LMFDB - Embedded newform 1280.3.g.e.1151.5

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			1151.5
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1280.1151
Dual form			1280.3.g.e.1151.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35114 q^{3} -2.23607i q^{5} -5.25731i q^{7} -3.47214 q^{9} +19.9192 q^{11} +8.47214i q^{13} -5.25731i q^{15} +11.8885 q^{17} +15.2169 q^{19} -12.3607i q^{21} -0.555029i q^{23} -5.00000 q^{25} -29.3238 q^{27} +10.9443i q^{29} +8.29451i q^{31} +46.8328 q^{33} -11.7557 q^{35} -18.3607i q^{37} +19.9192i q^{39} +14.5836 q^{41} +22.2703 q^{43} +7.76393i q^{45} -53.3902i q^{47} +21.3607 q^{49} +27.9516 q^{51} -66.3607i q^{53} -44.5407i q^{55} +35.7771 q^{57} +17.4370 q^{59} -90.1378i q^{61} +18.2541i q^{63} +18.9443 q^{65} -50.2220 q^{67} -1.30495i q^{69} -80.7868i q^{71} +5.55418 q^{73} -11.7557 q^{75} -104.721i q^{77} +13.8448i q^{79} -37.6950 q^{81} +76.2155 q^{83} -26.5836i q^{85} +25.7315i q^{87} +111.443 q^{89} +44.5407 q^{91} +19.5016i q^{93} -34.0260i q^{95} -92.8328 q^{97} -69.1621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.35114	0.783714	0.391857	−	0.920026i	$-0.371833\pi$
$3$	2.35114	0.783714	0.391857	+	0.920026i	$0.371833\pi$
$4$	0	0
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	− 5.25731i	− 0.751044i	−0.926813	−	0.375522i	$-0.877463\pi$
$7$	− 5.25731i	− 0.751044i	0.926813	−	0.375522i	$-0.122537\pi$
$8$	0	0
$9$	−3.47214	−0.385793
$10$	0	0
$11$	19.9192	1.81084	0.905418	−	0.424522i	$-0.139558\pi$
$11$	19.9192	1.81084	0.905418	+	0.424522i	$0.139558\pi$
$12$	0	0
$13$	8.47214i	0.651703i	0.945421	+	0.325851i	$0.105651\pi$
$13$	8.47214i	0.651703i	−0.945421	+	0.325851i	$0.894349\pi$
$14$	0	0
$15$	− 5.25731i	− 0.350487i
$16$	0	0
$17$	11.8885	0.699326	0.349663	−	0.936876i	$-0.386296\pi$
$17$	11.8885	0.699326	0.349663	+	0.936876i	$0.386296\pi$
$18$	0	0
$19$	15.2169	0.800890	0.400445	−	0.916321i	$-0.368856\pi$
$19$	15.2169	0.800890	0.400445	+	0.916321i	$0.368856\pi$
$20$	0	0
$21$	− 12.3607i	− 0.588604i
$22$	0	0
$23$	− 0.555029i	− 0.0241317i	−0.999927	−	0.0120659i	$-0.996159\pi$
$23$	− 0.555029i	− 0.0241317i	0.999927	−	0.0120659i	$-0.00384077\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−29.3238	−1.08606
$28$	0	0
$29$	10.9443i	0.377389i	0.982036	+	0.188694i	$0.0604255\pi$
$29$	10.9443i	0.377389i	−0.982036	+	0.188694i	$0.939574\pi$
$30$	0	0
$31$	8.29451i	0.267565i	0.991011	+	0.133782i	$0.0427123\pi$
$31$	8.29451i	0.267565i	−0.991011	+	0.133782i	$0.957288\pi$
$32$	0	0
$33$	46.8328	1.41918
$34$	0	0
$35$	−11.7557	−0.335877
$36$	0	0
$37$	− 18.3607i	− 0.496235i	−0.968730	−	0.248117i	$-0.920188\pi$
$37$	− 18.3607i	− 0.496235i	0.968730	−	0.248117i	$-0.0798118\pi$
$38$	0	0
$39$	19.9192i	0.510748i
$40$	0	0
$41$	14.5836	0.355697	0.177849	−	0.984058i	$-0.443086\pi$
$41$	14.5836	0.355697	0.177849	+	0.984058i	$0.443086\pi$
$42$	0	0
$43$	22.2703	0.517915	0.258957	−	0.965889i	$-0.416621\pi$
$43$	22.2703	0.517915	0.258957	+	0.965889i	$0.416621\pi$
$44$	0	0
$45$	7.76393i	0.172532i
$46$	0	0
$47$	− 53.3902i	− 1.13596i	−0.823042	−	0.567981i	$-0.807725\pi$
$47$	− 53.3902i	− 1.13596i	0.823042	−	0.567981i	$-0.192275\pi$
$48$	0	0
$49$	21.3607	0.435932
$50$	0	0
$51$	27.9516	0.548071
$52$	0	0
$53$	− 66.3607i	− 1.25209i	−0.779788	−	0.626044i	$-0.784673\pi$
$53$	− 66.3607i	− 1.25209i	0.779788	−	0.626044i	$-0.215327\pi$
$54$	0	0
$55$	− 44.5407i	− 0.809830i
$56$	0	0
$57$	35.7771	0.627668
$58$	0	0
$59$	17.4370	0.295543	0.147771	−	0.989022i	$-0.452790\pi$
$59$	17.4370	0.295543	0.147771	+	0.989022i	$0.452790\pi$
$60$	0	0
$61$	− 90.1378i	− 1.47767i	−0.673887	−	0.738834i	$-0.735377\pi$
$61$	− 90.1378i	− 1.47767i	0.673887	−	0.738834i	$-0.264623\pi$
$62$	0	0
$63$	18.2541i	0.289748i
$64$	0	0
$65$	18.9443	0.291450
$66$	0	0
$67$	−50.2220	−0.749582	−0.374791	−	0.927109i	$-0.622285\pi$
$67$	−50.2220	−0.749582	−0.374791	+	0.927109i	$0.622285\pi$
$68$	0	0
$69$	− 1.30495i	− 0.0189123i
$70$	0	0
$71$	− 80.7868i	− 1.13784i	−0.822392	−	0.568921i	$-0.807361\pi$
$71$	− 80.7868i	− 1.13784i	0.822392	−	0.568921i	$-0.192639\pi$
$72$	0	0
$73$	5.55418	0.0760846	0.0380423	−	0.999276i	$-0.487888\pi$
$73$	5.55418	0.0760846	0.0380423	+	0.999276i	$0.487888\pi$
$74$	0	0
$75$	−11.7557	−0.156743
$76$	0	0
$77$	− 104.721i	− 1.36002i
$78$	0	0
$79$	13.8448i	0.175251i	0.996154	+	0.0876253i	$0.0279278\pi$
$79$	13.8448i	0.175251i	−0.996154	+	0.0876253i	$0.972072\pi$
$80$	0	0
$81$	−37.6950	−0.465371
$82$	0	0
$83$	76.2155	0.918260	0.459130	−	0.888369i	$-0.348161\pi$
$83$	76.2155	0.918260	0.459130	+	0.888369i	$0.348161\pi$
$84$	0	0
$85$	− 26.5836i	− 0.312748i
$86$	0	0
$87$	25.7315i	0.295765i
$88$	0	0
$89$	111.443	1.25217	0.626083	−	0.779757i	$-0.284657\pi$
$89$	111.443	1.25217	0.626083	+	0.779757i	$0.284657\pi$
$90$	0	0
$91$	44.5407	0.489458
$92$	0	0
$93$	19.5016i	0.209694i
$94$	0	0
