LMFDB - Embedded newform 252.6.k.e.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(37,252)]

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, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, 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:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7081})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 $ x^4 - x^3 + 1771x^2 + 1770x + 3132900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$21.2872 - 36.8705i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.6.k.e.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.28717 + 16.0858i) q^{5} +(-127.649 + 22.6454i) q^{7} +(-80.7128 - 139.799i) q^{11} +14.1283 q^{13} +(-382.851 - 663.118i) q^{17} +(-707.146 + 1224.81i) q^{19} +(2092.72 - 3624.69i) q^{23} +(1390.00 + 2407.55i) q^{25} +4202.60 q^{29} +(1193.68 + 2067.51i) q^{31} +(821.224 - 2263.65i) q^{35} +(336.469 - 582.782i) q^{37} +4173.45 q^{41} -5430.94 q^{43} +(3151.34 - 5458.28i) q^{47} +(15781.4 - 5781.32i) q^{49} +(8208.34 + 14217.3i) q^{53} +2998.37 q^{55} +(1983.24 + 3435.08i) q^{59} +(25169.4 - 43594.6i) q^{61} +(-131.212 + 227.266i) q^{65} +(-6822.63 - 11817.1i) q^{67} +83957.2 q^{71} +(14289.2 + 24749.6i) q^{73} +(13468.7 + 16017.3i) q^{77} +(29977.7 - 51923.0i) q^{79} +61583.0 q^{83} +14222.4 q^{85} +(21149.2 - 36631.4i) q^{89} +(-1803.46 + 319.941i) q^{91} +(-13134.8 - 22750.1i) q^{95} +44638.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 47 q^{5} - 174 q^{7} - 407 q^{11} + 898 q^{13} - 1868 q^{17} + 1463 q^{19} - 44 q^{23} + 1605 q^{25} - 1534 q^{29} + 11170 q^{31} - 9674 q^{35} + 3113 q^{37} + 15684 q^{41} - 25258 q^{43} + 9576 q^{47}+ \cdots + 369570 q^{97}+O(q^{100}) $ 4 * q + 47 * q^5 - 174 * q^7 - 407 * q^11 + 898 * q^13 - 1868 * q^17 + 1463 * q^19 - 44 * q^23 + 1605 * q^25 - 1534 * q^29 + 11170 * q^31 - 9674 * q^35 + 3113 * q^37 + 15684 * q^41 - 25258 * q^43 + 9576 * q^47 + 4558 * q^49 + 13395 * q^53 - 26210 * q^55 - 47521 * q^59 + 63652 * q^61 + 28254 * q^65 - 44541 * q^67 + 251680 * q^71 - 6039 * q^73 - 35407 * q^77 - 17588 * q^79 - 78650 * q^83 - 116120 * q^85 + 83082 * q^89 + 31747 * q^91 - 214946 * q^95 + 369570 * q^97

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−9.28717	+	16.0858i	−0.166134	+	0.287752i	−0.937057	−	0.349175i	$-0.886462\pi$
$5$	−9.28717	+	16.0858i	−0.166134	+	0.287752i	0.770923	+	0.636928i	$0.219795\pi$
$6$		0			0
$7$	−127.649	+	22.6454i	−0.984626	+	0.174677i
$7$	−127.649	+	22.6454i	−0.984626	+	0.174677i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−80.7128	−	139.799i	−0.201123	−	0.348355i	0.747768	−	0.663960i	$-0.231126\pi$
$11$	−80.7128	−	139.799i	−0.201123	−	0.348355i	−0.948890	+	0.315606i	$0.897792\pi$
$12$		0			0
$13$	14.1283			0.0231863			0.0115932	−	0.999933i	$-0.496310\pi$
$13$	14.1283			0.0231863			0.0115932	+	0.999933i	$0.496310\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−382.851	−	663.118i	−0.321298	−	0.556504i	0.659458	−	0.751741i	$-0.270786\pi$
$17$	−382.851	−	663.118i	−0.321298	−	0.556504i	−0.980756	+	0.195237i	$0.937452\pi$
$18$		0			0
$19$	−707.146	+	1224.81i	−0.449392	+	0.778369i	−0.998346	−	0.0574830i	$-0.981693\pi$
$19$	−707.146	+	1224.81i	−0.449392	+	0.778369i	0.548955	+	0.835852i	$0.315026\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2092.72	−	3624.69i	0.824880	−	1.42873i	−0.0771305	−	0.997021i	$-0.524576\pi$
$23$	2092.72	−	3624.69i	0.824880	−	1.42873i	0.902011	−	0.431713i	$-0.142091\pi$
$24$		0			0
$25$	1390.00	+	2407.55i	0.444799	+	0.770415i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4202.60			0.927947			0.463974	−	0.885849i	$-0.346423\pi$
$29$	4202.60			0.927947			0.463974	+	0.885849i	$0.346423\pi$
$30$		0			0
$31$	1193.68	+	2067.51i	0.223091	+	0.386405i	0.955745	−	0.294196i	$-0.0950520\pi$
$31$	1193.68	+	2067.51i	0.223091	+	0.386405i	−0.732654	+	0.680601i	$0.761719\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	821.224	−	2263.65i	0.113316	−	0.312348i
$36$		0			0
$37$	336.469	−	582.782i	0.0404056	−	0.0699845i	−0.845115	−	0.534584i	$-0.820468\pi$
$37$	336.469	−	582.782i	0.0404056	−	0.0699845i	0.885521	+	0.464599i	$0.153802\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4173.45			0.387735			0.193868	−	0.981028i	$-0.437897\pi$
$41$	4173.45			0.387735			0.193868	+	0.981028i	$0.437897\pi$
$42$		0			0
$43$	−5430.94			−0.447923			−0.223962	−	0.974598i	$-0.571899\pi$
$43$	−5430.94			−0.447923			−0.223962	+	0.974598i	$0.571899\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3151.34	−	5458.28i	0.208090	−	0.360422i	−0.743023	−	0.669266i	$-0.766609\pi$
$47$	3151.34	−	5458.28i	0.208090	−	0.360422i	0.951113	+	0.308844i	$0.0999421\pi$
$48$		0			0
$49$	15781.4	−	5781.32i	0.938976	−	0.343983i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8208.34	+	14217.3i	0.401389	+	0.695226i	0.993894	−	0.110341i	$-0.0351942\pi$
