LMFDB - Embedded newform 9196.2.a.v.1.13

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 = [14,0,-2,0,-6,0,4,0,12,0,0,0,4] 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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 25 x^{12} + 52 x^{11} + 222 x^{10} - 492 x^{9} - 800 x^{8} + 1984 x^{7} + 854 x^{6} + \cdots + 13 $ x^14 - 2x^13 - 25x^12 + 52x^11 + 222x^10 - 492x^9 - 800x^8 + 1984x^7 + 854x^6 - 2946x^5 + 300x^4 + 878x^3 - 194x^2 - 52*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.80947$ 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)

$f(q)$	$=$	$q+2.80947 q^{3} +1.54127 q^{5} +1.86523 q^{7} +4.89312 q^{9} +5.24687 q^{13} +4.33016 q^{15} +7.79245 q^{17} -1.00000 q^{19} +5.24030 q^{21} +1.79343 q^{23} -2.62448 q^{25} +5.31867 q^{27} -2.99075 q^{29} +0.235398 q^{31} +2.87482 q^{35} +2.81816 q^{37} +14.7409 q^{39} +9.72869 q^{41} -2.90049 q^{43} +7.54163 q^{45} -7.40622 q^{47} -3.52093 q^{49} +21.8927 q^{51} -10.4680 q^{53} -2.80947 q^{57} +0.554631 q^{59} -3.56898 q^{61} +9.12678 q^{63} +8.08685 q^{65} +1.59694 q^{67} +5.03858 q^{69} -4.80573 q^{71} -5.12586 q^{73} -7.37340 q^{75} +7.05506 q^{79} +0.263273 q^{81} -12.5690 q^{83} +12.0103 q^{85} -8.40241 q^{87} +10.2773 q^{89} +9.78660 q^{91} +0.661345 q^{93} -1.54127 q^{95} -17.6824 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} + 4 q^{13} + 2 q^{15} + 8 q^{17} - 14 q^{19} + 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} - 4 q^{29} - 2 q^{31} + 24 q^{35} - 10 q^{37} + 24 q^{39} + 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) $ 14 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 + 4 * q^13 + 2 * q^15 + 8 * q^17 - 14 * q^19 + 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 - 4 * q^29 - 2 * q^31 + 24 * q^35 - 10 * q^37 + 24 * q^39 + 4 * q^41 + 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 + 22 * q^51 + 6 * q^53 + 2 * q^57 - 6 * q^59 - 4 * q^61 - 16 * q^63 + 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 - 24 * q^73 - 10 * q^75 + 48 * q^79 + 2 * q^81 - 40 * q^85 + 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 + 6 * q^95 - 20 * 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
$3$	2.80947	1.62205	0.811024	−	0.585013i	$-0.198910\pi$
$3$	2.80947	1.62205	0.811024	+	0.585013i	$0.198910\pi$
$4$	0	0
$5$	1.54127	0.689278	0.344639	−	0.938735i	$-0.388001\pi$
$5$	1.54127	0.689278	0.344639	+	0.938735i	$0.388001\pi$
$6$	0	0
$7$	1.86523	0.704990	0.352495	−	0.935814i	$-0.385333\pi$
$7$	1.86523	0.704990	0.352495	+	0.935814i	$0.385333\pi$
$8$	0	0
$9$	4.89312	1.63104
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.24687	1.45522	0.727610	−	0.685991i	$-0.240631\pi$
$13$	5.24687	1.45522	0.727610	+	0.685991i	$0.240631\pi$
$14$	0	0
$15$	4.33016	1.11804
$16$	0	0
$17$	7.79245	1.88995	0.944973	−	0.327147i	$-0.106087\pi$
$17$	7.79245	1.88995	0.944973	+	0.327147i	$0.106087\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.24030	1.14353
$22$	0	0
$23$	1.79343	0.373956	0.186978	−	0.982364i	$-0.440131\pi$
$23$	1.79343	0.373956	0.186978	+	0.982364i	$0.440131\pi$
$24$	0	0
$25$	−2.62448	−0.524896
$26$	0	0
$27$	5.31867	1.02358
$28$	0	0
$29$	−2.99075	−0.555368	−0.277684	−	0.960672i	$-0.589567\pi$
$29$	−2.99075	−0.555368	−0.277684	+	0.960672i	$0.589567\pi$
$30$	0	0
$31$	0.235398	0.0422788	0.0211394	−	0.999777i	$-0.493271\pi$
$31$	0.235398	0.0422788	0.0211394	+	0.999777i	$0.493271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.87482	0.485934
$36$	0	0
$37$	2.81816	0.463302	0.231651	−	0.972799i	$-0.425587\pi$
