LMFDB - Embedded newform 312.2.bp.a.89.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,2,Mod(41,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 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("312.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	312.bp (of order $12$, degree $4$, 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:	$2.49133254306$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.5
Character	$\chi$	$=$	312.89
Dual form			312.2.bp.a.305.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.19941 + 1.24957i) q^{3} +(1.44076 - 1.44076i) q^{5} +(0.918532 + 3.42801i) q^{7} +(-0.122851 - 2.99748i) q^{9} +O(q^{10})$	\(q+(-1.19941 + 1.24957i) q^{3} +(1.44076 - 1.44076i) q^{5} +(0.918532 + 3.42801i) q^{7} +(-0.122851 - 2.99748i) q^{9} +(0.0392149 - 0.146352i) q^{11} +(1.92366 + 3.04952i) q^{13} +(0.0722747 + 3.52839i) q^{15} +(-1.60706 + 2.78351i) q^{17} +(1.90607 - 0.510729i) q^{19} +(-5.38523 - 2.96380i) q^{21} +(4.19640 + 7.26837i) q^{23} +0.848416i q^{25} +(3.89291 + 3.44169i) q^{27} +(-0.0238088 + 0.0137460i) q^{29} +(-5.54595 - 5.54595i) q^{31} +(0.135842 + 0.224537i) q^{33} +(6.26232 + 3.61555i) q^{35} +(3.84429 + 1.03007i) q^{37} +(-6.11783 - 1.25387i) q^{39} +(2.53382 + 0.678935i) q^{41} +(-5.90548 - 3.40953i) q^{43} +(-4.49566 - 4.14166i) q^{45} +(-4.77303 - 4.77303i) q^{47} +(-4.84536 + 2.79747i) q^{49} +(-1.55068 - 5.34670i) q^{51} -13.3755i q^{53} +(-0.154359 - 0.267357i) q^{55} +(-1.64796 + 2.99434i) q^{57} +(8.09489 - 2.16902i) q^{59} +(6.61697 - 11.4609i) q^{61} +(10.1626 - 3.17442i) q^{63} +(7.16515 + 1.62210i) q^{65} +(1.18370 - 4.41762i) q^{67} +(-14.1155 - 3.47404i) q^{69} +(1.29021 + 4.81512i) q^{71} +(6.22036 - 6.22036i) q^{73} +(-1.06016 - 1.01760i) q^{75} +0.537715 q^{77} -13.3852 q^{79} +(-8.96982 + 0.736488i) q^{81} +(1.94085 - 1.94085i) q^{83} +(1.69499 + 6.32577i) q^{85} +(0.0113798 - 0.0462379i) q^{87} +(-2.87300 + 10.7222i) q^{89} +(-8.68683 + 9.39538i) q^{91} +(13.5819 - 0.278209i) q^{93} +(2.01035 - 3.48203i) q^{95} +(-17.7357 + 4.75227i) q^{97} +(-0.443505 - 0.0995664i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 4 q^{7}+O(q^{10}) $ 56 * q + 4 * q^7	$ 56 q + 4 q^{7} + 8 q^{13} - 8 q^{15} + 4 q^{19} + 16 q^{21} + 24 q^{27} - 36 q^{31} + 28 q^{33} + 20 q^{37} + 16 q^{39} - 84 q^{43} + 12 q^{45} - 12 q^{49} - 24 q^{55} - 36 q^{57} - 24 q^{61} - 12 q^{63} - 32 q^{67} - 36 q^{69} - 20 q^{73} - 60 q^{75} - 32 q^{79} - 88 q^{85} - 16 q^{87} + 28 q^{91} - 88 q^{93} - 36 q^{97} + 44 q^{99}+O(q^{100}) $ 56 * q + 4 * q^7 + 8 * q^13 - 8 * q^15 + 4 * q^19 + 16 * q^21 + 24 * q^27 - 36 * q^31 + 28 * q^33 + 20 * q^37 + 16 * q^39 - 84 * q^43 + 12 * q^45 - 12 * q^49 - 24 * q^55 - 36 * q^57 - 24 * q^61 - 12 * q^63 - 32 * q^67 - 36 * q^69 - 20 * q^73 - 60 * q^75 - 32 * q^79 - 88 * q^85 - 16 * q^87 + 28 * q^91 - 88 * q^93 - 36 * q^97 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$e\left(\frac{7}{12}\right)$	$1$	$-1$

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
$3$	−1.19941	+	1.24957i	−0.692477	+	0.721440i
$3$	−1.19941	+	1.24957i	−0.692477	+	0.721440i
$4$		0			0
$5$	1.44076	−	1.44076i	0.644328	−	0.644328i	−0.307289	−	0.951616i	$-0.599422\pi$
$5$	1.44076	−	1.44076i	0.644328	−	0.644328i	0.951616	+	0.307289i	$0.0994217\pi$
$6$		0			0
$7$	0.918532	+	3.42801i	0.347172	+	1.29567i	0.890054	+	0.455856i	$0.150667\pi$
$7$	0.918532	+	3.42801i	0.347172	+	1.29567i	−0.542881	+	0.839809i	$0.682667\pi$
$8$		0			0
$9$	−0.122851	−	2.99748i	−0.0409503	−	0.999161i
$10$		0			0
$11$	0.0392149	−	0.146352i	0.0118237	−	0.0441267i	−0.959762	−	0.280815i	$-0.909395\pi$
$11$	0.0392149	−	0.146352i	0.0118237	−	0.0441267i	0.971586	+	0.236688i	$0.0760620\pi$
$12$		0			0
$13$	1.92366	+	3.04952i	0.533526	+	0.845784i
$13$	1.92366	+	3.04952i	0.533526	+	0.845784i
$14$		0			0
$15$	0.0722747	+	3.52839i	0.0186612	+	0.911026i
$16$		0			0
$17$	−1.60706	+	2.78351i	−0.389770	+	0.675101i	−0.992418	−	0.122905i	$-0.960779\pi$
$17$	−1.60706	+	2.78351i	−0.389770	+	0.675101i	0.602648	+	0.798007i	$0.294112\pi$
$18$		0			0
$19$	1.90607	−	0.510729i	0.437282	−	0.117169i	−0.0334605	−	0.999440i	$-0.510653\pi$
$19$	1.90607	−	0.510729i	0.437282	−	0.117169i	0.470742	+	0.882271i	$0.343986\pi$
$20$		0			0
$21$	−5.38523	−	2.96380i	−1.17515	−	0.646755i
$22$		0			0
$23$	4.19640	+	7.26837i	0.875009	+	1.51556i	0.856753	+	0.515727i	$0.172478\pi$
$23$	4.19640	+	7.26837i	0.875009	+	1.51556i	0.0182561	+	0.999833i	$0.494189\pi$
$24$		0			0
$25$			0.848416i			0.169683i
$26$		0			0
$27$	3.89291	+	3.44169i	0.749192	+	0.662353i
$28$		0			0
$29$	−0.0238088	+	0.0137460i	−0.00442119	+	0.00255258i	−0.502209	−	0.864746i	$-0.667479\pi$
$29$	−0.0238088	+	0.0137460i	−0.00442119	+	0.00255258i	0.497788	+	0.867299i	$0.334146\pi$
$30$		0			0
$31$	−5.54595	−	5.54595i	−0.996083	−	0.996083i	0.00390967	−	0.999992i	$-0.498756\pi$
$31$	−5.54595	−	5.54595i	−0.996083	−	0.996083i	−0.999992	+	0.00390967i	$0.998756\pi$
$32$		0			0
$33$	0.135842	+	0.224537i	0.0236471	+	0.0390869i
$34$		0			0
$35$	6.26232	+	3.61555i	1.05853	+	0.611140i
$36$		0			0
$37$	3.84429	+	1.03007i	0.631997	+	0.169343i	0.560576	−	0.828103i	$-0.310580\pi$
$37$	3.84429	+	1.03007i	0.631997	+	0.169343i	0.0714212	+	0.997446i	$0.477247\pi$
$38$		0			0
$39$	−6.11783	−	1.25387i	−0.979637	−	0.200779i
$40$		0			0
$41$	2.53382	+	0.678935i	0.395716	+	0.106032i	0.451189	−	0.892428i	$-0.351000\pi$
$41$	2.53382	+	0.678935i	0.395716	+	0.106032i	−0.0554728	+	0.998460i	$0.517667\pi$
$42$		0			0
$43$	−5.90548	−	3.40953i	−0.900577	−	0.519948i	−0.0231896	−	0.999731i	$-0.507382\pi$
$43$	−5.90548	−	3.40953i	−0.900577	−	0.519948i	−0.877387	+	0.479783i	$0.840715\pi$
$44$		0			0
$45$	−4.49566	−	4.14166i	−0.670173	−	0.617402i
$46$		0			0
$47$	−4.77303	−	4.77303i	−0.696218	−	0.696218i	0.267374	−	0.963593i	$-0.413844\pi$
$47$	−4.77303	−	4.77303i	−0.696218	−	0.696218i	−0.963593	+	0.267374i	$0.913844\pi$
$48$		0			0
$49$	−4.84536	+	2.79747i	−0.692194	+	0.399638i
$50$		0			0
$51$	−1.55068	−	5.34670i	−0.217138	−	0.748688i
$52$		0			0
$53$		−	13.3755i		−	1.83726i	−0.395119	−	0.918630i	$-0.629297\pi$
$53$		−	13.3755i		−	1.83726i	0.395119	−	0.918630i	$-0.370703\pi$
$54$		0			0
$55$	−0.154359	−	0.267357i	−0.0208137	−	0.0360504i
$56$		0			0
$57$	−1.64796	+	2.99434i	−0.218277	+	0.396610i
$58$		0			0
$59$	8.09489	−	2.16902i	1.05386	−	0.282382i	0.310017	−	0.950731i	$-0.399665\pi$
$59$	8.09489	−	2.16902i	1.05386	−	0.282382i	0.743848	+	0.668349i	$0.232999\pi$
$60$		0			0
$61$	6.61697	−	11.4609i	0.847216	−	1.46742i	−0.0364673	−	0.999335i	$-0.511610\pi$
$61$	6.61697	−	11.4609i	0.847216	−	1.46742i	0.883683	−	0.468086i	$-0.155056\pi$
$62$		0			0
$63$	10.1626	−	3.17442i	1.28036	−	0.399939i
$64$		0			0
$65$	7.16515	+	1.62210i	0.888728	+	0.201196i
$66$		0			0
$67$	1.18370	−	4.41762i	0.144612	−	0.539698i	−0.855161	−	0.518363i	$-0.826542\pi$
$67$	1.18370	−	4.41762i	0.144612	−	0.539698i	0.999772	−	0.0213353i	$-0.00679175\pi$
