LMFDB - Embedded newform 490.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,6,Mod(1,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, 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("490.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	490.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:	$78.5880717084$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	490.1

$q$ -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} +23.0000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -92.0000 q^{6} -64.0000 q^{8} +286.000 q^{9} +100.000 q^{10} +555.000 q^{11} +368.000 q^{12} +241.000 q^{13} -575.000 q^{15} +256.000 q^{16} +1491.00 q^{17} -1144.00 q^{18} +2038.00 q^{19} -400.000 q^{20} -2220.00 q^{22} -1230.00 q^{23} -1472.00 q^{24} +625.000 q^{25} -964.000 q^{26} +989.000 q^{27} -5001.00 q^{29} +2300.00 q^{30} -5696.00 q^{31} -1024.00 q^{32} +12765.0 q^{33} -5964.00 q^{34} +4576.00 q^{36} -5602.00 q^{37} -8152.00 q^{38} +5543.00 q^{39} +1600.00 q^{40} +2424.00 q^{41} +602.000 q^{43} +8880.00 q^{44} -7150.00 q^{45} +4920.00 q^{46} +23163.0 q^{47} +5888.00 q^{48} -2500.00 q^{50} +34293.0 q^{51} +3856.00 q^{52} -25296.0 q^{53} -3956.00 q^{54} -13875.0 q^{55} +46874.0 q^{57} +20004.0 q^{58} -5724.00 q^{59} -9200.00 q^{60} +36112.0 q^{61} +22784.0 q^{62} +4096.00 q^{64} -6025.00 q^{65} -51060.0 q^{66} +66104.0 q^{67} +23856.0 q^{68} -28290.0 q^{69} +16080.0 q^{71} -18304.0 q^{72} +80482.0 q^{73} +22408.0 q^{74} +14375.0 q^{75} +32608.0 q^{76} -22172.0 q^{78} -64147.0 q^{79} -6400.00 q^{80} -46751.0 q^{81} -9696.00 q^{82} +106284. q^{83} -37275.0 q^{85} -2408.00 q^{86} -115023. q^{87} -35520.0 q^{88} +71676.0 q^{89} +28600.0 q^{90} -19680.0 q^{92} -131008. q^{93} -92652.0 q^{94} -50950.0 q^{95} -23552.0 q^{96} -151025. q^{97} +158730. 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
$3$	23.0000	1.47545	0.737725	−	0.675101i	$-0.235900\pi$
$3$	23.0000	1.47545	0.737725	+	0.675101i	$0.235900\pi$
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−92.0000	−1.04330
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	286.000	1.17695
$10$	100.000	0.316228
$11$	555.000	1.38297	0.691483	−	0.722393i	$-0.256958\pi$
$11$	555.000	1.38297	0.691483	+	0.722393i	$0.256958\pi$
$12$	368.000	0.737725
$13$	241.000	0.395511	0.197756	−	0.980251i	$-0.436635\pi$
$13$	241.000	0.395511	0.197756	+	0.980251i	$0.436635\pi$
$14$	0	0
$15$	−575.000	−0.659842
$16$	256.000	0.250000
$17$	1491.00	1.25128	0.625641	−	0.780111i	$-0.284837\pi$
$17$	1491.00	1.25128	0.625641	+	0.780111i	$0.284837\pi$
$18$	−1144.00	−0.832233
$19$	2038.00	1.29515	0.647575	−	0.762002i	$-0.275783\pi$
$19$	2038.00	1.29515	0.647575	+	0.762002i	$0.275783\pi$
$20$	−400.000	−0.223607
$21$	0	0
$22$	−2220.00	−0.977904
$23$	−1230.00	−0.484826	−0.242413	−	0.970173i	$-0.577939\pi$
$23$	−1230.00	−0.484826	−0.242413	+	0.970173i	$0.577939\pi$
$24$	−1472.00	−0.521651
$25$	625.000	0.200000
$26$	−964.000	−0.279669
$27$	989.000	0.261088
$28$	0	0
$29$	−5001.00	−1.10424	−0.552118	−	0.833766i	$-0.686180\pi$
$29$	−5001.00	−1.10424	−0.552118	+	0.833766i	$0.686180\pi$
$30$	2300.00	0.466578
$31$	−5696.00	−1.06455	−0.532275	−	0.846572i	$-0.678663\pi$
$31$	−5696.00	−1.06455	−0.532275	+	0.846572i	$0.678663\pi$
$32$	−1024.00	−0.176777
$33$	12765.0	2.04050
$34$	−5964.00	−0.884790
$35$	0	0
$36$	4576.00	0.588477
$37$	−5602.00	−0.672727	−0.336363	−	0.941732i	$-0.609197\pi$
$37$	−5602.00	−0.672727	−0.336363	+	0.941732i	$0.609197\pi$
$38$	−8152.00	−0.915810
$39$	5543.00	0.583557
$40$	1600.00	0.158114
$41$	2424.00	0.225202	0.112601	−	0.993640i	$-0.464082\pi$
$41$	2424.00	0.225202	0.112601	+	0.993640i	$0.464082\pi$
$42$	0	0
$43$	602.000	0.0496507	0.0248253	−	0.999692i	$-0.492097\pi$
$43$	602.000	0.0496507	0.0248253	+	0.999692i	$0.492097\pi$
$44$	8880.00	0.691483
$45$	−7150.00	−0.526350
$46$	4920.00	0.342823
$47$	23163.0	1.52950	0.764751	−	0.644326i	$-0.222862\pi$
$47$	23163.0	1.52950	0.764751	+	0.644326i	$0.222862\pi$
$48$	5888.00	0.368863
$49$	0	0
$50$	−2500.00	−0.141421
$51$	34293.0	1.84621
$52$	3856.00	0.197756
$53$	−25296.0	−1.23698	−0.618489	−	0.785793i	$-0.712255\pi$
$53$	−25296.0	−1.23698	−0.618489	+	0.785793i	$0.712255\pi$
$54$	−3956.00	−0.184617
$55$	−13875.0	−0.618481
$56$	0	0
$57$	46874.0	1.91093
$58$	20004.0	0.780813
$59$	−5724.00	−0.214077	−0.107038	−	0.994255i	$-0.534137\pi$
$59$	−5724.00	−0.214077	−0.107038	+	0.994255i	$0.534137\pi$
$60$	−9200.00	−0.329921
$61$	36112.0	1.24259	0.621294	−	0.783578i	$-0.286607\pi$
$61$	36112.0	1.24259	0.621294	+	0.783578i	$0.286607\pi$
$62$	22784.0	0.752750
$63$	0	0
$64$	4096.00	0.125000
$65$	−6025.00	−0.176878
$66$	−51060.0	−1.44285
$67$	66104.0	1.79904	0.899520	−	0.436880i	$-0.143917\pi$
$67$	66104.0	1.79904	0.899520	+	0.436880i	$0.143917\pi$
$68$	23856.0	0.625641
$69$	−28290.0	−0.715336
$70$	0	0
$71$	16080.0	0.378565	0.189282	−	0.981923i	$-0.439384\pi$
$71$	16080.0	0.378565	0.189282	+	0.981923i	$0.439384\pi$
$72$	−18304.0	−0.416116
$73$	80482.0	1.76763	0.883816	−	0.467836i	$-0.154966\pi$
$73$	80482.0	1.76763	0.883816	+	0.467836i	$0.154966\pi$
$74$	22408.0	0.475690
$75$	14375.0	0.295090
$76$	32608.0	0.647575
$77$	0	0
$78$	−22172.0	−0.412637
$79$	−64147.0	−1.15640	−0.578201	−	0.815895i	$-0.696245\pi$
$79$	−64147.0	−1.15640	−0.578201	+	0.815895i	$0.696245\pi$