$53$	8208.34	+	14217.3i	0.401389	+	0.695226i	−0.592505	+	0.805567i	$0.701861\pi$
$54$		0			0
$55$	2998.37			0.133653
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1983.24	+	3435.08i	0.0741731	+	0.128472i	0.900726	−	0.434387i	$-0.143035\pi$
$59$	1983.24	+	3435.08i	0.0741731	+	0.128472i	−0.826553	+	0.562858i	$0.809702\pi$
$60$		0			0
$61$	25169.4	−	43594.6i	0.866059	−	1.50006i	6.72450e−5	−	1.00000i	$-0.499979\pi$
$61$	25169.4	−	43594.6i	0.866059	−	1.50006i	0.865992	−	0.500058i	$-0.166688\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−131.212	+	227.266i	−0.00385203	+	0.00667192i
$66$		0			0
$67$	−6822.63	−	11817.1i	−0.185680	−	0.321607i	0.758126	−	0.652109i	$-0.226115\pi$
$67$	−6822.63	−	11817.1i	−0.185680	−	0.321607i	−0.943805	+	0.330502i	$0.892782\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	83957.2			1.97657			0.988284	−	0.152624i	$-0.0487723\pi$
$71$	83957.2			1.97657			0.988284	+	0.152624i	$0.0487723\pi$
$72$		0			0
$73$	14289.2	+	24749.6i	0.313834	+	0.543576i	0.979189	−	0.202951i	$-0.0650532\pi$
$73$	14289.2	+	24749.6i	0.313834	+	0.543576i	−0.665355	+	0.746527i	$0.731720\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13468.7	+	16017.3i	0.258880	+	0.307867i
$78$		0			0
$79$	29977.7	−	51923.0i	0.540420	−	0.936034i	−0.458460	−	0.888715i	$-0.651599\pi$
$79$	29977.7	−	51923.0i	0.540420	−	0.936034i	0.998880	−	0.0473193i	$-0.0150678\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	61583.0			0.981219			0.490610	−	0.871380i	$-0.336774\pi$
$83$	61583.0			0.981219			0.490610	+	0.871380i	$0.336774\pi$
$84$		0			0
$85$	14222.4			0.213514
$86$		0			0
$87$		0			0
$88$		0			0
$89$	21149.2	−	36631.4i	0.283021	−	0.490206i	−0.689107	−	0.724660i	$-0.741997\pi$
$89$	21149.2	−	36631.4i	0.283021	−	0.490206i	0.972127	+	0.234454i	$0.0753301\pi$
$90$		0			0
$91$	−1803.46	+	319.941i	−0.0228298	+	0.00405011i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13134.8	−	22750.1i	−0.149318	−	0.258627i
$96$		0			0
$97$	44638.1			0.481700			0.240850	−	0.970562i	$-0.422574\pi$
$97$	44638.1			0.481700			0.240850	+	0.970562i	$0.422574\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−55962.2	−	96929.4i	−0.545873	−	0.945480i	−0.998551	−	0.0538058i	$-0.982865\pi$
$101$	−55962.2	−	96929.4i	−0.545873	−	0.945480i	0.452678	−	0.891674i	$-0.350469\pi$
$102$		0			0
$103$	−83474.6	+	144582.i	−0.775285	+	1.34283i	0.159350	+	0.987222i	$0.449060\pi$
$103$	−83474.6	+	144582.i	−0.775285	+	1.34283i	−0.934634	+	0.355610i	$0.884273\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	51492.8	−	89188.2i	0.434798	−	0.753092i	−0.562481	−	0.826810i	$-0.690153\pi$
$107$	51492.8	−	89188.2i	0.434798	−	0.753092i	0.997279	+	0.0737182i	$0.0234865\pi$
$108$		0			0
$109$	68910.6	+	119357.i	0.555546	+	0.962233i	0.997861	+	0.0653736i	$0.0208239\pi$
$109$	68910.6	+	119357.i	0.555546	+	0.962233i	−0.442315	+	0.896860i	$0.645843\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	129967.			0.957495			0.478748	−	0.877953i	$-0.341091\pi$
$113$	129967.			0.957495			0.478748	+	0.877953i	$0.341091\pi$
$114$		0			0
$115$	38870.8	+	67326.3i	0.274081	+	0.474723i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	63887.0	+	75976.3i	0.413567	+	0.491825i
$120$		0			0
$121$	67496.4	−	116907.i	0.419099	−	0.725901i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−109681.			−0.627853
$126$		0			0
$127$	−243265.			−1.33835			−0.669177	−	0.743103i	$-0.733353\pi$
$127$	−243265.			−1.33835			−0.669177	+	0.743103i	$0.733353\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	97167.8	−	168300.i	0.494703	−	0.856850i	−0.505279	−	0.862956i	$-0.668610\pi$
$131$	97167.8	−	168300.i	0.494703	−	0.856850i	0.999981	+	0.00610601i	$0.00194361\pi$
$132$		0			0
$133$	62529.8	−	172359.i	0.306519	−	0.844900i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1546.85	−	2679.22i	−0.00704121	−	0.0121957i	0.862483	−	0.506085i	$-0.168908\pi$
$137$	−1546.85	−	2679.22i	−0.00704121	−	0.0121957i	−0.869525	+	0.493890i	$0.835575\pi$
$138$		0			0
$139$	22600.4			0.0992155			0.0496078	−	0.998769i	$-0.484203\pi$
$139$	22600.4			0.0992155			0.0496078	+	0.998769i	$0.484203\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1140.34	−	1975.12i	−0.00466329	−	0.00807706i
$144$		0			0
$145$	−39030.3	+	67602.4i	−0.154164	+	0.267019i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−176397.	+	305529.i	−0.650917	+	1.12742i	0.331983	+	0.943285i	$0.392282\pi$
$149$	−176397.	+	305529.i	−0.650917	+	1.12742i	−0.982901	+	0.184137i	$0.941051\pi$
$150$		0			0
$151$	−72548.2	−	125657.i	−0.258931	−	0.448482i	0.707025	−	0.707189i	$-0.250037\pi$
$151$	−72548.2	−	125657.i	−0.258931	−	0.448482i	−0.965956	+	0.258707i	$0.916704\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−44343.5			−0.148252
$156$		0			0
