LMFDB - Embedded newform 1280.3.g.e.1151.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1280,3,Mod(1151,1280)] 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("1280.1151"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.7
Root			$-0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1280.1151
Dual form			1280.3.g.e.1151.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.80423 q^{3} -2.23607i q^{5} -8.50651i q^{7} +5.47214 q^{9} -1.79611 q^{11} +0.472136i q^{13} -8.50651i q^{15} -23.8885 q^{17} -9.40456 q^{19} -32.3607i q^{21} -16.1150i q^{23} -5.00000 q^{25} -13.4208 q^{27} +6.94427i q^{29} -47.4468i q^{31} -6.83282 q^{33} -19.0211 q^{35} -26.3607i q^{37} +1.79611i q^{39} +41.4164 q^{41} +2.00811 q^{43} -12.2361i q^{45} +35.3481i q^{47} -23.3607 q^{49} -90.8774 q^{51} +21.6393i q^{53} +4.01623i q^{55} -35.7771 q^{57} -73.8644 q^{59} -26.1378i q^{61} -46.5488i q^{63} +1.05573 q^{65} +88.8693 q^{67} -61.3050i q^{69} -39.4144i q^{71} -137.554 q^{73} -19.0211 q^{75} +15.2786i q^{77} +113.703i q^{79} -100.305 q^{81} +21.2412 q^{83} +53.4164i q^{85} +26.4176i q^{87} -67.4427 q^{89} +4.01623 q^{91} -180.498i q^{93} +21.0292i q^{95} -39.1672 q^{97} -9.82857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} - 48 q^{17} - 40 q^{25} + 160 q^{33} + 224 q^{41} - 8 q^{49} + 80 q^{65} - 528 q^{73} - 552 q^{81} + 176 q^{89} - 528 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 - 48 * q^17 - 40 * q^25 + 160 * q^33 + 224 * q^41 - 8 * q^49 + 80 * q^65 - 528 * q^73 - 552 * q^81 + 176 * q^89 - 528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	3.80423	1.26808	0.634038	−	0.773302i	$-0.281396\pi$
$3$	3.80423	1.26808	0.634038	+	0.773302i	$0.281396\pi$
$4$	0	0
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	− 8.50651i	− 1.21522i	−0.794237	−	0.607608i	$-0.792129\pi$
$7$	− 8.50651i	− 1.21522i	0.794237	−	0.607608i	$-0.207871\pi$
$8$	0	0
$9$	5.47214	0.608015
$10$	0	0
$11$	−1.79611	−0.163283	−0.0816415	−	0.996662i	$-0.526016\pi$
$11$	−1.79611	−0.163283	−0.0816415	+	0.996662i	$0.526016\pi$
$12$	0	0
$13$	0.472136i	0.0363182i	0.999835	+	0.0181591i	$0.00578053\pi$
$13$	0.472136i	0.0363182i	−0.999835	+	0.0181591i	$0.994219\pi$
$14$	0	0
$15$	− 8.50651i	− 0.567101i
$16$	0	0
$17$	−23.8885	−1.40521	−0.702604	−	0.711581i	$-0.747980\pi$
$17$	−23.8885	−1.40521	−0.702604	+	0.711581i	$0.747980\pi$
$18$	0	0
$19$	−9.40456	−0.494977	−0.247489	−	0.968891i	$-0.579605\pi$
$19$	−9.40456	−0.494977	−0.247489	+	0.968891i	$0.579605\pi$
$20$	0	0
$21$	− 32.3607i	− 1.54098i
$22$	0	0
$23$	− 16.1150i	− 0.700650i	−0.936628	−	0.350325i	$-0.886071\pi$
$23$	− 16.1150i	− 0.700650i	0.936628	−	0.350325i	$-0.113929\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−13.4208	−0.497066
$28$	0	0
$29$	6.94427i	0.239458i	0.992807	+	0.119729i	$0.0382025\pi$
$29$	6.94427i	0.239458i	−0.992807	+	0.119729i	$0.961797\pi$
$30$	0	0
$31$	− 47.4468i	− 1.53054i	−0.643708	−	0.765271i	$-0.722605\pi$
$31$	− 47.4468i	− 1.53054i	0.643708	−	0.765271i	$-0.277395\pi$
$32$	0	0
$33$	−6.83282	−0.207055
$34$	0	0
$35$	−19.0211	−0.543461
$36$	0	0
$37$	− 26.3607i	− 0.712451i	−0.934400	−	0.356225i	$-0.884064\pi$
$37$	− 26.3607i	− 0.712451i	0.934400	−	0.356225i	$-0.115936\pi$
$38$	0	0
$39$	1.79611i	0.0460542i
$40$	0	0
$41$	41.4164	1.01016	0.505078	−	0.863074i	$-0.331464\pi$
$41$	41.4164	1.01016	0.505078	+	0.863074i	$0.331464\pi$
$42$	0	0
$43$	2.00811	0.0467003	0.0233502	−	0.999727i	$-0.492567\pi$
$43$	2.00811	0.0467003	0.0233502	+	0.999727i	$0.492567\pi$
$44$	0	0
$45$	− 12.2361i	− 0.271913i
$46$	0	0
$47$	35.3481i	0.752087i	0.926602	+	0.376044i	$0.122716\pi$
$47$	35.3481i	0.752087i	−0.926602	+	0.376044i	$0.877284\pi$
$48$	0	0
$49$	−23.3607	−0.476749
$50$	0	0
$51$	−90.8774	−1.78191
$52$	0	0
$53$	21.6393i	0.408289i	0.978941	+	0.204145i	$0.0654413\pi$
$53$	21.6393i	0.408289i	−0.978941	+	0.204145i	$0.934559\pi$
$54$	0	0
$55$	4.01623i	0.0730223i
$56$	0	0
$57$	−35.7771	−0.627668
$58$	0	0
$59$	−73.8644	−1.25194	−0.625970	−	0.779848i	$-0.715297\pi$
$59$	−73.8644	−1.25194	−0.625970	+	0.779848i	$0.715297\pi$
$60$	0	0
$61$	− 26.1378i	− 0.428488i	−0.976780	−	0.214244i	$-0.931271\pi$
$61$	− 26.1378i	− 0.428488i	0.976780	−	0.214244i	$-0.0687288\pi$
$62$	0	0
$63$	− 46.5488i	− 0.738869i
$64$	0	0
$65$	1.05573	0.0162420
$66$	0	0
$67$	88.8693	1.32641	0.663204	−	0.748439i	$-0.269196\pi$
$67$	88.8693	1.32641	0.663204	+	0.748439i	$0.269196\pi$
$68$	0	0
$69$	− 61.3050i	− 0.888478i
$70$	0	0
$71$	− 39.4144i	− 0.555132i	−0.960707	−	0.277566i	$-0.910472\pi$
$71$	− 39.4144i	− 0.555132i	0.960707	−	0.277566i	$-0.0895277\pi$
$72$	0	0
$73$	−137.554	−1.88430	−0.942152	−	0.335186i	$-0.891201\pi$
$73$	−137.554	−1.88430	−0.942152	+	0.335186i	$0.891201\pi$
$74$	0	0
$75$	−19.0211	−0.253615
$76$	0	0
$77$	15.2786i	0.198424i
$78$	0	0
$79$	113.703i	1.43928i	0.694350	+	0.719638i	$0.255692\pi$
$79$	113.703i	1.43928i	−0.694350	+	0.719638i	$0.744308\pi$
$80$	0	0
