$		0			0
$49$	6.10291	+	3.42849i	0.871844	+	0.489784i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.46832	−	2.54321i	0.201689	−	0.349336i	−0.747383	−	0.664393i	$-0.768690\pi$
$53$	1.46832	−	2.54321i	0.201689	−	0.349336i	0.949073	+	0.315057i	$0.102023\pi$
$54$		0			0
$55$			7.22769i			0.974581i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.84763	−	5.68553i	−1.28205	−	0.740194i	−0.304830	−	0.952407i	$-0.598599\pi$
$59$	−9.84763	−	5.68553i	−1.28205	−	0.740194i	−0.977223	+	0.212213i	$0.931933\pi$
$60$		0			0
$61$	−5.60858	−	9.71434i	−0.718105	−	1.24379i	−0.961750	−	0.273929i	$-0.911677\pi$
$61$	−5.60858	−	9.71434i	−0.718105	−	1.24379i	0.243645	−	0.969864i	$-0.421657\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.26242	−	1.88356i	0.404654	−	0.233627i
$66$		0			0
$67$	10.7563	+	6.21014i	1.31409	+	0.758689i	0.982770	−	0.184830i	$-0.0591736\pi$
$67$	10.7563	+	6.21014i	1.31409	+	0.758689i	0.331317	+	0.943519i	$0.392507\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.79753i		−	0.925397i	−0.886516	−	0.462698i	$-0.846881\pi$
$71$		−	7.79753i		−	0.925397i	0.886516	−	0.462698i	$-0.153119\pi$
$72$		0			0
$73$	7.56066	+	4.36515i	0.884909	+	0.510902i	0.872274	−	0.489018i	$-0.162645\pi$
$73$	7.56066	+	4.36515i	0.884909	+	0.510902i	0.0126348	+	0.999920i	$0.495978\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.21757	−	6.29110i	0.708558	−	0.716938i
$78$		0			0
$79$	4.77913	+	8.27770i	0.537694	+	0.931314i	0.999028	+	0.0440870i	$0.0140379\pi$
$79$	4.77913	+	8.27770i	0.537694	+	0.931314i	−0.461333	+	0.887227i	$0.652629\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	15.9958i		−	1.75577i	−0.478875	−	0.877883i	$-0.658955\pi$
$83$		−	15.9958i		−	1.75577i	0.478875	−	0.877883i	$-0.341045\pi$
$84$		0			0
$85$			1.22771i			0.133164i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.30930	−	10.9280i	−0.668784	−	1.15837i	−0.978244	−	0.207456i	$-0.933482\pi$
$89$	−6.30930	−	10.9280i	−0.668784	−	1.15837i	0.309460	−	0.950912i	$-0.399852\pi$
$90$		0			0
$91$	−4.45999	−	1.16699i	−0.467534	−	0.122334i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.01144	+	0.583953i	0.103771	+	0.0599123i
$96$		0			0
$97$			16.4013i			1.66530i	0.553802	+	0.832649i	$0.313177\pi$
$97$			16.4013i			1.66530i	−0.553802	+	0.832649i	$0.686823\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.4514	−	7.76615i	−1.33846	−	0.772761i	−0.351882	−	0.936044i	$-0.614458\pi$
$101$	−13.4514	−	7.76615i	−1.33846	−	0.772761i	−0.986579	+	0.163283i	$0.947792\pi$
$102$		0			0
$103$	4.69281	−	2.70939i	0.462396	−	0.266964i	−0.250655	−	0.968076i	$-0.580646\pi$
$103$	4.69281	−	2.70939i	0.462396	−	0.266964i	0.713051	+	0.701112i	$0.247313\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.85983	+	3.22131i	0.179796	+	0.311416i	0.941811	−	0.336144i	$-0.109123\pi$
$107$	1.85983	+	3.22131i	0.179796	+	0.311416i	−0.762014	+	0.647560i	$0.775790\pi$
$108$		0			0
$109$	−5.72483	−	3.30523i	−0.548340	−	0.316584i	0.200112	−	0.979773i	$-0.435869\pi$
$109$	−5.72483	−	3.30523i	−0.548340	−	0.316584i	−0.748452	+	0.663189i	$0.769203\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.32161i			0.406543i	0.979122	+	0.203272i	$0.0651575\pi$
$113$			4.32161i			0.406543i	−0.979122	+	0.203272i	$0.934843\pi$
$114$		0			0
$115$	6.51081	−	11.2771i	0.607136	−	1.05159i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.05613	−	1.06862i	0.0968154	−	0.0979603i
$120$		0			0
$121$	−0.0882982	−	0.152937i	−0.00802711	−	0.0139034i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.5145i			1.02989i
$126$		0			0
$127$	−6.96976			−0.618467			−0.309233	−	0.950986i	$-0.600072\pi$
$127$	−6.96976			−0.618467			−0.309233	+	0.950986i	$0.600072\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.62582	−	4.40277i	0.666271	−	0.384672i	−0.128391	−	0.991724i	$-0.540981\pi$
$131$	7.62582	−	4.40277i	0.666271	−	0.384672i	0.794662	+	0.607052i	$0.207648\pi$
$132$		0			0
$133$	−0.378029	−	1.37836i	−0.0327793	−	0.119519i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.5256	+	8.38635i	1.24100	+	0.716494i	0.969299	−	0.245887i	$-0.0790791\pi$
$137$	14.5256	+	8.38635i	1.24100	+	0.716494i	0.271705	+	0.962381i	$0.412412\pi$
$138$		0			0
$139$	10.3308			0.876249			0.438124	−	0.898914i	$-0.355643\pi$
$139$	10.3308			0.876249			0.438124	+	0.898914i	$0.355643\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.91266	+	5.04487i	−0.243569	+	0.421873i
$144$		0			0
$145$	3.33105	−	1.92318i	0.276628	−	0.159711i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.10661	−	3.64876i	−0.172580	−	0.298918i	0.766741	−	0.641957i	$-0.221877\pi$
$149$	−2.10661	−	3.64876i	−0.172580	−	0.298918i	−0.939321	+	0.343039i	$0.888544\pi$
$150$		0			0
$151$	3.10493	−	5.37790i	0.252676	−	0.437647i	−0.711586	−	0.702599i	$-0.752023\pi$
$151$	3.10493	−	5.37790i	0.252676	−	0.437647i	0.964262	+	0.264952i	$0.0853561\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.3945			1.31684
$156$		0			0
$157$	−10.3210	+	17.8764i	−0.823702	+	1.42669i	0.0792057	+	0.996858i	$0.474762\pi$
$157$	−10.3210	+	17.8764i	−0.823702	+	1.42669i	−0.902907	+	0.429835i	$0.858572\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.3681	+	4.21486i	−1.21118	+	0.332177i
$162$		0			0
$163$	−3.56609	+	2.05888i	−0.279317	+	0.161264i	−0.633114	−	0.774058i	$-0.718224\pi$
