LMFDB - Embedded newform 390.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,6,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	390.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:	$62.5496897271$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} -36.0000 q^{11} -144.000 q^{12} +169.000 q^{13} -225.000 q^{15} +256.000 q^{16} +866.000 q^{17} +324.000 q^{18} -116.000 q^{19} +400.000 q^{20} -144.000 q^{22} -2008.00 q^{23} -576.000 q^{24} +625.000 q^{25} +676.000 q^{26} -729.000 q^{27} +4006.00 q^{29} -900.000 q^{30} +3752.00 q^{31} +1024.00 q^{32} +324.000 q^{33} +3464.00 q^{34} +1296.00 q^{36} -8490.00 q^{37} -464.000 q^{38} -1521.00 q^{39} +1600.00 q^{40} -6446.00 q^{41} +20916.0 q^{43} -576.000 q^{44} +2025.00 q^{45} -8032.00 q^{46} +24440.0 q^{47} -2304.00 q^{48} -16807.0 q^{49} +2500.00 q^{50} -7794.00 q^{51} +2704.00 q^{52} -18210.0 q^{53} -2916.00 q^{54} -900.000 q^{55} +1044.00 q^{57} +16024.0 q^{58} +39404.0 q^{59} -3600.00 q^{60} +48718.0 q^{61} +15008.0 q^{62} +4096.00 q^{64} +4225.00 q^{65} +1296.00 q^{66} +52732.0 q^{67} +13856.0 q^{68} +18072.0 q^{69} -52464.0 q^{71} +5184.00 q^{72} -61622.0 q^{73} -33960.0 q^{74} -5625.00 q^{75} -1856.00 q^{76} -6084.00 q^{78} +27544.0 q^{79} +6400.00 q^{80} +6561.00 q^{81} -25784.0 q^{82} +42132.0 q^{83} +21650.0 q^{85} +83664.0 q^{86} -36054.0 q^{87} -2304.00 q^{88} +111970. q^{89} +8100.00 q^{90} -32128.0 q^{92} -33768.0 q^{93} +97760.0 q^{94} -2900.00 q^{95} -9216.00 q^{96} +87586.0 q^{97} -67228.0 q^{98} -2916.00 q^{99} +O(q^{100})\)

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
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−36.0000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	100.000	0.316228
$11$	−36.0000	−0.0897059	−0.0448529	−	0.998994i	$-0.514282\pi$
$11$	−36.0000	−0.0897059	−0.0448529	+	0.998994i	$0.514282\pi$
$12$	−144.000	−0.288675
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	866.000	0.726768	0.363384	−	0.931640i	$-0.381621\pi$
$17$	866.000	0.726768	0.363384	+	0.931640i	$0.381621\pi$
$18$	324.000	0.235702
$19$	−116.000	−0.0737181	−0.0368590	−	0.999320i	$-0.511735\pi$
$19$	−116.000	−0.0737181	−0.0368590	+	0.999320i	$0.511735\pi$
$20$	400.000	0.223607
$21$	0	0
$22$	−144.000	−0.0634316
$23$	−2008.00	−0.791488	−0.395744	−	0.918361i	$-0.629513\pi$
$23$	−2008.00	−0.791488	−0.395744	+	0.918361i	$0.629513\pi$
$24$	−576.000	−0.204124
$25$	625.000	0.200000
$26$	676.000	0.196116
$27$	−729.000	−0.192450
$28$	0	0
$29$	4006.00	0.884537	0.442268	−	0.896883i	$-0.354174\pi$
$29$	4006.00	0.884537	0.442268	+	0.896883i	$0.354174\pi$
$30$	−900.000	−0.182574
$31$	3752.00	0.701227	0.350613	−	0.936520i	$-0.385973\pi$
$31$	3752.00	0.701227	0.350613	+	0.936520i	$0.385973\pi$
$32$	1024.00	0.176777
$33$	324.000	0.0517917
$34$	3464.00	0.513902
$35$	0	0
$36$	1296.00	0.166667
$37$	−8490.00	−1.01954	−0.509769	−	0.860311i	$-0.670269\pi$
$37$	−8490.00	−1.01954	−0.509769	+	0.860311i	$0.670269\pi$
$38$	−464.000	−0.0521266
$39$	−1521.00	−0.160128
$40$	1600.00	0.158114
$41$	−6446.00	−0.598867	−0.299434	−	0.954117i	$-0.596798\pi$
$41$	−6446.00	−0.598867	−0.299434	+	0.954117i	$0.596798\pi$
$42$	0	0
$43$	20916.0	1.72507	0.862537	−	0.505995i	$-0.168875\pi$
$43$	20916.0	1.72507	0.862537	+	0.505995i	$0.168875\pi$
$44$	−576.000	−0.0448529
$45$	2025.00	0.149071
$46$	−8032.00	−0.559666
$47$	24440.0	1.61383	0.806913	−	0.590671i	$-0.201137\pi$
$47$	24440.0	1.61383	0.806913	+	0.590671i	$0.201137\pi$
$48$	−2304.00	−0.144338
$49$	−16807.0	−1.00000
$50$	2500.00	0.141421
$51$	−7794.00	−0.419599
$52$	2704.00	0.138675
$53$	−18210.0	−0.890472	−0.445236	−	0.895413i	$-0.646880\pi$
$53$	−18210.0	−0.890472	−0.445236	+	0.895413i	$0.646880\pi$
$54$	−2916.00	−0.136083
$55$	−900.000	−0.0401177
$56$	0	0
$57$	1044.00	0.0425612
$58$	16024.0	0.625462
$59$	39404.0	1.47370	0.736852	−	0.676054i	$-0.236311\pi$
$59$	39404.0	1.47370	0.736852	+	0.676054i	$0.236311\pi$
$60$	−3600.00	−0.129099
$61$	48718.0	1.67635	0.838175	−	0.545401i	$-0.183623\pi$
$61$	48718.0	1.67635	0.838175	+	0.545401i	$0.183623\pi$
$62$	15008.0	0.495842
$63$	0	0
$64$	4096.00	0.125000
$65$	4225.00	0.124035
$66$	1296.00	0.0366223
$67$	52732.0	1.43512	0.717558	−	0.696498i	$-0.245260\pi$
$67$	52732.0	1.43512	0.717558	+	0.696498i	$0.245260\pi$
$68$	13856.0	0.363384
$69$	18072.0	0.456966
$70$	0	0
$71$	−52464.0	−1.23514	−0.617569	−	0.786517i	$-0.711882\pi$
$71$	−52464.0	−1.23514	−0.617569	+	0.786517i	$0.711882\pi$
$72$	5184.00	0.117851
$73$	−61622.0	−1.35341	−0.676704	−	0.736255i	$-0.736592\pi$
$73$	−61622.0	−1.35341	−0.676704	+	0.736255i	$0.736592\pi$
$74$	−33960.0	−0.720922
$75$	−5625.00	−0.115470
$76$	−1856.00	−0.0368590
$77$	0	0
$78$	−6084.00	−0.113228
$79$	27544.0	0.496546	0.248273	−	0.968690i	$-0.420137\pi$
$79$	27544.0	0.496546	0.248273	+	0.968690i	$0.420137\pi$
$80$	6400.00	0.111803
$81$	6561.00	0.111111
$82$	−25784.0	−0.423463
$83$	42132.0	0.671300	0.335650	−	0.941987i	$-0.391044\pi$
$83$	42132.0	0.671300	0.335650	+	0.941987i	$0.391044\pi$
$84$	0	0
$85$	21650.0	0.325020
$86$	83664.0	1.21981
$87$	−36054.0	−0.510688
$88$	−2304.00	−0.0317158
$89$	111970.	1.49840	0.749198	−	0.662346i	$-0.230439\pi$
