LMFDB - Embedded newform 9800.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.239575536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 8x^{3} + 35x^{2} - 15x - 8 $ x^6 - x^5 - 12x^4 + 8x^3 + 35x^2 - 15x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.10821$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10821 q^{3} +1.44457 q^{9} +O(q^{10})$	$q-2.10821 q^{3} +1.44457 q^{9} +1.77465 q^{11} -1.44457 q^{13} -5.86806 q^{17} -7.16205 q^{19} +0.663645 q^{23} +3.27918 q^{27} +6.45763 q^{29} +10.0732 q^{31} -3.74135 q^{33} +5.01705 q^{37} +3.04546 q^{39} +1.92363 q^{41} -5.81995 q^{43} -3.70322 q^{47} +12.3711 q^{51} +0.592363 q^{53} +15.0991 q^{57} +7.59076 q^{59} -8.72111 q^{61} +2.16832 q^{67} -1.39911 q^{69} +6.49826 q^{71} +15.5320 q^{73} -2.70720 q^{79} -11.2469 q^{81} -6.35340 q^{83} -13.6141 q^{87} -5.86157 q^{89} -21.2364 q^{93} -14.7748 q^{97} +2.56361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 7 q^{9}+O(q^{10}) $ 6 * q - q^3 + 7 * q^9	$ 6 q - q^{3} + 7 q^{9} + q^{11} - 7 q^{13} - 4 q^{17} + 5 q^{19} - 6 q^{23} - 7 q^{27} - 3 q^{29} + 2 q^{31} - 16 q^{33} - 9 q^{37} + 10 q^{39} - 6 q^{41} + 3 q^{43} - 27 q^{47} + 5 q^{53} + 26 q^{57} + 24 q^{59} - 9 q^{61} - 17 q^{67} - 15 q^{69} + 4 q^{71} - 18 q^{73} - 22 q^{79} - 6 q^{81} - 9 q^{83} - 39 q^{87} - 15 q^{89} - 10 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100}) $ 6 * q - q^3 + 7 * q^9 + q^11 - 7 * q^13 - 4 * q^17 + 5 * q^19 - 6 * q^23 - 7 * q^27 - 3 * q^29 + 2 * q^31 - 16 * q^33 - 9 * q^37 + 10 * q^39 - 6 * q^41 + 3 * q^43 - 27 * q^47 + 5 * q^53 + 26 * q^57 + 24 * q^59 - 9 * q^61 - 17 * q^67 - 15 * q^69 + 4 * q^71 - 18 * q^73 - 22 * q^79 - 6 * q^81 - 9 * q^83 - 39 * q^87 - 15 * q^89 - 10 * q^93 - 12 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.10821	−1.21718	−0.608589	−	0.793485i	$-0.708264\pi$
$3$	−2.10821	−1.21718	−0.608589	+	0.793485i	$0.708264\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.44457	0.481523
$10$	0	0
$11$	1.77465	0.535078	0.267539	−	0.963547i	$-0.413790\pi$
$11$	1.77465	0.535078	0.267539	+	0.963547i	$0.413790\pi$
$12$	0	0
$13$	−1.44457	−0.400652	−0.200326	−	0.979729i	$-0.564200\pi$
$13$	−1.44457	−0.400652	−0.200326	+	0.979729i	$0.564200\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.86806	−1.42321	−0.711607	−	0.702578i	$-0.752032\pi$
$17$	−5.86806	−1.42321	−0.711607	+	0.702578i	$0.752032\pi$
$18$	0	0
$19$	−7.16205	−1.64309	−0.821543	−	0.570146i	$-0.806886\pi$
$19$	−7.16205	−1.64309	−0.821543	+	0.570146i	$0.806886\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.663645	0.138380	0.0691898	−	0.997604i	$-0.477959\pi$
$23$	0.663645	0.138380	0.0691898	+	0.997604i	$0.477959\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.27918	0.631079
$28$	0	0
$29$	6.45763	1.19915	0.599576	−	0.800318i	$-0.295336\pi$
$29$	6.45763	1.19915	0.599576	+	0.800318i	$0.295336\pi$
$30$	0	0
$31$	10.0732	1.80919	0.904597	−	0.426267i	$-0.140172\pi$
$31$	10.0732	1.80919	0.904597	+	0.426267i	$0.140172\pi$
$32$	0	0
$33$	−3.74135	−0.651285
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.01705	0.824797	0.412399	−	0.911004i	$-0.364691\pi$
$37$	5.01705	0.824797	0.412399	+	0.911004i	$0.364691\pi$
$38$	0	0
$39$	3.04546	0.487664
$40$	0	0
$41$	1.92363	0.300421	0.150211	−	0.988654i	$-0.452005\pi$
$41$	1.92363	0.300421	0.150211	+	0.988654i	$0.452005\pi$
$42$	0	0
$43$	−5.81995	−0.887535	−0.443767	−	0.896142i	$-0.646358\pi$
$43$	−5.81995	−0.887535	−0.443767	+	0.896142i	$0.646358\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.70322	−0.540171	−0.270085	−	0.962836i	$-0.587052\pi$
$47$	−3.70322	−0.540171	−0.270085	+	0.962836i	$0.587052\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.3711	1.73231
$52$	0	0
$53$	0.592363	0.0813674	0.0406837	−	0.999172i	$-0.487046\pi$
$53$	0.592363	0.0813674	0.0406837	+	0.999172i	$0.487046\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.0991	1.99993
$58$	0	0
$59$	7.59076	0.988233	0.494116	−	0.869396i	$-0.335492\pi$
$59$	7.59076	0.988233	0.494116	+	0.869396i	$0.335492\pi$
$60$	0	0
$61$	−8.72111	−1.11662	−0.558312	−	0.829631i	$-0.688551\pi$
$61$	−8.72111	−1.11662	−0.558312	+	0.829631i	$0.688551\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.16832	0.264903	0.132451	−	0.991190i	$-0.457715\pi$
$67$	2.16832	0.264903	0.132451	+	0.991190i	$0.457715\pi$
$68$	0	0
$69$	−1.39911	−0.168433
$70$	0	0
$71$	6.49826	0.771201	0.385601	−	0.922666i	$-0.373994\pi$
$71$	6.49826	0.771201	0.385601	+	0.922666i	$0.373994\pi$
$72$	0	0
$73$	15.5320	1.81788	0.908941	−	0.416924i	$-0.136892\pi$
