LMFDB - Embedded newform 78.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,6,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, 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("78.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	78.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:	$12.5099379454$
Analytic rank:	$0$
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	$\chi$	$=$	78.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +76.0000 q^{5} -36.0000 q^{6} +100.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -304.000 q^{10} -106.000 q^{11} +144.000 q^{12} -169.000 q^{13} -400.000 q^{14} +684.000 q^{15} +256.000 q^{16} +234.000 q^{17} -324.000 q^{18} -276.000 q^{19} +1216.00 q^{20} +900.000 q^{21} +424.000 q^{22} +2548.00 q^{23} -576.000 q^{24} +2651.00 q^{25} +676.000 q^{26} +729.000 q^{27} +1600.00 q^{28} +8266.00 q^{29} -2736.00 q^{30} -608.000 q^{31} -1024.00 q^{32} -954.000 q^{33} -936.000 q^{34} +7600.00 q^{35} +1296.00 q^{36} -2010.00 q^{37} +1104.00 q^{38} -1521.00 q^{39} -4864.00 q^{40} +8844.00 q^{41} -3600.00 q^{42} -17636.0 q^{43} -1696.00 q^{44} +6156.00 q^{45} -10192.0 q^{46} +18770.0 q^{47} +2304.00 q^{48} -6807.00 q^{49} -10604.0 q^{50} +2106.00 q^{51} -2704.00 q^{52} -26970.0 q^{53} -2916.00 q^{54} -8056.00 q^{55} -6400.00 q^{56} -2484.00 q^{57} -33064.0 q^{58} -41966.0 q^{59} +10944.0 q^{60} +778.000 q^{61} +2432.00 q^{62} +8100.00 q^{63} +4096.00 q^{64} -12844.0 q^{65} +3816.00 q^{66} -12632.0 q^{67} +3744.00 q^{68} +22932.0 q^{69} -30400.0 q^{70} +40466.0 q^{71} -5184.00 q^{72} +54302.0 q^{73} +8040.00 q^{74} +23859.0 q^{75} -4416.00 q^{76} -10600.0 q^{77} +6084.00 q^{78} -44656.0 q^{79} +19456.0 q^{80} +6561.00 q^{81} -35376.0 q^{82} +69918.0 q^{83} +14400.0 q^{84} +17784.0 q^{85} +70544.0 q^{86} +74394.0 q^{87} +6784.00 q^{88} -44520.0 q^{89} -24624.0 q^{90} -16900.0 q^{91} +40768.0 q^{92} -5472.00 q^{93} -75080.0 q^{94} -20976.0 q^{95} -9216.00 q^{96} -86026.0 q^{97} +27228.0 q^{98} -8586.00 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	76.0000	1.35953	0.679765	−	0.733430i	$-0.262082\pi$
$5$	76.0000	1.35953	0.679765	+	0.733430i	$0.262082\pi$
$6$	−36.0000	−0.408248
$7$	100.000	0.771356	0.385678	−	0.922633i	$-0.373968\pi$
$7$	100.000	0.771356	0.385678	+	0.922633i	$0.373968\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−304.000	−0.961332
$11$	−106.000	−0.264134	−0.132067	−	0.991241i	$-0.542161\pi$
$11$	−106.000	−0.264134	−0.132067	+	0.991241i	$0.542161\pi$
$12$	144.000	0.288675
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−400.000	−0.545431
$15$	684.000	0.784925
$16$	256.000	0.250000
$17$	234.000	0.196378	0.0981892	−	0.995168i	$-0.468695\pi$
$17$	234.000	0.196378	0.0981892	+	0.995168i	$0.468695\pi$
$18$	−324.000	−0.235702
$19$	−276.000	−0.175398	−0.0876991	−	0.996147i	$-0.527951\pi$
$19$	−276.000	−0.175398	−0.0876991	+	0.996147i	$0.527951\pi$
$20$	1216.00	0.679765
$21$	900.000	0.445343
$22$	424.000	0.186771
$23$	2548.00	1.00434	0.502169	−	0.864770i	$-0.332536\pi$
$23$	2548.00	1.00434	0.502169	+	0.864770i	$0.332536\pi$
$24$	−576.000	−0.204124
$25$	2651.00	0.848320
$26$	676.000	0.196116
$27$	729.000	0.192450
$28$	1600.00	0.385678
$29$	8266.00	1.82516	0.912579	−	0.408901i	$-0.134088\pi$
$29$	8266.00	1.82516	0.912579	+	0.408901i	$0.134088\pi$
$30$	−2736.00	−0.555026
$31$	−608.000	−0.113632	−0.0568158	−	0.998385i	$-0.518095\pi$
$31$	−608.000	−0.113632	−0.0568158	+	0.998385i	$0.518095\pi$
$32$	−1024.00	−0.176777
$33$	−954.000	−0.152498
$34$	−936.000	−0.138860
$35$	7600.00	1.04868
$36$	1296.00	0.166667
$37$	−2010.00	−0.241375	−0.120687	−	0.992691i	$-0.538510\pi$
$37$	−2010.00	−0.241375	−0.120687	+	0.992691i	$0.538510\pi$
$38$	1104.00	0.124025
$39$	−1521.00	−0.160128
$40$	−4864.00	−0.480666
$41$	8844.00	0.821654	0.410827	−	0.911713i	$-0.365240\pi$
$41$	8844.00	0.821654	0.410827	+	0.911713i	$0.365240\pi$
$42$	−3600.00	−0.314905
$43$	−17636.0	−1.45455	−0.727275	−	0.686346i	$-0.759214\pi$
$43$	−17636.0	−1.45455	−0.727275	+	0.686346i	$0.759214\pi$
$44$	−1696.00	−0.132067
$45$	6156.00	0.453176
$46$	−10192.0	−0.710174
$47$	18770.0	1.23942	0.619712	−	0.784830i	$-0.287250\pi$
$47$	18770.0	1.23942	0.619712	+	0.784830i	$0.287250\pi$
$48$	2304.00	0.144338
$49$	−6807.00	−0.405010
$50$	−10604.0	−0.599853
$51$	2106.00	0.113379
$52$	−2704.00	−0.138675
$53$	−26970.0	−1.31884	−0.659419	−	0.751776i	$-0.729198\pi$
$53$	−26970.0	−1.31884	−0.659419	+	0.751776i	$0.729198\pi$
$54$	−2916.00	−0.136083
$55$	−8056.00	−0.359098
$56$	−6400.00	−0.272716
$57$	−2484.00	−0.101266
$58$	−33064.0	−1.29058
$59$	−41966.0	−1.56952	−0.784761	−	0.619798i	$-0.787214\pi$
$59$	−41966.0	−1.56952	−0.784761	+	0.619798i	$0.787214\pi$
$60$	10944.0	0.392462
$61$	778.000	0.0267704	0.0133852	−	0.999910i	$-0.495739\pi$
$61$	778.000	0.0267704	0.0133852	+	0.999910i	$0.495739\pi$
$62$	2432.00	0.0803497
$63$	8100.00	0.257119
$64$	4096.00	0.125000
$65$	−12844.0	−0.377066
$66$	3816.00	0.107832
$67$	−12632.0	−0.343784	−0.171892	−	0.985116i	$-0.554988\pi$
$67$	−12632.0	−0.343784	−0.171892	+	0.985116i	$0.554988\pi$
$68$	3744.00	0.0981892
$69$	22932.0	0.579855
$70$	−30400.0	−0.741530
$71$	40466.0	0.952674	0.476337	−	0.879263i	$-0.341964\pi$
$71$	40466.0	0.952674	0.476337	+	0.879263i	$0.341964\pi$
$72$	−5184.00	−0.117851
$73$	54302.0	1.19264	0.596319	−	0.802748i	$-0.296629\pi$
$73$	54302.0	1.19264	0.596319	+	0.802748i	$0.296629\pi$
$74$	8040.00	0.170678
$75$	23859.0	0.489778
$76$	−4416.00	−0.0876991
$77$	−10600.0	−0.203741
