LMFDB - Embedded newform 624.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,6,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("624.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$624 = 2^{4} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	624.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:	$100.079503563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-9.00000 q^{3} -74.0000 q^{5} +112.000 q^{7} +81.0000 q^{9} -164.000 q^{11} +169.000 q^{13} +666.000 q^{15} -1646.00 q^{17} +2052.00 q^{19} -1008.00 q^{21} -4152.00 q^{23} +2351.00 q^{25} -729.000 q^{27} +2638.00 q^{29} +8936.00 q^{31} +1476.00 q^{33} -8288.00 q^{35} +1846.00 q^{37} -1521.00 q^{39} +8010.00 q^{41} +19236.0 q^{43} -5994.00 q^{45} +12840.0 q^{47} -4263.00 q^{49} +14814.0 q^{51} -1434.00 q^{53} +12136.0 q^{55} -18468.0 q^{57} -1428.00 q^{59} -25202.0 q^{61} +9072.00 q^{63} -12506.0 q^{65} +22868.0 q^{67} +37368.0 q^{69} -17280.0 q^{71} +54410.0 q^{73} -21159.0 q^{75} -18368.0 q^{77} -65312.0 q^{79} +6561.00 q^{81} +70372.0 q^{83} +121804. q^{85} -23742.0 q^{87} -76390.0 q^{89} +18928.0 q^{91} -80424.0 q^{93} -151848. q^{95} -174398. q^{97} -13284.0 q^{99} +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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−74.0000	−1.32375	−0.661876	−	0.749613i	$-0.730240\pi$
$5$	−74.0000	−1.32375	−0.661876	+	0.749613i	$0.730240\pi$
$6$	0	0
$7$	112.000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	112.000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−164.000	−0.408660	−0.204330	−	0.978902i	$-0.565502\pi$
$11$	−164.000	−0.408660	−0.204330	+	0.978902i	$0.565502\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	666.000	0.764269
$16$	0	0
$17$	−1646.00	−1.38136	−0.690681	−	0.723160i	$-0.742689\pi$
$17$	−1646.00	−1.38136	−0.690681	+	0.723160i	$0.742689\pi$
$18$	0	0
$19$	2052.00	1.30405	0.652024	−	0.758199i	$-0.273920\pi$
$19$	2052.00	1.30405	0.652024	+	0.758199i	$0.273920\pi$
$20$	0	0
$21$	−1008.00	−0.498784
$22$	0	0
$23$	−4152.00	−1.63658	−0.818291	−	0.574804i	$-0.805078\pi$
$23$	−4152.00	−1.63658	−0.818291	+	0.574804i	$0.805078\pi$
$24$	0	0
$25$	2351.00	0.752320
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	2638.00	0.582478	0.291239	−	0.956650i	$-0.405932\pi$
$29$	2638.00	0.582478	0.291239	+	0.956650i	$0.405932\pi$
$30$	0	0
$31$	8936.00	1.67009	0.835043	−	0.550184i	$-0.185443\pi$
$31$	8936.00	1.67009	0.835043	+	0.550184i	$0.185443\pi$
$32$	0	0
$33$	1476.00	0.235940
$34$	0	0
$35$	−8288.00	−1.14361
$36$	0	0
$37$	1846.00	0.221680	0.110840	−	0.993838i	$-0.464646\pi$
$37$	1846.00	0.221680	0.110840	+	0.993838i	$0.464646\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	8010.00	0.744171	0.372086	−	0.928198i	$-0.378643\pi$
$41$	8010.00	0.744171	0.372086	+	0.928198i	$0.378643\pi$
$42$	0	0
$43$	19236.0	1.58651	0.793256	−	0.608888i	$-0.208384\pi$
$43$	19236.0	1.58651	0.793256	+	0.608888i	$0.208384\pi$
$44$	0	0
$45$	−5994.00	−0.441251
$46$	0	0
$47$	12840.0	0.847853	0.423926	−	0.905697i	$-0.360652\pi$
$47$	12840.0	0.847853	0.423926	+	0.905697i	$0.360652\pi$
$48$	0	0
$49$	−4263.00	−0.253644
$50$	0	0
$51$	14814.0	0.797530
$52$	0	0
$53$	−1434.00	−0.0701228	−0.0350614	−	0.999385i	$-0.511163\pi$
$53$	−1434.00	−0.0701228	−0.0350614	+	0.999385i	$0.511163\pi$
$54$	0	0
$55$	12136.0	0.540965
$56$	0	0
$57$	−18468.0	−0.752892
$58$	0	0
$59$	−1428.00	−0.0534070	−0.0267035	−	0.999643i	$-0.508501\pi$
$59$	−1428.00	−0.0534070	−0.0267035	+	0.999643i	$0.508501\pi$
$60$	0	0
$61$	−25202.0	−0.867182	−0.433591	−	0.901110i	$-0.642754\pi$
$61$	−25202.0	−0.867182	−0.433591	+	0.901110i	$0.642754\pi$
$62$	0	0
$63$	9072.00	0.287973
$64$	0	0
$65$	−12506.0	−0.367143
$66$	0	0
$67$	22868.0	0.622359	0.311180	−	0.950351i	$-0.399276\pi$
$67$	22868.0	0.622359	0.311180	+	0.950351i	$0.399276\pi$
$68$	0	0
$69$	37368.0	0.944881
$70$	0	0
$71$	−17280.0	−0.406816	−0.203408	−	0.979094i	$-0.565202\pi$
$71$	−17280.0	−0.406816	−0.203408	+	0.979094i	$0.565202\pi$
$72$	0	0
$73$	54410.0	1.19501	0.597505	−	0.801865i	$-0.296159\pi$
$73$	54410.0	1.19501	0.597505	+	0.801865i	$0.296159\pi$
$74$	0	0
$75$	−21159.0	−0.434352
$76$	0	0
$77$	−18368.0	−0.353049
$78$	0	0
$79$	−65312.0	−1.17740	−0.588702	−	0.808350i	$-0.700361\pi$
$79$	−65312.0	−1.17740	−0.588702	+	0.808350i	$0.700361\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	70372.0	1.12126	0.560628	−	0.828068i	$-0.310560\pi$
$83$	70372.0	1.12126	0.560628	+	0.828068i	$0.310560\pi$
$84$	0	0