$68$		0			0
$69$	−14.1155	−	3.47404i	−1.69931	−	0.418225i
$70$		0			0
$71$	1.29021	+	4.81512i	0.153119	+	0.571449i	0.999259	+	0.0384882i	$0.0122542\pi$
$71$	1.29021	+	4.81512i	0.153119	+	0.571449i	−0.846140	+	0.532961i	$0.821079\pi$
$72$		0			0
$73$	6.22036	−	6.22036i	0.728038	−	0.728038i	−0.242191	−	0.970229i	$-0.577866\pi$
$73$	6.22036	−	6.22036i	0.728038	−	0.728038i	0.970229	+	0.242191i	$0.0778660\pi$
$74$		0			0
$75$	−1.06016	−	1.01760i	−0.122416	−	0.117502i
$76$		0			0
$77$	0.537715			0.0612784
$78$		0			0
$79$	−13.3852			−1.50596			−0.752979	−	0.658045i	$-0.771384\pi$
$79$	−13.3852			−1.50596			−0.752979	+	0.658045i	$0.771384\pi$
$80$		0			0
$81$	−8.96982	+	0.736488i	−0.996646	+	0.0818320i
$82$		0			0
$83$	1.94085	−	1.94085i	0.213036	−	0.213036i	−0.592520	−	0.805556i	$-0.701867\pi$
$83$	1.94085	−	1.94085i	0.213036	−	0.213036i	0.805556	+	0.592520i	$0.201867\pi$
$84$		0			0
$85$	1.69499	+	6.32577i	0.183847	+	0.686126i
$86$		0			0
$87$	0.0113798	−	0.0462379i	0.00122005	−	0.00495722i
$88$		0			0
$89$	−2.87300	+	10.7222i	−0.304537	+	1.13655i	0.628805	+	0.777563i	$0.283544\pi$
$89$	−2.87300	+	10.7222i	−0.304537	+	1.13655i	−0.933343	+	0.358987i	$0.883122\pi$
$90$		0			0
$91$	−8.68683	+	9.39538i	−0.910627	+	0.984904i
$92$		0			0
$93$	13.5819	−	0.278209i	1.40838	−	0.0288489i
$94$		0			0
$95$	2.01035	−	3.48203i	0.206257	−	0.357248i
$96$		0			0
$97$	−17.7357	+	4.75227i	−1.80079	+	0.482520i	−0.994101	−	0.108458i	$-0.965409\pi$
$97$	−17.7357	+	4.75227i	−1.80079	+	0.482520i	−0.806687	+	0.590978i	$0.798742\pi$
$98$		0			0
$99$	−0.443505	−	0.0995664i	−0.0445739	−	0.0100068i
$100$		0			0
$101$	−0.226814	−	0.392853i	−0.0225688	−	0.0390903i	0.854520	−	0.519418i	$-0.173851\pi$
$101$	−0.226814	−	0.392853i	−0.0225688	−	0.0390903i	−0.877089	+	0.480327i	$0.840518\pi$
$102$		0			0
$103$		−	10.0525i		−	0.990499i	−0.868751	−	0.495250i	$-0.835077\pi$
$103$		−	10.0525i		−	0.990499i	0.868751	−	0.495250i	$-0.164923\pi$
$104$		0			0
$105$	−12.0290	+	3.48870i	−1.17391	+	0.340462i
$106$		0			0
$107$	−11.4165	+	6.59132i	−1.10368	+	0.637207i	−0.937184	−	0.348836i	$-0.886577\pi$
$107$	−11.4165	+	6.59132i	−1.10368	+	0.637207i	−0.166492	+	0.986043i	$0.553244\pi$
$108$		0			0
$109$	4.43289	+	4.43289i	0.424594	+	0.424594i	0.886782	−	0.462188i	$-0.152936\pi$
$109$	4.43289	+	4.43289i	0.424594	+	0.424594i	−0.462188	+	0.886782i	$0.652936\pi$
$110$		0			0
$111$	−5.89801	+	3.56823i	−0.559814	+	0.338681i
$112$		0			0
$113$	5.33256	+	3.07875i	0.501645	+	0.289625i	0.729393	−	0.684095i	$-0.239803\pi$
$113$	5.33256	+	3.07875i	0.501645	+	0.289625i	−0.227748	+	0.973720i	$0.573136\pi$
$114$		0			0
$115$	16.5180	+	4.42598i	1.54031	+	0.412725i
$116$		0			0
$117$	8.90455	−	6.14076i	0.823226	−	0.567714i
$118$		0			0
$119$	−11.0180	−	2.95228i	−1.01002	−	0.270635i
$120$		0			0
$121$	9.50640	+	5.48852i	0.864218	+	0.498957i
$122$		0			0
$123$	−3.88746	+	2.35187i	−0.350520	+	0.212061i
$124$		0			0
$125$	8.42617	+	8.42617i	0.753659	+	0.753659i
$126$		0			0
$127$	8.15516	−	4.70839i	0.723654	−	0.417802i	−0.0924423	−	0.995718i	$-0.529467\pi$
$127$	8.15516	−	4.70839i	0.723654	−	0.417802i	0.816096	+	0.577916i	$0.196134\pi$
$128$		0			0
$129$	11.3435	−	3.28990i	0.998740	−	0.289659i
$130$		0			0
$131$		−	14.8806i		−	1.30012i	−0.759881	−	0.650062i	$-0.774743\pi$
$131$		−	14.8806i		−	1.30012i	0.759881	−	0.650062i	$-0.225257\pi$
$132$		0			0
$133$	3.50157	+	6.06489i	0.303624	+	0.525893i
$134$		0			0
$135$	10.5674	−	0.650108i	0.909498	−	0.0559524i
$136$		0			0
$137$	−8.69061	+	2.32864i	−0.742489	+	0.198949i	−0.610184	−	0.792260i	$-0.708905\pi$
$137$	−8.69061	+	2.32864i	−0.742489	+	0.198949i	−0.132305	+	0.991209i	$0.542238\pi$
$138$		0			0
$139$	−0.0458925	+	0.0794881i	−0.00389255	+	0.00674210i	−0.867965	−	0.496625i	$-0.834572\pi$
$139$	−0.0458925	+	0.0794881i	−0.00389255	+	0.00674210i	0.864073	+	0.503367i	$0.167906\pi$
$140$		0			0
$141$	11.6890	−	0.239436i	0.984395	−	0.0201641i
$142$		0			0
$143$	0.521738	−	0.161944i	0.0436299	−	0.0135424i
$144$		0			0
$145$	−0.0144981	+	0.0541076i	−0.00120400	+	0.00449339i
$146$		0			0
$147$	2.31592	−	9.40992i	0.191014	−	0.776117i
$148$		0			0
$149$	−3.50013	−	13.0627i	−0.286742	−	1.07014i	−0.947557	−	0.319587i	$-0.896456\pi$
$149$	−3.50013	−	13.0627i	−0.286742	−	1.07014i	0.660815	−	0.750549i	$-0.270211\pi$
$150$		0			0
$151$	8.24642	−	8.24642i	0.671084	−	0.671084i	−0.286882	−	0.957966i	$-0.592619\pi$
$151$	8.24642	−	8.24642i	0.671084	−	0.671084i	0.957966	+	0.286882i	$0.0926187\pi$
$152$		0			0
$153$	8.54097	+	4.47519i	0.690496	+	0.361797i
$154$		0			0
$155$	−15.9808			−1.28361
$156$		0			0
$157$	−5.99636			−0.478562			−0.239281	−	0.970950i	$-0.576912\pi$
$157$	−5.99636			−0.478562			−0.239281	+	0.970950i	$0.576912\pi$
$158$		0			0
$159$	16.7136	+	16.0426i	1.32547	+	1.27226i
$160$		0			0
$161$	−21.0615	+	21.0615i	−1.65988	+	1.65988i
$162$		0			0
$163$	3.72133	+	13.8882i	0.291477	+	1.08781i	0.943975	+	0.330016i	$0.107054\pi$
$163$	3.72133	+	13.8882i	0.291477	+	1.08781i	−0.652499	+	0.757790i	$0.726279\pi$
$164$		0			0
$165$	0.519220	+	0.127788i	0.0404213	+	0.00994826i
$166$		0			0
$167$	1.36831	−	5.10660i	0.105883	−	0.395161i	−0.892561	−	0.450927i	$-0.851094\pi$
$167$	1.36831	−	5.10660i	0.105883	−	0.395161i	0.998444	+	0.0557663i	$0.0177602\pi$
$168$		0			0
$169$	−5.59910	+	11.7324i	−0.430700	+	0.902495i
$170$		0			0
$171$	−1.76506	−	5.65066i	−0.134978	−	0.432117i
$172$		0			0
$173$	3.05984	−	5.29979i	0.232635	−	0.402936i	−0.725948	−	0.687750i	$-0.758599\pi$
$173$	3.05984	−	5.29979i	0.232635	−	0.402936i	0.958583	+	0.284814i	$0.0919319\pi$
$174$		0			0
$175$	−2.90838	+	0.779297i	−0.219853	+	0.0589093i
$176$		0			0
$177$	−6.99872	+	12.7167i	−0.526056	+	0.955843i
$178$		0			0
$179$	−10.1144	−	17.5187i	−0.755986	−	1.30941i	−0.944883	−	0.327409i	$-0.893825\pi$
$179$	−10.1144	−	17.5187i	−0.755986	−	1.30941i	0.188897	−	0.981997i	$-0.439509\pi$
$180$		0			0
$181$		−	4.39016i		−	0.326318i	−0.986600	−	0.163159i	$-0.947832\pi$
$181$		−	4.39016i		−	0.326318i	0.986600	−	0.163159i	$-0.0521683\pi$
$182$		0			0
$183$	6.38480	+	22.0147i	0.471978	+	1.62737i
$184$		0			0
$185$	7.02279	−	4.05461i	0.516326	−	0.298101i
$186$		0			0
$187$	0.344352	+	0.344352i	0.0251815	+	0.0251815i
$188$		0			0
$189$	−8.22237	+	16.5062i	−0.598089	+	1.20065i
$190$		0			0
$191$	12.3027	+	7.10298i	0.890194	+	0.513954i	0.874006	−	0.485915i	$-0.161514\pi$
$191$	12.3027	+	7.10298i	0.890194	+	0.513954i	0.0161878	+	0.999869i	$0.494847\pi$
$192$		0			0
$193$	14.1074	+	3.78006i	1.01547	+	0.272095i	0.727914	−	0.685668i	$-0.240490\pi$
$193$	14.1074	+	3.78006i	1.01547	+	0.272095i	0.287558	+	0.957763i	$0.407157\pi$
$194$		0			0
$195$	−10.6208	+	7.00781i	−0.760575	+	0.501839i
$196$		0			0
$197$	−0.668505	−	0.179125i	−0.0476290	−	0.0127622i	0.234926	−	0.972013i	$-0.424515\pi$
$197$	−0.668505	−	0.179125i	−0.0476290	−	0.0127622i	−0.282555	+	0.959251i	$0.591182\pi$
$198$		0			0