$157$	108948.	+	188703.i	0.352751	+	0.610983i	0.986730	−	0.162367i	$-0.0519128\pi$
$157$	108948.	+	188703.i	0.352751	+	0.610983i	−0.633979	+	0.773350i	$0.718579\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−185050.	+	510078.i	−0.562632	+	1.55086i
$162$		0			0
$163$	161055.	−	278956.i	0.474795	−	0.822370i	−0.524788	−	0.851233i	$-0.675855\pi$
$163$	161055.	−	278956.i	0.474795	−	0.822370i	0.999583	+	0.0288633i	$0.00918874\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−707392.			−1.96277			−0.981384	−	0.192056i	$-0.938485\pi$
$167$	−707392.			−1.96277			−0.981384	+	0.192056i	$0.938485\pi$
$168$		0			0
$169$	−371093.			−0.999462
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−85253.7	+	147664.i	−0.216570	+	0.375110i	−0.953757	−	0.300579i	$-0.902820\pi$
$173$	−85253.7	+	147664.i	−0.216570	+	0.375110i	0.737187	+	0.675689i	$0.236154\pi$
$174$		0			0
$175$	−231951.	−	275843.i	−0.572534	−	0.680874i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−207703.	−	359752.i	−0.484519	−	0.839211i	0.515323	−	0.856996i	$-0.327672\pi$
$179$	−207703.	−	359752.i	−0.484519	−	0.839211i	−0.999842	+	0.0177851i	$0.994339\pi$
$180$		0			0
$181$	−1162.38			−0.00263726			−0.00131863	−	0.999999i	$-0.500420\pi$
$181$	−1162.38			−0.00263726			−0.00131863	+	0.999999i	$0.500420\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6249.70	+	10824.8i	0.0134255	+	0.0232536i
$186$		0			0
$187$	−61802.0	+	107044.i	−0.129241	+	0.223851i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−372807.	+	645721.i	−0.739436	+	1.28074i	0.213313	+	0.976984i	$0.431575\pi$
$191$	−372807.	+	645721.i	−0.739436	+	1.28074i	−0.952749	+	0.303757i	$0.901759\pi$
$192$		0			0
$193$	99186.5	+	171796.i	0.191672	+	0.331986i	0.945805	−	0.324737i	$-0.105276\pi$
$193$	99186.5	+	171796.i	0.191672	+	0.331986i	−0.754132	+	0.656722i	$0.771942\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	469368.			0.861684			0.430842	−	0.902427i	$-0.358217\pi$
$197$	469368.			0.861684			0.430842	+	0.902427i	$0.358217\pi$
$198$		0			0
$199$	96796.7	+	167657.i	0.173272	+	0.300116i	0.939562	−	0.342379i	$-0.111233\pi$
$199$	96796.7	+	167657.i	0.173272	+	0.300116i	−0.766290	+	0.642495i	$0.777899\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−536457.	+	95169.7i	−0.913681	+	0.162091i
$204$		0			0
$205$	−38759.5	+	67133.4i	−0.0644160	+	0.111572i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	228303.			0.361531
$210$		0			0
$211$	−298066.			−0.460900			−0.230450	−	0.973084i	$-0.574020\pi$
$211$	−298066.			−0.460900			−0.230450	+	0.973084i	$0.574020\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	50438.0	−	87361.3i	0.0744153	−	0.128891i
$216$		0			0
$217$	−199191.	−	236883.i	−0.287157	−	0.341495i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5409.04	−	9368.73i	−0.00744971	−	0.0129033i
$222$		0			0
$223$	−187215.			−0.252103			−0.126051	−	0.992024i	$-0.540230\pi$
$223$	−187215.			−0.252103			−0.126051	+	0.992024i	$0.540230\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−669482.	−	1.15958e6i	−0.862332	−	1.49360i	−0.869672	−	0.493630i	$-0.835670\pi$
$227$	−669482.	−	1.15958e6i	−0.862332	−	1.49360i	0.00734045	−	0.999973i	$-0.497663\pi$
$228$		0			0
$229$	−475828.	+	824158.i	−0.599600	+	1.03854i	0.393280	+	0.919419i	$0.371340\pi$
$229$	−475828.	+	824158.i	−0.599600	+	1.03854i	−0.992880	+	0.119119i	$0.961993\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	371709.	−	643820.i	0.448553	−	0.776917i	−0.549739	−	0.835336i	$-0.685273\pi$
$233$	371709.	−	643820.i	0.448553	−	0.776917i	0.998292	+	0.0584197i	$0.0186062\pi$
$234$		0			0
$235$	58534.0	+	101384.i	0.0691415	+	0.119757i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	625847.			0.708718			0.354359	−	0.935110i	$-0.384699\pi$
$239$	625847.			0.708718			0.354359	+	0.935110i	$0.384699\pi$
$240$		0			0
$241$	666411.	+	1.15426e6i	0.739094	+	1.28015i	0.952904	+	0.303273i	$0.0980793\pi$
$241$	666411.	+	1.15426e6i	0.739094	+	1.28015i	−0.213810	+	0.976875i	$0.568587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−53566.9	+	307549.i	−0.0570140	+	0.327340i
$246$		0			0
$247$	−9990.77	+	17304.5i	−0.0104197	+	0.0180475i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.78809e6			−1.79145			−0.895726	−	0.444606i	$-0.853344\pi$
$251$	−1.78809e6			−1.79145			−0.895726	+	0.444606i	$0.853344\pi$
$252$		0			0
$253$	−675636.			−0.663608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	254462.	−	440741.i	0.240320	−	0.416247i	−0.720485	−	0.693470i	$-0.756081\pi$
$257$	254462.	−	440741.i	0.240320	−	0.416247i	0.960805	+	0.277224i	$0.0894142\pi$
$258$		0			0
$259$	−29752.5	+	82010.9i	−0.0275597	+	0.0759665i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−168247.	−	291412.i	−0.149988	−	0.259787i	0.781235	−	0.624237i	$-0.214590\pi$
$263$	−168247.	−	291412.i	−0.149988	−	0.259787i	−0.931223	+	0.364450i	$0.881257\pi$