$85$	121804.	1.82858
$86$	0	0
$87$	−23742.0	−0.336294
$88$	0	0
$89$	−76390.0	−1.02226	−0.511130	−	0.859503i	$-0.670773\pi$
$89$	−76390.0	−1.02226	−0.511130	+	0.859503i	$0.670773\pi$
$90$	0	0
$91$	18928.0	0.239608
$92$	0	0
$93$	−80424.0	−0.964225
$94$	0	0
$95$	−151848.	−1.72624
$96$	0	0
$97$	−174398.	−1.88197	−0.940984	−	0.338451i	$-0.890097\pi$
$97$	−174398.	−1.88197	−0.940984	+	0.338451i	$0.890097\pi$
$98$	0	0
$99$	−13284.0	−0.136220
$100$	0	0
$101$	−2730.00	−0.0266293	−0.0133146	−	0.999911i	$-0.504238\pi$
$101$	−2730.00	−0.0266293	−0.0133146	+	0.999911i	$0.504238\pi$
$102$	0	0
$103$	−163464.	−1.51820	−0.759100	−	0.650974i	$-0.774361\pi$
$103$	−163464.	−1.51820	−0.759100	+	0.650974i	$0.774361\pi$
$104$	0	0
$105$	74592.0	0.660266
$106$	0	0
$107$	−127884.	−1.07983	−0.539917	−	0.841718i	$-0.681544\pi$
$107$	−127884.	−1.07983	−0.539917	+	0.841718i	$0.681544\pi$
$108$	0	0
$109$	9694.00	0.0781514	0.0390757	−	0.999236i	$-0.487559\pi$
$109$	9694.00	0.0781514	0.0390757	+	0.999236i	$0.487559\pi$
$110$	0	0
$111$	−16614.0	−0.127987
$112$	0	0
$113$	−161166.	−1.18735	−0.593673	−	0.804706i	$-0.702323\pi$
$113$	−161166.	−1.18735	−0.593673	+	0.804706i	$0.702323\pi$
$114$	0	0
$115$	307248.	2.16643
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	−184352.	−1.19338
$120$	0	0
$121$	−134155.	−0.832997
$122$	0	0
$123$	−72090.0	−0.429647
$124$	0	0
$125$	57276.0	0.327867
$126$	0	0
$127$	52000.0	0.286084	0.143042	−	0.989717i	$-0.454312\pi$
$127$	52000.0	0.286084	0.143042	+	0.989717i	$0.454312\pi$
$128$	0	0
$129$	−173124.	−0.915974
$130$	0	0
$131$	−284324.	−1.44756	−0.723778	−	0.690033i	$-0.757596\pi$
$131$	−284324.	−1.44756	−0.723778	+	0.690033i	$0.757596\pi$
$132$	0	0
$133$	229824.	1.12659
$134$	0	0
$135$	53946.0	0.254756
$136$	0	0
$137$	231978.	1.05595	0.527977	−	0.849258i	$-0.322951\pi$
$137$	231978.	1.05595	0.527977	+	0.849258i	$0.322951\pi$
$138$	0	0
$139$	−135084.	−0.593017	−0.296508	−	0.955030i	$-0.595822\pi$
$139$	−135084.	−0.593017	−0.296508	+	0.955030i	$0.595822\pi$
$140$	0	0
$141$	−115560.	−0.489508
$142$	0	0
$143$	−27716.0	−0.113342
$144$	0	0
$145$	−195212.	−0.771057
$146$	0	0
$147$	38367.0	0.146442
$148$	0	0
$149$	78438.0	0.289442	0.144721	−	0.989473i	$-0.453772\pi$
$149$	78438.0	0.289442	0.144721	+	0.989473i	$0.453772\pi$
$150$	0	0
$151$	131264.	0.468493	0.234247	−	0.972177i	$-0.424738\pi$
$151$	131264.	0.468493	0.234247	+	0.972177i	$0.424738\pi$
$152$	0	0
$153$	−133326.	−0.460454
$154$	0	0
$155$	−661264.	−2.21078
$156$	0	0
$157$	198030.	0.641183	0.320591	−	0.947218i	$-0.396118\pi$
$157$	198030.	0.641183	0.320591	+	0.947218i	$0.396118\pi$
$158$	0	0
$159$	12906.0	0.0404854
$160$	0	0
$161$	−465024.	−1.41387
$162$	0	0
$163$	−190172.	−0.560632	−0.280316	−	0.959908i	$-0.590439\pi$
$163$	−190172.	−0.560632	−0.280316	+	0.959908i	$0.590439\pi$
$164$	0	0
$165$	−109224.	−0.312326
$166$	0	0
$167$	135952.	0.377220	0.188610	−	0.982052i	$-0.439602\pi$
$167$	135952.	0.377220	0.188610	+	0.982052i	$0.439602\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	166212.	0.434682
$172$	0	0
$173$	743934.	1.88981	0.944907	−	0.327338i	$-0.106152\pi$
$173$	743934.	1.88981	0.944907	+	0.327338i	$0.106152\pi$
$174$	0	0
$175$	263312.	0.649943
$176$	0	0
$177$	12852.0	0.0308345
$178$	0	0
$179$	619708.	1.44562	0.722811	−	0.691046i	$-0.242850\pi$
$179$	619708.	1.44562	0.722811	+	0.691046i	$0.242850\pi$
$180$	0	0
$181$	−559162.	−1.26865	−0.634324	−	0.773067i	$-0.718722\pi$
$181$	−559162.	−1.26865	−0.634324	+	0.773067i	$0.718722\pi$
$182$	0	0
$183$	226818.	0.500668
$184$	0	0
$185$	−136604.	−0.293450
$186$	0	0
$187$	269944.	0.564507
$188$	0	0
$189$	−81648.0	−0.166261
$190$	0	0
$191$	154448.	0.306337	0.153168	−	0.988200i	$-0.451052\pi$
$191$	154448.	0.306337	0.153168	+	0.988200i	$0.451052\pi$
$192$	0	0
$193$	−268638.	−0.519128	−0.259564	−	0.965726i	$-0.583579\pi$
$193$	−268638.	−0.519128	−0.259564	+	0.965726i	$0.583579\pi$
$194$	0	0
$195$	112554.	0.211970
$196$	0	0
$197$	−825450.	−1.51539	−0.757696	−	0.652607i	$-0.773675\pi$
$197$	−825450.	−1.51539	−0.757696	+	0.652607i	$0.773675\pi$
$198$	0	0
$199$	−874184.	−1.56484	−0.782420	−	0.622751i	$-0.786015\pi$
$199$	−874184.	−1.56484	−0.782420	+	0.622751i	$0.786015\pi$
$200$	0	0
$201$	−205812.	−0.359319
$202$	0	0
$203$	295456.	0.503214
$204$	0	0
$205$	−592740.	−0.985098
$206$	0	0
$207$	−336312.	−0.545527
$208$	0	0
$209$	−336528.	−0.532912
$210$	0	0
$211$	−409876.	−0.633791	−0.316896	−	0.948460i	$-0.602641\pi$
