LMFDB - Embedded newform 45.8.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,8,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.0573261468$
Analytic rank:	$0$
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	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+22.0000 q^{2} +356.000 q^{4} +125.000 q^{5} -420.000 q^{7} +5016.00 q^{8} +2750.00 q^{10} +2944.00 q^{11} -11006.0 q^{13} -9240.00 q^{14} +64784.0 q^{16} +16546.0 q^{17} -25364.0 q^{19} +44500.0 q^{20} +64768.0 q^{22} +5880.00 q^{23} +15625.0 q^{25} -242132. q^{26} -149520. q^{28} -163042. q^{29} -201600. q^{31} +783200. q^{32} +364012. q^{34} -52500.0 q^{35} +120530. q^{37} -558008. q^{38} +627000. q^{40} +115910. q^{41} -19148.0 q^{43} +1.04806e6 q^{44} +129360. q^{46} -841016. q^{47} -647143. q^{49} +343750. q^{50} -3.91814e6 q^{52} -501890. q^{53} +368000. q^{55} -2.10672e6 q^{56} -3.58692e6 q^{58} +1.58618e6 q^{59} -372962. q^{61} -4.43520e6 q^{62} +8.93805e6 q^{64} -1.37575e6 q^{65} +4.56104e6 q^{67} +5.89038e6 q^{68} -1.15500e6 q^{70} -1.51283e6 q^{71} -1.52291e6 q^{73} +2.65166e6 q^{74} -9.02958e6 q^{76} -1.23648e6 q^{77} +4.23192e6 q^{79} +8.09800e6 q^{80} +2.55002e6 q^{82} +1.85420e6 q^{83} +2.06825e6 q^{85} -421256. q^{86} +1.47671e7 q^{88} +6.88817e6 q^{89} +4.62252e6 q^{91} +2.09328e6 q^{92} -1.85024e7 q^{94} -3.17050e6 q^{95} +3.70003e6 q^{97} -1.42371e7 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	22.0000	1.94454	0.972272	−	0.233854i	$-0.0751336\pi$
$2$	22.0000	1.94454	0.972272	+	0.233854i	$0.0751336\pi$
$3$	0	0
$3$	0	0
$4$	356.000	2.78125
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	−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
$9$	0	0
$10$	2750.00	0.869626
$11$	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	$0.744463\pi$
$14$	−9240.00	−0.899961
$15$	0	0
$16$	64784.0	3.95410
$17$	16546.0	0.816811	0.408406	−	0.912801i	$-0.366085\pi$
$17$	16546.0	0.816811	0.408406	+	0.912801i	$0.366085\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	64768.0	1.29682
$23$	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
$26$	−242132.	−2.70175
$27$	0	0
$28$	−149520.	−1.28720
$29$	−163042.	−1.24139	−0.620693	−	0.784054i	$-0.713148\pi$
$29$	−163042.	−1.24139	−0.620693	+	0.784054i	$0.713148\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	364012.	1.58833
$35$	−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	$0.437336\pi$
$38$	−558008.	−1.64967
$39$	0	0
$40$	627000.	1.54902
$41$	115910.	0.262650	0.131325	−	0.991339i	$-0.458077\pi$
$41$	115910.	0.262650	0.131325	+	0.991339i	$0.458077\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	129360.	0.195951
$47$	−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
$50$	343750.	0.388909
$51$	0	0
$52$	−3.91814e6	−3.86427
$53$	−501890.	−0.463066	−0.231533	−	0.972827i	$-0.574374\pi$
$53$	−501890.	−0.463066	−0.231533	+	0.972827i	$0.574374\pi$
$54$	0	0
$55$	368000.	0.298249
$56$	−2.10672e6	−1.60306
$57$	0	0
$58$	−3.58692e6	−2.41393
$59$	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	$0.533546\pi$
$62$	−4.43520e6	−2.36343
$63$	0	0
$64$	8.93805e6	4.26199
$65$	−1.37575e6	−0.621359
$66$	0	0
$67$	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
$69$	0	0
$70$	−1.15500e6	−0.402475
$71$	−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	$0.573576\pi$
$74$	2.65166e6	0.760688
$75$	0	0
$76$	−9.02958e6	−2.35950
$77$	−1.23648e6	−0.308652
