LMFDB - Embedded newform 9196.2.a.s.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,8,0,-5,0,2,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} - 3x^{7} + 84x^{6} + 16x^{5} - 174x^{4} - 16x^{3} + 122x^{2} - 9 $ x^10 - 16x^8 - 3x^7 + 84x^6 + 16x^5 - 174x^4 - 16x^3 + 122*x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.16122$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.16122 q^{3} +0.343496 q^{5} -0.282327 q^{7} +1.67087 q^{9} -2.09639 q^{13} +0.742372 q^{15} +4.01817 q^{17} +1.00000 q^{19} -0.610170 q^{21} -3.47332 q^{23} -4.88201 q^{25} -2.87254 q^{27} -2.22445 q^{29} -8.57304 q^{31} -0.0969782 q^{35} -4.79841 q^{37} -4.53077 q^{39} +9.22214 q^{41} -11.3823 q^{43} +0.573939 q^{45} +6.93022 q^{47} -6.92029 q^{49} +8.68415 q^{51} -2.14066 q^{53} +2.16122 q^{57} +2.04262 q^{59} +6.72000 q^{61} -0.471732 q^{63} -0.720104 q^{65} -14.4421 q^{67} -7.50661 q^{69} +0.318506 q^{71} -11.9824 q^{73} -10.5511 q^{75} +1.22714 q^{79} -11.2208 q^{81} -4.24900 q^{83} +1.38023 q^{85} -4.80754 q^{87} +5.72713 q^{89} +0.591867 q^{91} -18.5282 q^{93} +0.343496 q^{95} -10.0256 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} - 4 q^{17} + 10 q^{19} - 6 q^{21} - 8 q^{23} - 10 q^{25} - 9 q^{27} + 3 q^{29} - 12 q^{31} - 9 q^{35} - 23 q^{37} + 18 q^{39} + 5 q^{41} + 14 q^{43}+ \cdots + 22 q^{97}+O(q^{100}) $ 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 - 4 * q^17 + 10 * q^19 - 6 * q^21 - 8 * q^23 - 10 * q^25 - 9 * q^27 + 3 * q^29 - 12 * q^31 - 9 * q^35 - 23 * q^37 + 18 * q^39 + 5 * q^41 + 14 * q^43 - 27 * q^47 - 17 * q^49 + 12 * q^51 - 10 * q^53 - 9 * q^59 - 4 * q^61 - 14 * q^63 - 7 * q^65 - 17 * q^67 - 3 * q^69 - 34 * q^71 + 9 * q^73 - 26 * q^75 + 16 * q^79 - 22 * q^81 - 6 * q^83 - 24 * q^85 - q^87 - 35 * q^89 - 35 * q^91 - 44 * q^93 + 8 * q^95 + 22 * q^97

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
$3$	2.16122	1.24778	0.623891	−	0.781512i	$-0.285551\pi$
$3$	2.16122	1.24778	0.623891	+	0.781512i	$0.285551\pi$
$4$	0	0
$5$	0.343496	0.153616	0.0768082	−	0.997046i	$-0.475527\pi$
$5$	0.343496	0.153616	0.0768082	+	0.997046i	$0.475527\pi$
$6$	0	0
$7$	−0.282327	−0.106709	−0.0533547	−	0.998576i	$-0.516991\pi$
$7$	−0.282327	−0.106709	−0.0533547	+	0.998576i	$0.516991\pi$
$8$	0	0
$9$	1.67087	0.556957
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.09639	−0.581435	−0.290717	−	0.956809i	$-0.593894\pi$
$13$	−2.09639	−0.581435	−0.290717	+	0.956809i	$0.593894\pi$
$14$	0	0
$15$	0.742372	0.191680
$16$	0	0
$17$	4.01817	0.974550	0.487275	−	0.873249i	$-0.337991\pi$
$17$	4.01817	0.974550	0.487275	+	0.873249i	$0.337991\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.610170	−0.133150
$22$	0	0
$23$	−3.47332	−0.724238	−0.362119	−	0.932132i	$-0.617947\pi$
$23$	−3.47332	−0.724238	−0.362119	+	0.932132i	$0.617947\pi$
$24$	0	0
$25$	−4.88201	−0.976402
$26$	0	0
$27$	−2.87254	−0.552820
$28$	0	0
$29$	−2.22445	−0.413071	−0.206535	−	0.978439i	$-0.566219\pi$
$29$	−2.22445	−0.413071	−0.206535	+	0.978439i	$0.566219\pi$
$30$	0	0
$31$	−8.57304	−1.53976	−0.769881	−	0.638187i	$-0.779685\pi$
$31$	−8.57304	−1.53976	−0.769881	+	0.638187i	$0.779685\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.0969782	−0.0163923
$36$	0	0
$37$	−4.79841	−0.788853	−0.394427	−	0.918927i	$-0.629057\pi$
$37$	−4.79841	−0.788853	−0.394427	+	0.918927i	$0.629057\pi$
$38$	0	0
$39$	−4.53077	−0.725504
$40$	0	0
$41$	9.22214	1.44026	0.720129	−	0.693841i	$-0.244083\pi$
$41$	9.22214	1.44026	0.720129	+	0.693841i	$0.244083\pi$
$42$	0	0
$43$	−11.3823	−1.73578	−0.867889	−	0.496758i	$-0.834524\pi$
$43$	−11.3823	−1.73578	−0.867889	+	0.496758i	$0.834524\pi$
$44$	0	0
$45$	0.573939	0.0855577
$46$	0	0
$47$	6.93022	1.01088	0.505438	−	0.862863i	$-0.331331\pi$
$47$	6.93022	1.01088	0.505438	+	0.862863i	$0.331331\pi$
$48$	0	0
$49$	−6.92029	−0.988613
$50$	0	0
$51$	8.68415	1.21602
$52$	0	0
$53$	−2.14066	−0.294043	−0.147021	−	0.989133i	$-0.546969\pi$
$53$	−2.14066	−0.294043	−0.147021	+	0.989133i	$0.546969\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.16122	0.286261
$58$	0	0
$59$	2.04262	0.265927	0.132963	−	0.991121i	$-0.457551\pi$
$59$	2.04262	0.265927	0.132963	+	0.991121i	$0.457551\pi$
$60$	0	0
$61$	6.72000	0.860408	0.430204	−	0.902732i	$-0.358442\pi$
$61$	6.72000	0.860408	0.430204	+	0.902732i	$0.358442\pi$
$62$	0	0
$63$	−0.471732	−0.0594326
$64$	0	0
$65$	−0.720104	−0.0893179
$66$	0	0
$67$	−14.4421	−1.76438	−0.882191	−	0.470893i	$-0.843932\pi$
$67$	−14.4421	−1.76438	−0.882191	+	0.470893i	$0.843932\pi$
$68$	0	0
$69$	−7.50661	−0.903690
$70$	0	0
$71$	0.318506	0.0377997	0.0188999	−	0.999821i	$-0.493984\pi$