$80$	−6400.00	−0.111803
$81$	−46751.0	−0.791732
$82$	−9696.00	−0.159242
$83$	106284.	1.69345	0.846726	−	0.532030i	$-0.178571\pi$
$83$	106284.	1.69345	0.846726	+	0.532030i	$0.178571\pi$
$84$	0	0
$85$	−37275.0	−0.559591
$86$	−2408.00	−0.0351083
$87$	−115023.	−1.62925
$88$	−35520.0	−0.488952
$89$	71676.0	0.959177	0.479588	−	0.877494i	$-0.340786\pi$
$89$	71676.0	0.959177	0.479588	+	0.877494i	$0.340786\pi$
$90$	28600.0	0.372186
$91$	0	0
$92$	−19680.0	−0.242413
$93$	−131008.	−1.57069
$94$	−92652.0	−1.08152
$95$	−50950.0	−0.579209
$96$	−23552.0	−0.260825
$97$	−151025.	−1.62974	−0.814872	−	0.579641i	$-0.803193\pi$
$97$	−151025.	−1.62974	−0.814872	+	0.579641i	$0.803193\pi$
$98$	0	0
$99$	158730.	1.62769
$100$	10000.0	0.100000
$101$	57150.0	0.557459	0.278729	−	0.960370i	$-0.410087\pi$
$101$	57150.0	0.557459	0.278729	+	0.960370i	$0.410087\pi$
$102$	−137172.	−1.30546
$103$	−115889.	−1.07634	−0.538170	−	0.842837i	$-0.680884\pi$
$103$	−115889.	−1.07634	−0.538170	+	0.842837i	$0.680884\pi$
$104$	−15424.0	−0.139834
$105$	0	0
$106$	101184.	0.874676
$107$	−137862.	−1.16409	−0.582043	−	0.813158i	$-0.697747\pi$
$107$	−137862.	−1.16409	−0.582043	+	0.813158i	$0.697747\pi$
$108$	15824.0	0.130544
$109$	88397.0	0.712642	0.356321	−	0.934364i	$-0.384031\pi$
$109$	88397.0	0.712642	0.356321	+	0.934364i	$0.384031\pi$
$110$	55500.0	0.437332
$111$	−128846.	−0.992575
$112$	0	0
$113$	205554.	1.51436	0.757181	−	0.653205i	$-0.226576\pi$
$113$	205554.	1.51436	0.757181	+	0.653205i	$0.226576\pi$
$114$	−187496.	−1.35123
$115$	30750.0	0.216821
$116$	−80016.0	−0.552118
$117$	68926.0	0.465499
$118$	22896.0	0.151375
$119$	0	0
$120$	36800.0	0.233289
$121$	146974.	0.912593
$122$	−144448.	−0.878642
$123$	55752.0	0.332275
$124$	−91136.0	−0.532275
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	250916.	1.38044	0.690222	−	0.723597i	$-0.257513\pi$
$127$	250916.	1.38044	0.690222	+	0.723597i	$0.257513\pi$
$128$	−16384.0	−0.0883883
$129$	13846.0	0.0732572
$130$	24100.0	0.125072
$131$	52122.0	0.265365	0.132682	−	0.991159i	$-0.457641\pi$
$131$	52122.0	0.265365	0.132682	+	0.991159i	$0.457641\pi$
$132$	204240.	1.02025
$133$	0	0
$134$	−264416.	−1.27211
$135$	−24725.0	−0.116762
$136$	−95424.0	−0.442395
$137$	−135468.	−0.616645	−0.308323	−	0.951282i	$-0.599768\pi$
$137$	−135468.	−0.616645	−0.308323	+	0.951282i	$0.599768\pi$
$138$	113160.	0.505819
$139$	349486.	1.53424	0.767119	−	0.641505i	$-0.221690\pi$
$139$	349486.	1.53424	0.767119	+	0.641505i	$0.221690\pi$
$140$	0	0
$141$	532749.	2.25671
$142$	−64320.0	−0.267686
$143$	133755.	0.546978
$144$	73216.0	0.294239
$145$	125025.	0.493829
$146$	−321928.	−1.24990
$147$	0	0
$148$	−89632.0	−0.336363
$149$	176082.	0.649754	0.324877	−	0.945756i	$-0.394677\pi$
$149$	176082.	0.649754	0.324877	+	0.945756i	$0.394677\pi$
$150$	−57500.0	−0.208660
$151$	383333.	1.36815	0.684075	−	0.729411i	$-0.260206\pi$
$151$	383333.	1.36815	0.684075	+	0.729411i	$0.260206\pi$
$152$	−130432.	−0.457905
$153$	426426.	1.47270
$154$	0	0
$155$	142400.	0.476081
$156$	88688.0	0.291779
$157$	−345914.	−1.12000	−0.560001	−	0.828492i	$-0.689199\pi$
$157$	−345914.	−1.12000	−0.560001	+	0.828492i	$0.689199\pi$
$158$	256588.	0.817699
$159$	−581808.	−1.82510
$160$	25600.0	0.0790569
$161$	0	0
$162$	187004.	0.559839
$163$	91586.0	0.269998	0.134999	−	0.990846i	$-0.456897\pi$
$163$	91586.0	0.269998	0.134999	+	0.990846i	$0.456897\pi$
$164$	38784.0	0.112601
$165$	−319125.	−0.912538
$166$	−425136.	−1.19745
$167$	−38097.0	−0.105706	−0.0528530	−	0.998602i	$-0.516831\pi$
$167$	−38097.0	−0.105706	−0.0528530	+	0.998602i	$0.516831\pi$
$168$	0	0
$169$	−313212.	−0.843571
$170$	149100.	0.395690
$171$	582868.	1.52433
$172$	9632.00	0.0248253
$173$	541443.	1.37543	0.687713	−	0.725982i	$-0.258615\pi$
$173$	541443.	1.37543	0.687713	+	0.725982i	$0.258615\pi$
$174$	460092.	1.15205
$175$	0	0
$176$	142080.	0.345741
$177$	−131652.	−0.315860
$178$	−286704.	−0.678241
$179$	166188.	0.387674	0.193837	−	0.981034i	$-0.437907\pi$
$179$	166188.	0.387674	0.193837	+	0.981034i	$0.437907\pi$
$180$	−114400.	−0.263175
$181$	197320.	0.447687	0.223844	−	0.974625i	$-0.428140\pi$
$181$	197320.	0.447687	0.223844	+	0.974625i	$0.428140\pi$
$182$	0	0
$183$	830576.	1.83338
$184$	78720.0	0.171412
$185$	140050.	0.300853
$186$	524032.	1.11065
$187$	827505.	1.73048
$188$	370608.	0.764751
$189$	0	0
$190$	203800.	0.409562
$191$	−337221.	−0.668854	−0.334427	−	0.942422i	$-0.608543\pi$
$191$	−337221.	−0.668854	−0.334427	+	0.942422i	$0.608543\pi$
$192$	94208.0	0.184431
$193$	260516.	0.503432	0.251716	−	0.967801i	$-0.419005\pi$
$193$	260516.	0.503432	0.251716	+	0.967801i	$0.419005\pi$
$194$	604100.	1.15240
$195$	−138575.	−0.260975
$196$	0	0
$197$	−409212.	−0.751247	−0.375624	−	0.926772i	$-0.622571\pi$
$197$	−409212.	−0.751247	−0.375624	+	0.926772i	$0.622571\pi$
$198$	−634920.	−1.15095
$199$	−300980.	−0.538772	−0.269386	−	0.963032i	$-0.586821\pi$
$199$	−300980.	−0.538772	−0.269386	+	0.963032i	$0.586821\pi$
$200$	−40000.0	−0.0707107
$201$	1.52039e6	2.65439
$202$	−228600.	−0.394183
$203$	0	0
$204$	548688.	0.923103
$205$	−60600.0	−0.100714
$206$	463556.	0.761087
$207$	−351780.	−0.570618
$208$	61696.0	0.0988778
$209$	1.13109e6	1.79115
$210$	0	0
$211$	−1.22618e6	−1.89604	−0.948021	−	0.318209i	$-0.896919\pi$
