LMFDB - Embedded newform 98.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,8,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	98.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:	$30.6137324974$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +82.0000 q^{3} +64.0000 q^{4} -448.000 q^{5} -656.000 q^{6} -512.000 q^{8} +4537.00 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +82.0000 q^{3} +64.0000 q^{4} -448.000 q^{5} -656.000 q^{6} -512.000 q^{8} +4537.00 q^{9} +3584.00 q^{10} +2408.00 q^{11} +5248.00 q^{12} -7116.00 q^{13} -36736.0 q^{15} +4096.00 q^{16} -2486.00 q^{17} -36296.0 q^{18} -36482.0 q^{19} -28672.0 q^{20} -19264.0 q^{22} -12880.0 q^{23} -41984.0 q^{24} +122579. q^{25} +56928.0 q^{26} +192700. q^{27} -88094.0 q^{29} +293888. q^{30} -282636. q^{31} -32768.0 q^{32} +197456. q^{33} +19888.0 q^{34} +290368. q^{36} -214534. q^{37} +291856. q^{38} -583512. q^{39} +229376. q^{40} +140874. q^{41} +36464.0 q^{43} +154112. q^{44} -2.03258e6 q^{45} +103040. q^{46} -716868. q^{47} +335872. q^{48} -980632. q^{50} -203852. q^{51} -455424. q^{52} -56946.0 q^{53} -1.54160e6 q^{54} -1.07878e6 q^{55} -2.99152e6 q^{57} +704752. q^{58} +2.14986e6 q^{59} -2.35110e6 q^{60} -3.08436e6 q^{61} +2.26109e6 q^{62} +262144. q^{64} +3.18797e6 q^{65} -1.57965e6 q^{66} -3.03436e6 q^{67} -159104. q^{68} -1.05616e6 q^{69} -106624. q^{71} -2.32294e6 q^{72} -988930. q^{73} +1.71627e6 q^{74} +1.00515e7 q^{75} -2.33485e6 q^{76} +4.66810e6 q^{78} +3.41590e6 q^{79} -1.83501e6 q^{80} +5.87898e6 q^{81} -1.12699e6 q^{82} +15142.0 q^{83} +1.11373e6 q^{85} -291712. q^{86} -7.22371e6 q^{87} -1.23290e6 q^{88} -174810. q^{89} +1.62606e7 q^{90} -824320. q^{92} -2.31762e7 q^{93} +5.73494e6 q^{94} +1.63439e7 q^{95} -2.68698e6 q^{96} -1.35068e7 q^{97} +1.09251e7 q^{99} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	82.0000	1.75343	0.876717	−	0.481006i	$-0.159729\pi$
$3$	82.0000	1.75343	0.876717	+	0.481006i	$0.159729\pi$
$4$	64.0000	0.500000
$5$	−448.000	−1.60281	−0.801407	−	0.598120i	$-0.795915\pi$
$5$	−448.000	−1.60281	−0.801407	+	0.598120i	$0.795915\pi$
$6$	−656.000	−1.23987
$7$	0	0
$7$	0	0
$8$	−512.000	−0.353553
$9$	4537.00	2.07453
$10$	3584.00	1.13336
$11$	2408.00	0.545484	0.272742	−	0.962087i	$-0.412069\pi$
$11$	2408.00	0.545484	0.272742	+	0.962087i	$0.412069\pi$
$12$	5248.00	0.876717
$13$	−7116.00	−0.898326	−0.449163	−	0.893450i	$-0.648278\pi$
$13$	−7116.00	−0.898326	−0.449163	+	0.893450i	$0.648278\pi$
$14$	0	0
$15$	−36736.0	−2.81043
$16$	4096.00	0.250000
$17$	−2486.00	−0.122724	−0.0613621	−	0.998116i	$-0.519544\pi$
$17$	−2486.00	−0.122724	−0.0613621	+	0.998116i	$0.519544\pi$
$18$	−36296.0	−1.46692
$19$	−36482.0	−1.22023	−0.610114	−	0.792314i	$-0.708876\pi$
$19$	−36482.0	−1.22023	−0.610114	+	0.792314i	$0.708876\pi$
$20$	−28672.0	−0.801407
$21$	0	0
$22$	−19264.0	−0.385715
$23$	−12880.0	−0.220734	−0.110367	−	0.993891i	$-0.535203\pi$
$23$	−12880.0	−0.220734	−0.110367	+	0.993891i	$0.535203\pi$
$24$	−41984.0	−0.619933
$25$	122579.	1.56901
$26$	56928.0	0.635213
$27$	192700.	1.88412
$28$	0	0
$29$	−88094.0	−0.670739	−0.335369	−	0.942087i	$-0.608861\pi$
$29$	−88094.0	−0.670739	−0.335369	+	0.942087i	$0.608861\pi$
$30$	293888.	1.98727
$31$	−282636.	−1.70397	−0.851984	−	0.523567i	$-0.824601\pi$
$31$	−282636.	−1.70397	−0.851984	+	0.523567i	$0.824601\pi$
$32$	−32768.0	−0.176777
$33$	197456.	0.956470
$34$	19888.0	0.0867791
$35$	0	0
$36$	290368.	1.03727
$37$	−214534.	−0.696290	−0.348145	−	0.937441i	$-0.613188\pi$
$37$	−214534.	−0.696290	−0.348145	+	0.937441i	$0.613188\pi$
$38$	291856.	0.862832
$39$	−583512.	−1.57516
$40$	229376.	0.566680
$41$	140874.	0.319218	0.159609	−	0.987180i	$-0.448977\pi$
$41$	140874.	0.319218	0.159609	+	0.987180i	$0.448977\pi$
$42$	0	0
$43$	36464.0	0.0699399	0.0349699	−	0.999388i	$-0.488866\pi$
$43$	36464.0	0.0699399	0.0349699	+	0.999388i	$0.488866\pi$
$44$	154112.	0.272742
$45$	−2.03258e6	−3.32509
$46$	103040.	0.156082
$47$	−716868.	−1.00716	−0.503578	−	0.863950i	$-0.667983\pi$
$47$	−716868.	−1.00716	−0.503578	+	0.863950i	$0.667983\pi$
$48$	335872.	0.438359
$49$	0	0
$50$	−980632.	−1.10946
$51$	−203852.	−0.215189
$52$	−455424.	−0.449163
$53$	−56946.0	−0.0525409	−0.0262705	−	0.999655i	$-0.508363\pi$
$53$	−56946.0	−0.0525409	−0.0262705	+	0.999655i	$0.508363\pi$
$54$	−1.54160e6	−1.33227
$55$	−1.07878e6	−0.874309
$56$	0	0
$57$	−2.99152e6	−2.13959
$58$	704752.	0.474284
$59$	2.14986e6	1.36279	0.681394	−	0.731916i	$-0.261374\pi$
$59$	2.14986e6	1.36279	0.681394	+	0.731916i	$0.261374\pi$
$60$	−2.35110e6	−1.40521
$61$	−3.08436e6	−1.73985	−0.869923	−	0.493188i	$-0.835831\pi$
$61$	−3.08436e6	−1.73985	−0.869923	+	0.493188i	$0.835831\pi$
$62$	2.26109e6	1.20489
$63$	0	0
$64$	262144.	0.125000
$65$	3.18797e6	1.43985
$66$	−1.57965e6	−0.676327
$67$	−3.03436e6	−1.23255	−0.616277	−	0.787530i	$-0.711360\pi$
$67$	−3.03436e6	−1.23255	−0.616277	+	0.787530i	$0.711360\pi$
$68$	−159104.	−0.0613621
$69$	−1.05616e6	−0.387042
$70$	0	0
$71$	−106624.	−0.0353550	−0.0176775	−	0.999844i	$-0.505627\pi$
$71$	−106624.	−0.0353550	−0.0176775	+	0.999844i	$0.505627\pi$
$72$	−2.32294e6	−0.733458
$73$	−988930.	−0.297533	−0.148767	−	0.988872i	$-0.547530\pi$
$73$	−988930.	−0.297533	−0.148767	+	0.988872i	$0.547530\pi$
$74$	1.71627e6	0.492351
$75$	1.00515e7	2.75116
$76$	−2.33485e6	−0.610114
$77$	0	0
$78$	4.66810e6	1.11380
$79$	3.41590e6	0.779489	0.389744	−	0.920923i	$-0.372563\pi$