$163$	−3.56609	+	2.05888i	−0.279317	+	0.161264i	0.353797	+	0.935322i	$0.384890\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.2654			1.02651			0.513253	−	0.858237i	$-0.328440\pi$
$167$	13.2654			1.02651			0.513253	+	0.858237i	$0.328440\pi$
$168$		0			0
$169$	−9.96381			−0.766447
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−19.4886	+	11.2517i	−1.48169	+	0.855453i	−0.999784	−	0.0207720i	$-0.993388\pi$
$173$	−19.4886	+	11.2517i	−1.48169	+	0.855453i	−0.481903	+	0.876225i	$0.660054\pi$
$174$		0			0
$175$	0.606283	−	0.613453i	0.0458307	−	0.0463727i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.63774	+	16.6931i	−0.720358	+	1.24770i	0.240498	+	0.970650i	$0.422689\pi$
$179$	−9.63774	+	16.6931i	−0.720358	+	1.24770i	−0.960856	+	0.277048i	$0.910644\pi$
$180$		0			0
$181$	22.3500			1.66126			0.830631	−	0.556823i	$-0.187980\pi$
$181$	22.3500			1.66126			0.830631	+	0.556823i	$0.187980\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.9831	−	20.7553i	0.881013	−	1.52596i
$186$		0			0
$187$	−0.949241	−	1.64413i	−0.0694154	−	0.120231i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.69175	−	3.28613i	0.411841	−	0.237776i	−0.279740	−	0.960076i	$-0.590248\pi$
$191$	5.69175	−	3.28613i	0.411841	−	0.237776i	0.691580	+	0.722300i	$0.256915\pi$
$192$		0			0
$193$	3.70334	−	6.41438i	0.266572	−	0.461717i	−0.701402	−	0.712766i	$-0.747442\pi$
$193$	3.70334	−	6.41438i	0.266572	−	0.461717i	0.967974	+	0.251049i	$0.0807755\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.80322			0.128474			0.0642371	−	0.997935i	$-0.479539\pi$
$197$	1.80322			0.128474			0.0642371	+	0.997935i	$0.479539\pi$
$198$		0			0
$199$	7.72728	+	4.46135i	0.547772	+	0.316257i	0.748223	−	0.663447i	$-0.230907\pi$
$199$	7.72728	+	4.46135i	0.547772	+	0.316257i	−0.200451	+	0.979704i	$0.564241\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.55380	−	1.19154i	−0.319614	−	0.0836299i
$204$		0			0
$205$	14.5515	−	8.40132i	1.01632	−	0.586774i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.80600			−0.124924
$210$		0			0
$211$		−	11.3878i		−	0.783969i	−0.919972	−	0.391985i	$-0.871789\pi$
$211$		−	11.3878i		−	0.783969i	0.919972	−	0.391985i	$-0.128211\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.94740	−	3.37299i	−0.132811	−	0.230036i
$216$		0			0
$217$	−14.2701	−	14.1033i	−0.968715	−	0.957393i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.494751	+	0.856933i	−0.0332805	+	0.0576436i
$222$		0			0
$223$			4.90035i			0.328151i	0.986448	+	0.164076i	$0.0524641\pi$
$223$			4.90035i			0.328151i	−0.986448	+	0.164076i	$0.947536\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	18.4585	+	10.6570i	1.22514	+	0.707332i	0.966008	−	0.258511i	$-0.0832319\pi$
$227$	18.4585	+	10.6570i	1.22514	+	0.707332i	0.259127	+	0.965843i	$0.416565\pi$
$228$		0			0
$229$	1.59915	+	2.76981i	0.105675	+	0.183034i	0.914014	−	0.405683i	$-0.132966\pi$
$229$	1.59915	+	2.76981i	0.105675	+	0.183034i	−0.808339	+	0.588717i	$0.799633\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.35808	−	1.93879i	0.219995	−	0.127014i	−0.385953	−	0.922518i	$-0.626127\pi$
$233$	3.35808	−	1.93879i	0.219995	−	0.127014i	0.605948	+	0.795504i	$0.292794\pi$
$234$		0			0
$235$	−2.54445	−	1.46904i	−0.165982	−	0.0958295i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			22.1432i			1.43233i	0.697933	+	0.716163i	$0.254103\pi$
$239$			22.1432i			1.43233i	−0.697933	+	0.716163i	$0.745897\pi$
$240$		0			0
$241$	6.23397	+	3.59918i	0.401565	+	0.231844i	0.687159	−	0.726507i	$-0.258858\pi$
$241$	6.23397	+	3.59918i	0.401565	+	0.231844i	−0.285594	+	0.958351i	$0.592191\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.1326	−	0.177912i	0.966784	−	0.0113664i
$246$		0			0
$247$	0.470650	+	0.815190i	0.0299467	+	0.0518693i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	9.58773i		−	0.605172i	−0.953122	−	0.302586i	$-0.902150\pi$
$251$		−	9.58773i		−	0.605172i	0.953122	−	0.302586i	$-0.0978500\pi$
$252$		0			0
$253$			20.1361i			1.26595i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.33124	+	7.50192i	0.270175	+	0.467957i	0.968907	−	0.247427i	$-0.0795851\pi$
$257$	4.33124	+	7.50192i	0.270175	+	0.467957i	−0.698731	+	0.715384i	$0.746252\pi$
$258$		0			0
$259$	−28.2849	+	7.75739i	−1.75754	+	0.482021i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.39801	−	3.11654i	−0.332855	−	0.192174i	0.324253	−	0.945970i	$-0.394887\pi$
$263$	−5.39801	−	3.11654i	−0.332855	−	0.192174i	−0.657108	+	0.753796i	$0.728220\pi$
$264$		0			0
$265$		−	6.34886i		−	0.390007i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.46950	−	5.46722i	−0.577366	−	0.333342i	0.182720	−	0.983165i	$-0.441510\pi$
$269$	−9.46950	−	5.46722i	−0.577366	−	0.333342i	−0.760086	+	0.649823i	$0.774843\pi$
$270$		0			0
$271$	18.1322	−	10.4687i	1.10145	−	0.635925i	0.164852	−	0.986318i	$-0.447285\pi$
$271$	18.1322	−	10.4687i	1.10145	−	0.635925i	0.936603	+	0.350393i	$0.113952\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.544922	−	0.943833i	−0.0328600	−	0.0569152i
$276$		0			0
$277$	−3.80313	−	2.19574i	−0.228508	−	0.131929i	0.381376	−	0.924420i	$-0.375450\pi$
$277$	−3.80313	−	2.19574i	−0.228508	−	0.131929i	−0.609884	+	0.792491i	$0.708784\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.93386i		−	0.473295i	−0.971596	−	0.236647i	$-0.923951\pi$
$281$		−	7.93386i		−	0.473295i	0.971596	−	0.236647i	$-0.0760486\pi$
