LMFDB - Embedded newform 1028.6.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.0570 q^{3} -9.45543 q^{5} -38.2403 q^{7} +120.170 q^{9} +O(q^{10})$	$q-19.0570 q^{3} -9.45543 q^{5} -38.2403 q^{7} +120.170 q^{9} +7.22540 q^{11} +889.805 q^{13} +180.192 q^{15} +1471.44 q^{17} +276.005 q^{19} +728.747 q^{21} +984.169 q^{23} -3035.59 q^{25} +2340.77 q^{27} +769.025 q^{29} -668.974 q^{31} -137.695 q^{33} +361.579 q^{35} +6278.91 q^{37} -16957.0 q^{39} +13961.2 q^{41} -14575.6 q^{43} -1136.26 q^{45} +18334.3 q^{47} -15344.7 q^{49} -28041.2 q^{51} -19674.9 q^{53} -68.3193 q^{55} -5259.84 q^{57} -41926.3 q^{59} +9344.06 q^{61} -4595.35 q^{63} -8413.49 q^{65} -7933.62 q^{67} -18755.3 q^{69} +71897.5 q^{71} +3548.88 q^{73} +57849.4 q^{75} -276.301 q^{77} -53468.8 q^{79} -73809.5 q^{81} -34874.4 q^{83} -13913.1 q^{85} -14655.3 q^{87} +7579.35 q^{89} -34026.4 q^{91} +12748.7 q^{93} -2609.75 q^{95} +3491.44 q^{97} +868.280 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−19.0570	−1.22251	−0.611255	−	0.791434i	$-0.709335\pi$
$3$	−19.0570	−1.22251	−0.611255	+	0.791434i	$0.709335\pi$
$4$	0	0
$5$	−9.45543	−0.169144	−0.0845720	−	0.996417i	$-0.526952\pi$
$5$	−9.45543	−0.169144	−0.0845720	+	0.996417i	$0.526952\pi$
$6$	0	0
$7$	−38.2403	−0.294969	−0.147484	−	0.989064i	$-0.547118\pi$
$7$	−38.2403	−0.294969	−0.147484	+	0.989064i	$0.547118\pi$
$8$	0	0
$9$	120.170	0.494529
$10$	0	0
$11$	7.22540	0.0180045	0.00900223	−	0.999959i	$-0.497134\pi$
$11$	7.22540	0.0180045	0.00900223	+	0.999959i	$0.497134\pi$
$12$	0	0
$13$	889.805	1.46028	0.730140	−	0.683297i	$-0.239455\pi$
$13$	889.805	1.46028	0.730140	+	0.683297i	$0.239455\pi$
$14$	0	0
$15$	180.192	0.206780
$16$	0	0
$17$	1471.44	1.23486	0.617432	−	0.786624i	$-0.288173\pi$
$17$	1471.44	1.23486	0.617432	+	0.786624i	$0.288173\pi$
$18$	0	0
$19$	276.005	0.175401	0.0877007	−	0.996147i	$-0.472048\pi$
$19$	276.005	0.175401	0.0877007	+	0.996147i	$0.472048\pi$
$20$	0	0
$21$	728.747	0.360602
$22$	0	0
$23$	984.169	0.387927	0.193964	−	0.981009i	$-0.437866\pi$
$23$	984.169	0.387927	0.193964	+	0.981009i	$0.437866\pi$
$24$	0	0
$25$	−3035.59	−0.971390
$26$	0	0
$27$	2340.77	0.617943
$28$	0	0
$29$	769.025	0.169803	0.0849015	−	0.996389i	$-0.472942\pi$
$29$	769.025	0.169803	0.0849015	+	0.996389i	$0.472942\pi$
$30$	0	0
$31$	−668.974	−0.125027	−0.0625137	−	0.998044i	$-0.519912\pi$
$31$	−668.974	−0.125027	−0.0625137	+	0.998044i	$0.519912\pi$
$32$	0	0
$33$	−137.695	−0.0220106
$34$	0	0
$35$	361.579	0.0498922
$36$	0	0
$37$	6278.91	0.754014	0.377007	−	0.926210i	$-0.376953\pi$
$37$	6278.91	0.754014	0.377007	+	0.926210i	$0.376953\pi$
$38$	0	0
$39$	−16957.0	−1.78521
$40$	0	0
$41$	13961.2	1.29707	0.648535	−	0.761185i	$-0.275382\pi$
$41$	13961.2	1.29707	0.648535	+	0.761185i	$0.275382\pi$
$42$	0	0
$43$	−14575.6	−1.20214	−0.601071	−	0.799195i	$-0.705259\pi$
$43$	−14575.6	−1.20214	−0.601071	+	0.799195i	$0.705259\pi$
$44$	0	0
$45$	−1136.26	−0.0836465
$46$	0	0
$47$	18334.3	1.21066	0.605328	−	0.795976i	$-0.293042\pi$
$47$	18334.3	1.21066	0.605328	+	0.795976i	$0.293042\pi$
$48$	0	0
$49$	−15344.7	−0.912993
$50$	0	0
$51$	−28041.2	−1.50963
$52$	0	0
$53$	−19674.9	−0.962107	−0.481053	−	0.876691i	$-0.659746\pi$
$53$	−19674.9	−0.962107	−0.481053	+	0.876691i	$0.659746\pi$
$54$	0	0
$55$	−68.3193	−0.00304534
$56$	0	0
$57$	−5259.84	−0.214430
$58$	0	0
$59$	−41926.3	−1.56804	−0.784019	−	0.620737i	$-0.786833\pi$
$59$	−41926.3	−1.56804	−0.784019	+	0.620737i	$0.786833\pi$
$60$	0	0
$61$	9344.06	0.321522	0.160761	−	0.986993i	$-0.448605\pi$
$61$	9344.06	0.321522	0.160761	+	0.986993i	$0.448605\pi$
$62$	0	0
$63$	−4595.35	−0.145871
$64$	0	0
$65$	−8413.49	−0.246998
$66$	0	0
$67$	−7933.62	−0.215916	−0.107958	−	0.994155i	$-0.534431\pi$
$67$	−7933.62	−0.215916	−0.107958	+	0.994155i	$0.534431\pi$
$68$	0	0
$69$	−18755.3	−0.474245
$70$	0	0
$71$	71897.5	1.69265	0.846326	−	0.532665i	$-0.178809\pi$
$71$	71897.5	1.69265	0.846326	+	0.532665i	$0.178809\pi$
$72$	0	0
$73$	3548.88	0.0779442	0.0389721	−	0.999240i	$-0.487592\pi$
$73$	3548.88	0.0779442	0.0389721	+	0.999240i	$0.487592\pi$
$74$	0	0
$75$	57849.4	1.18753
$76$	0	0
$77$	−276.301	−0.00531076
$78$	0	0
$79$	−53468.8	−0.963901	−0.481950	−	0.876199i	$-0.660071\pi$
$79$	−53468.8	−0.963901	−0.481950	+	0.876199i	$0.660071\pi$
$80$	0	0
$81$	−73809.5	−1.24997
$82$	0	0
$83$	−34874.4	−0.555664	−0.277832	−	0.960630i	$-0.589616\pi$
$83$	−34874.4	−0.555664	−0.277832	+	0.960630i	$0.589616\pi$
$84$	0	0
$85$	−13913.1	−0.208870
$86$	0	0
$87$	−14655.3	−0.207586
$88$	0	0