$71$	0.318506	0.0377997	0.0188999	+	0.999821i	$0.493984\pi$
$72$	0	0
$73$	−11.9824	−1.40243	−0.701215	−	0.712950i	$-0.747359\pi$
$73$	−11.9824	−1.40243	−0.701215	+	0.712950i	$0.747359\pi$
$74$	0	0
$75$	−10.5511	−1.21834
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.22714	0.138064	0.0690320	−	0.997614i	$-0.478009\pi$
$79$	1.22714	0.138064	0.0690320	+	0.997614i	$0.478009\pi$
$80$	0	0
$81$	−11.2208	−1.24676
$82$	0	0
$83$	−4.24900	−0.466388	−0.233194	−	0.972430i	$-0.574918\pi$
$83$	−4.24900	−0.466388	−0.233194	+	0.972430i	$0.574918\pi$
$84$	0	0
$85$	1.38023	0.149707
$86$	0	0
$87$	−4.80754	−0.515422
$88$	0	0
$89$	5.72713	0.607075	0.303537	−	0.952820i	$-0.401832\pi$
$89$	5.72713	0.607075	0.303537	+	0.952820i	$0.401832\pi$
$90$	0	0
$91$	0.591867	0.0620446
$92$	0	0
$93$	−18.5282	−1.92129
$94$	0	0
$95$	0.343496	0.0352420
$96$	0	0
$97$	−10.0256	−1.01795	−0.508974	−	0.860782i	$-0.669975\pi$
$97$	−10.0256	−1.01795	−0.508974	+	0.860782i	$0.669975\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.7952	1.57168	0.785840	−	0.618430i	$-0.212231\pi$
$101$	15.7952	1.57168	0.785840	+	0.618430i	$0.212231\pi$
$102$	0	0
$103$	−17.0641	−1.68138	−0.840688	−	0.541520i	$-0.817849\pi$
$103$	−17.0641	−1.68138	−0.840688	+	0.541520i	$0.817849\pi$
$104$	0	0
$105$	−0.209591	−0.0204540
$106$	0	0
$107$	1.06155	0.102624	0.0513120	−	0.998683i	$-0.483660\pi$
$107$	1.06155	0.102624	0.0513120	+	0.998683i	$0.483660\pi$
$108$	0	0
$109$	1.34834	0.129147	0.0645736	−	0.997913i	$-0.479431\pi$
$109$	1.34834	0.129147	0.0645736	+	0.997913i	$0.479431\pi$
$110$	0	0
$111$	−10.3704	−0.984316
$112$	0	0
$113$	13.1339	1.23553	0.617766	−	0.786362i	$-0.288038\pi$
$113$	13.1339	1.23553	0.617766	+	0.786362i	$0.288038\pi$
$114$	0	0
$115$	−1.19307	−0.111255
$116$	0	0
$117$	−3.50281	−0.323835
$118$	0	0
$119$	−1.13444	−0.103994
$120$	0	0
$121$	0	0
$122$	0	0
$123$	19.9311	1.79713
$124$	0	0
$125$	−3.39444	−0.303608
$126$	0	0
$127$	10.5842	0.939193	0.469597	−	0.882881i	$-0.344399\pi$
$127$	10.5842	0.939193	0.469597	+	0.882881i	$0.344399\pi$
$128$	0	0
$129$	−24.5996	−2.16587
$130$	0	0
$131$	1.79175	0.156546	0.0782729	−	0.996932i	$-0.475059\pi$
$131$	1.79175	0.156546	0.0782729	+	0.996932i	$0.475059\pi$
$132$	0	0
$133$	−0.282327	−0.0244808
$134$	0	0
$135$	−0.986707	−0.0849222
$136$	0	0
$137$	−1.15047	−0.0982915	−0.0491457	−	0.998792i	$-0.515650\pi$
$137$	−1.15047	−0.0982915	−0.0491457	+	0.998792i	$0.515650\pi$
$138$	0	0
$139$	−12.3731	−1.04947	−0.524735	−	0.851266i	$-0.675836\pi$
$139$	−12.3731	−1.04947	−0.524735	+	0.851266i	$0.675836\pi$
$140$	0	0
$141$	14.9777	1.26135
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.764092	−0.0634544
$146$	0	0
$147$	−14.9563	−1.23357
$148$	0	0
$149$	−13.6634	−1.11935	−0.559674	−	0.828713i	$-0.689074\pi$
$149$	−13.6634	−1.11935	−0.559674	+	0.828713i	$0.689074\pi$
$150$	0	0
$151$	−4.73390	−0.385239	−0.192620	−	0.981274i	$-0.561698\pi$
$151$	−4.73390	−0.385239	−0.192620	+	0.981274i	$0.561698\pi$
$152$	0	0
$153$	6.71385	0.542783
$154$	0	0
$155$	−2.94481	−0.236533
$156$	0	0
$157$	18.0696	1.44211	0.721055	−	0.692878i	$-0.243657\pi$
$157$	18.0696	1.44211	0.721055	+	0.692878i	$0.243657\pi$
$158$	0	0
$159$	−4.62645	−0.366901
$160$	0	0
$161$	0.980611	0.0772830
$162$	0	0
$163$	19.6426	1.53853	0.769264	−	0.638931i	$-0.220623\pi$
$163$	19.6426	1.53853	0.769264	+	0.638931i	$0.220623\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.0630	1.93943	0.969716	−	0.244234i	$-0.0785365\pi$
$167$	25.0630	1.93943	0.969716	+	0.244234i	$0.0785365\pi$
$168$	0	0
$169$	−8.60513	−0.661933
$170$	0	0
$171$	1.67087	0.127775
$172$	0	0
$173$	−6.22867	−0.473557	−0.236778	−	0.971564i	$-0.576092\pi$
$173$	−6.22867	−0.473557	−0.236778	+	0.971564i	$0.576092\pi$
$174$	0	0
$175$	1.37832	0.104191
$176$	0	0
$177$	4.41456	0.331819
$178$	0	0
$179$	−13.3782	−0.999934	−0.499967	−	0.866044i	$-0.666655\pi$
$179$	−13.3782	−0.999934	−0.499967	+	0.866044i	$0.666655\pi$
$180$	0	0
$181$	5.76994	0.428876	0.214438	−	0.976738i	$-0.431208\pi$
$181$	5.76994	0.428876	0.214438	+	0.976738i	$0.431208\pi$
$182$	0	0
$183$	14.5234	1.07360
$184$	0	0
$185$	−1.64824	−0.121181
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.810993	0.0589911
$190$	0	0
$191$	−0.131397	−0.00950757	−0.00475379	−	0.999989i	$-0.501513\pi$
$191$	−0.131397	−0.00950757	−0.00475379	+	0.999989i	$0.501513\pi$
$192$	0	0
$193$	8.34437	0.600641	0.300320	−	0.953838i	$-0.402906\pi$
$193$	8.34437	0.600641	0.300320	+	0.953838i	$0.402906\pi$
$194$	0	0
$195$	−1.55630	−0.111449
$196$	0	0
$197$	14.3089	1.01947	0.509735	−	0.860332i	$-0.329744\pi$
$197$	14.3089	1.01947	0.509735	+	0.860332i	$0.329744\pi$