$199$	−13.8158	−	7.97656i	−0.979377	−	0.565443i	−0.0772948	−	0.997008i	$-0.524628\pi$
$199$	−13.8158	−	7.97656i	−0.979377	−	0.565443i	−0.902082	+	0.431565i	$0.857962\pi$
$200$		0			0
$201$	4.10039	+	6.77764i	0.289219	+	0.478058i
$202$		0			0
$203$	−0.0689907	−	0.0689907i	−0.00484220	−	0.00484220i
$204$		0			0
$205$	4.62881	−	2.67245i	0.323290	−	0.186652i
$206$		0			0
$207$	21.2713	−	13.4716i	1.47846	−	0.936338i
$208$		0			0
$209$		−	0.298985i		−	0.0206812i
$210$		0			0
$211$	3.28525	+	5.69022i	0.226166	+	0.391731i	0.956669	−	0.291179i	$-0.0940475\pi$
$211$	3.28525	+	5.69022i	0.226166	+	0.391731i	−0.730503	+	0.682910i	$0.760714\pi$
$212$		0			0
$213$	−7.56431	−	4.16308i	−0.518298	−	0.285249i
$214$		0			0
$215$	−13.4207	+	3.59607i	−0.915284	+	0.245250i
$216$		0			0
$217$	13.9174	−	24.1057i	0.944777	−	1.63640i
$218$		0			0
$219$	0.312040	+	15.2335i	0.0210857	+	1.02938i
$220$		0			0
$221$	−11.5798	+	0.453757i	−0.778942	+	0.0305230i
$222$		0			0
$223$	1.79575	−	6.70184i	0.120252	−	0.448788i	−0.879374	−	0.476132i	$-0.842038\pi$
$223$	1.79575	−	6.70184i	0.120252	−	0.448788i	0.999626	+	0.0273441i	$0.00870497\pi$
$224$		0			0
$225$	2.54311	−	0.104229i	0.169541	−	0.00694859i
$226$		0			0
$227$	3.00698	+	11.2222i	0.199580	+	0.744843i	0.991033	+	0.133614i	$0.0426581\pi$
$227$	3.00698	+	11.2222i	0.199580	+	0.744843i	−0.791453	+	0.611230i	$0.790675\pi$
$228$		0			0
$229$	−18.7056	+	18.7056i	−1.23610	+	1.23610i	−0.274523	+	0.961580i	$0.588520\pi$
$229$	−18.7056	+	18.7056i	−1.23610	+	1.23610i	−0.961580	+	0.274523i	$0.911480\pi$
$230$		0			0
$231$	−0.644939	+	0.671913i	−0.0424339	+	0.0442086i
$232$		0			0
$233$	4.96187			0.325063			0.162531	−	0.986703i	$-0.448034\pi$
$233$	4.96187			0.325063			0.162531	+	0.986703i	$0.448034\pi$
$234$		0			0
$235$	−13.7536			−0.897186
$236$		0			0
$237$	16.0543	−	16.7258i	1.04284	−	1.08646i
$238$		0			0
$239$	19.8475	−	19.8475i	1.28382	−	1.28382i	0.345352	−	0.938473i	$-0.387760\pi$
$239$	19.8475	−	19.8475i	1.28382	−	1.28382i	0.938473	−	0.345352i	$-0.112240\pi$
$240$		0			0
$241$	1.59927	+	5.96857i	0.103018	+	0.384469i	0.998113	−	0.0614071i	$-0.0195588\pi$
$241$	1.59927	+	5.96857i	0.103018	+	0.384469i	−0.895095	+	0.445876i	$0.852892\pi$
$242$		0			0
$243$	9.83816	−	12.0918i	0.631118	−	0.775687i
$244$		0			0
$245$	−2.95052	+	11.0115i	−0.188502	+	0.703498i
$246$		0			0
$247$	5.22409	+	4.83012i	0.332401	+	0.307333i
$248$		0			0
$249$	0.0973611	+	4.75309i	0.00617001	+	0.301215i
$250$		0			0
$251$	2.16622	−	3.75201i	0.136731	−	0.236825i	−0.789526	−	0.613717i	$-0.789674\pi$
$251$	2.16622	−	3.75201i	0.136731	−	0.236825i	0.926257	+	0.376892i	$0.123007\pi$
$252$		0			0
$253$	1.22830	−	0.329122i	0.0772226	−	0.0206917i
$254$		0			0
$255$	−9.93747	−	5.46917i	−0.622309	−	0.342492i
$256$		0			0
$257$	−10.6302	−	18.4120i	−0.663093	−	1.14851i	−0.979799	−	0.199986i	$-0.935910\pi$
$257$	−10.6302	−	18.4120i	−0.663093	−	1.14851i	0.316706	−	0.948524i	$-0.397423\pi$
$258$		0			0
$259$			14.1244i			0.877648i
$260$		0			0
$261$	0.0441285	+	0.0696779i	0.00273148	+	0.00431295i
$262$		0			0
$263$	−15.0017	+	8.66123i	−0.925044	+	0.534074i	−0.885241	−	0.465133i	$-0.846006\pi$
$263$	−15.0017	+	8.66123i	−0.925044	+	0.534074i	−0.0398032	+	0.999208i	$0.512673\pi$
$264$		0			0
$265$	−19.2708	−	19.2708i	−1.18380	−	1.18380i
$266$		0			0
$267$	−9.95223	−	16.4503i	−0.609067	−	1.00674i
$268$		0			0
$269$	−27.3500	−	15.7905i	−1.66756	−	0.962766i	−0.968948	−	0.247266i	$-0.920468\pi$
$269$	−27.3500	−	15.7905i	−1.66756	−	0.962766i	−0.698612	−	0.715500i	$-0.746199\pi$
$270$		0			0
$271$	−8.18818	−	2.19402i	−0.497397	−	0.133277i	0.00139487	−	0.999999i	$-0.499556\pi$
$271$	−8.18818	−	2.19402i	−0.497397	−	0.133277i	−0.498792	+	0.866722i	$0.666223\pi$
$272$		0			0
$273$	−1.32116	−	22.1237i	−0.0799601	−	1.33899i
$274$		0			0
$275$	0.124167	+	0.0332705i	0.00748757	+	0.00200629i
$276$		0			0
$277$	−16.6692	−	9.62397i	−1.00156	−	0.578248i	−0.0928470	−	0.995680i	$-0.529597\pi$
$277$	−16.6692	−	9.62397i	−1.00156	−	0.578248i	−0.908708	+	0.417432i	$0.862930\pi$
$278$		0			0
$279$	−15.9426	+	17.3052i	−0.954457	+	1.03604i
$280$		0			0
$281$	16.6481	+	16.6481i	0.993141	+	0.993141i	0.999977	−	0.00683587i	$-0.00217594\pi$
$281$	16.6481	+	16.6481i	0.993141	+	0.993141i	−0.00683587	+	0.999977i	$0.502176\pi$
$282$		0			0
$283$	8.15184	−	4.70646i	0.484576	−	0.279770i	−0.237745	−	0.971328i	$-0.576408\pi$
$283$	8.15184	−	4.70646i	0.484576	−	0.279770i	0.722322	+	0.691557i	$0.243075\pi$
$284$		0			0
$285$	1.93981	+	6.68843i	0.114905	+	0.396189i
$286$		0			0
$287$			9.30958i			0.549527i
$288$		0			0
$289$	3.33470	+	5.77587i	0.196159	+	0.339757i
$290$		0			0
$291$	15.3340	−	27.8619i	0.898896	−	1.63329i
$292$		0			0
$293$	25.3285	−	6.78675i	1.47971	−	0.396486i	0.573460	−	0.819234i	$-0.305601\pi$
$293$	25.3285	−	6.78675i	1.47971	−	0.396486i	0.906248	+	0.422747i	$0.138934\pi$
$294$		0			0
$295$	8.53776	−	14.7878i	0.497088	−	0.860981i
$296$		0			0
$297$	0.656358	−	0.434770i	0.0380857	−	0.0252279i
$298$		0			0
$299$	−14.0926	+	26.7788i	−0.814996	+	1.54866i
$300$		0			0
$301$	6.26352	−	23.3758i	0.361023	−	1.34736i
$302$		0			0
$303$	0.762939	+	0.187770i	0.0438297	+	0.0107871i
$304$		0			0
$305$	−6.97898	−	26.0459i	−0.399615	−	1.49138i
$306$		0			0
$307$	−15.7215	+	15.7215i	−0.897271	+	0.897271i	−0.995194	−	0.0979227i	$-0.968780\pi$
$307$	−15.7215	+	15.7215i	−0.897271	+	0.897271i	0.0979227	+	0.995194i	$0.468780\pi$
$308$		0			0
$309$	12.5613	+	12.0570i	0.714585	+	0.685898i
$310$		0			0
$311$	−4.34210			−0.246218			−0.123109	−	0.992393i	$-0.539286\pi$
$311$	−4.34210			−0.246218			−0.123109	+	0.992393i	$0.539286\pi$
$312$		0			0
$313$	18.6341			1.05326			0.526630	−	0.850095i	$-0.323455\pi$
$313$	18.6341			1.05326			0.526630	+	0.850095i	$0.323455\pi$
$314$		0			0
$315$	10.0682	−	19.2154i	0.567281	−	1.08266i
$316$		0			0
$317$	−9.07889	+	9.07889i	−0.509921	+	0.509921i	−0.914502	−	0.404581i	$-0.867417\pi$
$317$	−9.07889	+	9.07889i	−0.509921	+	0.509921i	0.404581	+	0.914502i	$0.367417\pi$
$318$		0			0
$319$	0.00107810	+	0.00402352i	6.03619e−5	+	0.000225274i
$320$		0			0
$321$	5.45671	−	22.1714i	0.304564	−	1.23749i
$322$		0			0
$323$	−1.64155	+	6.12634i	−0.0913382	+	0.340879i
$324$		0			0
$325$	−2.58726	+	1.63206i	−0.143515	+	0.0905304i
$326$		0			0
$327$	−10.8560	+	0.222372i	−0.600340	+	0.0122972i
$328$		0			0
$329$	11.9778	−	20.7462i	0.660358	−	1.14377i
$330$		0			0
$331$	11.3587	−	3.04354i	0.624328	−	0.167288i	0.0672335	−	0.997737i	$-0.478583\pi$
$331$	11.3587	−	3.04354i	0.624328	−	0.167288i	0.557095	+	0.830449i	$0.311916\pi$
$332$		0			0
$333$	2.61535	−	11.6497i	0.143321	−	0.638401i
$334$		0			0
$335$	−4.65931	−	8.07016i	−0.254565	−	0.440920i
$336$		0			0
$337$		−	19.7236i		−	1.07441i	−0.843451	−	0.537206i	$-0.819480\pi$
$337$		−	19.7236i		−	1.07441i	0.843451	−	0.537206i	$-0.180520\pi$
$338$		0			0
$339$	−10.2430	+	2.97073i	−0.556324	+	0.161348i
$340$		0			0
$341$	−1.02914	+	0.594177i	−0.0557313	+	0.0321765i