$89$	111970.	1.49840	0.749198	+	0.662346i	$0.230439\pi$
$90$	8100.00	0.105409
$91$	0	0
$92$	−32128.0	−0.395744
$93$	−33768.0	−0.404854
$94$	97760.0	1.14115
$95$	−2900.00	−0.0329677
$96$	−9216.00	−0.102062
$97$	87586.0	0.945160	0.472580	−	0.881288i	$-0.343323\pi$
$97$	87586.0	0.945160	0.472580	+	0.881288i	$0.343323\pi$
$98$	−67228.0	−0.707107
$99$	−2916.00	−0.0299020
$100$	10000.0	0.100000
$101$	46158.0	0.450239	0.225120	−	0.974331i	$-0.427723\pi$
$101$	46158.0	0.450239	0.225120	+	0.974331i	$0.427723\pi$
$102$	−31176.0	−0.296702
$103$	166400.	1.54547	0.772734	−	0.634729i	$-0.218888\pi$
$103$	166400.	1.54547	0.772734	+	0.634729i	$0.218888\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	−72840.0	−0.629659
$107$	−9468.00	−0.0799464	−0.0399732	−	0.999201i	$-0.512727\pi$
$107$	−9468.00	−0.0799464	−0.0399732	+	0.999201i	$0.512727\pi$
$108$	−11664.0	−0.0962250
$109$	99070.0	0.798686	0.399343	−	0.916802i	$-0.369238\pi$
$109$	99070.0	0.798686	0.399343	+	0.916802i	$0.369238\pi$
$110$	−3600.00	−0.0283675
$111$	76410.0	0.588630
$112$	0	0
$113$	166690.	1.22804	0.614021	−	0.789289i	$-0.289551\pi$
$113$	166690.	1.22804	0.614021	+	0.789289i	$0.289551\pi$
$114$	4176.00	0.0300953
$115$	−50200.0	−0.353964
$116$	64096.0	0.442268
$117$	13689.0	0.0924500
$118$	157616.	1.04207
$119$	0	0
$120$	−14400.0	−0.0912871
$121$	−159755.	−0.991953
$122$	194872.	1.18536
$123$	58014.0	0.345756
$124$	60032.0	0.350613
$125$	15625.0	0.0894427
$126$	0	0
$127$	61432.0	0.337976	0.168988	−	0.985618i	$-0.445950\pi$
$127$	61432.0	0.337976	0.168988	+	0.985618i	$0.445950\pi$
$128$	16384.0	0.0883883
$129$	−188244.	−0.995971
$130$	16900.0	0.0877058
$131$	5212.00	0.0265354	0.0132677	−	0.999912i	$-0.495777\pi$
$131$	5212.00	0.0265354	0.0132677	+	0.999912i	$0.495777\pi$
$132$	5184.00	0.0258958
$133$	0	0
$134$	210928.	1.01478
$135$	−18225.0	−0.0860663
$136$	55424.0	0.256951
$137$	−43710.0	−0.198966	−0.0994831	−	0.995039i	$-0.531719\pi$
$137$	−43710.0	−0.198966	−0.0994831	+	0.995039i	$0.531719\pi$
$138$	72288.0	0.323123
$139$	−67148.0	−0.294779	−0.147389	−	0.989079i	$-0.547087\pi$
$139$	−67148.0	−0.294779	−0.147389	+	0.989079i	$0.547087\pi$
$140$	0	0
$141$	−219960.	−0.931743
$142$	−209856.	−0.873375
$143$	−6084.00	−0.0248799
$144$	20736.0	0.0833333
$145$	100150.	0.395577
$146$	−246488.	−0.957004
$147$	151263.	0.577350
$148$	−135840.	−0.509769
$149$	189366.	0.698773	0.349387	−	0.936979i	$-0.386390\pi$
$149$	189366.	0.698773	0.349387	+	0.936979i	$0.386390\pi$
$150$	−22500.0	−0.0816497
$151$	460720.	1.64435	0.822176	−	0.569234i	$-0.192760\pi$
$151$	460720.	1.64435	0.822176	+	0.569234i	$0.192760\pi$
$152$	−7424.00	−0.0260633
$153$	70146.0	0.242256
$154$	0	0
$155$	93800.0	0.313598
$156$	−24336.0	−0.0800641
$157$	−572418.	−1.85338	−0.926689	−	0.375828i	$-0.877358\pi$
$157$	−572418.	−1.85338	−0.926689	+	0.375828i	$0.877358\pi$
$158$	110176.	0.351111
$159$	163890.	0.514114
$160$	25600.0	0.0790569
$161$	0	0
$162$	26244.0	0.0785674
$163$	−445444.	−1.31318	−0.656590	−	0.754248i	$-0.728002\pi$
$163$	−445444.	−1.31318	−0.656590	+	0.754248i	$0.728002\pi$
$164$	−103136.	−0.299434
$165$	8100.00	0.0231620
$166$	168528.	0.474681
$167$	−535568.	−1.48602	−0.743008	−	0.669283i	$-0.766602\pi$
$167$	−535568.	−1.48602	−0.743008	+	0.669283i	$0.766602\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	86600.0	0.229824
$171$	−9396.00	−0.0245727
$172$	334656.	0.862537
$173$	−356202.	−0.904859	−0.452430	−	0.891800i	$-0.649443\pi$
$173$	−356202.	−0.904859	−0.452430	+	0.891800i	$0.649443\pi$
$174$	−144216.	−0.361111
$175$	0	0
$176$	−9216.00	−0.0224265
$177$	−354636.	−0.850843
$178$	447880.	1.05953
$179$	−15908.0	−0.0371093	−0.0185547	−	0.999828i	$-0.505906\pi$
$179$	−15908.0	−0.0371093	−0.0185547	+	0.999828i	$0.505906\pi$
$180$	32400.0	0.0745356
$181$	−200714.	−0.455388	−0.227694	−	0.973733i	$-0.573119\pi$
$181$	−200714.	−0.455388	−0.227694	+	0.973733i	$0.573119\pi$
$182$	0	0
$183$	−438462.	−0.967841
$184$	−128512.	−0.279833
$185$	−212250.	−0.455951
$186$	−135072.	−0.286275
$187$	−31176.0	−0.0651953
$188$	391040.	0.806913
$189$	0	0
$190$	−11600.0	−0.0233117
$191$	−168480.	−0.334168	−0.167084	−	0.985943i	$-0.553435\pi$
$191$	−168480.	−0.334168	−0.167084	+	0.985943i	$0.553435\pi$
$192$	−36864.0	−0.0721688
$193$	924210.	1.78598	0.892992	−	0.450073i	$-0.148602\pi$
$193$	924210.	1.78598	0.892992	+	0.450073i	$0.148602\pi$
$194$	350344.	0.668329
$195$	−38025.0	−0.0716115
$196$	−268912.	−0.500000
$197$	−516986.	−0.949103	−0.474551	−	0.880228i	$-0.657390\pi$
$197$	−516986.	−0.949103	−0.474551	+	0.880228i	$0.657390\pi$
$198$	−11664.0	−0.0211439
$199$	63312.0	0.113332	0.0566661	−	0.998393i	$-0.481953\pi$
$199$	63312.0	0.113332	0.0566661	+	0.998393i	$0.481953\pi$
$200$	40000.0	0.0707107
$201$	−474588.	−0.828565
$202$	184632.	0.318367
$203$	0	0
$204$	−124704.	−0.209800
$205$	−161150.	−0.267822
$206$	665600.	1.09281
$207$	−162648.	−0.263829
$208$	43264.0	0.0693375
$209$	4176.00	0.00661294
$210$	0	0
$211$	−277892.	−0.429705	−0.214852	−	0.976647i	$-0.568927\pi$
$211$	−277892.	−0.429705	−0.214852	+	0.976647i	$0.568927\pi$
$212$	−291360.	−0.445236
$213$	472176.	0.713107
$214$	−37872.0	−0.0565306
