LMFDB - Embedded newform 245.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,8,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	245.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:	$76.5343312436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-14.0000 q^{2} +48.0000 q^{3} +68.0000 q^{4} -125.000 q^{5} -672.000 q^{6} +840.000 q^{8} +117.000 q^{9} +O(q^{10})$

\(q-14.0000 q^{2} +48.0000 q^{3} +68.0000 q^{4} -125.000 q^{5} -672.000 q^{6} +840.000 q^{8} +117.000 q^{9} +1750.00 q^{10} +172.000 q^{11} +3264.00 q^{12} -3862.00 q^{13} -6000.00 q^{15} -20464.0 q^{16} +12254.0 q^{17} -1638.00 q^{18} +25940.0 q^{19} -8500.00 q^{20} -2408.00 q^{22} +12972.0 q^{23} +40320.0 q^{24} +15625.0 q^{25} +54068.0 q^{26} -99360.0 q^{27} -81610.0 q^{29} +84000.0 q^{30} +156888. q^{31} +178976. q^{32} +8256.00 q^{33} -171556. q^{34} +7956.00 q^{36} +110126. q^{37} -363160. q^{38} -185376. q^{39} -105000. q^{40} -467882. q^{41} -499208. q^{43} +11696.0 q^{44} -14625.0 q^{45} -181608. q^{46} +396884. q^{47} -982272. q^{48} -218750. q^{50} +588192. q^{51} -262616. q^{52} -1.28050e6 q^{53} +1.39104e6 q^{54} -21500.0 q^{55} +1.24512e6 q^{57} +1.14254e6 q^{58} +1.33742e6 q^{59} -408000. q^{60} +923978. q^{61} -2.19643e6 q^{62} +113728. q^{64} +482750. q^{65} -115584. q^{66} -797304. q^{67} +833272. q^{68} +622656. q^{69} +5.10339e6 q^{71} +98280.0 q^{72} +4.26748e6 q^{73} -1.54176e6 q^{74} +750000. q^{75} +1.76392e6 q^{76} +2.59526e6 q^{78} -960.000 q^{79} +2.55800e6 q^{80} -5.02516e6 q^{81} +6.55035e6 q^{82} -6.14083e6 q^{83} -1.53175e6 q^{85} +6.98891e6 q^{86} -3.91728e6 q^{87} +144480. q^{88} -2.01057e6 q^{89} +204750. q^{90} +882096. q^{92} +7.53062e6 q^{93} -5.55638e6 q^{94} -3.24250e6 q^{95} +8.59085e6 q^{96} +4.88193e6 q^{97} +20124.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−14.0000	−1.23744	−0.618718	−	0.785613i	$-0.712348\pi$
$2$	−14.0000	−1.23744	−0.618718	+	0.785613i	$0.712348\pi$
$3$	48.0000	1.02640	0.513200	−	0.858269i	$-0.328460\pi$
$3$	48.0000	1.02640	0.513200	+	0.858269i	$0.328460\pi$
$4$	68.0000	0.531250
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	−672.000	−1.27011
$7$	0	0
$7$	0	0
$8$	840.000	0.580049
$9$	117.000	0.0534979
$10$	1750.00	0.553399
$11$	172.000	0.0389631	0.0194816	−	0.999810i	$-0.493798\pi$
$11$	172.000	0.0389631	0.0194816	+	0.999810i	$0.493798\pi$
$12$	3264.00	0.545275
$13$	−3862.00	−0.487540	−0.243770	−	0.969833i	$-0.578384\pi$
$13$	−3862.00	−0.487540	−0.243770	+	0.969833i	$0.578384\pi$
$14$	0	0
$15$	−6000.00	−0.459020
$16$	−20464.0	−1.24902
$17$	12254.0	0.604932	0.302466	−	0.953160i	$-0.402190\pi$
$17$	12254.0	0.604932	0.302466	+	0.953160i	$0.402190\pi$
$18$	−1638.00	−0.0662003
$19$	25940.0	0.867626	0.433813	−	0.901003i	$-0.357168\pi$
$19$	25940.0	0.867626	0.433813	+	0.901003i	$0.357168\pi$
$20$	−8500.00	−0.237582
$21$	0	0
$22$	−2408.00	−0.0482144
$23$	12972.0	0.222310	0.111155	−	0.993803i	$-0.464545\pi$
$23$	12972.0	0.222310	0.111155	+	0.993803i	$0.464545\pi$
$24$	40320.0	0.595362
$25$	15625.0	0.200000
$26$	54068.0	0.603300
$27$	−99360.0	−0.971490
$28$	0	0
$29$	−81610.0	−0.621370	−0.310685	−	0.950513i	$-0.600558\pi$
$29$	−81610.0	−0.621370	−0.310685	+	0.950513i	$0.600558\pi$
$30$	84000.0	0.568009
$31$	156888.	0.945853	0.472927	−	0.881102i	$-0.343198\pi$
$31$	156888.	0.945853	0.472927	+	0.881102i	$0.343198\pi$
$32$	178976.	0.965539
$33$	8256.00	0.0399918
$34$	−171556.	−0.748565
$35$	0	0
$36$	7956.00	0.0284208
$37$	110126.	0.357424	0.178712	−	0.983901i	$-0.442807\pi$
$37$	110126.	0.357424	0.178712	+	0.983901i	$0.442807\pi$
$38$	−363160.	−1.07363
$39$	−185376.	−0.500412
$40$	−105000.	−0.259406
$41$	−467882.	−1.06021	−0.530106	−	0.847931i	$-0.677848\pi$
$41$	−467882.	−1.06021	−0.530106	+	0.847931i	$0.677848\pi$
$42$	0	0
$43$	−499208.	−0.957507	−0.478753	−	0.877949i	$-0.658911\pi$
$43$	−499208.	−0.957507	−0.478753	+	0.877949i	$0.658911\pi$
$44$	11696.0	0.0206992
$45$	−14625.0	−0.0239250
$46$	−181608.	−0.275095
$47$	396884.	0.557598	0.278799	−	0.960349i	$-0.410064\pi$
$47$	396884.	0.557598	0.278799	+	0.960349i	$0.410064\pi$
$48$	−982272.	−1.28200
$49$	0	0
$50$	−218750.	−0.247487
$51$	588192.	0.620903
$52$	−262616.	−0.259006
$53$	−1.28050e6	−1.18144	−0.590722	−	0.806875i	$-0.701157\pi$
$53$	−1.28050e6	−1.18144	−0.590722	+	0.806875i	$0.701157\pi$
$54$	1.39104e6	1.20216
$55$	−21500.0	−0.0174248
$56$	0	0
$57$	1.24512e6	0.890531
$58$	1.14254e6	0.768906
$59$	1.33742e6	0.847785	0.423893	−	0.905712i	$-0.360663\pi$
$59$	1.33742e6	0.847785	0.423893	+	0.905712i	$0.360663\pi$
$60$	−408000.	−0.243855
$61$	923978.	0.521203	0.260602	−	0.965446i	$-0.416079\pi$
$61$	923978.	0.521203	0.260602	+	0.965446i	$0.416079\pi$
$62$	−2.19643e6	−1.17043
$63$	0	0
$64$	113728.	0.0542297
$65$	482750.	0.218035
$66$	−115584.	−0.0494873
$67$	−797304.	−0.323864	−0.161932	−	0.986802i	$-0.551772\pi$
$67$	−797304.	−0.323864	−0.161932	+	0.986802i	$0.551772\pi$
$68$	833272.	0.321370
$69$	622656.	0.228179
$70$	0	0
$71$	5.10339e6	1.69221	0.846106	−	0.533015i	$-0.178941\pi$
$71$	5.10339e6	1.69221	0.846106	+	0.533015i	$0.178941\pi$
$72$	98280.0	0.0310314
$73$	4.26748e6	1.28393	0.641965	−	0.766734i	$-0.278119\pi$
$73$	4.26748e6	1.28393	0.641965	+	0.766734i	$0.278119\pi$
$74$	−1.54176e6	−0.442290
$75$	750000.	0.205280
$76$	1.76392e6	0.460926
$77$	0	0
$78$	2.59526e6	0.619228
$79$	−960.000	−0.000219067 0	−0.000109533	−	1.00000i	$-0.500035\pi$
