LMFDB - Embedded newform 15.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,8,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, 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("15.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	15.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:	$4.68577538226$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-22.0000 q^{2} +27.0000 q^{3} +356.000 q^{4} -125.000 q^{5} -594.000 q^{6} -420.000 q^{7} -5016.00 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-22.0000 q^{2} +27.0000 q^{3} +356.000 q^{4} -125.000 q^{5} -594.000 q^{6} -420.000 q^{7} -5016.00 q^{8} +729.000 q^{9} +2750.00 q^{10} -2944.00 q^{11} +9612.00 q^{12} -11006.0 q^{13} +9240.00 q^{14} -3375.00 q^{15} +64784.0 q^{16} -16546.0 q^{17} -16038.0 q^{18} -25364.0 q^{19} -44500.0 q^{20} -11340.0 q^{21} +64768.0 q^{22} -5880.00 q^{23} -135432. q^{24} +15625.0 q^{25} +242132. q^{26} +19683.0 q^{27} -149520. q^{28} +163042. q^{29} +74250.0 q^{30} -201600. q^{31} -783200. q^{32} -79488.0 q^{33} +364012. q^{34} +52500.0 q^{35} +259524. q^{36} +120530. q^{37} +558008. q^{38} -297162. q^{39} +627000. q^{40} -115910. q^{41} +249480. q^{42} -19148.0 q^{43} -1.04806e6 q^{44} -91125.0 q^{45} +129360. q^{46} +841016. q^{47} +1.74917e6 q^{48} -647143. q^{49} -343750. q^{50} -446742. q^{51} -3.91814e6 q^{52} +501890. q^{53} -433026. q^{54} +368000. q^{55} +2.10672e6 q^{56} -684828. q^{57} -3.58692e6 q^{58} -1.58618e6 q^{59} -1.20150e6 q^{60} -372962. q^{61} +4.43520e6 q^{62} -306180. q^{63} +8.93805e6 q^{64} +1.37575e6 q^{65} +1.74874e6 q^{66} +4.56104e6 q^{67} -5.89038e6 q^{68} -158760. q^{69} -1.15500e6 q^{70} +1.51283e6 q^{71} -3.65666e6 q^{72} -1.52291e6 q^{73} -2.65166e6 q^{74} +421875. q^{75} -9.02958e6 q^{76} +1.23648e6 q^{77} +6.53756e6 q^{78} +4.23192e6 q^{79} -8.09800e6 q^{80} +531441. q^{81} +2.55002e6 q^{82} -1.85420e6 q^{83} -4.03704e6 q^{84} +2.06825e6 q^{85} +421256. q^{86} +4.40213e6 q^{87} +1.47671e7 q^{88} -6.88817e6 q^{89} +2.00475e6 q^{90} +4.62252e6 q^{91} -2.09328e6 q^{92} -5.44320e6 q^{93} -1.85024e7 q^{94} +3.17050e6 q^{95} -2.11464e7 q^{96} +3.70003e6 q^{97} +1.42371e7 q^{98} -2.14618e6 q^{99} +O(q^{100})\)

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 $

$2$	−22.0000	−1.94454	−0.972272	−	0.233854i	$-0.924866\pi$
$2$	−22.0000	−1.94454	−0.972272	+	0.233854i	$0.924866\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	356.000	2.78125
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	−594.000	−1.12268
$7$	−420.000	−0.462814	−0.231407	−	0.972857i	$-0.574333\pi$
$7$	−420.000	−0.462814	−0.231407	+	0.972857i	$0.574333\pi$
$8$	−5016.00	−3.46372
$9$	729.000	0.333333
$10$	2750.00	0.869626
$11$	−2944.00	−0.666904	−0.333452	−	0.942767i	$-0.608214\pi$
$11$	−2944.00	−0.666904	−0.333452	+	0.942767i	$0.608214\pi$
$12$	9612.00	1.60576
$13$	−11006.0	−1.38940	−0.694701	−	0.719299i	$-0.744463\pi$
$13$	−11006.0	−1.38940	−0.694701	+	0.719299i	$0.744463\pi$
$14$	9240.00	0.899961
$15$	−3375.00	−0.258199
$16$	64784.0	3.95410
$17$	−16546.0	−0.816811	−0.408406	−	0.912801i	$-0.633915\pi$
$17$	−16546.0	−0.816811	−0.408406	+	0.912801i	$0.633915\pi$
$18$	−16038.0	−0.648181
$19$	−25364.0	−0.848360	−0.424180	−	0.905578i	$-0.639438\pi$
$19$	−25364.0	−0.848360	−0.424180	+	0.905578i	$0.639438\pi$
$20$	−44500.0	−1.24381
$21$	−11340.0	−0.267206
$22$	64768.0	1.29682
$23$	−5880.00	−0.100770	−0.0503848	−	0.998730i	$-0.516045\pi$
$23$	−5880.00	−0.100770	−0.0503848	+	0.998730i	$0.516045\pi$
$24$	−135432.	−1.99978
$25$	15625.0	0.200000
$26$	242132.	2.70175
$27$	19683.0	0.192450
$28$	−149520.	−1.28720
$29$	163042.	1.24139	0.620693	−	0.784054i	$-0.286852\pi$
$29$	163042.	1.24139	0.620693	+	0.784054i	$0.286852\pi$
$30$	74250.0	0.502079
$31$	−201600.	−1.21542	−0.607708	−	0.794161i	$-0.707911\pi$
$31$	−201600.	−1.21542	−0.607708	+	0.794161i	$0.707911\pi$
$32$	−783200.	−4.22520
$33$	−79488.0	−0.385037
$34$	364012.	1.58833
$35$	52500.0	0.206977
$36$	259524.	0.927083
$37$	120530.	0.391191	0.195596	−	0.980685i	$-0.437336\pi$
$37$	120530.	0.391191	0.195596	+	0.980685i	$0.437336\pi$
$38$	558008.	1.64967
$39$	−297162.	−0.802171
$40$	627000.	1.54902
$41$	−115910.	−0.262650	−0.131325	−	0.991339i	$-0.541923\pi$
$41$	−115910.	−0.262650	−0.131325	+	0.991339i	$0.541923\pi$
$42$	249480.	0.519593
$43$	−19148.0	−0.0367269	−0.0183634	−	0.999831i	$-0.505846\pi$
$43$	−19148.0	−0.0367269	−0.0183634	+	0.999831i	$0.505846\pi$
$44$	−1.04806e6	−1.85483
$45$	−91125.0	−0.149071
$46$	129360.	0.195951
$47$	841016.	1.18158	0.590788	−	0.806827i	$-0.298817\pi$
$47$	841016.	1.18158	0.590788	+	0.806827i	$0.298817\pi$
$48$	1.74917e6	2.28290
$49$	−647143.	−0.785804
$50$	−343750.	−0.388909
$51$	−446742.	−0.471586
$52$	−3.91814e6	−3.86427
$53$	501890.	0.463066	0.231533	−	0.972827i	$-0.425626\pi$
$53$	501890.	0.463066	0.231533	+	0.972827i	$0.425626\pi$
$54$	−433026.	−0.374228
$55$	368000.	0.298249
$56$	2.10672e6	1.60306
$57$	−684828.	−0.489801
$58$	−3.58692e6	−2.41393
$59$	−1.58618e6	−1.00547	−0.502735	−	0.864440i	$-0.667673\pi$
$59$	−1.58618e6	−1.00547	−0.502735	+	0.864440i	$0.667673\pi$
$60$	−1.20150e6	−0.718116
$61$	−372962.	−0.210383	−0.105191	−	0.994452i	$-0.533546\pi$
$61$	−372962.	−0.210383	−0.105191	+	0.994452i	$0.533546\pi$
$62$	4.43520e6	2.36343
$63$	−306180.	−0.154271
$64$	8.93805e6	4.26199
$65$	1.37575e6	0.621359
$66$	1.74874e6	0.748722
$67$	4.56104e6	1.85269	0.926344	−	0.376678i	$-0.122934\pi$
$67$	4.56104e6	1.85269	0.926344	+	0.376678i	$0.122934\pi$
$68$	−5.89038e6	−2.27176
$69$	−158760.	−0.0581794
$70$	−1.15500e6	−0.402475
$71$	1.51283e6	0.501633	0.250817	−	0.968035i	$-0.419301\pi$
$71$	1.51283e6	0.501633	0.250817	+	0.968035i	$0.419301\pi$
