LMFDB - Embedded newform 675.2.k.b.424.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(199,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.k (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			424.4
Root			$-1.50511 + 0.403293i$ of defining polynomial
Character	$\chi$	$=$	675.424
Dual form			675.2.k.b.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.495361 + 0.285997i) q^{2} +(-0.836412 - 1.44871i) q^{4} +(-1.23669 - 0.714003i) q^{7} -2.10083i q^{8} +O(q^{10})$	\(q+(0.495361 + 0.285997i) q^{2} +(-0.836412 - 1.44871i) q^{4} +(-1.23669 - 0.714003i) q^{7} -2.10083i q^{8} +(1.33641 - 2.31473i) q^{11} +(-4.04678 + 2.33641i) q^{13} +(-0.408405 - 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} -2.67282i q^{17} -4.67282 q^{19} +(1.32401 - 0.764419i) q^{22} +(-5.12483 + 2.95882i) q^{23} -2.67282 q^{26} +2.38880i q^{28} +(4.74482 - 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(-4.70079 + 2.71400i) q^{32} +(0.764419 - 1.32401i) q^{34} -1.81681i q^{37} +(-2.31473 - 1.33641i) q^{38} +(-0.735581 - 1.27406i) q^{41} +(0.408039 + 0.235581i) q^{43} -4.47116 q^{44} -3.38485 q^{46} +(6.02480 + 3.47842i) q^{47} +(-2.48040 - 4.29618i) q^{49} +(6.76956 + 3.90841i) q^{52} -1.14399i q^{53} +(-1.50000 + 2.59808i) q^{56} +(4.70079 - 2.71400i) q^{58} +(0.571993 + 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} -3.98153i q^{62} +1.18319 q^{64} +(-5.70751 + 3.29523i) q^{67} +(-3.87214 + 2.23558i) q^{68} +12.8745 q^{71} +1.71203i q^{73} +(0.519602 - 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(-3.30545 + 1.90841i) q^{77} +(-0.143987 + 0.249392i) q^{79} -0.841495i q^{82} +(3.71007 + 2.14201i) q^{83} +(0.134751 + 0.233396i) q^{86} +(-4.86286 - 2.80757i) q^{88} -3.00000 q^{89} +6.67282 q^{91} +(8.57293 + 4.94958i) q^{92} +(1.98963 + 3.44615i) q^{94} +(-6.78555 - 3.91764i) q^{97} -2.83754i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{4}+O(q^{10}) $ 12 * q + 10 * q^4	$ 12 q + 10 q^{4} - 4 q^{11} + 18 q^{14} - 10 q^{16} - 16 q^{19} + 8 q^{26} + 14 q^{29} - 16 q^{31} - 8 q^{34} - 26 q^{41} - 88 q^{44} - 12 q^{46} - 4 q^{49} - 18 q^{56} + 4 q^{59} - 2 q^{61} + 60 q^{64} + 40 q^{71} + 32 q^{74} + 24 q^{76} + 4 q^{79} + 56 q^{86} - 36 q^{89} + 40 q^{91} - 62 q^{94}+O(q^{100}) $ 12 * q + 10 * q^4 - 4 * q^11 + 18 * q^14 - 10 * q^16 - 16 * q^19 + 8 * q^26 + 14 * q^29 - 16 * q^31 - 8 * q^34 - 26 * q^41 - 88 * q^44 - 12 * q^46 - 4 * q^49 - 18 * q^56 + 4 * q^59 - 2 * q^61 + 60 * q^64 + 40 * q^71 + 32 * q^74 + 24 * q^76 + 4 * q^79 + 56 * q^86 - 36 * q^89 + 40 * q^91 - 62 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	0.495361	+	0.285997i	0.350273	+	0.202230i	0.664805	−	0.747017i	$-0.268514\pi$
$2$	0.495361	+	0.285997i	0.350273	+	0.202230i	−0.314533	+	0.949247i	$0.601848\pi$
$3$		0			0
$3$		0			0
$4$	−0.836412	−	1.44871i	−0.418206	−	0.724354i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.23669	−	0.714003i	−0.467425	−	0.269868i	0.247736	−	0.968828i	$-0.420313\pi$
$7$	−1.23669	−	0.714003i	−0.467425	−	0.269868i	−0.715161	+	0.698960i	$0.753647\pi$
$8$		−	2.10083i		−	0.742756i
$9$		0			0
$10$		0			0
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	−0.591136	−	0.806572i	$-0.701321\pi$
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	0.994080	+	0.108653i	$0.0346538\pi$
$12$		0			0
$13$	−4.04678	+	2.33641i	−1.12238	+	0.648004i	−0.942006	−	0.335595i	$-0.891063\pi$
$13$	−4.04678	+	2.33641i	−1.12238	+	0.648004i	−0.180370	+	0.983599i	$0.557729\pi$
$14$	−0.408405	−	0.707378i	−0.109151	−	0.189055i
$15$		0			0
$16$	−1.07199	+	1.85675i	−0.267998	+	0.464187i
$17$		−	2.67282i		−	0.648255i	−0.946013	−	0.324127i	$-0.894929\pi$
$17$		−	2.67282i		−	0.648255i	0.946013	−	0.324127i	$-0.105071\pi$
$18$		0			0
$19$	−4.67282			−1.07202			−0.536010	−	0.844212i	$-0.680069\pi$
$19$	−4.67282			−1.07202			−0.536010	+	0.844212i	$0.680069\pi$
$20$		0			0
$21$		0			0
$22$	1.32401	−	0.764419i	0.282280	−	0.162975i
$23$	−5.12483	+	2.95882i	−1.06860	+	0.616957i	−0.927799	−	0.373081i	$-0.878301\pi$
$23$	−5.12483	+	2.95882i	−1.06860	+	0.616957i	−0.140802	+	0.990038i	$0.544968\pi$
$24$		0			0
$25$		0			0
$26$	−2.67282			−0.524184
$27$		0			0
$28$			2.38880i			0.451441i
$29$	4.74482	−	8.21826i	0.881090	−	1.52609i	0.0309603	−	0.999521i	$-0.490143\pi$
$29$	4.74482	−	8.21826i	0.881090	−	1.52609i	0.850130	−	0.526573i	$-0.176523\pi$
$30$		0			0
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	−0.988522	−	0.151078i	$-0.951726\pi$
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	0.363424	−	0.931624i	$-0.381608\pi$
$32$	−4.70079	+	2.71400i	−0.830990	+	0.479773i
$33$		0			0
$34$	0.764419	−	1.32401i	0.131097	−	0.227066i
$35$		0			0
$36$		0			0
$37$		−	1.81681i		−	0.298682i	−0.988786	−	0.149341i	$-0.952285\pi$
$37$		−	1.81681i		−	0.298682i	0.988786	−	0.149341i	$-0.0477152\pi$
$38$	−2.31473	−	1.33641i	−0.375499	−	0.216795i
$39$		0			0
$40$		0			0
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	0.802853	−	0.596177i	$-0.203315\pi$
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	−0.917731	+	0.397202i	$0.869981\pi$
$42$		0			0
$43$	0.408039	+	0.235581i	0.0622254	+	0.0359258i	0.530790	−	0.847503i	$-0.321895\pi$
$43$	0.408039	+	0.235581i	0.0622254	+	0.0359258i	−0.468565	+	0.883429i	$0.655229\pi$
$44$	−4.47116			−0.674053
$45$		0			0
$46$	−3.38485			−0.499069
$47$	6.02480	+	3.47842i	0.878808	+	0.507380i	0.870265	−	0.492584i	$-0.163947\pi$
$47$	6.02480	+	3.47842i	0.878808	+	0.507380i	0.00854274	+	0.999964i	$0.497281\pi$
$48$		0			0
$49$	−2.48040	−	4.29618i	−0.354343	−	0.613739i
$50$		0			0
$51$		0			0
$52$	6.76956	+	3.90841i	0.938769	+	0.541998i
$53$		−	1.14399i		−	0.157139i	−0.996909	−	0.0785693i	$-0.974965\pi$
$53$		−	1.14399i		−	0.157139i	0.996909	−	0.0785693i	$-0.0250352\pi$
$54$		0			0
$55$		0			0
$56$	−1.50000	+	2.59808i	−0.200446	+	0.347183i
$57$		0			0
$58$	4.70079	−	2.71400i	0.617244	−	0.356366i
$59$	0.571993	+	0.990721i	0.0744672	+	0.128981i	0.900854	−	0.434121i	$-0.142941\pi$
$59$	0.571993	+	0.990721i	0.0744672	+	0.128981i	−0.826387	+	0.563102i	$0.809608\pi$
$60$		0			0
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	−0.773655	−	0.633607i	$-0.781574\pi$
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	0.935547	+	0.353201i	$0.114907\pi$
$62$		−	3.98153i		−	0.505655i
$63$		0			0
$64$	1.18319			0.147899
$65$		0			0
$66$		0			0
$67$	−5.70751	+	3.29523i	−0.697283	+	0.402577i	−0.806335	−	0.591459i	$-0.798552\pi$
$67$	−5.70751	+	3.29523i	−0.697283	+	0.402577i	0.109051	+	0.994036i	$0.465219\pi$
$68$	−3.87214	+	2.23558i	−0.469566	+	0.271104i
$69$		0			0
$70$		0			0
$71$	12.8745			1.52792			0.763960	−	0.645263i	$-0.223252\pi$
$71$	12.8745			1.52792			0.763960	+	0.645263i	$0.223252\pi$
$72$		0			0
$73$			1.71203i			0.200378i	0.994968	+	0.100189i	$0.0319447\pi$