$215$	522900.	0.771476
$216$	−46656.0	−0.0680414
$217$	0	0
$218$	396280.	0.564756
$219$	554598.	0.781390
$220$	−14400.0	−0.0200588
$221$	146354.	0.201569
$222$	305640.	0.416225
$223$	−952088.	−1.28208	−0.641040	−	0.767508i	$-0.721497\pi$
$223$	−952088.	−1.28208	−0.641040	+	0.767508i	$0.721497\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	666760.	0.868357
$227$	−25804.0	−0.0332370	−0.0166185	−	0.999862i	$-0.505290\pi$
$227$	−25804.0	−0.0332370	−0.0166185	+	0.999862i	$0.505290\pi$
$228$	16704.0	0.0212806
$229$	−162826.	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$229$	−162826.	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$230$	−200800.	−0.250290
$231$	0	0
$232$	256384.	0.312731
$233$	−720006.	−0.868853	−0.434427	−	0.900707i	$-0.643049\pi$
$233$	−720006.	−0.868853	−0.434427	+	0.900707i	$0.643049\pi$
$234$	54756.0	0.0653720
$235$	611000.	0.721725
$236$	630464.	0.736852
$237$	−247896.	−0.286681
$238$	0	0
$239$	−311368.	−0.352597	−0.176299	−	0.984337i	$-0.556412\pi$
$239$	−311368.	−0.352597	−0.176299	+	0.984337i	$0.556412\pi$
$240$	−57600.0	−0.0645497
$241$	−388702.	−0.431096	−0.215548	−	0.976493i	$-0.569154\pi$
$241$	−388702.	−0.431096	−0.215548	+	0.976493i	$0.569154\pi$
$242$	−639020.	−0.701417
$243$	−59049.0	−0.0641500
$244$	779488.	0.838175
$245$	−420175.	−0.447214
$246$	232056.	0.244487
$247$	−19604.0	−0.0204457
$248$	240128.	0.247921
$249$	−379188.	−0.387575
$250$	62500.0	0.0632456
$251$	1.65237e6	1.65548	0.827739	−	0.561113i	$-0.189627\pi$
$251$	1.65237e6	1.65548	0.827739	+	0.561113i	$0.189627\pi$
$252$	0	0
$253$	72288.0	0.0710011
$254$	245728.	0.238985
$255$	−194850.	−0.187651
$256$	65536.0	0.0625000
$257$	−247054.	−0.233324	−0.116662	−	0.993172i	$-0.537219\pi$
$257$	−247054.	−0.233324	−0.116662	+	0.993172i	$0.537219\pi$
$258$	−752976.	−0.704258
$259$	0	0
$260$	67600.0	0.0620174
$261$	324486.	0.294846
$262$	20848.0	0.0187634
$263$	1.48014e6	1.31951	0.659754	−	0.751481i	$-0.270660\pi$
$263$	1.48014e6	1.31951	0.659754	+	0.751481i	$0.270660\pi$
$264$	20736.0	0.0183111
$265$	−455250.	−0.398231
$266$	0	0
$267$	−1.00773e6	−0.865099
$268$	843712.	0.717558
$269$	568854.	0.479314	0.239657	−	0.970858i	$-0.422965\pi$
$269$	568854.	0.479314	0.239657	+	0.970858i	$0.422965\pi$
$270$	−72900.0	−0.0608581
$271$	−1.84622e6	−1.52707	−0.763536	−	0.645766i	$-0.776538\pi$
$271$	−1.84622e6	−1.52707	−0.763536	+	0.645766i	$0.776538\pi$
$272$	221696.	0.181692
$273$	0	0
$274$	−174840.	−0.140690
$275$	−22500.0	−0.0179412
$276$	289152.	0.228483
$277$	−1.51193e6	−1.18395	−0.591974	−	0.805957i	$-0.701651\pi$
$277$	−1.51193e6	−1.18395	−0.591974	+	0.805957i	$0.701651\pi$
$278$	−268592.	−0.208440
$279$	303912.	0.233742
$280$	0	0
$281$	−1.11931e6	−0.845638	−0.422819	−	0.906214i	$-0.638959\pi$
$281$	−1.11931e6	−0.845638	−0.422819	+	0.906214i	$0.638959\pi$
$282$	−879840.	−0.658841
$283$	1.10362e6	0.819131	0.409566	−	0.912281i	$-0.365680\pi$
$283$	1.10362e6	0.819131	0.409566	+	0.912281i	$0.365680\pi$
$284$	−839424.	−0.617569
$285$	26100.0	0.0190339
$286$	−24336.0	−0.0175928
$287$	0	0
$288$	82944.0	0.0589256
$289$	−669901.	−0.471809
$290$	400600.	0.279715
$291$	−788274.	−0.545688
$292$	−985952.	−0.676704
$293$	1.69705e6	1.15485	0.577424	−	0.816445i	$-0.304058\pi$
$293$	1.69705e6	1.15485	0.577424	+	0.816445i	$0.304058\pi$
$294$	605052.	0.408248
$295$	985100.	0.659060
$296$	−543360.	−0.360461
$297$	26244.0	0.0172639
$298$	757464.	0.494107
$299$	−339352.	−0.219519
$300$	−90000.0	−0.0577350
$301$	0	0
$302$	1.84288e6	1.16273
$303$	−415422.	−0.259946
$304$	−29696.0	−0.0184295
$305$	1.21795e6	0.749687
$306$	280584.	0.171301
$307$	−1.78255e6	−1.07943	−0.539716	−	0.841847i	$-0.681468\pi$
$307$	−1.78255e6	−1.07943	−0.539716	+	0.841847i	$0.681468\pi$
$308$	0	0
$309$	−1.49760e6	−0.892277
$310$	375200.	0.221747
$311$	−371304.	−0.217685	−0.108843	−	0.994059i	$-0.534714\pi$
$311$	−371304.	−0.217685	−0.108843	+	0.994059i	$0.534714\pi$
$312$	−97344.0	−0.0566139
$313$	−2.83652e6	−1.63653	−0.818266	−	0.574839i	$-0.805065\pi$
$313$	−2.83652e6	−1.63653	−0.818266	+	0.574839i	$0.805065\pi$
$314$	−2.28967e6	−1.31054
$315$	0	0
$316$	440704.	0.248273
$317$	−1.12762e6	−0.630251	−0.315126	−	0.949050i	$-0.602047\pi$
$317$	−1.12762e6	−0.630251	−0.315126	+	0.949050i	$0.602047\pi$
$318$	655560.	0.363534
$319$	−144216.	−0.0793481
$320$	102400.	0.0559017
$321$	85212.0	0.0461571
$322$	0	0
$323$	−100456.	−0.0535759
$324$	104976.	0.0555556
$325$	105625.	0.0554700
$326$	−1.78178e6	−0.928558
$327$	−891630.	−0.461121
$328$	−412544.	−0.211732
$329$	0	0
$330$	32400.0	0.0163780
$331$	2.25634e6	1.13197	0.565985	−	0.824416i	$-0.308496\pi$
$331$	2.25634e6	1.13197	0.565985	+	0.824416i	$0.308496\pi$
$332$	674112.	0.335650
$333$	−687690.	−0.339846
$334$	−2.14227e6	−1.05077
$335$	1.31830e6	0.641804
$336$	0	0
$337$	1.36802e6	0.656171	0.328086	−	0.944648i	$-0.393597\pi$
$337$	1.36802e6	0.656171	0.328086	+	0.944648i	$0.393597\pi$
$338$	114244.	0.0543928
$339$	−1.50021e6	−0.709011
$340$	346400.	0.162510
$341$	−135072.	−0.0629042
$342$	−37584.0	−0.0173755
$343$	0	0
$344$	1.33862e6	0.609905
$345$	451800.	0.204361
$346$	−1.42481e6	−0.639832
$347$	−3.10798e6	−1.38565	−0.692827	−	0.721104i	$-0.743635\pi$