$72$	−3.65666e6	−1.15457
$73$	−1.52291e6	−0.458189	−0.229094	−	0.973404i	$-0.573576\pi$
$73$	−1.52291e6	−0.458189	−0.229094	+	0.973404i	$0.573576\pi$
$74$	−2.65166e6	−0.760688
$75$	421875.	0.115470
$76$	−9.02958e6	−2.35950
$77$	1.23648e6	0.308652
$78$	6.53756e6	1.55986
$79$	4.23192e6	0.965701	0.482850	−	0.875703i	$-0.339601\pi$
$79$	4.23192e6	0.965701	0.482850	+	0.875703i	$0.339601\pi$
$80$	−8.09800e6	−1.76833
$81$	531441.	0.111111
$82$	2.55002e6	0.510734
$83$	−1.85420e6	−0.355946	−0.177973	−	0.984035i	$-0.556954\pi$
$83$	−1.85420e6	−0.355946	−0.177973	+	0.984035i	$0.556954\pi$
$84$	−4.03704e6	−0.743166
$85$	2.06825e6	0.365289
$86$	421256.	0.0714170
$87$	4.40213e6	0.716714
$88$	1.47671e7	2.30997
$89$	−6.88817e6	−1.03571	−0.517856	−	0.855468i	$-0.673270\pi$
$89$	−6.88817e6	−1.03571	−0.517856	+	0.855468i	$0.673270\pi$
$90$	2.00475e6	0.289875
$91$	4.62252e6	0.643034
$92$	−2.09328e6	−0.280266
$93$	−5.44320e6	−0.701720
$94$	−1.85024e7	−2.29763
$95$	3.17050e6	0.379398
$96$	−2.11464e7	−2.43942
$97$	3.70003e6	0.411628	0.205814	−	0.978591i	$-0.434016\pi$
$97$	3.70003e6	0.411628	0.205814	+	0.978591i	$0.434016\pi$
$98$	1.42371e7	1.52803
$99$	−2.14618e6	−0.222301
$100$	5.56250e6	0.556250
$101$	−1.80025e7	−1.73863	−0.869314	−	0.494259i	$-0.835439\pi$
$101$	−1.80025e7	−1.73863	−0.869314	+	0.494259i	$0.835439\pi$
$102$	9.82832e6	0.917020
$103$	−5.37207e6	−0.484408	−0.242204	−	0.970225i	$-0.577870\pi$
$103$	−5.37207e6	−0.484408	−0.242204	+	0.970225i	$0.577870\pi$
$104$	5.52061e7	4.81250
$105$	1.41750e6	0.119498
$106$	−1.10416e7	−0.900452
$107$	−1.15398e7	−0.910655	−0.455327	−	0.890324i	$-0.650478\pi$
$107$	−1.15398e7	−0.910655	−0.455327	+	0.890324i	$0.650478\pi$
$108$	7.00715e6	0.535252
$109$	−1.57179e6	−0.116253	−0.0581263	−	0.998309i	$-0.518513\pi$
$109$	−1.57179e6	−0.116253	−0.0581263	+	0.998309i	$0.518513\pi$
$110$	−8.09600e6	−0.579957
$111$	3.25431e6	0.225854
$112$	−2.72093e7	−1.83001
$113$	−2.52050e7	−1.64328	−0.821640	−	0.570007i	$-0.806940\pi$
$113$	−2.52050e7	−1.64328	−0.821640	+	0.570007i	$0.806940\pi$
$114$	1.50662e7	0.952439
$115$	735000.	0.0450656
$116$	5.80430e7	3.45260
$117$	−8.02337e6	−0.463134
$118$	3.48959e7	1.95518
$119$	6.94932e6	0.378031
$120$	1.69290e7	0.894328
$121$	−1.08200e7	−0.555239
$122$	8.20516e6	0.409098
$123$	−3.12957e6	−0.151641
$124$	−7.17696e7	−3.38037
$125$	−1.95312e6	−0.0894427
$126$	6.73596e6	0.299987
$127$	3.94080e7	1.70715	0.853574	−	0.520971i	$-0.174430\pi$
$127$	3.94080e7	1.70715	0.853574	+	0.520971i	$0.174430\pi$
$128$	−9.63875e7	−4.06243
$129$	−516996.	−0.0212043
$130$	−3.02665e7	−1.20826
$131$	1.41082e7	0.548305	0.274153	−	0.961686i	$-0.411603\pi$
$131$	1.41082e7	0.548305	0.274153	+	0.961686i	$0.411603\pi$
$132$	−2.82977e7	−1.07088
$133$	1.06529e7	0.392633
$134$	−1.00343e8	−3.60263
$135$	−2.46038e6	−0.0860663
$136$	8.29947e7	2.82920
$137$	−8.00512e6	−0.265978	−0.132989	−	0.991118i	$-0.542458\pi$
$137$	−8.00512e6	−0.265978	−0.132989	+	0.991118i	$0.542458\pi$
$138$	3.49272e6	0.113132
$139$	4.60716e7	1.45506	0.727532	−	0.686074i	$-0.240667\pi$
$139$	4.60716e7	1.45506	0.727532	+	0.686074i	$0.240667\pi$
$140$	1.86900e7	0.575654
$141$	2.27074e7	0.682183
$142$	−3.32823e7	−0.975448
$143$	3.24017e7	0.926598
$144$	4.72275e7	1.31803
$145$	−2.03802e7	−0.555164
$146$	3.35040e7	0.890968
$147$	−1.74729e7	−0.453684
$148$	4.29087e7	1.08800
$149$	7.23525e7	1.79185	0.895925	−	0.444206i	$-0.146514\pi$
$149$	7.23525e7	1.79185	0.895925	+	0.444206i	$0.146514\pi$
$150$	−9.28125e6	−0.224537
$151$	−3.70062e7	−0.874692	−0.437346	−	0.899293i	$-0.644082\pi$
$151$	−3.70062e7	−0.874692	−0.437346	+	0.899293i	$0.644082\pi$
$152$	1.27226e8	2.93848
$153$	−1.20620e7	−0.272270
$154$	−2.72026e7	−0.600188
$155$	2.52000e7	0.543550
$156$	−1.05790e8	−2.23104
$157$	−7.85080e7	−1.61907	−0.809534	−	0.587073i	$-0.800280\pi$
$157$	−7.85080e7	−1.61907	−0.809534	+	0.587073i	$0.800280\pi$
$158$	−9.31022e7	−1.87785
$159$	1.35510e7	0.267351
$160$	9.79000e7	1.88957
$161$	2.46960e6	0.0466376
$162$	−1.16917e7	−0.216060
$163$	−4.68184e7	−0.846759	−0.423380	−	0.905952i	$-0.639156\pi$
$163$	−4.68184e7	−0.846759	−0.423380	+	0.905952i	$0.639156\pi$
$164$	−4.12640e7	−0.730495
$165$	9.93600e6	0.172194
$166$	4.07925e7	0.692153
$167$	−2.50043e7	−0.415438	−0.207719	−	0.978188i	$-0.566604\pi$
$167$	−2.50043e7	−0.415438	−0.207719	+	0.978188i	$0.566604\pi$
$168$	5.68814e7	0.925525
$169$	5.83835e7	0.930437
$170$	−4.55015e7	−0.710321
$171$	−1.84904e7	−0.282787
$172$	−6.81669e6	−0.102147
$173$	5.30671e7	0.779227	0.389613	−	0.920979i	$-0.372609\pi$
$173$	5.30671e7	0.779227	0.389613	+	0.920979i	$0.372609\pi$
$174$	−9.68469e7	−1.39368
$175$	−6.56250e6	−0.0925627
$176$	−1.90724e8	−2.63701
$177$	−4.28268e7	−0.580509
$178$	1.51540e8	2.01399
$179$	4.22054e7	0.550025	0.275012	−	0.961441i	$-0.411318\pi$
$179$	4.22054e7	0.550025	0.275012	+	0.961441i	$0.411318\pi$
$180$	−3.24405e7	−0.414604
$181$	−1.00020e8	−1.25376	−0.626879	−	0.779116i	$-0.715668\pi$
$181$	−1.00020e8	−1.25376	−0.626879	+	0.779116i	$0.715668\pi$
$182$	−1.01695e8	−1.25041
$183$	−1.00700e7	−0.121465
$184$	2.94941e7	0.349038
$185$	−1.50662e7	−0.174946
$186$	1.19750e8	1.36453
$187$	4.87114e7	0.544735
$188$	2.99402e8	3.28626
$189$	−8.26686e6	−0.0890685
$190$	−6.97510e7	−0.737756
$191$	6.17610e7	0.641354	0.320677	−	0.947189i	$-0.396090\pi$
$191$	6.17610e7	0.641354	0.320677	+	0.947189i	$0.396090\pi$
$192$	2.41327e8	2.46066
$193$	−7.67419e7	−0.768390	−0.384195	−	0.923252i	$-0.625521\pi$
$193$	−7.67419e7	−0.768390	−0.384195	+	0.923252i	$0.625521\pi$
$194$	−8.14007e7	−0.800428
$195$	3.71452e7	0.358742
$196$	−2.30383e8	−2.18552
$197$	−1.81032e8	−1.68703	−0.843516	−	0.537105i	$-0.819518\pi$