$78$	0	0
$79$	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
$81$	0	0
$82$	2.55002e6	0.510734
$83$	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
$86$	−421256.	−0.0714170
$87$	0	0
$88$	1.47671e7	2.30997
$89$	6.88817e6	1.03571	0.517856	−	0.855468i	$-0.326730\pi$
$89$	6.88817e6	1.03571	0.517856	+	0.855468i	$0.326730\pi$
$90$	0	0
$91$	4.62252e6	0.643034
$92$	2.09328e6	0.280266
$93$	0	0
$94$	−1.85024e7	−2.29763
$95$	−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	$0.434016\pi$
$98$	−1.42371e7	−1.52803
$99$	0	0
$100$	5.56250e6	0.556250
$101$	1.80025e7	1.73863	0.869314	−	0.494259i	$-0.164561\pi$
$101$	1.80025e7	1.73863	0.869314	+	0.494259i	$0.164561\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	−1.10416e7	−0.900452
$107$	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	$0.518513\pi$
$110$	8.09600e6	0.579957
$111$	0	0
$112$	−2.72093e7	−1.83001
$113$	2.52050e7	1.64328	0.821640	−	0.570007i	$-0.193060\pi$
$113$	2.52050e7	1.64328	0.821640	+	0.570007i	$0.193060\pi$
$114$	0	0
$115$	735000.	0.0450656
$116$	−5.80430e7	−3.45260
$117$	0	0
$118$	3.48959e7	1.95518
$119$	−6.94932e6	−0.378031
$120$	0	0
$121$	−1.08200e7	−0.555239
$122$	−8.20516e6	−0.409098
$123$	0	0
$124$	−7.17696e7	−3.38037
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	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
$129$	0	0
$130$	−3.02665e7	−1.20826
$131$	−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
$134$	1.00343e8	3.60263
$135$	0	0
$136$	8.29947e7	2.82920
$137$	8.00512e6	0.265978	0.132989	−	0.991118i	$-0.457542\pi$
$137$	8.00512e6	0.265978	0.132989	+	0.991118i	$0.457542\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−3.32823e7	−0.975448
$143$	−3.24017e7	−0.926598
$144$	0	0
$145$	−2.03802e7	−0.555164
$146$	−3.35040e7	−0.890968
$147$	0	0
$148$	4.29087e7	1.08800
$149$	−7.23525e7	−1.79185	−0.895925	−	0.444206i	$-0.853486\pi$
$149$	−7.23525e7	−1.79185	−0.895925	+	0.444206i	$0.853486\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	−2.72026e7	−0.600188
$155$	−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	$0.800280\pi$
$158$	9.31022e7	1.87785
$159$	0	0
$160$	9.79000e7	1.88957
$161$	−2.46960e6	−0.0466376
$162$	0	0
$163$	−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
$165$	0	0
$166$	4.07925e7	0.692153
$167$	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
$170$	4.55015e7	0.710321
$171$	0	0
$172$	−6.81669e6	−0.102147
$173$	−5.30671e7	−0.779227	−0.389613	−	0.920979i	$-0.627391\pi$
$173$	−5.30671e7	−0.779227	−0.389613	+	0.920979i	$0.627391\pi$
$174$	0	0
$175$	−6.56250e6	−0.0925627
$176$	1.90724e8	2.63701
$177$	0	0
$178$	1.51540e8	2.01399
$179$	−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	$0.715668\pi$
$182$	1.01695e8	1.25041
$183$	0	0
$184$	2.94941e7	0.349038
$185$	1.50662e7	0.174946
$186$	0	0
$187$	4.87114e7	0.544735
$188$	−2.99402e8	−3.28626
$189$	0	0
$190$	−6.97510e7	−0.737756
$191$	−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	$0.625521\pi$
$194$	8.14007e7	0.800428
$195$	0	0
$196$	−2.30383e8	−2.18552
$197$	1.81032e8	1.68703	0.843516	−	0.537105i	$-0.180482\pi$
$197$	1.81032e8	1.68703	0.843516	+	0.537105i	$0.180482\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	3.96054e8	3.38084
$203$	6.84776e7	0.574530
$204$	0	0
$205$	1.44888e7	0.117461
$206$	−1.18185e8	−0.941952
$207$	0	0
$208$	−7.13013e8	−5.49383
$209$	−7.46716e7	−0.565775
$210$	0	0