$347$	−3.10798e6	−1.38565	−0.692827	+	0.721104i	$0.743635\pi$
$348$	−576864.	−0.255344
$349$	−3.25083e6	−1.42867	−0.714334	−	0.699805i	$-0.753270\pi$
$349$	−3.25083e6	−1.42867	−0.714334	+	0.699805i	$0.753270\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	−36864.0	−0.0158579
$353$	1.22721e6	0.524182	0.262091	−	0.965043i	$-0.415588\pi$
$353$	1.22721e6	0.524182	0.262091	+	0.965043i	$0.415588\pi$
$354$	−1.41854e6	−0.601637
$355$	−1.31160e6	−0.552371
$356$	1.79152e6	0.749198
$357$	0	0
$358$	−63632.0	−0.0262403
$359$	3.14162e6	1.28652	0.643261	−	0.765647i	$-0.277581\pi$
$359$	3.14162e6	1.28652	0.643261	+	0.765647i	$0.277581\pi$
$360$	129600.	0.0527046
$361$	−2.46264e6	−0.994566
$362$	−802856.	−0.322008
$363$	1.43780e6	0.572704
$364$	0	0
$365$	−1.54055e6	−0.605262
$366$	−1.75385e6	−0.684367
$367$	−2.02822e6	−0.786048	−0.393024	−	0.919528i	$-0.628571\pi$
$367$	−2.02822e6	−0.786048	−0.393024	+	0.919528i	$0.628571\pi$
$368$	−514048.	−0.197872
$369$	−522126.	−0.199622
$370$	−849000.	−0.322406
$371$	0	0
$372$	−540288.	−0.202427
$373$	−3.52756e6	−1.31281	−0.656406	−	0.754408i	$-0.727924\pi$
$373$	−3.52756e6	−1.31281	−0.656406	+	0.754408i	$0.727924\pi$
$374$	−124704.	−0.0461000
$375$	−140625.	−0.0516398
$376$	1.56416e6	0.570573
$377$	677014.	0.245326
$378$	0	0
$379$	−1.96556e6	−0.702893	−0.351446	−	0.936208i	$-0.614310\pi$
$379$	−1.96556e6	−0.702893	−0.351446	+	0.936208i	$0.614310\pi$
$380$	−46400.0	−0.0164839
$381$	−552888.	−0.195130
$382$	−673920.	−0.236293
$383$	2.64577e6	0.921626	0.460813	−	0.887497i	$-0.347558\pi$
$383$	2.64577e6	0.921626	0.460813	+	0.887497i	$0.347558\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	3.69684e6	1.26288
$387$	1.69420e6	0.575024
$388$	1.40138e6	0.472580
$389$	2.75077e6	0.921679	0.460839	−	0.887484i	$-0.347548\pi$
$389$	2.75077e6	0.921679	0.460839	+	0.887484i	$0.347548\pi$
$390$	−152100.	−0.0506370
$391$	−1.73893e6	−0.575228
$392$	−1.07565e6	−0.353553
$393$	−46908.0	−0.0153202
$394$	−2.06794e6	−0.671117
$395$	688600.	0.222062
$396$	−46656.0	−0.0149510
$397$	−2.36349e6	−0.752623	−0.376312	−	0.926493i	$-0.622808\pi$
$397$	−2.36349e6	−0.752623	−0.376312	+	0.926493i	$0.622808\pi$
$398$	253248.	0.0801380
$399$	0	0
$400$	160000.	0.0500000
$401$	3.90359e6	1.21228	0.606141	−	0.795357i	$-0.292717\pi$
$401$	3.90359e6	1.21228	0.606141	+	0.795357i	$0.292717\pi$
$402$	−1.89835e6	−0.585884
$403$	634088.	0.194485
$404$	738528.	0.225120
$405$	164025.	0.0496904
$406$	0	0
$407$	305640.	0.0914585
$408$	−498816.	−0.148351
$409$	−4.59122e6	−1.35713	−0.678563	−	0.734542i	$-0.737397\pi$
$409$	−4.59122e6	−1.35713	−0.678563	+	0.734542i	$0.737397\pi$
$410$	−644600.	−0.189378
$411$	393390.	0.114873
$412$	2.66240e6	0.772734
$413$	0	0
$414$	−650592.	−0.186555
$415$	1.05330e6	0.300215
$416$	173056.	0.0490290
$417$	604332.	0.170191
$418$	16704.0	0.00467606
$419$	−971604.	−0.270367	−0.135184	−	0.990821i	$-0.543162\pi$
$419$	−971604.	−0.270367	−0.135184	+	0.990821i	$0.543162\pi$
$420$	0	0
$421$	1.79513e6	0.493617	0.246808	−	0.969064i	$-0.420618\pi$
$421$	1.79513e6	0.493617	0.246808	+	0.969064i	$0.420618\pi$
$422$	−1.11157e6	−0.303847
$423$	1.97964e6	0.537942
$424$	−1.16544e6	−0.314829
$425$	541250.	0.145354
$426$	1.88870e6	0.504243
$427$	0	0
$428$	−151488.	−0.0399732
$429$	54756.0	0.0143644
$430$	2.09160e6	0.545516
$431$	81336.0	0.0210906	0.0105453	−	0.999944i	$-0.496643\pi$
$431$	81336.0	0.0210906	0.0105453	+	0.999944i	$0.496643\pi$
$432$	−186624.	−0.0481125
$433$	6.27167e6	1.60755	0.803773	−	0.594937i	$-0.202823\pi$
$433$	6.27167e6	1.60755	0.803773	+	0.594937i	$0.202823\pi$
$434$	0	0
$435$	−901350.	−0.228386
$436$	1.58512e6	0.399343
$437$	232928.	0.0583469
$438$	2.21839e6	0.552526
$439$	338160.	0.0837454	0.0418727	−	0.999123i	$-0.486668\pi$
$439$	338160.	0.0837454	0.0418727	+	0.999123i	$0.486668\pi$
$440$	−57600.0	−0.0141837
$441$	−1.36137e6	−0.333333
$442$	585416.	0.142531
$443$	−990876.	−0.239889	−0.119944	−	0.992781i	$-0.538272\pi$
$443$	−990876.	−0.239889	−0.119944	+	0.992781i	$0.538272\pi$
$444$	1.22256e6	0.294315
$445$	2.79925e6	0.670103
$446$	−3.80835e6	−0.906567
$447$	−1.70429e6	−0.403437
$448$	0	0
$449$	−2.39434e6	−0.560493	−0.280247	−	0.959928i	$-0.590416\pi$
$449$	−2.39434e6	−0.560493	−0.280247	+	0.959928i	$0.590416\pi$
$450$	202500.	0.0471405
$451$	232056.	0.0537219
$452$	2.66704e6	0.614021
$453$	−4.14648e6	−0.949367
$454$	−103216.	−0.0235021
$455$	0	0
$456$	66816.0	0.0150476
$457$	−5.94229e6	−1.33096	−0.665478	−	0.746418i	$-0.731772\pi$
$457$	−5.94229e6	−1.33096	−0.665478	+	0.746418i	$0.731772\pi$
$458$	−651304.	−0.145084
$459$	−631314.	−0.139866
$460$	−803200.	−0.176982
$461$	6.32502e6	1.38615	0.693074	−	0.720866i	$-0.256256\pi$
$461$	6.32502e6	1.38615	0.693074	+	0.720866i	$0.256256\pi$
$462$	0	0
$463$	−7.22697e6	−1.56676	−0.783382	−	0.621540i	$-0.786507\pi$
$463$	−7.22697e6	−1.56676	−0.783382	+	0.621540i	$0.786507\pi$
$464$	1.02554e6	0.221134
$465$	−844200.	−0.181056
$466$	−2.88002e6	−0.614372
$467$	8.15611e6	1.73058	0.865288	−	0.501275i	$-0.167135\pi$
$467$	8.15611e6	1.73058	0.865288	+	0.501275i	$0.167135\pi$
$468$	219024.	0.0462250
$469$	0	0
$470$	2.44400e6	0.510336