$78$	6084.00	0.113228
$79$	−44656.0	−0.805030	−0.402515	−	0.915413i	$-0.631864\pi$
$79$	−44656.0	−0.805030	−0.402515	+	0.915413i	$0.631864\pi$
$80$	19456.0	0.339882
$81$	6561.00	0.111111
$82$	−35376.0	−0.580997
$83$	69918.0	1.11402	0.557011	−	0.830505i	$-0.311948\pi$
$83$	69918.0	1.11402	0.557011	+	0.830505i	$0.311948\pi$
$84$	14400.0	0.222671
$85$	17784.0	0.266982
$86$	70544.0	1.02852
$87$	74394.0	1.05376
$88$	6784.00	0.0933854
$89$	−44520.0	−0.595772	−0.297886	−	0.954601i	$-0.596282\pi$
$89$	−44520.0	−0.595772	−0.297886	+	0.954601i	$0.596282\pi$
$90$	−24624.0	−0.320444
$91$	−16900.0	−0.213936
$92$	40768.0	0.502169
$93$	−5472.00	−0.0656053
$94$	−75080.0	−0.876405
$95$	−20976.0	−0.238459
$96$	−9216.00	−0.102062
$97$	−86026.0	−0.928326	−0.464163	−	0.885750i	$-0.653645\pi$
$97$	−86026.0	−0.928326	−0.464163	+	0.885750i	$0.653645\pi$
$98$	27228.0	0.286385
$99$	−8586.00	−0.0880446
$100$	42416.0	0.424160
$101$	22418.0	0.218672	0.109336	−	0.994005i	$-0.465128\pi$
$101$	22418.0	0.218672	0.109336	+	0.994005i	$0.465128\pi$
$102$	−8424.00	−0.0801711
$103$	−137600.	−1.27798	−0.638992	−	0.769213i	$-0.720648\pi$
$103$	−137600.	−1.27798	−0.638992	+	0.769213i	$0.720648\pi$
$104$	10816.0	0.0980581
$105$	68400.0	0.605456
$106$	107880.	0.932559
$107$	−102152.	−0.862556	−0.431278	−	0.902219i	$-0.641937\pi$
$107$	−102152.	−0.862556	−0.431278	+	0.902219i	$0.641937\pi$
$108$	11664.0	0.0962250
$109$	156250.	1.25966	0.629831	−	0.776732i	$-0.283124\pi$
$109$	156250.	1.25966	0.629831	+	0.776732i	$0.283124\pi$
$110$	32224.0	0.253920
$111$	−18090.0	−0.139358
$112$	25600.0	0.192839
$113$	−72890.0	−0.536997	−0.268498	−	0.963280i	$-0.586527\pi$
$113$	−72890.0	−0.536997	−0.268498	+	0.963280i	$0.586527\pi$
$114$	9936.00	0.0716060
$115$	193648.	1.36543
$116$	132256.	0.912579
$117$	−13689.0	−0.0924500
$118$	167864.	1.10982
$119$	23400.0	0.151478
$120$	−43776.0	−0.277513
$121$	−149815.	−0.930233
$122$	−3112.00	−0.0189295
$123$	79596.0	0.474382
$124$	−9728.00	−0.0568158
$125$	−36024.0	−0.206213
$126$	−32400.0	−0.181810
$127$	36568.0	0.201183	0.100592	−	0.994928i	$-0.467926\pi$
$127$	36568.0	0.201183	0.100592	+	0.994928i	$0.467926\pi$
$128$	−16384.0	−0.0883883
$129$	−158724.	−0.839785
$130$	51376.0	0.266626
$131$	−304208.	−1.54879	−0.774395	−	0.632703i	$-0.781946\pi$
$131$	−304208.	−1.54879	−0.774395	+	0.632703i	$0.781946\pi$
$132$	−15264.0	−0.0762489
$133$	−27600.0	−0.135294
$134$	50528.0	0.243092
$135$	55404.0	0.261642
$136$	−14976.0	−0.0694302
$137$	−211140.	−0.961101	−0.480551	−	0.876967i	$-0.659563\pi$
$137$	−211140.	−0.961101	−0.480551	+	0.876967i	$0.659563\pi$
$138$	−91728.0	−0.410019
$139$	−274748.	−1.20614	−0.603070	−	0.797688i	$-0.706056\pi$
$139$	−274748.	−1.20614	−0.603070	+	0.797688i	$0.706056\pi$
$140$	121600.	0.524341
$141$	168930.	0.715581
$142$	−161864.	−0.673642
$143$	17914.0	0.0732576
$144$	20736.0	0.0833333
$145$	628216.	2.48136
$146$	−217208.	−0.843322
$147$	−61263.0	−0.233833
$148$	−32160.0	−0.120687
$149$	377976.	1.39476	0.697379	−	0.716703i	$-0.254350\pi$
$149$	377976.	1.39476	0.697379	+	0.716703i	$0.254350\pi$
$150$	−95436.0	−0.346325
$151$	−480960.	−1.71659	−0.858295	−	0.513157i	$-0.828476\pi$
$151$	−480960.	−1.71659	−0.858295	+	0.513157i	$0.828476\pi$
$152$	17664.0	0.0620126
$153$	18954.0	0.0654594
$154$	42400.0	0.144067
$155$	−46208.0	−0.154486
$156$	−24336.0	−0.0800641
$157$	381038.	1.23373	0.616864	−	0.787070i	$-0.288403\pi$
$157$	381038.	1.23373	0.616864	+	0.787070i	$0.288403\pi$
$158$	178624.	0.569242
$159$	−242730.	−0.761431
$160$	−77824.0	−0.240333
$161$	254800.	0.774702
$162$	−26244.0	−0.0785674
$163$	−292636.	−0.862698	−0.431349	−	0.902185i	$-0.641962\pi$
$163$	−292636.	−0.862698	−0.431349	+	0.902185i	$0.641962\pi$
$164$	141504.	0.410827
$165$	−72504.0	−0.207325
$166$	−279672.	−0.787733
$167$	−463302.	−1.28550	−0.642751	−	0.766075i	$-0.722207\pi$
$167$	−463302.	−1.28550	−0.642751	+	0.766075i	$0.722207\pi$
$168$	−57600.0	−0.157452
$169$	28561.0	0.0769231
$170$	−71136.0	−0.188785
$171$	−22356.0	−0.0584661
$172$	−282176.	−0.727275
$173$	−12518.0	−0.0317995	−0.0158997	−	0.999874i	$-0.505061\pi$
$173$	−12518.0	−0.0317995	−0.0158997	+	0.999874i	$0.505061\pi$
$174$	−297576.	−0.745118
$175$	265100.	0.654357
$176$	−27136.0	−0.0660335
$177$	−377694.	−0.906164
$178$	178080.	0.421274
$179$	−609108.	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$179$	−609108.	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$180$	98496.0	0.226588
$181$	217206.	0.492805	0.246403	−	0.969168i	$-0.420751\pi$
$181$	217206.	0.492805	0.246403	+	0.969168i	$0.420751\pi$
$182$	67600.0	0.151275
$183$	7002.00	0.0154559
$184$	−163072.	−0.355087
$185$	−152760.	−0.328156
$186$	21888.0	0.0463899
$187$	−24804.0	−0.0518702
$188$	300320.	0.619712
$189$	72900.0	0.148448
$190$	83904.0	0.168616
$191$	−702160.	−1.39268	−0.696342	−	0.717710i	$-0.745190\pi$
$191$	−702160.	−1.39268	−0.696342	+	0.717710i	$0.745190\pi$
$192$	36864.0	0.0721688
$193$	744170.	1.43807	0.719033	−	0.694976i	$-0.244585\pi$
$193$	744170.	1.43807	0.719033	+	0.694976i	$0.244585\pi$
$194$	344104.	0.656425
$195$	−115596.	−0.217699
$196$	−108912.	−0.202505
$197$	476676.	0.875100	0.437550	−	0.899194i	$-0.355846\pi$
$197$	476676.	0.875100	0.437550	+	0.899194i	$0.355846\pi$
$198$	34344.0	0.0622570
$199$	496232.	0.888284	0.444142	−	0.895956i	$-0.353508\pi$
$199$	496232.	0.888284	0.444142	+	0.895956i	$0.353508\pi$
$200$	−169664.	−0.299926
$201$	−113688.	−0.198484
$202$	−89672.0	−0.154625
$203$	826600.	1.40785