$198$	0	0
$199$	17.3174	1.22759	0.613797	−	0.789464i	$-0.289641\pi$
$199$	17.3174	1.22759	0.613797	+	0.789464i	$0.289641\pi$
$200$	0	0
$201$	−31.2125	−2.20156
$202$	0	0
$203$	0.628023	0.0440785
$204$	0	0
$205$	3.16777	0.221247
$206$	0	0
$207$	−5.80348	−0.403370
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.5272	−0.793567	−0.396783	−	0.917912i	$-0.629874\pi$
$211$	−11.5272	−0.793567	−0.396783	+	0.917912i	$0.629874\pi$
$212$	0	0
$213$	0.688362	0.0471658
$214$	0	0
$215$	−3.90977	−0.266644
$216$	0	0
$217$	2.42040	0.164307
$218$	0	0
$219$	−25.8965	−1.74993
$220$	0	0
$221$	−8.42367	−0.566637
$222$	0	0
$223$	11.2165	0.751115	0.375557	−	0.926799i	$-0.377451\pi$
$223$	11.2165	0.751115	0.375557	+	0.926799i	$0.377451\pi$
$224$	0	0
$225$	−8.15722	−0.543814
$226$	0	0
$227$	14.9270	0.990739	0.495370	−	0.868682i	$-0.335033\pi$
$227$	14.9270	0.990739	0.495370	+	0.868682i	$0.335033\pi$
$228$	0	0
$229$	5.58402	0.369003	0.184501	−	0.982832i	$-0.440933\pi$
$229$	5.58402	0.369003	0.184501	+	0.982832i	$0.440933\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.0587	−1.37960	−0.689801	−	0.723999i	$-0.742302\pi$
$233$	−21.0587	−1.37960	−0.689801	+	0.723999i	$0.742302\pi$
$234$	0	0
$235$	2.38051	0.155287
$236$	0	0
$237$	2.65212	0.172274
$238$	0	0
$239$	−29.1025	−1.88248	−0.941241	−	0.337737i	$-0.890339\pi$
$239$	−29.1025	−1.88248	−0.941241	+	0.337737i	$0.890339\pi$
$240$	0	0
$241$	−25.4808	−1.64136	−0.820682	−	0.571385i	$-0.806406\pi$
$241$	−25.4808	−1.64136	−0.820682	+	0.571385i	$0.806406\pi$
$242$	0	0
$243$	−15.6330	−1.00286
$244$	0	0
$245$	−2.37710	−0.151867
$246$	0	0
$247$	−2.09639	−0.133390
$248$	0	0
$249$	−9.18302	−0.581951
$250$	0	0
$251$	−23.7627	−1.49989	−0.749945	−	0.661500i	$-0.769920\pi$
$251$	−23.7627	−1.49989	−0.749945	+	0.661500i	$0.769920\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.98298	0.186801
$256$	0	0
$257$	−2.30352	−0.143689	−0.0718447	−	0.997416i	$-0.522889\pi$
$257$	−2.30352	−0.143689	−0.0718447	+	0.997416i	$0.522889\pi$
$258$	0	0
$259$	1.35472	0.0841780
$260$	0	0
$261$	−3.71678	−0.230063
$262$	0	0
$263$	−10.1003	−0.622813	−0.311406	−	0.950277i	$-0.600800\pi$
$263$	−10.1003	−0.622813	−0.311406	+	0.950277i	$0.600800\pi$
$264$	0	0
$265$	−0.735311	−0.0451698
$266$	0	0
$267$	12.3776	0.757496
$268$	0	0
$269$	−31.2688	−1.90649	−0.953246	−	0.302196i	$-0.902280\pi$
$269$	−31.2688	−1.90649	−0.953246	+	0.302196i	$0.902280\pi$
$270$	0	0
$271$	−27.7601	−1.68630	−0.843152	−	0.537675i	$-0.819303\pi$
$271$	−27.7601	−1.68630	−0.843152	+	0.537675i	$0.819303\pi$
$272$	0	0
$273$	1.27916	0.0774180
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.3367	1.40217	0.701084	−	0.713079i	$-0.252700\pi$
$277$	23.3367	1.40217	0.701084	+	0.713079i	$0.252700\pi$
$278$	0	0
$279$	−14.3244	−0.857582
$280$	0	0
$281$	5.04071	0.300704	0.150352	−	0.988633i	$-0.451959\pi$
$281$	5.04071	0.300704	0.150352	+	0.988633i	$0.451959\pi$
$282$	0	0
$283$	5.16600	0.307087	0.153543	−	0.988142i	$-0.450932\pi$
$283$	5.16600	0.307087	0.153543	+	0.988142i	$0.450932\pi$
$284$	0	0
$285$	0.742372	0.0439743
$286$	0	0
$287$	−2.60366	−0.153689
$288$	0	0
$289$	−0.854304	−0.0502532
$290$	0	0
$291$	−21.6676	−1.27018
$292$	0	0
$293$	−2.31169	−0.135050	−0.0675252	−	0.997718i	$-0.521510\pi$
$293$	−2.31169	−0.135050	−0.0675252	+	0.997718i	$0.521510\pi$
$294$	0	0
$295$	0.701634	0.0408507
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.28145	0.421097
$300$	0	0
$301$	3.21351	0.185224
$302$	0	0
$303$	34.1369	1.96111
$304$	0	0
$305$	2.30830	0.132173
$306$	0	0
$307$	−0.556178	−0.0317427	−0.0158714	−	0.999874i	$-0.505052\pi$
$307$	−0.556178	−0.0317427	−0.0158714	+	0.999874i	$0.505052\pi$
$308$	0	0
$309$	−36.8793	−2.09799
$310$	0	0
$311$	22.6961	1.28698	0.643489	−	0.765455i	$-0.277486\pi$
$311$	22.6961	1.28698	0.643489	+	0.765455i	$0.277486\pi$
$312$	0	0
$313$	−10.1132	−0.571633	−0.285816	−	0.958284i	$-0.592265\pi$
$313$	−10.1132	−0.571633	−0.285816	+	0.958284i	$0.592265\pi$
$314$	0	0
$315$	−0.162038	−0.00912981
$316$	0	0
$317$	−16.2584	−0.913163	−0.456581	−	0.889682i	$-0.650926\pi$
$317$	−16.2584	−0.913163	−0.456581	+	0.889682i	$0.650926\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.29424	0.128052
$322$	0	0
$323$	4.01817	0.223577
$324$	0	0
$325$	10.2346	0.567714
$326$	0	0
$327$	2.91405	0.161148
$328$	0	0
$329$	−1.95658	−0.107870
$330$	0	0
$331$	−19.6622	−1.08073	−0.540365	−	0.841431i	$-0.681714\pi$
$331$	−19.6622	−1.08073	−0.540365	+	0.841431i	$0.681714\pi$
$332$	0	0
$333$	−8.01752	−0.439358
$334$	0	0
$335$	−4.96080	−0.271038
$336$	0	0
$337$	−22.7452	−1.23901	−0.619505	−	0.784993i	$-0.712667\pi$
