LMFDB - Embedded newform 416.2.bk.a.111.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(15,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 6, 1]))

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

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

chi := DirichletCharacter("416.15");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.bk (of order $12$, degree $4$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			111.1
Character	$\chi$	$=$	416.111
Dual form			416.2.bk.a.15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.49630 + 2.59167i) q^{3} +(-1.40823 + 1.40823i) q^{5} +(-1.02175 + 3.81323i) q^{7} +(-2.97784 - 5.15776i) q^{9} +O(q^{10})$	\(q+(-1.49630 + 2.59167i) q^{3} +(-1.40823 + 1.40823i) q^{5} +(-1.02175 + 3.81323i) q^{7} +(-2.97784 - 5.15776i) q^{9} +(-0.756644 - 2.82384i) q^{11} +(2.47643 + 2.62055i) q^{13} +(-1.54254 - 5.75682i) q^{15} +(0.373696 - 0.215754i) q^{17} +(0.960219 - 3.58359i) q^{19} +(-8.35378 - 8.35378i) q^{21} +(0.264259 - 0.457711i) q^{23} +1.03375i q^{25} +8.84516 q^{27} +(-1.96425 - 1.13406i) q^{29} +(-0.525538 + 0.525538i) q^{31} +(8.45062 + 2.26434i) q^{33} +(-3.93105 - 6.80878i) q^{35} +(-4.50702 + 1.20765i) q^{37} +(-10.4971 + 2.49694i) q^{39} +(-1.02295 + 0.274098i) q^{41} +(-5.78795 + 3.34168i) q^{43} +(11.4568 + 3.06985i) q^{45} +(-3.10880 - 3.10880i) q^{47} +(-7.43454 - 4.29233i) q^{49} +1.29133i q^{51} +5.89363i q^{53} +(5.04215 + 2.91109i) q^{55} +(7.85070 + 7.85070i) q^{57} +(5.50874 + 1.47606i) q^{59} +(4.67379 - 2.69841i) q^{61} +(22.7103 - 6.08522i) q^{63} +(-7.17774 - 0.202965i) q^{65} +(-1.76688 + 0.473433i) q^{67} +(0.790823 + 1.36975i) q^{69} +(-4.80671 - 1.28795i) q^{71} +(-8.61849 + 8.61849i) q^{73} +(-2.67915 - 1.54681i) q^{75} +11.5410 q^{77} +17.5395i q^{79} +(-4.30151 + 7.45044i) q^{81} +(9.64657 + 9.64657i) q^{83} +(-0.222420 + 0.830084i) q^{85} +(5.87822 - 3.39379i) q^{87} +(-2.41190 - 9.00133i) q^{89} +(-12.5231 + 6.76562i) q^{91} +(-0.575658 - 2.14838i) q^{93} +(3.69432 + 6.39874i) q^{95} +(0.871452 - 3.25230i) q^{97} +(-12.3115 + 12.3115i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{3} - 20 q^{9}+O(q^{10}) $ 48 * q + 4 * q^3 - 20 * q^9	$ 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83} + 12 q^{89} - 104 q^{91} + 4 q^{97} - 168 q^{99}+O(q^{100}) $ 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{11}{12}\right)$

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
$3$	−1.49630	+	2.59167i	−0.863890	+	1.49630i	0.00425488	+	0.999991i	$0.498646\pi$
$3$	−1.49630	+	2.59167i	−0.863890	+	1.49630i	−0.868145	+	0.496311i	$0.834688\pi$
$4$		0			0
$5$	−1.40823	+	1.40823i	−0.629781	+	0.629781i	−0.948013	−	0.318232i	$-0.896911\pi$
$5$	−1.40823	+	1.40823i	−0.629781	+	0.629781i	0.318232	+	0.948013i	$0.396911\pi$
$6$		0			0
$7$	−1.02175	+	3.81323i	−0.386186	+	1.44126i	0.450104	+	0.892976i	$0.351387\pi$
$7$	−1.02175	+	3.81323i	−0.386186	+	1.44126i	−0.836290	+	0.548288i	$0.815280\pi$
$8$		0			0
$9$	−2.97784	−	5.15776i	−0.992612	−	1.71925i
$10$		0			0
$11$	−0.756644	−	2.82384i	−0.228137	−	0.851418i	−0.981123	−	0.193382i	$-0.938054\pi$
$11$	−0.756644	−	2.82384i	−0.228137	−	0.851418i	0.752987	−	0.658036i	$-0.228612\pi$
$12$		0			0
$13$	2.47643	+	2.62055i	0.686837	+	0.726811i
$13$	2.47643	+	2.62055i	0.686837	+	0.726811i
$14$		0			0
$15$	−1.54254	−	5.75682i	−0.398281	−	1.48640i
$16$		0			0
$17$	0.373696	−	0.215754i	0.0906347	−	0.0523280i	−0.453998	−	0.891003i	$-0.650003\pi$
$17$	0.373696	−	0.215754i	0.0906347	−	0.0523280i	0.544632	+	0.838675i	$0.316669\pi$
$18$		0			0
$19$	0.960219	−	3.58359i	0.220289	−	0.822131i	−0.763948	−	0.645278i	$-0.776742\pi$
$19$	0.960219	−	3.58359i	0.220289	−	0.822131i	0.984237	−	0.176853i	$-0.0565918\pi$
$20$		0			0
$21$	−8.35378	−	8.35378i	−1.82294	−	1.82294i
$22$		0			0
$23$	0.264259	−	0.457711i	0.0551019	−	0.0954393i	−0.837159	−	0.546960i	$-0.815785\pi$
$23$	0.264259	−	0.457711i	0.0551019	−	0.0954393i	0.892261	+	0.451521i	$0.149118\pi$
$24$		0			0
$25$			1.03375i			0.206751i
$26$		0			0
$27$	8.84516			1.70225
$28$		0			0
$29$	−1.96425	−	1.13406i	−0.364752	−	0.210590i	0.306411	−	0.951899i	$-0.400872\pi$
$29$	−1.96425	−	1.13406i	−0.364752	−	0.210590i	−0.671163	+	0.741310i	$0.734205\pi$
$30$		0			0
$31$	−0.525538	+	0.525538i	−0.0943894	+	0.0943894i	−0.752725	−	0.658335i	$-0.771261\pi$
$31$	−0.525538	+	0.525538i	−0.0943894	+	0.0943894i	0.658335	+	0.752725i	$0.271261\pi$
$32$		0			0
$33$	8.45062	+	2.26434i	1.47106	+	0.394170i
$34$		0			0
$35$	−3.93105	−	6.80878i	−0.664469	−	1.15089i
$36$		0			0
$37$	−4.50702	+	1.20765i	−0.740950	+	0.198537i	−0.609500	−	0.792786i	$-0.708630\pi$
$37$	−4.50702	+	1.20765i	−0.740950	+	0.198537i	−0.131450	+	0.991323i	$0.541963\pi$
$38$		0			0
$39$	−10.4971	+	2.49694i	−1.68088	+	0.399831i
$40$		0			0
$41$	−1.02295	+	0.274098i	−0.159757	+	0.0428069i	−0.337811	−	0.941214i	$-0.609687\pi$
$41$	−1.02295	+	0.274098i	−0.159757	+	0.0428069i	0.178054	+	0.984021i	$0.443020\pi$
$42$		0			0
$43$	−5.78795	+	3.34168i	−0.882654	+	0.509601i	−0.871533	−	0.490337i	$-0.836874\pi$
$43$	−5.78795	+	3.34168i	−0.882654	+	0.509601i	−0.0111217	+	0.999938i	$0.503540\pi$
$44$		0			0
$45$	11.4568	+	3.06985i	1.70788	+	0.457626i
$46$		0			0
$47$	−3.10880	−	3.10880i	−0.453465	−	0.453465i	0.443038	−	0.896503i	$-0.353901\pi$
$47$	−3.10880	−	3.10880i	−0.453465	−	0.453465i	−0.896503	+	0.443038i	$0.853901\pi$
$48$		0			0
$49$	−7.43454	−	4.29233i	−1.06208	−	0.613191i
$50$		0			0
$51$			1.29133i			0.180822i
$52$		0			0
$53$			5.89363i			0.809552i	0.914416	+	0.404776i	$0.132650\pi$
$53$			5.89363i			0.809552i	−0.914416	+	0.404776i	$0.867350\pi$
$54$		0			0
$55$	5.04215	+	2.91109i	0.679884	+	0.392531i
$56$		0			0
$57$	7.85070	+	7.85070i	1.03985	+	1.03985i
$58$		0			0
$59$	5.50874	+	1.47606i	0.717176	+	0.192167i	0.598911	−	0.800815i	$-0.295600\pi$
$59$	5.50874	+	1.47606i	0.717176	+	0.192167i	0.118265	+	0.992982i	$0.462267\pi$
$60$		0			0
$61$	4.67379	−	2.69841i	0.598417	−	0.345496i	−0.170002	−	0.985444i	$-0.554377\pi$
$61$	4.67379	−	2.69841i	0.598417	−	0.345496i	0.768419	+	0.639948i	$0.221044\pi$
$62$		0			0
$63$	22.7103	−	6.08522i	2.86123	−	0.766665i
$64$		0			0
$65$	−7.17774	−	0.202965i	−0.890290	−	0.0251747i
$66$		0			0
$67$	−1.76688	+	0.473433i	−0.215858	+	0.0578391i	−0.365127	−	0.930958i	$-0.618975\pi$
$67$	−1.76688	+	0.473433i	−0.215858	+	0.0578391i	0.149269	+	0.988797i	$0.452308\pi$
$68$		0			0
$69$	0.790823	+	1.36975i	0.0952040	+	0.164898i
$70$		0			0
$71$	−4.80671	−	1.28795i	−0.570452	−	0.152852i	−0.0379481	−	0.999280i	$-0.512082\pi$
$71$	−4.80671	−	1.28795i	−0.570452	−	0.152852i	−0.532504	+	0.846428i	$0.678749\pi$
$72$		0			0
$73$	−8.61849	+	8.61849i	−1.00872	+	1.00872i	−0.00875645	+	0.999962i	$0.502787\pi$
