LMFDB - Embedded newform 252.7.z.f.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,7,Mod(73,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("252.73");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	252.z (of order $6$, degree $2$, 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:	$57.9736290722$
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} + 6586 x^{10} - 4380 x^{9} + 29743412 x^{8} - 22316460 x^{7} + 72097469286 x^{6} + \cdots + 78\!\cdots\!61 $ x^12 + 6586x^10 - 4380x^9 + 29743412x^8 - 22316460x^7 + 72097469286x^6 - 74114131680x^5 + 127603094377722x^4 - 68863099720860x^3 + 120620660396154896x^2 + 69805069827473280x + 78212563531160784961
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{9}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Root			$-17.5255 - 30.3551i$ of defining polynomial
Character	$\chi$	$=$	252.145
Dual form			252.7.z.f.73.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(111.254 + 64.2323i) q^{5} +(55.4571 - 338.487i) q^{7} +O(q^{10})$	$q+(111.254 + 64.2323i) q^{5} +(55.4571 - 338.487i) q^{7} +(-578.419 - 1001.85i) q^{11} -1644.18i q^{13} +(-6301.05 + 3637.91i) q^{17} +(-870.055 - 502.327i) q^{19} +(-3764.87 + 6520.94i) q^{23} +(439.085 + 760.517i) q^{25} -12913.2 q^{29} +(-8719.05 + 5033.95i) q^{31} +(27911.6 - 34095.8i) q^{35} +(-16874.4 + 29227.3i) q^{37} -18172.4i q^{41} -6598.85 q^{43} +(173182. + 99986.5i) q^{47} +(-111498. - 37543.0i) q^{49} +(-60090.5 - 104080. i) q^{53} -148613. i q^{55} +(-185327. + 106999. i) q^{59} +(-205526. - 118660. i) q^{61} +(105609. - 182921. i) q^{65} +(-95226.0 - 164936. i) q^{67} +49009.2 q^{71} +(-456525. + 263575. i) q^{73} +(-371191. + 140228. i) q^{77} +(-169200. + 293062. i) q^{79} +357882. i q^{83} -934687. q^{85} +(-229350. - 132415. i) q^{89} +(-556532. - 91181.3i) q^{91} +(-64531.2 - 111771. i) q^{95} -213971. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 714 q^{7}+O(q^{10}) $ 12 * q + 714 * q^7	$ 12 q + 714 q^{7} - 59466 q^{19} + 64086 q^{25} + 219834 q^{31} - 166242 q^{37} - 326916 q^{43} - 132006 q^{49} + 736008 q^{61} + 176238 q^{67} - 2931018 q^{73} + 449490 q^{79} + 2174592 q^{85} - 808542 q^{91}+O(q^{100}) $ 12 * q + 714 * q^7 - 59466 * q^19 + 64086 * q^25 + 219834 * q^31 - 166242 * q^37 - 326916 * q^43 - 132006 * q^49 + 736008 * q^61 + 176238 * q^67 - 2931018 * q^73 + 449490 * q^79 + 2174592 * q^85 - 808542 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	111.254	+	64.2323i	0.890029	+	0.513859i	0.873952	−	0.486012i	$-0.161549\pi$
$5$	111.254	+	64.2323i	0.890029	+	0.513859i	0.0160772	+	0.999871i	$0.494882\pi$
$6$		0			0
$7$	55.4571	−	338.487i	0.161683	−	0.986843i
$7$	55.4571	−	338.487i	0.161683	−	0.986843i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−578.419	−	1001.85i	−0.434575	−	0.752705i	0.562686	−	0.826671i	$-0.309768\pi$
$11$	−578.419	−	1001.85i	−0.434575	−	0.752705i	−0.997261	+	0.0739652i	$0.976435\pi$
$12$		0			0
$13$		−	1644.18i		−	0.748373i	−0.927353	−	0.374187i	$-0.877922\pi$
$13$		−	1644.18i		−	0.748373i	0.927353	−	0.374187i	$-0.122078\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6301.05	+	3637.91i	−1.28253	+	0.740467i	−0.977310	−	0.211816i	$-0.932062\pi$
$17$	−6301.05	+	3637.91i	−1.28253	+	0.740467i	−0.305216	+	0.952283i	$0.598729\pi$
$18$		0			0
$19$	−870.055	−	502.327i	−0.126849	−	0.0732361i	0.435233	−	0.900318i	$-0.356666\pi$
$19$	−870.055	−	502.327i	−0.126849	−	0.0732361i	−0.562082	+	0.827082i	$0.689999\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3764.87	+	6520.94i	−0.309433	+	0.535953i	−0.978238	−	0.207484i	$-0.933472\pi$
$23$	−3764.87	+	6520.94i	−0.309433	+	0.535953i	0.668806	+	0.743437i	$0.266806\pi$
$24$		0			0
$25$	439.085	+	760.517i	0.0281014	+	0.0486731i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−12913.2			−0.529469			−0.264734	−	0.964321i	$-0.585284\pi$
$29$	−12913.2			−0.529469			−0.264734	+	0.964321i	$0.585284\pi$
$30$		0			0
$31$	−8719.05	+	5033.95i	−0.292674	+	0.168975i	−0.639147	−	0.769085i	$-0.720713\pi$
$31$	−8719.05	+	5033.95i	−0.292674	+	0.168975i	0.346473	+	0.938060i	$0.387379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	27911.6	−	34095.8i	0.651000	−	0.795237i
$36$		0			0
$37$	−16874.4	+	29227.3i	−0.333137	+	0.577010i	−0.983125	−	0.182934i	$-0.941441\pi$
$37$	−16874.4	+	29227.3i	−0.333137	+	0.577010i	0.649988	+	0.759944i	$0.274774\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	18172.4i		−	0.263670i	−0.991272	−	0.131835i	$-0.957913\pi$
$41$		−	18172.4i		−	0.263670i	0.991272	−	0.131835i	$-0.0420870\pi$
$42$		0			0
$43$	−6598.85			−0.0829971			−0.0414986	−	0.999139i	$-0.513213\pi$
$43$	−6598.85			−0.0829971			−0.0414986	+	0.999139i	$0.513213\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	173182.	+	99986.5i	1.66805	+	0.963048i	0.968689	+	0.248277i	$0.0798643\pi$
$47$	173182.	+	99986.5i	1.66805	+	0.963048i	0.699359	+	0.714771i	$0.253469\pi$
$48$		0			0
$49$	−111498.	−	37543.0i	−0.947717	−	0.319111i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−60090.5	−	104080.i	−0.403625	−	0.699099i	0.590535	−	0.807012i	$-0.298917\pi$
$53$	−60090.5	−	104080.i	−0.403625	−	0.699099i	−0.994160	+	0.107913i	$0.965583\pi$
$54$		0			0
$55$		−	148613.i		−	0.893240i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−185327.	+	106999.i	−0.902367	+	0.520982i	−0.877967	−	0.478720i	$-0.841101\pi$
$59$	−185327.	+	106999.i	−0.902367	+	0.520982i	−0.0243997	+	0.999702i	$0.507767\pi$
$60$		0			0
$61$	−205526.	−	118660.i	−0.905475	−	0.522776i	−0.0265026	−	0.999649i	$-0.508437\pi$
$61$	−205526.	−	118660.i	−0.905475	−	0.522776i	−0.878972	+	0.476872i	$0.841770\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	105609.	−	182921.i	0.384558	−	0.666074i
$66$		0			0
$67$	−95226.0	−	164936.i	−0.316615	−	0.548393i	0.663165	−	0.748474i	$-0.269213\pi$
$67$	−95226.0	−	164936.i	−0.316615	−	0.548393i	−0.979779	+	0.200081i	$0.935880\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	49009.2			0.136931			0.0684656	−	0.997653i	$-0.478190\pi$
$71$	49009.2			0.136931			0.0684656	+	0.997653i	$0.478190\pi$
