LMFDB - Embedded newform 252.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$252 = 2^{2} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	252.a (trivial)

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:	yes
Analytic conductor:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	252.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+96.0000 q^{5} +49.0000 q^{7} +720.000 q^{11} +572.000 q^{13} -1254.00 q^{17} -94.0000 q^{19} -96.0000 q^{23} +6091.00 q^{25} +4374.00 q^{29} -6244.00 q^{31} +4704.00 q^{35} -10798.0 q^{37} -12006.0 q^{41} -9160.00 q^{43} +25836.0 q^{47} +2401.00 q^{49} -1014.00 q^{53} +69120.0 q^{55} -1242.00 q^{59} +7592.00 q^{61} +54912.0 q^{65} +41132.0 q^{67} +37632.0 q^{71} -13438.0 q^{73} +35280.0 q^{77} +6248.00 q^{79} +25254.0 q^{83} -120384. q^{85} +45126.0 q^{89} +28028.0 q^{91} -9024.00 q^{95} +107222. q^{97} +O(q^{100})

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
$5$	96.0000	1.71730	0.858650	−	0.512562i	$-0.171304\pi$
$5$	96.0000	1.71730	0.858650	+	0.512562i	$0.171304\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	720.000	1.79412	0.897059	−	0.441912i	$-0.145700\pi$
$11$	720.000	1.79412	0.897059	+	0.441912i	$0.145700\pi$
$12$	0	0
$13$	572.000	0.938723	0.469362	−	0.883006i	$-0.344484\pi$
$13$	572.000	0.938723	0.469362	+	0.883006i	$0.344484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1254.00	−1.05239	−0.526193	−	0.850365i	$-0.676381\pi$
$17$	−1254.00	−1.05239	−0.526193	+	0.850365i	$0.676381\pi$
$18$	0	0
$19$	−94.0000	−0.0597371	−0.0298685	−	0.999554i	$-0.509509\pi$
$19$	−94.0000	−0.0597371	−0.0298685	+	0.999554i	$0.509509\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−96.0000	−0.0378400	−0.0189200	−	0.999821i	$-0.506023\pi$
$23$	−96.0000	−0.0378400	−0.0189200	+	0.999821i	$0.506023\pi$
$24$	0	0
$25$	6091.00	1.94912
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4374.00	0.965792	0.482896	−	0.875678i	$-0.339585\pi$
$29$	4374.00	0.965792	0.482896	+	0.875678i	$0.339585\pi$
$30$	0	0
$31$	−6244.00	−1.16697	−0.583484	−	0.812125i	$-0.698311\pi$
$31$	−6244.00	−1.16697	−0.583484	+	0.812125i	$0.698311\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4704.00	0.649078
$36$	0	0
$37$	−10798.0	−1.29670	−0.648349	−	0.761343i	$-0.724540\pi$
$37$	−10798.0	−1.29670	−0.648349	+	0.761343i	$0.724540\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12006.0	−1.11542	−0.557710	−	0.830036i	$-0.688320\pi$
$41$	−12006.0	−1.11542	−0.557710	+	0.830036i	$0.688320\pi$
$42$	0	0
$43$	−9160.00	−0.755482	−0.377741	−	0.925911i	$-0.623299\pi$
$43$	−9160.00	−0.755482	−0.377741	+	0.925911i	$0.623299\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	25836.0	1.70601	0.853003	−	0.521906i	$-0.174779\pi$
$47$	25836.0	1.70601	0.853003	+	0.521906i	$0.174779\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1014.00	−0.0495848	−0.0247924	−	0.999693i	$-0.507892\pi$
$53$	−1014.00	−0.0495848	−0.0247924	+	0.999693i	$0.507892\pi$
$54$	0	0
$55$	69120.0	3.08104
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1242.00	−0.0464506	−0.0232253	−	0.999730i	$-0.507394\pi$
$59$	−1242.00	−0.0464506	−0.0232253	+	0.999730i	$0.507394\pi$
$60$	0	0
$61$	7592.00	0.261235	0.130618	−	0.991433i	$-0.458304\pi$
$61$	7592.00	0.261235	0.130618	+	0.991433i	$0.458304\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	54912.0	1.61207
$66$	0	0
$67$	41132.0	1.11942	0.559710	−	0.828689i	$-0.310913\pi$
$67$	41132.0	1.11942	0.559710	+	0.828689i	$0.310913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	37632.0	0.885955	0.442977	−	0.896533i	$-0.353922\pi$
$71$	37632.0	0.885955	0.442977	+	0.896533i	$0.353922\pi$
$72$	0	0
$73$	−13438.0	−0.295140	−0.147570	−	0.989052i	$-0.547145\pi$
$73$	−13438.0	−0.295140	−0.147570	+	0.989052i	$0.547145\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	35280.0	0.678113
$78$	0	0
$79$	6248.00	0.112635	0.0563175	−	0.998413i	$-0.482064\pi$
$79$	6248.00	0.112635	0.0563175	+	0.998413i	$0.482064\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	25254.0	0.402379	0.201189	−	0.979552i	$-0.435519\pi$
$83$	25254.0	0.402379	0.201189	+	0.979552i	$0.435519\pi$
$84$	0	0
$85$	−120384.	−1.80726
$86$	0	0
$87$	0	0
$88$	0	0
$89$	45126.0	0.603882	0.301941	−	0.953327i	$-0.402365\pi$
$89$	45126.0	0.603882	0.301941	+	0.953327i	$0.402365\pi$
$90$	0	0
$91$	28028.0	0.354804
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−9024.00	−0.102586
$96$	0	0