$81$	−100.305	−1.23833
$82$	0	0
$83$	21.2412	0.255919	0.127959	−	0.991779i	$-0.459157\pi$
$83$	21.2412	0.255919	0.127959	+	0.991779i	$0.459157\pi$
$84$	0	0
$85$	53.4164i	0.628428i
$86$	0	0
$87$	26.4176i	0.303650i
$88$	0	0
$89$	−67.4427	−0.757783	−0.378892	−	0.925441i	$-0.623695\pi$
$89$	−67.4427	−0.757783	−0.378892	+	0.925441i	$0.623695\pi$
$90$	0	0
$91$	4.01623	0.0441344
$92$	0	0
$93$	− 180.498i	− 1.94084i
$94$	0	0
$95$	21.0292i	0.221360i
$96$	0	0
$97$	−39.1672	−0.403785	−0.201893	−	0.979408i	$-0.564709\pi$
$97$	−39.1672	−0.403785	−0.201893	+	0.979408i	$0.564709\pi$
$98$	0	0
$99$	−9.82857	−0.0992785
$100$	0	0
$101$	− 99.8885i	− 0.988995i	−0.869179	−	0.494498i	$-0.835352\pi$
$101$	− 99.8885i	− 0.988995i	0.869179	−	0.494498i	$-0.164648\pi$
$102$	0	0
$103$	− 35.7721i	− 0.347302i	−0.984807	−	0.173651i	$-0.944444\pi$
$103$	− 35.7721i	− 0.347302i	0.984807	−	0.173651i	$-0.0555565\pi$
$104$	0	0
$105$	−72.3607	−0.689149
$106$	0	0
$107$	121.099	1.13177	0.565884	−	0.824485i	$-0.308535\pi$
$107$	121.099	1.13177	0.565884	+	0.824485i	$0.308535\pi$
$108$	0	0
$109$	− 197.469i	− 1.81164i	−0.423660	−	0.905821i	$-0.639255\pi$
$109$	− 197.469i	− 1.81164i	0.423660	−	0.905821i	$-0.360745\pi$
$110$	0	0
$111$	− 100.282i	− 0.903441i
$112$	0	0
$113$	81.2786	0.719280	0.359640	−	0.933091i	$-0.382900\pi$
$113$	81.2786	0.719280	0.359640	+	0.933091i	$0.382900\pi$
$114$	0	0
$115$	−36.0341	−0.313340
$116$	0	0
$117$	2.58359i	0.0220820i
$118$	0	0
$119$	203.208i	1.70763i
$120$	0	0
$121$	−117.774	−0.973339
$122$	0	0
$123$	157.557	1.28095
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	1.84616i	0.0145367i	0.999974	+	0.00726834i	$0.00231361\pi$
$127$	1.84616i	0.0145367i	−0.999974	+	0.00726834i	$0.997686\pi$
$128$	0	0
$129$	7.63932	0.0592195
$130$	0	0
$131$	225.609	1.72221	0.861105	−	0.508428i	$-0.169773\pi$
$131$	225.609	1.72221	0.861105	+	0.508428i	$0.169773\pi$
$132$	0	0
$133$	80.0000i	0.601504i
$134$	0	0
$135$	30.0098i	0.222295i
$136$	0	0
$137$	52.8328	0.385641	0.192820	−	0.981234i	$-0.438236\pi$
$137$	52.8328	0.385641	0.192820	+	0.981234i	$0.438236\pi$
$138$	0	0
$139$	125.852	0.905407	0.452703	−	0.891661i	$-0.350460\pi$
$139$	125.852	0.905407	0.452703	+	0.891661i	$0.350460\pi$
$140$	0	0
$141$	134.472i	0.953703i
$142$	0	0
$143$	− 0.848009i	− 0.00593013i
$144$	0	0
$145$	15.5279	0.107089
$146$	0	0
$147$	−88.8693	−0.604553
$148$	0	0
$149$	− 132.971i	− 0.892420i	−0.894928	−	0.446210i	$-0.852773\pi$
$149$	− 132.971i	− 0.892420i	0.894928	−	0.446210i	$-0.147227\pi$
$150$	0	0
$151$	− 151.221i	− 1.00146i	−0.865603	−	0.500732i	$-0.833064\pi$
$151$	− 151.221i	− 1.00146i	0.865603	−	0.500732i	$-0.166936\pi$
$152$	0	0
$153$	−130.721	−0.854388
$154$	0	0
$155$	−106.094	−0.684480
$156$	0	0
$157$	− 36.7477i	− 0.234062i	−0.993128	−	0.117031i	$-0.962662\pi$
$157$	− 36.7477i	− 0.234062i	0.993128	−	0.117031i	$-0.0373376\pi$
$158$	0	0
$159$	82.3209i	0.517741i
$160$	0	0
$161$	−137.082	−0.851441
$162$	0	0
$163$	302.854	1.85800	0.929000	−	0.370079i	$-0.120669\pi$
$163$	302.854	1.85800	0.929000	+	0.370079i	$0.120669\pi$
$164$	0	0
$165$	15.2786i	0.0925978i
$166$	0	0
$167$	99.3839i	0.595113i	0.954704	+	0.297557i	$0.0961717\pi$
$167$	99.3839i	0.595113i	−0.954704	+	0.297557i	$0.903828\pi$
$168$	0	0
$169$	168.777	0.998681
$170$	0	0
$171$	−51.4631	−0.300954
$172$	0	0
$173$	− 181.639i	− 1.04994i	−0.851121	−	0.524969i	$-0.824077\pi$
$173$	− 181.639i	− 1.04994i	0.851121	−	0.524969i	$-0.175923\pi$
$174$	0	0
$175$	42.5325i	0.243043i
$176$	0	0
$177$	−280.997	−1.58755
$178$	0	0
$179$	−260.907	−1.45758	−0.728792	−	0.684735i	$-0.759918\pi$
$179$	−260.907	−1.45758	−0.728792	+	0.684735i	$0.759918\pi$
$180$	0	0
$181$	− 157.777i	− 0.871697i	−0.900020	−	0.435848i	$-0.856448\pi$
$181$	− 157.777i	− 0.871697i	0.900020	−	0.435848i	$-0.143552\pi$
$182$	0	0
$183$	− 99.4340i	− 0.543355i
$184$	0	0
$185$	−58.9443	−0.318618
$186$	0	0
$187$	42.9065	0.229447
$188$	0	0
$189$	114.164i	0.604043i
$190$	0	0
$191$	324.095i	1.69683i	0.529328	+	0.848417i	$0.322444\pi$
$191$	324.095i	1.69683i	−0.529328	+	0.848417i	$0.677556\pi$
$192$	0	0
$193$	181.777	0.941850	0.470925	−	0.882173i	$-0.343920\pi$
$193$	181.777	0.941850	0.470925	+	0.882173i	$0.343920\pi$
$194$	0	0
$195$	4.01623	0.0205960
$196$	0	0
$197$	− 140.525i	− 0.713324i	−0.934234	−	0.356662i	$-0.883915\pi$
$197$	− 140.525i	− 0.713324i	0.934234	−	0.356662i	$-0.116085\pi$
$198$	0	0
$199$	168.234i	0.845397i	0.906270	+	0.422698i	$0.138917\pi$
$199$	168.234i	0.845397i	−0.906270	+	0.422698i	$0.861083\pi$
$200$	0	0
$201$	338.079	1.68198
$202$	0	0
$203$	59.0715	0.290993
$204$	0	0
$205$	− 92.6099i	− 0.451756i
$206$	0	0
$207$	− 88.1833i	− 0.426006i
$208$	0	0
$209$	16.8916	0.0808213
$210$	0	0
$211$	93.9455	0.445240	0.222620	−	0.974905i	$-0.428539\pi$
$211$	93.9455	0.445240	0.222620	+	0.974905i	$0.428539\pi$
$212$	0	0
$213$	− 149.941i	− 0.703949i
$214$	0	0
$215$	− 4.49028i	− 0.0208850i
$216$	0	0
$217$	−403.607	−1.85994
$218$	0	0