$471$	5.15176e6	1.07005
$472$	2.52186e6	0.521033
$473$	−752976.	−0.154749
$474$	−991584.	−0.202714
$475$	−72500.0	−0.0147436
$476$	0	0
$477$	−1.47501e6	−0.296824
$478$	−1.24547e6	−0.249324
$479$	6.76215e6	1.34662	0.673312	−	0.739359i	$-0.264871\pi$
$479$	6.76215e6	1.34662	0.673312	+	0.739359i	$0.264871\pi$
$480$	−230400.	−0.0456435
$481$	−1.43481e6	−0.282769
$482$	−1.55481e6	−0.304831
$483$	0	0
$484$	−2.55608e6	−0.495976
$485$	2.18965e6	0.422688
$486$	−236196.	−0.0453609
$487$	920656.	0.175904	0.0879519	−	0.996125i	$-0.471968\pi$
$487$	920656.	0.175904	0.0879519	+	0.996125i	$0.471968\pi$
$488$	3.11795e6	0.592679
$489$	4.00900e6	0.758165
$490$	−1.68070e6	−0.316228
$491$	8.58232e6	1.60658	0.803288	−	0.595591i	$-0.203082\pi$
$491$	8.58232e6	1.60658	0.803288	+	0.595591i	$0.203082\pi$
$492$	928224.	0.172878
$493$	3.46920e6	0.642853
$494$	−78416.0	−0.0144573
$495$	−72900.0	−0.0133726
$496$	960512.	0.175307
$497$	0	0
$498$	−1.51675e6	−0.274057
$499$	−4.52194e6	−0.812968	−0.406484	−	0.913658i	$-0.633245\pi$
$499$	−4.52194e6	−0.812968	−0.406484	+	0.913658i	$0.633245\pi$
$500$	250000.	0.0447214
$501$	4.82011e6	0.857952
$502$	6.60949e6	1.17060
$503$	3.09735e6	0.545847	0.272923	−	0.962036i	$-0.412009\pi$
$503$	3.09735e6	0.545847	0.272923	+	0.962036i	$0.412009\pi$
$504$	0	0
$505$	1.15395e6	0.201353
$506$	289152.	0.0502053
$507$	−257049.	−0.0444116
$508$	982912.	0.168988
$509$	5.11731e6	0.875482	0.437741	−	0.899101i	$-0.355779\pi$
$509$	5.11731e6	0.875482	0.437741	+	0.899101i	$0.355779\pi$
$510$	−779400.	−0.132689
$511$	0	0
$512$	262144.	0.0441942
$513$	84564.0	0.0141871
$514$	−988216.	−0.164985
$515$	4.16000e6	0.691155
$516$	−3.01190e6	−0.497986
$517$	−879840.	−0.144770
$518$	0	0
$519$	3.20582e6	0.522421
$520$	270400.	0.0438529
$521$	1.53607e6	0.247924	0.123962	−	0.992287i	$-0.460440\pi$
$521$	1.53607e6	0.247924	0.123962	+	0.992287i	$0.460440\pi$
$522$	1.29794e6	0.208487
$523$	7.57464e6	1.21090	0.605450	−	0.795884i	$-0.292993\pi$
$523$	7.57464e6	1.21090	0.605450	+	0.795884i	$0.292993\pi$
$524$	83392.0	0.0132677
$525$	0	0
$526$	5.92054e6	0.933034
$527$	3.24923e6	0.509629
$528$	82944.0	0.0129479
$529$	−2.40428e6	−0.373547
$530$	−1.82100e6	−0.281592
$531$	3.19172e6	0.491235
$532$	0	0
$533$	−1.08937e6	−0.166096
$534$	−4.03092e6	−0.611718
$535$	−236700.	−0.0357531
$536$	3.37485e6	0.507390
$537$	143172.	0.0214251
$538$	2.27542e6	0.338926
$539$	605052.	0.0897059
$540$	−291600.	−0.0430331
$541$	6.74942e6	0.991456	0.495728	−	0.868478i	$-0.334901\pi$
$541$	6.74942e6	0.991456	0.495728	+	0.868478i	$0.334901\pi$
$542$	−7.38486e6	−1.07980
$543$	1.80643e6	0.262918
$544$	886784.	0.128476
$545$	2.47675e6	0.357183
$546$	0	0
$547$	317180.	0.0453250	0.0226625	−	0.999743i	$-0.492786\pi$
$547$	317180.	0.0453250	0.0226625	+	0.999743i	$0.492786\pi$
$548$	−699360.	−0.0994831
$549$	3.94616e6	0.558784
$550$	−90000.0	−0.0126863
$551$	−464696.	−0.0652064
$552$	1.15661e6	0.161562
$553$	0	0
$554$	−6.04772e6	−0.837177
$555$	1.91025e6	0.263244
$556$	−1.07437e6	−0.147389
$557$	−3.05624e6	−0.417397	−0.208699	−	0.977980i	$-0.566923\pi$
$557$	−3.05624e6	−0.417397	−0.208699	+	0.977980i	$0.566923\pi$
$558$	1.21565e6	0.165281
$559$	3.53480e6	0.478449
$560$	0	0
$561$	280584.	0.0376405
$562$	−4.47724e6	−0.597956
$563$	−6.53293e6	−0.868635	−0.434317	−	0.900760i	$-0.643010\pi$
$563$	−6.53293e6	−0.868635	−0.434317	+	0.900760i	$0.643010\pi$
$564$	−3.51936e6	−0.465871
$565$	4.16725e6	0.549197
$566$	4.41448e6	0.579213
$567$	0	0
$568$	−3.35770e6	−0.436687
$569$	1.17458e7	1.52091	0.760455	−	0.649390i	$-0.224976\pi$
$569$	1.17458e7	1.52091	0.760455	+	0.649390i	$0.224976\pi$
$570$	104400.	0.0134590
$571$	6.69919e6	0.859868	0.429934	−	0.902860i	$-0.358537\pi$
$571$	6.69919e6	0.859868	0.429934	+	0.902860i	$0.358537\pi$
$572$	−97344.0	−0.0124400
$573$	1.51632e6	0.192932
$574$	0	0
$575$	−1.25500e6	−0.158298
$576$	331776.	0.0416667
$577$	−1.12177e7	−1.40270	−0.701352	−	0.712815i	$-0.747420\pi$
$577$	−1.12177e7	−1.40270	−0.701352	+	0.712815i	$0.747420\pi$
$578$	−2.67960e6	−0.333619
$579$	−8.31789e6	−1.03114
$580$	1.60240e6	0.197788
$581$	0	0
$582$	−3.15310e6	−0.385860
$583$	655560.	0.0798806
$584$	−3.94381e6	−0.478502
$585$	342225.	0.0413449
$586$	6.78818e6	0.816600
$587$	−5.15445e6	−0.617429	−0.308715	−	0.951155i	$-0.599899\pi$
$587$	−5.15445e6	−0.617429	−0.308715	+	0.951155i	$0.599899\pi$
$588$	2.42021e6	0.288675
$589$	−435232.	−0.0516931
$590$	3.94040e6	0.466026
$591$	4.65287e6	0.547965
$592$	−2.17344e6	−0.254884
$593$	−1.65909e6	−0.193747	−0.0968733	−	0.995297i	$-0.530884\pi$
$593$	−1.65909e6	−0.193747	−0.0968733	+	0.995297i	$0.530884\pi$
$594$	104976.	0.0122074
$595$	0	0
$596$	3.02986e6	0.349387
$597$	−569808.	−0.0654324
$598$	−1.35741e6	−0.155223
$599$	1.11675e7	1.27171	0.635856	−	0.771808i	$-0.280647\pi$
$599$	1.11675e7	1.27171	0.635856	+	0.771808i	$0.280647\pi$
$600$	−360000.	−0.0408248
$601$	−1.21782e7	−1.37530	−0.687652	−	0.726041i	$-0.741358\pi$
$601$	−1.21782e7	−1.37530	−0.687652	+	0.726041i	$0.741358\pi$
$602$	0	0
$603$	4.27129e6	0.478372
$604$	7.37152e6	0.822176
$605$	−3.99388e6	−0.443615
$606$	−1.66169e6	−0.183809
$607$	1.21542e7	1.33893	0.669463	−	0.742846i	$-0.266524\pi$