$97$	107222.	1.15706	0.578528	−	0.815662i	$-0.303627\pi$
$97$	107222.	1.15706	0.578528	+	0.815662i	$0.303627\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−47136.0	−0.459779	−0.229890	−	0.973217i	$-0.573837\pi$
$101$	−47136.0	−0.459779	−0.229890	+	0.973217i	$0.573837\pi$
$102$	0	0
$103$	122204.	1.13499	0.567495	−	0.823377i	$-0.307912\pi$
$103$	122204.	1.13499	0.567495	+	0.823377i	$0.307912\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	129636.	1.09463	0.547314	−	0.836928i	$-0.315651\pi$
$107$	129636.	1.09463	0.547314	+	0.836928i	$0.315651\pi$
$108$	0	0
$109$	−220558.	−1.77810	−0.889051	−	0.457809i	$-0.848635\pi$
$109$	−220558.	−1.77810	−0.889051	+	0.457809i	$0.848635\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−170694.	−1.25754	−0.628770	−	0.777591i	$-0.716441\pi$
$113$	−170694.	−1.25754	−0.628770	+	0.777591i	$0.716441\pi$
$114$	0	0
$115$	−9216.00	−0.0649827
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−61446.0	−0.397765
$120$	0	0
$121$	357349.	2.21886
$122$	0	0
$123$	0	0
$124$	0	0
$125$	284736.	1.62992
$126$	0	0
$127$	−249808.	−1.37435	−0.687175	−	0.726492i	$-0.741149\pi$
$127$	−249808.	−1.37435	−0.687175	+	0.726492i	$0.741149\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12210.0	−0.0621638	−0.0310819	−	0.999517i	$-0.509895\pi$
$131$	−12210.0	−0.0621638	−0.0310819	+	0.999517i	$0.509895\pi$
$132$	0	0
$133$	−4606.00	−0.0225785
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13902.0	0.0632814	0.0316407	−	0.999499i	$-0.489927\pi$
$137$	13902.0	0.0632814	0.0316407	+	0.999499i	$0.489927\pi$
$138$	0	0
$139$	−431794.	−1.89557	−0.947785	−	0.318911i	$-0.896683\pi$
$139$	−431794.	−1.89557	−0.947785	+	0.318911i	$0.896683\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	411840.	1.68418
$144$	0	0
$145$	419904.	1.65856
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−326814.	−1.20597	−0.602983	−	0.797754i	$-0.706021\pi$
$149$	−326814.	−1.20597	−0.602983	+	0.797754i	$0.706021\pi$
$150$	0	0
$151$	173480.	0.619166	0.309583	−	0.950872i	$-0.399811\pi$
$151$	173480.	0.619166	0.309583	+	0.950872i	$0.399811\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−599424.	−2.00403
$156$	0	0
$157$	−54532.0	−0.176564	−0.0882820	−	0.996096i	$-0.528138\pi$
$157$	−54532.0	−0.176564	−0.0882820	+	0.996096i	$0.528138\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4704.00	−0.0143022
$162$	0	0
$163$	104960.	0.309425	0.154712	−	0.987960i	$-0.450555\pi$
$163$	104960.	0.309425	0.154712	+	0.987960i	$0.450555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−160788.	−0.446131	−0.223066	−	0.974803i	$-0.571606\pi$
$167$	−160788.	−0.446131	−0.223066	+	0.974803i	$0.571606\pi$
$168$	0	0
$169$	−44109.0	−0.118798
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−360564.	−0.915940	−0.457970	−	0.888968i	$-0.651423\pi$
$173$	−360564.	−0.915940	−0.457970	+	0.888968i	$0.651423\pi$
$174$	0	0
$175$	298459.	0.736698
$176$	0	0
$177$	0	0
$178$	0	0
$179$	312732.	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$179$	312732.	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$180$	0	0
$181$	−123820.	−0.280928	−0.140464	−	0.990086i	$-0.544859\pi$
$181$	−123820.	−0.280928	−0.140464	+	0.990086i	$0.544859\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.03661e6	−2.22682
$186$	0	0
$187$	−902880.	−1.88810
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−323448.	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$191$	−323448.	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$192$	0	0
$193$	−619954.	−1.19803	−0.599013	−	0.800739i	$-0.704440\pi$
$193$	−619954.	−1.19803	−0.599013	+	0.800739i	$0.704440\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	499362.	0.916748	0.458374	−	0.888759i	$-0.348432\pi$
$197$	499362.	0.916748	0.458374	+	0.888759i	$0.348432\pi$
$198$	0	0
$199$	−785932.	−1.40686	−0.703432	−	0.710762i	$-0.748350\pi$
$199$	−785932.	−1.40686	−0.703432	+	0.710762i	$0.748350\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	214326.	0.365035
$204$	0	0
$205$	−1.15258e6	−1.91551
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−67680.0	−0.107175
$210$	0	0
$211$	1.06276e6	1.64335	0.821676	−	0.569955i	$-0.193039\pi$
$211$	1.06276e6	1.64335	0.821676	+	0.569955i	$0.193039\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−879360.	−1.29739
$216$	0	0
$217$	−305956.	−0.441072
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−717288.	−0.987900
$222$	0	0
$223$	707720.	0.953014	0.476507	−	0.879171i	$-0.341903\pi$