$95$	− 34.0260i	− 0.358169i
$96$	0	0
$97$	−92.8328	−0.957039	−0.478520	−	0.878077i	$-0.658826\pi$
$97$	−92.8328	−0.957039	−0.478520	+	0.878077i	$0.658826\pi$
$98$	0	0
$99$	−69.1621	−0.698607
$100$	0	0
$101$	64.1115i	0.634767i	0.948297	+	0.317383i	$0.102804\pi$
$101$	64.1115i	0.634767i	−0.948297	+	0.317383i	$0.897196\pi$
$102$	0	0
$103$	137.769i	1.33757i	0.743458	+	0.668783i	$0.233184\pi$
$103$	137.769i	1.33757i	−0.743458	+	0.668783i	$0.766816\pi$
$104$	0	0
$105$	−27.6393	−0.263232
$106$	0	0
$107$	−51.3320	−0.479739	−0.239869	−	0.970805i	$-0.577105\pi$
$107$	−51.3320	−0.479739	−0.239869	+	0.970805i	$0.577105\pi$
$108$	0	0
$109$	− 133.469i	− 1.22449i	−0.790669	−	0.612243i	$-0.790267\pi$
$109$	− 133.469i	− 1.22449i	0.790669	−	0.612243i	$-0.209733\pi$
$110$	0	0
$111$	− 43.1685i	− 0.388906i
$112$	0	0
$113$	170.721	1.51081	0.755404	−	0.655259i	$-0.227441\pi$
$113$	170.721	1.51081	0.755404	+	0.655259i	$0.227441\pi$
$114$	0	0
$115$	−1.24108	−0.0107920
$116$	0	0
$117$	− 29.4164i	− 0.251422i
$118$	0	0
$119$	− 62.5018i	− 0.525225i
$120$	0	0
$121$	275.774	2.27912
$122$	0	0
$123$	34.2881	0.278765
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	198.637i	1.56407i	0.623235	+	0.782035i	$0.285818\pi$
$127$	198.637i	1.56407i	−0.623235	+	0.782035i	$0.714182\pi$
$128$	0	0
$129$	52.3607	0.405897
$130$	0	0
$131$	−7.77041	−0.0593161	−0.0296580	−	0.999560i	$-0.509442\pi$
$131$	−7.77041	−0.0593161	−0.0296580	+	0.999560i	$0.509442\pi$
$132$	0	0
$133$	− 80.0000i	− 0.601504i
$134$	0	0
$135$	65.5699i	0.485703i
$136$	0	0
$137$	−0.832816	−0.00607895	−0.00303947	−	0.999995i	$-0.500967\pi$
$137$	−0.832816	−0.00607895	−0.00303947	+	0.999995i	$0.500967\pi$
$138$	0	0
$139$	−237.658	−1.70977	−0.854885	−	0.518817i	$-0.826373\pi$
$139$	−237.658	−1.70977	−0.854885	+	0.518817i	$0.826373\pi$
$140$	0	0
$141$	− 125.528i	− 0.890269i
$142$	0	0
$143$	168.758i	1.18013i
$144$	0	0
$145$	24.4721	0.168773
$146$	0	0
$147$	50.2220	0.341646
$148$	0	0
$149$	− 36.9706i	− 0.248125i	−0.992274	−	0.124062i	$-0.960408\pi$
$149$	− 36.9706i	− 0.248125i	0.992274	−	0.124062i	$-0.0395923\pi$
$150$	0	0
$151$	282.723i	1.87234i	0.351552	+	0.936168i	$0.385654\pi$
$151$	282.723i	1.87234i	−0.351552	+	0.936168i	$0.614346\pi$
$152$	0	0
$153$	−41.2786	−0.269795
$154$	0	0
$155$	18.5471	0.119659
$156$	0	0
$157$	− 204.748i	− 1.30413i	−0.758165	−	0.652063i	$-0.773904\pi$
$157$	− 204.748i	− 1.30413i	0.758165	−	0.652063i	$-0.226096\pi$
$158$	0	0
$159$	− 156.023i	− 0.981279i
$160$	0	0
$161$	−2.91796	−0.0181240
$162$	0	0
$163$	−107.235	−0.657885	−0.328943	−	0.944350i	$-0.606692\pi$
$163$	−107.235	−0.657885	−0.328943	+	0.944350i	$0.606692\pi$
$164$	0	0
$165$	− 104.721i	− 0.634675i
$166$	0	0
$167$	33.2090i	0.198856i	0.995045	+	0.0994280i	$0.0317013\pi$
$167$	33.2090i	0.198856i	−0.995045	+	0.0994280i	$0.968299\pi$
$168$	0	0
$169$	97.2229	0.575284
$170$	0	0
$171$	−52.8352	−0.308978
$172$	0	0
$173$	226.361i	1.30844i	0.756303	+	0.654222i	$0.227004\pi$
$173$	226.361i	1.30844i	−0.756303	+	0.654222i	$0.772996\pi$
$174$	0	0
$175$	26.2866i	0.150209i
$176$	0	0
$177$	40.9969	0.231621
$178$	0	0
$179$	−224.337	−1.25328	−0.626641	−	0.779308i	$-0.715571\pi$
$179$	−224.337	−1.25328	−0.626641	+	0.779308i	$0.715571\pi$
$180$	0	0
$181$	86.2229i	0.476370i	0.971220	+	0.238185i	$0.0765524\pi$
$181$	86.2229i	0.476370i	−0.971220	+	0.238185i	$0.923448\pi$
$182$	0	0
$183$	− 211.927i	− 1.15807i
$184$	0	0
$185$	−41.0557	−0.221923
$186$	0	0
$187$	236.810	1.26636
$188$	0	0
$189$	154.164i	0.815683i
$190$	0	0
$191$	31.0198i	0.162407i	0.996698	+	0.0812036i	$0.0258764\pi$
$191$	31.0198i	0.162407i	−0.996698	+	0.0812036i	$0.974124\pi$
$192$	0	0
$193$	110.223	0.571103	0.285552	−	0.958363i	$-0.407823\pi$
$193$	110.223	0.571103	0.285552	+	0.958363i	$0.407823\pi$
$194$	0	0
$195$	44.5407	0.228414
$196$	0	0
$197$	− 172.525i	− 0.875760i	−0.899033	−	0.437880i	$-0.855729\pi$
$197$	− 172.525i	− 0.875760i	0.899033	−	0.437880i	$-0.144271\pi$
$198$	0	0
$199$	− 272.208i	− 1.36788i	−0.729538	−	0.683940i	$-0.760265\pi$
$199$	− 272.208i	− 1.36788i	0.729538	−	0.683940i	$-0.239735\pi$
$200$	0	0
$201$	−118.079	−0.587457
$202$	0	0
$203$	57.5374	0.283436
$204$	0	0
$205$	− 32.6099i	− 0.159073i
$206$	0	0
$207$	1.92714i	0.00930984i
$208$	0	0
$209$	303.108	1.45028
$210$	0	0
$211$	205.266	0.972826	0.486413	−	0.873729i	$-0.338305\pi$
$211$	205.266	0.972826	0.486413	+	0.873729i	$0.338305\pi$
$212$	0	0
$213$	− 189.941i	− 0.891743i
$214$	0	0
$215$	− 49.7980i	− 0.231618i
$216$	0	0
$217$	43.6068	0.200953
$218$	0	0
$219$	13.0586	0.0596285
$220$	0	0
$221$	100.721i	0.455753i
$222$	0	0
$223$	− 235.731i	− 1.05709i	−0.848905	−	0.528545i	$-0.822738\pi$
$223$	− 235.731i	− 1.05709i	0.848905	−	0.528545i	$-0.177262\pi$
$224$	0	0
$225$	17.3607	0.0771586
$226$	0	0
$227$	58.5165	0.257782	0.128891	−	0.991659i	$-0.458858\pi$