$607$	1.21542e7	1.33893	0.669463	+	0.742846i	$0.266524\pi$
$608$	−118784.	−0.0130316
$609$	0	0
$610$	4.87180e6	0.530109
$611$	4.13036e6	0.447595
$612$	1.12234e6	0.121128
$613$	−9.65761e6	−1.03805	−0.519025	−	0.854759i	$-0.673705\pi$
$613$	−9.65761e6	−1.03805	−0.519025	+	0.854759i	$0.673705\pi$
$614$	−7.13019e6	−0.763274
$615$	1.45035e6	0.154627
$616$	0	0
$617$	6.21346e6	0.657083	0.328542	−	0.944489i	$-0.393443\pi$
$617$	6.21346e6	0.657083	0.328542	+	0.944489i	$0.393443\pi$
$618$	−5.99040e6	−0.630935
$619$	−1.85348e6	−0.194430	−0.0972148	−	0.995263i	$-0.530993\pi$
$619$	−1.85348e6	−0.194430	−0.0972148	+	0.995263i	$0.530993\pi$
$620$	1.50080e6	0.156799
$621$	1.46383e6	0.152322
$622$	−1.48522e6	−0.153927
$623$	0	0
$624$	−389376.	−0.0400320
$625$	390625.	0.0400000
$626$	−1.13461e7	−1.15720
$627$	−37584.0	−0.00381798
$628$	−9.15869e6	−0.926689
$629$	−7.35234e6	−0.740967
$630$	0	0
$631$	1.19584e7	1.19564	0.597818	−	0.801632i	$-0.296034\pi$
$631$	1.19584e7	1.19564	0.597818	+	0.801632i	$0.296034\pi$
$632$	1.76282e6	0.175555
$633$	2.50103e6	0.248090
$634$	−4.51047e6	−0.445655
$635$	1.53580e6	0.151147
$636$	2.62224e6	0.257057
$637$	−2.84038e6	−0.277350
$638$	−576864.	−0.0561076
$639$	−4.24958e6	−0.411713
$640$	409600.	0.0395285
$641$	7.26405e6	0.698287	0.349143	−	0.937069i	$-0.386473\pi$
$641$	7.26405e6	0.698287	0.349143	+	0.937069i	$0.386473\pi$
$642$	340848.	0.0326380
$643$	−725204.	−0.0691724	−0.0345862	−	0.999402i	$-0.511011\pi$
$643$	−725204.	−0.0691724	−0.0345862	+	0.999402i	$0.511011\pi$
$644$	0	0
$645$	−4.70610e6	−0.445412
$646$	−401824.	−0.0378839
$647$	−1.89184e7	−1.77674	−0.888371	−	0.459127i	$-0.848162\pi$
$647$	−1.89184e7	−1.77674	−0.888371	+	0.459127i	$0.848162\pi$
$648$	419904.	0.0392837
$649$	−1.41854e6	−0.132200
$650$	422500.	0.0392232
$651$	0	0
$652$	−7.12710e6	−0.656590
$653$	−1.05204e7	−0.965496	−0.482748	−	0.875759i	$-0.660361\pi$
$653$	−1.05204e7	−0.965496	−0.482748	+	0.875759i	$0.660361\pi$
$654$	−3.56652e6	−0.326062
$655$	130300.	0.0118670
$656$	−1.65018e6	−0.149717
$657$	−4.99138e6	−0.451136
$658$	0	0
$659$	−7.36794e6	−0.660895	−0.330448	−	0.943824i	$-0.607200\pi$
$659$	−7.36794e6	−0.660895	−0.330448	+	0.943824i	$0.607200\pi$
$660$	129600.	0.0115810
$661$	1.94357e7	1.73020	0.865098	−	0.501602i	$-0.167256\pi$
$661$	1.94357e7	1.73020	0.865098	+	0.501602i	$0.167256\pi$
$662$	9.02536e6	0.800423
$663$	−1.31719e6	−0.116376
$664$	2.69645e6	0.237340
$665$	0	0
$666$	−2.75076e6	−0.240307
$667$	−8.04405e6	−0.700100
$668$	−8.56909e6	−0.743008
$669$	8.56879e6	0.740209
$670$	5.27320e6	0.453824
$671$	−1.75385e6	−0.150378
$672$	0	0
$673$	8.61671e6	0.733337	0.366669	−	0.930352i	$-0.380498\pi$
$673$	8.61671e6	0.733337	0.366669	+	0.930352i	$0.380498\pi$
$674$	5.47207e6	0.463983
$675$	−455625.	−0.0384900
$676$	456976.	0.0384615
$677$	−1.12435e7	−0.942824	−0.471412	−	0.881913i	$-0.656256\pi$
$677$	−1.12435e7	−0.942824	−0.471412	+	0.881913i	$0.656256\pi$
$678$	−6.00084e6	−0.501346
$679$	0	0
$680$	1.38560e6	0.114912
$681$	232236.	0.0191894
$682$	−540288.	−0.0444800
$683$	−1.76204e7	−1.44532	−0.722659	−	0.691204i	$-0.757081\pi$
$683$	−1.76204e7	−1.44532	−0.722659	+	0.691204i	$0.757081\pi$
$684$	−150336.	−0.0122863
$685$	−1.09275e6	−0.0889804
$686$	0	0
$687$	1.46543e6	0.118461
$688$	5.35450e6	0.431268
$689$	−3.07749e6	−0.246972
$690$	1.80720e6	0.144505
$691$	1.07776e7	0.858674	0.429337	−	0.903144i	$-0.358747\pi$
$691$	1.07776e7	0.858674	0.429337	+	0.903144i	$0.358747\pi$
$692$	−5.69923e6	−0.452430
$693$	0	0
$694$	−1.24319e7	−0.979805
$695$	−1.67870e6	−0.131829
$696$	−2.30746e6	−0.180555
$697$	−5.58224e6	−0.435237
$698$	−1.30033e7	−1.01022
$699$	6.48005e6	0.501633
$700$	0	0
$701$	1.34370e6	0.103278	0.0516390	−	0.998666i	$-0.483555\pi$
$701$	1.34370e6	0.103278	0.0516390	+	0.998666i	$0.483555\pi$
$702$	−492804.	−0.0377426
$703$	984840.	0.0751584
$704$	−147456.	−0.0112132
$705$	−5.49900e6	−0.416688
$706$	4.90884e6	0.370653
$707$	0	0
$708$	−5.67418e6	−0.425422
$709$	8.49695e6	0.634815	0.317408	−	0.948289i	$-0.397188\pi$
$709$	8.49695e6	0.634815	0.317408	+	0.948289i	$0.397188\pi$
$710$	−5.24640e6	−0.390585
$711$	2.23106e6	0.165515
$712$	7.16608e6	0.529763
$713$	−7.53402e6	−0.555012
$714$	0	0
$715$	−152100.	−0.0111266
$716$	−254528.	−0.0185547
$717$	2.80231e6	0.203572
$718$	1.25665e7	0.909708
$719$	−387920.	−0.0279847	−0.0139923	−	0.999902i	$-0.504454\pi$
$719$	−387920.	−0.0279847	−0.0139923	+	0.999902i	$0.504454\pi$
$720$	518400.	0.0372678
$721$	0	0
$722$	−9.85057e6	−0.703264
$723$	3.49832e6	0.248893
$724$	−3.21142e6	−0.227694
$725$	2.50375e6	0.176907
$726$	5.75118e6	0.404963
$727$	−1.05219e7	−0.738343	−0.369172	−	0.929361i	$-0.620359\pi$
$727$	−1.05219e7	−0.738343	−0.369172	+	0.929361i	$0.620359\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	−6.16220e6	−0.427985
$731$	1.81133e7	1.25373
$732$	−7.01539e6	−0.483921
$733$	−2.81010e7	−1.93180	−0.965900	−	0.258917i	$-0.916634\pi$
$733$	−2.81010e7	−1.93180	−0.965900	+	0.258917i	$0.916634\pi$
$734$	−8.11286e6	−0.555820
$735$	3.78158e6	0.258199
$736$	−2.05619e6	−0.139917
$737$	−1.89835e6	−0.128738
$738$	−2.08850e6	−0.141154
$739$	−1.41619e7	−0.953914	−0.476957	−	0.878927i	$-0.658260\pi$