$223$	707720.	0.953014	0.476507	+	0.879171i	$0.341903\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.04437e6	1.34520	0.672602	−	0.740005i	$-0.265177\pi$
$227$	1.04437e6	1.34520	0.672602	+	0.740005i	$0.265177\pi$
$228$	0	0
$229$	−539716.	−0.680106	−0.340053	−	0.940406i	$-0.610445\pi$
$229$	−539716.	−0.680106	−0.340053	+	0.940406i	$0.610445\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−177114.	−0.213729	−0.106864	−	0.994274i	$-0.534081\pi$
$233$	−177114.	−0.213729	−0.106864	+	0.994274i	$0.534081\pi$
$234$	0	0
$235$	2.48026e6	2.92972
$236$	0	0
$237$	0	0
$238$	0	0
$239$	655464.	0.742257	0.371128	−	0.928582i	$-0.378971\pi$
$239$	655464.	0.742257	0.371128	+	0.928582i	$0.378971\pi$
$240$	0	0
$241$	1.38709e6	1.53838	0.769189	−	0.639021i	$-0.220660\pi$
$241$	1.38709e6	1.53838	0.769189	+	0.639021i	$0.220660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	230496.	0.245329
$246$	0	0
$247$	−53768.0	−0.0560766
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.88811e6	−1.89166	−0.945830	−	0.324663i	$-0.894749\pi$
$251$	−1.88811e6	−1.89166	−0.945830	+	0.324663i	$0.894749\pi$
$252$	0	0
$253$	−69120.0	−0.0678895
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−346194.	−0.326954	−0.163477	−	0.986547i	$-0.552271\pi$
$257$	−346194.	−0.326954	−0.163477	+	0.986547i	$0.552271\pi$
$258$	0	0
$259$	−529102.	−0.490106
$260$	0	0
$261$	0	0
$262$	0	0
$263$	929088.	0.828262	0.414131	−	0.910217i	$-0.364086\pi$
$263$	929088.	0.828262	0.414131	+	0.910217i	$0.364086\pi$
$264$	0	0
$265$	−97344.0	−0.0851519
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−382068.	−0.321929	−0.160964	−	0.986960i	$-0.551460\pi$
$269$	−382068.	−0.321929	−0.160964	+	0.986960i	$0.551460\pi$
$270$	0	0
$271$	−1.58056e6	−1.30734	−0.653669	−	0.756781i	$-0.726771\pi$
$271$	−1.58056e6	−1.30734	−0.653669	+	0.756781i	$0.726771\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.38552e6	3.49695
$276$	0	0
$277$	−1.36911e6	−1.07211	−0.536056	−	0.844182i	$-0.680086\pi$
$277$	−1.36911e6	−1.07211	−0.536056	+	0.844182i	$0.680086\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	394854.	0.298312	0.149156	−	0.988814i	$-0.452344\pi$
$281$	394854.	0.298312	0.149156	+	0.988814i	$0.452344\pi$
$282$	0	0
$283$	673034.	0.499541	0.249770	−	0.968305i	$-0.419645\pi$
$283$	673034.	0.499541	0.249770	+	0.968305i	$0.419645\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−588294.	−0.421589
$288$	0	0
$289$	152659.	0.107517
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.83468e6	−1.24851	−0.624254	−	0.781222i	$-0.714597\pi$
$293$	−1.83468e6	−1.24851	−0.624254	+	0.781222i	$0.714597\pi$
$294$	0	0
$295$	−119232.	−0.0797697
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−54912.0	−0.0355213
$300$	0	0
$301$	−448840.	−0.285545
$302$	0	0
$303$	0	0
$304$	0	0
$305$	728832.	0.448619
$306$	0	0
$307$	−1.51056e6	−0.914727	−0.457363	−	0.889280i	$-0.651206\pi$
$307$	−1.51056e6	−0.914727	−0.457363	+	0.889280i	$0.651206\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.87529e6	1.09943	0.549714	−	0.835353i	$-0.314737\pi$
$311$	1.87529e6	1.09943	0.549714	+	0.835353i	$0.314737\pi$
$312$	0	0
$313$	−1.51076e6	−0.871636	−0.435818	−	0.900035i	$-0.643541\pi$
$313$	−1.51076e6	−0.871636	−0.435818	+	0.900035i	$0.643541\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.02709e6	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$317$	−2.02709e6	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$318$	0	0
$319$	3.14928e6	1.73274
$320$	0	0
$321$	0	0
$322$	0	0
$323$	117876.	0.0628665
$324$	0	0
$325$	3.48405e6	1.82968
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.26596e6	0.644810
$330$	0	0
$331$	1.54009e6	0.772637	0.386319	−	0.922365i	$-0.373747\pi$
$331$	1.54009e6	0.772637	0.386319	+	0.922365i	$0.373747\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.94867e6	1.92238
$336$	0	0
$337$	1.01166e6	0.485245	0.242622	−	0.970121i	$-0.421992\pi$
$337$	1.01166e6	0.485245	0.242622	+	0.970121i	$0.421992\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.49568e6	−2.09368
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.15748e6	0.961885	0.480942	−	0.876752i	$-0.340295\pi$
$347$	2.15748e6	0.961885	0.480942	+	0.876752i	$0.340295\pi$
$348$	0	0
$349$	−1.15798e6	−0.508906	−0.254453	−	0.967085i	$-0.581895\pi$
$349$	−1.15798e6	−0.508906	−0.254453	+	0.967085i	$0.581895\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.17566e6	1.35643	0.678215	−	0.734863i	$-0.262754\pi$