$37$	2.81816	0.463302	0.231651	+	0.972799i	$0.425587\pi$
$38$	0	0
$39$	14.7409	2.36044
$40$	0	0
$41$	9.72869	1.51937	0.759683	−	0.650293i	$-0.225354\pi$
$41$	9.72869	1.51937	0.759683	+	0.650293i	$0.225354\pi$
$42$	0	0
$43$	−2.90049	−0.442320	−0.221160	−	0.975237i	$-0.570984\pi$
$43$	−2.90049	−0.442320	−0.221160	+	0.975237i	$0.570984\pi$
$44$	0	0
$45$	7.54163	1.12424
$46$	0	0
$47$	−7.40622	−1.08031	−0.540154	−	0.841566i	$-0.681634\pi$
$47$	−7.40622	−1.08031	−0.540154	+	0.841566i	$0.681634\pi$
$48$	0	0
$49$	−3.52093	−0.502990
$50$	0	0
$51$	21.8927	3.06558
$52$	0	0
$53$	−10.4680	−1.43789	−0.718947	−	0.695064i	$-0.755376\pi$
$53$	−10.4680	−1.43789	−0.718947	+	0.695064i	$0.755376\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.80947	−0.372123
$58$	0	0
$59$	0.554631	0.0722068	0.0361034	−	0.999348i	$-0.488505\pi$
$59$	0.554631	0.0722068	0.0361034	+	0.999348i	$0.488505\pi$
$60$	0	0
$61$	−3.56898	−0.456961	−0.228481	−	0.973548i	$-0.573376\pi$
$61$	−3.56898	−0.456961	−0.228481	+	0.973548i	$0.573376\pi$
$62$	0	0
$63$	9.12678	1.14987
$64$	0	0
$65$	8.08685	1.00305
$66$	0	0
$67$	1.59694	0.195097	0.0975487	−	0.995231i	$-0.468900\pi$
$67$	1.59694	0.195097	0.0975487	+	0.995231i	$0.468900\pi$
$68$	0	0
$69$	5.03858	0.606574
$70$	0	0
$71$	−4.80573	−0.570336	−0.285168	−	0.958478i	$-0.592049\pi$
$71$	−4.80573	−0.570336	−0.285168	+	0.958478i	$0.592049\pi$
$72$	0	0
$73$	−5.12586	−0.599937	−0.299968	−	0.953949i	$-0.596976\pi$
$73$	−5.12586	−0.599937	−0.299968	+	0.953949i	$0.596976\pi$
$74$	0	0
$75$	−7.37340	−0.851407
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.05506	0.793757	0.396878	−	0.917871i	$-0.370094\pi$
$79$	7.05506	0.793757	0.396878	+	0.917871i	$0.370094\pi$
$80$	0	0
$81$	0.263273	0.0292526
$82$	0	0
$83$	−12.5690	−1.37962	−0.689812	−	0.723988i	$-0.742307\pi$
$83$	−12.5690	−1.37962	−0.689812	+	0.723988i	$0.742307\pi$
$84$	0	0
$85$	12.0103	1.30270
$86$	0	0
$87$	−8.40241	−0.900833
$88$	0	0
$89$	10.2773	1.08939	0.544697	−	0.838633i	$-0.316645\pi$
$89$	10.2773	1.08939	0.544697	+	0.838633i	$0.316645\pi$
$90$	0	0
$91$	9.78660	1.02591
$92$	0	0
$93$	0.661345	0.0685782
$94$	0	0
$95$	−1.54127	−0.158131
$96$	0	0
$97$	−17.6824	−1.79537	−0.897686	−	0.440636i	$-0.854753\pi$
$97$	−17.6824	−1.79537	−0.897686	+	0.440636i	$0.854753\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.19286	0.616212	0.308106	−	0.951352i	$-0.400305\pi$
$101$	6.19286	0.616212	0.308106	+	0.951352i	$0.400305\pi$
$102$	0	0
$103$	11.6680	1.14969	0.574843	−	0.818264i	$-0.305063\pi$
$103$	11.6680	1.14969	0.574843	+	0.818264i	$0.305063\pi$
$104$	0	0
$105$	8.07673	0.788208
$106$	0	0
$107$	2.50561	0.242227	0.121113	−	0.992639i	$-0.461354\pi$
$107$	2.50561	0.242227	0.121113	+	0.992639i	$0.461354\pi$
$108$	0	0
$109$	−8.15730	−0.781327	−0.390664	−	0.920533i	$-0.627754\pi$
$109$	−8.15730	−0.781327	−0.390664	+	0.920533i	$0.627754\pi$
$110$	0	0
$111$	7.91752	0.751498
$112$	0	0
$113$	−4.12795	−0.388325	−0.194162	−	0.980969i	$-0.562199\pi$
$113$	−4.12795	−0.388325	−0.194162	+	0.980969i	$0.562199\pi$
$114$	0	0
$115$	2.76416	0.257759
$116$	0	0
$117$	25.6736	2.37352
$118$	0	0
$119$	14.5347	1.33239
$120$	0	0
$121$	0	0
$122$	0	0
$123$	27.3325	2.46449
$124$	0	0
$125$	−11.7514	−1.05108
$126$	0	0
$127$	−10.1520	−0.900845	−0.450423	−	0.892815i	$-0.648727\pi$
$127$	−10.1520	−0.900845	−0.450423	+	0.892815i	$0.648727\pi$
$128$	0	0
$129$	−8.14884	−0.717465
$130$	0	0
$131$	−6.59368	−0.576093	−0.288046	−	0.957616i	$-0.593006\pi$
$131$	−6.59368	−0.576093	−0.288046	+	0.957616i	$0.593006\pi$
$132$	0	0
$133$	−1.86523	−0.161736
$134$	0	0