$227$	58.5165	0.257782	0.128891	+	0.991659i	$0.458858\pi$
$228$	0	0
$229$	162.721i	0.710574i	0.934757	+	0.355287i	$0.115617\pi$
$229$	162.721i	0.710574i	−0.934757	+	0.355287i	$0.884383\pi$
$230$	0	0
$231$	− 246.215i	− 1.06586i
$232$	0	0
$233$	−319.050	−1.36931	−0.684656	−	0.728867i	$-0.740047\pi$
$233$	−319.050	−1.36931	−0.684656	+	0.728867i	$0.740047\pi$
$234$	0	0
$235$	−119.384	−0.508017
$236$	0	0
$237$	32.5511i	0.137346i
$238$	0	0
$239$	− 236.810i	− 0.990837i	−0.868654	−	0.495419i	$-0.835015\pi$
$239$	− 236.810i	− 0.990837i	0.868654	−	0.495419i	$-0.164985\pi$
$240$	0	0
$241$	−0.917961	−0.00380897	−0.00190448	−	0.999998i	$-0.500606\pi$
$241$	−0.917961	−0.00380897	−0.00190448	+	0.999998i	$0.500606\pi$
$242$	0	0
$243$	175.287	0.721347
$244$	0	0
$245$	− 47.7639i	− 0.194955i
$246$	0	0
$247$	128.920i	0.521942i
$248$	0	0
$249$	179.193	0.719653
$250$	0	0
$251$	−136.690	−0.544582	−0.272291	−	0.962215i	$-0.587781\pi$
$251$	−136.690	−0.544582	−0.272291	+	0.962215i	$0.587781\pi$
$252$	0	0
$253$	− 11.0557i	− 0.0436985i
$254$	0	0
$255$	− 62.5018i	− 0.245105i
$256$	0	0
$257$	−274.944	−1.06982	−0.534911	−	0.844908i	$-0.679655\pi$
$257$	−274.944	−1.06982	−0.534911	+	0.844908i	$0.679655\pi$
$258$	0	0
$259$	−96.5278	−0.372694
$260$	0	0
$261$	− 38.0000i	− 0.145594i
$262$	0	0
$263$	406.385i	1.54519i	0.634899	+	0.772596i	$0.281042\pi$
$263$	406.385i	1.54519i	−0.634899	+	0.772596i	$0.718958\pi$
$264$	0	0
$265$	−148.387	−0.559951
$266$	0	0
$267$	262.018	0.981339
$268$	0	0
$269$	348.525i	1.29563i	0.761797	+	0.647816i	$0.224317\pi$
$269$	348.525i	1.29563i	−0.761797	+	0.647816i	$0.775683\pi$
$270$	0	0
$271$	247.849i	0.914571i	0.889320	+	0.457286i	$0.151178\pi$
$271$	247.849i	0.914571i	−0.889320	+	0.457286i	$0.848822\pi$
$272$	0	0
$273$	104.721	0.383595
$274$	0	0
$275$	−99.5959	−0.362167
$276$	0	0
$277$	− 54.7539i	− 0.197667i	−0.995104	−	0.0988337i	$-0.968489\pi$
$277$	− 54.7539i	− 0.197667i	0.995104	−	0.0988337i	$-0.0315112\pi$
$278$	0	0
$279$	− 28.7997i	− 0.103225i
$280$	0	0
$281$	50.3607	0.179220	0.0896098	−	0.995977i	$-0.471438\pi$
$281$	50.3607	0.179220	0.0896098	+	0.995977i	$0.471438\pi$
$282$	0	0
$283$	−147.336	−0.520621	−0.260310	−	0.965525i	$-0.583825\pi$
$283$	−147.336	−0.520621	−0.260310	+	0.965525i	$0.583825\pi$
$284$	0	0
$285$	− 80.0000i	− 0.280702i
$286$	0	0
$287$	− 76.6705i	− 0.267145i
$288$	0	0
$289$	−147.663	−0.510943
$290$	0	0
$291$	−218.263	−0.750045
$292$	0	0
$293$	178.859i	0.610441i	0.952282	+	0.305220i	$0.0987301\pi$
$293$	178.859i	0.610441i	−0.952282	+	0.305220i	$0.901270\pi$
$294$	0	0
$295$	− 38.9904i	− 0.132171i
$296$	0	0
$297$	−584.105	−1.96668
$298$	0	0
$299$	4.70228	0.0157267
$300$	0	0
$301$	− 117.082i	− 0.388977i
$302$	0	0
$303$	150.735i	0.497475i
$304$	0	0
$305$	−201.554	−0.660833
$306$	0	0
$307$	284.550	0.926873	0.463436	−	0.886130i	$-0.346616\pi$
$307$	284.550	0.926873	0.463436	+	0.886130i	$0.346616\pi$
$308$	0	0
$309$	323.915i	1.04827i
$310$	0	0
$311$	282.199i	0.907392i	0.891157	+	0.453696i	$0.149895\pi$
$311$	282.199i	0.907392i	−0.891157	+	0.453696i	$0.850105\pi$
$312$	0	0
$313$	567.548	1.81325	0.906626	−	0.421935i	$-0.138649\pi$
$313$	567.548	1.81325	0.906626	+	0.421935i	$0.138649\pi$
$314$	0	0
$315$	40.8174	0.129579
$316$	0	0
$317$	− 161.141i	− 0.508331i	−0.967161	−	0.254165i	$-0.918199\pi$
$317$	− 161.141i	− 0.508331i	0.967161	−	0.254165i	$-0.0818008\pi$
$318$	0	0
$319$	218.001i	0.683389i
$320$	0	0
$321$	−120.689	−0.375978
$322$	0	0
$323$	180.907	0.560083
$324$	0	0
$325$	− 42.3607i	− 0.130341i
$326$	0	0
$327$	− 313.805i	− 0.959647i
$328$	0	0
$329$	−280.689	−0.853158
$330$	0	0
$331$	331.966	1.00292	0.501459	−	0.865181i	$-0.332797\pi$
$331$	331.966	1.00292	0.501459	+	0.865181i	$0.332797\pi$
$332$	0	0
$333$	63.7508i	0.191444i
$334$	0	0
$335$	112.300i	0.335223i
$336$	0	0
$337$	−269.108	−0.798541	−0.399271	−	0.916833i	$-0.630737\pi$
$337$	−269.108	−0.798541	−0.399271	+	0.916833i	$0.630737\pi$
$338$	0	0
$339$	401.390	1.18404
$340$	0	0
$341$	165.220i	0.484516i
$342$	0	0
$343$	− 369.908i	− 1.07845i
$344$	0	0
$345$	−2.91796	−0.00845786
$346$	0	0
$347$	−503.075	−1.44978	−0.724892	−	0.688863i	$-0.758110\pi$
$347$	−503.075	−1.44978	−0.724892	+	0.688863i	$0.758110\pi$
$348$	0	0
$349$	0.504658i	0.00144601i	1.00000		0.000723006i	$0.000230140\pi$
$349$	0.504658i	0.00144601i	−1.00000		0.000723006i	$0.999770\pi$
$350$	0	0
$351$	− 248.435i	− 0.707791i
$352$	0	0
$353$	−335.994	−0.951824	−0.475912	−	0.879493i	$-0.657882\pi$
$353$	−335.994	−0.951824	−0.475912	+	0.879493i	$0.657882\pi$
$354$	0	0
$355$	−180.645	−0.508859
$356$	0	0
$357$	− 146.950i	− 0.411626i
$358$	0	0
$359$	98.4859i	0.274334i	0.990548	+	0.137167i	$0.0437997\pi$
$359$	98.4859i	0.274334i	−0.990548	+	0.137167i	$0.956200\pi$
$360$	0	0
$361$	−129.446	−0.358576
$362$	0	0
$363$	648.384	1.78618
$364$	0	0
$365$	− 12.4195i	− 0.0340261i
