LMFDB - Embedded newform 9280.2.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,3,0,7,0,2,0,2,0,9,0,-3,0,1,0,6,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.745898 q^{3} +1.00000 q^{5} +4.42723 q^{7} -2.44364 q^{9} +5.87086 q^{11} +3.25410 q^{13} -0.745898 q^{15} +2.93543 q^{17} +2.50820 q^{19} -3.30226 q^{21} -0.745898 q^{23} +1.00000 q^{25} +4.06040 q^{27} -1.00000 q^{29} +4.29809 q^{31} -4.37907 q^{33} +4.42723 q^{35} -6.85446 q^{37} -2.42723 q^{39} +7.87086 q^{41} +5.44364 q^{43} -2.44364 q^{45} +10.3791 q^{47} +12.6004 q^{49} -2.18953 q^{51} +0.0481609 q^{53} +5.87086 q^{55} -1.87086 q^{57} -11.5040 q^{59} -1.60036 q^{61} -10.8185 q^{63} +3.25410 q^{65} -14.3791 q^{67} +0.556364 q^{69} +7.74173 q^{71} +9.66075 q^{73} -0.745898 q^{75} +25.9917 q^{77} -3.97942 q^{79} +4.30226 q^{81} +3.87086 q^{83} +2.93543 q^{85} +0.745898 q^{87} +0.766474 q^{89} +14.4067 q^{91} -3.20594 q^{93} +2.50820 q^{95} -0.270508 q^{97} -14.3463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} - 3 q^{15} + q^{17} + 6 q^{19} - 20 q^{21} - 3 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} - 9 q^{31} + 4 q^{33} + 7 q^{35} - 8 q^{37} - q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 7 * q^7 + 2 * q^9 + 2 * q^11 + 9 * q^13 - 3 * q^15 + q^17 + 6 * q^19 - 20 * q^21 - 3 * q^23 + 3 * q^25 - 12 * q^27 - 3 * q^29 - 9 * q^31 + 4 * q^33 + 7 * q^35 - 8 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 2 * q^45 + 14 * q^47 + 26 * q^49 + 2 * q^51 + 11 * q^53 + 2 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + 26 * q^63 + 9 * q^65 - 26 * q^67 + 11 * q^69 - 8 * q^71 - 7 * q^73 - 3 * q^75 + 14 * q^77 + 17 * q^79 + 23 * q^81 - 4 * q^83 + q^85 + 3 * q^87 + 32 * q^89 + 8 * q^91 + 2 * q^93 + 6 * q^95 + 3 * q^97 - 32 * q^99

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
$3$	−0.745898	−0.430645	−0.215322	−	0.976543i	$-0.569080\pi$
$3$	−0.745898	−0.430645	−0.215322	+	0.976543i	$0.569080\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.42723	1.67334	0.836668	−	0.547711i	$-0.184501\pi$
$7$	4.42723	1.67334	0.836668	+	0.547711i	$0.184501\pi$
$8$	0	0
$9$	−2.44364	−0.814545
$10$	0	0
$11$	5.87086	1.77013	0.885066	−	0.465465i	$-0.154113\pi$
$11$	5.87086	1.77013	0.885066	+	0.465465i	$0.154113\pi$
$12$	0	0
$13$	3.25410	0.902525	0.451263	−	0.892391i	$-0.350974\pi$
$13$	3.25410	0.902525	0.451263	+	0.892391i	$0.350974\pi$
$14$	0	0
$15$	−0.745898	−0.192590
$16$	0	0
$17$	2.93543	0.711947	0.355973	−	0.934496i	$-0.384149\pi$
$17$	2.93543	0.711947	0.355973	+	0.934496i	$0.384149\pi$
$18$	0	0
$19$	2.50820	0.575421	0.287711	−	0.957717i	$-0.407106\pi$
$19$	2.50820	0.575421	0.287711	+	0.957717i	$0.407106\pi$
$20$	0	0
$21$	−3.30226	−0.720613
$22$	0	0
$23$	−0.745898	−0.155531	−0.0777653	−	0.996972i	$-0.524778\pi$
$23$	−0.745898	−0.155531	−0.0777653	+	0.996972i	$0.524778\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.06040	0.781424
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.29809	0.771960	0.385980	−	0.922507i	$-0.373863\pi$
$31$	4.29809	0.771960	0.385980	+	0.922507i	$0.373863\pi$
$32$	0	0
$33$	−4.37907	−0.762298
$34$	0	0
$35$	4.42723	0.748338
$36$	0	0
$37$	−6.85446	−1.12687	−0.563433	−	0.826162i	$-0.690520\pi$
$37$	−6.85446	−1.12687	−0.563433	+	0.826162i	$0.690520\pi$
$38$	0	0
$39$	−2.42723	−0.388668
$40$	0	0
$41$	7.87086	1.22922	0.614611	−	0.788830i	$-0.289313\pi$
$41$	7.87086	1.22922	0.614611	+	0.788830i	$0.289313\pi$
$42$	0	0
$43$	5.44364	0.830147	0.415073	−	0.909788i	$-0.363756\pi$
$43$	5.44364	0.830147	0.415073	+	0.909788i	$0.363756\pi$
$44$	0	0
$45$	−2.44364	−0.364276
$46$	0	0
$47$	10.3791	1.51394	0.756971	−	0.653448i	$-0.226678\pi$
$47$	10.3791	1.51394	0.756971	+	0.653448i	$0.226678\pi$
$48$	0	0
$49$	12.6004	1.80005
$50$	0	0
$51$	−2.18953	−0.306596
$52$	0	0
$53$	0.0481609	0.00661541	0.00330771	−	0.999995i	$-0.498947\pi$
$53$	0.0481609	0.00661541	0.00330771	+	0.999995i	$0.498947\pi$
$54$	0	0
$55$	5.87086	0.791627
$56$	0	0
$57$	−1.87086	−0.247802
$58$	0	0
$59$	−11.5040	−1.49770	−0.748849	−	0.662741i	$-0.769393\pi$
$59$	−11.5040	−1.49770	−0.748849	+	0.662741i	$0.769393\pi$
$60$	0	0
$61$	−1.60036	−0.204905	−0.102452	−	0.994738i	$-0.532669\pi$
$61$	−1.60036	−0.204905	−0.102452	+	0.994738i	$0.532669\pi$
$62$	0	0
$63$	−10.8185	−1.36301
$64$	0	0
$65$	3.25410	0.403622
$66$	0	0
$67$	−14.3791	−1.75668	−0.878341	−	0.478034i	$-0.841350\pi$
$67$	−14.3791	−1.75668	−0.878341	+	0.478034i	$0.841350\pi$
$68$	0	0
$69$	0.556364	0.0669784
$70$	0	0
$71$	7.74173	0.918774	0.459387	−	0.888236i	$-0.348069\pi$
$71$	7.74173	0.918774	0.459387	+	0.888236i	$0.348069\pi$
$72$	0	0