$204$	33696.0	0.0566895
$205$	672144.	1.11706
$206$	550400.	0.903671
$207$	206388.	0.334779
$208$	−43264.0	−0.0693375
$209$	29256.0	0.0463286
$210$	−273600.	−0.428122
$211$	672068.	1.03922	0.519610	−	0.854404i	$-0.326077\pi$
$211$	672068.	1.03922	0.519610	+	0.854404i	$0.326077\pi$
$212$	−431520.	−0.659419
$213$	364194.	0.550027
$214$	408608.	0.609919
$215$	−1.34034e6	−1.97750
$216$	−46656.0	−0.0680414
$217$	−60800.0	−0.0876505
$218$	−625000.	−0.890715
$219$	488718.	0.688570
$220$	−128896.	−0.179549
$221$	−39546.0	−0.0544655
$222$	72360.0	0.0985408
$223$	327408.	0.440887	0.220443	−	0.975400i	$-0.429250\pi$
$223$	327408.	0.440887	0.220443	+	0.975400i	$0.429250\pi$
$224$	−102400.	−0.136358
$225$	214731.	0.282773
$226$	291560.	0.379714
$227$	490674.	0.632016	0.316008	−	0.948756i	$-0.397657\pi$
$227$	490674.	0.632016	0.316008	+	0.948756i	$0.397657\pi$
$228$	−39744.0	−0.0506331
$229$	601674.	0.758180	0.379090	−	0.925360i	$-0.376237\pi$
$229$	601674.	0.758180	0.379090	+	0.925360i	$0.376237\pi$
$230$	−774592.	−0.965503
$231$	−95400.0	−0.117630
$232$	−529024.	−0.645291
$233$	1.42557e6	1.72027	0.860137	−	0.510063i	$-0.170378\pi$
$233$	1.42557e6	1.72027	0.860137	+	0.510063i	$0.170378\pi$
$234$	54756.0	0.0653720
$235$	1.42652e6	1.68503
$236$	−671456.	−0.784761
$237$	−401904.	−0.464784
$238$	−93600.0	−0.107111
$239$	412482.	0.467100	0.233550	−	0.972345i	$-0.424966\pi$
$239$	412482.	0.467100	0.233550	+	0.972345i	$0.424966\pi$
$240$	175104.	0.196231
$241$	−1.59256e6	−1.76626	−0.883128	−	0.469132i	$-0.844567\pi$
$241$	−1.59256e6	−1.76626	−0.883128	+	0.469132i	$0.844567\pi$
$242$	599260.	0.657774
$243$	59049.0	0.0641500
$244$	12448.0	0.0133852
$245$	−517332.	−0.550623
$246$	−318384.	−0.335439
$247$	46644.0	0.0486467
$248$	38912.0	0.0401749
$249$	629262.	0.643181
$250$	144096.	0.145815
$251$	−1.28333e6	−1.28574	−0.642870	−	0.765975i	$-0.722257\pi$
$251$	−1.28333e6	−1.28574	−0.642870	+	0.765975i	$0.722257\pi$
$252$	129600.	0.128559
$253$	−270088.	−0.265280
$254$	−146272.	−0.142258
$255$	160056.	0.154142
$256$	65536.0	0.0625000
$257$	88014.0	0.0831226	0.0415613	−	0.999136i	$-0.486767\pi$
$257$	88014.0	0.0831226	0.0415613	+	0.999136i	$0.486767\pi$
$258$	634896.	0.593818
$259$	−201000.	−0.186186
$260$	−205504.	−0.188533
$261$	669546.	0.608386
$262$	1.21683e6	1.09516
$263$	1.16708e6	1.04043	0.520215	−	0.854035i	$-0.325852\pi$
$263$	1.16708e6	1.04043	0.520215	+	0.854035i	$0.325852\pi$
$264$	61056.0	0.0539161
$265$	−2.04972e6	−1.79300
$266$	110400.	0.0956676
$267$	−400680.	−0.343969
$268$	−202112.	−0.171892
$269$	−206406.	−0.173917	−0.0869584	−	0.996212i	$-0.527715\pi$
$269$	−206406.	−0.173917	−0.0869584	+	0.996212i	$0.527715\pi$
$270$	−221616.	−0.185009
$271$	−1.36692e6	−1.13063	−0.565313	−	0.824877i	$-0.691244\pi$
$271$	−1.36692e6	−1.13063	−0.565313	+	0.824877i	$0.691244\pi$
$272$	59904.0	0.0490946
$273$	−152100.	−0.123516
$274$	844560.	0.679601
$275$	−281006.	−0.224070
$276$	366912.	0.289927
$277$	−2.28813e6	−1.79177	−0.895883	−	0.444290i	$-0.853456\pi$
$277$	−2.28813e6	−1.79177	−0.895883	+	0.444290i	$0.853456\pi$
$278$	1.09899e6	0.852869
$279$	−49248.0	−0.0378772
$280$	−486400.	−0.370765
$281$	2.00462e6	1.51449	0.757245	−	0.653131i	$-0.226545\pi$
$281$	2.00462e6	1.51449	0.757245	+	0.653131i	$0.226545\pi$
$282$	−675720.	−0.505992
$283$	276340.	0.205106	0.102553	−	0.994728i	$-0.467299\pi$
$283$	276340.	0.205106	0.102553	+	0.994728i	$0.467299\pi$
$284$	647456.	0.476337
$285$	−188784.	−0.137674
$286$	−71656.0	−0.0518009
$287$	884400.	0.633788
$288$	−82944.0	−0.0589256
$289$	−1.36510e6	−0.961436
$290$	−2.51286e6	−1.75458
$291$	−774234.	−0.535969
$292$	868832.	0.596319
$293$	−526216.	−0.358092	−0.179046	−	0.983841i	$-0.557301\pi$
$293$	−526216.	−0.358092	−0.179046	+	0.983841i	$0.557301\pi$
$294$	245052.	0.165345
$295$	−3.18942e6	−2.13381
$296$	128640.	0.0853388
$297$	−77274.0	−0.0508326
$298$	−1.51190e6	−0.986242
$299$	−430612.	−0.278553
$300$	381744.	0.244889
$301$	−1.76360e6	−1.12198
$302$	1.92384e6	1.21381
$303$	201762.	0.126250
$304$	−70656.0	−0.0438495
$305$	59128.0	0.0363952
$306$	−75816.0	−0.0462868
$307$	−621592.	−0.376409	−0.188204	−	0.982130i	$-0.560267\pi$
$307$	−621592.	−0.376409	−0.188204	+	0.982130i	$0.560267\pi$
$308$	−169600.	−0.101871
$309$	−1.23840e6	−0.737844
$310$	184832.	0.109238
$311$	−1.89260e6	−1.10958	−0.554790	−	0.831990i	$-0.687201\pi$
$311$	−1.89260e6	−1.10958	−0.554790	+	0.831990i	$0.687201\pi$
$312$	97344.0	0.0566139
$313$	2.36212e6	1.36283	0.681414	−	0.731899i	$-0.261366\pi$
$313$	2.36212e6	1.36283	0.681414	+	0.731899i	$0.261366\pi$
$314$	−1.52415e6	−0.872377
$315$	615600.	0.349560
$316$	−714496.	−0.402515
$317$	−674292.	−0.376877	−0.188439	−	0.982085i	$-0.560343\pi$
$317$	−674292.	−0.376877	−0.188439	+	0.982085i	$0.560343\pi$
$318$	970920.	0.538413
$319$	−876196.	−0.482086
$320$	311296.	0.169941
$321$	−919368.	−0.497997
$322$	−1.01920e6	−0.547797
$323$	−64584.0	−0.0344444
$324$	104976.	0.0555556
$325$	−448019.	−0.235282
$326$	1.17054e6	0.610020
$327$	1.40625e6	0.727266
$328$	−566016.	−0.290499
$329$	1.87700e6	0.956037
$330$	290016.	0.146601
$331$	1.76000e6	0.882963	0.441482	−	0.897270i	$-0.354453\pi$
$331$	1.76000e6	0.882963	0.441482	+	0.897270i	$0.354453\pi$
$332$	1.11869e6	0.557011
$333$	−162810.	−0.0804582
$334$	1.85321e6	0.908988
$335$	−960032.	−0.467384
$336$	230400.	0.111336
$337$	3.57150e6	1.71307	0.856537	−	0.516086i	$-0.172611\pi$
$337$	3.57150e6	1.71307	0.856537	+	0.516086i	$0.172611\pi$
$338$	−114244.	−0.0543928