$73$	−8.61849	+	8.61849i	−1.00872	+	1.00872i	−0.999962	+	0.00875645i	$0.997213\pi$
$74$		0			0
$75$	−2.67915	−	1.54681i	−0.309361	−	0.178610i
$76$		0			0
$77$	11.5410			1.31522
$78$		0			0
$79$			17.5395i			1.97334i	0.162727	+	0.986671i	$0.447971\pi$
$79$			17.5395i			1.97334i	−0.162727	+	0.986671i	$0.552029\pi$
$80$		0			0
$81$	−4.30151	+	7.45044i	−0.477946	+	0.827827i
$82$		0			0
$83$	9.64657	+	9.64657i	1.05885	+	1.05885i	0.998157	+	0.0606926i	$0.0193309\pi$
$83$	9.64657	+	9.64657i	1.05885	+	1.05885i	0.0606926	+	0.998157i	$0.480669\pi$
$84$		0			0
$85$	−0.222420	+	0.830084i	−0.0241249	+	0.0900352i
$86$		0			0
$87$	5.87822	−	3.39379i	0.630211	−	0.363853i
$88$		0			0
$89$	−2.41190	−	9.00133i	−0.255661	−	0.954139i	−0.967722	−	0.252022i	$-0.918905\pi$
$89$	−2.41190	−	9.00133i	−0.255661	−	0.954139i	0.712061	−	0.702118i	$-0.247762\pi$
$90$		0			0
$91$	−12.5231	+	6.76562i	−1.31277	+	0.709230i
$92$		0			0
$93$	−0.575658	−	2.14838i	−0.0596929	−	0.222777i
$94$		0			0
$95$	3.69432	+	6.39874i	0.379029	+	0.656497i
$96$		0			0
$97$	0.871452	−	3.25230i	0.0884825	−	0.330221i	−0.907468	−	0.420120i	$-0.861988\pi$
$97$	0.871452	−	3.25230i	0.0884825	−	0.330221i	0.995951	+	0.0898992i	$0.0286545\pi$
$98$		0			0
$99$	−12.3115	+	12.3115i	−1.23735	+	1.23735i
$100$		0			0
$101$	−0.895975	+	1.55187i	−0.0891529	+	0.154417i	−0.907153	−	0.420800i	$-0.861749\pi$
$101$	−0.895975	+	1.55187i	−0.0891529	+	0.154417i	0.818000	+	0.575218i	$0.195083\pi$
$102$		0			0
$103$	−6.26322			−0.617133			−0.308567	−	0.951203i	$-0.599849\pi$
$103$	−6.26322			−0.617133			−0.308567	+	0.951203i	$0.599849\pi$
$104$		0			0
$105$	23.5282			2.29611
$106$		0			0
$107$	3.81015	−	6.59937i	0.368341	−	0.637985i	−0.620965	−	0.783838i	$-0.713259\pi$
$107$	3.81015	−	6.59937i	0.368341	−	0.637985i	0.989306	+	0.145853i	$0.0465926\pi$
$108$		0			0
$109$	−12.4705	+	12.4705i	−1.19445	+	1.19445i	−0.218650	+	0.975803i	$0.570165\pi$
$109$	−12.4705	+	12.4705i	−1.19445	+	1.19445i	−0.975803	+	0.218650i	$0.929835\pi$
$110$		0			0
$111$	3.61403	−	13.4877i	0.343028	−	1.28020i
$112$		0			0
$113$	5.82301	+	10.0857i	0.547783	+	0.948787i	0.998426	+	0.0560833i	$0.0178612\pi$
$113$	5.82301	+	10.0857i	0.547783	+	0.948787i	−0.450643	+	0.892704i	$0.648805\pi$
$114$		0			0
$115$	0.272425	+	1.01670i	0.0254037	+	0.0948080i
$116$		0			0
$117$	6.14181	−	20.5764i	0.567810	−	1.90229i
$118$		0			0
$119$	0.440893	+	1.64544i	0.0404166	+	0.150837i
$120$		0			0
$121$	2.12474	−	1.22672i	0.193159	−	0.111520i
$122$		0			0
$123$	0.820266	−	3.06127i	0.0739609	−	0.276026i
$124$		0			0
$125$	−8.49694	−	8.49694i	−0.759989	−	0.759989i
$126$		0			0
$127$	9.77681	−	16.9339i	0.867552	−	1.50264i	0.00306091	−	0.999995i	$-0.499026\pi$
$127$	9.77681	−	16.9339i	0.867552	−	1.50264i	0.864491	−	0.502648i	$-0.167641\pi$
$128$		0			0
$129$		−	20.0006i		−	1.76096i
$130$		0			0
$131$	16.1468			1.41075			0.705375	−	0.708834i	$-0.250778\pi$
$131$	16.1468			1.41075			0.705375	+	0.708834i	$0.250778\pi$
$132$		0			0
$133$	12.6839	+	7.32306i	1.09984	+	0.634990i
$134$		0			0
$135$	−12.4561	+	12.4561i	−1.07205	+	1.07205i
$136$		0			0
$137$	−12.8147	−	3.43369i	−1.09483	−	0.293360i	−0.334175	−	0.942511i	$-0.608458\pi$
$137$	−12.8147	−	3.43369i	−1.09483	−	0.293360i	−0.760660	+	0.649151i	$0.775124\pi$
$138$		0			0
$139$	6.07441	+	10.5212i	0.515225	+	0.892396i	0.999844	+	0.0176705i	$0.00562500\pi$
$139$	6.07441	+	10.5212i	0.515225	+	0.892396i	−0.484619	+	0.874725i	$0.661042\pi$
$140$		0			0
$141$	12.7087	−	3.40528i	1.07026	−	0.286776i
$142$		0			0
$143$	5.52624	−	8.97585i	0.462127	−	0.750598i
$144$		0			0
$145$	4.36315	−	1.16910i	0.362340	−	0.0970886i
$146$		0			0
$147$	22.2486	−	12.8453i	1.83504	−	1.05946i
$148$		0			0
$149$	−11.0487	−	2.96049i	−0.905146	−	0.242533i	−0.223921	−	0.974607i	$-0.571886\pi$
$149$	−11.0487	−	2.96049i	−0.905146	−	0.242533i	−0.681225	+	0.732074i	$0.738552\pi$
$150$		0			0
$151$	4.47584	+	4.47584i	0.364239	+	0.364239i	0.865371	−	0.501132i	$-0.167083\pi$
$151$	4.47584	+	4.47584i	0.364239	+	0.364239i	−0.501132	+	0.865371i	$0.667083\pi$
$152$		0			0
$153$	−2.22561	−	1.28496i	−0.179930	−	0.103883i
$154$		0			0
$155$		−	1.48016i		−	0.118889i
$156$		0			0
$157$			2.29073i			0.182820i	0.995813	+	0.0914101i	$0.0291374\pi$
$157$			2.29073i			0.182820i	−0.995813	+	0.0914101i	$0.970863\pi$
$158$		0			0
$159$	−15.2743	−	8.81864i	−1.21133	−	0.699364i
$160$		0			0
$161$	1.47535	+	1.47535i	0.116274	+	0.116274i
$162$		0			0
$163$	−18.8514	−	5.05123i	−1.47656	−	0.395643i	−0.571385	−	0.820682i	$-0.693594\pi$
$163$	−18.8514	−	5.05123i	−1.47656	−	0.395643i	−0.905175	+	0.425040i	$0.860260\pi$
$164$		0			0
$165$	−15.0892	+	8.71173i	−1.17469	+	0.678208i
$166$		0			0
$167$	−19.6986	+	5.27823i	−1.52433	+	0.408442i	−0.921163	−	0.389177i	$-0.872760\pi$
$167$	−19.6986	+	5.27823i	−1.52433	+	0.408442i	−0.603162	+	0.797618i	$0.706093\pi$
$168$		0			0
$169$	−0.734614	+	12.9792i	−0.0565087	+	0.998402i
$170$		0			0
$171$	−21.3427	+	5.71875i	−1.63211	+	0.437324i
$172$		0			0
$173$	−11.5483	−	20.0022i	−0.877999	−	1.52074i	−0.853534	−	0.521038i	$-0.825545\pi$
$173$	−11.5483	−	20.0022i	−0.877999	−	1.52074i	−0.0244651	−	0.999701i	$-0.507788\pi$
$174$		0			0
$175$	−3.94194	−	1.05624i	−0.297982	−	0.0798441i
$176$		0			0
$177$	−12.0682	+	12.0682i	−0.907101	+	0.907101i
$178$		0			0
$179$	−8.55591	−	4.93976i	−0.639499	−	0.369215i	0.144923	−	0.989443i	$-0.453707\pi$
$179$	−8.55591	−	4.93976i	−0.639499	−	0.369215i	−0.784421	+	0.620228i	$0.787040\pi$
$180$		0			0
$181$	14.6104			1.08598			0.542991	−	0.839738i	$-0.317292\pi$
$181$	14.6104			1.08598			0.542991	+	0.839738i	$0.317292\pi$
$182$		0			0
$183$			16.1505i			1.19388i
$184$		0			0
$185$	4.64629	−	8.04760i	0.341602	−	0.591672i
$186$		0			0
$187$	−0.892008	−	0.892008i	−0.0652301	−	0.0652301i
$188$		0			0
$189$	−9.03755	+	33.7286i	−0.657385	+	2.45339i
$190$		0			0
$191$	9.51822	−	5.49535i	0.688714	−	0.397629i	−0.114416	−	0.993433i	$-0.536500\pi$
$191$	9.51822	−	5.49535i	0.688714	−	0.397629i	0.803130	+	0.595804i	$0.203166\pi$
$192$		0			0
$193$	−1.07308	−	4.00480i	−0.0772422	−	0.288272i	0.916490	−	0.400057i	$-0.131010\pi$
$193$	−1.07308	−	4.00480i	−0.0772422	−	0.288272i	−0.993732	+	0.111785i	$0.964343\pi$
$194$		0			0
$195$	11.2661	−	18.2986i	0.806781	−	1.31039i
$196$		0			0
$197$	3.39832	+	12.6827i	0.242120	+	0.903606i	0.974809	+	0.223042i	$0.0715986\pi$
$197$	3.39832	+	12.6827i	0.242120	+	0.903606i	−0.732688	+	0.680564i	$0.761735\pi$
$198$		0			0
$199$	6.83125	+	11.8321i	0.484255	+	0.838754i	0.999836	−	0.0180866i	$-0.00575744\pi$
$199$	6.83125	+	11.8321i	0.484255	+	0.838754i	−0.515582	+	0.856840i	$0.672424\pi$
$200$		0			0
$201$	1.41680	−	5.28756i	0.0999332	−	0.372956i
$202$		0			0
$203$	6.33140	−	6.33140i	0.444377	−	0.444377i