$197$	−1.81032e8	−1.68703	−0.843516	+	0.537105i	$0.819518\pi$
$198$	4.72159e7	0.432275
$199$	6.16084e7	0.554185	0.277092	−	0.960843i	$-0.410629\pi$
$199$	6.16084e7	0.554185	0.277092	+	0.960843i	$0.410629\pi$
$200$	−7.83750e7	−0.692744
$201$	1.23148e8	1.06965
$202$	3.96054e8	3.38084
$203$	−6.84776e7	−0.574530
$204$	−1.59040e8	−1.31160
$205$	1.44888e7	0.117461
$206$	1.18185e8	0.941952
$207$	−4.28652e6	−0.0335899
$208$	−7.13013e8	−5.49383
$209$	7.46716e7	0.565775
$210$	−3.11850e7	−0.232369
$211$	−1.69917e8	−1.24523	−0.622613	−	0.782530i	$-0.713929\pi$
$211$	−1.69917e8	−1.24523	−0.622613	+	0.782530i	$0.713929\pi$
$212$	1.78673e8	1.28790
$213$	4.08465e7	0.289618
$214$	2.53875e8	1.77081
$215$	2.39350e6	0.0164248
$216$	−9.87299e7	−0.666593
$217$	8.46720e7	0.562511
$218$	3.45795e7	0.226058
$219$	−4.11186e7	−0.264535
$220$	1.31008e8	0.829504
$221$	1.82105e8	1.13488
$222$	−7.15948e7	−0.439184
$223$	1.48129e8	0.894484	0.447242	−	0.894413i	$-0.352406\pi$
$223$	1.48129e8	0.894484	0.447242	+	0.894413i	$0.352406\pi$
$224$	3.28944e8	1.95548
$225$	1.13906e7	0.0666667
$226$	5.54509e8	3.19543
$227$	−3.96127e7	−0.224773	−0.112387	−	0.993665i	$-0.535849\pi$
$227$	−3.96127e7	−0.224773	−0.112387	+	0.993665i	$0.535849\pi$
$228$	−2.43799e8	−1.36226
$229$	−3.71816e7	−0.204599	−0.102300	−	0.994754i	$-0.532620\pi$
$229$	−3.71816e7	−0.204599	−0.102300	+	0.994754i	$0.532620\pi$
$230$	−1.61700e7	−0.0876320
$231$	3.33850e7	0.178201
$232$	−8.17819e8	−4.29981
$233$	−1.79591e8	−0.930122	−0.465061	−	0.885279i	$-0.653968\pi$
$233$	−1.79591e8	−0.930122	−0.465061	+	0.885279i	$0.653968\pi$
$234$	1.76514e8	0.900584
$235$	−1.05127e8	−0.528417
$236$	−5.64679e8	−2.79646
$237$	1.14262e8	0.557548
$238$	−1.52885e8	−0.735099
$239$	−3.73328e8	−1.76888	−0.884439	−	0.466655i	$-0.845459\pi$
$239$	−3.73328e8	−1.76888	−0.884439	+	0.466655i	$0.845459\pi$
$240$	−2.18646e8	−1.02094
$241$	−2.57022e8	−1.18280	−0.591398	−	0.806380i	$-0.701424\pi$
$241$	−2.57022e8	−1.18280	−0.591398	+	0.806380i	$0.701424\pi$
$242$	2.38041e8	1.07969
$243$	1.43489e7	0.0641500
$244$	−1.32774e8	−0.585127
$245$	8.08929e7	0.351422
$246$	6.88505e7	0.294873
$247$	2.79156e8	1.17871
$248$	1.01123e9	4.20986
$249$	−5.00635e7	−0.205506
$250$	4.29688e7	0.173925
$251$	−1.27344e8	−0.508302	−0.254151	−	0.967165i	$-0.581796\pi$
$251$	−1.27344e8	−0.508302	−0.254151	+	0.967165i	$0.581796\pi$
$252$	−1.09000e8	−0.429067
$253$	1.73107e7	0.0672037
$254$	−8.66976e8	−3.31963
$255$	5.58427e7	0.210900
$256$	9.76454e8	3.63757
$257$	1.30682e8	0.480230	0.240115	−	0.970744i	$-0.422815\pi$
$257$	1.30682e8	0.480230	0.240115	+	0.970744i	$0.422815\pi$
$258$	1.13739e7	0.0412326
$259$	−5.06226e7	−0.181049
$260$	4.89767e8	1.72816
$261$	1.18858e8	0.413795
$262$	−3.10381e8	−1.06620
$263$	−2.67747e8	−0.907568	−0.453784	−	0.891112i	$-0.649926\pi$
$263$	−2.67747e8	−0.907568	−0.453784	+	0.891112i	$0.649926\pi$
$264$	3.98712e8	1.33366
$265$	−6.27363e7	−0.207089
$266$	−2.34363e8	−0.763491
$267$	−1.85981e8	−0.597969
$268$	1.62373e9	5.15279
$269$	1.49432e8	0.468070	0.234035	−	0.972228i	$-0.424807\pi$
$269$	1.49432e8	0.468070	0.234035	+	0.972228i	$0.424807\pi$
$270$	5.41282e7	0.167360
$271$	−1.53185e8	−0.467545	−0.233773	−	0.972291i	$-0.575107\pi$
$271$	−1.53185e8	−0.467545	−0.233773	+	0.972291i	$0.575107\pi$
$272$	−1.07192e9	−3.22976
$273$	1.24808e8	0.371256
$274$	1.76113e8	0.517206
$275$	−4.60000e7	−0.133381
$276$	−5.65186e7	−0.161811
$277$	6.54462e8	1.85014	0.925072	−	0.379792i	$-0.124004\pi$
$277$	6.54462e8	1.85014	0.925072	+	0.379792i	$0.124004\pi$
$278$	−1.01358e9	−2.82943
$279$	−1.46966e8	−0.405138
$280$	−2.63340e8	−0.716908
$281$	5.51493e8	1.48275	0.741375	−	0.671091i	$-0.234174\pi$
$281$	5.51493e8	1.48275	0.741375	+	0.671091i	$0.234174\pi$
$282$	−4.99564e8	−1.32654
$283$	−2.10200e8	−0.551291	−0.275646	−	0.961259i	$-0.588892\pi$
$283$	−2.10200e8	−0.551291	−0.275646	+	0.961259i	$0.588892\pi$
$284$	5.38568e8	1.39517
$285$	8.56035e7	0.219046
$286$	−7.12837e8	−1.80181
$287$	4.86822e7	0.121558
$288$	−5.70953e8	−1.40840
$289$	−1.36569e8	−0.332819
$290$	4.48366e8	1.07954
$291$	9.99009e7	0.237653
$292$	−5.42156e8	−1.27434
$293$	5.42402e8	1.25975	0.629875	−	0.776697i	$-0.283106\pi$
$293$	5.42402e8	1.25975	0.629875	+	0.776697i	$0.283106\pi$
$294$	3.84403e8	0.882208
$295$	1.98272e8	0.449660
$296$	−6.04578e8	−1.35498
$297$	−5.79468e7	−0.128346
$298$	−1.59175e9	−3.48433
$299$	6.47153e7	0.140010
$300$	1.50188e8	0.321151
$301$	8.04216e6	0.0169977
$302$	8.14137e8	1.70088
$303$	−4.86066e8	−1.00380
$304$	−1.64318e9	−3.35450
$305$	4.66202e7	0.0940860
$306$	2.65365e8	0.529442
$307$	9.26477e8	1.82747	0.913736	−	0.406310i	$-0.133185\pi$
$307$	9.26477e8	1.82747	0.913736	+	0.406310i	$0.133185\pi$
$308$	4.40187e8	0.858439
$309$	−1.45046e8	−0.279673
$310$	−5.54400e8	−1.05696
$311$	−2.12976e8	−0.401485	−0.200743	−	0.979644i	$-0.564335\pi$
$311$	−2.12976e8	−0.401485	−0.200743	+	0.979644i	$0.564335\pi$
$312$	1.49056e9	2.77850
$313$	3.63896e8	0.670768	0.335384	−	0.942081i	$-0.391134\pi$
$313$	3.63896e8	0.670768	0.335384	+	0.942081i	$0.391134\pi$
$314$	1.72718e9	3.14835
$315$	3.82725e7	0.0689922
$316$	1.50656e9	2.68586
$317$	−3.17049e8	−0.559009	−0.279505	−	0.960144i	$-0.590170\pi$
$317$	−3.17049e8	−0.559009	−0.279505	+	0.960144i	$0.590170\pi$
$318$	−2.98123e8	−0.519876
$319$	−4.79996e8	−0.827885
$320$	−1.11726e9	−1.90602
$321$	−3.11574e8	−0.525767
$322$	−5.43312e7	−0.0906888
$323$	4.19673e8	0.692950
$324$	1.89193e8	0.309028
$325$	−1.71969e8	−0.277880
$326$	1.03001e9	1.64656
$327$	−4.24384e7	−0.0671185
$328$	5.81405e8	0.909746
$329$	−3.53227e8	−0.546850
$330$	−2.18592e8	−0.334839
$331$	2.24556e8	0.340351	0.170176	−	0.985414i	$-0.445566\pi$
$331$	2.24556e8	0.340351	0.170176	+	0.985414i	$0.445566\pi$