$353$	3.17566e6	1.35643	0.678215	+	0.734863i	$0.262754\pi$
$354$	0	0
$355$	3.61267e6	1.52145
$356$	0	0
$357$	0	0
$358$	0	0
$359$	74616.0	0.0305560	0.0152780	−	0.999883i	$-0.495137\pi$
$359$	74616.0	0.0305560	0.0152780	+	0.999883i	$0.495137\pi$
$360$	0	0
$361$	−2.46726e6	−0.996431
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.29005e6	−0.506843
$366$	0	0
$367$	−1.79807e6	−0.696854	−0.348427	−	0.937336i	$-0.613284\pi$
$367$	−1.79807e6	−0.696854	−0.348427	+	0.937336i	$0.613284\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−49686.0	−0.0187413
$372$	0	0
$373$	2.20461e6	0.820463	0.410231	−	0.911981i	$-0.365448\pi$
$373$	2.20461e6	0.820463	0.410231	+	0.911981i	$0.365448\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.50193e6	0.906612
$378$	0	0
$379$	−177568.	−0.0634990	−0.0317495	−	0.999496i	$-0.510108\pi$
$379$	−177568.	−0.0634990	−0.0317495	+	0.999496i	$0.510108\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.87468e6	1.00137	0.500683	−	0.865630i	$-0.333082\pi$
$383$	2.87468e6	1.00137	0.500683	+	0.865630i	$0.333082\pi$
$384$	0	0
$385$	3.38688e6	1.16452
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.79965e6	1.60818	0.804091	−	0.594506i	$-0.202652\pi$
$389$	4.79965e6	1.60818	0.804091	+	0.594506i	$0.202652\pi$
$390$	0	0
$391$	120384.	0.0398223
$392$	0	0
$393$	0	0
$394$	0	0
$395$	599808.	0.193428
$396$	0	0
$397$	−2.81643e6	−0.896855	−0.448428	−	0.893819i	$-0.648016\pi$
$397$	−2.81643e6	−0.896855	−0.448428	+	0.893819i	$0.648016\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.83797e6	0.881347	0.440673	−	0.897667i	$-0.354740\pi$
$401$	2.83797e6	0.881347	0.440673	+	0.897667i	$0.354740\pi$
$402$	0	0
$403$	−3.57157e6	−1.09546
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.77456e6	−2.32643
$408$	0	0
$409$	154286.	0.0456056	0.0228028	−	0.999740i	$-0.492741\pi$
$409$	154286.	0.0456056	0.0228028	+	0.999740i	$0.492741\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−60858.0	−0.0175567
$414$	0	0
$415$	2.42438e6	0.691005
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.72865e6	−1.03757	−0.518783	−	0.854906i	$-0.673615\pi$
$419$	−3.72865e6	−1.03757	−0.518783	+	0.854906i	$0.673615\pi$
$420$	0	0
$421$	−2.32623e6	−0.639658	−0.319829	−	0.947475i	$-0.603626\pi$
$421$	−2.32623e6	−0.639658	−0.319829	+	0.947475i	$0.603626\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.63811e6	−2.05123
$426$	0	0
$427$	372008.	0.0987376
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.61482e6	0.678031	0.339015	−	0.940781i	$-0.389906\pi$
$431$	2.61482e6	0.678031	0.339015	+	0.940781i	$0.389906\pi$
$432$	0	0
$433$	−1.19226e6	−0.305598	−0.152799	−	0.988257i	$-0.548829\pi$
$433$	−1.19226e6	−0.305598	−0.152799	+	0.988257i	$0.548829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9024.00	0.00226045
$438$	0	0
$439$	1.05793e6	0.261996	0.130998	−	0.991383i	$-0.458182\pi$
$439$	1.05793e6	0.261996	0.130998	+	0.991383i	$0.458182\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.12756e6	−0.999272	−0.499636	−	0.866235i	$-0.666533\pi$
$443$	−4.12756e6	−0.999272	−0.499636	+	0.866235i	$0.666533\pi$
$444$	0	0
$445$	4.33210e6	1.03705
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.75823e6	−0.879766	−0.439883	−	0.898055i	$-0.644980\pi$
$449$	−3.75823e6	−0.879766	−0.439883	+	0.898055i	$0.644980\pi$
$450$	0	0
$451$	−8.64432e6	−2.00120
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.69069e6	0.609305
$456$	0	0
$457$	−451114.	−0.101041	−0.0505203	−	0.998723i	$-0.516088\pi$
$457$	−451114.	−0.101041	−0.0505203	+	0.998723i	$0.516088\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.95186e6	0.427756	0.213878	−	0.976860i	$-0.431390\pi$
$461$	1.95186e6	0.427756	0.213878	+	0.976860i	$0.431390\pi$
$462$	0	0
$463$	−7.20218e6	−1.56139	−0.780695	−	0.624913i	$-0.785135\pi$
$463$	−7.20218e6	−1.56139	−0.780695	+	0.624913i	$0.785135\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.17801e6	−1.52304	−0.761521	−	0.648140i	$-0.775547\pi$
$467$	−7.17801e6	−1.52304	−0.761521	+	0.648140i	$0.775547\pi$
$468$	0	0
$469$	2.01547e6	0.423101
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.59520e6	−1.35542
$474$	0	0
$475$	−572554.	−0.116435
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.17632e6	−1.22996	−0.614980	−	0.788543i	$-0.710836\pi$
$479$	−6.17632e6	−1.22996	−0.614980	+	0.788543i	$0.710836\pi$
$480$	0	0
$481$	−6.17646e6	−1.21724
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.02933e7	1.98701