$264$		0			0
$265$	−304929.			−0.266737
$266$		0			0
$267$		0			0
$268$		0			0
$269$	747765.	+	1.29517e6i	0.630064	+	1.09130i	0.987538	+	0.157379i	$0.0503045\pi$
$269$	747765.	+	1.29517e6i	0.630064	+	1.09130i	−0.357475	+	0.933923i	$0.616362\pi$
$270$		0			0
$271$	888484.	−	1.53890e6i	0.734897	−	1.27288i	−0.219871	−	0.975529i	$-0.570564\pi$
$271$	888484.	−	1.53890e6i	0.734897	−	1.27288i	0.954768	−	0.297350i	$-0.0961029\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	224381.	−	388640.i	0.178918	−	0.309896i
$276$		0			0
$277$	−440632.	−	763198.i	−0.345046	−	0.597637i	0.640316	−	0.768112i	$-0.278803\pi$
$277$	−440632.	−	763198.i	−0.345046	−	0.597637i	−0.985362	+	0.170474i	$0.945470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.13932e6			0.860756			0.430378	−	0.902649i	$-0.358380\pi$
$281$	1.13932e6			0.860756			0.430378	+	0.902649i	$0.358380\pi$
$282$		0			0
$283$	448733.	+	777228.i	0.333059	+	0.576876i	0.983110	−	0.183015i	$-0.0585857\pi$
$283$	448733.	+	777228.i	0.333059	+	0.576876i	−0.650051	+	0.759891i	$0.725252\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−532735.	+	94509.4i	−0.381774	+	0.0677283i
$288$		0			0
$289$	416778.	−	721881.i	0.293535	−	0.508418i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.84614e6			1.25631			0.628153	−	0.778090i	$-0.283811\pi$
$293$	1.84614e6			1.25631			0.628153	+	0.778090i	$0.283811\pi$
$294$		0			0
$295$	−73674.9			−0.0492907
$296$		0			0
$297$		0			0
$298$		0			0
$299$	29566.5	−	51210.8i	0.0191259	−	0.0331271i
$300$		0			0
$301$	693252.	−	122986.i	0.441037	−	0.0782418i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	467504.	+	809741.i	0.287764	+	0.498421i
$306$		0			0
$307$	1.38565e6			0.839089			0.419544	−	0.907735i	$-0.362190\pi$
$307$	1.38565e6			0.839089			0.419544	+	0.907735i	$0.362190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.15624e6	+	2.00267e6i	0.677871	+	1.17411i	0.975621	+	0.219463i	$0.0704305\pi$
$311$	1.15624e6	+	2.00267e6i	0.677871	+	1.17411i	−0.297750	+	0.954644i	$0.596236\pi$
$312$		0			0
$313$	584400.	−	1.01221e6i	0.337171	−	0.583996i	−0.646729	−	0.762720i	$-0.723863\pi$
$313$	584400.	−	1.01221e6i	0.337171	−	0.583996i	0.983899	+	0.178724i	$0.0571968\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	717166.	−	1.24217e6i	0.400840	−	0.694276i	−0.592987	−	0.805212i	$-0.702052\pi$
$317$	717166.	−	1.24217e6i	0.400840	−	0.694276i	0.993828	+	0.110936i	$0.0353849\pi$
$318$		0			0
$319$	−339204.	−	587519.i	−0.186631	−	0.323255i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.08293e6			0.577554
$324$		0			0
$325$	19638.3	+	34014.5i	0.0103132	+	0.0178631i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−278659.	+	768105.i	−0.141933	+	0.391229i
$330$		0			0
$331$	1.11709e6	−	1.93486e6i	0.560426	−	0.970686i	−0.437033	−	0.899445i	$-0.643971\pi$
$331$	1.11709e6	−	1.93486e6i	0.560426	−	0.970686i	0.997459	−	0.0712406i	$-0.0226958\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	253452.			0.123391
$336$		0			0
$337$	3.08787e6			1.48110			0.740549	−	0.672002i	$-0.234565\pi$
$337$	3.08787e6			1.48110			0.740549	+	0.672002i	$0.234565\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	192690.	−	333749.i	0.0897373	−	0.155429i
$342$		0			0
$343$	−1.88355e6	+	1.09535e6i	−0.864454	+	0.502711i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.55877e6	−	2.69987e6i	−0.694959	−	1.20370i	−0.970195	−	0.242327i	$-0.922089\pi$
$347$	−1.55877e6	−	2.69987e6i	−0.694959	−	1.20370i	0.275236	−	0.961377i	$-0.411244\pi$
$348$		0			0
$349$	613026.			0.269411			0.134706	−	0.990886i	$-0.456991\pi$
$349$	613026.			0.269411			0.134706	+	0.990886i	$0.456991\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.89185e6	+	3.27678e6i	0.808071	+	1.39962i	0.914198	+	0.405268i	$0.132822\pi$
$353$	1.89185e6	+	3.27678e6i	0.808071	+	1.39962i	−0.106127	+	0.994353i	$0.533845\pi$
$354$		0			0
$355$	−779724.	+	1.35052e6i	−0.328375	+	0.568762i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.86107e6	−	3.22346e6i	0.762125	−	1.32004i	−0.179629	−	0.983734i	$-0.557490\pi$
$359$	1.86107e6	−	3.22346e6i	0.762125	−	1.32004i	0.941753	−	0.336304i	$-0.109177\pi$
$360$		0			0
$361$	237940.	+	412123.i	0.0960945	+	0.166441i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−530824.			−0.208554
$366$		0			0
$367$	−2.06898e6	−	3.58357e6i	−0.801845	−	1.38884i	−0.918400	−	0.395652i	$-0.870519\pi$
$367$	−2.06898e6	−	3.58357e6i	−0.801845	−	1.38884i	0.116555	−	0.993184i	$-0.462815\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.36974e6	−	1.62893e6i	−0.516658	−	0.614424i
$372$		0			0
$373$	−1.47209e6	+	2.54973e6i	−0.547851	+	0.948905i	0.450571	+	0.892741i	$0.351220\pi$
$373$	−1.47209e6	+	2.54973e6i	−0.547851	+	0.948905i	−0.998422	+	0.0561644i	$0.982113\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	59375.7			0.0215157