$73$	9.66075	1.13071	0.565353	−	0.824849i	$-0.308740\pi$
$73$	9.66075	1.13071	0.565353	+	0.824849i	$0.308740\pi$
$74$	0	0
$75$	−0.745898	−0.0861289
$76$	0	0
$77$	25.9917	2.96202
$78$	0	0
$79$	−3.97942	−0.447720	−0.223860	−	0.974621i	$-0.571866\pi$
$79$	−3.97942	−0.447720	−0.223860	+	0.974621i	$0.571866\pi$
$80$	0	0
$81$	4.30226	0.478029
$82$	0	0
$83$	3.87086	0.424883	0.212441	−	0.977174i	$-0.431859\pi$
$83$	3.87086	0.424883	0.212441	+	0.977174i	$0.431859\pi$
$84$	0	0
$85$	2.93543	0.318392
$86$	0	0
$87$	0.745898	0.0799687
$88$	0	0
$89$	0.766474	0.0812461	0.0406230	−	0.999175i	$-0.487066\pi$
$89$	0.766474	0.0812461	0.0406230	+	0.999175i	$0.487066\pi$
$90$	0	0
$91$	14.4067	1.51023
$92$	0	0
$93$	−3.20594	−0.332441
$94$	0	0
$95$	2.50820	0.257336
$96$	0	0
$97$	−0.270508	−0.0274660	−0.0137330	−	0.999906i	$-0.504371\pi$
$97$	−0.270508	−0.0274660	−0.0137330	+	0.999906i	$0.504371\pi$
$98$	0	0
$99$	−14.3463	−1.44185
$100$	0	0
$101$	−11.8503	−1.17915	−0.589574	−	0.807714i	$-0.700704\pi$
$101$	−11.8503	−1.17915	−0.589574	+	0.807714i	$0.700704\pi$
$102$	0	0
$103$	6.37907	0.628548	0.314274	−	0.949332i	$-0.398239\pi$
$103$	6.37907	0.628548	0.314274	+	0.949332i	$0.398239\pi$
$104$	0	0
$105$	−3.30226	−0.322268
$106$	0	0
$107$	−8.88727	−0.859165	−0.429582	−	0.903028i	$-0.641339\pi$
$107$	−8.88727	−0.859165	−0.429582	+	0.903028i	$0.641339\pi$
$108$	0	0
$109$	−7.61259	−0.729154	−0.364577	−	0.931173i	$-0.618786\pi$
$109$	−7.61259	−0.729154	−0.364577	+	0.931173i	$0.618786\pi$
$110$	0	0
$111$	5.11273	0.485279
$112$	0	0
$113$	−18.7376	−1.76268	−0.881341	−	0.472481i	$-0.843358\pi$
$113$	−18.7376	−1.76268	−0.881341	+	0.472481i	$0.843358\pi$
$114$	0	0
$115$	−0.745898	−0.0695554
$116$	0	0
$117$	−7.95184	−0.735148
$118$	0	0
$119$	12.9958	1.19133
$120$	0	0
$121$	23.4671	2.13337
$122$	0	0
$123$	−5.87086	−0.529358
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.65375	0.324217	0.162109	−	0.986773i	$-0.448170\pi$
$127$	3.65375	0.324217	0.162109	+	0.986773i	$0.448170\pi$
$128$	0	0
$129$	−4.06040	−0.357498
$130$	0	0
$131$	−0.379068	−0.0331193	−0.0165597	−	0.999863i	$-0.505271\pi$
$131$	−0.379068	−0.0331193	−0.0165597	+	0.999863i	$0.505271\pi$
$132$	0	0
$133$	11.1044	0.962873
$134$	0	0
$135$	4.06040	0.349463
$136$	0	0
$137$	−12.9630	−1.10751	−0.553753	−	0.832681i	$-0.686805\pi$
$137$	−12.9630	−1.10751	−0.553753	+	0.832681i	$0.686805\pi$
$138$	0	0
$139$	−8.46004	−0.717571	−0.358786	−	0.933420i	$-0.616809\pi$
$139$	−8.46004	−0.717571	−0.358786	+	0.933420i	$0.616809\pi$
$140$	0	0
$141$	−7.74173	−0.651971
$142$	0	0
$143$	19.1044	1.59759
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−9.39858	−0.775182
$148$	0	0
$149$	20.2499	1.65894	0.829470	−	0.558552i	$-0.188643\pi$
$149$	20.2499	1.65894	0.829470	+	0.558552i	$0.188643\pi$
$150$	0	0
$151$	−1.49180	−0.121401	−0.0607003	−	0.998156i	$-0.519333\pi$
$151$	−1.49180	−0.121401	−0.0607003	+	0.998156i	$0.519333\pi$
$152$	0	0
$153$	−7.17313	−0.579913
$154$	0	0
$155$	4.29809	0.345231
$156$	0	0
$157$	8.98359	0.716969	0.358484	−	0.933536i	$-0.383294\pi$
$157$	8.98359	0.716969	0.358484	+	0.933536i	$0.383294\pi$
$158$	0	0
$159$	−0.0359231	−0.00284889
$160$	0	0
$161$	−3.30226	−0.260255
$162$	0	0
$163$	−13.3627	−1.04664	−0.523322	−	0.852135i	$-0.675308\pi$
$163$	−13.3627	−1.04664	−0.523322	+	0.852135i	$0.675308\pi$
$164$	0	0
$165$	−4.37907	−0.340910
$166$	0	0
$167$	−19.0562	−1.47462	−0.737308	−	0.675557i	$-0.763903\pi$
$167$	−19.0562	−1.47462	−0.737308	+	0.675557i	$0.763903\pi$
$168$	0	0
$169$	−2.41082	−0.185448
$170$	0	0
$171$	−6.12914	−0.468707
$172$	0	0
$173$	12.4876	0.949417	0.474708	−	0.880143i	$-0.342554\pi$
$173$	12.4876	0.949417	0.474708	+	0.880143i	$0.342554\pi$
$174$	0	0
$175$	4.42723	0.334667
$176$	0	0
$177$	8.58084	0.644975
$178$	0	0
$179$	−17.9466	−1.34139	−0.670696	−	0.741732i	$-0.734005\pi$
$179$	−17.9466	−1.34139	−0.670696	+	0.741732i	$0.734005\pi$
$180$	0	0
$181$	7.61259	0.565840	0.282920	−	0.959144i	$-0.408697\pi$
$181$	7.61259	0.565840	0.282920	+	0.959144i	$0.408697\pi$
$182$	0	0
$183$	1.19370	0.0882411
$184$	0	0
$185$	−6.85446	−0.503950
$186$	0	0
$187$	17.2335	1.26024
$188$	0	0
$189$	17.9763	1.30758
$190$	0	0
$191$	−14.3309	−1.03695	−0.518474	−	0.855093i	$-0.673500\pi$
$191$	−14.3309	−1.03695	−0.518474	+	0.855093i	$0.673500\pi$
$192$	0	0
$193$	−16.2294	−1.16821	−0.584107	−	0.811676i	$-0.698555\pi$
$193$	−16.2294	−1.16821	−0.584107	+	0.811676i	$0.698555\pi$
$194$	0	0
$195$	−2.42723	−0.173817
$196$	0	0
$197$	8.04816	0.573408	0.286704	−	0.958019i	$-0.407440\pi$
$197$	8.04816	0.573408	0.286704	+	0.958019i	$0.407440\pi$
$198$	0	0
$199$	−12.8545	−0.911228	−0.455614	−	0.890177i	$-0.650580\pi$