$486$	0	0
$487$	7.59330e6	1.45080	0.725401	−	0.688327i	$-0.241655\pi$
$487$	7.59330e6	1.45080	0.725401	+	0.688327i	$0.241655\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.51878e6	0.284309	0.142155	−	0.989844i	$-0.454597\pi$
$491$	1.51878e6	0.284309	0.142155	+	0.989844i	$0.454597\pi$
$492$	0	0
$493$	−5.48500e6	−1.01639
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.84397e6	0.334859
$498$	0	0
$499$	1.47576e6	0.265316	0.132658	−	0.991162i	$-0.457649\pi$
$499$	1.47576e6	0.265316	0.132658	+	0.991162i	$0.457649\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.31309e6	0.231406	0.115703	−	0.993284i	$-0.463088\pi$
$503$	1.31309e6	0.231406	0.115703	+	0.993284i	$0.463088\pi$
$504$	0	0
$505$	−4.52506e6	−0.789579
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.40932e6	−0.754357	−0.377178	−	0.926141i	$-0.623106\pi$
$509$	−4.40932e6	−0.754357	−0.377178	+	0.926141i	$0.623106\pi$
$510$	0	0
$511$	−658462.	−0.111552
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.17316e7	1.94912
$516$	0	0
$517$	1.86019e7	3.06078
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.97629e6	−0.480376	−0.240188	−	0.970726i	$-0.577209\pi$
$521$	−2.97629e6	−0.480376	−0.240188	+	0.970726i	$0.577209\pi$
$522$	0	0
$523$	6.34627e6	1.01453	0.507265	−	0.861790i	$-0.330657\pi$
$523$	6.34627e6	1.01453	0.507265	+	0.861790i	$0.330657\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.82998e6	1.22810
$528$	0	0
$529$	−6.42713e6	−0.998568
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.86743e6	−1.04707
$534$	0	0
$535$	1.24451e7	1.87980
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.72872e6	0.256302
$540$	0	0
$541$	−1.36667e6	−0.200756	−0.100378	−	0.994949i	$-0.532005\pi$
$541$	−1.36667e6	−0.200756	−0.100378	+	0.994949i	$0.532005\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.11736e7	−3.05353
$546$	0	0
$547$	−9.55818e6	−1.36586	−0.682931	−	0.730483i	$-0.739295\pi$
$547$	−9.55818e6	−1.36586	−0.682931	+	0.730483i	$0.739295\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−411156.	−0.0576936
$552$	0	0
$553$	306152.	0.0425720
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.94287e6	−0.948202	−0.474101	−	0.880470i	$-0.657227\pi$
$557$	−6.94287e6	−0.948202	−0.474101	+	0.880470i	$0.657227\pi$
$558$	0	0
$559$	−5.23952e6	−0.709189
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.24662e6	−0.697604	−0.348802	−	0.937196i	$-0.613411\pi$
$563$	−5.24662e6	−0.697604	−0.348802	+	0.937196i	$0.613411\pi$
$564$	0	0
$565$	−1.63866e7	−2.15958
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.46551e6	−0.448731	−0.224366	−	0.974505i	$-0.572031\pi$
$569$	−3.46551e6	−0.448731	−0.224366	+	0.974505i	$0.572031\pi$
$570$	0	0
$571$	4.90069e6	0.629023	0.314512	−	0.949254i	$-0.398159\pi$
$571$	4.90069e6	0.629023	0.314512	+	0.949254i	$0.398159\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−584736.	−0.0737548
$576$	0	0
$577$	2.28346e6	0.285531	0.142766	−	0.989757i	$-0.454401\pi$
$577$	2.28346e6	0.285531	0.142766	+	0.989757i	$0.454401\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.23745e6	0.152085
$582$	0	0
$583$	−730080.	−0.0889609
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.03157e7	−1.23568	−0.617838	−	0.786305i	$-0.711991\pi$
$587$	−1.03157e7	−1.23568	−0.617838	+	0.786305i	$0.711991\pi$
$588$	0	0
$589$	586936.	0.0697112
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.52838e6	−0.412039	−0.206020	−	0.978548i	$-0.566051\pi$
$593$	−3.52838e6	−0.412039	−0.206020	+	0.978548i	$0.566051\pi$
$594$	0	0
$595$	−5.89882e6	−0.683081
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.92260e6	−0.332815	−0.166407	−	0.986057i	$-0.553217\pi$
$599$	−2.92260e6	−0.332815	−0.166407	+	0.986057i	$0.553217\pi$
$600$	0	0
$601$	−1.17567e7	−1.32770	−0.663849	−	0.747866i	$-0.731078\pi$
$601$	−1.17567e7	−1.32770	−0.663849	+	0.747866i	$0.731078\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.43055e7	3.81044
$606$	0	0
$607$	−4.71491e6	−0.519400	−0.259700	−	0.965689i	$-0.583624\pi$
$607$	−4.71491e6	−0.519400	−0.259700	+	0.965689i	$0.583624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.47782e7	1.60147
$612$	0	0
$613$	213842.	0.0229849	0.0114924	−	0.999934i	$-0.496342\pi$
$613$	213842.	0.0229849	0.0114924	+	0.999934i	$0.496342\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	336666.	0.0356030	0.0178015	−	0.999842i	$-0.494333\pi$
$617$	336666.	0.0356030	0.0178015	+	0.999842i	$0.494333\pi$