$73$			1.71203i			0.200378i	−0.994968	+	0.100189i	$0.968055\pi$
$74$	0.519602	−	0.899976i	0.0604025	−	0.104620i
$75$		0			0
$76$	3.90841	+	6.76956i	0.448325	+	0.776521i
$77$	−3.30545	+	1.90841i	−0.376692	+	0.217483i
$78$		0			0
$79$	−0.143987	+	0.249392i	−0.0161998	+	0.0280588i	−0.874012	−	0.485905i	$-0.838490\pi$
$79$	−0.143987	+	0.249392i	−0.0161998	+	0.0280588i	0.857812	+	0.513964i	$0.171823\pi$
$80$		0			0
$81$		0			0
$82$		−	0.841495i		−	0.0929276i
$83$	3.71007	+	2.14201i	0.407233	+	0.235116i	0.689600	−	0.724190i	$-0.257786\pi$
$83$	3.71007	+	2.14201i	0.407233	+	0.235116i	−0.282367	+	0.959306i	$0.591120\pi$
$84$		0			0
$85$		0			0
$86$	0.134751	+	0.233396i	0.0145306	+	0.0251677i
$87$		0			0
$88$	−4.86286	−	2.80757i	−0.518383	−	0.299288i
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	6.67282			0.699502
$92$	8.57293	+	4.94958i	0.893790	+	0.516030i
$93$		0			0
$94$	1.98963	+	3.44615i	0.205215	+	0.355443i
$95$		0			0
$96$		0			0
$97$	−6.78555	−	3.91764i	−0.688968	−	0.397776i	0.114257	−	0.993451i	$-0.463551\pi$
$97$	−6.78555	−	3.91764i	−0.688968	−	0.397776i	−0.803225	+	0.595675i	$0.796885\pi$
$98$		−	2.83754i		−	0.286635i
$99$		0			0
$100$		0			0
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	−0.951413	−	0.307919i	$-0.900367\pi$
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	0.742372	+	0.669988i	$0.233701\pi$
$102$		0			0
$103$	1.57340	−	0.908405i	0.155032	−	0.0895078i	−0.420477	−	0.907303i	$-0.638137\pi$
$103$	1.57340	−	0.908405i	0.155032	−	0.0895078i	0.575509	+	0.817795i	$0.304804\pi$
$104$	4.90841	+	8.50161i	0.481309	+	0.833651i
$105$		0			0
$106$	0.327176	−	0.566686i	0.0317782	−	0.0550414i
$107$		−	11.9176i		−	1.15212i	−0.817407	−	0.576061i	$-0.804589\pi$
$107$		−	11.9176i		−	1.15212i	0.817407	−	0.576061i	$-0.195411\pi$
$108$		0			0
$109$	16.6521			1.59498			0.797491	−	0.603331i	$-0.206160\pi$
$109$	16.6521			1.59498			0.797491	+	0.603331i	$0.206160\pi$
$110$		0			0
$111$		0			0
$112$	2.65145	−	1.53081i	0.250538	−	0.144648i
$113$	17.4272	−	10.0616i	1.63942	−	0.946518i	0.658384	−	0.752682i	$-0.271240\pi$
$113$	17.4272	−	10.0616i	1.63942	−	0.946518i	0.981034	−	0.193836i	$-0.0620929\pi$
$114$		0			0
$115$		0			0
$116$	−15.8745			−1.47391
$117$		0			0
$118$			0.654353i			0.0602380i
$119$	−1.90841	+	3.30545i	−0.174943	+	0.303011i
$120$		0			0
$121$	1.92801	+	3.33941i	0.175273	+	0.303582i
$122$	1.25269	−	0.723239i	0.113413	−	0.0654790i
$123$		0			0
$124$	−5.82209	+	10.0842i	−0.522839	+	0.905584i
$125$		0			0
$126$		0			0
$127$			2.18714i			0.194078i	0.995281	+	0.0970388i	$0.0309371\pi$
$127$			2.18714i			0.194078i	−0.995281	+	0.0970388i	$0.969063\pi$
$128$	9.98769	+	5.76640i	0.882795	+	0.509682i
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	5.77883	+	3.33641i	0.501089	+	0.289304i
$134$	−3.76970			−0.325653
$135$		0			0
$136$	−5.61515			−0.481495
$137$	−8.83490	−	5.10083i	−0.754816	−	0.435793i	0.0726153	−	0.997360i	$-0.476865\pi$
$137$	−8.83490	−	5.10083i	−0.754816	−	0.435793i	−0.827431	+	0.561567i	$0.810199\pi$
$138$		0			0
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	0.984297	−	0.176522i	$-0.0564848\pi$
$139$	4.00000	+	6.92820i	0.339276	+	0.587643i	−0.645021	+	0.764165i	$0.723151\pi$
$140$		0			0
$141$		0			0
$142$	6.37751	+	3.68206i	0.535189	+	0.308992i
$143$			12.4896i			1.04444i
$144$		0			0
$145$		0			0
$146$	−0.489634	+	0.848071i	−0.0405224	+	0.0701868i
$147$		0			0
$148$	−2.63203	+	1.51960i	−0.216351	+	0.124910i
$149$	10.0381	+	17.3865i	0.822351	+	1.42435i	0.903927	+	0.427687i	$0.140671\pi$
$149$	10.0381	+	17.3865i	0.822351	+	1.42435i	−0.0815762	+	0.996667i	$0.525995\pi$
$150$		0			0
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	−0.921210	−	0.389066i	$-0.872798\pi$
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	0.797546	+	0.603258i	$0.206131\pi$
$152$			9.81681i			0.796248i
$153$		0			0
$154$	−2.18319			−0.175926
$155$		0			0
$156$		0			0
$157$	0.174643	−	0.100830i	0.0139381	−	0.00804714i	−0.493015	−	0.870021i	$-0.664105\pi$
$157$	0.174643	−	0.100830i	0.0139381	−	0.00804714i	0.506953	+	0.861974i	$0.330772\pi$
$158$	−0.142651	+	0.0823593i	−0.0113487	+	0.00655216i
$159$		0			0
$160$		0			0
$161$	8.45043			0.665987
$162$		0			0
$163$		−	17.8168i		−	1.39552i	−0.716331	−	0.697760i	$-0.754180\pi$
$163$		−	17.8168i		−	1.39552i	0.716331	−	0.697760i	$-0.245820\pi$
$164$	−1.23050	+	2.13129i	−0.0960858	+	0.166425i
$165$		0			0
$166$	1.22522	+	2.12214i	0.0950952	+	0.164710i
$167$	−12.2117	+	7.05042i	−0.944968	+	0.545578i	−0.891514	−	0.452993i	$-0.850356\pi$
$167$	−12.2117	+	7.05042i	−0.944968	+	0.545578i	−0.0534538	+	0.998570i	$0.517023\pi$
$168$		0			0
$169$	4.41764	−	7.65158i	0.339819	−	0.588583i
$170$		0			0
$171$		0			0
$172$		−	0.788172i		−	0.0600976i
$173$	−3.78140	−	2.18319i	−0.287494	−	0.165985i	0.349317	−	0.937005i	$-0.386414\pi$
$173$	−3.78140	−	2.18319i	−0.287494	−	0.165985i	−0.636811	+	0.771020i	$0.719747\pi$
$174$		0			0
$175$		0			0
$176$	2.86525	+	4.96276i	0.215976	+	0.374082i
$177$		0			0
$178$	−1.48608	−	0.857990i	−0.111387	−	0.0643091i
$179$	−15.1625			−1.13330			−0.566648	−	0.823960i	$-0.691760\pi$
$179$	−15.1625			−1.13330			−0.566648	+	0.823960i	$0.691760\pi$
$180$		0			0
$181$	3.20166			0.237978			0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166			0.237978			0.118989	+	0.992896i	$0.462035\pi$
$182$	3.30545	+	1.90841i	0.245017	+	0.141460i
$183$		0			0
$184$	6.21598	+	10.7664i	0.458248	+	0.793709i
$185$		0			0
$186$		0			0
$187$	−6.18687	−	3.57199i	−0.452429	−	0.261210i
$188$		−	11.6376i		−	0.848757i
$189$		0			0
$190$		0			0
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	−0.912779	−	0.408453i	$-0.866068\pi$
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	0.810121	+	0.586263i	$0.199402\pi$
$192$		0			0
$193$	16.2710	−	9.39409i	1.17121	−	0.676201i	0.217249	−	0.976116i	$-0.430292\pi$
$193$	16.2710	−	9.39409i	1.17121	−	0.676201i	0.953966	+	0.299915i	$0.0969584\pi$
$194$	−2.24086	−	3.88129i	−0.160885	−	0.278660i
$195$		0			0
$196$	−4.14927	+	7.18675i	−0.296376	+	0.513339i
$197$		−	5.83528i		−	0.415747i	−0.978156	−	0.207873i	$-0.933346\pi$
$197$		−	5.83528i		−	0.415747i	0.978156	−	0.207873i	$-0.0666542\pi$
$198$		0			0
$199$	−13.0761			−0.926943			−0.463472	−	0.886112i	$-0.653396\pi$
$199$	−13.0761			−0.926943			−0.463472	+	0.886112i	$0.653396\pi$
$200$		0			0
$201$		0			0
$202$	−2.08134	+	1.20166i	−0.146442	+	0.0845486i
$203$	−11.7357	+	6.77563i	−0.823687	+	0.475556i
$204$		0			0
$205$		0			0
$206$	1.03920			0.0724047
$207$		0			0
$208$		−	10.0185i		−	0.694656i
$209$	−6.24482	+	10.8163i	−0.431963	+	0.748182i
$210$		0			0
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	0.684862	−	0.728673i	$-0.259863\pi$
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	−0.973480	+	0.228771i	$0.926529\pi$