$339$	−656010.	−0.310035
$340$	284544.	0.133491
$341$	64448.0	0.0300140
$342$	89424.0	0.0413417
$343$	−2.36140e6	−1.08376
$344$	1.12870e6	0.514261
$345$	1.74283e6	0.788330
$346$	50072.0	0.0224856
$347$	2.26974e6	1.01193	0.505967	−	0.862553i	$-0.331136\pi$
$347$	2.26974e6	1.01193	0.505967	+	0.862553i	$0.331136\pi$
$348$	1.19030e6	0.526878
$349$	−28874.0	−0.0126895	−0.00634473	−	0.999980i	$-0.502020\pi$
$349$	−28874.0	−0.0126895	−0.00634473	+	0.999980i	$0.502020\pi$
$350$	−1.06040e6	−0.462700
$351$	−123201.	−0.0533761
$352$	108544.	0.0466927
$353$	3.38366e6	1.44527	0.722637	−	0.691228i	$-0.242930\pi$
$353$	3.38366e6	1.44527	0.722637	+	0.691228i	$0.242930\pi$
$354$	1.51078e6	0.640755
$355$	3.07542e6	1.29519
$356$	−712320.	−0.297886
$357$	210600.	0.0874556
$358$	2.43643e6	1.00472
$359$	1.05979e6	0.433992	0.216996	−	0.976172i	$-0.430374\pi$
$359$	1.05979e6	0.433992	0.216996	+	0.976172i	$0.430374\pi$
$360$	−393984.	−0.160222
$361$	−2.39992e6	−0.969235
$362$	−868824.	−0.348466
$363$	−1.34834e6	−0.537070
$364$	−270400.	−0.106968
$365$	4.12695e6	1.62143
$366$	−28008.0	−0.0109290
$367$	−926104.	−0.358917	−0.179459	−	0.983766i	$-0.557435\pi$
$367$	−926104.	−0.358917	−0.179459	+	0.983766i	$0.557435\pi$
$368$	652288.	0.251084
$369$	716364.	0.273885
$370$	611040.	0.232041
$371$	−2.69700e6	−1.01729
$372$	−87552.0	−0.0328026
$373$	−4.41324e6	−1.64242	−0.821212	−	0.570623i	$-0.806702\pi$
$373$	−4.41324e6	−1.64242	−0.821212	+	0.570623i	$0.806702\pi$
$374$	99216.0	0.0366778
$375$	−324216.	−0.119057
$376$	−1.20128e6	−0.438202
$377$	−1.39695e6	−0.506208
$378$	−291600.	−0.104968
$379$	4.12124e6	1.47377	0.736885	−	0.676019i	$-0.236296\pi$
$379$	4.12124e6	1.47377	0.736885	+	0.676019i	$0.236296\pi$
$380$	−335616.	−0.119229
$381$	329112.	0.116153
$382$	2.80864e6	0.984776
$383$	3.85666e6	1.34343	0.671714	−	0.740810i	$-0.265558\pi$
$383$	3.85666e6	1.34343	0.671714	+	0.740810i	$0.265558\pi$
$384$	−147456.	−0.0510310
$385$	−805600.	−0.276992
$386$	−2.97668e6	−1.01687
$387$	−1.42852e6	−0.484850
$388$	−1.37642e6	−0.464163
$389$	−578154.	−0.193718	−0.0968589	−	0.995298i	$-0.530880\pi$
$389$	−578154.	−0.193718	−0.0968589	+	0.995298i	$0.530880\pi$
$390$	462384.	0.153936
$391$	596232.	0.197230
$392$	435648.	0.143193
$393$	−2.73787e6	−0.894194
$394$	−1.90670e6	−0.618789
$395$	−3.39386e6	−1.09446
$396$	−137376.	−0.0440223
$397$	3.11975e6	0.993444	0.496722	−	0.867910i	$-0.334537\pi$
$397$	3.11975e6	0.993444	0.496722	+	0.867910i	$0.334537\pi$
$398$	−1.98493e6	−0.628112
$399$	−248400.	−0.0781123
$400$	678656.	0.212080
$401$	4.00850e6	1.24486	0.622431	−	0.782674i	$-0.286145\pi$
$401$	4.00850e6	1.24486	0.622431	+	0.782674i	$0.286145\pi$
$402$	454752.	0.140349
$403$	102752.	0.0315158
$404$	358688.	0.109336
$405$	498636.	0.151059
$406$	−3.30640e6	−0.995498
$407$	213060.	0.0637552
$408$	−134784.	−0.0400856
$409$	1.37484e6	0.406390	0.203195	−	0.979138i	$-0.434867\pi$
$409$	1.37484e6	0.406390	0.203195	+	0.979138i	$0.434867\pi$
$410$	−2.68858e6	−0.789883
$411$	−1.90026e6	−0.554892
$412$	−2.20160e6	−0.638992
$413$	−4.19660e6	−1.21066
$414$	−825552.	−0.236725
$415$	5.31377e6	1.51455
$416$	173056.	0.0490290
$417$	−2.47273e6	−0.696365
$418$	−117024.	−0.0327593
$419$	3.58834e6	0.998523	0.499261	−	0.866451i	$-0.333605\pi$
$419$	3.58834e6	0.998523	0.499261	+	0.866451i	$0.333605\pi$
$420$	1.09440e6	0.302728
$421$	−4.01005e6	−1.10267	−0.551334	−	0.834284i	$-0.685881\pi$
$421$	−4.01005e6	−1.10267	−0.551334	+	0.834284i	$0.685881\pi$
$422$	−2.68827e6	−0.734839
$423$	1.52037e6	0.413141
$424$	1.72608e6	0.466279
$425$	620334.	0.166592
$426$	−1.45678e6	−0.388928
$427$	77800.0	0.0206495
$428$	−1.63443e6	−0.431278
$429$	161226.	0.0422953
$430$	5.36134e6	1.39831
$431$	2.95249e6	0.765587	0.382794	−	0.923834i	$-0.374962\pi$
$431$	2.95249e6	0.765587	0.382794	+	0.923834i	$0.374962\pi$
$432$	186624.	0.0481125
$433$	2.48889e6	0.637950	0.318975	−	0.947763i	$-0.396661\pi$
$433$	2.48889e6	0.637950	0.318975	+	0.947763i	$0.396661\pi$
$434$	243200.	0.0619782
$435$	5.65394e6	1.43261
$436$	2.50000e6	0.629831
$437$	−703248.	−0.176159
$438$	−1.95487e6	−0.486892
$439$	−2.49604e6	−0.618145	−0.309072	−	0.951039i	$-0.600019\pi$
$439$	−2.49604e6	−0.618145	−0.309072	+	0.951039i	$0.600019\pi$
$440$	515584.	0.126960
$441$	−551367.	−0.135003
$442$	158184.	0.0385130
$443$	855216.	0.207046	0.103523	−	0.994627i	$-0.466988\pi$
$443$	855216.	0.207046	0.103523	+	0.994627i	$0.466988\pi$
$444$	−289440.	−0.0696789
$445$	−3.38352e6	−0.809970
$446$	−1.30963e6	−0.311754
$447$	3.40178e6	0.805263
$448$	409600.	0.0964195
$449$	−1.47275e6	−0.344758	−0.172379	−	0.985031i	$-0.555145\pi$
$449$	−1.47275e6	−0.344758	−0.172379	+	0.985031i	$0.555145\pi$
$450$	−858924.	−0.199951
$451$	−937464.	−0.217027
$452$	−1.16624e6	−0.268498
$453$	−4.32864e6	−0.991074
$454$	−1.96270e6	−0.446903
$455$	−1.28440e6	−0.290852
$456$	158976.	0.0358030
$457$	5.56125e6	1.24561	0.622805	−	0.782377i	$-0.285993\pi$
$457$	5.56125e6	1.24561	0.622805	+	0.782377i	$0.285993\pi$
$458$	−2.40670e6	−0.536114
$459$	170586.	0.0377930
$460$	3.09837e6	0.682713
$461$	−8.04203e6	−1.76244	−0.881218	−	0.472710i	$-0.843276\pi$
$461$	−8.04203e6	−1.76244	−0.881218	+	0.472710i	$0.843276\pi$
$462$	381600.	0.0831770
$463$	1.63479e6	0.354412	0.177206	−	0.984174i	$-0.443294\pi$
$463$	1.63479e6	0.354412	0.177206	+	0.984174i	$0.443294\pi$
$464$	2.11610e6	0.456289
$465$	−415872.	−0.0891923
$466$	−5.70226e6	−1.21642
$467$	−3.69145e6	−0.783257	−0.391629	−	0.920123i	$-0.628088\pi$
