LMFDB - Embedded newform 240.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("240.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	240.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:	$74.9724061162$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -125.000 q^{5} +420.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -125.000 q^{5} +420.000 q^{7} +729.000 q^{9} +2944.00 q^{11} -11006.0 q^{13} +3375.00 q^{15} -16546.0 q^{17} +25364.0 q^{19} -11340.0 q^{21} +5880.00 q^{23} +15625.0 q^{25} -19683.0 q^{27} +163042. q^{29} +201600. q^{31} -79488.0 q^{33} -52500.0 q^{35} +120530. q^{37} +297162. q^{39} -115910. q^{41} +19148.0 q^{43} -91125.0 q^{45} -841016. q^{47} -647143. q^{49} +446742. q^{51} +501890. q^{53} -368000. q^{55} -684828. q^{57} +1.58618e6 q^{59} -372962. q^{61} +306180. q^{63} +1.37575e6 q^{65} -4.56104e6 q^{67} -158760. q^{69} -1.51283e6 q^{71} -1.52291e6 q^{73} -421875. q^{75} +1.23648e6 q^{77} -4.23192e6 q^{79} +531441. q^{81} +1.85420e6 q^{83} +2.06825e6 q^{85} -4.40213e6 q^{87} -6.88817e6 q^{89} -4.62252e6 q^{91} -5.44320e6 q^{93} -3.17050e6 q^{95} +3.70003e6 q^{97} +2.14618e6 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	420.000	0.462814	0.231407	−	0.972857i	$-0.425667\pi$
$7$	420.000	0.462814	0.231407	+	0.972857i	$0.425667\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	2944.00	0.666904	0.333452	−	0.942767i	$-0.391786\pi$
$11$	2944.00	0.666904	0.333452	+	0.942767i	$0.391786\pi$
$12$	0	0
$13$	−11006.0	−1.38940	−0.694701	−	0.719299i	$-0.744463\pi$
$13$	−11006.0	−1.38940	−0.694701	+	0.719299i	$0.744463\pi$
$14$	0	0
$15$	3375.00	0.258199
$16$	0	0
$17$	−16546.0	−0.816811	−0.408406	−	0.912801i	$-0.633915\pi$
$17$	−16546.0	−0.816811	−0.408406	+	0.912801i	$0.633915\pi$
$18$	0	0
$19$	25364.0	0.848360	0.424180	−	0.905578i	$-0.360562\pi$
$19$	25364.0	0.848360	0.424180	+	0.905578i	$0.360562\pi$
$20$	0	0
$21$	−11340.0	−0.267206
$22$	0	0
$23$	5880.00	0.100770	0.0503848	−	0.998730i	$-0.483955\pi$
$23$	5880.00	0.100770	0.0503848	+	0.998730i	$0.483955\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	163042.	1.24139	0.620693	−	0.784054i	$-0.286852\pi$
$29$	163042.	1.24139	0.620693	+	0.784054i	$0.286852\pi$
$30$	0	0
$31$	201600.	1.21542	0.607708	−	0.794161i	$-0.292089\pi$
$31$	201600.	1.21542	0.607708	+	0.794161i	$0.292089\pi$
$32$	0	0
$33$	−79488.0	−0.385037
$34$	0	0
$35$	−52500.0	−0.206977
$36$	0	0
$37$	120530.	0.391191	0.195596	−	0.980685i	$-0.437336\pi$
$37$	120530.	0.391191	0.195596	+	0.980685i	$0.437336\pi$
$38$	0	0
$39$	297162.	0.802171
$40$	0	0
$41$	−115910.	−0.262650	−0.131325	−	0.991339i	$-0.541923\pi$
$41$	−115910.	−0.262650	−0.131325	+	0.991339i	$0.541923\pi$
$42$	0	0
$43$	19148.0	0.0367269	0.0183634	−	0.999831i	$-0.494154\pi$
$43$	19148.0	0.0367269	0.0183634	+	0.999831i	$0.494154\pi$
$44$	0	0
$45$	−91125.0	−0.149071
$46$	0	0
$47$	−841016.	−1.18158	−0.590788	−	0.806827i	$-0.701183\pi$
$47$	−841016.	−1.18158	−0.590788	+	0.806827i	$0.701183\pi$
$48$	0	0
$49$	−647143.	−0.785804
$50$	0	0
$51$	446742.	0.471586
$52$	0	0
$53$	501890.	0.463066	0.231533	−	0.972827i	$-0.425626\pi$
$53$	501890.	0.463066	0.231533	+	0.972827i	$0.425626\pi$
$54$	0	0
$55$	−368000.	−0.298249
$56$	0	0
$57$	−684828.	−0.489801
$58$	0	0
$59$	1.58618e6	1.00547	0.502735	−	0.864440i	$-0.332327\pi$
$59$	1.58618e6	1.00547	0.502735	+	0.864440i	$0.332327\pi$
$60$	0	0
$61$	−372962.	−0.210383	−0.105191	−	0.994452i	$-0.533546\pi$
$61$	−372962.	−0.210383	−0.105191	+	0.994452i	$0.533546\pi$
$62$	0	0
$63$	306180.	0.154271
$64$	0	0
$65$	1.37575e6	0.621359
$66$	0	0
$67$	−4.56104e6	−1.85269	−0.926344	−	0.376678i	$-0.877066\pi$
$67$	−4.56104e6	−1.85269	−0.926344	+	0.376678i	$0.877066\pi$
$68$	0	0
$69$	−158760.	−0.0581794
$70$	0	0
$71$	−1.51283e6	−0.501633	−0.250817	−	0.968035i	$-0.580699\pi$
$71$	−1.51283e6	−0.501633	−0.250817	+	0.968035i	$0.580699\pi$
$72$	0	0
$73$	−1.52291e6	−0.458189	−0.229094	−	0.973404i	$-0.573576\pi$
$73$	−1.52291e6	−0.458189	−0.229094	+	0.973404i	$0.573576\pi$
$74$	0	0
$75$	−421875.	−0.115470
$76$	0	0
$77$	1.23648e6	0.308652
$78$	0	0
$79$	−4.23192e6	−0.965701	−0.482850	−	0.875703i	$-0.660399\pi$
$79$	−4.23192e6	−0.965701	−0.482850	+	0.875703i	$0.660399\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	1.85420e6	0.355946	0.177973	−	0.984035i	$-0.443046\pi$
$83$	1.85420e6	0.355946	0.177973	+	0.984035i	$0.443046\pi$
$84$	0	0
$85$	2.06825e6	0.365289
$86$	0	0
$87$	−4.40213e6	−0.716714
$88$	0	0
$89$	−6.88817e6	−1.03571	−0.517856	−	0.855468i	$-0.673270\pi$
