LMFDB - Embedded newform 546.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,6,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	546.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:	$87.5695656179$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -54.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -54.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -216.000 q^{10} -192.000 q^{11} +144.000 q^{12} +169.000 q^{13} +196.000 q^{14} -486.000 q^{15} +256.000 q^{16} -1422.00 q^{17} +324.000 q^{18} +1748.00 q^{19} -864.000 q^{20} +441.000 q^{21} -768.000 q^{22} -3792.00 q^{23} +576.000 q^{24} -209.000 q^{25} +676.000 q^{26} +729.000 q^{27} +784.000 q^{28} -954.000 q^{29} -1944.00 q^{30} -568.000 q^{31} +1024.00 q^{32} -1728.00 q^{33} -5688.00 q^{34} -2646.00 q^{35} +1296.00 q^{36} +7886.00 q^{37} +6992.00 q^{38} +1521.00 q^{39} -3456.00 q^{40} -14802.0 q^{41} +1764.00 q^{42} +4964.00 q^{43} -3072.00 q^{44} -4374.00 q^{45} -15168.0 q^{46} -18948.0 q^{47} +2304.00 q^{48} +2401.00 q^{49} -836.000 q^{50} -12798.0 q^{51} +2704.00 q^{52} -426.000 q^{53} +2916.00 q^{54} +10368.0 q^{55} +3136.00 q^{56} +15732.0 q^{57} -3816.00 q^{58} +34872.0 q^{59} -7776.00 q^{60} -25618.0 q^{61} -2272.00 q^{62} +3969.00 q^{63} +4096.00 q^{64} -9126.00 q^{65} -6912.00 q^{66} -67060.0 q^{67} -22752.0 q^{68} -34128.0 q^{69} -10584.0 q^{70} -28428.0 q^{71} +5184.00 q^{72} -22894.0 q^{73} +31544.0 q^{74} -1881.00 q^{75} +27968.0 q^{76} -9408.00 q^{77} +6084.00 q^{78} -1408.00 q^{79} -13824.0 q^{80} +6561.00 q^{81} -59208.0 q^{82} -17304.0 q^{83} +7056.00 q^{84} +76788.0 q^{85} +19856.0 q^{86} -8586.00 q^{87} -12288.0 q^{88} -93690.0 q^{89} -17496.0 q^{90} +8281.00 q^{91} -60672.0 q^{92} -5112.00 q^{93} -75792.0 q^{94} -94392.0 q^{95} +9216.00 q^{96} +16826.0 q^{97} +9604.00 q^{98} -15552.0 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−54.0000	−0.965981	−0.482991	−	0.875625i	$-0.660450\pi$
$5$	−54.0000	−0.965981	−0.482991	+	0.875625i	$0.660450\pi$
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−216.000	−0.683052
$11$	−192.000	−0.478431	−0.239216	−	0.970966i	$-0.576890\pi$
$11$	−192.000	−0.478431	−0.239216	+	0.970966i	$0.576890\pi$
$12$	144.000	0.288675
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	196.000	0.267261
$15$	−486.000	−0.557710
$16$	256.000	0.250000
$17$	−1422.00	−1.19338	−0.596688	−	0.802473i	$-0.703517\pi$
$17$	−1422.00	−1.19338	−0.596688	+	0.802473i	$0.703517\pi$
$18$	324.000	0.235702
$19$	1748.00	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	1748.00	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	−864.000	−0.482991
$21$	441.000	0.218218
$22$	−768.000	−0.338302
$23$	−3792.00	−1.49468	−0.747341	−	0.664441i	$-0.768670\pi$
$23$	−3792.00	−1.49468	−0.747341	+	0.664441i	$0.768670\pi$
$24$	576.000	0.204124
$25$	−209.000	−0.0668800
$26$	676.000	0.196116
$27$	729.000	0.192450
$28$	784.000	0.188982
$29$	−954.000	−0.210646	−0.105323	−	0.994438i	$-0.533588\pi$
$29$	−954.000	−0.210646	−0.105323	+	0.994438i	$0.533588\pi$
$30$	−1944.00	−0.394360
$31$	−568.000	−0.106156	−0.0530779	−	0.998590i	$-0.516903\pi$
$31$	−568.000	−0.106156	−0.0530779	+	0.998590i	$0.516903\pi$
$32$	1024.00	0.176777
$33$	−1728.00	−0.276222
$34$	−5688.00	−0.843844
$35$	−2646.00	−0.365107
$36$	1296.00	0.166667
$37$	7886.00	0.947005	0.473503	−	0.880792i	$-0.342989\pi$
$37$	7886.00	0.947005	0.473503	+	0.880792i	$0.342989\pi$
$38$	6992.00	0.785493
$39$	1521.00	0.160128
$40$	−3456.00	−0.341526
$41$	−14802.0	−1.37518	−0.687592	−	0.726097i	$-0.741332\pi$
$41$	−14802.0	−1.37518	−0.687592	+	0.726097i	$0.741332\pi$
$42$	1764.00	0.154303
$43$	4964.00	0.409412	0.204706	−	0.978823i	$-0.434376\pi$
$43$	4964.00	0.409412	0.204706	+	0.978823i	$0.434376\pi$
$44$	−3072.00	−0.239216
$45$	−4374.00	−0.321994
$46$	−15168.0	−1.05690
$47$	−18948.0	−1.25118	−0.625588	−	0.780153i	$-0.715141\pi$
$47$	−18948.0	−1.25118	−0.625588	+	0.780153i	$0.715141\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	−836.000	−0.0472913
$51$	−12798.0	−0.688996
$52$	2704.00	0.138675
$53$	−426.000	−0.0208315	−0.0104157	−	0.999946i	$-0.503315\pi$
$53$	−426.000	−0.0208315	−0.0104157	+	0.999946i	$0.503315\pi$
$54$	2916.00	0.136083
$55$	10368.0	0.462156
$56$	3136.00	0.133631
$57$	15732.0	0.641353
$58$	−3816.00	−0.148949
$59$	34872.0	1.30421	0.652104	−	0.758130i	$-0.273887\pi$
$59$	34872.0	1.30421	0.652104	+	0.758130i	$0.273887\pi$
$60$	−7776.00	−0.278855
$61$	−25618.0	−0.881497	−0.440748	−	0.897631i	$-0.645287\pi$
$61$	−25618.0	−0.881497	−0.440748	+	0.897631i	$0.645287\pi$
$62$	−2272.00	−0.0750636
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	−9126.00	−0.267915
$66$	−6912.00	−0.195319
$67$	−67060.0	−1.82506	−0.912529	−	0.409013i	$-0.865873\pi$
$67$	−67060.0	−1.82506	−0.912529	+	0.409013i	$0.865873\pi$
$68$	−22752.0	−0.596688
$69$	−34128.0	−0.862955
$70$	−10584.0	−0.258169
$71$	−28428.0	−0.669269	−0.334634	−	0.942348i	$-0.608613\pi$
$71$	−28428.0	−0.669269	−0.334634	+	0.942348i	$0.608613\pi$
$72$	5184.00	0.117851
$73$	−22894.0	−0.502822	−0.251411	−	0.967880i	$-0.580895\pi$
$73$	−22894.0	−0.502822	−0.251411	+	0.967880i	$0.580895\pi$
$74$	31544.0	0.669634
$75$	−1881.00	−0.0386132
$76$	27968.0	0.555428
$77$	−9408.00	−0.180830
$78$	6084.00	0.113228
$79$	−1408.00	−0.0253825	−0.0126913	−	0.999919i	$-0.504040\pi$
$79$	−1408.00	−0.0253825	−0.0126913	+	0.999919i	$0.504040\pi$
$80$	−13824.0	−0.241495
$81$	6561.00	0.111111
$82$	−59208.0	−0.972402