$89$	7579.35	0.101428	0.0507139	−	0.998713i	$-0.483850\pi$
$89$	7579.35	0.101428	0.0507139	+	0.998713i	$0.483850\pi$
$90$	0	0
$91$	−34026.4	−0.430737
$92$	0	0
$93$	12748.7	0.152847
$94$	0	0
$95$	−2609.75	−0.0296681
$96$	0	0
$97$	3491.44	0.0376769	0.0188384	−	0.999823i	$-0.494003\pi$
$97$	3491.44	0.0376769	0.0188384	+	0.999823i	$0.494003\pi$
$98$	0	0
$99$	868.280	0.00890372
$100$	0	0
$101$	170097.	1.65918	0.829590	−	0.558373i	$-0.188574\pi$
$101$	170097.	1.65918	0.829590	+	0.558373i	$0.188574\pi$
$102$	0	0
$103$	−150119.	−1.39425	−0.697126	−	0.716948i	$-0.745538\pi$
$103$	−150119.	−1.39425	−0.697126	+	0.716948i	$0.745538\pi$
$104$	0	0
$105$	−6890.61	−0.0609937
$106$	0	0
$107$	7702.01	0.0650346	0.0325173	−	0.999471i	$-0.489648\pi$
$107$	7702.01	0.0650346	0.0325173	+	0.999471i	$0.489648\pi$
$108$	0	0
$109$	241559.	1.94741	0.973703	−	0.227820i	$-0.0731597\pi$
$109$	241559.	1.94741	0.973703	+	0.227820i	$0.0731597\pi$
$110$	0	0
$111$	−119657.	−0.921789
$112$	0	0
$113$	−75183.0	−0.553890	−0.276945	−	0.960886i	$-0.589322\pi$
$113$	−75183.0	−0.553890	−0.276945	+	0.960886i	$0.589322\pi$
$114$	0	0
$115$	−9305.75	−0.0656155
$116$	0	0
$117$	106928.	0.722151
$118$	0	0
$119$	−56268.1	−0.364246
$120$	0	0
$121$	−160999.	−0.999676
$122$	0	0
$123$	−266059.	−1.58568
$124$	0	0
$125$	58251.1	0.333449
$126$	0	0
$127$	−177659.	−0.977414	−0.488707	−	0.872448i	$-0.662531\pi$
$127$	−177659.	−0.977414	−0.488707	+	0.872448i	$0.662531\pi$
$128$	0	0
$129$	277768.	1.46963
$130$	0	0
$131$	−70956.7	−0.361256	−0.180628	−	0.983551i	$-0.557813\pi$
$131$	−70956.7	−0.361256	−0.180628	+	0.983551i	$0.557813\pi$
$132$	0	0
$133$	−10554.5	−0.0517379
$134$	0	0
$135$	−22133.0	−0.104521
$136$	0	0
$137$	−67518.9	−0.307344	−0.153672	−	0.988122i	$-0.549110\pi$
$137$	−67518.9	−0.307344	−0.153672	+	0.988122i	$0.549110\pi$
$138$	0	0
$139$	367658.	1.61401	0.807006	−	0.590543i	$-0.201086\pi$
$139$	367658.	1.61401	0.807006	+	0.590543i	$0.201086\pi$
$140$	0	0
$141$	−349398.	−1.48004
$142$	0	0
$143$	6429.20	0.0262916
$144$	0	0
$145$	−7271.46	−0.0287211
$146$	0	0
$147$	292424.	1.11614
$148$	0	0
$149$	144468.	0.533097	0.266549	−	0.963821i	$-0.414117\pi$
$149$	144468.	0.533097	0.266549	+	0.963821i	$0.414117\pi$
$150$	0	0
$151$	−300221.	−1.07152	−0.535759	−	0.844371i	$-0.679974\pi$
$151$	−300221.	−1.07152	−0.535759	+	0.844371i	$0.679974\pi$
$152$	0	0
$153$	176823.	0.610676
$154$	0	0
$155$	6325.44	0.0211476
$156$	0	0
$157$	114341.	0.370212	0.185106	−	0.982719i	$-0.440737\pi$
$157$	114341.	0.370212	0.185106	+	0.982719i	$0.440737\pi$
$158$	0	0
$159$	374946.	1.17618
$160$	0	0
$161$	−37634.9	−0.114426
$162$	0	0
$163$	−587692.	−1.73253	−0.866266	−	0.499584i	$-0.833486\pi$
$163$	−587692.	−1.73253	−0.866266	+	0.499584i	$0.833486\pi$
$164$	0	0
$165$	1301.96	0.00372296
$166$	0	0
$167$	306287.	0.849841	0.424920	−	0.905231i	$-0.360302\pi$
$167$	306287.	0.849841	0.424920	+	0.905231i	$0.360302\pi$
$168$	0	0
$169$	420460.	1.13242
$170$	0	0
$171$	33167.6	0.0867410
$172$	0	0
$173$	−524601.	−1.33264	−0.666321	−	0.745665i	$-0.732132\pi$
$173$	−524601.	−1.33264	−0.666321	+	0.745665i	$0.732132\pi$
$174$	0	0
$175$	116082.	0.286530
$176$	0	0
$177$	798991.	1.91694
$178$	0	0
$179$	−516995.	−1.20602	−0.603009	−	0.797734i	$-0.706032\pi$
$179$	−516995.	−1.20602	−0.603009	+	0.797734i	$0.706032\pi$
$180$	0	0
$181$	172517.	0.391413	0.195707	−	0.980663i	$-0.437300\pi$
$181$	172517.	0.391413	0.195707	+	0.980663i	$0.437300\pi$
$182$	0	0
$183$	−178070.	−0.393064
$184$	0	0
$185$	−59369.8	−0.127537
$186$	0	0
$187$	10631.7	0.0222331
$188$	0	0
$189$	−89511.6	−0.182274
$190$	0	0
$191$	558916.	1.10857	0.554285	−	0.832327i	$-0.312992\pi$
$191$	558916.	1.10857	0.554285	+	0.832327i	$0.312992\pi$
$192$	0	0
$193$	−45147.2	−0.0872443	−0.0436222	−	0.999048i	$-0.513890\pi$
$193$	−45147.2	−0.0872443	−0.0436222	+	0.999048i	$0.513890\pi$
$194$	0	0
$195$	160336.	0.301957
$196$	0	0
$197$	−184842.	−0.339340	−0.169670	−	0.985501i	$-0.554270\pi$
$197$	−184842.	−0.339340	−0.169670	+	0.985501i	$0.554270\pi$
$198$	0	0
$199$	948699.	1.69823	0.849113	−	0.528211i	$-0.177137\pi$
$199$	948699.	1.69823	0.849113	+	0.528211i	$0.177137\pi$
$200$	0	0
$201$	151191.	0.263959
$202$	0	0
$203$	−29407.7	−0.0500866
$204$	0	0
$205$	−132009.	−0.219391
$206$	0	0
$207$	118268.	0.191841
$208$	0	0
$209$	1994.25	0.00315801
$210$	0	0
$211$	929103.	1.43667	0.718336	−	0.695696i	$-0.244904\pi$
$211$	929103.	1.43667	0.718336	+	0.695696i	$0.244904\pi$
$212$	0	0
$213$	−1.37015e6	−2.06928
$214$	0	0
$215$	137819.	0.203335
$216$	0	0