$332$	−6.60097e8	−0.989975
$333$	8.78664e7	0.130397
$334$	5.50094e8	0.807838
$335$	−5.70130e8	−0.828548
$336$	−7.34651e8	−1.05656
$337$	−1.23886e9	−1.76327	−0.881633	−	0.471935i	$-0.843556\pi$
$337$	−1.23886e9	−1.76327	−0.881633	+	0.471935i	$0.843556\pi$
$338$	−1.28444e9	−1.80927
$339$	−6.80534e8	−0.948748
$340$	7.36297e8	1.01596
$341$	5.93510e8	0.810565
$342$	4.06788e8	0.549891
$343$	6.17688e8	0.826494
$344$	9.60464e7	0.127212
$345$	1.98450e7	0.0260186
$346$	−1.16748e9	−1.51524
$347$	−5.83643e8	−0.749884	−0.374942	−	0.927048i	$-0.622337\pi$
$347$	−5.83643e8	−0.749884	−0.374942	+	0.927048i	$0.622337\pi$
$348$	1.56716e9	1.99336
$349$	−4.69471e8	−0.591180	−0.295590	−	0.955315i	$-0.595516\pi$
$349$	−4.69471e8	−0.591180	−0.295590	+	0.955315i	$0.595516\pi$
$350$	1.44375e8	0.179992
$351$	−2.16631e8	−0.267390
$352$	2.30574e9	2.81781
$353$	6.18559e7	0.0748461	0.0374231	−	0.999300i	$-0.488085\pi$
$353$	6.18559e7	0.0748461	0.0374231	+	0.999300i	$0.488085\pi$
$354$	9.42189e8	1.12882
$355$	−1.89104e8	−0.224337
$356$	−2.45219e9	−2.88057
$357$	1.87632e8	0.218257
$358$	−9.28519e8	−1.06955
$359$	−1.23537e7	−0.0140918	−0.00704592	−	0.999975i	$-0.502243\pi$
$359$	−1.23537e7	−0.0140918	−0.00704592	+	0.999975i	$0.502243\pi$
$360$	4.57083e8	0.516341
$361$	−2.50539e8	−0.280285
$362$	2.20045e9	2.43799
$363$	−2.92141e8	−0.320567
$364$	1.64562e9	1.78844
$365$	1.90364e8	0.204908
$366$	2.21539e8	0.236193
$367$	−1.25514e7	−0.0132544	−0.00662721	−	0.999978i	$-0.502110\pi$
$367$	−1.25514e7	−0.0132544	−0.00662721	+	0.999978i	$0.502110\pi$
$368$	−3.80930e8	−0.398454
$369$	−8.44984e7	−0.0875500
$370$	3.31458e8	0.340190
$371$	−2.10794e8	−0.214313
$372$	−1.93778e9	−1.95166
$373$	−5.65994e8	−0.564717	−0.282359	−	0.959309i	$-0.591117\pi$
$373$	−5.65994e8	−0.564717	−0.282359	+	0.959309i	$0.591117\pi$
$374$	−1.07165e9	−1.05926
$375$	−5.27344e7	−0.0516398
$376$	−4.21854e9	−4.09265
$377$	−1.79444e9	−1.72478
$378$	1.81871e8	0.173198
$379$	1.77776e9	1.67740	0.838698	−	0.544597i	$-0.183317\pi$
$379$	1.77776e9	1.67740	0.838698	+	0.544597i	$0.183317\pi$
$380$	1.12870e9	1.05520
$381$	1.06402e9	0.985623
$382$	−1.35874e9	−1.24714
$383$	1.18195e9	1.07498	0.537492	−	0.843269i	$-0.319372\pi$
$383$	1.18195e9	1.07498	0.537492	+	0.843269i	$0.319372\pi$
$384$	−2.60246e9	−2.34544
$385$	−1.54560e8	−0.138034
$386$	1.68832e9	1.49417
$387$	−1.39589e7	−0.0122423
$388$	1.31721e9	1.14484
$389$	6.02482e8	0.518944	0.259472	−	0.965751i	$-0.416451\pi$
$389$	6.02482e8	0.518944	0.259472	+	0.965751i	$0.416451\pi$
$390$	−8.17196e8	−0.697589
$391$	9.72905e7	0.0823098
$392$	3.24607e9	2.72180
$393$	3.80922e8	0.316564
$394$	3.98270e9	3.28051
$395$	−5.28990e8	−0.431875
$396$	−7.64039e8	−0.618276
$397$	−1.86765e8	−0.149806	−0.0749029	−	0.997191i	$-0.523865\pi$
$397$	−1.86765e8	−0.149806	−0.0749029	+	0.997191i	$0.523865\pi$
$398$	−1.35539e9	−1.07764
$399$	2.87628e8	0.226687
$400$	1.01225e9	0.790820
$401$	−9.96333e8	−0.771613	−0.385806	−	0.922580i	$-0.626077\pi$
$401$	−9.96333e8	−0.771613	−0.385806	+	0.922580i	$0.626077\pi$
$402$	−2.70926e9	−2.07998
$403$	2.21881e9	1.68870
$404$	−6.40887e9	−4.83556
$405$	−6.64301e7	−0.0496904
$406$	1.50651e9	1.11720
$407$	−3.54840e8	−0.260887
$408$	2.24086e9	1.63344
$409$	−2.38644e9	−1.72472	−0.862362	−	0.506293i	$-0.831015\pi$
$409$	−2.38644e9	−1.72472	−0.862362	+	0.506293i	$0.831015\pi$
$410$	−3.18752e8	−0.228407
$411$	−2.16138e8	−0.153563
$412$	−1.91246e9	−1.34726
$413$	6.66194e8	0.465345
$414$	9.43034e7	0.0653170
$415$	2.31775e8	0.159184
$416$	8.61990e9	5.87051
$417$	1.24393e9	0.840081
$418$	−1.64278e9	−1.10017
$419$	2.59644e9	1.72437	0.862183	−	0.506597i	$-0.169097\pi$
$419$	2.59644e9	1.72437	0.862183	+	0.506597i	$0.169097\pi$
$420$	5.04630e8	0.332354
$421$	2.83850e9	1.85396	0.926981	−	0.375108i	$-0.122394\pi$
$421$	2.83850e9	1.85396	0.926981	+	0.375108i	$0.122394\pi$
$422$	3.73817e9	2.42139
$423$	6.13101e8	0.393859
$424$	−2.51748e9	−1.60393
$425$	−2.58531e8	−0.163362
$426$	−8.98622e8	−0.563175
$427$	1.56644e8	0.0973680
$428$	−4.10816e9	−2.53276
$429$	8.74845e8	0.534971
$430$	−5.26570e7	−0.0319386
$431$	−1.52808e9	−0.919342	−0.459671	−	0.888089i	$-0.652033\pi$
$431$	−1.52808e9	−0.919342	−0.459671	+	0.888089i	$0.652033\pi$
$432$	1.27514e9	0.760967
$433$	−2.66061e9	−1.57498	−0.787488	−	0.616330i	$-0.788619\pi$
$433$	−2.66061e9	−1.57498	−0.787488	+	0.616330i	$0.788619\pi$
$434$	−1.86278e9	−1.09383
$435$	−5.50267e8	−0.320524
$436$	−5.59559e8	−0.323328
$437$	1.49140e8	0.0854890
$438$	9.04609e8	0.514400
$439$	2.29222e9	1.29310	0.646549	−	0.762873i	$-0.276212\pi$
$439$	2.29222e9	1.29310	0.646549	+	0.762873i	$0.276212\pi$
$440$	−1.84589e9	−1.03305
$441$	−4.71767e8	−0.261935
$442$	−4.00632e9	−2.20682
$443$	−9.34043e8	−0.510451	−0.255225	−	0.966882i	$-0.582150\pi$
$443$	−9.34043e8	−0.510451	−0.255225	+	0.966882i	$0.582150\pi$
$444$	1.15853e9	0.628157
$445$	8.61022e8	0.463185
$446$	−3.25884e9	−1.73936
$447$	1.95352e9	1.03452
$448$	−3.75398e9	−1.97251
$449$	−8.85012e7	−0.0461410	−0.0230705	−	0.999734i	$-0.507344\pi$
$449$	−8.85012e7	−0.0461410	−0.0230705	+	0.999734i	$0.507344\pi$
$450$	−2.50594e8	−0.129636
$451$	3.41239e8	0.175162
$452$	−8.97296e9	−4.57037
$453$	−9.99168e8	−0.505004
$454$	8.71480e8	0.437081
$455$	−5.77815e8	−0.287574
$456$	3.43510e9	1.69653
$457$	−1.21064e9	−0.593347	−0.296674	−	0.954979i	$-0.595877\pi$
$457$	−1.21064e9	−0.593347	−0.296674	+	0.954979i	$0.595877\pi$
$458$	8.17995e8	0.397852
$459$	−3.25675e8	−0.157195
$460$	2.61660e8	0.125339
$461$	8.57428e8	0.407610	0.203805	−	0.979012i	$-0.434669\pi$
$461$	8.57428e8	0.407610	0.203805	+	0.979012i	$0.434669\pi$
$462$	−7.34469e8	−0.346519
$463$	−1.94662e9	−0.911481	−0.455741	−	0.890113i	$-0.650625\pi$