$73$	15.5320	1.81788	0.908941	+	0.416924i	$0.136892\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.70720	−0.304584	−0.152292	−	0.988336i	$-0.548665\pi$
$79$	−2.70720	−0.304584	−0.152292	+	0.988336i	$0.548665\pi$
$80$	0	0
$81$	−11.2469	−1.24966
$82$	0	0
$83$	−6.35340	−0.697376	−0.348688	−	0.937239i	$-0.613373\pi$
$83$	−6.35340	−0.697376	−0.348688	+	0.937239i	$0.613373\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−13.6141	−1.45958
$88$	0	0
$89$	−5.86157	−0.621326	−0.310663	−	0.950520i	$-0.600551\pi$
$89$	−5.86157	−0.621326	−0.310663	+	0.950520i	$0.600551\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−21.2364	−2.20211
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.7748	−1.50015	−0.750076	−	0.661351i	$-0.769983\pi$
$97$	−14.7748	−1.50015	−0.750076	+	0.661351i	$0.769983\pi$
$98$	0	0
$99$	2.56361	0.257652
$100$	0	0
$101$	−3.48186	−0.346458	−0.173229	−	0.984882i	$-0.555420\pi$
$101$	−3.48186	−0.346458	−0.173229	+	0.984882i	$0.555420\pi$
$102$	0	0
$103$	−8.07861	−0.796009	−0.398004	−	0.917384i	$-0.630297\pi$
$103$	−8.07861	−0.796009	−0.398004	+	0.917384i	$0.630297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.129645	−0.0125333	−0.00626664	−	0.999980i	$-0.501995\pi$
$107$	−0.129645	−0.0125333	−0.00626664	+	0.999980i	$0.501995\pi$
$108$	0	0
$109$	12.2648	1.17476	0.587379	−	0.809312i	$-0.300160\pi$
$109$	12.2648	1.17476	0.587379	+	0.809312i	$0.300160\pi$
$110$	0	0
$111$	−10.5770	−1.00393
$112$	0	0
$113$	−11.7467	−1.10503	−0.552517	−	0.833501i	$-0.686333\pi$
$113$	−11.7467	−1.10503	−0.552517	+	0.833501i	$0.686333\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.08678	−0.192923
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.85061	−0.713692
$122$	0	0
$123$	−4.05543	−0.365666
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.6983	1.12679	0.563395	−	0.826187i	$-0.309495\pi$
$127$	12.6983	1.12679	0.563395	+	0.826187i	$0.309495\pi$
$128$	0	0
$129$	12.2697	1.08029
$130$	0	0
$131$	0.230932	0.0201766	0.0100883	−	0.999949i	$-0.496789\pi$
$131$	0.230932	0.0201766	0.0100883	+	0.999949i	$0.496789\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.8117	−1.43632	−0.718161	−	0.695877i	$-0.755016\pi$
$137$	−16.8117	−1.43632	−0.718161	+	0.695877i	$0.755016\pi$
$138$	0	0
$139$	20.5531	1.74329	0.871644	−	0.490139i	$-0.163054\pi$
$139$	20.5531	1.74329	0.871644	+	0.490139i	$0.163054\pi$
$140$	0	0
$141$	7.80719	0.657484
$142$	0	0
$143$	−2.56361	−0.214380
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.2725	1.74271	0.871356	−	0.490652i	$-0.163241\pi$
$149$	21.2725	1.74271	0.871356	+	0.490652i	$0.163241\pi$
$150$	0	0
$151$	6.47990	0.527327	0.263663	−	0.964615i	$-0.415069\pi$
$151$	6.47990	0.527327	0.263663	+	0.964615i	$0.415069\pi$
$152$	0	0
$153$	−8.47683	−0.685311
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.44990	0.434950	0.217475	−	0.976066i	$-0.430218\pi$
$157$	5.44990	0.434950	0.217475	+	0.976066i	$0.430218\pi$
$158$	0	0
$159$	−1.24883	−0.0990386
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.26921	0.491042	0.245521	−	0.969391i	$-0.421041\pi$
$163$	6.26921	0.491042	0.245521	+	0.969391i	$0.421041\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.5728	1.43720	0.718602	−	0.695422i	$-0.244782\pi$
$167$	18.5728	1.43720	0.718602	+	0.695422i	$0.244782\pi$
$168$	0	0
$169$	−10.9132	−0.839478
$170$	0	0
$171$	−10.3461	−0.791184
$172$	0	0
$173$	−13.9672	−1.06191	−0.530954	−	0.847401i	$-0.678166\pi$
$173$	−13.9672	−1.06191	−0.530954	+	0.847401i	$0.678166\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.0030	−1.20286
$178$	0	0
$179$	7.92539	0.592372	0.296186	−	0.955130i	$-0.404285\pi$
$179$	7.92539	0.592372	0.296186	+	0.955130i	$0.404285\pi$
$180$	0	0
$181$	11.6887	0.868815	0.434407	−	0.900717i	$-0.356958\pi$
$181$	11.6887	0.868815	0.434407	+	0.900717i	$0.356958\pi$
$182$	0	0
$183$	18.3860	1.35913
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−10.4138	−0.761530
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.9782	−0.939066	−0.469533	−	0.882915i	$-0.655578\pi$
$191$	−12.9782	−0.939066	−0.469533	+	0.882915i	$0.655578\pi$
$192$	0	0
$193$	−21.4053	−1.54079	−0.770393	−	0.637570i	$-0.779940\pi$
$193$	−21.4053	−1.54079	−0.770393	+	0.637570i	$0.779940\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.42871	0.600521	0.300261	−	0.953857i	$-0.402926\pi$
$197$	8.42871	0.600521	0.300261	+	0.953857i	$0.402926\pi$
$198$	0	0
$199$	−4.21481	−0.298780	−0.149390	−	0.988778i	$-0.547731\pi$
$199$	−4.21481	−0.298780	−0.149390	+	0.988778i	$0.547731\pi$
$200$	0	0
$201$	−4.57129	−0.322434
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.958682	0.0666330
$208$	0	0