$83$	−17304.0	−0.275709	−0.137855	−	0.990452i	$-0.544021\pi$
$83$	−17304.0	−0.275709	−0.137855	+	0.990452i	$0.544021\pi$
$84$	7056.00	0.109109
$85$	76788.0	1.15278
$86$	19856.0	0.289498
$87$	−8586.00	−0.121617
$88$	−12288.0	−0.169151
$89$	−93690.0	−1.25377	−0.626886	−	0.779111i	$-0.715671\pi$
$89$	−93690.0	−1.25377	−0.626886	+	0.779111i	$0.715671\pi$
$90$	−17496.0	−0.227684
$91$	8281.00	0.104828
$92$	−60672.0	−0.747341
$93$	−5112.00	−0.0612891
$94$	−75792.0	−0.884716
$95$	−94392.0	−1.07307
$96$	9216.00	0.102062
$97$	16826.0	0.181573	0.0907866	−	0.995870i	$-0.471062\pi$
$97$	16826.0	0.181573	0.0907866	+	0.995870i	$0.471062\pi$
$98$	9604.00	0.101015
$99$	−15552.0	−0.159477
$100$	−3344.00	−0.0334400
$101$	−129090.	−1.25918	−0.629592	−	0.776926i	$-0.716778\pi$
$101$	−129090.	−1.25918	−0.629592	+	0.776926i	$0.716778\pi$
$102$	−51192.0	−0.487194
$103$	−5536.00	−0.0514166	−0.0257083	−	0.999669i	$-0.508184\pi$
$103$	−5536.00	−0.0514166	−0.0257083	+	0.999669i	$0.508184\pi$
$104$	10816.0	0.0980581
$105$	−23814.0	−0.210794
$106$	−1704.00	−0.0147301
$107$	−201996.	−1.70562	−0.852812	−	0.522218i	$-0.825105\pi$
$107$	−201996.	−1.70562	−0.852812	+	0.522218i	$0.825105\pi$
$108$	11664.0	0.0962250
$109$	32510.0	0.262090	0.131045	−	0.991376i	$-0.458167\pi$
$109$	32510.0	0.262090	0.131045	+	0.991376i	$0.458167\pi$
$110$	41472.0	0.326793
$111$	70974.0	0.546754
$112$	12544.0	0.0944911
$113$	−36918.0	−0.271983	−0.135992	−	0.990710i	$-0.543422\pi$
$113$	−36918.0	−0.271983	−0.135992	+	0.990710i	$0.543422\pi$
$114$	62928.0	0.453505
$115$	204768.	1.44383
$116$	−15264.0	−0.105323
$117$	13689.0	0.0924500
$118$	139488.	0.922214
$119$	−69678.0	−0.451054
$120$	−31104.0	−0.197180
$121$	−124187.	−0.771104
$122$	−102472.	−0.623312
$123$	−133218.	−0.793963
$124$	−9088.00	−0.0530779
$125$	180036.	1.03059
$126$	15876.0	0.0890871
$127$	−169792.	−0.934131	−0.467066	−	0.884223i	$-0.654689\pi$
$127$	−169792.	−0.934131	−0.467066	+	0.884223i	$0.654689\pi$
$128$	16384.0	0.0883883
$129$	44676.0	0.236374
$130$	−36504.0	−0.189445
$131$	177660.	0.904506	0.452253	−	0.891890i	$-0.350620\pi$
$131$	177660.	0.904506	0.452253	+	0.891890i	$0.350620\pi$
$132$	−27648.0	−0.138111
$133$	85652.0	0.419864
$134$	−268240.	−1.29051
$135$	−39366.0	−0.185903
$136$	−91008.0	−0.421922
$137$	141582.	0.644476	0.322238	−	0.946659i	$-0.395565\pi$
$137$	141582.	0.644476	0.322238	+	0.946659i	$0.395565\pi$
$138$	−136512.	−0.610201
$139$	282356.	1.23954	0.619769	−	0.784784i	$-0.287226\pi$
$139$	282356.	1.23954	0.619769	+	0.784784i	$0.287226\pi$
$140$	−42336.0	−0.182553
$141$	−170532.	−0.722367
$142$	−113712.	−0.473244
$143$	−32448.0	−0.132693
$144$	20736.0	0.0833333
$145$	51516.0	0.203480
$146$	−91576.0	−0.355549
$147$	21609.0	0.0824786
$148$	126176.	0.473503
$149$	203682.	0.751600	0.375800	−	0.926701i	$-0.377368\pi$
$149$	203682.	0.751600	0.375800	+	0.926701i	$0.377368\pi$
$150$	−7524.00	−0.0273036
$151$	114368.	0.408190	0.204095	−	0.978951i	$-0.434575\pi$
$151$	114368.	0.408190	0.204095	+	0.978951i	$0.434575\pi$
$152$	111872.	0.392747
$153$	−115182.	−0.397792
$154$	−37632.0	−0.127866
$155$	30672.0	0.102545
$156$	24336.0	0.0800641
$157$	48734.0	0.157791	0.0788956	−	0.996883i	$-0.474861\pi$
$157$	48734.0	0.157791	0.0788956	+	0.996883i	$0.474861\pi$
$158$	−5632.00	−0.0179482
$159$	−3834.00	−0.0120271
$160$	−55296.0	−0.170763
$161$	−185808.	−0.564937
$162$	26244.0	0.0785674
$163$	−18124.0	−0.0534300	−0.0267150	−	0.999643i	$-0.508505\pi$
$163$	−18124.0	−0.0534300	−0.0267150	+	0.999643i	$0.508505\pi$
$164$	−236832.	−0.687592
$165$	93312.0	0.266826
$166$	−69216.0	−0.194956
$167$	261300.	0.725017	0.362509	−	0.931980i	$-0.381920\pi$
$167$	261300.	0.725017	0.362509	+	0.931980i	$0.381920\pi$
$168$	28224.0	0.0771517
$169$	28561.0	0.0769231
$170$	307152.	0.815138
$171$	141588.	0.370285
$172$	79424.0	0.204706
$173$	550710.	1.39897	0.699484	−	0.714648i	$-0.253413\pi$
$173$	550710.	1.39897	0.699484	+	0.714648i	$0.253413\pi$
$174$	−34344.0	−0.0859959
$175$	−10241.0	−0.0252783
$176$	−49152.0	−0.119608
$177$	313848.	0.752985
$178$	−374760.	−0.886550
$179$	−264444.	−0.616881	−0.308440	−	0.951244i	$-0.599807\pi$
$179$	−264444.	−0.616881	−0.308440	+	0.951244i	$0.599807\pi$
$180$	−69984.0	−0.160997
$181$	−31930.0	−0.0724440	−0.0362220	−	0.999344i	$-0.511532\pi$
$181$	−31930.0	−0.0724440	−0.0362220	+	0.999344i	$0.511532\pi$
$182$	33124.0	0.0741249
$183$	−230562.	−0.508932
$184$	−242688.	−0.528450
$185$	−425844.	−0.914790
$186$	−20448.0	−0.0433380
$187$	273024.	0.570948
$188$	−303168.	−0.625588
$189$	35721.0	0.0727393
$190$	−377568.	−0.758772
$191$	−210240.	−0.416996	−0.208498	−	0.978023i	$-0.566857\pi$
$191$	−210240.	−0.416996	−0.208498	+	0.978023i	$0.566857\pi$
$192$	36864.0	0.0721688
$193$	−488062.	−0.943152	−0.471576	−	0.881825i	$-0.656315\pi$
$193$	−488062.	−0.943152	−0.471576	+	0.881825i	$0.656315\pi$
$194$	67304.0	0.128392
$195$	−82134.0	−0.154681
$196$	38416.0	0.0714286
$197$	−165558.	−0.303938	−0.151969	−	0.988385i	$-0.548561\pi$
$197$	−165558.	−0.303938	−0.151969	+	0.988385i	$0.548561\pi$
$198$	−62208.0	−0.112767
$199$	−800152.	−1.43232	−0.716160	−	0.697937i	$-0.754102\pi$
$199$	−800152.	−1.43232	−0.716160	+	0.697937i	$0.754102\pi$
$200$	−13376.0	−0.0236457
$201$	−603540.	−1.05370
$202$	−516360.	−0.890377
$203$	−46746.0	−0.0796167
$204$	−204768.	−0.344498
$205$	799308.	1.32840
$206$	−22144.0	−0.0363570
$207$	−307152.	−0.498227
$208$	43264.0	0.0693375
$209$	−335616.	−0.531468