$337$	−22.7452	−1.23901	−0.619505	+	0.784993i	$0.712667\pi$
$338$	0	0
$339$	28.3852	1.54167
$340$	0	0
$341$	0	0
$342$	0	0
$343$	3.93007	0.212204
$344$	0	0
$345$	−2.57850	−0.138822
$346$	0	0
$347$	6.50158	0.349023	0.174512	−	0.984655i	$-0.444165\pi$
$347$	6.50158	0.349023	0.174512	+	0.984655i	$0.444165\pi$
$348$	0	0
$349$	−12.7889	−0.684572	−0.342286	−	0.939596i	$-0.611201\pi$
$349$	−12.7889	−0.684572	−0.342286	+	0.939596i	$0.611201\pi$
$350$	0	0
$351$	6.02197	0.321429
$352$	0	0
$353$	−1.79369	−0.0954683	−0.0477341	−	0.998860i	$-0.515200\pi$
$353$	−1.79369	−0.0954683	−0.0477341	+	0.998860i	$0.515200\pi$
$354$	0	0
$355$	0.109406	0.00580666
$356$	0	0
$357$	−2.45177	−0.129761
$358$	0	0
$359$	6.87402	0.362797	0.181398	−	0.983410i	$-0.441938\pi$
$359$	6.87402	0.362797	0.181398	+	0.983410i	$0.441938\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.11590	−0.215436
$366$	0	0
$367$	−29.3242	−1.53071	−0.765354	−	0.643609i	$-0.777436\pi$
$367$	−29.3242	−1.53071	−0.765354	+	0.643609i	$0.777436\pi$
$368$	0	0
$369$	15.4090	0.802162
$370$	0	0
$371$	0.604366	0.0313771
$372$	0	0
$373$	31.8947	1.65145	0.825723	−	0.564076i	$-0.190767\pi$
$373$	31.8947	1.65145	0.825723	+	0.564076i	$0.190767\pi$
$374$	0	0
$375$	−7.33612	−0.378836
$376$	0	0
$377$	4.66333	0.240174
$378$	0	0
$379$	19.0456	0.978307	0.489153	−	0.872198i	$-0.337306\pi$
$379$	19.0456	0.978307	0.489153	+	0.872198i	$0.337306\pi$
$380$	0	0
$381$	22.8747	1.17191
$382$	0	0
$383$	−10.0891	−0.515531	−0.257765	−	0.966208i	$-0.582986\pi$
$383$	−10.0891	−0.515531	−0.257765	+	0.966208i	$0.582986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19.0183	−0.966755
$388$	0	0
$389$	24.2773	1.23091	0.615455	−	0.788172i	$-0.288972\pi$
$389$	24.2773	1.23091	0.615455	+	0.788172i	$0.288972\pi$
$390$	0	0
$391$	−13.9564	−0.705806
$392$	0	0
$393$	3.87236	0.195335
$394$	0	0
$395$	0.421518	0.0212089
$396$	0	0
$397$	5.36816	0.269420	0.134710	−	0.990885i	$-0.456990\pi$
$397$	5.36816	0.269420	0.134710	+	0.990885i	$0.456990\pi$
$398$	0	0
$399$	−0.610170	−0.0305467
$400$	0	0
$401$	0.481544	0.0240472	0.0120236	−	0.999928i	$-0.496173\pi$
$401$	0.481544	0.0240472	0.0120236	+	0.999928i	$0.496173\pi$
$402$	0	0
$403$	17.9725	0.895272
$404$	0	0
$405$	−3.85431	−0.191522
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−3.55105	−0.175588	−0.0877940	−	0.996139i	$-0.527982\pi$
$409$	−3.55105	−0.175588	−0.0877940	+	0.996139i	$0.527982\pi$
$410$	0	0
$411$	−2.48642	−0.122646
$412$	0	0
$413$	−0.576687	−0.0283769
$414$	0	0
$415$	−1.45952	−0.0716449
$416$	0	0
$417$	−26.7409	−1.30951
$418$	0	0
$419$	−8.56948	−0.418647	−0.209323	−	0.977846i	$-0.567126\pi$
$419$	−8.56948	−0.418647	−0.209323	+	0.977846i	$0.567126\pi$
$420$	0	0
$421$	3.55252	0.173139	0.0865696	−	0.996246i	$-0.472410\pi$
$421$	3.55252	0.173139	0.0865696	+	0.996246i	$0.472410\pi$
$422$	0	0
$423$	11.5795	0.563015
$424$	0	0
$425$	−19.6168	−0.951552
$426$	0	0
$427$	−1.89724	−0.0918136
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.6587	1.28411	0.642053	−	0.766661i	$-0.278083\pi$
$431$	26.6587	1.28411	0.642053	+	0.766661i	$0.278083\pi$
$432$	0	0
$433$	0.385551	0.0185284	0.00926421	−	0.999957i	$-0.497051\pi$
$433$	0.385551	0.0185284	0.00926421	+	0.999957i	$0.497051\pi$
$434$	0	0
$435$	−1.65137	−0.0791772
$436$	0	0
$437$	−3.47332	−0.166152
$438$	0	0
$439$	8.14358	0.388672	0.194336	−	0.980935i	$-0.437745\pi$
$439$	8.14358	0.388672	0.194336	+	0.980935i	$0.437745\pi$
$440$	0	0
$441$	−11.5629	−0.550615
$442$	0	0
$443$	−15.9326	−0.756978	−0.378489	−	0.925606i	$-0.623556\pi$
$443$	−15.9326	−0.756978	−0.378489	+	0.925606i	$0.623556\pi$
$444$	0	0
$445$	1.96725	0.0932566
$446$	0	0
$447$	−29.5296	−1.39670
$448$	0	0
$449$	26.4276	1.24720	0.623598	−	0.781745i	$-0.285670\pi$
$449$	26.4276	1.24720	0.623598	+	0.781745i	$0.285670\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−10.2310	−0.480694
$454$	0	0
$455$	0.203304	0.00953106
$456$	0	0
$457$	13.0798	0.611845	0.305923	−	0.952056i	$-0.401035\pi$
$457$	13.0798	0.611845	0.305923	+	0.952056i	$0.401035\pi$
$458$	0	0
$459$	−11.5423	−0.538751
$460$	0	0
$461$	5.64540	0.262932	0.131466	−	0.991321i	$-0.458032\pi$
$461$	5.64540	0.262932	0.131466	+	0.991321i	$0.458032\pi$
$462$	0	0
$463$	5.30598	0.246590	0.123295	−	0.992370i	$-0.460654\pi$
$463$	5.30598	0.246590	0.123295	+	0.992370i	$0.460654\pi$
$464$	0	0
$465$	−6.36438	−0.295141
$466$	0	0
$467$	−19.0238	−0.880317	−0.440158	−	0.897920i	$-0.645078\pi$
$467$	−19.0238	−0.880317	−0.440158	+	0.897920i	$0.645078\pi$
$468$	0	0
$469$	4.07738	0.188276
$470$	0	0
$471$	39.0524	1.79944
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.88201	−0.224002