$217$	25581.8	0.0368792
$218$	0	0
$219$	−67631.1	−0.0952875
$220$	0	0
$221$	1.30929e6	1.80325
$222$	0	0
$223$	1.18838e6	1.60026	0.800132	−	0.599823i	$-0.204762\pi$
$223$	1.18838e6	1.60026	0.800132	+	0.599823i	$0.204762\pi$
$224$	0	0
$225$	−364789.	−0.480380
$226$	0	0
$227$	−342101.	−0.440646	−0.220323	−	0.975427i	$-0.570711\pi$
$227$	−342101.	−0.440646	−0.220323	+	0.975427i	$0.570711\pi$
$228$	0	0
$229$	−276381.	−0.348273	−0.174137	−	0.984721i	$-0.555713\pi$
$229$	−276381.	−0.348273	−0.174137	+	0.984721i	$0.555713\pi$
$230$	0	0
$231$	5265.48	0.00649245
$232$	0	0
$233$	68000.2	0.0820579	0.0410289	−	0.999158i	$-0.486936\pi$
$233$	68000.2	0.0820579	0.0410289	+	0.999158i	$0.486936\pi$
$234$	0	0
$235$	−173359.	−0.204775
$236$	0	0
$237$	1.01896e6	1.17838
$238$	0	0
$239$	−185349.	−0.209892	−0.104946	−	0.994478i	$-0.533467\pi$
$239$	−185349.	−0.209892	−0.104946	+	0.994478i	$0.533467\pi$
$240$	0	0
$241$	1.40060e6	1.55335	0.776676	−	0.629900i	$-0.216904\pi$
$241$	1.40060e6	1.55335	0.776676	+	0.629900i	$0.216904\pi$
$242$	0	0
$243$	837783.	0.910157
$244$	0	0
$245$	145091.	0.154427
$246$	0	0
$247$	245591.	0.256135
$248$	0	0
$249$	664603.	0.679304
$250$	0	0
$251$	983817.	0.985666	0.492833	−	0.870124i	$-0.335961\pi$
$251$	983817.	0.985666	0.492833	+	0.870124i	$0.335961\pi$
$252$	0	0
$253$	7111.02	0.00698442
$254$	0	0
$255$	265142.	0.255345
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−240107.	−0.222411
$260$	0	0
$261$	92414.1	0.0839725
$262$	0	0
$263$	1.05607e6	0.941467	0.470734	−	0.882275i	$-0.343989\pi$
$263$	1.05607e6	0.941467	0.470734	+	0.882275i	$0.343989\pi$
$264$	0	0
$265$	186035.	0.162734
$266$	0	0
$267$	−144440.	−0.123996
$268$	0	0
$269$	−771532.	−0.650090	−0.325045	−	0.945699i	$-0.605379\pi$
$269$	−771532.	−0.650090	−0.325045	+	0.945699i	$0.605379\pi$
$270$	0	0
$271$	−234008.	−0.193556	−0.0967782	−	0.995306i	$-0.530854\pi$
$271$	−234008.	−0.193556	−0.0967782	+	0.995306i	$0.530854\pi$
$272$	0	0
$273$	648442.	0.526580
$274$	0	0
$275$	−21933.4	−0.0174894
$276$	0	0
$277$	194342.	0.152183	0.0760916	−	0.997101i	$-0.475756\pi$
$277$	194342.	0.152183	0.0760916	+	0.997101i	$0.475756\pi$
$278$	0	0
$279$	−80390.9	−0.0618296
$280$	0	0
$281$	440998.	0.333174	0.166587	−	0.986027i	$-0.446725\pi$
$281$	440998.	0.333174	0.166587	+	0.986027i	$0.446725\pi$
$282$	0	0
$283$	−219545.	−0.162951	−0.0814754	−	0.996675i	$-0.525963\pi$
$283$	−219545.	−0.162951	−0.0814754	+	0.996675i	$0.525963\pi$
$284$	0	0
$285$	49734.0	0.0362695
$286$	0	0
$287$	−533881.	−0.382595
$288$	0	0
$289$	745266.	0.524888
$290$	0	0
$291$	−66536.4	−0.0460603
$292$	0	0
$293$	2.50763e6	1.70645	0.853227	−	0.521540i	$-0.174642\pi$
$293$	2.50763e6	1.70645	0.853227	+	0.521540i	$0.174642\pi$
$294$	0	0
$295$	396431.	0.265224
$296$	0	0
$297$	16913.0	0.0111257
$298$	0	0
$299$	875719.	0.566483
$300$	0	0
$301$	557376.	0.354595
$302$	0	0
$303$	−3.24155e6	−2.02836
$304$	0	0
$305$	−88352.1	−0.0543835
$306$	0	0
$307$	1.33870e6	0.810659	0.405329	−	0.914171i	$-0.367157\pi$
$307$	1.33870e6	0.810659	0.405329	+	0.914171i	$0.367157\pi$
$308$	0	0
$309$	2.86081e6	1.70449
$310$	0	0
$311$	2.32757e6	1.36459	0.682295	−	0.731077i	$-0.260982\pi$
$311$	2.32757e6	1.36459	0.682295	+	0.731077i	$0.260982\pi$
$312$	0	0
$313$	2.89291e6	1.66907	0.834535	−	0.550954i	$-0.185736\pi$
$313$	2.89291e6	1.66907	0.834535	+	0.550954i	$0.185736\pi$
$314$	0	0
$315$	43451.1	0.0246731
$316$	0	0
$317$	3.50202e6	1.95736	0.978679	−	0.205397i	$-0.0658487\pi$
$317$	3.50202e6	1.95736	0.978679	+	0.205397i	$0.0658487\pi$
$318$	0	0
$319$	5556.51	0.00305721
$320$	0	0
$321$	−146777.	−0.0795055
$322$	0	0
$323$	406124.	0.216597
$324$	0	0
$325$	−2.70109e6	−1.41850
$326$	0	0
$327$	−4.60339e6	−2.38072
$328$	0	0
$329$	−701110.	−0.357106
$330$	0	0
$331$	−2.93257e6	−1.47122	−0.735611	−	0.677404i	$-0.763105\pi$
$331$	−2.93257e6	−1.47122	−0.735611	+	0.677404i	$0.763105\pi$
$332$	0	0
$333$	754539.	0.372882
$334$	0	0
$335$	75015.8	0.0365208
$336$	0	0
$337$	4.03293e6	1.93440	0.967198	−	0.254023i	$-0.0817540\pi$
$337$	4.03293e6	1.93440	0.967198	+	0.254023i	$0.0817540\pi$
$338$	0	0
$339$	1.43276e6	0.677135
$340$	0	0
$341$	−4833.60	−0.00225105
$342$	0	0
$343$	1.22949e6	0.564273
$344$	0	0
$345$	177340.	0.0802156
$346$	0	0
$347$	−2.27067e6	−1.01235	−0.506173	−	0.862432i	$-0.668940\pi$
$347$	−2.27067e6	−1.01235	−0.506173	+	0.862432i	$0.668940\pi$
$348$	0	0
$349$	3.16049e6	1.38897	0.694483	−	0.719509i	$-0.255633\pi$
$349$	3.16049e6	1.38897	0.694483	+	0.719509i	$0.255633\pi$
$350$	0	0
$351$	2.08283e6	0.902371