$467$	−3.69145e6	−0.783257	−0.391629	+	0.920123i	$0.628088\pi$
$468$	−219024.	−0.0462250
$469$	−1.26320e6	−0.265180
$470$	−5.70608e6	−1.19150
$471$	3.42934e6	0.712293
$472$	2.68582e6	0.554910
$473$	1.86942e6	0.384196
$474$	1.60762e6	0.328652
$475$	−731676.	−0.148794
$476$	374400.	0.0757388
$477$	−2.18457e6	−0.439612
$478$	−1.64993e6	−0.330290
$479$	−8.12522e6	−1.61807	−0.809033	−	0.587763i	$-0.800009\pi$
$479$	−8.12522e6	−1.61807	−0.809033	+	0.587763i	$0.800009\pi$
$480$	−700416.	−0.138756
$481$	339690.	0.0669453
$482$	6.37025e6	1.24893
$483$	2.29320e6	0.447274
$484$	−2.39704e6	−0.465117
$485$	−6.53798e6	−1.26209
$486$	−236196.	−0.0453609
$487$	3.52078e6	0.672693	0.336347	−	0.941738i	$-0.390809\pi$
$487$	3.52078e6	0.672693	0.336347	+	0.941738i	$0.390809\pi$
$488$	−49792.0	−0.00946477
$489$	−2.63372e6	−0.498079
$490$	2.06933e6	0.389349
$491$	−31576.0	−0.00591090	−0.00295545	−	0.999996i	$-0.500941\pi$
$491$	−31576.0	−0.00591090	−0.00295545	+	0.999996i	$0.500941\pi$
$492$	1.27354e6	0.237191
$493$	1.93424e6	0.358421
$494$	−186576.	−0.0343984
$495$	−652536.	−0.119699
$496$	−155648.	−0.0284079
$497$	4.04660e6	0.734851
$498$	−2.51705e6	−0.454798
$499$	1.59040e6	0.285927	0.142963	−	0.989728i	$-0.454337\pi$
$499$	1.59040e6	0.285927	0.142963	+	0.989728i	$0.454337\pi$
$500$	−576384.	−0.103107
$501$	−4.16972e6	−0.742185
$502$	5.13331e6	0.909156
$503$	9.81007e6	1.72883	0.864415	−	0.502780i	$-0.167689\pi$
$503$	9.81007e6	1.72883	0.864415	+	0.502780i	$0.167689\pi$
$504$	−518400.	−0.0909052
$505$	1.70377e6	0.297291
$506$	1.08035e6	0.187581
$507$	257049.	0.0444116
$508$	585088.	0.100592
$509$	205940.	0.0352327	0.0176164	−	0.999845i	$-0.494392\pi$
$509$	205940.	0.0352327	0.0176164	+	0.999845i	$0.494392\pi$
$510$	−640224.	−0.108995
$511$	5.43020e6	0.919949
$512$	−262144.	−0.0441942
$513$	−201204.	−0.0337554
$514$	−352056.	−0.0587765
$515$	−1.04576e7	−1.73746
$516$	−2.53958e6	−0.419893
$517$	−1.98962e6	−0.327374
$518$	804000.	0.131653
$519$	−112662.	−0.0183594
$520$	822016.	0.133313
$521$	−7.96711e6	−1.28590	−0.642949	−	0.765909i	$-0.722289\pi$
$521$	−7.96711e6	−1.28590	−0.642949	+	0.765909i	$0.722289\pi$
$522$	−2.67818e6	−0.430194
$523$	9.14536e6	1.46200	0.730998	−	0.682379i	$-0.239055\pi$
$523$	9.14536e6	1.46200	0.730998	+	0.682379i	$0.239055\pi$
$524$	−4.86733e6	−0.774395
$525$	2.38590e6	0.377793
$526$	−4.66834e6	−0.735695
$527$	−142272.	−0.0223148
$528$	−244224.	−0.0381244
$529$	55961.0	0.00869453
$530$	8.19888e6	1.26784
$531$	−3.39925e6	−0.523174
$532$	−441600.	−0.0676472
$533$	−1.49464e6	−0.227886
$534$	1.60272e6	0.243223
$535$	−7.76355e6	−1.17267
$536$	808448.	0.121546
$537$	−5.48197e6	−0.820354
$538$	825624.	0.122978
$539$	721542.	0.106977
$540$	886464.	0.130821
$541$	3.46356e6	0.508780	0.254390	−	0.967102i	$-0.418125\pi$
$541$	3.46356e6	0.508780	0.254390	+	0.967102i	$0.418125\pi$
$542$	5.46766e6	0.799473
$543$	1.95485e6	0.284521
$544$	−239616.	−0.0347151
$545$	1.18750e7	1.71255
$546$	608400.	0.0873389
$547$	1.51606e6	0.216645	0.108322	−	0.994116i	$-0.465452\pi$
$547$	1.51606e6	0.216645	0.108322	+	0.994116i	$0.465452\pi$
$548$	−3.37824e6	−0.480551
$549$	63018.0	0.00892347
$550$	1.12402e6	0.158441
$551$	−2.28142e6	−0.320129
$552$	−1.46765e6	−0.205010
$553$	−4.46560e6	−0.620965
$554$	9.15252e6	1.26697
$555$	−1.37484e6	−0.189461
$556$	−4.39597e6	−0.603070
$557$	1.30872e7	1.78734	0.893670	−	0.448725i	$-0.148122\pi$
$557$	1.30872e7	1.78734	0.893670	+	0.448725i	$0.148122\pi$
$558$	196992.	0.0267832
$559$	2.98048e6	0.403420
$560$	1.94560e6	0.262170
$561$	−223236.	−0.0299473
$562$	−8.01848e6	−1.07091
$563$	−5.79391e6	−0.770372	−0.385186	−	0.922839i	$-0.625863\pi$
$563$	−5.79391e6	−0.770372	−0.385186	+	0.922839i	$0.625863\pi$
$564$	2.70288e6	0.357791
$565$	−5.53964e6	−0.730063
$566$	−1.10536e6	−0.145032
$567$	656100.	0.0857062
$568$	−2.58982e6	−0.336821
$569$	7.35779e6	0.952723	0.476362	−	0.879249i	$-0.341955\pi$
$569$	7.35779e6	0.952723	0.476362	+	0.879249i	$0.341955\pi$
$570$	755136.	0.0973505
$571$	−4.28317e6	−0.549763	−0.274881	−	0.961478i	$-0.588639\pi$
$571$	−4.28317e6	−0.549763	−0.274881	+	0.961478i	$0.588639\pi$
$572$	286624.	0.0366288
$573$	−6.31944e6	−0.804067
$574$	−3.53760e6	−0.448156
$575$	6.75475e6	0.852000
$576$	331776.	0.0416667
$577$	−4.11440e6	−0.514478	−0.257239	−	0.966348i	$-0.582813\pi$
$577$	−4.11440e6	−0.514478	−0.257239	+	0.966348i	$0.582813\pi$
$578$	5.46040e6	0.679838
$579$	6.69753e6	0.830268
$580$	1.00515e7	1.24068
$581$	6.99180e6	0.859308
$582$	3.09694e6	0.378987
$583$	2.85882e6	0.348350
$584$	−3.47533e6	−0.421661
$585$	−1.04036e6	−0.125689
$586$	2.10486e6	0.253210
$587$	−102218.	−0.0122442	−0.00612212	−	0.999981i	$-0.501949\pi$
$587$	−102218.	−0.0122442	−0.00612212	+	0.999981i	$0.501949\pi$
$588$	−980208.	−0.116916
$589$	167808.	0.0199308
$590$	1.27577e7	1.50883
$591$	4.29008e6	0.505239
$592$	−514560.	−0.0603437
$593$	2.27550e6	0.265730	0.132865	−	0.991134i	$-0.457582\pi$
$593$	2.27550e6	0.265730	0.132865	+	0.991134i	$0.457582\pi$
$594$	309096.	0.0359441
$595$	1.77840e6	0.205938
$596$	6.04762e6	0.697379
$597$	4.46609e6	0.512851
$598$	1.72245e6	0.196967
$599$	−1.05767e7	−1.20443	−0.602216	−	0.798333i	$-0.705716\pi$
$599$	−1.05767e7	−1.20443	−0.602216	+	0.798333i	$0.705716\pi$
$600$	−1.52698e6	−0.173163
$601$	−1.32670e7	−1.49826	−0.749128	−	0.662425i	$-0.769527\pi$
$601$	−1.32670e7	−1.49826	−0.749128	+	0.662425i	$0.769527\pi$
$602$	7.05440e6	0.793357
$603$	−1.02319e6	−0.114595
$604$	−7.69536e6	−0.858295
$605$	−1.13859e7	−1.26468