$79$	3.41590e6	0.779489	0.389744	+	0.920923i	$0.372563\pi$
$80$	−1.83501e6	−0.400703
$81$	5.87898e6	1.22915
$82$	−1.12699e6	−0.225721
$83$	15142.0	0.00290676	0.00145338	−	0.999999i	$-0.499537\pi$
$83$	15142.0	0.00290676	0.00145338	+	0.999999i	$0.499537\pi$
$84$	0	0
$85$	1.11373e6	0.196704
$86$	−291712.	−0.0494549
$87$	−7.22371e6	−1.17610
$88$	−1.23290e6	−0.192858
$89$	−174810.	−0.0262846	−0.0131423	−	0.999914i	$-0.504183\pi$
$89$	−174810.	−0.0262846	−0.0131423	+	0.999914i	$0.504183\pi$
$90$	1.62606e7	2.35119
$91$	0	0
$92$	−824320.	−0.110367
$93$	−2.31762e7	−2.98780
$94$	5.73494e6	0.712167
$95$	1.63439e7	1.95580
$96$	−2.68698e6	−0.309966
$97$	−1.35068e7	−1.50263	−0.751313	−	0.659946i	$-0.770579\pi$
$97$	−1.35068e7	−1.50263	−0.751313	+	0.659946i	$0.770579\pi$
$98$	0	0
$99$	1.09251e7	1.13162
$100$	7.84506e6	0.784506
$101$	1.87645e7	1.81222	0.906112	−	0.423039i	$-0.139037\pi$
$101$	1.87645e7	1.81222	0.906112	+	0.423039i	$0.139037\pi$
$102$	1.63082e6	0.152161
$103$	1.62080e7	1.46150	0.730751	−	0.682644i	$-0.239170\pi$
$103$	1.62080e7	1.46150	0.730751	+	0.682644i	$0.239170\pi$
$104$	3.64339e6	0.317606
$105$	0	0
$106$	455568.	0.0371520
$107$	6.96580e6	0.549702	0.274851	−	0.961487i	$-0.411371\pi$
$107$	6.96580e6	0.549702	0.274851	+	0.961487i	$0.411371\pi$
$108$	1.23328e7	0.942060
$109$	−1.49039e7	−1.10232	−0.551159	−	0.834400i	$-0.685814\pi$
$109$	−1.49039e7	−1.10232	−0.551159	+	0.834400i	$0.685814\pi$
$110$	8.63027e6	0.618230
$111$	−1.75918e7	−1.22090
$112$	0	0
$113$	−2.60684e7	−1.69958	−0.849788	−	0.527125i	$-0.823270\pi$
$113$	−2.60684e7	−1.69958	−0.849788	+	0.527125i	$0.823270\pi$
$114$	2.39322e7	1.51292
$115$	5.77024e6	0.353795
$116$	−5.63802e6	−0.335369
$117$	−3.22853e7	−1.86361
$118$	−1.71989e7	−0.963637
$119$	0	0
$120$	1.88088e7	0.993636
$121$	−1.36887e7	−0.702447
$122$	2.46749e7	1.23026
$123$	1.15517e7	0.559728
$124$	−1.80887e7	−0.851984
$125$	−1.99154e7	−0.912019
$126$	0	0
$127$	2.13136e7	0.923301	0.461651	−	0.887062i	$-0.347257\pi$
$127$	2.13136e7	0.923301	0.461651	+	0.887062i	$0.347257\pi$
$128$	−2.09715e6	−0.0883883
$129$	2.99005e6	0.122635
$130$	−2.55037e7	−1.01813
$131$	−1.04773e7	−0.407194	−0.203597	−	0.979055i	$-0.565263\pi$
$131$	−1.04773e7	−0.407194	−0.203597	+	0.979055i	$0.565263\pi$
$132$	1.26372e7	0.478235
$133$	0	0
$134$	2.42749e7	0.871547
$135$	−8.63296e7	−3.01989
$136$	1.27283e6	0.0433895
$137$	3.61981e7	1.20272	0.601359	−	0.798979i	$-0.294626\pi$
$137$	3.61981e7	1.20272	0.601359	+	0.798979i	$0.294626\pi$
$138$	8.44928e6	0.273680
$139$	5.27655e7	1.66647	0.833237	−	0.552916i	$-0.186485\pi$
$139$	5.27655e7	1.66647	0.833237	+	0.552916i	$0.186485\pi$
$140$	0	0
$141$	−5.87832e7	−1.76598
$142$	852992.	0.0249998
$143$	−1.71353e7	−0.490023
$144$	1.85836e7	0.518633
$145$	3.94661e7	1.07507
$146$	7.91144e6	0.210388
$147$	0	0
$148$	−1.37302e7	−0.348145
$149$	7.36003e7	1.82275	0.911376	−	0.411575i	$-0.135021\pi$
$149$	7.36003e7	1.82275	0.911376	+	0.411575i	$0.135021\pi$
$150$	−8.04118e7	−1.94536
$151$	1.74289e7	0.411956	0.205978	−	0.978557i	$-0.433963\pi$
$151$	1.74289e7	0.411956	0.205978	+	0.978557i	$0.433963\pi$
$152$	1.86788e7	0.431416
$153$	−1.12790e7	−0.254595
$154$	0	0
$155$	1.26621e8	2.73114
$156$	−3.73448e7	−0.787578
$157$	2.22157e7	0.458154	0.229077	−	0.973408i	$-0.426429\pi$
$157$	2.22157e7	0.458154	0.229077	+	0.973408i	$0.426429\pi$
$158$	−2.73272e7	−0.551182
$159$	−4.66957e6	−0.0921270
$160$	1.46801e7	0.283340
$161$	0	0
$162$	−4.70318e7	−0.869140
$163$	6.23796e7	1.12820	0.564099	−	0.825707i	$-0.309223\pi$
$163$	6.23796e7	1.12820	0.564099	+	0.825707i	$0.309223\pi$
$164$	9.01594e6	0.159609
$165$	−8.84603e7	−1.53304
$166$	−121136.	−0.00205539
$167$	−3.31238e7	−0.550343	−0.275171	−	0.961395i	$-0.588735\pi$
$167$	−3.31238e7	−0.550343	−0.275171	+	0.961395i	$0.588735\pi$
$168$	0	0
$169$	−1.21111e7	−0.193010
$170$	−8.90982e6	−0.139091
$171$	−1.65519e8	−2.53140
$172$	2.33370e6	0.0349699
$173$	7.05799e7	1.03638	0.518191	−	0.855265i	$-0.326606\pi$
$173$	7.05799e7	1.03638	0.518191	+	0.855265i	$0.326606\pi$
$174$	5.77897e7	0.831626
$175$	0	0
$176$	9.86317e6	0.136371
$177$	1.76289e8	2.38956
$178$	1.39848e6	0.0185860
$179$	−1.21176e8	−1.57918	−0.789590	−	0.613634i	$-0.789707\pi$
$179$	−1.21176e8	−1.57918	−0.789590	+	0.613634i	$0.789707\pi$
$180$	−1.30085e8	−1.66254
$181$	−2.30550e7	−0.288995	−0.144498	−	0.989505i	$-0.546157\pi$
$181$	−2.30550e7	−0.288995	−0.144498	+	0.989505i	$0.546157\pi$
$182$	0	0
$183$	−2.52918e8	−3.05070
$184$	6.59456e6	0.0780411
$185$	9.61112e7	1.11602
$186$	1.85409e8	2.11269
$187$	−5.98629e6	−0.0669441
$188$	−4.58796e7	−0.503578
$189$	0	0
$190$	−1.30751e8	−1.38296
$191$	−2.02927e7	−0.210728	−0.105364	−	0.994434i	$-0.533601\pi$
$191$	−2.02927e7	−0.210728	−0.105364	+	0.994434i	$0.533601\pi$
$192$	2.14958e7	0.219179
$193$	2.26081e7	0.226367	0.113184	−	0.993574i	$-0.463895\pi$
$193$	2.26081e7	0.226367	0.113184	+	0.993574i	$0.463895\pi$
$194$	1.08054e8	1.06252
$195$	2.61413e8	2.52468
$196$	0	0
$197$	−3.27180e7	−0.304898	−0.152449	−	0.988311i	$-0.548716\pi$
$197$	−3.27180e7	−0.304898	−0.152449	+	0.988311i	$0.548716\pi$
$198$	−8.74008e7	−0.800179
$199$	−7.24472e7	−0.651682	−0.325841	−	0.945424i	$-0.605647\pi$
$199$	−7.24472e7	−0.651682	−0.325841	+	0.945424i	$0.605647\pi$
$200$	−6.27604e7	−0.554729
$201$	−2.48818e8	−2.16120
$202$	−1.50116e8	−1.28144
$203$	0	0
$204$	−1.30465e7	−0.107594