$211$	−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
$213$	0	0
$214$	2.53875e8	1.77081
$215$	−2.39350e6	−0.0164248
$216$	0	0
$217$	8.46720e7	0.562511
$218$	−3.45795e7	−0.226058
$219$	0	0
$220$	1.31008e8	0.829504
$221$	−1.82105e8	−1.13488
$222$	0	0
$223$	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
$225$	0	0
$226$	5.54509e8	3.19543
$227$	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	$0.532620\pi$
$230$	1.61700e7	0.0876320
$231$	0	0
$232$	−8.17819e8	−4.29981
$233$	1.79591e8	0.930122	0.465061	−	0.885279i	$-0.346032\pi$
$233$	1.79591e8	0.930122	0.465061	+	0.885279i	$0.346032\pi$
$234$	0	0
$235$	−1.05127e8	−0.528417
$236$	5.64679e8	2.79646
$237$	0	0
$238$	−1.52885e8	−0.735099
$239$	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	$0.701424\pi$
$242$	−2.38041e8	−1.07969
$243$	0	0
$244$	−1.32774e8	−0.585127
$245$	−8.08929e7	−0.351422
$246$	0	0
$247$	2.79156e8	1.17871
$248$	−1.01123e9	−4.20986
$249$	0	0
$250$	4.29688e7	0.173925
$251$	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
$254$	8.66976e8	3.31963
$255$	0	0
$256$	9.76454e8	3.63757
$257$	−1.30682e8	−0.480230	−0.240115	−	0.970744i	$-0.577185\pi$
$257$	−1.30682e8	−0.480230	−0.240115	+	0.970744i	$0.577185\pi$
$258$	0	0
$259$	−5.06226e7	−0.181049
$260$	−4.89767e8	−1.72816
$261$	0	0
$262$	−3.10381e8	−1.06620
$263$	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
$266$	2.34363e8	0.763491
$267$	0	0
$268$	1.62373e9	5.15279
$269$	−1.49432e8	−0.468070	−0.234035	−	0.972228i	$-0.575193\pi$
$269$	−1.49432e8	−0.468070	−0.234035	+	0.972228i	$0.575193\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	1.76113e8	0.517206
$275$	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	$0.124004\pi$
$278$	1.01358e9	2.82943
$279$	0	0
$280$	−2.63340e8	−0.716908
$281$	−5.51493e8	−1.48275	−0.741375	−	0.671091i	$-0.765826\pi$
$281$	−5.51493e8	−1.48275	−0.741375	+	0.671091i	$0.765826\pi$
$282$	0	0
$283$	−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
$285$	0	0
$286$	−7.12837e8	−1.80181
$287$	−4.86822e7	−0.121558
$288$	0	0
$289$	−1.36569e8	−0.332819
$290$	−4.48366e8	−1.07954
$291$	0	0
$292$	−5.42156e8	−1.27434
$293$	−5.42402e8	−1.25975	−0.629875	−	0.776697i	$-0.716894\pi$
$293$	−5.42402e8	−1.25975	−0.629875	+	0.776697i	$0.716894\pi$
$294$	0	0
$295$	1.98272e8	0.449660
$296$	6.04578e8	1.35498
$297$	0	0
$298$	−1.59175e9	−3.48433
$299$	−6.47153e7	−0.140010
$300$	0	0
$301$	8.04216e6	0.0169977
$302$	−8.14137e8	−1.70088
$303$	0	0
$304$	−1.64318e9	−3.35450
$305$	−4.66202e7	−0.0940860
$306$	0	0
$307$	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
$309$	0	0
$310$	−5.54400e8	−1.05696
$311$	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	$0.391134\pi$
$314$	−1.72718e9	−3.14835
$315$	0	0
$316$	1.50656e9	2.68586
$317$	3.17049e8	0.559009	0.279505	−	0.960144i	$-0.409830\pi$
$317$	3.17049e8	0.559009	0.279505	+	0.960144i	$0.409830\pi$
$318$	0	0
$319$	−4.79996e8	−0.827885
$320$	1.11726e9	1.90602
$321$	0	0
$322$	−5.43312e7	−0.0906888
$323$	−4.19673e8	−0.692950
$324$	0	0
$325$	−1.71969e8	−0.277880
$326$	−1.03001e9	−1.64656
$327$	0	0
$328$	5.81405e8	0.909746
$329$	3.53227e8	0.546850
$330$	0	0
$331$	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
$333$	0	0
$334$	5.50094e8	0.807838
$335$	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	$0.843556\pi$
$338$	1.28444e9	1.80927
$339$	0	0
$340$	7.36297e8	1.01596
$341$	−5.93510e8	−0.810565