$79$	−960.000	−0.000219067 0	−0.000109533		1.00000i	$0.500035\pi$
$80$	2.55800e6	0.558580
$81$	−5.02516e6	−1.05064
$82$	6.55035e6	1.31195
$83$	−6.14083e6	−1.17884	−0.589419	−	0.807828i	$-0.700643\pi$
$83$	−6.14083e6	−1.17884	−0.589419	+	0.807828i	$0.700643\pi$
$84$	0	0
$85$	−1.53175e6	−0.270534
$86$	6.98891e6	1.18485
$87$	−3.91728e6	−0.637775
$88$	144480.	0.0226005
$89$	−2.01057e6	−0.302311	−0.151156	−	0.988510i	$-0.548299\pi$
$89$	−2.01057e6	−0.302311	−0.151156	+	0.988510i	$0.548299\pi$
$90$	204750.	0.0296057
$91$	0	0
$92$	882096.	0.118102
$93$	7.53062e6	0.970824
$94$	−5.55638e6	−0.689992
$95$	−3.24250e6	−0.388014
$96$	8.59085e6	0.991030
$97$	4.88193e6	0.543114	0.271557	−	0.962422i	$-0.412461\pi$
$97$	4.88193e6	0.543114	0.271557	+	0.962422i	$0.412461\pi$
$98$	0	0
$99$	20124.0	0.00208445
$100$	1.06250e6	0.106250
$101$	−9.72670e6	−0.939379	−0.469689	−	0.882832i	$-0.655634\pi$
$101$	−9.72670e6	−0.939379	−0.469689	+	0.882832i	$0.655634\pi$
$102$	−8.23469e6	−0.768328
$103$	−1.63151e7	−1.47115	−0.735577	−	0.677441i	$-0.763089\pi$
$103$	−1.63151e7	−1.47115	−0.735577	+	0.677441i	$0.763089\pi$
$104$	−3.24408e6	−0.282797
$105$	0	0
$106$	1.79270e7	1.46196
$107$	−4.08974e6	−0.322740	−0.161370	−	0.986894i	$-0.551591\pi$
$107$	−4.08974e6	−0.322740	−0.161370	+	0.986894i	$0.551591\pi$
$108$	−6.75648e6	−0.516104
$109$	−2.68318e7	−1.98453	−0.992263	−	0.124158i	$-0.960377\pi$
$109$	−2.68318e7	−1.98453	−0.992263	+	0.124158i	$0.960377\pi$
$110$	301000.	0.0215622
$111$	5.28605e6	0.366860
$112$	0	0
$113$	−1.74810e7	−1.13971	−0.569853	−	0.821747i	$-0.693000\pi$
$113$	−1.74810e7	−1.13971	−0.569853	+	0.821747i	$0.693000\pi$
$114$	−1.74317e7	−1.10198
$115$	−1.62150e6	−0.0994202
$116$	−5.54948e6	−0.330103
$117$	−451854.	−0.0260824
$118$	−1.87239e7	−1.04908
$119$	0	0
$120$	−5.04000e6	−0.266254
$121$	−1.94576e7	−0.998482
$122$	−1.29357e7	−0.644956
$123$	−2.24583e7	−1.08820
$124$	1.06684e7	0.502485
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−1.25018e7	−0.541575	−0.270787	−	0.962639i	$-0.587284\pi$
$127$	−1.25018e7	−0.541575	−0.270787	+	0.962639i	$0.587284\pi$
$128$	−2.45011e7	−1.03264
$129$	−2.39620e7	−0.982786
$130$	−6.75850e6	−0.269804
$131$	7.75619e6	0.301439	0.150719	−	0.988577i	$-0.451841\pi$
$131$	7.75619e6	0.301439	0.150719	+	0.988577i	$0.451841\pi$
$132$	561408.	0.0212456
$133$	0	0
$134$	1.11623e7	0.400761
$135$	1.24200e7	0.434464
$136$	1.02934e7	0.350890
$137$	3.61720e7	1.20185	0.600926	−	0.799305i	$-0.294799\pi$
$137$	3.61720e7	1.20185	0.600926	+	0.799305i	$0.294799\pi$
$138$	−8.71718e6	−0.282358
$139$	−1.09092e7	−0.344542	−0.172271	−	0.985050i	$-0.555110\pi$
$139$	−1.09092e7	−0.344542	−0.172271	+	0.985050i	$0.555110\pi$
$140$	0	0
$141$	1.90504e7	0.572319
$142$	−7.14475e7	−2.09401
$143$	−664264.	−0.0189961
$144$	−2.39429e6	−0.0668202
$145$	1.02012e7	0.277885
$146$	−5.97447e7	−1.58878
$147$	0	0
$148$	7.48857e6	0.189882
$149$	−3.64580e7	−0.902904	−0.451452	−	0.892295i	$-0.649094\pi$
$149$	−3.64580e7	−0.902904	−0.451452	+	0.892295i	$0.649094\pi$
$150$	−1.05000e7	−0.254021
$151$	7.18955e6	0.169935	0.0849674	−	0.996384i	$-0.472921\pi$
$151$	7.18955e6	0.169935	0.0849674	+	0.996384i	$0.472921\pi$
$152$	2.17896e7	0.503265
$153$	1.43372e6	0.0323626
$154$	0	0
$155$	−1.96110e7	−0.422998
$156$	−1.26056e7	−0.265844
$157$	−8.79932e7	−1.81468	−0.907341	−	0.420396i	$-0.861891\pi$
$157$	−8.79932e7	−1.81468	−0.907341	+	0.420396i	$0.861891\pi$
$158$	13440.0	0.000271081 0
$159$	−6.14639e7	−1.21264
$160$	−2.23720e7	−0.431802
$161$	0	0
$162$	7.03522e7	1.30010
$163$	−5.48875e7	−0.992697	−0.496349	−	0.868123i	$-0.665326\pi$
$163$	−5.48875e7	−0.992697	−0.496349	+	0.868123i	$0.665326\pi$
$164$	−3.18160e7	−0.563238
$165$	−1.03200e6	−0.0178849
$166$	8.59716e7	1.45874
$167$	−8.61460e6	−0.143129	−0.0715644	−	0.997436i	$-0.522799\pi$
$167$	−8.61460e6	−0.143129	−0.0715644	+	0.997436i	$0.522799\pi$
$168$	0	0
$169$	−4.78335e7	−0.762304
$170$	2.14445e7	0.334769
$171$	3.03498e6	0.0464162
$172$	−3.39461e7	−0.508676
$173$	5.12524e7	0.752580	0.376290	−	0.926502i	$-0.377200\pi$
$173$	5.12524e7	0.752580	0.376290	+	0.926502i	$0.377200\pi$
$174$	5.48419e7	0.789206
$175$	0	0
$176$	−3.51981e6	−0.0486659
$177$	6.41962e7	0.870167
$178$	2.81480e7	0.374091
$179$	−5.01627e7	−0.653725	−0.326862	−	0.945072i	$-0.605991\pi$
$179$	−5.01627e7	−0.653725	−0.326862	+	0.945072i	$0.605991\pi$
$180$	−994500.	−0.0127102
$181$	−6.90817e7	−0.865940	−0.432970	−	0.901408i	$-0.642534\pi$
$181$	−6.90817e7	−0.865940	−0.432970	+	0.901408i	$0.642534\pi$
$182$	0	0
$183$	4.43509e7	0.534963
$184$	1.08965e7	0.128951
$185$	−1.37658e7	−0.159845
$186$	−1.05429e8	−1.20133
$187$	2.10769e6	0.0235701
$188$	2.69881e7	0.296224
$189$	0	0
$190$	4.53950e7	0.480143
$191$	1.54745e8	1.60695	0.803473	−	0.595342i	$-0.202983\pi$
$191$	1.54745e8	1.60695	0.803473	+	0.595342i	$0.202983\pi$
$192$	5.45894e6	0.0556614
$193$	−1.59406e7	−0.159607	−0.0798037	−	0.996811i	$-0.525429\pi$
$193$	−1.59406e7	−0.159607	−0.0798037	+	0.996811i	$0.525429\pi$
$194$	−6.83471e7	−0.672069
$195$	2.31720e7	0.223791
$196$	0	0
$197$	−1.68188e8	−1.56734	−0.783670	−	0.621177i	$-0.786655\pi$
$197$	−1.68188e8	−1.56734	−0.783670	+	0.621177i	$0.786655\pi$
$198$	−281736.	−0.00257937
$199$	−1.77773e8	−1.59911	−0.799556	−	0.600591i	$-0.794932\pi$
$199$	−1.77773e8	−1.59911	−0.799556	+	0.600591i	$0.794932\pi$
$200$	1.31250e7	0.116010
$201$	−3.82706e7	−0.332414
$202$	1.36174e8	1.16242
$203$	0	0
$204$	3.99971e7	0.329855
$205$	5.84852e7	0.474141