$476$	0	0
$477$	−3.57678	−0.163769
$478$	0	0
$479$	18.0959	0.826823	0.413411	−	0.910544i	$-0.364337\pi$
$479$	18.0959	0.826823	0.413411	+	0.910544i	$0.364337\pi$
$480$	0	0
$481$	10.0593	0.458667
$482$	0	0
$483$	2.11932	0.0964322
$484$	0	0
$485$	−3.44377	−0.156373
$486$	0	0
$487$	29.1164	1.31939	0.659695	−	0.751533i	$-0.270685\pi$
$487$	29.1164	1.31939	0.659695	+	0.751533i	$0.270685\pi$
$488$	0	0
$489$	42.4520	1.91975
$490$	0	0
$491$	14.5832	0.658128	0.329064	−	0.944308i	$-0.393267\pi$
$491$	14.5832	0.658128	0.329064	+	0.944308i	$0.393267\pi$
$492$	0	0
$493$	−8.93824	−0.402558
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.0899228	−0.00403359
$498$	0	0
$499$	−1.01335	−0.0453639	−0.0226820	−	0.999743i	$-0.507221\pi$
$499$	−1.01335	−0.0453639	−0.0226820	+	0.999743i	$0.507221\pi$
$500$	0	0
$501$	54.1666	2.41999
$502$	0	0
$503$	−44.0282	−1.96312	−0.981560	−	0.191156i	$-0.938776\pi$
$503$	−44.0282	−1.96312	−0.981560	+	0.191156i	$0.938776\pi$
$504$	0	0
$505$	5.42559	0.241436
$506$	0	0
$507$	−18.5976	−0.825948
$508$	0	0
$509$	−26.7147	−1.18411	−0.592053	−	0.805899i	$-0.701682\pi$
$509$	−26.7147	−1.18411	−0.592053	+	0.805899i	$0.701682\pi$
$510$	0	0
$511$	3.38294	0.149652
$512$	0	0
$513$	−2.87254	−0.126826
$514$	0	0
$515$	−5.86146	−0.258287
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−13.4615	−0.590895
$520$	0	0
$521$	14.8432	0.650292	0.325146	−	0.945664i	$-0.394587\pi$
$521$	14.8432	0.650292	0.325146	+	0.945664i	$0.394587\pi$
$522$	0	0
$523$	6.13837	0.268412	0.134206	−	0.990953i	$-0.457152\pi$
$523$	6.13837	0.268412	0.134206	+	0.990953i	$0.457152\pi$
$524$	0	0
$525$	2.97885	0.130008
$526$	0	0
$527$	−34.4479	−1.50058
$528$	0	0
$529$	−10.9360	−0.475480
$530$	0	0
$531$	3.41296	0.148110
$532$	0	0
$533$	−19.3332	−0.837416
$534$	0	0
$535$	0.364639	0.0157647
$536$	0	0
$537$	−28.9132	−1.24770
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0.173931	0.00747786	0.00373893	−	0.999993i	$-0.498810\pi$
$541$	0.173931	0.00747786	0.00373893	+	0.999993i	$0.498810\pi$
$542$	0	0
$543$	12.4701	0.535143
$544$	0	0
$545$	0.463149	0.0198391
$546$	0	0
$547$	−15.0048	−0.641558	−0.320779	−	0.947154i	$-0.603945\pi$
$547$	−15.0048	−0.641558	−0.320779	+	0.947154i	$0.603945\pi$
$548$	0	0
$549$	11.2283	0.479211
$550$	0	0
$551$	−2.22445	−0.0947650
$552$	0	0
$553$	−0.346454	−0.0147327
$554$	0	0
$555$	−3.56220	−0.151207
$556$	0	0
$557$	−8.31914	−0.352493	−0.176247	−	0.984346i	$-0.556396\pi$
$557$	−8.31914	−0.352493	−0.176247	+	0.984346i	$0.556396\pi$
$558$	0	0
$559$	23.8617	1.00924
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.49149	−0.231439	−0.115719	−	0.993282i	$-0.536917\pi$
$563$	−5.49149	−0.231439	−0.115719	+	0.993282i	$0.536917\pi$
$564$	0	0
$565$	4.51144	0.189798
$566$	0	0
$567$	3.16793	0.133041
$568$	0	0
$569$	10.1783	0.426695	0.213348	−	0.976976i	$-0.431563\pi$
$569$	10.1783	0.426695	0.213348	+	0.976976i	$0.431563\pi$
$570$	0	0
$571$	9.23522	0.386482	0.193241	−	0.981151i	$-0.438100\pi$
$571$	9.23522	0.386482	0.193241	+	0.981151i	$0.438100\pi$
$572$	0	0
$573$	−0.283978	−0.0118634
$574$	0	0
$575$	16.9568	0.707147
$576$	0	0
$577$	−22.4835	−0.936002	−0.468001	−	0.883728i	$-0.655026\pi$
$577$	−22.4835	−0.936002	−0.468001	+	0.883728i	$0.655026\pi$
$578$	0	0
$579$	18.0340	0.749468
$580$	0	0
$581$	1.19961	0.0497680
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.20320	−0.0497463
$586$	0	0
$587$	−6.15496	−0.254043	−0.127021	−	0.991900i	$-0.540542\pi$
$587$	−6.15496	−0.254043	−0.127021	+	0.991900i	$0.540542\pi$
$588$	0	0
$589$	−8.57304	−0.353246
$590$	0	0
$591$	30.9248	1.27207
$592$	0	0
$593$	34.4716	1.41558	0.707790	−	0.706423i	$-0.249692\pi$
$593$	34.4716	1.41558	0.707790	+	0.706423i	$0.249692\pi$
$594$	0	0
$595$	−0.389675	−0.0159751
$596$	0	0
$597$	37.4266	1.53177
$598$	0	0
$599$	−10.0985	−0.412613	−0.206306	−	0.978487i	$-0.566144\pi$
$599$	−10.0985	−0.412613	−0.206306	+	0.978487i	$0.566144\pi$
$600$	0	0
$601$	−23.8750	−0.973881	−0.486941	−	0.873435i	$-0.661887\pi$
$601$	−23.8750	−0.973881	−0.486941	+	0.873435i	$0.661887\pi$
$602$	0	0
$603$	−24.1309	−0.982685
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.4581	−0.424481	−0.212241	−	0.977217i	$-0.568076\pi$
$607$	−10.4581	−0.424481	−0.212241	+	0.977217i	$0.568076\pi$
$608$	0	0
$609$	1.35730	0.0550004
$610$	0	0
$611$	−14.5285	−0.587759
$612$	0	0
$613$	11.8593	0.478994	0.239497	−	0.970897i	$-0.423018\pi$
$613$	11.8593	0.478994	0.239497	+	0.970897i	$0.423018\pi$
$614$	0	0
$615$	6.84626	0.276068
$616$	0	0
$617$	−47.9792	−1.93157	−0.965785	−	0.259344i	$-0.916494\pi$
$617$	−47.9792	−1.93157	−0.965785	+	0.259344i	$0.916494\pi$