$282$		0			0
$283$	15.1600	−	26.2580i	0.901171	−	1.56087i	0.0751951	−	0.997169i	$-0.476042\pi$
$283$	15.1600	−	26.2580i	0.901171	−	1.56087i	0.825976	−	0.563705i	$-0.190625\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.8931	−	5.20520i	−1.17425	−	0.307253i
$288$		0			0
$289$	8.33876	+	14.4432i	0.490515	+	0.849597i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.34894i		−	0.254068i	−0.991898	−	0.127034i	$-0.959454\pi$
$293$		−	4.34894i		−	0.254068i	0.991898	−	0.127034i	$-0.0405457\pi$
$294$		0			0
$295$	−24.5836			−1.43131
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.08899	−	5.24753i	0.525630	−	0.303473i
$300$		0			0
$301$	−1.20655	+	4.61115i	−0.0695442	+	0.265782i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.0019	−	12.1254i	−1.20256	−	0.694300i
$306$		0			0
$307$	−4.81287			−0.274685			−0.137343	−	0.990524i	$-0.543856\pi$
$307$	−4.81287			−0.274685			−0.137343	+	0.990524i	$0.543856\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.96879	+	6.87414i	−0.225049	+	0.389797i	−0.956334	−	0.292275i	$-0.905588\pi$
$311$	−3.96879	+	6.87414i	−0.225049	+	0.389797i	0.731285	+	0.682072i	$0.238921\pi$
$312$		0			0
$313$	1.95587	−	1.12922i	0.110552	−	0.0638273i	−0.443704	−	0.896173i	$-0.646336\pi$
$313$	1.95587	−	1.12922i	0.110552	−	0.0638273i	0.554257	+	0.832346i	$0.313003\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.72820	−	6.45744i	−0.209397	−	0.362686i	0.742128	−	0.670258i	$-0.233817\pi$
$317$	−3.72820	−	6.45744i	−0.209397	−	0.362686i	−0.951525	+	0.307572i	$0.900483\pi$
$318$		0			0
$319$	−2.97393	+	5.15099i	−0.166508	+	0.288400i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.306771			−0.0170692
$324$		0			0
$325$	−0.284017	+	0.491932i	−0.0157544	+	0.0272875i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.951001	+	3.46752i	0.0524304	+	0.191171i
$330$		0			0
$331$	−19.5260	+	11.2733i	−1.07325	+	0.619639i	−0.929067	−	0.369912i	$-0.879388\pi$
$331$	−19.5260	+	11.2733i	−1.07325	+	0.619639i	−0.144180	+	0.989551i	$0.546054\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	26.8519			1.46708
$336$		0			0
$337$	−25.9907			−1.41580			−0.707901	−	0.706312i	$-0.750358\pi$
$337$	−25.9907			−1.41580			−0.707901	+	0.706312i	$0.750358\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−21.9553	+	12.6759i	−1.18895	+	0.686438i
$342$		0			0
$343$	−13.3247	−	12.8628i	−0.719466	−	0.694528i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.86526	+	13.6230i	−0.422229	+	0.731322i	−0.996157	−	0.0875835i	$-0.972086\pi$
$347$	−7.86526	+	13.6230i	−0.422229	+	0.731322i	0.573928	+	0.818906i	$0.305419\pi$
$348$		0			0
$349$	−1.79135			−0.0958889			−0.0479444	−	0.998850i	$-0.515267\pi$
$349$	−1.79135			−0.0958889			−0.0479444	+	0.998850i	$0.515267\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.0108	+	22.5353i	−0.692493	+	1.19943i	0.278526	+	0.960429i	$0.410154\pi$
$353$	−13.0108	+	22.5353i	−0.692493	+	1.19943i	−0.971019	+	0.239004i	$0.923179\pi$
$354$		0			0
$355$	−8.42892	−	14.5993i	−0.447361	−	0.774851i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.8932	+	8.59857i	−0.786031	+	0.453815i	−0.838563	−	0.544804i	$-0.816604\pi$
$359$	−14.8932	+	8.59857i	−0.786031	+	0.453815i	0.0525323	+	0.998619i	$0.483271\pi$
$360$		0			0
$361$	9.35409	−	16.2018i	0.492320	−	0.852724i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.8744			0.987933
$366$		0			0
$367$	−11.5047	−	6.64226i	−0.600542	−	0.346723i	0.168713	−	0.985665i	$-0.446039\pi$
$367$	−11.5047	−	6.64226i	−0.600542	−	0.346723i	−0.769255	+	0.638942i	$0.779372\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.46157	+	5.52616i	−0.283550	+	0.286904i
$372$		0			0
$373$	−32.2609	+	18.6259i	−1.67041	+	0.964410i	−0.702999	+	0.711191i	$0.748156\pi$
$373$	−32.2609	+	18.6259i	−1.67041	+	0.964410i	−0.967409	+	0.253219i	$0.918511\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.10006			0.159661
$378$		0			0
$379$			33.4030i			1.71580i	0.513821	+	0.857898i	$0.328230\pi$
$379$			33.4030i			1.71580i	−0.513821	+	0.857898i	$0.671770\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.1288	−	17.5435i	−0.517556	−	0.896433i	−0.999792	−	0.0203917i	$-0.993509\pi$
$383$	−10.1288	−	17.5435i	−0.517556	−	0.896433i	0.482236	−	0.876041i	$-0.339825\pi$
$384$		0			0
$385$	4.84065	−	18.4999i	0.246702	−	0.942840i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.87868	−	17.1104i	0.500869	−	0.867530i	−0.499131	−	0.866527i	$-0.666347\pi$
$389$	9.87868	−	17.1104i	0.500869	−	0.867530i	0.999999	−	0.00100341i	$-0.000319396\pi$
$390$		0			0
$391$			3.42036i			0.172975i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.8959	+	10.3322i	0.900442	+	0.519870i
$396$		0			0
$397$	−6.15364	−	10.6584i	−0.308842	−	0.534930i	0.669267	−	0.743022i	$-0.266608\pi$
$397$	−6.15364	−	10.6584i	−0.308842	−	0.534930i	−0.978109	+	0.208092i	$0.933275\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−29.5658	+	17.0698i	−1.47645	+	0.852427i	−0.999647	−	0.0265843i	$-0.991537\pi$
$401$	−29.5658	+	17.0698i	−1.47645	+	0.852427i	−0.476801	+	0.879011i	$0.658204\pi$
$402$		0			0
$403$	11.4432	+	6.60676i	0.570029	+	0.329106i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			37.0603i			1.83701i
$408$		0			0
$409$	−23.4733	−	13.5523i	−1.16068	−	0.670119i	−0.209213	−	0.977870i	$-0.567090\pi$