$211$	−409876.	−0.633791	−0.316896	+	0.948460i	$0.602641\pi$
$212$	0	0
$213$	155520.	0.234875
$214$	0	0
$215$	−1.42346e6	−2.10015
$216$	0	0
$217$	1.00083e6	1.44282
$218$	0	0
$219$	−489690.	−0.689939
$220$	0	0
$221$	−278174.	−0.383121
$222$	0	0
$223$	−488248.	−0.657474	−0.328737	−	0.944422i	$-0.606623\pi$
$223$	−488248.	−0.657474	−0.328737	+	0.944422i	$0.606623\pi$
$224$	0	0
$225$	190431.	0.250773
$226$	0	0
$227$	1.03311e6	1.33070	0.665351	−	0.746530i	$-0.268282\pi$
$227$	1.03311e6	1.33070	0.665351	+	0.746530i	$0.268282\pi$
$228$	0	0
$229$	−784042.	−0.987986	−0.493993	−	0.869466i	$-0.664463\pi$
$229$	−784042.	−0.987986	−0.493993	+	0.869466i	$0.664463\pi$
$230$	0	0
$231$	165312.	0.203833
$232$	0	0
$233$	−366262.	−0.441979	−0.220990	−	0.975276i	$-0.570929\pi$
$233$	−366262.	−0.441979	−0.220990	+	0.975276i	$0.570929\pi$
$234$	0	0
$235$	−950160.	−1.12235
$236$	0	0
$237$	587808.	0.679774
$238$	0	0
$239$	−1.51639e6	−1.71718	−0.858592	−	0.512660i	$-0.828660\pi$
$239$	−1.51639e6	−1.71718	−0.858592	+	0.512660i	$0.828660\pi$
$240$	0	0
$241$	−1.53438e6	−1.70173	−0.850865	−	0.525384i	$-0.823922\pi$
$241$	−1.53438e6	−1.70173	−0.850865	+	0.525384i	$0.823922\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	315462.	0.335762
$246$	0	0
$247$	346788.	0.361678
$248$	0	0
$249$	−633348.	−0.647357
$250$	0	0
$251$	419700.	0.420489	0.210245	−	0.977649i	$-0.432574\pi$
$251$	419700.	0.420489	0.210245	+	0.977649i	$0.432574\pi$
$252$	0	0
$253$	680928.	0.668806
$254$	0	0
$255$	−1.09624e6	−1.05573
$256$	0	0
$257$	450562.	0.425522	0.212761	−	0.977104i	$-0.431754\pi$
$257$	450562.	0.425522	0.212761	+	0.977104i	$0.431754\pi$
$258$	0	0
$259$	206752.	0.191514
$260$	0	0
$261$	213678.	0.194159
$262$	0	0
$263$	368328.	0.328356	0.164178	−	0.986431i	$-0.447503\pi$
$263$	368328.	0.328356	0.164178	+	0.986431i	$0.447503\pi$
$264$	0	0
$265$	106116.	0.0928253
$266$	0	0
$267$	687510.	0.590202
$268$	0	0
$269$	−1.65459e6	−1.39415	−0.697077	−	0.716996i	$-0.745516\pi$
$269$	−1.65459e6	−1.39415	−0.697077	+	0.716996i	$0.745516\pi$
$270$	0	0
$271$	2.00225e6	1.65613	0.828065	−	0.560631i	$-0.189442\pi$
$271$	2.00225e6	1.65613	0.828065	+	0.560631i	$0.189442\pi$
$272$	0	0
$273$	−170352.	−0.138338
$274$	0	0
$275$	−385564.	−0.307443
$276$	0	0
$277$	−1.15398e6	−0.903646	−0.451823	−	0.892108i	$-0.649226\pi$
$277$	−1.15398e6	−0.903646	−0.451823	+	0.892108i	$0.649226\pi$
$278$	0	0
$279$	723816.	0.556695
$280$	0	0
$281$	−1.76087e6	−1.33034	−0.665168	−	0.746694i	$-0.731640\pi$
$281$	−1.76087e6	−1.33034	−0.665168	+	0.746694i	$0.731640\pi$
$282$	0	0
$283$	−1.78523e6	−1.32504	−0.662518	−	0.749046i	$-0.730512\pi$
$283$	−1.78523e6	−1.32504	−0.662518	+	0.749046i	$0.730512\pi$
$284$	0	0
$285$	1.36663e6	0.996643
$286$	0	0
$287$	897120.	0.642904
$288$	0	0
$289$	1.28946e6	0.908161
$290$	0	0
$291$	1.56958e6	1.08655
$292$	0	0
$293$	172854.	0.117628	0.0588140	−	0.998269i	$-0.481268\pi$
$293$	172854.	0.117628	0.0588140	+	0.998269i	$0.481268\pi$
$294$	0	0
$295$	105672.	0.0706976
$296$	0	0
$297$	119556.	0.0786467
$298$	0	0
$299$	−701688.	−0.453906
$300$	0	0
$301$	2.15443e6	1.37062
$302$	0	0
$303$	24570.0	0.0153744
$304$	0	0
$305$	1.86495e6	1.14793
$306$	0	0
$307$	−1.06286e6	−0.643621	−0.321810	−	0.946804i	$-0.604291\pi$
$307$	−1.06286e6	−0.643621	−0.321810	+	0.946804i	$0.604291\pi$
$308$	0	0
$309$	1.47118e6	0.876533
$310$	0	0
$311$	379656.	0.222582	0.111291	−	0.993788i	$-0.464501\pi$
$311$	379656.	0.222582	0.111291	+	0.993788i	$0.464501\pi$
$312$	0	0
$313$	−3.17463e6	−1.83161	−0.915803	−	0.401627i	$-0.868445\pi$
$313$	−3.17463e6	−1.83161	−0.915803	+	0.401627i	$0.868445\pi$
$314$	0	0
$315$	−671328.	−0.381205
$316$	0	0
$317$	920398.	0.514431	0.257216	−	0.966354i	$-0.417195\pi$
$317$	920398.	0.514431	0.257216	+	0.966354i	$0.417195\pi$
$318$	0	0
$319$	−432632.	−0.238036
$320$	0	0
$321$	1.15096e6	0.623442
$322$	0	0
$323$	−3.37759e6	−1.80136
$324$	0	0
$325$	397319.	0.208656
$326$	0	0
$327$	−87246.0	−0.0451207
$328$	0	0
$329$	1.43808e6	0.732476
$330$	0	0
$331$	411820.	0.206603	0.103302	−	0.994650i	$-0.467059\pi$
$331$	411820.	0.206603	0.103302	+	0.994650i	$0.467059\pi$
$332$	0	0
$333$	149526.	0.0738935
$334$	0	0
$335$	−1.69223e6	−0.823850
$336$	0	0
$337$	749394.	0.359447	0.179724	−	0.983717i	$-0.442480\pi$
$337$	749394.	0.359447	0.179724	+	0.983717i	$0.442480\pi$
$338$	0	0
$339$	1.45049e6	0.685515
$340$	0	0
$341$	−1.46550e6	−0.682497
$342$	0	0
$343$	−2.35984e6	−1.08305
$344$	0	0
$345$	−2.76523e6	−1.25079
$346$	0	0