$352$	0	0
$353$	−2.73526e6	−1.16832	−0.584160	−	0.811639i	$-0.698576\pi$
$353$	−2.73526e6	−1.16832	−0.584160	+	0.811639i	$0.698576\pi$
$354$	0	0
$355$	−679822.	−0.286302
$356$	0	0
$357$	1.07230e6	0.445295
$358$	0	0
$359$	−2.87840e6	−1.17873	−0.589367	−	0.807866i	$-0.700623\pi$
$359$	−2.87840e6	−1.17873	−0.589367	+	0.807866i	$0.700623\pi$
$360$	0	0
$361$	−2.39992e6	−0.969234
$362$	0	0
$363$	3.06816e6	1.22211
$364$	0	0
$365$	−33556.2	−0.0131838
$366$	0	0
$367$	4.52400e6	1.75330	0.876652	−	0.481125i	$-0.159772\pi$
$367$	4.52400e6	1.75330	0.876652	+	0.481125i	$0.159772\pi$
$368$	0	0
$369$	1.67772e6	0.641438
$370$	0	0
$371$	752375.	0.283792
$372$	0	0
$373$	−2.85830e6	−1.06374	−0.531870	−	0.846826i	$-0.678511\pi$
$373$	−2.85830e6	−1.06374	−0.531870	+	0.846826i	$0.678511\pi$
$374$	0	0
$375$	−1.11009e6	−0.407644
$376$	0	0
$377$	684282.	0.247960
$378$	0	0
$379$	−224897.	−0.0804240	−0.0402120	−	0.999191i	$-0.512803\pi$
$379$	−224897.	−0.0804240	−0.0402120	+	0.999191i	$0.512803\pi$
$380$	0	0
$381$	3.38566e6	1.19490
$382$	0	0
$383$	2.24747e6	0.782884	0.391442	−	0.920203i	$-0.371976\pi$
$383$	2.24747e6	0.782884	0.391442	+	0.920203i	$0.371976\pi$
$384$	0	0
$385$	2612.55	0.000898282 0
$386$	0	0
$387$	−1.75156e6	−0.594494
$388$	0	0
$389$	−1.80088e6	−0.603407	−0.301704	−	0.953402i	$-0.597555\pi$
$389$	−1.80088e6	−0.603407	−0.301704	+	0.953402i	$0.597555\pi$
$390$	0	0
$391$	1.44814e6	0.479037
$392$	0	0
$393$	1.35222e6	0.441639
$394$	0	0
$395$	505570.	0.163038
$396$	0	0
$397$	1.54300e6	0.491348	0.245674	−	0.969353i	$-0.420991\pi$
$397$	1.54300e6	0.491348	0.245674	+	0.969353i	$0.420991\pi$
$398$	0	0
$399$	201138.	0.0632501
$400$	0	0
$401$	693937.	0.215506	0.107753	−	0.994178i	$-0.465634\pi$
$401$	693937.	0.215506	0.107753	+	0.994178i	$0.465634\pi$
$402$	0	0
$403$	−595256.	−0.182575
$404$	0	0
$405$	697901.	0.211425
$406$	0	0
$407$	45367.6	0.0135756
$408$	0	0
$409$	−675164.	−0.199573	−0.0997864	−	0.995009i	$-0.531816\pi$
$409$	−675164.	−0.199573	−0.0997864	+	0.995009i	$0.531816\pi$
$410$	0	0
$411$	1.28671e6	0.375730
$412$	0	0
$413$	1.60327e6	0.462522
$414$	0	0
$415$	329753.	0.0939871
$416$	0	0
$417$	−7.00647e6	−1.97314
$418$	0	0
$419$	−2.04016e6	−0.567713	−0.283856	−	0.958867i	$-0.591614\pi$
$419$	−2.04016e6	−0.567713	−0.283856	+	0.958867i	$0.591614\pi$
$420$	0	0
$421$	−724187.	−0.199134	−0.0995671	−	0.995031i	$-0.531746\pi$
$421$	−724187.	−0.199134	−0.0995671	+	0.995031i	$0.531746\pi$
$422$	0	0
$423$	2.20325e6	0.598704
$424$	0	0
$425$	−4.46668e6	−1.19953
$426$	0	0
$427$	−357320.	−0.0948391
$428$	0	0
$429$	−122521.	−0.0321417
$430$	0	0
$431$	−7.06362e6	−1.83161	−0.915807	−	0.401618i	$-0.868448\pi$
$431$	−7.06362e6	−1.83161	−0.915807	+	0.401618i	$0.868448\pi$
$432$	0	0
$433$	−1.42999e6	−0.366534	−0.183267	−	0.983063i	$-0.558667\pi$
$433$	−1.42999e6	−0.366534	−0.183267	+	0.983063i	$0.558667\pi$
$434$	0	0
$435$	138572.	0.0351119
$436$	0	0
$437$	271636.	0.0680430
$438$	0	0
$439$	2.28042e6	0.564747	0.282374	−	0.959304i	$-0.408878\pi$
$439$	2.28042e6	0.564747	0.282374	+	0.959304i	$0.408878\pi$
$440$	0	0
$441$	−1.84398e6	−0.451501
$442$	0	0
$443$	−21457.5	−0.00519480	−0.00259740	−	0.999997i	$-0.500827\pi$
$443$	−21457.5	−0.00519480	−0.00259740	+	0.999997i	$0.500827\pi$
$444$	0	0
$445$	−71666.0	−0.0171559
$446$	0	0
$447$	−2.75313e6	−0.651716
$448$	0	0
$449$	1.91419e6	0.448093	0.224047	−	0.974578i	$-0.428073\pi$
$449$	1.91419e6	0.448093	0.224047	+	0.974578i	$0.428073\pi$
$450$	0	0
$451$	100875.	0.0233530
$452$	0	0
$453$	5.72133e6	1.30994
$454$	0	0
$455$	321734.	0.0728566
$456$	0	0
$457$	−3.24706e6	−0.727278	−0.363639	−	0.931540i	$-0.618466\pi$
$457$	−3.24706e6	−0.727278	−0.363639	+	0.931540i	$0.618466\pi$
$458$	0	0
$459$	3.44429e6	0.763076
$460$	0	0
$461$	6.35630e6	1.39300	0.696502	−	0.717555i	$-0.254739\pi$
$461$	6.35630e6	1.39300	0.696502	+	0.717555i	$0.254739\pi$
$462$	0	0
$463$	−7.96907e6	−1.72765	−0.863823	−	0.503795i	$-0.831937\pi$
$463$	−7.96907e6	−1.72765	−0.863823	+	0.503795i	$0.831937\pi$
$464$	0	0
$465$	−120544.	−0.0258531
$466$	0	0
$467$	−8.99985e6	−1.90960	−0.954802	−	0.297244i	$-0.903932\pi$
$467$	−8.99985e6	−1.90960	−0.954802	+	0.297244i	$0.903932\pi$
$468$	0	0
$469$	303384.	0.0636884
$470$	0	0
$471$	−2.17899e6	−0.452588
$472$	0	0
$473$	−105315.	−0.0216439
$474$	0	0
$475$	−837839.	−0.170383
$476$	0	0
$477$	−2.36434e6	−0.475789
$478$	0	0
$479$	−2.49269e6	−0.496398	−0.248199	−	0.968709i	$-0.579839\pi$
$479$	−2.49269e6	−0.496398	−0.248199	+	0.968709i	$0.579839\pi$
$480$	0	0
$481$	5.58700e6	1.10107