$210$	−95256.0	−0.149054
$211$	−382060.	−0.590780	−0.295390	−	0.955377i	$-0.595450\pi$
$211$	−382060.	−0.590780	−0.295390	+	0.955377i	$0.595450\pi$
$212$	−6816.00	−0.0104157
$213$	−255852.	−0.386402
$214$	−807984.	−1.20606
$215$	−268056.	−0.395484
$216$	46656.0	0.0680414
$217$	−27832.0	−0.0401232
$218$	130040.	0.185326
$219$	−206046.	−0.290305
$220$	165888.	0.231078
$221$	−240318.	−0.330983
$222$	283896.	0.386613
$223$	−303016.	−0.408041	−0.204020	−	0.978967i	$-0.565401\pi$
$223$	−303016.	−0.408041	−0.204020	+	0.978967i	$0.565401\pi$
$224$	50176.0	0.0668153
$225$	−16929.0	−0.0222933
$226$	−147672.	−0.192321
$227$	383376.	0.493810	0.246905	−	0.969040i	$-0.420586\pi$
$227$	383376.	0.493810	0.246905	+	0.969040i	$0.420586\pi$
$228$	251712.	0.320676
$229$	226070.	0.284875	0.142437	−	0.989804i	$-0.454506\pi$
$229$	226070.	0.284875	0.142437	+	0.989804i	$0.454506\pi$
$230$	819072.	1.02095
$231$	−84672.0	−0.104402
$232$	−61056.0	−0.0744746
$233$	1.16387e6	1.40448	0.702241	−	0.711939i	$-0.252183\pi$
$233$	1.16387e6	1.40448	0.702241	+	0.711939i	$0.252183\pi$
$234$	54756.0	0.0653720
$235$	1.02319e6	1.20861
$236$	557952.	0.652104
$237$	−12672.0	−0.0146546
$238$	−278712.	−0.318943
$239$	−661332.	−0.748902	−0.374451	−	0.927247i	$-0.622169\pi$
$239$	−661332.	−0.748902	−0.374451	+	0.927247i	$0.622169\pi$
$240$	−124416.	−0.139427
$241$	−662854.	−0.735149	−0.367574	−	0.929994i	$-0.619812\pi$
$241$	−662854.	−0.735149	−0.367574	+	0.929994i	$0.619812\pi$
$242$	−496748.	−0.545253
$243$	59049.0	0.0641500
$244$	−409888.	−0.440748
$245$	−129654.	−0.137997
$246$	−532872.	−0.561416
$247$	295412.	0.308096
$248$	−36352.0	−0.0375318
$249$	−155736.	−0.159181
$250$	720144.	0.728734
$251$	−221100.	−0.221516	−0.110758	−	0.993847i	$-0.535328\pi$
$251$	−221100.	−0.221516	−0.110758	+	0.993847i	$0.535328\pi$
$252$	63504.0	0.0629941
$253$	728064.	0.715102
$254$	−679168.	−0.660531
$255$	691092.	0.665557
$256$	65536.0	0.0625000
$257$	1.12972e6	1.06694	0.533469	−	0.845820i	$-0.320888\pi$
$257$	1.12972e6	1.06694	0.533469	+	0.845820i	$0.320888\pi$
$258$	178704.	0.167142
$259$	386414.	0.357934
$260$	−146016.	−0.133958
$261$	−77274.0	−0.0702154
$262$	710640.	0.639582
$263$	1.01544e6	0.905242	0.452621	−	0.891703i	$-0.350489\pi$
$263$	1.01544e6	0.905242	0.452621	+	0.891703i	$0.350489\pi$
$264$	−110592.	−0.0976594
$265$	23004.0	0.0201228
$266$	342608.	0.296889
$267$	−843210.	−0.723865
$268$	−1.07296e6	−0.912529
$269$	1.39282e6	1.17359	0.586793	−	0.809737i	$-0.300390\pi$
$269$	1.39282e6	1.17359	0.586793	+	0.809737i	$0.300390\pi$
$270$	−157464.	−0.131453
$271$	−1.48739e6	−1.23028	−0.615138	−	0.788420i	$-0.710900\pi$
$271$	−1.48739e6	−1.23028	−0.615138	+	0.788420i	$0.710900\pi$
$272$	−364032.	−0.298344
$273$	74529.0	0.0605228
$274$	566328.	0.455713
$275$	40128.0	0.0319975
$276$	−546048.	−0.431477
$277$	195782.	0.153311	0.0766555	−	0.997058i	$-0.475576\pi$
$277$	195782.	0.153311	0.0766555	+	0.997058i	$0.475576\pi$
$278$	1.12942e6	0.876486
$279$	−46008.0	−0.0353853
$280$	−169344.	−0.129085
$281$	−1.50376e6	−1.13609	−0.568046	−	0.822997i	$-0.692300\pi$
$281$	−1.50376e6	−1.13609	−0.568046	+	0.822997i	$0.692300\pi$
$282$	−682128.	−0.510791
$283$	365660.	0.271401	0.135700	−	0.990750i	$-0.456672\pi$
$283$	365660.	0.271401	0.135700	+	0.990750i	$0.456672\pi$
$284$	−454848.	−0.334634
$285$	−849528.	−0.619535
$286$	−129792.	−0.0938281
$287$	−725298.	−0.519771
$288$	82944.0	0.0589256
$289$	602227.	0.424146
$290$	206064.	0.143882
$291$	151434.	0.104831
$292$	−366304.	−0.251411
$293$	1.87936e6	1.27891	0.639457	−	0.768827i	$-0.279159\pi$
$293$	1.87936e6	1.27891	0.639457	+	0.768827i	$0.279159\pi$
$294$	86436.0	0.0583212
$295$	−1.88309e6	−1.25984
$296$	504704.	0.334817
$297$	−139968.	−0.0920741
$298$	814728.	0.531462
$299$	−640848.	−0.414550
$300$	−30096.0	−0.0193066
$301$	243236.	0.154743
$302$	457472.	0.288634
$303$	−1.16181e6	−0.726990
$304$	447488.	0.277714
$305$	1.38337e6	0.851509
$306$	−460728.	−0.281281
$307$	1.18737e6	0.719020	0.359510	−	0.933141i	$-0.382944\pi$
$307$	1.18737e6	0.719020	0.359510	+	0.933141i	$0.382944\pi$
$308$	−150528.	−0.0904150
$309$	−49824.0	−0.0296854
$310$	122688.	0.0725100
$311$	−539808.	−0.316474	−0.158237	−	0.987401i	$-0.550581\pi$
$311$	−539808.	−0.316474	−0.158237	+	0.987401i	$0.550581\pi$
$312$	97344.0	0.0566139
$313$	3.30535e6	1.90703	0.953514	−	0.301348i	$-0.0974367\pi$
$313$	3.30535e6	1.90703	0.953514	+	0.301348i	$0.0974367\pi$
$314$	194936.	0.111575
$315$	−214326.	−0.121702
$316$	−22528.0	−0.0126913
$317$	1.92640e6	1.07671	0.538355	−	0.842718i	$-0.319046\pi$
$317$	1.92640e6	1.07671	0.538355	+	0.842718i	$0.319046\pi$
$318$	−15336.0	−0.00850441
$319$	183168.	0.100780
$320$	−221184.	−0.120748
$321$	−1.81796e6	−0.984743
$322$	−743232.	−0.399471
$323$	−2.48566e6	−1.32567
$324$	104976.	0.0555556
$325$	−35321.0	−0.0185492
$326$	−72496.0	−0.0377807
$327$	292590.	0.151318
$328$	−947328.	−0.486201
$329$	−928452.	−0.472900
$330$	373248.	0.188674
$331$	−967612.	−0.485435	−0.242718	−	0.970097i	$-0.578039\pi$
$331$	−967612.	−0.485435	−0.242718	+	0.970097i	$0.578039\pi$
$332$	−276864.	−0.137855
$333$	638766.	0.315668
$334$	1.04520e6	0.512664
$335$	3.62124e6	1.76297
$336$	112896.	0.0545545
$337$	−763966.	−0.366437	−0.183218	−	0.983072i	$-0.558652\pi$
$337$	−763966.	−0.366437	−0.183218	+	0.983072i	$0.558652\pi$
$338$	114244.	0.0543928
$339$	−332262.	−0.157030
$340$	1.22861e6	0.576389
$341$	109056.	0.0507883