$135$	8.19751	0.705530
$136$	0	0
$137$	−17.5770	−1.50170	−0.750852	−	0.660471i	$-0.770357\pi$
$137$	−17.5770	−1.50170	−0.750852	+	0.660471i	$0.770357\pi$
$138$	0	0
$139$	20.5855	1.74604	0.873018	−	0.487688i	$-0.162160\pi$
$139$	20.5855	1.74604	0.873018	+	0.487688i	$0.162160\pi$
$140$	0	0
$141$	−20.8076	−1.75231
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.60956	−0.382803
$146$	0	0
$147$	−9.89194	−0.815874
$148$	0	0
$149$	3.76525	0.308461	0.154231	−	0.988035i	$-0.450710\pi$
$149$	3.76525	0.308461	0.154231	+	0.988035i	$0.450710\pi$
$150$	0	0
$151$	14.5935	1.18760	0.593801	−	0.804612i	$-0.297627\pi$
$151$	14.5935	1.18760	0.593801	+	0.804612i	$0.297627\pi$
$152$	0	0
$153$	38.1294	3.08258
$154$	0	0
$155$	0.362813	0.0291418
$156$	0	0
$157$	−17.8442	−1.42412	−0.712060	−	0.702119i	$-0.752237\pi$
$157$	−17.8442	−1.42412	−0.712060	+	0.702119i	$0.752237\pi$
$158$	0	0
$159$	−29.4096	−2.33233
$160$	0	0
$161$	3.34515	0.263635
$162$	0	0
$163$	16.4706	1.29007	0.645037	−	0.764151i	$-0.276842\pi$
$163$	16.4706	1.29007	0.645037	+	0.764151i	$0.276842\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8569	−1.07228	−0.536141	−	0.844129i	$-0.680118\pi$
$167$	−13.8569	−1.07228	−0.536141	+	0.844129i	$0.680118\pi$
$168$	0	0
$169$	14.5296	1.11766
$170$	0	0
$171$	−4.89312	−0.374186
$172$	0	0
$173$	−15.0657	−1.14542	−0.572710	−	0.819758i	$-0.694108\pi$
$173$	−15.0657	−1.14542	−0.572710	+	0.819758i	$0.694108\pi$
$174$	0	0
$175$	−4.89525	−0.370046
$176$	0	0
$177$	1.55822	0.117123
$178$	0	0
$179$	−24.2405	−1.81182	−0.905911	−	0.423468i	$-0.860813\pi$
$179$	−24.2405	−1.81182	−0.905911	+	0.423468i	$0.860813\pi$
$180$	0	0
$181$	−20.5837	−1.52998	−0.764988	−	0.644045i	$-0.777255\pi$
$181$	−20.5837	−1.52998	−0.764988	+	0.644045i	$0.777255\pi$
$182$	0	0
$183$	−10.0269	−0.741213
$184$	0	0
$185$	4.34355	0.319344
$186$	0	0
$187$	0	0
$188$	0	0
$189$	9.92052	0.721612
$190$	0	0
$191$	−3.28957	−0.238025	−0.119012	−	0.992893i	$-0.537973\pi$
$191$	−3.28957	−0.238025	−0.119012	+	0.992893i	$0.537973\pi$
$192$	0	0
$193$	−21.9797	−1.58213	−0.791066	−	0.611731i	$-0.790473\pi$
$193$	−21.9797	−1.58213	−0.791066	+	0.611731i	$0.790473\pi$
$194$	0	0
$195$	22.7198	1.62700
$196$	0	0
$197$	8.53727	0.608255	0.304128	−	0.952631i	$-0.401635\pi$
$197$	8.53727	0.608255	0.304128	+	0.952631i	$0.401635\pi$
$198$	0	0
$199$	19.4322	1.37751	0.688755	−	0.724994i	$-0.258158\pi$
$199$	19.4322	1.37751	0.688755	+	0.724994i	$0.258158\pi$
$200$	0	0
$201$	4.48656	0.316457
$202$	0	0
$203$	−5.57842	−0.391528
$204$	0	0
$205$	14.9946	1.04727
$206$	0	0
$207$	8.77546	0.609937
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.342104	0.0235514	0.0117757	−	0.999931i	$-0.496252\pi$
$211$	0.342104	0.0235514	0.0117757	+	0.999931i	$0.496252\pi$
$212$	0	0
$213$	−13.5016	−0.925112
$214$	0	0
$215$	−4.47044	−0.304882
$216$	0	0
$217$	0.439071	0.0298061
$218$	0	0
$219$	−14.4010	−0.973127
$220$	0	0
$221$	40.8860	2.75029
$222$	0	0
$223$	17.1816	1.15057	0.575284	−	0.817954i	$-0.304892\pi$
$223$	17.1816	1.15057	0.575284	+	0.817954i	$0.304892\pi$
$224$	0	0
$225$	−12.8419	−0.856127
$226$	0	0
$227$	7.47515	0.496143	0.248072	−	0.968742i	$-0.420203\pi$
$227$	7.47515	0.496143	0.248072	+	0.968742i	$0.420203\pi$
$228$	0	0
$229$	−22.7824	−1.50550	−0.752752	−	0.658304i	$-0.771274\pi$
$229$	−22.7824	−1.50550	−0.752752	+	0.658304i	$0.771274\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.12655	−0.0738026	−0.0369013	−	0.999319i	$-0.511749\pi$
$233$	−1.12655	−0.0738026	−0.0369013	+	0.999319i	$0.511749\pi$
$234$	0	0
$235$	−11.4150	−0.744633