$606$	−807048.	−0.0892725
$607$	1.56567e7	1.72476	0.862380	−	0.506261i	$-0.168973\pi$
$607$	1.56567e7	1.72476	0.862380	+	0.506261i	$0.168973\pi$
$608$	282624.	0.0310063
$609$	7.43940e6	0.812821
$610$	−236512.	−0.0257353
$611$	−3.17213e6	−0.343754
$612$	303264.	0.0327297
$613$	3.48923e6	0.375041	0.187520	−	0.982261i	$-0.439955\pi$
$613$	3.48923e6	0.375041	0.187520	+	0.982261i	$0.439955\pi$
$614$	2.48637e6	0.266161
$615$	6.04930e6	0.644937
$616$	678400.	0.0720334
$617$	2.73237e6	0.288953	0.144476	−	0.989508i	$-0.453850\pi$
$617$	2.73237e6	0.288953	0.144476	+	0.989508i	$0.453850\pi$
$618$	4.95360e6	0.521735
$619$	1.04378e6	0.109492	0.0547458	−	0.998500i	$-0.482565\pi$
$619$	1.04378e6	0.109492	0.0547458	+	0.998500i	$0.482565\pi$
$620$	−739328.	−0.0772428
$621$	1.85749e6	0.193285
$622$	7.57042e6	0.784592
$623$	−4.45200e6	−0.459552
$624$	−389376.	−0.0400320
$625$	−1.10222e7	−1.12867
$626$	−9.44847e6	−0.963664
$627$	263304.	0.0267478
$628$	6.09661e6	0.616864
$629$	−470340.	−0.0474008
$630$	−2.46240e6	−0.247177
$631$	−1.21693e7	−1.21672	−0.608360	−	0.793661i	$-0.708172\pi$
$631$	−1.21693e7	−1.21672	−0.608360	+	0.793661i	$0.708172\pi$
$632$	2.85798e6	0.284621
$633$	6.04861e6	0.599993
$634$	2.69717e6	0.266492
$635$	2.77917e6	0.273515
$636$	−3.88368e6	−0.380716
$637$	1.15038e6	0.112330
$638$	3.50478e6	0.340886
$639$	3.27775e6	0.317558
$640$	−1.24518e6	−0.120167
$641$	−1.69525e7	−1.62963	−0.814813	−	0.579724i	$-0.803160\pi$
$641$	−1.69525e7	−1.62963	−0.814813	+	0.579724i	$0.803160\pi$
$642$	3.67747e6	0.352137
$643$	−1.46819e7	−1.40041	−0.700204	−	0.713943i	$-0.746908\pi$
$643$	−1.46819e7	−1.40041	−0.700204	+	0.713943i	$0.746908\pi$
$644$	4.07680e6	0.387351
$645$	−1.20630e7	−1.14171
$646$	258336.	0.0243559
$647$	−4.59194e6	−0.431256	−0.215628	−	0.976476i	$-0.569180\pi$
$647$	−4.59194e6	−0.431256	−0.215628	+	0.976476i	$0.569180\pi$
$648$	−419904.	−0.0392837
$649$	4.44840e6	0.414564
$650$	1.79208e6	0.166369
$651$	−547200.	−0.0506050
$652$	−4.68218e6	−0.431349
$653$	9.17275e6	0.841815	0.420907	−	0.907104i	$-0.361712\pi$
$653$	9.17275e6	0.841815	0.420907	+	0.907104i	$0.361712\pi$
$654$	−5.62500e6	−0.514255
$655$	−2.31198e7	−2.10562
$656$	2.26406e6	0.205414
$657$	4.39846e6	0.397546
$658$	−7.50800e6	−0.676020
$659$	−1.06207e7	−0.952660	−0.476330	−	0.879267i	$-0.658033\pi$
$659$	−1.06207e7	−0.952660	−0.476330	+	0.879267i	$0.658033\pi$
$660$	−1.16006e6	−0.103663
$661$	−7.02779e6	−0.625626	−0.312813	−	0.949815i	$-0.601271\pi$
$661$	−7.02779e6	−0.625626	−0.312813	+	0.949815i	$0.601271\pi$
$662$	−7.04000e6	−0.624349
$663$	−355914.	−0.0314457
$664$	−4.47475e6	−0.393866
$665$	−2.09760e6	−0.183937
$666$	651240.	0.0568926
$667$	2.10618e7	1.83308
$668$	−7.41283e6	−0.642751
$669$	2.94667e6	0.254546
$670$	3.84013e6	0.330490
$671$	−82468.0	−0.00707097
$672$	−921600.	−0.0787262
$673$	1.24935e7	1.06327	0.531637	−	0.846972i	$-0.321577\pi$
$673$	1.24935e7	1.06327	0.531637	+	0.846972i	$0.321577\pi$
$674$	−1.42860e7	−1.21133
$675$	1.93258e6	0.163259
$676$	456976.	0.0384615
$677$	−1.84440e7	−1.54662	−0.773310	−	0.634028i	$-0.781401\pi$
$677$	−1.84440e7	−1.54662	−0.773310	+	0.634028i	$0.781401\pi$
$678$	2.62404e6	0.219228
$679$	−8.60260e6	−0.716070
$680$	−1.13818e6	−0.0943924
$681$	4.41607e6	0.364895
$682$	−257792.	−0.0212231
$683$	−2.20988e6	−0.181266	−0.0906332	−	0.995884i	$-0.528889\pi$
$683$	−2.20988e6	−0.181266	−0.0906332	+	0.995884i	$0.528889\pi$
$684$	−357696.	−0.0292330
$685$	−1.60466e7	−1.30665
$686$	9.44560e6	0.766336
$687$	5.41507e6	0.437736
$688$	−4.51482e6	−0.363638
$689$	4.55793e6	0.365780
$690$	−6.97133e6	−0.557433
$691$	6.70893e6	0.534513	0.267256	−	0.963625i	$-0.413883\pi$
$691$	6.70893e6	0.534513	0.267256	+	0.963625i	$0.413883\pi$
$692$	−200288.	−0.0158997
$693$	−858600.	−0.0679138
$694$	−9.07896e6	−0.715546
$695$	−2.08808e7	−1.63978
$696$	−4.76122e6	−0.372559
$697$	2.06950e6	0.161355
$698$	115496.	0.00897281
$699$	1.28301e7	0.993200
$700$	4.24160e6	0.327178
$701$	1.74858e7	1.34397	0.671986	−	0.740564i	$-0.265442\pi$
$701$	1.74858e7	1.34397	0.671986	+	0.740564i	$0.265442\pi$
$702$	492804.	0.0377426
$703$	554760.	0.0423367
$704$	−434176.	−0.0330167
$705$	1.28387e7	0.972854
$706$	−1.35346e7	−1.02196
$707$	2.24180e6	0.168674
$708$	−6.04310e6	−0.453082
$709$	1.01508e7	0.758380	0.379190	−	0.925319i	$-0.376203\pi$
$709$	1.01508e7	0.758380	0.379190	+	0.925319i	$0.376203\pi$
$710$	−1.23017e7	−0.915837
$711$	−3.61714e6	−0.268343
$712$	2.84928e6	0.210637
$713$	−1.54918e6	−0.114125
$714$	−842400.	−0.0618405
$715$	1.36146e6	0.0995958
$716$	−9.74573e6	−0.710447
$717$	3.71234e6	0.269681
$718$	−4.23914e6	−0.306879
$719$	2.55785e7	1.84524	0.922619	−	0.385714i	$-0.126045\pi$
$719$	2.55785e7	1.84524	0.922619	+	0.385714i	$0.126045\pi$
$720$	1.57594e6	0.113294
$721$	−1.37600e7	−0.985781
$722$	9.59969e6	0.685353
$723$	−1.43331e7	−1.01975
$724$	3.47530e6	0.246403
$725$	2.19132e7	1.54832
$726$	5.39334e6	0.379766
$727$	1.10507e7	0.775453	0.387727	−	0.921774i	$-0.373261\pi$
$727$	1.10507e7	0.775453	0.387727	+	0.921774i	$0.373261\pi$
$728$	1.08160e6	0.0756377
$729$	531441.	0.0370370
$730$	−1.65078e7	−1.14652
$731$	−4.12682e6	−0.285642
$732$	112032.	0.00772795
$733$	−1.69360e7	−1.16426	−0.582132	−	0.813094i	$-0.697781\pi$
$733$	−1.69360e7	−1.16426	−0.582132	+	0.813094i	$0.697781\pi$
$734$	3.70442e6	0.253793
$735$	−4.65599e6	−0.317902
$736$	−2.60915e6	−0.177544
$737$	1.33899e6	0.0908049
$738$	−2.86546e6	−0.193666
$739$	−2.26764e6	−0.152744	−0.0763718	−	0.997079i	$-0.524334\pi$