$205$	−6.31116e7	−0.511647
$206$	−1.29664e8	−1.03344
$207$	−5.84366e7	−0.457919
$208$	−2.91471e7	−0.224582
$209$	−8.78487e7	−0.665615
$210$	0	0
$211$	9.38006e7	0.687412	0.343706	−	0.939077i	$-0.388318\pi$
$211$	9.38006e7	0.687412	0.343706	+	0.939077i	$0.388318\pi$
$212$	−3.64454e6	−0.0262705
$213$	−8.74317e6	−0.0619927
$214$	−5.57264e7	−0.388698
$215$	−1.63359e7	−0.112101
$216$	−9.86624e7	−0.666137
$217$	0	0
$218$	1.19231e8	0.779457
$219$	−8.10923e7	−0.521705
$220$	−6.90422e7	−0.437155
$221$	1.76904e7	0.110246
$222$	1.40734e8	0.863306
$223$	−1.62567e6	−0.00981671	−0.00490835	−	0.999988i	$-0.501562\pi$
$223$	−1.62567e6	−0.00981671	−0.00490835	+	0.999988i	$0.501562\pi$
$224$	0	0
$225$	5.56141e8	3.25496
$226$	2.08548e8	1.20178
$227$	2.46153e8	1.39673	0.698367	−	0.715740i	$-0.253910\pi$
$227$	2.46153e8	1.39673	0.698367	+	0.715740i	$0.253910\pi$
$228$	−1.91458e8	−1.06979
$229$	1.06041e8	0.583510	0.291755	−	0.956493i	$-0.405761\pi$
$229$	1.06041e8	0.583510	0.291755	+	0.956493i	$0.405761\pi$
$230$	−4.61619e7	−0.250171
$231$	0	0
$232$	4.51041e7	0.237142
$233$	−2.02206e8	−1.04725	−0.523623	−	0.851950i	$-0.675420\pi$
$233$	−2.02206e8	−1.04725	−0.523623	+	0.851950i	$0.675420\pi$
$234$	2.58282e8	1.31777
$235$	3.21157e8	1.61428
$236$	1.37591e8	0.681394
$237$	2.80103e8	1.36678
$238$	0	0
$239$	−3.67523e7	−0.174137	−0.0870687	−	0.996202i	$-0.527750\pi$
$239$	−3.67523e7	−0.174137	−0.0870687	+	0.996202i	$0.527750\pi$
$240$	−1.50471e8	−0.702607
$241$	−5.76415e7	−0.265262	−0.132631	−	0.991165i	$-0.542343\pi$
$241$	−5.76415e7	−0.265262	−0.132631	+	0.991165i	$0.542343\pi$
$242$	1.09510e8	0.496705
$243$	6.06415e7	0.271112
$244$	−1.97399e8	−0.869923
$245$	0	0
$246$	−9.24133e7	−0.395787
$247$	2.59606e8	1.09616
$248$	1.44710e8	0.602444
$249$	1.24164e6	0.00509682
$250$	1.59323e8	0.644895
$251$	2.61332e8	1.04312	0.521561	−	0.853214i	$-0.325350\pi$
$251$	2.61332e8	1.04312	0.521561	+	0.853214i	$0.325350\pi$
$252$	0	0
$253$	−3.10150e7	−0.120407
$254$	−1.70509e8	−0.652873
$255$	9.13257e7	0.344907
$256$	1.67772e7	0.0625000
$257$	−3.47625e8	−1.27745	−0.638727	−	0.769433i	$-0.720539\pi$
$257$	−3.47625e8	−1.27745	−0.638727	+	0.769433i	$0.720539\pi$
$258$	−2.39204e7	−0.0867160
$259$	0	0
$260$	2.04030e8	0.719925
$261$	−3.99682e8	−1.39147
$262$	8.38186e7	0.287929
$263$	−1.76501e8	−0.598278	−0.299139	−	0.954210i	$-0.596699\pi$
$263$	−1.76501e8	−0.598278	−0.299139	+	0.954210i	$0.596699\pi$
$264$	−1.01097e8	−0.338163
$265$	2.55118e7	0.0842133
$266$	0	0
$267$	−1.43344e7	−0.0460883
$268$	−1.94199e8	−0.616277
$269$	4.07889e8	1.27764	0.638821	−	0.769356i	$-0.279423\pi$
$269$	4.07889e8	1.27764	0.638821	+	0.769356i	$0.279423\pi$
$270$	6.90637e8	2.13539
$271$	−3.17618e8	−0.969422	−0.484711	−	0.874674i	$-0.661075\pi$
$271$	−3.17618e8	−0.969422	−0.484711	+	0.874674i	$0.661075\pi$
$272$	−1.01827e7	−0.0306810
$273$	0	0
$274$	−2.89585e8	−0.850450
$275$	2.95170e8	0.855871
$276$	−6.75942e7	−0.193521
$277$	2.93727e8	0.830356	0.415178	−	0.909740i	$-0.363719\pi$
$277$	2.93727e8	0.830356	0.415178	+	0.909740i	$0.363719\pi$
$278$	−4.22124e8	−1.17838
$279$	−1.28232e9	−3.53494
$280$	0	0
$281$	−5.45027e7	−0.146536	−0.0732682	−	0.997312i	$-0.523343\pi$
$281$	−5.45027e7	−0.146536	−0.0732682	+	0.997312i	$0.523343\pi$
$282$	4.70265e8	1.24874
$283$	−2.05409e8	−0.538725	−0.269362	−	0.963039i	$-0.586813\pi$
$283$	−2.05409e8	−0.538725	−0.269362	+	0.963039i	$0.586813\pi$
$284$	−6.82394e6	−0.0176775
$285$	1.34020e9	3.42936
$286$	1.37083e8	0.346498
$287$	0	0
$288$	−1.48668e8	−0.366729
$289$	−4.04158e8	−0.984939
$290$	−3.15729e8	−0.760189
$291$	−1.10756e9	−2.63476
$292$	−6.32915e7	−0.148767
$293$	1.18887e8	0.276119	0.138059	−	0.990424i	$-0.455913\pi$
$293$	1.18887e8	0.276119	0.138059	+	0.990424i	$0.455913\pi$
$294$	0	0
$295$	−9.63138e8	−2.18430
$296$	1.09841e8	0.246176
$297$	4.64022e8	1.02776
$298$	−5.88802e8	−1.28888
$299$	9.16541e7	0.198291
$300$	6.43295e8	1.37558
$301$	0	0
$302$	−1.39431e8	−0.291297
$303$	1.53869e9	3.17761
$304$	−1.49430e8	−0.305057
$305$	1.38179e9	2.78865
$306$	9.02319e7	0.180026
$307$	−1.63581e8	−0.322663	−0.161332	−	0.986900i	$-0.551579\pi$
$307$	−1.63581e8	−0.322663	−0.161332	+	0.986900i	$0.551579\pi$
$308$	0	0
$309$	1.32906e9	2.56265
$310$	−1.01297e9	−1.93121
$311$	4.32480e8	0.815276	0.407638	−	0.913144i	$-0.366352\pi$
$311$	4.32480e8	0.815276	0.407638	+	0.913144i	$0.366352\pi$
$312$	2.98758e8	0.556902
$313$	4.27033e8	0.787148	0.393574	−	0.919293i	$-0.371239\pi$
$313$	4.27033e8	0.787148	0.393574	+	0.919293i	$0.371239\pi$
$314$	−1.77726e8	−0.323964
$315$	0	0
$316$	2.18617e8	0.389744
$317$	−3.47502e8	−0.612703	−0.306352	−	0.951918i	$-0.599108\pi$
$317$	−3.47502e8	−0.612703	−0.306352	+	0.951918i	$0.599108\pi$
$318$	3.73566e7	0.0651437
$319$	−2.12130e8	−0.365877
$320$	−1.17441e8	−0.200352
$321$	5.71195e8	0.963867
$322$	0	0
$323$	9.06943e7	0.149751
$324$	3.76255e8	0.614574
$325$	−8.72272e8	−1.40948
$326$	−4.99036e8	−0.797757
$327$	−1.22212e9	−1.93284
$328$	−7.21275e7	−0.112861
$329$	0	0
$330$	7.07682e8	1.08403
$331$	−8.15413e8	−1.23589	−0.617945	−	0.786221i	$-0.712035\pi$
$331$	−8.15413e8	−1.23589	−0.617945	+	0.786221i	$0.712035\pi$
$332$	969088.	0.00145338
$333$	−9.73341e8	−1.44448
$334$	2.64991e8	0.389151
$335$	1.35940e9	1.97555
$336$	0	0
$337$	−5.55790e8	−0.791054	−0.395527	−	0.918454i	$-0.629438\pi$
$337$	−5.55790e8	−0.791054	−0.395527	+	0.918454i	$0.629438\pi$
$338$	9.68885e7	0.136478