$236$	0	0
$237$	19.8210	1.28751
$238$	0	0
$239$	14.1200	0.913346	0.456673	−	0.889635i	$-0.349041\pi$
$239$	14.1200	0.913346	0.456673	+	0.889635i	$0.349041\pi$
$240$	0	0
$241$	10.8532	0.699115	0.349558	−	0.936915i	$-0.386332\pi$
$241$	10.8532	0.699115	0.349558	+	0.936915i	$0.386332\pi$
$242$	0	0
$243$	−15.2163	−0.976129
$244$	0	0
$245$	−5.42671	−0.346700
$246$	0	0
$247$	−5.24687	−0.333850
$248$	0	0
$249$	−35.3122	−2.23782
$250$	0	0
$251$	24.9764	1.57650	0.788249	−	0.615356i	$-0.210988\pi$
$251$	24.9764	1.57650	0.788249	+	0.615356i	$0.210988\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	33.7425	2.11304
$256$	0	0
$257$	−8.89768	−0.555022	−0.277511	−	0.960722i	$-0.589510\pi$
$257$	−8.89768	−0.555022	−0.277511	+	0.960722i	$0.589510\pi$
$258$	0	0
$259$	5.25650	0.326623
$260$	0	0
$261$	−14.6341	−0.905827
$262$	0	0
$263$	−4.61780	−0.284746	−0.142373	−	0.989813i	$-0.545473\pi$
$263$	−4.61780	−0.284746	−0.142373	+	0.989813i	$0.545473\pi$
$264$	0	0
$265$	−16.1341	−0.991109
$266$	0	0
$267$	28.8738	1.76705
$268$	0	0
$269$	20.2974	1.23756	0.618778	−	0.785566i	$-0.287628\pi$
$269$	20.2974	1.23756	0.618778	+	0.785566i	$0.287628\pi$
$270$	0	0
$271$	18.8874	1.14733	0.573663	−	0.819092i	$-0.305522\pi$
$271$	18.8874	1.14733	0.573663	+	0.819092i	$0.305522\pi$
$272$	0	0
$273$	27.4952	1.66408
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.0953	−0.846904	−0.423452	−	0.905919i	$-0.639182\pi$
$277$	−14.0953	−0.846904	−0.423452	+	0.905919i	$0.639182\pi$
$278$	0	0
$279$	1.15183	0.0689584
$280$	0	0
$281$	−9.63700	−0.574895	−0.287448	−	0.957796i	$-0.592807\pi$
$281$	−9.63700	−0.574895	−0.287448	+	0.957796i	$0.592807\pi$
$282$	0	0
$283$	−16.7048	−0.992999	−0.496500	−	0.868037i	$-0.665382\pi$
$283$	−16.7048	−0.992999	−0.496500	+	0.868037i	$0.665382\pi$
$284$	0	0
$285$	−4.33016	−0.256496
$286$	0	0
$287$	18.1462	1.07114
$288$	0	0
$289$	43.7223	2.57190
$290$	0	0
$291$	−49.6781	−2.91218
$292$	0	0
$293$	−11.7168	−0.684503	−0.342251	−	0.939608i	$-0.611189\pi$
$293$	−11.7168	−0.684503	−0.342251	+	0.939608i	$0.611189\pi$
$294$	0	0
$295$	0.854838	0.0497706
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.40988	0.544188
$300$	0	0
$301$	−5.41007	−0.311831
$302$	0	0
$303$	17.3986	0.999526
$304$	0	0
$305$	−5.50077	−0.314973
$306$	0	0
$307$	13.9523	0.796302	0.398151	−	0.917320i	$-0.369652\pi$
$307$	13.9523	0.796302	0.398151	+	0.917320i	$0.369652\pi$
$308$	0	0
$309$	32.7810	1.86484
$310$	0	0
$311$	20.7977	1.17933	0.589664	−	0.807649i	$-0.299260\pi$
$311$	20.7977	1.17933	0.589664	+	0.807649i	$0.299260\pi$
$312$	0	0
$313$	−13.2692	−0.750018	−0.375009	−	0.927021i	$-0.622360\pi$
$313$	−13.2692	−0.750018	−0.375009	+	0.927021i	$0.622360\pi$
$314$	0	0
$315$	14.0669	0.792577
$316$	0	0
$317$	−8.65780	−0.486270	−0.243135	−	0.969992i	$-0.578176\pi$
$317$	−8.65780	−0.486270	−0.243135	+	0.969992i	$0.578176\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	7.03944	0.392903
$322$	0	0
$323$	−7.79245	−0.433584
$324$	0	0
$325$	−13.7703	−0.763839
$326$	0	0
$327$	−22.9177	−1.26735
$328$	0	0
$329$	−13.8143	−0.761606
$330$	0	0
$331$	1.75889	0.0966771	0.0483385	−	0.998831i	$-0.484607\pi$
$331$	1.75889	0.0966771	0.0483385	+	0.998831i	$0.484607\pi$
$332$	0	0
$333$	13.7896	0.755664
$334$	0	0
$335$	2.46132	0.134476
$336$	0	0
$337$	20.0982	1.09482	0.547410	−	0.836865i	$-0.315614\pi$
$337$	20.0982	1.09482	0.547410	+	0.836865i	$0.315614\pi$
$338$	0	0
$339$	−11.5974	−0.629882
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.6239	−1.05959
$344$	0	0
$345$	7.76583	0.418098
$346$	0	0