$342$		0			0
$343$	3.52597	+	3.52597i	0.190384	+	0.190384i
$344$		0			0
$345$	−25.3423	+	15.3318i	−1.36439	+	0.825438i
$346$		0			0
$347$	30.5245	+	17.6233i	1.63864	+	0.946071i	0.981301	+	0.192478i	$0.0616525\pi$
$347$	30.5245	+	17.6233i	1.63864	+	0.946071i	0.657342	+	0.753593i	$0.271681\pi$
$348$		0			0
$349$	−22.9572	−	6.15136i	−1.22887	−	0.329275i	−0.414730	−	0.909944i	$-0.636124\pi$
$349$	−22.9572	−	6.15136i	−1.22887	−	0.329275i	−0.814139	+	0.580670i	$0.802791\pi$
$350$		0			0
$351$	−3.00686	+	18.4921i	−0.160494	+	0.987037i
$352$		0			0
$353$	−7.81588	−	2.09426i	−0.415997	−	0.111466i	0.0447484	−	0.998998i	$-0.485751\pi$
$353$	−7.81588	−	2.09426i	−0.415997	−	0.111466i	−0.460746	+	0.887532i	$0.652418\pi$
$354$		0			0
$355$	8.79631	+	5.07855i	0.466860	+	0.269542i
$356$		0			0
$357$	16.9042	−	10.2268i	0.894665	−	0.541262i
$358$		0			0
$359$	9.77619	+	9.77619i	0.515968	+	0.515968i	0.916349	−	0.400381i	$-0.131122\pi$
$359$	9.77619	+	9.77619i	0.515968	+	0.515968i	−0.400381	+	0.916349i	$0.631122\pi$
$360$		0			0
$361$	−13.0822	+	7.55303i	−0.688539	+	0.397528i
$362$		0			0
$363$	−18.2603	+	5.29595i	−0.958418	+	0.277965i
$364$		0			0
$365$		−	17.9241i		−	0.938190i
$366$		0			0
$367$	6.84234	+	11.8513i	0.357167	+	0.618632i	0.987486	−	0.157704i	$-0.0504093\pi$
$367$	6.84234	+	11.8513i	0.357167	+	0.618632i	−0.630319	+	0.776336i	$0.717076\pi$
$368$		0			0
$369$	1.72382	−	7.67850i	0.0897382	−	0.399727i
$370$		0			0
$371$	45.8512	−	12.2858i	2.38047	−	0.637846i
$372$		0			0
$373$	−1.97549	+	3.42164i	−0.102287	+	0.177166i	−0.912626	−	0.408794i	$-0.865949\pi$
$373$	−1.97549	+	3.42164i	−0.102287	+	0.177166i	0.810340	+	0.585960i	$0.199283\pi$
$374$		0			0
$375$	−20.6355	+	0.422692i	−1.06561	+	0.0218277i
$376$		0			0
$377$	−0.0877188	−	0.0461628i	−0.00451775	−	0.00237751i
$378$		0			0
$379$	−4.23059	+	15.7888i	−0.217311	+	0.811014i	0.768030	+	0.640414i	$0.221237\pi$
$379$	−4.23059	+	15.7888i	−0.217311	+	0.811014i	−0.985340	+	0.170600i	$0.945429\pi$
$380$		0			0
$381$	−3.89789	+	15.8377i	−0.199695	+	0.811391i
$382$		0			0
$383$	−5.85081	−	21.8355i	−0.298963	−	1.11574i	−0.938019	−	0.346584i	$-0.887342\pi$
$383$	−5.85081	−	21.8355i	−0.298963	−	1.11574i	0.639057	−	0.769160i	$-0.279325\pi$
$384$		0			0
$385$	0.774719	−	0.774719i	0.0394834	−	0.0394834i
$386$		0			0
$387$	−9.49451	+	18.1204i	−0.482633	+	0.921114i
$388$		0			0
$389$	−1.80850			−0.0916947			−0.0458474	−	0.998948i	$-0.514599\pi$
$389$	−1.80850			−0.0916947			−0.0458474	+	0.998948i	$0.514599\pi$
$390$		0			0
$391$	−26.9755			−1.36421
$392$		0			0
$393$	18.5943	+	17.8479i	0.937961	+	0.900306i
$394$		0			0
$395$	−19.2849	+	19.2849i	−0.970331	+	0.970331i
$396$		0			0
$397$	−1.04738	−	3.90888i	−0.0525665	−	0.196181i	0.934649	−	0.355572i	$-0.115714\pi$
$397$	−1.04738	−	3.90888i	−0.0525665	−	0.196181i	−0.987216	+	0.159391i	$0.949047\pi$
$398$		0			0
$399$	−11.7783	−	2.89881i	−0.589653	−	0.145122i
$400$		0			0
$401$	−0.540214	+	2.01611i	−0.0269770	+	0.100680i	−0.978102	−	0.208128i	$-0.933263\pi$
$401$	−0.540214	+	2.01611i	−0.0269770	+	0.100680i	0.951125	+	0.308807i	$0.0999298\pi$
$402$		0			0
$403$	6.24398	−	27.5810i	0.311035	−	1.37391i
$404$		0			0
$405$	−11.8623	+	13.9845i	−0.589440	+	0.694894i
$406$		0			0
$407$	0.301506	−	0.522224i	0.0149451	−	0.0258857i
$408$		0			0
$409$	−8.15394	+	2.18484i	−0.403186	+	0.108033i	−0.454713	−	0.890638i	$-0.650258\pi$
$409$	−8.15394	+	2.18484i	−0.403186	+	0.108033i	0.0515263	+	0.998672i	$0.483591\pi$
$410$		0			0
$411$	7.51377	−	13.6525i	0.370627	−	0.673429i
$412$		0			0
$413$	14.8708	+	25.7570i	0.731746	+	1.26742i
$414$		0			0
$415$		−	5.59259i		−	0.274529i
$416$		0			0
$417$	−0.0442823	−	0.152684i	−0.00216851	−	0.00747699i
$418$		0			0
$419$	0.585414	−	0.337989i	0.0285993	−	0.0165118i	−0.485632	−	0.874163i	$-0.661411\pi$
$419$	0.585414	−	0.337989i	0.0285993	−	0.0165118i	0.514232	+	0.857651i	$0.328077\pi$
$420$		0			0
$421$	6.17588	+	6.17588i	0.300994	+	0.300994i	0.841403	−	0.540409i	$-0.181730\pi$
$421$	6.17588	+	6.17588i	0.300994	+	0.300994i	−0.540409	+	0.841403i	$0.681730\pi$
$422$		0			0
$423$	−13.7207	+	14.8935i	−0.667124	+	0.724145i
$424$		0			0
$425$	−2.36158	−	1.36346i	−0.114553	−	0.0661374i
$426$		0			0
$427$	45.3660	+	12.1558i	2.19542	+	0.588260i
$428$		0			0
$429$	−0.423415	+	0.846185i	−0.0204427	+	0.0408542i
$430$		0			0
$431$	9.78631	+	2.62223i	0.471390	+	0.126309i	0.486691	−	0.873574i	$-0.338204\pi$
$431$	9.78631	+	2.62223i	0.471390	+	0.126309i	−0.0153006	+	0.999883i	$0.504871\pi$
$432$		0			0
$433$	−11.0298	−	6.36809i	−0.530061	−	0.306031i	0.210980	−	0.977490i	$-0.432334\pi$
$433$	−11.0298	−	6.36809i	−0.530061	−	0.306031i	−0.741041	+	0.671460i	$0.765668\pi$
$434$		0			0
$435$	−0.0502222	−	0.0830134i	−0.00240797	−	0.00398019i
$436$		0			0
$437$	11.7108	+	11.7108i	0.560203	+	0.560203i
$438$		0			0
$439$	3.07428	−	1.77494i	0.146727	−	0.0847131i	−0.424839	−	0.905269i	$-0.639669\pi$
$439$	3.07428	−	1.77494i	0.146727	−	0.0847131i	0.571566	+	0.820556i	$0.306336\pi$
$440$		0			0
$441$	8.98063	+	14.1802i	0.427649	+	0.675248i
$442$		0			0
$443$			13.7327i			0.652461i	0.945290	+	0.326231i	$0.105779\pi$
$443$			13.7327i			0.652461i	−0.945290	+	0.326231i	$0.894221\pi$
$444$		0			0
$445$	11.3088	+	19.5874i	0.536088	+	0.928532i
$446$		0			0
$447$	20.5208	+	11.2938i	0.970601	+	0.534178i
$448$		0			0
$449$	12.7370	−	3.41288i	0.601098	−	0.161064i	0.0545774	−	0.998510i	$-0.482619\pi$
$449$	12.7370	−	3.41288i	0.601098	−	0.161064i	0.546520	+	0.837446i	$0.315952\pi$
$450$		0			0
$451$	0.198727	−	0.344205i	0.00935769	−	0.0162080i
$452$		0			0
$453$	0.413675	+	20.1953i	0.0194362	+	0.948857i
$454$		0			0
$455$	1.02086	+	26.0521i	0.0478586	+	1.22134i
$456$		0			0
$457$	−4.80678	+	17.9391i	−0.224852	+	0.839157i	0.757612	+	0.652705i	$0.226366\pi$
$457$	−4.80678	+	17.9391i	−0.224852	+	0.839157i	−0.982464	+	0.186453i	$0.940301\pi$
$458$		0			0
$459$	−15.8361	+	5.30497i	−0.739168	+	0.247615i
$460$		0			0
$461$	1.15902	+	4.32554i	0.0539812	+	0.201460i	0.987650	−	0.156678i	$-0.0500784\pi$
$461$	1.15902	+	4.32554i	0.0539812	+	0.201460i	−0.933669	+	0.358138i	$0.883412\pi$
$462$		0			0
$463$	8.53278	−	8.53278i	0.396552	−	0.396552i	−0.480463	−	0.877015i	$-0.659531\pi$
$463$	8.53278	−	8.53278i	0.396552	−	0.396552i	0.877015	+	0.480463i	$0.159531\pi$
$464$		0			0
$465$	19.1674	−	19.9691i	0.888869	−	0.926045i
$466$		0			0
$467$	−9.69885			−0.448809			−0.224405	−	0.974496i	$-0.572044\pi$
$467$	−9.69885			−0.448809			−0.224405	+	0.974496i	$0.572044\pi$
$468$		0			0
$469$	16.2309			0.749474
$470$		0			0
$471$	7.19207	−	7.49288i	0.331393	−	0.345254i
$472$		0			0
$473$	−0.730573	+	0.730573i	−0.0335918	+	0.0335918i
$474$		0			0
$475$	0.433311	+	1.61714i	0.0198817	+	0.0741994i
$476$		0			0
$477$	−40.0927	+	1.64319i	−1.83572	+	0.0752364i
$478$		0			0
$479$	6.64461	−	24.7980i	0.303600	−	1.13305i	−0.630544	−	0.776154i	$-0.717168\pi$
$479$	6.64461	−	24.7980i	0.303600	−	1.13305i	0.934144	−	0.356897i	$-0.116165\pi$