$342$	566352.	0.261831
$343$	117649.	0.0539949
$344$	317696.	0.144749
$345$	1.84291e6	0.833598
$346$	2.20284e6	0.989220
$347$	−3.32893e6	−1.48416	−0.742081	−	0.670310i	$-0.766161\pi$
$347$	−3.32893e6	−1.48416	−0.742081	+	0.670310i	$0.766161\pi$
$348$	−137376.	−0.0608083
$349$	2.02761e6	0.891087	0.445543	−	0.895260i	$-0.353011\pi$
$349$	2.02761e6	0.891087	0.445543	+	0.895260i	$0.353011\pi$
$350$	−40964.0	−0.0178744
$351$	123201.	0.0533761
$352$	−196608.	−0.0845755
$353$	2.94337e6	1.25721	0.628605	−	0.777725i	$-0.283626\pi$
$353$	2.94337e6	1.25721	0.628605	+	0.777725i	$0.283626\pi$
$354$	1.25539e6	0.532441
$355$	1.53511e6	0.646501
$356$	−1.49904e6	−0.626886
$357$	−627102.	−0.260416
$358$	−1.05778e6	−0.436200
$359$	4.25456e6	1.74228	0.871142	−	0.491031i	$-0.163380\pi$
$359$	4.25456e6	1.74228	0.871142	+	0.491031i	$0.163380\pi$
$360$	−279936.	−0.113842
$361$	579405.	0.233999
$362$	−127720.	−0.0512257
$363$	−1.11768e6	−0.445197
$364$	132496.	0.0524142
$365$	1.23628e6	0.485717
$366$	−922248.	−0.359869
$367$	−3.30786e6	−1.28198	−0.640992	−	0.767548i	$-0.721477\pi$
$367$	−3.30786e6	−1.28198	−0.640992	+	0.767548i	$0.721477\pi$
$368$	−970752.	−0.373670
$369$	−1.19896e6	−0.458395
$370$	−1.70338e6	−0.646854
$371$	−20874.0	−0.00787356
$372$	−81792.0	−0.0306446
$373$	2.86357e6	1.06570	0.532852	−	0.846209i	$-0.321120\pi$
$373$	2.86357e6	1.06570	0.532852	+	0.846209i	$0.321120\pi$
$374$	1.09210e6	0.403721
$375$	1.62032e6	0.595009
$376$	−1.21267e6	−0.442358
$377$	−161226.	−0.0584227
$378$	142884.	0.0514344
$379$	1.41360e6	0.505507	0.252754	−	0.967531i	$-0.418664\pi$
$379$	1.41360e6	0.505507	0.252754	+	0.967531i	$0.418664\pi$
$380$	−1.51027e6	−0.536533
$381$	−1.52813e6	−0.539321
$382$	−840960.	−0.294861
$383$	−4.39148e6	−1.52973	−0.764864	−	0.644191i	$-0.777194\pi$
$383$	−4.39148e6	−1.52973	−0.764864	+	0.644191i	$0.777194\pi$
$384$	147456.	0.0510310
$385$	508032.	0.174678
$386$	−1.95225e6	−0.666909
$387$	402084.	0.136471
$388$	269216.	0.0907866
$389$	−1.33886e6	−0.448601	−0.224301	−	0.974520i	$-0.572010\pi$
$389$	−1.33886e6	−0.448601	−0.224301	+	0.974520i	$0.572010\pi$
$390$	−328536.	−0.109376
$391$	5.39222e6	1.78372
$392$	153664.	0.0505076
$393$	1.59894e6	0.522217
$394$	−662232.	−0.214916
$395$	76032.0	0.0245191
$396$	−248832.	−0.0797385
$397$	3.38994e6	1.07948	0.539742	−	0.841831i	$-0.318522\pi$
$397$	3.38994e6	1.07948	0.539742	+	0.841831i	$0.318522\pi$
$398$	−3.20061e6	−1.01280
$399$	770868.	0.242408
$400$	−53504.0	−0.0167200
$401$	−2.84695e6	−0.884134	−0.442067	−	0.896982i	$-0.645755\pi$
$401$	−2.84695e6	−0.884134	−0.442067	+	0.896982i	$0.645755\pi$
$402$	−2.41416e6	−0.745077
$403$	−95992.0	−0.0294423
$404$	−2.06544e6	−0.629592
$405$	−354294.	−0.107331
$406$	−186984.	−0.0562975
$407$	−1.51411e6	−0.453077
$408$	−819072.	−0.243597
$409$	811970.	0.240011	0.120006	−	0.992773i	$-0.461709\pi$
$409$	811970.	0.240011	0.120006	+	0.992773i	$0.461709\pi$
$410$	3.19723e6	0.939322
$411$	1.27424e6	0.372088
$412$	−88576.0	−0.0257083
$413$	1.70873e6	0.492944
$414$	−1.22861e6	−0.352300
$415$	934416.	0.266330
$416$	173056.	0.0490290
$417$	2.54120e6	0.715648
$418$	−1.34246e6	−0.375804
$419$	4.41160e6	1.22761	0.613805	−	0.789457i	$-0.289638\pi$
$419$	4.41160e6	1.22761	0.613805	+	0.789457i	$0.289638\pi$
$420$	−381024.	−0.105397
$421$	5.62170e6	1.54583	0.772917	−	0.634508i	$-0.218797\pi$
$421$	5.62170e6	1.54583	0.772917	+	0.634508i	$0.218797\pi$
$422$	−1.52824e6	−0.417744
$423$	−1.53479e6	−0.417059
$424$	−27264.0	−0.00736504
$425$	297198.	0.0798130
$426$	−1.02341e6	−0.273228
$427$	−1.25528e6	−0.333174
$428$	−3.23194e6	−0.852812
$429$	−292032.	−0.0766103
$430$	−1.07222e6	−0.279650
$431$	3.35442e6	0.869810	0.434905	−	0.900476i	$-0.356782\pi$
$431$	3.35442e6	0.869810	0.434905	+	0.900476i	$0.356782\pi$
$432$	186624.	0.0481125
$433$	−691486.	−0.177241	−0.0886204	−	0.996065i	$-0.528246\pi$
$433$	−691486.	−0.177241	−0.0886204	+	0.996065i	$0.528246\pi$
$434$	−111328.	−0.0283714
$435$	463644.	0.117479
$436$	520160.	0.131045
$437$	−6.62842e6	−1.66037
$438$	−824184.	−0.205276
$439$	7.77128e6	1.92456	0.962280	−	0.272063i	$-0.0877058\pi$
$439$	7.77128e6	1.92456	0.962280	+	0.272063i	$0.0877058\pi$
$440$	663552.	0.163397
$441$	194481.	0.0476190
$442$	−961272.	−0.234040
$443$	−1.39940e6	−0.338793	−0.169396	−	0.985548i	$-0.554182\pi$
$443$	−1.39940e6	−0.338793	−0.169396	+	0.985548i	$0.554182\pi$
$444$	1.13558e6	0.273377
$445$	5.05926e6	1.21112
$446$	−1.21206e6	−0.288528
$447$	1.83314e6	0.433937
$448$	200704.	0.0472456
$449$	246846.	0.0577844	0.0288922	−	0.999583i	$-0.490802\pi$
$449$	246846.	0.0577844	0.0288922	+	0.999583i	$0.490802\pi$
$450$	−67716.0	−0.0157638
$451$	2.84198e6	0.657931
$452$	−590688.	−0.135992
$453$	1.02931e6	0.235668
$454$	1.53350e6	0.349177
$455$	−447174.	−0.101262
$456$	1.00685e6	0.226752
$457$	−2.55912e6	−0.573192	−0.286596	−	0.958052i	$-0.592524\pi$
$457$	−2.55912e6	−0.573192	−0.286596	+	0.958052i	$0.592524\pi$
$458$	904280.	0.201437
$459$	−1.03664e6	−0.229665
$460$	3.27629e6	0.721917
$461$	−8.34037e6	−1.82782	−0.913909	−	0.405920i	$-0.866951\pi$
$461$	−8.34037e6	−1.82782	−0.913909	+	0.405920i	$0.866951\pi$
$462$	−338688.	−0.0738235
$463$	−3.17066e6	−0.687380	−0.343690	−	0.939083i	$-0.611677\pi$
$463$	−3.17066e6	−0.687380	−0.343690	+	0.939083i	$0.611677\pi$
$464$	−244224.	−0.0526615
$465$	276048.	0.0592042
$466$	4.65550e6	0.993119
$467$	7.53467e6	1.59872	0.799359	−	0.600854i	$-0.205173\pi$
$467$	7.53467e6	1.59872	0.799359	+	0.600854i	$0.205173\pi$