$342$	0	0
$343$	6.17688e8	0.826494
$344$	−9.60464e7	−0.127212
$345$	0	0
$346$	−1.16748e9	−1.51524
$347$	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	$0.595516\pi$
$350$	−1.44375e8	−0.179992
$351$	0	0
$352$	2.30574e9	2.81781
$353$	−6.18559e7	−0.0748461	−0.0374231	−	0.999300i	$-0.511915\pi$
$353$	−6.18559e7	−0.0748461	−0.0374231	+	0.999300i	$0.511915\pi$
$354$	0	0
$355$	−1.89104e8	−0.224337
$356$	2.45219e9	2.88057
$357$	0	0
$358$	−9.28519e8	−1.06955
$359$	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
$362$	−2.20045e9	−2.43799
$363$	0	0
$364$	1.64562e9	1.78844
$365$	−1.90364e8	−0.204908
$366$	0	0
$367$	−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
$369$	0	0
$370$	3.31458e8	0.340190
$371$	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	$0.591117\pi$
$374$	1.07165e9	1.05926
$375$	0	0
$376$	−4.21854e9	−4.09265
$377$	1.79444e9	1.72478
$378$	0	0
$379$	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
$381$	0	0
$382$	−1.35874e9	−1.24714
$383$	−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
$386$	−1.68832e9	−1.49417
$387$	0	0
$388$	1.31721e9	1.14484
$389$	−6.02482e8	−0.518944	−0.259472	−	0.965751i	$-0.583549\pi$
$389$	−6.02482e8	−0.518944	−0.259472	+	0.965751i	$0.583549\pi$
$390$	0	0
$391$	9.72905e7	0.0823098
$392$	−3.24607e9	−2.72180
$393$	0	0
$394$	3.98270e9	3.28051
$395$	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	$0.523865\pi$
$398$	1.35539e9	1.07764
$399$	0	0
$400$	1.01225e9	0.790820
$401$	9.96333e8	0.771613	0.385806	−	0.922580i	$-0.373923\pi$
$401$	9.96333e8	0.771613	0.385806	+	0.922580i	$0.373923\pi$
$402$	0	0
$403$	2.21881e9	1.68870
$404$	6.40887e9	4.83556
$405$	0	0
$406$	1.50651e9	1.11720
$407$	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	$0.831015\pi$
$410$	3.18752e8	0.228407
$411$	0	0
$412$	−1.91246e9	−1.34726
$413$	−6.66194e8	−0.465345
$414$	0	0
$415$	2.31775e8	0.159184
$416$	−8.61990e9	−5.87051
$417$	0	0
$418$	−1.64278e9	−1.10017
$419$	−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	$0.122394\pi$
$422$	−3.73817e9	−2.42139
$423$	0	0
$424$	−2.51748e9	−1.60393
$425$	2.58531e8	0.163362
$426$	0	0
$427$	1.56644e8	0.0973680
$428$	4.10816e9	2.53276
$429$	0	0
$430$	−5.26570e7	−0.0319386
$431$	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	$0.788619\pi$
$434$	1.86278e9	1.09383
$435$	0	0
$436$	−5.59559e8	−0.323328
$437$	−1.49140e8	−0.0854890
$438$	0	0
$439$	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
$441$	0	0
$442$	−4.00632e9	−2.20682
$443$	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
$446$	3.25884e9	1.73936
$447$	0	0
$448$	−3.75398e9	−1.97251
$449$	8.85012e7	0.0461410	0.0230705	−	0.999734i	$-0.492656\pi$
$449$	8.85012e7	0.0461410	0.0230705	+	0.999734i	$0.492656\pi$
$450$	0	0
$451$	3.41239e8	0.175162
$452$	8.97296e9	4.57037
$453$	0	0
$454$	8.71480e8	0.437081
$455$	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	$0.595877\pi$
$458$	−8.17995e8	−0.397852
$459$	0	0
$460$	2.61660e8	0.125339
$461$	−8.57428e8	−0.407610	−0.203805	−	0.979012i	$-0.565331\pi$
$461$	−8.57428e8	−0.407610	−0.203805	+	0.979012i	$0.565331\pi$
$462$	0	0
$463$	−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
$465$	0	0
$466$	3.95101e9	1.80866
$467$	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
$470$	−2.31279e9	−1.02753
$471$	0	0
$472$	7.95626e9	3.48267
$473$	−5.63717e7	−0.0244933
$474$	0	0
$475$	−3.96312e8	−0.169672