$211$	−1.22618e6	−1.89604	−0.948021	+	0.318209i	$0.896919\pi$
$212$	−404736.	−0.618489
$213$	369840.	0.558554
$214$	551448.	0.823133
$215$	−15050.0	−0.0222045
$216$	−63296.0	−0.0923085
$217$	0	0
$218$	−353588.	−0.503914
$219$	1.85109e6	2.60805
$220$	−222000.	−0.309240
$221$	359331.	0.494896
$222$	515384.	0.701857
$223$	−621257.	−0.836583	−0.418292	−	0.908313i	$-0.637371\pi$
$223$	−621257.	−0.836583	−0.418292	+	0.908313i	$0.637371\pi$
$224$	0	0
$225$	178750.	0.235391
$226$	−822216.	−1.07082
$227$	−1.29768e6	−1.67148	−0.835742	−	0.549123i	$-0.814962\pi$
$227$	−1.29768e6	−1.67148	−0.835742	+	0.549123i	$0.814962\pi$
$228$	749984.	0.955465
$229$	124264.	0.156587	0.0782937	−	0.996930i	$-0.475053\pi$
$229$	124264.	0.156587	0.0782937	+	0.996930i	$0.475053\pi$
$230$	−123000.	−0.153315
$231$	0	0
$232$	320064.	0.390406
$233$	1.08742e6	1.31222	0.656109	−	0.754666i	$-0.272201\pi$
$233$	1.08742e6	1.31222	0.656109	+	0.754666i	$0.272201\pi$
$234$	−275704.	−0.329157
$235$	−579075.	−0.684014
$236$	−91584.0	−0.107038
$237$	−1.47538e6	−1.70621
$238$	0	0
$239$	−545631.	−0.617880	−0.308940	−	0.951081i	$-0.599974\pi$
$239$	−545631.	−0.617880	−0.308940	+	0.951081i	$0.599974\pi$
$240$	−147200.	−0.164960
$241$	−811310.	−0.899796	−0.449898	−	0.893080i	$-0.648540\pi$
$241$	−811310.	−0.899796	−0.449898	+	0.893080i	$0.648540\pi$
$242$	−587896.	−0.645301
$243$	−1.31560e6	−1.42925
$244$	577792.	0.621294
$245$	0	0
$246$	−223008.	−0.234954
$247$	491158.	0.512246
$248$	364544.	0.376375
$249$	2.44453e6	2.49860
$250$	62500.0	0.0632456
$251$	897738.	0.899426	0.449713	−	0.893173i	$-0.351526\pi$
$251$	897738.	0.899426	0.449713	+	0.893173i	$0.351526\pi$
$252$	0	0
$253$	−682650.	−0.670497
$254$	−1.00366e6	−0.976122
$255$	−857325.	−0.825648
$256$	65536.0	0.0625000
$257$	594678.	0.561628	0.280814	−	0.959762i	$-0.409396\pi$
$257$	594678.	0.561628	0.280814	+	0.959762i	$0.409396\pi$
$258$	−55384.0	−0.0518006
$259$	0	0
$260$	−96400.0	−0.0884390
$261$	−1.43029e6	−1.29964
$262$	−208488.	−0.187641
$263$	1.02837e6	0.916769	0.458385	−	0.888754i	$-0.348428\pi$
$263$	1.02837e6	0.916769	0.458385	+	0.888754i	$0.348428\pi$
$264$	−816960.	−0.721425
$265$	632400.	0.553194
$266$	0	0
$267$	1.64855e6	1.41522
$268$	1.05766e6	0.899520
$269$	1.24390e6	1.04811	0.524053	−	0.851685i	$-0.324419\pi$
$269$	1.24390e6	1.04811	0.524053	+	0.851685i	$0.324419\pi$
$270$	98900.0	0.0825633
$271$	−737624.	−0.610115	−0.305058	−	0.952334i	$-0.598676\pi$
$271$	−737624.	−0.610115	−0.305058	+	0.952334i	$0.598676\pi$
$272$	381696.	0.312821
$273$	0	0
$274$	541872.	0.436034
$275$	346875.	0.276593
$276$	−452640.	−0.357668
$277$	−2.20063e6	−1.72325	−0.861624	−	0.507548i	$-0.830552\pi$
$277$	−2.20063e6	−1.72325	−0.861624	+	0.507548i	$0.830552\pi$
$278$	−1.39794e6	−1.08487
$279$	−1.62906e6	−1.25293
$280$	0	0
$281$	173979.	0.131441	0.0657205	−	0.997838i	$-0.479065\pi$
$281$	173979.	0.131441	0.0657205	+	0.997838i	$0.479065\pi$
$282$	−2.13100e6	−1.59573
$283$	551053.	0.409004	0.204502	−	0.978866i	$-0.434443\pi$
$283$	551053.	0.409004	0.204502	+	0.978866i	$0.434443\pi$
$284$	257280.	0.189282
$285$	−1.17185e6	−0.854594
$286$	−535020.	−0.386772
$287$	0	0
$288$	−292864.	−0.208058
$289$	803224.	0.565708
$290$	−500100.	−0.349190
$291$	−3.47358e6	−2.40461
$292$	1.28771e6	0.883816
$293$	1.67512e6	1.13993	0.569963	−	0.821670i	$-0.306958\pi$
$293$	1.67512e6	1.13993	0.569963	+	0.821670i	$0.306958\pi$
$294$	0	0
$295$	143100.	0.0957381
$296$	358528.	0.237845
$297$	548895.	0.361076
$298$	−704328.	−0.459446
$299$	−296430.	−0.191754
$300$	230000.	0.147545
$301$	0	0
$302$	−1.53333e6	−0.967428
$303$	1.31445e6	0.822503
$304$	521728.	0.323788
$305$	−902800.	−0.555702
$306$	−1.70570e6	−1.04136
$307$	−2.33060e6	−1.41131	−0.705655	−	0.708556i	$-0.749347\pi$
$307$	−2.33060e6	−1.41131	−0.705655	+	0.708556i	$0.749347\pi$
$308$	0	0
$309$	−2.66545e6	−1.58809
$310$	−569600.	−0.336640
$311$	−706266.	−0.414064	−0.207032	−	0.978334i	$-0.566380\pi$
$311$	−706266.	−0.414064	−0.207032	+	0.978334i	$0.566380\pi$
$312$	−354752.	−0.206319
$313$	183565.	0.105908	0.0529540	−	0.998597i	$-0.483136\pi$
$313$	183565.	0.105908	0.0529540	+	0.998597i	$0.483136\pi$
$314$	1.38366e6	0.791961
$315$	0	0
$316$	−1.02635e6	−0.578201
$317$	2.70665e6	1.51281	0.756405	−	0.654103i	$-0.226954\pi$
$317$	2.70665e6	1.51281	0.756405	+	0.654103i	$0.226954\pi$
$318$	2.32723e6	1.29054
$319$	−2.77556e6	−1.52712
$320$	−102400.	−0.0559017
$321$	−3.17083e6	−1.71755
$322$	0	0
$323$	3.03866e6	1.62060
$324$	−748016.	−0.395866
$325$	150625.	0.0791022
$326$	−366344.	−0.190917
$327$	2.03313e6	1.05147
$328$	−155136.	−0.0796211
$329$	0	0
$330$	1.27650e6	0.645262
$331$	−2.14337e6	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$331$	−2.14337e6	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$332$	1.70054e6	0.846726
$333$	−1.60217e6	−0.791769
$334$	152388.	0.0747454
$335$	−1.65260e6	−0.804555
$336$	0	0
$337$	655346.	0.314337	0.157169	−	0.987572i	$-0.449763\pi$
$337$	655346.	0.314337	0.157169	+	0.987572i	$0.449763\pi$
$338$	1.25285e6	0.596495
$339$	4.72774e6	2.23437
$340$	−596400.	−0.279795
$341$	−3.16128e6	−1.47223
$342$	−2.33147e6	−1.07787
$343$	0	0
$344$	−38528.0	−0.0175542
$345$	707250.	0.319908
$346$	−2.16577e6	−0.972574
$347$	−4.22275e6	−1.88266	−0.941329	−	0.337491i	$-0.890422\pi$