$480$		0			0
$481$	4.25386	+	13.7047i	0.193959	+	0.624882i
$482$		0			0
$483$	−1.05653	−	51.5791i	−0.0480740	−	2.34693i
$484$		0			0
$485$	−18.7060	+	32.3998i	−0.849397	+	1.47120i
$486$		0			0
$487$	−15.4146	+	4.13033i	−0.698501	+	0.187163i	−0.590559	−	0.806994i	$-0.701093\pi$
$487$	−15.4146	+	4.13033i	−0.698501	+	0.187163i	−0.107942	+	0.994157i	$0.534426\pi$
$488$		0			0
$489$	−21.8176	−	12.0075i	−0.986627	−	0.542998i
$490$		0			0
$491$	−6.38912	−	11.0663i	−0.288337	−	0.499414i	0.685076	−	0.728472i	$-0.259769\pi$
$491$	−6.38912	−	11.0663i	−0.288337	−	0.499414i	−0.973413	+	0.229057i	$0.926436\pi$
$492$		0			0
$493$		−	0.0883630i		−	0.00397967i
$494$		0			0
$495$	−0.782436	+	0.495533i	−0.0351679	+	0.0222726i
$496$		0			0
$497$	−15.3212	+	8.84568i	−0.687248	+	0.396783i
$498$		0			0
$499$	26.5255	+	26.5255i	1.18744	+	1.18744i	0.977772	+	0.209671i	$0.0672393\pi$
$499$	26.5255	+	26.5255i	1.18744	+	1.18744i	0.209671	+	0.977772i	$0.432761\pi$
$500$		0			0
$501$	4.73990	+	7.83468i	0.211763	+	0.350028i
$502$		0			0
$503$	3.29946	+	1.90495i	0.147116	+	0.0849374i	0.571751	−	0.820427i	$-0.306264\pi$
$503$	3.29946	+	1.90495i	0.147116	+	0.0849374i	−0.424635	+	0.905365i	$0.639598\pi$
$504$		0			0
$505$	−0.892792	−	0.239223i	−0.0397287	−	0.0106453i
$506$		0			0
$507$	−7.94491	−	21.0684i	−0.352846	−	0.935682i
$508$		0			0
$509$	37.5368	+	10.0580i	1.66379	+	0.445811i	0.963426	−	0.267975i	$-0.0863545\pi$
$509$	37.5368	+	10.0580i	1.66379	+	0.445811i	0.700364	+	0.713786i	$0.253021\pi$
$510$		0			0
$511$	27.0370	+	15.6098i	1.19605	+	0.690539i
$512$		0			0
$513$	9.17793	+	4.57186i	0.405215	+	0.201853i
$514$		0			0
$515$	−14.4832	−	14.4832i	−0.638206	−	0.638206i
$516$		0			0
$517$	−0.885716	+	0.511368i	−0.0389537	+	0.0224900i
$518$		0			0
$519$	2.95248	+	10.1801i	0.129599	+	0.446856i
$520$		0			0
$521$		−	41.6988i		−	1.82686i	−0.406996	−	0.913430i	$-0.633424\pi$
$521$		−	41.6988i		−	1.82686i	0.406996	−	0.913430i	$-0.366576\pi$
$522$		0			0
$523$	−13.1004	−	22.6906i	−0.572842	−	0.992192i	−0.996272	−	0.0862632i	$-0.972507\pi$
$523$	−13.1004	−	22.6906i	−0.572842	−	0.992192i	0.423430	−	0.905929i	$-0.360826\pi$
$524$		0			0
$525$	2.51454	−	4.56892i	0.109743	−	0.199404i
$526$		0			0
$527$	24.3499	−	6.52455i	1.06070	−	0.284214i
$528$		0			0
$529$	−23.7195	+	41.0833i	−1.03128	+	1.78623i
$530$		0			0
$531$	−7.49606	−	23.9978i	−0.325301	−	1.04142i
$532$		0			0
$533$	2.80377	+	9.03297i	0.121445	+	0.391261i
$534$		0			0
$535$	−6.95194	+	25.9450i	−0.300558	+	1.12170i
$536$		0			0
$537$	34.0221	+	8.37333i	1.46816	+	0.361336i
$538$		0			0
$539$	0.219405	+	0.818830i	0.00945043	+	0.0352695i
$540$		0			0
$541$	14.9894	−	14.9894i	0.644447	−	0.644447i	−0.307199	−	0.951645i	$-0.599392\pi$
$541$	14.9894	−	14.9894i	0.644447	−	0.644447i	0.951645	+	0.307199i	$0.0993916\pi$
$542$		0			0
$543$	5.48581	+	5.26558i	0.235419	+	0.225968i
$544$		0			0
$545$	12.7735			0.547155
$546$		0			0
$547$	23.7046			1.01354			0.506768	−	0.862082i	$-0.330840\pi$
$547$	23.7046			1.01354			0.506768	+	0.862082i	$0.330840\pi$
$548$		0			0
$549$	−35.1668	−	18.4263i	−1.50088	−	0.786414i
$550$		0			0
$551$	−0.0383608	+	0.0383608i	−0.00163422	+	0.00163422i
$552$		0			0
$553$	−12.2948	−	45.8847i	−0.522827	−	1.95122i
$554$		0			0
$555$	−3.35666	+	13.6386i	−0.142482	+	0.578926i
$556$		0			0
$557$	−10.2264	+	38.1653i	−0.433304	+	1.61711i	0.311787	+	0.950152i	$0.399072\pi$
$557$	−10.2264	+	38.1653i	−0.433304	+	1.61711i	−0.745091	+	0.666962i	$0.767594\pi$
$558$		0			0
$559$	−0.962687	−	24.5676i	−0.0407173	−	1.03910i
$560$		0			0
$561$	−0.843309	+	0.0172741i	−0.0356045	+	0.000729315i
$562$		0			0
$563$	8.48383	−	14.6944i	0.357551	−	0.619296i	−0.630000	−	0.776595i	$-0.716945\pi$
$563$	8.48383	−	14.6944i	0.357551	−	0.619296i	0.987551	+	0.157299i	$0.0502785\pi$
$564$		0			0
$565$	12.1187	−	3.24719i	0.509837	−	0.136610i
$566$		0			0
$567$	−10.7637	−	30.0721i	−0.452035	−	1.26291i
$568$		0			0
$569$	−15.2347	−	26.3873i	−0.638674	−	1.10622i	−0.985724	−	0.168369i	$-0.946150\pi$
$569$	−15.2347	−	26.3873i	−0.638674	−	1.10622i	0.347051	−	0.937846i	$-0.387183\pi$
$570$		0			0
$571$			5.03996i			0.210916i	0.994424	+	0.105458i	$0.0336309\pi$
$571$			5.03996i			0.210916i	−0.994424	+	0.105458i	$0.966369\pi$
$572$		0			0
$573$	−23.6316	+	6.85376i	−0.987225	+	0.286320i
$574$		0			0
$575$	−6.16660	+	3.56029i	−0.257165	+	0.148474i
$576$		0			0
$577$	15.8862	+	15.8862i	0.661350	+	0.661350i	0.955698	−	0.294348i	$-0.0951024\pi$
$577$	15.8862	+	15.8862i	0.661350	+	0.661350i	−0.294348	+	0.955698i	$0.595102\pi$
$578$		0			0
$579$	−21.6439	+	13.0943i	−0.899492	+	0.544182i
$580$		0			0
$581$	8.43596	+	4.87051i	0.349983	+	0.202063i
$582$		0			0
$583$	−1.95752	−	0.524516i	−0.0810723	−	0.0217233i
$584$		0			0
$585$	3.98196	−	21.6767i	0.164634	−	0.896221i
$586$		0			0
$587$	−8.06056	−	2.15982i	−0.332695	−	0.0891453i	0.0886047	−	0.996067i	$-0.471759\pi$
$587$	−8.06056	−	2.15982i	−0.332695	−	0.0891453i	−0.421300	+	0.906922i	$0.638426\pi$
$588$		0			0
$589$	−13.4034	−	7.73848i	−0.552279	−	0.318859i
$590$		0			0
$591$	1.02564	−	0.620500i	0.0421892	−	0.0255240i
$592$		0			0
$593$	30.0266	+	30.0266i	1.23305	+	1.23305i	0.962788	+	0.270257i	$0.0871086\pi$
$593$	30.0266	+	30.0266i	1.23305	+	1.23305i	0.270257	+	0.962788i	$0.412891\pi$
$594$		0			0
$595$	−20.1279	+	11.6208i	−0.825163	+	0.476408i
$596$		0			0
$597$	26.5380	−	7.69669i	1.08613	−	0.315004i
$598$		0			0
$599$		−	17.0119i		−	0.695088i	−0.937664	−	0.347544i	$-0.887016\pi$
$599$		−	17.0119i		−	0.695088i	0.937664	−	0.347544i	$-0.112984\pi$
$600$		0			0
$601$	−12.8220	−	22.2084i	−0.523022	−	0.905901i	−0.999641	−	0.0267907i	$-0.991471\pi$
$601$	−12.8220	−	22.2084i	−0.523022	−	0.905901i	0.476619	−	0.879110i	$-0.341862\pi$
$602$		0			0
$603$	−13.3872	−	3.00541i	−0.545168	−	0.122390i
$604$		0			0
$605$	21.6041	−	5.78880i	0.878331	−	0.235348i
$606$		0			0
$607$	−8.99964	+	15.5878i	−0.365284	+	0.632691i	−0.988822	−	0.149103i	$-0.952362\pi$
$607$	−8.99964	+	15.5878i	−0.365284	+	0.632691i	0.623537	+	0.781793i	$0.285695\pi$
$608$		0			0
$609$	0.168957	−	0.00346087i	0.00684647	−	0.000140241i
$610$		0			0
$611$	5.37377	−	23.7371i	0.217400	−	0.960301i
$612$		0			0
$613$	6.40516	−	23.9044i	0.258702	−	0.965489i	−0.707292	−	0.706922i	$-0.750083\pi$
$613$	6.40516	−	23.9044i	0.258702	−	0.965489i	0.965993	−	0.258567i	$-0.0832502\pi$
$614$		0			0
$615$	−2.21242	+	8.98938i	−0.0892133	+	0.362487i
$616$		0			0
$617$	9.32585	+	34.8045i	0.375444	+	1.40118i	0.852694	+	0.522410i	$0.174967\pi$
$617$	9.32585	+	34.8045i	0.375444	+	1.40118i	−0.477250	+	0.878768i	$0.658366\pi$
$618$		0			0
$619$	−8.29611	+	8.29611i	−0.333449	+	0.333449i	−0.853895	−	0.520446i	$-0.825766\pi$
$619$	−8.29611	+	8.29611i	−0.333449	+	0.333449i	0.520446	+	0.853895i	$0.325766\pi$
$620$		0			0
$621$	−8.67926	+	42.7378i	−0.348287	+	1.71501i
$622$		0			0
$623$	−39.3947			−1.57831
$624$		0			0
$625$	20.0381			0.801524