$199$	−12.8545	−0.911228	−0.455614	+	0.890177i	$0.650580\pi$
$200$	0	0
$201$	10.7253	0.756506
$202$	0	0
$203$	−4.42723	−0.310731
$204$	0	0
$205$	7.87086	0.549725
$206$	0	0
$207$	1.82270	0.126687
$208$	0	0
$209$	14.7253	1.01857
$210$	0	0
$211$	−24.3379	−1.67549	−0.837746	−	0.546061i	$-0.816127\pi$
$211$	−24.3379	−1.67549	−0.837746	+	0.546061i	$0.816127\pi$
$212$	0	0
$213$	−5.77454	−0.395665
$214$	0	0
$215$	5.44364	0.371253
$216$	0	0
$217$	19.0286	1.29175
$218$	0	0
$219$	−7.20594	−0.486932
$220$	0	0
$221$	9.55220	0.642550
$222$	0	0
$223$	−13.7623	−0.921592	−0.460796	−	0.887506i	$-0.652436\pi$
$223$	−13.7623	−0.921592	−0.460796	+	0.887506i	$0.652436\pi$
$224$	0	0
$225$	−2.44364	−0.162909
$226$	0	0
$227$	−13.7417	−0.912071	−0.456035	−	0.889962i	$-0.650731\pi$
$227$	−13.7417	−0.912071	−0.456035	+	0.889962i	$0.650731\pi$
$228$	0	0
$229$	−15.4108	−1.01838	−0.509188	−	0.860655i	$-0.670054\pi$
$229$	−15.4108	−1.01838	−0.509188	+	0.860655i	$0.670054\pi$
$230$	0	0
$231$	−19.3871	−1.27558
$232$	0	0
$233$	−12.7253	−0.833664	−0.416832	−	0.908984i	$-0.636860\pi$
$233$	−12.7253	−0.833664	−0.416832	+	0.908984i	$0.636860\pi$
$234$	0	0
$235$	10.3791	0.677056
$236$	0	0
$237$	2.96825	0.192808
$238$	0	0
$239$	−17.8297	−1.15331	−0.576654	−	0.816988i	$-0.695642\pi$
$239$	−17.8297	−1.15331	−0.576654	+	0.816988i	$0.695642\pi$
$240$	0	0
$241$	16.8667	1.08648	0.543240	−	0.839578i	$-0.317197\pi$
$241$	16.8667	1.08648	0.543240	+	0.839578i	$0.317197\pi$
$242$	0	0
$243$	−15.3902	−0.987285
$244$	0	0
$245$	12.6004	0.805007
$246$	0	0
$247$	8.16195	0.519332
$248$	0	0
$249$	−2.88727	−0.182973
$250$	0	0
$251$	27.9588	1.76475	0.882373	−	0.470550i	$-0.155944\pi$
$251$	27.9588	1.76475	0.882373	+	0.470550i	$0.155944\pi$
$252$	0	0
$253$	−4.37907	−0.275310
$254$	0	0
$255$	−2.18953	−0.137114
$256$	0	0
$257$	−12.5634	−0.783682	−0.391841	−	0.920033i	$-0.628162\pi$
$257$	−12.5634	−0.783682	−0.391841	+	0.920033i	$0.628162\pi$
$258$	0	0
$259$	−30.3463	−1.88562
$260$	0	0
$261$	2.44364	0.151257
$262$	0	0
$263$	−12.8461	−0.792126	−0.396063	−	0.918223i	$-0.629624\pi$
$263$	−12.8461	−0.792126	−0.396063	+	0.918223i	$0.629624\pi$
$264$	0	0
$265$	0.0481609	0.00295850
$266$	0	0
$267$	−0.571712	−0.0349882
$268$	0	0
$269$	13.3473	0.813800	0.406900	−	0.913473i	$-0.366610\pi$
$269$	13.3473	0.813800	0.406900	+	0.913473i	$0.366610\pi$
$270$	0	0
$271$	17.0164	1.03367	0.516837	−	0.856084i	$-0.327109\pi$
$271$	17.0164	1.03367	0.516837	+	0.856084i	$0.327109\pi$
$272$	0	0
$273$	−10.7459	−0.650371
$274$	0	0
$275$	5.87086	0.354026
$276$	0	0
$277$	−12.7909	−0.768534	−0.384267	−	0.923222i	$-0.625546\pi$
$277$	−12.7909	−0.768534	−0.384267	+	0.923222i	$0.625546\pi$
$278$	0	0
$279$	−10.5030	−0.628797
$280$	0	0
$281$	25.4436	1.51784	0.758920	−	0.651184i	$-0.225727\pi$
$281$	25.4436	1.51784	0.758920	+	0.651184i	$0.225727\pi$
$282$	0	0
$283$	−1.14554	−0.0680954	−0.0340477	−	0.999420i	$-0.510840\pi$
$283$	−1.14554	−0.0680954	−0.0340477	+	0.999420i	$0.510840\pi$
$284$	0	0
$285$	−1.87086	−0.110820
$286$	0	0
$287$	34.8461	2.05690
$288$	0	0
$289$	−8.38324	−0.493132
$290$	0	0
$291$	0.201772	0.0118281
$292$	0	0
$293$	−10.5962	−0.619036	−0.309518	−	0.950894i	$-0.600168\pi$
$293$	−10.5962	−0.619036	−0.309518	+	0.950894i	$0.600168\pi$
$294$	0	0
$295$	−11.5040	−0.669791
$296$	0	0
$297$	23.8381	1.38322
$298$	0	0
$299$	−2.42723	−0.140370
$300$	0	0
$301$	24.1002	1.38911
$302$	0	0
$303$	8.83911	0.507794
$304$	0	0
$305$	−1.60036	−0.0916361
$306$	0	0
$307$	−30.8133	−1.75861	−0.879304	−	0.476261i	$-0.841992\pi$
$307$	−30.8133	−1.75861	−0.879304	+	0.476261i	$0.841992\pi$
$308$	0	0
$309$	−4.75814	−0.270681
$310$	0	0
$311$	13.9466	0.790840	0.395420	−	0.918500i	$-0.370599\pi$
$311$	13.9466	0.790840	0.395420	+	0.918500i	$0.370599\pi$
$312$	0	0
$313$	20.9836	1.18606	0.593031	−	0.805179i	$-0.297931\pi$
$313$	20.9836	1.18606	0.593031	+	0.805179i	$0.297931\pi$
$314$	0	0
$315$	−10.8185	−0.609555
$316$	0	0
$317$	12.7665	0.717037	0.358518	−	0.933523i	$-0.383282\pi$
$317$	12.7665	0.717037	0.358518	+	0.933523i	$0.383282\pi$
$318$	0	0
$319$	−5.87086	−0.328705
$320$	0	0
$321$	6.62900	0.369995
$322$	0	0
$323$	7.36266	0.409669
$324$	0	0
$325$	3.25410	0.180505
$326$	0	0
$327$	5.67822	0.314006
$328$	0	0
$329$	45.9505	2.53333
$330$	0	0
$331$	7.90368	0.434425	0.217213	−	0.976124i	$-0.430303\pi$
$331$	7.90368	0.434425	0.217213	+	0.976124i	$0.430303\pi$
$332$	0	0
$333$	16.7498	0.917883
$334$	0	0
$335$	−14.3791	−0.785612
$336$	0	0
$337$	−2.67716	−0.145834	−0.0729171	−	0.997338i	$-0.523231\pi$
$337$	−2.67716	−0.145834	−0.0729171	+	0.997338i	$0.523231\pi$
$338$	0	0
$339$	13.9763	0.759089