$347$	32.4866	1.74397	0.871986	−	0.489531i	$-0.162832\pi$
$347$	32.4866	1.74397	0.871986	+	0.489531i	$0.162832\pi$
$348$	0	0
$349$	−8.48961	−0.454438	−0.227219	−	0.973844i	$-0.572963\pi$
$349$	−8.48961	−0.454438	−0.227219	+	0.973844i	$0.572963\pi$
$350$	0	0
$351$	27.9064	1.48953
$352$	0	0
$353$	9.66484	0.514408	0.257204	−	0.966357i	$-0.417199\pi$
$353$	9.66484	0.514408	0.257204	+	0.966357i	$0.417199\pi$
$354$	0	0
$355$	−7.40694	−0.393120
$356$	0	0
$357$	40.8348	2.16121
$358$	0	0
$359$	32.9359	1.73829	0.869146	−	0.494556i	$-0.164669\pi$
$359$	32.9359	1.73829	0.869146	+	0.494556i	$0.164669\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.90035	−0.413523
$366$	0	0
$367$	−11.8270	−0.617365	−0.308683	−	0.951165i	$-0.599888\pi$
$367$	−11.8270	−0.617365	−0.308683	+	0.951165i	$0.599888\pi$
$368$	0	0
$369$	47.6037	2.47815
$370$	0	0
$371$	−19.5253	−1.01370
$372$	0	0
$373$	7.09287	0.367255	0.183628	−	0.982996i	$-0.441216\pi$
$373$	7.09287	0.367255	0.183628	+	0.982996i	$0.441216\pi$
$374$	0	0
$375$	−33.0152	−1.70490
$376$	0	0
$377$	−15.6921	−0.808182
$378$	0	0
$379$	−3.01618	−0.154931	−0.0774654	−	0.996995i	$-0.524683\pi$
$379$	−3.01618	−0.154931	−0.0774654	+	0.996995i	$0.524683\pi$
$380$	0	0
$381$	−28.5218	−1.46121
$382$	0	0
$383$	−9.43358	−0.482033	−0.241017	−	0.970521i	$-0.577481\pi$
$383$	−9.43358	−0.482033	−0.241017	+	0.970521i	$0.577481\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.1924	−0.721442
$388$	0	0
$389$	24.5734	1.24592	0.622960	−	0.782254i	$-0.285930\pi$
$389$	24.5734	1.24592	0.622960	+	0.782254i	$0.285930\pi$
$390$	0	0
$391$	13.9752	0.706756
$392$	0	0
$393$	−18.5247	−0.934450
$394$	0	0
$395$	10.8738	0.547119
$396$	0	0
$397$	13.3211	0.668567	0.334283	−	0.942473i	$-0.391506\pi$
$397$	13.3211	0.668567	0.334283	+	0.942473i	$0.391506\pi$
$398$	0	0
$399$	−5.24030	−0.262343
$400$	0	0
$401$	21.3127	1.06431	0.532153	−	0.846648i	$-0.321383\pi$
$401$	21.3127	1.06431	0.532153	+	0.846648i	$0.321383\pi$
$402$	0	0
$403$	1.23510	0.0615249
$404$	0	0
$405$	0.405776	0.0201632
$406$	0	0
$407$	0	0
$408$	0	0
$409$	19.1599	0.947397	0.473699	−	0.880687i	$-0.342919\pi$
$409$	19.1599	0.947397	0.473699	+	0.880687i	$0.342919\pi$
$410$	0	0
$411$	−49.3820	−2.43583
$412$	0	0
$413$	1.03451	0.0509051
$414$	0	0
$415$	−19.3722	−0.950945
$416$	0	0
$417$	57.8342	2.83215
$418$	0	0
$419$	2.56201	0.125162	0.0625811	−	0.998040i	$-0.480067\pi$
$419$	2.56201	0.125162	0.0625811	+	0.998040i	$0.480067\pi$
$420$	0	0
$421$	12.7793	0.622825	0.311413	−	0.950275i	$-0.399198\pi$
$421$	12.7793	0.622825	0.311413	+	0.950275i	$0.399198\pi$
$422$	0	0
$423$	−36.2395	−1.76203
$424$	0	0
$425$	−20.4511	−0.992026
$426$	0	0
$427$	−6.65696	−0.322153
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.1639	−1.35661	−0.678303	−	0.734782i	$-0.737284\pi$
$431$	−28.1639	−1.35661	−0.678303	+	0.734782i	$0.737284\pi$
$432$	0	0
$433$	11.6009	0.557505	0.278752	−	0.960363i	$-0.410079\pi$
$433$	11.6009	0.557505	0.278752	+	0.960363i	$0.410079\pi$
$434$	0	0
$435$	−12.9504	−0.620925
$436$	0	0
$437$	−1.79343	−0.0857913
$438$	0	0
$439$	−1.47386	−0.0703435	−0.0351717	−	0.999381i	$-0.511198\pi$
$439$	−1.47386	−0.0703435	−0.0351717	+	0.999381i	$0.511198\pi$
$440$	0	0
$441$	−17.2283	−0.820397
$442$	0	0
$443$	13.4441	0.638749	0.319374	−	0.947629i	$-0.396527\pi$
$443$	13.4441	0.638749	0.319374	+	0.947629i	$0.396527\pi$
$444$	0	0
$445$	15.8401	0.750895
$446$	0	0
$447$	10.5783	0.500339
$448$	0	0
$449$	21.8671	1.03197	0.515985	−	0.856598i	$-0.327426\pi$