$618$	0	0
$619$	1.17365	0.0471730	0.0235865	−	0.999722i	$-0.492491\pi$
$619$	1.17365	0.0471730	0.0235865	+	0.999722i	$0.492491\pi$
$620$	0	0
$621$	9.97725	0.400373
$622$	0	0
$623$	−1.61692	−0.0647806
$624$	0	0
$625$	23.2441	0.929763
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.2808	−0.768776
$630$	0	0
$631$	1.87488	0.0746377	0.0373188	−	0.999303i	$-0.488118\pi$
$631$	1.87488	0.0746377	0.0373188	+	0.999303i	$0.488118\pi$
$632$	0	0
$633$	−24.9129	−0.990198
$634$	0	0
$635$	3.63562	0.144275
$636$	0	0
$637$	14.5077	0.574814
$638$	0	0
$639$	0.532183	0.0210528
$640$	0	0
$641$	−25.5646	−1.00974	−0.504871	−	0.863195i	$-0.668460\pi$
$641$	−25.5646	−1.00974	−0.504871	+	0.863195i	$0.668460\pi$
$642$	0	0
$643$	−7.44931	−0.293772	−0.146886	−	0.989153i	$-0.546925\pi$
$643$	−7.44931	−0.293772	−0.146886	+	0.989153i	$0.546925\pi$
$644$	0	0
$645$	−8.44987	−0.332713
$646$	0	0
$647$	46.2066	1.81657	0.908285	−	0.418352i	$-0.137392\pi$
$647$	46.2066	1.81657	0.908285	+	0.418352i	$0.137392\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	5.23101	0.205019
$652$	0	0
$653$	25.2244	0.987105	0.493553	−	0.869716i	$-0.335698\pi$
$653$	25.2244	0.987105	0.493553	+	0.869716i	$0.335698\pi$
$654$	0	0
$655$	0.615460	0.0240480
$656$	0	0
$657$	−20.0210	−0.781094
$658$	0	0
$659$	32.0888	1.25000	0.625000	−	0.780625i	$-0.285099\pi$
$659$	32.0888	1.25000	0.625000	+	0.780625i	$0.285099\pi$
$660$	0	0
$661$	31.5125	1.22570	0.612848	−	0.790201i	$-0.290024\pi$
$661$	31.5125	1.22570	0.612848	+	0.790201i	$0.290024\pi$
$662$	0	0
$663$	−18.2054	−0.707039
$664$	0	0
$665$	−0.0969782	−0.00376065
$666$	0	0
$667$	7.72625	0.299162
$668$	0	0
$669$	24.2414	0.937227
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0.524385	0.0202136	0.0101068	−	0.999949i	$-0.496783\pi$
$673$	0.524385	0.0202136	0.0101068	+	0.999949i	$0.496783\pi$
$674$	0	0
$675$	14.0238	0.539775
$676$	0	0
$677$	30.6943	1.17968	0.589840	−	0.807520i	$-0.299191\pi$
$677$	30.6943	1.17968	0.589840	+	0.807520i	$0.299191\pi$
$678$	0	0
$679$	2.83050	0.108625
$680$	0	0
$681$	32.2605	1.23623
$682$	0	0
$683$	−5.94316	−0.227409	−0.113704	−	0.993515i	$-0.536272\pi$
$683$	−5.94316	−0.227409	−0.113704	+	0.993515i	$0.536272\pi$
$684$	0	0
$685$	−0.395183	−0.0150992
$686$	0	0
$687$	12.0683	0.460435
$688$	0	0
$689$	4.48767	0.170967
$690$	0	0
$691$	−43.9944	−1.67362	−0.836812	−	0.547490i	$-0.815584\pi$
$691$	−43.9944	−1.67362	−0.836812	+	0.547490i	$0.815584\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.25011	−0.161216
$696$	0	0
$697$	37.0562	1.40360
$698$	0	0
$699$	−45.5125	−1.72144
$700$	0	0
$701$	−14.1305	−0.533701	−0.266851	−	0.963738i	$-0.585983\pi$
$701$	−14.1305	−0.533701	−0.266851	+	0.963738i	$0.585983\pi$
$702$	0	0
$703$	−4.79841	−0.180975
$704$	0	0
$705$	5.14480	0.193764
$706$	0	0
$707$	−4.45940	−0.167713
$708$	0	0
$709$	2.02185	0.0759321	0.0379660	−	0.999279i	$-0.487912\pi$
$709$	2.02185	0.0759321	0.0379660	+	0.999279i	$0.487912\pi$
$710$	0	0
$711$	2.05039	0.0768957
$712$	0	0
$713$	29.7769	1.11515
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−62.8968	−2.34892
$718$	0	0
$719$	−27.9954	−1.04405	−0.522026	−	0.852930i	$-0.674824\pi$
$719$	−27.9954	−1.04405	−0.522026	+	0.852930i	$0.674824\pi$
$720$	0	0
$721$	4.81765	0.179419
$722$	0	0
$723$	−55.0697	−2.04806
$724$	0	0
$725$	10.8598	0.403323
$726$	0	0
$727$	18.8375	0.698643	0.349322	−	0.937003i	$-0.386412\pi$
$727$	18.8375	0.698643	0.349322	+	0.937003i	$0.386412\pi$
$728$	0	0
$729$	−0.123972	−0.00459157
$730$	0	0
$731$	−45.7359	−1.69160
$732$	0	0
$733$	−17.7717	−0.656414	−0.328207	−	0.944606i	$-0.606444\pi$
$733$	−17.7717	−0.656414	−0.328207	+	0.944606i	$0.606444\pi$
$734$	0	0
$735$	−5.13743	−0.189497
$736$	0	0
$737$	0	0
$738$	0	0
$739$	36.6947	1.34984	0.674918	−	0.737893i	$-0.264179\pi$
$739$	36.6947	1.34984	0.674918	+	0.737893i	$0.264179\pi$
$740$	0	0
$741$	−4.53077	−0.166442
$742$	0	0
$743$	38.9915	1.43046	0.715230	−	0.698889i	$-0.246322\pi$
$743$	38.9915	1.43046	0.715230	+	0.698889i	$0.246322\pi$
$744$	0	0
$745$	−4.69333	−0.171950
$746$	0	0
$747$	−7.09954	−0.259758
$748$	0	0
$749$	−0.299704	−0.0109509
$750$	0	0
$751$	−25.7396	−0.939250	−0.469625	−	0.882866i	$-0.655611\pi$
$751$	−25.7396	−0.939250	−0.469625	+	0.882866i	$0.655611\pi$
$752$	0	0
$753$	−51.3565	−1.87153
$754$	0	0
$755$	−1.62608	−0.0591790
$756$	0	0
$757$	6.60603	0.240100	0.120050	−	0.992768i	$-0.461694\pi$
$757$	6.60603	0.240100	0.120050	+	0.992768i	$0.461694\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.57948	−0.0935059	−0.0467530	−	0.998906i	$-0.514887\pi$
$761$	−2.57948	−0.0935059	−0.0467530	+	0.998906i	$0.514887\pi$