$204$		0			0
$205$	1.05455	−	1.82654i	0.0736533	−	0.127571i
$206$		0			0
$207$	−3.14769			−0.218779
$208$		0			0
$209$	−10.8460			−0.750234
$210$		0			0
$211$	−4.28662	+	7.42464i	−0.295103	+	0.511133i	−0.975009	−	0.222166i	$-0.928687\pi$
$211$	−4.28662	+	7.42464i	−0.295103	+	0.511133i	0.679906	+	0.733299i	$0.262021\pi$
$212$		0			0
$213$	10.5302	−	10.5302i	0.721521	−	0.721521i
$214$		0			0
$215$	3.44493	−	12.8567i	0.234942	−	0.876816i
$216$		0			0
$217$	−1.46703	−	2.54096i	−0.0995882	−	0.172492i
$218$		0			0
$219$	−9.44043	−	35.2322i	−0.637925	−	2.38077i
$220$		0			0
$221$	1.49083	+	0.444993i	0.100284	+	0.0299335i
$222$		0			0
$223$	−0.481305	−	1.79626i	−0.0322306	−	0.120286i	0.947936	−	0.318460i	$-0.103166\pi$
$223$	−0.481305	−	1.79626i	−0.0322306	−	0.120286i	−0.980167	+	0.198174i	$0.936499\pi$
$224$		0			0
$225$	5.33186	−	3.07835i	0.355457	−	0.205223i
$226$		0			0
$227$	−5.15967	+	19.2561i	−0.342459	+	1.27808i	0.553093	+	0.833119i	$0.313447\pi$
$227$	−5.15967	+	19.2561i	−0.342459	+	1.27808i	−0.895552	+	0.444956i	$0.853219\pi$
$228$		0			0
$229$	5.88318	+	5.88318i	0.388772	+	0.388772i	0.874249	−	0.485478i	$-0.161354\pi$
$229$	5.88318	+	5.88318i	0.388772	+	0.388772i	−0.485478	+	0.874249i	$0.661354\pi$
$230$		0			0
$231$	−17.2689	+	29.9105i	−1.13621	+	1.96797i
$232$		0			0
$233$			14.9520i			0.979537i	0.871853	+	0.489768i	$0.162919\pi$
$233$			14.9520i			0.979537i	−0.871853	+	0.489768i	$0.837081\pi$
$234$		0			0
$235$	8.75583			0.571167
$236$		0			0
$237$	−45.4565	−	26.2443i	−2.95272	−	1.70475i
$238$		0			0
$239$	2.11437	−	2.11437i	0.136767	−	0.136767i	−0.635409	−	0.772176i	$-0.719168\pi$
$239$	2.11437	−	2.11437i	0.136767	−	0.136767i	0.772176	+	0.635409i	$0.219168\pi$
$240$		0			0
$241$	13.8151	+	3.70175i	0.889911	+	0.238451i	0.674678	−	0.738112i	$-0.264282\pi$
$241$	13.8151	+	3.70175i	0.889911	+	0.238451i	0.215233	+	0.976563i	$0.430949\pi$
$242$		0			0
$243$	0.395012	+	0.684181i	0.0253400	+	0.0438902i
$244$		0			0
$245$	16.5142	−	4.42496i	1.05505	−	0.282701i
$246$		0			0
$247$	11.7689	−	6.35819i	0.748837	−	0.404562i
$248$		0			0
$249$	−39.4349	+	10.5666i	−2.49909	+	0.669628i
$250$		0			0
$251$	−4.77498	+	2.75684i	−0.301394	+	0.174010i	−0.643069	−	0.765808i	$-0.722339\pi$
$251$	−4.77498	+	2.75684i	−0.301394	+	0.174010i	0.341675	+	0.939818i	$0.389006\pi$
$252$		0			0
$253$	−1.49245	−	0.399901i	−0.0938295	−	0.0251415i
$254$		0			0
$255$	−1.81850	−	1.81850i	−0.113879	−	0.113879i
$256$		0			0
$257$	24.7355	+	14.2811i	1.54296	+	0.890828i	0.998650	+	0.0519442i	$0.0165418\pi$
$257$	24.7355	+	14.2811i	1.54296	+	0.890828i	0.544310	+	0.838884i	$0.316792\pi$
$258$		0			0
$259$		−	18.4202i		−	1.14458i
$260$		0			0
$261$			13.5082i			0.836136i
$262$		0			0
$263$	−22.3279	−	12.8910i	−1.37680	−	0.794896i	−0.385027	−	0.922905i	$-0.625808\pi$
$263$	−22.3279	−	12.8910i	−1.37680	−	0.794896i	−0.991773	+	0.128009i	$0.959141\pi$
$264$		0			0
$265$	−8.29960	−	8.29960i	−0.509841	−	0.509841i
$266$		0			0
$267$	26.9374	+	7.21786i	1.64854	+	0.441726i
$268$		0			0
$269$	−13.1981	+	7.61995i	−0.804704	+	0.464596i	−0.845113	−	0.534587i	$-0.820467\pi$
$269$	−13.1981	+	7.61995i	−0.804704	+	0.464596i	0.0404091	+	0.999183i	$0.487134\pi$
$270$		0			0
$271$	2.30836	−	0.618524i	0.140223	−	0.0375727i	−0.188025	−	0.982164i	$-0.560209\pi$
$271$	2.30836	−	0.618524i	0.140223	−	0.0375727i	0.328248	+	0.944592i	$0.393542\pi$
$272$		0			0
$273$	1.20401	−	42.5791i	0.0728698	−	2.57700i
$274$		0			0
$275$	2.91915	−	0.782184i	0.176031	−	0.0471675i
$276$		0			0
$277$	9.66247	+	16.7359i	0.580561	+	1.00556i	0.995413	+	0.0956726i	$0.0305002\pi$
$277$	9.66247	+	16.7359i	0.580561	+	1.00556i	−0.414852	+	0.909889i	$0.636166\pi$
$278$		0			0
$279$	4.27557	+	1.14563i	0.255971	+	0.0685873i
$280$		0			0
$281$	21.9119	−	21.9119i	1.30715	−	1.30715i	0.383691	−	0.923462i	$-0.374653\pi$
$281$	21.9119	−	21.9119i	1.30715	−	1.30715i	0.923462	−	0.383691i	$-0.125347\pi$
$282$		0			0
$283$	9.50892	+	5.48998i	0.565247	+	0.326345i	0.755249	−	0.655438i	$-0.227516\pi$
$283$	9.50892	+	5.48998i	0.565247	+	0.326345i	−0.190002	+	0.981784i	$0.560849\pi$
$284$		0			0
$285$	−22.1112			−1.30976
$286$		0			0
$287$		−	4.18079i		−	0.246784i
$288$		0			0
$289$	−8.40690	+	14.5612i	−0.494524	+	0.856540i
$290$		0			0
$291$	7.12494	+	7.12494i	0.417671	+	0.417671i
$292$		0			0
$293$	1.28916	−	4.81120i	0.0753133	−	0.281073i	−0.917991	−	0.396601i	$-0.870189\pi$
$293$	1.28916	−	4.81120i	0.0753133	−	0.281073i	0.993304	+	0.115528i	$0.0368562\pi$
$294$		0			0
$295$	−9.83623	+	5.67895i	−0.572687	+	0.330641i
$296$		0			0
$297$	−6.69264	−	24.9773i	−0.388346	−	1.44933i
$298$		0			0
$299$	1.85388	−	0.440981i	0.107212	−	0.0255026i
$300$		0			0
$301$	−6.82872	−	25.4851i	−0.393601	−	1.46894i
$302$		0			0
$303$	−2.68130	−	4.64414i	−0.154037	−	0.266799i
$304$		0			0
$305$	−2.78179	+	10.3818i	−0.159285	+	0.594459i
$306$		0			0
$307$	−9.32161	+	9.32161i	−0.532013	+	0.532013i	−0.921171	−	0.389158i	$-0.872766\pi$
$307$	−9.32161	+	9.32161i	−0.532013	+	0.532013i	0.389158	+	0.921171i	$0.372766\pi$
$308$		0			0
$309$	9.37166	−	16.2322i	0.533135	−	0.923418i
$310$		0			0
$311$	7.84329			0.444752			0.222376	−	0.974961i	$-0.428619\pi$
$311$	7.84329			0.444752			0.222376	+	0.974961i	$0.428619\pi$
$312$		0			0
$313$	17.5938			0.994462			0.497231	−	0.867618i	$-0.334350\pi$
$313$	17.5938			0.994462			0.497231	+	0.867618i	$0.334350\pi$
$314$		0			0
$315$	−23.4121	+	40.5509i	−1.31912	+	2.28478i
$316$		0			0
$317$	14.7327	−	14.7327i	0.827472	−	0.827472i	−0.159695	−	0.987166i	$-0.551051\pi$
$317$	14.7327	−	14.7327i	0.827472	−	0.827472i	0.987166	+	0.159695i	$0.0510510\pi$
$318$		0			0
$319$	−1.71616	+	6.40480i	−0.0960865	+	0.358600i
$320$		0			0
$321$	11.4023	+	19.7493i	0.636412	+	1.10230i
$322$		0			0
$323$	−0.414342	−	1.54634i	−0.0230546	−	0.0860409i
$324$		0			0
$325$	−2.70901	+	2.56002i	−0.150269	+	0.142004i
$326$		0			0
$327$	−13.6598	−	50.9789i	−0.755386	−	2.81914i
$328$		0			0
$329$	15.0310	−	8.67813i	0.828684	−	0.478441i
$330$		0			0
$331$	2.45117	−	9.14790i	0.134729	−	0.502814i	−0.865270	−	0.501306i	$-0.832853\pi$
$331$	2.45117	−	9.14790i	0.134729	−	0.502814i	0.999999	−	0.00150826i	$-0.000480094\pi$
$332$		0			0
$333$	19.6500	+	19.6500i	1.07681	+	1.07681i
$334$		0			0
$335$	1.82147	−	3.15488i	0.0995176	−	0.172370i
$336$		0			0
$337$		−	21.1551i		−	1.15239i	−0.817312	−	0.576196i	$-0.804537\pi$
$337$		−	21.1551i		−	1.15239i	0.817312	−	0.576196i	$-0.195463\pi$
$338$		0			0
$339$	−34.8519			−1.89290
$340$		0			0
$341$	1.88168	+	1.08639i	0.101899	+	0.0588311i
$342$		0			0
$343$	4.42357	−	4.42357i	0.238850	−	0.238850i
$344$		0			0
$345$	−3.04259	−	0.815259i	−0.163807	−	0.0438921i
$346$		0			0
$347$	8.19610	+	14.1961i	0.439990	+	0.762085i	0.997688	−	0.0679589i	$-0.0216487\pi$