$206$	2.28411e8	1.82046
$207$	1.51772e6	0.0118931
$208$	7.90320e7	0.608949
$209$	4.46168e6	0.0338054
$210$	0	0
$211$	−1.61996e8	−1.18718	−0.593590	−	0.804767i	$-0.702290\pi$
$211$	−1.61996e8	−1.18718	−0.593590	+	0.804767i	$0.702290\pi$
$212$	−8.70739e7	−0.627642
$213$	2.44963e8	1.73689
$214$	5.72564e7	0.399370
$215$	6.24010e7	0.428210
$216$	−8.34624e7	−0.563511
$217$	0	0
$218$	3.75645e8	2.45572
$219$	2.04839e8	1.31783
$220$	−1.46200e6	−0.00925695
$221$	−4.73249e7	−0.294929
$222$	−7.40047e7	−0.453966
$223$	−1.75932e7	−0.106237	−0.0531187	−	0.998588i	$-0.516916\pi$
$223$	−1.75932e7	−0.106237	−0.0531187	+	0.998588i	$0.516916\pi$
$224$	0	0
$225$	1.82812e6	0.0106996
$226$	2.44735e8	1.41031
$227$	2.03036e8	1.15208	0.576039	−	0.817422i	$-0.304597\pi$
$227$	2.03036e8	1.15208	0.576039	+	0.817422i	$0.304597\pi$
$228$	8.46682e7	0.473095
$229$	−1.59559e8	−0.878005	−0.439003	−	0.898486i	$-0.644668\pi$
$229$	−1.59559e8	−0.878005	−0.439003	+	0.898486i	$0.644668\pi$
$230$	2.27010e7	0.123026
$231$	0	0
$232$	−6.85524e7	−0.360425
$233$	−1.94985e7	−0.100985	−0.0504924	−	0.998724i	$-0.516079\pi$
$233$	−1.94985e7	−0.100985	−0.0504924	+	0.998724i	$0.516079\pi$
$234$	6.32596e6	0.0322753
$235$	−4.96105e7	−0.249365
$236$	9.09446e7	0.450386
$237$	−46080.0	−0.000224850 0
$238$	0	0
$239$	−1.60220e8	−0.759146	−0.379573	−	0.925162i	$-0.623929\pi$
$239$	−1.60220e8	−0.759146	−0.379573	+	0.925162i	$0.623929\pi$
$240$	1.22784e8	0.573327
$241$	3.78779e8	1.74311	0.871557	−	0.490294i	$-0.163110\pi$
$241$	3.78779e8	1.74311	0.871557	+	0.490294i	$0.163110\pi$
$242$	2.72406e8	1.23556
$243$	−2.39073e7	−0.106883
$244$	6.28305e7	0.276889
$245$	0	0
$246$	3.14417e8	1.34658
$247$	−1.00180e8	−0.423002
$248$	1.31786e8	0.548641
$249$	−2.94760e8	−1.20996
$250$	2.73438e7	0.110680
$251$	−1.61304e8	−0.643855	−0.321927	−	0.946764i	$-0.604331\pi$
$251$	−1.61304e8	−0.643855	−0.321927	+	0.946764i	$0.604331\pi$
$252$	0	0
$253$	2.23118e6	0.00866191
$254$	1.75025e8	0.670164
$255$	−7.35240e7	−0.277676
$256$	3.28458e8	1.22360
$257$	−2.27387e8	−0.835603	−0.417801	−	0.908538i	$-0.637199\pi$
$257$	−2.27387e8	−0.835603	−0.417801	+	0.908538i	$0.637199\pi$
$258$	3.35468e8	1.21614
$259$	0	0
$260$	3.28270e7	0.115831
$261$	−9.54837e6	−0.0332420
$262$	−1.08587e8	−0.373011
$263$	4.57728e8	1.55154	0.775768	−	0.631018i	$-0.217363\pi$
$263$	4.57728e8	1.55154	0.775768	+	0.631018i	$0.217363\pi$
$264$	6.93504e6	0.0231972
$265$	1.60062e8	0.528358
$266$	0	0
$267$	−9.65074e7	−0.310292
$268$	−5.42167e7	−0.172053
$269$	−4.67286e8	−1.46369	−0.731847	−	0.681469i	$-0.761341\pi$
$269$	−4.67286e8	−1.46369	−0.731847	+	0.681469i	$0.761341\pi$
$270$	−1.73880e8	−0.537621
$271$	4.45932e7	0.136106	0.0680528	−	0.997682i	$-0.478321\pi$
$271$	4.45932e7	0.136106	0.0680528	+	0.997682i	$0.478321\pi$
$272$	−2.50766e8	−0.755574
$273$	0	0
$274$	−5.06408e8	−1.48722
$275$	2.68750e6	0.00779263
$276$	4.23406e7	0.121220
$277$	3.16657e8	0.895179	0.447590	−	0.894239i	$-0.352283\pi$
$277$	3.16657e8	0.895179	0.447590	+	0.894239i	$0.352283\pi$
$278$	1.52729e8	0.426349
$279$	1.83559e7	0.0506012
$280$	0	0
$281$	−2.25818e8	−0.607136	−0.303568	−	0.952810i	$-0.598178\pi$
$281$	−2.25818e8	−0.607136	−0.303568	+	0.952810i	$0.598178\pi$
$282$	−2.66706e8	−0.708208
$283$	−2.08210e7	−0.0546072	−0.0273036	−	0.999627i	$-0.508692\pi$
$283$	−2.08210e7	−0.0546072	−0.0273036	+	0.999627i	$0.508692\pi$
$284$	3.47031e8	0.898987
$285$	−1.55640e8	−0.398258
$286$	9.29970e6	0.0235065
$287$	0	0
$288$	2.09402e7	0.0516544
$289$	−2.60178e8	−0.634057
$290$	−1.42818e8	−0.343865
$291$	2.34333e8	0.557452
$292$	2.90189e8	0.682088
$293$	1.78825e8	0.415329	0.207665	−	0.978200i	$-0.433414\pi$
$293$	1.78825e8	0.415329	0.207665	+	0.978200i	$0.433414\pi$
$294$	0	0
$295$	−1.67178e8	−0.379141
$296$	9.25058e7	0.207323
$297$	−1.70899e7	−0.0378523
$298$	5.10413e8	1.11729
$299$	−5.00979e7	−0.108385
$300$	5.10000e7	0.109055
$301$	0	0
$302$	−1.00654e8	−0.210284
$303$	−4.66882e8	−0.964179
$304$	−5.30836e8	−1.08368
$305$	−1.15497e8	−0.233089
$306$	−2.00721e7	−0.0400467
$307$	8.55159e7	0.168680	0.0843398	−	0.996437i	$-0.473122\pi$
$307$	8.55159e7	0.168680	0.0843398	+	0.996437i	$0.473122\pi$
$308$	0	0
$309$	−7.83122e8	−1.50999
$310$	2.74554e8	0.523434
$311$	4.84706e8	0.913728	0.456864	−	0.889537i	$-0.348973\pi$
$311$	4.84706e8	0.913728	0.456864	+	0.889537i	$0.348973\pi$
$312$	−1.55716e8	−0.290263
$313$	−5.70821e8	−1.05219	−0.526096	−	0.850425i	$-0.676345\pi$
$313$	−5.70821e8	−1.05219	−0.526096	+	0.850425i	$0.676345\pi$
$314$	1.23190e9	2.24555
$315$	0	0
$316$	−65280.0	−0.000116379 0
$317$	−5.50191e8	−0.970076	−0.485038	−	0.874493i	$-0.661194\pi$
$317$	−5.50191e8	−0.970076	−0.485038	+	0.874493i	$0.661194\pi$
$318$	8.60495e8	1.50056
$319$	−1.40369e7	−0.0242105
$320$	−1.42160e7	−0.0242523
$321$	−1.96308e8	−0.331261
$322$	0	0
$323$	3.17869e8	0.524855
$324$	−3.41711e8	−0.558150
$325$	−6.03438e7	−0.0975081
$326$	7.68425e8	1.22840
$327$	−1.28792e9	−2.03692
$328$	−3.93021e8	−0.614975
$329$	0	0
$330$	1.44480e7	0.0221314
$331$	−9.39839e8	−1.42448	−0.712238	−	0.701938i	$-0.752319\pi$
$331$	−9.39839e8	−1.42448	−0.712238	+	0.701938i	$0.752319\pi$
$332$	−4.17577e8	−0.626257
$333$	1.28847e7	0.0191215
$334$	1.20604e8	0.177113
$335$	9.96630e7	0.144836
$336$	0	0
$337$	5.33632e8	0.759516	0.379758	−	0.925086i	$-0.376007\pi$
$337$	5.33632e8	0.759516	0.379758	+	0.925086i	$0.376007\pi$
$338$	6.69669e8	0.943304
$339$	−8.39090e8	−1.16979