$340$	0	0
$341$	25.2335	1.36647
$342$	0	0
$343$	24.7941	1.33875
$344$	0	0
$345$	0.556364	0.0299536
$346$	0	0
$347$	7.87086	0.422530	0.211265	−	0.977429i	$-0.432242\pi$
$347$	7.87086	0.422530	0.211265	+	0.977429i	$0.432242\pi$
$348$	0	0
$349$	−28.8461	−1.54410	−0.772049	−	0.635563i	$-0.780768\pi$
$349$	−28.8461	−1.54410	−0.772049	+	0.635563i	$0.780768\pi$
$350$	0	0
$351$	13.2130	0.705255
$352$	0	0
$353$	33.7969	1.79883	0.899414	−	0.437098i	$-0.143994\pi$
$353$	33.7969	1.79883	0.899414	+	0.437098i	$0.143994\pi$
$354$	0	0
$355$	7.74173	0.410888
$356$	0	0
$357$	−9.69357	−0.513038
$358$	0	0
$359$	29.2541	1.54397	0.771986	−	0.635639i	$-0.219263\pi$
$359$	29.2541	1.54397	0.771986	+	0.635639i	$0.219263\pi$
$360$	0	0
$361$	−12.7089	−0.668890
$362$	0	0
$363$	−17.5040	−0.918724
$364$	0	0
$365$	9.66075	0.505667
$366$	0	0
$367$	15.8709	0.828452	0.414226	−	0.910174i	$-0.364052\pi$
$367$	15.8709	0.828452	0.414226	+	0.910174i	$0.364052\pi$
$368$	0	0
$369$	−19.2335	−1.00126
$370$	0	0
$371$	0.213219	0.0110698
$372$	0	0
$373$	−6.67716	−0.345730	−0.172865	−	0.984945i	$-0.555302\pi$
$373$	−6.67716	−0.345730	−0.172865	+	0.984945i	$0.555302\pi$
$374$	0	0
$375$	−0.745898	−0.0385180
$376$	0	0
$377$	−3.25410	−0.167595
$378$	0	0
$379$	−14.8873	−0.764708	−0.382354	−	0.924016i	$-0.624886\pi$
$379$	−14.8873	−0.764708	−0.382354	+	0.924016i	$0.624886\pi$
$380$	0	0
$381$	−2.72532	−0.139623
$382$	0	0
$383$	2.46004	0.125702	0.0628511	−	0.998023i	$-0.479981\pi$
$383$	2.46004	0.125702	0.0628511	+	0.998023i	$0.479981\pi$
$384$	0	0
$385$	25.9917	1.32466
$386$	0	0
$387$	−13.3023	−0.676192
$388$	0	0
$389$	28.2088	1.43024	0.715121	−	0.699001i	$-0.246372\pi$
$389$	28.2088	1.43024	0.715121	+	0.699001i	$0.246372\pi$
$390$	0	0
$391$	−2.18953	−0.110729
$392$	0	0
$393$	0.282746	0.0142627
$394$	0	0
$395$	−3.97942	−0.200227
$396$	0	0
$397$	5.22652	0.262311	0.131156	−	0.991362i	$-0.458131\pi$
$397$	5.22652	0.262311	0.131156	+	0.991362i	$0.458131\pi$
$398$	0	0
$399$	−8.28275	−0.414656
$400$	0	0
$401$	−26.1002	−1.30338	−0.651691	−	0.758484i	$-0.725940\pi$
$401$	−26.1002	−1.30338	−0.651691	+	0.758484i	$0.725940\pi$
$402$	0	0
$403$	13.9864	0.696714
$404$	0	0
$405$	4.30226	0.213781
$406$	0	0
$407$	−40.2416	−1.99470
$408$	0	0
$409$	38.8789	1.92244	0.961220	−	0.275784i	$-0.0889373\pi$
$409$	38.8789	1.92244	0.961220	+	0.275784i	$0.0889373\pi$
$410$	0	0
$411$	9.66909	0.476941
$412$	0	0
$413$	−50.9310	−2.50615
$414$	0	0
$415$	3.87086	0.190013
$416$	0	0
$417$	6.31033	0.309018
$418$	0	0
$419$	−15.0234	−0.733942	−0.366971	−	0.930232i	$-0.619605\pi$
$419$	−15.0234	−0.733942	−0.366971	+	0.930232i	$0.619605\pi$
$420$	0	0
$421$	5.04922	0.246084	0.123042	−	0.992401i	$-0.460735\pi$
$421$	5.04922	0.246084	0.123042	+	0.992401i	$0.460735\pi$
$422$	0	0
$423$	−25.3627	−1.23317
$424$	0	0
$425$	2.93543	0.142389
$426$	0	0
$427$	−7.08514	−0.342874
$428$	0	0
$429$	−14.2499	−0.687993
$430$	0	0
$431$	−30.5082	−1.46953	−0.734764	−	0.678323i	$-0.762707\pi$
$431$	−30.5082	−1.46953	−0.734764	+	0.678323i	$0.762707\pi$
$432$	0	0
$433$	−16.4671	−0.791356	−0.395678	−	0.918389i	$-0.629490\pi$
$433$	−16.4671	−0.791356	−0.395678	+	0.918389i	$0.629490\pi$
$434$	0	0
$435$	0.745898	0.0357631
$436$	0	0
$437$	−1.87086	−0.0894956
$438$	0	0
$439$	26.7805	1.27816	0.639082	−	0.769139i	$-0.279315\pi$
$439$	26.7805	1.27816	0.639082	+	0.769139i	$0.279315\pi$
$440$	0	0
$441$	−30.7907	−1.46622
$442$	0	0
$443$	−19.0615	−0.905637	−0.452819	−	0.891603i	$-0.649581\pi$
$443$	−19.0615	−0.905637	−0.452819	+	0.891603i	$0.649581\pi$
$444$	0	0
$445$	0.766474	0.0363344
$446$	0	0
$447$	−15.1044	−0.714413
$448$	0	0
$449$	−21.9037	−1.03370	−0.516849	−	0.856076i	$-0.672895\pi$
$449$	−21.9037	−1.03370	−0.516849	+	0.856076i	$0.672895\pi$
$450$	0	0
$451$	46.2088	2.17589
$452$	0	0
$453$	1.11273	0.0522805
$454$	0	0
$455$	14.4067	0.675394
$456$	0	0
$457$	11.2747	0.527407	0.263704	−	0.964604i	$-0.415056\pi$
$457$	11.2747	0.527407	0.263704	+	0.964604i	$0.415056\pi$
$458$	0	0
$459$	11.9190	0.556332
$460$	0	0
$461$	−31.9763	−1.48929	−0.744643	−	0.667463i	$-0.767380\pi$
$461$	−31.9763	−1.48929	−0.744643	+	0.667463i	$0.767380\pi$
$462$	0	0
$463$	26.5550	1.23412	0.617059	−	0.786917i	$-0.288324\pi$
$463$	26.5550	1.23412	0.617059	+	0.786917i	$0.288324\pi$
$464$	0	0
$465$	−3.20594	−0.148672
$466$	0	0
$467$	28.8615	1.33555	0.667775	−	0.744363i	$-0.267247\pi$
$467$	28.8615	1.33555	0.667775	+	0.744363i	$0.267247\pi$
$468$	0	0
$469$	−63.6594	−2.93952
$470$	0	0
$471$	−6.70085	−0.308759
$472$	0	0
$473$	31.9588	1.46947
$474$	0	0
$475$	2.50820	0.115084
$476$	0	0
$477$	−0.117688	−0.00538855
$478$	0	0