$89$	−6.88817e6	−1.03571	−0.517856	+	0.855468i	$0.673270\pi$
$90$	0	0
$91$	−4.62252e6	−0.643034
$92$	0	0
$93$	−5.44320e6	−0.701720
$94$	0	0
$95$	−3.17050e6	−0.379398
$96$	0	0
$97$	3.70003e6	0.411628	0.205814	−	0.978591i	$-0.434016\pi$
$97$	3.70003e6	0.411628	0.205814	+	0.978591i	$0.434016\pi$
$98$	0	0
$99$	2.14618e6	0.222301
$100$	0	0
$101$	−1.80025e7	−1.73863	−0.869314	−	0.494259i	$-0.835439\pi$
$101$	−1.80025e7	−1.73863	−0.869314	+	0.494259i	$0.835439\pi$
$102$	0	0
$103$	5.37207e6	0.484408	0.242204	−	0.970225i	$-0.422130\pi$
$103$	5.37207e6	0.484408	0.242204	+	0.970225i	$0.422130\pi$
$104$	0	0
$105$	1.41750e6	0.119498
$106$	0	0
$107$	1.15398e7	0.910655	0.455327	−	0.890324i	$-0.349522\pi$
$107$	1.15398e7	0.910655	0.455327	+	0.890324i	$0.349522\pi$
$108$	0	0
$109$	−1.57179e6	−0.116253	−0.0581263	−	0.998309i	$-0.518513\pi$
$109$	−1.57179e6	−0.116253	−0.0581263	+	0.998309i	$0.518513\pi$
$110$	0	0
$111$	−3.25431e6	−0.225854
$112$	0	0
$113$	−2.52050e7	−1.64328	−0.821640	−	0.570007i	$-0.806940\pi$
$113$	−2.52050e7	−1.64328	−0.821640	+	0.570007i	$0.806940\pi$
$114$	0	0
$115$	−735000.	−0.0450656
$116$	0	0
$117$	−8.02337e6	−0.463134
$118$	0	0
$119$	−6.94932e6	−0.378031
$120$	0	0
$121$	−1.08200e7	−0.555239
$122$	0	0
$123$	3.12957e6	0.151641
$124$	0	0
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.94080e7	−1.70715	−0.853574	−	0.520971i	$-0.825570\pi$
$127$	−3.94080e7	−1.70715	−0.853574	+	0.520971i	$0.825570\pi$
$128$	0	0
$129$	−516996.	−0.0212043
$130$	0	0
$131$	−1.41082e7	−0.548305	−0.274153	−	0.961686i	$-0.588397\pi$
$131$	−1.41082e7	−0.548305	−0.274153	+	0.961686i	$0.588397\pi$
$132$	0	0
$133$	1.06529e7	0.392633
$134$	0	0
$135$	2.46038e6	0.0860663
$136$	0	0
$137$	−8.00512e6	−0.265978	−0.132989	−	0.991118i	$-0.542458\pi$
$137$	−8.00512e6	−0.265978	−0.132989	+	0.991118i	$0.542458\pi$
$138$	0	0
$139$	−4.60716e7	−1.45506	−0.727532	−	0.686074i	$-0.759333\pi$
$139$	−4.60716e7	−1.45506	−0.727532	+	0.686074i	$0.759333\pi$
$140$	0	0
$141$	2.27074e7	0.682183
$142$	0	0
$143$	−3.24017e7	−0.926598
$144$	0	0
$145$	−2.03802e7	−0.555164
$146$	0	0
$147$	1.74729e7	0.453684
$148$	0	0
$149$	7.23525e7	1.79185	0.895925	−	0.444206i	$-0.146514\pi$
$149$	7.23525e7	1.79185	0.895925	+	0.444206i	$0.146514\pi$
$150$	0	0
$151$	3.70062e7	0.874692	0.437346	−	0.899293i	$-0.355918\pi$
$151$	3.70062e7	0.874692	0.437346	+	0.899293i	$0.355918\pi$
$152$	0	0
$153$	−1.20620e7	−0.272270
$154$	0	0
$155$	−2.52000e7	−0.543550
$156$	0	0
$157$	−7.85080e7	−1.61907	−0.809534	−	0.587073i	$-0.800280\pi$
$157$	−7.85080e7	−1.61907	−0.809534	+	0.587073i	$0.800280\pi$
$158$	0	0
$159$	−1.35510e7	−0.267351
$160$	0	0
$161$	2.46960e6	0.0466376
$162$	0	0
$163$	4.68184e7	0.846759	0.423380	−	0.905952i	$-0.360844\pi$
$163$	4.68184e7	0.846759	0.423380	+	0.905952i	$0.360844\pi$
$164$	0	0
$165$	9.93600e6	0.172194
$166$	0	0
$167$	2.50043e7	0.415438	0.207719	−	0.978188i	$-0.433396\pi$
$167$	2.50043e7	0.415438	0.207719	+	0.978188i	$0.433396\pi$
$168$	0	0
$169$	5.83835e7	0.930437
$170$	0	0
$171$	1.84904e7	0.282787
$172$	0	0
$173$	5.30671e7	0.779227	0.389613	−	0.920979i	$-0.372609\pi$
$173$	5.30671e7	0.779227	0.389613	+	0.920979i	$0.372609\pi$
$174$	0	0
$175$	6.56250e6	0.0925627
$176$	0	0
$177$	−4.28268e7	−0.580509
$178$	0	0
$179$	−4.22054e7	−0.550025	−0.275012	−	0.961441i	$-0.588682\pi$
$179$	−4.22054e7	−0.550025	−0.275012	+	0.961441i	$0.588682\pi$
$180$	0	0
$181$	−1.00020e8	−1.25376	−0.626879	−	0.779116i	$-0.715668\pi$
$181$	−1.00020e8	−1.25376	−0.626879	+	0.779116i	$0.715668\pi$
$182$	0	0
$183$	1.00700e7	0.121465
$184$	0	0
$185$	−1.50662e7	−0.174946
$186$	0	0
$187$	−4.87114e7	−0.544735
$188$	0	0
$189$	−8.26686e6	−0.0890685
$190$	0	0
$191$	−6.17610e7	−0.641354	−0.320677	−	0.947189i	$-0.603910\pi$
$191$	−6.17610e7	−0.641354	−0.320677	+	0.947189i	$0.603910\pi$
$192$	0	0
$193$	−7.67419e7	−0.768390	−0.384195	−	0.923252i	$-0.625521\pi$
$193$	−7.67419e7	−0.768390	−0.384195	+	0.923252i	$0.625521\pi$
$194$	0	0
$195$	−3.71452e7	−0.358742
$196$	0	0
$197$	−1.81032e8	−1.68703	−0.843516	−	0.537105i	$-0.819518\pi$
$197$	−1.81032e8	−1.68703	−0.843516	+	0.537105i	$0.819518\pi$
$198$	0	0
$199$	−6.16084e7	−0.554185	−0.277092	−	0.960843i	$-0.589371\pi$
$199$	−6.16084e7	−0.554185	−0.277092	+	0.960843i	$0.589371\pi$
$200$	0	0
$201$	1.23148e8	1.06965
$202$	0	0
$203$	6.84776e7	0.574530
$204$	0	0
$205$	1.44888e7	0.117461
$206$	0	0
$207$	4.28652e6	0.0335899
$208$	0	0
$209$	7.46716e7	0.565775
$210$	0	0
$211$	1.69917e8	1.24523	0.622613	−	0.782530i	$-0.286071\pi$
$211$	1.69917e8	1.24523	0.622613	+	0.782530i	$0.286071\pi$