$347$	2.26448e6	1.00959	0.504796	−	0.863239i	$-0.331568\pi$
$347$	2.26448e6	1.00959	0.504796	+	0.863239i	$0.331568\pi$
$348$	0	0
$349$	751342.	0.330198	0.165099	−	0.986277i	$-0.447206\pi$
$349$	751342.	0.330198	0.165099	+	0.986277i	$0.447206\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	2.69175e6	1.14973	0.574867	−	0.818247i	$-0.305054\pi$
$353$	2.69175e6	1.14973	0.574867	+	0.818247i	$0.305054\pi$
$354$	0	0
$355$	1.27872e6	0.538523
$356$	0	0
$357$	1.65917e6	0.689001
$358$	0	0
$359$	1.15234e6	0.471892	0.235946	−	0.971766i	$-0.424181\pi$
$359$	1.15234e6	0.471892	0.235946	+	0.971766i	$0.424181\pi$
$360$	0	0
$361$	1.73460e6	0.700539
$362$	0	0
$363$	1.20740e6	0.480931
$364$	0	0
$365$	−4.02634e6	−1.58190
$366$	0	0
$367$	4.70854e6	1.82483	0.912413	−	0.409271i	$-0.134217\pi$
$367$	4.70854e6	1.82483	0.912413	+	0.409271i	$0.134217\pi$
$368$	0	0
$369$	648810.	0.248057
$370$	0	0
$371$	−160608.	−0.0605804
$372$	0	0
$373$	4.79181e6	1.78331	0.891657	−	0.452711i	$-0.149543\pi$
$373$	4.79181e6	1.78331	0.891657	+	0.452711i	$0.149543\pi$
$374$	0	0
$375$	−515484.	−0.189294
$376$	0	0
$377$	445822.	0.161550
$378$	0	0
$379$	−786756.	−0.281347	−0.140673	−	0.990056i	$-0.544927\pi$
$379$	−786756.	−0.281347	−0.140673	+	0.990056i	$0.544927\pi$
$380$	0	0
$381$	−468000.	−0.165171
$382$	0	0
$383$	−1.29100e6	−0.449707	−0.224853	−	0.974393i	$-0.572190\pi$
$383$	−1.29100e6	−0.449707	−0.224853	+	0.974393i	$0.572190\pi$
$384$	0	0
$385$	1.35923e6	0.467349
$386$	0	0
$387$	1.55812e6	0.528838
$388$	0	0
$389$	−3.38929e6	−1.13562	−0.567812	−	0.823158i	$-0.692210\pi$
$389$	−3.38929e6	−1.13562	−0.567812	+	0.823158i	$0.692210\pi$
$390$	0	0
$391$	6.83419e6	2.26071
$392$	0	0
$393$	2.55892e6	0.835747
$394$	0	0
$395$	4.83309e6	1.55859
$396$	0	0
$397$	−1.00899e6	−0.321301	−0.160651	−	0.987011i	$-0.551359\pi$
$397$	−1.00899e6	−0.321301	−0.160651	+	0.987011i	$0.551359\pi$
$398$	0	0
$399$	−2.06842e6	−0.650438
$400$	0	0
$401$	1.10213e6	0.342272	0.171136	−	0.985247i	$-0.445256\pi$
$401$	1.10213e6	0.342272	0.171136	+	0.985247i	$0.445256\pi$
$402$	0	0
$403$	1.51018e6	0.463199
$404$	0	0
$405$	−485514.	−0.147084
$406$	0	0
$407$	−302744.	−0.0905919
$408$	0	0
$409$	−3.18196e6	−0.940559	−0.470280	−	0.882517i	$-0.655847\pi$
$409$	−3.18196e6	−0.940559	−0.470280	+	0.882517i	$0.655847\pi$
$410$	0	0
$411$	−2.08780e6	−0.609656
$412$	0	0
$413$	−159936.	−0.0461393
$414$	0	0
$415$	−5.20753e6	−1.48426
$416$	0	0
$417$	1.21576e6	0.342378
$418$	0	0
$419$	5.56976e6	1.54989	0.774945	−	0.632028i	$-0.217777\pi$
$419$	5.56976e6	1.54989	0.774945	+	0.632028i	$0.217777\pi$
$420$	0	0
$421$	741014.	0.203761	0.101881	−	0.994797i	$-0.467514\pi$
$421$	741014.	0.203761	0.101881	+	0.994797i	$0.467514\pi$
$422$	0	0
$423$	1.04004e6	0.282618
$424$	0	0
$425$	−3.86975e6	−1.03923
$426$	0	0
$427$	−2.82262e6	−0.749175
$428$	0	0
$429$	249444.	0.0654380
$430$	0	0
$431$	−3.54670e6	−0.919667	−0.459834	−	0.888005i	$-0.652091\pi$
$431$	−3.54670e6	−0.919667	−0.459834	+	0.888005i	$0.652091\pi$
$432$	0	0
$433$	−2.35797e6	−0.604391	−0.302195	−	0.953246i	$-0.597719\pi$
$433$	−2.35797e6	−0.604391	−0.302195	+	0.953246i	$0.597719\pi$
$434$	0	0
$435$	1.75691e6	0.445170
$436$	0	0
$437$	−8.51990e6	−2.13418
$438$	0	0
$439$	849352.	0.210342	0.105171	−	0.994454i	$-0.466461\pi$
$439$	849352.	0.210342	0.105171	+	0.994454i	$0.466461\pi$
$440$	0	0
$441$	−345303.	−0.0845481
$442$	0	0
$443$	−6.16044e6	−1.49143	−0.745715	−	0.666265i	$-0.767892\pi$
$443$	−6.16044e6	−1.49143	−0.745715	+	0.666265i	$0.767892\pi$
$444$	0	0
$445$	5.65286e6	1.35322
$446$	0	0
$447$	−705942.	−0.167109
$448$	0	0
$449$	−8.02416e6	−1.87838	−0.939190	−	0.343397i	$-0.888422\pi$
$449$	−8.02416e6	−1.87838	−0.939190	+	0.343397i	$0.888422\pi$
$450$	0	0
$451$	−1.31364e6	−0.304113
$452$	0	0
$453$	−1.18138e6	−0.270485
$454$	0	0
$455$	−1.40067e6	−0.317182
$456$	0	0
$457$	4.85047e6	1.08641	0.543205	−	0.839600i	$-0.317211\pi$
$457$	4.85047e6	1.08641	0.543205	+	0.839600i	$0.317211\pi$
$458$	0	0
$459$	1.19993e6	0.265843
$460$	0	0
$461$	−583810.	−0.127944	−0.0639719	−	0.997952i	$-0.520377\pi$
$461$	−583810.	−0.127944	−0.0639719	+	0.997952i	$0.520377\pi$
$462$	0	0
$463$	−787672.	−0.170763	−0.0853813	−	0.996348i	$-0.527211\pi$
$463$	−787672.	−0.170763	−0.0853813	+	0.996348i	$0.527211\pi$
$464$	0	0
$465$	5.95138e6	1.27639
$466$	0	0
$467$	−7.67629e6	−1.62877	−0.814384	−	0.580326i	$-0.802925\pi$
$467$	−7.67629e6	−1.62877	−0.814384	+	0.580326i	$0.802925\pi$
$468$	0	0
$469$	2.56122e6	0.537668
$470$	0	0