$340$	−1.04159e8	−0.143721
$341$	2.69847e7	0.0368534
$342$	−4.24897e7	−0.0574371
$343$	0	0
$344$	−4.19335e8	−0.555401
$345$	−7.78320e7	−0.102045
$346$	−7.17533e8	−0.931270
$347$	1.07934e9	1.38677	0.693385	−	0.720567i	$-0.256118\pi$
$347$	1.07934e9	1.38677	0.693385	+	0.720567i	$0.256118\pi$
$348$	−2.66375e8	−0.338818
$349$	4.27217e8	0.537972	0.268986	−	0.963144i	$-0.413311\pi$
$349$	4.27217e8	0.537972	0.268986	+	0.963144i	$0.413311\pi$
$350$	0	0
$351$	3.83728e8	0.473641
$352$	3.07839e7	0.0376204
$353$	1.48966e9	1.80250	0.901250	−	0.433299i	$-0.142650\pi$
$353$	1.48966e9	1.80250	0.901250	+	0.433299i	$0.142650\pi$
$354$	−8.98746e8	−1.07678
$355$	−6.37924e8	−0.756780
$356$	−1.36719e8	−0.160603
$357$	0	0
$358$	7.02277e8	0.808943
$359$	8.41275e8	0.959638	0.479819	−	0.877367i	$-0.340702\pi$
$359$	8.41275e8	0.959638	0.479819	+	0.877367i	$0.340702\pi$
$360$	−1.22850e7	−0.0138777
$361$	−2.20988e8	−0.247226
$362$	9.67143e8	1.07155
$363$	−9.33964e8	−1.02484
$364$	0	0
$365$	−5.33435e8	−0.574191
$366$	−6.20913e8	−0.661983
$367$	7.50462e8	0.792496	0.396248	−	0.918143i	$-0.370312\pi$
$367$	7.50462e8	0.792496	0.396248	+	0.918143i	$0.370312\pi$
$368$	−2.65459e8	−0.277671
$369$	−5.47422e7	−0.0567192
$370$	1.92720e8	0.197798
$371$	0	0
$372$	5.12082e8	0.515750
$373$	1.71074e8	0.170688	0.0853439	−	0.996352i	$-0.472801\pi$
$373$	1.71074e8	0.170688	0.0853439	+	0.996352i	$0.472801\pi$
$374$	−2.95076e7	−0.0291665
$375$	−9.37500e7	−0.0918040
$376$	3.33383e8	0.323434
$377$	3.15178e8	0.302943
$378$	0	0
$379$	4.66239e7	0.0439918	0.0219959	−	0.999758i	$-0.492998\pi$
$379$	4.66239e7	0.0439918	0.0219959	+	0.999758i	$0.492998\pi$
$380$	−2.20490e8	−0.206132
$381$	−6.00085e8	−0.555872
$382$	−2.16644e9	−1.98849
$383$	4.42266e8	0.402242	0.201121	−	0.979566i	$-0.435542\pi$
$383$	4.42266e8	0.402242	0.201121	+	0.979566i	$0.435542\pi$
$384$	−1.17605e9	−1.05991
$385$	0	0
$386$	2.23168e8	0.197504
$387$	−5.84073e7	−0.0512247
$388$	3.31972e8	0.288529
$389$	−4.64033e8	−0.399691	−0.199846	−	0.979827i	$-0.564044\pi$
$389$	−4.64033e8	−0.399691	−0.199846	+	0.979827i	$0.564044\pi$
$390$	−3.24408e8	−0.276927
$391$	1.58959e8	0.134483
$392$	0	0
$393$	3.72297e8	0.309397
$394$	2.35463e9	1.93948
$395$	120000.	9.79696e−5 0
$396$	1.36843e6	0.00110736
$397$	3.17792e8	0.254904	0.127452	−	0.991845i	$-0.459320\pi$
$397$	3.17792e8	0.254904	0.127452	+	0.991845i	$0.459320\pi$
$398$	2.48882e9	1.97880
$399$	0	0
$400$	−3.19750e8	−0.249805
$401$	−1.19563e9	−0.925958	−0.462979	−	0.886369i	$-0.653219\pi$
$401$	−1.19563e9	−0.925958	−0.462979	+	0.886369i	$0.653219\pi$
$402$	5.35788e8	0.411341
$403$	−6.05901e8	−0.461142
$404$	−6.61416e8	−0.499045
$405$	6.28145e8	0.469859
$406$	0	0
$407$	1.89417e7	0.0139264
$408$	4.94081e8	0.360154
$409$	−2.21305e9	−1.59941	−0.799704	−	0.600395i	$-0.795010\pi$
$409$	−2.21305e9	−1.59941	−0.799704	+	0.600395i	$0.795010\pi$
$410$	−8.18794e8	−0.586720
$411$	1.73626e9	1.23358
$412$	−1.10942e9	−0.781551
$413$	0	0
$414$	−2.12481e7	−0.0147170
$415$	7.67604e8	0.527192
$416$	−6.91205e8	−0.470739
$417$	−5.23643e8	−0.353638
$418$	−6.24635e7	−0.0418321
$419$	8.02299e8	0.532828	0.266414	−	0.963859i	$-0.414161\pi$
$419$	8.02299e8	0.532828	0.266414	+	0.963859i	$0.414161\pi$
$420$	0	0
$421$	3.44713e7	0.0225149	0.0112575	−	0.999937i	$-0.496417\pi$
$421$	3.44713e7	0.0225149	0.0112575	+	0.999937i	$0.496417\pi$
$422$	2.26795e9	1.46906
$423$	4.64354e7	0.0298303
$424$	−1.07562e9	−0.685295
$425$	1.91469e8	0.120986
$426$	−3.42948e9	−2.14929
$427$	0	0
$428$	−2.78103e8	−0.171456
$429$	−3.18847e7	−0.0194976
$430$	−8.73614e8	−0.529883
$431$	1.72692e9	1.03897	0.519485	−	0.854480i	$-0.326124\pi$
$431$	1.72692e9	1.03897	0.519485	+	0.854480i	$0.326124\pi$
$432$	2.03330e9	1.21341
$433$	−4.88308e8	−0.289059	−0.144529	−	0.989501i	$-0.546167\pi$
$433$	−4.88308e8	−0.289059	−0.144529	+	0.989501i	$0.546167\pi$
$434$	0	0
$435$	4.89660e8	0.285221
$436$	−1.82456e9	−1.05428
$437$	3.36494e8	0.192882
$438$	−2.86775e9	−1.63073
$439$	2.88640e9	1.62828	0.814142	−	0.580665i	$-0.197207\pi$
$439$	2.88640e9	1.62828	0.814142	+	0.580665i	$0.197207\pi$
$440$	−1.80600e7	−0.0101073
$441$	0	0
$442$	6.62549e8	0.364956
$443$	9.26583e8	0.506374	0.253187	−	0.967417i	$-0.418521\pi$
$443$	9.26583e8	0.506374	0.253187	+	0.967417i	$0.418521\pi$
$444$	3.59451e8	0.194895
$445$	2.51321e8	0.135198
$446$	2.46304e8	0.131462
$447$	−1.74999e9	−0.926741
$448$	0	0
$449$	1.35535e9	0.706627	0.353313	−	0.935505i	$-0.385055\pi$
$449$	1.35535e9	0.706627	0.353313	+	0.935505i	$0.385055\pi$
$450$	−2.55938e7	−0.0132401
$451$	−8.04757e7	−0.0413092
$452$	−1.18871e9	−0.605469
$453$	3.45098e8	0.174421
$454$	−2.84250e9	−1.42563
$455$	0	0
$456$	1.04590e9	0.516551
$457$	4.63429e7	0.0227131	0.0113566	−	0.999936i	$-0.496385\pi$
$457$	4.63429e7	0.0227131	0.0113566	+	0.999936i	$0.496385\pi$
$458$	2.23383e9	1.08648
$459$	−1.21756e9	−0.587686
$460$	−1.10262e8	−0.0528170
$461$	1.52117e8	0.0723144	0.0361572	−	0.999346i	$-0.488488\pi$
$461$	1.52117e8	0.0723144	0.0361572	+	0.999346i	$0.488488\pi$
$462$	0	0
$463$	1.63450e9	0.765337	0.382668	−	0.923886i	$-0.375005\pi$
$463$	1.63450e9	0.765337	0.382668	+	0.923886i	$0.375005\pi$
$464$	1.67007e9	0.776106
$465$	−9.41328e8	−0.434166
$466$	2.72979e8	0.124962
$467$	1.11380e9	0.506057	0.253029	−	0.967459i	$-0.418573\pi$
$467$	1.11380e9	0.506057	0.253029	+	0.967459i	$0.418573\pi$
$468$	−3.07261e7	−0.0138563
$469$	0	0
$470$	6.94547e8	0.308574
$471$	−4.22367e9	−1.86259