$219$	−523.287	−2.38944
$220$	0	0
$221$	− 11.2786i	− 0.0510346i
$222$	0	0
$223$	− 214.035i	− 0.959797i	−0.877324	−	0.479899i	$-0.840673\pi$
$223$	− 214.035i	− 0.959797i	0.877324	−	0.479899i	$-0.159327\pi$
$224$	0	0
$225$	−27.3607	−0.121603
$226$	0	0
$227$	−41.4225	−0.182478	−0.0912389	−	0.995829i	$-0.529083\pi$
$227$	−41.4225	−0.182478	−0.0912389	+	0.995829i	$0.529083\pi$
$228$	0	0
$229$	− 73.2786i	− 0.319994i	−0.987117	−	0.159997i	$-0.948852\pi$
$229$	− 73.2786i	− 0.319994i	0.987117	−	0.159997i	$-0.0511485\pi$
$230$	0	0
$231$	58.1234i	0.251616i
$232$	0	0
$233$	307.050	1.31781	0.658905	−	0.752227i	$-0.271020\pi$
$233$	307.050	1.31781	0.658905	+	0.752227i	$0.271020\pi$
$234$	0	0
$235$	79.0407	0.336344
$236$	0	0
$237$	432.551i	1.82511i
$238$	0	0
$239$	42.9065i	0.179525i	0.995963	+	0.0897625i	$0.0286108\pi$
$239$	42.9065i	0.179525i	−0.995963	+	0.0897625i	$0.971389\pi$
$240$	0	0
$241$	−135.082	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$241$	−135.082	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$242$	0	0
$243$	−260.796	−1.07323
$244$	0	0
$245$	52.2361i	0.213208i
$246$	0	0
$247$	− 4.44023i	− 0.0179767i
$248$	0	0
$249$	80.8065	0.324524
$250$	0	0
$251$	221.169	0.881152	0.440576	−	0.897715i	$-0.354774\pi$
$251$	221.169	0.881152	0.440576	+	0.897715i	$0.354774\pi$
$252$	0	0
$253$	28.9443i	0.114404i
$254$	0	0
$255$	203.208i	0.796894i
$256$	0	0
$257$	−257.056	−1.00022	−0.500108	−	0.865963i	$-0.666707\pi$
$257$	−257.056	−1.00022	−0.500108	+	0.865963i	$0.666707\pi$
$258$	0	0
$259$	−224.237	−0.865781
$260$	0	0
$261$	38.0000i	0.145594i
$262$	0	0
$263$	− 164.168i	− 0.624212i	−0.950047	−	0.312106i	$-0.898966\pi$
$263$	− 164.168i	− 0.624212i	0.950047	−	0.312106i	$-0.101034\pi$
$264$	0	0
$265$	48.3870	0.182592
$266$	0	0
$267$	−256.567	−0.960926
$268$	0	0
$269$	− 35.4752i	− 0.131878i	−0.997824	−	0.0659391i	$-0.978996\pi$
$269$	− 35.4752i	− 0.131878i	0.997824	−	0.0659391i	$-0.0210043\pi$
$270$	0	0
$271$	− 298.950i	− 1.10314i	−0.834130	−	0.551568i	$-0.814030\pi$
$271$	− 298.950i	− 1.10314i	0.834130	−	0.551568i	$-0.185970\pi$
$272$	0	0
$273$	15.2786	0.0559657
$274$	0	0
$275$	8.98056	0.0326566
$276$	0	0
$277$	457.246i	1.65071i	0.564616	+	0.825354i	$0.309024\pi$
$277$	457.246i	1.65071i	−0.564616	+	0.825354i	$0.690976\pi$
$278$	0	0
$279$	− 259.635i	− 0.930593i
$280$	0	0
$281$	5.63932	0.0200688	0.0100344	−	0.999950i	$-0.496806\pi$
$281$	5.63932	0.0200688	0.0100344	+	0.999950i	$0.496806\pi$
$282$	0	0
$283$	169.918	0.600418	0.300209	−	0.953874i	$-0.402944\pi$
$283$	169.918	0.600418	0.300209	+	0.953874i	$0.402944\pi$
$284$	0	0
$285$	80.0000i	0.280702i
$286$	0	0
$287$	− 352.309i	− 1.22756i
$288$	0	0
$289$	281.663	0.974611
$290$	0	0
$291$	−149.001	−0.512030
$292$	0	0
$293$	26.8591i	0.0916694i	0.998949	+	0.0458347i	$0.0145947\pi$
$293$	26.8591i	0.0916694i	−0.998949	+	0.0458347i	$0.985405\pi$
$294$	0	0
$295$	165.166i	0.559884i
$296$	0	0
$297$	24.1052	0.0811624
$298$	0	0
$299$	7.60845	0.0254463
$300$	0	0
$301$	− 17.0820i	− 0.0567510i
$302$	0	0
$303$	− 379.999i	− 1.25412i
$304$	0	0
$305$	−58.4458	−0.191626
$306$	0	0
$307$	−118.031	−0.384466	−0.192233	−	0.981349i	$-0.561573\pi$
$307$	−118.031	−0.384466	−0.192233	+	0.981349i	$0.561573\pi$
$308$	0	0
$309$	− 136.085i	− 0.440405i
$310$	0	0
$311$	121.835i	0.391753i	0.980629	+	0.195877i	$0.0627552\pi$
$311$	121.835i	0.391753i	−0.980629	+	0.195877i	$0.937245\pi$
$312$	0	0
$313$	−219.548	−0.701431	−0.350716	−	0.936482i	$-0.614062\pi$
$313$	−219.548	−0.701431	−0.350716	+	0.936482i	$0.614062\pi$
$314$	0	0
$315$	−104.086	−0.330432
$316$	0	0
$317$	366.859i	1.15728i	0.815582	+	0.578642i	$0.196417\pi$
$317$	366.859i	1.15728i	−0.815582	+	0.578642i	$0.803583\pi$
$318$	0	0
$319$	− 12.4727i	− 0.0390993i
$320$	0	0
$321$	460.689	1.43517
$322$	0	0
$323$	224.661	0.695546
$324$	0	0
$325$	− 2.36068i	− 0.00726363i
$326$	0	0
$327$	− 751.217i	− 2.29730i
$328$	0	0
$329$	300.689	0.913948
$330$	0	0
$331$	162.846	0.491981	0.245990	−	0.969272i	$-0.420887\pi$
$331$	162.846	0.491981	0.245990	+	0.969272i	$0.420887\pi$
$332$	0	0
$333$	− 144.249i	− 0.433181i
$334$	0	0
$335$	− 198.718i	− 0.593187i
$336$	0	0
$337$	17.1084	0.0507666	0.0253833	−	0.999678i	$-0.491919\pi$
$337$	17.1084	0.0507666	0.0253833	+	0.999678i	$0.491919\pi$
$338$	0	0
$339$	309.202	0.912101
$340$	0	0
$341$	85.2198i	0.249911i
$342$	0	0
$343$	− 218.101i	− 0.635863i
$344$	0	0
$345$	−137.082	−0.397339
$346$	0	0
$347$	−167.498	−0.482703	−0.241351	−	0.970438i	$-0.577591\pi$
$347$	−167.498	−0.482703	−0.241351	+	0.970438i	$0.577591\pi$
$348$	0	0
$349$	− 483.495i	− 1.38537i	−0.721239	−	0.692687i	$-0.756427\pi$
$349$	− 483.495i	− 1.38537i	0.721239	−	0.692687i	$-0.243573\pi$
$350$	0	0
$351$	− 6.33644i	− 0.0180525i
$352$	0	0
$353$	307.994	0.872504	0.436252	−	0.899825i	$-0.356306\pi$
$353$	307.994	0.872504	0.436252	+	0.899825i	$0.356306\pi$
$354$	0	0
$355$	−88.1332	−0.248263
$356$	0	0
$357$	773.050i	2.16540i
$358$	0	0