$409$	−23.4733	−	13.5523i	−1.16068	−	0.670119i	−0.951467	+	0.307752i	$0.900423\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	21.3980	+	21.1479i	1.05293	+	1.04062i
$414$		0			0
$415$	−17.2910	−	29.9489i	−0.848782	−	1.47013i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	6.84933i		−	0.334612i	−0.985905	−	0.167306i	$-0.946493\pi$
$419$		−	6.84933i		−	0.334612i	0.985905	−	0.167306i	$-0.0535067\pi$
$420$		0			0
$421$			10.8560i			0.529089i	0.964374	+	0.264544i	$0.0852215\pi$
$421$			10.8560i			0.529089i	−0.964374	+	0.264544i	$0.914778\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0925617	−	0.160322i	−0.00448990	−	0.00777674i
$426$		0			0
$427$	7.84955	+	28.6209i	0.379866	+	1.38506i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−19.5579	−	11.2917i	−0.942069	−	0.543904i	−0.0514606	−	0.998675i	$-0.516388\pi$
$431$	−19.5579	−	11.2917i	−0.942069	−	0.543904i	−0.890608	+	0.454771i	$0.849721\pi$
$432$		0			0
$433$			33.9522i			1.63164i	0.578307	+	0.815819i	$0.303713\pi$
$433$			33.9522i			1.63164i	−0.578307	+	0.815819i	$0.696287\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.81783	+	1.62687i	0.134795	+	0.0778239i
$438$		0			0
$439$	21.4344	−	12.3752i	1.02301	−	0.590634i	0.108034	−	0.994147i	$-0.465544\pi$
$439$	21.4344	−	12.3752i	1.02301	−	0.590634i	0.914974	+	0.403513i	$0.132211\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.24079	−	16.0055i	−0.439043	−	0.760445i	0.558573	−	0.829455i	$-0.311349\pi$
$443$	−9.24079	−	16.0055i	−0.439043	−	0.760445i	−0.997616	+	0.0690105i	$0.978016\pi$
$444$		0			0
$445$	−23.6258	−	13.6404i	−1.11997	−	0.646615i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			30.4888i			1.43886i	0.694567	+	0.719428i	$0.255596\pi$
$449$			30.4888i			1.43886i	−0.694567	+	0.719428i	$0.744404\pi$
$450$		0			0
$451$	−12.9915	+	22.5019i	−0.611744	+	1.05957i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.61192	+	2.63616i	−0.450614	+	0.123585i
$456$		0			0
$457$	−13.2873	−	23.0142i	−0.621553	−	1.07656i	−0.989197	−	0.146594i	$-0.953169\pi$
$457$	−13.2873	−	23.0142i	−0.621553	−	1.07656i	0.367644	−	0.929967i	$-0.380165\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	1.26742i		−	0.0590294i	−0.999564	−	0.0295147i	$-0.990604\pi$
$461$		−	1.26742i		−	0.0590294i	0.999564	−	0.0295147i	$-0.00939619\pi$
$462$		0			0
$463$	−9.82295			−0.456511			−0.228256	−	0.973601i	$-0.573302\pi$
$463$	−9.82295			−0.456511			−0.228256	+	0.973601i	$0.573302\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.5816	+	10.7281i	−0.859855	+	0.496438i	−0.863964	−	0.503554i	$-0.832025\pi$
$467$	−18.5816	+	10.7281i	−0.859855	+	0.496438i	0.00410868	+	0.999992i	$0.498692\pi$
$468$		0			0
$469$	−23.3724	−	23.0992i	−1.07924	−	1.06662i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.21585	+	3.01137i	0.239825	+	0.138463i
$474$		0			0
$475$	−0.176105			−0.00808027
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.5560	−	23.4798i	0.619391	−	1.07282i	−0.370206	−	0.928950i	$-0.620713\pi$
$479$	13.5560	−	23.4798i	0.619391	−	1.07282i	0.989597	−	0.143867i	$-0.0459539\pi$
$480$		0			0
$481$	16.7282	−	9.65803i	0.762740	−	0.440368i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.7293	+	30.7081i	0.805047	+	1.39438i
$486$		0			0
$487$	−4.06126	+	7.03430i	−0.184033	+	0.318755i	−0.943250	−	0.332083i	$-0.892249\pi$
$487$	−4.06126	+	7.03430i	−0.184033	+	0.318755i	0.759217	+	0.650837i	$0.225582\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.11572			0.275998			0.137999	−	0.990432i	$-0.455933\pi$
$491$	6.11572			0.275998			0.137999	+	0.990432i	$0.455933\pi$
$492$		0			0
$493$	−0.505158	+	0.874959i	−0.0227512	+	0.0394062i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.22229	+	19.9584i	−0.234252	+	0.895257i
$498$		0			0
$499$	16.4251	−	9.48302i	0.735287	−	0.424518i	−0.0850663	−	0.996375i	$-0.527110\pi$
$499$	16.4251	−	9.48302i	0.735287	−	0.424518i	0.820353	+	0.571857i	$0.193777\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.6249			−1.09797			−0.548985	−	0.835832i	$-0.684985\pi$
$503$	−24.6249			−1.09797			−0.548985	+	0.835832i	$0.684985\pi$
$504$		0			0
$505$	−33.5800			−1.49429
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.66857	−	4.42745i	0.339904	−	0.196243i	−0.320326	−	0.947307i	$-0.603792\pi$
$509$	7.66857	−	4.42745i	0.339904	−	0.196243i	0.660229	+	0.751064i	$0.270459\pi$
$510$		0			0
$511$	−16.4286	−	16.2366i	−0.726759	−	0.718265i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.85756	−	10.1456i	0.258115	−	0.447068i
$516$		0			0
$517$	4.54332			0.199815
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0185	−	20.8167i	0.526540	−	0.911995i	−0.472981	−	0.881072i	$-0.656822\pi$
$521$	12.0185	−	20.8167i	0.526540	−	0.911995i	0.999522	−	0.0309222i	$-0.00984441\pi$
$522$		0			0
$523$	−1.89548	−	3.28306i	−0.0828835	−	0.143558i	0.821604	−	0.570059i	$-0.193080\pi$
$523$	−1.89548	−	3.28306i	−0.0828835	−	0.143558i	−0.904487	+	0.426500i	$0.859746\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.72937	+	2.15316i	−0.162454	+	0.0937929i
$528$		0			0
$529$	6.63889	−	11.4989i	0.288648	−	0.499952i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	13.5425			0.586589
$534$		0			0
$535$	6.96430	+	4.02084i	0.301093	+	0.173836i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.1278	+	11.9384i	−0.866964	+	0.514225i
$540$		0			0