$339$	−2.13761e9	−2.98009
$340$	7.12786e7	0.0983519
$341$	−6.80587e8	−0.929488
$342$	1.32415e9	1.78997
$343$	0	0
$344$	−1.86696e7	−0.0247275
$345$	4.73160e8	0.620356
$346$	−5.64639e8	−0.732832
$347$	1.07708e9	1.38387	0.691934	−	0.721961i	$-0.256759\pi$
$347$	1.07708e9	1.38387	0.691934	+	0.721961i	$0.256759\pi$
$348$	−4.62317e8	−0.588048
$349$	−1.44167e9	−1.81541	−0.907707	−	0.419605i	$-0.862169\pi$
$349$	−1.44167e9	−1.81541	−0.907707	+	0.419605i	$0.862169\pi$
$350$	0	0
$351$	−1.37125e9	−1.69255
$352$	−7.89053e7	−0.0964289
$353$	−5.73266e8	−0.693657	−0.346829	−	0.937929i	$-0.612741\pi$
$353$	−5.73266e8	−0.693657	−0.346829	+	0.937929i	$0.612741\pi$
$354$	−1.41031e9	−1.68967
$355$	4.77676e7	0.0566675
$356$	−1.11878e7	−0.0131423
$357$	0	0
$358$	9.69410e8	1.11665
$359$	−9.49335e8	−1.08290	−0.541451	−	0.840733i	$-0.682125\pi$
$359$	−9.49335e8	−1.08290	−0.541451	+	0.840733i	$0.682125\pi$
$360$	1.04068e9	1.17560
$361$	4.37065e8	0.488957
$362$	1.84440e8	0.204350
$363$	−1.12247e9	−1.23169
$364$	0	0
$365$	4.43041e8	0.476890
$366$	2.02334e9	2.15717
$367$	3.01771e8	0.318674	0.159337	−	0.987224i	$-0.449064\pi$
$367$	3.01771e8	0.318674	0.159337	+	0.987224i	$0.449064\pi$
$368$	−5.27565e7	−0.0551834
$369$	6.39145e8	0.662228
$370$	−7.68890e8	−0.789147
$371$	0	0
$372$	−1.48327e9	−1.49390
$373$	−1.36182e9	−1.35875	−0.679374	−	0.733792i	$-0.737749\pi$
$373$	−1.36182e9	−1.35875	−0.679374	+	0.733792i	$0.737749\pi$
$374$	4.78903e7	0.0473366
$375$	−1.63306e9	−1.59916
$376$	3.67036e8	0.356083
$377$	6.26877e8	0.602542
$378$	0	0
$379$	−1.51839e9	−1.43267	−0.716333	−	0.697758i	$-0.754181\pi$
$379$	−1.51839e9	−1.43267	−0.716333	+	0.697758i	$0.754181\pi$
$380$	1.04601e9	0.977899
$381$	1.74771e9	1.61895
$382$	1.62342e8	0.149007
$383$	−4.54114e8	−0.413018	−0.206509	−	0.978445i	$-0.566210\pi$
$383$	−4.54114e8	−0.413018	−0.206509	+	0.978445i	$0.566210\pi$
$384$	−1.71966e8	−0.154983
$385$	0	0
$386$	−1.80865e8	−0.160066
$387$	1.65437e8	0.145092
$388$	−8.64435e8	−0.751313
$389$	−1.83700e9	−1.58229	−0.791143	−	0.611632i	$-0.790514\pi$
$389$	−1.83700e9	−1.58229	−0.791143	+	0.611632i	$0.790514\pi$
$390$	−2.09131e9	−1.78522
$391$	3.20197e7	0.0270893
$392$	0	0
$393$	−8.59141e8	−0.713987
$394$	2.61744e8	0.215595
$395$	−1.53032e9	−1.24938
$396$	6.99206e8	0.565812
$397$	−1.23668e9	−0.991950	−0.495975	−	0.868337i	$-0.665189\pi$
$397$	−1.23668e9	−0.991950	−0.495975	+	0.868337i	$0.665189\pi$
$398$	5.79578e8	0.460809
$399$	0	0
$400$	5.02084e8	0.392253
$401$	1.57948e9	1.22323	0.611615	−	0.791156i	$-0.290520\pi$
$401$	1.57948e9	1.22323	0.611615	+	0.791156i	$0.290520\pi$
$402$	1.99054e9	1.52820
$403$	2.01124e9	1.53072
$404$	1.20093e9	0.906112
$405$	−2.63378e9	−1.97010
$406$	0	0
$407$	−5.16598e8	−0.379815
$408$	1.04372e8	0.0760807
$409$	1.92558e9	1.39165	0.695823	−	0.718213i	$-0.255040\pi$
$409$	1.92558e9	1.39165	0.695823	+	0.718213i	$0.255040\pi$
$410$	5.04892e8	0.361789
$411$	2.96824e9	2.10889
$412$	1.03731e9	0.730751
$413$	0	0
$414$	4.67492e8	0.323797
$415$	−6.78362e6	−0.00465900
$416$	2.33177e8	0.158803
$417$	4.32677e9	2.92205
$418$	7.02789e8	0.470661
$419$	2.39996e9	1.59388	0.796938	−	0.604062i	$-0.206452\pi$
$419$	2.39996e9	1.59388	0.796938	+	0.604062i	$0.206452\pi$
$420$	0	0
$421$	1.20940e9	0.789919	0.394960	−	0.918698i	$-0.370759\pi$
$421$	1.20940e9	0.789919	0.394960	+	0.918698i	$0.370759\pi$
$422$	−7.50405e8	−0.486074
$423$	−3.25243e9	−2.08938
$424$	2.91564e7	0.0185760
$425$	−3.04731e8	−0.192556
$426$	6.99453e7	0.0438354
$427$	0	0
$428$	4.45811e8	0.274851
$429$	−1.40510e9	−0.859223
$430$	1.30687e8	0.0792671
$431$	−2.16285e9	−1.30124	−0.650618	−	0.759405i	$-0.725490\pi$
$431$	−2.16285e9	−1.30124	−0.650618	+	0.759405i	$0.725490\pi$
$432$	7.89299e8	0.471030
$433$	−2.25750e9	−1.33635	−0.668174	−	0.744005i	$-0.732924\pi$
$433$	−2.25750e9	−1.33635	−0.668174	+	0.744005i	$0.732924\pi$
$434$	0	0
$435$	3.23622e9	1.88506
$436$	−9.53849e8	−0.551159
$437$	4.69888e8	0.269345
$438$	6.48738e8	0.368901
$439$	7.45724e8	0.420680	0.210340	−	0.977628i	$-0.432543\pi$
$439$	7.45724e8	0.420680	0.210340	+	0.977628i	$0.432543\pi$
$440$	5.52337e8	0.309115
$441$	0	0
$442$	−1.41523e8	−0.0779559
$443$	−1.10215e9	−0.602322	−0.301161	−	0.953573i	$-0.597374\pi$
$443$	−1.10215e9	−0.602322	−0.301161	+	0.953573i	$0.597374\pi$
$444$	−1.12587e9	−0.610449
$445$	7.83149e7	0.0421293
$446$	1.30054e7	0.00694146
$447$	6.03522e9	3.19608
$448$	0	0
$449$	3.41600e9	1.78097	0.890483	−	0.455017i	$-0.150367\pi$
$449$	3.41600e9	1.78097	0.890483	+	0.455017i	$0.150367\pi$
$450$	−4.44913e9	−2.30161
$451$	3.39225e8	0.174128
$452$	−1.66838e9	−0.849788
$453$	1.42917e9	0.722337
$454$	−1.96922e9	−0.987640
$455$	0	0
$456$	1.53166e9	0.756459
$457$	4.27310e8	0.209429	0.104714	−	0.994502i	$-0.466607\pi$
$457$	4.27310e8	0.209429	0.104714	+	0.994502i	$0.466607\pi$
$458$	−8.48326e8	−0.412604
$459$	−4.79052e8	−0.231227
$460$	3.69295e8	0.176897
$461$	−2.18116e9	−1.03690	−0.518448	−	0.855109i	$-0.673490\pi$
$461$	−2.18116e9	−1.03690	−0.518448	+	0.855109i	$0.673490\pi$
$462$	0	0
$463$	1.88134e8	0.0880916	0.0440458	−	0.999030i	$-0.485975\pi$
$463$	1.88134e8	0.0880916	0.0440458	+	0.999030i	$0.485975\pi$
$464$	−3.60833e8	−0.167685
$465$	1.03829e10	4.78888
$466$	1.61765e9	0.740515
$467$	−8.54534e8	−0.388258	−0.194129	−	0.980976i	$-0.562188\pi$
$467$	−8.54534e8	−0.388258	−0.194129	+	0.980976i	$0.562188\pi$
$468$	−2.06626e9	−0.931803