$468$	219024.	0.0462250
$469$	−3.28594e6	−0.689807
$470$	4.09277e6	0.854619
$471$	438606.	0.0911008
$472$	2.23181e6	0.461107
$473$	−953088.	−0.195876
$474$	−50688.0	−0.0103624
$475$	−365332.	−0.0742940
$476$	−1.11485e6	−0.225527
$477$	−34506.0	−0.00694382
$478$	−2.64533e6	−0.529553
$479$	−3.18068e6	−0.633405	−0.316703	−	0.948525i	$-0.602576\pi$
$479$	−3.18068e6	−0.633405	−0.316703	+	0.948525i	$0.602576\pi$
$480$	−497664.	−0.0985901
$481$	1.33273e6	0.262652
$482$	−2.65142e6	−0.519829
$483$	−1.67227e6	−0.326166
$484$	−1.98699e6	−0.385552
$485$	−908604.	−0.175396
$486$	236196.	0.0453609
$487$	2.09245e6	0.399790	0.199895	−	0.979817i	$-0.435940\pi$
$487$	2.09245e6	0.399790	0.199895	+	0.979817i	$0.435940\pi$
$488$	−1.63955e6	−0.311656
$489$	−163116.	−0.0308478
$490$	−518616.	−0.0975789
$491$	1.21314e6	0.227095	0.113547	−	0.993533i	$-0.463779\pi$
$491$	1.21314e6	0.227095	0.113547	+	0.993533i	$0.463779\pi$
$492$	−2.13149e6	−0.396981
$493$	1.35659e6	0.251380
$494$	1.18165e6	0.217857
$495$	839808.	0.154052
$496$	−145408.	−0.0265390
$497$	−1.39297e6	−0.252960
$498$	−622944.	−0.112558
$499$	−63316.0	−0.0113831	−0.00569157	−	0.999984i	$-0.501812\pi$
$499$	−63316.0	−0.0113831	−0.00569157	+	0.999984i	$0.501812\pi$
$500$	2.88058e6	0.515293
$501$	2.35170e6	0.418589
$502$	−884400.	−0.156635
$503$	1.06032e6	0.186860	0.0934301	−	0.995626i	$-0.470217\pi$
$503$	1.06032e6	0.186860	0.0934301	+	0.995626i	$0.470217\pi$
$504$	254016.	0.0445435
$505$	6.97086e6	1.21635
$506$	2.91226e6	0.505654
$507$	257049.	0.0444116
$508$	−2.71667e6	−0.467066
$509$	−3.46136e6	−0.592178	−0.296089	−	0.955160i	$-0.595682\pi$
$509$	−3.46136e6	−0.592178	−0.296089	+	0.955160i	$0.595682\pi$
$510$	2.76437e6	0.470620
$511$	−1.12181e6	−0.190049
$512$	262144.	0.0441942
$513$	1.27429e6	0.213784
$514$	4.51889e6	0.754438
$515$	298944.	0.0496674
$516$	714816.	0.118187
$517$	3.63802e6	0.598602
$518$	1.54566e6	0.253098
$519$	4.95639e6	0.807694
$520$	−584064.	−0.0947223
$521$	1.05956e7	1.71014	0.855072	−	0.518509i	$-0.173513\pi$
$521$	1.05956e7	1.71014	0.855072	+	0.518509i	$0.173513\pi$
$522$	−309096.	−0.0496498
$523$	2.49885e6	0.399472	0.199736	−	0.979850i	$-0.435992\pi$
$523$	2.49885e6	0.399472	0.199736	+	0.979850i	$0.435992\pi$
$524$	2.84256e6	0.452253
$525$	−92169.0	−0.0145944
$526$	4.06176e6	0.640103
$527$	807696.	0.126684
$528$	−442368.	−0.0690556
$529$	7.94292e6	1.23407
$530$	92016.0	0.0142290
$531$	2.82463e6	0.434736
$532$	1.37043e6	0.209932
$533$	−2.50154e6	−0.381407
$534$	−3.37284e6	−0.511850
$535$	1.09078e7	1.64760
$536$	−4.29184e6	−0.645255
$537$	−2.38000e6	−0.356156
$538$	5.57129e6	0.829851
$539$	−460992.	−0.0683473
$540$	−629856.	−0.0929516
$541$	−9.36147e6	−1.37515	−0.687577	−	0.726112i	$-0.741325\pi$
$541$	−9.36147e6	−1.37515	−0.687577	+	0.726112i	$0.741325\pi$
$542$	−5.94957e6	−0.869936
$543$	−287370.	−0.0418256
$544$	−1.45613e6	−0.210961
$545$	−1.75554e6	−0.253174
$546$	298116.	0.0427960
$547$	1.00395e7	1.43465	0.717324	−	0.696740i	$-0.245367\pi$
$547$	1.00395e7	1.43465	0.717324	+	0.696740i	$0.245367\pi$
$548$	2.26531e6	0.322238
$549$	−2.07506e6	−0.293832
$550$	160512.	0.0226256
$551$	−1.66759e6	−0.233997
$552$	−2.18419e6	−0.305101
$553$	−68992.0	−0.00959369
$554$	783128.	0.108407
$555$	−3.83260e6	−0.528154
$556$	4.51770e6	0.619769
$557$	3.85718e6	0.526783	0.263391	−	0.964689i	$-0.415159\pi$
$557$	3.85718e6	0.526783	0.263391	+	0.964689i	$0.415159\pi$
$558$	−184032.	−0.0250212
$559$	838916.	0.113550
$560$	−677376.	−0.0912767
$561$	2.45722e6	0.329637
$562$	−6.01505e6	−0.803338
$563$	1.62056e6	0.215474	0.107737	−	0.994179i	$-0.465640\pi$
$563$	1.62056e6	0.215474	0.107737	+	0.994179i	$0.465640\pi$
$564$	−2.72851e6	−0.361184
$565$	1.99357e6	0.262731
$566$	1.46264e6	0.191909
$567$	321489.	0.0419961
$568$	−1.81939e6	−0.236622
$569$	−4.09976e6	−0.530857	−0.265429	−	0.964131i	$-0.585513\pi$
$569$	−4.09976e6	−0.530857	−0.265429	+	0.964131i	$0.585513\pi$
$570$	−3.39811e6	−0.438077
$571$	−826660.	−0.106105	−0.0530526	−	0.998592i	$-0.516895\pi$
$571$	−826660.	−0.106105	−0.0530526	+	0.998592i	$0.516895\pi$
$572$	−519168.	−0.0663465
$573$	−1.89216e6	−0.240753
$574$	−2.90119e6	−0.367533
$575$	792528.	0.0999643
$576$	331776.	0.0416667
$577$	−9.54662e6	−1.19374	−0.596871	−	0.802338i	$-0.703589\pi$
$577$	−9.54662e6	−1.19374	−0.596871	+	0.802338i	$0.703589\pi$
$578$	2.40891e6	0.299917
$579$	−4.39256e6	−0.544529
$580$	824256.	0.101740
$581$	−847896.	−0.104208
$582$	605736.	0.0741269
$583$	81792.0	0.00996643
$584$	−1.46522e6	−0.177775
$585$	−739206.	−0.0893050
$586$	7.51745e6	0.904329
$587$	8.65829e6	1.03714	0.518569	−	0.855036i	$-0.326465\pi$
$587$	8.65829e6	1.03714	0.518569	+	0.855036i	$0.326465\pi$
$588$	345744.	0.0412393
$589$	−992864.	−0.117924
$590$	−7.53235e6	−0.890842
$591$	−1.49002e6	−0.175479
$592$	2.01882e6	0.236751
$593$	−1.07810e7	−1.25899	−0.629493	−	0.777006i	$-0.716737\pi$
$593$	−1.07810e7	−1.25899	−0.629493	+	0.777006i	$0.716737\pi$
$594$	−559872.	−0.0651062
$595$	3.76261e6	0.435709
$596$	3.25891e6	0.375800
$597$	−7.20137e6	−0.826950
$598$	−2.56339e6	−0.293131
$599$	−7.08360e6	−0.806653	−0.403327	−	0.915056i	$-0.632146\pi$
$599$	−7.08360e6	−0.806653	−0.403327	+	0.915056i	$0.632146\pi$
$600$	−120384.	−0.0136518
$601$	1.59307e7	1.79907	0.899536	−	0.436846i	$-0.143905\pi$
$601$	1.59307e7	1.79907	0.899536	+	0.436846i	$0.143905\pi$
$602$	972944.	0.109420
$603$	−5.43186e6	−0.608352
$604$	1.82989e6	0.204095