$541$	16.4954	−	9.52363i	0.709193	−	0.409453i	−0.101569	−	0.994828i	$-0.532386\pi$
$541$	16.4954	−	9.52363i	0.709193	−	0.409453i	0.810762	+	0.585376i	$0.199053\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.2915			−0.612179
$546$		0			0
$547$			46.3065i			1.97992i	0.141342	+	0.989961i	$0.454858\pi$
$547$			46.3065i			1.97992i	−0.141342	+	0.989961i	$0.545142\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.480550	+	0.832337i	0.0204721	+	0.0354587i
$552$		0			0
$553$	−6.68869	−	24.3882i	−0.284432	−	1.03709i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.1463	−	19.3059i	0.472283	−	0.818019i	−0.527214	−	0.849733i	$-0.676763\pi$
$557$	11.1463	−	19.3059i	0.472283	−	0.818019i	0.999497	+	0.0317140i	$0.0100966\pi$
$558$		0			0
$559$		−	3.13910i		−	0.132770i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.42370	+	1.39932i	0.102147	+	0.0589745i	0.550203	−	0.835031i	$-0.314550\pi$
$563$	2.42370	+	1.39932i	0.102147	+	0.0589745i	−0.448056	+	0.894005i	$0.647884\pi$
$564$		0			0
$565$	4.67155	+	8.09136i	0.196533	+	0.340406i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	35.6988	−	20.6107i	1.49657	−	0.864046i	0.496579	−	0.867992i	$-0.334589\pi$
$569$	35.6988	−	20.6107i	1.49657	−	0.864046i	0.999992	+	0.00394571i	$0.00125596\pi$
$570$		0			0
$571$	−24.3886	−	14.0808i	−1.02063	−	0.589262i	−0.106345	−	0.994329i	$-0.533915\pi$
$571$	−24.3886	−	14.0808i	−1.02063	−	0.589262i	−0.914287	+	0.405068i	$0.867248\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.96350i			0.0818834i
$576$		0			0
$577$	8.23601	+	4.75506i	0.342870	+	0.197956i	0.661540	−	0.749910i	$-0.269903\pi$
$577$	8.23601	+	4.75506i	0.342870	+	0.197956i	−0.318671	+	0.947866i	$0.603236\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.7130	+	40.9425i	−0.444449	+	1.69858i
$582$		0			0
$583$	4.90881	+	8.50230i	0.203302	+	0.352129i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	6.47220i		−	0.267136i	−0.991040	−	0.133568i	$-0.957357\pi$
$587$		−	6.47220i		−	0.267136i	0.991040	−	0.133568i	$-0.0426435\pi$
$588$		0			0
$589$			4.09654i			0.168795i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14.5174	−	25.1449i	−0.596159	−	1.03258i	−0.993382	−	0.114855i	$-0.963360\pi$
$593$	−14.5174	−	25.1449i	−0.596159	−	1.03258i	0.397224	−	0.917722i	$-0.369974\pi$
$594$		0			0
$595$	0.822243	−	3.14243i	0.0337087	−	0.128827i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.06979	−	2.34969i	−0.166287	−	0.0960058i	0.414547	−	0.910028i	$-0.363940\pi$
$599$	−4.06979	−	2.34969i	−0.166287	−	0.0960058i	−0.580834	+	0.814022i	$0.697273\pi$
$600$		0			0
$601$			20.3259i			0.829111i	0.910024	+	0.414555i	$0.136063\pi$
$601$			20.3259i			0.829111i	−0.910024	+	0.414555i	$0.863937\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.330641	−	0.190896i	−0.0134425	−	0.00776102i
$606$		0			0
$607$	−26.3222	+	15.1971i	−1.06838	+	0.616832i	−0.927740	−	0.373228i	$-0.878251\pi$
$607$	−26.3222	+	15.1971i	−1.06838	+	0.616832i	−0.140645	+	0.990060i	$0.544918\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.18400	−	2.05076i	−0.0478997	−	0.0829647i
$612$		0			0
$613$	−24.4975	−	14.1436i	−0.989444	−	0.571256i	−0.0843357	−	0.996437i	$-0.526877\pi$
$613$	−24.4975	−	14.1436i	−0.989444	−	0.571256i	−0.905108	+	0.425182i	$0.860210\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.3569i			0.859795i	0.902878	+	0.429898i	$0.141450\pi$
$617$			21.3569i			0.859795i	−0.902878	+	0.429898i	$0.858550\pi$
$618$		0			0
$619$	10.6447	−	18.4372i	0.427847	−	0.741053i	−0.568834	−	0.822452i	$-0.692605\pi$
$619$	10.6447	−	18.4372i	0.427847	−	0.741053i	0.996682	+	0.0813988i	$0.0259388\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.83026	+	32.1967i	0.353777	+	1.28993i
$624$		0			0
$625$	11.6319	+	20.1470i	0.465275	+	0.805880i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.29514i			0.251004i
$630$		0			0
$631$	−8.31457			−0.330998			−0.165499	−	0.986210i	$-0.552923\pi$
$631$	−8.31457			−0.330998			−0.165499	+	0.986210i	$0.552923\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.0495	+	7.53412i	−0.517853	+	0.298983i
$636$		0			0
$637$	10.6341	+	5.97403i	0.421339	+	0.236700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.98896	+	2.88038i	0.197052	+	0.113768i	0.595280	−	0.803519i	$-0.297041\pi$
$641$	4.98896	+	2.88038i	0.197052	+	0.113768i	−0.398228	+	0.917287i	$0.630375\pi$
$642$		0			0
$643$	18.7692			0.740184			0.370092	−	0.928995i	$-0.379326\pi$
$643$	18.7692			0.740184			0.370092	+	0.928995i	$0.379326\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.705648	−	1.22222i	0.0277419	−	0.0480503i	−0.851821	−	0.523833i	$-0.824502\pi$
$647$	0.705648	−	1.22222i	0.0277419	−	0.0480503i	0.879563	+	0.475782i	$0.157835\pi$
$648$		0			0
$649$	32.9220	−	19.0075i	1.29230	−	0.746112i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.21537	−	5.56919i	−0.125827	−	0.217939i	0.796229	−	0.604996i	$-0.206825\pi$
$653$	−3.21537	−	5.56919i	−0.125827	−	0.217939i	−0.922056	+	0.387057i	$0.873492\pi$
$654$		0			0
$655$	9.51855	−	16.4866i	0.371920	−	0.644185i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.9229			−0.620269			−0.310134	−	0.950693i	$-0.600374\pi$
$659$	−15.9229			−0.620269			−0.310134	+	0.950693i	$0.600374\pi$
$660$		0			0
$661$	13.8309	−	23.9557i	0.537958	−	0.931770i	−0.461056	−	0.887371i	$-0.652529\pi$