$72$		0			0
$73$	−456525.	+	263575.i	−1.17354	+	0.677541i	−0.954510	−	0.298178i	$-0.903621\pi$
$73$	−456525.	+	263575.i	−1.17354	+	0.677541i	−0.219025	+	0.975719i	$0.570288\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−371191.	+	140228.i	−0.813065	+	0.307158i
$78$		0			0
$79$	−169200.	+	293062.i	−0.343177	+	0.594400i	−0.985021	−	0.172436i	$-0.944836\pi$
$79$	−169200.	+	293062.i	−0.343177	+	0.594400i	0.641844	+	0.766835i	$0.278170\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			357882.i			0.625901i	0.949769	+	0.312950i	$0.101317\pi$
$83$			357882.i			0.625901i	−0.949769	+	0.312950i	$0.898683\pi$
$84$		0			0
$85$	−934687.			−1.52198
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−229350.	−	132415.i	−0.325334	−	0.187832i	0.328434	−	0.944527i	$-0.393479\pi$
$89$	−229350.	−	132415.i	−0.325334	−	0.187832i	−0.653768	+	0.756695i	$0.726813\pi$
$90$		0			0
$91$	−556532.	−	91181.3i	−0.738527	−	0.120999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−64531.2	−	111771.i	−0.0752661	−	0.130365i
$96$		0			0
$97$		−	213971.i		−	0.234445i	−0.993106	−	0.117222i	$-0.962601\pi$
$97$		−	213971.i		−	0.234445i	0.993106	−	0.117222i	$-0.0373990\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−894256.	+	516299.i	−0.867956	+	0.501115i	−0.866669	−	0.498884i	$-0.833743\pi$
$101$	−894256.	+	516299.i	−0.867956	+	0.501115i	−0.00128777	+	0.999999i	$0.500410\pi$
$102$		0			0
$103$	−1.46801e6	−	847557.i	−1.34344	−	0.775635i	−0.356128	−	0.934437i	$-0.615903\pi$
$103$	−1.46801e6	−	847557.i	−1.34344	−	0.775635i	−0.987310	+	0.158803i	$0.949237\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	909496.	−	1.57529e6i	0.742420	−	1.28591i	−0.208970	−	0.977922i	$-0.567011\pi$
$107$	909496.	−	1.57529e6i	0.742420	−	1.28591i	0.951390	−	0.307987i	$-0.0996554\pi$
$108$		0			0
$109$	−984716.	−	1.70558e6i	−0.760381	−	1.31702i	−0.942654	−	0.333771i	$-0.891679\pi$
$109$	−984716.	−	1.70558e6i	−0.760381	−	1.31702i	0.182273	−	0.983248i	$-0.441655\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.43057e6			1.68451			0.842254	−	0.539080i	$-0.181228\pi$
$113$	2.43057e6			1.68451			0.842254	+	0.539080i	$0.181228\pi$
$114$		0			0
$115$	−837711.	+	483652.i	−0.550808	+	0.318009i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	881948.	+	2.33457e6i	0.523362	+	1.38537i
$120$		0			0
$121$	216644.	−	375238.i	0.122290	−	0.211812i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.89445e6i		−	0.969957i
$126$		0			0
$127$	−3.48846e6			−1.70303			−0.851515	−	0.524330i	$-0.824316\pi$
$127$	−3.48846e6			−1.70303			−0.851515	+	0.524330i	$0.824316\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	274696.	+	158596.i	0.122191	+	0.0705469i	0.559850	−	0.828594i	$-0.310859\pi$
$131$	274696.	+	158596.i	0.122191	+	0.0705469i	−0.437659	+	0.899141i	$0.644192\pi$
$132$		0			0
$133$	−218282.	+	266645.i	−0.0927818	+	0.113339i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.84772e6	−	3.20034e6i	−0.718578	−	1.24461i	−0.961563	−	0.274584i	$-0.911460\pi$
$137$	−1.84772e6	−	3.20034e6i	−0.718578	−	1.24461i	0.242985	−	0.970030i	$-0.421873\pi$
$138$		0			0
$139$			1.05298e6i			0.392081i	0.980596	+	0.196041i	$0.0628084\pi$
$139$			1.05298e6i			0.392081i	−0.980596	+	0.196041i	$0.937192\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.64722e6	+	951022.i	−0.563305	+	0.325224i
$144$		0			0
$145$	−1.43664e6	−	829446.i	−0.471243	−	0.272072i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.77890e6	+	4.81319e6i	−0.840067	+	1.45504i	0.0497711	+	0.998761i	$0.484151\pi$
$149$	−2.77890e6	+	4.81319e6i	−0.840067	+	1.45504i	−0.889838	+	0.456277i	$0.849183\pi$
$150$		0			0
$151$	637770.	+	1.10465e6i	0.185239	+	0.320844i	0.943657	−	0.330925i	$-0.107361\pi$
$151$	637770.	+	1.10465e6i	0.185239	+	0.320844i	−0.758418	+	0.651769i	$0.774027\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.29337e6			−0.347318
$156$		0			0
$157$	−470128.	+	271428.i	−0.121483	+	0.0701385i	−0.559510	−	0.828823i	$-0.689011\pi$
$157$	−470128.	+	271428.i	−0.121483	+	0.0701385i	0.438027	+	0.898962i	$0.355677\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.99847e6	+	1.63599e6i	0.478872	+	0.392016i
$162$		0			0
$163$	−415026.	+	718846.i	−0.0958324	+	0.165987i	−0.909956	−	0.414705i	$-0.863885\pi$
$163$	−415026.	+	718846.i	−0.0958324	+	0.165987i	0.814123	+	0.580692i	$0.197218\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	483248.i		−	0.103758i	−0.998653	−	0.0518788i	$-0.983479\pi$
$167$		−	483248.i		−	0.103758i	0.998653	−	0.0518788i	$-0.0165210\pi$
$168$		0			0
$169$	2.12349e6			0.439938
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.46815e6	+	1.42499e6i	0.476686	+	0.275215i	0.719035	−	0.694974i	$-0.244584\pi$
$173$	2.46815e6	+	1.42499e6i	0.476686	+	0.275215i	−0.242348	+	0.970189i	$0.577918\pi$
$174$		0			0
$175$	281775.	−	106448.i	0.0525762	−	0.0198621i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.51050e6	−	7.81241e6i	−0.786440	−	1.36215i	−0.928135	−	0.372243i	$-0.878589\pi$
$179$	−4.51050e6	−	7.81241e6i	−0.786440	−	1.36215i	0.141696	−	0.989910i	$-0.454745\pi$
$180$		0			0
$181$		−	4.43370e6i		−	0.747706i	−0.927488	−	0.373853i	$-0.878036\pi$
$181$		−	4.43370e6i		−	0.747706i	0.927488	−	0.373853i	$-0.121964\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.75468e6	+	2.16776e6i	−0.593004	+	0.342371i
$186$		0			0
$187$	7.28929e6	+	4.20848e6i	1.11471	+	0.643576i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.78786e6	−	4.82872e6i	0.400102	−	0.692997i	−0.593636	−	0.804734i	$-0.702308\pi$
$191$	2.78786e6	−	4.82872e6i	0.400102	−	0.692997i	0.993738	+	0.111737i	$0.0356413\pi$
$192$		0			0
$193$	186986.	+	323869.i	0.0260098	+	0.0450503i	0.878737	−	0.477306i	$-0.158387\pi$
$193$	186986.	+	323869.i	0.0260098	+	0.0450503i	−0.852728	+	0.522356i	$0.825053\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.19111e7			1.55795			0.778974	−	0.627056i	$-0.215740\pi$
$197$	1.19111e7			1.55795			0.778974	+	0.627056i	$0.215740\pi$
$198$		0			0
$199$	6.62202e6	−	3.82323e6i	0.840294	−	0.485144i	−0.0170699	−	0.999854i	$-0.505434\pi$