$212$	−1.65730	+	0.956844i	−0.113824	+	0.0657163i
$213$		0			0
$214$	3.40841	−	5.90353i	0.232994	−	0.403557i
$215$		0			0
$216$		0			0
$217$			9.94006i			0.674776i
$218$	8.24879	+	4.76244i	0.558679	+	0.322553i
$219$		0			0
$220$		0			0
$221$	6.24482	+	10.8163i	0.420072	+	0.727586i
$222$		0			0
$223$	7.93834	+	4.58321i	0.531591	+	0.306914i	0.741664	−	0.670772i	$-0.234037\pi$
$223$	7.93834	+	4.58321i	0.531591	+	0.306914i	−0.210073	+	0.977686i	$0.567370\pi$
$224$	7.75123			0.517901
$225$		0			0
$226$	11.5104			0.765658
$227$	−2.31473	−	1.33641i	−0.153634	−	0.0887008i	0.421212	−	0.906962i	$-0.361605\pi$
$227$	−2.31473	−	1.33641i	−0.153634	−	0.0887008i	−0.574846	+	0.818261i	$0.694938\pi$
$228$		0			0
$229$	1.27365	+	2.20603i	0.0841654	+	0.145779i	0.905035	−	0.425336i	$-0.139844\pi$
$229$	1.27365	+	2.20603i	0.0841654	+	0.145779i	−0.820870	+	0.571115i	$0.806511\pi$
$230$		0			0
$231$		0			0
$232$	−17.2652	−	9.96806i	−1.13351	−	0.654435i
$233$		−	6.22013i		−	0.407494i	−0.979024	−	0.203747i	$-0.934688\pi$
$233$		−	6.22013i		−	0.407494i	0.979024	−	0.203747i	$-0.0653121\pi$
$234$		0			0
$235$		0			0
$236$	0.956844	−	1.65730i	0.0622852	−	0.107881i
$237$		0			0
$238$	−1.89070	+	1.09159i	−0.122556	+	0.0707576i
$239$	−4.06163	−	7.03494i	−0.262725	−	0.455053i	0.704240	−	0.709962i	$-0.251288\pi$
$239$	−4.06163	−	7.03494i	−0.262725	−	0.455053i	−0.966965	+	0.254909i	$0.917954\pi$
$240$		0			0
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	−0.0328536	−	0.999460i	$-0.510460\pi$
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	0.881985	−	0.471278i	$-0.156207\pi$
$242$			2.20561i			0.141782i
$243$		0			0
$244$	−4.23030			−0.270817
$245$		0			0
$246$		0			0
$247$	18.9099	−	10.9176i	1.20321	−	0.694673i
$248$	−12.6643	+	7.31173i	−0.804183	+	0.464295i
$249$		0			0
$250$		0			0
$251$	0.549569			0.0346885			0.0173443	−	0.999850i	$-0.494479\pi$
$251$	0.549569			0.0346885			0.0173443	+	0.999850i	$0.494479\pi$
$252$		0			0
$253$			15.8168i			0.994394i
$254$	−0.625515	+	1.08342i	−0.0392483	+	0.0679801i
$255$		0			0
$256$	2.11515	+	3.66355i	0.132197	+	0.228972i
$257$	15.5885	−	9.00000i	0.972381	−	0.561405i	0.0724199	−	0.997374i	$-0.476928\pi$
$257$	15.5885	−	9.00000i	0.972381	−	0.561405i	0.899961	+	0.435970i	$0.143595\pi$
$258$		0			0
$259$	−1.29721	+	2.24683i	−0.0806046	+	0.139611i
$260$		0			0
$261$		0			0
$262$		−	3.43196i		−	0.212027i
$263$	−10.3016	−	5.94761i	−0.635221	−	0.366745i	0.147550	−	0.989055i	$-0.452861\pi$
$263$	−10.3016	−	5.94761i	−0.635221	−	0.366745i	−0.782771	+	0.622309i	$0.786195\pi$
$264$		0			0
$265$		0			0
$266$	1.90841	+	3.30545i	0.117012	+	0.202670i
$267$		0			0
$268$	9.54766	+	5.51234i	0.583216	+	0.336720i
$269$	28.5737			1.74217			0.871084	−	0.491134i	$-0.163417\pi$
$269$	28.5737			1.74217			0.871084	+	0.491134i	$0.163417\pi$
$270$		0			0
$271$	−23.3641			−1.41927			−0.709635	−	0.704570i	$-0.751140\pi$
$271$	−23.3641			−1.41927			−0.709635	+	0.704570i	$0.751140\pi$
$272$	4.96276	+	2.86525i	0.300911	+	0.173731i
$273$		0			0
$274$	−2.91764	−	5.05350i	−0.176261	−	0.305293i
$275$		0			0
$276$		0			0
$277$	13.0563	+	7.53807i	0.784479	+	0.452919i	0.838015	−	0.545647i	$-0.183716\pi$
$277$	13.0563	+	7.53807i	0.784479	+	0.452919i	−0.0535366	+	0.998566i	$0.517049\pi$
$278$			4.57595i			0.274447i
$279$		0			0
$280$		0			0
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	−0.749599	−	0.661892i	$-0.769754\pi$
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	0.948015	+	0.318226i	$0.103087\pi$
$282$		0			0
$283$	−23.2934	+	13.4485i	−1.38465	+	0.799428i	−0.992706	−	0.120562i	$-0.961530\pi$
$283$	−23.2934	+	13.4485i	−1.38465	+	0.799428i	−0.391943	+	0.919989i	$0.628197\pi$
$284$	−10.7684	−	18.6514i	−0.638985	−	1.10675i
$285$		0			0
$286$	−3.57199	+	6.18687i	−0.211216	+	0.365838i
$287$			2.10083i			0.124008i
$288$		0			0
$289$	9.85601			0.579765
$290$		0			0
$291$		0			0
$292$	2.48023	−	1.43196i	0.145144	−	0.0837991i
$293$	10.7256	−	6.19243i	0.626596	−	0.361765i	−0.152837	−	0.988251i	$-0.548841\pi$
$293$	10.7256	−	6.19243i	0.626596	−	0.361765i	0.779433	+	0.626486i	$0.215507\pi$
$294$		0			0
$295$		0			0
$296$	−3.81681			−0.221848
$297$		0			0
$298$			11.4834i			0.665217i
$299$	13.8260	−	23.9474i	0.799581	−	1.38491i
$300$		0			0
$301$	−0.336412	−	0.582682i	−0.0193905	−	0.0335853i
$302$	−1.50550	+	0.869202i	−0.0866319	+	0.0500169i
$303$		0			0
$304$	5.00924	−	8.67625i	0.287299	−	0.497617i
$305$		0			0
$306$		0			0
$307$		−	2.49359i		−	0.142317i	−0.997465	−	0.0711583i	$-0.977330\pi$
$307$		−	2.49359i		−	0.142317i	0.997465	−	0.0711583i	$-0.0226695\pi$
$308$	5.52944	+	3.19243i	0.315069	+	0.181905i
$309$		0			0
$310$		0			0
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	−0.972900	−	0.231228i	$-0.925726\pi$
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	0.286201	−	0.958170i	$-0.407608\pi$
$312$		0			0
$313$	−30.3837	−	17.5420i	−1.71739	−	0.991534i	−0.923618	−	0.383315i	$-0.874782\pi$
$313$	−30.3837	−	17.5420i	−1.71739	−	0.991534i	−0.793770	−	0.608219i	$-0.791884\pi$
$314$	0.115349			0.00650950
$315$		0			0
$316$	0.481728			0.0270993
$317$	−9.06829	−	5.23558i	−0.509326	−	0.294060i	0.223231	−	0.974766i	$-0.428340\pi$
$317$	−9.06829	−	5.23558i	−0.509326	−	0.294060i	−0.732557	+	0.680706i	$0.761673\pi$
$318$		0			0
$319$	−12.6821	−	21.9660i	−0.710059	−	1.22986i
$320$		0			0
$321$		0			0
$322$	4.18601	+	2.41679i	0.233277	+	0.134683i
$323$			12.4896i			0.694942i
$324$		0			0
$325$		0			0
$326$	5.09555	−	8.82575i	0.282216	−	0.488813i
$327$		0			0
$328$	−2.67659	+	1.54533i	−0.147790	+	0.0853267i
$329$	−4.96721	−	8.60346i	−0.273851	−	0.474324i
$330$		0			0
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	−0.999016	−	0.0443606i	$-0.985875\pi$
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	0.537925	+	0.842993i	$0.319208\pi$
$332$		−	7.16641i		−	0.393308i
$333$		0			0
$334$	−8.06558			−0.441329
$335$		0			0
$336$		0			0
$337$	−23.5394	+	13.5905i	−1.28227	+	0.740320i	−0.977263	−	0.212030i	$-0.931993\pi$
$337$	−23.5394	+	13.5905i	−1.28227	+	0.740320i	−0.305008	+	0.952350i	$0.598659\pi$
$338$	4.37665	−	2.52686i	0.238058	−	0.137443i
$339$		0			0
$340$		0			0
$341$	−18.6050			−1.00752
$342$		0			0
$343$			17.0801i			0.922239i
$344$	0.494917	−	0.857221i	0.0266841	−	0.0462182i
$345$		0			0
$346$	−1.24877	−	2.16293i	−0.0671343	−	0.116280i
$347$	20.4086	−	11.7829i	1.09559	−	0.632539i	0.160530	−	0.987031i	$-0.448680\pi$
$347$	20.4086	−	11.7829i	1.09559	−	0.632539i	0.935059	+	0.354492i	$0.115346\pi$
$348$		0			0
$349$	−5.35601	+	9.27689i	−0.286701	+	0.496580i	−0.973020	−	0.230720i	$-0.925892\pi$
$349$	−5.35601	+	9.27689i	−0.286701	+	0.496580i	0.686319	+	0.727300i	$0.259225\pi$
$350$		0			0
$351$		0			0
$352$			14.5081i			0.773285i