$605$	6.70610e6	0.744872
$606$	−4.64724e6	−0.514060
$607$	5.97212e6	0.657895	0.328948	−	0.944348i	$-0.393306\pi$
$607$	5.97212e6	0.657895	0.328948	+	0.944348i	$0.393306\pi$
$608$	1.78995e6	0.196373
$609$	−420714.	−0.0459667
$610$	5.53349e6	0.602108
$611$	−3.20221e6	−0.347014
$612$	−1.84291e6	−0.198896
$613$	−877426.	−0.0943103	−0.0471552	−	0.998888i	$-0.515016\pi$
$613$	−877426.	−0.0943103	−0.0471552	+	0.998888i	$0.515016\pi$
$614$	4.74949e6	0.508424
$615$	7.19377e6	0.766953
$616$	−602112.	−0.0639331
$617$	−9.60568e6	−1.01582	−0.507908	−	0.861411i	$-0.669581\pi$
$617$	−9.60568e6	−1.01582	−0.507908	+	0.861411i	$0.669581\pi$
$618$	−199296.	−0.0209907
$619$	818276.	0.0858367	0.0429184	−	0.999079i	$-0.486334\pi$
$619$	818276.	0.0858367	0.0429184	+	0.999079i	$0.486334\pi$
$620$	490752.	0.0512723
$621$	−2.76437e6	−0.287652
$622$	−2.15923e6	−0.223781
$623$	−4.59081e6	−0.473881
$624$	389376.	0.0400320
$625$	−9.06882e6	−0.928647
$626$	1.32214e7	1.34847
$627$	−3.02054e6	−0.306843
$628$	779744.	0.0788956
$629$	−1.12139e7	−1.13013
$630$	−857304.	−0.0860565
$631$	−2.73947e6	−0.273901	−0.136950	−	0.990578i	$-0.543730\pi$
$631$	−2.73947e6	−0.273901	−0.136950	+	0.990578i	$0.543730\pi$
$632$	−90112.0	−0.00897408
$633$	−3.43854e6	−0.341087
$634$	7.70561e6	0.761349
$635$	9.16877e6	0.902353
$636$	−61344.0	−0.00601353
$637$	405769.	0.0396214
$638$	732672.	0.0712620
$639$	−2.30267e6	−0.223090
$640$	−884736.	−0.0853815
$641$	−1.39383e7	−1.33987	−0.669937	−	0.742418i	$-0.733679\pi$
$641$	−1.39383e7	−1.33987	−0.669937	+	0.742418i	$0.733679\pi$
$642$	−7.27186e6	−0.696318
$643$	−3.92770e6	−0.374637	−0.187319	−	0.982299i	$-0.559980\pi$
$643$	−3.92770e6	−0.374637	−0.187319	+	0.982299i	$0.559980\pi$
$644$	−2.97293e6	−0.282468
$645$	−2.41250e6	−0.228333
$646$	−9.94262e6	−0.937389
$647$	−1.85072e7	−1.73812	−0.869060	−	0.494706i	$-0.835276\pi$
$647$	−1.85072e7	−1.73812	−0.869060	+	0.494706i	$0.835276\pi$
$648$	419904.	0.0392837
$649$	−6.69542e6	−0.623974
$650$	−141284.	−0.0131162
$651$	−250488.	−0.0231651
$652$	−289984.	−0.0267150
$653$	−5.05771e6	−0.464163	−0.232082	−	0.972696i	$-0.574554\pi$
$653$	−5.05771e6	−0.464163	−0.232082	+	0.972696i	$0.574554\pi$
$654$	1.17036e6	0.106998
$655$	−9.59364e6	−0.873736
$656$	−3.78931e6	−0.343796
$657$	−1.85441e6	−0.167607
$658$	−3.71381e6	−0.334391
$659$	−1.03811e7	−0.931173	−0.465587	−	0.885002i	$-0.654157\pi$
$659$	−1.03811e7	−0.931173	−0.465587	+	0.885002i	$0.654157\pi$
$660$	1.49299e6	0.133413
$661$	−1.40680e7	−1.25236	−0.626180	−	0.779678i	$-0.715383\pi$
$661$	−1.40680e7	−1.25236	−0.626180	+	0.779678i	$0.715383\pi$
$662$	−3.87045e6	−0.343255
$663$	−2.16286e6	−0.191093
$664$	−1.10746e6	−0.0974779
$665$	−4.62521e6	−0.405581
$666$	2.55506e6	0.223211
$667$	3.61757e6	0.314849
$668$	4.18080e6	0.362509
$669$	−2.72714e6	−0.235582
$670$	1.44850e7	1.24661
$671$	4.91866e6	0.421735
$672$	451584.	0.0385758
$673$	1.92279e7	1.63642	0.818211	−	0.574919i	$-0.194966\pi$
$673$	1.92279e7	1.63642	0.818211	+	0.574919i	$0.194966\pi$
$674$	−3.05586e6	−0.259110
$675$	−152361.	−0.0128711
$676$	456976.	0.0384615
$677$	−9.59831e6	−0.804865	−0.402433	−	0.915450i	$-0.631835\pi$
$677$	−9.59831e6	−0.804865	−0.402433	+	0.915450i	$0.631835\pi$
$678$	−1.32905e6	−0.111037
$679$	824474.	0.0686282
$680$	4.91443e6	0.407569
$681$	3.45038e6	0.285102
$682$	436224.	0.0359127
$683$	−1.88472e7	−1.54594	−0.772972	−	0.634440i	$-0.781231\pi$
$683$	−1.88472e7	−1.54594	−0.772972	+	0.634440i	$0.781231\pi$
$684$	2.26541e6	0.185143
$685$	−7.64543e6	−0.622552
$686$	470596.	0.0381802
$687$	2.03463e6	0.164473
$688$	1.27078e6	0.102353
$689$	−71994.0	−0.00577761
$690$	7.37165e6	0.589443
$691$	−1.96469e7	−1.56531	−0.782654	−	0.622457i	$-0.786134\pi$
$691$	−1.96469e7	−1.56531	−0.782654	+	0.622457i	$0.786134\pi$
$692$	8.81136e6	0.699484
$693$	−762048.	−0.0602767
$694$	−1.33157e7	−1.04946
$695$	−1.52472e7	−1.19737
$696$	−549504.	−0.0429980
$697$	2.10484e7	1.64111
$698$	8.11042e6	0.630094
$699$	1.04749e7	0.810878
$700$	−163856.	−0.0126391
$701$	−4.95664e6	−0.380972	−0.190486	−	0.981690i	$-0.561006\pi$
$701$	−4.95664e6	−0.380972	−0.190486	+	0.981690i	$0.561006\pi$
$702$	492804.	0.0377426
$703$	1.37847e7	1.05199
$704$	−786432.	−0.0598039
$705$	9.20873e6	0.697793
$706$	1.17735e7	0.888981
$707$	−6.32541e6	−0.475927
$708$	5.02157e6	0.376492
$709$	−8.40260e6	−0.627767	−0.313883	−	0.949462i	$-0.601630\pi$
$709$	−8.40260e6	−0.627767	−0.313883	+	0.949462i	$0.601630\pi$
$710$	6.14045e6	0.457145
$711$	−114048.	−0.00846084
$712$	−5.99616e6	−0.443275
$713$	2.15386e6	0.158669
$714$	−2.50841e6	−0.184142
$715$	1.75219e6	0.128179
$716$	−4.23110e6	−0.308440
$717$	−5.95199e6	−0.432379
$718$	1.70183e7	1.23198
$719$	7.19556e6	0.519090	0.259545	−	0.965731i	$-0.416427\pi$
$719$	7.19556e6	0.519090	0.259545	+	0.965731i	$0.416427\pi$
$720$	−1.11974e6	−0.0804984
$721$	−271264.	−0.0194336
$722$	2.31762e6	0.165462
$723$	−5.96569e6	−0.424438
$724$	−510880.	−0.0362220
$725$	199386.	0.0140880
$726$	−4.47073e6	−0.314802
$727$	4.22593e6	0.296542	0.148271	−	0.988947i	$-0.452629\pi$
$727$	4.22593e6	0.296542	0.148271	+	0.988947i	$0.452629\pi$
$728$	529984.	0.0370625
$729$	531441.	0.0370370
$730$	4.94510e6	0.343454
$731$	−7.05881e6	−0.488583
$732$	−3.68899e6	−0.254466
$733$	−7.64403e6	−0.525488	−0.262744	−	0.964866i	$-0.584627\pi$
$733$	−7.64403e6	−0.525488	−0.262744	+	0.964866i	$0.584627\pi$
$734$	−1.32315e7	−0.906499
$735$	−1.16689e6	−0.0796728
$736$	−3.88301e6	−0.264225
