LMFDB - Embedded newform 294.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,6,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	294.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:	$47.1528430250$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$ -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} -24.0000 q^{5} -36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} -96.0000 q^{10} +66.0000 q^{11} -144.000 q^{12} -98.0000 q^{13} +216.000 q^{15} +256.000 q^{16} +216.000 q^{17} +324.000 q^{18} +340.000 q^{19} -384.000 q^{20} +264.000 q^{22} -1038.00 q^{23} -576.000 q^{24} -2549.00 q^{25} -392.000 q^{26} -729.000 q^{27} -2490.00 q^{29} +864.000 q^{30} +7048.00 q^{31} +1024.00 q^{32} -594.000 q^{33} +864.000 q^{34} +1296.00 q^{36} -12238.0 q^{37} +1360.00 q^{38} +882.000 q^{39} -1536.00 q^{40} -6468.00 q^{41} -15412.0 q^{43} +1056.00 q^{44} -1944.00 q^{45} -4152.00 q^{46} -20604.0 q^{47} -2304.00 q^{48} -10196.0 q^{50} -1944.00 q^{51} -1568.00 q^{52} +32490.0 q^{53} -2916.00 q^{54} -1584.00 q^{55} -3060.00 q^{57} -9960.00 q^{58} -34224.0 q^{59} +3456.00 q^{60} -35654.0 q^{61} +28192.0 q^{62} +4096.00 q^{64} +2352.00 q^{65} -2376.00 q^{66} +12680.0 q^{67} +3456.00 q^{68} +9342.00 q^{69} -42642.0 q^{71} +5184.00 q^{72} -33734.0 q^{73} -48952.0 q^{74} +22941.0 q^{75} +5440.00 q^{76} +3528.00 q^{78} -85108.0 q^{79} -6144.00 q^{80} +6561.00 q^{81} -25872.0 q^{82} +106764. q^{83} -5184.00 q^{85} -61648.0 q^{86} +22410.0 q^{87} +4224.00 q^{88} -34884.0 q^{89} -7776.00 q^{90} -16608.0 q^{92} -63432.0 q^{93} -82416.0 q^{94} -8160.00 q^{95} -9216.00 q^{96} -18662.0 q^{97} +5346.00 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
$5$	−24.0000	−0.429325	−0.214663	−	0.976688i	$-0.568865\pi$
$5$	−24.0000	−0.429325	−0.214663	+	0.976688i	$0.568865\pi$
$6$	−36.0000	−0.408248
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−96.0000	−0.303579
$11$	66.0000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	66.0000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	−144.000	−0.288675
$13$	−98.0000	−0.160830	−0.0804151	−	0.996761i	$-0.525625\pi$
$13$	−98.0000	−0.160830	−0.0804151	+	0.996761i	$0.525625\pi$
$14$	0	0
$15$	216.000	0.247871
$16$	256.000	0.250000
$17$	216.000	0.181272	0.0906362	−	0.995884i	$-0.471110\pi$
$17$	216.000	0.181272	0.0906362	+	0.995884i	$0.471110\pi$
$18$	324.000	0.235702
$19$	340.000	0.216070	0.108035	−	0.994147i	$-0.465544\pi$
$19$	340.000	0.216070	0.108035	+	0.994147i	$0.465544\pi$
$20$	−384.000	−0.214663
$21$	0	0
$22$	264.000	0.116291
$23$	−1038.00	−0.409145	−0.204573	−	0.978851i	$-0.565580\pi$
$23$	−1038.00	−0.409145	−0.204573	+	0.978851i	$0.565580\pi$
$24$	−576.000	−0.204124
$25$	−2549.00	−0.815680
$26$	−392.000	−0.113724
$27$	−729.000	−0.192450
$28$	0	0
$29$	−2490.00	−0.549800	−0.274900	−	0.961473i	$-0.588645\pi$
$29$	−2490.00	−0.549800	−0.274900	+	0.961473i	$0.588645\pi$
$30$	864.000	0.175271
$31$	7048.00	1.31723	0.658615	−	0.752480i	$-0.271143\pi$
$31$	7048.00	1.31723	0.658615	+	0.752480i	$0.271143\pi$
$32$	1024.00	0.176777
$33$	−594.000	−0.0949514
$34$	864.000	0.128179
$35$	0	0
$36$	1296.00	0.166667
$37$	−12238.0	−1.46962	−0.734812	−	0.678271i	$-0.762730\pi$
$37$	−12238.0	−1.46962	−0.734812	+	0.678271i	$0.762730\pi$
$38$	1360.00	0.152785
$39$	882.000	0.0928554
$40$	−1536.00	−0.151789
$41$	−6468.00	−0.600911	−0.300456	−	0.953796i	$-0.597139\pi$
$41$	−6468.00	−0.600911	−0.300456	+	0.953796i	$0.597139\pi$
$42$	0	0
$43$	−15412.0	−1.27112	−0.635562	−	0.772050i	$-0.719232\pi$
$43$	−15412.0	−1.27112	−0.635562	+	0.772050i	$0.719232\pi$
$44$	1056.00	0.0822304
$45$	−1944.00	−0.143108
$46$	−4152.00	−0.289310
$47$	−20604.0	−1.36053	−0.680263	−	0.732968i	$-0.738134\pi$
$47$	−20604.0	−1.36053	−0.680263	+	0.732968i	$0.738134\pi$
$48$	−2304.00	−0.144338
$49$	0	0
$50$	−10196.0	−0.576773
$51$	−1944.00	−0.104658
$52$	−1568.00	−0.0804151
$53$	32490.0	1.58877	0.794383	−	0.607417i	$-0.207794\pi$
$53$	32490.0	1.58877	0.794383	+	0.607417i	$0.207794\pi$
$54$	−2916.00	−0.136083
$55$	−1584.00	−0.0706071
$56$	0	0
$57$	−3060.00	−0.124748
$58$	−9960.00	−0.388767
$59$	−34224.0	−1.27997	−0.639986	−	0.768386i	$-0.721060\pi$
$59$	−34224.0	−1.27997	−0.639986	+	0.768386i	$0.721060\pi$
$60$	3456.00	0.123935
$61$	−35654.0	−1.22683	−0.613414	−	0.789762i	$-0.710204\pi$
$61$	−35654.0	−1.22683	−0.613414	+	0.789762i	$0.710204\pi$
$62$	28192.0	0.931422
$63$	0	0
$64$	4096.00	0.125000
$65$	2352.00	0.0690484
$66$	−2376.00	−0.0671408
$67$	12680.0	0.345090	0.172545	−	0.985002i	$-0.444801\pi$
$67$	12680.0	0.345090	0.172545	+	0.985002i	$0.444801\pi$
$68$	3456.00	0.0906362
$69$	9342.00	0.236220
$70$	0	0
$71$	−42642.0	−1.00390	−0.501951	−	0.864896i	$-0.667384\pi$
$71$	−42642.0	−1.00390	−0.501951	+	0.864896i	$0.667384\pi$
$72$	5184.00	0.117851
$73$	−33734.0	−0.740902	−0.370451	−	0.928852i	$-0.620797\pi$
$73$	−33734.0	−0.740902	−0.370451	+	0.928852i	$0.620797\pi$
$74$	−48952.0	−1.03918
$75$	22941.0	0.470933
$76$	5440.00	0.108035
$77$	0	0
$78$	3528.00	0.0656587
$79$	−85108.0	−1.53427	−0.767137	−	0.641484i	$-0.778319\pi$
$79$	−85108.0	−1.53427	−0.767137	+	0.641484i	$0.778319\pi$
$80$	−6144.00	−0.107331
$81$	6561.00	0.111111
$82$	−25872.0	−0.424908
$83$	106764.	1.70110	0.850550	−	0.525895i	$-0.176270\pi$