$347$	8.19610	+	14.1961i	0.439990	+	0.762085i	−0.557698	+	0.830044i	$0.688315\pi$
$348$		0			0
$349$	5.13248	−	1.37525i	0.274736	−	0.0736152i	−0.118821	−	0.992916i	$-0.537911\pi$
$349$	5.13248	−	1.37525i	0.274736	−	0.0736152i	0.393556	+	0.919301i	$0.371245\pi$
$350$		0			0
$351$	21.9044	+	23.1792i	1.16917	+	1.23722i
$352$		0			0
$353$	6.04366	−	1.61939i	0.321671	−	0.0861916i	−0.0943702	−	0.995537i	$-0.530084\pi$
$353$	6.04366	−	1.61939i	0.321671	−	0.0861916i	0.416042	+	0.909346i	$0.363417\pi$
$354$		0			0
$355$	8.58272	−	4.95524i	0.455523	−	0.262997i
$356$		0			0
$357$	−4.92414	−	1.31942i	−0.260613	−	0.0698310i
$358$		0			0
$359$	9.28036	+	9.28036i	0.489799	+	0.489799i	0.908243	−	0.418444i	$-0.137424\pi$
$359$	9.28036	+	9.28036i	0.489799	+	0.489799i	−0.418444	+	0.908243i	$0.637424\pi$
$360$		0			0
$361$	4.53441	+	2.61794i	0.238653	+	0.137787i
$362$		0			0
$363$			7.34218i			0.385365i
$364$		0			0
$365$		−	24.2737i		−	1.27054i
$366$		0			0
$367$	21.4767	+	12.3996i	1.12107	+	0.647252i	0.941675	−	0.336524i	$-0.109251\pi$
$367$	21.4767	+	12.3996i	1.12107	+	0.647252i	0.179399	+	0.983776i	$0.442585\pi$
$368$		0			0
$369$	4.45990	+	4.45990i	0.232173	+	0.232173i
$370$		0			0
$371$	−22.4737	−	6.02182i	−1.16678	−	0.312637i
$372$		0			0
$373$	−1.91414	+	1.10513i	−0.0991107	+	0.0572216i	−0.548736	−	0.835996i	$-0.684891\pi$
$373$	−1.91414	+	1.10513i	−0.0991107	+	0.0572216i	0.449625	+	0.893217i	$0.351557\pi$
$374$		0			0
$375$	34.7352	−	9.30728i	1.79372	−	0.480626i
$376$		0			0
$377$	−1.89246	−	7.95584i	−0.0974664	−	0.409747i
$378$		0			0
$379$	11.5125	−	3.08477i	0.591358	−	0.158454i	0.0492842	−	0.998785i	$-0.484306\pi$
$379$	11.5125	−	3.08477i	0.591358	−	0.158454i	0.542074	+	0.840331i	$0.317639\pi$
$380$		0			0
$381$	29.2581	+	50.6765i	1.49894	+	2.59624i
$382$		0			0
$383$	23.5153	+	6.30090i	1.20158	+	0.321961i	0.803451	−	0.595370i	$-0.202995\pi$
$383$	23.5153	+	6.30090i	1.20158	+	0.321961i	0.398124	+	0.917332i	$0.369661\pi$
$384$		0			0
$385$	−16.2525	+	16.2525i	−0.828302	+	0.828302i
$386$		0			0
$387$	34.4712	+	19.9019i	1.75227	+	1.01167i
$388$		0			0
$389$	20.5350			1.04117			0.520584	−	0.853811i	$-0.325714\pi$
$389$	20.5350			1.04117			0.520584	+	0.853811i	$0.325714\pi$
$390$		0			0
$391$		−	0.228060i		−	0.0115335i
$392$		0			0
$393$	−24.1604	+	41.8471i	−1.21873	+	2.11091i
$394$		0			0
$395$	−24.6997	−	24.6997i	−1.24277	−	1.24277i
$396$		0			0
$397$	7.08669	−	26.4479i	0.355671	−	1.32738i	−0.523967	−	0.851738i	$-0.675549\pi$
$397$	7.08669	−	26.4479i	0.355671	−	1.32738i	0.879638	−	0.475643i	$-0.157785\pi$
$398$		0			0
$399$	−37.9579	+	21.9150i	−1.90027	+	1.09712i
$400$		0			0
$401$	1.38519	+	5.16959i	0.0691729	+	0.258157i	0.991849	−	0.127419i	$-0.0406694\pi$
$401$	1.38519	+	5.16959i	0.0691729	+	0.258157i	−0.922676	+	0.385576i	$0.874003\pi$
$402$		0			0
$403$	−2.67866	−	0.0757443i	−0.133433	−	0.00377309i
$404$		0			0
$405$	−4.43443	−	16.5495i	−0.220348	−	0.822352i
$406$		0			0
$407$	6.82043	+	11.8133i	0.338076	+	0.585565i
$408$		0			0
$409$	−8.27483	+	30.8821i	−0.409164	+	1.52702i	0.387081	+	0.922046i	$0.373483\pi$
$409$	−8.27483	+	30.8821i	−0.409164	+	1.52702i	−0.796245	+	0.604974i	$0.793183\pi$
$410$		0			0
$411$	28.0737	−	28.0737i	1.38477	−	1.38477i
$412$		0			0
$413$	−11.2571	+	19.4979i	−0.553926	+	0.959428i
$414$		0			0
$415$	−27.1693			−1.33369
$416$		0			0
$417$	−36.3566			−1.78039
$418$		0			0
$419$	−4.06151	+	7.03474i	−0.198418	+	0.343670i	−0.948016	−	0.318224i	$-0.896914\pi$
$419$	−4.06151	+	7.03474i	−0.198418	+	0.343670i	0.749598	+	0.661894i	$0.230247\pi$
$420$		0			0
$421$	−23.7134	+	23.7134i	−1.15572	+	1.15572i	−0.170333	+	0.985387i	$0.554484\pi$
$421$	−23.7134	+	23.7134i	−1.15572	+	1.15572i	−0.985387	+	0.170333i	$0.945516\pi$
$422$		0			0
$423$	−6.77696	+	25.2919i	−0.329507	+	1.22974i
$424$		0			0
$425$	0.223036	+	0.386310i	0.0108188	+	0.0187388i
$426$		0			0
$427$	5.51421	+	20.5793i	0.266851	+	0.995902i
$428$		0			0
$429$	14.9935	+	27.7528i	0.723894	+	1.33992i
$430$		0			0
$431$	8.20438	+	30.6192i	0.395191	+	1.47487i	0.821455	+	0.570274i	$0.193163\pi$
$431$	8.20438	+	30.6192i	0.395191	+	1.47487i	−0.426263	+	0.904599i	$0.640170\pi$
$432$		0			0
$433$	−16.7664	+	9.68007i	−0.805740	+	0.465194i	−0.845474	−	0.534016i	$-0.820682\pi$
$433$	−16.7664	+	9.68007i	−0.805740	+	0.465194i	0.0397341	+	0.999210i	$0.487349\pi$
$434$		0			0
$435$	−3.49866	+	13.0572i	−0.167748	+	0.626043i
$436$		0			0
$437$	−1.38650	−	1.38650i	−0.0663252	−	0.0663252i
$438$		0			0
$439$	9.29759	−	16.1039i	0.443750	−	0.768597i	−0.554214	−	0.832374i	$-0.686981\pi$
$439$	9.29759	−	16.1039i	0.443750	−	0.768597i	0.997964	+	0.0637766i	$0.0203145\pi$
$440$		0			0
$441$			51.1275i			2.43464i
$442$		0			0
$443$	3.81566			0.181288			0.0906439	−	0.995883i	$-0.471108\pi$
$443$	3.81566			0.181288			0.0906439	+	0.995883i	$0.471108\pi$
$444$		0			0
$445$	16.0725	+	9.27946i	0.761910	+	0.439889i
$446$		0			0
$447$	24.2048	−	24.2048i	1.14485	−	1.14485i
$448$		0			0
$449$	−7.39494	−	1.98147i	−0.348989	−	0.0935112i	0.0800665	−	0.996790i	$-0.474487\pi$
$449$	−7.39494	−	1.98147i	−0.348989	−	0.0935112i	−0.429055	+	0.903278i	$0.641153\pi$
$450$		0			0
$451$	1.54801	+	2.68124i	0.0728931	+	0.126255i
$452$		0			0
$453$	−18.2971	+	4.90269i	−0.859673	+	0.230349i
$454$		0			0
$455$	8.10782	−	27.1630i	0.380100	−	1.27342i
$456$		0			0
$457$	17.5841	−	4.71165i	0.822550	−	0.220402i	0.177089	−	0.984195i	$-0.443332\pi$
$457$	17.5841	−	4.71165i	0.822550	−	0.220402i	0.645461	+	0.763793i	$0.276665\pi$
$458$		0			0
$459$	3.30540	−	1.90838i	0.154283	−	0.0890754i
$460$		0			0
$461$	−3.90436	−	1.04617i	−0.181844	−	0.0487250i	0.166748	−	0.986000i	$-0.446673\pi$
$461$	−3.90436	−	1.04617i	−0.181844	−	0.0487250i	−0.348592	+	0.937275i	$0.613340\pi$
$462$		0			0
$463$	−0.0778804	−	0.0778804i	−0.00361941	−	0.00361941i	0.705295	−	0.708914i	$-0.250815\pi$
$463$	−0.0778804	−	0.0778804i	−0.00361941	−	0.00361941i	−0.708914	+	0.705295i	$0.750815\pi$
$464$		0			0
$465$	3.83609	+	2.21477i	0.177894	+	0.102707i
$466$		0			0
$467$		−	0.605322i		−	0.0280110i	−0.999902	−	0.0140055i	$-0.995542\pi$
$467$		−	0.605322i		−	0.0280110i	0.999902	−	0.0140055i	$-0.00445823\pi$
$468$		0			0
$469$		−	7.22123i		−	0.333445i
$470$		0			0
$471$	−5.93682	−	3.42762i	−0.273554	−	0.157937i
$472$		0			0
$473$	13.8158	+	13.8158i	0.635249	+	0.635249i
$474$		0			0
$475$	3.70455	+	0.992630i	0.169976	+	0.0455450i
$476$		0			0
$477$	30.3979	−	17.5503i	1.39183	−	0.803571i
$478$		0			0
$479$	−15.3267	+	4.10679i	−0.700297	+	0.187644i	−0.591364	−	0.806405i	$-0.701410\pi$
$479$	−15.3267	+	4.10679i	−0.700297	+	0.187644i	−0.108933	+	0.994049i	$0.534743\pi$
$480$		0			0
$481$	−14.3260	−	8.82024i	−0.653211	−	0.402168i
