LMFDB - Embedded newform 525.8.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,8,Mod(1,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$164.002138379$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +27.0000 q^{3} -124.000 q^{4} +54.0000 q^{6} -343.000 q^{7} -504.000 q^{8} +729.000 q^{9} +2724.00 q^{11} -3348.00 q^{12} -2874.00 q^{13} -686.000 q^{14} +14864.0 q^{16} -9278.00 q^{17} +1458.00 q^{18} -4304.00 q^{19} -9261.00 q^{21} +5448.00 q^{22} +41500.0 q^{23} -13608.0 q^{24} -5748.00 q^{26} +19683.0 q^{27} +42532.0 q^{28} -35498.0 q^{29} -52940.0 q^{31} +94240.0 q^{32} +73548.0 q^{33} -18556.0 q^{34} -90396.0 q^{36} +84098.0 q^{37} -8608.00 q^{38} -77598.0 q^{39} +180342. q^{41} -18522.0 q^{42} +33452.0 q^{43} -337776. q^{44} +83000.0 q^{46} +136120. q^{47} +401328. q^{48} +117649. q^{49} -250506. q^{51} +356376. q^{52} +1.27062e6 q^{53} +39366.0 q^{54} +172872. q^{56} -116208. q^{57} -70996.0 q^{58} -1.55325e6 q^{59} +213598. q^{61} -105880. q^{62} -250047. q^{63} -1.71411e6 q^{64} +147096. q^{66} -487228. q^{67} +1.15047e6 q^{68} +1.12050e6 q^{69} +1.08600e6 q^{71} -367416. q^{72} +5.92198e6 q^{73} +168196. q^{74} +533696. q^{76} -934332. q^{77} -155196. q^{78} -5.42982e6 q^{79} +531441. q^{81} +360684. q^{82} -6.93340e6 q^{83} +1.14836e6 q^{84} +66904.0 q^{86} -958446. q^{87} -1.37290e6 q^{88} +262614. q^{89} +985782. q^{91} -5.14600e6 q^{92} -1.42938e6 q^{93} +272240. q^{94} +2.54448e6 q^{96} +522234. q^{97} +235298. q^{98} +1.98580e6 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	2.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−124.000	−0.968750
$5$	0	0
$5$	0	0
$6$	54.0000	0.102062
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−504.000	−0.348029
$9$	729.000	0.333333
$10$	0	0
$11$	2724.00	0.617068	0.308534	−	0.951213i	$-0.400162\pi$
$11$	2724.00	0.617068	0.308534	+	0.951213i	$0.400162\pi$
$12$	−3348.00	−0.559308
$13$	−2874.00	−0.362815	−0.181407	−	0.983408i	$-0.558065\pi$
$13$	−2874.00	−0.362815	−0.181407	+	0.983408i	$0.558065\pi$
$14$	−686.000	−0.0668153
$15$	0	0
$16$	14864.0	0.907227
$17$	−9278.00	−0.458019	−0.229009	−	0.973424i	$-0.573549\pi$
$17$	−9278.00	−0.458019	−0.229009	+	0.973424i	$0.573549\pi$
$18$	1458.00	0.0589256
$19$	−4304.00	−0.143958	−0.0719788	−	0.997406i	$-0.522931\pi$
$19$	−4304.00	−0.143958	−0.0719788	+	0.997406i	$0.522931\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	5448.00	0.109083
$23$	41500.0	0.711215	0.355607	−	0.934635i	$-0.384274\pi$
$23$	41500.0	0.711215	0.355607	+	0.934635i	$0.384274\pi$
$24$	−13608.0	−0.200935
$25$	0	0
$26$	−5748.00	−0.0641372
$27$	19683.0	0.192450
$28$	42532.0	0.366153
$29$	−35498.0	−0.270278	−0.135139	−	0.990827i	$-0.543148\pi$
$29$	−35498.0	−0.270278	−0.135139	+	0.990827i	$0.543148\pi$
$30$	0	0
$31$	−52940.0	−0.319167	−0.159584	−	0.987184i	$-0.551015\pi$
$31$	−52940.0	−0.319167	−0.159584	+	0.987184i	$0.551015\pi$
$32$	94240.0	0.508406
$33$	73548.0	0.356264
$34$	−18556.0	−0.0809670
$35$	0	0
$36$	−90396.0	−0.322917
$37$	84098.0	0.272948	0.136474	−	0.990644i	$-0.456423\pi$
$37$	84098.0	0.272948	0.136474	+	0.990644i	$0.456423\pi$
$38$	−8608.00	−0.0254484
$39$	−77598.0	−0.209471
$40$	0	0
$41$	180342.	0.408652	0.204326	−	0.978903i	$-0.434500\pi$
$41$	180342.	0.408652	0.204326	+	0.978903i	$0.434500\pi$
$42$	−18522.0	−0.0385758
$43$	33452.0	0.0641627	0.0320813	−	0.999485i	$-0.489786\pi$
$43$	33452.0	0.0641627	0.0320813	+	0.999485i	$0.489786\pi$
$44$	−337776.	−0.597784
$45$	0	0
$46$	83000.0	0.125726
$47$	136120.	0.191240	0.0956202	−	0.995418i	$-0.469517\pi$
$47$	136120.	0.191240	0.0956202	+	0.995418i	$0.469517\pi$
$48$	401328.	0.523788
$49$	117649.	0.142857
$50$	0	0
$51$	−250506.	−0.264437
$52$	356376.	0.351477
$53$	1.27062e6	1.17233	0.586166	−	0.810191i	$-0.300637\pi$
$53$	1.27062e6	1.17233	0.586166	+	0.810191i	$0.300637\pi$
$54$	39366.0	0.0340207
$55$	0	0
$56$	172872.	0.131543
$57$	−116208.	−0.0831140
$58$	−70996.0	−0.0477789
$59$	−1.55325e6	−0.984600	−0.492300	−	0.870426i	$-0.663844\pi$
$59$	−1.55325e6	−0.984600	−0.492300	+	0.870426i	$0.663844\pi$
$60$	0	0
$61$	213598.	0.120488	0.0602439	−	0.998184i	$-0.480812\pi$
$61$	213598.	0.120488	0.0602439	+	0.998184i	$0.480812\pi$
$62$	−105880.	−0.0564213
$63$	−250047.	−0.125988
$64$	−1.71411e6	−0.817352
$65$	0	0
$66$	147096.	0.0629792
$67$	−487228.	−0.197911	−0.0989556	−	0.995092i	$-0.531550\pi$
$67$	−487228.	−0.197911	−0.0989556	+	0.995092i	$0.531550\pi$
$68$	1.15047e6	0.443706
$69$	1.12050e6	0.410620
$70$	0	0
$71$	1.08600e6	0.360102	0.180051	−	0.983657i	$-0.442374\pi$
$71$	1.08600e6	0.360102	0.180051	+	0.983657i	$0.442374\pi$
$72$	−367416.	−0.116010
$73$	5.92198e6	1.78171	0.890855	−	0.454289i	$-0.150107\pi$
$73$	5.92198e6	1.78171	0.890855	+	0.454289i	$0.150107\pi$
$74$	168196.	0.0482508
$75$	0	0
$76$	533696.	0.139459
$77$	−934332.	−0.233230
$78$	−155196.	−0.0370296
$79$	−5.42982e6	−1.23906	−0.619528	−	0.784975i	$-0.712676\pi$
$79$	−5.42982e6	−1.23906	−0.619528	+	0.784975i	$0.712676\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	360684.	0.0722401
$83$	−6.93340e6	−1.33099	−0.665493	−	0.746405i	$-0.731778\pi$
$83$	−6.93340e6	−1.33099	−0.665493	+	0.746405i	$0.731778\pi$
$84$	1.14836e6	0.211399