$347$	−4.22275e6	−1.88266	−0.941329	+	0.337491i	$0.890422\pi$
$348$	−1.84037e6	−0.814623
$349$	−3.01710e6	−1.32595	−0.662974	−	0.748643i	$-0.730706\pi$
$349$	−3.01710e6	−1.32595	−0.662974	+	0.748643i	$0.730706\pi$
$350$	0	0
$351$	238349.	0.103263
$352$	−568320.	−0.244476
$353$	−2.25258e6	−0.962150	−0.481075	−	0.876679i	$-0.659754\pi$
$353$	−2.25258e6	−0.962150	−0.481075	+	0.876679i	$0.659754\pi$
$354$	526608.	0.223347
$355$	−402000.	−0.169299
$356$	1.14682e6	0.479588
$357$	0	0
$358$	−664752.	−0.274127
$359$	−1.83950e6	−0.753294	−0.376647	−	0.926357i	$-0.622923\pi$
$359$	−1.83950e6	−0.753294	−0.376647	+	0.926357i	$0.622923\pi$
$360$	457600.	0.186093
$361$	1.67735e6	0.677414
$362$	−789280.	−0.316563
$363$	3.38040e6	1.34649
$364$	0	0
$365$	−2.01205e6	−0.790509
$366$	−3.32230e6	−1.29639
$367$	1.68832e6	0.654320	0.327160	−	0.944969i	$-0.393908\pi$
$367$	1.68832e6	0.654320	0.327160	+	0.944969i	$0.393908\pi$
$368$	−314880.	−0.121206
$369$	693264.	0.265053
$370$	−560200.	−0.212735
$371$	0	0
$372$	−2.09613e6	−0.785345
$373$	1.81212e6	0.674394	0.337197	−	0.941434i	$-0.390521\pi$
$373$	1.81212e6	0.674394	0.337197	+	0.941434i	$0.390521\pi$
$374$	−3.31002e6	−1.22363
$375$	−359375.	−0.131968
$376$	−1.48243e6	−0.540761
$377$	−1.20524e6	−0.436738
$378$	0	0
$379$	−4.76708e6	−1.70472	−0.852362	−	0.522952i	$-0.824831\pi$
$379$	−4.76708e6	−1.70472	−0.852362	+	0.522952i	$0.824831\pi$
$380$	−815200.	−0.289604
$381$	5.77107e6	2.03678
$382$	1.34888e6	0.472951
$383$	69996.0	0.0243824	0.0121912	−	0.999926i	$-0.496119\pi$
$383$	69996.0	0.0243824	0.0121912	+	0.999926i	$0.496119\pi$
$384$	−376832.	−0.130413
$385$	0	0
$386$	−1.04206e6	−0.355980
$387$	172172.	0.0584366
$388$	−2.41640e6	−0.814872
$389$	3.98895e6	1.33655	0.668275	−	0.743915i	$-0.267033\pi$
$389$	3.98895e6	1.33655	0.668275	+	0.743915i	$0.267033\pi$
$390$	554300.	0.184537
$391$	−1.83393e6	−0.606654
$392$	0	0
$393$	1.19881e6	0.391532
$394$	1.63685e6	0.531212
$395$	1.60367e6	0.517158
$396$	2.53968e6	0.813844
$397$	3.05904e6	0.974110	0.487055	−	0.873371i	$-0.338071\pi$
$397$	3.05904e6	0.974110	0.487055	+	0.873371i	$0.338071\pi$
$398$	1.20392e6	0.380969
$399$	0	0
$400$	160000.	0.0500000
$401$	4.30794e6	1.33785	0.668927	−	0.743329i	$-0.266754\pi$
$401$	4.30794e6	1.33785	0.668927	+	0.743329i	$0.266754\pi$
$402$	−6.08157e6	−1.87694
$403$	−1.37274e6	−0.421041
$404$	914400.	0.278729
$405$	1.16878e6	0.354073
$406$	0	0
$407$	−3.10911e6	−0.930358
$408$	−2.19475e6	−0.652732
$409$	239206.	0.0707072	0.0353536	−	0.999375i	$-0.488744\pi$
$409$	239206.	0.0707072	0.0353536	+	0.999375i	$0.488744\pi$
$410$	242400.	0.0712152
$411$	−3.11576e6	−0.909829
$412$	−1.85422e6	−0.538170
$413$	0	0
$414$	1.40712e6	0.403488
$415$	−2.65710e6	−0.757334
$416$	−246784.	−0.0699171
$417$	8.03818e6	2.26369
$418$	−4.52436e6	−1.26653
$419$	−4.63462e6	−1.28967	−0.644835	−	0.764322i	$-0.723074\pi$
$419$	−4.63462e6	−1.28967	−0.644835	+	0.764322i	$0.723074\pi$
$420$	0	0
$421$	−2.10108e6	−0.577745	−0.288873	−	0.957368i	$-0.593280\pi$
$421$	−2.10108e6	−0.577745	−0.288873	+	0.957368i	$0.593280\pi$
$422$	4.90472e6	1.34070
$423$	6.62462e6	1.80016
$424$	1.61894e6	0.437338
$425$	931875.	0.250256
$426$	−1.47936e6	−0.394957
$427$	0	0
$428$	−2.20579e6	−0.582043
$429$	3.07636e6	0.807039
$430$	60200.0	0.0157009
$431$	1.65484e6	0.429104	0.214552	−	0.976713i	$-0.431171\pi$
$431$	1.65484e6	0.429104	0.214552	+	0.976713i	$0.431171\pi$
$432$	253184.	0.0652720
$433$	1.84031e6	0.471705	0.235852	−	0.971789i	$-0.424212\pi$
$433$	1.84031e6	0.471705	0.235852	+	0.971789i	$0.424212\pi$
$434$	0	0
$435$	2.87558e6	0.728621
$436$	1.41435e6	0.356321
$437$	−2.50674e6	−0.627922
$438$	−7.40434e6	−1.84417
$439$	−5.83684e6	−1.44549	−0.722747	−	0.691113i	$-0.757121\pi$
$439$	−5.83684e6	−1.44549	−0.722747	+	0.691113i	$0.757121\pi$
$440$	888000.	0.218666
$441$	0	0
$442$	−1.43732e6	−0.349944
$443$	1.19704e6	0.289801	0.144901	−	0.989446i	$-0.453714\pi$
$443$	1.19704e6	0.289801	0.144901	+	0.989446i	$0.453714\pi$
$444$	−2.06154e6	−0.496288
$445$	−1.79190e6	−0.428957
$446$	2.48503e6	0.591554
$447$	4.04989e6	0.958681
$448$	0	0
$449$	−3.42570e6	−0.801924	−0.400962	−	0.916095i	$-0.631324\pi$
$449$	−3.42570e6	−0.801924	−0.400962	+	0.916095i	$0.631324\pi$
$450$	−715000.	−0.166447
$451$	1.34532e6	0.311447
$452$	3.28886e6	0.757181
$453$	8.81666e6	2.01864
$454$	5.19071e6	1.18192
$455$	0	0
$456$	−2.99994e6	−0.675616
$457$	5.29742e6	1.18652	0.593258	−	0.805012i	$-0.297841\pi$
$457$	5.29742e6	1.18652	0.593258	+	0.805012i	$0.297841\pi$
$458$	−497056.	−0.110724
$459$	1.47460e6	0.326695
$460$	492000.	0.108410
$461$	−8.87731e6	−1.94549	−0.972745	−	0.231876i	$-0.925514\pi$
$461$	−8.87731e6	−1.94549	−0.972745	+	0.231876i	$0.925514\pi$
$462$	0	0
$463$	−2.17475e6	−0.471473	−0.235737	−	0.971817i	$-0.575750\pi$
$463$	−2.17475e6	−0.471473	−0.235737	+	0.971817i	$0.575750\pi$
$464$	−1.28026e6	−0.276059
$465$	3.27520e6	0.702434
$466$	−4.34966e6	−0.927878
$467$	378969.	0.0804103	0.0402051	−	0.999191i	$-0.487199\pi$
$467$	378969.	0.0804103	0.0402051	+	0.999191i	$0.487199\pi$
$468$	1.10282e6	0.232749
$469$	0	0
$470$	2.31630e6	0.483671
$471$	−7.95602e6	−1.65251
$472$	366336.	0.0756876
$473$	334110.	0.0686652
$474$	5.90152e6	1.20647
$475$	1.27375e6	0.259030
$476$	0	0
$477$	−7.23466e6	−1.45587
$478$	2.18252e6	0.436907