$471$	−1.78227e6	−0.370187
$472$	0	0
$473$	−3.15470e6	−0.648344
$474$	0	0
$475$	4.82425e6	0.981061
$476$	0	0
$477$	−116154.	−0.0233743
$478$	0	0
$479$	2.16406e6	0.430953	0.215476	−	0.976509i	$-0.430870\pi$
$479$	2.16406e6	0.430953	0.215476	+	0.976509i	$0.430870\pi$
$480$	0	0
$481$	311974.	0.0614831
$482$	0	0
$483$	4.18522e6	0.816300
$484$	0	0
$485$	1.29055e7	2.49126
$486$	0	0
$487$	5.35363e6	1.02288	0.511442	−	0.859318i	$-0.329112\pi$
$487$	5.35363e6	1.02288	0.511442	+	0.859318i	$0.329112\pi$
$488$	0	0
$489$	1.71155e6	0.323681
$490$	0	0
$491$	387012.	0.0724470	0.0362235	−	0.999344i	$-0.488467\pi$
$491$	387012.	0.0724470	0.0362235	+	0.999344i	$0.488467\pi$
$492$	0	0
$493$	−4.34215e6	−0.804614
$494$	0	0
$495$	983016.	0.180322
$496$	0	0
$497$	−1.93536e6	−0.351456
$498$	0	0
$499$	1.93856e6	0.348521	0.174260	−	0.984700i	$-0.444247\pi$
$499$	1.93856e6	0.348521	0.174260	+	0.984700i	$0.444247\pi$
$500$	0	0
$501$	−1.22357e6	−0.217788
$502$	0	0
$503$	6.21407e6	1.09511	0.547553	−	0.836771i	$-0.315559\pi$
$503$	6.21407e6	1.09511	0.547553	+	0.836771i	$0.315559\pi$
$504$	0	0
$505$	202020.	0.0352506
$506$	0	0
$507$	−257049.	−0.0444116
$508$	0	0
$509$	1.03214e7	1.76580	0.882902	−	0.469558i	$-0.155587\pi$
$509$	1.03214e7	1.76580	0.882902	+	0.469558i	$0.155587\pi$
$510$	0	0
$511$	6.09392e6	1.03239
$512$	0	0
$513$	−1.49591e6	−0.250964
$514$	0	0
$515$	1.20963e7	2.00972
$516$	0	0
$517$	−2.10576e6	−0.346483
$518$	0	0
$519$	−6.69541e6	−1.09108
$520$	0	0
$521$	−4.25813e6	−0.687266	−0.343633	−	0.939104i	$-0.611658\pi$
$521$	−4.25813e6	−0.687266	−0.343633	+	0.939104i	$0.611658\pi$
$522$	0	0
$523$	7.11960e6	1.13816	0.569078	−	0.822284i	$-0.307300\pi$
$523$	7.11960e6	1.13816	0.569078	+	0.822284i	$0.307300\pi$
$524$	0	0
$525$	−2.36981e6	−0.375245
$526$	0	0
$527$	−1.47087e7	−2.30699
$528$	0	0
$529$	1.08028e7	1.67840
$530$	0	0
$531$	−115668.	−0.0178023
$532$	0	0
$533$	1.35369e6	0.206396
$534$	0	0
$535$	9.46342e6	1.42943
$536$	0	0
$537$	−5.57737e6	−0.834630
$538$	0	0
$539$	699132.	0.103654
$540$	0	0
$541$	9.82971e6	1.44393	0.721967	−	0.691927i	$-0.243238\pi$
$541$	9.82971e6	1.44393	0.721967	+	0.691927i	$0.243238\pi$
$542$	0	0
$543$	5.03246e6	0.732454
$544$	0	0
$545$	−717356.	−0.103453
$546$	0	0
$547$	−5.75751e6	−0.822747	−0.411373	−	0.911467i	$-0.634951\pi$
$547$	−5.75751e6	−0.822747	−0.411373	+	0.911467i	$0.634951\pi$
$548$	0	0
$549$	−2.04136e6	−0.289061
$550$	0	0
$551$	5.41318e6	0.759579
$552$	0	0
$553$	−7.31494e6	−1.01718
$554$	0	0
$555$	1.22944e6	0.169423
$556$	0	0
$557$	3.71523e6	0.507397	0.253698	−	0.967283i	$-0.418353\pi$
$557$	3.71523e6	0.507397	0.253698	+	0.967283i	$0.418353\pi$
$558$	0	0
$559$	3.25088e6	0.440020
$560$	0	0
$561$	−2.42950e6	−0.325919
$562$	0	0
$563$	−1.10492e7	−1.46912	−0.734561	−	0.678542i	$-0.762612\pi$
$563$	−1.10492e7	−1.46912	−0.734561	+	0.678542i	$0.762612\pi$
$564$	0	0
$565$	1.19263e7	1.57175
$566$	0	0
$567$	734832.	0.0959910
$568$	0	0
$569$	−5.87287e6	−0.760448	−0.380224	−	0.924894i	$-0.624153\pi$
$569$	−5.87287e6	−0.760448	−0.380224	+	0.924894i	$0.624153\pi$
$570$	0	0
$571$	−5.16868e6	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$571$	−5.16868e6	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$572$	0	0
$573$	−1.39003e6	−0.176864
$574$	0	0
$575$	−9.76135e6	−1.23123
$576$	0	0
$577$	−5.37369e6	−0.671944	−0.335972	−	0.941872i	$-0.609065\pi$
$577$	−5.37369e6	−0.671944	−0.335972	+	0.941872i	$0.609065\pi$
$578$	0	0
$579$	2.41774e6	0.299718
$580$	0	0
$581$	7.88166e6	0.968674
$582$	0	0
$583$	235176.	0.0286564
$584$	0	0
$585$	−1.01299e6	−0.122381
$586$	0	0
$587$	−1.50540e7	−1.80325	−0.901625	−	0.432519i	$-0.857625\pi$
$587$	−1.50540e7	−1.80325	−0.901625	+	0.432519i	$0.857625\pi$
$588$	0	0
$589$	1.83367e7	2.17787
$590$	0	0
$591$	7.42905e6	0.874912
$592$	0	0
$593$	7.27003e6	0.848984	0.424492	−	0.905432i	$-0.360453\pi$
$593$	7.27003e6	0.848984	0.424492	+	0.905432i	$0.360453\pi$
$594$	0	0
$595$	1.36420e7	1.57975
$596$	0	0
$597$	7.86766e6	0.903461
$598$	0	0
$599$	6.39058e6	0.727735	0.363868	−	0.931451i	$-0.381456\pi$
$599$	6.39058e6	0.727735	0.363868	+	0.931451i	$0.381456\pi$
$600$	0	0
$601$	7.89990e6	0.892145	0.446072	−	0.894997i	$-0.352822\pi$
$601$	7.89990e6	0.892145	0.446072	+	0.894997i	$0.352822\pi$
$602$	0	0
$603$	1.85231e6	0.207453
$604$	0	0
$605$	9.92747e6	1.10268
$606$	0	0
$607$	−1.31701e6	−0.145083	−0.0725415	−	0.997365i	$-0.523111\pi$
$607$	−1.31701e6	−0.145083	−0.0725415	+	0.997365i	$0.523111\pi$
$608$	0	0
$609$	−2.65910e6	−0.290531