$739$	−2.26764e6	−0.152744	−0.0763718	+	0.997079i	$0.524334\pi$
$740$	−2.44416e6	−0.164078
$741$	419796.	0.0280862
$742$	1.07880e7	0.719335
$743$	1.03705e7	0.689175	0.344588	−	0.938754i	$-0.388019\pi$
$743$	1.03705e7	0.689175	0.344588	+	0.938754i	$0.388019\pi$
$744$	350208.	0.0231950
$745$	2.87262e7	1.89621
$746$	1.76530e7	1.16137
$747$	5.66336e6	0.371341
$748$	−396864.	−0.0259351
$749$	−1.02152e7	−0.665338
$750$	1.29686e6	0.0841863
$751$	8.19751e6	0.530374	0.265187	−	0.964197i	$-0.414566\pi$
$751$	8.19751e6	0.530374	0.265187	+	0.964197i	$0.414566\pi$
$752$	4.80512e6	0.309856
$753$	−1.15500e7	−0.742323
$754$	5.58782e6	0.357943
$755$	−3.65530e7	−2.33375
$756$	1.16640e6	0.0742238
$757$	−2.55308e7	−1.61929	−0.809646	−	0.586919i	$-0.800341\pi$
$757$	−2.55308e7	−1.61929	−0.809646	+	0.586919i	$0.800341\pi$
$758$	−1.64849e7	−1.04211
$759$	−2.43079e6	−0.153159
$760$	1.34246e6	0.0843080
$761$	3.05669e7	1.91333	0.956664	−	0.291193i	$-0.0940520\pi$
$761$	3.05669e7	1.91333	0.956664	+	0.291193i	$0.0940520\pi$
$762$	−1.31645e6	−0.0821327
$763$	1.56250e7	0.971647
$764$	−1.12346e7	−0.696342
$765$	1.44050e6	0.0889940
$766$	−1.54266e7	−0.949948
$767$	7.09225e6	0.435307
$768$	589824.	0.0360844
$769$	1.53170e7	0.934026	0.467013	−	0.884251i	$-0.345330\pi$
$769$	1.53170e7	0.934026	0.467013	+	0.884251i	$0.345330\pi$
$770$	3.22240e6	0.195863
$771$	792126.	0.0479908
$772$	1.19067e7	0.719033
$773$	−2.36631e6	−0.142437	−0.0712186	−	0.997461i	$-0.522689\pi$
$773$	−2.36631e6	−0.142437	−0.0712186	+	0.997461i	$0.522689\pi$
$774$	5.71406e6	0.342841
$775$	−1.61181e6	−0.0963960
$776$	5.50566e6	0.328213
$777$	−1.80900e6	−0.107494
$778$	2.31262e6	0.136979
$779$	−2.44094e6	−0.144117
$780$	−1.84954e6	−0.108849
$781$	−4.28940e6	−0.251634
$782$	−2.38493e6	−0.139463
$783$	6.02591e6	0.351252
$784$	−1.74259e6	−0.101252
$785$	2.89589e7	1.67729
$786$	1.09515e7	0.632291
$787$	1.39758e7	0.804339	0.402170	−	0.915565i	$-0.368256\pi$
$787$	1.39758e7	0.804339	0.402170	+	0.915565i	$0.368256\pi$
$788$	7.62682e6	0.437550
$789$	1.05038e7	0.600692
$790$	1.35754e7	0.773901
$791$	−7.28900e6	−0.414216
$792$	549504.	0.0311285
$793$	−131482.	−0.00742478
$794$	−1.24790e7	−0.702471
$795$	−1.84475e7	−1.03519
$796$	7.93971e6	0.444142
$797$	1.87784e7	1.04716	0.523580	−	0.851977i	$-0.324596\pi$
$797$	1.87784e7	1.04716	0.523580	+	0.851977i	$0.324596\pi$
$798$	993600.	0.0552337
$799$	4.39218e6	0.243396
$800$	−2.71462e6	−0.149963
$801$	−3.60612e6	−0.198591
$802$	−1.60340e7	−0.880251
$803$	−5.75601e6	−0.315016
$804$	−1.81901e6	−0.0992418
$805$	1.93648e7	1.05323
$806$	−411008.	−0.0222850
$807$	−1.85765e6	−0.100411
$808$	−1.43475e6	−0.0773123
$809$	1.48824e7	0.799468	0.399734	−	0.916631i	$-0.369102\pi$
$809$	1.48824e7	0.799468	0.399734	+	0.916631i	$0.369102\pi$
$810$	−1.99454e6	−0.106815
$811$	−5.40879e6	−0.288767	−0.144384	−	0.989522i	$-0.546120\pi$
$811$	−5.40879e6	−0.288767	−0.144384	+	0.989522i	$0.546120\pi$
$812$	1.32256e7	0.703923
$813$	−1.23022e7	−0.652767
$814$	−852240.	−0.0450818
$815$	−2.22403e7	−1.17286
$816$	539136.	0.0283448
$817$	4.86754e6	0.255126
$818$	−5.49935e6	−0.287361
$819$	−1.36890e6	−0.0713119
$820$	1.07543e7	0.558532
$821$	−1.00651e7	−0.521147	−0.260574	−	0.965454i	$-0.583912\pi$
$821$	−1.00651e7	−0.521147	−0.260574	+	0.965454i	$0.583912\pi$
$822$	7.60104e6	0.392368
$823$	1.22916e7	0.632569	0.316285	−	0.948664i	$-0.397565\pi$
$823$	1.22916e7	0.632569	0.316285	+	0.948664i	$0.397565\pi$
$824$	8.80640e6	0.451836
$825$	−2.52905e6	−0.129367
$826$	1.67864e7	0.856066
$827$	1.20837e7	0.614378	0.307189	−	0.951648i	$-0.400612\pi$
$827$	1.20837e7	0.614378	0.307189	+	0.951648i	$0.400612\pi$
$828$	3.30221e6	0.167390
$829$	2.23351e7	1.12876	0.564379	−	0.825516i	$-0.309116\pi$
$829$	2.23351e7	1.12876	0.564379	+	0.825516i	$0.309116\pi$
$830$	−2.12551e7	−1.07095
$831$	−2.05932e7	−1.03448
$832$	−692224.	−0.0346688
$833$	−1.59284e6	−0.0795352
$834$	9.89093e6	0.492404
$835$	−3.52110e7	−1.74768
$836$	468096.	0.0231643
$837$	−443232.	−0.0218684
$838$	−1.43533e7	−0.706062
$839$	9.89083e6	0.485096	0.242548	−	0.970139i	$-0.422017\pi$
$839$	9.89083e6	0.485096	0.242548	+	0.970139i	$0.422017\pi$
$840$	−4.37760e6	−0.214061
$841$	4.78156e7	2.33120
$842$	1.60402e7	0.779704
$843$	1.80416e7	0.874391
$844$	1.07531e7	0.519610
$845$	2.17064e6	0.104579
$846$	−6.08148e6	−0.292135
$847$	−1.49815e7	−0.717541
$848$	−6.90432e6	−0.329709
$849$	2.48706e6	0.118418
$850$	−2.48134e6	−0.117798
$851$	−5.12148e6	−0.242422
$852$	5.82710e6	0.275013
$853$	4.84677e6	0.228076	0.114038	−	0.993476i	$-0.463621\pi$
$853$	4.84677e6	0.228076	0.114038	+	0.993476i	$0.463621\pi$
$854$	−311200.	−0.0146014
$855$	−1.69906e6	−0.0794863
$856$	6.53773e6	0.304960
$857$	−1.60904e6	−0.0748368	−0.0374184	−	0.999300i	$-0.511913\pi$
$857$	−1.60904e6	−0.0748368	−0.0374184	+	0.999300i	$0.511913\pi$
$858$	−644904.	−0.0299073
$859$	1.19283e7	0.551562	0.275781	−	0.961221i	$-0.411064\pi$
$859$	1.19283e7	0.551562	0.275781	+	0.961221i	$0.411064\pi$
$860$	−2.14454e7	−0.988752
$861$	7.95960e6	0.365918
$862$	−1.18099e7	−0.541352
$863$	−1.04897e7	−0.479442	−0.239721	−	0.970842i	$-0.577056\pi$
$863$	−1.04897e7	−0.479442	−0.239721	+	0.970842i	$0.577056\pi$
$864$	−746496.	−0.0340207
$865$	−951368.	−0.0432323
$866$	−9.95558e6	−0.451099
$867$	−1.22859e7	−0.555085
$868$	−972800.	−0.0438252
$869$	4.73354e6	0.212636
$870$	−2.26158e7	−1.01301
$871$	2.13481e6	0.0953484
$872$	−1.00000e7	−0.445358
$873$	−6.96811e6	−0.309442
$874$	2.81299e6	0.124563
$875$	−3.60240e6	−0.159064