$479$	−4.55636	−0.208186	−0.104093	−	0.994568i	$-0.533194\pi$
$479$	−4.55636	−0.208186	−0.104093	+	0.994568i	$0.533194\pi$
$480$	0	0
$481$	−22.3051	−1.01703
$482$	0	0
$483$	2.46315	0.112077
$484$	0	0
$485$	−0.270508	−0.0122832
$486$	0	0
$487$	13.2458	0.600223	0.300111	−	0.953904i	$-0.402976\pi$
$487$	13.2458	0.600223	0.300111	+	0.953904i	$0.402976\pi$
$488$	0	0
$489$	9.96719	0.450732
$490$	0	0
$491$	14.9836	0.676200	0.338100	−	0.941110i	$-0.390216\pi$
$491$	14.9836	0.676200	0.338100	+	0.941110i	$0.390216\pi$
$492$	0	0
$493$	−2.93543	−0.132205
$494$	0	0
$495$	−14.3463	−0.644816
$496$	0	0
$497$	34.2744	1.53742
$498$	0	0
$499$	−40.9958	−1.83523	−0.917613	−	0.397476i	$-0.869886\pi$
$499$	−40.9958	−1.83523	−0.917613	+	0.397476i	$0.869886\pi$
$500$	0	0
$501$	14.2140	0.635035
$502$	0	0
$503$	31.5714	1.40770	0.703851	−	0.710348i	$-0.251462\pi$
$503$	31.5714	1.40770	0.703851	+	0.710348i	$0.251462\pi$
$504$	0	0
$505$	−11.8503	−0.527331
$506$	0	0
$507$	1.79823	0.0798621
$508$	0	0
$509$	22.3379	0.990111	0.495055	−	0.868861i	$-0.335148\pi$
$509$	22.3379	0.990111	0.495055	+	0.868861i	$0.335148\pi$
$510$	0	0
$511$	42.7704	1.89205
$512$	0	0
$513$	10.1843	0.449648
$514$	0	0
$515$	6.37907	0.281095
$516$	0	0
$517$	60.9341	2.67988
$518$	0	0
$519$	−9.31450	−0.408861
$520$	0	0
$521$	−36.6995	−1.60784	−0.803918	−	0.594741i	$-0.797255\pi$
$521$	−36.6995	−1.60784	−0.803918	+	0.594741i	$0.797255\pi$
$522$	0	0
$523$	−23.3543	−1.02121	−0.510607	−	0.859814i	$-0.670579\pi$
$523$	−23.3543	−1.02121	−0.510607	+	0.859814i	$0.670579\pi$
$524$	0	0
$525$	−3.30226	−0.144123
$526$	0	0
$527$	12.6168	0.549595
$528$	0	0
$529$	−22.4436	−0.975810
$530$	0	0
$531$	28.1117	1.21994
$532$	0	0
$533$	25.6126	1.10940
$534$	0	0
$535$	−8.88727	−0.384230
$536$	0	0
$537$	13.3863	0.577663
$538$	0	0
$539$	73.9750	3.18633
$540$	0	0
$541$	20.9578	0.901046	0.450523	−	0.892765i	$-0.351238\pi$
$541$	20.9578	0.901046	0.450523	+	0.892765i	$0.351238\pi$
$542$	0	0
$543$	−5.67822	−0.243676
$544$	0	0
$545$	−7.61259	−0.326088
$546$	0	0
$547$	−12.8216	−0.548214	−0.274107	−	0.961699i	$-0.588382\pi$
$547$	−12.8216	−0.548214	−0.274107	+	0.961699i	$0.588382\pi$
$548$	0	0
$549$	3.91069	0.166904
$550$	0	0
$551$	−2.50820	−0.106853
$552$	0	0
$553$	−17.6178	−0.749186
$554$	0	0
$555$	5.11273	0.217023
$556$	0	0
$557$	15.4712	0.655537	0.327768	−	0.944758i	$-0.393703\pi$
$557$	15.4712	0.655537	0.327768	+	0.944758i	$0.393703\pi$
$558$	0	0
$559$	17.7141	0.749228
$560$	0	0
$561$	−12.8545	−0.542716
$562$	0	0
$563$	29.6555	1.24983	0.624916	−	0.780692i	$-0.285133\pi$
$563$	29.6555	1.24983	0.624916	+	0.780692i	$0.285133\pi$
$564$	0	0
$565$	−18.7376	−0.788295
$566$	0	0
$567$	19.0471	0.799903
$568$	0	0
$569$	−30.2968	−1.27011	−0.635053	−	0.772468i	$-0.719022\pi$
$569$	−30.2968	−1.27011	−0.635053	+	0.772468i	$0.719022\pi$
$570$	0	0
$571$	9.72949	0.407167	0.203583	−	0.979058i	$-0.434741\pi$
$571$	9.72949	0.407167	0.203583	+	0.979058i	$0.434741\pi$
$572$	0	0
$573$	10.6894	0.446556
$574$	0	0
$575$	−0.745898	−0.0311061
$576$	0	0
$577$	37.8367	1.57516	0.787582	−	0.616210i	$-0.211333\pi$
$577$	37.8367	1.57516	0.787582	+	0.616210i	$0.211333\pi$
$578$	0	0
$579$	12.1054	0.503085
$580$	0	0
$581$	17.1372	0.710971
$582$	0	0
$583$	0.282746	0.0117102
$584$	0	0
$585$	−7.95184	−0.328768
$586$	0	0
$587$	−34.2416	−1.41330	−0.706651	−	0.707562i	$-0.749795\pi$
$587$	−34.2416	−1.41330	−0.706651	+	0.707562i	$0.749795\pi$
$588$	0	0
$589$	10.7805	0.444202
$590$	0	0
$591$	−6.00311	−0.246935
$592$	0	0
$593$	30.9753	1.27200	0.636001	−	0.771689i	$-0.280588\pi$
$593$	30.9753	1.27200	0.636001	+	0.771689i	$0.280588\pi$
$594$	0	0
$595$	12.9958	0.532777
$596$	0	0
$597$	9.58812	0.392416
$598$	0	0
$599$	15.7816	0.644817	0.322408	−	0.946601i	$-0.395508\pi$
$599$	15.7816	0.644817	0.322408	+	0.946601i	$0.395508\pi$
$600$	0	0
$601$	−20.5082	−0.836547	−0.418274	−	0.908321i	$-0.637365\pi$
$601$	−20.5082	−0.836547	−0.418274	+	0.908321i	$0.637365\pi$
$602$	0	0
$603$	35.1372	1.43090
$604$	0	0
$605$	23.4671	0.954071
$606$	0	0
$607$	−40.3296	−1.63693	−0.818464	−	0.574558i	$-0.805174\pi$
$607$	−40.3296	−1.63693	−0.818464	+	0.574558i	$0.805174\pi$
$608$	0	0
$609$	3.30226	0.133814
$610$	0	0
$611$	33.7745	1.36637
$612$	0	0
$613$	34.3226	1.38628	0.693138	−	0.720805i	$-0.256228\pi$
$613$	34.3226	1.38628	0.693138	+	0.720805i	$0.256228\pi$
$614$	0	0
$615$	−5.87086	−0.236736
$616$	0	0
$617$	19.8555	0.799353	0.399677	−	0.916656i	$-0.369122\pi$
$617$	19.8555	0.799353	0.399677	+	0.916656i	$0.369122\pi$
$618$	0	0
$619$	−22.6045	−0.908553	−0.454276	−	0.890861i	$-0.650102\pi$
$619$	−22.6045	−0.908553	−0.454276	+	0.890861i	$0.650102\pi$
$620$	0	0