$212$	0	0
$213$	4.08465e7	0.289618
$214$	0	0
$215$	−2.39350e6	−0.0164248
$216$	0	0
$217$	8.46720e7	0.562511
$218$	0	0
$219$	4.11186e7	0.264535
$220$	0	0
$221$	1.82105e8	1.13488
$222$	0	0
$223$	−1.48129e8	−0.894484	−0.447242	−	0.894413i	$-0.647594\pi$
$223$	−1.48129e8	−0.894484	−0.447242	+	0.894413i	$0.647594\pi$
$224$	0	0
$225$	1.13906e7	0.0666667
$226$	0	0
$227$	3.96127e7	0.224773	0.112387	−	0.993665i	$-0.464151\pi$
$227$	3.96127e7	0.224773	0.112387	+	0.993665i	$0.464151\pi$
$228$	0	0
$229$	−3.71816e7	−0.204599	−0.102300	−	0.994754i	$-0.532620\pi$
$229$	−3.71816e7	−0.204599	−0.102300	+	0.994754i	$0.532620\pi$
$230$	0	0
$231$	−3.33850e7	−0.178201
$232$	0	0
$233$	−1.79591e8	−0.930122	−0.465061	−	0.885279i	$-0.653968\pi$
$233$	−1.79591e8	−0.930122	−0.465061	+	0.885279i	$0.653968\pi$
$234$	0	0
$235$	1.05127e8	0.528417
$236$	0	0
$237$	1.14262e8	0.557548
$238$	0	0
$239$	3.73328e8	1.76888	0.884439	−	0.466655i	$-0.154541\pi$
$239$	3.73328e8	1.76888	0.884439	+	0.466655i	$0.154541\pi$
$240$	0	0
$241$	−2.57022e8	−1.18280	−0.591398	−	0.806380i	$-0.701424\pi$
$241$	−2.57022e8	−1.18280	−0.591398	+	0.806380i	$0.701424\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	8.08929e7	0.351422
$246$	0	0
$247$	−2.79156e8	−1.17871
$248$	0	0
$249$	−5.00635e7	−0.205506
$250$	0	0
$251$	1.27344e8	0.508302	0.254151	−	0.967165i	$-0.418204\pi$
$251$	1.27344e8	0.508302	0.254151	+	0.967165i	$0.418204\pi$
$252$	0	0
$253$	1.73107e7	0.0672037
$254$	0	0
$255$	−5.58427e7	−0.210900
$256$	0	0
$257$	1.30682e8	0.480230	0.240115	−	0.970744i	$-0.422815\pi$
$257$	1.30682e8	0.480230	0.240115	+	0.970744i	$0.422815\pi$
$258$	0	0
$259$	5.06226e7	0.181049
$260$	0	0
$261$	1.18858e8	0.413795
$262$	0	0
$263$	2.67747e8	0.907568	0.453784	−	0.891112i	$-0.350074\pi$
$263$	2.67747e8	0.907568	0.453784	+	0.891112i	$0.350074\pi$
$264$	0	0
$265$	−6.27363e7	−0.207089
$266$	0	0
$267$	1.85981e8	0.597969
$268$	0	0
$269$	1.49432e8	0.468070	0.234035	−	0.972228i	$-0.424807\pi$
$269$	1.49432e8	0.468070	0.234035	+	0.972228i	$0.424807\pi$
$270$	0	0
$271$	1.53185e8	0.467545	0.233773	−	0.972291i	$-0.424893\pi$
$271$	1.53185e8	0.467545	0.233773	+	0.972291i	$0.424893\pi$
$272$	0	0
$273$	1.24808e8	0.371256
$274$	0	0
$275$	4.60000e7	0.133381
$276$	0	0
$277$	6.54462e8	1.85014	0.925072	−	0.379792i	$-0.124004\pi$
$277$	6.54462e8	1.85014	0.925072	+	0.379792i	$0.124004\pi$
$278$	0	0
$279$	1.46966e8	0.405138
$280$	0	0
$281$	5.51493e8	1.48275	0.741375	−	0.671091i	$-0.234174\pi$
$281$	5.51493e8	1.48275	0.741375	+	0.671091i	$0.234174\pi$
$282$	0	0
$283$	2.10200e8	0.551291	0.275646	−	0.961259i	$-0.411108\pi$
$283$	2.10200e8	0.551291	0.275646	+	0.961259i	$0.411108\pi$
$284$	0	0
$285$	8.56035e7	0.219046
$286$	0	0
$287$	−4.86822e7	−0.121558
$288$	0	0
$289$	−1.36569e8	−0.332819
$290$	0	0
$291$	−9.99009e7	−0.237653
$292$	0	0
$293$	5.42402e8	1.25975	0.629875	−	0.776697i	$-0.283106\pi$
$293$	5.42402e8	1.25975	0.629875	+	0.776697i	$0.283106\pi$
$294$	0	0
$295$	−1.98272e8	−0.449660
$296$	0	0
$297$	−5.79468e7	−0.128346
$298$	0	0
$299$	−6.47153e7	−0.140010
$300$	0	0
$301$	8.04216e6	0.0169977
$302$	0	0
$303$	4.86066e8	1.00380
$304$	0	0
$305$	4.66202e7	0.0940860
$306$	0	0
$307$	−9.26477e8	−1.82747	−0.913736	−	0.406310i	$-0.866815\pi$
$307$	−9.26477e8	−1.82747	−0.913736	+	0.406310i	$0.866815\pi$
$308$	0	0
$309$	−1.45046e8	−0.279673
$310$	0	0
$311$	2.12976e8	0.401485	0.200743	−	0.979644i	$-0.435665\pi$
$311$	2.12976e8	0.401485	0.200743	+	0.979644i	$0.435665\pi$
$312$	0	0
$313$	3.63896e8	0.670768	0.335384	−	0.942081i	$-0.391134\pi$
$313$	3.63896e8	0.670768	0.335384	+	0.942081i	$0.391134\pi$
$314$	0	0
$315$	−3.82725e7	−0.0689922
$316$	0	0
$317$	−3.17049e8	−0.559009	−0.279505	−	0.960144i	$-0.590170\pi$
$317$	−3.17049e8	−0.559009	−0.279505	+	0.960144i	$0.590170\pi$
$318$	0	0
$319$	4.79996e8	0.827885
$320$	0	0
$321$	−3.11574e8	−0.525767
$322$	0	0
$323$	−4.19673e8	−0.692950
$324$	0	0
$325$	−1.71969e8	−0.277880
$326$	0	0
$327$	4.24384e7	0.0671185
$328$	0	0
$329$	−3.53227e8	−0.546850
$330$	0	0
$331$	−2.24556e8	−0.340351	−0.170176	−	0.985414i	$-0.554434\pi$
$331$	−2.24556e8	−0.340351	−0.170176	+	0.985414i	$0.554434\pi$
$332$	0	0
$333$	8.78664e7	0.130397
$334$	0	0
$335$	5.70130e8	0.828548
$336$	0	0
$337$	−1.23886e9	−1.76327	−0.881633	−	0.471935i	$-0.843556\pi$
$337$	−1.23886e9	−1.76327	−0.881633	+	0.471935i	$0.843556\pi$
$338$	0	0
$339$	6.80534e8	0.948748
$340$	0	0
$341$	5.93510e8	0.810565
$342$	0	0
$343$	−6.17688e8	−0.826494
$344$	0	0
$345$	1.98450e7	0.0260186
$346$	0	0