$472$	1.12343e9	0.491756
$473$	−8.58638e7	−0.0373075
$474$	645120.	0.000278238 0
$475$	4.05312e8	0.173525
$476$	0	0
$477$	−1.49818e8	−0.0632049
$478$	2.24309e9	0.939395
$479$	−1.27745e9	−0.531091	−0.265546	−	0.964098i	$-0.585552\pi$
$479$	−1.27745e9	−0.531091	−0.265546	+	0.964098i	$0.585552\pi$
$480$	−1.07386e9	−0.443202
$481$	−4.25307e8	−0.174259
$482$	−5.30290e9	−2.15699
$483$	0	0
$484$	−1.32312e9	−0.530443
$485$	−6.10242e8	−0.242888
$486$	3.34702e8	0.132261
$487$	9.79673e8	0.384352	0.192176	−	0.981360i	$-0.438445\pi$
$487$	9.79673e8	0.384352	0.192176	+	0.981360i	$0.438445\pi$
$488$	7.76142e8	0.302323
$489$	−2.63460e9	−1.01890
$490$	0	0
$491$	−4.92125e9	−1.87625	−0.938124	−	0.346298i	$-0.887438\pi$
$491$	−4.92125e9	−1.87625	−0.938124	+	0.346298i	$0.887438\pi$
$492$	−1.52717e9	−0.578108
$493$	−1.00005e9	−0.375887
$494$	1.40252e9	0.523439
$495$	−2.51550e6	−0.000932194 0
$496$	−3.21056e9	−1.18139
$497$	0	0
$498$	4.12664e9	1.49725
$499$	−3.65786e9	−1.31788	−0.658940	−	0.752196i	$-0.728995\pi$
$499$	−3.65786e9	−1.31788	−0.658940	+	0.752196i	$0.728995\pi$
$500$	−1.32812e8	−0.0475164
$501$	−4.13501e8	−0.146908
$502$	2.25826e9	0.796729
$503$	3.88358e9	1.36064	0.680322	−	0.732914i	$-0.261840\pi$
$503$	3.88358e9	1.36064	0.680322	+	0.732914i	$0.261840\pi$
$504$	0	0
$505$	1.21584e9	0.420103
$506$	−3.12366e7	−0.0107186
$507$	−2.29601e9	−0.782430
$508$	−8.50120e8	−0.287711
$509$	−3.90072e9	−1.31109	−0.655545	−	0.755156i	$-0.727561\pi$
$509$	−3.90072e9	−1.31109	−0.655545	+	0.755156i	$0.727561\pi$
$510$	1.02934e9	0.343607
$511$	0	0
$512$	−1.46228e9	−0.481487
$513$	−2.57740e9	−0.842890
$514$	3.18342e9	1.03401
$515$	2.03938e9	0.657920
$516$	−1.62941e9	−0.522105
$517$	6.82640e7	0.0217258
$518$	0	0
$519$	2.46011e9	0.772448
$520$	4.05510e8	0.126471
$521$	−2.88399e9	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$521$	−2.88399e9	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$522$	1.33677e8	0.0411349
$523$	−8.77188e8	−0.268125	−0.134062	−	0.990973i	$-0.542802\pi$
$523$	−8.77188e8	−0.268125	−0.134062	+	0.990973i	$0.542802\pi$
$524$	5.27421e8	0.160139
$525$	0	0
$526$	−6.40819e9	−1.91993
$527$	1.92251e9	0.572177
$528$	−1.68951e8	−0.0499507
$529$	−3.23655e9	−0.950578
$530$	−2.24087e9	−0.653810
$531$	1.56478e8	0.0453548
$532$	0	0
$533$	1.80696e9	0.516896
$534$	1.35110e9	0.383967
$535$	5.11218e8	0.144334
$536$	−6.69735e8	−0.187857
$537$	−2.40781e9	−0.670983
$538$	6.54201e9	1.81123
$539$	0	0
$540$	8.44560e8	0.230809
$541$	−6.53485e8	−0.177437	−0.0887187	−	0.996057i	$-0.528277\pi$
$541$	−6.53485e8	−0.177437	−0.0887187	+	0.996057i	$0.528277\pi$
$542$	−6.24305e8	−0.168422
$543$	−3.31592e9	−0.888801
$544$	2.19317e9	0.584086
$545$	3.35397e9	0.887507
$546$	0	0
$547$	−4.59299e9	−1.19988	−0.599942	−	0.800043i	$-0.704810\pi$
$547$	−4.59299e9	−1.19988	−0.599942	+	0.800043i	$0.704810\pi$
$548$	2.45970e9	0.638484
$549$	1.08105e8	0.0278833
$550$	−3.76250e7	−0.00964289
$551$	−2.11696e9	−0.539117
$552$	5.23031e8	0.132355
$553$	0	0
$554$	−4.43320e9	−1.10773
$555$	−6.60756e8	−0.164065
$556$	−7.41827e8	−0.183038
$557$	6.83164e9	1.67507	0.837533	−	0.546387i	$-0.183997\pi$
$557$	6.83164e9	1.67507	0.837533	+	0.546387i	$0.183997\pi$
$558$	−2.56983e8	−0.0626158
$559$	1.92794e9	0.466823
$560$	0	0
$561$	1.01169e8	0.0241923
$562$	3.16145e9	0.751292
$563$	−3.42509e9	−0.808897	−0.404449	−	0.914561i	$-0.632537\pi$
$563$	−3.42509e9	−0.808897	−0.404449	+	0.914561i	$0.632537\pi$
$564$	1.29543e9	0.304044
$565$	2.18513e9	0.509692
$566$	2.91494e8	0.0675729
$567$	0	0
$568$	4.28685e9	0.981565
$569$	−7.50930e9	−1.70886	−0.854430	−	0.519566i	$-0.826094\pi$
$569$	−7.50930e9	−1.70886	−0.854430	+	0.519566i	$0.826094\pi$
$570$	2.17896e9	0.492819
$571$	−1.35841e8	−0.0305355	−0.0152677	−	0.999883i	$-0.504860\pi$
$571$	−1.35841e8	−0.0305355	−0.0152677	+	0.999883i	$0.504860\pi$
$572$	−4.51700e7	−0.0100917
$573$	7.42778e9	1.64937
$574$	0	0
$575$	2.02688e8	0.0444621
$576$	1.33062e7	0.00290118
$577$	−1.63775e9	−0.354922	−0.177461	−	0.984128i	$-0.556788\pi$
$577$	−1.63775e9	−0.354922	−0.177461	+	0.984128i	$0.556788\pi$
$578$	3.64249e9	0.784606
$579$	−7.65147e8	−0.163821
$580$	6.93685e8	0.147627
$581$	0	0
$582$	−3.28066e9	−0.689812
$583$	−2.20246e8	−0.0460328
$584$	3.58468e9	0.744742
$585$	5.64818e7	0.0116644
$586$	−2.50356e9	−0.513944
$587$	5.97205e9	1.21868	0.609341	−	0.792909i	$-0.291434\pi$
$587$	5.97205e9	1.21868	0.609341	+	0.792909i	$0.291434\pi$
$588$	0	0
$589$	4.06967e9	0.820647
$590$	2.34048e9	0.469163
$591$	−8.07303e9	−1.60872
$592$	−2.25362e9	−0.446431
$593$	−8.31347e9	−1.63716	−0.818579	−	0.574394i	$-0.805238\pi$
$593$	−8.31347e9	−1.63716	−0.818579	+	0.574394i	$0.805238\pi$
$594$	2.39259e8	0.0468399
$595$	0	0
$596$	−2.47915e9	−0.479668
$597$	−8.53308e9	−1.64133
$598$	7.01370e8	0.134120
$599$	9.78368e9	1.85998	0.929990	−	0.367585i	$-0.119815\pi$
$599$	9.78368e9	1.85998	0.929990	+	0.367585i	$0.119815\pi$
$600$	6.30000e8	0.119072
$601$	−5.40159e9	−1.01499	−0.507494	−	0.861655i	$-0.669428\pi$
$601$	−5.40159e9	−1.01499	−0.507494	+	0.861655i	$0.669428\pi$
$602$	0	0
$603$	−9.32846e7	−0.0173260
$604$	4.88890e8	0.0902779
$605$	2.43220e9	0.446535
$606$	6.53634e9	1.19311
$607$	2.84439e9	0.516214	0.258107	−	0.966116i	$-0.416901\pi$
$607$	2.84439e9	0.516214	0.258107	+	0.966116i	$0.416901\pi$
$608$	4.64264e9	0.837726
$609$	0	0
$610$	1.61696e9	0.288433
$611$	−1.53277e9	−0.271851
$612$	9.74928e7	0.0171926
$613$	−7.02106e9	−1.23109	−0.615547	−	0.788101i	$-0.711065\pi$