$366$	0	0
$367$	− 498.473i	− 1.35824i	−0.734029	−	0.679118i	$-0.762362\pi$
$367$	− 498.473i	− 1.35824i	0.734029	−	0.679118i	$-0.237638\pi$
$368$	0	0
$369$	−50.6362	−0.137226
$370$	0	0
$371$	−348.879	−0.940374
$372$	0	0
$373$	600.354i	1.60953i	0.593594	+	0.804765i	$0.297709\pi$
$373$	600.354i	1.60953i	−0.593594	+	0.804765i	$0.702291\pi$
$374$	0	0
$375$	26.2866i	0.0700975i
$376$	0	0
$377$	−92.7214	−0.245945
$378$	0	0
$379$	303.490	0.800765	0.400383	−	0.916348i	$-0.368877\pi$
$379$	303.490	0.800765	0.400383	+	0.916348i	$0.368877\pi$
$380$	0	0
$381$	467.023i	1.22578i
$382$	0	0
$383$	332.583i	0.868362i	0.900826	+	0.434181i	$0.142962\pi$
$383$	332.583i	0.868362i	−0.900826	+	0.434181i	$0.857038\pi$
$384$	0	0
$385$	−234.164	−0.608218
$386$	0	0
$387$	−77.3256	−0.199808
$388$	0	0
$389$	392.354i	1.00862i	0.863522	+	0.504312i	$0.168254\pi$
$389$	392.354i	1.00862i	−0.863522	+	0.504312i	$0.831746\pi$
$390$	0	0
$391$	− 6.59849i	− 0.0168759i
$392$	0	0
$393$	−18.2693	−0.0464868
$394$	0	0
$395$	30.9579	0.0783744
$396$	0	0
$397$	− 334.190i	− 0.841789i	−0.907110	−	0.420895i	$-0.861716\pi$
$397$	− 334.190i	− 0.841789i	0.907110	−	0.420895i	$-0.138284\pi$
$398$	0	0
$399$	− 188.091i	− 0.471407i
$400$	0	0
$401$	121.003	0.301753	0.150877	−	0.988553i	$-0.451790\pi$
$401$	121.003	0.301753	0.150877	+	0.988553i	$0.451790\pi$
$402$	0	0
$403$	−70.2722	−0.174373
$404$	0	0
$405$	84.2887i	0.208120i
$406$	0	0
$407$	− 365.730i	− 0.898599i
$408$	0	0
$409$	607.410	1.48511	0.742555	−	0.669785i	$-0.233614\pi$
$409$	607.410	1.48511	0.742555	+	0.669785i	$0.233614\pi$
$410$	0	0
$411$	−1.95807	−0.00476415
$412$	0	0
$413$	− 91.6718i	− 0.221966i
$414$	0	0
$415$	− 170.423i	− 0.410658i
$416$	0	0
$417$	−558.768	−1.33997
$418$	0	0
$419$	466.760	1.11398	0.556992	−	0.830518i	$-0.311955\pi$
$419$	466.760	1.11398	0.556992	+	0.830518i	$0.311955\pi$
$420$	0	0
$421$	− 73.0883i	− 0.173606i	−0.996225	−	0.0868031i	$-0.972335\pi$
$421$	− 73.0883i	− 0.173606i	0.996225	−	0.0868031i	$-0.0276651\pi$
$422$	0	0
$423$	185.378i	0.438246i
$424$	0	0
$425$	−59.4427	−0.139865
$426$	0	0
$427$	−473.882	−1.10979
$428$	0	0
$429$	396.774i	0.924881i
$430$	0	0
$431$	463.630i	1.07571i	0.843038	+	0.537853i	$0.180765\pi$
$431$	463.630i	1.07571i	−0.843038	+	0.537853i	$0.819235\pi$
$432$	0	0
$433$	−99.8359	−0.230568	−0.115284	−	0.993333i	$-0.536778\pi$
$433$	−99.8359	−0.230568	−0.115284	+	0.993333i	$0.536778\pi$
$434$	0	0
$435$	57.5374	0.132270
$436$	0	0
$437$	− 8.44582i	− 0.0193268i
$438$	0	0
$439$	374.086i	0.852133i	0.904692	+	0.426066i	$0.140101\pi$
$439$	374.086i	0.852133i	−0.904692	+	0.426066i	$0.859899\pi$
$440$	0	0
$441$	−74.1672	−0.168180
$442$	0	0
$443$	−290.100	−0.654854	−0.327427	−	0.944877i	$-0.606182\pi$
$443$	−290.100	−0.654854	−0.327427	+	0.944877i	$0.606182\pi$
$444$	0	0
$445$	− 249.193i	− 0.559985i
$446$	0	0
$447$	− 86.9231i	− 0.194459i
$448$	0	0
$449$	299.921	0.667976	0.333988	−	0.942577i	$-0.391606\pi$
$449$	299.921	0.667976	0.333988	+	0.942577i	$0.391606\pi$
$450$	0	0
$451$	290.493	0.644109
$452$	0	0
$453$	664.721i	1.46738i
$454$	0	0
$455$	− 99.5959i	− 0.218892i
$456$	0	0
$457$	−822.328	−1.79941	−0.899703	−	0.436503i	$-0.856217\pi$
$457$	−822.328	−1.79941	−0.899703	+	0.436503i	$0.856217\pi$
$458$	0	0
$459$	−348.617	−0.759513
$460$	0	0
$461$	456.885i	0.991075i	0.868587	+	0.495537i	$0.165029\pi$
$461$	456.885i	0.991075i	−0.868587	+	0.495537i	$0.834971\pi$
$462$	0	0
$463$	− 400.249i	− 0.864469i	−0.901761	−	0.432234i	$-0.857725\pi$
$463$	− 400.249i	− 0.864469i	0.901761	−	0.432234i	$-0.142275\pi$
$464$	0	0
$465$	43.6068	0.0937781
$466$	0	0
$467$	−913.145	−1.95534	−0.977672	−	0.210139i	$-0.932608\pi$
$467$	−913.145	−1.95534	−0.977672	+	0.210139i	$0.932608\pi$
$468$	0	0
$469$	264.033i	0.562969i
$470$	0	0
$471$	− 481.391i	− 1.02206i
$472$	0	0
$473$	443.607	0.937858
$474$	0	0
$475$	−76.0845	−0.160178
$476$	0	0
$477$	230.413i	0.483047i
$478$	0	0
$479$	526.131i	1.09840i	0.835692	+	0.549198i	$0.185067\pi$
$479$	526.131i	1.09840i	−0.835692	+	0.549198i	$0.814933\pi$
$480$	0	0
$481$	155.554	0.323397
$482$	0	0
$483$	−6.86054	−0.0142040
$484$	0	0
$485$	207.580i	0.428001i
$486$	0	0
$487$	443.541i	0.910762i	0.890297	+	0.455381i	$0.150497\pi$
$487$	443.541i	0.910762i	−0.890297	+	0.455381i	$0.849503\pi$
$488$	0	0
$489$	−252.125	−0.515594
$490$	0	0
$491$	−287.163	−0.584854	−0.292427	−	0.956288i	$-0.594463\pi$
$491$	−287.163	−0.584854	−0.292427	+	0.956288i	$0.594463\pi$
$492$	0	0
$493$	130.111i	0.263918i
$494$	0	0
$495$	154.651i	0.312427i
$496$	0	0
$497$	−424.721	−0.854570
$498$	0	0
$499$	−810.936	−1.62512	−0.812561	−	0.582876i	$-0.801927\pi$
$499$	−810.936	−1.62512	−0.812561	+	0.582876i	$0.801927\pi$
$500$	0	0
$501$	78.0789i	0.155846i
$502$	0	0
$503$	642.471i	1.27728i	0.769506	+	0.638639i	$0.220502\pi$
$503$	642.471i	1.27728i	−0.769506	+	0.638639i	$0.779498\pi$