$353$	−23.6141	−	13.6336i	−1.25685	−	0.725644i	−0.284392	−	0.958708i	$-0.591792\pi$
$353$	−23.6141	−	13.6336i	−1.25685	−	0.725644i	−0.972461	+	0.233064i	$0.925125\pi$
$354$		0			0
$355$		0			0
$356$	2.50924	+	4.34612i	0.132989	+	0.230344i
$357$		0			0
$358$	−7.51089	−	4.33641i	−0.396963	−	0.229186i
$359$	−10.6807			−0.563707			−0.281854	−	0.959457i	$-0.590949\pi$
$359$	−10.6807			−0.563707			−0.281854	+	0.959457i	$0.590949\pi$
$360$		0			0
$361$	2.83528			0.149225
$362$	1.58598	+	0.915664i	0.0833571	+	0.0481262i
$363$		0			0
$364$	−5.58123	−	9.66697i	−0.292536	−	0.506687i
$365$		0			0
$366$		0			0
$367$	7.33624	+	4.23558i	0.382949	+	0.221096i	0.679100	−	0.734045i	$-0.262370\pi$
$367$	7.33624	+	4.23558i	0.382949	+	0.221096i	−0.296152	+	0.955141i	$0.595703\pi$
$368$		−	12.6873i		−	0.661373i
$369$		0			0
$370$		0			0
$371$	−0.816810	+	1.41476i	−0.0424067	+	0.0734505i
$372$		0			0
$373$	−8.76700	+	5.06163i	−0.453938	+	0.262081i	−0.709492	−	0.704714i	$-0.751075\pi$
$373$	−8.76700	+	5.06163i	−0.453938	+	0.262081i	0.255554	+	0.966795i	$0.417742\pi$
$374$	−2.04316	−	3.53885i	−0.105649	−	0.182990i
$375$		0			0
$376$	7.30757	−	12.6571i	0.376859	−	0.652740i
$377$			44.3434i			2.28380i
$378$		0			0
$379$	−11.9216			−0.612371			−0.306186	−	0.951972i	$-0.599053\pi$
$379$	−11.9216			−0.612371			−0.306186	+	0.951972i	$0.599053\pi$
$380$		0			0
$381$		0			0
$382$	−1.40561	+	0.811528i	−0.0719171	+	0.0415214i
$383$	8.50161	−	4.90841i	0.434412	−	0.250808i	−0.266813	−	0.963748i	$-0.585970\pi$
$383$	8.50161	−	4.90841i	0.434412	−	0.250808i	0.701224	+	0.712941i	$0.252637\pi$
$384$		0			0
$385$		0			0
$386$	10.7467			0.546993
$387$		0			0
$388$			13.1070i			0.665409i
$389$	−4.61007	+	7.98487i	−0.233740	+	0.404849i	−0.958906	−	0.283725i	$-0.908430\pi$
$389$	−4.61007	+	7.98487i	−0.233740	+	0.404849i	0.725166	+	0.688574i	$0.241763\pi$
$390$		0			0
$391$	7.90841	+	13.6978i	0.399945	+	0.692725i
$392$	−9.02554	+	5.21090i	−0.455858	+	0.263190i
$393$		0			0
$394$	1.66887	−	2.89057i	0.0840765	−	0.145625i
$395$		0			0
$396$		0			0
$397$		−	22.9793i		−	1.15330i	−0.816993	−	0.576648i	$-0.804360\pi$
$397$		−	22.9793i		−	1.15330i	0.816993	−	0.576648i	$-0.195640\pi$
$398$	−6.47741	−	3.73973i	−0.324683	−	0.187456i
$399$		0			0
$400$		0			0
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	0.694167	−	0.719814i	$-0.255773\pi$
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	−0.970461	+	0.241259i	$0.922440\pi$
$402$		0			0
$403$	28.1688	+	16.2633i	1.40319	+	0.810132i
$404$	7.02864			0.349688
$405$		0			0
$406$	−7.75123			−0.384687
$407$	−4.20543	−	2.42801i	−0.208455	−	0.120352i
$408$		0			0
$409$	−8.81681	−	15.2712i	−0.435963	−	0.755110i	0.561411	−	0.827537i	$-0.310259\pi$
$409$	−8.81681	−	15.2712i	−0.435963	−	0.755110i	−0.997374	+	0.0724270i	$0.976926\pi$
$410$		0			0
$411$		0			0
$412$	−2.63203	−	1.51960i	−0.129671	−	0.0748654i
$413$		−	1.63362i		−	0.0803852i
$414$		0			0
$415$		0			0
$416$	12.6821	−	21.9660i	0.621789	−	1.07697i
$417$		0			0
$418$	−6.18687	+	3.57199i	−0.302610	+	0.174712i
$419$	−18.5173	−	32.0730i	−0.904631	−	1.56687i	−0.821411	−	0.570336i	$-0.806813\pi$
$419$	−18.5173	−	32.0730i	−0.904631	−	1.56687i	−0.0832199	−	0.996531i	$-0.526520\pi$
$420$		0			0
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	−0.921047	−	0.389452i	$-0.872664\pi$
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	0.797799	+	0.602924i	$0.205998\pi$
$422$		−	4.79608i		−	0.233469i
$423$		0			0
$424$	−2.40332			−0.116716
$425$		0			0
$426$		0			0
$427$	−3.12739	+	1.80560i	−0.151345	+	0.0873790i
$428$	−17.2652	+	9.96806i	−0.834544	+	0.481824i
$429$		0			0
$430$		0			0
$431$	5.23030			0.251935			0.125967	−	0.992034i	$-0.459797\pi$
$431$	5.23030			0.251935			0.125967	+	0.992034i	$0.459797\pi$
$432$		0			0
$433$		−	34.3434i		−	1.65044i	−0.564813	−	0.825219i	$-0.691052\pi$
$433$		−	34.3434i		−	1.65044i	0.564813	−	0.825219i	$-0.308948\pi$
$434$	−2.84283	+	4.92392i	−0.136460	+	0.236356i
$435$		0			0
$436$	−13.9280	−	24.1240i	−0.667031	−	1.15533i
$437$	23.9474	−	13.8260i	1.14556	−	0.661389i
$438$		0			0
$439$	−9.77365	+	16.9285i	−0.466471	+	0.807952i	−0.999267	−	0.0382924i	$-0.987808\pi$
$439$	−9.77365	+	16.9285i	−0.466471	+	0.807952i	0.532796	+	0.846244i	$0.321141\pi$
$440$		0			0
$441$		0			0
$442$			7.14399i			0.339805i
$443$	−9.09686	−	5.25208i	−0.432205	−	0.249534i	0.268081	−	0.963396i	$-0.413611\pi$
$443$	−9.09686	−	5.25208i	−0.432205	−	0.249534i	−0.700286	+	0.713863i	$0.746944\pi$
$444$		0			0
$445$		0			0
$446$	2.62156	+	4.54068i	0.124135	+	0.215007i
$447$		0			0
$448$	−1.46324	−	0.844801i	−0.0691315	−	0.0399131i
$449$	22.8560			1.07864			0.539321	−	0.842100i	$-0.318681\pi$
$449$	22.8560			1.07864			0.539321	+	0.842100i	$0.318681\pi$
$450$		0			0
$451$	−3.93216			−0.185158
$452$	−29.1527	−	16.8313i	−1.37123	−	0.791679i
$453$		0			0
$454$	−0.764419	−	1.32401i	−0.0358759	−	0.0621390i
$455$		0			0
$456$		0			0
$457$	12.1472	+	7.01319i	0.568222	+	0.328063i	0.756439	−	0.654064i	$-0.226937\pi$
$457$	12.1472	+	7.01319i	0.568222	+	0.328063i	−0.188217	+	0.982127i	$0.560271\pi$
$458$			1.45704i			0.0680832i
$459$		0			0
$460$		0			0
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	0.436102	+	0.899897i	$0.356359\pi$
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	−0.997385	+	0.0722736i	$0.976975\pi$
$462$		0			0
$463$	29.3930	−	16.9700i	1.36601	−	0.788664i	0.375591	−	0.926785i	$-0.377440\pi$
$463$	29.3930	−	16.9700i	1.36601	−	0.788664i	0.990415	+	0.138121i	$0.0441063\pi$
$464$	10.1728	+	17.6198i	0.472261	+	0.817981i
$465$		0			0
$466$	1.77894	−	3.08121i	0.0824077	−	0.142734i
$467$			27.3720i			1.26663i	0.773896	+	0.633313i	$0.218305\pi$
$467$			27.3720i			1.26663i	−0.773896	+	0.633313i	$0.781695\pi$
$468$		0			0
$469$	9.41123			0.434570
$470$		0			0
$471$		0			0
$472$	2.08134	−	1.20166i	0.0958013	−	0.0553109i
$473$	1.09062	−	0.629668i	0.0501466	−	0.0289521i
$474$		0			0
$475$		0			0
$476$	6.38485			0.292649
$477$		0			0
$478$		−	4.64645i		−	0.212524i
$479$	−2.61515	+	4.52957i	−0.119489	+	0.206961i	−0.919565	−	0.392937i	$-0.871459\pi$
$479$	−2.61515	+	4.52957i	−0.119489	+	0.206961i	0.800076	+	0.599898i	$0.204792\pi$
$480$		0			0
$481$	4.24482	+	7.35224i	0.193547	+	0.335233i
$482$	13.0597	−	7.54005i	0.594855	−	0.343440i
$483$		0			0
$484$	3.22522	−	5.58624i	0.146601	−	0.253920i
$485$		0			0
$486$		0			0
$487$		−	24.0185i		−	1.08838i	−0.838962	−	0.544190i	$-0.816837\pi$
$487$		−	24.0185i		−	1.08838i	0.838962	−	0.544190i	$-0.183163\pi$
$488$	−4.60090	−	2.65633i	−0.208273	−	0.120246i
$489$		0			0
$490$		0			0
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	0.983186	−	0.182606i	$-0.0584531\pi$
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	−0.649734	+	0.760161i	$0.725120\pi$
$492$		0			0
$493$	−21.9660	−	12.6821i	−0.989298	−	0.571171i