$83$	106764.	1.70110	0.850550	+	0.525895i	$0.176270\pi$
$84$	0	0
$85$	−5184.00	−0.0778247
$86$	−61648.0	−0.898820
$87$	22410.0	0.317427
$88$	4224.00	0.0581456
$89$	−34884.0	−0.466822	−0.233411	−	0.972378i	$-0.574989\pi$
$89$	−34884.0	−0.466822	−0.233411	+	0.972378i	$0.574989\pi$
$90$	−7776.00	−0.101193
$91$	0	0
$92$	−16608.0	−0.204573
$93$	−63432.0	−0.760503
$94$	−82416.0	−0.962037
$95$	−8160.00	−0.0927644
$96$	−9216.00	−0.102062
$97$	−18662.0	−0.201386	−0.100693	−	0.994918i	$-0.532106\pi$
$97$	−18662.0	−0.201386	−0.100693	+	0.994918i	$0.532106\pi$
$98$	0	0
$99$	5346.00	0.0548202
$100$	−40784.0	−0.407840
$101$	−153084.	−1.49323	−0.746614	−	0.665257i	$-0.768322\pi$
$101$	−153084.	−1.49323	−0.746614	+	0.665257i	$0.768322\pi$
$102$	−7776.00	−0.0740041
$103$	−35864.0	−0.333093	−0.166547	−	0.986034i	$-0.553262\pi$
$103$	−35864.0	−0.333093	−0.166547	+	0.986034i	$0.553262\pi$
$104$	−6272.00	−0.0568621
$105$	0	0
$106$	129960.	1.12343
$107$	−95454.0	−0.805999	−0.403000	−	0.915200i	$-0.632032\pi$
$107$	−95454.0	−0.805999	−0.403000	+	0.915200i	$0.632032\pi$
$108$	−11664.0	−0.0962250
$109$	212222.	1.71090	0.855449	−	0.517887i	$-0.173281\pi$
$109$	212222.	1.71090	0.855449	+	0.517887i	$0.173281\pi$
$110$	−6336.00	−0.0499268
$111$	110142.	0.848488
$112$	0	0
$113$	62106.0	0.457549	0.228774	−	0.973479i	$-0.426528\pi$
$113$	62106.0	0.457549	0.228774	+	0.973479i	$0.426528\pi$
$114$	−12240.0	−0.0882103
$115$	24912.0	0.175656
$116$	−39840.0	−0.274900
$117$	−7938.00	−0.0536101
$118$	−136896.	−0.905077
$119$	0	0
$120$	13824.0	0.0876356
$121$	−156695.	−0.972953
$122$	−142616.	−0.867498
$123$	58212.0	0.346936
$124$	112768.	0.658615
$125$	136176.	0.779517
$126$	0	0
$127$	−53044.0	−0.291828	−0.145914	−	0.989297i	$-0.546612\pi$
$127$	−53044.0	−0.291828	−0.145914	+	0.989297i	$0.546612\pi$
$128$	16384.0	0.0883883
$129$	138708.	0.733884
$130$	9408.00	0.0488246
$131$	−69324.0	−0.352944	−0.176472	−	0.984306i	$-0.556468\pi$
$131$	−69324.0	−0.352944	−0.176472	+	0.984306i	$0.556468\pi$
$132$	−9504.00	−0.0474757
$133$	0	0
$134$	50720.0	0.244015
$135$	17496.0	0.0826236
$136$	13824.0	0.0640894
$137$	129846.	0.591054	0.295527	−	0.955334i	$-0.404505\pi$
$137$	129846.	0.591054	0.295527	+	0.955334i	$0.404505\pi$
$138$	37368.0	0.167033
$139$	104356.	0.458121	0.229061	−	0.973412i	$-0.426435\pi$
$139$	104356.	0.458121	0.229061	+	0.973412i	$0.426435\pi$
$140$	0	0
$141$	185436.	0.785500
$142$	−170568.	−0.709867
$143$	−6468.00	−0.0264503
$144$	20736.0	0.0833333
$145$	59760.0	0.236043
$146$	−134936.	−0.523897
$147$	0	0
$148$	−195808.	−0.734812
$149$	217194.	0.801461	0.400730	−	0.916196i	$-0.368756\pi$
$149$	217194.	0.801461	0.400730	+	0.916196i	$0.368756\pi$
$150$	91764.0	0.333000
$151$	221000.	0.788769	0.394385	−	0.918945i	$-0.370958\pi$
$151$	221000.	0.788769	0.394385	+	0.918945i	$0.370958\pi$
$152$	21760.0	0.0763924
$153$	17496.0	0.0604241
$154$	0	0
$155$	−169152.	−0.565520
$156$	14112.0	0.0464277
$157$	378370.	1.22509	0.612544	−	0.790436i	$-0.290146\pi$
$157$	378370.	1.22509	0.612544	+	0.790436i	$0.290146\pi$
$158$	−340432.	−1.08489
$159$	−292410.	−0.917275
$160$	−24576.0	−0.0758947
$161$	0	0
$162$	26244.0	0.0785674
$163$	104816.	0.309000	0.154500	−	0.987993i	$-0.450623\pi$
$163$	104816.	0.309000	0.154500	+	0.987993i	$0.450623\pi$
$164$	−103488.	−0.300456
$165$	14256.0	0.0407650
$166$	427056.	1.20286
$167$	426972.	1.18470	0.592350	−	0.805681i	$-0.298200\pi$
$167$	426972.	1.18470	0.592350	+	0.805681i	$0.298200\pi$
$168$	0	0
$169$	−361689.	−0.974134
$170$	−20736.0	−0.0550304
$171$	27540.0	0.0720234
$172$	−246592.	−0.635562
$173$	−331068.	−0.841012	−0.420506	−	0.907290i	$-0.638147\pi$
$173$	−331068.	−0.841012	−0.420506	+	0.907290i	$0.638147\pi$
$174$	89640.0	0.224455
$175$	0	0
$176$	16896.0	0.0411152
$177$	308016.	0.738993
$178$	−139536.	−0.330093
$179$	−400194.	−0.933551	−0.466775	−	0.884376i	$-0.654584\pi$
$179$	−400194.	−0.933551	−0.466775	+	0.884376i	$0.654584\pi$
$180$	−31104.0	−0.0715542
$181$	−588098.	−1.33430	−0.667150	−	0.744924i	$-0.732486\pi$
$181$	−588098.	−1.33430	−0.667150	+	0.744924i	$0.732486\pi$
$182$	0	0
$183$	320886.	0.708309
$184$	−66432.0	−0.144655
$185$	293712.	0.630946
$186$	−253728.	−0.537757
$187$	14256.0	0.0298122
$188$	−329664.	−0.680263
$189$	0	0
$190$	−32640.0	−0.0655943
$191$	939342.	1.86312	0.931559	−	0.363590i	$-0.118449\pi$
$191$	939342.	1.86312	0.931559	+	0.363590i	$0.118449\pi$
$192$	−36864.0	−0.0721688
$193$	338390.	0.653919	0.326960	−	0.945038i	$-0.393976\pi$
$193$	338390.	0.653919	0.326960	+	0.945038i	$0.393976\pi$
$194$	−74648.0	−0.142401
$195$	−21168.0	−0.0398651
$196$	0	0
$197$	−237942.	−0.436823	−0.218412	−	0.975857i	$-0.570088\pi$
$197$	−237942.	−0.436823	−0.218412	+	0.975857i	$0.570088\pi$
$198$	21384.0	0.0387638
$199$	−204464.	−0.366003	−0.183001	−	0.983113i	$-0.558581\pi$
$199$	−204464.	−0.366003	−0.183001	+	0.983113i	$0.558581\pi$
$200$	−163136.	−0.288386
$201$	−114120.	−0.199238
$202$	−612336.	−1.05587
$203$	0	0
$204$	−31104.0	−0.0523288
$205$	155232.	0.257986
$206$	−143456.	−0.235532
$207$	−84078.0	−0.136382
$208$	−25088.0	−0.0402076
$209$	22440.0	0.0355351
$210$	0	0
$211$	−348724.	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$211$	−348724.	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$212$	519840.	0.794383