$504$	0	0
$505$	143.358	0.283876
$506$	0	0
$507$	228.585	0.450858
$508$	0	0
$509$	915.050i	1.79774i	0.438216	+	0.898870i	$0.355611\pi$
$509$	915.050i	1.79774i	−0.438216	+	0.898870i	$0.644389\pi$
$510$	0	0
$511$	− 29.2000i	− 0.0571429i
$512$	0	0
$513$	−446.217	−0.869818
$514$	0	0
$515$	308.061	0.598177
$516$	0	0
$517$	− 1063.49i	− 2.05704i
$518$	0	0
$519$	532.206i	1.02544i
$520$	0	0
$521$	−1006.98	−1.93279	−0.966396	−	0.257058i	$-0.917247\pi$
$521$	−1006.98	−1.93279	−0.966396	+	0.257058i	$0.917247\pi$
$522$	0	0
$523$	−774.173	−1.48025	−0.740127	−	0.672467i	$-0.765235\pi$
$523$	−774.173	−1.48025	−0.740127	+	0.672467i	$0.765235\pi$
$524$	0	0
$525$	61.8034i	0.117721i
$526$	0	0
$527$	98.6096i	0.187115i
$528$	0	0
$529$	528.692	0.999418
$530$	0	0
$531$	−60.5437	−0.114018
$532$	0	0
$533$	123.554i	0.231809i
$534$	0	0
$535$	114.782i	0.214546i
$536$	0	0
$537$	−527.449	−0.982214
$538$	0	0
$539$	425.487	0.789401
$540$	0	0
$541$	259.115i	0.478955i	0.970902	+	0.239477i	$0.0769761\pi$
$541$	259.115i	0.478955i	−0.970902	+	0.239477i	$0.923024\pi$
$542$	0	0
$543$	202.722i	0.373337i
$544$	0	0
$545$	−298.446	−0.547607
$546$	0	0
$547$	149.818	0.273890	0.136945	−	0.990579i	$-0.456272\pi$
$547$	149.818	0.273890	0.136945	+	0.990579i	$0.456272\pi$
$548$	0	0
$549$	312.971i	0.570074i
$550$	0	0
$551$	166.538i	0.302247i
$552$	0	0
$553$	72.7864	0.131621
$554$	0	0
$555$	−96.5278	−0.173924
$556$	0	0
$557$	− 511.698i	− 0.918668i	−0.888264	−	0.459334i	$-0.848088\pi$
$557$	− 511.698i	− 0.918668i	0.888264	−	0.459334i	$-0.151912\pi$
$558$	0	0
$559$	188.677i	0.337526i
$560$	0	0
$561$	556.774	0.992467
$562$	0	0
$563$	−490.726	−0.871627	−0.435814	−	0.900037i	$-0.643539\pi$
$563$	−490.726	−0.871627	−0.435814	+	0.900037i	$0.643539\pi$
$564$	0	0
$565$	− 381.745i	− 0.675654i
$566$	0	0
$567$	198.175i	0.349514i
$568$	0	0
$569$	232.748	0.409047	0.204523	−	0.978862i	$-0.434436\pi$
$569$	232.748	0.409047	0.204523	+	0.978862i	$0.434436\pi$
$570$	0	0
$571$	−210.755	−0.369098	−0.184549	−	0.982823i	$-0.559082\pi$
$571$	−210.755	−0.369098	−0.184549	+	0.982823i	$0.559082\pi$
$572$	0	0
$573$	72.9318i	0.127281i
$574$	0	0
$575$	2.77515i	0.00482634i
$576$	0	0
$577$	341.712	0.592222	0.296111	−	0.955154i	$-0.404310\pi$
$577$	341.712	0.592222	0.296111	+	0.955154i	$0.404310\pi$
$578$	0	0
$579$	259.150	0.447581
$580$	0	0
$581$	− 400.689i	− 0.689654i
$582$	0	0
$583$	− 1321.85i	− 2.26733i
$584$	0	0
$585$	−65.7771	−0.112439
$586$	0	0
$587$	618.412	1.05351	0.526756	−	0.850016i	$-0.323408\pi$
$587$	618.412	1.05351	0.526756	+	0.850016i	$0.323408\pi$
$588$	0	0
$589$	126.217i	0.214290i
$590$	0	0
$591$	− 405.630i	− 0.686345i
$592$	0	0
$593$	120.663	0.203478	0.101739	−	0.994811i	$-0.467559\pi$
$593$	120.663	0.203478	0.101739	+	0.994811i	$0.467559\pi$
$594$	0	0
$595$	−139.758	−0.234888
$596$	0	0
$597$	− 640.000i	− 1.07203i
$598$	0	0
$599$	− 849.927i	− 1.41891i	−0.704751	−	0.709455i	$-0.748941\pi$
$599$	− 849.927i	− 1.41891i	0.704751	−	0.709455i	$-0.251059\pi$
$600$	0	0
$601$	11.3576	0.0188978	0.00944890	−	0.999955i	$-0.496992\pi$
$601$	11.3576	0.0188978	0.00944890	+	0.999955i	$0.496992\pi$
$602$	0	0
$603$	174.378	0.289183
$604$	0	0
$605$	− 616.649i	− 1.01926i
$606$	0	0
$607$	1115.12i	1.83710i	0.395305	+	0.918550i	$0.370639\pi$
$607$	1115.12i	1.83710i	−0.395305	+	0.918550i	$0.629361\pi$
$608$	0	0
$609$	135.279	0.222132
$610$	0	0
$611$	452.329	0.740309
$612$	0	0
$613$	499.475i	0.814805i	0.913249	+	0.407402i	$0.133565\pi$
$613$	499.475i	0.814805i	−0.913249	+	0.407402i	$0.866435\pi$
$614$	0	0
$615$	− 76.6705i	− 0.124667i
$616$	0	0
$617$	−545.935	−0.884822	−0.442411	−	0.896813i	$-0.645877\pi$
$617$	−545.935	−0.884822	−0.442411	+	0.896813i	$0.645877\pi$
$618$	0	0
$619$	455.011	0.735075	0.367537	−	0.930009i	$-0.380201\pi$
$619$	455.011	0.735075	0.367537	+	0.930009i	$0.380201\pi$
$620$	0	0
$621$	16.2755i	0.0262086i
$622$	0	0
$623$	− 585.889i	− 0.940432i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	712.650	1.13660
$628$	0	0
$629$	− 218.282i	− 0.347030i
$630$	0	0
$631$	267.706i	0.424257i	0.977242	+	0.212128i	$0.0680395\pi$
$631$	267.706i	0.424257i	−0.977242	+	0.212128i	$0.931960\pi$
$632$	0	0
$633$	482.610	0.762417
$634$	0	0
$635$	444.165	0.699473
$636$	0	0
$637$	180.971i	0.284098i
$638$	0	0
$639$	280.503i	0.438971i
$640$	0	0
$641$	−418.571	−0.652997	−0.326499	−	0.945198i	$-0.605869\pi$
$641$	−418.571	−0.652997	−0.326499	+	0.945198i	$0.605869\pi$
$642$	0	0
$643$	−439.339	−0.683265	−0.341633	−	0.939834i	$-0.610980\pi$
$643$	−439.339	−0.683265	−0.341633	+	0.939834i	$0.610980\pi$
$644$	0	0
$645$	− 117.082i	− 0.181523i
$646$	0	0
$647$	419.644i	0.648600i	0.945954	+	0.324300i	$0.105129\pi$
$647$	419.644i	0.648600i	−0.945954	+	0.324300i	$0.894871\pi$
$648$	0	0
$649$	347.331	0.535179
$650$	0	0
$651$	102.526	0.157490
$652$	0	0