$359$	− 23.2494i	− 0.0647615i	−0.999476	−	0.0323807i	$-0.989691\pi$
$359$	− 23.2494i	− 0.0647615i	0.999476	−	0.0323807i	$-0.0103089\pi$
$360$	0	0
$361$	−272.554	−0.754998
$362$	0	0
$363$	−448.039	−1.23427
$364$	0	0
$365$	307.580i	0.842686i
$366$	0	0
$367$	517.325i	1.40960i	0.709404	+	0.704802i	$0.248964\pi$
$367$	517.325i	1.40960i	−0.709404	+	0.704802i	$0.751036\pi$
$368$	0	0
$369$	226.636	0.614190
$370$	0	0
$371$	184.075	0.496159
$372$	0	0
$373$	88.3545i	0.236875i	0.992961	+	0.118438i	$0.0377886\pi$
$373$	88.3545i	0.236875i	−0.992961	+	0.118438i	$0.962211\pi$
$374$	0	0
$375$	42.5325i	0.113420i
$376$	0	0
$377$	−3.27864	−0.00869666
$378$	0	0
$379$	−19.3332	−0.0510112	−0.0255056	−	0.999675i	$-0.508120\pi$
$379$	−19.3332	−0.0510112	−0.0255056	+	0.999675i	$0.508120\pi$
$380$	0	0
$381$	7.02321i	0.0184336i
$382$	0	0
$383$	431.612i	1.12692i	0.826142	+	0.563462i	$0.190531\pi$
$383$	431.612i	1.12692i	−0.826142	+	0.563462i	$0.809469\pi$
$384$	0	0
$385$	34.1641	0.0887379
$386$	0	0
$387$	10.9887	0.0283945
$388$	0	0
$389$	296.354i	0.761837i	0.924609	+	0.380918i	$0.124392\pi$
$389$	296.354i	0.761837i	−0.924609	+	0.380918i	$0.875608\pi$
$390$	0	0
$391$	384.963i	0.984560i
$392$	0	0
$393$	858.269	2.18389
$394$	0	0
$395$	254.247	0.643664
$396$	0	0
$397$	− 86.1904i	− 0.217104i	−0.994091	−	0.108552i	$-0.965379\pi$
$397$	− 86.1904i	− 0.217104i	0.994091	−	0.108552i	$-0.0346214\pi$
$398$	0	0
$399$	304.338i	0.762752i
$400$	0	0
$401$	442.997	1.10473	0.552365	−	0.833602i	$-0.313725\pi$
$401$	442.997	1.10473	0.552365	+	0.833602i	$0.313725\pi$
$402$	0	0
$403$	22.4014	0.0555865
$404$	0	0
$405$	224.289i	0.553799i
$406$	0	0
$407$	47.3467i	0.116331i
$408$	0	0
$409$	−63.4102	−0.155037	−0.0775186	−	0.996991i	$-0.524700\pi$
$409$	−63.4102	−0.155037	−0.0775186	+	0.996991i	$0.524700\pi$
$410$	0	0
$411$	200.988	0.489022
$412$	0	0
$413$	628.328i	1.52138i
$414$	0	0
$415$	− 47.4969i	− 0.114450i
$416$	0	0
$417$	478.768	1.14812
$418$	0	0
$419$	−435.678	−1.03980	−0.519902	−	0.854226i	$-0.674032\pi$
$419$	−435.678	−1.03980	−0.519902	+	0.854226i	$0.674032\pi$
$420$	0	0
$421$	582.912i	1.38459i	0.721615	+	0.692294i	$0.243400\pi$
$421$	582.912i	1.38459i	−0.721615	+	0.692294i	$0.756600\pi$
$422$	0	0
$423$	193.430i	0.457280i
$424$	0	0
$425$	119.443	0.281042
$426$	0	0
$427$	−222.341	−0.520705
$428$	0	0
$429$	− 3.22602i	− 0.00751986i
$430$	0	0
$431$	− 375.882i	− 0.872117i	−0.899918	−	0.436058i	$-0.856374\pi$
$431$	− 375.882i	− 0.872117i	0.899918	−	0.436058i	$-0.143626\pi$
$432$	0	0
$433$	−368.164	−0.850263	−0.425132	−	0.905131i	$-0.639772\pi$
$433$	−368.164	−0.850263	−0.425132	+	0.905131i	$0.639772\pi$
$434$	0	0
$435$	59.0715	0.135797
$436$	0	0
$437$	151.554i	0.346806i
$438$	0	0
$439$	483.549i	1.10148i	0.834677	+	0.550739i	$0.185654\pi$
$439$	483.549i	1.10148i	−0.834677	+	0.550739i	$0.814346\pi$
$440$	0	0
$441$	−127.833	−0.289870
$442$	0	0
$443$	279.181	0.630205	0.315102	−	0.949058i	$-0.397961\pi$
$443$	279.181	0.630205	0.315102	+	0.949058i	$0.397961\pi$
$444$	0	0
$445$	150.807i	0.338891i
$446$	0	0
$447$	− 505.850i	− 1.13166i
$448$	0	0
$449$	756.079	1.68392	0.841959	−	0.539542i	$-0.181403\pi$
$449$	756.079	1.68392	0.841959	+	0.539542i	$0.181403\pi$
$450$	0	0
$451$	−74.3885	−0.164941
$452$	0	0
$453$	− 575.279i	− 1.26993i
$454$	0	0
$455$	− 8.98056i	− 0.0197375i
$456$	0	0
$457$	−285.672	−0.625103	−0.312551	−	0.949901i	$-0.601184\pi$
$457$	−285.672	−0.625103	−0.312551	+	0.949901i	$0.601184\pi$
$458$	0	0
$459$	320.603	0.698482
$460$	0	0
$461$	− 99.1146i	− 0.214999i	−0.994205	−	0.107500i	$-0.965716\pi$
$461$	− 99.1146i	− 0.214999i	0.994205	−	0.107500i	$-0.0342844\pi$
$462$	0	0
$463$	630.603i	1.36199i	0.732286	+	0.680997i	$0.238453\pi$
$463$	630.603i	1.36199i	−0.732286	+	0.680997i	$0.761547\pi$
$464$	0	0
$465$	−403.607	−0.867972
$466$	0	0
$467$	496.010	1.06212	0.531060	−	0.847334i	$-0.321794\pi$
$467$	496.010	1.06212	0.531060	+	0.847334i	$0.321794\pi$
$468$	0	0
$469$	− 755.967i	− 1.61187i
$470$	0	0
$471$	− 139.796i	− 0.296808i
$472$	0	0
$473$	−3.60680	−0.00762537
$474$	0	0
$475$	47.0228	0.0989954
$476$	0	0
$477$	118.413i	0.248246i
$478$	0	0
$479$	− 579.090i	− 1.20896i	−0.796621	−	0.604478i	$-0.793382\pi$
$479$	− 579.090i	− 1.20896i	0.796621	−	0.604478i	$-0.206618\pi$
$480$	0	0
$481$	12.4458	0.0258749
$482$	0	0
$483$	−521.491	−1.07969
$484$	0	0
$485$	87.5805i	0.180578i
$486$	0	0
$487$	626.363i	1.28617i	0.765796	+	0.643084i	$0.222345\pi$
$487$	626.363i	1.28617i	−0.765796	+	0.643084i	$0.777655\pi$
$488$	0	0
$489$	1152.13	2.35608
$490$	0	0
$491$	−22.3013	−0.0454201	−0.0227100	−	0.999742i	$-0.507229\pi$
$491$	−22.3013	−0.0454201	−0.0227100	+	0.999742i	$0.507229\pi$
$492$	0	0
$493$	− 165.889i	− 0.336488i
$494$	0	0
$495$	21.9773i	0.0443987i
$496$	0	0
$497$	−335.279	−0.674605
$498$	0	0
$499$	627.362	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$499$	627.362	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$500$	0	0