$621$	−3.02864	−0.121535
$622$	0	0
$623$	3.39336	0.135952
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.9836	−0.438642
$628$	0	0
$629$	−20.1208	−0.802269
$630$	0	0
$631$	−2.41188	−0.0960155	−0.0480077	−	0.998847i	$-0.515287\pi$
$631$	−2.41188	−0.0960155	−0.0480077	+	0.998847i	$0.515287\pi$
$632$	0	0
$633$	18.1536	0.721541
$634$	0	0
$635$	3.65375	0.144994
$636$	0	0
$637$	41.0028	1.62459
$638$	0	0
$639$	−18.9180	−0.748383
$640$	0	0
$641$	−21.1455	−0.835199	−0.417599	−	0.908631i	$-0.637128\pi$
$641$	−21.1455	−0.835199	−0.417599	+	0.908631i	$0.637128\pi$
$642$	0	0
$643$	22.0245	0.868561	0.434280	−	0.900778i	$-0.357003\pi$
$643$	22.0245	0.868561	0.434280	+	0.900778i	$0.357003\pi$
$644$	0	0
$645$	−4.06040	−0.159878
$646$	0	0
$647$	26.0552	1.02433	0.512167	−	0.858886i	$-0.328843\pi$
$647$	26.0552	1.02433	0.512167	+	0.858886i	$0.328843\pi$
$648$	0	0
$649$	−67.5386	−2.65112
$650$	0	0
$651$	−14.1934	−0.556285
$652$	0	0
$653$	−42.3931	−1.65897	−0.829485	−	0.558529i	$-0.811366\pi$
$653$	−42.3931	−1.65897	−0.829485	+	0.558529i	$0.811366\pi$
$654$	0	0
$655$	−0.379068	−0.0148114
$656$	0	0
$657$	−23.6074	−0.921011
$658$	0	0
$659$	27.9037	1.08697	0.543486	−	0.839418i	$-0.317104\pi$
$659$	27.9037	1.08697	0.543486	+	0.839418i	$0.317104\pi$
$660$	0	0
$661$	9.02474	0.351022	0.175511	−	0.984477i	$-0.443842\pi$
$661$	9.02474	0.351022	0.175511	+	0.984477i	$0.443842\pi$
$662$	0	0
$663$	−7.12497	−0.276711
$664$	0	0
$665$	11.1044	0.430610
$666$	0	0
$667$	0.745898	0.0288813
$668$	0	0
$669$	10.2653	0.396879
$670$	0	0
$671$	−9.39547	−0.362708
$672$	0	0
$673$	1.83805	0.0708517	0.0354258	−	0.999372i	$-0.488721\pi$
$673$	1.83805	0.0708517	0.0354258	+	0.999372i	$0.488721\pi$
$674$	0	0
$675$	4.06040	0.156285
$676$	0	0
$677$	33.8625	1.30144	0.650721	−	0.759317i	$-0.274467\pi$
$677$	33.8625	1.30144	0.650721	+	0.759317i	$0.274467\pi$
$678$	0	0
$679$	−1.19760	−0.0459598
$680$	0	0
$681$	10.2499	0.392778
$682$	0	0
$683$	−31.3543	−1.19974	−0.599870	−	0.800098i	$-0.704781\pi$
$683$	−31.3543	−1.19974	−0.599870	+	0.800098i	$0.704781\pi$
$684$	0	0
$685$	−12.9630	−0.495291
$686$	0	0
$687$	11.4949	0.438558
$688$	0	0
$689$	0.156721	0.00597058
$690$	0	0
$691$	5.24887	0.199677	0.0998383	−	0.995004i	$-0.468167\pi$
$691$	5.24887	0.199677	0.0998383	+	0.995004i	$0.468167\pi$
$692$	0	0
$693$	−63.5142	−2.41270
$694$	0	0
$695$	−8.46004	−0.320908
$696$	0	0
$697$	23.1044	0.875141
$698$	0	0
$699$	9.49180	0.359013
$700$	0	0
$701$	−28.1843	−1.06451	−0.532253	−	0.846585i	$-0.678655\pi$
$701$	−28.1843	−1.06451	−0.532253	+	0.846585i	$0.678655\pi$
$702$	0	0
$703$	−17.1924	−0.648423
$704$	0	0
$705$	−7.74173	−0.291570
$706$	0	0
$707$	−52.4639	−1.97311
$708$	0	0
$709$	−17.4590	−0.655686	−0.327843	−	0.944732i	$-0.606322\pi$
$709$	−17.4590	−0.655686	−0.327843	+	0.944732i	$0.606322\pi$
$710$	0	0
$711$	9.72426	0.364688
$712$	0	0
$713$	−3.20594	−0.120063
$714$	0	0
$715$	19.1044	0.714464
$716$	0	0
$717$	13.2992	0.496666
$718$	0	0
$719$	−24.7170	−0.921788	−0.460894	−	0.887455i	$-0.652471\pi$
$719$	−24.7170	−0.921788	−0.460894	+	0.887455i	$0.652471\pi$
$720$	0	0
$721$	28.2416	1.05177
$722$	0	0
$723$	−12.5808	−0.467886
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−2.47539	−0.0918071	−0.0459036	−	0.998946i	$-0.514617\pi$
$727$	−2.47539	−0.0918071	−0.0459036	+	0.998946i	$0.514617\pi$
$728$	0	0
$729$	−1.42723	−0.0528603
$730$	0	0
$731$	15.9794	0.591020
$732$	0	0
$733$	43.8953	1.62131	0.810656	−	0.585523i	$-0.199111\pi$
$733$	43.8953	1.62131	0.810656	+	0.585523i	$0.199111\pi$
$734$	0	0
$735$	−9.39858	−0.346672
$736$	0	0
$737$	−84.4176	−3.10956
$738$	0	0
$739$	7.24186	0.266396	0.133198	−	0.991089i	$-0.457475\pi$
$739$	7.24186	0.266396	0.133198	+	0.991089i	$0.457475\pi$
$740$	0	0
$741$	−6.08798	−0.223648
$742$	0	0
$743$	14.8956	0.546467	0.273233	−	0.961948i	$-0.411907\pi$
$743$	14.8956	0.546467	0.273233	+	0.961948i	$0.411907\pi$
$744$	0	0
$745$	20.2499	0.741900
$746$	0	0
$747$	−9.45898	−0.346086
$748$	0	0
$749$	−39.3460	−1.43767
$750$	0	0
$751$	23.4178	0.854529	0.427264	−	0.904127i	$-0.359477\pi$
$751$	23.4178	0.854529	0.427264	+	0.904127i	$0.359477\pi$
$752$	0	0
$753$	−20.8545	−0.759979
$754$	0	0
$755$	−1.49180	−0.0542920
$756$	0	0
$757$	−45.9037	−1.66840	−0.834199	−	0.551464i	$-0.814069\pi$
$757$	−45.9037	−1.66840	−0.834199	+	0.551464i	$0.814069\pi$
$758$	0	0
$759$	3.26634	0.118561
$760$	0	0
$761$	25.0234	0.907098	0.453549	−	0.891231i	$-0.350158\pi$
$761$	25.0234	0.907098	0.453549	+	0.891231i	$0.350158\pi$
$762$	0	0
$763$	−33.7027	−1.22012
$764$	0	0
$765$	−7.17313	−0.259345
$766$	0	0
$767$	−37.4353	−1.35171
$768$	0	0
$769$	5.26634	0.189909	0.0949545	−	0.995482i	$-0.469729\pi$