$476$	−2.47396e9	−1.05140
$477$	0	0
$478$	8.21322e9	3.43966
$479$	−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
$482$	−5.65447e9	−2.30000
$483$	0	0
$484$	−3.85193e9	−1.54426
$485$	4.62504e8	0.184086
$486$	0	0
$487$	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
$489$	0	0
$490$	−1.77964e9	−0.683355
$491$	−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
$494$	6.14144e9	2.29206
$495$	0	0
$496$	−1.30605e10	−4.80587
$497$	6.35389e8	0.232163
$498$	0	0
$499$	−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
$501$	0	0
$502$	2.80158e9	0.988416
$503$	−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
$506$	3.80836e8	0.130681
$507$	0	0
$508$	1.40292e10	4.74801
$509$	−3.51209e9	−1.18047	−0.590233	−	0.807233i	$-0.700964\pi$
$509$	−3.51209e9	−1.18047	−0.590233	+	0.807233i	$0.700964\pi$
$510$	0	0
$511$	6.39622e8	0.212056
$512$	9.14439e9	3.01099
$513$	0	0
$514$	−2.87500e9	−0.933829
$515$	−6.71508e8	−0.216634
$516$	0	0
$517$	−2.47595e9	−0.787998
$518$	−1.11370e9	−0.352057
$519$	0	0
$520$	−6.90076e9	−2.15221
$521$	−9.90013e8	−0.306697	−0.153348	−	0.988172i	$-0.549006\pi$
$521$	−9.90013e8	−0.306697	−0.153348	+	0.988172i	$0.549006\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	5.89043e9	1.76481
$527$	−3.33567e9	−0.992765
$528$	0	0
$529$	−3.37025e9	−0.989845
$530$	−1.38020e9	−0.402694
$531$	0	0
$532$	3.79243e9	1.09201
$533$	−1.27571e9	−0.364926
$534$	0	0
$535$	1.44247e9	0.407257
$536$	2.28782e10	6.41719
$537$	0	0
$538$	−3.28751e9	−0.910183
$539$	−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	$0.373654\pi$
$542$	−3.37007e9	−0.909162
$543$	0	0
$544$	1.29588e10	3.45120
$545$	−1.96474e8	−0.0519898
$546$	0	0
$547$	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
$549$	0	0
$550$	1.01200e9	0.259365
$551$	4.13540e9	1.05314
$552$	0	0
$553$	−1.77741e9	−0.446940
$554$	1.43982e10	3.59769
$555$	0	0
$556$	1.64015e10	4.04690
$557$	−3.00124e9	−0.735880	−0.367940	−	0.929849i	$-0.619937\pi$
$557$	−3.00124e9	−0.735880	−0.367940	+	0.929849i	$0.619937\pi$
$558$	0	0
$559$	2.10743e8	0.0510284
$560$	−3.40116e9	−0.818406
$561$	0	0
$562$	−1.21328e10	−2.88327
$563$	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
$566$	−4.62441e9	−1.07201
$567$	0	0
$568$	−7.58837e9	−1.73752
$569$	7.20171e8	0.163886	0.0819431	−	0.996637i	$-0.473887\pi$
$569$	7.20171e8	0.163886	0.0819431	+	0.996637i	$0.473887\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	−1.07101e9	−0.236375
$575$	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	$0.794566\pi$
$578$	−3.00451e9	−0.647181
$579$	0	0
$580$	−7.25537e9	−1.54405
$581$	−7.78766e8	−0.164737
$582$	0	0
$583$	−1.47756e9	−0.308821
$584$	−7.63892e9	−1.58704
$585$	0	0
$586$	−1.19328e10	−2.44964
$587$	−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
$590$	4.36198e9	0.874384
$591$	0	0
$592$	7.80842e9	1.54681
$593$	5.08539e9	1.00146	0.500729	−	0.865604i	$-0.333065\pi$
$593$	5.08539e9	1.00146	0.500729	+	0.865604i	$0.333065\pi$
$594$	0	0
$595$	−8.68665e8	−0.169061
$596$	−2.57575e10	−4.98358
$597$	0	0
$598$	−1.42374e9	−0.272255
$599$	−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	$0.279994\pi$
$602$	1.76928e8	0.0330528
$603$	0	0
$604$	−1.31742e10	−2.43274
$605$	−1.35250e9	−0.248310
$606$	0	0
$607$	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
$609$	0	0
$610$	−1.02565e9	−0.182954
$611$	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	$0.262039\pi$