$199$	6.62202e6	−	3.82323e6i	0.840294	−	0.485144i	0.857364	+	0.514710i	$0.172100\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−716129.	+	4.37095e6i	−0.0856058	+	0.522502i
$204$		0			0
$205$	1.16726e6	−	2.02175e6i	0.135489	−	0.234674i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			1.16222e6i			0.127306i
$210$		0			0
$211$	1.28583e7			1.36879			0.684396	−	0.729110i	$-0.260066\pi$
$211$	1.28583e7			1.36879			0.684396	+	0.729110i	$0.260066\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−734146.	−	423860.i	−0.0738699	−	0.0426488i
$216$		0			0
$217$	1.22039e6	+	3.23045e6i	0.119432	+	0.316144i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.98137e6	+	1.03600e7i	0.554145	+	0.959808i
$222$		0			0
$223$			2.94393e6i			0.265468i	0.991152	+	0.132734i	$0.0423757\pi$
$223$			2.94393e6i			0.265468i	−0.991152	+	0.132734i	$0.957624\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.46513e6	−	1.42324e6i	0.210747	−	0.121675i	−0.390911	−	0.920428i	$-0.627840\pi$
$227$	2.46513e6	−	1.42324e6i	0.210747	−	0.121675i	0.601659	+	0.798753i	$0.294507\pi$
$228$		0			0
$229$	1.10108e7	+	6.35709e6i	0.916880	+	0.529361i	0.882638	−	0.470053i	$-0.155765\pi$
$229$	1.10108e7	+	6.35709e6i	0.916880	+	0.529361i	0.0342416	+	0.999414i	$0.489098\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.41366e6	−	1.11088e7i	0.507035	−	0.878210i	−0.492932	−	0.870068i	$-0.664075\pi$
$233$	6.41366e6	−	1.11088e7i	0.507035	−	0.878210i	0.999967	−	0.00814234i	$-0.00259182\pi$
$234$		0			0
$235$	1.28447e7	+	2.22477e7i	0.989741	+	1.71428i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.45787e7			−1.06789			−0.533943	−	0.845521i	$-0.679290\pi$
$239$	−1.45787e7			−1.06789			−0.533943	+	0.845521i	$0.679290\pi$
$240$		0			0
$241$	5.55801e6	−	3.20892e6i	0.397071	−	0.229249i	−0.288148	−	0.957586i	$-0.593040\pi$
$241$	5.55801e6	−	3.20892e6i	0.397071	−	0.229249i	0.685219	+	0.728337i	$0.259706\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.99309e6	−	1.13386e7i	−0.679519	−	0.771011i
$246$		0			0
$247$	−825914.	+	1.43052e6i	−0.0548080	+	0.0949302i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2.60913e6i			0.164996i	0.996591	+	0.0824982i	$0.0262899\pi$
$251$			2.60913e6i			0.164996i	−0.996591	+	0.0824982i	$0.973710\pi$
$252$		0			0
$253$	8.71068e6			0.537886
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.55273e6	−	3.78322e6i	−0.386032	−	0.222875i	0.294408	−	0.955680i	$-0.404878\pi$
$257$	−6.55273e6	−	3.78322e6i	−0.386032	−	0.222875i	−0.680439	+	0.732804i	$0.738211\pi$
$258$		0			0
$259$	8.95726e6	+	7.33263e6i	0.515556	+	0.422046i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.17078e7	+	2.02785e7i	0.643589	+	1.11473i	0.984626	+	0.174678i	$0.0558885\pi$
$263$	1.17078e7	+	2.02785e7i	0.643589	+	1.11473i	−0.341037	+	0.940050i	$0.610778\pi$
$264$		0			0
$265$		−	1.54390e7i		−	0.829625i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.75169e6	+	5.05279e6i	−0.449609	+	0.259582i	−0.707665	−	0.706548i	$-0.750251\pi$
$269$	−8.75169e6	+	5.05279e6i	−0.449609	+	0.259582i	0.258056	+	0.966130i	$0.416918\pi$
$270$		0			0
$271$	1.79802e7	+	1.03809e7i	0.903414	+	0.521586i	0.878306	−	0.478099i	$-0.158674\pi$
$271$	1.79802e7	+	1.03809e7i	0.903414	+	0.521586i	0.0251076	+	0.999685i	$0.492007\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	507950.	−	879795.i	0.0244243	−	0.0423042i
$276$		0			0
$277$	−7.96849e6	−	1.38018e7i	−0.374918	−	0.649378i	0.615396	−	0.788218i	$-0.288996\pi$
$277$	−7.96849e6	−	1.38018e7i	−0.374918	−	0.649378i	−0.990315	+	0.138840i	$0.955663\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.50879e7			−0.680000			−0.340000	−	0.940425i	$-0.610427\pi$
$281$	−1.50879e7			−0.680000			−0.340000	+	0.940425i	$0.610427\pi$
$282$		0			0
$283$	1.60161e7	−	9.24692e6i	0.706640	−	0.407979i	−0.103175	−	0.994663i	$-0.532900\pi$
$283$	1.60161e7	−	9.24692e6i	0.706640	−	0.407979i	0.809816	+	0.586684i	$0.199567\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.15113e6	−	1.00779e6i	−0.260201	−	0.0426309i
$288$		0			0
$289$	1.44000e7	−	2.49416e7i	0.596582	−	1.03331i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.39230e7i			0.553514i	0.960940	+	0.276757i	$0.0892597\pi$
$293$			1.39230e7i			0.553514i	−0.960940	+	0.276757i	$0.910740\pi$
$294$		0			0
$295$	−2.74911e7			−1.07084
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.07216e7	+	6.19010e6i	0.401093	+	0.231571i
$300$		0			0
$301$	−365953.	+	2.23363e6i	−0.0134192	+	0.0819051i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.52437e7	−	2.64028e7i	−0.537266	−	0.930572i
$306$		0			0
$307$			2.48744e7i			0.859680i	0.902905	+	0.429840i	$0.141430\pi$
$307$			2.48744e7i			0.859680i	−0.902905	+	0.429840i	$0.858570\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.28127e7	−	2.47179e7i	1.42328	−	0.821733i	0.426705	−	0.904391i	$-0.359674\pi$
$311$	4.28127e7	−	2.47179e7i	1.42328	−	0.821733i	0.996578	+	0.0826583i	$0.0263410\pi$
$312$		0			0
$313$	−9.40126e6	−	5.42782e6i	−0.306587	−	0.177008i	0.338811	−	0.940854i	$-0.389975\pi$
$313$	−9.40126e6	−	5.42782e6i	−0.306587	−	0.177008i	−0.645398	+	0.763846i	$0.723308\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.50745e7	+	4.34303e7i	−0.787144	+	1.36337i	0.140566	+	0.990071i	$0.455108\pi$
$317$	−2.50745e7	+	4.34303e7i	−0.787144	+	1.36337i	−0.927710	+	0.373302i	$0.878225\pi$
$318$		0			0
$319$	7.46924e6	+	1.29371e7i	0.230094	+	0.398534i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.30968e6			0.216916
$324$		0			0
$325$	1.25042e6	−	721932.i	0.0364256	−	0.0210303i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.34483e7	−	5.30748e7i	1.22007	−	1.49039i
$330$		0			0
$331$	−1.82665e7	+	3.16384e7i	−0.503698	+	0.872431i	0.496293	+	0.868155i	$0.334694\pi$
$331$	−1.82665e7	+	3.16384e7i	−0.503698	+	0.872431i	−0.999991	+	0.00427536i	$0.998639\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	2.44664e7i		−	0.650781i
$336$		0			0
$337$	−5.98140e7			−1.56283			−0.781417	−	0.624009i	$-0.785503\pi$