$613$	−7.02106e9	−1.23109	−0.615547	+	0.788101i	$0.711065\pi$
$614$	−1.19722e9	−0.208730
$615$	2.80729e9	0.486659
$616$	0	0
$617$	3.35166e9	0.574462	0.287231	−	0.957861i	$-0.407265\pi$
$617$	3.35166e9	0.574462	0.287231	+	0.957861i	$0.407265\pi$
$618$	1.09637e10	1.86852
$619$	3.92362e9	0.664921	0.332461	−	0.943117i	$-0.392121\pi$
$619$	3.92362e9	0.664921	0.332461	+	0.943117i	$0.392121\pi$
$620$	−1.33355e9	−0.224718
$621$	−1.28890e9	−0.215972
$622$	−6.78588e9	−1.13068
$623$	0	0
$624$	3.79353e9	0.625026
$625$	2.44141e8	0.0400000
$626$	7.99150e9	1.30202
$627$	2.14161e8	0.0346979
$628$	−5.98354e9	−0.964049
$629$	1.34948e9	0.216217
$630$	0	0
$631$	−6.81545e8	−0.107992	−0.0539960	−	0.998541i	$-0.517196\pi$
$631$	−6.81545e8	−0.107992	−0.0539960	+	0.998541i	$0.517196\pi$
$632$	−806400.	−0.000127069 0
$633$	−7.77583e9	−1.21852
$634$	7.70267e9	1.20041
$635$	1.56272e9	0.242200
$636$	−4.17955e9	−0.644213
$637$	0	0
$638$	1.96517e8	0.0299590
$639$	5.97097e8	0.0905298
$640$	3.06264e9	0.461813
$641$	9.65199e9	1.44748	0.723742	−	0.690071i	$-0.242421\pi$
$641$	9.65199e9	1.44748	0.723742	+	0.690071i	$0.242421\pi$
$642$	2.74831e9	0.409914
$643$	5.07826e9	0.753315	0.376657	−	0.926353i	$-0.377073\pi$
$643$	5.07826e9	0.753315	0.376657	+	0.926353i	$0.377073\pi$
$644$	0	0
$645$	2.99525e9	0.439515
$646$	−4.45016e9	−0.649474
$647$	2.08330e9	0.302404	0.151202	−	0.988503i	$-0.451686\pi$
$647$	2.08330e9	0.302404	0.151202	+	0.988503i	$0.451686\pi$
$648$	−4.22113e9	−0.609420
$649$	2.30036e8	0.0330324
$650$	8.44812e8	0.120660
$651$	0	0
$652$	−3.73235e9	−0.527370
$653$	6.84881e9	0.962540	0.481270	−	0.876572i	$-0.340176\pi$
$653$	6.84881e9	0.962540	0.481270	+	0.876572i	$0.340176\pi$
$654$	1.80309e10	2.52056
$655$	−9.69524e8	−0.134807
$656$	9.57474e9	1.32423
$657$	4.99295e8	0.0686876
$658$	0	0
$659$	6.99913e9	0.952676	0.476338	−	0.879262i	$-0.341964\pi$
$659$	6.99913e9	0.952676	0.476338	+	0.879262i	$0.341964\pi$
$660$	−7.01760e7	−0.00950134
$661$	1.99594e9	0.268809	0.134404	−	0.990927i	$-0.457088\pi$
$661$	1.99594e9	0.268809	0.134404	+	0.990927i	$0.457088\pi$
$662$	1.31577e10	1.76270
$663$	−2.27160e9	−0.302715
$664$	−5.15830e9	−0.683783
$665$	0	0
$666$	−1.80386e8	−0.0236616
$667$	−1.05864e9	−0.138137
$668$	−5.85793e8	−0.0760372
$669$	−8.44472e8	−0.109042
$670$	−1.39528e9	−0.179226
$671$	1.58924e8	0.0203077
$672$	0	0
$673$	−1.17939e10	−1.49144	−0.745718	−	0.666261i	$-0.767894\pi$
$673$	−1.17939e10	−1.49144	−0.745718	+	0.666261i	$0.767894\pi$
$674$	−7.47085e9	−0.939853
$675$	−1.55250e9	−0.194298
$676$	−3.25268e9	−0.404974
$677$	−7.80222e9	−0.966402	−0.483201	−	0.875509i	$-0.660526\pi$
$677$	−7.80222e9	−0.966402	−0.483201	+	0.875509i	$0.660526\pi$
$678$	1.17473e10	1.44755
$679$	0	0
$680$	−1.28667e9	−0.156923
$681$	9.74572e9	1.18249
$682$	−3.77786e8	−0.0456038
$683$	4.51153e9	0.541815	0.270908	−	0.962605i	$-0.412676\pi$
$683$	4.51153e9	0.541815	0.270908	+	0.962605i	$0.412676\pi$
$684$	2.06379e8	0.0246586
$685$	−4.52150e9	−0.537484
$686$	0	0
$687$	−7.65883e9	−0.901185
$688$	1.02158e10	1.19595
$689$	4.94528e9	0.576002
$690$	1.08965e9	0.126274
$691$	1.06331e10	1.22599	0.612997	−	0.790086i	$-0.289964\pi$
$691$	1.06331e10	1.22599	0.612997	+	0.790086i	$0.289964\pi$
$692$	3.48516e9	0.399808
$693$	0	0
$694$	−1.51107e10	−1.71604
$695$	1.36365e9	0.154084
$696$	−3.29052e9	−0.369940
$697$	−5.73343e9	−0.641356
$698$	−5.98104e9	−0.665707
$699$	−9.35929e8	−0.103651
$700$	0	0
$701$	−4.38514e9	−0.480807	−0.240403	−	0.970673i	$-0.577280\pi$
$701$	−4.38514e9	−0.480807	−0.240403	+	0.970673i	$0.577280\pi$
$702$	−5.37220e9	−0.586100
$703$	2.85667e9	0.310110
$704$	1.95612e7	0.00211296
$705$	−2.38130e9	−0.255949
$706$	−2.08552e10	−2.23048
$707$	0	0
$708$	4.36534e9	0.462276
$709$	−5.98805e9	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$709$	−5.98805e9	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$710$	8.93094e9	0.936468
$711$	−112320.	−1.17196e−5 0
$712$	−1.68888e9	−0.175355
$713$	2.03515e9	0.210273
$714$	0	0
$715$	8.30330e7	0.00849532
$716$	−3.41106e9	−0.347291
$717$	−7.69058e9	−0.779188
$718$	−1.17779e10	−1.18749
$719$	1.17768e10	1.18161	0.590807	−	0.806813i	$-0.298809\pi$
$719$	1.17768e10	1.18161	0.590807	+	0.806813i	$0.298809\pi$
$720$	2.99286e8	0.0298829
$721$	0	0
$722$	3.09383e9	0.305926
$723$	1.81814e10	1.78913
$724$	−4.69755e9	−0.460031
$725$	−1.27516e9	−0.124274
$726$	1.30755e10	1.26818
$727$	−8.41051e9	−0.811805	−0.405902	−	0.913916i	$-0.633043\pi$
$727$	−8.41051e9	−0.811805	−0.405902	+	0.913916i	$0.633043\pi$
$728$	0	0
$729$	9.84247e9	0.940931
$730$	7.46809e9	0.710525
$731$	−6.11729e9	−0.579227
$732$	3.01586e9	0.284199
$733$	−1.44084e10	−1.35130	−0.675650	−	0.737223i	$-0.736137\pi$
$733$	−1.44084e10	−1.35130	−0.675650	+	0.737223i	$0.736137\pi$
$734$	−1.05065e10	−0.980664
$735$	0	0
$736$	2.32168e9	0.214649
$737$	−1.37136e8	−0.0126187
$738$	7.66391e8	0.0701864
$739$	8.21708e9	0.748966	0.374483	−	0.927234i	$-0.377820\pi$
$739$	8.21708e9	0.748966	0.374483	+	0.927234i	$0.377820\pi$
$740$	−9.36071e8	−0.0849176
$741$	−4.80865e9	−0.434170
$742$	0	0
$743$	−1.72531e10	−1.54314	−0.771570	−	0.636144i	$-0.780528\pi$
$743$	−1.72531e10	−1.54314	−0.771570	+	0.636144i	$0.780528\pi$
$744$	6.32572e9	0.563125
$745$	4.55726e9	0.403791
$746$	−2.39503e9	−0.211215
$747$	−7.18477e8	−0.0630654
$748$	1.43323e8	0.0125216
$749$	0	0
$750$	1.31250e9	0.113602
$751$	1.58498e10	1.36548	0.682739	−	0.730662i	$-0.260789\pi$