$378$		0			0
$379$	2.97504e6			1.06388			0.531942	−	0.846781i	$-0.321462\pi$
$379$	2.97504e6			1.06388			0.531942	+	0.846781i	$0.321462\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.03052e6	−	1.78491e6i	0.358971	−	0.621756i	−0.628818	−	0.777552i	$-0.716461\pi$
$383$	1.03052e6	−	1.78491e6i	0.358971	−	0.621756i	0.987789	+	0.155796i	$0.0497944\pi$
$384$		0			0
$385$	−382739.	+	67899.5i	−0.131598	+	0.0233461i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.04542e6	+	1.81072e6i	0.350281	+	0.606705i	0.986299	−	0.164970i	$-0.0527527\pi$
$389$	1.04542e6	+	1.81072e6i	0.350281	+	0.606705i	−0.636018	+	0.771675i	$0.719419\pi$
$390$		0			0
$391$	−3.20480e6			−1.06013
$392$		0			0
$393$		0			0
$394$		0			0
$395$	556817.	+	964435.i	0.179564	+	0.311014i
$396$		0			0
$397$	−585647.	+	1.01437e6i	−0.186492	+	0.323013i	−0.944078	−	0.329722i	$-0.893045\pi$
$397$	−585647.	+	1.01437e6i	−0.186492	+	0.323013i	0.757586	+	0.652735i	$0.226378\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.54574e6	−	2.67731e6i	0.480039	−	0.831452i	−0.519699	−	0.854349i	$-0.673956\pi$
$401$	1.54574e6	−	2.67731e6i	0.480039	−	0.831452i	0.999738	+	0.0228979i	$0.00728925\pi$
$402$		0			0
$403$	16864.6	+	29210.4i	0.00517266	+	0.00895930i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−108630.			−0.0325059
$408$		0			0
$409$	2.17426e6	+	3.76593e6i	0.642693	+	1.11318i	0.984829	+	0.173527i	$0.0555163\pi$
$409$	2.17426e6	+	3.76593e6i	0.642693	+	1.11318i	−0.342136	+	0.939650i	$0.611150\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−330947.	−	393572.i	−0.0954737	−	0.113540i
$414$		0			0
$415$	−571932.	+	990616.i	−0.163014	+	0.282348i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.13660e6			−0.872818			−0.436409	−	0.899748i	$-0.643750\pi$
$419$	−3.13660e6			−0.872818			−0.436409	+	0.899748i	$0.643750\pi$
$420$		0			0
$421$	6.02560e6			1.65690			0.828448	−	0.560066i	$-0.189224\pi$
$421$	6.02560e6			1.65690			0.828448	+	0.560066i	$0.189224\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.06432e6	−	1.84346e6i	0.285826	−	0.495065i
$426$		0			0
$427$	−2.22562e6	+	6.13476e6i	−0.590719	+	1.62828i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.14931e6	−	1.99067e6i	−0.298020	−	0.516186i	0.677663	−	0.735373i	$-0.262993\pi$
$431$	−1.14931e6	−	1.99067e6i	−0.298020	−	0.516186i	−0.975683	+	0.219187i	$0.929660\pi$
$432$		0			0
$433$	−5.62982e6			−1.44303			−0.721515	−	0.692399i	$-0.756554\pi$
$433$	−5.62982e6			−1.44303			−0.721515	+	0.692399i	$0.756554\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.95971e6	+	5.12637e6i	0.741388	+	1.28412i
$438$		0			0
$439$	2.72482e6	−	4.71952e6i	0.674801	−	1.16879i	−0.301725	−	0.953395i	$-0.597563\pi$
$439$	2.72482e6	−	4.71952e6i	0.674801	−	1.16879i	0.976527	−	0.215396i	$-0.0691041\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.85467e6	−	4.94443e6i	0.691108	−	1.19703i	−0.280367	−	0.959893i	$-0.590456\pi$
$443$	2.85467e6	−	4.94443e6i	0.691108	−	1.19703i	0.971475	−	0.237142i	$-0.0762105\pi$
$444$		0			0
$445$	392832.	+	680405.i	0.0940387	+	0.162880i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.38234e6			−1.25996			−0.629978	−	0.776613i	$-0.716936\pi$
$449$	−5.38234e6			−1.25996			−0.629978	+	0.776613i	$0.716936\pi$
$450$		0			0
$451$	−336851.	−	583442.i	−0.0779823	−	0.135069i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	11602.5	−	31981.5i	0.00262738	−	0.00724220i
$456$		0			0
$457$	3.19525e6	−	5.53434e6i	0.715673	−	1.23958i	−0.247026	−	0.969009i	$-0.579453\pi$
$457$	3.19525e6	−	5.53434e6i	0.715673	−	1.23958i	0.962699	−	0.270573i	$-0.0872133\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.34511e6			−1.60970			−0.804851	−	0.593476i	$-0.797755\pi$
$461$	−7.34511e6			−1.60970			−0.804851	+	0.593476i	$0.797755\pi$
$462$		0			0
$463$	−4.63416e6			−1.00466			−0.502329	−	0.864677i	$-0.667523\pi$
$463$	−4.63416e6			−1.00466			−0.502329	+	0.864677i	$0.667523\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.81598e6	+	6.60946e6i	−0.809680	+	1.40241i	0.103406	+	0.994639i	$0.467026\pi$
$467$	−3.81598e6	+	6.60946e6i	−0.809680	+	1.40241i	−0.913086	+	0.407768i	$0.866307\pi$
$468$		0			0
$469$	1.13850e6	+	1.35394e6i	0.239002	+	0.284229i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	438346.	+	759238.i	0.0900875	+	0.156036i
$474$		0			0
$475$	−3.93172e6			−0.799556
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.80631e6	+	8.32477e6i	0.957135	+	1.65781i	0.729406	+	0.684081i	$0.239796\pi$
$479$	4.80631e6	+	8.32477e6i	0.957135	+	1.65781i	0.227728	+	0.973725i	$0.426870\pi$
$480$		0			0
$481$	4753.74	−	8233.72i	0.000936856	−	0.00162268i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−414562.	+	718042.i	−0.0800267	+	0.138610i
$486$		0			0
$487$	−1.40112e6	−	2.42680e6i	−0.267702	−	0.463673i	0.700566	−	0.713588i	$-0.252931\pi$