$469$	0	0
$470$	−2.56925e9	−1.14147
$471$	1.82169e9	0.803342
$472$	−1.10073e9	−0.481819
$473$	8.78053e7	0.0381511
$474$	−2.24083e9	−0.966461
$475$	−4.47193e9	−1.91455
$476$	0	0
$477$	−2.58364e8	−0.108998
$478$	2.94019e8	0.123134
$479$	6.82814e8	0.283876	0.141938	−	0.989876i	$-0.454667\pi$
$479$	6.82814e8	0.283876	0.141938	+	0.989876i	$0.454667\pi$
$480$	1.20377e9	0.496818
$481$	1.52662e9	0.625496
$482$	4.61132e8	0.187569
$483$	0	0
$484$	−8.76077e8	−0.351224
$485$	6.05104e9	2.40843
$486$	−4.85132e8	−0.191705
$487$	4.15119e9	1.62863	0.814313	−	0.580425i	$-0.197114\pi$
$487$	4.15119e9	1.62863	0.814313	+	0.580425i	$0.197114\pi$
$488$	1.57919e9	0.615128
$489$	5.11512e9	1.97822
$490$	0	0
$491$	8.99538e8	0.342953	0.171476	−	0.985188i	$-0.445146\pi$
$491$	8.99538e8	0.342953	0.171476	+	0.985188i	$0.445146\pi$
$492$	7.39307e8	0.279864
$493$	2.19002e8	0.0823158
$494$	−2.07685e9	−0.775104
$495$	−4.89444e9	−1.81378
$496$	−1.15768e9	−0.425992
$497$	0	0
$498$	−9.93315e6	−0.00360400
$499$	−6.11110e8	−0.220175	−0.110087	−	0.993922i	$-0.535113\pi$
$499$	−6.11110e8	−0.220175	−0.110087	+	0.993922i	$0.535113\pi$
$500$	−1.27459e9	−0.456009
$501$	−2.71616e9	−0.964989
$502$	−2.09066e9	−0.737598
$503$	−5.23241e9	−1.83322	−0.916608	−	0.399787i	$-0.869084\pi$
$503$	−5.23241e9	−1.83322	−0.916608	+	0.399787i	$0.869084\pi$
$504$	0	0
$505$	−8.40649e9	−2.90466
$506$	2.48120e8	0.0851404
$507$	−9.93107e8	−0.338429
$508$	1.36407e9	0.461651
$509$	−2.33531e8	−0.0784931	−0.0392465	−	0.999230i	$-0.512496\pi$
$509$	−2.33531e8	−0.0784931	−0.0392465	+	0.999230i	$0.512496\pi$
$510$	−7.30606e8	−0.243886
$511$	0	0
$512$	−1.34218e8	−0.0441942
$513$	−7.03008e9	−2.29906
$514$	2.78100e9	0.903297
$515$	−7.26119e9	−2.34252
$516$	1.91363e8	0.0613175
$517$	−1.72622e9	−0.549388
$518$	0	0
$519$	5.78755e9	1.81723
$520$	−1.63224e9	−0.509064
$521$	−2.23029e9	−0.690922	−0.345461	−	0.938433i	$-0.612277\pi$
$521$	−2.23029e9	−0.690922	−0.345461	+	0.938433i	$0.612277\pi$
$522$	3.19746e9	0.983917
$523$	−4.74795e8	−0.145128	−0.0725638	−	0.997364i	$-0.523118\pi$
$523$	−4.74795e8	−0.145128	−0.0725638	+	0.997364i	$0.523118\pi$
$524$	−6.70549e8	−0.203597
$525$	0	0
$526$	1.41201e9	0.423047
$527$	7.02633e8	0.209118
$528$	8.08780e8	0.239118
$529$	−3.23893e9	−0.951277
$530$	−2.04094e8	−0.0595478
$531$	9.75392e9	2.82715
$532$	0	0
$533$	−1.00246e9	−0.286762
$534$	1.14675e8	0.0325894
$535$	−3.12068e9	−0.881070
$536$	1.55359e9	0.435774
$537$	−9.93645e9	−2.76899
$538$	−3.26311e9	−0.903429
$539$	0	0
$540$	−5.52509e9	−1.50995
$541$	−8.47970e8	−0.230245	−0.115122	−	0.993351i	$-0.536726\pi$
$541$	−8.47970e8	−0.230245	−0.115122	+	0.993351i	$0.536726\pi$
$542$	2.54095e9	0.685485
$543$	−1.89051e9	−0.506734
$544$	8.14612e7	0.0216948
$545$	6.67694e9	1.76681
$546$	0	0
$547$	−6.55694e7	−0.0171295	−0.00856477	−	0.999963i	$-0.502726\pi$
$547$	−6.55694e7	−0.0171295	−0.00856477	+	0.999963i	$0.502726\pi$
$548$	2.31668e9	0.601359
$549$	−1.39937e10	−3.60936
$550$	−2.36136e9	−0.605192
$551$	3.21385e9	0.818454
$552$	5.40754e8	0.136840
$553$	0	0
$554$	−2.34981e9	−0.587150
$555$	7.88112e9	1.95687
$556$	3.37699e9	0.833237
$557$	6.12524e8	0.150186	0.0750930	−	0.997177i	$-0.476075\pi$
$557$	6.12524e8	0.150186	0.0750930	+	0.997177i	$0.476075\pi$
$558$	1.02586e10	2.49958
$559$	−2.59478e8	−0.0628288
$560$	0	0
$561$	−4.90876e8	−0.117382
$562$	4.36021e8	0.103617
$563$	−7.02101e8	−0.165814	−0.0829068	−	0.996557i	$-0.526420\pi$
$563$	−7.02101e8	−0.165814	−0.0829068	+	0.996557i	$0.526420\pi$
$564$	−3.76212e9	−0.882991
$565$	1.16787e10	2.72410
$566$	1.64327e9	0.380936
$567$	0	0
$568$	5.45915e7	0.0124999
$569$	2.43577e8	0.0554298	0.0277149	−	0.999616i	$-0.491177\pi$
$569$	2.43577e8	0.0554298	0.0277149	+	0.999616i	$0.491177\pi$
$570$	−1.07216e10	−2.42493
$571$	5.28919e9	1.18895	0.594474	−	0.804115i	$-0.297360\pi$
$571$	5.28919e9	1.18895	0.594474	+	0.804115i	$0.297360\pi$
$572$	−1.09666e9	−0.245011
$573$	−1.66400e9	−0.369498
$574$	0	0
$575$	−1.57882e9	−0.346333
$576$	1.18935e9	0.259316
$577$	3.68455e9	0.798490	0.399245	−	0.916844i	$-0.369272\pi$
$577$	3.68455e9	0.798490	0.399245	+	0.916844i	$0.369272\pi$
$578$	3.23327e9	0.696457
$579$	1.85387e9	0.396920
$580$	2.52583e9	0.537534
$581$	0	0
$582$	8.86045e9	1.86305
$583$	−1.37126e8	−0.0286602
$584$	5.06332e8	0.105194
$585$	1.44638e10	2.98701
$586$	−9.51093e8	−0.195246
$587$	4.23510e9	0.864232	0.432116	−	0.901818i	$-0.357767\pi$
$587$	4.23510e9	0.864232	0.432116	+	0.901818i	$0.357767\pi$
$588$	0	0
$589$	1.03111e10	2.07923
$590$	7.70511e9	1.54453
$591$	−2.68287e9	−0.534618
$592$	−8.78731e8	−0.174072
$593$	−9.70955e9	−1.91209	−0.956043	−	0.293225i	$-0.905271\pi$
$593$	−9.70955e9	−1.91209	−0.956043	+	0.293225i	$0.905271\pi$
$594$	−3.71217e9	−0.726734
$595$	0	0
$596$	4.71042e9	0.911376
$597$	−5.94067e9	−1.14268
$598$	−7.33233e8	−0.140213
$599$	−4.04176e9	−0.768382	−0.384191	−	0.923254i	$-0.625520\pi$
$599$	−4.04176e9	−0.768382	−0.384191	+	0.923254i	$0.625520\pi$
$600$	−5.14636e9	−0.972681
$601$	2.62572e7	0.00493387	0.00246694	−	0.999997i	$-0.499215\pi$
$601$	2.62572e7	0.00493387	0.00246694	+	0.999997i	$0.499215\pi$
$602$	0	0
$603$	−1.37669e10	−2.55697
$604$	1.11545e9	0.205978
$605$	6.13254e9	1.12589
$606$	−1.23095e10	−2.24691
$607$	7.26489e9	1.31846	0.659232	−	0.751939i	$-0.270881\pi$
$607$	7.26489e9	1.31846	0.659232	+	0.751939i	$0.270881\pi$
$608$	1.19544e9	0.215708
$609$	0	0
$610$	−1.10543e10	−1.97187