$213$	383778.	0.579604
$214$	−381816.	−0.569928
$215$	369888.	0.545725
$216$	−46656.0	−0.0680414
$217$	0	0
$218$	848888.	1.20979
$219$	303606.	0.427760
$220$	−25344.0	−0.0353036
$221$	−21168.0	−0.0291541
$222$	440568.	0.599971
$223$	−1.47006e6	−1.97957	−0.989787	−	0.142554i	$-0.954468\pi$
$223$	−1.47006e6	−1.97957	−0.989787	+	0.142554i	$0.954468\pi$
$224$	0	0
$225$	−206469.	−0.271893
$226$	248424.	0.323536
$227$	589560.	0.759387	0.379694	−	0.925112i	$-0.376029\pi$
$227$	589560.	0.759387	0.379694	+	0.925112i	$0.376029\pi$
$228$	−48960.0	−0.0623741
$229$	1.04534e6	1.31725	0.658627	−	0.752469i	$-0.271137\pi$
$229$	1.04534e6	1.31725	0.658627	+	0.752469i	$0.271137\pi$
$230$	99648.0	0.124208
$231$	0	0
$232$	−159360.	−0.194383
$233$	651222.	0.785849	0.392925	−	0.919571i	$-0.371463\pi$
$233$	651222.	0.785849	0.392925	+	0.919571i	$0.371463\pi$
$234$	−31752.0	−0.0379080
$235$	494496.	0.584108
$236$	−547584.	−0.639986
$237$	765972.	0.885813
$238$	0	0
$239$	−513462.	−0.581452	−0.290726	−	0.956806i	$-0.593897\pi$
$239$	−513462.	−0.581452	−0.290726	+	0.956806i	$0.593897\pi$
$240$	55296.0	0.0619677
$241$	694714.	0.770484	0.385242	−	0.922816i	$-0.374118\pi$
$241$	694714.	0.770484	0.385242	+	0.922816i	$0.374118\pi$
$242$	−626780.	−0.687981
$243$	−59049.0	−0.0641500
$244$	−570464.	−0.613414
$245$	0	0
$246$	232848.	0.245321
$247$	−33320.0	−0.0347506
$248$	451072.	0.465711
$249$	−960876.	−0.982130
$250$	544704.	0.551202
$251$	1.39608e6	1.39870	0.699352	−	0.714777i	$-0.253472\pi$
$251$	1.39608e6	1.39870	0.699352	+	0.714777i	$0.253472\pi$
$252$	0	0
$253$	−68508.0	−0.0672884
$254$	−212176.	−0.206354
$255$	46656.0	0.0449321
$256$	65536.0	0.0625000
$257$	1.00520e6	0.949339	0.474670	−	0.880164i	$-0.342568\pi$
$257$	1.00520e6	0.949339	0.474670	+	0.880164i	$0.342568\pi$
$258$	554832.	0.518934
$259$	0	0
$260$	37632.0	0.0345242
$261$	−201690.	−0.183267
$262$	−277296.	−0.249569
$263$	1.25301e6	1.11703	0.558515	−	0.829494i	$-0.311371\pi$
$263$	1.25301e6	1.11703	0.558515	+	0.829494i	$0.311371\pi$
$264$	−38016.0	−0.0335704
$265$	−779760.	−0.682097
$266$	0	0
$267$	313956.	0.269520
$268$	202880.	0.172545
$269$	1.76069e6	1.48355	0.741774	−	0.670650i	$-0.233985\pi$
$269$	1.76069e6	1.48355	0.741774	+	0.670650i	$0.233985\pi$
$270$	69984.0	0.0584237
$271$	−770528.	−0.637331	−0.318666	−	0.947867i	$-0.603235\pi$
$271$	−770528.	−0.637331	−0.318666	+	0.947867i	$0.603235\pi$
$272$	55296.0	0.0453181
$273$	0	0
$274$	519384.	0.417938
$275$	−168234.	−0.134147
$276$	149472.	0.118110
$277$	707738.	0.554208	0.277104	−	0.960840i	$-0.410625\pi$
$277$	707738.	0.554208	0.277104	+	0.960840i	$0.410625\pi$
$278$	417424.	0.323941
$279$	570888.	0.439077
$280$	0	0
$281$	2.30432e6	1.74091	0.870456	−	0.492247i	$-0.163824\pi$
$281$	2.30432e6	1.74091	0.870456	+	0.492247i	$0.163824\pi$
$282$	741744.	0.555432
$283$	−1.60903e6	−1.19426	−0.597128	−	0.802146i	$-0.703692\pi$
$283$	−1.60903e6	−1.19426	−0.597128	+	0.802146i	$0.703692\pi$
$284$	−682272.	−0.501951
$285$	73440.0	0.0535575
$286$	−25872.0	−0.0187032
$287$	0	0
$288$	82944.0	0.0589256
$289$	−1.37320e6	−0.967140
$290$	239040.	0.166907
$291$	167958.	0.116270
$292$	−539744.	−0.370451
$293$	−517020.	−0.351834	−0.175917	−	0.984405i	$-0.556289\pi$
$293$	−517020.	−0.351834	−0.175917	+	0.984405i	$0.556289\pi$
$294$	0	0
$295$	821376.	0.549524
$296$	−783232.	−0.519590
$297$	−48114.0	−0.0316505
$298$	868776.	0.566718
$299$	101724.	0.0658030
$300$	367056.	0.235467
$301$	0	0
$302$	884000.	0.557744
$303$	1.37776e6	0.862116
$304$	87040.0	0.0540176
$305$	855696.	0.526708
$306$	69984.0	0.0427263
$307$	−1.35002e6	−0.817512	−0.408756	−	0.912644i	$-0.634037\pi$
$307$	−1.35002e6	−0.817512	−0.408756	+	0.912644i	$0.634037\pi$
$308$	0	0
$309$	322776.	0.192311
$310$	−676608.	−0.399883
$311$	−1.34538e6	−0.788758	−0.394379	−	0.918948i	$-0.629040\pi$
$311$	−1.34538e6	−0.788758	−0.394379	+	0.918948i	$0.629040\pi$
$312$	56448.0	0.0328293
$313$	−256154.	−0.147788	−0.0738942	−	0.997266i	$-0.523543\pi$
$313$	−256154.	−0.147788	−0.0738942	+	0.997266i	$0.523543\pi$
$314$	1.51348e6	0.866269
$315$	0	0
$316$	−1.36173e6	−0.767137
$317$	1.84629e6	1.03193	0.515967	−	0.856609i	$-0.327433\pi$
$317$	1.84629e6	1.03193	0.515967	+	0.856609i	$0.327433\pi$
$318$	−1.16964e6	−0.648611
$319$	−164340.	−0.0904204
$320$	−98304.0	−0.0536656
$321$	859086.	0.465344
$322$	0	0
$323$	73440.0	0.0391675
$324$	104976.	0.0555556
$325$	249802.	0.131186
$326$	419264.	0.218496
$327$	−1.91000e6	−0.987788
$328$	−413952.	−0.212454
$329$	0	0
$330$	57024.0	0.0288252
$331$	−3.33238e6	−1.67180	−0.835900	−	0.548881i	$-0.815054\pi$
$331$	−3.33238e6	−1.67180	−0.835900	+	0.548881i	$0.815054\pi$
$332$	1.70822e6	0.850550
$333$	−991278.	−0.489875
$334$	1.70789e6	0.837709
$335$	−304320.	−0.148156
$336$	0	0
$337$	−1.63481e6	−0.784136	−0.392068	−	0.919936i	$-0.628240\pi$
$337$	−1.63481e6	−0.784136	−0.392068	+	0.919936i	$0.628240\pi$
$338$	−1.44676e6	−0.688816
$339$	−558954.	−0.264166
$340$	−82944.0	−0.0389124
$341$	465168.	0.216633
$342$	110160.	0.0509282
$343$	0	0
$344$	−986368.	−0.449410
$345$	−224208.	−0.101415
$346$	−1.32427e6	−0.594685
$347$	−841530.	−0.375185	−0.187593	−	0.982247i	$-0.560068\pi$
$347$	−841530.	−0.375185	−0.187593	+	0.982247i	$0.560068\pi$
$348$	358560.	0.158713