$487$	−1.40112e6	−	2.42680e6i	−0.267702	−	0.463673i	−0.968268	+	0.249914i	$0.919598\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.82008e6			−0.902299			−0.451149	−	0.892448i	$-0.648986\pi$
$491$	−4.82008e6			−0.902299			−0.451149	+	0.892448i	$0.648986\pi$
$492$		0			0
$493$	−1.60897e6	−	2.78682e6i	−0.298148	−	0.516407i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.07170e7	+	1.90125e6i	−1.94618	+	0.345261i
$498$		0			0
$499$	−2.29829e6	+	3.98076e6i	−0.413194	+	0.715673i	−0.995237	−	0.0974853i	$-0.968920\pi$
$499$	−2.29829e6	+	3.98076e6i	−0.413194	+	0.715673i	0.582043	+	0.813158i	$0.302253\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.80055e6			−0.317311			−0.158656	−	0.987334i	$-0.550716\pi$
$503$	−1.80055e6			−0.317311			−0.158656	+	0.987334i	$0.550716\pi$
$504$		0			0
$505$	2.07892e6			0.362752
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.41602e6	+	4.18467e6i	−0.413339	+	0.715924i	−0.995253	−	0.0973263i	$-0.968971\pi$
$509$	−2.41602e6	+	4.18467e6i	−0.413339	+	0.715924i	0.581913	+	0.813251i	$0.302304\pi$
$510$		0			0
$511$	−2.38446e6	−	2.83566e6i	−0.403959	−	0.480400i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.55048e6	−	2.68552e6i	−0.257602	−	0.446180i
$516$		0			0
$517$	−1.01741e6			−0.167406
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.40562e6	+	2.43461e6i	0.226869	+	0.392948i	0.956878	−	0.290489i	$-0.0938178\pi$
$521$	1.40562e6	+	2.43461e6i	0.226869	+	0.392948i	−0.730010	+	0.683437i	$0.760484\pi$
$522$		0			0
$523$	1.22715e6	−	2.12548e6i	0.196174	−	0.339784i	−0.751110	−	0.660177i	$-0.770481\pi$
$523$	1.22715e6	−	2.12548e6i	0.196174	−	0.339784i	0.947285	+	0.320392i	$0.103815\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	914000.	−	1.58309e6i	0.143357	−	0.248302i
$528$		0			0
$529$	−5.54076e6	−	9.59687e6i	−0.860855	−	1.49104i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	58963.7			0.00899015
$534$		0			0
$535$	956445.	+	1.65661e6i	0.144469	+	0.250228i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.08198e6	−	1.73959e6i	−0.308677	−	0.257914i
$540$		0			0
$541$	579232.	−	1.00326e6i	0.0850862	−	0.147374i	−0.820342	−	0.571874i	$-0.806217\pi$
$541$	579232.	−	1.00326e6i	0.0850862	−	0.147374i	0.905428	+	0.424500i	$0.139550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.55994e6			−0.369180
$546$		0			0
$547$	4.63638e6			0.662538			0.331269	−	0.943536i	$-0.392523\pi$
$547$	4.63638e6			0.662538			0.331269	+	0.943536i	$0.392523\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.97185e6	+	5.14740e6i	−0.417012	+	0.722285i
$552$		0			0
$553$	−2.65080e6	+	7.30676e6i	−0.368608	+	1.01604i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	135916.	+	235414.i	0.0185624	+	0.0321510i	0.875157	−	0.483838i	$-0.160758\pi$
$557$	135916.	+	235414.i	0.0185624	+	0.0321510i	−0.856595	+	0.515989i	$0.827424\pi$
$558$		0			0
$559$	−76730.0			−0.0103857
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.69122e6	+	2.92928e6i	0.224869	+	0.389485i	0.956280	−	0.292452i	$-0.0944712\pi$
$563$	1.69122e6	+	2.92928e6i	0.224869	+	0.389485i	−0.731411	+	0.681937i	$0.761138\pi$
$564$		0			0
$565$	−1.20702e6	+	2.09063e6i	−0.159072	+	0.275522i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.56130e6	−	1.13645e7i	0.849590	−	1.47153i	−0.0319843	−	0.999488i	$-0.510183\pi$
$569$	6.56130e6	−	1.13645e7i	0.849590	−	1.47153i	0.881574	−	0.472045i	$-0.156484\pi$
$570$		0			0
$571$	7.35811e6	+	1.27446e7i	0.944443	+	1.63582i	0.756862	+	0.653575i	$0.226731\pi$
$571$	7.35811e6	+	1.27446e7i	0.944443	+	1.63582i	0.187581	+	0.982249i	$0.439935\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.16355e7			1.46762
$576$		0			0
$577$	1.94148e6	+	3.36273e6i	0.242769	+	0.420487i	0.961502	−	0.274798i	$-0.0886111\pi$
$577$	1.94148e6	+	3.36273e6i	0.242769	+	0.420487i	−0.718733	+	0.695286i	$0.755278\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.86099e6	+	1.39457e6i	−0.966134	+	0.171396i
$582$		0			0
$583$	1.32504e6	−	2.29503e6i	0.161457	−	0.279651i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.97913e6			−0.596428			−0.298214	−	0.954499i	$-0.596391\pi$
$587$	−4.97913e6			−0.596428			−0.298214	+	0.954499i	$0.596391\pi$
$588$		0			0
$589$	−3.37641e6			−0.401021
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.66344e6	−	1.32735e7i	0.894926	−	1.55006i	0.0610292	−	0.998136i	$-0.480562\pi$
$593$	7.66344e6	−	1.32735e7i	0.894926	−	1.55006i	0.833896	−	0.551921i	$-0.186105\pi$
$594$		0			0
$595$	−1.81547e6	+	322073.i	−0.210231	+	0.0372959i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.31480e6	−	5.74141e6i	−0.377477	−	0.653810i	0.613217	−	0.789914i	$-0.289875\pi$
$599$	−3.31480e6	−	5.74141e6i	−0.377477	−	0.653810i	−0.990694	+	0.136104i	$0.956542\pi$
$600$		0			0
$601$	−1.45010e6			−0.163762			−0.0818808	−	0.996642i	$-0.526093\pi$