$463$	−1.94662e9	−0.911481	−0.455741	+	0.890113i	$0.650625\pi$
$464$	1.05625e10	4.90856
$465$	6.80400e8	0.313819
$466$	3.95101e9	1.80866
$467$	−9.59955e8	−0.436156	−0.218078	−	0.975931i	$-0.569979\pi$
$467$	−9.59955e8	−0.436156	−0.218078	+	0.975931i	$0.569979\pi$
$468$	−2.85632e9	−1.28809
$469$	−1.91564e9	−0.857450
$470$	2.31279e9	1.02753
$471$	−2.11972e9	−0.934769
$472$	7.95626e9	3.48267
$473$	5.63717e7	0.0244933
$474$	−2.51376e9	−1.08418
$475$	−3.96312e8	−0.169672
$476$	2.47396e9	1.05140
$477$	3.65878e8	0.154355
$478$	8.21322e9	3.43966
$479$	3.03579e8	0.126211	0.0631054	−	0.998007i	$-0.479900\pi$
$479$	3.03579e8	0.126211	0.0631054	+	0.998007i	$0.479900\pi$
$480$	2.64330e9	1.09094
$481$	−1.32655e9	−0.543522
$482$	5.65447e9	2.30000
$483$	6.66792e7	0.0269262
$484$	−3.85193e9	−1.54426
$485$	−4.62504e8	−0.184086
$486$	−3.15676e8	−0.124743
$487$	4.36059e8	0.171078	0.0855389	−	0.996335i	$-0.472739\pi$
$487$	4.36059e8	0.171078	0.0855389	+	0.996335i	$0.472739\pi$
$488$	1.87078e9	0.728707
$489$	−1.26410e9	−0.488877
$490$	−1.77964e9	−0.683355
$491$	8.34813e8	0.318276	0.159138	−	0.987256i	$-0.449128\pi$
$491$	8.34813e8	0.318276	0.159138	+	0.987256i	$0.449128\pi$
$492$	−1.11413e9	−0.421752
$493$	−2.69769e9	−1.01398
$494$	−6.14144e9	−2.29206
$495$	2.68272e8	0.0994162
$496$	−1.30605e10	−4.80587
$497$	−6.35389e8	−0.232163
$498$	1.10140e9	0.399615
$499$	−4.50230e9	−1.62212	−0.811059	−	0.584964i	$-0.801109\pi$
$499$	−4.50230e9	−1.62212	−0.811059	+	0.584964i	$0.801109\pi$
$500$	−6.95312e8	−0.248763
$501$	−6.75116e8	−0.239854
$502$	2.80158e9	0.988416
$503$	2.41700e9	0.846815	0.423407	−	0.905939i	$-0.360834\pi$
$503$	2.41700e9	0.846815	0.423407	+	0.905939i	$0.360834\pi$
$504$	1.53580e9	0.534352
$505$	2.25031e9	0.777538
$506$	−3.80836e8	−0.130681
$507$	1.57636e9	0.537188
$508$	1.40292e10	4.74801
$509$	3.51209e9	1.18047	0.590233	−	0.807233i	$-0.299036\pi$
$509$	3.51209e9	1.18047	0.590233	+	0.807233i	$0.299036\pi$
$510$	−1.22854e9	−0.410104
$511$	6.39622e8	0.212056
$512$	−9.14439e9	−3.01099
$513$	−4.99240e8	−0.163267
$514$	−2.87500e9	−0.933829
$515$	6.71508e8	0.216634
$516$	−1.84051e8	−0.0589744
$517$	−2.47595e9	−0.787998
$518$	1.11370e9	0.352057
$519$	1.43281e9	0.449887
$520$	−6.90076e9	−2.15221
$521$	9.90013e8	0.306697	0.153348	−	0.988172i	$-0.450994\pi$
$521$	9.90013e8	0.306697	0.153348	+	0.988172i	$0.450994\pi$
$522$	−2.61487e9	−0.804642
$523$	4.45926e8	0.136303	0.0681517	−	0.997675i	$-0.478290\pi$
$523$	4.45926e8	0.136303	0.0681517	+	0.997675i	$0.478290\pi$
$524$	5.02252e9	1.52497
$525$	−1.77188e8	−0.0534411
$526$	5.89043e9	1.76481
$527$	3.33567e9	0.992765
$528$	−5.14955e9	−1.52248
$529$	−3.37025e9	−0.989845
$530$	1.38020e9	0.402694
$531$	−1.15632e9	−0.335157
$532$	3.79243e9	1.09201
$533$	1.27571e9	0.364926
$534$	4.09158e9	1.16278
$535$	1.44247e9	0.407257
$536$	−2.28782e10	−6.41719
$537$	1.13955e9	0.317557
$538$	−3.28751e9	−0.910183
$539$	1.90519e9	0.524056
$540$	−8.75893e8	−0.239372
$541$	2.84753e9	0.773175	0.386588	−	0.922253i	$-0.373654\pi$
$541$	2.84753e9	0.773175	0.386588	+	0.922253i	$0.373654\pi$
$542$	3.37007e9	0.909162
$543$	−2.70055e9	−0.723858
$544$	1.29588e10	3.45120
$545$	1.96474e8	0.0519898
$546$	−2.74578e9	−0.721923
$547$	4.34116e9	1.13410	0.567048	−	0.823684i	$-0.308085\pi$
$547$	4.34116e9	1.13410	0.567048	+	0.823684i	$0.308085\pi$
$548$	−2.84982e9	−0.739752
$549$	−2.71889e8	−0.0701276
$550$	1.01200e9	0.259365
$551$	−4.13540e9	−1.05314
$552$	7.96340e8	0.201517
$553$	−1.77741e9	−0.446940
$554$	−1.43982e10	−3.59769
$555$	−4.06789e8	−0.101005
$556$	1.64015e10	4.04690
$557$	3.00124e9	0.735880	0.367940	−	0.929849i	$-0.380063\pi$
$557$	3.00124e9	0.735880	0.367940	+	0.929849i	$0.380063\pi$
$558$	3.23326e9	0.787809
$559$	2.10743e8	0.0510284
$560$	3.40116e9	0.818406
$561$	1.31521e9	0.314503
$562$	−1.21328e10	−2.88327
$563$	−7.43886e9	−1.75682	−0.878409	−	0.477909i	$-0.841395\pi$
$563$	−7.43886e9	−1.75682	−0.878409	+	0.477909i	$0.841395\pi$
$564$	8.08385e9	1.89732
$565$	3.15062e9	0.734897
$566$	4.62441e9	1.07201
$567$	−2.23205e8	−0.0514237
$568$	−7.58837e9	−1.73752
$569$	−7.20171e8	−0.163886	−0.0819431	−	0.996637i	$-0.526113\pi$
$569$	−7.20171e8	−0.163886	−0.0819431	+	0.996637i	$0.526113\pi$
$570$	−1.88328e9	−0.425944
$571$	−2.30186e9	−0.517431	−0.258716	−	0.965954i	$-0.583299\pi$
$571$	−2.30186e9	−0.517431	−0.258716	+	0.965954i	$0.583299\pi$
$572$	1.15350e10	2.57710
$573$	1.66755e9	0.370286
$574$	−1.07101e9	−0.236375
$575$	−9.18750e7	−0.0201539
$576$	6.51584e9	1.42066
$577$	−7.37257e9	−1.59773	−0.798865	−	0.601511i	$-0.794566\pi$
$577$	−7.37257e9	−1.59773	−0.798865	+	0.601511i	$0.794566\pi$
$578$	3.00451e9	0.647181
$579$	−2.07203e9	−0.443630
$580$	−7.25537e9	−1.54405
$581$	7.78766e8	0.164737
$582$	−2.19782e9	−0.462127
$583$	−1.47756e9	−0.308821
$584$	7.63892e9	1.58704
$585$	1.00292e9	0.207120
$586$	−1.19328e10	−2.44964
$587$	7.36500e8	0.150293	0.0751467	−	0.997172i	$-0.476058\pi$
$587$	7.36500e8	0.150293	0.0751467	+	0.997172i	$0.476058\pi$
$588$	−6.22034e9	−1.26181
$589$	5.11338e9	1.03111
$590$	−4.36198e9	−0.874384
$591$	−4.88786e9	−0.974008
$592$	7.80842e9	1.54681
$593$	−5.08539e9	−1.00146	−0.500729	−	0.865604i	$-0.666935\pi$
$593$	−5.08539e9	−1.00146	−0.500729	+	0.865604i	$0.666935\pi$
$594$	1.27483e9	0.249574
$595$	−8.68665e8	−0.169061
$596$	2.57575e10	4.98358
$597$	1.66343e9	0.319959
$598$	−1.42374e9	−0.272255
$599$	4.86023e9	0.923980	0.461990	−	0.886885i	$-0.347136\pi$
$599$	4.86023e9	0.923980	0.461990	+	0.886885i	$0.347136\pi$
$600$	−2.11613e9	−0.399956
$601$	6.78466e9	1.27488	0.637438	−	0.770502i	$-0.279994\pi$
$601$	6.78466e9	1.27488	0.637438	+	0.770502i	$0.279994\pi$
$602$	−1.76928e8	−0.0330528