$762$	0	0
$763$	−0.380671	−0.0137812
$764$	0	0
$765$	2.30618	0.0833803
$766$	0	0
$767$	−4.28214	−0.154619
$768$	0	0
$769$	19.6295	0.707859	0.353930	−	0.935272i	$-0.384845\pi$
$769$	19.6295	0.707859	0.353930	+	0.935272i	$0.384845\pi$
$770$	0	0
$771$	−4.97841	−0.179293
$772$	0	0
$773$	2.86174	0.102930	0.0514648	−	0.998675i	$-0.483611\pi$
$773$	2.86174	0.102930	0.0514648	+	0.998675i	$0.483611\pi$
$774$	0	0
$775$	41.8536	1.50343
$776$	0	0
$777$	2.92784	0.105036
$778$	0	0
$779$	9.22214	0.330418
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.38983	0.228354
$784$	0	0
$785$	6.20684	0.221532
$786$	0	0
$787$	−23.3121	−0.830986	−0.415493	−	0.909596i	$-0.636391\pi$
$787$	−23.3121	−0.830986	−0.415493	+	0.909596i	$0.636391\pi$
$788$	0	0
$789$	−21.8290	−0.777134
$790$	0	0
$791$	−3.70805	−0.131843
$792$	0	0
$793$	−14.0878	−0.500271
$794$	0	0
$795$	−1.58917	−0.0563620
$796$	0	0
$797$	31.5637	1.11804	0.559022	−	0.829153i	$-0.311177\pi$
$797$	31.5637	1.11804	0.559022	+	0.829153i	$0.311177\pi$
$798$	0	0
$799$	27.8468	0.985149
$800$	0	0
$801$	9.56931	0.338115
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0.336836	0.0118719
$806$	0	0
$807$	−67.5787	−2.37888
$808$	0	0
$809$	−32.8575	−1.15521	−0.577605	−	0.816317i	$-0.696012\pi$
$809$	−32.8575	−1.15521	−0.577605	+	0.816317i	$0.696012\pi$
$810$	0	0
$811$	−55.3275	−1.94281	−0.971405	−	0.237428i	$-0.923696\pi$
$811$	−55.3275	−1.94281	−0.971405	+	0.237428i	$0.923696\pi$
$812$	0	0
$813$	−59.9956	−2.10414
$814$	0	0
$815$	6.74717	0.236343
$816$	0	0
$817$	−11.3823	−0.398215
$818$	0	0
$819$	0.988935	0.0345562
$820$	0	0
$821$	−46.4218	−1.62013	−0.810066	−	0.586339i	$-0.800569\pi$
$821$	−46.4218	−1.62013	−0.810066	+	0.586339i	$0.800569\pi$
$822$	0	0
$823$	13.1444	0.458184	0.229092	−	0.973405i	$-0.426424\pi$
$823$	13.1444	0.458184	0.229092	+	0.973405i	$0.426424\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.6733	1.06661	0.533307	−	0.845922i	$-0.320949\pi$
$827$	30.6733	1.06661	0.533307	+	0.845922i	$0.320949\pi$
$828$	0	0
$829$	26.8231	0.931603	0.465802	−	0.884889i	$-0.345766\pi$
$829$	26.8231	0.931603	0.465802	+	0.884889i	$0.345766\pi$
$830$	0	0
$831$	50.4358	1.74960
$832$	0	0
$833$	−27.8069	−0.963452
$834$	0	0
$835$	8.60905	0.297928
$836$	0	0
$837$	24.6264	0.851212
$838$	0	0
$839$	−46.7254	−1.61314	−0.806570	−	0.591139i	$-0.798679\pi$
$839$	−46.7254	−1.61314	−0.806570	+	0.591139i	$0.798679\pi$
$840$	0	0
$841$	−24.0518	−0.829372
$842$	0	0
$843$	10.8941	0.375212
$844$	0	0
$845$	−2.95583	−0.101684
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.1649	0.383177
$850$	0	0
$851$	16.6664	0.571317
$852$	0	0
$853$	−15.5638	−0.532893	−0.266447	−	0.963850i	$-0.585850\pi$
$853$	−15.5638	−0.532893	−0.266447	+	0.963850i	$0.585850\pi$
$854$	0	0
$855$	0.573939	0.0196283
$856$	0	0
$857$	39.6061	1.35292	0.676459	−	0.736481i	$-0.263514\pi$
$857$	39.6061	1.35292	0.676459	+	0.736481i	$0.263514\pi$
$858$	0	0
$859$	11.2278	0.383087	0.191544	−	0.981484i	$-0.438651\pi$
$859$	11.2278	0.383087	0.191544	+	0.981484i	$0.438651\pi$
$860$	0	0
$861$	−5.62707	−0.191770
$862$	0	0
$863$	38.5644	1.31275	0.656373	−	0.754436i	$-0.272090\pi$
$863$	38.5644	1.31275	0.656373	+	0.754436i	$0.272090\pi$
$864$	0	0
$865$	−2.13952	−0.0727460
$866$	0	0
$867$	−1.84634	−0.0627049
$868$	0	0
$869$	0	0
$870$	0	0
$871$	30.2763	1.02587
$872$	0	0
$873$	−16.7515	−0.566954
$874$	0	0
$875$	0.958339	0.0323978
$876$	0	0
$877$	−1.91490	−0.0646616	−0.0323308	−	0.999477i	$-0.510293\pi$
$877$	−1.91490	−0.0646616	−0.0323308	+	0.999477i	$0.510293\pi$
$878$	0	0
$879$	−4.99607	−0.168513
$880$	0	0
$881$	−41.5986	−1.40149	−0.700747	−	0.713410i	$-0.747150\pi$
$881$	−41.5986	−1.40149	−0.700747	+	0.713410i	$0.747150\pi$
$882$	0	0
$883$	27.2729	0.917806	0.458903	−	0.888486i	$-0.348242\pi$
$883$	27.2729	0.917806	0.458903	+	0.888486i	$0.348242\pi$
$884$	0	0
$885$	1.51639	0.0509727
$886$	0	0
$887$	−19.3828	−0.650809	−0.325405	−	0.945575i	$-0.605501\pi$
$887$	−19.3828	−0.650809	−0.325405	+	0.945575i	$0.605501\pi$
$888$	0	0
$889$	−2.98819	−0.100221
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.93022	0.231911
$894$	0	0
$895$	−4.59537	−0.153606
$896$	0	0
$897$	15.7368	0.525437
$898$	0	0
$899$	19.0703	0.636031
$900$	0	0
$901$	−8.60155	−0.286559
$902$	0	0
$903$	6.94511	0.231119
$904$	0	0
$905$	1.98195	0.0658823
$906$	0	0
$907$	29.4808	0.978894	0.489447	−	0.872033i	$-0.337199\pi$
$907$	29.4808	0.978894	0.489447	+	0.872033i	$0.337199\pi$
$908$	0	0
$909$	26.3917	0.875359
$910$	0	0
$911$	16.8987	0.559879	0.279939	−	0.960018i	$-0.409686\pi$