$611$	5.10123e9	0.904755
$612$	−7.21855e8	−0.127298
$613$	−1.10037e10	−1.92942	−0.964712	−	0.263308i	$-0.915186\pi$
$613$	−1.10037e10	−1.92942	−0.964712	+	0.263308i	$0.915186\pi$
$614$	1.30865e9	0.228157
$615$	−5.17515e9	−0.897139
$616$	0	0
$617$	8.50181e8	0.145718	0.0728590	−	0.997342i	$-0.476788\pi$
$617$	8.50181e8	0.145718	0.0728590	+	0.997342i	$0.476788\pi$
$618$	−1.06325e10	−1.81207
$619$	−5.79271e9	−0.981668	−0.490834	−	0.871253i	$-0.663308\pi$
$619$	−5.79271e9	−0.981668	−0.490834	+	0.871253i	$0.663308\pi$
$620$	8.10374e9	1.36557
$621$	−2.48198e9	−0.415889
$622$	−3.45984e9	−0.576488
$623$	0	0
$624$	−2.39007e9	−0.393789
$625$	−6.54389e8	−0.107215
$626$	−3.41626e9	−0.556597
$627$	−7.20359e9	−1.16711
$628$	1.42180e9	0.229077
$629$	5.33332e8	0.0854516
$630$	0	0
$631$	−6.69052e9	−1.06012	−0.530062	−	0.847959i	$-0.677831\pi$
$631$	−6.69052e9	−1.06012	−0.530062	+	0.847959i	$0.677831\pi$
$632$	−1.74894e9	−0.275591
$633$	7.69165e9	1.20533
$634$	2.78002e9	0.433246
$635$	−9.54849e9	−1.47988
$636$	−2.98853e8	−0.0460635
$637$	0	0
$638$	1.69704e9	0.258714
$639$	−4.83753e8	−0.0733450
$640$	9.39524e8	0.141670
$641$	4.77194e9	0.715636	0.357818	−	0.933791i	$-0.383521\pi$
$641$	4.77194e9	0.715636	0.357818	+	0.933791i	$0.383521\pi$
$642$	−4.56956e9	−0.681557
$643$	−4.09295e9	−0.607153	−0.303576	−	0.952807i	$-0.598181\pi$
$643$	−4.09295e9	−0.607153	−0.303576	+	0.952807i	$0.598181\pi$
$644$	0	0
$645$	−1.33954e9	−0.196561
$646$	−7.25554e8	−0.105890
$647$	5.39902e9	0.783701	0.391850	−	0.920029i	$-0.371835\pi$
$647$	5.39902e9	0.783701	0.391850	+	0.920029i	$0.371835\pi$
$648$	−3.01004e9	−0.434570
$649$	5.17687e9	0.743380
$650$	6.97818e9	0.996656
$651$	0	0
$652$	3.99229e9	0.564099
$653$	−4.09787e9	−0.575919	−0.287960	−	0.957643i	$-0.592977\pi$
$653$	−4.09787e9	−0.575919	−0.287960	+	0.957643i	$0.592977\pi$
$654$	9.77695e9	1.36673
$655$	4.69384e9	0.652656
$656$	5.77020e8	0.0798045
$657$	−4.48678e9	−0.617242
$658$	0	0
$659$	−3.77455e9	−0.513766	−0.256883	−	0.966442i	$-0.582696\pi$
$659$	−3.77455e9	−0.513766	−0.256883	+	0.966442i	$0.582696\pi$
$660$	−5.66146e9	−0.766522
$661$	−9.81906e9	−1.32241	−0.661203	−	0.750207i	$-0.729954\pi$
$661$	−9.81906e9	−1.32241	−0.661203	+	0.750207i	$0.729954\pi$
$662$	6.52331e9	0.873906
$663$	1.45061e9	0.193310
$664$	−7.75270e6	−0.00102770
$665$	0	0
$666$	7.78673e9	1.02140
$667$	1.13465e9	0.148055
$668$	−2.11993e9	−0.275171
$669$	−1.33305e8	−0.0172130
$670$	−1.08752e10	−1.39693
$671$	−7.42714e9	−0.949058
$672$	0	0
$673$	9.98739e9	1.26299	0.631494	−	0.775381i	$-0.282442\pi$
$673$	9.98739e9	1.26299	0.631494	+	0.775381i	$0.282442\pi$
$674$	4.44632e9	0.559359
$675$	2.36210e10	2.95621
$676$	−7.75108e8	−0.0965048
$677$	−4.75559e8	−0.0589040	−0.0294520	−	0.999566i	$-0.509376\pi$
$677$	−4.75559e8	−0.0589040	−0.0294520	+	0.999566i	$0.509376\pi$
$678$	1.71009e10	2.10724
$679$	0	0
$680$	−5.70229e8	−0.0695453
$681$	2.01845e10	2.44908
$682$	5.44470e9	0.657247
$683$	2.76365e8	0.0331902	0.0165951	−	0.999862i	$-0.494717\pi$
$683$	2.76365e8	0.0331902	0.0165951	+	0.999862i	$0.494717\pi$
$684$	−1.05932e10	−1.26570
$685$	−1.62167e10	−1.92773
$686$	0	0
$687$	8.69534e9	1.02315
$688$	1.49357e8	0.0174850
$689$	4.05228e8	0.0471989
$690$	−3.78528e9	−0.438658
$691$	−8.82482e9	−1.01750	−0.508748	−	0.860915i	$-0.669892\pi$
$691$	−8.82482e9	−1.01750	−0.508748	+	0.860915i	$0.669892\pi$
$692$	4.51711e9	0.518191
$693$	0	0
$694$	−8.61664e9	−0.978542
$695$	−2.36390e10	−2.67105
$696$	3.69854e9	0.415813
$697$	−3.50213e8	−0.0391757
$698$	1.15333e10	1.28369
$699$	−1.65809e10	−1.83628
$700$	0	0
$701$	1.61615e10	1.77202	0.886012	−	0.463662i	$-0.153465\pi$
$701$	1.61615e10	1.77202	0.886012	+	0.463662i	$0.153465\pi$
$702$	1.09700e10	1.19682
$703$	7.82663e9	0.849633
$704$	6.31243e8	0.0681855
$705$	2.63349e10	2.83054
$706$	4.58613e9	0.490490
$707$	0	0
$708$	1.12825e10	1.19478
$709$	−8.36102e9	−0.881043	−0.440522	−	0.897742i	$-0.645207\pi$
$709$	−8.36102e9	−0.881043	−0.440522	+	0.897742i	$0.645207\pi$
$710$	−3.82140e8	−0.0400699
$711$	1.54979e10	1.61707
$712$	8.95027e7	0.00929301
$713$	3.64035e9	0.376123
$714$	0	0
$715$	7.67663e9	0.785415
$716$	−7.75528e9	−0.789590
$717$	−3.01369e9	−0.305339
$718$	7.59468e9	0.765727
$719$	8.18891e9	0.821627	0.410814	−	0.911719i	$-0.365245\pi$
$719$	8.18891e9	0.821627	0.410814	+	0.911719i	$0.365245\pi$
$720$	−8.32543e9	−0.831272
$721$	0	0
$722$	−3.49652e9	−0.345745
$723$	−4.72660e9	−0.465120
$724$	−1.47552e9	−0.144498
$725$	−1.07985e10	−1.05240
$726$	8.97979e9	0.870940
$727$	1.40657e10	1.35766	0.678832	−	0.734294i	$-0.262486\pi$
$727$	1.40657e10	1.35766	0.678832	+	0.734294i	$0.262486\pi$
$728$	0	0
$729$	−7.88473e9	−0.753772
$730$	−3.54433e9	−0.337212
$731$	−9.06495e7	−0.00858331
$732$	−1.61867e10	−1.52535
$733$	9.53632e9	0.894370	0.447185	−	0.894442i	$-0.352427\pi$
$733$	9.53632e9	0.894370	0.447185	+	0.894442i	$0.352427\pi$
$734$	−2.41417e9	−0.225336
$735$	0	0
$736$	4.22052e8	0.0390206
$737$	−7.30675e9	−0.672338
$738$	−5.11316e9	−0.468266
$739$	1.45133e10	1.32285	0.661425	−	0.750011i	$-0.269952\pi$
$739$	1.45133e10	1.32285	0.661425	+	0.750011i	$0.269952\pi$
$740$	6.15112e9	0.558011
$741$	2.12877e10	1.92205
$742$	0	0
$743$	−1.88434e9	−0.168538	−0.0842692	−	0.996443i	$-0.526856\pi$
$743$	−1.88434e9	−0.168538	−0.0842692	+	0.996443i	$0.526856\pi$
$744$	1.18662e10	1.05635
$745$	−3.29729e10	−2.92153
$746$	1.08946e10	0.960780