$653$	370.085i	0.566746i	0.959010	+	0.283373i	$0.0914534\pi$
$653$	370.085i	0.566746i	−0.959010	+	0.283373i	$0.908547\pi$
$654$	0	0
$655$	17.3752i	0.0265270i
$656$	0	0
$657$	−19.2849	−0.0293529
$658$	0	0
$659$	322.823	0.489868	0.244934	−	0.969540i	$-0.421234\pi$
$659$	322.823	0.489868	0.244934	+	0.969540i	$0.421234\pi$
$660$	0	0
$661$	− 812.735i	− 1.22955i	−0.788701	−	0.614777i	$-0.789246\pi$
$661$	− 812.735i	− 1.22955i	0.788701	−	0.614777i	$-0.210754\pi$
$662$	0	0
$663$	236.810i	0.357180i
$664$	0	0
$665$	−178.885	−0.269001
$666$	0	0
$667$	6.07439	0.00910703
$668$	0	0
$669$	− 554.237i	− 0.828456i
$670$	0	0
$671$	− 1795.47i	− 2.67581i
$672$	0	0
$673$	467.378	0.694469	0.347235	−	0.937778i	$-0.387121\pi$
$673$	467.378	0.694469	0.347235	+	0.937778i	$0.387121\pi$
$674$	0	0
$675$	146.619	0.217213
$676$	0	0
$677$	548.237i	0.809803i	0.914360	+	0.404902i	$0.132694\pi$
$677$	548.237i	0.809803i	−0.914360	+	0.404902i	$0.867306\pi$
$678$	0	0
$679$	488.051i	0.718779i
$680$	0	0
$681$	137.580	0.202027
$682$	0	0
$683$	23.9663	0.0350898	0.0175449	−	0.999846i	$-0.494415\pi$
$683$	23.9663	0.0350898	0.0175449	+	0.999846i	$0.494415\pi$
$684$	0	0
$685$	1.86223i	0.00271859i
$686$	0	0
$687$	382.581i	0.556886i
$688$	0	0
$689$	562.217	0.815989
$690$	0	0
$691$	186.981	0.270595	0.135298	−	0.990805i	$-0.456801\pi$
$691$	186.981	0.270595	0.135298	+	0.990805i	$0.456801\pi$
$692$	0	0
$693$	363.607i	0.524685i
$694$	0	0
$695$	531.420i	0.764633i
$696$	0	0
$697$	173.378	0.248748
$698$	0	0
$699$	−750.130	−1.07315
$700$	0	0
$701$	706.636i	1.00804i	0.863692	+	0.504020i	$0.168146\pi$
$701$	706.636i	1.00804i	−0.863692	+	0.504020i	$0.831854\pi$
$702$	0	0
$703$	− 279.393i	− 0.397429i
$704$	0	0
$705$	−280.689	−0.398140
$706$	0	0
$707$	337.054	0.476738
$708$	0	0
$709$	− 188.597i	− 0.266005i	−0.991116	−	0.133002i	$-0.957538\pi$
$709$	− 188.597i	− 0.266005i	0.991116	−	0.133002i	$-0.0424618\pi$
$710$	0	0
$711$	− 48.0710i	− 0.0676104i
$712$	0	0
$713$	4.60369	0.00645679
$714$	0	0
$715$	377.354	0.527769
$716$	0	0
$717$	− 556.774i	− 0.776533i
$718$	0	0
$719$	− 156.085i	− 0.217086i	−0.994092	−	0.108543i	$-0.965381\pi$
$719$	− 156.085i	− 0.217086i	0.994092	−	0.108543i	$-0.0346186\pi$
$720$	0	0
$721$	724.296	1.00457
$722$	0	0
$723$	−2.15825	−0.00298514
$724$	0	0
$725$	− 54.7214i	− 0.0754777i
$726$	0	0
$727$	− 715.164i	− 0.983719i	−0.870675	−	0.491859i	$-0.836317\pi$
$727$	− 715.164i	− 0.983719i	0.870675	−	0.491859i	$-0.163683\pi$
$728$	0	0
$729$	751.381	1.03070
$730$	0	0
$731$	264.762	0.362191
$732$	0	0
$733$	− 1233.29i	− 1.68252i	−0.540632	−	0.841259i	$-0.681815\pi$
$733$	− 1233.29i	− 1.68252i	0.540632	−	0.841259i	$-0.318185\pi$
$734$	0	0
$735$	− 112.300i	− 0.152789i
$736$	0	0
$737$	−1000.38	−1.35737
$738$	0	0
$739$	−8.55656	−0.0115786	−0.00578928	−	0.999983i	$-0.501843\pi$
$739$	−8.55656	−0.0115786	−0.00578928	+	0.999983i	$0.501843\pi$
$740$	0	0
$741$	303.108i	0.409053i
$742$	0	0
$743$	− 1010.56i	− 1.36011i	−0.733163	−	0.680053i	$-0.761957\pi$
$743$	− 1010.56i	− 1.36011i	0.733163	−	0.680053i	$-0.238043\pi$
$744$	0	0
$745$	−82.6687	−0.110965
$746$	0	0
$747$	−264.631	−0.354258
$748$	0	0
$749$	269.868i	0.360305i
$750$	0	0
$751$	1104.31i	1.47046i	0.677820	+	0.735228i	$0.262925\pi$
$751$	1104.31i	1.47046i	−0.677820	+	0.735228i	$0.737075\pi$
$752$	0	0
$753$	−321.378	−0.426796
$754$	0	0
$755$	632.188	0.837335
$756$	0	0
$757$	− 875.633i	− 1.15671i	−0.815783	−	0.578357i	$-0.803694\pi$
$757$	− 875.633i	− 1.15671i	0.815783	−	0.578357i	$-0.196306\pi$
$758$	0	0
$759$	− 25.9936i	− 0.0342471i
$760$	0	0
$761$	647.207	0.850470	0.425235	−	0.905083i	$-0.360192\pi$
$761$	647.207	0.850470	0.425235	+	0.905083i	$0.360192\pi$
$762$	0	0
$763$	−701.688	−0.919644
$764$	0	0
$765$	92.3018i	0.120656i
$766$	0	0
$767$	147.729i	0.192606i
$768$	0	0
$769$	631.430	0.821106	0.410553	−	0.911837i	$-0.365336\pi$
$769$	631.430	0.821106	0.410553	+	0.911837i	$0.365336\pi$
$770$	0	0
$771$	−646.433	−0.838434
$772$	0	0
$773$	− 421.522i	− 0.545306i	−0.962112	−	0.272653i	$-0.912099\pi$
$773$	− 421.522i	− 0.545306i	0.962112	−	0.272653i	$-0.0879011\pi$
$774$	0	0
$775$	− 41.4725i	− 0.0535129i
$776$	0	0
$777$	−226.950	−0.292086
$778$	0	0
$779$	221.917	0.284874
$780$	0	0
$781$	− 1609.21i	− 2.06044i
$782$	0	0
$783$	− 320.927i	− 0.409869i
$784$	0	0
$785$	−457.830	−0.583223
$786$	0	0
$787$	−838.633	−1.06561	−0.532804	−	0.846239i	$-0.678862\pi$
$787$	−838.633	−1.06561	−0.532804	+	0.846239i	$0.678862\pi$
$788$	0	0
$789$	955.469i	1.21099i
$790$	0	0
$791$	− 897.535i	− 1.13468i
$792$	0	0
$793$	763.659	0.963001
$794$	0	0
$795$	−348.879	−0.438841
$796$	0	0
$797$	− 1213.57i	− 1.52268i	−0.648354	−	0.761339i	$-0.724542\pi$
$797$	− 1213.57i	− 1.52268i	0.648354	−	0.761339i	$-0.275458\pi$
$798$	0	0
$799$	− 634.732i	− 0.794408i
$800$	0	0
$801$	−386.944	−0.483076
$802$	0	0
$803$	110.635	0.137777
$804$	0	0