$876$	7.81949e6	0.344285
$877$	−4.41328e7	−1.93759	−0.968797	−	0.247856i	$-0.920274\pi$
$877$	−4.41328e7	−1.93759	−0.968797	+	0.247856i	$0.920274\pi$
$878$	9.98416e6	0.437094
$879$	−4.73594e6	−0.206745
$880$	−2.06234e6	−0.0897744
$881$	−3.06299e7	−1.32955	−0.664777	−	0.747042i	$-0.731473\pi$
$881$	−3.06299e7	−1.32955	−0.664777	+	0.747042i	$0.731473\pi$
$882$	2.20547e6	0.0954617
$883$	2.84920e7	1.22976	0.614880	−	0.788620i	$-0.289204\pi$
$883$	2.84920e7	1.22976	0.614880	+	0.788620i	$0.289204\pi$
$884$	−632736.	−0.0272328
$885$	−2.87047e7	−1.23196
$886$	−3.42086e6	−0.146404
$887$	4.68877e6	0.200101	0.100051	−	0.994982i	$-0.468100\pi$
$887$	4.68877e6	0.200101	0.100051	+	0.994982i	$0.468100\pi$
$888$	1.15776e6	0.0492704
$889$	3.65680e6	0.155184
$890$	1.35341e7	0.572735
$891$	−695466.	−0.0293482
$892$	5.23853e6	0.220443
$893$	−5.18052e6	−0.217393
$894$	−1.36071e7	−0.569407
$895$	−4.62922e7	−1.93175
$896$	−1.63840e6	−0.0681789
$897$	−3.87551e6	−0.160823
$898$	5.89101e6	0.243780
$899$	−5.02573e6	−0.207396
$900$	3.43570e6	0.141387
$901$	−6.31098e6	−0.258991
$902$	3.74986e6	0.153461
$903$	−1.58724e7	−0.647774
$904$	4.66496e6	0.189857
$905$	1.65077e7	0.669983
$906$	1.73146e7	0.700795
$907$	−3.71845e7	−1.50087	−0.750437	−	0.660942i	$-0.770157\pi$
$907$	−3.71845e7	−1.50087	−0.750437	+	0.660942i	$0.770157\pi$
$908$	7.85078e6	0.316008
$909$	1.81586e6	0.0728907
$910$	5.13760e6	0.205663
$911$	1.04275e7	0.416280	0.208140	−	0.978099i	$-0.433259\pi$
$911$	1.04275e7	0.416280	0.208140	+	0.978099i	$0.433259\pi$
$912$	−635904.	−0.0253165
$913$	−7.41131e6	−0.294251
$914$	−2.22450e7	−0.880780
$915$	532152.	0.0210128
$916$	9.62678e6	0.379090
$917$	−3.04208e7	−1.19467
$918$	−682344.	−0.0267237
$919$	3.27595e7	1.27953	0.639763	−	0.768572i	$-0.279033\pi$
$919$	3.27595e7	1.27953	0.639763	+	0.768572i	$0.279033\pi$
$920$	−1.23935e7	−0.482751
$921$	−5.59433e6	−0.217320
$922$	3.21681e7	1.24623
$923$	−6.83875e6	−0.264224
$924$	−1.52640e6	−0.0588150
$925$	−5.32851e6	−0.204763
$926$	−6.53915e6	−0.250607
$927$	−1.11456e7	−0.425995
$928$	−8.46438e6	−0.322645
$929$	1.27643e7	0.485242	0.242621	−	0.970121i	$-0.421993\pi$
$929$	1.27643e7	0.485242	0.242621	+	0.970121i	$0.421993\pi$
$930$	1.66349e6	0.0630685
$931$	1.87873e6	0.0710380
$932$	2.28091e7	0.860137
$933$	−1.70334e7	−0.640617
$934$	1.47658e7	0.553847
$935$	−1.88510e6	−0.0705190
$936$	876096.	0.0326860
$937$	1.78729e7	0.665037	0.332518	−	0.943097i	$-0.392102\pi$
$937$	1.78729e7	0.665037	0.332518	+	0.943097i	$0.392102\pi$
$938$	5.05280e6	0.187510
$939$	2.12591e7	0.786829
$940$	2.28243e7	0.842516
$941$	6.47876e6	0.238516	0.119258	−	0.992863i	$-0.461948\pi$
$941$	6.47876e6	0.238516	0.119258	+	0.992863i	$0.461948\pi$
$942$	−1.37174e7	−0.503667
$943$	2.25345e7	0.825218
$944$	−1.07433e7	−0.392381
$945$	5.54040e6	0.201819
$946$	−7.47766e6	−0.271668
$947$	−3.14448e7	−1.13939	−0.569696	−	0.821855i	$-0.692939\pi$
$947$	−3.14448e7	−1.13939	−0.569696	+	0.821855i	$0.692939\pi$
$948$	−6.43046e6	−0.232392
$949$	−9.17704e6	−0.330778
$950$	2.92670e6	0.105213
$951$	−6.06863e6	−0.217590
$952$	−1.49760e6	−0.0535554
$953$	2.24057e7	0.799145	0.399572	−	0.916702i	$-0.369159\pi$
$953$	2.24057e7	0.799145	0.399572	+	0.916702i	$0.369159\pi$
$954$	8.73828e6	0.310853
$955$	−5.33642e7	−1.89340
$956$	6.59971e6	0.233550
$957$	−7.88576e6	−0.278333
$958$	3.25009e7	1.14415
$959$	−2.11140e7	−0.741351
$960$	2.80166e6	0.0981156
$961$	−2.82595e7	−0.987088
$962$	−1.35876e6	−0.0473375
$963$	−8.27431e6	−0.287519
$964$	−2.54810e7	−0.883128
$965$	5.65569e7	1.95509
$966$	−9.17280e6	−0.316271
$967$	−3.08409e7	−1.06062	−0.530311	−	0.847803i	$-0.677925\pi$
$967$	−3.08409e7	−1.06062	−0.530311	+	0.847803i	$0.677925\pi$
$968$	9.58816e6	0.328887
$969$	−581256.	−0.0198865
$970$	2.61519e7	0.892430
$971$	1.92695e7	0.655877	0.327939	−	0.944699i	$-0.393646\pi$
$971$	1.92695e7	0.655877	0.327939	+	0.944699i	$0.393646\pi$
$972$	944784.	0.0320750
$973$	−2.74748e7	−0.930363
$974$	−1.40831e7	−0.475666
$975$	−4.03217e6	−0.135840
$976$	199168.	0.00669260
$977$	1.57140e6	0.0526684	0.0263342	−	0.999653i	$-0.491617\pi$
$977$	1.57140e6	0.0526684	0.0263342	+	0.999653i	$0.491617\pi$
$978$	1.05349e7	0.352195
$979$	4.71912e6	0.157364
$980$	−8.27731e6	−0.275311
$981$	1.26562e7	0.419887
$982$	126304.	0.00417964
$983$	−3.62448e7	−1.19636	−0.598180	−	0.801362i	$-0.704109\pi$
$983$	−3.62448e7	−1.19636	−0.598180	+	0.801362i	$0.704109\pi$
$984$	−5.09414e6	−0.167719
$985$	3.62274e7	1.18972
$986$	−7.73698e6	−0.253442
$987$	1.68930e7	0.551968
$988$	746304.	0.0243234
$989$	−4.49365e7	−1.46086
$990$	2.61014e6	0.0846402
$991$	−2.93799e7	−0.950313	−0.475157	−	0.879901i	$-0.657609\pi$
$991$	−2.93799e7	−0.950313	−0.475157	+	0.879901i	$0.657609\pi$
$992$	622592.	0.0200874
$993$	1.58400e7	0.509779
$994$	−1.61864e7	−0.519618
$995$	3.77136e7	1.20765
$996$	1.00682e7	0.321590
$997$	1.72567e7	0.549818	0.274909	−	0.961470i	$-0.411352\pi$
$997$	1.72567e7	0.549818	0.274909	+	0.961470i	$0.411352\pi$
$998$	−6.36160e6	−0.202181
$999$	−1.46529e6	−0.0464526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.6.a.c.1.1	✓	1
3.2	odd	2		234.6.a.d.1.1		1
4.3	odd	2		624.6.a.d.1.1		1
13.12	even	2		1014.6.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.a.c.1.1	✓	1	1.1	even	1	trivial
234.6.a.d.1.1		1	3.2	odd	2
624.6.a.d.1.1		1	4.3	odd	2
1014.6.a.f.1.1		1	13.12	even	2

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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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