$618$	0	0
$619$	1.42655e7	1.49645	0.748223	−	0.663447i	$-0.230907\pi$
$619$	1.42655e7	1.49645	0.748223	+	0.663447i	$0.230907\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.21117e6	0.228246
$624$	0	0
$625$	8.30028e6	0.849949
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.35407e7	1.36463
$630$	0	0
$631$	−6.59637e6	−0.659525	−0.329763	−	0.944064i	$-0.606969\pi$
$631$	−6.59637e6	−0.659525	−0.329763	+	0.944064i	$0.606969\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.39816e7	−2.36017
$636$	0	0
$637$	1.37337e6	0.134103
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.02490e7	0.985225	0.492613	−	0.870249i	$-0.336042\pi$
$641$	1.02490e7	0.985225	0.492613	+	0.870249i	$0.336042\pi$
$642$	0	0
$643$	−4.16543e6	−0.397312	−0.198656	−	0.980069i	$-0.563658\pi$
$643$	−4.16543e6	−0.397312	−0.198656	+	0.980069i	$0.563658\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.35051e6	0.314666	0.157333	−	0.987546i	$-0.449710\pi$
$647$	3.35051e6	0.314666	0.157333	+	0.987546i	$0.449710\pi$
$648$	0	0
$649$	−894240.	−0.0833379
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.05408e6	0.830924	0.415462	−	0.909611i	$-0.363620\pi$
$653$	9.05408e6	0.830924	0.415462	+	0.909611i	$0.363620\pi$
$654$	0	0
$655$	−1.17216e6	−0.106754
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.45382e6	−0.578899	−0.289450	−	0.957193i	$-0.593472\pi$
$659$	−6.45382e6	−0.578899	−0.289450	+	0.957193i	$0.593472\pi$
$660$	0	0
$661$	1.43167e7	1.27450	0.637250	−	0.770657i	$-0.280072\pi$
$661$	1.43167e7	1.27450	0.637250	+	0.770657i	$0.280072\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−442176.	−0.0387740
$666$	0	0
$667$	−419904.	−0.0365456
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.46624e6	0.468686
$672$	0	0
$673$	2.27250e7	1.93404	0.967020	−	0.254701i	$-0.0819771\pi$
$673$	2.27250e7	1.93404	0.967020	+	0.254701i	$0.0819771\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.53249e6	0.715491	0.357746	−	0.933819i	$-0.383546\pi$
$677$	8.53249e6	0.715491	0.357746	+	0.933819i	$0.383546\pi$
$678$	0	0
$679$	5.25388e6	0.437326
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.24921e7	1.84492	0.922461	−	0.386090i	$-0.126175\pi$
$683$	2.24921e7	1.84492	0.922461	+	0.386090i	$0.126175\pi$
$684$	0	0
$685$	1.33459e6	0.108673
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−580008.	−0.0465464
$690$	0	0
$691$	−1.26894e7	−1.01099	−0.505495	−	0.862830i	$-0.668690\pi$
$691$	−1.26894e7	−1.01099	−0.505495	+	0.862830i	$0.668690\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.14522e7	−3.25526
$696$	0	0
$697$	1.50555e7	1.17385
$698$	0	0
$699$	0	0
$700$	0	0
$701$	5.13939e6	0.395018	0.197509	−	0.980301i	$-0.436715\pi$
$701$	5.13939e6	0.395018	0.197509	+	0.980301i	$0.436715\pi$
$702$	0	0
$703$	1.01501e6	0.0774610
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.30966e6	−0.173780
$708$	0	0
$709$	−1.16065e7	−0.867132	−0.433566	−	0.901122i	$-0.642745\pi$
$709$	−1.16065e7	−0.867132	−0.433566	+	0.901122i	$0.642745\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	599424.	0.0441581
$714$	0	0
$715$	3.95366e7	2.89224
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.50998e7	−1.08930	−0.544650	−	0.838663i	$-0.683338\pi$
$719$	−1.50998e7	−1.08930	−0.544650	+	0.838663i	$0.683338\pi$
$720$	0	0
$721$	5.98800e6	0.428986
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.66420e7	1.88245
$726$	0	0
$727$	−2.32536e7	−1.63175	−0.815874	−	0.578229i	$-0.803744\pi$
$727$	−2.32536e7	−1.63175	−0.815874	+	0.578229i	$0.803744\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.14866e7	0.795059
$732$	0	0
$733$	2.37814e7	1.63485	0.817423	−	0.576038i	$-0.195402\pi$
$733$	2.37814e7	1.63485	0.817423	+	0.576038i	$0.195402\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.96150e7	2.00837
$738$	0	0
$739$	−2.51392e6	−0.169333	−0.0846663	−	0.996409i	$-0.526982\pi$
$739$	−2.51392e6	−0.169333	−0.0846663	+	0.996409i	$0.526982\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.22646e7	1.47959	0.739797	−	0.672830i	$-0.234922\pi$
$743$	2.22646e7	1.47959	0.739797	+	0.672830i	$0.234922\pi$
$744$	0	0
$745$	−3.13741e7	−2.07101
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.35216e6	0.413730
$750$	0	0
$751$	2.30108e7	1.48878	0.744392	−	0.667743i	$-0.232740\pi$
$751$	2.30108e7	1.48878	0.744392	+	0.667743i	$0.232740\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.66541e7	1.06329
$756$	0	0
$757$	1.59335e7	1.01058	0.505290	−	0.862950i	$-0.331386\pi$