$626$		0			0
$627$	0.373602	+	0.358604i	0.0149202	+	0.0143213i
$628$		0			0
$629$	−9.04524	+	9.04524i	−0.360657	+	0.360657i
$630$		0			0
$631$	5.40401	+	20.1680i	0.215130	+	0.802877i	0.986121	+	0.166030i	$0.0530949\pi$
$631$	5.40401	+	20.1680i	0.215130	+	0.802877i	−0.770990	+	0.636847i	$0.780238\pi$
$632$		0			0
$633$	−11.0507	−	2.71973i	−0.439225	−	0.108100i
$634$		0			0
$635$	4.96598	−	18.5333i	0.197069	−	0.735471i
$636$		0			0
$637$	−17.8517	−	9.39464i	−0.707311	−	0.372229i
$638$		0			0
$639$	14.2747	−	4.45892i	0.564700	−	0.176392i
$640$		0			0
$641$	−5.39416	+	9.34296i	−0.213056	+	0.369025i	−0.952670	−	0.304008i	$-0.901675\pi$
$641$	−5.39416	+	9.34296i	−0.213056	+	0.369025i	0.739613	+	0.673032i	$0.235008\pi$
$642$		0			0
$643$	−8.87352	+	2.37765i	−0.349937	+	0.0937655i	−0.429506	−	0.903064i	$-0.641312\pi$
$643$	−8.87352	+	2.37765i	−0.349937	+	0.0937655i	0.0795687	+	0.996829i	$0.474646\pi$
$644$		0			0
$645$	11.6033	−	21.0832i	0.456881	−	0.830152i
$646$		0			0
$647$	12.5642	+	21.7619i	0.493951	+	0.855549i	0.999976	−	0.00697039i	$-0.00221876\pi$
$647$	12.5642	+	21.7619i	0.493951	+	0.855549i	−0.506024	+	0.862519i	$0.668885\pi$
$648$		0			0
$649$		−	1.26976i		−	0.0498424i
$650$		0			0
$651$	13.4291	+	46.3033i	0.526329	+	1.81477i
$652$		0			0
$653$	−12.6456	+	7.30093i	−0.494860	+	0.285708i	−0.726588	−	0.687073i	$-0.758895\pi$
$653$	−12.6456	+	7.30093i	−0.494860	+	0.285708i	0.231728	+	0.972781i	$0.425562\pi$
$654$		0			0
$655$	−21.4394	−	21.4394i	−0.837706	−	0.837706i
$656$		0			0
$657$	−19.4096	−	17.8812i	−0.757240	−	0.697614i
$658$		0			0
$659$	9.24365	+	5.33682i	0.360081	+	0.207893i	0.669116	−	0.743158i	$-0.266673\pi$
$659$	9.24365	+	5.33682i	0.360081	+	0.207893i	−0.309035	+	0.951051i	$0.600006\pi$
$660$		0			0
$661$	−39.5548	−	10.5987i	−1.53850	−	0.412241i	−0.612723	−	0.790298i	$-0.709926\pi$
$661$	−39.5548	−	10.5987i	−1.53850	−	0.412241i	−0.925782	+	0.378057i	$0.876592\pi$
$662$		0			0
$663$	13.3219	−	15.0140i	0.517379	−	0.583096i
$664$		0			0
$665$	13.7830	+	3.69314i	0.534481	+	0.143214i
$666$		0			0
$667$	−0.199823	−	0.115368i	−0.00773717	−	0.00446706i
$668$		0			0
$669$	6.22058	+	10.2821i	0.240502	+	0.397531i
$670$		0			0
$671$	−1.41784	−	1.41784i	−0.0547352	−	0.0547352i
$672$		0			0
$673$	2.95408	−	1.70554i	0.113871	−	0.0657437i	−0.441983	−	0.897024i	$-0.645725\pi$
$673$	2.95408	−	1.70554i	0.113871	−	0.0657437i	0.555854	+	0.831280i	$0.312391\pi$
$674$		0			0
$675$	−2.91998	+	3.30281i	−0.112390	+	0.127125i
$676$		0			0
$677$			9.33387i			0.358730i	0.983783	+	0.179365i	$0.0574043\pi$
$677$			9.33387i			0.358730i	−0.983783	+	0.179365i	$0.942596\pi$
$678$		0			0
$679$	−32.5816	−	56.4330i	−1.25037	−	2.16570i
$680$		0			0
$681$	−17.6295	−	9.70254i	−0.675564	−	0.371802i
$682$		0			0
$683$	−20.2277	+	5.42001i	−0.773993	+	0.207391i	−0.624135	−	0.781317i	$-0.714548\pi$
$683$	−20.2277	+	5.42001i	−0.773993	+	0.207391i	−0.149858	+	0.988708i	$0.547882\pi$
$684$		0			0
$685$	−9.16608	+	15.8761i	−0.350218	+	0.606595i
$686$		0			0
$687$	−0.938354	−	45.8097i	−0.0358005	−	1.74775i
$688$		0			0
$689$	40.7887	−	25.7298i	1.55392	−	0.980226i
$690$		0			0
$691$	−0.861578	+	3.21545i	−0.0327760	+	0.122322i	−0.980376	−	0.197139i	$-0.936835\pi$
$691$	−0.861578	+	3.21545i	−0.0327760	+	0.122322i	0.947600	+	0.319461i	$0.103502\pi$
$692$		0			0
$693$	−0.0660589	−	1.61179i	−0.00250937	−	0.0612270i
$694$		0			0
$695$	0.0484033	+	0.180643i	0.00183604	+	0.00685220i
$696$		0			0
$697$	−5.96184	+	5.96184i	−0.225821	+	0.225821i
$698$		0			0
$699$	−5.95130	+	6.20021i	−0.225099	+	0.234513i
$700$		0			0
$701$	−11.9388			−0.450923			−0.225462	−	0.974252i	$-0.572389\pi$
$701$	−11.9388			−0.450923			−0.225462	+	0.974252i	$0.572389\pi$
$702$		0			0
$703$	7.85356			0.296203
$704$		0			0
$705$	16.4961	−	17.1861i	0.621281	−	0.647265i
$706$		0			0
$707$	1.13837	−	1.13837i	0.0428127	−	0.0428127i
$708$		0			0
$709$	−1.31499	−	4.90761i	−0.0493855	−	0.184309i	0.936827	−	0.349793i	$-0.113748\pi$
$709$	−1.31499	−	4.90761i	−0.0493855	−	0.184309i	−0.986212	+	0.165484i	$0.947081\pi$
$710$		0			0
$711$	1.64439	+	40.1221i	0.0616695	+	1.50469i
$712$		0			0
$713$	17.0370	−	63.5831i	0.638042	−	2.38120i
$714$		0			0
$715$	0.518377	−	0.985023i	0.0193862	−	0.0368378i
$716$		0			0
$717$	0.995633	+	48.6059i	0.0371826	+	1.81522i
$718$		0			0
$719$	1.78171	−	3.08602i	0.0664468	−	0.115089i	−0.830888	−	0.556440i	$-0.812167\pi$
$719$	1.78171	−	3.08602i	0.0664468	−	0.115089i	0.897335	+	0.441350i	$0.145500\pi$
$720$		0			0
$721$	34.4599	−	9.23351i	1.28336	−	0.343874i
$722$		0			0
$723$	−9.37632	−	5.16033i	−0.348709	−	0.191915i
$724$		0			0
$725$	−0.0116624	−	0.0201998i	−0.000433129	−	0.000750202i
$726$		0			0
$727$		−	30.2149i		−	1.12061i	−0.828287	−	0.560304i	$-0.810684\pi$
$727$		−	30.2149i		−	1.12061i	0.828287	−	0.560304i	$-0.189316\pi$
$728$		0			0
$729$	3.30956	+	26.7964i	0.122576	+	0.992459i
$730$		0			0
$731$	18.9809	−	10.9587i	0.702036	−	0.405321i
$732$		0			0
$733$	−4.27805	−	4.27805i	−0.158013	−	0.158013i	0.623672	−	0.781686i	$-0.285640\pi$
$733$	−4.27805	−	4.27805i	−0.158013	−	0.158013i	−0.781686	+	0.623672i	$0.785640\pi$
$734$		0			0
$735$	−10.2208	−	16.8941i	−0.376998	−	0.623149i
$736$		0			0
$737$	−0.600109	−	0.346473i	−0.0221053	−	0.0127625i
$738$		0			0
$739$	−22.8637	−	6.12632i	−0.841056	−	0.225360i	−0.187525	−	0.982260i	$-0.560046\pi$
$739$	−22.8637	−	6.12632i	−0.841056	−	0.225360i	−0.653531	+	0.756900i	$0.726713\pi$
$740$		0			0
$741$	−12.3014	+	0.734600i	−0.451902	+	0.0269862i
$742$		0			0
$743$	−45.0215	−	12.0635i	−1.65168	−	0.442566i	−0.691597	−	0.722284i	$-0.743093\pi$
$743$	−45.0215	−	12.0635i	−1.65168	−	0.442566i	−0.960082	+	0.279717i	$0.909759\pi$
$744$		0			0
$745$	−23.8630	−	13.7773i	−0.874274	−	0.504762i
$746$		0			0
$747$	−6.05609	−	5.57922i	−0.221581	−	0.204133i
$748$		0			0
$749$	−33.0815	−	33.0815i	−1.20877	−	1.20877i
$750$		0			0
$751$	38.1177	−	22.0072i	1.39093	−	0.803056i	0.397514	−	0.917596i	$-0.369873\pi$
$751$	38.1177	−	22.0072i	1.39093	−	0.803056i	0.993419	+	0.114540i	$0.0365395\pi$
$752$		0			0
$753$	2.09022	+	7.20703i	0.0761718	+	0.262639i
$754$		0			0
$755$		−	23.7622i		−	0.864796i
$756$		0			0
$757$	15.5726	+	26.9726i	0.565996	+	0.980334i	0.996956	+	0.0779632i	$0.0248417\pi$
$757$	15.5726	+	26.9726i	0.565996	+	0.980334i	−0.430960	+	0.902371i	$0.641825\pi$
$758$		0			0
$759$	−1.06197	+	1.92960i	−0.0385471	+	0.0700400i
$760$		0			0
$761$	−35.3991	+	9.48515i	−1.28322	+	0.343836i	−0.835080	−	0.550129i	$-0.814578\pi$
$761$	−35.3991	+	9.48515i	−1.28322	+	0.343836i	−0.448136	+	0.893966i	$0.647912\pi$
$762$		0			0
$763$	−11.1242	+	19.2677i	−0.402724	+	0.697538i
$764$		0			0
$765$	18.7532	−	5.85782i	0.678022	−	0.211790i
$766$		0			0
$767$	22.1862	+	20.5131i	0.801098	+	0.740683i
$768$		0			0
$769$	−0.493875	+	1.84317i	−0.0178096	+	0.0664663i	−0.974259	−	0.225433i	$-0.927620\pi$
$769$	−0.493875	+	1.84317i	−0.0178096	+	0.0664663i	0.956449	+	0.291899i	$0.0942871\pi$