$347$	5.83643e8	0.749884	0.374942	−	0.927048i	$-0.377663\pi$
$347$	5.83643e8	0.749884	0.374942	+	0.927048i	$0.377663\pi$
$348$	0	0
$349$	−4.69471e8	−0.591180	−0.295590	−	0.955315i	$-0.595516\pi$
$349$	−4.69471e8	−0.591180	−0.295590	+	0.955315i	$0.595516\pi$
$350$	0	0
$351$	2.16631e8	0.267390
$352$	0	0
$353$	6.18559e7	0.0748461	0.0374231	−	0.999300i	$-0.488085\pi$
$353$	6.18559e7	0.0748461	0.0374231	+	0.999300i	$0.488085\pi$
$354$	0	0
$355$	1.89104e8	0.224337
$356$	0	0
$357$	1.87632e8	0.218257
$358$	0	0
$359$	1.23537e7	0.0140918	0.00704592	−	0.999975i	$-0.497757\pi$
$359$	1.23537e7	0.0140918	0.00704592	+	0.999975i	$0.497757\pi$
$360$	0	0
$361$	−2.50539e8	−0.280285
$362$	0	0
$363$	2.92141e8	0.320567
$364$	0	0
$365$	1.90364e8	0.204908
$366$	0	0
$367$	1.25514e7	0.0132544	0.00662721	−	0.999978i	$-0.497890\pi$
$367$	1.25514e7	0.0132544	0.00662721	+	0.999978i	$0.497890\pi$
$368$	0	0
$369$	−8.44984e7	−0.0875500
$370$	0	0
$371$	2.10794e8	0.214313
$372$	0	0
$373$	−5.65994e8	−0.564717	−0.282359	−	0.959309i	$-0.591117\pi$
$373$	−5.65994e8	−0.564717	−0.282359	+	0.959309i	$0.591117\pi$
$374$	0	0
$375$	5.27344e7	0.0516398
$376$	0	0
$377$	−1.79444e9	−1.72478
$378$	0	0
$379$	−1.77776e9	−1.67740	−0.838698	−	0.544597i	$-0.816683\pi$
$379$	−1.77776e9	−1.67740	−0.838698	+	0.544597i	$0.816683\pi$
$380$	0	0
$381$	1.06402e9	0.985623
$382$	0	0
$383$	−1.18195e9	−1.07498	−0.537492	−	0.843269i	$-0.680628\pi$
$383$	−1.18195e9	−1.07498	−0.537492	+	0.843269i	$0.680628\pi$
$384$	0	0
$385$	−1.54560e8	−0.138034
$386$	0	0
$387$	1.39589e7	0.0122423
$388$	0	0
$389$	6.02482e8	0.518944	0.259472	−	0.965751i	$-0.416451\pi$
$389$	6.02482e8	0.518944	0.259472	+	0.965751i	$0.416451\pi$
$390$	0	0
$391$	−9.72905e7	−0.0823098
$392$	0	0
$393$	3.80922e8	0.316564
$394$	0	0
$395$	5.28990e8	0.431875
$396$	0	0
$397$	−1.86765e8	−0.149806	−0.0749029	−	0.997191i	$-0.523865\pi$
$397$	−1.86765e8	−0.149806	−0.0749029	+	0.997191i	$0.523865\pi$
$398$	0	0
$399$	−2.87628e8	−0.226687
$400$	0	0
$401$	−9.96333e8	−0.771613	−0.385806	−	0.922580i	$-0.626077\pi$
$401$	−9.96333e8	−0.771613	−0.385806	+	0.922580i	$0.626077\pi$
$402$	0	0
$403$	−2.21881e9	−1.68870
$404$	0	0
$405$	−6.64301e7	−0.0496904
$406$	0	0
$407$	3.54840e8	0.260887
$408$	0	0
$409$	−2.38644e9	−1.72472	−0.862362	−	0.506293i	$-0.831015\pi$
$409$	−2.38644e9	−1.72472	−0.862362	+	0.506293i	$0.831015\pi$
$410$	0	0
$411$	2.16138e8	0.153563
$412$	0	0
$413$	6.66194e8	0.465345
$414$	0	0
$415$	−2.31775e8	−0.159184
$416$	0	0
$417$	1.24393e9	0.840081
$418$	0	0
$419$	−2.59644e9	−1.72437	−0.862183	−	0.506597i	$-0.830903\pi$
$419$	−2.59644e9	−1.72437	−0.862183	+	0.506597i	$0.830903\pi$
$420$	0	0
$421$	2.83850e9	1.85396	0.926981	−	0.375108i	$-0.122394\pi$
$421$	2.83850e9	1.85396	0.926981	+	0.375108i	$0.122394\pi$
$422$	0	0
$423$	−6.13101e8	−0.393859
$424$	0	0
$425$	−2.58531e8	−0.163362
$426$	0	0
$427$	−1.56644e8	−0.0973680
$428$	0	0
$429$	8.74845e8	0.534971
$430$	0	0
$431$	1.52808e9	0.919342	0.459671	−	0.888089i	$-0.347967\pi$
$431$	1.52808e9	0.919342	0.459671	+	0.888089i	$0.347967\pi$
$432$	0	0
$433$	−2.66061e9	−1.57498	−0.787488	−	0.616330i	$-0.788619\pi$
$433$	−2.66061e9	−1.57498	−0.787488	+	0.616330i	$0.788619\pi$
$434$	0	0
$435$	5.50267e8	0.320524
$436$	0	0
$437$	1.49140e8	0.0854890
$438$	0	0
$439$	−2.29222e9	−1.29310	−0.646549	−	0.762873i	$-0.723788\pi$
$439$	−2.29222e9	−1.29310	−0.646549	+	0.762873i	$0.723788\pi$
$440$	0	0
$441$	−4.71767e8	−0.261935
$442$	0	0
$443$	9.34043e8	0.510451	0.255225	−	0.966882i	$-0.417850\pi$
$443$	9.34043e8	0.510451	0.255225	+	0.966882i	$0.417850\pi$
$444$	0	0
$445$	8.61022e8	0.463185
$446$	0	0
$447$	−1.95352e9	−1.03452
$448$	0	0
$449$	−8.85012e7	−0.0461410	−0.0230705	−	0.999734i	$-0.507344\pi$
$449$	−8.85012e7	−0.0461410	−0.0230705	+	0.999734i	$0.507344\pi$
$450$	0	0
$451$	−3.41239e8	−0.175162
$452$	0	0
$453$	−9.99168e8	−0.505004
$454$	0	0
$455$	5.77815e8	0.287574
$456$	0	0
$457$	−1.21064e9	−0.593347	−0.296674	−	0.954979i	$-0.595877\pi$
$457$	−1.21064e9	−0.593347	−0.296674	+	0.954979i	$0.595877\pi$
$458$	0	0
$459$	3.25675e8	0.157195
$460$	0	0
$461$	8.57428e8	0.407610	0.203805	−	0.979012i	$-0.434669\pi$
$461$	8.57428e8	0.407610	0.203805	+	0.979012i	$0.434669\pi$
$462$	0	0
$463$	1.94662e9	0.911481	0.455741	−	0.890113i	$-0.349375\pi$
$463$	1.94662e9	0.911481	0.455741	+	0.890113i	$0.349375\pi$
$464$	0	0
$465$	6.80400e8	0.313819
$466$	0	0
$467$	9.59955e8	0.436156	0.218078	−	0.975931i	$-0.430021\pi$
$467$	9.59955e8	0.436156	0.218078	+	0.975931i	$0.430021\pi$
$468$	0	0
$469$	−1.91564e9	−0.857450