$349$	977242.	0.429476	0.214738	−	0.976672i	$-0.431110\pi$
$349$	977242.	0.429476	0.214738	+	0.976672i	$0.431110\pi$
$350$	0	0
$351$	71442.0	0.0309518
$352$	67584.0	0.0290728
$353$	−3.45857e6	−1.47727	−0.738634	−	0.674106i	$-0.764529\pi$
$353$	−3.45857e6	−1.47727	−0.738634	+	0.674106i	$0.764529\pi$
$354$	1.23206e6	0.522547
$355$	1.02341e6	0.431001
$356$	−558144.	−0.233411
$357$	0	0
$358$	−1.60078e6	−0.660120
$359$	−3.47301e6	−1.42223	−0.711115	−	0.703076i	$-0.751810\pi$
$359$	−3.47301e6	−1.42223	−0.711115	+	0.703076i	$0.751810\pi$
$360$	−124416.	−0.0505964
$361$	−2.36050e6	−0.953314
$362$	−2.35239e6	−0.943492
$363$	1.41026e6	0.561734
$364$	0	0
$365$	809616.	0.318088
$366$	1.28354e6	0.500850
$367$	−3.11994e6	−1.20915	−0.604575	−	0.796548i	$-0.706657\pi$
$367$	−3.11994e6	−1.20915	−0.604575	+	0.796548i	$0.706657\pi$
$368$	−265728.	−0.102286
$369$	−523908.	−0.200304
$370$	1.17485e6	0.446146
$371$	0	0
$372$	−1.01491e6	−0.380252
$373$	−2.01673e6	−0.750543	−0.375272	−	0.926915i	$-0.622451\pi$
$373$	−2.01673e6	−0.750543	−0.375272	+	0.926915i	$0.622451\pi$
$374$	57024.0	0.0210804
$375$	−1.22558e6	−0.450054
$376$	−1.31866e6	−0.481019
$377$	244020.	0.0884244
$378$	0	0
$379$	−5.38083e6	−1.92420	−0.962102	−	0.272690i	$-0.912087\pi$
$379$	−5.38083e6	−1.92420	−0.962102	+	0.272690i	$0.912087\pi$
$380$	−130560.	−0.0463822
$381$	477396.	0.168487
$382$	3.75737e6	1.31742
$383$	−807432.	−0.281261	−0.140630	−	0.990062i	$-0.544913\pi$
$383$	−807432.	−0.281261	−0.140630	+	0.990062i	$0.544913\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	1.35356e6	0.462391
$387$	−1.24837e6	−0.423708
$388$	−298592.	−0.100693
$389$	891390.	0.298671	0.149336	−	0.988787i	$-0.452286\pi$
$389$	891390.	0.298671	0.149336	+	0.988787i	$0.452286\pi$
$390$	−84672.0	−0.0281889
$391$	−224208.	−0.0741667
$392$	0	0
$393$	623916.	0.203772
$394$	−951768.	−0.308881
$395$	2.04259e6	0.658702
$396$	85536.0	0.0274101
$397$	−1.12345e6	−0.357749	−0.178875	−	0.983872i	$-0.557246\pi$
$397$	−1.12345e6	−0.357749	−0.178875	+	0.983872i	$0.557246\pi$
$398$	−817856.	−0.258803
$399$	0	0
$400$	−652544.	−0.203920
$401$	1.72037e6	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$401$	1.72037e6	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$402$	−456480.	−0.140882
$403$	−690704.	−0.211850
$404$	−2.44934e6	−0.746614
$405$	−157464.	−0.0477028
$406$	0	0
$407$	−807708.	−0.241695
$408$	−124416.	−0.0370021
$409$	−77246.0	−0.0228332	−0.0114166	−	0.999935i	$-0.503634\pi$
$409$	−77246.0	−0.0228332	−0.0114166	+	0.999935i	$0.503634\pi$
$410$	620928.	0.182424
$411$	−1.16861e6	−0.341245
$412$	−573824.	−0.166547
$413$	0	0
$414$	−336312.	−0.0964365
$415$	−2.56234e6	−0.730324
$416$	−100352.	−0.0284310
$417$	−939204.	−0.264496
$418$	89760.0	0.0251271
$419$	5.20615e6	1.44871	0.724356	−	0.689427i	$-0.242137\pi$
$419$	5.20615e6	1.44871	0.724356	+	0.689427i	$0.242137\pi$
$420$	0	0
$421$	1.71847e6	0.472539	0.236270	−	0.971688i	$-0.424075\pi$
$421$	1.71847e6	0.472539	0.236270	+	0.971688i	$0.424075\pi$
$422$	−1.39490e6	−0.381295
$423$	−1.66892e6	−0.453509
$424$	2.07936e6	0.561714
$425$	−550584.	−0.147860
$426$	1.53511e6	0.409842
$427$	0	0
$428$	−1.52726e6	−0.403000
$429$	58212.0	0.0152711
$430$	1.47955e6	0.385886
$431$	−580626.	−0.150558	−0.0752789	−	0.997163i	$-0.523985\pi$
$431$	−580626.	−0.150558	−0.0752789	+	0.997163i	$0.523985\pi$
$432$	−186624.	−0.0481125
$433$	−4.15087e6	−1.06395	−0.531973	−	0.846761i	$-0.678549\pi$
$433$	−4.15087e6	−1.06395	−0.531973	+	0.846761i	$0.678549\pi$
$434$	0	0
$435$	−537840.	−0.136279
$436$	3.39555e6	0.855449
$437$	−352920.	−0.0884042
$438$	1.21442e6	0.302472
$439$	−3.88407e6	−0.961891	−0.480946	−	0.876750i	$-0.659707\pi$
$439$	−3.88407e6	−0.961891	−0.480946	+	0.876750i	$0.659707\pi$
$440$	−101376.	−0.0249634
$441$	0	0
$442$	−84672.0	−0.0206150
$443$	−2.31499e6	−0.560453	−0.280226	−	0.959934i	$-0.590410\pi$
$443$	−2.31499e6	−0.560453	−0.280226	+	0.959934i	$0.590410\pi$
$444$	1.76227e6	0.424244
$445$	837216.	0.200418
$446$	−5.88022e6	−1.39977
$447$	−1.95475e6	−0.462723
$448$	0	0
$449$	−1.92281e6	−0.450113	−0.225056	−	0.974346i	$-0.572257\pi$
$449$	−1.92281e6	−0.450113	−0.225056	+	0.974346i	$0.572257\pi$
$450$	−825876.	−0.192258
$451$	−426888.	−0.0988263
$452$	993696.	0.228774
$453$	−1.98900e6	−0.455396
$454$	2.35824e6	0.536968
$455$	0	0
$456$	−195840.	−0.0441051
$457$	6.86215e6	1.53699	0.768493	−	0.639858i	$-0.221007\pi$
$457$	6.86215e6	1.53699	0.768493	+	0.639858i	$0.221007\pi$
$458$	4.18137e6	0.931440
$459$	−157464.	−0.0348859
$460$	398592.	0.0878282
$461$	−2.97167e6	−0.651250	−0.325625	−	0.945499i	$-0.605575\pi$
$461$	−2.97167e6	−0.651250	−0.325625	+	0.945499i	$0.605575\pi$
$462$	0	0
$463$	4.87423e6	1.05670	0.528352	−	0.849025i	$-0.322810\pi$
$463$	4.87423e6	1.05670	0.528352	+	0.849025i	$0.322810\pi$
$464$	−637440.	−0.137450
$465$	1.52237e6	0.326503
$466$	2.60489e6	0.555679
$467$	8.17301e6	1.73416	0.867081	−	0.498167i	$-0.165993\pi$
$467$	8.17301e6	1.73416	0.867081	+	0.498167i	$0.165993\pi$
$468$	−127008.	−0.0268050
$469$	0	0
$470$	1.97798e6	0.413027
$471$	−3.40533e6	−0.707305
$472$	−2.19034e6	−0.452539
$473$	−1.01719e6	−0.209050
$474$	3.06389e6	0.626364
$475$	−866660.	−0.176244
$476$	0	0
$477$	2.63169e6	0.529589
$478$	−2.05385e6	−0.411148
$479$	−2.34397e6	−0.466782	−0.233391	−	0.972383i	$-0.574982\pi$