$85$	0	0
$86$	66904.0	0.0113425
$87$	−958446.	−0.156045
$88$	−1.37290e6	−0.214757
$89$	262614.	0.0394869	0.0197434	−	0.999805i	$-0.493715\pi$
$89$	262614.	0.0394869	0.0197434	+	0.999805i	$0.493715\pi$
$90$	0	0
$91$	985782.	0.137131
$92$	−5.14600e6	−0.688989
$93$	−1.42938e6	−0.184271
$94$	272240.	0.0338068
$95$	0	0
$96$	2.54448e6	0.293528
$97$	522234.	0.0580984	0.0290492	−	0.999578i	$-0.490752\pi$
$97$	522234.	0.0580984	0.0290492	+	0.999578i	$0.490752\pi$
$98$	235298.	0.0252538
$99$	1.98580e6	0.205689
$100$	0	0
$101$	−1.63961e6	−0.158349	−0.0791744	−	0.996861i	$-0.525228\pi$
$101$	−1.63961e6	−0.158349	−0.0791744	+	0.996861i	$0.525228\pi$
$102$	−501012.	−0.0467463
$103$	6.57506e6	0.592883	0.296442	−	0.955051i	$-0.404200\pi$
$103$	6.57506e6	0.592883	0.296442	+	0.955051i	$0.404200\pi$
$104$	1.44850e6	0.126270
$105$	0	0
$106$	2.54124e6	0.207241
$107$	−1.21480e7	−0.958654	−0.479327	−	0.877636i	$-0.659119\pi$
$107$	−1.21480e7	−0.958654	−0.479327	+	0.877636i	$0.659119\pi$
$108$	−2.44069e6	−0.186436
$109$	−9.68452e6	−0.716284	−0.358142	−	0.933667i	$-0.616590\pi$
$109$	−9.68452e6	−0.716284	−0.358142	+	0.933667i	$0.616590\pi$
$110$	0	0
$111$	2.27065e6	0.157586
$112$	−5.09835e6	−0.342899
$113$	−2.50296e7	−1.63184	−0.815922	−	0.578162i	$-0.803770\pi$
$113$	−2.50296e7	−1.63184	−0.815922	+	0.578162i	$0.803770\pi$
$114$	−232416.	−0.0146926
$115$	0	0
$116$	4.40175e6	0.261832
$117$	−2.09515e6	−0.120938
$118$	−3.10650e6	−0.174054
$119$	3.18235e6	0.173115
$120$	0	0
$121$	−1.20670e7	−0.619228
$122$	427196.	0.0212994
$123$	4.86923e6	0.235935
$124$	6.56456e6	0.309193
$125$	0	0
$126$	−500094.	−0.0222718
$127$	−3.05960e6	−0.132541	−0.0662707	−	0.997802i	$-0.521110\pi$
$127$	−3.05960e6	−0.132541	−0.0662707	+	0.997802i	$0.521110\pi$
$128$	−1.54909e7	−0.652894
$129$	903204.	0.0370443
$130$	0	0
$131$	−3.56912e6	−0.138711	−0.0693555	−	0.997592i	$-0.522094\pi$
$131$	−3.56912e6	−0.138711	−0.0693555	+	0.997592i	$0.522094\pi$
$132$	−9.11995e6	−0.345131
$133$	1.47627e6	0.0544109
$134$	−974456.	−0.0349861
$135$	0	0
$136$	4.67611e6	0.159404
$137$	−4.83736e6	−0.160726	−0.0803630	−	0.996766i	$-0.525608\pi$
$137$	−4.83736e6	−0.160726	−0.0803630	+	0.996766i	$0.525608\pi$
$138$	2.24100e6	0.0725880
$139$	2.48573e6	0.0785058	0.0392529	−	0.999229i	$-0.487502\pi$
$139$	2.48573e6	0.0785058	0.0392529	+	0.999229i	$0.487502\pi$
$140$	0	0
$141$	3.67524e6	0.110413
$142$	2.17200e6	0.0636577
$143$	−7.82878e6	−0.223881
$144$	1.08359e7	0.302409
$145$	0	0
$146$	1.18440e7	0.314965
$147$	3.17652e6	0.0824786
$148$	−1.04282e7	−0.264418
$149$	−3.99895e7	−0.990362	−0.495181	−	0.868790i	$-0.664898\pi$
$149$	−3.99895e7	−0.990362	−0.495181	+	0.868790i	$0.664898\pi$
$150$	0	0
$151$	−5.37919e7	−1.27145	−0.635723	−	0.771918i	$-0.719298\pi$
$151$	−5.37919e7	−1.27145	−0.635723	+	0.771918i	$0.719298\pi$
$152$	2.16922e6	0.0501014
$153$	−6.76366e6	−0.152673
$154$	−1.86866e6	−0.0412296
$155$	0	0
$156$	9.62215e6	0.202925
$157$	−9.64019e6	−0.198809	−0.0994046	−	0.995047i	$-0.531694\pi$
$157$	−9.64019e6	−0.198809	−0.0994046	+	0.995047i	$0.531694\pi$
$158$	−1.08596e7	−0.219036
$159$	3.43068e7	0.676847
$160$	0	0
$161$	−1.42345e7	−0.268814
$162$	1.06288e6	0.0196419
$163$	7.67343e7	1.38782	0.693910	−	0.720062i	$-0.255887\pi$
$163$	7.67343e7	1.38782	0.693910	+	0.720062i	$0.255887\pi$
$164$	−2.23624e7	−0.395881
$165$	0	0
$166$	−1.38668e7	−0.235287
$167$	−1.03856e7	−0.172554	−0.0862770	−	0.996271i	$-0.527497\pi$
$167$	−1.03856e7	−0.172554	−0.0862770	+	0.996271i	$0.527497\pi$
$168$	4.66754e6	0.0759462
$169$	−5.44886e7	−0.868365
$170$	0	0
$171$	−3.13762e6	−0.0479859
$172$	−4.14805e6	−0.0621576
$173$	5.64927e6	0.0829528	0.0414764	−	0.999139i	$-0.486794\pi$
$173$	5.64927e6	0.0829528	0.0414764	+	0.999139i	$0.486794\pi$
$174$	−1.91689e6	−0.0275851
$175$	0	0
$176$	4.04895e7	0.559820
$177$	−4.19378e7	−0.568459
$178$	525228.	0.00698036
$179$	−1.04373e8	−1.36020	−0.680098	−	0.733121i	$-0.738063\pi$
$179$	−1.04373e8	−1.36020	−0.680098	+	0.733121i	$0.738063\pi$
$180$	0	0
$181$	−1.42262e8	−1.78326	−0.891628	−	0.452769i	$-0.850436\pi$
$181$	−1.42262e8	−1.78326	−0.891628	+	0.452769i	$0.850436\pi$
$182$	1.97156e6	0.0242416
$183$	5.76715e6	0.0695636
$184$	−2.09160e7	−0.247523
$185$	0	0
$186$	−2.85876e6	−0.0325748
$187$	−2.52733e7	−0.282628
$188$	−1.68789e7	−0.185264
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	−3.35533e7	−0.348433	−0.174216	−	0.984707i	$-0.555739\pi$
$191$	−3.35533e7	−0.348433	−0.174216	+	0.984707i	$0.555739\pi$
$192$	−4.62810e7	−0.471899
$193$	1.70599e8	1.70815	0.854073	−	0.520153i	$-0.174125\pi$
$193$	1.70599e8	1.70815	0.854073	+	0.520153i	$0.174125\pi$
$194$	1.04447e6	0.0102704
$195$	0	0
$196$	−1.45885e7	−0.138393
$197$	−6.92904e7	−0.645716	−0.322858	−	0.946447i	$-0.604644\pi$
$197$	−6.92904e7	−0.645716	−0.322858	+	0.946447i	$0.604644\pi$
$198$	3.97159e6	0.0363611
$199$	−6.71910e7	−0.604401	−0.302201	−	0.953244i	$-0.597721\pi$
$199$	−6.71910e7	−0.604401	−0.302201	+	0.953244i	$0.597721\pi$
$200$	0	0
$201$	−1.31552e7	−0.114264
$202$	−3.27921e6	−0.0279924
$203$	1.21758e7	0.102156
$204$	3.10627e7	0.256174
$205$	0	0
$206$	1.31501e7	0.104808
$207$	3.02535e7	0.237072
$208$	−4.27191e7	−0.329155