$603$	3.32500e9	0.617563
$604$	−1.31742e10	−2.43274
$605$	1.35250e9	0.248310
$606$	1.06935e10	1.95193
$607$	2.85250e9	0.517685	0.258842	−	0.965920i	$-0.416659\pi$
$607$	2.85250e9	0.517685	0.258842	+	0.965920i	$0.416659\pi$
$608$	1.98651e10	3.58449
$609$	−1.84890e9	−0.331705
$610$	−1.02565e9	−0.182954
$611$	−9.25622e9	−1.64168
$612$	−4.29408e9	−0.757252
$613$	7.75467e9	1.35973	0.679864	−	0.733339i	$-0.262039\pi$
$613$	7.75467e9	1.35973	0.679864	+	0.733339i	$0.262039\pi$
$614$	−2.03825e10	−3.55360
$615$	3.91196e8	0.0678159
$616$	−6.20218e9	−1.06908
$617$	−6.34097e9	−1.08682	−0.543410	−	0.839468i	$-0.682867\pi$
$617$	−6.34097e9	−1.08682	−0.543410	+	0.839468i	$0.682867\pi$
$618$	3.19101e9	0.543836
$619$	3.19831e9	0.542005	0.271003	−	0.962579i	$-0.412645\pi$
$619$	3.19831e9	0.542005	0.271003	+	0.962579i	$0.412645\pi$
$620$	8.97120e9	1.51175
$621$	−1.15736e8	−0.0193931
$622$	4.68548e9	0.780706
$623$	2.89303e9	0.479342
$624$	−1.92513e10	−3.17187
$625$	2.44141e8	0.0400000
$626$	−8.00572e9	−1.30434
$627$	2.01613e9	0.326650
$628$	−2.79488e10	−4.50303
$629$	−1.99429e9	−0.319529
$630$	−8.41995e8	−0.134158
$631$	−3.79459e9	−0.601260	−0.300630	−	0.953741i	$-0.597197\pi$
$631$	−3.79459e9	−0.601260	−0.300630	+	0.953741i	$0.597197\pi$
$632$	−2.12273e10	−3.34492
$633$	−4.58776e9	−0.718931
$634$	6.97508e9	1.08702
$635$	−4.92600e9	−0.763460
$636$	4.82417e9	0.743571
$637$	7.12246e9	1.09180
$638$	1.05599e10	1.60986
$639$	1.10285e9	0.167211
$640$	1.20484e10	1.81677
$641$	6.77964e9	1.01672	0.508362	−	0.861143i	$-0.330251\pi$
$641$	6.77964e9	1.01672	0.508362	+	0.861143i	$0.330251\pi$
$642$	6.85462e9	1.02238
$643$	−1.13203e9	−0.167927	−0.0839635	−	0.996469i	$-0.526758\pi$
$643$	−1.13203e9	−0.167927	−0.0839635	+	0.996469i	$0.526758\pi$
$644$	8.79178e8	0.129711
$645$	6.46245e7	0.00948284
$646$	−9.23280e9	−1.34747
$647$	−5.54265e9	−0.804549	−0.402274	−	0.915519i	$-0.631780\pi$
$647$	−5.54265e9	−0.804549	−0.402274	+	0.915519i	$0.631780\pi$
$648$	−2.66571e9	−0.384858
$649$	4.66970e9	0.670552
$650$	3.78331e9	0.540350
$651$	2.28614e9	0.324766
$652$	−1.66674e10	−2.35505
$653$	9.41765e9	1.32357	0.661784	−	0.749695i	$-0.269800\pi$
$653$	9.41765e9	1.32357	0.661784	+	0.749695i	$0.269800\pi$
$654$	9.33646e8	0.130515
$655$	−1.76353e9	−0.245210
$656$	−7.50911e9	−1.03854
$657$	−1.11020e9	−0.152730
$658$	7.77099e9	1.06337
$659$	7.46390e9	1.01594	0.507969	−	0.861376i	$-0.330397\pi$
$659$	7.46390e9	1.01594	0.507969	+	0.861376i	$0.330397\pi$
$660$	3.53722e9	0.478914
$661$	5.58309e9	0.751917	0.375958	−	0.926637i	$-0.377314\pi$
$661$	5.58309e9	0.751917	0.375958	+	0.926637i	$0.377314\pi$
$662$	−4.94024e9	−0.661828
$663$	4.91684e9	0.655223
$664$	9.30069e9	1.23290
$665$	−1.33161e9	−0.175591
$666$	−1.93306e9	−0.253563
$667$	−9.58687e8	−0.125094
$668$	−8.90152e9	−1.15544
$669$	3.99948e9	0.516431
$670$	1.25429e10	1.61115
$671$	1.09800e9	0.140305
$672$	8.88149e9	1.12900
$673$	3.39933e9	0.429873	0.214936	−	0.976628i	$-0.431046\pi$
$673$	3.39933e9	0.429873	0.214936	+	0.976628i	$0.431046\pi$
$674$	2.72549e10	3.42875
$675$	3.07547e8	0.0384900
$676$	2.07845e10	2.58778
$677$	−1.38930e10	−1.72082	−0.860409	−	0.509604i	$-0.829792\pi$
$677$	−1.38930e10	−1.72082	−0.860409	+	0.509604i	$0.829792\pi$
$678$	1.49717e10	1.84488
$679$	−1.55401e9	−0.190507
$680$	−1.03743e10	−1.26526
$681$	−1.06954e9	−0.129773
$682$	−1.30572e10	−1.57618
$683$	−7.22576e9	−0.867782	−0.433891	−	0.900965i	$-0.642860\pi$
$683$	−7.22576e9	−0.867782	−0.433891	+	0.900965i	$0.642860\pi$
$684$	−6.58257e9	−0.786500
$685$	1.00064e9	0.118949
$686$	−1.35891e10	−1.60715
$687$	−1.00390e9	−0.118125
$688$	−1.24048e9	−0.145222
$689$	−5.52380e9	−0.643385
$690$	−4.36590e8	−0.0505943
$691$	−5.83386e9	−0.672641	−0.336320	−	0.941748i	$-0.609182\pi$
$691$	−5.83386e9	−0.672641	−0.336320	+	0.941748i	$0.609182\pi$
$692$	1.88919e10	2.16722
$693$	9.01394e8	0.102884
$694$	1.28401e10	1.45818
$695$	−5.75896e9	−0.650724
$696$	−2.20811e10	−2.48250
$697$	1.91785e9	0.214536
$698$	1.03284e10	1.14958
$699$	−4.84897e9	−0.537006
$700$	−2.33625e9	−0.257440
$701$	4.37486e9	0.479680	0.239840	−	0.970812i	$-0.422905\pi$
$701$	4.37486e9	0.479680	0.239840	+	0.970812i	$0.422905\pi$
$702$	4.76588e9	0.519952
$703$	−3.05712e9	−0.331871
$704$	−2.63136e10	−2.84234
$705$	−2.83843e9	−0.305082
$706$	−1.36083e9	−0.145542
$707$	7.56103e9	0.804661
$708$	−1.52463e10	−1.61454
$709$	−3.54685e9	−0.373750	−0.186875	−	0.982384i	$-0.559836\pi$
$709$	−3.54685e9	−0.373750	−0.186875	+	0.982384i	$0.559836\pi$
$710$	4.16029e9	0.436234
$711$	3.08507e9	0.321900
$712$	3.45511e10	3.58742
$713$	1.18541e9	0.122477
$714$	−4.12790e9	−0.424409
$715$	−4.05021e9	−0.414387
$716$	1.50251e10	1.52976
$717$	−1.00799e10	−1.02126
$718$	2.71782e8	0.0274022
$719$	−1.06545e10	−1.06901	−0.534507	−	0.845164i	$-0.679503\pi$
$719$	−1.06545e10	−1.06901	−0.534507	+	0.845164i	$0.679503\pi$
$720$	−5.90344e9	−0.589443
$721$	2.25627e9	0.224191
$722$	5.51186e9	0.545027
$723$	−6.93958e9	−0.682888
$724$	−3.56073e10	−3.48702
$725$	2.54753e9	0.248277
$726$	6.42710e9	0.623357
$727$	−9.21169e9	−0.889138	−0.444569	−	0.895745i	$-0.646643\pi$
$727$	−9.21169e9	−0.889138	−0.444569	+	0.895745i	$0.646643\pi$
$728$	−2.31866e10	−2.22729
$729$	3.87420e8	0.0370370
$730$	−4.18800e9	−0.398453
$731$	3.16823e8	0.0299989
$732$	−3.58491e9	−0.337823
$733$	−5.81770e8	−0.0545616	−0.0272808	−	0.999628i	$-0.508685\pi$
$733$	−5.81770e8	−0.0545616	−0.0272808	+	0.999628i	$0.508685\pi$
$734$	2.76131e8	0.0257738
$735$	2.18411e9	0.202894
$736$	4.60522e9	0.425773
$737$	−1.34277e10	−1.23557
$738$	1.85896e9	0.170245
$739$	1.43208e8	0.0130531	0.00652654	−	0.999979i	$-0.497923\pi$
$739$	1.43208e8	0.0130531	0.00652654	+	0.999979i	$0.497923\pi$