$479$	−2.34397e6	−0.466782	−0.233391	+	0.972383i	$0.574982\pi$
$480$	221184.	0.0438178
$481$	1.19932e6	0.236360
$482$	2.77886e6	0.544814
$483$	0	0
$484$	−2.50712e6	−0.486476
$485$	447888.	0.0864600
$486$	−236196.	−0.0453609
$487$	316928.	0.0605534	0.0302767	−	0.999542i	$-0.490361\pi$
$487$	316928.	0.0605534	0.0302767	+	0.999542i	$0.490361\pi$
$488$	−2.28186e6	−0.433749
$489$	−943344.	−0.178401
$490$	0	0
$491$	−5.20041e6	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$491$	−5.20041e6	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$492$	931392.	0.173468
$493$	−537840.	−0.0996634
$494$	−133280.	−0.0245724
$495$	−128304.	−0.0235357
$496$	1.80429e6	0.329308
$497$	0	0
$498$	−3.84350e6	−0.694471
$499$	−4.86773e6	−0.875135	−0.437568	−	0.899185i	$-0.644160\pi$
$499$	−4.86773e6	−0.875135	−0.437568	+	0.899185i	$0.644160\pi$
$500$	2.17882e6	0.389758
$501$	−3.84275e6	−0.683987
$502$	5.58432e6	0.989034
$503$	−426888.	−0.0752305	−0.0376153	−	0.999292i	$-0.511976\pi$
$503$	−426888.	−0.0752305	−0.0376153	+	0.999292i	$0.511976\pi$
$504$	0	0
$505$	3.67402e6	0.641081
$506$	−274032.	−0.0475801
$507$	3.25520e6	0.562416
$508$	−848704.	−0.145914
$509$	9.41621e6	1.61095	0.805474	−	0.592631i	$-0.201911\pi$
$509$	9.41621e6	1.61095	0.805474	+	0.592631i	$0.201911\pi$
$510$	186624.	0.0317718
$511$	0	0
$512$	262144.	0.0441942
$513$	−247860.	−0.0415827
$514$	4.02082e6	0.671284
$515$	860736.	0.143005
$516$	2.21933e6	0.366942
$517$	−1.35986e6	−0.223753
$518$	0	0
$519$	2.97961e6	0.485558
$520$	150528.	0.0244123
$521$	−1.84039e6	−0.297041	−0.148520	−	0.988909i	$-0.547451\pi$
$521$	−1.84039e6	−0.297041	−0.148520	+	0.988909i	$0.547451\pi$
$522$	−806760.	−0.129589
$523$	979108.	0.156522	0.0782612	−	0.996933i	$-0.475063\pi$
$523$	979108.	0.156522	0.0782612	+	0.996933i	$0.475063\pi$
$524$	−1.10918e6	−0.176472
$525$	0	0
$526$	5.01204e6	0.789860
$527$	1.52237e6	0.238777
$528$	−152064.	−0.0237379
$529$	−5.35890e6	−0.832600
$530$	−3.11904e6	−0.482316
$531$	−2.77214e6	−0.426658
$532$	0	0
$533$	633864.	0.0966447
$534$	1.25582e6	0.190579
$535$	2.29090e6	0.346036
$536$	811520.	0.122008
$537$	3.60175e6	0.538986
$538$	7.04275e6	1.04903
$539$	0	0
$540$	279936.	0.0413118
$541$	5.96117e6	0.875666	0.437833	−	0.899056i	$-0.355746\pi$
$541$	5.96117e6	0.875666	0.437833	+	0.899056i	$0.355746\pi$
$542$	−3.08211e6	−0.450661
$543$	5.29288e6	0.770358
$544$	221184.	0.0320447
$545$	−5.09333e6	−0.734531
$546$	0	0
$547$	8.73025e6	1.24755	0.623775	−	0.781604i	$-0.285598\pi$
$547$	8.73025e6	1.24755	0.623775	+	0.781604i	$0.285598\pi$
$548$	2.07754e6	0.295527
$549$	−2.88797e6	−0.408943
$550$	−672936.	−0.0948565
$551$	−846600.	−0.118795
$552$	597888.	0.0835165
$553$	0	0
$554$	2.83095e6	0.391885
$555$	−2.64341e6	−0.364277
$556$	1.66970e6	0.229061
$557$	−3.01066e6	−0.411172	−0.205586	−	0.978639i	$-0.565910\pi$
$557$	−3.01066e6	−0.411172	−0.205586	+	0.978639i	$0.565910\pi$
$558$	2.28355e6	0.310474
$559$	1.51038e6	0.204435
$560$	0	0
$561$	−128304.	−0.0172121
$562$	9.21727e6	1.23101
$563$	−1.17573e7	−1.56327	−0.781637	−	0.623733i	$-0.785615\pi$
$563$	−1.17573e7	−1.56327	−0.781637	+	0.623733i	$0.785615\pi$
$564$	2.96698e6	0.392750
$565$	−1.49054e6	−0.196437
$566$	−6.43611e6	−0.844467
$567$	0	0
$568$	−2.72909e6	−0.354933
$569$	1.31578e7	1.70374	0.851870	−	0.523754i	$-0.175469\pi$
$569$	1.31578e7	1.70374	0.851870	+	0.523754i	$0.175469\pi$
$570$	293760.	0.0378709
$571$	−1.03344e7	−1.32647	−0.663234	−	0.748412i	$-0.730817\pi$
$571$	−1.03344e7	−1.32647	−0.663234	+	0.748412i	$0.730817\pi$
$572$	−103488.	−0.0132251
$573$	−8.45408e6	−1.07567
$574$	0	0
$575$	2.64586e6	0.333732
$576$	331776.	0.0416667
$577$	7.88133e6	0.985508	0.492754	−	0.870169i	$-0.335990\pi$
$577$	7.88133e6	0.985508	0.492754	+	0.870169i	$0.335990\pi$
$578$	−5.49280e6	−0.683872
$579$	−3.04551e6	−0.377541
$580$	956160.	0.118021
$581$	0	0
$582$	671832.	0.0822154
$583$	2.14434e6	0.261290
$584$	−2.15898e6	−0.261948
$585$	190512.	0.0230161
$586$	−2.06808e6	−0.248784
$587$	554568.	0.0664293	0.0332146	−	0.999448i	$-0.489426\pi$
$587$	554568.	0.0664293	0.0332146	+	0.999448i	$0.489426\pi$
$588$	0	0
$589$	2.39632e6	0.284614
$590$	3.28550e6	0.388572
$591$	2.14148e6	0.252200
$592$	−3.13293e6	−0.367406
$593$	9.20369e6	1.07479	0.537397	−	0.843329i	$-0.319408\pi$
$593$	9.20369e6	1.07479	0.537397	+	0.843329i	$0.319408\pi$
$594$	−192456.	−0.0223803
$595$	0	0
$596$	3.47510e6	0.400730
$597$	1.84018e6	0.211312
$598$	406896.	0.0465297
$599$	8.54295e6	0.972839	0.486419	−	0.873725i	$-0.338303\pi$
$599$	8.54295e6	0.972839	0.486419	+	0.873725i	$0.338303\pi$
$600$	1.46822e6	0.166500
$601$	9.61555e6	1.08590	0.542948	−	0.839767i	$-0.317308\pi$
$601$	9.61555e6	1.08590	0.542948	+	0.839767i	$0.317308\pi$
$602$	0	0
$603$	1.02708e6	0.115030
$604$	3.53600e6	0.394385
$605$	3.76068e6	0.417713
$606$	5.51102e6	0.609608
$607$	−2.21264e6	−0.243747	−0.121873	−	0.992546i	$-0.538890\pi$
$607$	−2.21264e6	−0.243747	−0.121873	+	0.992546i	$0.538890\pi$
$608$	348160.	0.0381962
$609$	0	0
$610$	3.42278e6	0.372439
$611$	2.01919e6	0.218814
$612$	279936.	0.0302121
$613$	−7.96215e6	−0.855814	−0.427907	−	0.903823i	$-0.640749\pi$
$613$	−7.96215e6	−0.855814	−0.427907	+	0.903823i	$0.640749\pi$
$614$	−5.40008e6	−0.578068
$615$	−1.39709e6	−0.148948
$616$	0	0