$209$	−1.17241e7	−0.0888316
$210$	0	0
$211$	−5.14714e7	−0.377205	−0.188602	−	0.982054i	$-0.560396\pi$
$211$	−5.14714e7	−0.377205	−0.188602	+	0.982054i	$0.560396\pi$
$212$	−1.57557e8	−1.13570
$213$	2.93220e7	0.207905
$214$	−2.42960e7	−0.169468
$215$	0	0
$216$	−9.92023e6	−0.0669782
$217$	1.81584e7	0.120634
$218$	−1.93690e7	−0.126622
$219$	1.59893e8	1.02867
$220$	0	0
$221$	2.66650e7	0.166176
$222$	4.54129e6	0.0278576
$223$	2.49372e8	1.50585	0.752923	−	0.658109i	$-0.228643\pi$
$223$	2.49372e8	1.50585	0.752923	+	0.658109i	$0.228643\pi$
$224$	−3.23243e7	−0.192159
$225$	0	0
$226$	−5.00591e7	−0.288472
$227$	−3.93928e7	−0.223525	−0.111762	−	0.993735i	$-0.535650\pi$
$227$	−3.93928e7	−0.223525	−0.111762	+	0.993735i	$0.535650\pi$
$228$	1.44098e7	0.0805167
$229$	1.25945e8	0.693035	0.346517	−	0.938044i	$-0.387364\pi$
$229$	1.25945e8	0.693035	0.346517	+	0.938044i	$0.387364\pi$
$230$	0	0
$231$	−2.52270e7	−0.134655
$232$	1.78910e7	0.0940647
$233$	1.55290e8	0.804265	0.402132	−	0.915582i	$-0.368269\pi$
$233$	1.55290e8	0.804265	0.402132	+	0.915582i	$0.368269\pi$
$234$	−4.19029e6	−0.0213791
$235$	0	0
$236$	1.92603e8	0.953831
$237$	−1.46605e8	−0.715369
$238$	6.36471e6	0.0306027
$239$	−1.53561e8	−0.727593	−0.363797	−	0.931478i	$-0.618520\pi$
$239$	−1.53561e8	−0.727593	−0.363797	+	0.931478i	$0.618520\pi$
$240$	0	0
$241$	1.67184e8	0.769369	0.384685	−	0.923048i	$-0.374310\pi$
$241$	1.67184e8	0.769369	0.384685	+	0.923048i	$0.374310\pi$
$242$	−2.41340e7	−0.109465
$243$	1.43489e7	0.0641500
$244$	−2.64862e7	−0.116722
$245$	0	0
$246$	9.73847e6	0.0417078
$247$	1.23697e7	0.0522300
$248$	2.66818e7	0.111079
$249$	−1.87202e8	−0.768445
$250$	0	0
$251$	−4.29028e7	−0.171249	−0.0856245	−	0.996327i	$-0.527289\pi$
$251$	−4.29028e7	−0.171249	−0.0856245	+	0.996327i	$0.527289\pi$
$252$	3.10058e7	0.122051
$253$	1.13046e8	0.438867
$254$	−6.11920e6	−0.0234302
$255$	0	0
$256$	1.88424e8	0.701936
$257$	−4.70837e8	−1.73023	−0.865116	−	0.501571i	$-0.832755\pi$
$257$	−4.70837e8	−1.73023	−0.865116	+	0.501571i	$0.832755\pi$
$258$	1.80641e6	0.00654858
$259$	−2.88456e7	−0.103165
$260$	0	0
$261$	−2.58780e7	−0.0900927
$262$	−7.13823e6	−0.0245209
$263$	1.02607e8	0.347802	0.173901	−	0.984763i	$-0.444363\pi$
$263$	1.02607e8	0.347802	0.173901	+	0.984763i	$0.444363\pi$
$264$	−3.70682e7	−0.123990
$265$	0	0
$266$	2.95254e6	0.00961857
$267$	7.09058e6	0.0227978
$268$	6.04163e7	0.191727
$269$	−5.49478e8	−1.72114	−0.860571	−	0.509330i	$-0.829893\pi$
$269$	−5.49478e8	−1.72114	−0.860571	+	0.509330i	$0.829893\pi$
$270$	0	0
$271$	3.38205e8	1.03225	0.516127	−	0.856512i	$-0.327373\pi$
$271$	3.38205e8	1.03225	0.516127	+	0.856512i	$0.327373\pi$
$272$	−1.37908e8	−0.415527
$273$	2.66161e7	0.0791727
$274$	−9.67472e6	−0.0284126
$275$	0	0
$276$	−1.38942e8	−0.397788
$277$	−3.85029e8	−1.08846	−0.544232	−	0.838934i	$-0.683179\pi$
$277$	−3.85029e8	−1.08846	−0.544232	+	0.838934i	$0.683179\pi$
$278$	4.97146e6	0.0138780
$279$	−3.85933e7	−0.106389
$280$	0	0
$281$	3.58731e8	0.964488	0.482244	−	0.876037i	$-0.339822\pi$
$281$	3.58731e8	0.964488	0.482244	+	0.876037i	$0.339822\pi$
$282$	7.35048e6	0.0195184
$283$	−5.81811e7	−0.152591	−0.0762955	−	0.997085i	$-0.524309\pi$
$283$	−5.81811e7	−0.152591	−0.0762955	+	0.997085i	$0.524309\pi$
$284$	−1.34664e8	−0.348849
$285$	0	0
$286$	−1.56576e7	−0.0395770
$287$	−6.18573e7	−0.154456
$288$	6.87010e7	0.169469
$289$	−3.24257e8	−0.790219
$290$	0	0
$291$	1.41003e7	0.0335431
$292$	−7.34325e8	−1.72603
$293$	−5.57054e8	−1.29378	−0.646890	−	0.762583i	$-0.723931\pi$
$293$	−5.57054e8	−1.29378	−0.646890	+	0.762583i	$0.723931\pi$
$294$	6.35305e6	0.0145803
$295$	0	0
$296$	−4.23854e7	−0.0949938
$297$	5.36165e7	0.118755
$298$	−7.99790e7	−0.175073
$299$	−1.19271e8	−0.258039
$300$	0	0
$301$	−1.14740e7	−0.0242512
$302$	−1.07584e8	−0.224762
$303$	−4.42694e7	−0.0914227
$304$	−6.39747e7	−0.130602
$305$	0	0
$306$	−1.35273e7	−0.0269890
$307$	−5.19798e8	−1.02530	−0.512649	−	0.858598i	$-0.671336\pi$
$307$	−5.19798e8	−1.02530	−0.512649	+	0.858598i	$0.671336\pi$
$308$	1.15857e8	0.225941
$309$	1.77527e8	0.342301
$310$	0	0
$311$	−3.93391e8	−0.741589	−0.370794	−	0.928715i	$-0.620915\pi$
$311$	−3.93391e8	−0.741589	−0.370794	+	0.928715i	$0.620915\pi$
$312$	3.91094e7	0.0729021
$313$	−7.46643e8	−1.37628	−0.688142	−	0.725576i	$-0.741573\pi$
$313$	−7.46643e8	−1.37628	−0.688142	+	0.725576i	$0.741573\pi$
$314$	−1.92804e7	−0.0351448
$315$	0	0
$316$	6.73298e8	1.20034
$317$	−1.90220e8	−0.335389	−0.167694	−	0.985839i	$-0.553632\pi$
$317$	−1.90220e8	−0.335389	−0.167694	+	0.985839i	$0.553632\pi$
$318$	6.86136e7	0.119651
$319$	−9.66966e7	−0.166780
$320$	0	0
$321$	−3.27996e8	−0.553479
$322$	−2.84690e7	−0.0475200
$323$	3.99325e7	0.0659353
$324$	−6.58987e7	−0.107639
$325$	0	0
$326$	1.53469e8	0.245334
$327$	−2.61482e8	−0.413547
$328$	−9.08924e7	−0.142223
$329$	−4.66892e7	−0.0722820
$330$	0	0
$331$	−1.09993e9	−1.66712	−0.833559	−	0.552430i	$-0.813701\pi$
$331$	−1.09993e9	−1.66712	−0.833559	+	0.552430i	$0.813701\pi$
$332$	8.59742e8	1.28939
$333$	6.13074e7	0.0909826
$334$	−2.07712e7	−0.0305035
$335$	0	0
$336$	−1.37656e8	−0.197973
$337$	−4.94657e8	−0.704043	−0.352022	−	0.935992i	$-0.614506\pi$
$337$	−4.94657e8	−0.704043	−0.352022	+	0.935992i	$0.614506\pi$