$501$	378.079i	0.754649i
$502$	0	0
$503$	780.853i	1.55239i	0.630492	+	0.776196i	$0.282853\pi$
$503$	780.853i	1.55239i	−0.630492	+	0.776196i	$0.717147\pi$
$504$	0	0
$505$	−223.358	−0.442292
$506$	0	0
$507$	642.066	1.26640
$508$	0	0
$509$	− 288.950i	− 0.567683i	−0.958871	−	0.283841i	$-0.908391\pi$
$509$	− 288.950i	− 0.567683i	0.958871	−	0.283841i	$-0.0916089\pi$
$510$	0	0
$511$	1170.11i	2.28984i
$512$	0	0
$513$	126.217	0.246036
$514$	0	0
$515$	−79.9888	−0.155318
$516$	0	0
$517$	− 63.4891i	− 0.122803i
$518$	0	0
$519$	− 690.997i	− 1.33140i
$520$	0	0
$521$	602.984	1.15736	0.578680	−	0.815555i	$-0.303568\pi$
$521$	602.984	1.15736	0.578680	+	0.815555i	$0.303568\pi$
$522$	0	0
$523$	−367.962	−0.703560	−0.351780	−	0.936083i	$-0.614423\pi$
$523$	−367.962	−0.703560	−0.351780	+	0.936083i	$0.614423\pi$
$524$	0	0
$525$	161.803i	0.308197i
$526$	0	0
$527$	1133.44i	2.15073i
$528$	0	0
$529$	269.308	0.509089
$530$	0	0
$531$	−404.196	−0.761198
$532$	0	0
$533$	19.5542i	0.0366870i
$534$	0	0
$535$	− 270.786i	− 0.506142i
$536$	0	0
$537$	−992.551	−1.84833
$538$	0	0
$539$	41.9584	0.0778449
$540$	0	0
$541$	− 616.885i	− 1.14027i	−0.821551	−	0.570134i	$-0.806891\pi$
$541$	− 616.885i	− 1.14027i	0.821551	−	0.570134i	$-0.193109\pi$
$542$	0	0
$543$	− 600.220i	− 1.10538i
$544$	0	0
$545$	−441.554	−0.810191
$546$	0	0
$547$	−97.8499	−0.178885	−0.0894423	−	0.995992i	$-0.528508\pi$
$547$	−97.8499	−0.178885	−0.0894423	+	0.995992i	$0.528508\pi$
$548$	0	0
$549$	− 143.029i	− 0.260527i
$550$	0	0
$551$	− 65.3078i	− 0.118526i
$552$	0	0
$553$	967.214	1.74903
$554$	0	0
$555$	−224.237	−0.404031
$556$	0	0
$557$	896.302i	1.60916i	0.593845	+	0.804580i	$0.297609\pi$
$557$	896.302i	1.60916i	−0.593845	+	0.804580i	$0.702391\pi$
$558$	0	0
$559$	0.948103i	0.00169607i
$560$	0	0
$561$	163.226	0.290955
$562$	0	0
$563$	771.186	1.36978	0.684890	−	0.728647i	$-0.259850\pi$
$563$	771.186	1.36978	0.684890	+	0.728647i	$0.259850\pi$
$564$	0	0
$565$	− 181.745i	− 0.321672i
$566$	0	0
$567$	853.245i	1.50484i
$568$	0	0
$569$	−8.74767	−0.0153738	−0.00768688	−	0.999970i	$-0.502447\pi$
$569$	−8.74767	−0.0153738	−0.00768688	+	0.999970i	$0.502447\pi$
$570$	0	0
$571$	−511.138	−0.895164	−0.447582	−	0.894243i	$-0.647715\pi$
$571$	−511.138	−0.895164	−0.447582	+	0.894243i	$0.647715\pi$
$572$	0	0
$573$	1232.93i	2.15171i
$574$	0	0
$575$	80.5748i	0.140130i
$576$	0	0
$577$	−713.712	−1.23694	−0.618468	−	0.785810i	$-0.712246\pi$
$577$	−713.712	−1.23694	−0.618468	+	0.785810i	$0.712246\pi$
$578$	0	0
$579$	691.521	1.19434
$580$	0	0
$581$	− 180.689i	− 0.310996i
$582$	0	0
$583$	− 38.8666i	− 0.0666666i
$584$	0	0
$585$	5.77709	0.00987536
$586$	0	0
$587$	422.169	0.719198	0.359599	−	0.933107i	$-0.382914\pi$
$587$	422.169	0.719198	0.359599	+	0.933107i	$0.382914\pi$
$588$	0	0
$589$	446.217i	0.757584i
$590$	0	0
$591$	− 534.588i	− 0.904548i
$592$	0	0
$593$	−308.663	−0.520510	−0.260255	−	0.965540i	$-0.583807\pi$
$593$	−308.663	−0.520510	−0.260255	+	0.965540i	$0.583807\pi$
$594$	0	0
$595$	454.387	0.763676
$596$	0	0
$597$	640.000i	1.07203i
$598$	0	0
$599$	− 462.196i	− 0.771612i	−0.922580	−	0.385806i	$-0.873923\pi$
$599$	− 462.196i	− 0.771612i	0.922580	−	0.385806i	$-0.126077\pi$
$600$	0	0
$601$	−355.358	−0.591277	−0.295639	−	0.955300i	$-0.595532\pi$
$601$	−355.358	−0.591277	−0.295639	+	0.955300i	$0.595532\pi$
$602$	0	0
$603$	486.305	0.806476
$604$	0	0
$605$	263.351i	0.435290i
$606$	0	0
$607$	− 630.403i	− 1.03856i	−0.854605	−	0.519278i	$-0.826201\pi$
$607$	− 630.403i	− 1.03856i	0.854605	−	0.519278i	$-0.173799\pi$
$608$	0	0
$609$	224.721	0.369001
$610$	0	0
$611$	−16.6891	−0.0273144
$612$	0	0
$613$	− 812.525i	− 1.32549i	−0.748846	−	0.662745i	$-0.769392\pi$
$613$	− 812.525i	− 1.32549i	0.748846	−	0.662745i	$-0.230608\pi$
$614$	0	0
$615$	− 352.309i	− 0.572860i
$616$	0	0
$617$	437.935	0.709781	0.354891	−	0.934908i	$-0.384518\pi$
$617$	437.935	0.709781	0.354891	+	0.934908i	$0.384518\pi$
$618$	0	0
$619$	770.250	1.24435	0.622173	−	0.782880i	$-0.286250\pi$
$619$	770.250	1.24435	0.622173	+	0.782880i	$0.286250\pi$
$620$	0	0
$621$	216.276i	0.348270i
$622$	0	0
$623$	573.702i	0.920870i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	64.2597	0.102487
$628$	0	0
$629$	629.718i	1.00114i
$630$	0	0
$631$	− 875.496i	− 1.38747i	−0.720228	−	0.693737i	$-0.755963\pi$
$631$	− 875.496i	− 1.38747i	0.720228	−	0.693737i	$-0.244037\pi$
$632$	0	0
$633$	357.390	0.564597
$634$	0	0
$635$	4.12814	0.00650100
$636$	0	0
$637$	− 11.0294i	− 0.0173146i
$638$	0	0
$639$	− 215.681i	− 0.337529i
$640$	0	0
$641$	842.571	1.31446	0.657232	−	0.753689i	$-0.271727\pi$
$641$	842.571	1.31446	0.657232	+	0.753689i	$0.271727\pi$
$642$	0	0
$643$	−1153.20	−1.79348	−0.896738	−	0.442563i	$-0.854069\pi$
$643$	−1153.20	−1.79348	−0.896738	+	0.442563i	$0.854069\pi$
$644$	0	0
$645$	− 17.0820i	− 0.0264838i
$646$	0	0
$647$	− 355.751i	− 0.549847i	−0.961466	−	0.274924i	$-0.911347\pi$
$647$	− 355.751i	− 0.549847i	0.961466	−	0.274924i	$-0.0886526\pi$