$617$	−1.37397e7	−1.45299	−0.726497	−	0.687170i	$-0.758853\pi$
$617$	−1.37397e7	−1.45299	−0.726497	+	0.687170i	$0.758853\pi$
$618$	1.29110e6	0.135985
$619$	8.70113e6	0.912744	0.456372	−	0.889789i	$-0.349149\pi$
$619$	8.70113e6	0.912744	0.456372	+	0.889789i	$0.349149\pi$
$620$	−2.70643e6	−0.282760
$621$	756702.	0.0787401
$622$	−5.38152e6	−0.557736
$623$	0	0
$624$	225792.	0.0232138
$625$	4.69740e6	0.481014
$626$	−1.02462e6	−0.104502
$627$	−201960.	−0.0205162
$628$	6.05392e6	0.612544
$629$	−2.64341e6	−0.266402
$630$	0	0
$631$	445412.	0.0445337	0.0222668	−	0.999752i	$-0.492912\pi$
$631$	445412.	0.0445337	0.0222668	+	0.999752i	$0.492912\pi$
$632$	−5.44691e6	−0.542447
$633$	3.13852e6	0.311326
$634$	7.38516e6	0.729687
$635$	1.27306e6	0.125289
$636$	−4.67856e6	−0.458637
$637$	0	0
$638$	−657360.	−0.0639369
$639$	−3.45400e6	−0.334634
$640$	−393216.	−0.0379473
$641$	−8.00119e6	−0.769147	−0.384573	−	0.923094i	$-0.625651\pi$
$641$	−8.00119e6	−0.769147	−0.384573	+	0.923094i	$0.625651\pi$
$642$	3.43634e6	0.329048
$643$	1.58402e7	1.51090	0.755448	−	0.655209i	$-0.227419\pi$
$643$	1.58402e7	1.51090	0.755448	+	0.655209i	$0.227419\pi$
$644$	0	0
$645$	−3.32899e6	−0.315075
$646$	293760.	0.0276956
$647$	−1.30187e6	−0.122266	−0.0611331	−	0.998130i	$-0.519471\pi$
$647$	−1.30187e6	−0.122266	−0.0611331	+	0.998130i	$0.519471\pi$
$648$	419904.	0.0392837
$649$	−2.25878e6	−0.210505
$650$	999208.	0.0927625
$651$	0	0
$652$	1.67706e6	0.154500
$653$	7.34149e6	0.673753	0.336877	−	0.941549i	$-0.390629\pi$
$653$	7.34149e6	0.673753	0.336877	+	0.941549i	$0.390629\pi$
$654$	−7.63999e6	−0.698471
$655$	1.66378e6	0.151528
$656$	−1.65581e6	−0.150228
$657$	−2.73245e6	−0.246967
$658$	0	0
$659$	−6.18934e6	−0.555176	−0.277588	−	0.960700i	$-0.589535\pi$
$659$	−6.18934e6	−0.555176	−0.277588	+	0.960700i	$0.589535\pi$
$660$	228096.	0.0203825
$661$	1.96690e7	1.75097	0.875484	−	0.483248i	$-0.160543\pi$
$661$	1.96690e7	1.75097	0.875484	+	0.483248i	$0.160543\pi$
$662$	−1.33295e7	−1.18214
$663$	190512.	0.0168321
$664$	6.83290e6	0.601429
$665$	0	0
$666$	−3.96511e6	−0.346394
$667$	2.58462e6	0.224948
$668$	6.83155e6	0.592350
$669$	1.32305e7	1.14291
$670$	−1.21728e6	−0.104762
$671$	−2.35316e6	−0.201765
$672$	0	0
$673$	7.18259e6	0.611285	0.305642	−	0.952146i	$-0.401129\pi$
$673$	7.18259e6	0.611285	0.305642	+	0.952146i	$0.401129\pi$
$674$	−6.53922e6	−0.554468
$675$	1.85822e6	0.156978
$676$	−5.78702e6	−0.487067
$677$	1.89192e7	1.58647	0.793234	−	0.608917i	$-0.208396\pi$
$677$	1.89192e7	1.58647	0.793234	+	0.608917i	$0.208396\pi$
$678$	−2.23582e6	−0.186794
$679$	0	0
$680$	−331776.	−0.0275152
$681$	−5.30604e6	−0.438432
$682$	1.86067e6	0.153182
$683$	2.12204e7	1.74061	0.870306	−	0.492512i	$-0.163921\pi$
$683$	2.12204e7	1.74061	0.870306	+	0.492512i	$0.163921\pi$
$684$	440640.	0.0360117
$685$	−3.11630e6	−0.253754
$686$	0	0
$687$	−9.40808e6	−0.760517
$688$	−3.94547e6	−0.317781
$689$	−3.18402e6	−0.255522
$690$	−896832.	−0.0717114
$691$	−1.63276e7	−1.30085	−0.650424	−	0.759571i	$-0.725409\pi$
$691$	−1.63276e7	−1.30085	−0.650424	+	0.759571i	$0.725409\pi$
$692$	−5.29709e6	−0.420506
$693$	0	0
$694$	−3.36612e6	−0.265296
$695$	−2.50454e6	−0.196683
$696$	1.43424e6	0.112227
$697$	−1.39709e6	−0.108929
$698$	3.90897e6	0.303685
$699$	−5.86100e6	−0.453710
$700$	0	0
$701$	−5.40470e6	−0.415409	−0.207705	−	0.978192i	$-0.566599\pi$
$701$	−5.40470e6	−0.415409	−0.207705	+	0.978192i	$0.566599\pi$
$702$	285768.	0.0218862
$703$	−4.16092e6	−0.317542
$704$	270336.	0.0205576
$705$	−4.45046e6	−0.337235
$706$	−1.38343e7	−1.04459
$707$	0	0
$708$	4.92826e6	0.369496
$709$	2.21195e7	1.65257	0.826284	−	0.563253i	$-0.190450\pi$
$709$	2.21195e7	1.65257	0.826284	+	0.563253i	$0.190450\pi$
$710$	4.09363e6	0.304763
$711$	−6.89375e6	−0.511424
$712$	−2.23258e6	−0.165046
$713$	−7.31582e6	−0.538939
$714$	0	0
$715$	155232.	0.0113558
$716$	−6.40310e6	−0.466775
$717$	4.62116e6	0.335701
$718$	−1.38920e7	−1.00567
$719$	−2.55819e7	−1.84548	−0.922742	−	0.385418i	$-0.874057\pi$
$719$	−2.55819e7	−1.84548	−0.922742	+	0.385418i	$0.874057\pi$
$720$	−497664.	−0.0357771
$721$	0	0
$722$	−9.44200e6	−0.674095
$723$	−6.25243e6	−0.444839
$724$	−9.40957e6	−0.667150
$725$	6.34701e6	0.448460
$726$	5.64102e6	0.397206
$727$	9.29438e6	0.652205	0.326103	−	0.945334i	$-0.394265\pi$
$727$	9.29438e6	0.652205	0.326103	+	0.945334i	$0.394265\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	3.23846e6	0.224922
$731$	−3.32899e6	−0.230420
$732$	5.13418e6	0.354155
$733$	−3.40699e6	−0.234213	−0.117107	−	0.993119i	$-0.537362\pi$
$733$	−3.40699e6	−0.234213	−0.117107	+	0.993119i	$0.537362\pi$
$734$	−1.24797e7	−0.854999
$735$	0	0
$736$	−1.06291e6	−0.0723274
$737$	836880.	0.0567537
$738$	−2.09563e6	−0.141636
$739$	2.18135e7	1.46932	0.734658	−	0.678438i	$-0.237343\pi$
$739$	2.18135e7	1.46932	0.734658	+	0.678438i	$0.237343\pi$
$740$	4.69939e6	0.315473
$741$	299880.	0.0200633
$742$	0	0
$743$	3.79246e6	0.252028	0.126014	−	0.992028i	$-0.459782\pi$
$743$	3.79246e6	0.252028	0.126014	+	0.992028i	$0.459782\pi$
$744$	−4.05965e6	−0.268878
$745$	−5.21266e6	−0.344087
$746$	−8.06692e6	−0.530714
$747$	8.64788e6	0.567033
$748$	228096.	0.0149061
$749$	0	0
$750$	−4.90234e6	−0.318236
$751$	−2.01483e7	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$751$	−2.01483e7	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$752$	−5.27462e6	−0.340132