$338$	−1.08977e8	−0.153507
$339$	−6.75798e8	−0.942146
$340$	0	0
$341$	−1.44209e8	−0.196948
$342$	−6.27523e6	−0.00848278
$343$	−4.03536e7	−0.0539949
$344$	−1.68598e7	−0.0223305
$345$	0	0
$346$	1.12985e7	0.0146641
$347$	9.31091e8	1.19630	0.598148	−	0.801385i	$-0.295903\pi$
$347$	9.31091e8	1.19630	0.598148	+	0.801385i	$0.295903\pi$
$348$	1.18847e8	0.151169
$349$	−4.56646e8	−0.575031	−0.287515	−	0.957776i	$-0.592829\pi$
$349$	−4.56646e8	−0.575031	−0.287515	+	0.957776i	$0.592829\pi$
$350$	0	0
$351$	−5.65689e7	−0.0698237
$352$	2.56710e8	0.313721
$353$	5.02682e8	0.608250	0.304125	−	0.952632i	$-0.401636\pi$
$353$	5.02682e8	0.608250	0.304125	+	0.952632i	$0.401636\pi$
$354$	−8.38756e7	−0.100490
$355$	0	0
$356$	−3.25641e7	−0.0382529
$357$	8.59236e7	0.0999479
$358$	−2.08746e8	−0.240451
$359$	−1.39665e9	−1.59316	−0.796578	−	0.604536i	$-0.793359\pi$
$359$	−1.39665e9	−1.59316	−0.796578	+	0.604536i	$0.793359\pi$
$360$	0	0
$361$	−8.75347e8	−0.979276
$362$	−2.84524e8	−0.315238
$363$	−3.25809e8	−0.357511
$364$	−1.22237e8	−0.132846
$365$	0	0
$366$	1.15343e7	0.0122972
$367$	2.70500e8	0.285651	0.142826	−	0.989748i	$-0.454381\pi$
$367$	2.70500e8	0.285651	0.142826	+	0.989748i	$0.454381\pi$
$368$	6.16856e8	0.645233
$369$	1.31469e8	0.136217
$370$	0	0
$371$	−4.35823e8	−0.443100
$372$	1.77243e8	0.178513
$373$	1.06422e9	1.06182	0.530912	−	0.847427i	$-0.321850\pi$
$373$	1.06422e9	1.06182	0.530912	+	0.847427i	$0.321850\pi$
$374$	−5.05465e7	−0.0499621
$375$	0	0
$376$	−6.86045e7	−0.0665572
$377$	1.02021e8	0.0980609
$378$	−1.35025e7	−0.0128586
$379$	−3.95456e8	−0.373131	−0.186565	−	0.982443i	$-0.559736\pi$
$379$	−3.95456e8	−0.373131	−0.186565	+	0.982443i	$0.559736\pi$
$380$	0	0
$381$	−8.26092e7	−0.0765228
$382$	−6.71067e7	−0.0615948
$383$	−1.64793e8	−0.149880	−0.0749400	−	0.997188i	$-0.523877\pi$
$383$	−1.64793e8	−0.149880	−0.0749400	+	0.997188i	$0.523877\pi$
$384$	−4.18255e8	−0.376949
$385$	0	0
$386$	3.41197e8	0.301960
$387$	2.43865e7	0.0213876
$388$	−6.47570e7	−0.0562828
$389$	−1.71125e8	−0.147398	−0.0736989	−	0.997281i	$-0.523480\pi$
$389$	−1.71125e8	−0.147398	−0.0736989	+	0.997281i	$0.523480\pi$
$390$	0	0
$391$	−3.85037e8	−0.325750
$392$	−5.92951e7	−0.0497184
$393$	−9.63661e7	−0.0800849
$394$	−1.38581e8	−0.114147
$395$	0	0
$396$	−2.46239e8	−0.199261
$397$	−1.73552e9	−1.39207	−0.696037	−	0.718006i	$-0.745055\pi$
$397$	−1.73552e9	−1.39207	−0.696037	+	0.718006i	$0.745055\pi$
$398$	−1.34382e8	−0.106844
$399$	3.98593e7	0.0314141
$400$	0	0
$401$	2.26343e8	0.175292	0.0876458	−	0.996152i	$-0.472066\pi$
$401$	2.26343e8	0.175292	0.0876458	+	0.996152i	$0.472066\pi$
$402$	−2.63103e7	−0.0201992
$403$	1.52150e8	0.115799
$404$	2.03311e8	0.153400
$405$	0	0
$406$	2.43516e7	0.0180587
$407$	2.29083e8	0.168427
$408$	1.26255e8	0.0920318
$409$	1.14869e9	0.830176	0.415088	−	0.909781i	$-0.363751\pi$
$409$	1.14869e9	0.830176	0.415088	+	0.909781i	$0.363751\pi$
$410$	0	0
$411$	−1.30609e8	−0.0927952
$412$	−8.15307e8	−0.574356
$413$	5.32765e8	0.372144
$414$	6.05070e7	0.0419087
$415$	0	0
$416$	−2.70846e8	−0.184457
$417$	6.71147e7	0.0453254
$418$	−2.34482e7	−0.0157034
$419$	1.98365e9	1.31740	0.658699	−	0.752407i	$-0.271107\pi$
$419$	1.98365e9	1.31740	0.658699	+	0.752407i	$0.271107\pi$
$420$	0	0
$421$	2.66045e9	1.73767	0.868835	−	0.495103i	$-0.164870\pi$
$421$	2.66045e9	1.73767	0.868835	+	0.495103i	$0.164870\pi$
$422$	−1.02943e8	−0.0666810
$423$	9.92315e7	0.0637468
$424$	−6.40393e8	−0.408006
$425$	0	0
$426$	5.86440e7	0.0367528
$427$	−7.32641e7	−0.0455401
$428$	1.50635e9	0.928696
$429$	−2.11377e8	−0.129258
$430$	0	0
$431$	−1.50478e9	−0.905322	−0.452661	−	0.891683i	$-0.649525\pi$
$431$	−1.50478e9	−0.905322	−0.452661	+	0.891683i	$0.649525\pi$
$432$	2.92568e8	0.174596
$433$	−1.37803e9	−0.815738	−0.407869	−	0.913040i	$-0.633728\pi$
$433$	−1.37803e9	−0.815738	−0.407869	+	0.913040i	$0.633728\pi$
$434$	3.63168e7	0.0213252
$435$	0	0
$436$	1.20088e9	0.693900
$437$	−1.78616e8	−0.102385
$438$	3.19787e8	0.181845
$439$	2.15016e9	1.21296	0.606478	−	0.795100i	$-0.292582\pi$
$439$	2.15016e9	1.21296	0.606478	+	0.795100i	$0.292582\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	5.33299e7	0.0293760
$443$	1.01094e9	0.552474	0.276237	−	0.961090i	$-0.410913\pi$
$443$	1.01094e9	0.552474	0.276237	+	0.961090i	$0.410913\pi$
$444$	−2.81560e8	−0.152662
$445$	0	0
$446$	4.98744e8	0.266198
$447$	−1.07972e9	−0.571786
$448$	5.87940e8	0.308930
$449$	1.82648e8	0.0952256	0.0476128	−	0.998866i	$-0.484839\pi$
$449$	1.82648e8	0.0952256	0.0476128	+	0.998866i	$0.484839\pi$
$450$	0	0
$451$	4.91252e8	0.252166
$452$	3.10367e9	1.58085
$453$	−1.45238e9	−0.734069
$454$	−7.87855e7	−0.0395140
$455$	0	0
$456$	5.85688e7	0.0289261
$457$	1.51611e6	0.000743060 0	0.000371530	−	1.00000i	$-0.499882\pi$
$457$	1.51611e6	0.000743060 0	0.000371530		1.00000i	$0.499882\pi$
$458$	2.51889e8	0.122512
$459$	−1.82619e8	−0.0881457
$460$	0	0
$461$	−4.79940e8	−0.228157	−0.114079	−	0.993472i	$-0.536392\pi$
$461$	−4.79940e8	−0.228157	−0.114079	+	0.993472i	$0.536392\pi$
$462$	−5.04539e7	−0.0238039
$463$	6.86812e8	0.321591	0.160796	−	0.986988i	$-0.448594\pi$
$463$	6.86812e8	0.321591	0.160796	+	0.986988i	$0.448594\pi$
$464$	−5.27642e8	−0.245203