$610$	0	0
$611$	2.16996e6	0.235152
$612$	0	0
$613$	−5.47284e6	−0.588250	−0.294125	−	0.955767i	$-0.595028\pi$
$613$	−5.47284e6	−0.588250	−0.294125	+	0.955767i	$0.595028\pi$
$614$	0	0
$615$	5.33466e6	0.568747
$616$	0	0
$617$	1.28342e7	1.35724	0.678620	−	0.734490i	$-0.262578\pi$
$617$	1.28342e7	1.35724	0.678620	+	0.734490i	$0.262578\pi$
$618$	0	0
$619$	−1.62292e6	−0.170243	−0.0851215	−	0.996371i	$-0.527128\pi$
$619$	−1.62292e6	−0.170243	−0.0851215	+	0.996371i	$0.527128\pi$
$620$	0	0
$621$	3.02681e6	0.314960
$622$	0	0
$623$	−8.55568e6	−0.883150
$624$	0	0
$625$	−1.15853e7	−1.18633
$626$	0	0
$627$	3.02875e6	0.307677
$628$	0	0
$629$	−3.03852e6	−0.306221
$630$	0	0
$631$	−2.71568e6	−0.271522	−0.135761	−	0.990742i	$-0.543348\pi$
$631$	−2.71568e6	−0.271522	−0.135761	+	0.990742i	$0.543348\pi$
$632$	0	0
$633$	3.68888e6	0.365920
$634$	0	0
$635$	−3.84800e6	−0.378705
$636$	0	0
$637$	−720447.	−0.0703483
$638$	0	0
$639$	−1.39968e6	−0.135605
$640$	0	0
$641$	−1.04180e7	−1.00147	−0.500736	−	0.865600i	$-0.666937\pi$
$641$	−1.04180e7	−1.00147	−0.500736	+	0.865600i	$0.666937\pi$
$642$	0	0
$643$	−4.64510e6	−0.443065	−0.221533	−	0.975153i	$-0.571106\pi$
$643$	−4.64510e6	−0.443065	−0.221533	+	0.975153i	$0.571106\pi$
$644$	0	0
$645$	1.28112e7	1.21252
$646$	0	0
$647$	7.10745e6	0.667503	0.333751	−	0.942661i	$-0.391685\pi$
$647$	7.10745e6	0.667503	0.333751	+	0.942661i	$0.391685\pi$
$648$	0	0
$649$	234192.	0.0218253
$650$	0	0
$651$	−9.00749e6	−0.833012
$652$	0	0
$653$	−9.26906e6	−0.850653	−0.425327	−	0.905040i	$-0.639841\pi$
$653$	−9.26906e6	−0.850653	−0.425327	+	0.905040i	$0.639841\pi$
$654$	0	0
$655$	2.10400e7	1.91621
$656$	0	0
$657$	4.40721e6	0.398337
$658$	0	0
$659$	8.23630e6	0.738786	0.369393	−	0.929273i	$-0.379566\pi$
$659$	8.23630e6	0.738786	0.369393	+	0.929273i	$0.379566\pi$
$660$	0	0
$661$	−1.19099e7	−1.06024	−0.530122	−	0.847921i	$-0.677854\pi$
$661$	−1.19099e7	−1.06024	−0.530122	+	0.847921i	$0.677854\pi$
$662$	0	0
$663$	2.50357e6	0.221195
$664$	0	0
$665$	−1.70070e7	−1.49133
$666$	0	0
$667$	−1.09530e7	−0.953274
$668$	0	0
$669$	4.39423e6	0.379593
$670$	0	0
$671$	4.13313e6	0.354383
$672$	0	0
$673$	−7.15869e6	−0.609250	−0.304625	−	0.952472i	$-0.598531\pi$
$673$	−7.15869e6	−0.609250	−0.304625	+	0.952472i	$0.598531\pi$
$674$	0	0
$675$	−1.71388e6	−0.144784
$676$	0	0
$677$	7.95484e6	0.667052	0.333526	−	0.942741i	$-0.391762\pi$
$677$	7.95484e6	0.667052	0.333526	+	0.942741i	$0.391762\pi$
$678$	0	0
$679$	−1.95326e7	−1.62587
$680$	0	0
$681$	−9.29797e6	−0.768282
$682$	0	0
$683$	−6.35956e6	−0.521645	−0.260822	−	0.965387i	$-0.583994\pi$
$683$	−6.35956e6	−0.521645	−0.260822	+	0.965387i	$0.583994\pi$
$684$	0	0
$685$	−1.71664e7	−1.39782
$686$	0	0
$687$	7.05638e6	0.570414
$688$	0	0
$689$	−242346.	−0.0194486
$690$	0	0
$691$	−1.57740e7	−1.25674	−0.628372	−	0.777913i	$-0.716279\pi$
$691$	−1.57740e7	−1.25674	−0.628372	+	0.777913i	$0.716279\pi$
$692$	0	0
$693$	−1.48781e6	−0.117683
$694$	0	0
$695$	9.99622e6	0.785007
$696$	0	0
$697$	−1.31845e7	−1.02797
$698$	0	0
$699$	3.29636e6	0.255177
$700$	0	0
$701$	−1.23423e7	−0.948638	−0.474319	−	0.880353i	$-0.657306\pi$
$701$	−1.23423e7	−0.948638	−0.474319	+	0.880353i	$0.657306\pi$
$702$	0	0
$703$	3.78799e6	0.289082
$704$	0	0
$705$	8.55144e6	0.647987
$706$	0	0
$707$	−305760.	−0.0230055
$708$	0	0
$709$	6.72837e6	0.502683	0.251342	−	0.967898i	$-0.419128\pi$
$709$	6.72837e6	0.502683	0.251342	+	0.967898i	$0.419128\pi$
$710$	0	0
$711$	−5.29027e6	−0.392468
$712$	0	0
$713$	−3.71023e7	−2.73323
$714$	0	0
$715$	2.05098e6	0.150037
$716$	0	0
$717$	1.36475e7	0.991416
$718$	0	0
$719$	−1.72837e7	−1.24685	−0.623425	−	0.781883i	$-0.714259\pi$
$719$	−1.72837e7	−1.24685	−0.623425	+	0.781883i	$0.714259\pi$
$720$	0	0
$721$	−1.83080e7	−1.31160
$722$	0	0
$723$	1.38094e7	0.982495
$724$	0	0
$725$	6.20194e6	0.438210
$726$	0	0
$727$	−3.18889e6	−0.223771	−0.111885	−	0.993721i	$-0.535689\pi$
$727$	−3.18889e6	−0.223771	−0.111885	+	0.993721i	$0.535689\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.16625e7	−2.19155
$732$	0	0
$733$	1.09760e7	0.754546	0.377273	−	0.926102i	$-0.376862\pi$
$733$	1.09760e7	0.754546	0.377273	+	0.926102i	$0.376862\pi$
$734$	0	0
$735$	−2.83916e6	−0.193852
$736$	0	0
$737$	−3.75035e6	−0.254333
$738$	0	0
$739$	1.92371e7	1.29577	0.647885	−	0.761739i	$-0.275654\pi$
$739$	1.92371e7	1.29577	0.647885	+	0.761739i	$0.275654\pi$
$740$	0	0
$741$	−3.12109e6	−0.208815
$742$	0	0
$743$	−5.44474e6	−0.361830	−0.180915	−	0.983499i	$-0.557906\pi$