$479$	−1.88489e6	−0.375360	−0.187680	−	0.982230i	$-0.560097\pi$
$479$	−1.88489e6	−0.375360	−0.187680	+	0.982230i	$0.560097\pi$
$480$	588800.	0.116645
$481$	−1.35008e6	−0.266071
$482$	3.24524e6	0.636252
$483$	0	0
$484$	2.35158e6	0.456296
$485$	3.77562e6	0.728844
$486$	5.26240e6	1.01063
$487$	3.67689e6	0.702518	0.351259	−	0.936278i	$-0.385754\pi$
$487$	3.67689e6	0.702518	0.351259	+	0.936278i	$0.385754\pi$
$488$	−2.31117e6	−0.439321
$489$	2.10648e6	0.398368
$490$	0	0
$491$	9.54015e6	1.78588	0.892939	−	0.450178i	$-0.148640\pi$
$491$	9.54015e6	1.78588	0.892939	+	0.450178i	$0.148640\pi$
$492$	892032.	0.166138
$493$	−7.45649e6	−1.38171
$494$	−1.96463e6	−0.362213
$495$	−3.96825e6	−0.727924
$496$	−1.45818e6	−0.266137
$497$	0	0
$498$	−9.77813e6	−1.76678
$499$	4.78243e6	0.859800	0.429900	−	0.902877i	$-0.358549\pi$
$499$	4.78243e6	0.859800	0.429900	+	0.902877i	$0.358549\pi$
$500$	−250000.	−0.0447214
$501$	−876231.	−0.155964
$502$	−3.59095e6	−0.635990
$503$	1.08395e7	1.91024	0.955120	−	0.296220i	$-0.0957260\pi$
$503$	1.08395e7	1.91024	0.955120	+	0.296220i	$0.0957260\pi$
$504$	0	0
$505$	−1.42875e6	−0.249303
$506$	2.73060e6	0.474113
$507$	−7.20388e6	−1.24465
$508$	4.01466e6	0.690222
$509$	7.64177e6	1.30737	0.653687	−	0.756765i	$-0.273221\pi$
$509$	7.64177e6	1.30737	0.653687	+	0.756765i	$0.273221\pi$
$510$	3.42930e6	0.583821
$511$	0	0
$512$	−262144.	−0.0441942
$513$	2.01558e6	0.338148
$514$	−2.37871e6	−0.397131
$515$	2.89723e6	0.481354
$516$	221536.	0.0366286
$517$	1.28555e7	2.11525
$518$	0	0
$519$	1.24532e7	2.02937
$520$	385600.	0.0625358
$521$	−6.44011e6	−1.03944	−0.519719	−	0.854337i	$-0.673963\pi$
$521$	−6.44011e6	−1.03944	−0.519719	+	0.854337i	$0.673963\pi$
$522$	5.72114e6	0.918981
$523$	4.77929e6	0.764028	0.382014	−	0.924157i	$-0.375231\pi$
$523$	4.77929e6	0.764028	0.382014	+	0.924157i	$0.375231\pi$
$524$	833952.	0.132682
$525$	0	0
$526$	−4.11348e6	−0.648254
$527$	−8.49274e6	−1.33205
$528$	3.26784e6	0.510124
$529$	−4.92344e6	−0.764944
$530$	−2.52960e6	−0.391167
$531$	−1.63706e6	−0.251959
$532$	0	0
$533$	584184.	0.0890700
$534$	−6.59419e6	−1.00071
$535$	3.44655e6	0.520595
$536$	−4.23066e6	−0.636057
$537$	3.82232e6	0.571994
$538$	−4.97561e6	−0.741123
$539$	0	0
$540$	−395600.	−0.0583810
$541$	−2.05678e6	−0.302130	−0.151065	−	0.988524i	$-0.548270\pi$
$541$	−2.05678e6	−0.302130	−0.151065	+	0.988524i	$0.548270\pi$
$542$	2.95050e6	0.431417
$543$	4.53836e6	0.660540
$544$	−1.52678e6	−0.221198
$545$	−2.20992e6	−0.318703
$546$	0	0
$547$	−1.20189e7	−1.71750	−0.858751	−	0.512393i	$-0.828759\pi$
$547$	−1.20189e7	−1.71750	−0.858751	+	0.512393i	$0.828759\pi$
$548$	−2.16749e6	−0.308323
$549$	1.03280e7	1.46247
$550$	−1.38750e6	−0.195581
$551$	−1.01920e7	−1.43015
$552$	1.81056e6	0.252910
$553$	0	0
$554$	8.80252e6	1.21852
$555$	3.22115e6	0.443893
$556$	5.59178e6	0.767119
$557$	8.69942e6	1.18810	0.594049	−	0.804429i	$-0.297528\pi$
$557$	8.69942e6	1.18810	0.594049	+	0.804429i	$0.297528\pi$
$558$	6.51622e6	0.885953
$559$	145082.	0.0196374
$560$	0	0
$561$	1.90326e7	2.55324
$562$	−695916.	−0.0929429
$563$	7.35942e6	0.978527	0.489263	−	0.872136i	$-0.337266\pi$
$563$	7.35942e6	0.978527	0.489263	+	0.872136i	$0.337266\pi$
$564$	8.52398e6	1.12835
$565$	−5.13885e6	−0.677243
$566$	−2.20421e6	−0.289209
$567$	0	0
$568$	−1.02912e6	−0.133843
$569$	−7.50029e6	−0.971175	−0.485588	−	0.874188i	$-0.661394\pi$
$569$	−7.50029e6	−0.971175	−0.485588	+	0.874188i	$0.661394\pi$
$570$	4.68740e6	0.604289
$571$	−2.22879e6	−0.286074	−0.143037	−	0.989717i	$-0.545687\pi$
$571$	−2.22879e6	−0.286074	−0.143037	+	0.989717i	$0.545687\pi$
$572$	2.14008e6	0.273489
$573$	−7.75608e6	−0.986861
$574$	0	0
$575$	−768750.	−0.0969651
$576$	1.17146e6	0.147119
$577$	5.10946e6	0.638903	0.319452	−	0.947603i	$-0.396501\pi$
$577$	5.10946e6	0.638903	0.319452	+	0.947603i	$0.396501\pi$
$578$	−3.21290e6	−0.400016
$579$	5.99187e6	0.742790
$580$	2.00040e6	0.246915
$581$	0	0
$582$	1.38943e7	1.70031
$583$	−1.40393e7	−1.71070
$584$	−5.15085e6	−0.624952
$585$	−1.72315e6	−0.208177
$586$	−6.70048e6	−0.806049
$587$	−1.10646e7	−1.32537	−0.662687	−	0.748896i	$-0.730584\pi$
$587$	−1.10646e7	−1.32537	−0.662687	+	0.748896i	$0.730584\pi$
$588$	0	0
$589$	−1.16084e7	−1.37875
$590$	−572400.	−0.0676970
$591$	−9.41188e6	−1.10843
$592$	−1.43411e6	−0.168182
$593$	−9.10043e6	−1.06274	−0.531368	−	0.847141i	$-0.678322\pi$
$593$	−9.10043e6	−1.06274	−0.531368	+	0.847141i	$0.678322\pi$
$594$	−2.19558e6	−0.255319
$595$	0	0
$596$	2.81731e6	0.324877
$597$	−6.92254e6	−0.794931
$598$	1.18572e6	0.135590
$599$	−1.28615e7	−1.46462	−0.732309	−	0.680973i	$-0.761557\pi$
$599$	−1.28615e7	−1.46462	−0.732309	+	0.680973i	$0.761557\pi$
$600$	−920000.	−0.104330
$601$	−1.58163e6	−0.178615	−0.0893074	−	0.996004i	$-0.528465\pi$
$601$	−1.58163e6	−0.178615	−0.0893074	+	0.996004i	$0.528465\pi$
$602$	0	0
$603$	1.89057e7	2.11739
$604$	6.13333e6	0.684075
$605$	−3.67435e6	−0.408124
$606$	−5.25780e6	−0.581597
$607$	688297.	0.0758236	0.0379118	−	0.999281i	$-0.487929\pi$
$607$	688297.	0.0758236	0.0379118	+	0.999281i	$0.487929\pi$
$608$	−2.08691e6	−0.228952
$609$	0	0
$610$	3.61120e6	0.392941
$611$	5.58228e6	0.604935
$612$	6.82282e6	0.736351
$613$	−6.02150e6	−0.647223	−0.323611	−	0.946190i	$-0.604897\pi$
$613$	−6.02150e6	−0.647223	−0.323611	+	0.946190i	$0.604897\pi$