$465$	0	0
$466$	3.10581e8	0.142175
$467$	−1.69004e9	−0.767869	−0.383934	−	0.923360i	$-0.625431\pi$
$467$	−1.69004e9	−0.767869	−0.383934	+	0.923360i	$0.625431\pi$
$468$	2.59798e8	0.117159
$469$	1.67119e8	0.0748034
$470$	0	0
$471$	−2.60285e8	−0.114783
$472$	7.82839e8	0.342670
$473$	9.11232e7	0.0395927
$474$	−2.93210e8	−0.126461
$475$	0	0
$476$	−3.94612e8	−0.167705
$477$	9.26283e8	0.390778
$478$	−3.07122e8	−0.128622
$479$	2.39843e9	0.997132	0.498566	−	0.866852i	$-0.333860\pi$
$479$	2.39843e9	0.997132	0.498566	+	0.866852i	$0.333860\pi$
$480$	0	0
$481$	−2.41698e8	−0.0990295
$482$	3.34368e8	0.136007
$483$	−3.84332e8	−0.155200
$484$	1.49631e9	0.599877
$485$	0	0
$486$	2.86978e7	0.0113402
$487$	6.76814e8	0.265533	0.132766	−	0.991147i	$-0.457614\pi$
$487$	6.76814e8	0.265533	0.132766	+	0.991147i	$0.457614\pi$
$488$	−1.07653e8	−0.0419332
$489$	2.07183e9	0.801258
$490$	0	0
$491$	3.97038e9	1.51372	0.756862	−	0.653574i	$-0.226731\pi$
$491$	3.97038e9	1.51372	0.756862	+	0.653574i	$0.226731\pi$
$492$	−6.03785e8	−0.228562
$493$	3.29350e8	0.123792
$494$	2.47394e7	0.00923304
$495$	0	0
$496$	−7.86900e8	−0.289557
$497$	−3.72498e8	−0.136106
$498$	−3.74404e8	−0.135843
$499$	1.59757e9	0.575584	0.287792	−	0.957693i	$-0.407079\pi$
$499$	1.59757e9	0.575584	0.287792	+	0.957693i	$0.407079\pi$
$500$	0	0
$501$	−2.80412e8	−0.0996241
$502$	−8.58056e7	−0.0302728
$503$	−1.72479e9	−0.604295	−0.302148	−	0.953261i	$-0.597704\pi$
$503$	−1.72479e9	−0.604295	−0.302148	+	0.953261i	$0.597704\pi$
$504$	1.26024e8	0.0438475
$505$	0	0
$506$	2.26092e8	0.0775815
$507$	−1.47119e9	−0.501351
$508$	3.79390e8	0.128400
$509$	5.83626e9	1.96165	0.980827	−	0.194880i	$-0.0624316\pi$
$509$	5.83626e9	1.96165	0.980827	+	0.194880i	$0.0624316\pi$
$510$	0	0
$511$	−2.03124e9	−0.673423
$512$	2.35969e9	0.776980
$513$	−8.47156e7	−0.0277047
$514$	−9.41674e8	−0.305865
$515$	0	0
$516$	−1.11997e8	−0.0358867
$517$	3.70791e8	0.118008
$518$	−5.76912e7	−0.0182371
$519$	1.52530e8	0.0478928
$520$	0	0
$521$	4.99457e9	1.54727	0.773634	−	0.633632i	$-0.218437\pi$
$521$	4.99457e9	1.54727	0.773634	+	0.633632i	$0.218437\pi$
$522$	−5.17561e7	−0.0159263
$523$	−1.03723e9	−0.317045	−0.158522	−	0.987355i	$-0.550673\pi$
$523$	−1.03723e9	−0.317045	−0.158522	+	0.987355i	$0.550673\pi$
$524$	4.42570e8	0.134376
$525$	0	0
$526$	2.05214e8	0.0614834
$527$	4.91177e8	0.146184
$528$	1.09322e9	0.323212
$529$	−1.68258e9	−0.494174
$530$	0	0
$531$	−1.13232e9	−0.328200
$532$	−1.83058e8	−0.0527105
$533$	−5.18303e8	−0.148265
$534$	1.41812e7	0.00403011
$535$	0	0
$536$	2.45563e8	0.0688789
$537$	−2.81807e9	−0.785310
$538$	−1.09896e9	−0.304258
$539$	3.20476e8	0.0881525
$540$	0	0
$541$	−2.48336e9	−0.674295	−0.337147	−	0.941452i	$-0.609462\pi$
$541$	−2.48336e9	−0.674295	−0.337147	+	0.941452i	$0.609462\pi$
$542$	6.76409e8	0.182479
$543$	−3.84107e9	−1.02956
$544$	−8.74359e8	−0.232859
$545$	0	0
$546$	5.32322e7	0.0139959
$547$	3.74874e9	0.979331	0.489665	−	0.871910i	$-0.337119\pi$
$547$	3.74874e9	0.979331	0.489665	+	0.871910i	$0.337119\pi$
$548$	5.99832e8	0.155703
$549$	1.55713e8	0.0401626
$550$	0	0
$551$	1.52783e8	0.0389086
$552$	−5.64732e8	−0.142908
$553$	1.86243e9	0.468319
$554$	−7.70058e8	−0.192415
$555$	0	0
$556$	−3.08230e8	−0.0760525
$557$	2.18032e9	0.534596	0.267298	−	0.963614i	$-0.413869\pi$
$557$	2.18032e9	0.534596	0.267298	+	0.963614i	$0.413869\pi$
$558$	−7.71865e7	−0.0188071
$559$	−9.61410e7	−0.0232792
$560$	0	0
$561$	−6.82378e8	−0.163176
$562$	7.17462e8	0.170499
$563$	−2.56548e9	−0.605883	−0.302941	−	0.953009i	$-0.597969\pi$
$563$	−2.56548e9	−0.605883	−0.302941	+	0.953009i	$0.597969\pi$
$564$	−4.55730e8	−0.106962
$565$	0	0
$566$	−1.16362e8	−0.0269745
$567$	−1.82284e8	−0.0419961
$568$	−5.47344e8	−0.125326
$569$	1.65916e8	0.0377568	0.0188784	−	0.999822i	$-0.493990\pi$
$569$	1.65916e8	0.0377568	0.0188784	+	0.999822i	$0.493990\pi$
$570$	0	0
$571$	−5.11928e9	−1.15075	−0.575377	−	0.817888i	$-0.695145\pi$
$571$	−5.11928e9	−1.15075	−0.575377	+	0.817888i	$0.695145\pi$
$572$	9.70768e8	0.216885
$573$	−9.05940e8	−0.201168
$574$	−1.23715e8	−0.0273042
$575$	0	0
$576$	−1.24959e9	−0.272451
$577$	6.90669e9	1.49677	0.748384	−	0.663265i	$-0.230830\pi$
$577$	6.90669e9	1.49677	0.748384	+	0.663265i	$0.230830\pi$
$578$	−6.48515e8	−0.139692
$579$	4.60616e9	0.986199
$580$	0	0
$581$	2.37816e9	0.503065
$582$	2.82006e7	0.00592964
$583$	3.46117e9	0.723408
$584$	−2.98468e9	−0.620087
$585$	0	0
$586$	−1.11411e9	−0.228710
$587$	4.92686e9	1.00540	0.502698	−	0.864462i	$-0.332341\pi$
$587$	4.92686e9	1.00540	0.502698	+	0.864462i	$0.332341\pi$
$588$	−3.93889e8	−0.0799012
$589$	2.27854e8	0.0459465
$590$	0	0
$591$	−1.87084e9	−0.372804
$592$	1.25003e9	0.247626
$593$	−2.48990e9	−0.490332	−0.245166	−	0.969481i	$-0.578842\pi$
$593$	−2.48990e9	−0.490332	−0.245166	+	0.969481i	$0.578842\pi$
$594$	1.07233e8	0.0209931
$595$	0	0
$596$	4.95870e9	0.959413
$597$	−1.81416e9	−0.348951
$598$	−2.38542e8	−0.0456153
$599$	−1.79975e9	−0.342151	−0.171076	−	0.985258i	$-0.554724\pi$
$599$	−1.79975e9	−0.342151	−0.171076	+	0.985258i	$0.554724\pi$
$600$	0	0
$601$	−9.64250e9	−1.81188	−0.905939	−	0.423409i	$-0.860834\pi$
$601$	−9.64250e9	−1.81188	−0.905939	+	0.423409i	$0.860834\pi$