$209$	−12.7101	−0.879179
$210$	0	0
$211$	−0.629004	−0.0433025	−0.0216512	−	0.999766i	$-0.506892\pi$
$211$	−0.629004	−0.0433025	−0.0216512	+	0.999766i	$0.506892\pi$
$212$	0	0
$213$	−13.6997	−0.938689
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−32.7448	−2.21269
$220$	0	0
$221$	8.47683	0.570213
$222$	0	0
$223$	21.4138	1.43398	0.716988	−	0.697086i	$-0.245520\pi$
$223$	21.4138	1.43398	0.716988	+	0.697086i	$0.245520\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.0493	−1.46346	−0.731732	−	0.681593i	$-0.761288\pi$
$227$	−22.0493	−1.46346	−0.731732	+	0.681593i	$0.761288\pi$
$228$	0	0
$229$	−11.6359	−0.768925	−0.384462	−	0.923141i	$-0.625613\pi$
$229$	−11.6359	−0.768925	−0.384462	+	0.923141i	$0.625613\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.2193	0.735004	0.367502	−	0.930023i	$-0.380213\pi$
$233$	11.2193	0.735004	0.367502	+	0.930023i	$0.380213\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.70737	0.370733
$238$	0	0
$239$	22.8279	1.47662	0.738308	−	0.674464i	$-0.235625\pi$
$239$	22.8279	1.47662	0.738308	+	0.674464i	$0.235625\pi$
$240$	0	0
$241$	−4.20270	−0.270720	−0.135360	−	0.990796i	$-0.543219\pi$
$241$	−4.20270	−0.270720	−0.135360	+	0.990796i	$0.543219\pi$
$242$	0	0
$243$	13.8734	0.889979
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.3461	0.658305
$248$	0	0
$249$	13.3943	0.848831
$250$	0	0
$251$	16.9587	1.07042	0.535211	−	0.844718i	$-0.320232\pi$
$251$	16.9587	1.07042	0.535211	+	0.844718i	$0.320232\pi$
$252$	0	0
$253$	1.17774	0.0740438
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.9303	−1.67986	−0.839932	−	0.542692i	$-0.817405\pi$
$257$	−26.9303	−1.67986	−0.839932	+	0.542692i	$0.817405\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.32850	0.577420
$262$	0	0
$263$	21.9520	1.35362	0.676809	−	0.736158i	$-0.263362\pi$
$263$	21.9520	1.35362	0.676809	+	0.736158i	$0.263362\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.3575	0.756264
$268$	0	0
$269$	−24.0854	−1.46851	−0.734256	−	0.678872i	$-0.762469\pi$
$269$	−24.0854	−1.46851	−0.734256	+	0.678872i	$0.762469\pi$
$270$	0	0
$271$	−2.06731	−0.125580	−0.0627900	−	0.998027i	$-0.520000\pi$
$271$	−2.06731	−0.125580	−0.0627900	+	0.998027i	$0.520000\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.0757	1.02598	0.512990	−	0.858395i	$-0.328538\pi$
$277$	17.0757	1.02598	0.512990	+	0.858395i	$0.328538\pi$
$278$	0	0
$279$	14.5514	0.871169
$280$	0	0
$281$	−13.2035	−0.787655	−0.393828	−	0.919184i	$-0.628849\pi$
$281$	−13.2035	−0.787655	−0.393828	+	0.919184i	$0.628849\pi$
$282$	0	0
$283$	−26.5989	−1.58114	−0.790570	−	0.612372i	$-0.790216\pi$
$283$	−26.5989	−1.58114	−0.790570	+	0.612372i	$0.790216\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.4342	1.02554
$290$	0	0
$291$	31.1484	1.82595
$292$	0	0
$293$	0.472681	0.0276143	0.0138071	−	0.999905i	$-0.495605\pi$
$293$	0.472681	0.0276143	0.0138071	+	0.999905i	$0.495605\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.81940	0.337676
$298$	0	0
$299$	−0.958682	−0.0554420
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.34050	0.421701
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.2073	−0.867929	−0.433965	−	0.900930i	$-0.642886\pi$
$307$	−15.2073	−0.867929	−0.433965	+	0.900930i	$0.642886\pi$
$308$	0	0
$309$	17.0314	0.968885
$310$	0	0
$311$	16.5806	0.940200	0.470100	−	0.882613i	$-0.344218\pi$
$311$	16.5806	0.940200	0.470100	+	0.882613i	$0.344218\pi$
$312$	0	0
$313$	1.17950	0.0666695	0.0333348	−	0.999444i	$-0.489387\pi$
$313$	1.17950	0.0666695	0.0333348	+	0.999444i	$0.489387\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.4288	−1.54056	−0.770278	−	0.637709i	$-0.779882\pi$
$317$	−27.4288	−1.54056	−0.770278	+	0.637709i	$0.779882\pi$
$318$	0	0
$319$	11.4600	0.641640
$320$	0	0
$321$	0.273320	0.0152552
$322$	0	0
$323$	42.0273	2.33846
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−25.8569	−1.42989
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.21160	0.286455	0.143228	−	0.989690i	$-0.454252\pi$
$331$	5.21160	0.286455	0.143228	+	0.989690i	$0.454252\pi$
$332$	0	0
$333$	7.24747	0.397159
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.52496	−0.518858	−0.259429	−	0.965762i	$-0.583534\pi$
$337$	−9.52496	−0.518858	−0.259429	+	0.965762i	$0.583534\pi$
$338$	0	0
$339$	24.7645	1.34502
$340$	0	0
$341$	17.8764	0.968059
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.2781	−1.46436	−0.732182	−	0.681109i	$-0.761498\pi$
$347$	−27.2781	−1.46436	−0.732182	+	0.681109i	$0.761498\pi$
$348$	0	0
$349$	−6.78145	−0.363003	−0.181501	−	0.983391i	$-0.558096\pi$
$349$	−6.78145	−0.363003	−0.181501	+	0.983391i	$0.558096\pi$