$614$	9.32241e6	0.997947
$615$	−1.39380e6	−0.148598
$616$	0	0
$617$	3.36137e6	0.355471	0.177735	−	0.984078i	$-0.443123\pi$
$617$	3.36137e6	0.355471	0.177735	+	0.984078i	$0.443123\pi$
$618$	1.06618e7	1.12295
$619$	−1.31769e7	−1.38225	−0.691126	−	0.722734i	$-0.742885\pi$
$619$	−1.31769e7	−1.38225	−0.691126	+	0.722734i	$0.742885\pi$
$620$	2.27840e6	0.238040
$621$	−1.21647e6	−0.126582
$622$	2.82506e6	0.292787
$623$	0	0
$624$	1.41901e6	0.145889
$625$	390625.	0.0400000
$626$	−734260.	−0.0748883
$627$	2.60151e7	2.64275
$628$	−5.53462e6	−0.560001
$629$	−8.35258e6	−0.841771
$630$	0	0
$631$	1.26264e6	0.126243	0.0631213	−	0.998006i	$-0.479895\pi$
$631$	1.26264e6	0.126243	0.0631213	+	0.998006i	$0.479895\pi$
$632$	4.10541e6	0.408850
$633$	−2.82021e7	−2.79752
$634$	−1.08266e7	−1.06972
$635$	−6.27290e6	−0.617354
$636$	−9.30893e6	−0.912550
$637$	0	0
$638$	1.11022e7	1.07984
$639$	4.59888e6	0.445554
$640$	409600.	0.0395285
$641$	−1.58859e7	−1.52710	−0.763550	−	0.645749i	$-0.776545\pi$
$641$	−1.58859e7	−1.52710	−0.763550	+	0.645749i	$0.776545\pi$
$642$	1.26833e7	1.21449
$643$	−1.80880e6	−0.172529	−0.0862647	−	0.996272i	$-0.527493\pi$
$643$	−1.80880e6	−0.172529	−0.0862647	+	0.996272i	$0.527493\pi$
$644$	0	0
$645$	−346150.	−0.0327616
$646$	−1.21546e7	−1.14594
$647$	−95712.0	−0.00898888	−0.00449444	−	0.999990i	$-0.501431\pi$
$647$	−95712.0	−0.00898888	−0.00449444	+	0.999990i	$0.501431\pi$
$648$	2.99206e6	0.279920
$649$	−3.17682e6	−0.296061
$650$	−602500.	−0.0559337
$651$	0	0
$652$	1.46538e6	0.134999
$653$	3.06736e6	0.281502	0.140751	−	0.990045i	$-0.455048\pi$
$653$	3.06736e6	0.281502	0.140751	+	0.990045i	$0.455048\pi$
$654$	−8.13252e6	−0.743500
$655$	−1.30305e6	−0.118675
$656$	620544.	0.0563006
$657$	2.30179e7	2.08042
$658$	0	0
$659$	−1.32961e6	−0.119264	−0.0596321	−	0.998220i	$-0.518993\pi$
$659$	−1.32961e6	−0.119264	−0.0596321	+	0.998220i	$0.518993\pi$
$660$	−5.10600e6	−0.456269
$661$	7.37188e6	0.656258	0.328129	−	0.944633i	$-0.393582\pi$
$661$	7.37188e6	0.656258	0.328129	+	0.944633i	$0.393582\pi$
$662$	8.57349e6	0.760348
$663$	8.26461e6	0.730195
$664$	−6.80218e6	−0.598725
$665$	0	0
$666$	6.40869e6	0.559865
$667$	6.15123e6	0.535362
$668$	−609552.	−0.0528530
$669$	−1.42889e7	−1.23434
$670$	6.61040e6	0.568906
$671$	2.00422e7	1.71846
$672$	0	0
$673$	8.48476e6	0.722108	0.361054	−	0.932545i	$-0.382417\pi$
$673$	8.48476e6	0.722108	0.361054	+	0.932545i	$0.382417\pi$
$674$	−2.62138e6	−0.222270
$675$	618125.	0.0522176
$676$	−5.01139e6	−0.421785
$677$	4.35891e6	0.365516	0.182758	−	0.983158i	$-0.441497\pi$
$677$	4.35891e6	0.365516	0.182758	+	0.983158i	$0.441497\pi$
$678$	−1.89110e7	−1.57994
$679$	0	0
$680$	2.38560e6	0.197845
$681$	−2.98466e7	−2.46619
$682$	1.26451e7	1.04103
$683$	−1.58732e7	−1.30200	−0.651001	−	0.759077i	$-0.725651\pi$
$683$	−1.58732e7	−1.30200	−0.651001	+	0.759077i	$0.725651\pi$
$684$	9.32589e6	0.762167
$685$	3.38670e6	0.275772
$686$	0	0
$687$	2.85807e6	0.231037
$688$	154112.	0.0124127
$689$	−6.09634e6	−0.489239
$690$	−2.82900e6	−0.226209
$691$	554956.	0.0442144	0.0221072	−	0.999756i	$-0.492962\pi$
$691$	554956.	0.0442144	0.0221072	+	0.999756i	$0.492962\pi$
$692$	8.66309e6	0.687713
$693$	0	0
$694$	1.68910e7	1.33124
$695$	−8.73715e6	−0.686132
$696$	7.36147e6	0.576025
$697$	3.61418e6	0.281792
$698$	1.20684e7	0.937587
$699$	2.50106e7	1.93611
$700$	0	0
$701$	−7.74720e6	−0.595456	−0.297728	−	0.954651i	$-0.596229\pi$
$701$	−7.74720e6	−0.595456	−0.297728	+	0.954651i	$0.596229\pi$
$702$	−953396.	−0.0730181
$703$	−1.14169e7	−0.871282
$704$	2.27328e6	0.172871
$705$	−1.33187e7	−1.00923
$706$	9.01031e6	0.680343
$707$	0	0
$708$	−2.10643e6	−0.157930
$709$	−1.89055e7	−1.41245	−0.706225	−	0.707987i	$-0.749603\pi$
$709$	−1.89055e7	−1.41245	−0.706225	+	0.707987i	$0.749603\pi$
$710$	1.60800e6	0.119713
$711$	−1.83460e7	−1.36103
$712$	−4.58726e6	−0.339120
$713$	7.00608e6	0.516121
$714$	0	0
$715$	−3.34388e6	−0.244616
$716$	2.65901e6	0.193837
$717$	−1.25495e7	−0.911652
$718$	7.35802e6	0.532659
$719$	1.83928e7	1.32686	0.663430	−	0.748238i	$-0.269100\pi$
$719$	1.83928e7	1.32686	0.663430	+	0.748238i	$0.269100\pi$
$720$	−1.83040e6	−0.131588
$721$	0	0
$722$	−6.70938e6	−0.479004
$723$	−1.86601e7	−1.32761
$724$	3.15712e6	0.223844
$725$	−3.12562e6	−0.220847
$726$	−1.35216e7	−0.952109
$727$	1.34259e7	0.942123	0.471061	−	0.882100i	$-0.343871\pi$
$727$	1.34259e7	0.942123	0.471061	+	0.882100i	$0.343871\pi$
$728$	0	0
$729$	−1.88983e7	−1.31706
$730$	8.04820e6	0.558974
$731$	897582.	0.0621270
$732$	1.32892e7	0.916688
$733$	−1.08473e7	−0.745697	−0.372848	−	0.927892i	$-0.621619\pi$
$733$	−1.08473e7	−0.745697	−0.372848	+	0.927892i	$0.621619\pi$
$734$	−6.75329e6	−0.462674
$735$	0	0
$736$	1.25952e6	0.0857059
$737$	3.66877e7	2.48801
$738$	−2.77306e6	−0.187421
$739$	2.64323e7	1.78043	0.890214	−	0.455542i	$-0.150555\pi$
$739$	2.64323e7	1.78043	0.890214	+	0.455542i	$0.150555\pi$
$740$	2.24080e6	0.150426
$741$	1.12966e7	0.755794
$742$	0	0
$743$	2.03120e7	1.34984	0.674918	−	0.737893i	$-0.264179\pi$
$743$	2.03120e7	1.34984	0.674918	+	0.737893i	$0.264179\pi$
$744$	8.38451e6	0.555323
$745$	−4.40205e6	−0.290579
$746$	−7.24846e6	−0.476869
$747$	3.03972e7	1.99312
$748$	1.32401e7	0.865240
$749$	0	0
$750$	1.43750e6	0.0933157
$751$	−3.95388e6	−0.255813	−0.127907	−	0.991786i	$-0.540826\pi$