$482$		0			0
$483$	−6.03118	+	1.61605i	−0.274428	+	0.0735328i
$484$		0			0
$485$	3.35279	+	5.80721i	0.152243	+	0.263692i
$486$		0			0
$487$	−1.60788	−	0.430831i	−0.0728601	−	0.0195228i	0.222205	−	0.975000i	$-0.428675\pi$
$487$	−1.60788	−	0.430831i	−0.0728601	−	0.0195228i	−0.295065	+	0.955477i	$0.595341\pi$
$488$		0			0
$489$	41.2986	−	41.2986i	1.86759	−	1.86759i
$490$		0			0
$491$	−19.1677	−	11.0665i	−0.865026	−	0.499423i	0.000666399	−	1.00000i	$-0.499788\pi$
$491$	−19.1677	−	11.0665i	−0.865026	−	0.499423i	−0.865692	+	0.500577i	$0.833121\pi$
$492$		0			0
$493$	−0.978711			−0.0440789
$494$		0			0
$495$		−	34.6750i		−	1.55852i
$496$		0			0
$497$	9.82253	−	17.0131i	0.440601	−	0.763143i
$498$		0			0
$499$	5.15371	+	5.15371i	0.230712	+	0.230712i	0.812990	−	0.582278i	$-0.197838\pi$
$499$	5.15371	+	5.15371i	0.230712	+	0.230712i	−0.582278	+	0.812990i	$0.697838\pi$
$500$		0			0
$501$	15.7957	−	58.9502i	0.705698	−	2.63370i
$502$		0			0
$503$	−37.9365	+	21.9026i	−1.69150	+	0.976590i	−0.738196	+	0.674586i	$0.764322\pi$
$503$	−37.9365	+	21.9026i	−1.69150	+	0.976590i	−0.953307	+	0.302003i	$0.902345\pi$
$504$		0			0
$505$	−0.923660	−	3.44715i	−0.0411023	−	0.153396i
$506$		0			0
$507$	−32.5387	−	21.3247i	−1.44509	−	0.947064i
$508$		0			0
$509$	1.64040	+	6.12207i	0.0727096	+	0.271356i	0.992704	−	0.120575i	$-0.0384738\pi$
$509$	1.64040	+	6.12207i	0.0727096	+	0.271356i	−0.919995	+	0.391931i	$0.871807\pi$
$510$		0			0
$511$	−24.0583	−	41.6702i	−1.06428	−	1.84338i
$512$		0			0
$513$	8.49329	−	31.6974i	0.374988	−	1.39947i
$514$		0			0
$515$	8.82008	−	8.82008i	0.388659	−	0.388659i
$516$		0			0
$517$	−6.42648	+	11.1310i	−0.282636	+	0.489540i
$518$		0			0
$519$	69.1188			3.03398
$520$		0			0
$521$	23.1087			1.01241			0.506206	−	0.862413i	$-0.331048\pi$
$521$	23.1087			1.01241			0.506206	+	0.862413i	$0.331048\pi$
$522$		0			0
$523$	10.4715	−	18.1372i	0.457887	−	0.793083i	−0.540962	−	0.841047i	$-0.681940\pi$
$523$	10.4715	−	18.1372i	0.457887	−	0.793083i	0.998849	+	0.0479636i	$0.0152731\pi$
$524$		0			0
$525$	8.63575	−	8.63575i	0.376895	−	0.376895i
$526$		0			0
$527$	−0.0830048	+	0.309778i	−0.00361575	+	0.0134942i
$528$		0			0
$529$	11.3603	+	19.6767i	0.493928	+	0.855508i
$530$		0			0
$531$	−8.79094	−	32.8082i	−0.381494	−	1.42376i
$532$		0			0
$533$	−3.25154	−	2.00190i	−0.140840	−	0.0867121i
$534$		0			0
$535$	3.92788	+	14.6590i	0.169817	+	0.633766i
$536$		0			0
$537$	25.6044	−	14.7827i	1.10491	−	0.637922i
$538$		0			0
$539$	−6.49554	+	24.2417i	−0.279783	+	1.04416i
$540$		0			0
$541$	−22.4938	−	22.4938i	−0.967084	−	0.967084i	0.0323917	−	0.999475i	$-0.489688\pi$
$541$	−22.4938	−	22.4938i	−0.967084	−	0.967084i	−0.999475	+	0.0323917i	$0.989688\pi$
$542$		0			0
$543$	−21.8616	+	37.8653i	−0.938170	+	1.62496i
$544$		0			0
$545$		−	35.1227i		−	1.50449i
$546$		0			0
$547$	14.1772			0.606172			0.303086	−	0.952963i	$-0.401983\pi$
$547$	14.1772			0.606172			0.303086	+	0.952963i	$0.401983\pi$
$548$		0			0
$549$	−27.8355	−	16.0709i	−1.18799	−	0.685887i
$550$		0			0
$551$	−5.95011	+	5.95011i	−0.253483	+	0.253483i
$552$		0			0
$553$	−66.8819	−	17.9209i	−2.84411	−	0.762076i
$554$		0			0
$555$	13.9045	+	24.0833i	0.590213	+	1.02228i
$556$		0			0
$557$	11.2083	−	3.00327i	0.474913	−	0.127253i	−0.0134200	−	0.999910i	$-0.504272\pi$
$557$	11.2083	−	3.00327i	0.474913	−	0.127253i	0.488333	+	0.872657i	$0.337605\pi$
$558$		0			0
$559$	−23.0905	−	6.89223i	−0.976624	−	0.291510i
$560$		0			0
$561$	3.64651	−	0.977078i	0.153956	−	0.0412523i
$562$		0			0
$563$	8.60923	−	4.97054i	0.362836	−	0.209483i	−0.307488	−	0.951552i	$-0.599488\pi$
$563$	8.60923	−	4.97054i	0.362836	−	0.209483i	0.670324	+	0.742069i	$0.266155\pi$
$564$		0			0
$565$	−22.4033	−	6.00293i	−0.942512	−	0.252545i
$566$		0			0
$567$	−24.0151	−	24.0151i	−1.00854	−	1.00854i
$568$		0			0
$569$	−31.2898	−	18.0652i	−1.31174	−	0.757332i	−0.329353	−	0.944207i	$-0.606831\pi$
$569$	−31.2898	−	18.0652i	−1.31174	−	0.757332i	−0.982384	+	0.186875i	$0.940164\pi$
$570$		0			0
$571$		−	3.28225i		−	0.137358i	−0.997639	−	0.0686790i	$-0.978122\pi$
$571$		−	3.28225i		−	0.137358i	0.997639	−	0.0686790i	$-0.0218784\pi$
$572$		0			0
$573$			32.8908i			1.37403i
$574$		0			0
$575$	0.473160	+	0.273179i	0.0197321	+	0.0113924i
$576$		0			0
$577$	−7.55694	−	7.55694i	−0.314600	−	0.314600i	0.532089	−	0.846689i	$-0.321407\pi$
$577$	−7.55694	−	7.55694i	−0.314600	−	0.314600i	−0.846689	+	0.532089i	$0.821407\pi$
$578$		0			0
$579$	11.9848	+	3.21131i	0.498070	+	0.133458i
$580$		0			0
$581$	−46.6410	+	26.9282i	−1.93499	+	1.11717i
$582$		0			0
$583$	16.6426	−	4.45938i	0.689267	−	0.184689i
$584$		0			0
$585$	20.3273	+	37.6255i	0.840431	+	1.55562i
$586$		0			0
$587$	−14.0208	+	3.75688i	−0.578702	+	0.155063i	−0.536285	−	0.844037i	$-0.680173\pi$
$587$	−14.0208	+	3.75688i	−0.578702	+	0.155063i	−0.0424176	+	0.999100i	$0.513506\pi$
$588$		0			0
$589$	1.37868	+	2.38794i	0.0568075	+	0.0983934i
$590$		0			0
$591$	−37.9543	−	10.1698i	−1.56123	−	0.418331i
$592$		0			0
$593$	−2.71248	+	2.71248i	−0.111388	+	0.111388i	−0.760604	−	0.649216i	$-0.775097\pi$
$593$	−2.71248	+	2.71248i	−0.111388	+	0.111388i	0.649216	+	0.760604i	$0.275097\pi$
$594$		0			0
$595$	−2.93804	−	1.69628i	−0.120448	−	0.0695406i
$596$		0			0
$597$	−40.8865			−1.67337
$598$		0			0
$599$		−	20.2262i		−	0.826421i	−0.910635	−	0.413211i	$-0.864407\pi$
$599$		−	20.2262i		−	0.826421i	0.910635	−	0.413211i	$-0.135593\pi$
$600$		0			0
$601$	20.9818	−	36.3415i	0.855865	−	1.48240i	−0.0199752	−	0.999800i	$-0.506359\pi$
$601$	20.9818	−	36.3415i	0.855865	−	1.48240i	0.875840	−	0.482601i	$-0.160308\pi$
$602$		0			0
$603$	7.70333	+	7.70333i	0.313704	+	0.313704i
$604$		0			0
$605$	−1.26463	+	4.71965i	−0.0514144	+	0.191881i
$606$		0			0
$607$	−14.4059	+	8.31726i	−0.584718	+	0.337587i	−0.763006	−	0.646391i	$-0.776277\pi$
$607$	−14.4059	+	8.31726i	−0.584718	+	0.337587i	0.178288	+	0.983978i	$0.442944\pi$
$608$		0			0
$609$	6.93522	+	25.8826i	0.281029	+	1.04882i
$610$		0			0
$611$	0.448062	−	15.8455i	0.0181267	−	0.641040i
$612$		0			0
$613$	0.326274	+	1.21767i	0.0131781	+	0.0491812i	0.972202	−	0.234145i	$-0.0752289\pi$
$613$	0.326274	+	1.21767i	0.0131781	+	0.0491812i	−0.959024	+	0.283326i	$0.908562\pi$
$614$		0			0
$615$	3.15586	+	5.46612i	0.127257	+	0.220415i
$616$		0			0
$617$	−6.14320	+	22.9267i	−0.247316	+	0.922995i	0.724889	+	0.688865i	$0.241891\pi$
$617$	−6.14320	+	22.9267i	−0.247316	+	0.922995i	−0.972205	+	0.234130i	$0.924776\pi$
$618$		0			0
$619$	−7.13833	+	7.13833i	−0.286914	+	0.286914i	−0.835859	−	0.548945i	$-0.815030\pi$
$619$	−7.13833	+	7.13833i	−0.286914	+	0.286914i	0.548945	+	0.835859i	$0.315030\pi$
$620$		0			0
$621$	2.33742	−	4.04852i	0.0937973	−	0.162462i
$622$		0			0
$623$	36.7885			1.47390
$624$		0			0