$350$	0	0
$351$	−4.73701	−0.252843
$352$	0	0
$353$	15.6791	0.834517	0.417258	−	0.908788i	$-0.362991\pi$
$353$	15.6791	0.834517	0.417258	+	0.908788i	$0.362991\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.33467	−0.334331	−0.167165	−	0.985929i	$-0.553461\pi$
$359$	−6.33467	−0.334331	−0.167165	+	0.985929i	$0.553461\pi$
$360$	0	0
$361$	32.2949	1.69973
$362$	0	0
$363$	16.5508	0.868691
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.7599	−0.666059	−0.333030	−	0.942916i	$-0.608071\pi$
$367$	−12.7599	−0.666059	−0.333030	+	0.942916i	$0.608071\pi$
$368$	0	0
$369$	2.77882	0.144660
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.66915	−0.0864255	−0.0432127	−	0.999066i	$-0.513759\pi$
$373$	−1.66915	−0.0864255	−0.0432127	+	0.999066i	$0.513759\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.32850	−0.480442
$378$	0	0
$379$	−21.5557	−1.10724	−0.553621	−	0.832768i	$-0.686755\pi$
$379$	−21.5557	−1.10724	−0.553621	+	0.832768i	$0.686755\pi$
$380$	0	0
$381$	−26.7707	−1.37151
$382$	0	0
$383$	5.04425	0.257749	0.128875	−	0.991661i	$-0.458864\pi$
$383$	5.04425	0.257749	0.128875	+	0.991661i	$0.458864\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.40733	−0.427369
$388$	0	0
$389$	−12.5195	−0.634764	−0.317382	−	0.948298i	$-0.602804\pi$
$389$	−12.5195	−0.634764	−0.317382	+	0.948298i	$0.602804\pi$
$390$	0	0
$391$	−3.89431	−0.196944
$392$	0	0
$393$	−0.486853	−0.0245585
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.3935	−1.12390	−0.561950	−	0.827171i	$-0.689949\pi$
$397$	−22.3935	−1.12390	−0.561950	+	0.827171i	$0.689949\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.3931	−0.768694	−0.384347	−	0.923189i	$-0.625573\pi$
$401$	−15.3931	−0.768694	−0.384347	+	0.923189i	$0.625573\pi$
$402$	0	0
$403$	−14.5514	−0.724857
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.90351	0.441330
$408$	0	0
$409$	−30.9701	−1.53137	−0.765687	−	0.643213i	$-0.777601\pi$
$409$	−30.9701	−1.53137	−0.765687	+	0.643213i	$0.777601\pi$
$410$	0	0
$411$	35.4427	1.74826
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−43.3303	−2.12189
$418$	0	0
$419$	−0.434578	−0.0212305	−0.0106153	−	0.999944i	$-0.503379\pi$
$419$	−0.434578	−0.0212305	−0.0106153	+	0.999944i	$0.503379\pi$
$420$	0	0
$421$	−27.3152	−1.33126	−0.665630	−	0.746282i	$-0.731837\pi$
$421$	−27.3152	−1.33126	−0.665630	+	0.746282i	$0.731837\pi$
$422$	0	0
$423$	−5.34957	−0.260105
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.40464	0.260938
$430$	0	0
$431$	−20.5275	−0.988773	−0.494386	−	0.869242i	$-0.664607\pi$
$431$	−20.5275	−0.988773	−0.494386	+	0.869242i	$0.664607\pi$
$432$	0	0
$433$	−16.0082	−0.769304	−0.384652	−	0.923062i	$-0.625679\pi$
$433$	−16.0082	−0.769304	−0.384652	+	0.923062i	$0.625679\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.75306	−0.227370
$438$	0	0
$439$	−8.06958	−0.385140	−0.192570	−	0.981283i	$-0.561682\pi$
$439$	−8.06958	−0.385140	−0.192570	+	0.981283i	$0.561682\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.19214	0.341709	0.170854	−	0.985296i	$-0.445347\pi$
$443$	7.19214	0.341709	0.170854	+	0.985296i	$0.445347\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−44.8470	−2.12119
$448$	0	0
$449$	−4.16729	−0.196667	−0.0983334	−	0.995154i	$-0.531351\pi$
$449$	−4.16729	−0.196667	−0.0983334	+	0.995154i	$0.531351\pi$
$450$	0	0
$451$	3.41378	0.160749
$452$	0	0
$453$	−13.6610	−0.641851
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.17950	0.429399	0.214700	−	0.976680i	$-0.431123\pi$
$457$	9.17950	0.429399	0.214700	+	0.976680i	$0.431123\pi$
$458$	0	0
$459$	−19.2424	−0.898160
$460$	0	0
$461$	−27.3537	−1.27399	−0.636995	−	0.770868i	$-0.719823\pi$
$461$	−27.3537	−1.27399	−0.636995	+	0.770868i	$0.719823\pi$
$462$	0	0
$463$	12.8660	0.597932	0.298966	−	0.954264i	$-0.403358\pi$
$463$	12.8660	0.597932	0.298966	+	0.954264i	$0.403358\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.9010	−0.643260	−0.321630	−	0.946865i	$-0.604231\pi$
$467$	−13.9010	−0.643260	−0.321630	+	0.946865i	$0.604231\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.4896	−0.529411
$472$	0	0
$473$	−10.3284	−0.474900
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.855710	0.0391803
$478$	0	0
$479$	0.371944	0.0169946	0.00849729	−	0.999964i	$-0.497295\pi$
$479$	0.371944	0.0169946	0.00849729	+	0.999964i	$0.497295\pi$
$480$	0	0
$481$	−7.24747	−0.330456
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.2340	−0.871577	−0.435788	−	0.900049i	$-0.643530\pi$
$487$	−19.2340	−0.871577	−0.435788	+	0.900049i	$0.643530\pi$
$488$	0	0
$489$	−13.2168	−0.597686
$490$	0	0
$491$	15.2267	0.687170	0.343585	−	0.939122i	$-0.388359\pi$