$740$	−5.36358e9	−0.486569
$741$	7.53722e9	0.680530
$742$	4.63746e9	0.416742
$743$	1.76012e10	1.57428	0.787139	−	0.616775i	$-0.211561\pi$
$743$	1.76012e10	1.57428	0.787139	+	0.616775i	$0.211561\pi$
$744$	2.73031e10	2.43056
$745$	−9.04406e9	−0.801340
$746$	1.24519e10	1.09812
$747$	−1.35171e9	−0.118649
$748$	1.73413e10	1.51504
$749$	4.84670e9	0.421463
$750$	1.16016e9	0.100416
$751$	−2.10398e10	−1.81260	−0.906299	−	0.422636i	$-0.861105\pi$
$751$	−2.10398e10	−1.81260	−0.906299	+	0.422636i	$0.861105\pi$
$752$	5.44844e10	4.67207
$753$	−3.43830e9	−0.293468
$754$	3.94777e10	3.35391
$755$	4.62578e9	0.391174
$756$	−2.94300e9	−0.247722
$757$	−3.91015e9	−0.327610	−0.163805	−	0.986493i	$-0.552377\pi$
$757$	−3.91015e9	−0.327610	−0.163805	+	0.986493i	$0.552377\pi$
$758$	−3.91107e10	−3.26177
$759$	4.67389e8	0.0388001
$760$	−1.59032e10	−1.31413
$761$	1.20242e10	0.989032	0.494516	−	0.869169i	$-0.335345\pi$
$761$	1.20242e10	0.989032	0.494516	+	0.869169i	$0.335345\pi$
$762$	−2.34084e10	−1.91659
$763$	6.60153e8	0.0538033
$764$	2.19869e10	1.78377
$765$	1.50775e9	0.121763
$766$	−2.60028e10	−2.09035
$767$	1.74575e10	1.39700
$768$	2.63643e10	2.10015
$769$	1.18948e10	0.943223	0.471611	−	0.881807i	$-0.343673\pi$
$769$	1.18948e10	0.943223	0.471611	+	0.881807i	$0.343673\pi$
$770$	3.40032e9	0.268412
$771$	3.52841e9	0.277261
$772$	−2.73201e10	−2.13709
$773$	−7.77614e9	−0.605531	−0.302765	−	0.953065i	$-0.597910\pi$
$773$	−7.77614e9	−0.605531	−0.302765	+	0.953065i	$0.597910\pi$
$774$	3.07096e8	0.0238057
$775$	−3.15000e9	−0.243083
$776$	−1.85594e10	−1.42576
$777$	−1.36681e9	−0.104528
$778$	−1.32546e10	−1.00911
$779$	2.93994e9	0.222822
$780$	1.32237e10	0.997751
$781$	−4.45378e9	−0.334541
$782$	−2.14039e9	−0.160055
$783$	3.20916e9	0.238905
$784$	−4.19245e10	−3.10715
$785$	9.81350e9	0.724069
$786$	−8.38028e9	−0.615573
$787$	2.44365e10	1.78701	0.893507	−	0.449050i	$-0.148237\pi$
$787$	2.44365e10	1.78701	0.893507	+	0.449050i	$0.148237\pi$
$788$	−6.44473e10	−4.69206
$789$	−7.22916e9	−0.523985
$790$	1.16378e10	0.839799
$791$	1.05861e10	0.760532
$792$	1.07652e10	0.769989
$793$	4.10482e9	0.292306
$794$	4.10883e9	0.291304
$795$	−1.69388e9	−0.119563
$796$	2.19326e10	1.54133
$797$	−2.26970e9	−0.158805	−0.0794024	−	0.996843i	$-0.525301\pi$
$797$	−2.26970e9	−0.158805	−0.0794024	+	0.996843i	$0.525301\pi$
$798$	−6.32781e9	−0.440802
$799$	−1.39155e10	−0.965125
$800$	−1.22375e10	−0.845041
$801$	−5.02148e9	−0.345237
$802$	2.19193e10	1.50043
$803$	4.48345e9	0.305568
$804$	4.38408e10	2.97496
$805$	−3.08700e8	−0.0208570
$806$	−4.88138e10	−3.28375
$807$	4.03467e9	0.270240
$808$	9.03003e10	6.02212
$809$	−1.63200e10	−1.08368	−0.541838	−	0.840483i	$-0.682271\pi$
$809$	−1.63200e10	−1.08368	−0.541838	+	0.840483i	$0.682271\pi$
$810$	1.46146e9	0.0966252
$811$	6.99393e9	0.460414	0.230207	−	0.973142i	$-0.426060\pi$
$811$	6.99393e9	0.460414	0.230207	+	0.973142i	$0.426060\pi$
$812$	−2.43780e10	−1.59791
$813$	−4.13599e9	−0.269937
$814$	7.80649e9	0.507306
$815$	5.85230e9	0.378682
$816$	−2.89417e10	−1.86470
$817$	4.85670e8	0.0311576
$818$	5.25017e10	3.35380
$819$	3.36982e9	0.214345
$820$	5.15800e9	0.326687
$821$	4.00949e9	0.252865	0.126432	−	0.991975i	$-0.459647\pi$
$821$	4.00949e9	0.252865	0.126432	+	0.991975i	$0.459647\pi$
$822$	4.75504e9	0.298609
$823$	1.88572e10	1.17917	0.589586	−	0.807706i	$-0.299291\pi$
$823$	1.88572e10	1.17917	0.589586	+	0.807706i	$0.299291\pi$
$824$	2.69463e10	1.67785
$825$	−1.24200e9	−0.0770075
$826$	−1.46563e10	−0.904885
$827$	1.66386e10	1.02293	0.511466	−	0.859304i	$-0.329103\pi$
$827$	1.66386e10	1.02293	0.511466	+	0.859304i	$0.329103\pi$
$828$	−1.52600e9	−0.0934219
$829$	−1.37224e10	−0.836547	−0.418274	−	0.908321i	$-0.637365\pi$
$829$	−1.37224e10	−0.836547	−0.418274	+	0.908321i	$0.637365\pi$
$830$	−5.09906e9	−0.309540
$831$	1.76705e10	1.06818
$832$	−9.83722e10	−5.92162
$833$	1.07076e10	0.641853
$834$	−2.73666e10	−1.63357
$835$	3.12554e9	0.185790
$836$	2.65831e10	1.57356
$837$	−3.96809e9	−0.233907
$838$	−5.71217e10	−3.35311
$839$	−6.56954e8	−0.0384033	−0.0192016	−	0.999816i	$-0.506112\pi$
$839$	−6.56954e8	−0.0384033	−0.0192016	+	0.999816i	$0.506112\pi$
$840$	−7.11018e9	−0.413907
$841$	9.33282e9	0.541037
$842$	−6.24469e10	−3.60511
$843$	1.48903e10	0.856066
$844$	−6.04904e10	−3.46328
$845$	−7.29794e9	−0.416104
$846$	−1.34882e10	−0.765876
$847$	4.54441e9	0.256972
$848$	3.25144e10	1.83101
$849$	−5.67541e9	−0.318288
$850$	5.68769e9	0.317665
$851$	−7.08716e8	−0.0394202
$852$	1.45413e10	0.805501
$853$	−8.70997e9	−0.480502	−0.240251	−	0.970711i	$-0.577230\pi$
$853$	−8.70997e9	−0.480502	−0.240251	+	0.970711i	$0.577230\pi$
$854$	−3.44617e9	−0.189336
$855$	2.31129e9	0.126466
$856$	5.78835e10	3.15425
$857$	1.93825e9	0.105190	0.0525952	−	0.998616i	$-0.483251\pi$
$857$	1.93825e9	0.105190	0.0525952	+	0.998616i	$0.483251\pi$
$858$	−1.92466e10	−1.04028
$859$	−7.95332e8	−0.0428127	−0.0214063	−	0.999771i	$-0.506814\pi$
$859$	−7.95332e8	−0.0428127	−0.0214063	+	0.999771i	$0.506814\pi$
$860$	8.52086e8	0.0456813
$861$	1.31442e9	0.0701815
$862$	3.36179e10	1.78770
$863$	2.24172e9	0.118725	0.0593626	−	0.998236i	$-0.481093\pi$
$863$	2.24172e9	0.118725	0.0593626	+	0.998236i	$0.481093\pi$
$864$	−1.54157e10	−0.813141
$865$	−6.63338e9	−0.348481
$866$	5.85335e10	3.06261
$867$	−3.68735e9	−0.192153
$868$	3.01432e10	1.56448
$869$	−1.24588e10	−0.644030
$870$	1.21059e10	0.623273
$871$	−5.01989e10	−2.57413
$872$	7.88412e9	0.402666
$873$	2.69732e9	0.137209
$874$	−3.28109e9	−0.166237
$875$	8.20312e8	0.0413953
$876$	−1.46382e10	−0.735739
$877$	1.99169e10	0.997062	0.498531	−	0.866872i	$-0.333873\pi$
$877$	1.99169e10	0.997062	0.498531	+	0.866872i	$0.333873\pi$
$878$	−5.04289e10	−2.51449