$661$	13.8309	−	23.9557i	0.537958	−	0.931770i	0.999014	−	0.0443993i	$-0.0141374\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.19776	−	2.17207i	−0.0852254	−	0.0842293i
$666$		0			0
$667$	9.28018	−	5.35791i	0.359330	−	0.207459i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	37.5005			1.44769
$672$		0			0
$673$	16.8132			0.648103			0.324052	−	0.946039i	$-0.394955\pi$
$673$	16.8132			0.648103			0.324052	+	0.946039i	$0.394955\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.3325	−	19.2445i	1.28107	−	0.739626i	0.304026	−	0.952664i	$-0.401669\pi$
$677$	33.3325	−	19.2445i	1.28107	−	0.739626i	0.977044	+	0.213037i	$0.0683356\pi$
$678$		0			0
$679$	10.9845	−	41.9804i	0.421548	−	1.61106i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.8564	+	29.1961i	−0.644991	+	1.11716i	0.339313	+	0.940674i	$0.389806\pi$
$683$	−16.8564	+	29.1961i	−0.644991	+	1.11716i	−0.984304	+	0.176483i	$0.943528\pi$
$684$		0			0
$685$	36.2616			1.38549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.55850	−	4.43146i	0.0974712	−	0.168825i
$690$		0			0
$691$	−16.3070	−	28.2445i	−0.620346	−	1.07447i	−0.989421	−	0.145072i	$-0.953659\pi$
$691$	−16.3070	−	28.2445i	−0.620346	−	1.07447i	0.369075	−	0.929400i	$-0.379675\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	19.3424	−	11.1673i	0.733699	−	0.423601i
$696$		0			0
$697$	−2.20676	+	3.82222i	−0.0835869	+	0.144777i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.4784			−0.697919			−0.348959	−	0.937138i	$-0.613465\pi$
$701$	−18.4784			−0.697919			−0.348959	+	0.937138i	$0.613465\pi$
$702$		0			0
$703$	5.18618	+	2.99424i	0.195600	+	0.112930i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	29.2286	+	28.8870i	1.09925	+	1.08641i
$708$		0			0
$709$	−0.277961	+	0.160481i	−0.0104390	+	0.00602697i	−0.505210	−	0.862996i	$-0.668585\pi$
$709$	−0.277961	+	0.160481i	−0.0104390	+	0.00602697i	0.494771	+	0.869023i	$0.335252\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	45.6745			1.71053
$714$		0			0
$715$			12.5940i			0.470989i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.95557	−	12.0474i	−0.259399	−	0.449292i	0.706682	−	0.707531i	$-0.250191\pi$
$719$	−6.95557	−	12.0474i	−0.259399	−	0.449292i	−0.966081	+	0.258239i	$0.916858\pi$
$720$		0			0
$721$	−13.8262	+	3.79196i	−0.514914	+	0.141220i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.289991	+	0.502280i	−0.0107700	+	0.0186542i
$726$		0			0
$727$			42.1216i			1.56220i	0.624403	+	0.781102i	$0.285342\pi$
$727$			42.1216i			1.56220i	−0.624403	+	0.781102i	$0.714658\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.885977	+	0.511519i	0.0327690	+	0.0189192i
$732$		0			0
$733$	4.20815	+	7.28873i	0.155432	+	0.269215i	0.933216	−	0.359316i	$-0.116990\pi$
$733$	4.20815	+	7.28873i	0.155432	+	0.269215i	−0.777785	+	0.628531i	$0.783657\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−35.9597	+	20.7614i	−1.32459	+	0.764755i
$738$		0			0
$739$	−2.15443	−	1.24386i	−0.0792519	−	0.0457561i	0.459850	−	0.887996i	$-0.347903\pi$
$739$	−2.15443	−	1.24386i	−0.0792519	−	0.0457561i	−0.539102	+	0.842240i	$0.681236\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	27.2694i		−	1.00042i	−0.865904	−	0.500209i	$-0.833256\pi$
$743$		−	27.2694i		−	1.00042i	0.865904	−	0.500209i	$-0.166744\pi$
$744$		0			0
$745$	−7.88841	−	4.55438i	−0.289009	−	0.166859i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.60294	−	9.49080i	−0.0951094	−	0.346786i
$750$		0			0
$751$	15.7789	+	27.3299i	0.575781	+	0.997282i	0.995956	+	0.0898389i	$0.0286352\pi$
$751$	15.7789	+	27.3299i	0.575781	+	0.997282i	−0.420175	+	0.907443i	$0.638031\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	13.4254i		−	0.488599i
$756$		0			0
$757$			36.9092i			1.34149i	0.741689	+	0.670744i	$0.234025\pi$
$757$			36.9092i			1.34149i	−0.741689	+	0.670744i	$0.765975\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.14893	−	15.8464i	−0.331648	−	0.574432i	0.651187	−	0.758917i	$-0.274271\pi$
$761$	−9.14893	−	15.8464i	−0.331648	−	0.574432i	−0.982835	+	0.184486i	$0.940938\pi$
$762$		0			0
$763$	12.4395	+	12.2941i	0.450341	+	0.445078i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−17.1592	−	9.90686i	−0.619582	−	0.357716i
$768$		0			0
$769$			5.09495i			0.183729i	0.995772	+	0.0918644i	$0.0292826\pi$
$769$			5.09495i			0.183729i	−0.995772	+	0.0918644i	$0.970717\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.0448	−	15.6143i	−0.972735	−	0.561609i	−0.0726659	−	0.997356i	$-0.523151\pi$
$773$	−27.0448	−	15.6143i	−0.972735	−	0.561609i	−0.900069	+	0.435748i	$0.856484\pi$
$774$		0			0
$775$	−2.14089	+	1.23604i	−0.0769030	+	0.0444000i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.09926	+	3.63603i	0.0752138	+	0.130274i
$780$		0			0
$781$	22.5758	+	13.0341i	0.807825	+	0.466398i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			44.6267i			1.59279i
$786$		0			0
$787$	−9.96558	+	17.2609i	−0.355235	+	0.615284i	−0.987158	−	0.159746i	$-0.948932\pi$
$787$	−9.96558	+	17.2609i	−0.355235	+	0.615284i	0.631923	+	0.775031i	$0.282266\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.89434	−	11.0615i	0.102911	−	0.393302i
$792$		0			0
$793$	−9.77276	−	16.9269i	−0.347041	−	0.601093i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	46.5551i		−	1.64907i	−0.565813	−	0.824534i	$-0.691438\pi$
$797$		−	46.5551i		−	1.64907i	0.565813	−	0.824534i	$-0.308562\pi$
$798$		0			0