$770$		0			0
$771$	35.7570	+	8.80032i	1.28776	+	0.316936i
$772$		0			0
$773$	−7.52425	−	28.0809i	−0.270629	−	1.01000i	−0.958714	−	0.284371i	$-0.908215\pi$
$773$	−7.52425	−	28.0809i	−0.270629	−	1.01000i	0.688086	−	0.725629i	$-0.258451\pi$
$774$		0			0
$775$	4.70528	−	4.70528i	0.169019	−	0.169019i
$776$		0			0
$777$	−17.6494	−	16.9409i	−0.633170	−	0.607751i
$778$		0			0
$779$	5.17639			0.185463
$780$		0			0
$781$	0.755297			0.0270266
$782$		0			0
$783$	−0.139995	−	0.0284305i	−0.00500303	−	0.00101602i
$784$		0			0
$785$	−8.63933	+	8.63933i	−0.308351	+	0.308351i
$786$		0			0
$787$	−11.8946	−	44.3912i	−0.423996	−	1.58237i	−0.766106	−	0.642714i	$-0.777808\pi$
$787$	−11.8946	−	44.3912i	−0.423996	−	1.58237i	0.342110	−	0.939660i	$-0.388858\pi$
$788$		0			0
$789$	7.17030	−	29.1340i	0.255270	−	1.03720i
$790$		0			0
$791$	−5.65587	+	21.1080i	−0.201099	+	0.750513i
$792$		0			0
$793$	47.6790	−	1.86831i	1.69313	−	0.0663457i
$794$		0			0
$795$	47.1938	−	0.966707i	1.67379	−	0.0342856i
$796$		0			0
$797$	4.53756	−	7.85929i	0.160729	−	0.278390i	−0.774402	−	0.632694i	$-0.781949\pi$
$797$	4.53756	−	7.85929i	0.160729	−	0.278390i	0.935130	+	0.354304i	$0.115282\pi$
$798$		0			0
$799$	20.9564	−	5.61524i	0.741383	−	0.198653i
$800$		0			0
$801$	32.4925	+	7.29454i	1.14807	+	0.257740i
$802$		0			0
$803$	−0.666430	−	1.15429i	−0.0235178	−	0.0407341i
$804$		0			0
$805$			60.6892i			2.13901i
$806$		0			0
$807$	52.5352	−	15.2365i	1.84933	−	0.536350i
$808$		0			0
$809$	−47.1684	+	27.2327i	−1.65835	+	0.957451i	−0.684879	+	0.728656i	$0.740145\pi$
$809$	−47.1684	+	27.2327i	−1.65835	+	0.957451i	−0.973475	+	0.228795i	$0.926522\pi$
$810$		0			0
$811$	−10.2355	−	10.2355i	−0.359418	−	0.359418i	0.504180	−	0.863598i	$-0.331795\pi$
$811$	−10.2355	−	10.2355i	−0.359418	−	0.359418i	−0.863598	+	0.504180i	$0.831795\pi$
$812$		0			0
$813$	12.5625	−	7.60019i	0.440587	−	0.266550i
$814$		0			0
$815$	25.3711	+	14.6480i	0.888710	+	0.513097i
$816$		0			0
$817$	−12.9976	−	3.48269i	−0.454728	−	0.121844i
$818$		0			0
$819$	29.2297	+	24.8844i	1.02137	+	0.869531i
$820$		0			0
$821$	−38.1061	−	10.2105i	−1.32991	−	0.356349i	−0.477230	−	0.878778i	$-0.658359\pi$
$821$	−38.1061	−	10.2105i	−1.32991	−	0.356349i	−0.852683	+	0.522429i	$0.825026\pi$
$822$		0			0
$823$	23.7150	+	13.6918i	0.826652	+	0.477268i	0.852705	−	0.522393i	$-0.174960\pi$
$823$	23.7150	+	13.6918i	0.826652	+	0.477268i	−0.0260528	+	0.999661i	$0.508294\pi$
$824$		0			0
$825$	−0.190501	+	0.115251i	−0.00663239	+	0.00401252i
$826$		0			0
$827$	−4.19558	−	4.19558i	−0.145895	−	0.145895i	0.630387	−	0.776281i	$-0.282896\pi$
$827$	−4.19558	−	4.19558i	−0.145895	−	0.145895i	−0.776281	+	0.630387i	$0.782896\pi$
$828$		0			0
$829$	−4.55666	+	2.63079i	−0.158259	+	0.0913710i	−0.577038	−	0.816717i	$-0.695792\pi$
$829$	−4.55666	+	2.63079i	−0.158259	+	0.0913710i	0.418779	+	0.908088i	$0.362458\pi$
$830$		0			0
$831$	32.0190	−	9.28629i	1.11073	−	0.322138i
$832$		0			0
$833$		−	17.9828i		−	0.623068i
$834$		0			0
$835$	−5.38598	−	9.32880i	−0.186390	−	0.322836i
$836$		0			0
$837$	−2.50248	−	40.6774i	−0.0864983	−	1.40602i
$838$		0			0
$839$	10.4794	−	2.80796i	0.361790	−	0.0969414i	−0.0733447	−	0.997307i	$-0.523367\pi$
$839$	10.4794	−	2.80796i	0.361790	−	0.0969414i	0.435135	+	0.900365i	$0.356701\pi$
$840$		0			0
$841$	−14.4996	+	25.1141i	−0.499987	+	0.866003i
$842$		0			0
$843$	−40.7707	+	0.835138i	−1.40422	+	0.0287637i
$844$		0			0
$845$	8.83667	+	24.9706i	0.303991	+	0.859015i
$846$		0			0
$847$	−10.0828	+	37.6294i	−0.346448	+	1.29296i
$848$		0			0
$849$	−3.89630	+	15.8313i	−0.133721	+	0.543327i
$850$		0			0
$851$	8.64519	+	32.2643i	0.296353	+	1.10601i
$852$		0			0
$853$	−0.245921	+	0.245921i	−0.00842016	+	0.00842016i	−0.711304	−	0.702884i	$-0.751895\pi$
$853$	−0.245921	+	0.245921i	−0.00842016	+	0.00842016i	0.702884	+	0.711304i	$0.251895\pi$
$854$		0			0
$855$	−10.6843	−	5.59822i	−0.365395	−	0.191455i
$856$		0			0
$857$	−38.9362			−1.33004			−0.665018	−	0.746827i	$-0.731576\pi$
$857$	−38.9362			−1.33004			−0.665018	+	0.746827i	$0.731576\pi$
$858$		0			0
$859$	18.3610			0.626469			0.313234	−	0.949676i	$-0.398587\pi$
$859$	18.3610			0.626469			0.313234	+	0.949676i	$0.398587\pi$
$860$		0			0
$861$	−11.6330	−	11.1660i	−0.396451	−	0.380535i
$862$		0			0
$863$	−19.7327	+	19.7327i	−0.671710	+	0.671710i	−0.958110	−	0.286400i	$-0.907541\pi$
$863$	−19.7327	+	19.7327i	−0.671710	+	0.671710i	0.286400	+	0.958110i	$0.407541\pi$
$864$		0			0
$865$	−3.22724	−	12.0442i	−0.109729	−	0.409516i
$866$		0			0
$867$	−11.2170	−	2.76067i	−0.380949	−	0.0937572i
$868$		0			0
$869$	−0.524901	+	1.95896i	−0.0178060	+	0.0664530i
$870$		0			0
$871$	15.7486	−	4.88827i	0.533622	−	0.165633i
$872$		0			0
$873$	16.4237	+	52.5787i	0.555858	+	1.77952i
$874$		0			0
$875$	−21.1453	+	36.6247i	−0.714841	+	1.23814i
$876$		0			0
$877$	4.13636	−	1.10833i	0.139675	−	0.0374258i	−0.188304	−	0.982111i	$-0.560299\pi$
$877$	4.13636	−	1.10833i	0.139675	−	0.0374258i	0.327979	+	0.944685i	$0.393632\pi$
$878$		0			0
$879$	−21.8986	+	39.7898i	−0.738623	+	1.34208i
$880$		0			0
$881$	−13.4790	−	23.3463i	−0.454118	−	0.786556i	0.544519	−	0.838749i	$-0.316712\pi$
$881$	−13.4790	−	23.3463i	−0.454118	−	0.786556i	−0.998637	+	0.0521929i	$0.983379\pi$
$882$		0			0
$883$			34.0811i			1.14692i	0.819234	+	0.573460i	$0.194399\pi$
$883$			34.0811i			1.14692i	−0.819234	+	0.573460i	$0.805601\pi$
$884$		0			0
$885$	8.23820	+	28.4052i	0.276924	+	0.954829i
$886$		0			0
$887$	−20.2091	+	11.6677i	−0.678555	+	0.391764i	−0.799310	−	0.600918i	$-0.794802\pi$
$887$	−20.2091	+	11.6677i	−0.678555	+	0.391764i	0.120755	+	0.992682i	$0.461468\pi$
$888$		0			0
$889$	23.6312	+	23.6312i	0.792564	+	0.792564i
$890$		0			0
$891$	−0.243964	+	1.34163i	−0.00817309	+	0.0449463i
$892$		0			0
$893$	−11.5355	−	6.66000i	−0.386019	−	0.222868i
$894$		0			0
$895$	−39.8126	−	10.6678i	−1.33079	−	0.356584i
$896$		0			0
$897$	−16.5593	−	49.7284i	−0.552898	−	1.66038i
$898$		0			0
$899$	0.208278	+	0.0558078i	0.00694645	+	0.00186130i
$900$		0			0
$901$	37.2308	+	21.4952i	1.24034	+	0.716109i
$902$		0			0
$903$	21.6972	+	35.8638i	0.722037	+	1.19347i
$904$		0			0
$905$	−6.32517	−	6.32517i	−0.210256	−	0.210256i
$906$		0			0
$907$	−32.6995	+	18.8791i	−1.08577	+	0.626870i	−0.932447	−	0.361307i	$-0.882331\pi$
$907$	−32.6995	+	18.8791i	−1.08577	+	0.626870i	−0.153323	+	0.988176i	$0.548997\pi$
$908$		0			0
$909$	−1.14971	+	0.728133i	−0.0381333	+	0.0241506i
$910$		0			0
$911$		−	8.27053i		−	0.274015i	−0.990570	−	0.137007i	$-0.956252\pi$
$911$		−	8.27053i		−	0.274015i	0.990570	−	0.137007i	$-0.0437484\pi$
$912$		0			0
$913$	−0.207936	−	0.360156i	−0.00688169	−	0.0119194i
$914$		0			0
$915$	40.9168	+	22.5189i	1.35267	+	0.744452i
$916$		0			0
$917$	51.0108	−	13.6683i	1.68452	−	0.451367i
$918$		0			0
$919$	−4.64513	+	8.04561i	−0.153229	+	0.265400i	−0.932413	−	0.361395i	$-0.882301\pi$
$919$	−4.64513	+	8.04561i	−0.153229	+	0.265400i	0.779184	+	0.626795i	$0.215634\pi$