$743$	−5.44474e6	−0.361830	−0.180915	+	0.983499i	$0.557906\pi$
$744$	0	0
$745$	−5.80441e6	−0.383149
$746$	0	0
$747$	5.70013e6	0.373752
$748$	0	0
$749$	−1.43230e7	−0.932888
$750$	0	0
$751$	−1.51686e7	−0.981397	−0.490698	−	0.871330i	$-0.663258\pi$
$751$	−1.51686e7	−0.981397	−0.490698	+	0.871330i	$0.663258\pi$
$752$	0	0
$753$	−3.77730e6	−0.242769
$754$	0	0
$755$	−9.71354e6	−0.620169
$756$	0	0
$757$	1.03850e6	0.0658670	0.0329335	−	0.999458i	$-0.489515\pi$
$757$	1.03850e6	0.0658670	0.0329335	+	0.999458i	$0.489515\pi$
$758$	0	0
$759$	−6.12835e6	−0.386135
$760$	0	0
$761$	−1.47627e7	−0.924070	−0.462035	−	0.886862i	$-0.652881\pi$
$761$	−1.47627e7	−0.924070	−0.462035	+	0.886862i	$0.652881\pi$
$762$	0	0
$763$	1.08573e6	0.0675165
$764$	0	0
$765$	9.86612e6	0.609527
$766$	0	0
$767$	−241332.	−0.0148124
$768$	0	0
$769$	−6.77571e6	−0.413180	−0.206590	−	0.978428i	$-0.566237\pi$
$769$	−6.77571e6	−0.413180	−0.206590	+	0.978428i	$0.566237\pi$
$770$	0	0
$771$	−4.05506e6	−0.245675
$772$	0	0
$773$	−2.56935e6	−0.154659	−0.0773295	−	0.997006i	$-0.524639\pi$
$773$	−2.56935e6	−0.154659	−0.0773295	+	0.997006i	$0.524639\pi$
$774$	0	0
$775$	2.10085e7	1.25644
$776$	0	0
$777$	−1.86077e6	−0.110571
$778$	0	0
$779$	1.64365e7	0.970435
$780$	0	0
$781$	2.83392e6	0.166249
$782$	0	0
$783$	−1.92310e6	−0.112098
$784$	0	0
$785$	−1.46542e7	−0.848767
$786$	0	0
$787$	−2.22659e7	−1.28146	−0.640729	−	0.767768i	$-0.721368\pi$
$787$	−2.22659e7	−1.28146	−0.640729	+	0.767768i	$0.721368\pi$
$788$	0	0
$789$	−3.31495e6	−0.189577
$790$	0	0
$791$	−1.80506e7	−1.02577
$792$	0	0
$793$	−4.25914e6	−0.240513
$794$	0	0
$795$	−955044.	−0.0535927
$796$	0	0
$797$	201038.	0.0112107	0.00560535	−	0.999984i	$-0.498216\pi$
$797$	201038.	0.0112107	0.00560535	+	0.999984i	$0.498216\pi$
$798$	0	0
$799$	−2.11346e7	−1.17119
$800$	0	0
$801$	−6.18759e6	−0.340753
$802$	0	0
$803$	−8.92324e6	−0.488353
$804$	0	0
$805$	3.44118e7	1.87162
$806$	0	0
$807$	1.48913e7	0.804915
$808$	0	0
$809$	−9.77017e6	−0.524844	−0.262422	−	0.964953i	$-0.584521\pi$
$809$	−9.77017e6	−0.524844	−0.262422	+	0.964953i	$0.584521\pi$
$810$	0	0
$811$	−4.71324e6	−0.251633	−0.125816	−	0.992054i	$-0.540155\pi$
$811$	−4.71324e6	−0.251633	−0.125816	+	0.992054i	$0.540155\pi$
$812$	0	0
$813$	−1.80202e7	−0.956168
$814$	0	0
$815$	1.40727e7	0.742137
$816$	0	0
$817$	3.94723e7	2.06889
$818$	0	0
$819$	1.53317e6	0.0798693
$820$	0	0
$821$	2.87700e7	1.48964	0.744822	−	0.667264i	$-0.232535\pi$
$821$	2.87700e7	1.48964	0.744822	+	0.667264i	$0.232535\pi$
$822$	0	0
$823$	2.57812e7	1.32679	0.663396	−	0.748268i	$-0.269114\pi$
$823$	2.57812e7	1.32679	0.663396	+	0.748268i	$0.269114\pi$
$824$	0	0
$825$	3.47008e6	0.177502
$826$	0	0
$827$	−2.31260e7	−1.17581	−0.587905	−	0.808930i	$-0.700047\pi$
$827$	−2.31260e7	−1.17581	−0.587905	+	0.808930i	$0.700047\pi$
$828$	0	0
$829$	6.77051e6	0.342165	0.171082	−	0.985257i	$-0.445274\pi$
$829$	6.77051e6	0.342165	0.171082	+	0.985257i	$0.445274\pi$
$830$	0	0
$831$	1.03858e7	0.521720
$832$	0	0
$833$	7.01690e6	0.350375
$834$	0	0
$835$	−1.00604e7	−0.499345
$836$	0	0
$837$	−6.51434e6	−0.321408
$838$	0	0
$839$	3.47592e6	0.170477	0.0852383	−	0.996361i	$-0.472835\pi$
$839$	3.47592e6	0.170477	0.0852383	+	0.996361i	$0.472835\pi$
$840$	0	0
$841$	−1.35521e7	−0.660719
$842$	0	0
$843$	1.58478e7	0.768070
$844$	0	0
$845$	−2.11351e6	−0.101827
$846$	0	0
$847$	−1.50254e7	−0.719642
$848$	0	0
$849$	1.60671e7	0.765010
$850$	0	0
$851$	−7.66459e6	−0.362798
$852$	0	0
$853$	−2.01423e7	−0.947845	−0.473922	−	0.880567i	$-0.657162\pi$
$853$	−2.01423e7	−0.947845	−0.473922	+	0.880567i	$0.657162\pi$
$854$	0	0
$855$	−1.22997e7	−0.575412
$856$	0	0
$857$	−5.37441e6	−0.249965	−0.124982	−	0.992159i	$-0.539887\pi$
$857$	−5.37441e6	−0.249965	−0.124982	+	0.992159i	$0.539887\pi$
$858$	0	0
$859$	1.01896e7	0.471166	0.235583	−	0.971854i	$-0.424300\pi$
$859$	1.01896e7	0.471166	0.235583	+	0.971854i	$0.424300\pi$
$860$	0	0
$861$	−8.07408e6	−0.371181
$862$	0	0
$863$	−1.51496e7	−0.692428	−0.346214	−	0.938155i	$-0.612533\pi$
$863$	−1.51496e7	−0.692428	−0.346214	+	0.938155i	$0.612533\pi$
$864$	0	0
$865$	−5.50511e7	−2.50165
$866$	0	0
$867$	−1.16051e7	−0.524327
$868$	0	0
$869$	1.07112e7	0.481158
$870$	0	0
$871$	3.86469e6	0.172611
$872$	0	0
$873$	−1.41262e7	−0.627323
$874$	0	0
$875$	6.41491e6	0.283250
$876$	0	0
$877$	−7.57856e6	−0.332727	−0.166363	−	0.986065i	$-0.553202\pi$
$877$	−7.57856e6	−0.332727	−0.166363	+	0.986065i	$0.553202\pi$
$878$	0	0
$879$	−1.55569e6	−0.0679125
$880$	0	0