$799$	0.771739			0.0273022
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−25.2764	+	14.5933i	−0.891984	+	0.514987i
$804$		0			0
$805$	−24.2176	+	24.5040i	−0.853558	+	0.863652i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−35.1771	−	20.3095i	−1.23676	−	0.714045i	−0.268331	−	0.963327i	$-0.586472\pi$
$809$	−35.1771	−	20.3095i	−1.23676	−	0.714045i	−0.968431	+	0.249282i	$0.919805\pi$
$810$		0			0
$811$	−43.3104			−1.52083			−0.760417	−	0.649435i	$-0.775005\pi$
$811$	−43.3104			−1.52083			−0.760417	+	0.649435i	$0.775005\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.45119	+	7.70968i	−0.155918	+	0.270058i
$816$		0			0
$817$	0.842818	−	0.486601i	0.0294865	−	0.0170240i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.21375	−	2.10228i	−0.0423602	−	0.0733699i	0.844068	−	0.536236i	$-0.180154\pi$
$821$	−1.21375	−	2.10228i	−0.0423602	−	0.0733699i	−0.886428	+	0.462866i	$0.846821\pi$
$822$		0			0
$823$	8.90145	−	15.4178i	0.310285	−	0.537429i	−0.668139	−	0.744036i	$-0.732909\pi$
$823$	8.90145	−	15.4178i	0.310285	−	0.537429i	0.978424	+	0.206607i	$0.0662422\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.3153			1.36713			0.683564	−	0.729890i	$-0.260429\pi$
$827$	39.3153			1.36713			0.683564	+	0.729890i	$0.260429\pi$
$828$		0			0
$829$	21.4409	−	37.1368i	0.744675	−	1.28981i	−0.205672	−	0.978621i	$-0.565938\pi$
$829$	21.4409	−	37.1368i	0.744675	−	1.28981i	0.950346	−	0.311194i	$-0.100729\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.41895	+	2.02789i	−0.118459	+	0.0702623i
$834$		0			0
$835$	24.8368	−	14.3395i	0.859512	−	0.496239i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.03008			0.139134			0.0695668	−	0.997577i	$-0.477838\pi$
$839$	4.03008			0.139134			0.0695668	+	0.997577i	$0.477838\pi$
$840$		0			0
$841$	−25.8347			−0.890853
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−18.6552	+	10.7706i	−0.641759	+	0.370520i
$846$		0			0
$847$	0.123579	+	0.450591i	0.00424622	+	0.0154825i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	33.3844	−	57.8235i	1.14440	−	1.98217i
$852$		0			0
$853$	1.89394			0.0648474			0.0324237	−	0.999474i	$-0.489677\pi$
$853$	1.89394			0.0648474			0.0324237	+	0.999474i	$0.489677\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.3144	−	17.8650i	0.352333	−	0.610258i	−0.634325	−	0.773066i	$-0.718722\pi$
$857$	10.3144	−	17.8650i	0.352333	−	0.610258i	0.986658	+	0.162808i	$0.0520552\pi$
$858$		0			0
$859$	−25.2260	−	43.6928i	−0.860701	−	1.49078i	−0.871253	−	0.490833i	$-0.836692\pi$
$859$	−25.2260	−	43.6928i	−0.860701	−	1.49078i	0.0105526	−	0.999944i	$-0.496641\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.95161	−	1.12676i	0.0664336	−	0.0383554i	−0.466415	−	0.884566i	$-0.654455\pi$
$863$	1.95161	−	1.12676i	0.0664336	−	0.0383554i	0.532849	+	0.846210i	$0.321121\pi$
$864$		0			0
$865$	−24.3256	+	42.1332i	−0.827095	+	1.43257i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−31.9546			−1.08399
$870$		0			0
$871$	18.7425	+	10.8210i	0.635064	+	0.366654i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.71169	−	29.4723i	0.260703	−	0.996346i
$876$		0			0
$877$	−22.7244	+	13.1200i	−0.767350	+	0.443029i	−0.831928	−	0.554883i	$-0.812763\pi$
$877$	−22.7244	+	13.1200i	−0.767350	+	0.443029i	0.0645788	+	0.997913i	$0.479430\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−13.9743			−0.470804			−0.235402	−	0.971898i	$-0.575641\pi$
$881$	−13.9743			−0.470804			−0.235402	+	0.971898i	$0.575641\pi$
$882$		0			0
$883$		−	40.0650i		−	1.34830i	−0.738596	−	0.674148i	$-0.764511\pi$
$883$		−	40.0650i		−	1.34830i	0.738596	−	0.674148i	$-0.235489\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	22.9798	+	39.8022i	0.771586	+	1.33643i	0.936694	+	0.350150i	$0.113869\pi$
$887$	22.9798	+	39.8022i	0.771586	+	1.33643i	−0.165108	+	0.986275i	$0.552797\pi$
$888$		0			0
$889$	17.8397	+	4.66791i	0.598323	+	0.156557i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.367073	−	0.635788i	0.0122836	−	0.0212758i
$894$		0			0
$895$			41.6725i			1.39296i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.6840	+	6.74573i	0.389682	+	0.224983i
$900$		0			0
$901$	0.833821	+	1.44422i	0.0277786	+	0.0481140i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41.8459	−	24.1597i	1.39100	−	0.803096i
$906$		0			0
$907$	−8.32291	−	4.80524i	−0.276358	−	0.159555i	0.355416	−	0.934708i	$-0.384339\pi$
$907$	−8.32291	−	4.80524i	−0.276358	−	0.159555i	−0.631773	+	0.775153i	$0.717673\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			55.0751i			1.82472i	0.409390	+	0.912359i	$0.365742\pi$
$911$			55.0751i			1.82472i	−0.409390	+	0.912359i	$0.634258\pi$
$912$		0			0
$913$	46.3117	+	26.7381i	1.53269	+	0.884902i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.4676	+	6.16195i	−0.741946	+	0.203486i
$918$		0			0
$919$	10.4392	+	18.0812i	0.344357	+	0.596444i	0.985237	−	0.171198i	$-0.0547636\pi$
$919$	10.4392	+	18.0812i	0.344357	+	0.596444i	−0.640880	+	0.767641i	$0.721430\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	13.5870i		−	0.447220i
$924$		0			0
$925$			3.61379i			0.118821i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.3495	−	33.5143i	−0.634837	−	1.09957i	−0.986550	−	0.163462i	$-0.947734\pi$
$929$	−19.3495	−	33.5143i	−0.634837	−	1.09957i	0.351713	−	0.936108i	$-0.385599\pi$
$930$		0			0