$470$	0	0
$471$	2.11972e9	0.934769
$472$	0	0
$473$	5.63717e7	0.0244933
$474$	0	0
$475$	3.96312e8	0.169672
$476$	0	0
$477$	3.65878e8	0.154355
$478$	0	0
$479$	−3.03579e8	−0.126211	−0.0631054	−	0.998007i	$-0.520100\pi$
$479$	−3.03579e8	−0.126211	−0.0631054	+	0.998007i	$0.520100\pi$
$480$	0	0
$481$	−1.32655e9	−0.543522
$482$	0	0
$483$	−6.66792e7	−0.0269262
$484$	0	0
$485$	−4.62504e8	−0.184086
$486$	0	0
$487$	−4.36059e8	−0.171078	−0.0855389	−	0.996335i	$-0.527261\pi$
$487$	−4.36059e8	−0.171078	−0.0855389	+	0.996335i	$0.527261\pi$
$488$	0	0
$489$	−1.26410e9	−0.488877
$490$	0	0
$491$	−8.34813e8	−0.318276	−0.159138	−	0.987256i	$-0.550872\pi$
$491$	−8.34813e8	−0.318276	−0.159138	+	0.987256i	$0.550872\pi$
$492$	0	0
$493$	−2.69769e9	−1.01398
$494$	0	0
$495$	−2.68272e8	−0.0994162
$496$	0	0
$497$	−6.35389e8	−0.232163
$498$	0	0
$499$	4.50230e9	1.62212	0.811059	−	0.584964i	$-0.198891\pi$
$499$	4.50230e9	1.62212	0.811059	+	0.584964i	$0.198891\pi$
$500$	0	0
$501$	−6.75116e8	−0.239854
$502$	0	0
$503$	−2.41700e9	−0.846815	−0.423407	−	0.905939i	$-0.639166\pi$
$503$	−2.41700e9	−0.846815	−0.423407	+	0.905939i	$0.639166\pi$
$504$	0	0
$505$	2.25031e9	0.777538
$506$	0	0
$507$	−1.57636e9	−0.537188
$508$	0	0
$509$	3.51209e9	1.18047	0.590233	−	0.807233i	$-0.299036\pi$
$509$	3.51209e9	1.18047	0.590233	+	0.807233i	$0.299036\pi$
$510$	0	0
$511$	−6.39622e8	−0.212056
$512$	0	0
$513$	−4.99240e8	−0.163267
$514$	0	0
$515$	−6.71508e8	−0.216634
$516$	0	0
$517$	−2.47595e9	−0.787998
$518$	0	0
$519$	−1.43281e9	−0.449887
$520$	0	0
$521$	9.90013e8	0.306697	0.153348	−	0.988172i	$-0.450994\pi$
$521$	9.90013e8	0.306697	0.153348	+	0.988172i	$0.450994\pi$
$522$	0	0
$523$	−4.45926e8	−0.136303	−0.0681517	−	0.997675i	$-0.521710\pi$
$523$	−4.45926e8	−0.136303	−0.0681517	+	0.997675i	$0.521710\pi$
$524$	0	0
$525$	−1.77188e8	−0.0534411
$526$	0	0
$527$	−3.33567e9	−0.992765
$528$	0	0
$529$	−3.37025e9	−0.989845
$530$	0	0
$531$	1.15632e9	0.335157
$532$	0	0
$533$	1.27571e9	0.364926
$534$	0	0
$535$	−1.44247e9	−0.407257
$536$	0	0
$537$	1.13955e9	0.317557
$538$	0	0
$539$	−1.90519e9	−0.524056
$540$	0	0
$541$	2.84753e9	0.773175	0.386588	−	0.922253i	$-0.373654\pi$
$541$	2.84753e9	0.773175	0.386588	+	0.922253i	$0.373654\pi$
$542$	0	0
$543$	2.70055e9	0.723858
$544$	0	0
$545$	1.96474e8	0.0519898
$546$	0	0
$547$	−4.34116e9	−1.13410	−0.567048	−	0.823684i	$-0.691915\pi$
$547$	−4.34116e9	−1.13410	−0.567048	+	0.823684i	$0.691915\pi$
$548$	0	0
$549$	−2.71889e8	−0.0701276
$550$	0	0
$551$	4.13540e9	1.05314
$552$	0	0
$553$	−1.77741e9	−0.446940
$554$	0	0
$555$	4.06789e8	0.101005
$556$	0	0
$557$	3.00124e9	0.735880	0.367940	−	0.929849i	$-0.380063\pi$
$557$	3.00124e9	0.735880	0.367940	+	0.929849i	$0.380063\pi$
$558$	0	0
$559$	−2.10743e8	−0.0510284
$560$	0	0
$561$	1.31521e9	0.314503
$562$	0	0
$563$	7.43886e9	1.75682	0.878409	−	0.477909i	$-0.158605\pi$
$563$	7.43886e9	1.75682	0.878409	+	0.477909i	$0.158605\pi$
$564$	0	0
$565$	3.15062e9	0.734897
$566$	0	0
$567$	2.23205e8	0.0514237
$568$	0	0
$569$	−7.20171e8	−0.163886	−0.0819431	−	0.996637i	$-0.526113\pi$
$569$	−7.20171e8	−0.163886	−0.0819431	+	0.996637i	$0.526113\pi$
$570$	0	0
$571$	2.30186e9	0.517431	0.258716	−	0.965954i	$-0.416701\pi$
$571$	2.30186e9	0.517431	0.258716	+	0.965954i	$0.416701\pi$
$572$	0	0
$573$	1.66755e9	0.370286
$574$	0	0
$575$	9.18750e7	0.0201539
$576$	0	0
$577$	−7.37257e9	−1.59773	−0.798865	−	0.601511i	$-0.794566\pi$
$577$	−7.37257e9	−1.59773	−0.798865	+	0.601511i	$0.794566\pi$
$578$	0	0
$579$	2.07203e9	0.443630
$580$	0	0
$581$	7.78766e8	0.164737
$582$	0	0
$583$	1.47756e9	0.308821
$584$	0	0
$585$	1.00292e9	0.207120
$586$	0	0
$587$	−7.36500e8	−0.150293	−0.0751467	−	0.997172i	$-0.523942\pi$
$587$	−7.36500e8	−0.150293	−0.0751467	+	0.997172i	$0.523942\pi$
$588$	0	0
$589$	5.11338e9	1.03111
$590$	0	0
$591$	4.88786e9	0.974008
$592$	0	0
$593$	−5.08539e9	−1.00146	−0.500729	−	0.865604i	$-0.666935\pi$
$593$	−5.08539e9	−1.00146	−0.500729	+	0.865604i	$0.666935\pi$
$594$	0	0
$595$	8.68665e8	0.169061
$596$	0	0
$597$	1.66343e9	0.319959
$598$	0	0
$599$	−4.86023e9	−0.923980	−0.461990	−	0.886885i	$-0.652864\pi$
$599$	−4.86023e9	−0.923980	−0.461990	+	0.886885i	$0.652864\pi$
$600$	0	0
$601$	6.78466e9	1.27488	0.637438	−	0.770502i	$-0.279994\pi$
$601$	6.78466e9	1.27488	0.637438	+	0.770502i	$0.279994\pi$
$602$	0	0
$603$	−3.32500e9	−0.617563
$604$	0	0
$605$	1.35250e9	0.248310
$606$	0	0
$607$	−2.85250e9	−0.517685	−0.258842	−	0.965920i	$-0.583341\pi$
$607$	−2.85250e9	−0.517685	−0.258842	+	0.965920i	$0.583341\pi$
$608$	0	0