$337$	−5.98140e7			−1.56283			−0.781417	+	0.624009i	$0.785503\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.00865e7	+	5.82346e6i	0.254377	+	0.146865i
$342$		0			0
$343$	−1.88912e7	+	3.56586e7i	−0.468141	+	0.883654i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.07628e7	+	5.32828e7i	0.736271	+	1.27526i	0.954163	+	0.299287i	$0.0967487\pi$
$347$	3.07628e7	+	5.32828e7i	0.736271	+	1.27526i	−0.217892	+	0.975973i	$0.569918\pi$
$348$		0			0
$349$		−	6.28884e7i		−	1.47943i	−0.672920	−	0.739715i	$-0.734960\pi$
$349$		−	6.28884e7i		−	1.47943i	0.672920	−	0.739715i	$-0.265040\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.93898e7	−	2.85152e7i	1.12283	−	0.648265i	0.180706	−	0.983537i	$-0.442162\pi$
$353$	4.93898e7	−	2.85152e7i	1.12283	−	0.648265i	0.942121	+	0.335272i	$0.108828\pi$
$354$		0			0
$355$	5.45245e6	+	3.14797e6i	0.121873	+	0.0703632i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.65683e7	+	2.86972e7i	−0.358092	+	0.620234i	−0.987642	−	0.156726i	$-0.949906\pi$
$359$	−1.65683e7	+	2.86972e7i	−0.358092	+	0.620234i	0.629550	+	0.776960i	$0.283239\pi$
$360$		0			0
$361$	−2.30183e7	−	3.98688e7i	−0.489273	−	0.847446i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.77201e7			−1.39264
$366$		0			0
$367$	−4.88323e7	+	2.81933e7i	−0.987891	+	0.570359i	−0.904643	−	0.426170i	$-0.859863\pi$
$367$	−4.88323e7	+	2.81933e7i	−0.987891	+	0.570359i	−0.0832474	+	0.996529i	$0.526529\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.85621e7	+	1.45679e7i	−0.755160	+	0.285282i
$372$		0			0
$373$	2.07150e7	−	3.58795e7i	0.399171	−	0.691385i	−0.594453	−	0.804131i	$-0.702631\pi$
$373$	2.07150e7	−	3.58795e7i	0.399171	−	0.691385i	0.993624	+	0.112746i	$0.0359645\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.12316e7i			0.396240i
$378$		0			0
$379$	−5.43244e7			−0.997877			−0.498939	−	0.866637i	$-0.666277\pi$
$379$	−5.43244e7			−0.997877			−0.498939	+	0.866637i	$0.666277\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.88492e7	−	4.55236e7i	−1.40346	−	0.810289i	−0.408716	−	0.912662i	$-0.634023\pi$
$383$	−7.88492e7	−	4.55236e7i	−1.40346	−	0.810289i	−0.994746	+	0.102372i	$0.967357\pi$
$384$		0			0
$385$	−5.03035e7	−	8.24164e6i	−0.881487	−	0.144421i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.96962e6	−	6.87559e6i	−0.0674373	−	0.116805i	0.830335	−	0.557264i	$-0.188149\pi$
$389$	−3.96962e6	−	6.87559e6i	−0.0674373	−	0.116805i	−0.897773	+	0.440459i	$0.854816\pi$
$390$		0			0
$391$		−	5.47850e7i		−	0.916498i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.76481e7	+	2.17362e7i	−0.610875	+	0.352689i
$396$		0			0
$397$	−6.87193e7	−	3.96751e7i	−1.09827	−	0.634084i	−0.162500	−	0.986709i	$-0.551956\pi$
$397$	−6.87193e7	−	3.96751e7i	−1.09827	−	0.634084i	−0.935765	+	0.352625i	$0.885289\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.28659e6	−	5.69255e6i	0.0509698	−	0.0882823i	−0.839415	−	0.543491i	$-0.817102\pi$
$401$	3.28659e6	−	5.69255e6i	0.0509698	−	0.0882823i	0.890385	+	0.455209i	$0.150435\pi$
$402$		0			0
$403$	8.27669e6	+	1.43357e7i	0.126457	+	0.219029i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.90419e7			0.579092
$408$		0			0
$409$	2.23226e7	−	1.28880e7i	0.326269	−	0.188371i	−0.327915	−	0.944707i	$-0.606346\pi$
$409$	2.23226e7	−	1.28880e7i	0.326269	−	0.188371i	0.654183	+	0.756336i	$0.273012\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.59400e7	+	6.86647e7i	0.368230	+	0.974728i
$414$		0			0
$415$	−2.29876e7	+	3.98157e7i	−0.321625	+	0.557070i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	3.61871e7i		−	0.491940i	−0.969277	−	0.245970i	$-0.920893\pi$
$419$		−	3.61871e7i		−	0.491940i	0.969277	−	0.245970i	$-0.0791065\pi$
$420$		0			0
$421$	−1.33346e8			−1.78704			−0.893521	−	0.449022i	$-0.851773\pi$
$421$	−1.33346e8			−1.78704			−0.893521	+	0.449022i	$0.851773\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.53339e6	−	3.19470e6i	−0.0720816	−	0.0416163i
$426$		0			0
$427$	−5.15628e7	+	6.29872e7i	−0.662298	+	0.809038i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.84104e7	+	3.18878e7i	0.229949	+	0.398283i	0.957793	−	0.287460i	$-0.0928107\pi$
$431$	1.84104e7	+	3.18878e7i	0.229949	+	0.398283i	−0.727844	+	0.685743i	$0.759477\pi$
$432$		0			0
$433$		−	1.34962e8i		−	1.66245i	−0.555939	−	0.831223i	$-0.687641\pi$
$433$		−	1.34962e8i		−	1.66245i	0.555939	−	0.831223i	$-0.312359\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.55129e6	−	3.78239e6i	0.0785023	−	0.0453233i
$438$		0			0
$439$	1.16712e8	+	6.73835e7i	1.37950	+	0.796452i	0.992099	−	0.125460i	$-0.0400408\pi$
$439$	1.16712e8	+	6.73835e7i	1.37950	+	0.796452i	0.387398	+	0.921913i	$0.373374\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.29386e7	−	9.16924e7i	0.608922	−	1.05468i	−0.382497	−	0.923957i	$-0.624936\pi$
$443$	5.29386e7	−	9.16924e7i	0.608922	−	1.05468i	0.991418	−	0.130727i	$-0.0417310\pi$
$444$		0			0
$445$	−1.70107e7	−	2.94634e7i	−0.193038	−	0.334351i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4.37596e7			0.483431			0.241716	−	0.970347i	$-0.422290\pi$
$449$	4.37596e7			0.483431			0.241716	+	0.970347i	$0.422290\pi$
$450$		0			0
$451$	−1.82061e7	+	1.05113e7i	−0.198466	+	0.114584i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.60595e7	−	4.58916e7i	−0.595134	−	0.487191i
$456$		0			0
$457$	−3.18617e7	+	5.51861e7i	−0.333826	+	0.578204i	−0.983259	−	0.182215i	$-0.941673\pi$
$457$	−3.18617e7	+	5.51861e7i	−0.333826	+	0.578204i	0.649432	+	0.760419i	$0.275007\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.68392e7i			0.376017i	0.982167	+	0.188009i	$0.0602033\pi$
$461$			3.68392e7i			0.376017i	−0.982167	+	0.188009i	$0.939797\pi$
$462$		0			0
$463$	−3.99828e6			−0.0402838			−0.0201419	−	0.999797i	$-0.506412\pi$
$463$	−3.99828e6			−0.0402838			−0.0201419	+	0.999797i	$0.506412\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.12708e7	+	6.50722e6i	0.110664	+	0.0638918i	0.554310	−	0.832310i	$-0.312982\pi$
$467$	1.12708e7	+	6.50722e6i	0.110664	+	0.0638918i	−0.443647	+	0.896202i	$0.646315\pi$
$468$		0			0