$648$	0	0
$649$	132.669	0.204420
$650$	0	0
$651$	−1535.41	−2.35854
$652$	0	0
$653$	− 557.915i	− 0.854387i	−0.904160	−	0.427194i	$-0.859502\pi$
$653$	− 557.915i	− 0.854387i	0.904160	−	0.427194i	$-0.140498\pi$
$654$	0	0
$655$	− 504.478i	− 0.770195i
$656$	0	0
$657$	−752.715	−1.14569
$658$	0	0
$659$	284.157	0.431194	0.215597	−	0.976482i	$-0.430830\pi$
$659$	284.157	0.431194	0.215597	+	0.976482i	$0.430830\pi$
$660$	0	0
$661$	− 716.735i	− 1.08432i	−0.840275	−	0.542160i	$-0.817607\pi$
$661$	− 716.735i	− 1.08432i	0.840275	−	0.542160i	$-0.182393\pi$
$662$	0	0
$663$	− 42.9065i	− 0.0647157i
$664$	0	0
$665$	178.885	0.269001
$666$	0	0
$667$	111.907	0.167776
$668$	0	0
$669$	− 814.237i	− 1.21710i
$670$	0	0
$671$	46.9464i	0.0699648i
$672$	0	0
$673$	−695.378	−1.03325	−0.516625	−	0.856212i	$-0.672812\pi$
$673$	−695.378	−1.03325	−0.516625	+	0.856212i	$0.672812\pi$
$674$	0	0
$675$	67.1040	0.0994133
$676$	0	0
$677$	820.237i	1.21158i	0.795626	+	0.605788i	$0.207142\pi$
$677$	820.237i	1.21158i	−0.795626	+	0.605788i	$0.792858\pi$
$678$	0	0
$679$	333.176i	0.490686i
$680$	0	0
$681$	−157.580	−0.231396
$682$	0	0
$683$	−335.508	−0.491227	−0.245613	−	0.969368i	$-0.578989\pi$
$683$	−335.508	−0.491227	−0.245613	+	0.969368i	$0.578989\pi$
$684$	0	0
$685$	− 118.138i	− 0.172464i
$686$	0	0
$687$	− 278.769i	− 0.405777i
$688$	0	0
$689$	−10.2167	−0.0148283
$690$	0	0
$691$	336.568	0.487074	0.243537	−	0.969892i	$-0.421692\pi$
$691$	336.568	0.487074	0.243537	+	0.969892i	$0.421692\pi$
$692$	0	0
$693$	83.6068i	0.120645i
$694$	0	0
$695$	− 281.413i	− 0.404910i
$696$	0	0
$697$	−989.378	−1.41948
$698$	0	0
$699$	1168.09	1.67108
$700$	0	0
$701$	− 429.364i	− 0.612502i	−0.951951	−	0.306251i	$-0.900925\pi$
$701$	− 429.364i	− 0.612502i	0.951951	−	0.306251i	$-0.0990747\pi$
$702$	0	0
$703$	247.911i	0.352647i
$704$	0	0
$705$	300.689	0.426509
$706$	0	0
$707$	−849.703	−1.20184
$708$	0	0
$709$	− 1224.60i	− 1.72722i	−0.504162	−	0.863609i	$-0.668199\pi$
$709$	− 1224.60i	− 1.72722i	0.504162	−	0.863609i	$-0.331801\pi$
$710$	0	0
$711$	622.197i	0.875101i
$712$	0	0
$713$	−764.604	−1.07238
$714$	0	0
$715$	−1.89621	−0.00265204
$716$	0	0
$717$	163.226i	0.227651i
$718$	0	0
$719$	− 496.022i	− 0.689877i	−0.938625	−	0.344939i	$-0.887900\pi$
$719$	− 496.022i	− 0.689877i	0.938625	−	0.344939i	$-0.112100\pi$
$720$	0	0
$721$	−304.296	−0.422047
$722$	0	0
$723$	−513.883	−0.710764
$724$	0	0
$725$	− 34.7214i	− 0.0478915i
$726$	0	0
$727$	− 152.843i	− 0.210238i	−0.994460	−	0.105119i	$-0.966478\pi$
$727$	− 152.843i	− 0.210238i	0.994460	−	0.105119i	$-0.0335224\pi$
$728$	0	0
$729$	−89.3808	−0.122607
$730$	0	0
$731$	−47.9709	−0.0656237
$732$	0	0
$733$	− 761.286i	− 1.03859i	−0.854595	−	0.519295i	$-0.826195\pi$
$733$	− 761.286i	− 1.03859i	0.854595	−	0.519295i	$-0.173805\pi$
$734$	0	0
$735$	198.718i	0.270364i
$736$	0	0
$737$	−159.619	−0.216580
$738$	0	0
$739$	−183.975	−0.248951	−0.124476	−	0.992223i	$-0.539725\pi$
$739$	−183.975	−0.248951	−0.124476	+	0.992223i	$0.539725\pi$
$740$	0	0
$741$	− 16.8916i	− 0.0227957i
$742$	0	0
$743$	495.247i	0.666551i	0.942830	+	0.333275i	$0.108154\pi$
$743$	495.247i	0.666551i	−0.942830	+	0.333275i	$0.891846\pi$
$744$	0	0
$745$	−297.331	−0.399102
$746$	0	0
$747$	116.235	0.155602
$748$	0	0
$749$	− 1030.13i	− 1.37534i
$750$	0	0
$751$	− 800.059i	− 1.06533i	−0.846328	−	0.532663i	$-0.821191\pi$
$751$	− 800.059i	− 1.06533i	0.846328	−	0.532663i	$-0.178809\pi$
$752$	0	0
$753$	841.378	1.11737
$754$	0	0
$755$	−338.140	−0.447868
$756$	0	0
$757$	276.367i	0.365082i	0.983198	+	0.182541i	$0.0584322\pi$
$757$	276.367i	0.365082i	−0.983198	+	0.182541i	$0.941568\pi$
$758$	0	0
$759$	110.111i	0.145073i
$760$	0	0
$761$	−891.207	−1.17110	−0.585550	−	0.810636i	$-0.699121\pi$
$761$	−891.207	−1.17110	−0.585550	+	0.810636i	$0.699121\pi$
$762$	0	0
$763$	−1679.77	−2.20154
$764$	0	0
$765$	292.302i	0.382094i
$766$	0	0
$767$	− 34.8740i	− 0.0454681i
$768$	0	0
$769$	−835.430	−1.08639	−0.543193	−	0.839608i	$-0.682785\pi$
$769$	−835.430	−1.08639	−0.543193	+	0.839608i	$0.682785\pi$
$770$	0	0
$771$	−977.898	−1.26835
$772$	0	0
$773$	− 213.522i	− 0.276225i	−0.990417	−	0.138112i	$-0.955897\pi$
$773$	− 213.522i	− 0.276225i	0.990417	−	0.138112i	$-0.0441035\pi$
$774$	0	0
$775$	237.234i	0.306109i
$776$	0	0
$777$	−853.050	−1.09788
$778$	0	0
$779$	−389.503	−0.500004
$780$	0	0
$781$	70.7926i	0.0906436i
$782$	0	0
$783$	− 93.1976i	− 0.119026i
$784$	0	0
$785$	−82.1703	−0.104676
$786$	0	0
$787$	−370.182	−0.470371	−0.235185	−	0.971951i	$-0.575570\pi$
$787$	−370.182	−0.470371	−0.235185	+	0.971951i	$0.575570\pi$
$788$	0	0
$789$	− 624.531i	− 0.791547i
$790$	0	0
$791$	− 691.397i	− 0.874080i
$792$	0	0
$793$	12.3406	0.0155619
$794$	0	0
$795$	184.075	0.231541
$796$	0	0
$797$	274.426i	0.344323i	0.985069	+	0.172162i	$0.0550752\pi$
$797$	274.426i	0.344323i	−0.985069	+	0.172162i	$0.944925\pi$
$798$	0	0
$799$	− 844.414i	− 1.05684i
$800$	0	0
$801$	−369.056	−0.460744
$802$	0	0