$482$	0	0
$483$	717210.	0.139887
$484$	0	0
$485$	−33013.0	−0.00637281
$486$	0	0
$487$	2.18919e6	0.418275	0.209137	−	0.977886i	$-0.432934\pi$
$487$	2.18919e6	0.418275	0.209137	+	0.977886i	$0.432934\pi$
$488$	0	0
$489$	1.11997e7	2.11804
$490$	0	0
$491$	7.49587e6	1.40320	0.701598	−	0.712573i	$-0.252470\pi$
$491$	7.49587e6	1.40320	0.701598	+	0.712573i	$0.252470\pi$
$492$	0	0
$493$	1.13157e6	0.209684
$494$	0	0
$495$	−8209.96	−0.00150601
$496$	0	0
$497$	−2.74938e6	−0.499280
$498$	0	0
$499$	−6.86744e6	−1.23465	−0.617324	−	0.786709i	$-0.711783\pi$
$499$	−6.86744e6	−1.23465	−0.617324	+	0.786709i	$0.711783\pi$
$500$	0	0
$501$	−5.83692e6	−1.03894
$502$	0	0
$503$	−9.74424e6	−1.71723	−0.858614	−	0.512623i	$-0.828674\pi$
$503$	−9.74424e6	−1.71723	−0.858614	+	0.512623i	$0.828674\pi$
$504$	0	0
$505$	−1.60834e6	−0.280640
$506$	0	0
$507$	−8.01272e6	−1.38439
$508$	0	0
$509$	−593354.	−0.101512	−0.0507562	−	0.998711i	$-0.516163\pi$
$509$	−593354.	−0.101512	−0.0507562	+	0.998711i	$0.516163\pi$
$510$	0	0
$511$	−135710.	−0.0229911
$512$	0	0
$513$	646063.	0.108388
$514$	0	0
$515$	1.41944e6	0.235829
$516$	0	0
$517$	132473.	0.0217972
$518$	0	0
$519$	9.99734e6	1.62917
$520$	0	0
$521$	8.30186e6	1.33993	0.669963	−	0.742394i	$-0.266310\pi$
$521$	8.30186e6	1.33993	0.669963	+	0.742394i	$0.266310\pi$
$522$	0	0
$523$	7.27268e6	1.16263	0.581313	−	0.813680i	$-0.302539\pi$
$523$	7.27268e6	1.16263	0.581313	+	0.813680i	$0.302539\pi$
$524$	0	0
$525$	−2.21218e6	−0.350285
$526$	0	0
$527$	−984352.	−0.154392
$528$	0	0
$529$	−5.46775e6	−0.849512
$530$	0	0
$531$	−5.03830e6	−0.775440
$532$	0	0
$533$	1.24228e7	1.89409
$534$	0	0
$535$	−72825.9	−0.0110002
$536$	0	0
$537$	9.85240e6	1.47437
$538$	0	0
$539$	−110871.	−0.0164380
$540$	0	0
$541$	−9.05883e6	−1.33070	−0.665348	−	0.746533i	$-0.731717\pi$
$541$	−9.05883e6	−1.33070	−0.665348	+	0.746533i	$0.731717\pi$
$542$	0	0
$543$	−3.28766e6	−0.478506
$544$	0	0
$545$	−2.28404e6	−0.329392
$546$	0	0
$547$	−1.09507e7	−1.56485	−0.782427	−	0.622743i	$-0.786018\pi$
$547$	−1.09507e7	−1.56485	−0.782427	+	0.622743i	$0.786018\pi$
$548$	0	0
$549$	1.12288e6	0.159002
$550$	0	0
$551$	212255.	0.0297837
$552$	0	0
$553$	2.04466e6	0.284321
$554$	0	0
$555$	1.13141e6	0.155915
$556$	0	0
$557$	3.27524e6	0.447306	0.223653	−	0.974669i	$-0.428202\pi$
$557$	3.27524e6	0.447306	0.223653	+	0.974669i	$0.428202\pi$
$558$	0	0
$559$	−1.29695e7	−1.75547
$560$	0	0
$561$	−202609.	−0.0271801
$562$	0	0
$563$	1.27701e7	1.69794	0.848969	−	0.528443i	$-0.177224\pi$
$563$	1.27701e7	1.69794	0.848969	+	0.528443i	$0.177224\pi$
$564$	0	0
$565$	710888.	0.0936871
$566$	0	0
$567$	2.82250e6	0.368702
$568$	0	0
$569$	−9.90709e6	−1.28282	−0.641409	−	0.767199i	$-0.721650\pi$
$569$	−9.90709e6	−1.28282	−0.641409	+	0.767199i	$0.721650\pi$
$570$	0	0
$571$	−3.04269e6	−0.390541	−0.195271	−	0.980749i	$-0.562558\pi$
$571$	−3.04269e6	−0.390541	−0.195271	+	0.980749i	$0.562558\pi$
$572$	0	0
$573$	−1.06513e7	−1.35524
$574$	0	0
$575$	−2.98754e6	−0.376829
$576$	0	0
$577$	1.55918e7	1.94966	0.974828	−	0.222958i	$-0.0715711\pi$
$577$	1.55918e7	1.94966	0.974828	+	0.222958i	$0.0715711\pi$
$578$	0	0
$579$	860371.	0.106657
$580$	0	0
$581$	1.33361e6	0.163903
$582$	0	0
$583$	−142159.	−0.0173222
$584$	0	0
$585$	−1.01105e6	−0.122147
$586$	0	0
$587$	−1.04572e6	−0.125263	−0.0626313	−	0.998037i	$-0.519949\pi$
$587$	−1.04572e6	−0.125263	−0.0626313	+	0.998037i	$0.519949\pi$
$588$	0	0
$589$	−184640.	−0.0219300
$590$	0	0
$591$	3.52254e6	0.414846
$592$	0	0
$593$	1.41049e7	1.64715	0.823573	−	0.567210i	$-0.191977\pi$
$593$	1.41049e7	1.64715	0.823573	+	0.567210i	$0.191977\pi$
$594$	0	0
$595$	532040.	0.0616101
$596$	0	0
$597$	−1.80794e7	−2.07610
$598$	0	0
$599$	1.67206e6	0.190407	0.0952037	−	0.995458i	$-0.469650\pi$
$599$	1.67206e6	0.190407	0.0952037	+	0.995458i	$0.469650\pi$
$600$	0	0
$601$	−4.69364e6	−0.530059	−0.265029	−	0.964240i	$-0.585382\pi$
$601$	−4.69364e6	−0.530059	−0.265029	+	0.964240i	$0.585382\pi$
$602$	0	0
$603$	−953387.	−0.106777
$604$	0	0
$605$	1.52231e6	0.169089
$606$	0	0
$607$	−2.06207e6	−0.227160	−0.113580	−	0.993529i	$-0.536232\pi$
$607$	−2.06207e6	−0.227160	−0.113580	+	0.993529i	$0.536232\pi$
$608$	0	0
$609$	560424.	0.0612313
$610$	0	0
$611$	1.63140e7	1.76790
$612$	0	0
$613$	1.11053e7	1.19365	0.596827	−	0.802370i	$-0.296428\pi$
$613$	1.11053e7	1.19365	0.596827	+	0.802370i	$0.296428\pi$
$614$	0	0
$615$	2.51570e6	0.268208
$616$	0	0
$617$	2.88021e6	0.304587	0.152294	−	0.988335i	$-0.451334\pi$
$617$	2.88021e6	0.304587	0.152294	+	0.988335i	$0.451334\pi$