$757$	1.59335e7	1.01058	0.505290	+	0.862950i	$0.331386\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.68629e7	−1.05553	−0.527764	−	0.849391i	$-0.676969\pi$
$761$	−1.68629e7	−1.05553	−0.527764	+	0.849391i	$0.676969\pi$
$762$	0	0
$763$	−1.08073e7	−0.672059
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−710424.	−0.0436043
$768$	0	0
$769$	−2.75402e7	−1.67939	−0.839694	−	0.543060i	$-0.817265\pi$
$769$	−2.75402e7	−1.67939	−0.839694	+	0.543060i	$0.817265\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.26820e7	−1.36532	−0.682658	−	0.730738i	$-0.739176\pi$
$773$	−2.26820e7	−1.36532	−0.682658	+	0.730738i	$0.739176\pi$
$774$	0	0
$775$	−3.80322e7	−2.27456
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.12856e6	0.0666320
$780$	0	0
$781$	2.70950e7	1.58951
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.23507e6	−0.303213
$786$	0	0
$787$	−2.34266e7	−1.34826	−0.674129	−	0.738614i	$-0.735481\pi$
$787$	−2.34266e7	−1.34826	−0.674129	+	0.738614i	$0.735481\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.36401e6	−0.475306
$792$	0	0
$793$	4.34262e6	0.245228
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.82051e7	1.01519	0.507594	−	0.861596i	$-0.330535\pi$
$797$	1.82051e7	1.01519	0.507594	+	0.861596i	$0.330535\pi$
$798$	0	0
$799$	−3.23983e7	−1.79538
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.67536e6	−0.529515
$804$	0	0
$805$	−451584.	−0.0245612
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.47411e7	−0.791878	−0.395939	−	0.918277i	$-0.629581\pi$
$809$	−1.47411e7	−0.791878	−0.395939	+	0.918277i	$0.629581\pi$
$810$	0	0
$811$	1.69629e7	0.905625	0.452812	−	0.891606i	$-0.350421\pi$
$811$	1.69629e7	0.905625	0.452812	+	0.891606i	$0.350421\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.00762e7	0.531375
$816$	0	0
$817$	861040.	0.0451303
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.03929e6	−0.416255	−0.208128	−	0.978102i	$-0.566737\pi$
$821$	−8.03929e6	−0.416255	−0.208128	+	0.978102i	$0.566737\pi$
$822$	0	0
$823$	386648.	0.0198983	0.00994915	−	0.999951i	$-0.496833\pi$
$823$	386648.	0.0198983	0.00994915	+	0.999951i	$0.496833\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.55021e7	−1.80505	−0.902526	−	0.430635i	$-0.858290\pi$
$827$	−3.55021e7	−1.80505	−0.902526	+	0.430635i	$0.858290\pi$
$828$	0	0
$829$	2.48814e7	1.25745	0.628723	−	0.777630i	$-0.283578\pi$
$829$	2.48814e7	1.25745	0.628723	+	0.777630i	$0.283578\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.01085e6	−0.150341
$834$	0	0
$835$	−1.54356e7	−0.766141
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.41458e7	−1.67468	−0.837340	−	0.546682i	$-0.815891\pi$
$839$	−3.41458e7	−1.67468	−0.837340	+	0.546682i	$0.815891\pi$
$840$	0	0
$841$	−1.37927e6	−0.0672450
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.23446e6	−0.204012
$846$	0	0
$847$	1.75101e7	0.838649
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.03661e6	0.0490671
$852$	0	0
$853$	2.50701e7	1.17973	0.589865	−	0.807502i	$-0.299181\pi$
$853$	2.50701e7	1.17973	0.589865	+	0.807502i	$0.299181\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.81938e7	0.846199	0.423099	−	0.906083i	$-0.360942\pi$
$857$	1.81938e7	0.846199	0.423099	+	0.906083i	$0.360942\pi$
$858$	0	0
$859$	6.91797e6	0.319886	0.159943	−	0.987126i	$-0.448869\pi$
$859$	6.91797e6	0.319886	0.159943	+	0.987126i	$0.448869\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.78069e7	1.27094	0.635471	−	0.772125i	$-0.280806\pi$
$863$	2.78069e7	1.27094	0.635471	+	0.772125i	$0.280806\pi$
$864$	0	0
$865$	−3.46141e7	−1.57294
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.49856e6	0.202080
$870$	0	0
$871$	2.35275e7	1.05083
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.39521e7	0.616053
$876$	0	0
$877$	3.79587e6	0.166653	0.0833263	−	0.996522i	$-0.473446\pi$
$877$	3.79587e6	0.166653	0.0833263	+	0.996522i	$0.473446\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.48904e7	−1.08042	−0.540210	−	0.841530i	$-0.681655\pi$
$881$	−2.48904e7	−1.08042	−0.540210	+	0.841530i	$0.681655\pi$
$882$	0	0
$883$	−3.13568e6	−0.135341	−0.0676705	−	0.997708i	$-0.521557\pi$
$883$	−3.13568e6	−0.135341	−0.0676705	+	0.997708i	$0.521557\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.02437e7	0.863933	0.431966	−	0.901890i	$-0.357820\pi$
$887$	2.02437e7	0.863933	0.431966	+	0.901890i	$0.357820\pi$
$888$	0	0
$889$	−1.22406e7	−0.519455
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.42858e6	−0.101912