$625$	18.7626			0.750503
$626$		0			0
$627$	16.2289	−	28.1093i	0.648119	−	1.12258i
$628$		0			0
$629$	−1.42370	+	1.42370i	−0.0567668	+	0.0567668i
$630$		0			0
$631$	−0.0485313	+	0.181121i	−0.00193200	+	0.00721032i	−0.966885	−	0.255212i	$-0.917855\pi$
$631$	−0.0485313	+	0.181121i	−0.00193200	+	0.00721032i	0.964953	+	0.262422i	$0.0845214\pi$
$632$		0			0
$633$	−12.8281	−	22.2190i	−0.509873	−	0.883126i
$634$		0			0
$635$	10.0789	+	37.6150i	0.399969	+	1.49271i
$636$		0			0
$637$	−7.16281	−	30.1123i	−0.283801	−	1.19309i
$638$		0			0
$639$	7.67064	+	28.6272i	0.303446	+	1.13248i
$640$		0			0
$641$	−11.5562	+	6.67195i	−0.456441	+	0.263526i	−0.710546	−	0.703650i	$-0.751552\pi$
$641$	−11.5562	+	6.67195i	−0.456441	+	0.263526i	0.254106	+	0.967176i	$0.418219\pi$
$642$		0			0
$643$	3.41461	−	12.7435i	0.134659	−	0.502554i	−0.865340	−	0.501185i	$-0.832898\pi$
$643$	3.41461	−	12.7435i	0.134659	−	0.502554i	0.999999	−	0.00136906i	$-0.000435785\pi$
$644$		0			0
$645$	28.1656	+	28.1656i	1.10902	+	1.10902i
$646$		0			0
$647$	−4.24409	+	7.35098i	−0.166852	+	0.288997i	−0.937312	−	0.348492i	$-0.886694\pi$
$647$	−4.24409	+	7.35098i	−0.166852	+	0.288997i	0.770459	+	0.637489i	$0.220027\pi$
$648$		0			0
$649$		−	16.6726i		−	0.654457i
$650$		0			0
$651$	8.78045			0.344133
$652$		0			0
$653$	39.7745	+	22.9638i	1.55649	+	0.898643i	0.997588	+	0.0694122i	$0.0221124\pi$
$653$	39.7745	+	22.9638i	1.55649	+	0.898643i	0.558907	+	0.829230i	$0.311221\pi$
$654$		0			0
$655$	−22.7384	+	22.7384i	−0.888464	+	0.888464i
$656$		0			0
$657$	70.1166	+	18.7877i	2.73551	+	0.732978i
$658$		0			0
$659$	12.3199	+	21.3386i	0.479913	+	0.831234i	0.999735	−	0.0230408i	$-0.00733476\pi$
$659$	12.3199	+	21.3386i	0.479913	+	0.831234i	−0.519821	+	0.854275i	$0.674001\pi$
$660$		0			0
$661$	−4.80194	+	1.28668i	−0.186774	+	0.0500459i	−0.350993	−	0.936378i	$-0.614156\pi$
$661$	−4.80194	+	1.28668i	−0.186774	+	0.0500459i	0.164220	+	0.986424i	$0.447489\pi$
$662$		0			0
$663$	−3.38400	+	3.19789i	−0.131424	+	0.124196i
$664$		0			0
$665$	−28.1745	+	7.54934i	−1.09256	+	0.292751i
$666$		0			0
$667$	−1.03814	+	0.599372i	−0.0401971	+	0.0232078i
$668$		0			0
$669$	5.37548	+	1.44036i	0.207828	+	0.0556874i
$670$		0			0
$671$	−11.1563	−	11.1563i	−0.430683	−	0.430683i
$672$		0			0
$673$	−18.9947	−	10.9666i	−0.732193	−	0.422732i	0.0870306	−	0.996206i	$-0.472262\pi$
$673$	−18.9947	−	10.9666i	−0.732193	−	0.422732i	−0.819224	+	0.573474i	$0.805596\pi$
$674$		0			0
$675$			9.14371i			0.351942i
$676$		0			0
$677$			27.2466i			1.04717i	0.851972	+	0.523587i	$0.175406\pi$
$677$			27.2466i			1.04717i	−0.851972	+	0.523587i	$0.824594\pi$
$678$		0			0
$679$	11.5114	+	6.64608i	0.441765	+	0.255053i
$680$		0			0
$681$	−42.1852	−	42.1852i	−1.61654	−	1.61654i
$682$		0			0
$683$	−36.4710	−	9.77237i	−1.39552	−	0.373929i	−0.518788	−	0.854903i	$-0.673617\pi$
$683$	−36.4710	−	9.77237i	−1.39552	−	0.373929i	−0.876735	+	0.480974i	$0.840283\pi$
$684$		0			0
$685$	22.8816	−	13.2107i	0.874259	−	0.504754i
$686$		0			0
$687$	−24.0503	+	6.44425i	−0.917576	+	0.245864i
$688$		0			0
$689$	−15.4446	+	14.5951i	−0.588391	+	0.556030i
$690$		0			0
$691$	−2.76976	+	0.742154i	−0.105367	+	0.0282329i	−0.311117	−	0.950372i	$-0.600703\pi$
$691$	−2.76976	+	0.742154i	−0.105367	+	0.0282329i	0.205751	+	0.978604i	$0.434036\pi$
$692$		0			0
$693$	−34.3673	−	59.5259i	−1.30551	−	2.26120i
$694$		0			0
$695$	−23.3705	−	6.26211i	−0.886494	−	0.237535i
$696$		0			0
$697$	−0.323134	+	0.323134i	−0.0122396	+	0.0122396i
$698$		0			0
$699$	−38.7506	−	22.3727i	−1.46568	−	0.846212i
$700$		0			0
$701$	−39.5968			−1.49555			−0.747775	−	0.663952i	$-0.768878\pi$
$701$	−39.5968			−1.49555			−0.747775	+	0.663952i	$0.768878\pi$
$702$		0			0
$703$			17.3109i			0.652894i
$704$		0			0
$705$	−13.1014	+	22.6922i	−0.493426	+	0.854639i
$706$		0			0
$707$	−5.00219	−	5.00219i	−0.188127	−	0.188127i
$708$		0			0
$709$	0.979951	−	3.65723i	0.0368028	−	0.137350i	−0.945080	−	0.326839i	$-0.894016\pi$
$709$	0.979951	−	3.65723i	0.0368028	−	0.137350i	0.981883	+	0.189489i	$0.0606831\pi$
$710$		0			0
$711$	90.4644	−	52.2296i	3.39268	−	1.95876i
$712$		0			0
$713$	0.101666	+	0.379422i	0.00380742	+	0.0142095i
$714$		0			0
$715$	4.85786	+	20.4223i	0.181674	+	0.763752i
$716$		0			0
$717$	2.31601	+	8.64346i	0.0864930	+	0.322796i
$718$		0			0
$719$	19.2355	+	33.3168i	0.717363	+	1.24251i	0.962041	+	0.272904i	$0.0879843\pi$
$719$	19.2355	+	33.3168i	0.717363	+	1.24251i	−0.244678	+	0.969604i	$0.578682\pi$
$720$		0			0
$721$	6.39945	−	23.8831i	0.238328	−	0.889452i
$722$		0			0
$723$	−30.2653	+	30.2653i	−1.12558	+	1.12558i
$724$		0			0
$725$	1.17234	−	2.03055i	0.0435396	−	0.0754127i
$726$		0			0
$727$	−4.98457			−0.184867			−0.0924337	−	0.995719i	$-0.529465\pi$
$727$	−4.98457			−0.184867			−0.0924337	+	0.995719i	$0.529465\pi$
$728$		0			0
$729$	−28.1733			−1.04346
$730$		0			0
$731$	−1.44196	+	2.49754i	−0.0533327	+	0.0923750i
$732$		0			0
$733$	−7.68903	+	7.68903i	−0.284001	+	0.284001i	−0.834702	−	0.550701i	$-0.814360\pi$
$733$	−7.68903	+	7.68903i	−0.284001	+	0.284001i	0.550701	+	0.834702i	$0.314360\pi$
$734$		0			0
$735$	−13.2422	+	49.4204i	−0.488444	+	1.82290i
$736$		0			0
$737$	2.67379	+	4.63115i	0.0984905	+	0.170591i
$738$		0			0
$739$	−3.25679	−	12.1545i	−0.119803	−	0.447111i	0.879798	−	0.475347i	$-0.157678\pi$
$739$	−3.25679	−	12.1545i	−0.119803	−	0.447111i	−0.999601	+	0.0282364i	$0.991011\pi$
$740$		0			0
$741$	−1.13150	+	40.0149i	−0.0415666	+	1.46998i
$742$		0			0
$743$	−1.24991	−	4.66472i	−0.0458547	−	0.171132i	0.939201	−	0.343368i	$-0.111568\pi$
$743$	−1.24991	−	4.66472i	−0.0458547	−	0.171132i	−0.985056	+	0.172235i	$0.944901\pi$
$744$		0			0
$745$	19.7282	−	11.3901i	0.722787	−	0.417301i
$746$		0			0
$747$	21.0288	−	78.4807i	0.769405	−	2.87146i
$748$		0			0
$749$	21.2719	+	21.2719i	0.777257	+	0.777257i
$750$		0			0
$751$	16.3389	−	28.2997i	0.596213	−	1.03267i	−0.397161	−	0.917749i	$-0.630005\pi$
$751$	16.3389	−	28.2997i	0.596213	−	1.03267i	0.993374	−	0.114923i	$-0.0366620\pi$
$752$		0			0
$753$		−	16.5002i		−	0.601302i
$754$		0			0
$755$	−12.6061			−0.458781
$756$		0			0
$757$	21.9250	+	12.6584i	0.796879	+	0.460079i	0.842379	−	0.538886i	$-0.181155\pi$
$757$	21.9250	+	12.6584i	0.796879	+	0.460079i	−0.0454994	+	0.998964i	$0.514488\pi$
$758$		0			0
$759$	3.26957	−	3.26957i	0.118678	−	0.118678i
$760$		0			0
$761$	−12.5595	−	3.36531i	−0.455282	−	0.121993i	0.0238877	−	0.999715i	$-0.492396\pi$
$761$	−12.5595	−	3.36531i	−0.455282	−	0.121993i	−0.479170	+	0.877722i	$0.659062\pi$
$762$		0			0
$763$	−34.8110	−	60.2944i	−1.26024	−	2.18280i
$764$		0			0
$765$	4.94371	−	1.32466i	0.178740	−	0.0478933i
$766$		0			0
$767$	9.77389	+	18.0913i	0.352915	+	0.653239i
$768$		0			0