$911$	16.8987	0.559879	0.279939	+	0.960018i	$0.409686\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	4.98874	0.164923
$916$	0	0
$917$	−0.505858	−0.0167049
$918$	0	0
$919$	50.1383	1.65391	0.826955	−	0.562268i	$-0.190071\pi$
$919$	50.1383	1.65391	0.826955	+	0.562268i	$0.190071\pi$
$920$	0	0
$921$	−1.20202	−0.0396080
$922$	0	0
$923$	−0.667714	−0.0219781
$924$	0	0
$925$	23.4259	0.770238
$926$	0	0
$927$	−28.5119	−0.936455
$928$	0	0
$929$	19.0065	0.623583	0.311791	−	0.950151i	$-0.399071\pi$
$929$	19.0065	0.623583	0.311791	+	0.950151i	$0.399071\pi$
$930$	0	0
$931$	−6.92029	−0.226803
$932$	0	0
$933$	49.0513	1.60587
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.9969	−0.947288	−0.473644	−	0.880717i	$-0.657062\pi$
$937$	−28.9969	−0.947288	−0.473644	+	0.880717i	$0.657062\pi$
$938$	0	0
$939$	−21.8569	−0.713273
$940$	0	0
$941$	−55.2715	−1.80180	−0.900900	−	0.434027i	$-0.857092\pi$
$941$	−55.2715	−1.80180	−0.900900	+	0.434027i	$0.857092\pi$
$942$	0	0
$943$	−32.0315	−1.04309
$944$	0	0
$945$	0.278573	0.00906199
$946$	0	0
$947$	15.0687	0.489667	0.244834	−	0.969565i	$-0.421267\pi$
$947$	15.0687	0.489667	0.244834	+	0.969565i	$0.421267\pi$
$948$	0	0
$949$	25.1198	0.815422
$950$	0	0
$951$	−35.1380	−1.13943
$952$	0	0
$953$	53.9429	1.74738	0.873691	−	0.486482i	$-0.161720\pi$
$953$	53.9429	1.74738	0.873691	+	0.486482i	$0.161720\pi$
$954$	0	0
$955$	−0.0451345	−0.00146052
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.324809	0.0104886
$960$	0	0
$961$	42.4970	1.37087
$962$	0	0
$963$	1.77371	0.0571571
$964$	0	0
$965$	2.86626	0.0922682
$966$	0	0
$967$	−40.1935	−1.29253	−0.646267	−	0.763111i	$-0.723671\pi$
$967$	−40.1935	−1.29253	−0.646267	+	0.763111i	$0.723671\pi$
$968$	0	0
$969$	8.68415	0.278975
$970$	0	0
$971$	57.3812	1.84145	0.920726	−	0.390211i	$-0.127598\pi$
$971$	57.3812	1.84145	0.920726	+	0.390211i	$0.127598\pi$
$972$	0	0
$973$	3.49325	0.111988
$974$	0	0
$975$	22.1193	0.708383
$976$	0	0
$977$	−25.7585	−0.824087	−0.412044	−	0.911164i	$-0.635185\pi$
$977$	−25.7585	−0.824087	−0.412044	+	0.911164i	$0.635185\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.25290	0.0719295
$982$	0	0
$983$	15.3546	0.489736	0.244868	−	0.969556i	$-0.421255\pi$
$983$	15.3546	0.489736	0.244868	+	0.969556i	$0.421255\pi$
$984$	0	0
$985$	4.91507	0.156607
$986$	0	0
$987$	−4.22861	−0.134598
$988$	0	0
$989$	39.5343	1.25712
$990$	0	0
$991$	17.7796	0.564787	0.282394	−	0.959299i	$-0.408872\pi$
$991$	17.7796	0.564787	0.282394	+	0.959299i	$0.408872\pi$
$992$	0	0
$993$	−42.4943	−1.34852
$994$	0	0
$995$	5.94845	0.188579
$996$	0	0
$997$	34.6667	1.09791	0.548953	−	0.835854i	$-0.315027\pi$
$997$	34.6667	1.09791	0.548953	+	0.835854i	$0.315027\pi$
$998$	0	0
$999$	13.7836	0.436094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.s.1.9		10
11.3	even	5		836.2.j.b.229.2	✓	20
11.4	even	5		836.2.j.b.533.2	yes	20
11.10	odd	2		9196.2.a.t.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.b.229.2	✓	20	11.3	even	5
836.2.j.b.533.2	yes	20	11.4	even	5
9196.2.a.s.1.9		10	1.1	even	1	trivial
9196.2.a.t.1.9		10	11.10	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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+2.16122 q^{3} +0.343496 q^{5} -0.282327 q^{7} +1.67087 q^{9} -2.09639 q^{13} +0.742372 q^{15} +4.01817 q^{17} +1.00000 q^{19} -0.610170 q^{21} -3.47332 q^{23} -4.88201 q^{25} -2.87254 q^{27} -2.22445 q^{29} -8.57304 q^{31} -0.0969782 q^{35} -4.79841 q^{37} -4.53077 q^{39} +9.22214 q^{41} -11.3823 q^{43} +0.573939 q^{45} +6.93022 q^{47} -6.92029 q^{49} +8.68415 q^{51} -2.14066 q^{53} +2.16122 q^{57} +2.04262 q^{59} +6.72000 q^{61} -0.471732 q^{63} -0.720104 q^{65} -14.4421 q^{67} -7.50661 q^{69} +0.318506 q^{71} -11.9824 q^{73} -10.5511 q^{75} +1.22714 q^{79} -11.2208 q^{81} -4.24900 q^{83} +1.38023 q^{85} -4.80754 q^{87} +5.72713 q^{89} +0.591867 q^{91} -18.5282 q^{93} +0.343496 q^{95} -10.0256 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} - 4 q^{17} + 10 q^{19} - 6 q^{21} - 8 q^{23} - 10 q^{25} - 9 q^{27} + 3 q^{29} - 12 q^{31} - 9 q^{35} - 23 q^{37} + 18 q^{39} + 5 q^{41} + 14 q^{43}+ \cdots + 22 q^{97}+O(q^{100}) \) 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 - 4 * q^17 + 10 * q^19 - 6 * q^21 - 8 * q^23 - 10 * q^25 - 9 * q^27 + 3 * q^29 - 12 * q^31 - 9 * q^35 - 23 * q^37 + 18 * q^39 + 5 * q^41 + 14 * q^43 - 27 * q^47 - 17 * q^49 + 12 * q^51 - 10 * q^53 - 9 * q^59 - 4 * q^61 - 14 * q^63 - 7 * q^65 - 17 * q^67 - 3 * q^69 - 34 * q^71 + 9 * q^73 - 26 * q^75 + 16 * q^79 - 22 * q^81 - 6 * q^83 - 24 * q^85 - q^87 - 35 * q^89 - 35 * q^91 - 44 * q^93 + 8 * q^95 + 22 * q^97

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