$931$	0.0444554	+	3.78121i	0.00145697	+	0.123924i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.55453	−	2.05221i	−0.116245	−	0.0671143i
$936$		0			0
$937$			2.23597i			0.0730458i	0.999333	+	0.0365229i	$0.0116282\pi$
$937$			2.23597i			0.0730458i	−0.999333	+	0.0365229i	$0.988372\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.23966	+	4.75717i	0.268605	+	0.155079i	0.628254	−	0.778009i	$-0.283770\pi$
$941$	8.23966	+	4.75717i	0.268605	+	0.155079i	−0.359648	+	0.933088i	$0.617103\pi$
$942$		0			0
$943$	40.5400	−	23.4058i	1.32017	−	0.762198i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.3346	−	33.4885i	−0.628290	−	1.08823i	−0.987895	−	0.155126i	$-0.950422\pi$
$947$	−19.3346	−	33.4885i	−0.628290	−	1.08823i	0.359605	−	0.933105i	$-0.382912\pi$
$948$		0			0
$949$	13.1742	+	7.60613i	0.427653	+	0.246906i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31.5906i		−	1.02332i	−0.859188	−	0.511659i	$-0.829031\pi$
$953$		−	31.5906i		−	1.02332i	0.859188	−	0.511659i	$-0.170969\pi$
$954$		0			0
$955$	7.10444	−	12.3053i	0.229894	−	0.398189i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−31.5627	−	31.1938i	−1.01921	−	1.00730i
$960$		0			0
$961$	13.2526	+	22.9542i	0.427504	+	0.740459i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	16.0128i		−	0.515472i
$966$		0			0
$967$	−54.1994			−1.74294			−0.871468	−	0.490452i	$-0.836832\pi$
$967$	−54.1994			−1.74294			−0.871468	+	0.490452i	$0.836832\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.92025	−	3.41806i	0.189990	−	0.109691i	−0.401988	−	0.915645i	$-0.631681\pi$
$971$	5.92025	−	3.41806i	0.189990	−	0.109691i	0.591978	+	0.805954i	$0.298347\pi$
$972$		0			0
$973$	−26.4426	−	6.91893i	−0.847710	−	0.221811i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.1650	+	11.0649i	0.613144	+	0.353999i	0.774195	−	0.632947i	$-0.218155\pi$
$977$	19.1650	+	11.0649i	0.613144	+	0.353999i	−0.161051	+	0.986946i	$0.551488\pi$
$978$		0			0
$979$	42.1858			1.34826
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.18986	−	5.52499i	0.101741	−	0.176220i	−0.810661	−	0.585515i	$-0.800892\pi$
$983$	3.18986	−	5.52499i	0.101741	−	0.176220i	0.912402	+	0.409296i	$0.134225\pi$
$984$		0			0
$985$	3.37617	−	1.94923i	0.107574	−	0.0621077i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.42538	−	9.39704i	−0.172517	−	0.298808i
$990$		0			0
$991$	−5.49218	+	9.51273i	−0.174465	+	0.302182i	−0.939976	−	0.341241i	$-0.889153\pi$
$991$	−5.49218	+	9.51273i	−0.174465	+	0.302182i	0.765511	+	0.643423i	$0.222486\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	19.2904			0.611546
$996$		0			0
$997$	3.65692	−	6.33397i	0.115816	−	0.200599i	−0.802290	−	0.596935i	$-0.796385\pi$
$997$	3.65692	−	6.33397i	0.115816	−	0.200599i	0.918106	+	0.396336i	$0.129718\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.cp.b.17.23		56
3.2	odd	2	inner	2016.2.cp.b.17.5		56
4.3	odd	2		504.2.ch.b.269.2	✓	56
7.5	odd	6	inner	2016.2.cp.b.593.24		56
8.3	odd	2		504.2.ch.b.269.10	yes	56
8.5	even	2	inner	2016.2.cp.b.17.6		56
12.11	even	2		504.2.ch.b.269.27	yes	56
21.5	even	6	inner	2016.2.cp.b.593.6		56
24.5	odd	2	inner	2016.2.cp.b.17.24		56
24.11	even	2		504.2.ch.b.269.19	yes	56
28.19	even	6		504.2.ch.b.341.19	yes	56
56.5	odd	6	inner	2016.2.cp.b.593.5		56
56.19	even	6		504.2.ch.b.341.27	yes	56
84.47	odd	6		504.2.ch.b.341.10	yes	56
168.5	even	6	inner	2016.2.cp.b.593.23		56
168.131	odd	6		504.2.ch.b.341.2	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.ch.b.269.2	✓	56	4.3	odd	2
504.2.ch.b.269.10	yes	56	8.3	odd	2
504.2.ch.b.269.19	yes	56	24.11	even	2
504.2.ch.b.269.27	yes	56	12.11	even	2
504.2.ch.b.341.2	yes	56	168.131	odd	6
504.2.ch.b.341.10	yes	56	84.47	odd	6
504.2.ch.b.341.19	yes	56	28.19	even	6
504.2.ch.b.341.27	yes	56	56.19	even	6
2016.2.cp.b.17.5		56	3.2	odd	2	inner
2016.2.cp.b.17.6		56	8.5	even	2	inner
2016.2.cp.b.17.23		56	1.1	even	1	trivial
2016.2.cp.b.17.24		56	24.5	odd	2	inner
2016.2.cp.b.593.5		56	56.5	odd	6	inner
2016.2.cp.b.593.6		56	21.5	even	6	inner
2016.2.cp.b.593.23		56	168.5	even	6	inner
2016.2.cp.b.593.24		56	7.5	odd	6	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.cp (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.87230 - 1.08097i) q^{5} +(-2.55958 - 0.669737i) q^{7} +O(q^{10})\)	\(q+(1.87230 - 1.08097i) q^{5} +(-2.55958 - 0.669737i) q^{7} +(-1.67157 + 2.89525i) q^{11} +1.74247 q^{13} +(-0.283937 + 0.491793i) q^{17} +(0.270105 + 0.467836i) q^{19} +(5.21616 - 3.01155i) q^{23} +(-0.162997 + 0.282319i) q^{25} +1.77912 q^{29} +(6.56726 + 3.79161i) q^{31} +(-5.51627 + 1.51289i) q^{35} +(9.60029 - 5.54273i) q^{37} +7.77201 q^{41} -1.80152i q^{43} +(-0.679499 - 1.17693i) q^{47} +(6.10291 + 3.42849i) q^{49} +(1.46832 - 2.54321i) q^{53} +7.22769i q^{55} +(-9.84763 - 5.68553i) q^{59} +(-5.60858 - 9.71434i) q^{61} +(3.26242 - 1.88356i) q^{65} +(10.7563 + 6.21014i) q^{67} -7.79753i q^{71} +(7.56066 + 4.36515i) q^{73} +(6.21757 - 6.29110i) q^{77} +(4.77913 + 8.27770i) q^{79} -15.9958i q^{83} +1.22771i q^{85} +(-6.30930 - 10.9280i) q^{89} +(-4.45999 - 1.16699i) q^{91} +(1.01144 + 0.583953i) q^{95} +16.4013i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 20 q^{7}+O(q^{10}) \) 56 * q + 20 * q^7	\( 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) \) 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

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