$753$	−1.25647e7	−0.807542
$754$	976080.	0.0625255
$755$	−5.30400e6	−0.338638
$756$	0	0
$757$	1.18427e7	0.751126	0.375563	−	0.926797i	$-0.377449\pi$
$757$	1.18427e7	0.751126	0.375563	+	0.926797i	$0.377449\pi$
$758$	−2.15233e7	−1.36062
$759$	616572.	0.0388490
$760$	−522240.	−0.0327972
$761$	−2.97791e6	−0.186402	−0.0932008	−	0.995647i	$-0.529710\pi$
$761$	−2.97791e6	−0.186402	−0.0932008	+	0.995647i	$0.529710\pi$
$762$	1.90958e6	0.119138
$763$	0	0
$764$	1.50295e7	0.931559
$765$	−419904.	−0.0259416
$766$	−3.22973e6	−0.198881
$767$	3.35395e6	0.205858
$768$	−589824.	−0.0360844
$769$	2.02441e7	1.23447	0.617237	−	0.786777i	$-0.288252\pi$
$769$	2.02441e7	1.23447	0.617237	+	0.786777i	$0.288252\pi$
$770$	0	0
$771$	−9.04684e6	−0.548101
$772$	5.41424e6	0.326960
$773$	7.37953e6	0.444202	0.222101	−	0.975024i	$-0.428709\pi$
$773$	7.37953e6	0.444202	0.222101	+	0.975024i	$0.428709\pi$
$774$	−4.99349e6	−0.299607
$775$	−1.79654e7	−1.07444
$776$	−1.19437e6	−0.0712006
$777$	0	0
$778$	3.56556e6	0.211193
$779$	−2.19912e6	−0.129839
$780$	−338688.	−0.0199326
$781$	−2.81437e6	−0.165103
$782$	−896832.	−0.0524438
$783$	1.81521e6	0.105809
$784$	0	0
$785$	−9.08088e6	−0.525961
$786$	2.49566e6	0.144089
$787$	−1.36289e7	−0.784377	−0.392188	−	0.919885i	$-0.628282\pi$
$787$	−1.36289e7	−0.784377	−0.392188	+	0.919885i	$0.628282\pi$
$788$	−3.80707e6	−0.218412
$789$	−1.12771e7	−0.644918
$790$	8.17037e6	0.465773
$791$	0	0
$792$	342144.	0.0193819
$793$	3.49409e6	0.197311
$794$	−4.49382e6	−0.252967
$795$	7.01784e6	0.393809
$796$	−3.27142e6	−0.183001
$797$	1.49548e7	0.833938	0.416969	−	0.908921i	$-0.363092\pi$
$797$	1.49548e7	0.833938	0.416969	+	0.908921i	$0.363092\pi$
$798$	0	0
$799$	−4.45046e6	−0.246626
$800$	−2.61018e6	−0.144193
$801$	−2.82560e6	−0.155607
$802$	6.88150e6	0.377787
$803$	−2.22644e6	−0.121849
$804$	−1.82592e6	−0.0996189
$805$	0	0
$806$	−2.76282e6	−0.149801
$807$	−1.58462e7	−0.856527
$808$	−9.79738e6	−0.527936
$809$	2.87242e7	1.54304	0.771519	−	0.636206i	$-0.219497\pi$
$809$	2.87242e7	1.54304	0.771519	+	0.636206i	$0.219497\pi$
$810$	−629856.	−0.0337310
$811$	1.52265e7	0.812922	0.406461	−	0.913668i	$-0.366763\pi$
$811$	1.52265e7	0.812922	0.406461	+	0.913668i	$0.366763\pi$
$812$	0	0
$813$	6.93475e6	0.367963
$814$	−3.23083e6	−0.170904
$815$	−2.51558e6	−0.132661
$816$	−497664.	−0.0261644
$817$	−5.24008e6	−0.274652
$818$	−308984.	−0.0161455
$819$	0	0
$820$	2.48371e6	0.128993
$821$	−3.31001e7	−1.71384	−0.856921	−	0.515447i	$-0.827626\pi$
$821$	−3.31001e7	−1.71384	−0.856921	+	0.515447i	$0.827626\pi$
$822$	−4.67446e6	−0.241297
$823$	−1.35915e7	−0.699470	−0.349735	−	0.936849i	$-0.613728\pi$
$823$	−1.35915e7	−0.699470	−0.349735	+	0.936849i	$0.613728\pi$
$824$	−2.29530e6	−0.117766
$825$	1.51411e6	0.0774500
$826$	0	0
$827$	3.13936e6	0.159616	0.0798082	−	0.996810i	$-0.474569\pi$
$827$	3.13936e6	0.159616	0.0798082	+	0.996810i	$0.474569\pi$
$828$	−1.34525e6	−0.0681909
$829$	−1.27081e7	−0.642234	−0.321117	−	0.947040i	$-0.604058\pi$
$829$	−1.27081e7	−0.642234	−0.321117	+	0.947040i	$0.604058\pi$
$830$	−1.02493e7	−0.516417
$831$	−6.36964e6	−0.319972
$832$	−401408.	−0.0201038
$833$	0	0
$834$	−3.75682e6	−0.187027
$835$	−1.02473e7	−0.508621
$836$	359040.	0.0177675
$837$	−5.13799e6	−0.253501
$838$	2.08246e7	1.02439
$839$	2.98312e7	1.46307	0.731536	−	0.681803i	$-0.238804\pi$
$839$	2.98312e7	1.46307	0.731536	+	0.681803i	$0.238804\pi$
$840$	0	0
$841$	−1.43110e7	−0.697720
$842$	6.87390e6	0.334136
$843$	−2.07389e7	−1.00512
$844$	−5.57958e6	−0.269616
$845$	8.68054e6	0.418220
$846$	−6.67570e6	−0.320679
$847$	0	0
$848$	8.31744e6	0.397192
$849$	1.44813e7	0.689504
$850$	−2.20234e6	−0.104553
$851$	1.27030e7	0.601290
$852$	6.14045e6	0.289802
$853$	1.92215e7	0.904515	0.452257	−	0.891888i	$-0.350619\pi$
$853$	1.92215e7	0.904515	0.452257	+	0.891888i	$0.350619\pi$
$854$	0	0
$855$	−660960.	−0.0309215
$856$	−6.10906e6	−0.284964
$857$	2.65655e7	1.23556	0.617782	−	0.786349i	$-0.288031\pi$
$857$	2.65655e7	1.23556	0.617782	+	0.786349i	$0.288031\pi$
$858$	232848.	0.0107983
$859$	9.16844e6	0.423948	0.211974	−	0.977275i	$-0.432011\pi$
$859$	9.16844e6	0.423948	0.211974	+	0.977275i	$0.432011\pi$
$860$	5.91821e6	0.272863
$861$	0	0
$862$	−2.32250e6	−0.106460
$863$	−2.92196e7	−1.33551	−0.667755	−	0.744381i	$-0.732745\pi$
$863$	−2.92196e7	−1.33551	−0.667755	+	0.744381i	$0.732745\pi$
$864$	−746496.	−0.0340207
$865$	7.94563e6	0.361067
$866$	−1.66035e7	−0.752324
$867$	1.23588e7	0.558379
$868$	0	0
$869$	−5.61713e6	−0.252328
$870$	−2.15136e6	−0.0963640
$871$	−1.24264e6	−0.0555009
$872$	1.35822e7	0.604894
$873$	−1.51162e6	−0.0671286
$874$	−1.41168e6	−0.0625112
$875$	0	0
$876$	4.85770e6	0.213880
$877$	9.71286e6	0.426430	0.213215	−	0.977005i	$-0.431606\pi$
$877$	9.71286e6	0.426430	0.213215	+	0.977005i	$0.431606\pi$
$878$	−1.55363e7	−0.680160
$879$	4.65318e6	0.203132
$880$	−405504.	−0.0176518
$881$	−1.65372e7	−0.717833	−0.358917	−	0.933370i	$-0.616854\pi$
$881$	−1.65372e7	−0.717833	−0.358917	+	0.933370i	$0.616854\pi$
$882$	0	0
$883$	−2.39487e7	−1.03367	−0.516833	−	0.856086i	$-0.672889\pi$
$883$	−2.39487e7	−1.03367	−0.516833	+	0.856086i	$0.672889\pi$
$884$	−338688.	−0.0145770
$885$	−7.39238e6	−0.317268
$886$	−9.25994e6	−0.396300