$769$	5.26634	0.189909	0.0949545	+	0.995482i	$0.469729\pi$
$770$	0	0
$771$	9.37100	0.337488
$772$	0	0
$773$	40.2744	1.44857	0.724285	−	0.689501i	$-0.242170\pi$
$773$	40.2744	1.44857	0.724285	+	0.689501i	$0.242170\pi$
$774$	0	0
$775$	4.29809	0.154392
$776$	0	0
$777$	22.6352	0.812034
$778$	0	0
$779$	19.7417	0.707321
$780$	0	0
$781$	45.4506	1.62635
$782$	0	0
$783$	−4.06040	−0.145107
$784$	0	0
$785$	8.98359	0.320638
$786$	0	0
$787$	19.0961	0.680701	0.340350	−	0.940299i	$-0.389454\pi$
$787$	19.0961	0.680701	0.340350	+	0.940299i	$0.389454\pi$
$788$	0	0
$789$	9.58190	0.341125
$790$	0	0
$791$	−82.9555	−2.94956
$792$	0	0
$793$	−5.20772	−0.184932
$794$	0	0
$795$	−0.0359231	−0.00127406
$796$	0	0
$797$	20.6045	0.729850	0.364925	−	0.931037i	$-0.381095\pi$
$797$	20.6045	0.729850	0.364925	+	0.931037i	$0.381095\pi$
$798$	0	0
$799$	30.4671	1.07785
$800$	0	0
$801$	−1.87298	−0.0661786
$802$	0	0
$803$	56.7170	2.00150
$804$	0	0
$805$	−3.30226	−0.116389
$806$	0	0
$807$	−9.95574	−0.350459
$808$	0	0
$809$	6.05517	0.212888	0.106444	−	0.994319i	$-0.466053\pi$
$809$	6.05517	0.212888	0.106444	+	0.994319i	$0.466053\pi$
$810$	0	0
$811$	−0.0398229	−0.00139837	−0.000699186	−	1.00000i	$-0.500223\pi$
$811$	−0.0398229	−0.00139837	−0.000699186		1.00000i	$0.500223\pi$
$812$	0	0
$813$	−12.6925	−0.445146
$814$	0	0
$815$	−13.3627	−0.468074
$816$	0	0
$817$	13.6537	0.477684
$818$	0	0
$819$	−35.2046	−1.23015
$820$	0	0
$821$	−8.57383	−0.299229	−0.149614	−	0.988744i	$-0.547803\pi$
$821$	−8.57383	−0.299229	−0.149614	+	0.988744i	$0.547803\pi$
$822$	0	0
$823$	−25.7417	−0.897300	−0.448650	−	0.893707i	$-0.648095\pi$
$823$	−25.7417	−0.897300	−0.448650	+	0.893707i	$0.648095\pi$
$824$	0	0
$825$	−4.37907	−0.152460
$826$	0	0
$827$	29.9742	1.04230	0.521152	−	0.853464i	$-0.325502\pi$
$827$	29.9742	1.04230	0.521152	+	0.853464i	$0.325502\pi$
$828$	0	0
$829$	−27.6524	−0.960408	−0.480204	−	0.877157i	$-0.659437\pi$
$829$	−27.6524	−0.960408	−0.480204	+	0.877157i	$0.659437\pi$
$830$	0	0
$831$	9.54075	0.330965
$832$	0	0
$833$	36.9875	1.28154
$834$	0	0
$835$	−19.0562	−0.659468
$836$	0	0
$837$	17.4520	0.603228
$838$	0	0
$839$	−14.2911	−0.493383	−0.246692	−	0.969094i	$-0.579343\pi$
$839$	−14.2911	−0.493383	−0.246692	+	0.969094i	$0.579343\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−18.9784	−0.653650
$844$	0	0
$845$	−2.41082	−0.0829348
$846$	0	0
$847$	103.894	3.56984
$848$	0	0
$849$	0.854458	0.0293249
$850$	0	0
$851$	5.11273	0.175262
$852$	0	0
$853$	−15.1784	−0.519697	−0.259848	−	0.965649i	$-0.583673\pi$
$853$	−15.1784	−0.519697	−0.259848	+	0.965649i	$0.583673\pi$
$854$	0	0
$855$	−6.12914	−0.209612
$856$	0	0
$857$	28.5082	0.973822	0.486911	−	0.873452i	$-0.338124\pi$
$857$	28.5082	0.973822	0.486911	+	0.873452i	$0.338124\pi$
$858$	0	0
$859$	2.70085	0.0921517	0.0460759	−	0.998938i	$-0.485328\pi$
$859$	2.70085	0.0921517	0.0460759	+	0.998938i	$0.485328\pi$
$860$	0	0
$861$	−25.9917	−0.885793
$862$	0	0
$863$	39.9711	1.36063	0.680316	−	0.732919i	$-0.261843\pi$
$863$	39.9711	1.36063	0.680316	+	0.732919i	$0.261843\pi$
$864$	0	0
$865$	12.4876	0.424592
$866$	0	0
$867$	6.25304	0.212364
$868$	0	0
$869$	−23.3627	−0.792524
$870$	0	0
$871$	−46.7909	−1.58545
$872$	0	0
$873$	0.661024	0.0223723
$874$	0	0
$875$	4.42723	0.149668
$876$	0	0
$877$	−17.9138	−0.604906	−0.302453	−	0.953164i	$-0.597806\pi$
$877$	−17.9138	−0.604906	−0.302453	+	0.953164i	$0.597806\pi$
$878$	0	0
$879$	7.90368	0.266584
$880$	0	0
$881$	23.6782	0.797740	0.398870	−	0.917008i	$-0.369403\pi$
$881$	23.6782	0.797740	0.398870	+	0.917008i	$0.369403\pi$
$882$	0	0
$883$	32.8873	1.10674	0.553372	−	0.832934i	$-0.313341\pi$
$883$	32.8873	1.10674	0.553372	+	0.832934i	$0.313341\pi$
$884$	0	0
$885$	8.58084	0.288442
$886$	0	0
$887$	27.8953	0.936634	0.468317	−	0.883561i	$-0.344861\pi$
$887$	27.8953	0.936634	0.468317	+	0.883561i	$0.344861\pi$
$888$	0	0
$889$	16.1760	0.542525
$890$	0	0
$891$	25.2580	0.846175
$892$	0	0
$893$	26.0328	0.871155
$894$	0	0
$895$	−17.9466	−0.599889
$896$	0	0
$897$	1.81047	0.0604497
$898$	0	0
$899$	−4.29809	−0.143349
$900$	0	0
$901$	0.141373	0.00470982
$902$	0	0
$903$	−17.9763	−0.598214
$904$	0	0
$905$	7.61259	0.253051
$906$	0	0
$907$	−25.5093	−0.847021	−0.423511	−	0.905891i	$-0.639202\pi$
$907$	−25.5093	−0.847021	−0.423511	+	0.905891i	$0.639202\pi$
$908$	0	0
$909$	28.9578	0.960469
$910$	0	0
$911$	−17.2953	−0.573017	−0.286509	−	0.958078i	$-0.592495\pi$
$911$	−17.2953	−0.573017	−0.286509	+	0.958078i	$0.592495\pi$
$912$	0	0
$913$	22.7253	0.752098
$914$	0	0
$915$	1.19370	0.0394626
$916$	0	0
$917$	−1.67822	−0.0554197
$918$	0	0
$919$	34.9836	1.15400	0.577001	−	0.816743i	$-0.304223\pi$