$491$	15.2267	0.687170	0.343585	+	0.939122i	$0.388359\pi$
$492$	0	0
$493$	−37.8938	−1.70665
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.9430	0.624173	0.312087	−	0.950054i	$-0.398972\pi$
$499$	13.9430	0.624173	0.312087	+	0.950054i	$0.398972\pi$
$500$	0	0
$501$	−39.1554	−1.74933
$502$	0	0
$503$	−41.6394	−1.85661	−0.928305	−	0.371821i	$-0.878734\pi$
$503$	−41.6394	−1.85661	−0.928305	+	0.371821i	$0.878734\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0074	1.02179
$508$	0	0
$509$	31.1663	1.38142	0.690712	−	0.723130i	$-0.257297\pi$
$509$	31.1663	1.38142	0.690712	+	0.723130i	$0.257297\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−23.4856	−1.03692
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.57193	−0.289033
$518$	0	0
$519$	29.4459	1.29253
$520$	0	0
$521$	9.91918	0.434567	0.217284	−	0.976109i	$-0.430280\pi$
$521$	9.91918	0.434567	0.217284	+	0.976109i	$0.430280\pi$
$522$	0	0
$523$	−19.2340	−0.841045	−0.420523	−	0.907282i	$-0.638153\pi$
$523$	−19.2340	−0.841045	−0.420523	+	0.907282i	$0.638153\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−59.1100	−2.57487
$528$	0	0
$529$	−22.5596	−0.980851
$530$	0	0
$531$	10.9654	0.475857
$532$	0	0
$533$	−2.77882	−0.120364
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.7084	−0.721022
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.60558	−0.155016	−0.0775081	−	0.996992i	$-0.524696\pi$
$541$	−3.60558	−0.155016	−0.0775081	+	0.996992i	$0.524696\pi$
$542$	0	0
$543$	−24.6423	−1.05750
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.1011	0.474647	0.237324	−	0.971431i	$-0.423730\pi$
$547$	11.1011	0.474647	0.237324	+	0.971431i	$0.423730\pi$
$548$	0	0
$549$	−12.5982	−0.537680
$550$	0	0
$551$	−46.2499	−1.97031
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.8346	1.05228	0.526138	−	0.850399i	$-0.323640\pi$
$557$	24.8346	1.05228	0.526138	+	0.850399i	$0.323640\pi$
$558$	0	0
$559$	8.40733	0.355592
$560$	0	0
$561$	21.9545	0.926918
$562$	0	0
$563$	−2.82819	−0.119194	−0.0595970	−	0.998223i	$-0.518982\pi$
$563$	−2.82819	−0.119194	−0.0595970	+	0.998223i	$0.518982\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.1318	−1.05358	−0.526791	−	0.849995i	$-0.676605\pi$
$569$	−25.1318	−1.05358	−0.526791	+	0.849995i	$0.676605\pi$
$570$	0	0
$571$	−35.2093	−1.47347	−0.736733	−	0.676184i	$-0.763632\pi$
$571$	−35.2093	−1.47347	−0.736733	+	0.676184i	$0.763632\pi$
$572$	0	0
$573$	27.3607	1.14301
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.8365	−1.15885	−0.579424	−	0.815027i	$-0.696722\pi$
$577$	−27.8365	−1.15885	−0.579424	+	0.815027i	$0.696722\pi$
$578$	0	0
$579$	45.1269	1.87541
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.05124	0.0435378
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.74640	0.195905	0.0979524	−	0.995191i	$-0.468771\pi$
$587$	4.74640	0.195905	0.0979524	+	0.995191i	$0.468771\pi$
$588$	0	0
$589$	−72.1445	−2.97266
$590$	0	0
$591$	−17.7695	−0.730941
$592$	0	0
$593$	−7.48084	−0.307201	−0.153601	−	0.988133i	$-0.549087\pi$
$593$	−7.48084	−0.307201	−0.153601	+	0.988133i	$0.549087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.88573	0.363669
$598$	0	0
$599$	−4.56165	−0.186384	−0.0931920	−	0.995648i	$-0.529707\pi$
$599$	−4.56165	−0.186384	−0.0931920	+	0.995648i	$0.529707\pi$
$600$	0	0
$601$	4.96538	0.202542	0.101271	−	0.994859i	$-0.467709\pi$
$601$	4.96538	0.202542	0.101271	+	0.994859i	$0.467709\pi$
$602$	0	0
$603$	3.13229	0.127557
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.40325	−0.178722	−0.0893612	−	0.995999i	$-0.528483\pi$
$607$	−4.40325	−0.178722	−0.0893612	+	0.995999i	$0.528483\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.34957	0.216420
$612$	0	0
$613$	42.6950	1.72444	0.862218	−	0.506538i	$-0.169075\pi$
$613$	42.6950	1.72444	0.862218	+	0.506538i	$0.169075\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.2135	−0.934541	−0.467270	−	0.884114i	$-0.654763\pi$
$617$	−23.2135	−0.934541	−0.467270	+	0.884114i	$0.654763\pi$
$618$	0	0
$619$	28.0134	1.12595	0.562977	−	0.826473i	$-0.309656\pi$
$619$	28.0134	1.12595	0.562977	+	0.826473i	$0.309656\pi$
$620$	0	0
$621$	2.17621	0.0873284
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	26.7957	1.07012
$628$	0	0
$629$	−29.4403	−1.17386
$630$	0	0
$631$	−28.8446	−1.14829	−0.574144	−	0.818755i	$-0.694665\pi$
$631$	−28.8446	−1.14829	−0.574144	+	0.818755i	$0.694665\pi$
$632$	0	0
$633$	1.32608	0.0527068
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.38718	0.371351
$640$	0	0
$641$	−41.1854	−1.62673	−0.813363	−	0.581757i	$-0.802366\pi$
$641$	−41.1854	−1.62673	−0.813363	+	0.581757i	$0.802366\pi$