$494$	12.4896			0.561935
$495$		0			0
$496$	14.9239			0.670101
$497$	−15.9217	−	9.19243i	−0.714188	−	0.412337i
$498$		0			0
$499$	12.4280	+	21.5259i	0.556354	+	0.963633i	0.997797	+	0.0663440i	$0.0211335\pi$
$499$	12.4280	+	21.5259i	0.556354	+	0.963633i	−0.441443	+	0.897289i	$0.645533\pi$
$500$		0			0
$501$		0			0
$502$	0.272235	+	0.157175i	0.0121504	+	0.00701506i
$503$			38.9154i			1.73515i	0.497305	+	0.867576i	$0.334323\pi$
$503$			38.9154i			1.73515i	−0.497305	+	0.867576i	$0.665677\pi$
$504$		0			0
$505$		0			0
$506$	−4.52355	+	7.83503i	−0.201097	+	0.348309i
$507$		0			0
$508$	3.16853	−	1.82935i	0.140581	−	0.0811644i
$509$	−1.01037	−	1.75001i	−0.0447837	−	0.0775676i	0.842765	−	0.538282i	$-0.180927\pi$
$509$	−1.01037	−	1.75001i	−0.0447837	−	0.0775676i	−0.887548	+	0.460715i	$0.847593\pi$
$510$		0			0
$511$	1.22239	−	2.11725i	0.0540755	−	0.0936615i
$512$		−	20.6459i		−	0.912428i
$513$		0			0
$514$	10.2959			0.454132
$515$		0			0
$516$		0			0
$517$	16.1032	−	9.29721i	0.708220	−	0.408891i
$518$	−1.28517	+	0.741995i	−0.0564672	+	0.0326014i
$519$		0			0
$520$		0			0
$521$	23.0290			1.00892			0.504460	−	0.863435i	$-0.331692\pi$
$521$	23.0290			1.00892			0.504460	+	0.863435i	$0.331692\pi$
$522$		0			0
$523$			41.1170i			1.79792i	0.438028	+	0.898961i	$0.355677\pi$
$523$			41.1170i			1.79792i	−0.438028	+	0.898961i	$0.644323\pi$
$524$	−5.01847	+	8.69225i	−0.219233	+	0.379723i
$525$		0			0
$526$	−3.40199	−	5.89242i	−0.148334	−	0.256922i
$527$	−16.1124	+	9.30249i	−0.701867	+	0.405223i
$528$		0			0
$529$	6.00924	−	10.4083i	0.261271	−	0.452535i
$530$		0			0
$531$		0			0
$532$		−	11.1625i		−	0.483954i
$533$	5.95348	+	3.43724i	0.257874	+	0.148883i
$534$		0			0
$535$		0			0
$536$	6.92272	+	11.9905i	0.299016	+	0.517911i
$537$		0			0
$538$	14.1543	+	8.17198i	0.610234	+	0.352319i
$539$	−13.2593			−0.571120
$540$		0			0
$541$	5.20957			0.223977			0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957			0.223977			0.111988	+	0.993710i	$0.464278\pi$
$542$	−11.5737	−	6.68206i	−0.497132	−	0.287019i
$543$		0			0
$544$	7.25405	+	12.5644i	0.311015	+	0.538694i
$545$		0			0
$546$		0			0
$547$	−34.6764	−	20.0204i	−1.48266	−	0.856013i	−0.482851	−	0.875702i	$-0.660399\pi$
$547$	−34.6764	−	20.0204i	−1.48266	−	0.856013i	−0.999806	+	0.0196900i	$0.993732\pi$
$548$			17.0656i			0.729005i
$549$		0			0
$550$		0			0
$551$	−22.1717	+	38.4025i	−0.944546	+	1.63600i
$552$		0			0
$553$	0.356133	−	0.205614i	0.0151443	−	0.00874359i
$554$	4.31173	+	7.46813i	0.183188	+	0.317290i
$555$		0			0
$556$	6.69129	−	11.5897i	0.283774	−	0.491511i
$557$			14.4033i			0.610288i	0.952306	+	0.305144i	$0.0987047\pi$
$557$			14.4033i			0.610288i	−0.952306	+	0.305144i	$0.901295\pi$
$558$		0			0
$559$	−2.20166			−0.0931203
$560$		0			0
$561$		0			0
$562$	3.29518	−	1.90248i	0.138999	−	0.0802511i
$563$	25.4335	−	14.6840i	1.07189	−	0.618858i	0.143196	−	0.989694i	$-0.454262\pi$
$563$	25.4335	−	14.6840i	1.07189	−	0.618858i	0.928698	+	0.370836i	$0.120929\pi$
$564$		0			0
$565$		0			0
$566$	−15.3849			−0.646674
$567$		0			0
$568$		−	27.0471i		−	1.13487i
$569$	−23.4033	+	40.5357i	−0.981118	+	1.69935i	−0.323062	+	0.946378i	$0.604712\pi$
$569$	−23.4033	+	40.5357i	−0.981118	+	1.69935i	−0.658056	+	0.752969i	$0.728621\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$	18.0938	−	10.4465i	0.756541	−	0.436789i
$573$		0			0
$574$	−0.600830	+	1.04067i	−0.0250782	+	0.0434367i
$575$		0			0
$576$		0			0
$577$			28.2386i			1.17559i	0.809010	+	0.587794i	$0.200004\pi$
$577$			28.2386i			1.17559i	−0.809010	+	0.587794i	$0.799996\pi$
$578$	4.88228	+	2.81879i	0.203076	+	0.117246i
$579$		0			0
$580$		0			0
$581$	−3.05880	−	5.29801i	−0.126901	−	0.219798i
$582$		0			0
$583$	−2.64802	−	1.52884i	−0.109670	−	0.0633180i
$584$	3.59668			0.148832
$585$		0			0
$586$	7.08405			0.292639
$587$	15.6598	+	9.04118i	0.646348	+	0.373169i	0.787056	−	0.616882i	$-0.211604\pi$
$587$	15.6598	+	9.04118i	0.646348	+	0.373169i	−0.140707	+	0.990051i	$0.544938\pi$
$588$		0			0
$589$	16.2633	+	28.1688i	0.670117	+	1.16068i
$590$		0			0
$591$		0			0
$592$	3.37336	+	1.94761i	0.138644	+	0.0800462i
$593$		−	7.73840i		−	0.317778i	−0.987296	−	0.158889i	$-0.949209\pi$
$593$		−	7.73840i		−	0.317778i	0.987296	−	0.158889i	$-0.0507912\pi$
$594$		0			0
$595$		0			0
$596$	16.7919	−	29.0845i	0.687824	−	1.19135i
$597$		0			0
$598$	13.6978	−	7.90841i	0.560143	−	0.323399i
$599$	13.9608	+	24.1808i	0.570423	+	0.988001i	0.996522	+	0.0833249i	$0.0265539\pi$
$599$	13.9608	+	24.1808i	0.570423	+	0.988001i	−0.426100	+	0.904676i	$0.640113\pi$
$600$		0			0
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	0.145586	+	0.989346i	$0.453493\pi$
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	−0.929591	+	0.368592i	$0.879840\pi$
$602$		−	0.384851i		−	0.0156853i
$603$		0			0
$604$	5.08405			0.206867
$605$		0			0
$606$		0			0
$607$	−0.554113	+	0.319917i	−0.0224907	+	0.0129850i	−0.511203	−	0.859460i	$-0.670800\pi$
$607$	−0.554113	+	0.319917i	−0.0224907	+	0.0129850i	0.488712	+	0.872445i	$0.337467\pi$
$608$	21.9660	−	12.6821i	0.890838	−	0.514325i
$609$		0			0
$610$		0			0
$611$	−32.5081			−1.31514
$612$		0			0
$613$		−	42.7467i		−	1.72652i	−0.504757	−	0.863262i	$-0.668418\pi$
$613$		−	42.7467i		−	1.72652i	0.504757	−	0.863262i	$-0.331582\pi$
$614$	0.713157	−	1.23522i	0.0287807	−	0.0498496i
$615$		0			0
$616$	4.00924	+	6.94420i	0.161537	+	0.279790i
$617$	−18.2753	+	10.5513i	−0.735737	+	0.424778i	−0.820517	−	0.571622i	$-0.806314\pi$
$617$	−18.2753	+	10.5513i	−0.735737	+	0.424778i	0.0847805	+	0.996400i	$0.472981\pi$
$618$		0			0
$619$	−6.82605	+	11.8231i	−0.274362	+	0.475209i	−0.969974	−	0.243209i	$-0.921800\pi$
$619$	−6.82605	+	11.8231i	−0.274362	+	0.475209i	0.695612	+	0.718418i	$0.255133\pi$
$620$		0			0
$621$		0			0
$622$		−	13.8538i		−	0.555485i
$623$	3.71007	+	2.14201i	0.148641	+	0.0858178i
$624$		0			0
$625$		0			0
$626$	−10.0339	−	17.3793i	−0.401036	−	0.694615i
$627$		0			0
$628$	−0.292148	−	0.168672i	−0.0116580	−	0.00673073i
$629$	−4.85601			−0.193622
$630$		0			0
$631$	33.2593			1.32403			0.662017	−	0.749489i	$-0.269701\pi$
$631$	33.2593			1.32403			0.662017	+	0.749489i	$0.269701\pi$
$632$	0.523930	+	0.302491i	0.0208408	+	0.0120325i
$633$		0			0
$634$	−2.99472	−	5.18700i	−0.118935	−	0.206002i
$635$		0			0
$636$		0			0
$637$	20.0753	+	11.5905i	0.795411	+	0.459231i
$638$		−	14.5081i		−	0.574381i
$639$		0			0
$640$		0			0
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	−0.480642	−	0.876917i	$-0.659596\pi$
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	0.999753	−	0.0222106i	$-0.00707044\pi$
$642$		0			0
$643$	−17.8250	+	10.2913i	−0.702950	+	0.405848i	−0.808445	−	0.588571i	$-0.799691\pi$
$643$	−17.8250	+	10.2913i	−0.702950	+	0.405848i	0.105495	+	0.994420i	$0.466357\pi$