$602$	−2.29481e7	−0.00428705
$603$	−3.55189e8	−0.0659704
$604$	6.67020e9	1.23171
$605$	0	0
$606$	−8.85387e7	−0.0161614
$607$	7.59240e9	1.37790	0.688951	−	0.724808i	$-0.258071\pi$
$607$	7.59240e9	1.37790	0.688951	+	0.724808i	$0.258071\pi$
$608$	−4.05609e8	−0.0731889
$609$	3.28747e8	0.0589795
$610$	0	0
$611$	−3.91209e8	−0.0693848
$612$	8.38694e8	0.147902
$613$	−6.21966e9	−1.09057	−0.545287	−	0.838249i	$-0.683579\pi$
$613$	−6.21966e9	−1.09057	−0.545287	+	0.838249i	$0.683579\pi$
$614$	−1.03960e9	−0.181249
$615$	0	0
$616$	4.70903e8	0.0811707
$617$	1.01931e10	1.74706	0.873529	−	0.486772i	$-0.161826\pi$
$617$	1.01931e10	1.74706	0.873529	+	0.486772i	$0.161826\pi$
$618$	3.55053e8	0.0605109
$619$	−1.86525e9	−0.316097	−0.158048	−	0.987431i	$-0.550520\pi$
$619$	−1.86525e9	−0.316097	−0.158048	+	0.987431i	$0.550520\pi$
$620$	0	0
$621$	8.16844e8	0.136873
$622$	−7.86782e8	−0.131096
$623$	−9.00766e7	−0.0149246
$624$	−1.15342e9	−0.190038
$625$	0	0
$626$	−1.49329e9	−0.243295
$627$	−3.16551e8	−0.0512869
$628$	1.19538e9	0.192596
$629$	−7.80261e8	−0.125015
$630$	0	0
$631$	3.03048e9	0.480185	0.240092	−	0.970750i	$-0.422822\pi$
$631$	3.03048e9	0.480185	0.240092	+	0.970750i	$0.422822\pi$
$632$	2.73663e9	0.431228
$633$	−1.38973e9	−0.217779
$634$	−3.80440e8	−0.0592889
$635$	0	0
$636$	−4.25404e9	−0.655695
$637$	−3.38123e8	−0.0518307
$638$	−1.93393e8	−0.0294828
$639$	7.91694e8	0.120034
$640$	0	0
$641$	1.08550e10	1.62789	0.813945	−	0.580942i	$-0.197316\pi$
$641$	1.08550e10	1.62789	0.813945	+	0.580942i	$0.197316\pi$
$642$	−6.55992e8	−0.0978422
$643$	−9.35593e8	−0.138787	−0.0693934	−	0.997589i	$-0.522106\pi$
$643$	−9.35593e8	−0.138787	−0.0693934	+	0.997589i	$0.522106\pi$
$644$	1.76508e9	0.260413
$645$	0	0
$646$	7.98650e7	0.0116558
$647$	8.82519e9	1.28103	0.640514	−	0.767946i	$-0.278721\pi$
$647$	8.82519e9	1.28103	0.640514	+	0.767946i	$0.278721\pi$
$648$	−2.67846e8	−0.0386699
$649$	−4.23106e9	−0.607565
$650$	0	0
$651$	4.90277e8	0.0696480
$652$	−9.51506e9	−1.34445
$653$	−2.99850e9	−0.421413	−0.210707	−	0.977549i	$-0.567576\pi$
$653$	−2.99850e9	−0.421413	−0.210707	+	0.977549i	$0.567576\pi$
$654$	−5.22964e8	−0.0731055
$655$	0	0
$656$	2.68060e9	0.370740
$657$	4.31712e9	0.593903
$658$	−9.33783e7	−0.0127778
$659$	−3.93595e9	−0.535736	−0.267868	−	0.963456i	$-0.586319\pi$
$659$	−3.93595e9	−0.535736	−0.267868	+	0.963456i	$0.586319\pi$
$660$	0	0
$661$	−6.26839e8	−0.0844211	−0.0422106	−	0.999109i	$-0.513440\pi$
$661$	−6.26839e8	−0.0844211	−0.0422106	+	0.999109i	$0.513440\pi$
$662$	−2.19986e9	−0.294708
$663$	7.19954e8	0.0959417
$664$	3.49444e9	0.463222
$665$	0	0
$666$	1.22615e8	0.0160836
$667$	−1.47317e9	−0.192226
$668$	1.28782e9	0.167162
$669$	6.73304e9	0.869400
$670$	0	0
$671$	5.81841e8	0.0743491
$672$	−8.72757e8	−0.110943
$673$	−4.74113e9	−0.599555	−0.299777	−	0.954009i	$-0.596912\pi$
$673$	−4.74113e9	−0.599555	−0.299777	+	0.954009i	$0.596912\pi$
$674$	−9.89313e8	−0.124458
$675$	0	0
$676$	6.75659e9	0.841229
$677$	8.34660e9	1.03383	0.516915	−	0.856037i	$-0.327080\pi$
$677$	8.34660e9	1.03383	0.516915	+	0.856037i	$0.327080\pi$
$678$	−1.35160e9	−0.166549
$679$	−1.79126e8	−0.0219591
$680$	0	0
$681$	−1.06360e9	−0.129052
$682$	−2.88417e8	−0.0348157
$683$	−1.00158e10	−1.20286	−0.601428	−	0.798927i	$-0.705401\pi$
$683$	−1.00158e10	−1.20286	−0.601428	+	0.798927i	$0.705401\pi$
$684$	3.89064e8	0.0464863
$685$	0	0
$686$	−8.07072e7	−0.00954504
$687$	3.40050e9	0.400124
$688$	4.97231e8	0.0582101
$689$	−3.65177e9	−0.425340
$690$	0	0
$691$	−1.16744e10	−1.34605	−0.673024	−	0.739621i	$-0.735005\pi$
$691$	−1.16744e10	−1.34605	−0.673024	+	0.739621i	$0.735005\pi$
$692$	−7.00509e8	−0.0803605
$693$	−6.81128e8	−0.0777432
$694$	1.86218e9	0.211477
$695$	0	0
$696$	4.83057e8	0.0543083
$697$	−1.67321e9	−0.187170
$698$	−9.13293e8	−0.101652
$699$	4.19284e9	0.464342
$700$	0	0
$701$	7.05584e9	0.773634	0.386817	−	0.922157i	$-0.373575\pi$
$701$	7.05584e9	0.773634	0.386817	+	0.922157i	$0.373575\pi$
$702$	−1.13138e8	−0.0123432
$703$	−3.61958e8	−0.0392929
$704$	−4.66924e9	−0.504362
$705$	0	0
$706$	1.00536e9	0.107524
$707$	5.62385e8	0.0598502
$708$	5.20029e9	0.550695
$709$	1.96477e8	0.0207038	0.0103519	−	0.999946i	$-0.496705\pi$
$709$	1.96477e8	0.0207038	0.0103519	+	0.999946i	$0.496705\pi$
$710$	0	0
$711$	−3.95834e9	−0.413019
$712$	−1.32357e8	−0.0137426
$713$	−2.19701e9	−0.226996
$714$	1.71847e8	0.0176685
$715$	0	0
$716$	1.29422e10	1.31769
$717$	−4.14615e9	−0.420076
$718$	−2.79331e9	−0.281633
$719$	−8.96572e9	−0.899569	−0.449784	−	0.893137i	$-0.648499\pi$
$719$	−8.96572e9	−0.899569	−0.449784	+	0.893137i	$0.648499\pi$
$720$	0	0
$721$	−2.25524e9	−0.224089
$722$	−1.75069e9	−0.173113
$723$	4.51397e9	0.444196
$724$	1.76405e10	1.72753
$725$	0	0
$726$	−6.51618e8	−0.0631997
$727$	4.44082e9	0.428640	0.214320	−	0.976763i	$-0.431246\pi$
$727$	4.44082e9	0.428640	0.214320	+	0.976763i	$0.431246\pi$
$728$	−4.96834e8	−0.0477256
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−3.10368e8	−0.0293877
$732$	−7.15126e8	−0.0673898
$733$	8.97113e9	0.841363	0.420681	−	0.907208i	$-0.361791\pi$
$733$	8.97113e9	0.841363	0.420681	+	0.907208i	$0.361791\pi$
$734$	5.41000e8	0.0504965
$735$	0	0
$736$	3.91096e9	0.361586
$737$	−1.32721e9	−0.122125