$469$	−6.11098e7	+	2.30859e7i	−0.592369	+	0.223783i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.81690e6	+	6.61107e6i	0.0360684	+	0.0624724i
$474$		0			0
$475$		−	882256.i		−	0.00823216i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.70145e6	+	1.55968e6i	−0.0245805	+	0.0141915i	−0.512240	−	0.858842i	$-0.671184\pi$
$479$	−2.70145e6	+	1.55968e6i	−0.0245805	+	0.0141915i	0.487659	+	0.873034i	$0.337851\pi$
$480$		0			0
$481$	4.80548e7	+	2.77445e7i	0.431819	+	0.249311i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.37439e7	−	2.38051e7i	0.120471	−	0.208663i
$486$		0			0
$487$	6.40950e7	+	1.11016e8i	0.554929	+	0.961165i	0.997909	+	0.0646339i	$0.0205880\pi$
$487$	6.40950e7	+	1.11016e8i	0.554929	+	0.961165i	−0.442980	+	0.896532i	$0.646079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.12183e7			0.601654			0.300827	−	0.953679i	$-0.402737\pi$
$491$	7.12183e7			0.601654			0.300827	+	0.953679i	$0.402737\pi$
$492$		0			0
$493$	8.13668e7	−	4.69771e7i	0.679057	−	0.392054i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.71791e6	−	1.65890e7i	0.0221394	−	0.135130i
$498$		0			0
$499$	6.74443e7	−	1.16817e8i	0.542805	−	0.940166i	−0.455937	−	0.890012i	$-0.650696\pi$
$499$	6.74443e7	−	1.16817e8i	0.542805	−	0.940166i	0.998742	−	0.0501533i	$-0.0159710\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.08199e8i			0.850198i	0.905147	+	0.425099i	$0.139761\pi$
$503$			1.08199e8i			0.850198i	−0.905147	+	0.425099i	$0.860239\pi$
$504$		0			0
$505$	−1.32652e8			−1.03001
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.03528e8	+	1.17507e8i	1.54337	+	0.891065i	0.998623	+	0.0524651i	$0.0167078\pi$
$509$	2.03528e8	+	1.17507e8i	1.54337	+	0.891065i	0.544747	+	0.838600i	$0.316625\pi$
$510$		0			0
$511$	6.38991e7	+	1.69145e8i	0.478886	+	1.26764i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.08881e8	−	1.88588e8i	−0.797133	−	1.38067i
$516$		0			0
$517$		−	2.31336e8i		−	1.67406i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.88469e8	+	1.08812e8i	−1.33268	+	0.769424i	−0.985710	−	0.168452i	$-0.946123\pi$
$521$	−1.88469e8	+	1.08812e8i	−1.33268	+	0.769424i	−0.346971	+	0.937876i	$0.612790\pi$
$522$		0			0
$523$	1.53060e8	+	8.83692e7i	1.06993	+	0.617726i	0.928163	−	0.372173i	$-0.121387\pi$
$523$	1.53060e8	+	8.83692e7i	1.06993	+	0.617726i	0.141770	+	0.989900i	$0.454721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.66261e7	−	6.34383e7i	0.250241	−	0.433431i
$528$		0			0
$529$	4.56695e7	+	7.91019e7i	0.308503	+	0.534343i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.98787e7			−0.197324
$534$		0			0
$535$	2.02370e8	−	1.16838e8i	1.32155	−	0.762998i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.68800e7	+	1.33420e8i	0.171658	+	0.852029i
$540$		0			0
$541$	1.05226e8	−	1.82257e8i	0.664556	−	1.15105i	−0.314849	−	0.949142i	$-0.601954\pi$
$541$	1.05226e8	−	1.82257e8i	0.664556	−	1.15105i	0.979405	−	0.201903i	$-0.0647127\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.53002e8i		−	1.56291i
$546$		0			0
$547$	−1.33467e8			−0.815480			−0.407740	−	0.913098i	$-0.633683\pi$
$547$	−1.33467e8			−0.815480			−0.407740	+	0.913098i	$0.633683\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.12352e7	+	6.48665e6i	0.0671624	+	0.0387762i
$552$		0			0
$553$	8.98144e7	+	7.35242e7i	0.531093	+	0.434766i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.52286e8	−	2.63767e8i	−0.881239	−	1.52635i	−0.849965	−	0.526839i	$-0.823377\pi$
$557$	−1.52286e8	−	2.63767e8i	−0.881239	−	1.52635i	−0.0312735	−	0.999511i	$-0.509956\pi$
$558$		0			0
$559$			1.08497e7i			0.0621128i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.31763e8	+	1.33808e8i	−1.29873	+	0.749821i	−0.980184	−	0.198087i	$-0.936527\pi$
$563$	−2.31763e8	+	1.33808e8i	−1.29873	+	0.749821i	−0.318544	+	0.947908i	$0.603194\pi$
$564$		0			0
$565$	2.70410e8	+	1.56121e8i	1.49926	+	0.865599i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.36080e8	+	2.35698e8i	−0.738684	+	1.27944i	0.214405	+	0.976745i	$0.431219\pi$
$569$	−1.36080e8	+	2.35698e8i	−0.738684	+	1.27944i	−0.953088	+	0.302692i	$0.902114\pi$
$570$		0			0
$571$	6.28995e7	+	1.08945e8i	0.337862	+	0.585194i	0.984030	−	0.178001i	$-0.0569630\pi$
$571$	6.28995e7	+	1.08945e8i	0.337862	+	0.585194i	−0.646168	+	0.763195i	$0.723630\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.61238e6			−0.0347820
$576$		0			0
$577$	1.85998e8	−	1.07386e8i	0.968236	−	0.559011i	0.0695377	−	0.997579i	$-0.477848\pi$
$577$	1.85998e8	−	1.07386e8i	0.968236	−	0.559011i	0.898698	+	0.438568i	$0.144514\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.21138e8	+	1.98471e7i	0.617666	+	0.101197i
$582$		0			0
$583$	−6.95149e7	+	1.20403e8i	−0.350810	+	0.607621i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.42103e8i			0.702570i	0.936269	+	0.351285i	$0.114255\pi$
$587$			1.42103e8i			0.702570i	−0.936269	+	0.351285i	$0.885745\pi$
$588$		0			0
$589$	1.01147e7			0.0495004
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.25553e8	+	7.24881e7i	0.602093	+	0.347618i	0.769864	−	0.638207i	$-0.220324\pi$
$593$	1.25553e8	+	7.24881e7i	0.602093	+	0.347618i	−0.167772	+	0.985826i	$0.553657\pi$
$594$		0			0
$595$	−5.18350e7	+	3.16379e8i	−0.246078	+	1.50196i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.31951e8	+	2.28546e8i	0.613951	+	1.06339i	0.990568	+	0.137025i	$0.0437539\pi$
$599$	1.31951e8	+	2.28546e8i	0.613951	+	1.06339i	−0.376617	+	0.926369i	$0.622913\pi$
$600$		0			0
$601$		−	6.75462e7i		−	0.311155i	−0.987824	−	0.155578i	$-0.950276\pi$
$601$		−	6.75462e7i		−	0.311155i	0.987824	−	0.155578i	$-0.0497239\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.82048e7	−	2.78311e7i	0.217683	−	0.125679i
$606$		0			0
$607$	3.59206e8	+	2.07388e8i	1.60612	+	0.927292i	0.990228	+	0.139460i	$0.0445368\pi$
$607$	3.59206e8	+	2.07388e8i	1.60612	+	0.927292i	0.615890	+	0.787832i	$0.288797\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.64395e8	−	2.84741e8i	0.720719	−	1.24832i
$612$		0			0
$613$	9.27340e7	+	1.60620e8i	0.402585	+	0.697298i	0.994037	−	0.109043i	$-0.0347785\pi$