$449$	21.8671	1.03197	0.515985	+	0.856598i	$0.327426\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	41.0000	1.92635
$454$	0	0
$455$	15.0838	0.707140
$456$	0	0
$457$	−5.70673	−0.266950	−0.133475	−	0.991052i	$-0.542614\pi$
$457$	−5.70673	−0.266950	−0.133475	+	0.991052i	$0.542614\pi$
$458$	0	0
$459$	41.4455	1.93451
$460$	0	0
$461$	−12.9195	−0.601722	−0.300861	−	0.953668i	$-0.597274\pi$
$461$	−12.9195	−0.601722	−0.300861	+	0.953668i	$0.597274\pi$
$462$	0	0
$463$	28.5096	1.32495	0.662477	−	0.749083i	$-0.269505\pi$
$463$	28.5096	1.32495	0.662477	+	0.749083i	$0.269505\pi$
$464$	0	0
$465$	1.01931	0.0472695
$466$	0	0
$467$	2.58383	0.119565	0.0597826	−	0.998211i	$-0.480959\pi$
$467$	2.58383	0.119565	0.0597826	+	0.998211i	$0.480959\pi$
$468$	0	0
$469$	2.97866	0.137542
$470$	0	0
$471$	−50.1326	−2.30999
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.62448	0.120419
$476$	0	0
$477$	−51.2214	−2.34526
$478$	0	0
$479$	−32.8637	−1.50158	−0.750791	−	0.660539i	$-0.770328\pi$
$479$	−32.8637	−1.50158	−0.750791	+	0.660539i	$0.770328\pi$
$480$	0	0
$481$	14.7865	0.674206
$482$	0	0
$483$	9.39810	0.427628
$484$	0	0
$485$	−27.2533	−1.23751
$486$	0	0
$487$	−40.5729	−1.83854	−0.919268	−	0.393633i	$-0.871218\pi$
$487$	−40.5729	−1.83854	−0.919268	+	0.393633i	$0.871218\pi$
$488$	0	0
$489$	46.2736	2.09256
$490$	0	0
$491$	−14.5913	−0.658495	−0.329247	−	0.944244i	$-0.606795\pi$
$491$	−14.5913	−0.658495	−0.329247	+	0.944244i	$0.606795\pi$
$492$	0	0
$493$	−23.3052	−1.04962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.96378	−0.402081
$498$	0	0
$499$	−37.3905	−1.67383	−0.836914	−	0.547334i	$-0.815643\pi$
$499$	−37.3905	−1.67383	−0.836914	+	0.547334i	$0.815643\pi$
$500$	0	0
$501$	−38.9306	−1.73929
$502$	0	0
$503$	−4.25395	−0.189674	−0.0948371	−	0.995493i	$-0.530233\pi$
$503$	−4.25395	−0.189674	−0.0948371	+	0.995493i	$0.530233\pi$
$504$	0	0
$505$	9.54488	0.424741
$506$	0	0
$507$	40.8206	1.81291
$508$	0	0
$509$	−0.379840	−0.0168361	−0.00841806	−	0.999965i	$-0.502680\pi$
$509$	−0.379840	−0.0168361	−0.00841806	+	0.999965i	$0.502680\pi$
$510$	0	0
$511$	−9.56090	−0.422949
$512$	0	0
$513$	−5.31867	−0.234825
$514$	0	0
$515$	17.9836	0.792452
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−42.3265	−1.85793
$520$	0	0
$521$	−16.6703	−0.730339	−0.365169	−	0.930941i	$-0.618989\pi$
$521$	−16.6703	−0.730339	−0.365169	+	0.930941i	$0.618989\pi$
$522$	0	0
$523$	9.60201	0.419867	0.209933	−	0.977716i	$-0.432675\pi$
$523$	9.60201	0.419867	0.209933	+	0.977716i	$0.432675\pi$
$524$	0	0
$525$	−13.7531	−0.600233
$526$	0	0
$527$	1.83433	0.0799047
$528$	0	0
$529$	−19.7836	−0.860157
$530$	0	0
$531$	2.71388	0.117772
$532$	0	0
$533$	51.0452	2.21101
$534$	0	0
$535$	3.86183	0.166962
$536$	0	0
$537$	−68.1030	−2.93886
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.1325	1.46747	0.733735	−	0.679436i	$-0.237775\pi$
$541$	34.1325	1.46747	0.733735	+	0.679436i	$0.237775\pi$
$542$	0	0
$543$	−57.8293	−2.48169
$544$	0	0
$545$	−12.5726	−0.538552
$546$	0	0
$547$	−32.0027	−1.36834	−0.684168	−	0.729324i	$-0.739835\pi$
$547$	−32.0027	−1.36834	−0.684168	+	0.729324i	$0.739835\pi$
$548$	0	0
$549$	−17.4635	−0.745322
$550$	0	0
$551$	2.99075	0.127410
$552$	0	0
$553$	13.1593	0.559590
$554$	0	0
$555$	12.2031	0.517991
$556$	0	0
$557$	−8.41274	−0.356459	−0.178230	−	0.983989i	$-0.557037\pi$
$557$	−8.41274	−0.356459	−0.178230	+	0.983989i	$0.557037\pi$
$558$	0	0
$559$	−15.2185	−0.643673
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.6071	1.75353	0.876766	−	0.480918i	$-0.159697\pi$
$563$	41.6071	1.75353	0.876766	+	0.480918i	$0.159697\pi$
$564$	0	0
$565$	−6.36229	−0.267664