$747$	6.86993e7	0.00603017
$748$	−3.83122e8	−0.0334720
$749$	0	0
$750$	1.30645e10	1.13078
$751$	−1.71941e10	−1.48129	−0.740646	−	0.671895i	$-0.765480\pi$
$751$	−1.71941e10	−1.48129	−0.740646	+	0.671895i	$0.765480\pi$
$752$	−2.93629e9	−0.251789
$753$	2.14292e10	1.82904
$754$	−5.01502e9	−0.426062
$755$	−7.80814e9	−0.660288
$756$	0	0
$757$	−1.02275e10	−0.856909	−0.428454	−	0.903563i	$-0.640942\pi$
$757$	−1.02275e10	−0.856909	−0.428454	+	0.903563i	$0.640942\pi$
$758$	1.21471e10	1.01305
$759$	−2.54323e9	−0.211125
$760$	−8.36810e9	−0.691479
$761$	−2.93806e9	−0.241666	−0.120833	−	0.992673i	$-0.538557\pi$
$761$	−2.93806e9	−0.241666	−0.120833	+	0.992673i	$0.538557\pi$
$762$	−1.39817e10	−1.14477
$763$	0	0
$764$	−1.29873e9	−0.105364
$765$	5.05298e9	0.408068
$766$	3.63291e9	0.292048
$767$	−1.52984e10	−1.22423
$768$	1.37573e9	0.109590
$769$	1.49710e10	1.18716	0.593579	−	0.804776i	$-0.297714\pi$
$769$	1.49710e10	1.18716	0.593579	+	0.804776i	$0.297714\pi$
$770$	0	0
$771$	−2.85053e10	−2.23993
$772$	1.44692e9	0.113184
$773$	1.53271e10	1.19352	0.596762	−	0.802419i	$-0.296454\pi$
$773$	1.53271e10	1.19352	0.596762	+	0.802419i	$0.296454\pi$
$774$	−1.32350e9	−0.102596
$775$	−3.46452e10	−2.67355
$776$	6.91548e9	0.531259
$777$	0	0
$778$	1.46960e10	1.11884
$779$	−5.13937e9	−0.389519
$780$	1.67305e10	1.26234
$781$	−2.56751e8	−0.0192856
$782$	−2.56157e8	−0.0191551
$783$	−1.69757e10	−1.26375
$784$	0	0
$785$	−9.95263e9	−0.734335
$786$	6.87313e9	0.504865
$787$	5.64462e8	0.0412784	0.0206392	−	0.999787i	$-0.493430\pi$
$787$	5.64462e8	0.0412784	0.0206392	+	0.999787i	$0.493430\pi$
$788$	−2.09395e9	−0.152449
$789$	−1.44731e10	−1.04904
$790$	1.22426e10	0.883442
$791$	0	0
$792$	−5.59365e9	−0.400089
$793$	2.19483e10	1.56295
$794$	9.89342e9	0.701414
$795$	2.09197e9	0.147662
$796$	−4.63662e9	−0.325841
$797$	−2.02786e10	−1.41884	−0.709421	−	0.704785i	$-0.751044\pi$
$797$	−2.02786e10	−1.41884	−0.709421	+	0.704785i	$0.751044\pi$
$798$	0	0
$799$	1.78213e9	0.123602
$800$	−4.01667e9	−0.277365
$801$	−7.93113e8	−0.0545282
$802$	−1.26358e10	−0.864954
$803$	−2.38134e9	−0.162300
$804$	−1.59243e10	−1.08060
$805$	0	0
$806$	−1.60899e10	−1.08238
$807$	3.34469e10	2.24026
$808$	−9.60741e9	−0.640718
$809$	3.20139e9	0.212578	0.106289	−	0.994335i	$-0.466103\pi$
$809$	3.20139e9	0.212578	0.106289	+	0.994335i	$0.466103\pi$
$810$	2.10703e10	1.39307
$811$	2.22219e10	1.46288	0.731439	−	0.681907i	$-0.238849\pi$
$811$	2.22219e10	1.46288	0.731439	+	0.681907i	$0.238849\pi$
$812$	0	0
$813$	−2.60447e10	−1.69982
$814$	4.13278e9	0.268570
$815$	−2.79460e10	−1.80829
$816$	−8.34978e8	−0.0537972
$817$	−1.33028e9	−0.0853426
$818$	−1.54046e10	−0.984043
$819$	0	0
$820$	−4.03914e9	−0.255823
$821$	2.57309e9	0.162276	0.0811381	−	0.996703i	$-0.474145\pi$
$821$	2.57309e9	0.162276	0.0811381	+	0.996703i	$0.474145\pi$
$822$	−2.37459e10	−1.49121
$823$	−5.15490e9	−0.322345	−0.161172	−	0.986926i	$-0.551528\pi$
$823$	−5.15490e9	−0.322345	−0.161172	+	0.986926i	$0.551528\pi$
$824$	−8.29851e9	−0.516719
$825$	2.42040e10	1.50071
$826$	0	0
$827$	2.02079e10	1.24237	0.621185	−	0.783664i	$-0.286651\pi$
$827$	2.02079e10	1.24237	0.621185	+	0.783664i	$0.286651\pi$
$828$	−3.73994e9	−0.228959
$829$	2.26920e10	1.38335	0.691676	−	0.722208i	$-0.256873\pi$
$829$	2.26920e10	1.38335	0.691676	+	0.722208i	$0.256873\pi$
$830$	5.42689e7	0.00329441
$831$	2.40856e10	1.45597
$832$	−1.86542e9	−0.112291
$833$	0	0
$834$	−3.46142e10	−2.06620
$835$	1.48395e10	0.882096
$836$	−5.62231e9	−0.332808
$837$	−5.44640e10	−3.21048
$838$	−1.91997e10	−1.12704
$839$	1.18283e10	0.691444	0.345722	−	0.938337i	$-0.387634\pi$
$839$	1.18283e10	0.691444	0.345722	+	0.938337i	$0.387634\pi$
$840$	0	0
$841$	−9.48932e9	−0.550110
$842$	−9.67520e9	−0.558557
$843$	−4.46922e9	−0.256942
$844$	6.00324e9	0.343706
$845$	5.42576e9	0.309358
$846$	2.60194e10	1.47741
$847$	0	0
$848$	−2.33251e8	−0.0131352
$849$	−1.68435e10	−0.944618
$850$	2.43785e9	0.136157
$851$	2.76320e9	0.153695
$852$	−5.59563e8	−0.0309963
$853$	−1.17141e10	−0.646230	−0.323115	−	0.946360i	$-0.604730\pi$
$853$	−1.17141e10	−0.646230	−0.323115	+	0.946360i	$0.604730\pi$
$854$	0	0
$855$	7.41524e10	4.05736
$856$	−3.56649e9	−0.194349
$857$	−1.67955e10	−0.911506	−0.455753	−	0.890106i	$-0.650630\pi$
$857$	−1.67955e10	−0.911506	−0.455753	+	0.890106i	$0.650630\pi$
$858$	1.12408e10	0.607562
$859$	1.86233e10	1.00249	0.501244	−	0.865306i	$-0.332876\pi$
$859$	1.86233e10	1.00249	0.501244	+	0.865306i	$0.332876\pi$
$860$	−1.04550e9	−0.0560503
$861$	0	0
$862$	1.73028e10	0.920113
$863$	−1.95683e10	−1.03637	−0.518184	−	0.855269i	$-0.673392\pi$
$863$	−1.95683e10	−1.03637	−0.518184	+	0.855269i	$0.673392\pi$
$864$	−6.31439e9	−0.333068
$865$	−3.16198e10	−1.66113
$866$	1.80600e10	0.944941
$867$	−3.31410e10	−1.72703
$868$	0	0
$869$	8.22548e9	0.425199
$870$	−2.58898e10	−1.33294
$871$	2.15925e10	1.10724
$872$	7.63079e9	0.389728
$873$	−6.12803e10	−3.11725
$874$	−3.75911e9	−0.190456
$875$	0	0
$876$	−5.18990e9	−0.260853
$877$	−8.76080e9	−0.438576	−0.219288	−	0.975660i	$-0.570373\pi$
$877$	−8.76080e9	−0.438576	−0.219288	+	0.975660i	$0.570373\pi$
$878$	−5.96579e9	−0.297466
$879$	9.74870e9	0.484156
$880$	−4.41870e9	−0.218577
$881$	−3.59103e10	−1.76931	−0.884653	−	0.466250i	$-0.845605\pi$
$881$	−3.59103e10	−1.76931	−0.884653	+	0.466250i	$0.845605\pi$
$882$	0	0
$883$	−3.42420e10	−1.67377	−0.836887	−	0.547375i	$-0.815627\pi$