$751$	−3.95388e6	−0.255813	−0.127907	+	0.991786i	$0.540826\pi$
$752$	5.92973e6	0.382376
$753$	2.06480e7	1.32706
$754$	4.82096e6	0.308820
$755$	−9.58332e6	−0.611855
$756$	0	0
$757$	−2.62165e7	−1.66278	−0.831391	−	0.555688i	$-0.812455\pi$
$757$	−2.62165e7	−1.66278	−0.831391	+	0.555688i	$0.812455\pi$
$758$	1.90683e7	1.20542
$759$	−1.57010e7	−0.989285
$760$	3.26080e6	0.204781
$761$	−1.14329e7	−0.715638	−0.357819	−	0.933791i	$-0.616479\pi$
$761$	−1.14329e7	−0.715638	−0.357819	+	0.933791i	$0.616479\pi$
$762$	−2.30843e7	−1.44022
$763$	0	0
$764$	−5.39554e6	−0.334427
$765$	−1.06606e7	−0.658613
$766$	−279984.	−0.0172410
$767$	−1.37948e6	−0.0846697
$768$	1.50733e6	0.0922157
$769$	−2.37076e7	−1.44568	−0.722840	−	0.691015i	$-0.757164\pi$
$769$	−2.37076e7	−1.44568	−0.722840	+	0.691015i	$0.757164\pi$
$770$	0	0
$771$	1.36776e7	0.828655
$772$	4.16826e6	0.251716
$773$	1.24180e7	0.747484	0.373742	−	0.927533i	$-0.378075\pi$
$773$	1.24180e7	0.747484	0.373742	+	0.927533i	$0.378075\pi$
$774$	−688688.	−0.0413209
$775$	−3.56000e6	−0.212910
$776$	9.66560e6	0.576202
$777$	0	0
$778$	−1.59558e7	−0.945083
$779$	4.94011e6	0.291671
$780$	−2.21720e6	−0.130487
$781$	8.92440e6	0.523542
$782$	7.33572e6	0.428969
$783$	−4.94599e6	−0.288303
$784$	0	0
$785$	8.64785e6	0.500880
$786$	−4.79522e6	−0.276855
$787$	−3.06553e7	−1.76428	−0.882142	−	0.470984i	$-0.843899\pi$
$787$	−3.06553e7	−1.76428	−0.882142	+	0.470984i	$0.843899\pi$
$788$	−6.54739e6	−0.375624
$789$	2.36525e7	1.35265
$790$	−6.41470e6	−0.365686
$791$	0	0
$792$	−1.01587e7	−0.575474
$793$	8.70299e6	0.491457
$794$	−1.22361e7	−0.688800
$795$	1.45452e7	0.816210
$796$	−4.81568e6	−0.269386
$797$	1.51870e7	0.846886	0.423443	−	0.905923i	$-0.360821\pi$
$797$	1.51870e7	0.846886	0.423443	+	0.905923i	$0.360821\pi$
$798$	0	0
$799$	3.45360e7	1.91384
$800$	−640000.	−0.0353553
$801$	2.04993e7	1.12891
$802$	−1.72317e7	−0.946005
$803$	4.46675e7	2.44457
$804$	2.43263e7	1.32720
$805$	0	0
$806$	5.49094e6	0.297721
$807$	2.86097e7	1.54643
$808$	−3.65760e6	−0.197091
$809$	539721.	0.0289933	0.0144967	−	0.999895i	$-0.495385\pi$
$809$	539721.	0.0289933	0.0144967	+	0.999895i	$0.495385\pi$
$810$	−4.67510e6	−0.250368
$811$	−1.39772e7	−0.746221	−0.373111	−	0.927787i	$-0.621709\pi$
$811$	−1.39772e7	−0.746221	−0.373111	+	0.927787i	$0.621709\pi$
$812$	0	0
$813$	−1.69654e7	−0.900195
$814$	1.24364e7	0.657862
$815$	−2.28965e6	−0.120747
$816$	8.77901e6	0.461551
$817$	1.22688e6	0.0643051
$818$	−956824.	−0.0499976
$819$	0	0
$820$	−969600.	−0.0503568
$821$	−1.78137e7	−0.922350	−0.461175	−	0.887309i	$-0.652572\pi$
$821$	−1.78137e7	−0.922350	−0.461175	+	0.887309i	$0.652572\pi$
$822$	1.24631e7	0.643347
$823$	1.91010e7	0.983005	0.491502	−	0.870876i	$-0.336448\pi$
$823$	1.91010e7	0.983005	0.491502	+	0.870876i	$0.336448\pi$
$824$	7.41690e6	0.380543
$825$	7.97813e6	0.408099
$826$	0	0
$827$	3.19225e6	0.162305	0.0811526	−	0.996702i	$-0.474140\pi$
$827$	3.19225e6	0.162305	0.0811526	+	0.996702i	$0.474140\pi$
$828$	−5.62848e6	−0.285309
$829$	−8.56842e6	−0.433026	−0.216513	−	0.976280i	$-0.569468\pi$
$829$	−8.56842e6	−0.433026	−0.216513	+	0.976280i	$0.569468\pi$
$830$	1.06284e7	0.535516
$831$	−5.06145e7	−2.54257
$832$	987136.	0.0494389
$833$	0	0
$834$	−3.21527e7	−1.60067
$835$	952425.	0.0472732
$836$	1.80974e7	0.895574
$837$	−5.63334e6	−0.277941
$838$	1.85385e7	0.911935
$839$	−3.56751e7	−1.74969	−0.874843	−	0.484407i	$-0.839036\pi$
$839$	−3.56751e7	−1.74969	−0.874843	+	0.484407i	$0.839036\pi$
$840$	0	0
$841$	4.49885e6	0.219337
$842$	8.40430e6	0.408528
$843$	4.00152e6	0.193935
$844$	−1.96189e7	−0.948021
$845$	7.83030e6	0.377256
$846$	−2.64985e7	−1.27290
$847$	0	0
$848$	−6.47578e6	−0.309245
$849$	1.26742e7	0.603465
$850$	−3.72750e6	−0.176958
$851$	6.89046e6	0.326155
$852$	5.91744e6	0.279277
$853$	−3.06355e7	−1.44163	−0.720814	−	0.693129i	$-0.756232\pi$
$853$	−3.06355e7	−1.44163	−0.720814	+	0.693129i	$0.756232\pi$
$854$	0	0
$855$	−1.45717e7	−0.681703
$856$	8.82317e6	0.411567
$857$	4.46188e6	0.207523	0.103761	−	0.994602i	$-0.466912\pi$
$857$	4.46188e6	0.207523	0.103761	+	0.994602i	$0.466912\pi$
$858$	−1.23055e7	−0.570663
$859$	−2.63974e7	−1.22061	−0.610307	−	0.792165i	$-0.708954\pi$
$859$	−2.63974e7	−1.22061	−0.610307	+	0.792165i	$0.708954\pi$
$860$	−240800.	−0.0111022
$861$	0	0
$862$	−6.61936e6	−0.303422
$863$	−2.17530e7	−0.994244	−0.497122	−	0.867681i	$-0.665610\pi$
$863$	−2.17530e7	−0.994244	−0.497122	+	0.867681i	$0.665610\pi$
$864$	−1.01274e6	−0.0461543
$865$	−1.35361e7	−0.615110
$866$	−7.36122e6	−0.333546
$867$	1.84742e7	0.834674
$868$	0	0
$869$	−3.56016e7	−1.59926
$870$	−1.15023e7	−0.515213
$871$	1.59311e7	0.711540
$872$	−5.65741e6	−0.251957
$873$	−4.31932e7	−1.91814
$874$	1.00270e7	0.444008
$875$	0	0
$876$	2.96174e7	1.30403
$877$	−1.58383e7	−0.695361	−0.347680	−	0.937613i	$-0.613031\pi$
$877$	−1.58383e7	−0.695361	−0.347680	+	0.937613i	$0.613031\pi$
$878$	2.33474e7	1.02212
$879$	3.85277e7	1.68190
$880$	−3.55200e6	−0.154620
$881$	−1.97427e7	−0.856974	−0.428487	−	0.903548i	$-0.640953\pi$
$881$	−1.97427e7	−0.856974	−0.428487	+	0.903548i	$0.640953\pi$
$882$	0	0
$883$	−1.72899e7	−0.746263	−0.373131	−	0.927779i	$-0.621716\pi$
$883$	−1.72899e7	−0.746263	−0.373131	+	0.927779i	$0.621716\pi$
$884$	5.74930e6	0.247448
$885$	3.29130e6	0.141257