$566$	0	0
$567$	0.491065	0.0206228
$568$	0	0
$569$	20.3380	0.852612	0.426306	−	0.904579i	$-0.359815\pi$
$569$	20.3380	0.852612	0.426306	+	0.904579i	$0.359815\pi$
$570$	0	0
$571$	11.0944	0.464286	0.232143	−	0.972682i	$-0.425426\pi$
$571$	11.0944	0.464286	0.232143	+	0.972682i	$0.425426\pi$
$572$	0	0
$573$	−9.24194	−0.386088
$574$	0	0
$575$	−4.70682	−0.196288
$576$	0	0
$577$	28.4516	1.18446	0.592228	−	0.805771i	$-0.298249\pi$
$577$	28.4516	1.18446	0.592228	+	0.805771i	$0.298249\pi$
$578$	0	0
$579$	−61.7512	−2.56629
$580$	0	0
$581$	−23.4440	−0.972621
$582$	0	0
$583$	0	0
$584$	0	0
$585$	39.5700	1.63602
$586$	0	0
$587$	−6.27687	−0.259074	−0.129537	−	0.991575i	$-0.541349\pi$
$587$	−6.27687	−0.259074	−0.129537	+	0.991575i	$0.541349\pi$
$588$	0	0
$589$	−0.235398	−0.00969942
$590$	0	0
$591$	23.9852	0.986619
$592$	0	0
$593$	43.8980	1.80267	0.901337	−	0.433118i	$-0.142587\pi$
$593$	43.8980	1.80267	0.901337	+	0.433118i	$0.142587\pi$
$594$	0	0
$595$	22.4019	0.918389
$596$	0	0
$597$	54.5941	2.23439
$598$	0	0
$599$	24.0381	0.982168	0.491084	−	0.871112i	$-0.336601\pi$
$599$	24.0381	0.982168	0.491084	+	0.871112i	$0.336601\pi$
$600$	0	0
$601$	−34.1183	−1.39171	−0.695857	−	0.718180i	$-0.744975\pi$
$601$	−34.1183	−1.39171	−0.695857	+	0.718180i	$0.744975\pi$
$602$	0	0
$603$	7.81403	0.318212
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.6867	0.636702	0.318351	−	0.947973i	$-0.396871\pi$
$607$	15.6867	0.636702	0.318351	+	0.947973i	$0.396871\pi$
$608$	0	0
$609$	−15.6724	−0.635078
$610$	0	0
$611$	−38.8595	−1.57209
$612$	0	0
$613$	−42.0977	−1.70031	−0.850156	−	0.526531i	$-0.823492\pi$
$613$	−42.0977	−1.70031	−0.850156	+	0.526531i	$0.823492\pi$
$614$	0	0
$615$	42.1268	1.69872
$616$	0	0
$617$	12.1355	0.488558	0.244279	−	0.969705i	$-0.421449\pi$
$617$	12.1355	0.488558	0.244279	+	0.969705i	$0.421449\pi$
$618$	0	0
$619$	−35.7908	−1.43855	−0.719277	−	0.694723i	$-0.755527\pi$
$619$	−35.7908	−1.43855	−0.719277	+	0.694723i	$0.755527\pi$
$620$	0	0
$621$	9.53865	0.382773
$622$	0	0
$623$	19.1695	0.768011
$624$	0	0
$625$	−4.98970	−0.199588
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.9603	0.875616
$630$	0	0
$631$	42.9721	1.71069	0.855347	−	0.518056i	$-0.173344\pi$
$631$	42.9721	1.71069	0.855347	+	0.518056i	$0.173344\pi$
$632$	0	0
$633$	0.961130	0.0382015
$634$	0	0
$635$	−15.6470	−0.620933
$636$	0	0
$637$	−18.4739	−0.731961
$638$	0	0
$639$	−23.5150	−0.930241
$640$	0	0
$641$	−33.7064	−1.33132	−0.665661	−	0.746254i	$-0.731850\pi$
$641$	−33.7064	−1.33132	−0.665661	+	0.746254i	$0.731850\pi$
$642$	0	0
$643$	13.7572	0.542530	0.271265	−	0.962505i	$-0.412558\pi$
$643$	13.7572	0.542530	0.271265	+	0.962505i	$0.412558\pi$
$644$	0	0
$645$	−12.5596	−0.494533
$646$	0	0
$647$	50.7127	1.99372	0.996860	−	0.0791837i	$-0.0252314\pi$
$647$	50.7127	1.99372	0.996860	+	0.0791837i	$0.0252314\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.23356	0.0483469
$652$	0	0
$653$	2.43762	0.0953913	0.0476956	−	0.998862i	$-0.484812\pi$
$653$	2.43762	0.0953913	0.0476956	+	0.998862i	$0.484812\pi$
$654$	0	0
$655$	−10.1627	−0.397088
$656$	0	0
$657$	−25.0815	−0.978521
$658$	0	0
$659$	23.0932	0.899582	0.449791	−	0.893134i	$-0.351498\pi$
$659$	23.0932	0.899582	0.449791	+	0.893134i	$0.351498\pi$
$660$	0	0
$661$	−9.25390	−0.359935	−0.179967	−	0.983673i	$-0.557599\pi$
$661$	−9.25390	−0.359935	−0.179967	+	0.983673i	$0.557599\pi$
$662$	0	0
$663$	114.868	4.46110
$664$	0	0
$665$	−2.87482	−0.111481
$666$	0	0
$667$	−5.36369	−0.207683
$668$	0	0
$669$	48.2713	1.86628