$803$	247.063	0.307675
$804$	0	0
$805$	306.525i	0.380776i
$806$	0	0
$807$	− 134.956i	− 0.167232i
$808$	0	0
$809$	−665.214	−0.822266	−0.411133	−	0.911575i	$-0.634867\pi$
$809$	−665.214	−0.822266	−0.411133	+	0.911575i	$0.634867\pi$
$810$	0	0
$811$	360.665	0.444717	0.222358	−	0.974965i	$-0.428624\pi$
$811$	360.665	0.444717	0.222358	+	0.974965i	$0.428624\pi$
$812$	0	0
$813$	− 1137.27i	− 1.39886i
$814$	0	0
$815$	− 677.202i	− 0.830923i
$816$	0	0
$817$	−18.8854	−0.0231156
$818$	0	0
$819$	21.9773	0.0268344
$820$	0	0
$821$	666.899i	0.812301i	0.913806	+	0.406151i	$0.133129\pi$
$821$	666.899i	0.812301i	−0.913806	+	0.406151i	$0.866871\pi$
$822$	0	0
$823$	− 122.433i	− 0.148764i	−0.997230	−	0.0743822i	$-0.976302\pi$
$823$	− 122.433i	− 0.148764i	0.997230	−	0.0743822i	$-0.0236985\pi$
$824$	0	0
$825$	34.1641	0.0414110
$826$	0	0
$827$	−1532.98	−1.85366	−0.926832	−	0.375477i	$-0.877479\pi$
$827$	−1532.98	−1.85366	−0.926832	+	0.375477i	$0.877479\pi$
$828$	0	0
$829$	− 195.475i	− 0.235796i	−0.993026	−	0.117898i	$-0.962384\pi$
$829$	− 195.475i	− 0.235796i	0.993026	−	0.117898i	$-0.0376157\pi$
$830$	0	0
$831$	1739.47i	2.09322i
$832$	0	0
$833$	558.053	0.669931
$834$	0	0
$835$	222.229	0.266143
$836$	0	0
$837$	636.774i	0.760781i
$838$	0	0
$839$	− 1325.97i	− 1.58041i	−0.612840	−	0.790207i	$-0.709973\pi$
$839$	− 1325.97i	− 1.58041i	0.612840	−	0.790207i	$-0.290027\pi$
$840$	0	0
$841$	792.777	0.942660
$842$	0	0
$843$	21.4532	0.0254487
$844$	0	0
$845$	− 377.397i	− 0.446624i
$846$	0	0
$847$	1001.85i	1.18282i
$848$	0	0
$849$	646.407	0.761375
$850$	0	0
$851$	−424.801	−0.499179
$852$	0	0
$853$	− 1055.28i	− 1.23714i	−0.785730	−	0.618570i	$-0.787712\pi$
$853$	− 1055.28i	− 1.23714i	0.785730	−	0.618570i	$-0.212288\pi$
$854$	0	0
$855$	115.075i	0.134591i
$856$	0	0
$857$	−155.378	−0.181304	−0.0906521	−	0.995883i	$-0.528895\pi$
$857$	−155.378	−0.181304	−0.0906521	+	0.995883i	$0.528895\pi$
$858$	0	0
$859$	−226.033	−0.263136	−0.131568	−	0.991307i	$-0.542001\pi$
$859$	−226.033	−0.263136	−0.131568	+	0.991307i	$0.542001\pi$
$860$	0	0
$861$	− 1340.26i	− 1.55664i
$862$	0	0
$863$	930.702i	1.07845i	0.842162	+	0.539225i	$0.181283\pi$
$863$	930.702i	1.07845i	−0.842162	+	0.539225i	$0.818717\pi$
$864$	0	0
$865$	−406.158	−0.469547
$866$	0	0
$867$	1071.51	1.23588
$868$	0	0
$869$	− 204.223i	− 0.235009i
$870$	0	0
$871$	41.9584i	0.0481727i
$872$	0	0
$873$	−214.328	−0.245508
$874$	0	0
$875$	95.1057	0.108692
$876$	0	0
$877$	− 33.5217i	− 0.0382231i	−0.999817	−	0.0191115i	$-0.993916\pi$
$877$	− 33.5217i	− 0.0382231i	0.999817	−	0.0191115i	$-0.00608376\pi$
$878$	0	0
$879$	102.178i	0.116244i
$880$	0	0
$881$	−933.850	−1.05999	−0.529994	−	0.848001i	$-0.677806\pi$
$881$	−933.850	−1.05999	−0.529994	+	0.848001i	$0.677806\pi$
$882$	0	0
$883$	−542.308	−0.614166	−0.307083	−	0.951683i	$-0.599353\pi$
$883$	−542.308	−0.614166	−0.307083	+	0.951683i	$0.599353\pi$
$884$	0	0
$885$	628.328i	0.709975i
$886$	0	0
$887$	714.720i	0.805773i	0.915250	+	0.402886i	$0.131993\pi$
$887$	714.720i	0.805773i	−0.915250	+	0.402886i	$0.868007\pi$
$888$	0	0
$889$	15.7044	0.0176652
$890$	0	0
$891$	180.159	0.202199
$892$	0	0
$893$	− 332.433i	− 0.372266i
$894$	0	0
$895$	583.407i	0.651851i
$896$	0	0
$897$	28.9443	0.0322679
$898$	0	0
$899$	329.484	0.366500
$900$	0	0
$901$	− 516.932i	− 0.573731i
$902$	0	0
$903$	− 64.9839i	− 0.0719645i
$904$	0	0
$905$	−352.800	−0.389835
$906$	0	0
$907$	−347.233	−0.382837	−0.191418	−	0.981509i	$-0.561309\pi$
$907$	−347.233	−0.382837	−0.191418	+	0.981509i	$0.561309\pi$
$908$	0	0
$909$	− 546.604i	− 0.601324i
$910$	0	0
$911$	− 1427.54i	− 1.56701i	−0.621386	−	0.783504i	$-0.713430\pi$
$911$	− 1427.54i	− 1.56701i	0.621386	−	0.783504i	$-0.286570\pi$
$912$	0	0
$913$	−38.1517	−0.0417871
$914$	0	0
$915$	−222.341	−0.242996
$916$	0	0
$917$	− 1919.15i	− 2.09286i
$918$	0	0
$919$	− 569.162i	− 0.619327i	−0.950846	−	0.309664i	$-0.899784\pi$
$919$	− 569.162i	− 0.619327i	0.950846	−	0.309664i	$-0.100216\pi$
$920$	0	0
$921$	−449.017	−0.487532
$922$	0	0
$923$	18.6089	0.0201614
$924$	0	0
$925$	131.803i	0.142490i
$926$	0	0
$927$	− 195.750i	− 0.211165i
$928$	0	0
$929$	1535.96	1.65335	0.826675	−	0.562680i	$-0.190230\pi$
$929$	1535.96	1.65335	0.826675	+	0.562680i	$0.190230\pi$
$930$	0	0
$931$	219.697	0.235980
$932$	0	0
$933$	463.489i	0.496773i
$934$	0	0
$935$	− 95.9418i	− 0.102612i
$936$	0	0
$937$	−338.721	−0.361496	−0.180748	−	0.983529i	$-0.557852\pi$
$937$	−338.721	−0.361496	−0.180748	+	0.983529i	$0.557852\pi$
$938$	0	0
$939$	−835.210	−0.889468
$940$	0	0
$941$	1439.77i	1.53004i	0.644004	+	0.765022i	$0.277272\pi$
$941$	1439.77i	1.53004i	−0.644004	+	0.765022i	$0.722728\pi$
$942$	0	0
$943$	− 667.424i	− 0.707766i
$944$	0	0
$945$	255.279	0.270136
$946$	0	0
$947$	656.135	0.692856	0.346428	−	0.938077i	$-0.387394\pi$
$947$	656.135	0.692856	0.346428	+	0.938077i	$0.387394\pi$
$948$	0	0
$949$	− 64.9443i	− 0.0684344i
$950$	0	0
$951$	1395.62i	1.46752i
$952$	0	0