$644$	−7.06804	−	12.2422i	−0.278520	−	0.482410i
$645$		0			0
$646$	−3.57199	+	6.18687i	−0.140538	+	0.243419i
$647$		−	23.2527i		−	0.914159i	−0.889426	−	0.457079i	$-0.848896\pi$
$647$		−	23.2527i		−	0.914159i	0.889426	−	0.457079i	$-0.151104\pi$
$648$		0			0
$649$	3.05767			0.120024
$650$		0			0
$651$		0			0
$652$	−25.8114	+	14.9022i	−1.01085	+	0.583615i
$653$	16.2459	−	9.37957i	0.635751	−	0.367051i	−0.147225	−	0.989103i	$-0.547034\pi$
$653$	16.2459	−	9.37957i	0.635751	−	0.367051i	0.782976	+	0.622052i	$0.213701\pi$
$654$		0			0
$655$		0			0
$656$	3.15415			0.123149
$657$		0			0
$658$		−	5.68242i		−	0.221524i
$659$	0.140034	−	0.242545i	0.00545494	−	0.00944823i	−0.863285	−	0.504717i	$-0.831597\pi$
$659$	0.140034	−	0.242545i	0.00545494	−	0.00944823i	0.868740	+	0.495268i	$0.164930\pi$
$660$		0			0
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	0.935496	+	0.353336i	$0.114953\pi$
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	−0.161750	+	0.986832i	$0.551714\pi$
$662$	−8.31097	+	4.79834i	−0.323015	+	0.186493i
$663$		0			0
$664$	4.50000	−	7.79423i	0.174634	−	0.302475i
$665$		0			0
$666$		0			0
$667$			56.1562i			2.17438i
$668$	20.4280	+	11.7941i	0.790382	+	0.456327i
$669$		0			0
$670$		0			0
$671$	−3.37957	−	5.85358i	−0.130467	−	0.225975i
$672$		0			0
$673$	29.0368	+	16.7644i	1.11929	+	0.646221i	0.941219	−	0.337797i	$-0.109682\pi$
$673$	29.0368	+	16.7644i	1.11929	+	0.646221i	0.178068	+	0.984018i	$0.443015\pi$
$674$	−15.5473			−0.598860
$675$		0			0
$676$	−14.7799			−0.568456
$677$	23.8048	+	13.7437i	0.914891	+	0.528213i	0.882002	−	0.471246i	$-0.156196\pi$
$677$	23.8048	+	13.7437i	0.914891	+	0.528213i	0.0328897	+	0.999459i	$0.489529\pi$
$678$		0			0
$679$	5.59442	+	9.68981i	0.214694	+	0.371861i
$680$		0			0
$681$		0			0
$682$	−9.21618	−	5.32096i	−0.352906	−	0.203750i
$683$		−	34.5865i		−	1.32342i	−0.749762	−	0.661708i	$-0.769832\pi$
$683$		−	34.5865i		−	1.32342i	0.749762	−	0.661708i	$-0.230168\pi$
$684$		0			0
$685$		0			0
$686$	−4.88485	+	8.46081i	−0.186504	+	0.323035i
$687$		0			0
$688$	−0.874830	+	0.505083i	−0.0333526	+	0.0192561i
$689$	2.67282	+	4.62947i	0.101826	+	0.176369i
$690$		0			0
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	0.160282	+	0.987071i	$0.448760\pi$
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	−0.934970	+	0.354727i	$0.884574\pi$
$692$			7.30418i			0.277663i
$693$		0			0
$694$	13.4795			0.511674
$695$		0			0
$696$		0			0
$697$	−3.40535	+	1.96608i	−0.128987	+	0.0744706i
$698$	−5.30632	+	3.06360i	−0.200847	+	0.115959i
$699$		0			0
$700$		0			0
$701$	19.4712			0.735416			0.367708	−	0.929941i	$-0.380143\pi$
$701$	19.4712			0.735416			0.367708	+	0.929941i	$0.380143\pi$
$702$		0			0
$703$			8.48963i			0.320193i
$704$	1.58123	−	2.73877i	0.0595948	−	0.103221i
$705$		0			0
$706$	−7.79834	−	13.5071i	−0.293494	−	0.508347i
$707$	5.19615	−	3.00000i	0.195421	−	0.112827i
$708$		0			0
$709$	7.54316	−	13.0651i	0.283289	−	0.490671i	−0.688904	−	0.724853i	$-0.741908\pi$
$709$	7.54316	−	13.0651i	0.283289	−	0.490671i	0.972193	+	0.234182i	$0.0752410\pi$
$710$		0			0
$711$		0			0
$712$			6.30249i			0.236196i
$713$	35.6729	+	20.5957i	1.33596	+	0.771317i
$714$		0			0
$715$		0			0
$716$	12.6821	+	21.9660i	0.473951	+	0.820907i
$717$		0			0
$718$	−5.29081	−	3.05465i	−0.197451	−	0.113999i
$719$	−3.43196			−0.127990			−0.0639952	−	0.997950i	$-0.520384\pi$
$719$	−3.43196			−0.127990			−0.0639952	+	0.997950i	$0.520384\pi$
$720$		0			0
$721$	−2.59442			−0.0966211
$722$	1.40449	+	0.810881i	0.0522696	+	0.0301779i
$723$		0			0
$724$	−2.67791	−	4.63827i	−0.0995236	−	0.172380i
$725$		0			0
$726$		0			0
$727$	30.9789	+	17.8857i	1.14895	+	0.663344i	0.948630	−	0.316388i	$-0.102470\pi$
$727$	30.9789	+	17.8857i	1.14895	+	0.663344i	0.200315	+	0.979732i	$0.435803\pi$
$728$		−	14.0185i		−	0.519559i
$729$		0			0
$730$		0			0
$731$	0.629668	−	1.09062i	0.0232891	−	0.0403379i
$732$		0			0
$733$	19.0526	−	11.0000i	0.703722	−	0.406294i	−0.105010	−	0.994471i	$-0.533487\pi$
$733$	19.0526	−	11.0000i	0.703722	−	0.406294i	0.808732	+	0.588177i	$0.200154\pi$
$734$	2.42272	+	4.19628i	0.0894244	+	0.154888i
$735$		0			0
$736$	16.0605	−	27.8176i	0.591998	−	1.02537i
$737$			17.6151i			0.648862i
$738$		0			0
$739$	−6.08631			−0.223889			−0.111944	−	0.993714i	$-0.535708\pi$
$739$	−6.08631			−0.223889			−0.111944	+	0.993714i	$0.535708\pi$
$740$		0			0
$741$		0			0
$742$	−0.809231	+	0.467210i	−0.0297078	+	0.0171518i
$743$	−22.0853	+	12.7509i	−0.810231	+	0.467787i	−0.847036	−	0.531536i	$-0.821615\pi$
$743$	−22.0853	+	12.7509i	−0.810231	+	0.467787i	0.0368054	+	0.999322i	$0.488282\pi$
$744$		0			0
$745$		0			0
$746$	−5.79043			−0.212003
$747$		0			0
$748$			11.9506i			0.436958i
$749$	−8.50924	+	14.7384i	−0.310921	+	0.538530i
$750$		0			0
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	0.983596	−	0.180384i	$-0.0577342\pi$
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	−0.648016	+	0.761627i	$0.724401\pi$
$752$	−12.9171	+	7.45769i	−0.471038	+	0.271954i
$753$		0			0
$754$	−12.6821	+	21.9660i	−0.461853	+	0.799953i
$755$		0			0
$756$		0			0
$757$		−	41.8986i		−	1.52283i	−0.648264	−	0.761415i	$-0.724505\pi$
$757$		−	41.8986i		−	1.52283i	0.648264	−	0.761415i	$-0.275495\pi$
$758$	−5.90549	−	3.40954i	−0.214497	−	0.123840i
$759$		0			0
$760$		0			0
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	0.784698	−	0.619878i	$-0.212818\pi$
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	−0.929179	+	0.369630i	$0.879485\pi$
$762$		0			0
$763$	−20.5935	−	11.8896i	−0.745534	−	0.430434i
$764$	4.74671			0.171730
$765$		0			0
$766$	5.61515			0.202884
$767$	−4.62947	−	2.67282i	−0.167160	−	0.0965101i
$768$		0			0
$769$	3.01432	+	5.22095i	0.108699	+	0.188272i	0.915244	−	0.402901i	$-0.131998\pi$
$769$	3.01432	+	5.22095i	0.108699	+	0.188272i	−0.806544	+	0.591174i	$0.798665\pi$
$770$		0			0
$771$		0			0
$772$	−27.2186	−	15.7147i	−0.979618	−	0.565583i
$773$			44.4033i			1.59708i	0.601944	+	0.798538i	$0.294393\pi$
$773$			44.4033i			1.59708i	−0.601944	+	0.798538i	$0.705607\pi$
$774$		0			0
$775$		0			0
$776$	−8.23030	+	14.2553i	−0.295451	+	0.511735i
$777$		0			0
$778$	−4.56729	+	2.63693i	−0.163745	+	0.0945384i
$779$	3.43724	+	5.95348i	0.123152	+	0.213305i
$780$		0			0
$781$	17.2056	−	29.8010i	0.615665	−	1.06636i
$782$			9.04711i			0.323524i
$783$		0			0
$784$	10.6359			0.379853
$785$		0			0
$786$		0			0
$787$	14.5817	−	8.41877i	0.519783	−	0.300097i	−0.217063	−	0.976158i	$-0.569648\pi$
$787$	14.5817	−	8.41877i	0.519783	−	0.300097i	0.736846	+	0.676061i	$0.236314\pi$
$788$	−8.45362	+	4.88070i	−0.301148	+	0.173868i
$789$		0			0
$790$		0			0
$791$	−28.7361			−1.02174
$792$		0			0
$793$			11.8168i			0.419627i
$794$	6.57199	−	11.3830i	0.233231	−	0.403968i
$795$		0			0
$796$	10.9370	+	18.9435i	0.387653	+	0.671435i