$738$	2.62939e8	0.0240800
$739$	1.27011e10	1.15767	0.578835	−	0.815445i	$-0.303508\pi$
$739$	1.27011e10	1.15767	0.578835	+	0.815445i	$0.303508\pi$
$740$	0	0
$741$	3.33982e8	0.0301550
$742$	−8.71647e8	−0.0783298
$743$	1.54200e10	1.37919	0.689593	−	0.724197i	$-0.257789\pi$
$743$	1.54200e10	1.37919	0.689593	+	0.724197i	$0.257789\pi$
$744$	7.20408e8	0.0641317
$745$	0	0
$746$	2.12845e9	0.187706
$747$	−5.05445e9	−0.443662
$748$	3.13389e9	0.273796
$749$	4.16677e9	0.362337
$750$	0	0
$751$	4.99561e9	0.430376	0.215188	−	0.976573i	$-0.430964\pi$
$751$	4.99561e9	0.430376	0.215188	+	0.976573i	$0.430964\pi$
$752$	2.02329e9	0.173498
$753$	−1.15838e9	−0.0988706
$754$	2.04043e8	0.0173349
$755$	0	0
$756$	8.37157e8	0.0704662
$757$	−1.13781e10	−0.953313	−0.476657	−	0.879090i	$-0.658152\pi$
$757$	−1.13781e10	−0.953313	−0.476657	+	0.879090i	$0.658152\pi$
$758$	−7.90912e8	−0.0659608
$759$	3.05224e9	0.253380
$760$	0	0
$761$	9.31372e9	0.766084	0.383042	−	0.923731i	$-0.374876\pi$
$761$	9.31372e9	0.766084	0.383042	+	0.923731i	$0.374876\pi$
$762$	−1.65218e8	−0.0135275
$763$	3.32179e9	0.270730
$764$	4.16061e9	0.337544
$765$	0	0
$766$	−3.29586e8	−0.0264953
$767$	4.46405e9	0.357228
$768$	5.08746e9	0.405263
$769$	−5.75726e9	−0.456535	−0.228267	−	0.973598i	$-0.573306\pi$
$769$	−5.75726e9	−0.456535	−0.228267	+	0.973598i	$0.573306\pi$
$770$	0	0
$771$	−1.27126e10	−0.998950
$772$	−2.11542e10	−1.65477
$773$	−1.45658e10	−1.13424	−0.567121	−	0.823634i	$-0.691943\pi$
$773$	−1.45658e10	−1.13424	−0.567121	+	0.823634i	$0.691943\pi$
$774$	4.87730e7	0.00378082
$775$	0	0
$776$	−2.63206e8	−0.0202199
$777$	−7.78832e8	−0.0595621
$778$	−3.42251e8	−0.0260565
$779$	−7.76192e8	−0.0588285
$780$	0	0
$781$	2.95826e9	0.222207
$782$	−7.70074e8	−0.0575849
$783$	−6.98707e8	−0.0520150
$784$	1.74873e9	0.129604
$785$	0	0
$786$	−1.92732e8	−0.0141571
$787$	−7.81273e9	−0.571336	−0.285668	−	0.958329i	$-0.592215\pi$
$787$	−7.81273e9	−0.571336	−0.285668	+	0.958329i	$0.592215\pi$
$788$	8.59201e9	0.625537
$789$	2.77039e9	0.200804
$790$	0	0
$791$	8.58514e9	0.616779
$792$	−1.00084e9	−0.0715858
$793$	−6.13881e8	−0.0437147
$794$	−3.47103e9	−0.246086
$795$	0	0
$796$	8.33168e9	0.585514
$797$	−5.72212e9	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$797$	−5.72212e9	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$798$	7.97187e7	0.00555329
$799$	−1.26292e9	−0.0875916
$800$	0	0
$801$	1.91446e8	0.0131623
$802$	4.52685e8	0.0309875
$803$	1.61315e10	1.09943
$804$	1.63124e9	0.110693
$805$	0	0
$806$	3.04299e8	0.0204705
$807$	−1.48359e10	−0.993702
$808$	8.26361e8	0.0551100
$809$	−9.16070e9	−0.608287	−0.304144	−	0.952626i	$-0.598370\pi$
$809$	−9.16070e9	−0.608287	−0.304144	+	0.952626i	$0.598370\pi$
$810$	0	0
$811$	−2.04091e10	−1.34354	−0.671771	−	0.740759i	$-0.734466\pi$
$811$	−2.04091e10	−1.34354	−0.671771	+	0.740759i	$0.734466\pi$
$812$	−1.50980e9	−0.0989632
$813$	9.13152e9	0.595973
$814$	4.58166e8	0.0297740
$815$	0	0
$816$	−3.72352e9	−0.239904
$817$	−1.43977e8	−0.00923671
$818$	2.29738e9	0.146756
$819$	7.18635e8	0.0457104
$820$	0	0
$821$	6.26680e9	0.395225	0.197613	−	0.980280i	$-0.436681\pi$
$821$	6.26680e9	0.395225	0.197613	+	0.980280i	$0.436681\pi$
$822$	−2.61217e8	−0.0164040
$823$	−4.37792e9	−0.273759	−0.136879	−	0.990588i	$-0.543707\pi$
$823$	−4.37792e9	−0.273759	−0.136879	+	0.990588i	$0.543707\pi$
$824$	−3.31383e9	−0.206341
$825$	0	0
$826$	1.06553e9	0.0657864
$827$	1.61665e10	0.993912	0.496956	−	0.867776i	$-0.334451\pi$
$827$	1.61665e10	0.993912	0.496956	+	0.867776i	$0.334451\pi$
$828$	−3.75143e9	−0.229663
$829$	2.76523e10	1.68574	0.842868	−	0.538120i	$-0.180865\pi$
$829$	2.76523e10	1.68574	0.842868	+	0.538120i	$0.180865\pi$
$830$	0	0
$831$	−1.03958e10	−0.628425
$832$	4.92636e9	0.296548
$833$	−1.09155e9	−0.0654312
$834$	1.34229e8	0.00801247
$835$	0	0
$836$	1.45379e9	0.0860556
$837$	−1.04202e9	−0.0614237
$838$	3.96731e9	0.232885
$839$	−1.72137e10	−1.00625	−0.503126	−	0.864213i	$-0.667817\pi$
$839$	−1.72137e10	−1.00625	−0.503126	+	0.864213i	$0.667817\pi$
$840$	0	0
$841$	−1.59898e10	−0.926950
$842$	5.32089e9	0.307179
$843$	9.68574e9	0.556847
$844$	6.38245e9	0.365417
$845$	0	0
$846$	1.98463e8	0.0112689
$847$	4.13898e9	0.234046
$848$	1.88865e10	1.06357
$849$	−1.57089e9	−0.0880985
$850$	0	0
$851$	3.49007e9	0.194124
$852$	−3.63593e9	−0.201408
$853$	−6.71439e9	−0.370412	−0.185206	−	0.982700i	$-0.559295\pi$
$853$	−6.71439e9	−0.370412	−0.185206	+	0.982700i	$0.559295\pi$
$854$	−1.46528e8	−0.00805042
$855$	0	0
$856$	6.12260e9	0.333639
$857$	9.89153e9	0.536822	0.268411	−	0.963304i	$-0.413501\pi$
$857$	9.89153e9	0.536822	0.268411	+	0.963304i	$0.413501\pi$
$858$	−4.22754e8	−0.0228498
$859$	−3.35625e10	−1.80667	−0.903335	−	0.428937i	$-0.858888\pi$
$859$	−3.35625e10	−1.80667	−0.903335	+	0.428937i	$0.858888\pi$
$860$	0	0
$861$	−1.67015e9	−0.0891751
$862$	−3.00956e9	−0.160040
$863$	−9.99427e9	−0.529314	−0.264657	−	0.964343i	$-0.585259\pi$
$863$	−9.99427e9	−0.529314	−0.264657	+	0.964343i	$0.585259\pi$
$864$	1.85493e9	0.0978427
$865$	0	0
$866$	−2.75606e9	−0.144203
$867$	−8.75495e9	−0.456233
$868$	−2.25164e9	−0.116864
$869$	−1.47908e10	−0.764581
$870$	0	0
$871$	1.40029e9	0.0718051
$872$	4.88100e9	0.249288