$618$	0	0
$619$	9.73994e6	1.02172	0.510858	−	0.859665i	$-0.329328\pi$
$619$	9.73994e6	1.02172	0.510858	+	0.859665i	$0.329328\pi$
$620$	0	0
$621$	2.30371e6	0.239717
$622$	0	0
$623$	−289837.	−0.0299180
$624$	0	0
$625$	8.93544e6	0.914990
$626$	0	0
$627$	−38004.4	−0.00386069
$628$	0	0
$629$	9.23901e6	0.931105
$630$	0	0
$631$	−9.21661e6	−0.921505	−0.460753	−	0.887529i	$-0.652420\pi$
$631$	−9.21661e6	−0.921505	−0.460753	+	0.887529i	$0.652420\pi$
$632$	0	0
$633$	−1.77059e7	−1.75634
$634$	0	0
$635$	1.67985e6	0.165324
$636$	0	0
$637$	−1.36538e7	−1.33323
$638$	0	0
$639$	8.63995e6	0.837065
$640$	0	0
$641$	−2.56772e6	−0.246832	−0.123416	−	0.992355i	$-0.539385\pi$
$641$	−2.56772e6	−0.246832	−0.123416	+	0.992355i	$0.539385\pi$
$642$	0	0
$643$	−1.22951e7	−1.17274	−0.586372	−	0.810042i	$-0.699444\pi$
$643$	−1.22951e7	−1.17274	−0.586372	+	0.810042i	$0.699444\pi$
$644$	0	0
$645$	−2.62642e6	−0.248579
$646$	0	0
$647$	1.26776e7	1.19063	0.595314	−	0.803494i	$-0.297028\pi$
$647$	1.26776e7	1.19063	0.595314	+	0.803494i	$0.297028\pi$
$648$	0	0
$649$	−302934.	−0.0282317
$650$	0	0
$651$	−487512.	−0.0450851
$652$	0	0
$653$	1.56079e7	1.43239	0.716194	−	0.697901i	$-0.245883\pi$
$653$	1.56079e7	1.43239	0.716194	+	0.697901i	$0.245883\pi$
$654$	0	0
$655$	670926.	0.0611043
$656$	0	0
$657$	426470.	0.0385457
$658$	0	0
$659$	−4.62936e6	−0.415247	−0.207624	−	0.978209i	$-0.566573\pi$
$659$	−4.62936e6	−0.415247	−0.207624	+	0.978209i	$0.566573\pi$
$660$	0	0
$661$	3.40160e6	0.302816	0.151408	−	0.988471i	$-0.451619\pi$
$661$	3.40160e6	0.302816	0.151408	+	0.988471i	$0.451619\pi$
$662$	0	0
$663$	−2.49512e7	−2.20449
$664$	0	0
$665$	99797.5	0.00875116
$666$	0	0
$667$	756851.	0.0658712
$668$	0	0
$669$	−2.26469e7	−1.95634
$670$	0	0
$671$	67514.6	0.00578883
$672$	0	0
$673$	3.39092e6	0.288589	0.144295	−	0.989535i	$-0.453909\pi$
$673$	3.39092e6	0.288589	0.144295	+	0.989535i	$0.453909\pi$
$674$	0	0
$675$	−7.10562e6	−0.600264
$676$	0	0
$677$	4.89868e6	0.410778	0.205389	−	0.978680i	$-0.434154\pi$
$677$	4.89868e6	0.410778	0.205389	+	0.978680i	$0.434154\pi$
$678$	0	0
$679$	−133514.	−0.0111135
$680$	0	0
$681$	6.51944e6	0.538694
$682$	0	0
$683$	−1.70258e7	−1.39655	−0.698276	−	0.715829i	$-0.746049\pi$
$683$	−1.70258e7	−1.39655	−0.698276	+	0.715829i	$0.746049\pi$
$684$	0	0
$685$	638421.	0.0519853
$686$	0	0
$687$	5.26701e6	0.425767
$688$	0	0
$689$	−1.75068e7	−1.40495
$690$	0	0
$691$	−6.56844e6	−0.523320	−0.261660	−	0.965160i	$-0.584270\pi$
$691$	−6.56844e6	−0.523320	−0.261660	+	0.965160i	$0.584270\pi$
$692$	0	0
$693$	−33203.3	−0.00262632
$694$	0	0
$695$	−3.47636e6	−0.273000
$696$	0	0
$697$	2.05430e7	1.60170
$698$	0	0
$699$	−1.29588e6	−0.100316
$700$	0	0
$701$	−6.46758e6	−0.497104	−0.248552	−	0.968619i	$-0.579955\pi$
$701$	−6.46758e6	−0.497104	−0.248552	+	0.968619i	$0.579955\pi$
$702$	0	0
$703$	1.73301e6	0.132255
$704$	0	0
$705$	3.30371e6	0.250339
$706$	0	0
$707$	−6.50457e6	−0.489407
$708$	0	0
$709$	2.53410e7	1.89325	0.946627	−	0.322331i	$-0.104466\pi$
$709$	2.53410e7	1.89325	0.946627	+	0.322331i	$0.104466\pi$
$710$	0	0
$711$	−6.42537e6	−0.476677
$712$	0	0
$713$	−658384.	−0.0485015
$714$	0	0
$715$	−60790.8	−0.00444706
$716$	0	0
$717$	3.53220e6	0.256595
$718$	0	0
$719$	−2.05783e7	−1.48452	−0.742261	−	0.670111i	$-0.766246\pi$
$719$	−2.05783e7	−1.48452	−0.742261	+	0.670111i	$0.766246\pi$
$720$	0	0
$721$	5.74058e6	0.411261
$722$	0	0
$723$	−2.66912e7	−1.89899
$724$	0	0
$725$	−2.33445e6	−0.164945
$726$	0	0
$727$	−6.98920e6	−0.490446	−0.245223	−	0.969467i	$-0.578861\pi$
$727$	−6.98920e6	−0.490446	−0.245223	+	0.969467i	$0.578861\pi$
$728$	0	0
$729$	1.97004e6	0.137295
$730$	0	0
$731$	−2.14471e7	−1.48448
$732$	0	0
$733$	4.33267e6	0.297849	0.148924	−	0.988849i	$-0.452419\pi$
$733$	4.33267e6	0.297849	0.148924	+	0.988849i	$0.452419\pi$
$734$	0	0
$735$	−2.76500e6	−0.188789
$736$	0	0
$737$	−57323.5	−0.00388745
$738$	0	0
$739$	1.61883e7	1.09041	0.545206	−	0.838302i	$-0.316452\pi$
$739$	1.61883e7	1.09041	0.545206	+	0.838302i	$0.316452\pi$
$740$	0	0
$741$	−4.68023e6	−0.313128
$742$	0	0
$743$	2.88216e7	1.91534	0.957669	−	0.287872i	$-0.0929478\pi$
$743$	2.88216e7	1.91534	0.957669	+	0.287872i	$0.0929478\pi$
$744$	0	0
$745$	−1.36601e6	−0.0901702
$746$	0	0
$747$	−4.19088e6	−0.274792
$748$	0	0
$749$	−294527.	−0.0191832
$750$	0	0
$751$	9.12406e6	0.590321	0.295161	−	0.955448i	$-0.404627\pi$
$751$	9.12406e6	0.590321	0.295161	+	0.955448i	$0.404627\pi$
$752$	0	0
$753$	−1.87486e7	−1.20499
$754$	0	0
$755$	2.83872e6	0.181241
$756$	0	0