$614$	2.03825e10	3.55360
$615$	0	0
$616$	−6.20218e9	−1.06908
$617$	6.34097e9	1.08682	0.543410	−	0.839468i	$-0.317133\pi$
$617$	6.34097e9	1.08682	0.543410	+	0.839468i	$0.317133\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	4.68548e9	0.780706
$623$	−2.89303e9	−0.479342
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	8.00572e9	1.30434
$627$	0	0
$628$	−2.79488e10	−4.50303
$629$	1.99429e9	0.319529
$630$	0	0
$631$	−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
$633$	0	0
$634$	6.97508e9	1.08702
$635$	4.92600e9	0.763460
$636$	0	0
$637$	7.12246e9	1.09180
$638$	−1.05599e10	−1.60986
$639$	0	0
$640$	1.20484e10	1.81677
$641$	−6.77964e9	−1.01672	−0.508362	−	0.861143i	$-0.669749\pi$
$641$	−6.77964e9	−1.01672	−0.508362	+	0.861143i	$0.669749\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	−9.23280e9	−1.34747
$647$	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
$650$	−3.78331e9	−0.540350
$651$	0	0
$652$	−1.66674e10	−2.35505
$653$	−9.41765e9	−1.32357	−0.661784	−	0.749695i	$-0.730200\pi$
$653$	−9.41765e9	−1.32357	−0.661784	+	0.749695i	$0.730200\pi$
$654$	0	0
$655$	−1.76353e9	−0.245210
$656$	7.50911e9	1.03854
$657$	0	0
$658$	7.77099e9	1.06337
$659$	−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	$0.377314\pi$
$662$	4.94024e9	0.661828
$663$	0	0
$664$	9.30069e9	1.23290
$665$	1.33161e9	0.175591
$666$	0	0
$667$	−9.58687e8	−0.125094
$668$	8.90152e9	1.15544
$669$	0	0
$670$	1.25429e10	1.61115
$671$	−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	$0.431046\pi$
$674$	−2.72549e10	−3.42875
$675$	0	0
$676$	2.07845e10	2.58778
$677$	1.38930e10	1.72082	0.860409	−	0.509604i	$-0.170208\pi$
$677$	1.38930e10	1.72082	0.860409	+	0.509604i	$0.170208\pi$
$678$	0	0
$679$	−1.55401e9	−0.190507
$680$	1.03743e10	1.26526
$681$	0	0
$682$	−1.30572e10	−1.57618
$683$	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
$686$	1.35891e10	1.60715
$687$	0	0
$688$	−1.24048e9	−0.145222
$689$	5.52380e9	0.643385
$690$	0	0
$691$	−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
$693$	0	0
$694$	1.28401e10	1.45818
$695$	5.75896e9	0.650724
$696$	0	0
$697$	1.91785e9	0.214536
$698$	−1.03284e10	−1.14958
$699$	0	0
$700$	−2.33625e9	−0.257440
$701$	−4.37486e9	−0.479680	−0.239840	−	0.970812i	$-0.577095\pi$
$701$	−4.37486e9	−0.479680	−0.239840	+	0.970812i	$0.577095\pi$
$702$	0	0
$703$	−3.05712e9	−0.331871
$704$	2.63136e10	2.84234
$705$	0	0
$706$	−1.36083e9	−0.145542
$707$	−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	$0.559836\pi$
$710$	−4.16029e9	−0.436234
$711$	0	0
$712$	3.45511e10	3.58742
$713$	−1.18541e9	−0.122477
$714$	0	0
$715$	−4.05021e9	−0.414387
$716$	−1.50251e10	−1.52976
$717$	0	0
$718$	2.71782e8	0.0274022
$719$	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
$722$	−5.51186e9	−0.545027
$723$	0	0
$724$	−3.56073e10	−3.48702
$725$	−2.54753e9	−0.248277
$726$	0	0
$727$	−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
$729$	0	0
$730$	−4.18800e9	−0.398453
$731$	−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	$0.508685\pi$
$734$	−2.76131e8	−0.0257738
$735$	0	0
$736$	4.60522e9	0.425773
$737$	1.34277e10	1.23557
$738$	0	0
$739$	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
$741$	0	0
$742$	4.63746e9	0.416742
$743$	−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
$746$	−1.24519e10	−1.09812
$747$	0	0
$748$	1.73413e10	1.51504
$749$	−4.84670e9	−0.421463
$750$	0	0