$609$	−1.84890e9	−0.331705
$610$	0	0
$611$	9.25622e9	1.64168
$612$	0	0
$613$	7.75467e9	1.35973	0.679864	−	0.733339i	$-0.262039\pi$
$613$	7.75467e9	1.35973	0.679864	+	0.733339i	$0.262039\pi$
$614$	0	0
$615$	−3.91196e8	−0.0678159
$616$	0	0
$617$	−6.34097e9	−1.08682	−0.543410	−	0.839468i	$-0.682867\pi$
$617$	−6.34097e9	−1.08682	−0.543410	+	0.839468i	$0.682867\pi$
$618$	0	0
$619$	−3.19831e9	−0.542005	−0.271003	−	0.962579i	$-0.587355\pi$
$619$	−3.19831e9	−0.542005	−0.271003	+	0.962579i	$0.587355\pi$
$620$	0	0
$621$	−1.15736e8	−0.0193931
$622$	0	0
$623$	−2.89303e9	−0.479342
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−2.01613e9	−0.326650
$628$	0	0
$629$	−1.99429e9	−0.319529
$630$	0	0
$631$	3.79459e9	0.601260	0.300630	−	0.953741i	$-0.402803\pi$
$631$	3.79459e9	0.601260	0.300630	+	0.953741i	$0.402803\pi$
$632$	0	0
$633$	−4.58776e9	−0.718931
$634$	0	0
$635$	4.92600e9	0.763460
$636$	0	0
$637$	7.12246e9	1.09180
$638$	0	0
$639$	−1.10285e9	−0.167211
$640$	0	0
$641$	6.77964e9	1.01672	0.508362	−	0.861143i	$-0.330251\pi$
$641$	6.77964e9	1.01672	0.508362	+	0.861143i	$0.330251\pi$
$642$	0	0
$643$	1.13203e9	0.167927	0.0839635	−	0.996469i	$-0.473242\pi$
$643$	1.13203e9	0.167927	0.0839635	+	0.996469i	$0.473242\pi$
$644$	0	0
$645$	6.46245e7	0.00948284
$646$	0	0
$647$	5.54265e9	0.804549	0.402274	−	0.915519i	$-0.368220\pi$
$647$	5.54265e9	0.804549	0.402274	+	0.915519i	$0.368220\pi$
$648$	0	0
$649$	4.66970e9	0.670552
$650$	0	0
$651$	−2.28614e9	−0.324766
$652$	0	0
$653$	9.41765e9	1.32357	0.661784	−	0.749695i	$-0.269800\pi$
$653$	9.41765e9	1.32357	0.661784	+	0.749695i	$0.269800\pi$
$654$	0	0
$655$	1.76353e9	0.245210
$656$	0	0
$657$	−1.11020e9	−0.152730
$658$	0	0
$659$	−7.46390e9	−1.01594	−0.507969	−	0.861376i	$-0.669603\pi$
$659$	−7.46390e9	−1.01594	−0.507969	+	0.861376i	$0.669603\pi$
$660$	0	0
$661$	5.58309e9	0.751917	0.375958	−	0.926637i	$-0.377314\pi$
$661$	5.58309e9	0.751917	0.375958	+	0.926637i	$0.377314\pi$
$662$	0	0
$663$	−4.91684e9	−0.655223
$664$	0	0
$665$	−1.33161e9	−0.175591
$666$	0	0
$667$	9.58687e8	0.125094
$668$	0	0
$669$	3.99948e9	0.516431
$670$	0	0
$671$	−1.09800e9	−0.140305
$672$	0	0
$673$	3.39933e9	0.429873	0.214936	−	0.976628i	$-0.431046\pi$
$673$	3.39933e9	0.429873	0.214936	+	0.976628i	$0.431046\pi$
$674$	0	0
$675$	−3.07547e8	−0.0384900
$676$	0	0
$677$	−1.38930e10	−1.72082	−0.860409	−	0.509604i	$-0.829792\pi$
$677$	−1.38930e10	−1.72082	−0.860409	+	0.509604i	$0.829792\pi$
$678$	0	0
$679$	1.55401e9	0.190507
$680$	0	0
$681$	−1.06954e9	−0.129773
$682$	0	0
$683$	7.22576e9	0.867782	0.433891	−	0.900965i	$-0.357140\pi$
$683$	7.22576e9	0.867782	0.433891	+	0.900965i	$0.357140\pi$
$684$	0	0
$685$	1.00064e9	0.118949
$686$	0	0
$687$	1.00390e9	0.118125
$688$	0	0
$689$	−5.52380e9	−0.643385
$690$	0	0
$691$	5.83386e9	0.672641	0.336320	−	0.941748i	$-0.390818\pi$
$691$	5.83386e9	0.672641	0.336320	+	0.941748i	$0.390818\pi$
$692$	0	0
$693$	9.01394e8	0.102884
$694$	0	0
$695$	5.75896e9	0.650724
$696$	0	0
$697$	1.91785e9	0.214536
$698$	0	0
$699$	4.84897e9	0.537006
$700$	0	0
$701$	4.37486e9	0.479680	0.239840	−	0.970812i	$-0.422905\pi$
$701$	4.37486e9	0.479680	0.239840	+	0.970812i	$0.422905\pi$
$702$	0	0
$703$	3.05712e9	0.331871
$704$	0	0
$705$	−2.83843e9	−0.305082
$706$	0	0
$707$	−7.56103e9	−0.804661
$708$	0	0
$709$	−3.54685e9	−0.373750	−0.186875	−	0.982384i	$-0.559836\pi$
$709$	−3.54685e9	−0.373750	−0.186875	+	0.982384i	$0.559836\pi$
$710$	0	0
$711$	−3.08507e9	−0.321900
$712$	0	0
$713$	1.18541e9	0.122477
$714$	0	0
$715$	4.05021e9	0.414387
$716$	0	0
$717$	−1.00799e10	−1.02126
$718$	0	0
$719$	1.06545e10	1.06901	0.534507	−	0.845164i	$-0.320497\pi$
$719$	1.06545e10	1.06901	0.534507	+	0.845164i	$0.320497\pi$
$720$	0	0
$721$	2.25627e9	0.224191
$722$	0	0
$723$	6.93958e9	0.682888
$724$	0	0
$725$	2.54753e9	0.248277
$726$	0	0
$727$	9.21169e9	0.889138	0.444569	−	0.895745i	$-0.353357\pi$
$727$	9.21169e9	0.889138	0.444569	+	0.895745i	$0.353357\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−3.16823e8	−0.0299989
$732$	0	0
$733$	−5.81770e8	−0.0545616	−0.0272808	−	0.999628i	$-0.508685\pi$
$733$	−5.81770e8	−0.0545616	−0.0272808	+	0.999628i	$0.508685\pi$
$734$	0	0
$735$	−2.18411e9	−0.202894
$736$	0	0
$737$	−1.34277e10	−1.23557
$738$	0	0
$739$	−1.43208e8	−0.0130531	−0.00652654	−	0.999979i	$-0.502077\pi$
$739$	−1.43208e8	−0.0130531	−0.00652654	+	0.999979i	$0.502077\pi$
$740$	0	0
$741$	7.53722e9	0.680530
$742$	0	0
$743$	−1.76012e10	−1.57428	−0.787139	−	0.616775i	$-0.788439\pi$
$743$	−1.76012e10	−1.57428	−0.787139	+	0.616775i	$0.788439\pi$