$739$	−1.41619e7	−0.953914	−0.476957	+	0.878927i	$0.658260\pi$
$740$	−3.39600e6	−0.227976
$741$	176436.	0.0118043
$742$	0	0
$743$	−4.74354e6	−0.315232	−0.157616	−	0.987500i	$-0.550381\pi$
$743$	−4.74354e6	−0.315232	−0.157616	+	0.987500i	$0.550381\pi$
$744$	−2.16115e6	−0.143137
$745$	4.73415e6	0.312501
$746$	−1.41102e7	−0.928298
$747$	3.41269e6	0.223767
$748$	−498816.	−0.0325977
$749$	0	0
$750$	−562500.	−0.0365148
$751$	4.34967e6	0.281421	0.140711	−	0.990051i	$-0.455061\pi$
$751$	4.34967e6	0.281421	0.140711	+	0.990051i	$0.455061\pi$
$752$	6.25664e6	0.403456
$753$	−1.48713e7	−0.955791
$754$	2.70806e6	0.173472
$755$	1.15180e7	0.735376
$756$	0	0
$757$	1.53022e7	0.970541	0.485270	−	0.874364i	$-0.338721\pi$
$757$	1.53022e7	0.970541	0.485270	+	0.874364i	$0.338721\pi$
$758$	−7.86226e6	−0.497020
$759$	−650592.	−0.0409925
$760$	−185600.	−0.0116559
$761$	−9.19664e6	−0.575662	−0.287831	−	0.957681i	$-0.592934\pi$
$761$	−9.19664e6	−0.575662	−0.287831	+	0.957681i	$0.592934\pi$
$762$	−2.21155e6	−0.137978
$763$	0	0
$764$	−2.69568e6	−0.167084
$765$	1.75365e6	0.108340
$766$	1.05831e7	0.651688
$767$	6.65928e6	0.408732
$768$	−589824.	−0.0360844
$769$	−1.26412e7	−0.770857	−0.385429	−	0.922738i	$-0.625946\pi$
$769$	−1.26412e7	−0.770857	−0.385429	+	0.922738i	$0.625946\pi$
$770$	0	0
$771$	2.22349e6	0.134710
$772$	1.47874e7	0.892992
$773$	1.68634e7	1.01507	0.507535	−	0.861631i	$-0.330557\pi$
$773$	1.68634e7	1.01507	0.507535	+	0.861631i	$0.330557\pi$
$774$	6.77678e6	0.406604
$775$	2.34500e6	0.140245
$776$	5.60550e6	0.334165
$777$	0	0
$778$	1.10031e7	0.651725
$779$	747736.	0.0441474
$780$	−608400.	−0.0358057
$781$	1.88870e6	0.110799
$782$	−6.95571e6	−0.406747
$783$	−2.92037e6	−0.170229
$784$	−4.30259e6	−0.250000
$785$	−1.43104e7	−0.828856
$786$	−187632.	−0.0108330
$787$	−2.63040e7	−1.51386	−0.756929	−	0.653497i	$-0.773301\pi$
$787$	−2.63040e7	−1.51386	−0.756929	+	0.653497i	$0.773301\pi$
$788$	−8.27178e6	−0.474551
$789$	−1.33212e7	−0.761819
$790$	2.75440e6	0.157022
$791$	0	0
$792$	−186624.	−0.0105719
$793$	8.23334e6	0.464936
$794$	−9.45396e6	−0.532185
$795$	4.09725e6	0.229919
$796$	1.01299e6	0.0566661
$797$	−2.83141e7	−1.57891	−0.789453	−	0.613811i	$-0.789636\pi$
$797$	−2.83141e7	−1.57891	−0.789453	+	0.613811i	$0.789636\pi$
$798$	0	0
$799$	2.11650e7	1.17288
$800$	640000.	0.0353553
$801$	9.06957e6	0.499465
$802$	1.56144e7	0.857213
$803$	2.21839e6	0.121409
$804$	−7.59341e6	−0.414283
$805$	0	0
$806$	2.53635e6	0.137522
$807$	−5.11969e6	−0.276732
$808$	2.95411e6	0.159184
$809$	1.62416e7	0.872486	0.436243	−	0.899829i	$-0.356309\pi$
$809$	1.62416e7	0.872486	0.436243	+	0.899829i	$0.356309\pi$
$810$	656100.	0.0351364
$811$	−2.03075e7	−1.08419	−0.542093	−	0.840319i	$-0.682368\pi$
$811$	−2.03075e7	−1.08419	−0.542093	+	0.840319i	$0.682368\pi$
$812$	0	0
$813$	1.66159e7	0.881655
$814$	1.22256e6	0.0646709
$815$	−1.11361e7	−0.587272
$816$	−1.99526e6	−0.104900
$817$	−2.42626e6	−0.127169
$818$	−1.83649e7	−0.959633
$819$	0	0
$820$	−2.57840e6	−0.133911
$821$	−1.05730e7	−0.547442	−0.273721	−	0.961809i	$-0.588255\pi$
$821$	−1.05730e7	−0.547442	−0.273721	+	0.961809i	$0.588255\pi$
$822$	1.57356e6	0.0812276
$823$	8.79574e6	0.452661	0.226330	−	0.974051i	$-0.427327\pi$
$823$	8.79574e6	0.452661	0.226330	+	0.974051i	$0.427327\pi$
$824$	1.06496e7	0.546406
$825$	202500.	0.0103583
$826$	0	0
$827$	−7.58624e6	−0.385712	−0.192856	−	0.981227i	$-0.561775\pi$
$827$	−7.58624e6	−0.385712	−0.192856	+	0.981227i	$0.561775\pi$
$828$	−2.60237e6	−0.131915
$829$	−1.80738e7	−0.913403	−0.456701	−	0.889620i	$-0.650969\pi$
$829$	−1.80738e7	−0.913403	−0.456701	+	0.889620i	$0.650969\pi$
$830$	4.21320e6	0.212284
$831$	1.36074e7	0.683552
$832$	692224.	0.0346688
$833$	−1.45549e7	−0.726768
$834$	2.41733e6	0.120343
$835$	−1.33892e7	−0.664566
$836$	66816.0	0.00330647
$837$	−2.73521e6	−0.134951
$838$	−3.88642e6	−0.191179
$839$	−1.22096e6	−0.0598820	−0.0299410	−	0.999552i	$-0.509532\pi$
$839$	−1.22096e6	−0.0598820	−0.0299410	+	0.999552i	$0.509532\pi$
$840$	0	0
$841$	−4.46311e6	−0.217594
$842$	7.18050e6	0.349040
$843$	1.00738e7	0.488229
$844$	−4.44627e6	−0.214852
$845$	714025.	0.0344010
$846$	7.91856e6	0.380382
$847$	0	0
$848$	−4.66176e6	−0.222618
$849$	−9.93258e6	−0.472926
$850$	2.16500e6	0.102780
$851$	1.70479e7	0.806952
$852$	7.55482e6	0.356554
$853$	9.34625e6	0.439810	0.219905	−	0.975521i	$-0.429425\pi$
$853$	9.34625e6	0.439810	0.219905	+	0.975521i	$0.429425\pi$
$854$	0	0
$855$	−234900.	−0.0109892
$856$	−605952.	−0.0282653
$857$	−3.56656e7	−1.65881	−0.829407	−	0.558644i	$-0.811322\pi$
$857$	−3.56656e7	−1.65881	−0.829407	+	0.558644i	$0.811322\pi$
$858$	219024.	0.0101572
$859$	7.49162e6	0.346412	0.173206	−	0.984886i	$-0.444587\pi$
$859$	7.49162e6	0.346412	0.173206	+	0.984886i	$0.444587\pi$
$860$	8.36640e6	0.385738
$861$	0	0
$862$	325344.	0.0149133
$863$	−1.32595e7	−0.606038	−0.303019	−	0.952985i	$-0.597994\pi$
$863$	−1.32595e7	−0.606038	−0.303019	+	0.952985i	$0.597994\pi$
$864$	−746496.	−0.0340207
$865$	−8.90505e6	−0.404665
$866$	2.50867e7	1.13671
$867$	6.02911e6	0.272399
$868$	0	0
$869$	−991584.	−0.0445431
$870$	−3.60540e6	−0.161494
$871$	8.91171e6	0.398030