$881$	−3.42977e7	−1.48876	−0.744380	−	0.667756i	$-0.767255\pi$
$881$	−3.42977e7	−1.48876	−0.744380	+	0.667756i	$0.767255\pi$
$882$	0	0
$883$	1.20211e7	0.518851	0.259426	−	0.965763i	$-0.416467\pi$
$883$	1.20211e7	0.518851	0.259426	+	0.965763i	$0.416467\pi$
$884$	0	0
$885$	−951048.	−0.0408173
$886$	0	0
$887$	1.36850e7	0.584031	0.292016	−	0.956414i	$-0.405674\pi$
$887$	1.36850e7	0.584031	0.292016	+	0.956414i	$0.405674\pi$
$888$	0	0
$889$	5.82400e6	0.247154
$890$	0	0
$891$	−1.07600e6	−0.0454067
$892$	0	0
$893$	2.63477e7	1.10564
$894$	0	0
$895$	−4.58584e7	−1.91364
$896$	0	0
$897$	6.31519e6	0.262063
$898$	0	0
$899$	2.35732e7	0.972789
$900$	0	0
$901$	2.36036e6	0.0968650
$902$	0	0
$903$	−1.93899e7	−0.791327
$904$	0	0
$905$	4.13780e7	1.67938
$906$	0	0
$907$	5.48096e6	0.221227	0.110614	−	0.993863i	$-0.464718\pi$
$907$	5.48096e6	0.221227	0.110614	+	0.993863i	$0.464718\pi$
$908$	0	0
$909$	−221130.	−0.00887642
$910$	0	0
$911$	6.16554e6	0.246136	0.123068	−	0.992398i	$-0.460727\pi$
$911$	6.16554e6	0.246136	0.123068	+	0.992398i	$0.460727\pi$
$912$	0	0
$913$	−1.15410e7	−0.458212
$914$	0	0
$915$	−1.67845e7	−0.662760
$916$	0	0
$917$	−3.18443e7	−1.25057
$918$	0	0
$919$	2.59526e7	1.01366	0.506829	−	0.862047i	$-0.330818\pi$
$919$	2.59526e7	1.01366	0.506829	+	0.862047i	$0.330818\pi$
$920$	0	0
$921$	9.56574e6	0.371595
$922$	0	0
$923$	−2.92032e6	−0.112830
$924$	0	0
$925$	4.33995e6	0.166775
$926$	0	0
$927$	−1.32406e7	−0.506067
$928$	0	0
$929$	−4.08585e7	−1.55326	−0.776628	−	0.629959i	$-0.783072\pi$
$929$	−4.08585e7	−1.55326	−0.776628	+	0.629959i	$0.783072\pi$
$930$	0	0
$931$	−8.74768e6	−0.330764
$932$	0	0
$933$	−3.41690e6	−0.128508
$934$	0	0
$935$	−1.99759e7	−0.747268
$936$	0	0
$937$	−9.29695e6	−0.345933	−0.172966	−	0.984928i	$-0.555335\pi$
$937$	−9.29695e6	−0.345933	−0.172966	+	0.984928i	$0.555335\pi$
$938$	0	0
$939$	2.85717e7	1.05748
$940$	0	0
$941$	2.57530e7	0.948100	0.474050	−	0.880498i	$-0.342792\pi$
$941$	2.57530e7	0.948100	0.474050	+	0.880498i	$0.342792\pi$
$942$	0	0
$943$	−3.32575e7	−1.21790
$944$	0	0
$945$	6.04195e6	0.220089
$946$	0	0
$947$	2.11004e7	0.764567	0.382284	−	0.924045i	$-0.375138\pi$
$947$	2.11004e7	0.764567	0.382284	+	0.924045i	$0.375138\pi$
$948$	0	0
$949$	9.19529e6	0.331436
$950$	0	0
$951$	−8.28358e6	−0.297007
$952$	0	0
$953$	−2.22437e7	−0.793368	−0.396684	−	0.917955i	$-0.629839\pi$
$953$	−2.22437e7	−0.793368	−0.396684	+	0.917955i	$0.629839\pi$
$954$	0	0
$955$	−1.14292e7	−0.405514
$956$	0	0
$957$	3.89369e6	0.137430
$958$	0	0
$959$	2.59815e7	0.912259
$960$	0	0
$961$	5.12229e7	1.78919
$962$	0	0
$963$	−1.03586e7	−0.359944
$964$	0	0
$965$	1.98792e7	0.687196
$966$	0	0
$967$	−28064.0	−0.000965125 0	−0.000482562	−	1.00000i	$-0.500154\pi$
$967$	−28064.0	−0.000965125 0	−0.000482562		1.00000i	$0.500154\pi$
$968$	0	0
$969$	3.03983e7	1.04002
$970$	0	0
$971$	−5.14613e7	−1.75159	−0.875795	−	0.482683i	$-0.839662\pi$
$971$	−5.14613e7	−1.75159	−0.875795	+	0.482683i	$0.839662\pi$
$972$	0	0
$973$	−1.51294e7	−0.512318
$974$	0	0
$975$	−3.57587e6	−0.120468
$976$	0	0
$977$	5.16124e7	1.72989	0.864943	−	0.501869i	$-0.167354\pi$
$977$	5.16124e7	1.72989	0.864943	+	0.501869i	$0.167354\pi$
$978$	0	0
$979$	1.25280e7	0.417757
$980$	0	0
$981$	785214.	0.0260505
$982$	0	0
$983$	2.78764e7	0.920136	0.460068	−	0.887884i	$-0.347825\pi$
$983$	2.78764e7	0.920136	0.460068	+	0.887884i	$0.347825\pi$
$984$	0	0
$985$	6.10833e7	2.00600
$986$	0	0
$987$	−1.29427e7	−0.422895
$988$	0	0
$989$	−7.98679e7	−2.59646
$990$	0	0
$991$	−9.30712e6	−0.301045	−0.150522	−	0.988607i	$-0.548096\pi$
$991$	−9.30712e6	−0.301045	−0.150522	+	0.988607i	$0.548096\pi$
$992$	0	0
$993$	−3.70638e6	−0.119283
$994$	0	0
$995$	6.46896e7	2.07146
$996$	0	0
$997$	3.65964e7	1.16600	0.583002	−	0.812471i	$-0.301878\pi$
$997$	3.65964e7	1.16600	0.583002	+	0.812471i	$0.301878\pi$
$998$	0	0
$999$	−1.34573e6	−0.0426624

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	624.6.a.b.1.1		1
4.3	odd	2		39.6.a.a.1.1	✓	1
12.11	even	2		117.6.a.a.1.1		1
20.19	odd	2		975.6.a.a.1.1		1
52.51	odd	2		507.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.6.a.a.1.1	✓	1	4.3	odd	2
117.6.a.a.1.1		1	12.11	even	2
507.6.a.a.1.1		1	52.51	odd	2
624.6.a.b.1.1		1	1.1	even	1	trivial
975.6.a.a.1.1		1	20.19	odd	2

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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	624.6.a.b.1.1

Level	$624$
Weight	$6$
Character	624.1
Self dual	yes
Analytic conductor	$100.080$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$