$744$	0	0
$745$	−9.04406e9	−0.801340
$746$	0	0
$747$	1.35171e9	0.118649
$748$	0	0
$749$	4.84670e9	0.421463
$750$	0	0
$751$	2.10398e10	1.81260	0.906299	−	0.422636i	$-0.138895\pi$
$751$	2.10398e10	1.81260	0.906299	+	0.422636i	$0.138895\pi$
$752$	0	0
$753$	−3.43830e9	−0.293468
$754$	0	0
$755$	−4.62578e9	−0.391174
$756$	0	0
$757$	−3.91015e9	−0.327610	−0.163805	−	0.986493i	$-0.552377\pi$
$757$	−3.91015e9	−0.327610	−0.163805	+	0.986493i	$0.552377\pi$
$758$	0	0
$759$	−4.67389e8	−0.0388001
$760$	0	0
$761$	1.20242e10	0.989032	0.494516	−	0.869169i	$-0.335345\pi$
$761$	1.20242e10	0.989032	0.494516	+	0.869169i	$0.335345\pi$
$762$	0	0
$763$	−6.60153e8	−0.0538033
$764$	0	0
$765$	1.50775e9	0.121763
$766$	0	0
$767$	−1.74575e10	−1.39700
$768$	0	0
$769$	1.18948e10	0.943223	0.471611	−	0.881807i	$-0.343673\pi$
$769$	1.18948e10	0.943223	0.471611	+	0.881807i	$0.343673\pi$
$770$	0	0
$771$	−3.52841e9	−0.277261
$772$	0	0
$773$	−7.77614e9	−0.605531	−0.302765	−	0.953065i	$-0.597910\pi$
$773$	−7.77614e9	−0.605531	−0.302765	+	0.953065i	$0.597910\pi$
$774$	0	0
$775$	3.15000e9	0.243083
$776$	0	0
$777$	−1.36681e9	−0.104528
$778$	0	0
$779$	−2.93994e9	−0.222822
$780$	0	0
$781$	−4.45378e9	−0.334541
$782$	0	0
$783$	−3.20916e9	−0.238905
$784$	0	0
$785$	9.81350e9	0.724069
$786$	0	0
$787$	−2.44365e10	−1.78701	−0.893507	−	0.449050i	$-0.851763\pi$
$787$	−2.44365e10	−1.78701	−0.893507	+	0.449050i	$0.851763\pi$
$788$	0	0
$789$	−7.22916e9	−0.523985
$790$	0	0
$791$	−1.05861e10	−0.760532
$792$	0	0
$793$	4.10482e9	0.292306
$794$	0	0
$795$	1.69388e9	0.119563
$796$	0	0
$797$	−2.26970e9	−0.158805	−0.0794024	−	0.996843i	$-0.525301\pi$
$797$	−2.26970e9	−0.158805	−0.0794024	+	0.996843i	$0.525301\pi$
$798$	0	0
$799$	1.39155e10	0.965125
$800$	0	0
$801$	−5.02148e9	−0.345237
$802$	0	0
$803$	−4.48345e9	−0.305568
$804$	0	0
$805$	−3.08700e8	−0.0208570
$806$	0	0
$807$	−4.03467e9	−0.270240
$808$	0	0
$809$	−1.63200e10	−1.08368	−0.541838	−	0.840483i	$-0.682271\pi$
$809$	−1.63200e10	−1.08368	−0.541838	+	0.840483i	$0.682271\pi$
$810$	0	0
$811$	−6.99393e9	−0.460414	−0.230207	−	0.973142i	$-0.573940\pi$
$811$	−6.99393e9	−0.460414	−0.230207	+	0.973142i	$0.573940\pi$
$812$	0	0
$813$	−4.13599e9	−0.269937
$814$	0	0
$815$	−5.85230e9	−0.378682
$816$	0	0
$817$	4.85670e8	0.0311576
$818$	0	0
$819$	−3.36982e9	−0.214345
$820$	0	0
$821$	4.00949e9	0.252865	0.126432	−	0.991975i	$-0.459647\pi$
$821$	4.00949e9	0.252865	0.126432	+	0.991975i	$0.459647\pi$
$822$	0	0
$823$	−1.88572e10	−1.17917	−0.589586	−	0.807706i	$-0.700709\pi$
$823$	−1.88572e10	−1.17917	−0.589586	+	0.807706i	$0.700709\pi$
$824$	0	0
$825$	−1.24200e9	−0.0770075
$826$	0	0
$827$	−1.66386e10	−1.02293	−0.511466	−	0.859304i	$-0.670897\pi$
$827$	−1.66386e10	−1.02293	−0.511466	+	0.859304i	$0.670897\pi$
$828$	0	0
$829$	−1.37224e10	−0.836547	−0.418274	−	0.908321i	$-0.637365\pi$
$829$	−1.37224e10	−0.836547	−0.418274	+	0.908321i	$0.637365\pi$
$830$	0	0
$831$	−1.76705e10	−1.06818
$832$	0	0
$833$	1.07076e10	0.641853
$834$	0	0
$835$	−3.12554e9	−0.185790
$836$	0	0
$837$	−3.96809e9	−0.233907
$838$	0	0
$839$	6.56954e8	0.0384033	0.0192016	−	0.999816i	$-0.493888\pi$
$839$	6.56954e8	0.0384033	0.0192016	+	0.999816i	$0.493888\pi$
$840$	0	0
$841$	9.33282e9	0.541037
$842$	0	0
$843$	−1.48903e10	−0.856066
$844$	0	0
$845$	−7.29794e9	−0.416104
$846$	0	0
$847$	−4.54441e9	−0.256972
$848$	0	0
$849$	−5.67541e9	−0.318288
$850$	0	0
$851$	7.08716e8	0.0394202
$852$	0	0
$853$	−8.70997e9	−0.480502	−0.240251	−	0.970711i	$-0.577230\pi$
$853$	−8.70997e9	−0.480502	−0.240251	+	0.970711i	$0.577230\pi$
$854$	0	0
$855$	−2.31129e9	−0.126466
$856$	0	0
$857$	1.93825e9	0.105190	0.0525952	−	0.998616i	$-0.483251\pi$
$857$	1.93825e9	0.105190	0.0525952	+	0.998616i	$0.483251\pi$
$858$	0	0
$859$	7.95332e8	0.0428127	0.0214063	−	0.999771i	$-0.493186\pi$
$859$	7.95332e8	0.0428127	0.0214063	+	0.999771i	$0.493186\pi$
$860$	0	0
$861$	1.31442e9	0.0701815
$862$	0	0
$863$	−2.24172e9	−0.118725	−0.0593626	−	0.998236i	$-0.518907\pi$
$863$	−2.24172e9	−0.118725	−0.0593626	+	0.998236i	$0.518907\pi$
$864$	0	0
$865$	−6.63338e9	−0.348481
$866$	0	0
$867$	3.68735e9	0.192153
$868$	0	0
$869$	−1.24588e10	−0.644030
$870$	0	0
$871$	5.01989e10	2.57413
$872$	0	0
$873$	2.69732e9	0.137209
$874$	0	0
$875$	−8.20312e8	−0.0413953
$876$	0	0
$877$	1.99169e10	0.997062	0.498531	−	0.866872i	$-0.333873\pi$
$877$	1.99169e10	0.997062	0.498531	+	0.866872i	$0.333873\pi$
$878$	0	0
$879$	−1.46448e10	−0.727317
$880$	0	0
$881$	−2.76906e10	−1.36432	−0.682161	−	0.731202i	$-0.738960\pi$