$872$	6.34048e6	0.282378
$873$	7.09447e6	0.315053
$874$	931712.	0.0412575
$875$	0	0
$876$	8.87357e6	0.390695
$877$	−3.44784e7	−1.51373	−0.756865	−	0.653571i	$-0.773270\pi$
$877$	−3.44784e7	−1.51373	−0.756865	+	0.653571i	$0.773270\pi$
$878$	1.35264e6	0.0592169
$879$	−1.52734e7	−0.666751
$880$	−230400.	−0.0100294
$881$	2.15136e6	0.0933843	0.0466921	−	0.998909i	$-0.485132\pi$
$881$	2.15136e6	0.0933843	0.0466921	+	0.998909i	$0.485132\pi$
$882$	−5.44547e6	−0.235702
$883$	3.88947e7	1.67876	0.839380	−	0.543544i	$-0.182918\pi$
$883$	3.88947e7	1.67876	0.839380	+	0.543544i	$0.182918\pi$
$884$	2.34166e6	0.100785
$885$	−8.86590e6	−0.380509
$886$	−3.96350e6	−0.169627
$887$	−3.37292e7	−1.43945	−0.719726	−	0.694258i	$-0.755733\pi$
$887$	−3.37292e7	−1.43945	−0.719726	+	0.694258i	$0.755733\pi$
$888$	4.89024e6	0.208112
$889$	0	0
$890$	1.11970e7	0.473834
$891$	−236196.	−0.00996732
$892$	−1.52334e7	−0.641040
$893$	−2.83504e6	−0.118968
$894$	−6.81718e6	−0.285273
$895$	−397700.	−0.0165958
$896$	0	0
$897$	3.05417e6	0.126739
$898$	−9.57737e6	−0.396329
$899$	1.50305e7	0.620261
$900$	810000.	0.0333333
$901$	−1.57699e7	−0.647166
$902$	928224.	0.0379871
$903$	0	0
$904$	1.06682e7	0.434179
$905$	−5.01785e6	−0.203656
$906$	−1.65859e7	−0.671304
$907$	−3.35640e7	−1.35474	−0.677370	−	0.735643i	$-0.736880\pi$
$907$	−3.35640e7	−1.35474	−0.677370	+	0.735643i	$0.736880\pi$
$908$	−412864.	−0.0166185
$909$	3.73880e6	0.150080
$910$	0	0
$911$	3.06696e6	0.122437	0.0612184	−	0.998124i	$-0.480501\pi$
$911$	3.06696e6	0.122437	0.0612184	+	0.998124i	$0.480501\pi$
$912$	267264.	0.0106403
$913$	−1.51675e6	−0.0602196
$914$	−2.37692e7	−0.941128
$915$	−1.09616e7	−0.432832
$916$	−2.60522e6	−0.102590
$917$	0	0
$918$	−2.52526e6	−0.0989006
$919$	1.82813e7	0.714032	0.357016	−	0.934098i	$-0.383794\pi$
$919$	1.82813e7	0.714032	0.357016	+	0.934098i	$0.383794\pi$
$920$	−3.21280e6	−0.125145
$921$	1.60429e7	0.623210
$922$	2.53001e7	0.980155
$923$	−8.86642e6	−0.342566
$924$	0	0
$925$	−5.30625e6	−0.203908
$926$	−2.89079e7	−1.10787
$927$	1.34784e7	0.515156
$928$	4.10214e6	0.156366
$929$	1.41207e7	0.536807	0.268404	−	0.963307i	$-0.413504\pi$
$929$	1.41207e7	0.536807	0.268404	+	0.963307i	$0.413504\pi$
$930$	−3.37680e6	−0.128026
$931$	1.94961e6	0.0737181
$932$	−1.15201e7	−0.434427
$933$	3.34174e6	0.125681
$934$	3.26244e7	1.22370
$935$	−779400.	−0.0291562
$936$	876096.	0.0326860
$937$	2.76594e7	1.02919	0.514593	−	0.857435i	$-0.327943\pi$
$937$	2.76594e7	1.02919	0.514593	+	0.857435i	$0.327943\pi$
$938$	0	0
$939$	2.55287e7	0.944853
$940$	9.77600e6	0.360862
$941$	1.04907e6	0.0386216	0.0193108	−	0.999814i	$-0.493853\pi$
$941$	1.04907e6	0.0386216	0.0193108	+	0.999814i	$0.493853\pi$
$942$	2.06070e7	0.756639
$943$	1.29436e7	0.473996
$944$	1.00874e7	0.368426
$945$	0	0
$946$	−3.01190e6	−0.109424
$947$	5.02304e7	1.82008	0.910042	−	0.414515i	$-0.136049\pi$
$947$	5.02304e7	1.82008	0.910042	+	0.414515i	$0.136049\pi$
$948$	−3.96634e6	−0.143340
$949$	−1.04141e7	−0.375368
$950$	−290000.	−0.0104253
$951$	1.01486e7	0.363876
$952$	0	0
$953$	−3.34080e7	−1.19157	−0.595783	−	0.803146i	$-0.703158\pi$
$953$	−3.34080e7	−1.19157	−0.595783	+	0.803146i	$0.703158\pi$
$954$	−5.90004e6	−0.209886
$955$	−4.21200e6	−0.149444
$956$	−4.98189e6	−0.176299
$957$	1.29794e6	0.0458117
$958$	2.70486e7	0.952207
$959$	0	0
$960$	−921600.	−0.0322749
$961$	−1.45516e7	−0.508281
$962$	−5.73924e6	−0.199948
$963$	−766908.	−0.0266488
$964$	−6.21923e6	−0.215548
$965$	2.31052e7	0.798716
$966$	0	0
$967$	−3.86391e7	−1.32880	−0.664402	−	0.747375i	$-0.731314\pi$
$967$	−3.86391e7	−1.32880	−0.664402	+	0.747375i	$0.731314\pi$
$968$	−1.02243e7	−0.350708
$969$	904104.	0.0309321
$970$	8.75860e6	0.298886
$971$	−7.63380e6	−0.259832	−0.129916	−	0.991525i	$-0.541471\pi$
$971$	−7.63380e6	−0.259832	−0.129916	+	0.991525i	$0.541471\pi$
$972$	−944784.	−0.0320750
$973$	0	0
$974$	3.68262e6	0.124383
$975$	−950625.	−0.0320256
$976$	1.24718e7	0.419088
$977$	−1.23479e7	−0.413863	−0.206932	−	0.978355i	$-0.566348\pi$
$977$	−1.23479e7	−0.413863	−0.206932	+	0.978355i	$0.566348\pi$
$978$	1.60360e7	0.536103
$979$	−4.03092e6	−0.134415
$980$	−6.72280e6	−0.223607
$981$	8.02467e6	0.266229
$982$	3.43293e7	1.13602
$983$	−1.44524e7	−0.477040	−0.238520	−	0.971138i	$-0.576662\pi$
$983$	−1.44524e7	−0.477040	−0.238520	+	0.971138i	$0.576662\pi$
$984$	3.71290e6	0.122243
$985$	−1.29246e7	−0.424452
$986$	1.38768e7	0.454566
$987$	0	0
$988$	−313664.	−0.0102229
$989$	−4.19993e7	−1.36537
$990$	−291600.	−0.00945583
$991$	3.52722e7	1.14090	0.570451	−	0.821332i	$-0.306769\pi$
$991$	3.52722e7	1.14090	0.570451	+	0.821332i	$0.306769\pi$
$992$	3.84205e6	0.123961
$993$	−2.03071e7	−0.653543
$994$	0	0
$995$	1.58280e6	0.0506837
$996$	−6.06701e6	−0.193788
$997$	−2.59772e7	−0.827666	−0.413833	−	0.910353i	$-0.635810\pi$
$997$	−2.59772e7	−0.827666	−0.413833	+	0.910353i	$0.635810\pi$
$998$	−1.80878e7	−0.574855
$999$	6.18921e6	0.196210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.6.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.6.a.c.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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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