$751$	1.58498e10	1.36548	0.682739	+	0.730662i	$0.260789\pi$
$752$	−8.12183e9	−0.696453
$753$	−7.74260e9	−0.660853
$754$	−4.41249e9	−0.374873
$755$	−8.98694e8	−0.0759972
$756$	0	0
$757$	7.13856e9	0.598102	0.299051	−	0.954237i	$-0.403330\pi$
$757$	7.13856e9	0.598102	0.299051	+	0.954237i	$0.403330\pi$
$758$	−6.52735e8	−0.0544371
$759$	1.07097e8	0.00889059
$760$	−2.72370e9	−0.225067
$761$	2.59993e9	0.213853	0.106926	−	0.994267i	$-0.465899\pi$
$761$	2.59993e9	0.213853	0.106926	+	0.994267i	$0.465899\pi$
$762$	8.40119e9	0.687857
$763$	0	0
$764$	1.05227e10	0.853690
$765$	−1.79215e8	−0.0144730
$766$	−6.19172e9	−0.497749
$767$	−5.16512e9	−0.413329
$768$	1.57660e10	1.25591
$769$	4.96477e9	0.393692	0.196846	−	0.980434i	$-0.436930\pi$
$769$	4.96477e9	0.393692	0.196846	+	0.980434i	$0.436930\pi$
$770$	0	0
$771$	−1.09146e10	−0.857663
$772$	−1.08396e9	−0.0847914
$773$	1.49681e10	1.16557	0.582786	−	0.812626i	$-0.301963\pi$
$773$	1.49681e10	1.16557	0.582786	+	0.812626i	$0.301963\pi$
$774$	8.17703e8	0.0633873
$775$	2.45138e9	0.189171
$776$	4.10082e9	0.315032
$777$	0	0
$778$	6.49646e9	0.494593
$779$	−1.21369e10	−0.919867
$780$	1.57570e9	0.118889
$781$	8.77783e8	0.0659339
$782$	−2.22542e9	−0.166414
$783$	8.10877e9	0.603655
$784$	0	0
$785$	1.09992e10	0.811550
$786$	−5.21216e9	−0.382859
$787$	9.79990e9	0.716655	0.358328	−	0.933596i	$-0.383347\pi$
$787$	9.79990e9	0.716655	0.358328	+	0.933596i	$0.383347\pi$
$788$	−1.14368e10	−0.832650
$789$	2.19709e10	1.59250
$790$	−1.68000e6	−0.000121231 0
$791$	0	0
$792$	1.69042e7	0.00120908
$793$	−3.56840e9	−0.254108
$794$	−4.44909e9	−0.315428
$795$	7.68299e9	0.542307
$796$	−1.20885e10	−0.849529
$797$	−3.93169e9	−0.275090	−0.137545	−	0.990495i	$-0.543921\pi$
$797$	−3.93169e9	−0.275090	−0.137545	+	0.990495i	$0.543921\pi$
$798$	0	0
$799$	4.86342e9	0.337309
$800$	2.79650e9	0.193108
$801$	−2.35237e8	−0.0161730
$802$	1.67388e10	1.14581
$803$	7.34006e8	0.0500259
$804$	−2.60240e9	−0.176595
$805$	0	0
$806$	8.48262e9	0.570634
$807$	−2.24297e10	−1.50234
$808$	−8.17043e9	−0.544885
$809$	1.26324e10	0.838816	0.419408	−	0.907798i	$-0.362238\pi$
$809$	1.26324e10	0.838816	0.419408	+	0.907798i	$0.362238\pi$
$810$	−8.79403e9	−0.581420
$811$	−1.16653e10	−0.767934	−0.383967	−	0.923347i	$-0.625442\pi$
$811$	−1.16653e10	−0.767934	−0.383967	+	0.923347i	$0.625442\pi$
$812$	0	0
$813$	2.14047e9	0.139699
$814$	−2.65183e8	−0.0172330
$815$	6.86094e9	0.443948
$816$	−1.20368e10	−0.775522
$817$	−1.29495e10	−0.830758
$818$	3.09827e10	1.97917
$819$	0	0
$820$	3.97700e9	0.251888
$821$	−8.17500e9	−0.515569	−0.257784	−	0.966202i	$-0.582992\pi$
$821$	−8.17500e9	−0.515569	−0.257784	+	0.966202i	$0.582992\pi$
$822$	−2.43076e10	−1.52648
$823$	−1.75211e10	−1.09563	−0.547813	−	0.836601i	$-0.684540\pi$
$823$	−1.75211e10	−1.09563	−0.547813	+	0.836601i	$0.684540\pi$
$824$	−1.37046e10	−0.853341
$825$	1.29000e8	0.00799836
$826$	0	0
$827$	1.22225e10	0.751437	0.375718	−	0.926734i	$-0.377396\pi$
$827$	1.22225e10	0.751437	0.375718	+	0.926734i	$0.377396\pi$
$828$	1.03205e8	0.00631823
$829$	1.06634e10	0.650063	0.325032	−	0.945703i	$-0.394625\pi$
$829$	1.06634e10	0.650063	0.325032	+	0.945703i	$0.394625\pi$
$830$	−1.07465e10	−0.652367
$831$	1.51995e10	0.918812
$832$	−4.39218e8	−0.0264392
$833$	0	0
$834$	7.33100e9	0.437605
$835$	1.07682e9	0.0640092
$836$	3.03394e8	0.0179591
$837$	−1.55884e10	−0.918887
$838$	−1.12322e10	−0.659341
$839$	2.31400e9	0.135268	0.0676342	−	0.997710i	$-0.478455\pi$
$839$	2.31400e9	0.135268	0.0676342	+	0.997710i	$0.478455\pi$
$840$	0	0
$841$	−1.05897e10	−0.613899
$842$	−4.82599e8	−0.0278608
$843$	−1.08392e10	−0.623164
$844$	−1.10158e10	−0.630690
$845$	5.97918e9	0.340913
$846$	−6.50096e8	−0.0369132
$847$	0	0
$848$	2.62041e10	1.47565
$849$	−9.99410e8	−0.0560488
$850$	−2.68056e9	−0.149713
$851$	1.42855e9	0.0794590
$852$	1.66575e10	0.922721
$853$	4.22377e9	0.233012	0.116506	−	0.993190i	$-0.462831\pi$
$853$	4.22377e9	0.233012	0.116506	+	0.993190i	$0.462831\pi$
$854$	0	0
$855$	−3.79372e8	−0.0207579
$856$	−3.43538e9	−0.187205
$857$	3.52104e9	0.191090	0.0955450	−	0.995425i	$-0.469541\pi$
$857$	3.52104e9	0.191090	0.0955450	+	0.995425i	$0.469541\pi$
$858$	4.46385e8	0.0241271
$859$	−2.44930e10	−1.31846	−0.659229	−	0.751943i	$-0.729117\pi$
$859$	−2.44930e10	−1.31846	−0.659229	+	0.751943i	$0.729117\pi$
$860$	4.24327e9	0.227487
$861$	0	0
$862$	−2.41769e10	−1.28566
$863$	−5.40573e9	−0.286297	−0.143148	−	0.989701i	$-0.545723\pi$
$863$	−5.40573e9	−0.286297	−0.143148	+	0.989701i	$0.545723\pi$
$864$	−1.77831e10	−0.938012
$865$	−6.40655e9	−0.336564
$866$	6.83631e9	0.357692
$867$	−1.24886e10	−0.650797
$868$	0	0
$869$	−165120.	−8.53553e−6 0
$870$	−6.85524e9	−0.352944
$871$	3.07919e9	0.157897
$872$	−2.25387e10	−1.15112
$873$	5.71186e8	0.0290555
$874$	−4.71091e9	−0.238679
$875$	0	0
$876$	1.39290e10	0.700095
$877$	−2.89155e10	−1.44755	−0.723773	−	0.690039i	$-0.757594\pi$
$877$	−2.89155e10	−1.44755	−0.723773	+	0.690039i	$0.757594\pi$
$878$	−4.04096e10	−2.01490
$879$	8.58362e9	0.426294
$880$	4.39976e8	0.0217640
$881$	−7.80643e8	−0.0384624	−0.0192312	−	0.999815i	$-0.506122\pi$
$881$	−7.80643e8	−0.0384624	−0.0192312	+	0.999815i	$0.506122\pi$
$882$	0	0
$883$	−1.36907e10	−0.669211	−0.334605	−	0.942358i	$-0.608603\pi$
$883$	−1.36907e10	−0.669211	−0.334605	+	0.942358i	$0.608603\pi$
$884$	−3.21810e9	−0.156681
$885$	−8.02452e9	−0.389151
$886$	−1.29722e10	−0.626606
$887$	−3.58403e9	−0.172441	−0.0862203	−	0.996276i	$-0.527479\pi$