$920$		0			0
$921$	−0.788655	−	38.5015i	−0.0259871	−	1.26867i
$922$		0			0
$923$	−12.2019	+	13.1971i	−0.401629	+	0.434389i
$924$		0			0
$925$	−0.873931	+	3.26156i	−0.0287347	+	0.107239i
$926$		0			0
$927$	−30.1321	+	1.23496i	−0.989668	+	0.0405613i
$928$		0			0
$929$	12.5588	+	46.8701i	0.412041	+	1.53776i	0.790689	+	0.612218i	$0.209722\pi$
$929$	12.5588	+	46.8701i	0.412041	+	1.53776i	−0.378648	+	0.925541i	$0.623611\pi$
$930$		0			0
$931$	−7.80683	+	7.80683i	−0.255859	+	0.255859i
$932$		0			0
$933$	5.20794	−	5.42576i	0.170500	−	0.177631i
$934$		0			0
$935$	0.992257			0.0324503
$936$		0			0
$937$	28.0209			0.915405			0.457702	−	0.889106i	$-0.348673\pi$
$937$	28.0209			0.915405			0.457702	+	0.889106i	$0.348673\pi$
$938$		0			0
$939$	−22.3498	+	23.2846i	−0.729358	+	0.759863i
$940$		0			0
$941$	−5.59288	+	5.59288i	−0.182323	+	0.182323i	−0.792367	−	0.610045i	$-0.791152\pi$
$941$	−5.59288	+	5.59288i	−0.182323	+	0.182323i	0.610045	+	0.792367i	$0.291152\pi$
$942$		0			0
$943$	5.69816	+	21.2658i	0.185558	+	0.692511i
$944$		0			0
$945$	11.9351	+	35.6280i	0.388248	+	1.15898i
$946$		0			0
$947$	−3.77906	+	14.1036i	−0.122803	+	0.458307i	−0.999752	−	0.0222770i	$-0.992908\pi$
$947$	−3.77906	+	14.1036i	−0.122803	+	0.458307i	0.876949	+	0.480583i	$0.159575\pi$
$948$		0			0
$949$	30.9349	+	7.00326i	1.00419	+	0.227335i
$950$		0			0
$951$	−0.455435	−	22.2340i	−0.0147685	−	0.720986i
$952$		0			0
$953$	21.0448	−	36.4507i	0.681709	−	1.18076i	−0.292750	−	0.956189i	$-0.594570\pi$
$953$	21.0448	−	36.4507i	0.681709	−	1.18076i	0.974459	−	0.224566i	$-0.0720964\pi$
$954$		0			0
$955$	27.9590	−	7.49158i	0.904731	−	0.242422i
$956$		0			0
$957$	−0.00632075	−	0.00347867i	−0.000204321	−	0.000112449i
$958$		0			0
$959$	−15.9652	−	27.6526i	−0.515544	−	0.892948i
$960$		0			0
$961$			30.5152i			0.984361i
$962$		0			0
$963$	21.1599	+	33.4110i	0.681869	+	1.07666i
$964$		0			0
$965$	25.7715	−	14.8792i	0.829615	−	0.478979i
$966$		0			0
$967$	−8.40927	−	8.40927i	−0.270424	−	0.270424i	0.558847	−	0.829271i	$-0.311244\pi$
$967$	−8.40927	−	8.40927i	−0.270424	−	0.270424i	−0.829271	+	0.558847i	$0.811244\pi$
$968$		0			0
$969$	−5.68641	−	9.39920i	−0.182674	−	0.301946i
$970$		0			0
$971$	−44.6595	−	25.7842i	−1.43319	−	0.827454i	−0.435830	−	0.900029i	$-0.643545\pi$
$971$	−44.6595	−	25.7842i	−1.43319	−	0.827454i	−0.997363	+	0.0725748i	$0.976878\pi$
$972$		0			0
$973$	−0.314640	−	0.0843074i	−0.0100869	−	0.00270277i
$974$		0			0
$975$	1.06380	−	5.19046i	0.0340689	−	0.166228i
$976$		0			0
$977$	−51.2573	−	13.7343i	−1.63987	−	0.439401i	−0.683115	−	0.730311i	$-0.739375\pi$
$977$	−51.2573	−	13.7343i	−1.63987	−	0.439401i	−0.956750	+	0.290910i	$0.906042\pi$
$978$		0			0
$979$	1.45655	+	0.840938i	0.0465515	+	0.0268765i
$980$		0			0
$981$	12.7429	−	13.8321i	0.406850	−	0.441625i
$982$		0			0
$983$	39.5229	+	39.5229i	1.26058	+	1.26058i	0.950810	+	0.309774i	$0.100253\pi$
$983$	39.5229	+	39.5229i	1.26058	+	1.26058i	0.309774	+	0.950810i	$0.399747\pi$
$984$		0			0
$985$	−1.22123	+	0.705079i	−0.0389117	+	0.0224657i
$986$		0			0
$987$	11.5575	+	39.8502i	0.367881	+	1.26845i
$988$		0			0
$989$		−	57.2309i		−	1.81984i
$990$		0			0
$991$	4.81441	+	8.33881i	0.152935	+	0.264891i	0.932305	−	0.361673i	$-0.117794\pi$
$991$	4.81441	+	8.33881i	0.152935	+	0.264891i	−0.779370	+	0.626564i	$0.784461\pi$
$992$		0			0
$993$	−9.82052	+	17.8439i	−0.311645	+	0.566258i
$994$		0			0
$995$	−31.3976	+	8.41296i	−0.995371	+	0.266709i
$996$		0			0
$997$	14.1749	−	24.5517i	0.448925	−	0.777561i	−0.549392	−	0.835565i	$-0.685140\pi$
$997$	14.1749	−	24.5517i	0.448925	−	0.777561i	0.998316	+	0.0580045i	$0.0184738\pi$
$998$		0			0
$999$	11.4203	+	17.2408i	0.361322	+	0.545476i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.2.bp.a.89.5	✓	56
3.2	odd	2	inner	312.2.bp.a.89.14	yes	56
4.3	odd	2		624.2.cn.f.401.10		56
12.11	even	2		624.2.cn.f.401.1		56
13.6	odd	12	inner	312.2.bp.a.305.14	yes	56
39.32	even	12	inner	312.2.bp.a.305.5	yes	56
52.19	even	12		624.2.cn.f.305.1		56
156.71	odd	12		624.2.cn.f.305.10		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bp.a.89.5	✓	56	1.1	even	1	trivial
312.2.bp.a.89.14	yes	56	3.2	odd	2	inner
312.2.bp.a.305.5	yes	56	39.32	even	12	inner
312.2.bp.a.305.14	yes	56	13.6	odd	12	inner
624.2.cn.f.305.1		56	52.19	even	12
624.2.cn.f.305.10		56	156.71	odd	12
624.2.cn.f.401.1		56	12.11	even	2
624.2.cn.f.401.10		56	4.3	odd	2

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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bp (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.19941 + 1.24957i) q^{3} +(1.44076 - 1.44076i) q^{5} +(0.918532 + 3.42801i) q^{7} +(-0.122851 - 2.99748i) q^{9} +O(q^{10})\)	\(q+(-1.19941 + 1.24957i) q^{3} +(1.44076 - 1.44076i) q^{5} +(0.918532 + 3.42801i) q^{7} +(-0.122851 - 2.99748i) q^{9} +(0.0392149 - 0.146352i) q^{11} +(1.92366 + 3.04952i) q^{13} +(0.0722747 + 3.52839i) q^{15} +(-1.60706 + 2.78351i) q^{17} +(1.90607 - 0.510729i) q^{19} +(-5.38523 - 2.96380i) q^{21} +(4.19640 + 7.26837i) q^{23} +0.848416i q^{25} +(3.89291 + 3.44169i) q^{27} +(-0.0238088 + 0.0137460i) q^{29} +(-5.54595 - 5.54595i) q^{31} +(0.135842 + 0.224537i) q^{33} +(6.26232 + 3.61555i) q^{35} +(3.84429 + 1.03007i) q^{37} +(-6.11783 - 1.25387i) q^{39} +(2.53382 + 0.678935i) q^{41} +(-5.90548 - 3.40953i) q^{43} +(-4.49566 - 4.14166i) q^{45} +(-4.77303 - 4.77303i) q^{47} +(-4.84536 + 2.79747i) q^{49} +(-1.55068 - 5.34670i) q^{51} -13.3755i q^{53} +(-0.154359 - 0.267357i) q^{55} +(-1.64796 + 2.99434i) q^{57} +(8.09489 - 2.16902i) q^{59} +(6.61697 - 11.4609i) q^{61} +(10.1626 - 3.17442i) q^{63} +(7.16515 + 1.62210i) q^{65} +(1.18370 - 4.41762i) q^{67} +(-14.1155 - 3.47404i) q^{69} +(1.29021 + 4.81512i) q^{71} +(6.22036 - 6.22036i) q^{73} +(-1.06016 - 1.01760i) q^{75} +0.537715 q^{77} -13.3852 q^{79} +(-8.96982 + 0.736488i) q^{81} +(1.94085 - 1.94085i) q^{83} +(1.69499 + 6.32577i) q^{85} +(0.0113798 - 0.0462379i) q^{87} +(-2.87300 + 10.7222i) q^{89} +(-8.68683 + 9.39538i) q^{91} +(13.5819 - 0.278209i) q^{93} +(2.01035 - 3.48203i) q^{95} +(-17.7357 + 4.75227i) q^{97} +(-0.443505 - 0.0995664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 4 q^{7}+O(q^{10}) \) 56 * q + 4 * q^7	\( 56 q + 4 q^{7} + 8 q^{13} - 8 q^{15} + 4 q^{19} + 16 q^{21} + 24 q^{27} - 36 q^{31} + 28 q^{33} + 20 q^{37} + 16 q^{39} - 84 q^{43} + 12 q^{45} - 12 q^{49} - 24 q^{55} - 36 q^{57} - 24 q^{61} - 12 q^{63} - 32 q^{67} - 36 q^{69} - 20 q^{73} - 60 q^{75} - 32 q^{79} - 88 q^{85} - 16 q^{87} + 28 q^{91} - 88 q^{93} - 36 q^{97} + 44 q^{99}+O(q^{100}) \) 56 * q + 4 * q^7 + 8 * q^13 - 8 * q^15 + 4 * q^19 + 16 * q^21 + 24 * q^27 - 36 * q^31 + 28 * q^33 + 20 * q^37 + 16 * q^39 - 84 * q^43 + 12 * q^45 - 12 * q^49 - 24 * q^55 - 36 * q^57 - 24 * q^61 - 12 * q^63 - 32 * q^67 - 36 * q^69 - 20 * q^73 - 60 * q^75 - 32 * q^79 - 88 * q^85 - 16 * q^87 + 28 * q^91 - 88 * q^93 - 36 * q^97 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more