$883$	−3.42420e10	−1.67377	−0.836887	+	0.547375i	$0.815627\pi$
$884$	1.13218e9	0.0551232
$885$	−7.89773e10	−3.83002
$886$	8.81722e9	0.425906
$887$	−1.33299e10	−0.641350	−0.320675	−	0.947189i	$-0.603910\pi$
$887$	−1.33299e10	−0.641350	−0.320675	+	0.947189i	$0.603910\pi$
$888$	9.00700e9	0.431653
$889$	0	0
$890$	−6.26519e8	−0.0297899
$891$	1.41566e10	0.670481
$892$	−1.04043e8	−0.00490835
$893$	2.61528e10	1.22896
$894$	−4.82818e10	−2.25997
$895$	5.42870e10	2.53113
$896$	0	0
$897$	7.51563e9	0.347690
$898$	−2.73280e10	−1.25933
$899$	2.48985e10	1.14292
$900$	3.55930e10	1.62748
$901$	1.41568e8	0.00644804
$902$	−2.71380e9	−0.123127
$903$	0	0
$904$	1.33470e10	0.600891
$905$	1.03286e10	0.463205
$906$	−1.14334e10	−0.510769
$907$	−3.95632e9	−0.176062	−0.0880310	−	0.996118i	$-0.528057\pi$
$907$	−3.95632e9	−0.176062	−0.0880310	+	0.996118i	$0.528057\pi$
$908$	1.57538e10	0.698367
$909$	8.51344e10	3.75951
$910$	0	0
$911$	1.77864e10	0.779426	0.389713	−	0.920936i	$-0.372574\pi$
$911$	1.77864e10	0.779426	0.389713	+	0.920936i	$0.372574\pi$
$912$	−1.22533e10	−0.534897
$913$	3.64619e7	0.00158559
$914$	−3.41848e9	−0.148088
$915$	1.13307e11	4.88971
$916$	6.78661e9	0.291755
$917$	0	0
$918$	3.83242e9	0.163502
$919$	3.79307e10	1.61208	0.806040	−	0.591862i	$-0.201607\pi$
$919$	3.79307e10	1.61208	0.806040	+	0.591862i	$0.201607\pi$
$920$	−2.95436e9	−0.125085
$921$	−1.34137e10	−0.565769
$922$	1.74493e10	0.733196
$923$	7.58736e8	0.0317603
$924$	0	0
$925$	−2.62974e10	−1.09249
$926$	−1.50508e9	−0.0622902
$927$	7.35358e10	3.03193
$928$	2.88666e9	0.118571
$929$	−4.23414e10	−1.73265	−0.866323	−	0.499485i	$-0.833523\pi$
$929$	−4.23414e10	−1.73265	−0.866323	+	0.499485i	$0.833523\pi$
$930$	−8.30633e10	−3.38625
$931$	0	0
$932$	−1.29412e10	−0.523623
$933$	3.54634e10	1.42953
$934$	6.83627e9	0.274540
$935$	2.68186e9	0.107299
$936$	1.65301e10	0.658884
$937$	−1.65147e10	−0.655817	−0.327909	−	0.944709i	$-0.606344\pi$
$937$	−1.65147e10	−0.655817	−0.327909	+	0.944709i	$0.606344\pi$
$938$	0	0
$939$	3.50167e10	1.38021
$940$	2.05540e10	0.807142
$941$	3.14329e10	1.22976	0.614881	−	0.788620i	$-0.289204\pi$
$941$	3.14329e10	1.22976	0.614881	+	0.788620i	$0.289204\pi$
$942$	−1.45735e10	−0.568049
$943$	−1.81446e9	−0.0704621
$944$	8.80583e9	0.340697
$945$	0	0
$946$	−7.02442e8	−0.0269769
$947$	−3.27114e10	−1.25163	−0.625813	−	0.779973i	$-0.715233\pi$
$947$	−3.27114e10	−1.25163	−0.625813	+	0.779973i	$0.715233\pi$
$948$	1.79266e10	0.683391
$949$	7.03723e9	0.267282
$950$	3.57754e10	1.35379
$951$	−2.84952e10	−1.07433
$952$	0	0
$953$	−2.62655e10	−0.983017	−0.491509	−	0.870873i	$-0.663554\pi$
$953$	−2.62655e10	−0.983017	−0.491509	+	0.870873i	$0.663554\pi$
$954$	2.06691e9	0.0770731
$955$	9.09113e9	0.337758
$956$	−2.35215e9	−0.0870687
$957$	−1.73947e10	−0.641542
$958$	−5.46251e9	−0.200730
$959$	0	0
$960$	−9.63012e9	−0.351303
$961$	5.23705e10	1.90351
$962$	−1.22130e10	−0.442292
$963$	3.16038e10	1.14037
$964$	−3.68906e9	−0.132631
$965$	−1.01284e10	−0.362825
$966$	0	0
$967$	2.58253e10	0.918443	0.459222	−	0.888322i	$-0.348128\pi$
$967$	2.58253e10	0.918443	0.459222	+	0.888322i	$0.348128\pi$
$968$	7.00862e9	0.248353
$969$	7.43693e9	0.262579
$970$	−4.84083e10	−1.70302
$971$	−2.24102e10	−0.785560	−0.392780	−	0.919632i	$-0.628487\pi$
$971$	−2.24102e10	−0.785560	−0.392780	+	0.919632i	$0.628487\pi$
$972$	3.88106e9	0.135556
$973$	0	0
$974$	−3.32095e10	−1.15161
$975$	−7.15263e10	−2.47144
$976$	−1.26335e10	−0.434961
$977$	−3.54582e10	−1.21643	−0.608214	−	0.793773i	$-0.708114\pi$
$977$	−3.54582e10	−1.21643	−0.608214	+	0.793773i	$0.708114\pi$
$978$	−4.09210e10	−1.39881
$979$	−4.20942e8	−0.0143378
$980$	0	0
$981$	−6.76190e10	−2.28679
$982$	−7.19631e9	−0.242504
$983$	1.45767e10	0.489467	0.244733	−	0.969590i	$-0.421300\pi$
$983$	1.45767e10	0.489467	0.244733	+	0.969590i	$0.421300\pi$
$984$	−5.91445e9	−0.197894
$985$	1.46576e10	0.488694
$986$	−1.75201e9	−0.0582061
$987$	0	0
$988$	1.66148e10	0.548082
$989$	−4.69656e8	−0.0154381
$990$	3.91555e10	1.28254
$991$	1.25713e10	0.410319	0.205160	−	0.978729i	$-0.434229\pi$
$991$	1.25713e10	0.410319	0.205160	+	0.978729i	$0.434229\pi$
$992$	9.26142e9	0.301222
$993$	−6.68639e10	−2.16705
$994$	0	0
$995$	3.24564e10	1.04453
$996$	7.94652e7	0.00254841
$997$	1.04037e10	0.332471	0.166236	−	0.986086i	$-0.446839\pi$
$997$	1.04037e10	0.332471	0.166236	+	0.986086i	$0.446839\pi$
$998$	4.88888e9	0.155687
$999$	−4.13407e10	−1.31189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.8.a.b.1.1		1
7.2	even	3		98.8.c.c.67.1		2
7.3	odd	6		98.8.c.f.79.1		2
7.4	even	3		98.8.c.c.79.1		2
7.5	odd	6		98.8.c.f.67.1		2
7.6	odd	2		14.8.a.a.1.1	✓	1
21.20	even	2		126.8.a.d.1.1		1
28.27	even	2		112.8.a.e.1.1		1
35.13	even	4		350.8.c.d.99.2		2
35.27	even	4		350.8.c.d.99.1		2
35.34	odd	2		350.8.a.h.1.1		1
56.13	odd	2		448.8.a.j.1.1		1
56.27	even	2		448.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.a.1.1	✓	1	7.6	odd	2
98.8.a.b.1.1		1	1.1	even	1	trivial
98.8.c.c.67.1		2	7.2	even	3
98.8.c.c.79.1		2	7.4	even	3
98.8.c.f.67.1		2	7.5	odd	6
98.8.c.f.79.1		2	7.3	odd	6
112.8.a.e.1.1		1	28.27	even	2
126.8.a.d.1.1		1	21.20	even	2
350.8.a.h.1.1		1	35.34	odd	2
350.8.c.d.99.1		2	35.27	even	4
350.8.c.d.99.2		2	35.13	even	4
448.8.a.a.1.1		1	56.27	even	2
448.8.a.j.1.1		1	56.13	odd	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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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