$757$	5.25235e6	0.333130	0.166565	−	0.986030i	$-0.446732\pi$
$757$	5.25235e6	0.333130	0.166565	+	0.986030i	$0.446732\pi$
$758$	0	0
$759$	−135515.	−0.00853852
$760$	0	0
$761$	1.52840e7	0.956697	0.478349	−	0.878170i	$-0.341236\pi$
$761$	1.52840e7	0.956697	0.478349	+	0.878170i	$0.341236\pi$
$762$	0	0
$763$	−9.23728e6	−0.574424
$764$	0	0
$765$	−1.67194e6	−0.103292
$766$	0	0
$767$	−3.73062e7	−2.28978
$768$	0	0
$769$	−2.79517e7	−1.70448	−0.852240	−	0.523151i	$-0.824756\pi$
$769$	−2.79517e7	−1.70448	−0.852240	+	0.523151i	$0.824756\pi$
$770$	0	0
$771$	1.25870e6	0.0762580
$772$	0	0
$773$	−1.02509e6	−0.0617042	−0.0308521	−	0.999524i	$-0.509822\pi$
$773$	−1.02509e6	−0.0617042	−0.0308521	+	0.999524i	$0.509822\pi$
$774$	0	0
$775$	2.03073e6	0.121450
$776$	0	0
$777$	4.57573e6	0.271899
$778$	0	0
$779$	3.85336e6	0.227508
$780$	0	0
$781$	519488.	0.0304753
$782$	0	0
$783$	1.80011e6	0.104929
$784$	0	0
$785$	−1.08114e6	−0.0626192
$786$	0	0
$787$	3.93886e6	0.226691	0.113345	−	0.993556i	$-0.463843\pi$
$787$	3.93886e6	0.226691	0.113345	+	0.993556i	$0.463843\pi$
$788$	0	0
$789$	−2.01256e7	−1.15095
$790$	0	0
$791$	2.87502e6	0.163380
$792$	0	0
$793$	8.31439e6	0.469513
$794$	0	0
$795$	−3.54527e6	−0.198944
$796$	0	0
$797$	−1.76853e7	−0.986205	−0.493103	−	0.869971i	$-0.664137\pi$
$797$	−1.76853e7	−0.986205	−0.493103	+	0.869971i	$0.664137\pi$
$798$	0	0
$799$	2.69778e7	1.49499
$800$	0	0
$801$	910814.	0.0501589
$802$	0	0
$803$	25642.1	0.00140334
$804$	0	0
$805$	355855.	0.0193545
$806$	0	0
$807$	1.47031e7	0.794741
$808$	0	0
$809$	2.05146e7	1.10203	0.551013	−	0.834497i	$-0.314242\pi$
$809$	2.05146e7	1.10203	0.551013	+	0.834497i	$0.314242\pi$
$810$	0	0
$811$	3.15482e7	1.68431	0.842155	−	0.539235i	$-0.181287\pi$
$811$	3.15482e7	1.68431	0.842155	+	0.539235i	$0.181287\pi$
$812$	0	0
$813$	4.45950e6	0.236624
$814$	0	0
$815$	5.55689e6	0.293047
$816$	0	0
$817$	−4.02294e6	−0.210857
$818$	0	0
$819$	−4.08897e6	−0.213012
$820$	0	0
$821$	8.86271e6	0.458890	0.229445	−	0.973322i	$-0.426309\pi$
$821$	8.86271e6	0.458890	0.229445	+	0.973322i	$0.426309\pi$
$822$	0	0
$823$	1.84664e7	0.950347	0.475174	−	0.879892i	$-0.342385\pi$
$823$	1.84664e7	0.950347	0.475174	+	0.879892i	$0.342385\pi$
$824$	0	0
$825$	417985.	0.0213809
$826$	0	0
$827$	1.03054e7	0.523964	0.261982	−	0.965073i	$-0.415624\pi$
$827$	1.03054e7	0.523964	0.261982	+	0.965073i	$0.415624\pi$
$828$	0	0
$829$	1.60382e7	0.810532	0.405266	−	0.914199i	$-0.367179\pi$
$829$	1.60382e7	0.810532	0.405266	+	0.914199i	$0.367179\pi$
$830$	0	0
$831$	−3.70358e6	−0.186045
$832$	0	0
$833$	−2.25787e7	−1.12742
$834$	0	0
$835$	−2.89608e6	−0.143745
$836$	0	0
$837$	−1.56591e6	−0.0772598
$838$	0	0
$839$	−7.42491e6	−0.364155	−0.182077	−	0.983284i	$-0.558282\pi$
$839$	−7.42491e6	−0.364155	−0.182077	+	0.983284i	$0.558282\pi$
$840$	0	0
$841$	−1.99197e7	−0.971167
$842$	0	0
$843$	−8.40411e6	−0.407308
$844$	0	0
$845$	−3.97563e6	−0.191542
$846$	0	0
$847$	6.15664e6	0.294873
$848$	0	0
$849$	4.18387e6	0.199209
$850$	0	0
$851$	6.17951e6	0.292503
$852$	0	0
$853$	2.58385e7	1.21589	0.607945	−	0.793979i	$-0.291994\pi$
$853$	2.58385e7	1.21589	0.607945	+	0.793979i	$0.291994\pi$
$854$	0	0
$855$	−313614.	−0.0146717
$856$	0	0
$857$	3.17274e6	0.147565	0.0737823	−	0.997274i	$-0.476493\pi$
$857$	3.17274e6	0.147565	0.0737823	+	0.997274i	$0.476493\pi$
$858$	0	0
$859$	−1.10268e6	−0.0509879	−0.0254939	−	0.999675i	$-0.508116\pi$
$859$	−1.10268e6	−0.0509879	−0.0254939	+	0.999675i	$0.508116\pi$
$860$	0	0
$861$	1.01742e7	0.467726
$862$	0	0
$863$	−1.86989e7	−0.854652	−0.427326	−	0.904098i	$-0.640544\pi$
$863$	−1.86989e7	−0.854652	−0.427326	+	0.904098i	$0.640544\pi$
$864$	0	0
$865$	4.96033e6	0.225408
$866$	0	0
$867$	−1.42026e7	−0.641681
$868$	0	0
$869$	−386333.	−0.0173545
$870$	0	0
$871$	−7.05937e6	−0.315298
$872$	0	0
$873$	419568.	0.0186323
$874$	0	0
$875$	−2.22754e6	−0.0983570
$876$	0	0
$877$	−8.98118e6	−0.394307	−0.197153	−	0.980373i	$-0.563170\pi$
$877$	−8.98118e6	−0.394307	−0.197153	+	0.980373i	$0.563170\pi$
$878$	0	0
$879$	−4.77880e7	−2.08615
$880$	0	0
$881$	−3.49086e7	−1.51528	−0.757640	−	0.652672i	$-0.773648\pi$
$881$	−3.49086e7	−1.51528	−0.757640	+	0.652672i	$0.773648\pi$
$882$	0	0
$883$	1.09747e6	0.0473687	0.0236844	−	0.999719i	$-0.492460\pi$
$883$	1.09747e6	0.0473687	0.0236844	+	0.999719i	$0.492460\pi$
$884$	0	0
$885$	−7.55480e6	−0.324239
$886$	0	0
$887$	−2.09071e7	−0.892245	−0.446122	−	0.894972i	$-0.647195\pi$
$887$	−2.09071e7	−0.892245	−0.446122	+	0.894972i	$0.647195\pi$
$888$	0	0
$889$	6.79375e6	0.288307
$890$	0	0
$891$	−533303.	−0.0225050