$805$	6.52476i	0.00810529i
$806$	0	0
$807$	819.431i	1.01540i
$808$	0	0
$809$	229.214	0.283330	0.141665	−	0.989915i	$-0.454754\pi$
$809$	229.214	0.283330	0.141665	+	0.989915i	$0.454754\pi$
$810$	0	0
$811$	454.225	0.560080	0.280040	−	0.959988i	$-0.409652\pi$
$811$	454.225	0.560080	0.280040	+	0.959988i	$0.409652\pi$
$812$	0	0
$813$	582.728i	0.716762i
$814$	0	0
$815$	239.785i	0.294215i
$816$	0	0
$817$	338.885	0.414792
$818$	0	0
$819$	−154.651	−0.188829
$820$	0	0
$821$	1130.90i	1.37747i	0.725015	+	0.688733i	$0.241833\pi$
$821$	1130.90i	1.37747i	−0.725015	+	0.688733i	$0.758167\pi$
$822$	0	0
$823$	780.148i	0.947931i	0.880543	+	0.473966i	$0.157178\pi$
$823$	780.148i	0.947931i	−0.880543	+	0.473966i	$0.842822\pi$
$824$	0	0
$825$	−234.164	−0.283835
$826$	0	0
$827$	209.175	0.252932	0.126466	−	0.991971i	$-0.459636\pi$
$827$	209.175	0.252932	0.126466	+	0.991971i	$0.459636\pi$
$828$	0	0
$829$	508.525i	0.613419i	0.951803	+	0.306710i	$0.0992281\pi$
$829$	508.525i	0.613419i	−0.951803	+	0.306710i	$0.900772\pi$
$830$	0	0
$831$	− 128.734i	− 0.154915i
$832$	0	0
$833$	253.947	0.304859
$834$	0	0
$835$	74.2575	0.0889311
$836$	0	0
$837$	− 243.226i	− 0.290593i
$838$	0	0
$839$	− 274.028i	− 0.326613i	−0.986575	−	0.163306i	$-0.947784\pi$
$839$	− 274.028i	− 0.326613i	0.986575	−	0.163306i	$-0.0522159\pi$
$840$	0	0
$841$	721.223	0.857578
$842$	0	0
$843$	118.405	0.140457
$844$	0	0
$845$	− 217.397i	− 0.257275i
$846$	0	0
$847$	− 1449.83i	− 1.71172i
$848$	0	0
$849$	−346.407	−0.408018
$850$	0	0
$851$	−10.1907	−0.0119750
$852$	0	0
$853$	− 1583.28i	− 1.85613i	−0.372416	−	0.928066i	$-0.621471\pi$
$853$	− 1583.28i	− 1.85613i	0.372416	−	0.928066i	$-0.378529\pi$
$854$	0	0
$855$	118.143i	0.138179i
$856$	0	0
$857$	1007.38	1.17547	0.587735	−	0.809054i	$-0.300020\pi$
$857$	1007.38	1.17547	0.587735	+	0.809054i	$0.300020\pi$
$858$	0	0
$859$	−76.6086	−0.0891835	−0.0445917	−	0.999005i	$-0.514199\pi$
$859$	−76.6086	−0.0891835	−0.0445917	+	0.999005i	$0.514199\pi$
$860$	0	0
$861$	− 180.263i	− 0.209365i
$862$	0	0
$863$	255.450i	0.296002i	0.988987	+	0.148001i	$0.0472839\pi$
$863$	255.450i	0.296002i	−0.988987	+	0.148001i	$0.952716\pi$
$864$	0	0
$865$	506.158	0.585154
$866$	0	0
$867$	−347.175	−0.400433
$868$	0	0
$869$	275.777i	0.317350i
$870$	0	0
$871$	− 425.487i	− 0.488504i
$872$	0	0
$873$	322.328	0.369219
$874$	0	0
$875$	58.7785	0.0671755
$876$	0	0
$877$	− 601.522i	− 0.685886i	−0.939356	−	0.342943i	$-0.888576\pi$
$877$	− 601.522i	− 0.685886i	0.939356	−	0.342943i	$-0.111424\pi$
$878$	0	0
$879$	420.523i	0.478411i
$880$	0	0
$881$	237.850	0.269977	0.134989	−	0.990847i	$-0.456900\pi$
$881$	237.850	0.269977	0.134989	+	0.990847i	$0.456900\pi$
$882$	0	0
$883$	1.30294	0.00147559	0.000737794	−	1.00000i	$-0.499765\pi$
$883$	1.30294	0.00147559	0.000737794		1.00000i	$0.499765\pi$
$884$	0	0
$885$	− 91.6718i	− 0.103584i
$886$	0	0
$887$	− 536.353i	− 0.604682i	−0.953200	−	0.302341i	$-0.902232\pi$
$887$	− 536.353i	− 0.604682i	0.953200	−	0.302341i	$-0.0977682\pi$
$888$	0	0
$889$	1044.30	1.17469
$890$	0	0
$891$	−750.855	−0.842710
$892$	0	0
$893$	− 812.433i	− 0.909780i
$894$	0	0
$895$	501.634i	0.560485i
$896$	0	0
$897$	11.0557	0.0123252
$898$	0	0
$899$	−90.7773	−0.100976
$900$	0	0
$901$	− 788.932i	− 0.875618i
$902$	0	0
$903$	− 275.276i	− 0.304846i
$904$	0	0
$905$	192.800	0.213039
$906$	0	0
$907$	332.159	0.366217	0.183108	−	0.983093i	$-0.441384\pi$
$907$	332.159	0.366217	0.183108	+	0.983093i	$0.441384\pi$
$908$	0	0
$909$	− 222.604i	− 0.244889i
$910$	0	0
$911$	1450.06i	1.59172i	0.605478	+	0.795862i	$0.292982\pi$
$911$	1450.06i	1.59172i	−0.605478	+	0.795862i	$0.707018\pi$
$912$	0	0
$913$	1518.15	1.66282
$914$	0	0
$915$	−473.882	−0.517904
$916$	0	0
$917$	40.8514i	0.0445490i
$918$	0	0
$919$	814.405i	0.886186i	0.896476	+	0.443093i	$0.146119\pi$
$919$	814.405i	0.886186i	−0.896476	+	0.443093i	$0.853881\pi$
$920$	0	0
$921$	669.017	0.726403
$922$	0	0
$923$	684.437	0.741535
$924$	0	0
$925$	91.8034i	0.0992469i
$926$	0	0
$927$	− 478.353i	− 0.516023i
$928$	0	0
$929$	400.039	0.430612	0.215306	−	0.976547i	$-0.430925\pi$
$929$	400.039	0.430612	0.215306	+	0.976547i	$0.430925\pi$
$930$	0	0
$931$	325.043	0.349134
$932$	0	0
$933$	663.489i	0.711135i
$934$	0	0
$935$	− 529.524i	− 0.566335i
$936$	0	0
$937$	−249.279	−0.266039	−0.133020	−	0.991113i	$-0.542467\pi$
$937$	−249.279	−0.266039	−0.133020	+	0.991113i	$0.542467\pi$
$938$	0	0
$939$	1334.39	1.42107
$940$	0	0
$941$	− 724.229i	− 0.769638i	−0.922992	−	0.384819i	$-0.874264\pi$
$941$	− 724.229i	− 0.769638i	0.922992	−	0.384819i	$-0.125736\pi$
$942$	0	0
$943$	− 8.09432i	− 0.00858358i
$944$	0	0
$945$	344.721	0.364785
$946$	0	0
$947$	1141.54	1.20542	0.602712	−	0.797959i	$-0.294087\pi$
$947$	1141.54	1.20542	0.602712	+	0.797959i	$0.294087\pi$
$948$	0	0
$949$	47.0557i	0.0495845i
$950$	0	0
$951$	− 378.865i	− 0.398386i
$952$	0	0