$953$	−436.675	−0.458211	−0.229105	−	0.973402i	$-0.573580\pi$
$953$	−436.675	−0.458211	−0.229105	+	0.973402i	$0.573580\pi$
$954$	0	0
$955$	724.699	0.758847
$956$	0	0
$957$	− 47.4489i	− 0.0495809i
$958$	0	0
$959$	− 449.423i	− 0.468637i
$960$	0	0
$961$	−1290.20	−1.34256
$962$	0	0
$963$	662.671	0.688132
$964$	0	0
$965$	− 406.466i	− 0.421208i
$966$	0	0
$967$	− 903.436i	− 0.934267i	−0.884187	−	0.467133i	$-0.845287\pi$
$967$	− 903.436i	− 0.934267i	0.884187	−	0.467133i	$-0.154713\pi$
$968$	0	0
$969$	854.663	0.882005
$970$	0	0
$971$	1866.89	1.92265	0.961324	−	0.275420i	$-0.0888168\pi$
$971$	1866.89	1.92265	0.961324	+	0.275420i	$0.0888168\pi$
$972$	0	0
$973$	− 1070.56i	− 1.10026i
$974$	0	0
$975$	− 8.98056i	− 0.00921083i
$976$	0	0
$977$	−1073.95	−1.09923	−0.549615	−	0.835418i	$-0.685226\pi$
$977$	−1073.95	−1.09923	−0.549615	+	0.835418i	$0.685226\pi$
$978$	0	0
$979$	121.135	0.123733
$980$	0	0
$981$	− 1080.58i	− 1.10151i
$982$	0	0
$983$	534.114i	0.543351i	0.962389	+	0.271675i	$0.0875777\pi$
$983$	534.114i	0.543351i	−0.962389	+	0.271675i	$0.912422\pi$
$984$	0	0
$985$	−314.223	−0.319008
$986$	0	0
$987$	1143.89	1.15895
$988$	0	0
$989$	− 32.3607i	− 0.0327206i
$990$	0	0
$991$	− 520.419i	− 0.525146i	−0.964912	−	0.262573i	$-0.915429\pi$
$991$	− 520.419i	− 0.525146i	0.964912	−	0.262573i	$-0.0845710\pi$
$992$	0	0
$993$	619.502	0.623869
$994$	0	0
$995$	376.183	0.378073
$996$	0	0
$997$	− 457.680i	− 0.459057i	−0.973302	−	0.229528i	$-0.926282\pi$
$997$	− 457.680i	− 0.459057i	0.973302	−	0.229528i	$-0.0737184\pi$
$998$	0	0
$999$	353.781i	0.354135i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.g.e.1151.7		8
4.3	odd	2	inner	1280.3.g.e.1151.1		8
8.3	odd	2	inner	1280.3.g.e.1151.8		8
8.5	even	2	inner	1280.3.g.e.1151.2		8
16.3	odd	4		320.3.b.c.191.1		4
16.5	even	4		20.3.b.a.11.1	✓	4
16.11	odd	4		20.3.b.a.11.2	yes	4
16.13	even	4		320.3.b.c.191.4		4
48.5	odd	4		180.3.c.a.91.4		4
48.11	even	4		180.3.c.a.91.3		4
48.29	odd	4		2880.3.e.e.2431.4		4
48.35	even	4		2880.3.e.e.2431.3		4
80.3	even	4		1600.3.h.n.1599.7		8
80.13	odd	4		1600.3.h.n.1599.2		8
80.19	odd	4		1600.3.b.s.1151.4		4
80.27	even	4		100.3.d.b.99.3		8
80.29	even	4		1600.3.b.s.1151.1		4
80.37	odd	4		100.3.d.b.99.5		8
80.43	even	4		100.3.d.b.99.6		8
80.53	odd	4		100.3.d.b.99.4		8
80.59	odd	4		100.3.b.f.51.3		4
80.67	even	4		1600.3.h.n.1599.1		8
80.69	even	4		100.3.b.f.51.4		4
80.77	odd	4		1600.3.h.n.1599.8		8
240.53	even	4		900.3.f.e.199.5		8
240.59	even	4		900.3.c.k.451.2		4
240.107	odd	4		900.3.f.e.199.6		8
240.149	odd	4		900.3.c.k.451.1		4
240.197	even	4		900.3.f.e.199.4		8
240.203	odd	4		900.3.f.e.199.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.1	✓	4	16.5	even	4
20.3.b.a.11.2	yes	4	16.11	odd	4
100.3.b.f.51.3		4	80.59	odd	4
100.3.b.f.51.4		4	80.69	even	4
100.3.d.b.99.3		8	80.27	even	4
100.3.d.b.99.4		8	80.53	odd	4
100.3.d.b.99.5		8	80.37	odd	4
100.3.d.b.99.6		8	80.43	even	4
180.3.c.a.91.3		4	48.11	even	4
180.3.c.a.91.4		4	48.5	odd	4
320.3.b.c.191.1		4	16.3	odd	4
320.3.b.c.191.4		4	16.13	even	4
900.3.c.k.451.1		4	240.149	odd	4
900.3.c.k.451.2		4	240.59	even	4
900.3.f.e.199.3		8	240.203	odd	4
900.3.f.e.199.4		8	240.197	even	4
900.3.f.e.199.5		8	240.53	even	4
900.3.f.e.199.6		8	240.107	odd	4
1280.3.g.e.1151.1		8	4.3	odd	2	inner
1280.3.g.e.1151.2		8	8.5	even	2	inner
1280.3.g.e.1151.7		8	1.1	even	1	trivial
1280.3.g.e.1151.8		8	8.3	odd	2	inner
1600.3.b.s.1151.1		4	80.29	even	4
1600.3.b.s.1151.4		4	80.19	odd	4
1600.3.h.n.1599.1		8	80.67	even	4
1600.3.h.n.1599.2		8	80.13	odd	4
1600.3.h.n.1599.7		8	80.3	even	4
1600.3.h.n.1599.8		8	80.77	odd	4
2880.3.e.e.2431.3		4	48.35	even	4
2880.3.e.e.2431.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.80423 q^{3} -2.23607i q^{5} -8.50651i q^{7} +5.47214 q^{9} -1.79611 q^{11} +0.472136i q^{13} -8.50651i q^{15} -23.8885 q^{17} -9.40456 q^{19} -32.3607i q^{21} -16.1150i q^{23} -5.00000 q^{25} -13.4208 q^{27} +6.94427i q^{29} -47.4468i q^{31} -6.83282 q^{33} -19.0211 q^{35} -26.3607i q^{37} +1.79611i q^{39} +41.4164 q^{41} +2.00811 q^{43} -12.2361i q^{45} +35.3481i q^{47} -23.3607 q^{49} -90.8774 q^{51} +21.6393i q^{53} +4.01623i q^{55} -35.7771 q^{57} -73.8644 q^{59} -26.1378i q^{61} -46.5488i q^{63} +1.05573 q^{65} +88.8693 q^{67} -61.3050i q^{69} -39.4144i q^{71} -137.554 q^{73} -19.0211 q^{75} +15.2786i q^{77} +113.703i q^{79} -100.305 q^{81} +21.2412 q^{83} +53.4164i q^{85} +26.4176i q^{87} -67.4427 q^{89} +4.01623 q^{91} -180.498i q^{93} +21.0292i q^{95} -39.1672 q^{97} -9.82857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} - 48 q^{17} - 40 q^{25} + 160 q^{33} + 224 q^{41} - 8 q^{49} + 80 q^{65} - 528 q^{73} - 552 q^{81} + 176 q^{89} - 528 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 - 48 * q^17 - 40 * q^25 + 160 * q^33 + 224 * q^41 - 8 * q^49 + 80 * q^65 - 528 * q^73 - 552 * q^81 + 176 * q^89 - 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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