$769$	47.5222	−	12.7335i	1.71369	−	0.459183i	0.737369	−	0.675490i	$-0.236068\pi$
$769$	47.5222	−	12.7335i	1.71369	−	0.459183i	0.976325	+	0.216308i	$0.0694014\pi$
$770$		0			0
$771$	−74.0236	+	42.7376i	−2.66590	+	1.53916i
$772$		0			0
$773$	4.85399	+	1.30062i	0.174586	+	0.0467802i	0.345053	−	0.938583i	$-0.387861\pi$
$773$	4.85399	+	1.30062i	0.174586	+	0.0467802i	−0.170467	+	0.985363i	$0.554528\pi$
$774$		0			0
$775$	−0.543276	−	0.543276i	−0.0195151	−	0.0195151i
$776$		0			0
$777$	47.7391	+	27.5622i	1.71263	+	0.988789i
$778$		0			0
$779$			3.92901i			0.140771i
$780$		0			0
$781$			14.5479i			0.520564i
$782$		0			0
$783$	−17.3741	−	10.0309i	−0.620900	−	0.358477i
$784$		0			0
$785$	−3.22588	−	3.22588i	−0.115137	−	0.115137i
$786$		0			0
$787$	45.4229	+	12.1710i	1.61915	+	0.433851i	0.950752	−	0.309951i	$-0.100313\pi$
$787$	45.4229	+	12.1710i	1.61915	+	0.433851i	0.668400	+	0.743802i	$0.266979\pi$
$788$		0			0
$789$	66.8187	−	38.5778i	2.37881	−	1.37341i
$790$		0			0
$791$	−44.4089	+	11.8993i	−1.57900	+	0.423091i
$792$		0			0
$793$	18.6456	+	5.56549i	0.662126	+	0.197636i
$794$		0			0
$795$	33.9285	−	9.09113i	1.20332	−	0.322429i
$796$		0			0
$797$	−23.4500	−	40.6165i	−0.830641	−	1.43871i	−0.897531	−	0.440951i	$-0.854641\pi$
$797$	−23.4500	−	40.6165i	−0.830641	−	1.43871i	0.0668904	−	0.997760i	$-0.478692\pi$
$798$		0			0
$799$	−1.83248	−	0.491012i	−0.0648285	−	0.0173708i
$800$		0			0
$801$	−39.2445	+	39.2445i	−1.38664	+	1.38664i
$802$		0			0
$803$	30.8583	+	17.8161i	1.08897	+	0.628715i
$804$		0			0
$805$	−4.15527			−0.146454
$806$		0			0
$807$		−	45.6069i		−	1.60544i
$808$		0			0
$809$	−5.37324	+	9.30673i	−0.188913	+	0.327207i	−0.944888	−	0.327393i	$-0.893830\pi$
$809$	−5.37324	+	9.30673i	−0.188913	+	0.327207i	0.755975	+	0.654600i	$0.227163\pi$
$810$		0			0
$811$	13.3668	+	13.3668i	0.469372	+	0.469372i	0.901711	−	0.432339i	$-0.142312\pi$
$811$	13.3668	+	13.3668i	0.469372	+	0.469372i	−0.432339	+	0.901711i	$0.642312\pi$
$812$		0			0
$813$	−1.85100	+	6.90802i	−0.0649173	+	0.242275i
$814$		0			0
$815$	33.6606	−	19.4339i	1.17908	−	0.680741i
$816$		0			0
$817$	6.41748	+	23.9504i	0.224519	+	0.837917i
$818$		0			0
$819$	72.1871	+	44.4441i	2.52242	+	1.55300i
$820$		0			0
$821$	−3.83338	−	14.3064i	−0.133786	−	0.499295i	0.866214	−	0.499673i	$-0.166546\pi$
$821$	−3.83338	−	14.3064i	−0.133786	−	0.499295i	−1.00000		0.000377692i	$0.999880\pi$
$822$		0			0
$823$	−18.8317	−	32.6174i	−0.656431	−	1.13697i	−0.981533	−	0.191292i	$-0.938732\pi$
$823$	−18.8317	−	32.6174i	−0.656431	−	1.13697i	0.325102	−	0.945679i	$-0.394601\pi$
$824$		0			0
$825$	−2.34077	+	8.73586i	−0.0814950	+	0.304144i
$826$		0			0
$827$	2.55841	−	2.55841i	0.0889646	−	0.0889646i	−0.661224	−	0.750189i	$-0.729963\pi$
$827$	2.55841	−	2.55841i	0.0889646	−	0.0889646i	0.750189	+	0.661224i	$0.229963\pi$
$828$		0			0
$829$	−3.90712	+	6.76733i	−0.135700	+	0.235039i	−0.925865	−	0.377855i	$-0.876662\pi$
$829$	−3.90712	+	6.76733i	−0.135700	+	0.235039i	0.790165	+	0.612895i	$0.209995\pi$
$830$		0			0
$831$	−57.8319			−2.00616
$832$		0			0
$833$	−3.70435			−0.128348
$834$		0			0
$835$	20.3073	−	35.1733i	0.702763	−	1.21722i
$836$		0			0
$837$	−4.64846	+	4.64846i	−0.160674	+	0.160674i
$838$		0			0
$839$	7.70458	−	28.7539i	0.265991	−	0.992694i	−0.695649	−	0.718382i	$-0.744883\pi$
$839$	7.70458	−	28.7539i	0.265991	−	0.992694i	0.961641	−	0.274312i	$-0.0884501\pi$
$840$		0			0
$841$	−11.9278	−	20.6596i	−0.411304	−	0.712399i
$842$		0			0
$843$	24.0016	+	89.5751i	0.826658	+	3.08513i
$844$		0			0
$845$	−17.2433	−	19.3123i	−0.593187	−	0.664363i
$846$		0			0
$847$	2.50681	+	9.35554i	0.0861349	+	0.321460i
$848$		0			0
$849$	−28.4564	+	16.4293i	−0.976622	+	0.563853i
$850$		0			0
$851$	−0.638268	+	2.38205i	−0.0218795	+	0.0816555i
$852$		0			0
$853$	−7.62369	−	7.62369i	−0.261030	−	0.261030i	0.564442	−	0.825473i	$-0.309091\pi$
$853$	−7.62369	−	7.62369i	−0.261030	−	0.261030i	−0.825473	+	0.564442i	$0.809091\pi$
$854$		0			0
$855$	22.0021	−	38.1088i	0.752457	−	1.30329i
$856$		0			0
$857$			56.9565i			1.94560i	0.231650	+	0.972799i	$0.425588\pi$
$857$			56.9565i			1.94560i	−0.231650	+	0.972799i	$0.574412\pi$
$858$		0			0
$859$	12.7259			0.434201			0.217101	−	0.976149i	$-0.430340\pi$
$859$	12.7259			0.434201			0.217101	+	0.976149i	$0.430340\pi$
$860$		0			0
$861$	10.8352	+	6.25572i	0.369263	+	0.213194i
$862$		0			0
$863$	−28.5737	+	28.5737i	−0.972660	+	0.972660i	−0.999636	−	0.0269758i	$-0.991412\pi$
$863$	−28.5737	+	28.5737i	−0.972660	+	0.972660i	0.0269758	+	0.999636i	$0.491412\pi$
$864$		0			0
$865$	44.4304	+	11.9051i	1.51068	+	0.404786i
$866$		0			0
$867$	−25.1585	−	43.5758i	−0.854428	−	1.47991i
$868$		0			0
$869$	49.5285	−	13.2711i	1.68014	−	0.450192i
$870$		0			0
$871$	−5.61620	−	3.45777i	−0.190298	−	0.117162i
$872$		0			0
$873$	−19.3696	+	5.19008i	−0.655563	+	0.175658i
$874$		0			0
$875$	41.0825	−	23.7190i	1.38884	−	0.801848i
$876$		0			0
$877$	−7.32275	−	1.96212i	−0.247272	−	0.0662562i	0.133054	−	0.991109i	$-0.457522\pi$
$877$	−7.32275	−	1.96212i	−0.247272	−	0.0662562i	−0.380326	+	0.924853i	$0.624188\pi$
$878$		0			0
$879$	10.5401	+	10.5401i	0.355508	+	0.355508i
$880$		0			0
$881$	−29.2464	−	16.8854i	−0.985336	−	0.568884i	−0.0814588	−	0.996677i	$-0.525958\pi$
$881$	−29.2464	−	16.8854i	−0.985336	−	0.568884i	−0.903877	+	0.427793i	$0.859291\pi$
$882$		0			0
$883$		−	33.0864i		−	1.11345i	−0.830698	−	0.556723i	$-0.812058\pi$
$883$		−	33.0864i		−	1.11345i	0.830698	−	0.556723i	$-0.187942\pi$
$884$		0			0
$885$		−	33.9897i		−	1.14255i
$886$		0			0
$887$	5.41849	+	3.12837i	0.181935	+	0.105040i	0.588202	−	0.808714i	$-0.299836\pi$
$887$	5.41849	+	3.12837i	0.181935	+	0.105040i	−0.406266	+	0.913755i	$0.633169\pi$
$888$		0			0
$889$	54.5835	+	54.5835i	1.83067	+	1.83067i
$890$		0			0
$891$	24.2935	+	6.50943i	0.813864	+	0.218074i
$892$		0			0
$893$	−14.1258	+	8.15552i	−0.472701	+	0.272914i
$894$		0			0
$895$	19.0051	−	5.09239i	0.635269	−	0.170220i
$896$		0			0
$897$	−1.63108	+	5.46447i	−0.0544601	+	0.182453i
$898$		0			0
$899$	1.62828	−	0.436296i	0.0543061	−	0.0145513i
$900$		0			0
$901$	1.27157	+	2.20243i	0.0423622	+	0.0733735i
$902$		0			0
$903$	76.2669	+	20.4357i	2.53800	+	0.680056i
$904$		0			0
$905$	−20.5749	+	20.5749i	−0.683932	+	0.683932i
$906$		0			0
$907$	−39.7698	−	22.9611i	−1.32053	−	0.762410i	−0.336719	−	0.941605i	$-0.609317\pi$
$907$	−39.7698	−	22.9611i	−1.32053	−	0.762410i	−0.983814	+	0.179195i	$0.942651\pi$
$908$		0			0
$909$	10.6723			0.353977
$910$		0			0
$911$			40.9298i			1.35607i	0.735032	+	0.678033i	$0.237167\pi$
$911$			40.9298i			1.35607i	−0.735032	+	0.678033i	$0.762833\pi$
$912$		0			0
$913$	19.9413	−	34.5394i	0.659961	−	1.14309i
$914$		0			0
$915$	−22.7438	−	22.7438i	−0.751885	−	0.751885i
$916$		0			0
$917$	−16.4980	+	61.5713i	−0.544811	+	2.03326i