$737$	1.28755e7	0.873164
$738$	−4.79585e6	−0.324134
$739$	−2.62718e7	−1.76962	−0.884809	−	0.465954i	$-0.845711\pi$
$739$	−2.62718e7	−1.76962	−0.884809	+	0.465954i	$0.845711\pi$
$740$	−6.81350e6	−0.457395
$741$	2.65871e6	0.177879
$742$	−83496.0	−0.00556744
$743$	−2.46314e7	−1.63688	−0.818441	−	0.574591i	$-0.805161\pi$
$743$	−2.46314e7	−1.63688	−0.818441	+	0.574591i	$0.805161\pi$
$744$	−327168.	−0.0216690
$745$	−1.09988e7	−0.726032
$746$	1.14543e7	0.753566
$747$	−1.40162e6	−0.0919031
$748$	4.36838e6	0.285474
$749$	−9.89780e6	−0.644665
$750$	6.48130e6	0.420735
$751$	1.96305e7	1.27008	0.635041	−	0.772478i	$-0.280983\pi$
$751$	1.96305e7	1.27008	0.635041	+	0.772478i	$0.280983\pi$
$752$	−4.85069e6	−0.312794
$753$	−1.98990e6	−0.127892
$754$	−644904.	−0.0413111
$755$	−6.17587e6	−0.394304
$756$	571536.	0.0363696
$757$	−8.87151e6	−0.562676	−0.281338	−	0.959609i	$-0.590778\pi$
$757$	−8.87151e6	−0.562676	−0.281338	+	0.959609i	$0.590778\pi$
$758$	5.65438e6	0.357448
$759$	6.55258e6	0.412865
$760$	−6.04109e6	−0.379386
$761$	1.97983e7	1.23927	0.619636	−	0.784890i	$-0.287280\pi$
$761$	1.97983e7	1.23927	0.619636	+	0.784890i	$0.287280\pi$
$762$	−6.11251e6	−0.381358
$763$	1.59299e6	0.0990608
$764$	−3.36384e6	−0.208498
$765$	6.21983e6	0.384260
$766$	−1.75659e7	−1.08168
$767$	5.89337e6	0.361722
$768$	589824.	0.0360844
$769$	−8.14829e6	−0.496879	−0.248439	−	0.968647i	$-0.579918\pi$
$769$	−8.14829e6	−0.496879	−0.248439	+	0.968647i	$0.579918\pi$
$770$	2.03213e6	0.123516
$771$	1.01675e7	0.615996
$772$	−7.80899e6	−0.471576
$773$	−7.69145e6	−0.462977	−0.231489	−	0.972838i	$-0.574360\pi$
$773$	−7.69145e6	−0.462977	−0.231489	+	0.972838i	$0.574360\pi$
$774$	1.60834e6	0.0964993
$775$	118712.	0.00709971
$776$	1.07686e6	0.0641958
$777$	3.47773e6	0.206654
$778$	−5.35543e6	−0.317209
$779$	−2.58739e7	−1.52763
$780$	−1.31414e6	−0.0773404
$781$	5.45818e6	0.320199
$782$	2.15689e7	1.26128
$783$	−695466.	−0.0405389
$784$	614656.	0.0357143
$785$	−2.63164e6	−0.152423
$786$	6.39576e6	0.369263
$787$	−2.49837e7	−1.43787	−0.718935	−	0.695078i	$-0.755370\pi$
$787$	−2.49837e7	−1.43787	−0.718935	+	0.695078i	$0.755370\pi$
$788$	−2.64893e6	−0.151969
$789$	9.13896e6	0.522642
$790$	304128.	0.0173376
$791$	−1.80898e6	−0.102800
$792$	−995328.	−0.0563837
$793$	−4.32944e6	−0.244483
$794$	1.35598e7	0.763310
$795$	207036.	0.0116179
$796$	−1.28024e7	−0.716160
$797$	6.19328e6	0.345362	0.172681	−	0.984978i	$-0.444757\pi$
$797$	6.19328e6	0.345362	0.172681	+	0.984978i	$0.444757\pi$
$798$	3.08347e6	0.171409
$799$	2.69441e7	1.49312
$800$	−214016.	−0.0118228
$801$	−7.58889e6	−0.417924
$802$	−1.13878e7	−0.625177
$803$	4.39565e6	0.240566
$804$	−9.65664e6	−0.526849
$805$	1.00336e7	0.545718
$806$	−383968.	−0.0208189
$807$	1.25354e7	0.677570
$808$	−8.26176e6	−0.445189
$809$	−5.09077e6	−0.273471	−0.136736	−	0.990608i	$-0.543661\pi$
$809$	−5.09077e6	−0.273471	−0.136736	+	0.990608i	$0.543661\pi$
$810$	−1.41718e6	−0.0758947
$811$	2.11099e6	0.112703	0.0563513	−	0.998411i	$-0.482053\pi$
$811$	2.11099e6	0.112703	0.0563513	+	0.998411i	$0.482053\pi$
$812$	−747936.	−0.0398084
$813$	−1.33865e7	−0.710300
$814$	−6.05645e6	−0.320374
$815$	978696.	0.0516124
$816$	−3.27629e6	−0.172249
$817$	8.67707e6	0.454797
$818$	3.24788e6	0.169714
$819$	670761.	0.0349428
$820$	1.27889e7	0.664201
$821$	−2.92262e7	−1.51326	−0.756631	−	0.653842i	$-0.773156\pi$
$821$	−2.92262e7	−1.51326	−0.756631	+	0.653842i	$0.773156\pi$
$822$	5.09695e6	0.263106
$823$	1.08712e7	0.559472	0.279736	−	0.960077i	$-0.409753\pi$
$823$	1.08712e7	0.559472	0.279736	+	0.960077i	$0.409753\pi$
$824$	−354304.	−0.0181785
$825$	361152.	0.0184738
$826$	6.83491e6	0.348564
$827$	7.79011e6	0.396077	0.198039	−	0.980194i	$-0.436543\pi$
$827$	7.79011e6	0.396077	0.198039	+	0.980194i	$0.436543\pi$
$828$	−4.91443e6	−0.249114
$829$	−1.49523e7	−0.755650	−0.377825	−	0.925877i	$-0.623328\pi$
$829$	−1.49523e7	−0.755650	−0.377825	+	0.925877i	$0.623328\pi$
$830$	3.73766e6	0.188324
$831$	1.76204e6	0.0885142
$832$	692224.	0.0346688
$833$	−3.41422e6	−0.170482
$834$	1.01648e7	0.506039
$835$	−1.41102e7	−0.700353
$836$	−5.36986e6	−0.265734
$837$	−414072.	−0.0204297
$838$	1.76464e7	0.868052
$839$	2.94083e7	1.44233	0.721166	−	0.692762i	$-0.243606\pi$
$839$	2.94083e7	1.44233	0.721166	+	0.692762i	$0.243606\pi$
$840$	−1.52410e6	−0.0745271
$841$	−1.96010e7	−0.955628
$842$	2.24868e7	1.09307
$843$	−1.35339e7	−0.655923
$844$	−6.11296e6	−0.295390
$845$	−1.54229e6	−0.0743063
$846$	−6.13915e6	−0.294905
$847$	−6.08516e6	−0.291450
$848$	−109056.	−0.00520787
$849$	3.29094e6	0.156693
$850$	1.18879e6	0.0564363
$851$	−2.99037e7	−1.41547
$852$	−4.09363e6	−0.193201
$853$	2.59454e7	1.22092	0.610460	−	0.792047i	$-0.290985\pi$
$853$	2.59454e7	1.22092	0.610460	+	0.792047i	$0.290985\pi$
$854$	−5.02113e6	−0.235590
$855$	−7.64575e6	−0.357688
$856$	−1.29277e7	−0.603029
$857$	−2.71981e7	−1.26499	−0.632495	−	0.774565i	$-0.717969\pi$
$857$	−2.71981e7	−1.26499	−0.632495	+	0.774565i	$0.717969\pi$
$858$	−1.16813e6	−0.0541717
$859$	1.55649e7	0.719718	0.359859	−	0.933007i	$-0.382825\pi$
$859$	1.55649e7	0.719718	0.359859	+	0.933007i	$0.382825\pi$
$860$	−4.28890e6	−0.197742
$861$	−6.52768e6	−0.300090
$862$	1.34177e7	0.615048
$863$	3.61264e7	1.65119	0.825596	−	0.564262i	$-0.190839\pi$
$863$	3.61264e7	1.65119	0.825596	+	0.564262i	$0.190839\pi$
$864$	746496.	0.0340207
$865$	−2.97383e7	−1.35138
$866$	−2.76594e6	−0.125328
$867$	5.42004e6	0.244881
$868$	−445312.	−0.0200616
$869$	270336.	0.0121438