$879$	1.46448e10	0.727317
$880$	2.38405e10	1.17931
$881$	−2.76906e10	−1.36432	−0.682161	−	0.731202i	$-0.738960\pi$
$881$	−2.76906e10	−1.36432	−0.682161	+	0.731202i	$0.738960\pi$
$882$	1.03789e10	0.509343
$883$	−1.65616e10	−0.809544	−0.404772	−	0.914418i	$-0.632649\pi$
$883$	−1.65616e10	−0.809544	−0.404772	+	0.914418i	$0.632649\pi$
$884$	6.48295e10	3.15638
$885$	5.35334e9	0.259611
$886$	2.05489e10	0.992593
$887$	6.69904e9	0.322314	0.161157	−	0.986929i	$-0.448477\pi$
$887$	6.69904e9	0.322314	0.161157	+	0.986929i	$0.448477\pi$
$888$	−1.63236e10	−0.782296
$889$	−1.65514e10	−0.790092
$890$	−1.89425e10	−0.900683
$891$	−1.56456e9	−0.0741005
$892$	5.27339e10	2.48778
$893$	−2.13315e10	−1.00240
$894$	−4.29774e10	−2.01168
$895$	−5.27568e9	−0.245979
$896$	4.04827e10	1.88015
$897$	1.74731e9	0.0808346
$898$	1.94703e9	0.0897232
$899$	−3.28693e10	−1.50880
$900$	4.05506e9	0.185417
$901$	−8.30427e9	−0.378238
$902$	−7.50726e9	−0.340611
$903$	2.17138e8	0.00981362
$904$	1.26428e11	5.69186
$905$	1.25026e10	0.560698
$906$	2.19817e10	0.982002
$907$	3.87814e9	0.172583	0.0862916	−	0.996270i	$-0.472498\pi$
$907$	3.87814e9	0.172583	0.0862916	+	0.996270i	$0.472498\pi$
$908$	−1.41021e10	−0.625150
$909$	−1.31238e10	−0.579543
$910$	1.27119e10	0.559199
$911$	−7.15870e9	−0.313704	−0.156852	−	0.987622i	$-0.550135\pi$
$911$	−7.15870e9	−0.313704	−0.156852	+	0.987622i	$0.550135\pi$
$912$	−4.43659e10	−1.93672
$913$	5.45878e9	0.237382
$914$	2.66341e10	1.15379
$915$	1.25875e9	0.0543206
$916$	−1.32366e10	−0.569041
$917$	−5.92545e9	−0.253763
$918$	7.16485e9	0.305673
$919$	2.54160e10	1.08020	0.540098	−	0.841602i	$-0.318387\pi$
$919$	2.54160e10	1.08020	0.540098	+	0.841602i	$0.318387\pi$
$920$	−3.68676e9	−0.156094
$921$	2.50149e10	1.05509
$922$	−1.88634e10	−0.792615
$923$	−1.66502e10	−0.696970
$924$	1.18850e10	0.495620
$925$	1.88328e9	0.0782382
$926$	4.28256e10	1.77241
$927$	−3.91624e9	−0.161469
$928$	−1.27694e11	−5.24511
$929$	−2.34566e10	−0.959865	−0.479932	−	0.877306i	$-0.659339\pi$
$929$	−2.34566e10	−0.959865	−0.479932	+	0.877306i	$0.659339\pi$
$930$	−1.49688e10	−0.610234
$931$	1.64141e10	0.666644
$932$	−6.39345e10	−2.58690
$933$	−5.75036e9	−0.231798
$934$	2.11190e10	0.848124
$935$	−6.08893e9	−0.243613
$936$	4.02452e10	1.60417
$937$	9.30070e9	0.369341	0.184670	−	0.982801i	$-0.440878\pi$
$937$	9.30070e9	0.369341	0.184670	+	0.982801i	$0.440878\pi$
$938$	4.21440e10	1.66735
$939$	9.82520e9	0.387268
$940$	−3.74252e10	−1.46966
$941$	−1.13117e10	−0.442553	−0.221276	−	0.975211i	$-0.571022\pi$
$941$	−1.13117e10	−0.442553	−0.221276	+	0.975211i	$0.571022\pi$
$942$	4.66337e10	1.81770
$943$	6.81551e8	0.0264672
$944$	−1.02759e11	−3.97573
$945$	1.03336e9	0.0398327
$946$	−1.24018e9	−0.0476283
$947$	4.46270e9	0.170755	0.0853774	−	0.996349i	$-0.472790\pi$
$947$	4.46270e9	0.170755	0.0853774	+	0.996349i	$0.472790\pi$
$948$	4.06772e10	1.55068
$949$	1.67611e10	0.636608
$950$	8.71888e9	0.329935
$951$	−8.56032e9	−0.322744
$952$	−3.48578e10	−1.30939
$953$	1.90872e10	0.714359	0.357179	−	0.934036i	$-0.383739\pi$
$953$	1.90872e10	0.714359	0.357179	+	0.934036i	$0.383739\pi$
$954$	−8.04931e9	−0.300151
$955$	−7.72013e9	−0.286822
$956$	−1.32905e11	−4.91969
$957$	−1.29599e10	−0.477980
$958$	−6.67873e9	−0.245423
$959$	3.36215e9	0.123098
$960$	−3.01659e10	−1.10044
$961$	1.31299e10	0.477234
$962$	2.91842e10	1.05690
$963$	−8.41249e9	−0.303552
$964$	−9.14997e10	−3.28965
$965$	9.59274e9	0.343635
$966$	−1.46694e9	−0.0523592
$967$	−1.34102e10	−0.476916	−0.238458	−	0.971153i	$-0.576642\pi$
$967$	−1.34102e10	−0.476916	−0.238458	+	0.971153i	$0.576642\pi$
$968$	5.42733e10	1.92319
$969$	1.13312e10	0.400075
$970$	1.01751e10	0.357962
$971$	4.64191e10	1.62716	0.813579	−	0.581455i	$-0.197516\pi$
$971$	4.64191e10	1.62716	0.813579	+	0.581455i	$0.197516\pi$
$972$	5.10821e9	0.178417
$973$	−1.93501e10	−0.673423
$974$	−9.59329e9	−0.332668
$975$	−4.64316e9	−0.160434
$976$	−2.41620e10	−0.831875
$977$	3.92972e10	1.34813	0.674064	−	0.738673i	$-0.264547\pi$
$977$	3.92972e10	1.34813	0.674064	+	0.738673i	$0.264547\pi$
$978$	2.78101e10	0.950642
$979$	2.02788e10	0.690721
$980$	2.87979e10	0.977393
$981$	−1.14584e9	−0.0387509
$982$	−1.83659e10	−0.618902
$983$	−3.69370e10	−1.24029	−0.620146	−	0.784486i	$-0.712927\pi$
$983$	−3.69370e10	−1.24029	−0.620146	+	0.784486i	$0.712927\pi$
$984$	1.56979e10	0.525242
$985$	2.26290e10	0.754463
$986$	5.93492e10	1.97172
$987$	−9.53712e9	−0.315724
$988$	9.93796e10	3.27829
$989$	1.12590e8	0.00370095
$990$	−5.90198e9	−0.193319
$991$	−1.02868e10	−0.335753	−0.167877	−	0.985808i	$-0.553691\pi$
$991$	−1.02868e10	−0.335753	−0.167877	+	0.985808i	$0.553691\pi$
$992$	1.57893e11	5.13538
$993$	6.06302e9	0.196502
$994$	1.39786e10	0.451451
$995$	−7.70106e9	−0.247839
$996$	−1.78226e10	−0.571562
$997$	−3.24439e10	−1.03681	−0.518407	−	0.855134i	$-0.673475\pi$
$997$	−3.24439e10	−1.03681	−0.518407	+	0.855134i	$0.673475\pi$
$998$	9.90506e10	3.15428
$999$	2.37239e9	0.0752848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.a.a.1.1	✓	1
3.2	odd	2		45.8.a.g.1.1		1
4.3	odd	2		240.8.a.c.1.1		1
5.2	odd	4		75.8.b.a.49.1		2
5.3	odd	4		75.8.b.a.49.2		2
5.4	even	2		75.8.a.c.1.1		1
15.2	even	4		225.8.b.a.199.2		2
15.8	even	4		225.8.b.a.199.1		2
15.14	odd	2		225.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.a.a.1.1	✓	1	1.1	even	1	trivial
45.8.a.g.1.1		1	3.2	odd	2
75.8.a.c.1.1		1	5.4	even	2
75.8.b.a.49.1		2	5.2	odd	4
75.8.b.a.49.2		2	5.3	odd	4
225.8.a.a.1.1		1	15.14	odd	2
225.8.b.a.199.1		2	15.8	even	4
225.8.b.a.199.2		2	15.2	even	4
240.8.a.c.1.1		1	4.3	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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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