$873$	3.80709e8	0.0193661
$874$	−3.57232e8	−0.0180992
$875$	0	0
$876$	−1.98268e10	−0.996524
$877$	−1.24239e8	−0.00621953	−0.00310977	−	0.999995i	$-0.500990\pi$
$877$	−1.24239e8	−0.00621953	−0.00310977	+	0.999995i	$0.500990\pi$
$878$	4.30032e9	0.214422
$879$	−1.50405e10	−0.746964
$880$	0	0
$881$	2.45438e10	1.20928	0.604640	−	0.796499i	$-0.293317\pi$
$881$	2.45438e10	1.20928	0.604640	+	0.796499i	$0.293317\pi$
$882$	1.71532e8	0.00841794
$883$	−9.35492e9	−0.457275	−0.228638	−	0.973512i	$-0.573427\pi$
$883$	−9.35492e9	−0.457275	−0.228638	+	0.973512i	$0.573427\pi$
$884$	−3.30646e9	−0.160983
$885$	0	0
$886$	2.02188e9	0.0976646
$887$	3.18320e10	1.53155	0.765776	−	0.643107i	$-0.222355\pi$
$887$	3.18320e10	1.53155	0.765776	+	0.643107i	$0.222355\pi$
$888$	−1.14441e9	−0.0548447
$889$	1.04944e9	0.0500959
$890$	0	0
$891$	1.44765e9	0.0685631
$892$	−3.09221e10	−1.45879
$893$	−5.85860e8	−0.0275305
$894$	−2.15943e9	−0.101078
$895$	0	0
$896$	5.31339e9	0.246771
$897$	−3.22032e9	−0.148979
$898$	3.65297e8	0.0168337
$899$	1.87926e9	0.0862639
$900$	0	0
$901$	−1.17888e10	−0.536950
$902$	9.82503e8	0.0445770
$903$	−3.09799e8	−0.0140014
$904$	1.26149e10	0.567929
$905$	0	0
$906$	−2.90476e9	−0.129766
$907$	−2.00342e10	−0.891552	−0.445776	−	0.895144i	$-0.647072\pi$
$907$	−2.00342e10	−0.891552	−0.445776	+	0.895144i	$0.647072\pi$
$908$	4.88470e9	0.216540
$909$	−1.19527e9	−0.0527829
$910$	0	0
$911$	−2.82239e10	−1.23681	−0.618405	−	0.785859i	$-0.712221\pi$
$911$	−2.82239e10	−1.23681	−0.618405	+	0.785859i	$0.712221\pi$
$912$	−1.72732e9	−0.0754032
$913$	−1.88866e10	−0.821308
$914$	3.03222e6	0.000131356 0
$915$	0	0
$916$	−1.56171e10	−0.671377
$917$	1.22421e9	0.0524279
$918$	−3.65238e8	−0.0155821
$919$	−3.40984e10	−1.44921	−0.724603	−	0.689167i	$-0.757977\pi$
$919$	−3.40984e10	−1.44921	−0.724603	+	0.689167i	$0.757977\pi$
$920$	0	0
$921$	−1.40345e10	−0.591956
$922$	−9.59881e8	−0.0403329
$923$	−3.12116e9	−0.130650
$924$	3.12814e9	0.130447
$925$	0	0
$926$	1.37362e9	0.0568499
$927$	4.79322e9	0.197628
$928$	−3.34533e9	−0.137411
$929$	−2.18397e10	−0.893698	−0.446849	−	0.894609i	$-0.647454\pi$
$929$	−2.18397e10	−0.893698	−0.446849	+	0.894609i	$0.647454\pi$
$930$	0	0
$931$	−5.06361e8	−0.0205654
$932$	−1.92560e10	−0.779131
$933$	−1.06216e10	−0.428157
$934$	−3.38007e9	−0.135741
$935$	0	0
$936$	1.05595e9	0.0420900
$937$	1.29359e10	0.513699	0.256849	−	0.966451i	$-0.417316\pi$
$937$	1.29359e10	0.513699	0.256849	+	0.966451i	$0.417316\pi$
$938$	3.34238e8	0.0132235
$939$	−2.01594e10	−0.794598
$940$	0	0
$941$	−1.70996e10	−0.668995	−0.334497	−	0.942397i	$-0.608567\pi$
$941$	−1.70996e10	−0.668995	−0.334497	+	0.942397i	$0.608567\pi$
$942$	−5.20570e8	−0.0202909
$943$	7.48419e9	0.290639
$944$	−2.30875e10	−0.893255
$945$	0	0
$946$	1.82246e8	0.00699907
$947$	1.95022e8	0.00746207	0.00373104	−	0.999993i	$-0.498812\pi$
$947$	1.95022e8	0.00746207	0.00373104	+	0.999993i	$0.498812\pi$
$948$	1.81791e10	0.693014
$949$	−1.70198e10	−0.646430
$950$	0	0
$951$	−5.13594e9	−0.193637
$952$	−1.60391e9	−0.0602490
$953$	1.97958e10	0.740878	0.370439	−	0.928857i	$-0.379207\pi$
$953$	1.97958e10	0.740878	0.370439	+	0.928857i	$0.379207\pi$
$954$	1.85257e9	0.0690804
$955$	0	0
$956$	1.90416e10	0.704856
$957$	−2.61081e9	−0.0962904
$958$	4.79686e9	0.176270
$959$	1.65921e9	0.0607487
$960$	0	0
$961$	−2.47100e10	−0.898132
$962$	−4.83395e8	−0.0175061
$963$	−8.85590e9	−0.319551
$964$	−2.07308e10	−0.745327
$965$	0	0
$966$	−7.68663e8	−0.0274357
$967$	−4.18825e10	−1.48950	−0.744748	−	0.667346i	$-0.767430\pi$
$967$	−4.18825e10	−1.48950	−0.744748	+	0.667346i	$0.767430\pi$
$968$	6.08177e9	0.215509
$969$	1.07818e9	0.0380677
$970$	0	0
$971$	−1.82766e9	−0.0640661	−0.0320331	−	0.999487i	$-0.510198\pi$
$971$	−1.82766e9	−0.0640661	−0.0320331	+	0.999487i	$0.510198\pi$
$972$	−1.77926e9	−0.0621453
$973$	−8.52605e8	−0.0296724
$974$	1.35363e9	0.0469400
$975$	0	0
$976$	3.17492e9	0.109310
$977$	−4.04873e10	−1.38895	−0.694477	−	0.719515i	$-0.744364\pi$
$977$	−4.04873e10	−1.38895	−0.694477	+	0.719515i	$0.744364\pi$
$978$	4.14365e9	0.141644
$979$	7.15361e8	0.0243661
$980$	0	0
$981$	−7.06002e9	−0.238761
$982$	7.94076e9	0.267591
$983$	−1.79913e9	−0.0604124	−0.0302062	−	0.999544i	$-0.509616\pi$
$983$	−1.79913e9	−0.0604124	−0.0302062	+	0.999544i	$0.509616\pi$
$984$	−2.45409e9	−0.0821123
$985$	0	0
$986$	6.58701e8	0.0218836
$987$	−1.26061e9	−0.0417321
$988$	−1.53384e9	−0.0505978
$989$	1.38826e9	0.0456334
$990$	0	0
$991$	−1.80926e10	−0.590532	−0.295266	−	0.955415i	$-0.595408\pi$
$991$	−1.80926e10	−0.590532	−0.295266	+	0.955415i	$0.595408\pi$
$992$	−4.98907e9	−0.162266
$993$	−2.96981e10	−0.962511
$994$	−7.44996e8	−0.0240603
$995$	0	0
$996$	2.32130e10	0.744431
$997$	−4.79210e10	−1.53141	−0.765707	−	0.643189i	$-0.777611\pi$
$997$	−4.79210e10	−1.53141	−0.765707	+	0.643189i	$0.777611\pi$
$998$	3.19514e9	0.101750
$999$	1.65530e9	0.0525288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.8.a.c.1.1		1
5.4	even	2		105.8.a.a.1.1	✓	1
15.14	odd	2		315.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.a.1.1	✓	1	5.4	even	2
315.8.a.b.1.1		1	15.14	odd	2
525.8.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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