$751$	−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
$753$	0	0
$754$	3.94777e10	3.35391
$755$	−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	$0.552377\pi$
$758$	3.91107e10	3.26177
$759$	0	0
$760$	−1.59032e10	−1.31413
$761$	−1.20242e10	−0.989032	−0.494516	−	0.869169i	$-0.664655\pi$
$761$	−1.20242e10	−0.989032	−0.494516	+	0.869169i	$0.664655\pi$
$762$	0	0
$763$	6.60153e8	0.0538033
$764$	−2.19869e10	−1.78377
$765$	0	0
$766$	−2.60028e10	−2.09035
$767$	−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	$0.343673\pi$
$770$	−3.40032e9	−0.268412
$771$	0	0
$772$	−2.73201e10	−2.13709
$773$	7.77614e9	0.605531	0.302765	−	0.953065i	$-0.402090\pi$
$773$	7.77614e9	0.605531	0.302765	+	0.953065i	$0.402090\pi$
$774$	0	0
$775$	−3.15000e9	−0.243083
$776$	1.85594e10	1.42576
$777$	0	0
$778$	−1.32546e10	−1.00911
$779$	−2.93994e9	−0.222822
$780$	0	0
$781$	−4.45378e9	−0.334541
$782$	2.14039e9	0.160055
$783$	0	0
$784$	−4.19245e10	−3.10715
$785$	−9.81350e9	−0.724069
$786$	0	0
$787$	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
$789$	0	0
$790$	1.16378e10	0.839799
$791$	−1.05861e10	−0.760532
$792$	0	0
$793$	4.10482e9	0.292306
$794$	−4.10883e9	−0.291304
$795$	0	0
$796$	2.19326e10	1.54133
$797$	2.26970e9	0.158805	0.0794024	−	0.996843i	$-0.474699\pi$
$797$	2.26970e9	0.158805	0.0794024	+	0.996843i	$0.474699\pi$
$798$	0	0
$799$	−1.39155e10	−0.965125
$800$	1.22375e10	0.845041
$801$	0	0
$802$	2.19193e10	1.50043
$803$	−4.48345e9	−0.305568
$804$	0	0
$805$	−3.08700e8	−0.0208570
$806$	4.88138e10	3.28375
$807$	0	0
$808$	9.03003e10	6.02212
$809$	1.63200e10	1.08368	0.541838	−	0.840483i	$-0.317729\pi$
$809$	1.63200e10	1.08368	0.541838	+	0.840483i	$0.317729\pi$
$810$	0	0
$811$	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
$813$	0	0
$814$	7.80649e9	0.507306
$815$	−5.85230e9	−0.378682
$816$	0	0
$817$	4.85670e8	0.0311576
$818$	−5.25017e10	−3.35380
$819$	0	0
$820$	5.15800e9	0.326687
$821$	−4.00949e9	−0.252865	−0.126432	−	0.991975i	$-0.540353\pi$
$821$	−4.00949e9	−0.252865	−0.126432	+	0.991975i	$0.540353\pi$
$822$	0	0
$823$	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
$825$	0	0
$826$	−1.46563e10	−0.904885
$827$	−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	$0.637365\pi$
$830$	5.09906e9	0.309540
$831$	0	0
$832$	−9.83722e10	−5.92162
$833$	−1.07076e10	−0.641853
$834$	0	0
$835$	3.12554e9	0.185790
$836$	−2.65831e10	−1.57356
$837$	0	0
$838$	−5.71217e10	−3.35311
$839$	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
$842$	6.24469e10	3.60511
$843$	0	0
$844$	−6.04904e10	−3.46328
$845$	7.29794e9	0.416104
$846$	0	0
$847$	4.54441e9	0.256972
$848$	−3.25144e10	−1.83101
$849$	0	0
$850$	5.68769e9	0.317665
$851$	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	$0.577230\pi$
$854$	3.44617e9	0.189336
$855$	0	0
$856$	5.78835e10	3.15425
$857$	−1.93825e9	−0.105190	−0.0525952	−	0.998616i	$-0.516749\pi$
$857$	−1.93825e9	−0.105190	−0.0525952	+	0.998616i	$0.516749\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	3.36179e10	1.78770
$863$	−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
$866$	−5.85335e10	−3.06261
$867$	0	0
$868$	3.01432e10	1.56448
$869$	1.24588e10	0.644030
$870$	0	0
$871$	−5.01989e10	−2.57413
$872$	−7.88412e9	−0.402666
$873$	0	0
$874$	−3.28109e9	−0.166237
$875$	−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	$0.333873\pi$
$878$	5.04289e10	2.51449
$879$	0	0
$880$	2.38405e10	1.17931