$797$	28.9010	−	16.6860i	1.02373	−	0.591049i	0.108545	−	0.994091i	$-0.465381\pi$
$797$	28.9010	−	16.6860i	1.02373	−	0.591049i	0.915181	+	0.403043i	$0.132047\pi$
$798$		0			0
$799$	9.29721	−	16.1032i	0.328912	−	0.569692i
$800$		0			0
$801$		0			0
$802$		−	6.32944i		−	0.223500i
$803$	3.96289	+	2.28797i	0.139847	+	0.0807408i
$804$		0			0
$805$		0			0
$806$	9.30249	+	16.1124i	0.327666	+	0.567535i
$807$		0			0
$808$	7.64439	+	4.41349i	0.268929	+	0.155266i
$809$	29.1809			1.02595			0.512973	−	0.858404i	$-0.328544\pi$
$809$	29.1809			1.02595			0.512973	+	0.858404i	$0.328544\pi$
$810$		0			0
$811$	−15.5552			−0.546217			−0.273109	−	0.961983i	$-0.588052\pi$
$811$	−15.5552			−0.546217			−0.273109	+	0.961983i	$0.588052\pi$
$812$	19.6318	+	11.3344i	0.688942	+	0.397761i
$813$		0			0
$814$	−1.38880	−	2.40548i	−0.0486775	−	0.0843120i
$815$		0			0
$816$		0			0
$817$	−1.90669	−	1.10083i	−0.0667068	−	0.0385132i
$818$		−	10.0863i		−	0.352660i
$819$		0			0
$820$		0			0
$821$	−11.8588	+	20.5401i	−0.413876	+	0.716855i	−0.995310	−	0.0967393i	$-0.969159\pi$
$821$	−11.8588	+	20.5401i	−0.413876	+	0.716855i	0.581434	+	0.813594i	$0.302492\pi$
$822$		0			0
$823$	−16.7732	+	9.68404i	−0.584678	+	0.337564i	−0.762990	−	0.646410i	$-0.776270\pi$
$823$	−16.7732	+	9.68404i	−0.584678	+	0.337564i	0.178312	+	0.983974i	$0.442936\pi$
$824$	−1.90841	−	3.30545i	−0.0664824	−	0.115151i
$825$		0			0
$826$	0.467210	−	0.809231i	0.0162563	−	0.0281568i
$827$		−	52.2241i		−	1.81601i	−0.418960	−	0.908005i	$-0.637605\pi$
$827$		−	52.2241i		−	1.81601i	0.418960	−	0.908005i	$-0.362395\pi$
$828$		0			0
$829$	26.6442			0.925391			0.462695	−	0.886517i	$-0.346882\pi$
$829$	26.6442			0.925391			0.462695	+	0.886517i	$0.346882\pi$
$830$		0			0
$831$		0			0
$832$	−4.78811	+	2.76442i	−0.165998	+	0.0958390i
$833$	−11.4829	+	6.62967i	−0.397860	+	0.229704i
$834$		0			0
$835$		0			0
$836$	20.8930			0.722598
$837$		0			0
$838$		−	21.1836i		−	0.731775i
$839$	12.5196	−	21.6846i	0.432225	−	0.748635i	−0.564840	−	0.825201i	$-0.691062\pi$
$839$	12.5196	−	21.6846i	0.432225	−	0.748635i	0.997065	+	0.0765655i	$0.0243954\pi$
$840$		0			0
$841$	−30.5266	−	52.8736i	−1.05264	−	1.82323i
$842$	−2.50537	+	1.44648i	−0.0863409	+	0.0498489i
$843$		0			0
$844$	−7.01319	+	12.1472i	−0.241404	+	0.418124i
$845$		0			0
$846$		0			0
$847$		−	5.50641i		−	0.189203i
$848$	2.12409	+	1.22635i	0.0729417	+	0.0421129i
$849$		0			0
$850$		0			0
$851$	5.37562	+	9.31084i	0.184274	+	0.319171i
$852$		0			0
$853$	−9.41758	−	5.43724i	−0.322452	−	0.186168i	0.330033	−	0.943969i	$-0.392940\pi$
$853$	−9.41758	−	5.43724i	−0.322452	−	0.186168i	−0.652485	+	0.757802i	$0.726273\pi$
$854$	−2.06558			−0.0706827
$855$		0			0
$856$	−25.0369			−0.855745
$857$	−16.1032	−	9.29721i	−0.550076	−	0.317587i	0.199077	−	0.979984i	$-0.436206\pi$
$857$	−16.1032	−	9.29721i	−0.550076	−	0.317587i	−0.749153	+	0.662397i	$0.769539\pi$
$858$		0			0
$859$	−2.33246	−	4.03994i	−0.0795825	−	0.137841i	0.823487	−	0.567335i	$-0.192025\pi$
$859$	−2.33246	−	4.03994i	−0.0795825	−	0.137841i	−0.903070	+	0.429494i	$0.858692\pi$
$860$		0			0
$861$		0			0
$862$	2.59088	+	1.49585i	0.0882459	+	0.0509488i
$863$		−	28.0594i		−	0.955152i	−0.878590	−	0.477576i	$-0.841516\pi$
$863$		−	28.0594i		−	0.955152i	0.878590	−	0.477576i	$-0.158484\pi$
$864$		0			0
$865$		0			0
$866$	9.82209	−	17.0124i	0.333768	−	0.578104i
$867$		0			0
$868$	14.4002	−	8.31399i	0.488776	−	0.282195i
$869$	0.384851	+	0.666581i	0.0130552	+	0.0226122i
$870$		0			0
$871$	15.3980	−	26.6702i	0.521743	−	0.903685i
$872$		−	34.9832i		−	1.18468i
$873$		0			0
$874$	15.8168			0.535012
$875$		0			0
$876$		0			0
$877$	30.0236	−	17.3342i	1.01383	−	0.585333i	0.101516	−	0.994834i	$-0.467631\pi$
$877$	30.0236	−	17.3342i	1.01383	−	0.585333i	0.912310	+	0.409501i	$0.134297\pi$
$878$	−9.68297	+	5.59046i	−0.326784	+	0.188669i
$879$		0			0
$880$		0			0
$881$	5.29854			0.178512			0.0892561	−	0.996009i	$-0.471551\pi$
$881$	5.29854			0.178512			0.0892561	+	0.996009i	$0.471551\pi$
$882$		0			0
$883$			10.3025i			0.346706i	0.984860	+	0.173353i	$0.0554602\pi$
$883$			10.3025i			0.346706i	−0.984860	+	0.173353i	$0.944540\pi$
$884$	10.4465	−	18.0938i	0.351353	−	0.608561i
$885$		0			0
$886$	−3.00415	−	5.20334i	−0.100926	−	0.174810i
$887$	−35.8971	+	20.7252i	−1.20531	+	0.695885i	−0.961731	−	0.273997i	$-0.911654\pi$
$887$	−35.8971	+	20.7252i	−1.20531	+	0.695885i	−0.243577	+	0.969882i	$0.578321\pi$
$888$		0			0
$889$	1.56163	−	2.70482i	0.0523753	−	0.0907167i
$890$		0			0
$891$		0			0
$892$		−	15.3338i		−	0.513413i
$893$	−28.1528	−	16.2541i	−0.942099	−	0.543921i
$894$		0			0
$895$		0			0
$896$	−8.23445	−	14.2625i	−0.275094	−	0.476476i
$897$		0			0
$898$	11.3220	+	6.53674i	0.377819	+	0.218134i
$899$	−66.0554			−2.20307
$900$		0			0
$901$	−3.05767			−0.101866
$902$	−1.94784	−	1.12458i	−0.0648559	−	0.0374446i
$903$		0			0
$904$	−21.1378	−	36.6117i	−0.703032	−	1.21769i
$905$		0			0
$906$		0			0
$907$	14.5424	+	8.39606i	0.482873	+	0.278787i	0.721613	−	0.692297i	$-0.243401\pi$
$907$	14.5424	+	8.39606i	0.482873	+	0.278787i	−0.238740	+	0.971083i	$0.576734\pi$
$908$			4.47116i			0.148381i
$909$		0			0
$910$		0			0
$911$	18.6768	−	32.3491i	0.618789	−	1.07177i	−0.370918	−	0.928666i	$-0.620957\pi$
$911$	18.6768	−	32.3491i	0.618789	−	1.07177i	0.989707	−	0.143109i	$-0.0457098\pi$
$912$		0			0
$913$	9.91636	−	5.72522i	0.328184	−	0.189477i
$914$	4.01150	+	6.94812i	0.132689	+	0.229823i
$915$		0			0
$916$	2.13060	−	3.69031i	0.0703970	−	0.121931i
$917$			8.56804i			0.282942i
$918$		0			0
$919$	−37.1316			−1.22486			−0.612429	−	0.790526i	$-0.709807\pi$
$919$	−37.1316			−1.22486			−0.612429	+	0.790526i	$0.709807\pi$
$920$		0			0
$921$		0			0
$922$	−11.9394	+	6.89324i	−0.393205	+	0.227017i
$923$	−52.1003	+	30.0801i	−1.71490	+	0.990098i
$924$		0			0
$925$		0			0
$926$	19.4135			0.637967
$927$		0			0
$928$			51.5098i			1.69089i
$929$	23.9977	−	41.5653i	0.787340	−	1.36371i	−0.140251	−	0.990116i	$-0.544791\pi$
$929$	23.9977	−	41.5653i	0.787340	−	1.36371i	0.927591	−	0.373597i	$-0.121876\pi$
$930$		0			0
$931$	11.5905	+	20.0753i	0.379862	+	0.657941i
$932$	−9.01115	+	5.20259i	−0.295170	+	0.170417i
$933$		0			0
$934$	−7.82831	+	13.5590i	−0.256150	+	0.443665i
$935$		0			0
$936$		0			0
$937$			22.0079i			0.718967i	0.933151	+	0.359483i	$0.117047\pi$
$937$			22.0079i			0.718967i	−0.933151	+	0.359483i	$0.882953\pi$
$938$	4.66195	+	2.69158i	0.152218	+	0.0878832i
$939$		0			0
$940$		0			0
$941$	20.2921	+	35.1470i	0.661504	+	1.14576i	0.980220	+	0.197909i	$0.0634150\pi$
$941$	20.2921	+	35.1470i	0.661504	+	1.14576i	−0.318716	+	0.947850i	$0.603252\pi$
$942$		0			0
$943$	7.53946	+	4.35291i	0.245518	+	0.141750i
$944$	−2.45269			−0.0798283
$945$		0			0
$946$	0.720331			0.0234200
$947$	26.1817	+	15.1160i	0.850790	+	0.491204i	0.860917	−	0.508745i	$-0.169890\pi$