$887$	4.62846e6	0.197527	0.0987637	−	0.995111i	$-0.468511\pi$
$887$	4.62846e6	0.197527	0.0987637	+	0.995111i	$0.468511\pi$
$888$	7.04909e6	0.299986
$889$	0	0
$890$	3.34886e6	0.141717
$891$	433026.	0.0182734
$892$	−2.35209e7	−0.989787
$893$	−7.00536e6	−0.293969
$894$	−7.81898e6	−0.327195
$895$	9.60466e6	0.400797
$896$	0	0
$897$	−915516.	−0.0379914
$898$	−7.69126e6	−0.318278
$899$	−1.75495e7	−0.724212
$900$	−3.30350e6	−0.135947
$901$	7.01784e6	0.287999
$902$	−1.70755e6	−0.0698808
$903$	0	0
$904$	3.97478e6	0.161768
$905$	1.41144e7	0.572848
$906$	−7.95600e6	−0.322014
$907$	2.06126e7	0.831983	0.415991	−	0.909369i	$-0.363435\pi$
$907$	2.06126e7	0.831983	0.415991	+	0.909369i	$0.363435\pi$
$908$	9.43296e6	0.379694
$909$	−1.23998e7	−0.497743
$910$	0	0
$911$	−3.46749e6	−0.138427	−0.0692133	−	0.997602i	$-0.522049\pi$
$911$	−3.46749e6	−0.138427	−0.0692133	+	0.997602i	$0.522049\pi$
$912$	−783360.	−0.0311870
$913$	7.04642e6	0.279764
$914$	2.74486e7	1.08681
$915$	−7.70126e6	−0.304095
$916$	1.67255e7	0.658627
$917$	0	0
$918$	−629856.	−0.0246680
$919$	−3.61227e7	−1.41088	−0.705442	−	0.708767i	$-0.749252\pi$
$919$	−3.61227e7	−1.41088	−0.705442	+	0.708767i	$0.749252\pi$
$920$	1.59437e6	0.0621039
$921$	1.21502e7	0.471991
$922$	−1.18867e7	−0.460504
$923$	4.17892e6	0.161458
$924$	0	0
$925$	3.11947e7	1.19874
$926$	1.94969e7	0.747203
$927$	−2.90498e6	−0.111031
$928$	−2.54976e6	−0.0971917
$929$	−1.29366e7	−0.491792	−0.245896	−	0.969296i	$-0.579082\pi$
$929$	−1.29366e7	−0.491792	−0.245896	+	0.969296i	$0.579082\pi$
$930$	6.08947e6	0.230873
$931$	0	0
$932$	1.04196e7	0.392925
$933$	1.21084e7	0.455390
$934$	3.26920e7	1.22624
$935$	−342144.	−0.0127991
$936$	−508032.	−0.0189540
$937$	−5.01394e7	−1.86565	−0.932824	−	0.360332i	$-0.882664\pi$
$937$	−5.01394e7	−1.86565	−0.932824	+	0.360332i	$0.882664\pi$
$938$	0	0
$939$	2.30539e6	0.0853257
$940$	7.91194e6	0.292054
$941$	1.05568e7	0.388651	0.194325	−	0.980937i	$-0.437748\pi$
$941$	1.05568e7	0.388651	0.194325	+	0.980937i	$0.437748\pi$
$942$	−1.36213e7	−0.500140
$943$	6.71378e6	0.245860
$944$	−8.76134e6	−0.319993
$945$	0	0
$946$	−4.06877e6	−0.147821
$947$	−3.14684e6	−0.114025	−0.0570124	−	0.998373i	$-0.518157\pi$
$947$	−3.14684e6	−0.114025	−0.0570124	+	0.998373i	$0.518157\pi$
$948$	1.22556e7	0.442906
$949$	3.30593e6	0.119159
$950$	−3.46664e6	−0.124623
$951$	−1.66166e7	−0.595787
$952$	0	0
$953$	5.22829e7	1.86478	0.932389	−	0.361455i	$-0.117720\pi$
$953$	5.22829e7	1.86478	0.932389	+	0.361455i	$0.117720\pi$
$954$	1.05268e7	0.374476
$955$	−2.25442e7	−0.799883
$956$	−8.21539e6	−0.290726
$957$	1.47906e6	0.0522043
$958$	−9.37589e6	−0.330064
$959$	0	0
$960$	884736.	0.0309839
$961$	2.10452e7	0.735095
$962$	4.79730e6	0.167132
$963$	−7.73177e6	−0.268666
$964$	1.11154e7	0.385242
$965$	−8.12136e6	−0.280744
$966$	0	0
$967$	−2.48235e7	−0.853682	−0.426841	−	0.904327i	$-0.640374\pi$
$967$	−2.48235e7	−0.853682	−0.426841	+	0.904327i	$0.640374\pi$
$968$	−1.00285e7	−0.343991
$969$	−660960.	−0.0226134
$970$	1.79155e6	0.0611364
$971$	−1.33077e7	−0.452956	−0.226478	−	0.974016i	$-0.572721\pi$
$971$	−1.33077e7	−0.452956	−0.226478	+	0.974016i	$0.572721\pi$
$972$	−944784.	−0.0320750
$973$	0	0
$974$	1.26771e6	0.0428177
$975$	−2.24822e6	−0.0757403
$976$	−9.12742e6	−0.306707
$977$	8.17705e6	0.274069	0.137035	−	0.990566i	$-0.456243\pi$
$977$	8.17705e6	0.274069	0.137035	+	0.990566i	$0.456243\pi$
$978$	−3.77338e6	−0.126149
$979$	−2.30234e6	−0.0767739
$980$	0	0
$981$	1.71900e7	0.570299
$982$	−2.08016e7	−0.688365
$983$	1.32465e7	0.437238	0.218619	−	0.975810i	$-0.429845\pi$
$983$	1.32465e7	0.437238	0.218619	+	0.975810i	$0.429845\pi$
$984$	3.72557e6	0.122661
$985$	5.71061e6	0.187539
$986$	−2.15136e6	−0.0704727
$987$	0	0
$988$	−533120.	−0.0173753
$989$	1.59977e7	0.520075
$990$	−513216.	−0.0166423
$991$	−1.48550e7	−0.480494	−0.240247	−	0.970712i	$-0.577228\pi$
$991$	−1.48550e7	−0.480494	−0.240247	+	0.970712i	$0.577228\pi$
$992$	7.21715e6	0.232856
$993$	2.99914e7	0.965215
$994$	0	0
$995$	4.90714e6	0.157134
$996$	−1.53740e7	−0.491065
$997$	3.33769e6	0.106343	0.0531714	−	0.998585i	$-0.483067\pi$
$997$	3.33769e6	0.106343	0.0531714	+	0.998585i	$0.483067\pi$
$998$	−1.94709e7	−0.618814
$999$	8.92150e6	0.282829

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.6.a.i.1.1		1
3.2	odd	2		882.6.a.i.1.1		1
7.2	even	3		294.6.e.f.67.1		2
7.3	odd	6		294.6.e.b.79.1		2
7.4	even	3		294.6.e.f.79.1		2
7.5	odd	6		294.6.e.b.67.1		2
7.6	odd	2		42.6.a.f.1.1	✓	1
21.20	even	2		126.6.a.b.1.1		1
28.27	even	2		336.6.a.g.1.1		1
35.13	even	4		1050.6.g.m.799.1		2
35.27	even	4		1050.6.g.m.799.2		2
35.34	odd	2		1050.6.a.a.1.1		1
84.83	odd	2		1008.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.f.1.1	✓	1	7.6	odd	2
126.6.a.b.1.1		1	21.20	even	2
294.6.a.i.1.1		1	1.1	even	1	trivial
294.6.e.b.67.1		2	7.5	odd	6
294.6.e.b.79.1		2	7.3	odd	6
294.6.e.f.67.1		2	7.2	even	3
294.6.e.f.79.1		2	7.4	even	3
336.6.a.g.1.1		1	28.27	even	2
882.6.a.i.1.1		1	3.2	odd	2
1008.6.a.k.1.1		1	84.83	odd	2
1050.6.a.a.1.1		1	35.34	odd	2
1050.6.g.m.799.1		2	35.13	even	4
1050.6.g.m.799.2		2	35.27	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	294.6.a.i.1.1

Level	$294$
Weight	$6$
Character	294.1
Self dual	yes
Analytic conductor	$47.153$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$