$601$	−1.45010e6			−0.163762			−0.0818808	+	0.996642i	$0.526093\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.25370e6	+	2.17147e6i	0.139253	+	0.241194i
$606$		0			0
$607$	−1.95555e6	+	3.38711e6i	−0.215425	+	0.373128i	−0.953404	−	0.301696i	$-0.902447\pi$
$607$	−1.95555e6	+	3.38711e6i	−0.215425	+	0.373128i	0.737979	+	0.674824i	$0.235780\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	44523.1	−	77116.2i	0.00482483	−	0.00835685i
$612$		0			0
$613$	−5.88765e6	−	1.01977e7i	−0.632835	−	1.09610i	−0.986969	−	0.160908i	$-0.948558\pi$
$613$	−5.88765e6	−	1.01977e7i	−0.632835	−	1.09610i	0.354134	−	0.935195i	$-0.384776\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.61462e6			0.488004			0.244002	−	0.969775i	$-0.421540\pi$
$617$	4.61462e6			0.488004			0.244002	+	0.969775i	$0.421540\pi$
$618$		0			0
$619$	2.83733e6	+	4.91440e6i	0.297634	+	0.515518i	0.975594	−	0.219581i	$-0.0704690\pi$
$619$	2.83733e6	+	4.91440e6i	0.297634	+	0.515518i	−0.677960	+	0.735099i	$0.737136\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.87013e6	+	5.15489e6i	−0.193042	+	0.532107i
$624$		0			0
$625$	−3.32511e6	+	5.75926e6i	−0.340491	+	0.589748i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−515271.			−0.0519289
$630$		0			0
$631$	5.67894e6			0.567798			0.283899	−	0.958854i	$-0.408372\pi$
$631$	5.67894e6			0.567798			0.283899	+	0.958854i	$0.408372\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.25925e6	−	3.91313e6i	0.222346	−	0.385114i
$636$		0			0
$637$	222964.	−	81680.2i	0.0217714	−	0.00797569i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.05369e6	−	1.82504e6i	−0.101290	−	0.175439i	0.810926	−	0.585148i	$-0.198964\pi$
$641$	−1.05369e6	−	1.82504e6i	−0.101290	−	0.175439i	−0.912216	+	0.409709i	$0.865630\pi$
$642$		0			0
$643$	−2.30987e6			−0.220323			−0.110162	−	0.993914i	$-0.535137\pi$
$643$	−2.30987e6			−0.220323			−0.110162	+	0.993914i	$0.535137\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.93033e6	+	8.53958e6i	0.463036	+	0.802003i	0.999111	−	0.0421683i	$-0.0134266\pi$
$647$	4.93033e6	+	8.53958e6i	0.463036	+	0.802003i	−0.536074	+	0.844171i	$0.680093\pi$
$648$		0			0
$649$	320147.	−	554510.i	0.0298358	−	0.0516771i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.01545e6	−	8.68701e6i	0.460285	−	0.797237i	−0.538690	−	0.842504i	$-0.681081\pi$
$653$	5.01545e6	−	8.68701e6i	0.460285	−	0.797237i	0.998975	+	0.0452673i	$0.0144139\pi$
$654$		0			0
$655$	1.80483e6	+	3.12605e6i	0.164374	+	0.284704i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.42828e7			1.28115			0.640575	−	0.767895i	$-0.278696\pi$
$659$	1.42828e7			1.28115			0.640575	+	0.767895i	$0.278696\pi$
$660$		0			0
$661$	9.82097e6	+	1.70104e7i	0.874280	+	1.51430i	0.857527	+	0.514438i	$0.172000\pi$
$661$	9.82097e6	+	1.70104e7i	0.874280	+	1.51430i	0.0167532	+	0.999860i	$0.494667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.19182e6	+	2.60657e6i	0.192199	+	0.228568i
$666$		0			0
$667$	8.79486e6	−	1.52331e7i	0.765445	−	1.32579i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.12596e6			−0.696736
$672$		0			0
$673$	−9.00150e6			−0.766086			−0.383043	−	0.923731i	$-0.625124\pi$
$673$	−9.00150e6			−0.766086			−0.383043	+	0.923731i	$0.625124\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.71181e6	−	1.16252e7i	0.562818	−	0.974830i	−0.434431	−	0.900705i	$-0.643050\pi$
$677$	6.71181e6	−	1.16252e7i	0.562818	−	0.974830i	0.997249	−	0.0741247i	$-0.0236163\pi$
$678$		0			0
$679$	−5.69800e6	+	1.01085e6i	−0.474294	+	0.0841418i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.00840e6	−	6.94276e6i	−0.328791	−	0.569483i	0.653481	−	0.756943i	$-0.273308\pi$
$683$	−4.00840e6	−	6.94276e6i	−0.328791	−	0.569483i	−0.982272	+	0.187460i	$0.939974\pi$
$684$		0			0
$685$	57463.5			0.00467913
$686$		0			0
$687$		0			0
$688$		0			0
$689$	115970.	+	200866.i	0.00930673	+	0.0161197i
$690$		0			0
$691$	1.20987e7	−	2.09555e7i	0.963923	−	1.66956i	0.251442	−	0.967872i	$-0.419095\pi$
$691$	1.20987e7	−	2.09555e7i	0.963923	−	1.66956i	0.712481	−	0.701691i	$-0.247572\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−209894.	+	363547.i	−0.0164831	+	0.0285495i
$696$		0			0
$697$	−1.59781e6	−	2.76749e6i	−0.124578	−	0.215776i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.56268e6			0.120109			0.0600543	−	0.998195i	$-0.480873\pi$
$701$	1.56268e6			0.120109			0.0600543	+	0.998195i	$0.480873\pi$
$702$		0			0
$703$	475866.	+	824224.i	0.0363158	+	0.0629009i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.33851e6	+	1.11056e7i	0.702634	+	0.835592i
$708$		0			0
$709$	9.76950e6	−	1.69213e7i	0.729889	−	1.26420i	−0.227041	−	0.973885i	$-0.572905\pi$
$709$	9.76950e6	−	1.69213e7i	0.729889	−	1.26420i	0.956930	−	0.290320i	$-0.0937616\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.99210e6			0.736093
$714$		0			0
$715$	42362.0			0.00309892