$947$	26.1817	+	15.1160i	0.850790	+	0.491204i	−0.0101273	+	0.999949i	$0.503224\pi$
$948$		0			0
$949$	−4.00000	−	6.92820i	−0.129845	−	0.224899i
$950$		0			0
$951$		0			0
$952$	6.94420	+	4.00924i	0.225063	+	0.129940i
$953$			2.50811i			0.0812455i	0.999175	+	0.0406227i	$0.0129342\pi$
$953$			2.50811i			0.0812455i	−0.999175	+	0.0406227i	$0.987066\pi$
$954$		0			0
$955$		0			0
$956$	−6.79439	+	11.7682i	−0.219746	+	0.380612i
$957$		0			0
$958$	−2.59088	+	1.49585i	−0.0837077	+	0.0483286i
$959$	7.28402	+	12.6163i	0.235213	+	0.407401i
$960$		0			0
$961$	−8.72635	+	15.1145i	−0.281495	+	0.487564i
$962$			4.85601i			0.156564i
$963$		0			0
$964$	−44.1025			−1.42045
$965$		0			0
$966$		0			0
$967$	−18.8386	+	10.8765i	−0.605808	+	0.349763i	−0.771323	−	0.636444i	$-0.780405\pi$
$967$	−18.8386	+	10.8765i	−0.605808	+	0.349763i	0.165515	+	0.986207i	$0.447071\pi$
$968$	7.01552	−	4.05042i	0.225488	−	0.130185i
$969$		0			0
$970$		0			0
$971$	−22.7512			−0.730122			−0.365061	−	0.930984i	$-0.618952\pi$
$971$	−22.7512			−0.730122			−0.365061	+	0.930984i	$0.618952\pi$
$972$		0			0
$973$		−	11.4241i		−	0.366238i
$974$	6.86920	−	11.8978i	0.220103	−	0.381230i
$975$		0			0
$976$	2.71090	+	4.69541i	0.0867737	+	0.150296i
$977$	−34.0156	+	19.6389i	−1.08825	+	0.628304i	−0.933111	−	0.359588i	$-0.882917\pi$
$977$	−34.0156	+	19.6389i	−1.08825	+	0.628304i	−0.155143	+	0.987892i	$0.549584\pi$
$978$		0			0
$979$	−4.00924	+	6.94420i	−0.128136	+	0.221938i
$980$		0			0
$981$		0			0
$982$			8.45269i			0.269736i
$983$	−29.2629	−	16.8949i	−0.933341	−	0.538865i	−0.0454743	−	0.998966i	$-0.514480\pi$
$983$	−29.2629	−	16.8949i	−0.933341	−	0.538865i	−0.887867	+	0.460101i	$0.847813\pi$
$984$		0			0
$985$		0			0
$986$	−7.25405	−	12.5644i	−0.231016	−	0.400132i
$987$		0			0
$988$	−31.6329	−	18.2633i	−1.00638	−	0.581033i
$989$	−2.78817			−0.0886587
$990$		0			0
$991$	−23.7983			−0.755979			−0.377990	−	0.925810i	$-0.623384\pi$
$991$	−23.7983			−0.755979			−0.377990	+	0.925810i	$0.623384\pi$
$992$	32.7213	+	18.8916i	1.03890	+	0.599810i
$993$		0			0
$994$	−5.25801	−	9.10713i	−0.166774	−	0.288861i
$995$		0			0
$996$		0			0
$997$	−44.7549	−	25.8392i	−1.41740	−	0.818337i	−0.421331	−	0.906907i	$-0.638437\pi$
$997$	−44.7549	−	25.8392i	−1.41740	−	0.818337i	−0.996070	+	0.0885702i	$0.971770\pi$
$998$			14.2175i			0.450046i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.k.b.424.4		12
3.2	odd	2		225.2.k.b.124.3		12
5.2	odd	4		675.2.e.b.451.2		6
5.3	odd	4		135.2.e.b.46.2		6
5.4	even	2	inner	675.2.k.b.424.3		12
9.2	odd	6		2025.2.b.l.649.4		6
9.4	even	3	inner	675.2.k.b.199.3		12
9.5	odd	6		225.2.k.b.49.4		12
9.7	even	3		2025.2.b.m.649.3		6
15.2	even	4		225.2.e.b.151.2		6
15.8	even	4		45.2.e.b.16.2	✓	6
15.14	odd	2		225.2.k.b.124.4		12
20.3	even	4		2160.2.q.k.721.2		6
45.2	even	12		2025.2.a.n.1.2		3
45.4	even	6	inner	675.2.k.b.199.4		12
45.7	odd	12		2025.2.a.o.1.2		3
45.13	odd	12		135.2.e.b.91.2		6
45.14	odd	6		225.2.k.b.49.3		12
45.22	odd	12		675.2.e.b.226.2		6
45.23	even	12		45.2.e.b.31.2	yes	6
45.29	odd	6		2025.2.b.l.649.3		6
45.32	even	12		225.2.e.b.76.2		6
45.34	even	6		2025.2.b.m.649.4		6
45.38	even	12		405.2.a.j.1.2		3
45.43	odd	12		405.2.a.i.1.2		3
60.23	odd	4		720.2.q.i.241.2		6
180.23	odd	12		720.2.q.i.481.2		6
180.43	even	12		6480.2.a.bs.1.2		3
180.83	odd	12		6480.2.a.bv.1.2		3
180.103	even	12		2160.2.q.k.1441.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.2	✓	6	15.8	even	4
45.2.e.b.31.2	yes	6	45.23	even	12
135.2.e.b.46.2		6	5.3	odd	4
135.2.e.b.91.2		6	45.13	odd	12
225.2.e.b.76.2		6	45.32	even	12
225.2.e.b.151.2		6	15.2	even	4
225.2.k.b.49.3		12	45.14	odd	6
225.2.k.b.49.4		12	9.5	odd	6
225.2.k.b.124.3		12	3.2	odd	2
225.2.k.b.124.4		12	15.14	odd	2
405.2.a.i.1.2		3	45.43	odd	12
405.2.a.j.1.2		3	45.38	even	12
675.2.e.b.226.2		6	45.22	odd	12
675.2.e.b.451.2		6	5.2	odd	4
675.2.k.b.199.3		12	9.4	even	3	inner
675.2.k.b.199.4		12	45.4	even	6	inner
675.2.k.b.424.3		12	5.4	even	2	inner
675.2.k.b.424.4		12	1.1	even	1	trivial
720.2.q.i.241.2		6	60.23	odd	4
720.2.q.i.481.2		6	180.23	odd	12
2025.2.a.n.1.2		3	45.2	even	12
2025.2.a.o.1.2		3	45.7	odd	12
2025.2.b.l.649.3		6	45.29	odd	6
2025.2.b.l.649.4		6	9.2	odd	6
2025.2.b.m.649.3		6	9.7	even	3
2025.2.b.m.649.4		6	45.34	even	6
2160.2.q.k.721.2		6	20.3	even	4
2160.2.q.k.1441.2		6	180.103	even	12
6480.2.a.bs.1.2		3	180.43	even	12
6480.2.a.bv.1.2		3	180.83	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.k (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.495361 + 0.285997i) q^{2} +(-0.836412 - 1.44871i) q^{4} +(-1.23669 - 0.714003i) q^{7} -2.10083i q^{8} +O(q^{10})\)	\(q+(0.495361 + 0.285997i) q^{2} +(-0.836412 - 1.44871i) q^{4} +(-1.23669 - 0.714003i) q^{7} -2.10083i q^{8} +(1.33641 - 2.31473i) q^{11} +(-4.04678 + 2.33641i) q^{13} +(-0.408405 - 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} -2.67282i q^{17} -4.67282 q^{19} +(1.32401 - 0.764419i) q^{22} +(-5.12483 + 2.95882i) q^{23} -2.67282 q^{26} +2.38880i q^{28} +(4.74482 - 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(-4.70079 + 2.71400i) q^{32} +(0.764419 - 1.32401i) q^{34} -1.81681i q^{37} +(-2.31473 - 1.33641i) q^{38} +(-0.735581 - 1.27406i) q^{41} +(0.408039 + 0.235581i) q^{43} -4.47116 q^{44} -3.38485 q^{46} +(6.02480 + 3.47842i) q^{47} +(-2.48040 - 4.29618i) q^{49} +(6.76956 + 3.90841i) q^{52} -1.14399i q^{53} +(-1.50000 + 2.59808i) q^{56} +(4.70079 - 2.71400i) q^{58} +(0.571993 + 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} -3.98153i q^{62} +1.18319 q^{64} +(-5.70751 + 3.29523i) q^{67} +(-3.87214 + 2.23558i) q^{68} +12.8745 q^{71} +1.71203i q^{73} +(0.519602 - 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(-3.30545 + 1.90841i) q^{77} +(-0.143987 + 0.249392i) q^{79} -0.841495i q^{82} +(3.71007 + 2.14201i) q^{83} +(0.134751 + 0.233396i) q^{86} +(-4.86286 - 2.80757i) q^{88} -3.00000 q^{89} +6.67282 q^{91} +(8.57293 + 4.94958i) q^{92} +(1.98963 + 3.44615i) q^{94} +(-6.78555 - 3.91764i) q^{97} -2.83754i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{4}+O(q^{10}) \) 12 * q + 10 * q^4	\( 12 q + 10 q^{4} - 4 q^{11} + 18 q^{14} - 10 q^{16} - 16 q^{19} + 8 q^{26} + 14 q^{29} - 16 q^{31} - 8 q^{34} - 26 q^{41} - 88 q^{44} - 12 q^{46} - 4 q^{49} - 18 q^{56} + 4 q^{59} - 2 q^{61} + 60 q^{64} + 40 q^{71} + 32 q^{74} + 24 q^{76} + 4 q^{79} + 56 q^{86} - 36 q^{89} + 40 q^{91} - 62 q^{94}+O(q^{100}) \) 12 * q + 10 * q^4 - 4 * q^11 + 18 * q^14 - 10 * q^16 - 16 * q^19 + 8 * q^26 + 14 * q^29 - 16 * q^31 - 8 * q^34 - 26 * q^41 - 88 * q^44 - 12 * q^46 - 4 * q^49 - 18 * q^56 + 4 * q^59 - 2 * q^61 + 60 * q^64 + 40 * q^71 + 32 * q^74 + 24 * q^76 + 4 * q^79 + 56 * q^86 - 36 * q^89 + 40 * q^91 - 62 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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