$886$	−4.78817e6	−0.204920
$887$	4.44693e7	1.89780	0.948901	−	0.315574i	$-0.102197\pi$
$887$	4.44693e7	1.89780	0.948901	+	0.315574i	$0.102197\pi$
$888$	8.24614e6	0.350928
$889$	0	0
$890$	7.16760e6	0.303318
$891$	−2.59468e7	−1.09494
$892$	−9.94011e6	−0.418292
$893$	4.72062e7	1.98094
$894$	−1.61995e7	−0.677890
$895$	−4.15470e6	−0.173373
$896$	0	0
$897$	−6.81789e6	−0.282923
$898$	1.37028e7	0.567046
$899$	2.84857e7	1.17551
$900$	2.86000e6	0.117695
$901$	−3.77163e7	−1.54781
$902$	−5.38128e6	−0.220226
$903$	0	0
$904$	−1.31555e7	−0.535408
$905$	−4.93300e6	−0.200212
$906$	−3.52666e7	−1.42739
$907$	2.56887e6	0.103687	0.0518434	−	0.998655i	$-0.483490\pi$
$907$	2.56887e6	0.103687	0.0518434	+	0.998655i	$0.483490\pi$
$908$	−2.07628e7	−0.835742
$909$	1.63449e7	0.656104
$910$	0	0
$911$	1.12692e7	0.449882	0.224941	−	0.974372i	$-0.427781\pi$
$911$	1.12692e7	0.449882	0.224941	+	0.974372i	$0.427781\pi$
$912$	1.19997e7	0.477733
$913$	5.89876e7	2.34198
$914$	−2.11897e7	−0.838994
$915$	−2.07644e7	−0.819911
$916$	1.98822e6	0.0782937
$917$	0	0
$918$	−5.89840e6	−0.231008
$919$	3.18378e7	1.24353	0.621763	−	0.783205i	$-0.286417\pi$
$919$	3.18378e7	1.24353	0.621763	+	0.783205i	$0.286417\pi$
$920$	−1.96800e6	−0.0766577
$921$	−5.36039e7	−2.08232
$922$	3.55092e7	1.37567
$923$	3.87528e6	0.149727
$924$	0	0
$925$	−3.50125e6	−0.134545
$926$	8.69901e6	0.333382
$927$	−3.31443e7	−1.26680
$928$	5.12102e6	0.195203
$929$	−1.88558e7	−0.716813	−0.358407	−	0.933566i	$-0.616680\pi$
$929$	−1.88558e7	−0.716813	−0.358407	+	0.933566i	$0.616680\pi$
$930$	−1.31008e7	−0.496696
$931$	0	0
$932$	1.73987e7	0.656109
$933$	−1.62441e7	−0.610931
$934$	−1.51588e6	−0.0568586
$935$	−2.06876e7	−0.773894
$936$	−4.41126e6	−0.164579
$937$	−1.12946e7	−0.420265	−0.210132	−	0.977673i	$-0.567390\pi$
$937$	−1.12946e7	−0.420265	−0.210132	+	0.977673i	$0.567390\pi$
$938$	0	0
$939$	4.22200e6	0.156262
$940$	−9.26520e6	−0.342007
$941$	2.91941e7	1.07478	0.537392	−	0.843333i	$-0.319410\pi$
$941$	2.91941e7	1.07478	0.537392	+	0.843333i	$0.319410\pi$
$942$	3.18241e7	1.16850
$943$	−2.98152e6	−0.109184
$944$	−1.46534e6	−0.0535192
$945$	0	0
$946$	−1.33644e6	−0.0485536
$947$	1.05892e7	0.383697	0.191848	−	0.981425i	$-0.438552\pi$
$947$	1.05892e7	0.383697	0.191848	+	0.981425i	$0.438552\pi$
$948$	−2.36061e7	−0.853107
$949$	1.93962e7	0.699118
$950$	−5.09500e6	−0.183162
$951$	6.22530e7	2.23208
$952$	0	0
$953$	−3.90317e7	−1.39215	−0.696074	−	0.717970i	$-0.745071\pi$
$953$	−3.90317e7	−1.39215	−0.696074	+	0.717970i	$0.745071\pi$
$954$	2.89386e7	1.02945
$955$	8.43052e6	0.299121
$956$	−8.73010e6	−0.308940
$957$	−6.38378e7	−2.25319
$958$	7.53958e6	0.265420
$959$	0	0
$960$	−2.35520e6	−0.0824802
$961$	3.81526e6	0.133265
$962$	5.40033e6	0.188141
$963$	−3.94285e7	−1.37008
$964$	−1.29810e7	−0.449898
$965$	−6.51290e6	−0.225142
$966$	0	0
$967$	−3.43395e7	−1.18094	−0.590470	−	0.807059i	$-0.701058\pi$
$967$	−3.43395e7	−1.18094	−0.590470	+	0.807059i	$0.701058\pi$
$968$	−9.40634e6	−0.322650
$969$	6.98891e7	2.39111
$970$	−1.51025e7	−0.515370
$971$	1.81464e7	0.617651	0.308826	−	0.951119i	$-0.400064\pi$
$971$	1.81464e7	0.617651	0.308826	+	0.951119i	$0.400064\pi$
$972$	−2.10496e7	−0.714625
$973$	0	0
$974$	−1.47075e7	−0.496756
$975$	3.46438e6	0.116711
$976$	9.24467e6	0.310647
$977$	−1.05223e7	−0.352675	−0.176338	−	0.984330i	$-0.556425\pi$
$977$	−1.05223e7	−0.352675	−0.176338	+	0.984330i	$0.556425\pi$
$978$	−8.42591e6	−0.281689
$979$	3.97802e7	1.32651
$980$	0	0
$981$	2.52815e7	0.838747
$982$	−3.81606e7	−1.26281
$983$	7.91353e6	0.261208	0.130604	−	0.991435i	$-0.458308\pi$
$983$	7.91353e6	0.261208	0.130604	+	0.991435i	$0.458308\pi$
$984$	−3.56813e6	−0.117477
$985$	1.02303e7	0.335968
$986$	2.98260e7	0.977017
$987$	0	0
$988$	7.85853e6	0.256123
$989$	−740460.	−0.0240719
$990$	1.58730e7	0.514720
$991$	4.01556e7	1.29886	0.649429	−	0.760422i	$-0.275008\pi$
$991$	4.01556e7	1.29886	0.649429	+	0.760422i	$0.275008\pi$
$992$	5.83270e6	0.188187
$993$	−4.92976e7	−1.58654
$994$	0	0
$995$	7.52450e6	0.240946
$996$	3.91125e7	1.24930
$997$	4.93478e7	1.57228	0.786140	−	0.618048i	$-0.212076\pi$
$997$	4.93478e7	1.57228	0.786140	+	0.618048i	$0.212076\pi$
$998$	−1.91297e7	−0.607970
$999$	−5.54038e6	−0.175641

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	490.6.a.i.1.1		1
7.6	odd	2		70.6.a.a.1.1	✓	1
21.20	even	2		630.6.a.j.1.1		1
28.27	even	2		560.6.a.i.1.1		1
35.13	even	4		350.6.c.h.99.2		2
35.27	even	4		350.6.c.h.99.1		2
35.34	odd	2		350.6.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.a.1.1	✓	1	7.6	odd	2
350.6.a.n.1.1		1	35.34	odd	2
350.6.c.h.99.1		2	35.27	even	4
350.6.c.h.99.2		2	35.13	even	4
490.6.a.i.1.1		1	1.1	even	1	trivial
560.6.a.i.1.1		1	28.27	even	2
630.6.a.j.1.1		1	21.20	even	2

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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 $\F_{q}$

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Fields

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Groups

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Properties

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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	490.6.a.i.1.1

Level	$490$
Weight	$6$
Character	490.1
Self dual	yes
Analytic conductor	$78.588$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$