$918$		0			0
$919$	6.79881	−	3.92529i	0.224272	−	0.129484i	−0.383655	−	0.923477i	$-0.625335\pi$
$919$	6.79881	−	3.92529i	0.224272	−	0.129484i	0.607927	+	0.793993i	$0.292001\pi$
$920$		0			0
$921$	−10.2106	−	38.1065i	−0.336451	−	1.25565i
$922$		0			0
$923$	−8.52832	−	15.7858i	−0.280713	−	0.519595i
$924$		0			0
$925$	−1.24842	−	4.65915i	−0.0410477	−	0.153192i
$926$		0			0
$927$	18.6508	+	32.3042i	0.612574	+	1.06101i
$928$		0			0
$929$	5.88675	−	21.9696i	0.193138	−	0.720800i	−0.799603	−	0.600529i	$-0.794957\pi$
$929$	5.88675	−	21.9696i	0.193138	−	0.720800i	0.992741	−	0.120272i	$-0.0383766\pi$
$930$		0			0
$931$	−22.5207	+	22.5207i	−0.738087	+	0.738087i
$932$		0			0
$933$	−11.7359	+	20.3272i	−0.384217	+	0.665483i
$934$		0			0
$935$	2.51231			0.0821614
$936$		0			0
$937$	35.1246			1.14747			0.573735	−	0.819041i	$-0.305494\pi$
$937$	35.1246			1.14747			0.573735	+	0.819041i	$0.305494\pi$
$938$		0			0
$939$	−26.3257	+	45.5974i	−0.859106	+	1.48801i
$940$		0			0
$941$	33.4717	−	33.4717i	1.09115	−	1.09115i	0.0957410	−	0.995406i	$-0.469478\pi$
$941$	33.4717	−	33.4717i	1.09115	−	1.09115i	0.995406	−	0.0957410i	$-0.0305221\pi$
$942$		0			0
$943$	−0.144866	+	0.540646i	−0.00471748	+	0.0176059i
$944$		0			0
$945$	−34.7708	−	60.2247i	−1.13109	−	1.95911i
$946$		0			0
$947$	−6.87571	−	25.6605i	−0.223430	−	0.833854i	−0.983027	−	0.183460i	$-0.941270\pi$
$947$	−6.87571	−	25.6605i	−0.223430	−	0.833854i	0.759597	−	0.650394i	$-0.225396\pi$
$948$		0			0
$949$	−43.9283	−	1.24216i	−1.42597	−	0.0403222i
$950$		0			0
$951$	16.1378	+	60.2269i	0.523303	+	1.95299i
$952$		0			0
$953$	−18.5794	+	10.7268i	−0.601845	+	0.347475i	−0.769767	−	0.638325i	$-0.779628\pi$
$953$	−18.5794	+	10.7268i	−0.601845	+	0.347475i	0.167922	+	0.985800i	$0.446294\pi$
$954$		0			0
$955$	−5.66515	+	21.1426i	−0.183320	+	0.684159i
$956$		0			0
$957$	−14.0312	−	14.0312i	−0.453565	−	0.453565i
$958$		0			0
$959$	26.1869	−	45.3570i	0.845618	−	1.46465i
$960$		0			0
$961$			30.4476i			0.982181i
$962$		0			0
$963$	−45.3840			−1.46248
$964$		0			0
$965$	7.15085	+	4.12854i	0.230194	+	0.132903i
$966$		0			0
$967$	28.4651	−	28.4651i	0.915375	−	0.915375i	−0.0813132	−	0.996689i	$-0.525911\pi$
$967$	28.4651	−	28.4651i	0.915375	−	0.915375i	0.996689	+	0.0813132i	$0.0259114\pi$
$968$		0			0
$969$	4.62759	+	1.23996i	0.148660	+	0.0398333i
$970$		0			0
$971$	2.57471	+	4.45954i	0.0826265	+	0.143113i	0.904377	−	0.426734i	$-0.140336\pi$
$971$	2.57471	+	4.45954i	0.0826265	+	0.143113i	−0.821751	+	0.569847i	$0.807002\pi$
$972$		0			0
$973$	−46.3262	+	12.4131i	−1.48515	+	0.397945i
$974$		0			0
$975$	−2.58122	−	10.8514i	−0.0826653	−	0.347523i
$976$		0			0
$977$	−12.7613	+	3.41938i	−0.408270	+	0.109396i	−0.457108	−	0.889411i	$-0.651115\pi$
$977$	−12.7613	+	3.41938i	−0.408270	+	0.109396i	0.0488381	+	0.998807i	$0.484448\pi$
$978$		0			0
$979$	−23.5933	+	13.6216i	−0.754046	+	0.435349i
$980$		0			0
$981$	101.455	+	27.1847i	3.23920	+	0.867941i
$982$		0			0
$983$	−6.17750	−	6.17750i	−0.197032	−	0.197032i	0.601695	−	0.798726i	$-0.294492\pi$
$983$	−6.17750	−	6.17750i	−0.197032	−	0.197032i	−0.798726	+	0.601695i	$0.794492\pi$
$984$		0			0
$985$	−22.6459	−	13.0746i	−0.721557	−	0.416591i
$986$		0			0
$987$			51.9404i			1.65328i
$988$		0			0
$989$			3.53228i			0.112320i
$990$		0			0
$991$	−23.9402	−	13.8219i	−0.760486	−	0.439067i	0.0689844	−	0.997618i	$-0.478024\pi$
$991$	−23.9402	−	13.8219i	−0.760486	−	0.439067i	−0.829470	+	0.558551i	$0.811357\pi$
$992$		0			0
$993$	20.0406	+	20.0406i	0.635971	+	0.635971i
$994$		0			0
$995$	−26.2823	−	7.04233i	−0.833206	−	0.223257i
$996$		0			0
$997$	52.0747	−	30.0653i	1.64922	−	0.952179i	0.671840	−	0.740696i	$-0.265504\pi$
$997$	52.0747	−	30.0653i	1.64922	−	0.952179i	0.977382	−	0.211483i	$-0.0678292\pi$
$998$		0			0
$999$	−39.8653	+	10.6819i	−1.26128	+	0.337960i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.111.1		48
4.3	odd	2		104.2.u.a.59.5	yes	48
8.3	odd	2	inner	416.2.bk.a.111.2		48
8.5	even	2		104.2.u.a.59.2	✓	48
12.11	even	2		936.2.ed.d.163.8		48
13.2	odd	12	inner	416.2.bk.a.15.2		48
24.5	odd	2		936.2.ed.d.163.11		48
52.15	even	12		104.2.u.a.67.2	yes	48
104.67	even	12	inner	416.2.bk.a.15.1		48
104.93	odd	12		104.2.u.a.67.5	yes	48
156.119	odd	12		936.2.ed.d.379.11		48
312.197	even	12		936.2.ed.d.379.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.59.2	✓	48	8.5	even	2
104.2.u.a.59.5	yes	48	4.3	odd	2
104.2.u.a.67.2	yes	48	52.15	even	12
104.2.u.a.67.5	yes	48	104.93	odd	12
416.2.bk.a.15.1		48	104.67	even	12	inner
416.2.bk.a.15.2		48	13.2	odd	12	inner
416.2.bk.a.111.1		48	1.1	even	1	trivial
416.2.bk.a.111.2		48	8.3	odd	2	inner
936.2.ed.d.163.8		48	12.11	even	2
936.2.ed.d.163.11		48	24.5	odd	2
936.2.ed.d.379.8		48	312.197	even	12
936.2.ed.d.379.11		48	156.119	odd	12

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bk (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.49630 + 2.59167i) q^{3} +(-1.40823 + 1.40823i) q^{5} +(-1.02175 + 3.81323i) q^{7} +(-2.97784 - 5.15776i) q^{9} +O(q^{10})\)	\(q+(-1.49630 + 2.59167i) q^{3} +(-1.40823 + 1.40823i) q^{5} +(-1.02175 + 3.81323i) q^{7} +(-2.97784 - 5.15776i) q^{9} +(-0.756644 - 2.82384i) q^{11} +(2.47643 + 2.62055i) q^{13} +(-1.54254 - 5.75682i) q^{15} +(0.373696 - 0.215754i) q^{17} +(0.960219 - 3.58359i) q^{19} +(-8.35378 - 8.35378i) q^{21} +(0.264259 - 0.457711i) q^{23} +1.03375i q^{25} +8.84516 q^{27} +(-1.96425 - 1.13406i) q^{29} +(-0.525538 + 0.525538i) q^{31} +(8.45062 + 2.26434i) q^{33} +(-3.93105 - 6.80878i) q^{35} +(-4.50702 + 1.20765i) q^{37} +(-10.4971 + 2.49694i) q^{39} +(-1.02295 + 0.274098i) q^{41} +(-5.78795 + 3.34168i) q^{43} +(11.4568 + 3.06985i) q^{45} +(-3.10880 - 3.10880i) q^{47} +(-7.43454 - 4.29233i) q^{49} +1.29133i q^{51} +5.89363i q^{53} +(5.04215 + 2.91109i) q^{55} +(7.85070 + 7.85070i) q^{57} +(5.50874 + 1.47606i) q^{59} +(4.67379 - 2.69841i) q^{61} +(22.7103 - 6.08522i) q^{63} +(-7.17774 - 0.202965i) q^{65} +(-1.76688 + 0.473433i) q^{67} +(0.790823 + 1.36975i) q^{69} +(-4.80671 - 1.28795i) q^{71} +(-8.61849 + 8.61849i) q^{73} +(-2.67915 - 1.54681i) q^{75} +11.5410 q^{77} +17.5395i q^{79} +(-4.30151 + 7.45044i) q^{81} +(9.64657 + 9.64657i) q^{83} +(-0.222420 + 0.830084i) q^{85} +(5.87822 - 3.39379i) q^{87} +(-2.41190 - 9.00133i) q^{89} +(-12.5231 + 6.76562i) q^{91} +(-0.575658 - 2.14838i) q^{93} +(3.69432 + 6.39874i) q^{95} +(0.871452 - 3.25230i) q^{97} +(-12.3115 + 12.3115i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9}+O(q^{10}) \) 48 * q + 4 * q^3 - 20 * q^9	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83} + 12 q^{89} - 104 q^{91} + 4 q^{97} - 168 q^{99}+O(q^{100}) \) 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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