$881$	−2.76906e10	−1.36432	−0.682161	+	0.731202i	$0.738960\pi$
$882$	0	0
$883$	1.65616e10	0.809544	0.404772	−	0.914418i	$-0.367351\pi$
$883$	1.65616e10	0.809544	0.404772	+	0.914418i	$0.367351\pi$
$884$	0	0
$885$	5.35334e9	0.259611
$886$	0	0
$887$	−6.69904e9	−0.322314	−0.161157	−	0.986929i	$-0.551523\pi$
$887$	−6.69904e9	−0.322314	−0.161157	+	0.986929i	$0.551523\pi$
$888$	0	0
$889$	−1.65514e10	−0.790092
$890$	0	0
$891$	1.56456e9	0.0741005
$892$	0	0
$893$	−2.13315e10	−1.00240
$894$	0	0
$895$	5.27568e9	0.245979
$896$	0	0
$897$	1.74731e9	0.0808346
$898$	0	0
$899$	3.28693e10	1.50880
$900$	0	0
$901$	−8.30427e9	−0.378238
$902$	0	0
$903$	−2.17138e8	−0.00981362
$904$	0	0
$905$	1.25026e10	0.560698
$906$	0	0
$907$	−3.87814e9	−0.172583	−0.0862916	−	0.996270i	$-0.527502\pi$
$907$	−3.87814e9	−0.172583	−0.0862916	+	0.996270i	$0.527502\pi$
$908$	0	0
$909$	−1.31238e10	−0.579543
$910$	0	0
$911$	7.15870e9	0.313704	0.156852	−	0.987622i	$-0.449865\pi$
$911$	7.15870e9	0.313704	0.156852	+	0.987622i	$0.449865\pi$
$912$	0	0
$913$	5.45878e9	0.237382
$914$	0	0
$915$	−1.25875e9	−0.0543206
$916$	0	0
$917$	−5.92545e9	−0.253763
$918$	0	0
$919$	−2.54160e10	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$919$	−2.54160e10	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$920$	0	0
$921$	2.50149e10	1.05509
$922$	0	0
$923$	1.66502e10	0.696970
$924$	0	0
$925$	1.88328e9	0.0782382
$926$	0	0
$927$	3.91624e9	0.161469
$928$	0	0
$929$	−2.34566e10	−0.959865	−0.479932	−	0.877306i	$-0.659339\pi$
$929$	−2.34566e10	−0.959865	−0.479932	+	0.877306i	$0.659339\pi$
$930$	0	0
$931$	−1.64141e10	−0.666644
$932$	0	0
$933$	−5.75036e9	−0.231798
$934$	0	0
$935$	6.08893e9	0.243613
$936$	0	0
$937$	9.30070e9	0.369341	0.184670	−	0.982801i	$-0.440878\pi$
$937$	9.30070e9	0.369341	0.184670	+	0.982801i	$0.440878\pi$
$938$	0	0
$939$	−9.82520e9	−0.387268
$940$	0	0
$941$	−1.13117e10	−0.442553	−0.221276	−	0.975211i	$-0.571022\pi$
$941$	−1.13117e10	−0.442553	−0.221276	+	0.975211i	$0.571022\pi$
$942$	0	0
$943$	−6.81551e8	−0.0264672
$944$	0	0
$945$	1.03336e9	0.0398327
$946$	0	0
$947$	−4.46270e9	−0.170755	−0.0853774	−	0.996349i	$-0.527210\pi$
$947$	−4.46270e9	−0.170755	−0.0853774	+	0.996349i	$0.527210\pi$
$948$	0	0
$949$	1.67611e10	0.636608
$950$	0	0
$951$	8.56032e9	0.322744
$952$	0	0
$953$	1.90872e10	0.714359	0.357179	−	0.934036i	$-0.383739\pi$
$953$	1.90872e10	0.714359	0.357179	+	0.934036i	$0.383739\pi$
$954$	0	0
$955$	7.72013e9	0.286822
$956$	0	0
$957$	−1.29599e10	−0.477980
$958$	0	0
$959$	−3.36215e9	−0.123098
$960$	0	0
$961$	1.31299e10	0.477234
$962$	0	0
$963$	8.41249e9	0.303552
$964$	0	0
$965$	9.59274e9	0.343635
$966$	0	0
$967$	1.34102e10	0.476916	0.238458	−	0.971153i	$-0.423358\pi$
$967$	1.34102e10	0.476916	0.238458	+	0.971153i	$0.423358\pi$
$968$	0	0
$969$	1.13312e10	0.400075
$970$	0	0
$971$	−4.64191e10	−1.62716	−0.813579	−	0.581455i	$-0.802484\pi$
$971$	−4.64191e10	−1.62716	−0.813579	+	0.581455i	$0.802484\pi$
$972$	0	0
$973$	−1.93501e10	−0.673423
$974$	0	0
$975$	4.64316e9	0.160434
$976$	0	0
$977$	3.92972e10	1.34813	0.674064	−	0.738673i	$-0.264547\pi$
$977$	3.92972e10	1.34813	0.674064	+	0.738673i	$0.264547\pi$
$978$	0	0
$979$	−2.02788e10	−0.690721
$980$	0	0
$981$	−1.14584e9	−0.0387509
$982$	0	0
$983$	3.69370e10	1.24029	0.620146	−	0.784486i	$-0.287073\pi$
$983$	3.69370e10	1.24029	0.620146	+	0.784486i	$0.287073\pi$
$984$	0	0
$985$	2.26290e10	0.754463
$986$	0	0
$987$	9.53712e9	0.315724
$988$	0	0
$989$	1.12590e8	0.00370095
$990$	0	0
$991$	1.02868e10	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$991$	1.02868e10	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$992$	0	0
$993$	6.06302e9	0.196502
$994$	0	0
$995$	7.70106e9	0.247839
$996$	0	0
$997$	−3.24439e10	−1.03681	−0.518407	−	0.855134i	$-0.673475\pi$
$997$	−3.24439e10	−1.03681	−0.518407	+	0.855134i	$0.673475\pi$
$998$	0	0
$999$	−2.37239e9	−0.0752848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.a.c.1.1		1
4.3	odd	2		15.8.a.a.1.1	✓	1
12.11	even	2		45.8.a.g.1.1		1
20.3	even	4		75.8.b.a.49.2		2
20.7	even	4		75.8.b.a.49.1		2
20.19	odd	2		75.8.a.c.1.1		1
60.23	odd	4		225.8.b.a.199.1		2
60.47	odd	4		225.8.b.a.199.2		2
60.59	even	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	4.3	odd	2
45.8.a.g.1.1		1	12.11	even	2
75.8.a.c.1.1		1	20.19	odd	2
75.8.b.a.49.1		2	20.7	even	4
75.8.b.a.49.2		2	20.3	even	4
225.8.a.a.1.1		1	60.59	even	2
225.8.b.a.199.1		2	60.23	odd	4
225.8.b.a.199.2		2	60.47	odd	4
240.8.a.c.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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