$894$	0	0
$895$	3.00223e7	1.25281
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.73113e7	−1.12705
$900$	0	0
$901$	1.27156e6	0.0521823
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.18867e7	−0.482437
$906$	0	0
$907$	−6.86324e6	−0.277020	−0.138510	−	0.990361i	$-0.544231\pi$
$907$	−6.86324e6	−0.277020	−0.138510	+	0.990361i	$0.544231\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.40661e6	0.375523	0.187762	−	0.982215i	$-0.439877\pi$
$911$	9.40661e6	0.375523	0.187762	+	0.982215i	$0.439877\pi$
$912$	0	0
$913$	1.81829e7	0.721914
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−598290.	−0.0234957
$918$	0	0
$919$	−2.10280e7	−0.821313	−0.410656	−	0.911790i	$-0.634700\pi$
$919$	−2.10280e7	−0.821313	−0.410656	+	0.911790i	$0.634700\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.15255e7	0.831666
$924$	0	0
$925$	−6.57706e7	−2.52742
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.30928e7	1.63819	0.819096	−	0.573656i	$-0.194475\pi$
$929$	4.30928e7	1.63819	0.819096	+	0.573656i	$0.194475\pi$
$930$	0	0
$931$	−225694.	−0.00853387
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.66765e7	−3.24244
$936$	0	0
$937$	−3.85862e6	−0.143576	−0.0717882	−	0.997420i	$-0.522871\pi$
$937$	−3.85862e6	−0.143576	−0.0717882	+	0.997420i	$0.522871\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.59601e7	−0.587572	−0.293786	−	0.955871i	$-0.594915\pi$
$941$	−1.59601e7	−0.587572	−0.293786	+	0.955871i	$0.594915\pi$
$942$	0	0
$943$	1.15258e6	0.0422076
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.66831e6	0.169155	0.0845775	−	0.996417i	$-0.473046\pi$
$947$	4.66831e6	0.169155	0.0845775	+	0.996417i	$0.473046\pi$
$948$	0	0
$949$	−7.68654e6	−0.277054
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.43457e7	1.22501	0.612505	−	0.790466i	$-0.290162\pi$
$953$	3.43457e7	1.22501	0.612505	+	0.790466i	$0.290162\pi$
$954$	0	0
$955$	−3.10510e7	−1.10171
$956$	0	0
$957$	0	0
$958$	0	0
$959$	681198.	0.0239181
$960$	0	0
$961$	1.03584e7	0.361813
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.95156e7	−2.05737
$966$	0	0
$967$	−2.28181e7	−0.784718	−0.392359	−	0.919812i	$-0.628341\pi$
$967$	−2.28181e7	−0.784718	−0.392359	+	0.919812i	$0.628341\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.94042e7	−1.68157	−0.840786	−	0.541367i	$-0.817907\pi$
$971$	−4.94042e7	−1.68157	−0.840786	+	0.541367i	$0.817907\pi$
$972$	0	0
$973$	−2.11579e7	−0.716458
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.17542e6	−0.240498	−0.120249	−	0.992744i	$-0.538369\pi$
$977$	−7.17542e6	−0.240498	−0.120249	+	0.992744i	$0.538369\pi$
$978$	0	0
$979$	3.24907e7	1.08343
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.22279e6	0.139385	0.0696924	−	0.997569i	$-0.477798\pi$
$983$	4.22279e6	0.139385	0.0696924	+	0.997569i	$0.477798\pi$
$984$	0	0
$985$	4.79388e7	1.57433
$986$	0	0
$987$	0	0
$988$	0	0
$989$	879360.	0.0285875
$990$	0	0
$991$	1.65645e7	0.535789	0.267895	−	0.963448i	$-0.413672\pi$
$991$	1.65645e7	0.535789	0.267895	+	0.963448i	$0.413672\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.54495e7	−2.41601
$996$	0	0
$997$	−4.40973e7	−1.40499	−0.702496	−	0.711687i	$-0.747931\pi$
$997$	−4.40973e7	−1.40499	−0.702496	+	0.711687i	$0.747931\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.a.d.1.1		1
3.2	odd	2		28.6.a.a.1.1	✓	1
4.3	odd	2		1008.6.a.bb.1.1		1
12.11	even	2		112.6.a.e.1.1		1
15.2	even	4		700.6.e.d.449.2		2
15.8	even	4		700.6.e.d.449.1		2
15.14	odd	2		700.6.a.d.1.1		1
21.2	odd	6		196.6.e.f.165.1		2
21.5	even	6		196.6.e.e.165.1		2
21.11	odd	6		196.6.e.f.177.1		2
21.17	even	6		196.6.e.e.177.1		2
21.20	even	2		196.6.a.d.1.1		1
24.5	odd	2		448.6.a.i.1.1		1
24.11	even	2		448.6.a.h.1.1		1
84.83	odd	2		784.6.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.a.1.1	✓	1	3.2	odd	2
112.6.a.e.1.1		1	12.11	even	2
196.6.a.d.1.1		1	21.20	even	2
196.6.e.e.165.1		2	21.5	even	6
196.6.e.e.177.1		2	21.17	even	6
196.6.e.f.165.1		2	21.2	odd	6
196.6.e.f.177.1		2	21.11	odd	6
252.6.a.d.1.1		1	1.1	even	1	trivial
448.6.a.h.1.1		1	24.11	even	2
448.6.a.i.1.1		1	24.5	odd	2
700.6.a.d.1.1		1	15.14	odd	2
700.6.e.d.449.1		2	15.8	even	4
700.6.e.d.449.2		2	15.2	even	4
784.6.a.f.1.1		1	84.83	odd	2
1008.6.a.bb.1.1		1	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	252.6.a.d.1.1

Level	$252$
Weight	$6$
Character	252.1
Self dual	yes
Analytic conductor	$40.417$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$