$642$	0	0
$643$	−7.64360	−0.301434	−0.150717	−	0.988577i	$-0.548158\pi$
$643$	−7.64360	−0.301434	−0.150717	+	0.988577i	$0.548158\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.1376	0.870318	0.435159	−	0.900354i	$-0.356692\pi$
$647$	22.1376	0.870318	0.435159	+	0.900354i	$0.356692\pi$
$648$	0	0
$649$	13.4710	0.528781
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.3026	0.716238	0.358119	−	0.933676i	$-0.383418\pi$
$653$	18.3026	0.716238	0.358119	+	0.933676i	$0.383418\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.4370	0.875353
$658$	0	0
$659$	−28.1377	−1.09609	−0.548044	−	0.836449i	$-0.684627\pi$
$659$	−28.1377	−1.09609	−0.548044	+	0.836449i	$0.684627\pi$
$660$	0	0
$661$	23.2742	0.905261	0.452630	−	0.891698i	$-0.350486\pi$
$661$	23.2742	0.905261	0.452630	+	0.891698i	$0.350486\pi$
$662$	0	0
$663$	−17.8710	−0.694051
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.28558	0.165938
$668$	0	0
$669$	−45.1449	−1.74540
$670$	0	0
$671$	−15.4769	−0.597480
$672$	0	0
$673$	19.0282	0.733484	0.366742	−	0.930323i	$-0.380473\pi$
$673$	19.0282	0.733484	0.366742	+	0.930323i	$0.380473\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−40.7526	−1.56625	−0.783124	−	0.621866i	$-0.786375\pi$
$677$	−40.7526	−1.56625	−0.783124	+	0.621866i	$0.786375\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	46.4847	1.78130
$682$	0	0
$683$	31.9224	1.22148	0.610738	−	0.791833i	$-0.290873\pi$
$683$	31.9224	1.22148	0.610738	+	0.791833i	$0.290873\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	24.5311	0.935918
$688$	0	0
$689$	−0.855710	−0.0326000
$690$	0	0
$691$	12.1124	0.460779	0.230389	−	0.973099i	$-0.426000\pi$
$691$	12.1124	0.460779	0.230389	+	0.973099i	$0.426000\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.2880	−0.427564
$698$	0	0
$699$	−23.6528	−0.894631
$700$	0	0
$701$	−9.83067	−0.371299	−0.185650	−	0.982616i	$-0.559439\pi$
$701$	−9.83067	−0.371299	−0.185650	+	0.982616i	$0.559439\pi$
$702$	0	0
$703$	−35.9323	−1.35521
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.22203	−0.346341	−0.173170	−	0.984892i	$-0.555401\pi$
$709$	−9.22203	−0.346341	−0.173170	+	0.984892i	$0.555401\pi$
$710$	0	0
$711$	−3.91075	−0.146664
$712$	0	0
$713$	6.68501	0.250356
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−48.1262	−1.79731
$718$	0	0
$719$	32.5773	1.21493	0.607464	−	0.794347i	$-0.292187\pi$
$719$	32.5773	1.21493	0.607464	+	0.794347i	$0.292187\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	8.86020	0.329515
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.4462	1.05501	0.527505	−	0.849552i	$-0.323128\pi$
$727$	28.4462	1.05501	0.527505	+	0.849552i	$0.323128\pi$
$728$	0	0
$729$	4.49268	0.166396
$730$	0	0
$731$	34.1518	1.26315
$732$	0	0
$733$	9.61408	0.355104	0.177552	−	0.984111i	$-0.443182\pi$
$733$	9.61408	0.355104	0.177552	+	0.984111i	$0.443182\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.84801	0.141743
$738$	0	0
$739$	3.88409	0.142879	0.0714393	−	0.997445i	$-0.477241\pi$
$739$	3.88409	0.142879	0.0714393	+	0.997445i	$0.477241\pi$
$740$	0	0
$741$	−21.8118	−0.801275
$742$	0	0
$743$	−16.8174	−0.616971	−0.308486	−	0.951229i	$-0.599822\pi$
$743$	−16.8174	−0.616971	−0.308486	+	0.951229i	$0.599822\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.17793	−0.335803
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−33.3654	−1.21752	−0.608760	−	0.793354i	$-0.708333\pi$
$751$	−33.3654	−1.21752	−0.608760	+	0.793354i	$0.708333\pi$
$752$	0	0
$753$	−35.7525	−1.30290
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.2781	1.13682	0.568412	−	0.822744i	$-0.307558\pi$
$757$	31.2781	1.13682	0.568412	+	0.822744i	$0.307558\pi$
$758$	0	0
$759$	−2.48293	−0.0901245
$760$	0	0
$761$	−2.82448	−0.102387	−0.0511937	−	0.998689i	$-0.516303\pi$
$761$	−2.82448	−0.102387	−0.0511937	+	0.998689i	$0.516303\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.9654	−0.395937
$768$	0	0
$769$	29.1637	1.05167	0.525836	−	0.850586i	$-0.323753\pi$
$769$	29.1637	1.05167	0.525836	+	0.850586i	$0.323753\pi$
$770$	0	0
$771$	56.7748	2.04469
$772$	0	0
$773$	40.7784	1.46670	0.733348	−	0.679853i	$-0.237957\pi$
$773$	40.7784	1.46670	0.733348	+	0.679853i	$0.237957\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.7772	−0.493618
$780$	0	0
$781$	11.5321	0.412652
$782$	0	0
$783$	21.1758	0.756760
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.25206	−0.294154	−0.147077	−	0.989125i	$-0.546987\pi$
$787$	−8.25206	−0.294154	−0.147077	+	0.989125i	$0.546987\pi$
$788$	0	0
$789$	−46.2795	−1.64760
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.5982	0.447377