$953$	−1295.33	−1.35921	−0.679604	−	0.733579i	$-0.737848\pi$
$953$	−1295.33	−1.35921	−0.679604	+	0.733579i	$0.737848\pi$
$954$	0	0
$955$	69.3623	0.0726307
$956$	0	0
$957$	512.551i	0.535581i
$958$	0	0
$959$	4.37837i	0.00456556i
$960$	0	0
$961$	892.201	0.928409
$962$	0	0
$963$	178.232	0.185080
$964$	0	0
$965$	− 246.466i	− 0.255405i
$966$	0	0
$967$	− 398.477i	− 0.412075i	−0.978544	−	0.206037i	$-0.933943\pi$
$967$	− 398.477i	− 0.412075i	0.978544	−	0.206037i	$-0.0660569\pi$
$968$	0	0
$969$	425.337	0.438945
$970$	0	0
$971$	−928.093	−0.955811	−0.477906	−	0.878411i	$-0.658604\pi$
$971$	−928.093	−0.955811	−0.477906	+	0.878411i	$0.658604\pi$
$972$	0	0
$973$	1249.44i	1.28411i
$974$	0	0
$975$	− 99.5959i	− 0.102150i
$976$	0	0
$977$	−1378.05	−1.41049	−0.705247	−	0.708962i	$-0.749164\pi$
$977$	−1378.05	−1.41049	−0.705247	+	0.708962i	$0.749164\pi$
$978$	0	0
$979$	2219.85	2.26747
$980$	0	0
$981$	463.423i	0.472398i
$982$	0	0
$983$	311.291i	0.316675i	0.987385	+	0.158337i	$0.0506134\pi$
$983$	311.291i	0.316675i	−0.987385	+	0.158337i	$0.949387\pi$
$984$	0	0
$985$	−385.777	−0.391652
$986$	0	0
$987$	−659.939	−0.668631
$988$	0	0
$989$	− 12.3607i	− 0.0124982i
$990$	0	0
$991$	− 961.147i	− 0.969876i	−0.874549	−	0.484938i	$-0.838842\pi$
$991$	− 961.147i	− 0.969876i	0.874549	−	0.484938i	$-0.161158\pi$
$992$	0	0
$993$	780.498	0.786000
$994$	0	0
$995$	−608.676	−0.611735
$996$	0	0
$997$	− 1089.68i	− 1.09296i	−0.837473	−	0.546479i	$-0.815968\pi$
$997$	− 1089.68i	− 1.09296i	0.837473	−	0.546479i	$-0.184032\pi$
$998$	0	0
$999$	538.404i	0.538943i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.g.e.1151.5		8
4.3	odd	2	inner	1280.3.g.e.1151.3		8
8.3	odd	2	inner	1280.3.g.e.1151.6		8
8.5	even	2	inner	1280.3.g.e.1151.4		8
16.3	odd	4		20.3.b.a.11.4	yes	4
16.5	even	4		320.3.b.c.191.2		4
16.11	odd	4		320.3.b.c.191.3		4
16.13	even	4		20.3.b.a.11.3	✓	4
48.5	odd	4		2880.3.e.e.2431.2		4
48.11	even	4		2880.3.e.e.2431.1		4
48.29	odd	4		180.3.c.a.91.2		4
48.35	even	4		180.3.c.a.91.1		4
80.3	even	4		100.3.d.b.99.7		8
80.13	odd	4		100.3.d.b.99.1		8
80.19	odd	4		100.3.b.f.51.1		4
80.27	even	4		1600.3.h.n.1599.5		8
80.29	even	4		100.3.b.f.51.2		4
80.37	odd	4		1600.3.h.n.1599.4		8
80.43	even	4		1600.3.h.n.1599.3		8
80.53	odd	4		1600.3.h.n.1599.6		8
80.59	odd	4		1600.3.b.s.1151.2		4
80.67	even	4		100.3.d.b.99.2		8
80.69	even	4		1600.3.b.s.1151.3		4
80.77	odd	4		100.3.d.b.99.8		8
240.29	odd	4		900.3.c.k.451.3		4
240.77	even	4		900.3.f.e.199.1		8
240.83	odd	4		900.3.f.e.199.2		8
240.173	even	4		900.3.f.e.199.8		8
240.179	even	4		900.3.c.k.451.4		4
240.227	odd	4		900.3.f.e.199.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.3	✓	4	16.13	even	4
20.3.b.a.11.4	yes	4	16.3	odd	4
100.3.b.f.51.1		4	80.19	odd	4
100.3.b.f.51.2		4	80.29	even	4
100.3.d.b.99.1		8	80.13	odd	4
100.3.d.b.99.2		8	80.67	even	4
100.3.d.b.99.7		8	80.3	even	4
100.3.d.b.99.8		8	80.77	odd	4
180.3.c.a.91.1		4	48.35	even	4
180.3.c.a.91.2		4	48.29	odd	4
320.3.b.c.191.2		4	16.5	even	4
320.3.b.c.191.3		4	16.11	odd	4
900.3.c.k.451.3		4	240.29	odd	4
900.3.c.k.451.4		4	240.179	even	4
900.3.f.e.199.1		8	240.77	even	4
900.3.f.e.199.2		8	240.83	odd	4
900.3.f.e.199.7		8	240.227	odd	4
900.3.f.e.199.8		8	240.173	even	4
1280.3.g.e.1151.3		8	4.3	odd	2	inner
1280.3.g.e.1151.4		8	8.5	even	2	inner
1280.3.g.e.1151.5		8	1.1	even	1	trivial
1280.3.g.e.1151.6		8	8.3	odd	2	inner
1600.3.b.s.1151.2		4	80.59	odd	4
1600.3.b.s.1151.3		4	80.69	even	4
1600.3.h.n.1599.3		8	80.43	even	4
1600.3.h.n.1599.4		8	80.37	odd	4
1600.3.h.n.1599.5		8	80.27	even	4
1600.3.h.n.1599.6		8	80.53	odd	4
2880.3.e.e.2431.1		4	48.11	even	4
2880.3.e.e.2431.2		4	48.5	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)

\(f(q)\)	\(=\)	\(q+2.35114 q^{3} -2.23607i q^{5} -5.25731i q^{7} -3.47214 q^{9} +19.9192 q^{11} +8.47214i q^{13} -5.25731i q^{15} +11.8885 q^{17} +15.2169 q^{19} -12.3607i q^{21} -0.555029i q^{23} -5.00000 q^{25} -29.3238 q^{27} +10.9443i q^{29} +8.29451i q^{31} +46.8328 q^{33} -11.7557 q^{35} -18.3607i q^{37} +19.9192i q^{39} +14.5836 q^{41} +22.2703 q^{43} +7.76393i q^{45} -53.3902i q^{47} +21.3607 q^{49} +27.9516 q^{51} -66.3607i q^{53} -44.5407i q^{55} +35.7771 q^{57} +17.4370 q^{59} -90.1378i q^{61} +18.2541i q^{63} +18.9443 q^{65} -50.2220 q^{67} -1.30495i q^{69} -80.7868i q^{71} +5.55418 q^{73} -11.7557 q^{75} -104.721i q^{77} +13.8448i q^{79} -37.6950 q^{81} +76.2155 q^{83} -26.5836i q^{85} +25.7315i q^{87} +111.443 q^{89} +44.5407 q^{91} +19.5016i q^{93} -34.0260i q^{95} -92.8328 q^{97} -69.1621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} - 48 q^{17} - 40 q^{25} + 160 q^{33} + 224 q^{41} - 8 q^{49} + 80 q^{65} - 528 q^{73} - 552 q^{81} + 176 q^{89} - 528 q^{97}+O(q^{100}) \) 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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