$881$	2.76906e10	1.36432	0.682161	−	0.731202i	$-0.261040\pi$
$881$	2.76906e10	1.36432	0.682161	+	0.731202i	$0.261040\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	2.05489e10	0.992593
$887$	−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
$890$	1.89425e10	0.900683
$891$	0	0
$892$	5.27339e10	2.48778
$893$	2.13315e10	1.00240
$894$	0	0
$895$	−5.27568e9	−0.245979
$896$	−4.04827e10	−1.88015
$897$	0	0
$898$	1.94703e9	0.0897232
$899$	3.28693e10	1.50880
$900$	0	0
$901$	−8.30427e9	−0.378238
$902$	7.50726e9	0.340611
$903$	0	0
$904$	1.26428e11	5.69186
$905$	−1.25026e10	−0.560698
$906$	0	0
$907$	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
$909$	0	0
$910$	1.27119e10	0.559199
$911$	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
$914$	−2.66341e10	−1.15379
$915$	0	0
$916$	−1.32366e10	−0.569041
$917$	5.92545e9	0.253763
$918$	0	0
$919$	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
$921$	0	0
$922$	−1.88634e10	−0.792615
$923$	1.66502e10	0.696970
$924$	0	0
$925$	1.88328e9	0.0782382
$926$	−4.28256e10	−1.77241
$927$	0	0
$928$	−1.27694e11	−5.24511
$929$	2.34566e10	0.959865	0.479932	−	0.877306i	$-0.340661\pi$
$929$	2.34566e10	0.959865	0.479932	+	0.877306i	$0.340661\pi$
$930$	0	0
$931$	1.64141e10	0.666644
$932$	6.39345e10	2.58690
$933$	0	0
$934$	2.11190e10	0.848124
$935$	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	$0.440878\pi$
$938$	−4.21440e10	−1.66735
$939$	0	0
$940$	−3.74252e10	−1.46966
$941$	1.13117e10	0.442553	0.221276	−	0.975211i	$-0.428978\pi$
$941$	1.13117e10	0.442553	0.221276	+	0.975211i	$0.428978\pi$
$942$	0	0
$943$	6.81551e8	0.0264672
$944$	1.02759e11	3.97573
$945$	0	0
$946$	−1.24018e9	−0.0476283
$947$	−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
$950$	−8.71888e9	−0.329935
$951$	0	0
$952$	−3.48578e10	−1.30939
$953$	−1.90872e10	−0.714359	−0.357179	−	0.934036i	$-0.616261\pi$
$953$	−1.90872e10	−0.714359	−0.357179	+	0.934036i	$0.616261\pi$
$954$	0	0
$955$	−7.72013e9	−0.286822
$956$	1.32905e11	4.91969
$957$	0	0
$958$	−6.67873e9	−0.245423
$959$	−3.36215e9	−0.123098
$960$	0	0
$961$	1.31299e10	0.477234
$962$	−2.91842e10	−1.05690
$963$	0	0
$964$	−9.14997e10	−3.28965
$965$	−9.59274e9	−0.343635
$966$	0	0
$967$	−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
$969$	0	0
$970$	1.01751e10	0.357962
$971$	−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
$974$	9.59329e9	0.332668
$975$	0	0
$976$	−2.41620e10	−0.831875
$977$	−3.92972e10	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$977$	−3.92972e10	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$978$	0	0
$979$	2.02788e10	0.690721
$980$	−2.87979e10	−0.977393
$981$	0	0
$982$	−1.83659e10	−0.618902
$983$	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
$986$	−5.93492e10	−1.97172
$987$	0	0
$988$	9.93796e10	3.27829
$989$	−1.12590e8	−0.00370095
$990$	0	0
$991$	−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
$993$	0	0
$994$	1.39786e10	0.451451
$995$	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	$0.673475\pi$
$998$	−9.90506e10	−3.15428
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.a.a.1.1	✓	1	3.2	odd	2
45.8.a.g.1.1		1	1.1	even	1	trivial
75.8.a.c.1.1		1	15.14	odd	2
75.8.b.a.49.1		2	15.2	even	4
75.8.b.a.49.2		2	15.8	even	4
225.8.a.a.1.1		1	5.4	even	2
225.8.b.a.199.1		2	5.3	odd	4
225.8.b.a.199.2		2	5.2	odd	4
240.8.a.c.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more