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.6687	1.40514	0.702569	−	0.711616i	$-0.252036\pi$
$797$	39.6687	1.40514	0.702569	+	0.711616i	$0.252036\pi$
$798$	0	0
$799$	21.7307	0.768779
$800$	0	0
$801$	−8.46745	−0.299183
$802$	0	0
$803$	27.5639	0.972708
$804$	0	0
$805$	0	0
$806$	0	0
$807$	50.7772	1.78744
$808$	0	0
$809$	−32.3156	−1.13616	−0.568078	−	0.822975i	$-0.692313\pi$
$809$	−32.3156	−1.13616	−0.568078	+	0.822975i	$0.692313\pi$
$810$	0	0
$811$	−7.67248	−0.269417	−0.134709	−	0.990885i	$-0.543010\pi$
$811$	−7.67248	−0.269417	−0.134709	+	0.990885i	$0.543010\pi$
$812$	0	0
$813$	4.35833	0.152853
$814$	0	0
$815$	0	0
$816$	0	0
$817$	41.6828	1.45830
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.0562	0.595266	0.297633	−	0.954680i	$-0.403803\pi$
$821$	17.0562	0.595266	0.297633	+	0.954680i	$0.403803\pi$
$822$	0	0
$823$	−33.1024	−1.15388	−0.576939	−	0.816787i	$-0.695753\pi$
$823$	−33.1024	−1.15388	−0.576939	+	0.816787i	$0.695753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.1051	0.699123	0.349561	−	0.936913i	$-0.386331\pi$
$827$	20.1051	0.699123	0.349561	+	0.936913i	$0.386331\pi$
$828$	0	0
$829$	−18.9667	−0.658740	−0.329370	−	0.944201i	$-0.606836\pi$
$829$	−18.9667	−0.658740	−0.329370	+	0.944201i	$0.606836\pi$
$830$	0	0
$831$	−35.9993	−1.24880
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	33.0317	1.14174
$838$	0	0
$839$	−4.15025	−0.143283	−0.0716413	−	0.997430i	$-0.522824\pi$
$839$	−4.15025	−0.143283	−0.0716413	+	0.997430i	$0.522824\pi$
$840$	0	0
$841$	12.7010	0.437967
$842$	0	0
$843$	27.8358	0.958717
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	56.0762	1.92453
$850$	0	0
$851$	3.32954	0.114135
$852$	0	0
$853$	−1.53140	−0.0524340	−0.0262170	−	0.999656i	$-0.508346\pi$
$853$	−1.53140	−0.0524340	−0.0262170	+	0.999656i	$0.508346\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.8235	−0.813794	−0.406897	−	0.913474i	$-0.633389\pi$
$857$	−23.8235	−0.813794	−0.406897	+	0.913474i	$0.633389\pi$
$858$	0	0
$859$	−14.6833	−0.500987	−0.250494	−	0.968118i	$-0.580593\pi$
$859$	−14.6833	−0.500987	−0.250494	+	0.968118i	$0.580593\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.3325	−1.44102	−0.720508	−	0.693446i	$-0.756091\pi$
$863$	−42.3325	−1.44102	−0.720508	+	0.693446i	$0.756091\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−36.7549	−1.24826
$868$	0	0
$869$	−4.80434	−0.162976
$870$	0	0
$871$	−3.13229	−0.106134
$872$	0	0
$873$	−21.3432	−0.722359
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.5940	0.357735	0.178868	−	0.983873i	$-0.442757\pi$
$877$	10.5940	0.357735	0.178868	+	0.983873i	$0.442757\pi$
$878$	0	0
$879$	−0.996512	−0.0336115
$880$	0	0
$881$	−27.1174	−0.913609	−0.456805	−	0.889567i	$-0.651006\pi$
$881$	−27.1174	−0.913609	−0.456805	+	0.889567i	$0.651006\pi$
$882$	0	0
$883$	−21.7240	−0.731072	−0.365536	−	0.930797i	$-0.619114\pi$
$883$	−21.7240	−0.731072	−0.365536	+	0.930797i	$0.619114\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.1605	0.710501	0.355251	−	0.934771i	$-0.384396\pi$
$887$	21.1605	0.710501	0.355251	+	0.934771i	$0.384396\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−19.9594	−0.668664
$892$	0	0
$893$	26.5227	0.887547
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.02111	0.0674828
$898$	0	0
$899$	65.0488	2.16950
$900$	0	0
$901$	−3.47602	−0.115803
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−37.8642	−1.25726	−0.628630	−	0.777705i	$-0.716384\pi$
$907$	−37.8642	−1.25726	−0.628630	+	0.777705i	$0.716384\pi$
$908$	0	0
$909$	−5.02978	−0.166827
$910$	0	0
$911$	−30.1221	−0.997990	−0.498995	−	0.866605i	$-0.666297\pi$
$911$	−30.1221	−0.997990	−0.498995	+	0.866605i	$0.666297\pi$
$912$	0	0
$913$	−11.2751	−0.373150
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−46.1132	−1.52113	−0.760566	−	0.649260i	$-0.775079\pi$
$919$	−46.1132	−1.52113	−0.760566	+	0.649260i	$0.775079\pi$
$920$	0	0
$921$	32.0604	1.05642
$922$	0	0
$923$	−9.38718	−0.308983
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−11.6701	−0.383297
$928$	0	0
$929$	−27.5174	−0.902816	−0.451408	−	0.892318i	$-0.649078\pi$
$929$	−27.5174	−0.902816	−0.451408	+	0.892318i	$0.649078\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−34.9555	−1.14439
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50.4704	−1.64880	−0.824399	−	0.566010i	$-0.808486\pi$
$937$	−50.4704	−1.64880	−0.824399	+	0.566010i	$0.808486\pi$
$938$	0	0
$939$	−2.48665	−0.0811487
$940$	0	0
$941$	−1.85663	−0.0605243	−0.0302622	−	0.999542i	$-0.509634\pi$
$941$	−1.85663	−0.0605243	−0.0302622	+	0.999542i	$0.509634\pi$
$942$	0	0
$943$	1.27661	0.0415721