$613$	9.27340e7	+	1.60620e8i	0.402585	+	0.697298i	−0.591452	+	0.806340i	$0.701445\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.06904e7			0.173235			0.0866177	−	0.996242i	$-0.472394\pi$
$617$	4.06904e7			0.173235			0.0866177	+	0.996242i	$0.472394\pi$
$618$		0			0
$619$	2.14992e7	−	1.24126e7i	0.0906465	−	0.0523348i	−0.453992	−	0.891006i	$-0.650000\pi$
$619$	2.14992e7	−	1.24126e7i	0.0906465	−	0.0523348i	0.544638	+	0.838671i	$0.316667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.75400e7	+	7.02887e7i	−0.237961	+	0.290684i
$624$		0			0
$625$	1.28545e8	−	2.22647e8i	0.526522	−	0.911963i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	2.45550e8i		−	0.986708i
$630$		0			0
$631$	−1.69134e8			−0.673197			−0.336599	−	0.941648i	$-0.609277\pi$
$631$	−1.69134e8			−0.673197			−0.336599	+	0.941648i	$0.609277\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.88104e8	−	2.24072e8i	−1.51575	−	0.875117i
$636$		0			0
$637$	−6.17274e7	+	1.83322e8i	−0.238814	+	0.709246i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.87335e7	+	3.24474e7i	0.0711288	+	0.123199i	0.899396	−	0.437134i	$-0.144007\pi$
$641$	1.87335e7	+	3.24474e7i	0.0711288	+	0.123199i	−0.828267	+	0.560333i	$0.810673\pi$
$642$		0			0
$643$		−	4.92999e8i		−	1.85444i	−0.374513	−	0.927221i	$-0.622190\pi$
$643$		−	4.92999e8i		−	1.85444i	0.374513	−	0.927221i	$-0.377810\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.01578e8	−	2.31851e8i	1.48271	−	0.856044i	0.482904	−	0.875673i	$-0.339582\pi$
$647$	4.01578e8	−	2.31851e8i	1.48271	−	0.856044i	0.999807	+	0.0196293i	$0.00624862\pi$
$648$		0			0
$649$	2.14394e8	+	1.23780e8i	0.784292	+	0.452811i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.21508e7	+	2.10458e7i	−0.0436379	+	0.0755831i	−0.887019	−	0.461732i	$-0.847228\pi$
$653$	−1.21508e7	+	2.10458e7i	−0.0436379	+	0.0755831i	0.843381	+	0.537315i	$0.180561\pi$
$654$		0			0
$655$	2.03740e7	+	3.52887e7i	0.0725023	+	0.125578i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.31281e8			1.85639			0.928193	−	0.372100i	$-0.121362\pi$
$659$	5.31281e8			1.85639			0.928193	+	0.372100i	$0.121362\pi$
$660$		0			0
$661$	3.20342e8	−	1.84949e8i	1.10920	−	0.640396i	0.170577	−	0.985344i	$-0.445437\pi$
$661$	3.20342e8	−	1.84949e8i	1.10920	−	0.640396i	0.938622	+	0.344949i	$0.112104\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.14119e7	+	1.56445e7i	−0.140819	+	0.0531981i
$666$		0			0
$667$	4.86165e7	−	8.42063e7i	0.163835	−	0.283770i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			2.74541e8i			0.908741i
$672$		0			0
$673$	−5.52287e8			−1.81184			−0.905919	−	0.423451i	$-0.860819\pi$
$673$	−5.52287e8			−1.81184			−0.905919	+	0.423451i	$0.860819\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.77661e7	−	1.02573e7i	−0.0572566	−	0.0330571i	0.471098	−	0.882081i	$-0.343858\pi$
$677$	−1.77661e7	−	1.02573e7i	−0.0572566	−	0.0330571i	−0.528355	+	0.849024i	$0.677191\pi$
$678$		0			0
$679$	−7.24265e7	−	1.18662e7i	−0.231360	−	0.0379056i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.97494e6	−	6.88480e6i	−0.0124758	−	0.0216087i	0.859720	−	0.510765i	$-0.170638\pi$
$683$	−3.97494e6	−	6.88480e6i	−0.0124758	−	0.0216087i	−0.872196	+	0.489157i	$0.837305\pi$
$684$		0			0
$685$		−	4.74733e8i		−	1.47699i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.71125e8	+	9.87993e7i	−0.523187	+	0.302062i
$690$		0			0
$691$	1.20180e7	+	6.93859e6i	0.0364249	+	0.0210299i	0.518102	−	0.855319i	$-0.326639\pi$
$691$	1.20180e7	+	6.93859e6i	0.0364249	+	0.0210299i	−0.481677	+	0.876349i	$0.659972\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.76354e7	+	1.17148e8i	−0.201474	+	0.348964i
$696$		0			0
$697$	6.61097e7	+	1.14505e8i	0.195239	+	0.338164i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.01622e8			−0.875606			−0.437803	−	0.899071i	$-0.644243\pi$
$701$	−3.01622e8			−0.875606			−0.437803	+	0.899071i	$0.644243\pi$
$702$		0			0
$703$	2.93633e7	−	1.69529e7i	0.0845160	−	0.0487954i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.25168e8	+	3.31327e8i	0.354188	+	0.937558i
$708$		0			0
$709$	−1.92778e8	+	3.33901e8i	−0.540902	+	0.936871i	0.457950	+	0.888978i	$0.348584\pi$
$709$	−1.92778e8	+	3.33901e8i	−0.540902	+	0.936871i	−0.998852	+	0.0478927i	$0.984749\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.58085e7i		−	0.209146i
$714$		0			0
$715$	−2.44346e8			−0.668477
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.81303e8	−	2.20145e8i	−1.02585	−	0.592274i	−0.110057	−	0.993925i	$-0.535103\pi$
$719$	−3.81303e8	−	2.20145e8i	−1.02585	−	0.592274i	−0.915793	+	0.401651i	$0.868437\pi$
$720$		0			0
$721$	−3.68299e8	+	4.49900e8i	−0.982640	+	1.20036i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.66999e6	−	9.82071e6i	−0.0148788	−	0.0257709i
$726$		0			0
$727$		−	4.63077e8i		−	1.20517i	−0.798053	−	0.602587i	$-0.794137\pi$
$727$		−	4.63077e8i		−	1.20517i	0.798053	−	0.602587i	$-0.205863\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.15797e7	−	2.40060e7i	0.106446	−	0.0614566i
$732$		0			0
$733$	1.80686e8	+	1.04319e8i	0.458789	+	0.264882i	0.711535	−	0.702651i	$-0.248000\pi$
$733$	1.80686e8	+	1.04319e8i	0.458789	+	0.264882i	−0.252746	+	0.967533i	$0.581334\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.10161e8	+	1.90805e8i	−0.275186	+	0.476635i
$738$		0			0
$739$	5.71544e7	+	9.89943e7i	0.141617	+	0.245288i	0.928106	−	0.372317i	$-0.121436\pi$
$739$	5.71544e7	+	9.89943e7i	0.141617	+	0.245288i	−0.786489	+	0.617605i	$0.788103\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.47529e8			−1.33488			−0.667438	−	0.744666i	$-0.732609\pi$
$743$	−5.47529e8			−1.33488			−0.667438	+	0.744666i	$0.732609\pi$
$744$		0			0
$745$	−6.18325e8	+	3.56990e8i	−1.49537	+	0.863351i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.82779e8	−	3.95214e8i	−1.14895	−	0.940561i
$750$		0			0
$751$	1.70241e8	−	2.94866e8i	0.401925	−	0.696154i	−0.592033	−	0.805914i	$-0.701675\pi$
$751$	1.70241e8	−	2.94866e8i	0.401925	−	0.696154i	0.993958	+	0.109759i	$0.0350079\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.63862e8i			0.380747i
$756$		0			0
$757$	−5.76851e8			−1.32977			−0.664885	−	0.746946i	$-0.731519\pi$