$887$	−3.58403e9	−0.172441	−0.0862203	+	0.996276i	$0.527479\pi$
$888$	4.44028e9	0.212797
$889$	0	0
$890$	−3.51850e9	−0.167299
$891$	−8.64327e8	−0.0409361
$892$	−1.19634e9	−0.0564386
$893$	1.02952e10	0.483786
$894$	2.44998e10	1.14678
$895$	6.27033e9	0.292355
$896$	0	0
$897$	−2.40470e9	−0.111247
$898$	−1.89749e10	−0.874406
$899$	−1.28036e10	−0.587725
$900$	1.24312e8	0.00568416
$901$	−1.56912e10	−0.714694
$902$	1.12666e9	0.0511175
$903$	0	0
$904$	−1.46841e10	−0.661085
$905$	8.63521e9	0.387260
$906$	−4.83138e9	−0.215835
$907$	−3.01108e10	−1.33998	−0.669988	−	0.742372i	$-0.733701\pi$
$907$	−3.01108e10	−1.33998	−0.669988	+	0.742372i	$0.733701\pi$
$908$	1.38064e10	0.612042
$909$	−1.13802e9	−0.0502548
$910$	0	0
$911$	−2.48800e10	−1.09027	−0.545137	−	0.838347i	$-0.683523\pi$
$911$	−2.48800e10	−1.09027	−0.545137	+	0.838347i	$0.683523\pi$
$912$	−2.54801e10	−1.11229
$913$	−1.05622e9	−0.0459312
$914$	−6.48801e8	−0.0281060
$915$	−5.54387e9	−0.239243
$916$	−1.08500e10	−0.466440
$917$	0	0
$918$	1.70458e10	0.727224
$919$	1.08420e10	0.460794	0.230397	−	0.973097i	$-0.425997\pi$
$919$	1.08420e10	0.460794	0.230397	+	0.973097i	$0.425997\pi$
$920$	−1.36206e9	−0.0576685
$921$	4.10477e9	0.173133
$922$	−2.12964e9	−0.0894846
$923$	−1.97093e10	−0.825021
$924$	0	0
$925$	1.72072e9	0.0714848
$926$	−2.28831e10	−0.947056
$927$	−1.90886e9	−0.0787037
$928$	−1.46062e10	−0.599957
$929$	−1.07045e10	−0.438038	−0.219019	−	0.975721i	$-0.570286\pi$
$929$	−1.07045e10	−0.438038	−0.219019	+	0.975721i	$0.570286\pi$
$930$	1.31786e10	0.537253
$931$	0	0
$932$	−1.32590e9	−0.0536482
$933$	2.32659e10	0.937851
$934$	−1.55933e10	−0.626214
$935$	−2.63461e8	−0.0105409
$936$	−3.79557e8	−0.0151291
$937$	−3.42787e10	−1.36124	−0.680621	−	0.732635i	$-0.738290\pi$
$937$	−3.42787e10	−1.36124	−0.680621	+	0.732635i	$0.738290\pi$
$938$	0	0
$939$	−2.73994e10	−1.07997
$940$	−3.37351e9	−0.132475
$941$	3.73695e9	0.146202	0.0731010	−	0.997325i	$-0.476710\pi$
$941$	3.73695e9	0.146202	0.0731010	+	0.997325i	$0.476710\pi$
$942$	5.91314e10	2.30484
$943$	−6.06937e9	−0.235696
$944$	−2.73690e10	−1.05890
$945$	0	0
$946$	1.20209e9	0.0461657
$947$	3.32150e10	1.27089	0.635447	−	0.772145i	$-0.280816\pi$
$947$	3.32150e10	1.27089	0.635447	+	0.772145i	$0.280816\pi$
$948$	−3.13344e6	−0.000119452 0
$949$	−1.64810e10	−0.625968
$950$	−5.67438e9	−0.214726
$951$	−2.64091e10	−0.995686
$952$	0	0
$953$	4.69895e10	1.75864	0.879318	−	0.476235i	$-0.157999\pi$
$953$	4.69895e10	1.75864	0.879318	+	0.476235i	$0.157999\pi$
$954$	2.09746e9	0.0782120
$955$	−1.93432e10	−0.718648
$956$	−1.08950e10	−0.403296
$957$	−6.73772e8	−0.0248497
$958$	1.78843e10	0.657192
$959$	0	0
$960$	−6.82368e8	−0.0248925
$961$	−2.89877e9	−0.105361
$962$	5.95429e9	0.215634
$963$	−4.78500e8	−0.0172659
$964$	2.57570e10	0.926030
$965$	1.99257e9	0.0713786
$966$	0	0
$967$	−1.42294e10	−0.506050	−0.253025	−	0.967460i	$-0.581425\pi$
$967$	−1.42294e10	−0.506050	−0.253025	+	0.967460i	$0.581425\pi$
$968$	−1.63444e10	−0.579168
$969$	1.52577e10	0.538711
$970$	8.54338e9	0.300558
$971$	−5.45474e8	−0.0191208	−0.00956041	−	0.999954i	$-0.503043\pi$
$971$	−5.45474e8	−0.0191208	−0.00956041	+	0.999954i	$0.503043\pi$
$972$	−1.62570e9	−0.0567816
$973$	0	0
$974$	−1.37154e10	−0.475612
$975$	−2.89650e9	−0.100082
$976$	−1.89083e10	−0.650995
$977$	−1.97916e10	−0.678968	−0.339484	−	0.940612i	$-0.610252\pi$
$977$	−1.97916e10	−0.678968	−0.339484	+	0.940612i	$0.610252\pi$
$978$	3.68844e10	1.26083
$979$	−3.45818e8	−0.0117790
$980$	0	0
$981$	−3.13932e9	−0.106168
$982$	6.88975e10	2.32174
$983$	4.71503e10	1.58324	0.791620	−	0.611013i	$-0.209238\pi$
$983$	4.71503e10	1.58324	0.791620	+	0.611013i	$0.209238\pi$
$984$	−1.88650e10	−0.631210
$985$	2.10235e10	0.700936
$986$	1.40007e10	0.465136
$987$	0	0
$988$	−6.81226e9	−0.224720
$989$	−6.47573e9	−0.212864
$990$	3.52170e7	0.00115353
$991$	3.87968e10	1.26631	0.633153	−	0.774027i	$-0.281761\pi$
$991$	3.87968e10	1.26631	0.633153	+	0.774027i	$0.281761\pi$
$992$	2.80792e10	0.913258
$993$	−4.51123e10	−1.46208
$994$	0	0
$995$	2.22216e10	0.715145
$996$	−2.00437e10	−0.642791
$997$	5.66394e10	1.81003	0.905015	−	0.425380i	$-0.139860\pi$
$997$	5.66394e10	1.81003	0.905015	+	0.425380i	$0.139860\pi$
$998$	5.12101e10	1.63079
$999$	−1.09421e10	−0.347234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.8.a.a.1.1		1
7.6	odd	2		5.8.a.a.1.1	✓	1
21.20	even	2		45.8.a.f.1.1		1
28.27	even	2		80.8.a.d.1.1		1
35.13	even	4		25.8.b.a.24.2		2
35.27	even	4		25.8.b.a.24.1		2
35.34	odd	2		25.8.a.a.1.1		1
56.13	odd	2		320.8.a.h.1.1		1
56.27	even	2		320.8.a.a.1.1		1
77.76	even	2		605.8.a.c.1.1		1
105.62	odd	4		225.8.b.b.199.2		2
105.83	odd	4		225.8.b.b.199.1		2
105.104	even	2		225.8.a.b.1.1		1
140.27	odd	4		400.8.c.e.49.1		2
140.83	odd	4		400.8.c.e.49.2		2
140.139	even	2		400.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.a.1.1	✓	1	7.6	odd	2
25.8.a.a.1.1		1	35.34	odd	2
25.8.b.a.24.1		2	35.27	even	4
25.8.b.a.24.2		2	35.13	even	4
45.8.a.f.1.1		1	21.20	even	2
80.8.a.d.1.1		1	28.27	even	2
225.8.a.b.1.1		1	105.104	even	2
225.8.b.b.199.1		2	105.83	odd	4
225.8.b.b.199.2		2	105.62	odd	4
245.8.a.a.1.1		1	1.1	even	1	trivial
320.8.a.a.1.1		1	56.27	even	2
320.8.a.h.1.1		1	56.13	odd	2
400.8.a.e.1.1		1	140.139	even	2
400.8.c.e.49.1		2	140.27	odd	4
400.8.c.e.49.2		2	140.83	odd	4
605.8.a.c.1.1		1	77.76	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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