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.05095e6	+	7.01645e6i	−0.292237	+	0.506169i	−0.974338	−	0.225089i	$-0.927733\pi$
$719$	−4.05095e6	+	7.01645e6i	−0.292237	+	0.506169i	0.682102	+	0.731257i	$0.261066\pi$
$720$		0			0
$721$	7.38129e6	−	2.03460e7i	0.528804	−	1.45761i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.84161e6	+	1.01180e7i	0.412750	+	0.714904i
$726$		0			0
$727$	2.52759e7			1.77366			0.886829	−	0.462098i	$-0.152903\pi$
$727$	2.52759e7			1.77366			0.886829	+	0.462098i	$0.152903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.07924e6	+	3.60135e6i	0.143917	+	0.249271i
$732$		0			0
$733$	−1.37207e7	+	2.37650e7i	−0.943228	+	1.63372i	−0.183969	+	0.982932i	$0.558894\pi$
$733$	−1.37207e7	+	2.37650e7i	−0.943228	+	1.63372i	−0.759260	+	0.650787i	$0.774439\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.10135e6	+	1.90759e6i	−0.0746888	+	0.129365i
$738$		0			0
$739$	3.26785e6	+	5.66008e6i	0.220115	+	0.381251i	0.954843	−	0.297111i	$-0.0960233\pi$
$739$	3.26785e6	+	5.66008e6i	0.220115	+	0.381251i	−0.734727	+	0.678362i	$0.762690\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.46784e7			1.64000			0.820001	−	0.572362i	$-0.193973\pi$
$743$	2.46784e7			1.64000			0.820001	+	0.572362i	$0.193973\pi$
$744$		0			0
$745$	−3.27646e6	−	5.67499e6i	−0.216279	−	0.374606i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.55329e6	+	1.25508e7i	−0.296565	+	0.817463i
$750$		0			0
$751$	1.63911e6	−	2.83902e6i	0.106049	−	0.183683i	−0.808117	−	0.589022i	$-0.799513\pi$
$751$	1.63911e6	−	2.83902e6i	0.106049	−	0.183683i	0.914166	+	0.405339i	$0.132847\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.69507e6			0.172069
$756$		0			0
$757$	3.68090e6			0.233461			0.116731	−	0.993164i	$-0.462759\pi$
$757$	3.68090e6			0.233461			0.116731	+	0.993164i	$0.462759\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.05099e7	−	1.82037e7i	0.657866	−	1.13946i	−0.323301	−	0.946296i	$-0.604792\pi$
$761$	1.05099e7	−	1.82037e7i	0.657866	−	1.13946i	0.981167	−	0.193162i	$-0.0618742\pi$
$762$		0			0
$763$	−1.14992e7	−	1.36752e7i	−0.715084	−	0.850399i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28019.9	+	48531.9i	0.00171980	+	0.00297878i
$768$		0			0
$769$	−4.15418e6			−0.253320			−0.126660	−	0.991946i	$-0.540426\pi$
$769$	−4.15418e6			−0.253320			−0.126660	+	0.991946i	$0.540426\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	731741.	+	1.26741e6i	0.0440462	+	0.0762903i	0.887208	−	0.461370i	$-0.152642\pi$
$773$	731741.	+	1.26741e6i	0.0440462	+	0.0762903i	−0.843162	+	0.537660i	$0.819308\pi$
$774$		0			0
$775$	−3.31841e6	+	5.74765e6i	−0.198461	+	0.343745i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.95123e6	+	5.11169e6i	−0.174245	+	0.301801i
$780$		0			0
$781$	−6.77642e6	−	1.17371e7i	−0.397533	−	0.688547i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.04726e6			−0.234416
$786$		0			0
$787$	1.00855e7	+	1.74687e7i	0.580446	+	1.00536i	0.995426	+	0.0955317i	$0.0304551\pi$
$787$	1.00855e7	+	1.74687e7i	0.580446	+	1.00536i	−0.414980	+	0.909830i	$0.636212\pi$
$788$		0			0
$789$		0			0
$790$		0			0

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.28717 + 16.0858i) q^{5} +(-127.649 + 22.6454i) q^{7} +(-80.7128 - 139.799i) q^{11} +14.1283 q^{13} +(-382.851 - 663.118i) q^{17} +(-707.146 + 1224.81i) q^{19} +(2092.72 - 3624.69i) q^{23} +(1390.00 + 2407.55i) q^{25} +4202.60 q^{29} +(1193.68 + 2067.51i) q^{31} +(821.224 - 2263.65i) q^{35} +(336.469 - 582.782i) q^{37} +4173.45 q^{41} -5430.94 q^{43} +(3151.34 - 5458.28i) q^{47} +(15781.4 - 5781.32i) q^{49} +(8208.34 + 14217.3i) q^{53} +2998.37 q^{55} +(1983.24 + 3435.08i) q^{59} +(25169.4 - 43594.6i) q^{61} +(-131.212 + 227.266i) q^{65} +(-6822.63 - 11817.1i) q^{67} +83957.2 q^{71} +(14289.2 + 24749.6i) q^{73} +(13468.7 + 16017.3i) q^{77} +(29977.7 - 51923.0i) q^{79} +61583.0 q^{83} +14222.4 q^{85} +(21149.2 - 36631.4i) q^{89} +(-1803.46 + 319.941i) q^{91} +(-13134.8 - 22750.1i) q^{95} +44638.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 47 q^{5} - 174 q^{7} - 407 q^{11} + 898 q^{13} - 1868 q^{17} + 1463 q^{19} - 44 q^{23} + 1605 q^{25} - 1534 q^{29} + 11170 q^{31} - 9674 q^{35} + 3113 q^{37} + 15684 q^{41} - 25258 q^{43} + 9576 q^{47}+ \cdots + 369570 q^{97}+O(q^{100}) \) 4 * q + 47 * q^5 - 174 * q^7 - 407 * q^11 + 898 * q^13 - 1868 * q^17 + 1463 * q^19 - 44 * q^23 + 1605 * q^25 - 1534 * q^29 + 11170 * q^31 - 9674 * q^35 + 3113 * q^37 + 15684 * q^41 - 25258 * q^43 + 9576 * q^47 + 4558 * q^49 + 13395 * q^53 - 26210 * q^55 - 47521 * q^59 + 63652 * q^61 + 28254 * q^65 - 44541 * q^67 + 251680 * q^71 - 6039 * q^73 - 35407 * q^77 - 17588 * q^79 - 78650 * q^83 - 116120 * q^85 + 83082 * q^89 + 31747 * q^91 - 214946 * q^95 + 369570 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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