$757$	−5.76851e8			−1.32977			−0.664885	+	0.746946i	$0.731519\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.58986e8	+	1.49526e8i	0.587655	+	0.339283i	0.764170	−	0.645015i	$-0.223149\pi$
$761$	2.58986e8	+	1.49526e8i	0.587655	+	0.339283i	−0.176514	+	0.984298i	$0.556482\pi$
$762$		0			0
$763$	−6.31926e8	+	2.38727e8i	−1.42263	+	0.537438i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.75925e8	+	3.04711e8i	0.389889	+	0.675307i
$768$		0			0
$769$		−	4.65905e8i		−	1.02451i	−0.858832	−	0.512257i	$-0.828809\pi$
$769$		−	4.65905e8i		−	1.02451i	0.858832	−	0.512257i	$-0.171191\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.58408e8	−	3.80132e8i	1.42547	−	0.822993i	0.428708	−	0.903443i	$-0.358969\pi$
$773$	6.58408e8	−	3.80132e8i	1.42547	−	0.822993i	0.996759	+	0.0804500i	$0.0256357\pi$
$774$		0			0
$775$	−7.65680e6	−	4.42066e6i	−0.0164491	−	0.00949689i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.12849e6	+	1.58110e7i	−0.0193102	+	0.0334462i
$780$		0			0
$781$	−2.83478e7	−	4.90999e7i	−0.0595068	−	0.103069i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.97379e7			−0.144165
$786$		0			0
$787$	6.82077e8	−	3.93797e8i	1.39929	−	0.807883i	0.404976	−	0.914327i	$-0.367280\pi$
$787$	6.82077e8	−	3.93797e8i	1.39929	−	0.807883i	0.994319	+	0.106444i	$0.0339466\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.34793e8	−	8.22717e8i	0.272356	−	1.66235i
$792$		0			0
$793$	−1.95098e8	+	3.37920e8i	−0.391232	+	0.677633i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.31019e8i			0.258796i	0.991593	+	0.129398i	$0.0413045\pi$
$797$			1.31019e8i			0.258796i	−0.991593	+	0.129398i	$0.958695\pi$
$798$		0			0
$799$	−1.45497e9			−2.85242
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.28126e8	+	3.04913e8i	1.01998	+	0.588884i
$804$		0			0
$805$	1.17253e8	+	3.10376e8i	0.224769	+	0.594978i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.91871e8	+	5.05536e8i	0.551247	+	0.954787i	0.998185	+	0.0602223i	$0.0191810\pi$
$809$	2.91871e8	+	5.05536e8i	0.551247	+	0.954787i	−0.446938	+	0.894565i	$0.647486\pi$
$810$		0			0
$811$		−	5.94780e8i		−	1.11505i	−0.830161	−	0.557524i	$-0.811751\pi$
$811$		−	5.94780e8i		−	1.11505i	0.830161	−	0.557524i	$-0.188249\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.23463e7	+	5.33162e7i	−0.170587	+	0.0984886i
$816$		0			0
$817$	5.74137e6	+	3.31478e6i	0.0105281	+	0.00607839i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.35596e8	+	2.34860e8i	−0.245030	+	0.424404i	−0.962140	−	0.272556i	$-0.912131\pi$
$821$	−1.35596e8	+	2.34860e8i	−0.245030	+	0.424404i	0.717110	+	0.696960i	$0.245464\pi$
$822$		0			0
$823$	−4.20224e8	−	7.27849e8i	−0.753844	−	1.30570i	−0.945947	−	0.324321i	$-0.894864\pi$
$823$	−4.20224e8	−	7.27849e8i	−0.753844	−	1.30570i	0.192103	−	0.981375i	$-0.438469\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.04592e8			−1.24572			−0.622861	−	0.782332i	$-0.714030\pi$
$827$	−7.04592e8			−1.24572			−0.622861	+	0.782332i	$0.714030\pi$
$828$		0			0
$829$	−1.07124e8	+	6.18482e7i	−0.188029	+	0.108558i	−0.591059	−	0.806628i	$-0.701290\pi$
$829$	−1.07124e8	+	6.18482e7i	−0.188029	+	0.108558i	0.403031	+	0.915186i	$0.367957\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.39133e8	−	1.69060e8i	1.45176	−	0.292486i
$834$		0			0
$835$	3.10401e7	−	5.37631e7i	0.0533168	−	0.0923474i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	8.75395e8i		−	1.48224i	−0.671373	−	0.741119i	$-0.734295\pi$
$839$		−	8.75395e8i		−	1.48224i	0.671373	−	0.741119i	$-0.265705\pi$
$840$		0			0
$841$	−4.28072e8			−0.719663
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.36247e8	+	1.36397e8i	0.391557	+	0.226066i
$846$		0			0
$847$	−1.14999e8	−	9.41407e7i	−0.189253	−	0.154927i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.27060e8	−	2.20074e8i	−0.206167	−	0.357092i
$852$		0			0
$853$			1.10580e7i			0.0178168i	0.999960	+	0.00890842i	$0.00283568\pi$
$853$			1.10580e7i			0.0178168i	−0.999960	+	0.00890842i	$0.997164\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.05874e8	+	1.76597e8i	−0.485960	+	0.280569i	−0.722897	−	0.690956i	$-0.757190\pi$
$857$	−3.05874e8	+	1.76597e8i	−0.485960	+	0.280569i	0.236937	+	0.971525i	$0.423857\pi$
$858$		0			0
$859$	−1.76199e7	−	1.01728e7i	−0.0277986	−	0.0160496i	0.486036	−	0.873939i	$-0.338442\pi$
$859$	−1.76199e7	−	1.01728e7i	−0.0277986	−	0.0160496i	−0.513835	+	0.857889i	$0.671776\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.77647e7	−	8.27308e7i	0.0743146	−	0.128717i	−0.826473	−	0.562976i	$-0.809656\pi$
$863$	4.77647e7	−	8.27308e7i	0.0743146	−	0.128717i	0.900788	+	0.434259i	$0.142990\pi$
$864$		0			0
$865$	1.83060e8	+	3.17070e8i	0.282843	+	0.489899i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.91473e8			0.596544
$870$		0			0
$871$	−2.71184e8	+	1.56568e8i	−0.410403	+	0.236946i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.41246e8	−	1.05061e8i	−0.957195	−	0.156825i
$876$		0			0
$877$	5.41322e8	−	9.37597e8i	0.802522	−	1.39001i	−0.115430	−	0.993316i	$-0.536825\pi$
$877$	5.41322e8	−	9.37597e8i	0.802522	−	1.39001i	0.917951	−	0.396693i	$-0.129842\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1.17071e9i			1.71208i	0.516913	+	0.856038i	$0.327081\pi$
$881$			1.17071e9i			1.71208i	−0.516913	+	0.856038i	$0.672919\pi$
$882$		0			0
$883$	−1.02937e9			−1.49517			−0.747587	−	0.664164i	$-0.768787\pi$
$883$	−1.02937e9			−1.49517			−0.747587	+	0.664164i	$0.768787\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.28771e7	−	3.05286e7i	−0.0757699	−	0.0437458i	0.461636	−	0.887069i	$-0.347262\pi$
$887$	−5.28771e7	−	3.05286e7i	−0.0757699	−	0.0437458i	−0.537406	+	0.843323i	$0.680596\pi$
$888$		0			0
$889$	−1.93460e8	+	1.18080e9i	−0.275350	+	1.68062i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.00452e8	−	1.73988e8i	−0.141060	−	0.244323i
$894$		0			0
$895$		−	1.15888e9i		−	1.61648i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.12591e8	−	6.50044e7i	0.154962	−	0.0894672i
$900$		0			0
$901$	7.57266e8	+	4.37208e8i	1.03532	+	0.597742i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.84787e8	−	4.93266e8i	0.384215	−	0.665480i