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.1689	−1.17533	−0.587666	−	0.809104i	$-0.699953\pi$
$947$	−36.1689	−1.17533	−0.587666	+	0.809104i	$0.699953\pi$
$948$	0	0
$949$	−22.4370	−0.728337
$950$	0	0
$951$	57.8258	1.87513
$952$	0	0
$953$	17.3367	0.561590	0.280795	−	0.959768i	$-0.409402\pi$
$953$	17.3367	0.561590	0.280795	+	0.959768i	$0.409402\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.1602	−0.780990
$958$	0	0
$959$	0	0
$960$	0	0
$961$	70.4688	2.27319
$962$	0	0
$963$	−0.187282	−0.00603506
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.4618	0.754480	0.377240	−	0.926116i	$-0.376873\pi$
$967$	23.4618	0.754480	0.377240	+	0.926116i	$0.376873\pi$
$968$	0	0
$969$	−88.6027	−2.84633
$970$	0	0
$971$	56.5553	1.81495	0.907473	−	0.420110i	$-0.138009\pi$
$971$	56.5553	1.81495	0.907473	+	0.420110i	$0.138009\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.9783	−0.927099	−0.463549	−	0.886071i	$-0.653424\pi$
$977$	−28.9783	−0.927099	−0.463549	+	0.886071i	$0.653424\pi$
$978$	0	0
$979$	−10.4022	−0.332457
$980$	0	0
$981$	17.7174	0.565673
$982$	0	0
$983$	41.3280	1.31816	0.659079	−	0.752074i	$-0.270946\pi$
$983$	41.3280	1.31816	0.659079	+	0.752074i	$0.270946\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.86238	−0.122817
$990$	0	0
$991$	−29.0897	−0.924065	−0.462032	−	0.886863i	$-0.652880\pi$
$991$	−29.0897	−0.924065	−0.462032	+	0.886863i	$0.652880\pi$
$992$	0	0
$993$	−10.9872	−0.348667
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21.4734	0.680069	0.340035	−	0.940413i	$-0.389561\pi$
$997$	21.4734	0.680069	0.340035	+	0.940413i	$0.389561\pi$
$998$	0	0
$999$	16.4518	0.520512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cw.1.2		6
5.2	odd	4		1960.2.g.e.1569.10		12
5.3	odd	4		1960.2.g.e.1569.3		12
5.4	even	2		9800.2.a.cy.1.5		6
7.3	odd	6		1400.2.q.n.401.2		12
7.5	odd	6		1400.2.q.n.1201.2		12
7.6	odd	2		9800.2.a.cx.1.5		6
35.3	even	12		280.2.bg.a.9.10	yes	24
35.12	even	12		280.2.bg.a.249.10	yes	24
35.13	even	4		1960.2.g.f.1569.10		12
35.17	even	12		280.2.bg.a.9.3	✓	24
35.19	odd	6		1400.2.q.o.1201.5		12
35.24	odd	6		1400.2.q.o.401.5		12
35.27	even	4		1960.2.g.f.1569.3		12
35.33	even	12		280.2.bg.a.249.3	yes	24
35.34	odd	2		9800.2.a.cv.1.2		6
140.3	odd	12		560.2.bw.f.289.3		24
140.47	odd	12		560.2.bw.f.529.3		24
140.87	odd	12		560.2.bw.f.289.10		24
140.103	odd	12		560.2.bw.f.529.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bg.a.9.3	✓	24	35.17	even	12
280.2.bg.a.9.10	yes	24	35.3	even	12
280.2.bg.a.249.3	yes	24	35.33	even	12
280.2.bg.a.249.10	yes	24	35.12	even	12
560.2.bw.f.289.3		24	140.3	odd	12
560.2.bw.f.289.10		24	140.87	odd	12
560.2.bw.f.529.3		24	140.47	odd	12
560.2.bw.f.529.10		24	140.103	odd	12
1400.2.q.n.401.2		12	7.3	odd	6
1400.2.q.n.1201.2		12	7.5	odd	6
1400.2.q.o.401.5		12	35.24	odd	6
1400.2.q.o.1201.5		12	35.19	odd	6
1960.2.g.e.1569.3		12	5.3	odd	4
1960.2.g.e.1569.10		12	5.2	odd	4
1960.2.g.f.1569.3		12	35.27	even	4
1960.2.g.f.1569.10		12	35.13	even	4
9800.2.a.cv.1.2		6	35.34	odd	2
9800.2.a.cw.1.2		6	1.1	even	1	trivial
9800.2.a.cx.1.5		6	7.6	odd	2
9800.2.a.cy.1.5		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10821 q^{3} +1.44457 q^{9} +O(q^{10})\)	\(q-2.10821 q^{3} +1.44457 q^{9} +1.77465 q^{11} -1.44457 q^{13} -5.86806 q^{17} -7.16205 q^{19} +0.663645 q^{23} +3.27918 q^{27} +6.45763 q^{29} +10.0732 q^{31} -3.74135 q^{33} +5.01705 q^{37} +3.04546 q^{39} +1.92363 q^{41} -5.81995 q^{43} -3.70322 q^{47} +12.3711 q^{51} +0.592363 q^{53} +15.0991 q^{57} +7.59076 q^{59} -8.72111 q^{61} +2.16832 q^{67} -1.39911 q^{69} +6.49826 q^{71} +15.5320 q^{73} -2.70720 q^{79} -11.2469 q^{81} -6.35340 q^{83} -13.6141 q^{87} -5.86157 q^{89} -21.2364 q^{93} -14.7748 q^{97} +2.56361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 7 q^{9}+O(q^{10}) \) 6 * q - q^3 + 7 * q^9	\( 6 q - q^{3} + 7 q^{9} + q^{11} - 7 q^{13} - 4 q^{17} + 5 q^{19} - 6 q^{23} - 7 q^{27} - 3 q^{29} + 2 q^{31} - 16 q^{33} - 9 q^{37} + 10 q^{39} - 6 q^{41} + 3 q^{43} - 27 q^{47} + 5 q^{53} + 26 q^{57} + 24 q^{59} - 9 q^{61} - 17 q^{67} - 15 q^{69} + 4 q^{71} - 18 q^{73} - 22 q^{79} - 6 q^{81} - 9 q^{83} - 39 q^{87} - 15 q^{89} - 10 q^{93} - 12 q^{97} + 11 q^{99}+O(q^{100}) \) 6 * q - q^3 + 7 * q^9 + q^11 - 7 * q^13 - 4 * q^17 + 5 * q^19 - 6 * q^23 - 7 * q^27 - 3 * q^29 + 2 * q^31 - 16 * q^33 - 9 * q^37 + 10 * q^39 - 6 * q^41 + 3 * q^43 - 27 * q^47 + 5 * q^53 + 26 * q^57 + 24 * q^59 - 9 * q^61 - 17 * q^67 - 15 * q^69 + 4 * q^71 - 18 * q^73 - 22 * q^79 - 6 * q^81 - 9 * q^83 - 39 * q^87 - 15 * q^89 - 10 * q^93 - 12 * q^97 + 11 * q^99

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