$892$	0	0
$893$	5.06037e6	0.212351
$894$	0	0
$895$	4.88842e6	0.203991
$896$	0	0
$897$	−1.66886e7	−0.692530
$898$	0	0
$899$	−514458.	−0.0212300
$900$	0	0
$901$	−2.89504e7	−1.18807
$902$	0	0
$903$	−1.06219e7	−0.433495
$904$	0	0
$905$	−1.63122e6	−0.0662052
$906$	0	0
$907$	−3.80701e7	−1.53662	−0.768309	−	0.640079i	$-0.778902\pi$
$907$	−3.80701e7	−1.53662	−0.768309	+	0.640079i	$0.778902\pi$
$908$	0	0
$909$	2.04407e7	0.820512
$910$	0	0
$911$	−3.71383e7	−1.48261	−0.741304	−	0.671169i	$-0.765792\pi$
$911$	−3.71383e7	−1.48261	−0.741304	+	0.671169i	$0.765792\pi$
$912$	0	0
$913$	−251982.	−0.0100044
$914$	0	0
$915$	1.68373e6	0.0664844
$916$	0	0
$917$	2.71341e6	0.106559
$918$	0	0
$919$	−1.36555e7	−0.533356	−0.266678	−	0.963786i	$-0.585926\pi$
$919$	−1.36555e7	−0.533356	−0.266678	+	0.963786i	$0.585926\pi$
$920$	0	0
$921$	−2.55117e7	−0.991038
$922$	0	0
$923$	6.39747e7	2.47175
$924$	0	0
$925$	−1.90602e7	−0.732442
$926$	0	0
$927$	−1.80398e7	−0.689498
$928$	0	0
$929$	−2.11028e7	−0.802235	−0.401117	−	0.916027i	$-0.631378\pi$
$929$	−2.11028e7	−0.802235	−0.401117	+	0.916027i	$0.631378\pi$
$930$	0	0
$931$	−4.23521e6	−0.160140
$932$	0	0
$933$	−4.43566e7	−1.66822
$934$	0	0
$935$	−100527.	−0.00376059
$936$	0	0
$937$	−6.57061e6	−0.244487	−0.122244	−	0.992500i	$-0.539009\pi$
$937$	−6.57061e6	−0.244487	−0.122244	+	0.992500i	$0.539009\pi$
$938$	0	0
$939$	−5.51304e7	−2.04045
$940$	0	0
$941$	2.76066e7	1.01634	0.508169	−	0.861257i	$-0.330323\pi$
$941$	2.76066e7	1.01634	0.508169	+	0.861257i	$0.330323\pi$
$942$	0	0
$943$	1.37402e7	0.503169
$944$	0	0
$945$	846371.	0.0308305
$946$	0	0
$947$	4.77883e7	1.73160	0.865799	−	0.500392i	$-0.166811\pi$
$947$	4.77883e7	1.73160	0.865799	+	0.500392i	$0.166811\pi$
$948$	0	0
$949$	3.15781e6	0.113820
$950$	0	0
$951$	−6.67381e7	−2.39289
$952$	0	0
$953$	−2.31731e7	−0.826517	−0.413258	−	0.910614i	$-0.635609\pi$
$953$	−2.31731e7	−0.826517	−0.413258	+	0.910614i	$0.635609\pi$
$954$	0	0
$955$	−5.28479e6	−0.187508
$956$	0	0
$957$	−105891.	−0.00373747
$958$	0	0
$959$	2.58194e6	0.0906568
$960$	0	0
$961$	−2.81816e7	−0.984368
$962$	0	0
$963$	925554.	0.0321615
$964$	0	0
$965$	426886.	0.0147568
$966$	0	0
$967$	2.68802e7	0.924413	0.462206	−	0.886772i	$-0.347058\pi$
$967$	2.68802e7	0.924413	0.462206	+	0.886772i	$0.347058\pi$
$968$	0	0
$969$	−7.73951e6	−0.264791
$970$	0	0
$971$	1.40794e7	0.479222	0.239611	−	0.970869i	$-0.422980\pi$
$971$	1.40794e7	0.479222	0.239611	+	0.970869i	$0.422980\pi$
$972$	0	0
$973$	−1.40593e7	−0.476083
$974$	0	0
$975$	5.14747e7	1.73413
$976$	0	0
$977$	−4.12190e7	−1.38153	−0.690767	−	0.723078i	$-0.742727\pi$
$977$	−4.12190e7	−1.38153	−0.690767	+	0.723078i	$0.742727\pi$
$978$	0	0
$979$	54763.8	0.00182615
$980$	0	0
$981$	2.90282e7	0.963048
$982$	0	0
$983$	−1.44851e6	−0.0478121	−0.0239061	−	0.999714i	$-0.507610\pi$
$983$	−1.44851e6	−0.0478121	−0.0239061	+	0.999714i	$0.507610\pi$
$984$	0	0
$985$	1.74776e6	0.0573972
$986$	0	0
$987$	1.33611e7	0.436565
$988$	0	0
$989$	−1.43449e7	−0.466344
$990$	0	0
$991$	−8.93415e6	−0.288981	−0.144490	−	0.989506i	$-0.546154\pi$
$991$	−8.93415e6	−0.288981	−0.144490	+	0.989506i	$0.546154\pi$
$992$	0	0
$993$	5.58861e7	1.79858
$994$	0	0
$995$	−8.97036e6	−0.287245
$996$	0	0
$997$	2.71109e7	0.863786	0.431893	−	0.901925i	$-0.357846\pi$
$997$	2.71109e7	0.863786	0.431893	+	0.901925i	$0.357846\pi$
$998$	0	0
$999$	1.46974e7	0.465938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.13	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.13	✓	57	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-19.0570 q^{3} -9.45543 q^{5} -38.2403 q^{7} +120.170 q^{9} +O(q^{10})\)	\(q-19.0570 q^{3} -9.45543 q^{5} -38.2403 q^{7} +120.170 q^{9} +7.22540 q^{11} +889.805 q^{13} +180.192 q^{15} +1471.44 q^{17} +276.005 q^{19} +728.747 q^{21} +984.169 q^{23} -3035.59 q^{25} +2340.77 q^{27} +769.025 q^{29} -668.974 q^{31} -137.695 q^{33} +361.579 q^{35} +6278.91 q^{37} -16957.0 q^{39} +13961.2 q^{41} -14575.6 q^{43} -1136.26 q^{45} +18334.3 q^{47} -15344.7 q^{49} -28041.2 q^{51} -19674.9 q^{53} -68.3193 q^{55} -5259.84 q^{57} -41926.3 q^{59} +9344.06 q^{61} -4595.35 q^{63} -8413.49 q^{65} -7933.62 q^{67} -18755.3 q^{69} +71897.5 q^{71} +3548.88 q^{73} +57849.4 q^{75} -276.301 q^{77} -53468.8 q^{79} -73809.5 q^{81} -34874.4 q^{83} -13913.1 q^{85} -14655.3 q^{87} +7579.35 q^{89} -34026.4 q^{91} +12748.7 q^{93} -2609.75 q^{95} +3491.44 q^{97} +868.280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

Varieties

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