$906$		0			0
$907$	2.14059e8	+	3.70762e8i	0.286888	+	0.496905i	0.973065	−	0.230530i	$-0.0740459\pi$
$907$	2.14059e8	+	3.70762e8i	0.286888	+	0.496905i	−0.686177	+	0.727434i	$0.740713\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.98250e8			0.526745			0.263373	−	0.964694i	$-0.415165\pi$
$911$	3.98250e8			0.526745			0.263373	+	0.964694i	$0.415165\pi$
$912$		0			0
$913$	3.58544e8	−	2.07006e8i	0.471119	−	0.272001i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.89165e7	−	8.41858e7i	0.0893748	−	0.109177i
$918$		0			0
$919$	−1.08859e8	+	1.88549e8i	−0.140255	+	0.242929i	−0.927593	−	0.373593i	$-0.878126\pi$
$919$	−1.08859e8	+	1.88549e8i	−0.140255	+	0.242929i	0.787338	+	0.616522i	$0.211459\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	8.05797e7i		−	0.102476i
$924$		0			0
$925$	−2.96371e7			−0.0374465
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.05112e8	−	4.64832e8i	−1.00417	−	0.579761i	−0.0946939	−	0.995506i	$-0.530187\pi$
$929$	−8.05112e8	−	4.64832e8i	−1.00417	−	0.579761i	−0.909481	+	0.415746i	$0.863521\pi$
$930$		0			0
$931$	7.81506e7	+	8.86730e7i	0.0968463	+	0.109886i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.40640e8	+	9.36416e8i	0.661414	+	1.14560i
$936$		0			0
$937$			4.95387e8i			0.602179i	0.953596	+	0.301089i	$0.0973503\pi$
$937$			4.95387e8i			0.602179i	−0.953596	+	0.301089i	$0.902650\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.51524e8	−	8.74825e7i	0.181850	−	0.104991i	−0.406312	−	0.913735i	$-0.633185\pi$
$941$	1.51524e8	−	8.74825e7i	0.181850	−	0.104991i	0.588161	+	0.808744i	$0.299852\pi$
$942$		0			0
$943$	1.18501e8	+	6.84168e7i	0.141315	+	0.0815882i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.03728e8	+	1.79662e8i	−0.122137	+	0.211547i	−0.920610	−	0.390483i	$-0.872308\pi$
$947$	−1.03728e8	+	1.79662e8i	−0.122137	+	0.211547i	0.798473	+	0.602030i	$0.205641\pi$
$948$		0			0
$949$	4.33364e8	+	7.50608e8i	0.507054	+	0.878243i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.31559e9			−1.51999			−0.759997	−	0.649927i	$-0.774799\pi$
$953$	−1.31559e9			−1.51999			−0.759997	+	0.649927i	$0.774799\pi$
$954$		0			0
$955$	6.20319e8	−	3.58142e8i	0.712205	−	0.411192i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.18574e9	+	4.47947e8i	−1.34442	+	0.507891i
$960$		0			0
$961$	−3.93071e8	+	6.80818e8i	−0.442895	+	0.767116i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			4.80422e7i			0.0534614i
$966$		0			0
$967$	3.56832e8			0.394625			0.197312	−	0.980341i	$-0.436779\pi$
$967$	3.56832e8			0.394625			0.197312	+	0.980341i	$0.436779\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.53380e8	−	2.04024e8i	−0.385997	−	0.222856i	0.294427	−	0.955674i	$-0.404871\pi$
$971$	−3.53380e8	−	2.04024e8i	−0.385997	−	0.222856i	−0.680424	+	0.732818i	$0.738204\pi$
$972$		0			0
$973$	3.56420e8	+	5.83953e7i	0.386923	+	0.0633927i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.47375e8	−	4.28465e8i	−0.265260	−	0.459443i	0.702372	−	0.711810i	$-0.252124\pi$
$977$	−2.47375e8	−	4.28465e8i	−0.265260	−	0.459443i	−0.967632	+	0.252367i	$0.918791\pi$
$978$		0			0
$979$			3.06366e8i			0.326507i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.35186e8	−	1.35785e8i	0.247600	−	0.142952i	−0.371065	−	0.928607i	$-0.621007\pi$
$983$	2.35186e8	−	1.35785e8i	0.247600	−	0.142952i	0.618665	+	0.785655i	$0.287674\pi$
$984$		0			0
$985$	1.32515e9	+	7.65077e8i	1.38662	+	0.800565i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.48438e7	−	4.30307e7i	0.0256820	−	0.0444826i
$990$		0			0
$991$	−3.84173e8	−	6.65407e8i	−0.394735	−	0.683702i	0.598332	−	0.801248i	$-0.295830\pi$
$991$	−3.84173e8	−	6.65407e8i	−0.394735	−	0.683702i	−0.993067	+	0.117547i	$0.962497\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.82299e8			0.997182
$996$		0			0
$997$	5.88765e8	−	3.39924e8i	0.594096	−	0.343001i	−0.172620	−	0.984989i	$-0.555223\pi$
$997$	5.88765e8	−	3.39924e8i	0.594096	−	0.343001i	0.766715	+	0.641987i	$0.221890\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.7.z.f.145.4	yes	12
3.2	odd	2	inner	252.7.z.f.145.3	yes	12
7.3	odd	6	inner	252.7.z.f.73.4	yes	12
21.17	even	6	inner	252.7.z.f.73.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.7.z.f.73.3	✓	12	21.17	even	6	inner
252.7.z.f.73.4	yes	12	7.3	odd	6	inner
252.7.z.f.145.3	yes	12	3.2	odd	2	inner
252.7.z.f.145.4	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	252.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(111.254 + 64.2323i) q^{5} +(55.4571 - 338.487i) q^{7} +O(q^{10})\)	\(q+(111.254 + 64.2323i) q^{5} +(55.4571 - 338.487i) q^{7} +(-578.419 - 1001.85i) q^{11} -1644.18i q^{13} +(-6301.05 + 3637.91i) q^{17} +(-870.055 - 502.327i) q^{19} +(-3764.87 + 6520.94i) q^{23} +(439.085 + 760.517i) q^{25} -12913.2 q^{29} +(-8719.05 + 5033.95i) q^{31} +(27911.6 - 34095.8i) q^{35} +(-16874.4 + 29227.3i) q^{37} -18172.4i q^{41} -6598.85 q^{43} +(173182. + 99986.5i) q^{47} +(-111498. - 37543.0i) q^{49} +(-60090.5 - 104080. i) q^{53} -148613. i q^{55} +(-185327. + 106999. i) q^{59} +(-205526. - 118660. i) q^{61} +(105609. - 182921. i) q^{65} +(-95226.0 - 164936. i) q^{67} +49009.2 q^{71} +(-456525. + 263575. i) q^{73} +(-371191. + 140228. i) q^{77} +(-169200. + 293062. i) q^{79} +357882. i q^{83} -934687. q^{85} +(-229350. - 132415. i) q^{89} +(-556532. - 91181.3i) q^{91} +(-64531.2 - 111771. i) q^{95} -213971. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 714 q^{7}+O(q^{10}) \) 12 * q + 714 * q^7	\( 12 q + 714 q^{7} - 59466 q^{19} + 64086 q^{25} + 219834 q^{31} - 166242 q^{37} - 326916 q^{43} - 132006 q^{49} + 736008 q^{61} + 176238 q^{67} - 2931018 q^{73} + 449490 q^{79} + 2174592 q^{85} - 808542 q^{91}+O(q^{100}) \) 12 * q + 714 * q^7 - 59466 * q^19 + 64086 * q^25 + 219834 * q^31 - 166242 * q^37 - 326916 * q^43 - 132006 * q^49 + 736008 * q^61 + 176238 * q^67 - 2931018 * q^73 + 449490 * q^79 + 2174592 * q^85 - 808542 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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