$919$	34.9836	1.15400	0.577001	+	0.816743i	$0.304223\pi$
$920$	0	0
$921$	22.9836	0.757335
$922$	0	0
$923$	25.1924	0.829217
$924$	0	0
$925$	−6.85446	−0.225373
$926$	0	0
$927$	−15.5881	−0.511981
$928$	0	0
$929$	39.4905	1.29564	0.647820	−	0.761793i	$-0.275681\pi$
$929$	39.4905	1.29564	0.647820	+	0.761793i	$0.275681\pi$
$930$	0	0
$931$	31.6043	1.03579
$932$	0	0
$933$	−10.4028	−0.340571
$934$	0	0
$935$	17.2335	0.563597
$936$	0	0
$937$	−37.8381	−1.23612	−0.618058	−	0.786133i	$-0.712080\pi$
$937$	−37.8381	−1.23612	−0.618058	+	0.786133i	$0.712080\pi$
$938$	0	0
$939$	−15.6516	−0.510772
$940$	0	0
$941$	23.8953	0.778966	0.389483	−	0.921034i	$-0.372654\pi$
$941$	23.8953	0.778966	0.389483	+	0.921034i	$0.372654\pi$
$942$	0	0
$943$	−5.87086	−0.191182
$944$	0	0
$945$	17.9763	0.584770
$946$	0	0
$947$	9.22129	0.299652	0.149826	−	0.988712i	$-0.452129\pi$
$947$	9.22129	0.299652	0.149826	+	0.988712i	$0.452129\pi$
$948$	0	0
$949$	31.4371	1.02049
$950$	0	0
$951$	−9.52249	−0.308788
$952$	0	0
$953$	37.2252	1.20584	0.602921	−	0.797801i	$-0.294003\pi$
$953$	37.2252	1.20584	0.602921	+	0.797801i	$0.294003\pi$
$954$	0	0
$955$	−14.3309	−0.463737
$956$	0	0
$957$	4.37907	0.141555
$958$	0	0
$959$	−57.3902	−1.85323
$960$	0	0
$961$	−12.5264	−0.404077
$962$	0	0
$963$	21.7173	0.699829
$964$	0	0
$965$	−16.2294	−0.522441
$966$	0	0
$967$	−6.97526	−0.224309	−0.112155	−	0.993691i	$-0.535775\pi$
$967$	−6.97526	−0.224309	−0.112155	+	0.993691i	$0.535775\pi$
$968$	0	0
$969$	−5.49180	−0.176422
$970$	0	0
$971$	−23.8932	−0.766770	−0.383385	−	0.923589i	$-0.625242\pi$
$971$	−23.8932	−0.766770	−0.383385	+	0.923589i	$0.625242\pi$
$972$	0	0
$973$	−37.4545	−1.20074
$974$	0	0
$975$	−2.42723	−0.0777335
$976$	0	0
$977$	27.9917	0.895533	0.447766	−	0.894151i	$-0.352220\pi$
$977$	27.9917	0.895533	0.447766	+	0.894151i	$0.352220\pi$
$978$	0	0
$979$	4.49987	0.143816
$980$	0	0
$981$	18.6024	0.593929
$982$	0	0
$983$	26.6758	0.850827	0.425413	−	0.904999i	$-0.360129\pi$
$983$	26.6758	0.850827	0.425413	+	0.904999i	$0.360129\pi$
$984$	0	0
$985$	8.04816	0.256436
$986$	0	0
$987$	−34.2744	−1.09097
$988$	0	0
$989$	−4.06040	−0.129113
$990$	0	0
$991$	−28.5962	−0.908388	−0.454194	−	0.890903i	$-0.650073\pi$
$991$	−28.5962	−0.908388	−0.454194	+	0.890903i	$0.650073\pi$
$992$	0	0
$993$	−5.89534	−0.187083
$994$	0	0
$995$	−12.8545	−0.407514
$996$	0	0
$997$	0.508203	0.0160950	0.00804748	−	0.999968i	$-0.497438\pi$
$997$	0.508203	0.0160950	0.00804748	+	0.999968i	$0.497438\pi$
$998$	0	0
$999$	−27.8318	−0.880560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.bg.1.2		3
4.3	odd	2		9280.2.a.bx.1.2		3
8.3	odd	2		4640.2.a.j.1.2	✓	3
8.5	even	2		4640.2.a.k.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.j.1.2	✓	3	8.3	odd	2
4640.2.a.k.1.2	yes	3	8.5	even	2
9280.2.a.bg.1.2		3	1.1	even	1	trivial
9280.2.a.bx.1.2		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-0.745898 q^{3} +1.00000 q^{5} +4.42723 q^{7} -2.44364 q^{9} +5.87086 q^{11} +3.25410 q^{13} -0.745898 q^{15} +2.93543 q^{17} +2.50820 q^{19} -3.30226 q^{21} -0.745898 q^{23} +1.00000 q^{25} +4.06040 q^{27} -1.00000 q^{29} +4.29809 q^{31} -4.37907 q^{33} +4.42723 q^{35} -6.85446 q^{37} -2.42723 q^{39} +7.87086 q^{41} +5.44364 q^{43} -2.44364 q^{45} +10.3791 q^{47} +12.6004 q^{49} -2.18953 q^{51} +0.0481609 q^{53} +5.87086 q^{55} -1.87086 q^{57} -11.5040 q^{59} -1.60036 q^{61} -10.8185 q^{63} +3.25410 q^{65} -14.3791 q^{67} +0.556364 q^{69} +7.74173 q^{71} +9.66075 q^{73} -0.745898 q^{75} +25.9917 q^{77} -3.97942 q^{79} +4.30226 q^{81} +3.87086 q^{83} +2.93543 q^{85} +0.745898 q^{87} +0.766474 q^{89} +14.4067 q^{91} -3.20594 q^{93} +2.50820 q^{95} -0.270508 q^{97} -14.3463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 7 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} - 3 q^{15} + q^{17} + 6 q^{19} - 20 q^{21} - 3 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} - 9 q^{31} + 4 q^{33} + 7 q^{35} - 8 q^{37} - q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 7 * q^7 + 2 * q^9 + 2 * q^11 + 9 * q^13 - 3 * q^15 + q^17 + 6 * q^19 - 20 * q^21 - 3 * q^23 + 3 * q^25 - 12 * q^27 - 3 * q^29 - 9 * q^31 + 4 * q^33 + 7 * q^35 - 8 * q^37 - q^39 + 8 * q^41 + 7 * q^43 + 2 * q^45 + 14 * q^47 + 26 * q^49 + 2 * q^51 + 11 * q^53 + 2 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + 26 * q^63 + 9 * q^65 - 26 * q^67 + 11 * q^69 - 8 * q^71 - 7 * q^73 - 3 * q^75 + 14 * q^77 + 17 * q^79 + 23 * q^81 - 4 * q^83 + q^85 + 3 * q^87 + 32 * q^89 + 8 * q^91 + 2 * q^93 + 6 * q^95 + 3 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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