$870$	1.85458e6	0.0830704
$871$	−1.13331e7	−0.506180
$872$	2.08064e6	0.0926629
$873$	1.36291e6	0.0605244
$874$	−2.65137e7	−1.17406
$875$	8.82176e6	0.389525
$876$	−3.29674e6	−0.145152
$877$	−6.96990e6	−0.306004	−0.153002	−	0.988226i	$-0.548894\pi$
$877$	−6.96990e6	−0.306004	−0.153002	+	0.988226i	$0.548894\pi$
$878$	3.10851e7	1.36087
$879$	1.69143e7	0.738381
$880$	2.65421e6	0.115539
$881$	−1.73080e7	−0.751291	−0.375645	−	0.926763i	$-0.622579\pi$
$881$	−1.73080e7	−0.751291	−0.375645	+	0.926763i	$0.622579\pi$
$882$	777924.	0.0336718
$883$	−1.97666e7	−0.853158	−0.426579	−	0.904450i	$-0.640281\pi$
$883$	−1.97666e7	−0.853158	−0.426579	+	0.904450i	$0.640281\pi$
$884$	−3.84509e6	−0.165491
$885$	−1.69478e7	−0.727369
$886$	−5.59762e6	−0.239562
$887$	500856.	0.0213749	0.0106874	−	0.999943i	$-0.496598\pi$
$887$	500856.	0.0213749	0.0106874	+	0.999943i	$0.496598\pi$
$888$	4.54234e6	0.193307
$889$	−8.31981e6	−0.353068
$890$	2.02370e7	0.856391
$891$	−1.25971e6	−0.0531590
$892$	−4.84826e6	−0.204020
$893$	−3.31211e7	−1.38988
$894$	7.33255e6	0.306840
$895$	1.42800e7	0.595895
$896$	802816.	0.0334077
$897$	−5.76763e6	−0.239341
$898$	987384.	0.0408597
$899$	541872.	0.0223613
$900$	−270864.	−0.0111467
$901$	605772.	0.0248598
$902$	1.13679e7	0.465227
$903$	2.18912e6	0.0893410
$904$	−2.36275e6	−0.0961606
$905$	1.72422e6	0.0699796
$906$	4.11725e6	0.166643
$907$	3.70905e7	1.49708	0.748539	−	0.663091i	$-0.230756\pi$
$907$	3.70905e7	1.49708	0.748539	+	0.663091i	$0.230756\pi$
$908$	6.13402e6	0.246905
$909$	−1.04563e7	−0.419728
$910$	−1.78870e6	−0.0716033
$911$	4.04155e7	1.61344	0.806719	−	0.590935i	$-0.201241\pi$
$911$	4.04155e7	1.61344	0.806719	+	0.590935i	$0.201241\pi$
$912$	4.02739e6	0.160338
$913$	3.32237e6	0.131908
$914$	−1.02365e7	−0.405308
$915$	1.24503e7	0.491619
$916$	3.61712e6	0.142437
$917$	8.70534e6	0.341871
$918$	−4.14655e6	−0.162398
$919$	−1.25534e7	−0.490313	−0.245157	−	0.969483i	$-0.578839\pi$
$919$	−1.25534e7	−0.490313	−0.245157	+	0.969483i	$0.578839\pi$
$920$	1.31052e7	0.510473
$921$	1.06863e7	0.415126
$922$	−3.33615e7	−1.29246
$923$	−4.80433e6	−0.185622
$924$	−1.35475e6	−0.0522011
$925$	−1.64817e6	−0.0633357
$926$	−1.26826e7	−0.486051
$927$	−448416.	−0.0171389
$928$	−976896.	−0.0372373
$929$	−8.47147e6	−0.322047	−0.161024	−	0.986951i	$-0.551479\pi$
$929$	−8.47147e6	−0.322047	−0.161024	+	0.986951i	$0.551479\pi$
$930$	1.10419e6	0.0418637
$931$	4.19695e6	0.158694
$932$	1.86220e7	0.702241
$933$	−4.85827e6	−0.182716
$934$	3.01387e7	1.13046
$935$	−1.47433e7	−0.551525
$936$	876096.	0.0326860
$937$	−2.13109e7	−0.792963	−0.396482	−	0.918043i	$-0.629769\pi$
$937$	−2.13109e7	−0.792963	−0.396482	+	0.918043i	$0.629769\pi$
$938$	−1.31438e7	−0.487767
$939$	2.97482e7	1.10102
$940$	1.63711e7	0.604307
$941$	−2.70629e7	−0.996325	−0.498162	−	0.867084i	$-0.665992\pi$
$941$	−2.70629e7	−0.996325	−0.498162	+	0.867084i	$0.665992\pi$
$942$	1.75442e6	0.0644180
$943$	5.61292e7	2.05546
$944$	8.92723e6	0.326052
$945$	−1.92893e6	−0.0702648
$946$	−3.81235e6	−0.138505
$947$	−2.34466e7	−0.849582	−0.424791	−	0.905292i	$-0.639652\pi$
$947$	−2.34466e7	−0.849582	−0.424791	+	0.905292i	$0.639652\pi$
$948$	−202752.	−0.00732731
$949$	−3.86909e6	−0.139458
$950$	−1.46133e6	−0.0525338
$951$	1.73376e7	0.621639
$952$	−4.45939e6	−0.159472
$953$	−3.26567e7	−1.16477	−0.582385	−	0.812913i	$-0.697880\pi$
$953$	−3.26567e7	−1.16477	−0.582385	+	0.812913i	$0.697880\pi$
$954$	−138024.	−0.00491002
$955$	1.13530e7	0.402810
$956$	−1.05813e7	−0.374451
$957$	1.64851e6	0.0581852
$958$	−1.27227e7	−0.447885
$959$	6.93752e6	0.243589
$960$	−1.99066e6	−0.0697137
$961$	−2.83065e7	−0.988731
$962$	5.33094e6	0.185723
$963$	−1.63617e7	−0.568541
$964$	−1.06057e7	−0.367574
$965$	2.63553e7	0.911067
$966$	−6.68909e6	−0.230634
$967$	−4.74790e7	−1.63281	−0.816405	−	0.577480i	$-0.804036\pi$
$967$	−4.74790e7	−1.63281	−0.816405	+	0.577480i	$0.804036\pi$
$968$	−7.94797e6	−0.272626
$969$	−2.23709e7	−0.765375
$970$	−3.63442e6	−0.124024
$971$	14748.0	0.000501978 0	0.000250989	−	1.00000i	$-0.499920\pi$
$971$	14748.0	0.000501978 0	0.000250989		1.00000i	$0.499920\pi$
$972$	944784.	0.0320750
$973$	1.38354e7	0.468502
$974$	8.36979e6	0.282694
$975$	−317889.	−0.0107094
$976$	−6.55821e6	−0.220374
$977$	−4.16012e7	−1.39434	−0.697171	−	0.716905i	$-0.745558\pi$
$977$	−4.16012e7	−1.39434	−0.697171	+	0.716905i	$0.745558\pi$
$978$	−652464.	−0.0218127
$979$	1.79885e7	0.599843
$980$	−2.07446e6	−0.0689987
$981$	2.63331e6	0.0873634
$982$	4.85256e6	0.160580
$983$	5.28488e7	1.74442	0.872211	−	0.489130i	$-0.162686\pi$
$983$	5.28488e7	1.74442	0.872211	+	0.489130i	$0.162686\pi$
$984$	−8.52595e6	−0.280708
$985$	8.94013e6	0.293598
$986$	5.42635e6	0.177752
$987$	−8.35607e6	−0.273029
$988$	4.72659e6	0.154048
$989$	−1.88235e7	−0.611941
$990$	3.35923e6	0.108931
$991$	1.09734e6	0.0354940	0.0177470	−	0.999843i	$-0.494351\pi$
$991$	1.09734e6	0.0354940	0.0177470	+	0.999843i	$0.494351\pi$
$992$	−581632.	−0.0187659
$993$	−8.70851e6	−0.280266
$994$	−5.57189e6	−0.178870
$995$	4.32082e7	1.38359
$996$	−2.49178e6	−0.0795904
$997$	3.63025e7	1.15664	0.578320	−	0.815810i	$-0.303708\pi$
$997$	3.63025e7	1.15664	0.578320	+	0.815810i	$0.303708\pi$
$998$	−253264.	−0.00804910
$999$	5.74889e6	0.182251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.6.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.6.a.f.1.1	✓	1	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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