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.4680	−1.17446	−0.587228	−	0.809421i	$-0.699781\pi$
$673$	−30.4680	−1.17446	−0.587228	+	0.809421i	$0.699781\pi$
$674$	0	0
$675$	−13.9587	−0.537272
$676$	0	0
$677$	−2.23854	−0.0860342	−0.0430171	−	0.999074i	$-0.513697\pi$
$677$	−2.23854	−0.0860342	−0.0430171	+	0.999074i	$0.513697\pi$
$678$	0	0
$679$	−32.9816	−1.26572
$680$	0	0
$681$	21.0012	0.804768
$682$	0	0
$683$	11.2964	0.432246	0.216123	−	0.976366i	$-0.430659\pi$
$683$	11.2964	0.432246	0.216123	+	0.976366i	$0.430659\pi$
$684$	0	0
$685$	−27.0909	−1.03509
$686$	0	0
$687$	−64.0065	−2.44200
$688$	0	0
$689$	−54.9244	−2.09245
$690$	0	0
$691$	13.9515	0.530742	0.265371	−	0.964146i	$-0.414506\pi$
$691$	13.9515	0.530742	0.265371	+	0.964146i	$0.414506\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.7278	1.20350
$696$	0	0
$697$	75.8104	2.87152
$698$	0	0
$699$	−3.16500	−0.119711
$700$	0	0
$701$	−34.6890	−1.31018	−0.655092	−	0.755549i	$-0.727370\pi$
$701$	−34.6890	−1.31018	−0.655092	+	0.755549i	$0.727370\pi$
$702$	0	0
$703$	−2.81816	−0.106289
$704$	0	0
$705$	−32.0701	−1.20783
$706$	0	0
$707$	11.5511	0.434423
$708$	0	0
$709$	−41.6961	−1.56593	−0.782964	−	0.622067i	$-0.786293\pi$
$709$	−41.6961	−1.56593	−0.782964	+	0.622067i	$0.786293\pi$
$710$	0	0
$711$	34.5213	1.29465
$712$	0	0
$713$	0.422170	0.0158104
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.6697	1.48149
$718$	0	0
$719$	−32.4310	−1.20947	−0.604737	−	0.796425i	$-0.706722\pi$
$719$	−32.4310	−1.20947	−0.604737	+	0.796425i	$0.706722\pi$
$720$	0	0
$721$	21.7635	0.810516
$722$	0	0
$723$	30.4917	1.13400
$724$	0	0
$725$	7.84916	0.291510
$726$	0	0
$727$	12.2214	0.453268	0.226634	−	0.973980i	$-0.427228\pi$
$727$	12.2214	0.453268	0.226634	+	0.973980i	$0.427228\pi$
$728$	0	0
$729$	−43.5397	−1.61258
$730$	0	0
$731$	−22.6019	−0.835962
$732$	0	0
$733$	−12.3011	−0.454353	−0.227176	−	0.973854i	$-0.572949\pi$
$733$	−12.3011	−0.454353	−0.227176	+	0.973854i	$0.572949\pi$
$734$	0	0
$735$	−15.2462	−0.562364
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.7857	−0.654258	−0.327129	−	0.944980i	$-0.606081\pi$
$739$	−17.7857	−0.654258	−0.327129	+	0.944980i	$0.606081\pi$
$740$	0	0
$741$	−14.7409	−0.541521
$742$	0	0
$743$	−40.9433	−1.50206	−0.751032	−	0.660266i	$-0.770443\pi$
$743$	−40.9433	−1.50206	−0.751032	+	0.660266i	$0.770443\pi$
$744$	0	0
$745$	5.80327	0.212615
$746$	0	0
$747$	−61.5015	−2.25022
$748$	0	0
$749$	4.67354	0.170767
$750$	0	0
$751$	42.3079	1.54384	0.771919	−	0.635721i	$-0.219297\pi$
$751$	42.3079	1.54384	0.771919	+	0.635721i	$0.219297\pi$
$752$	0	0
$753$	70.1706	2.55716
$754$	0	0
$755$	22.4926	0.818588
$756$	0	0
$757$	5.41121	0.196674	0.0983369	−	0.995153i	$-0.468648\pi$
$757$	5.41121	0.196674	0.0983369	+	0.995153i	$0.468648\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	43.7033	1.58424	0.792122	−	0.610362i	$-0.208976\pi$
$761$	43.7033	1.58424	0.792122	+	0.610362i	$0.208976\pi$
$762$	0	0
$763$	−15.2152	−0.550828
$764$	0	0
$765$	58.7678	2.12475
$766$	0	0
$767$	2.91008	0.105077
$768$	0	0
$769$	0.759135	0.0273751	0.0136875	−	0.999906i	$-0.495643\pi$
$769$	0.759135	0.0273751	0.0136875	+	0.999906i	$0.495643\pi$
$770$	0	0
$771$	−24.9978	−0.900273
$772$	0	0
$773$	20.8205	0.748862	0.374431	−	0.927255i	$-0.377838\pi$
$773$	20.8205	0.748862	0.374431	+	0.927255i	$0.377838\pi$
$774$	0	0
$775$	−0.617798	−0.0221920
$776$	0	0
$777$	14.7680	0.529798
$778$	0	0
$779$	−9.72869	−0.348567
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−15.9068	−0.568462
$784$	0	0
$785$	−27.5027	−0.981614
$786$	0	0
$787$	−7.22568	−0.257568	−0.128784	−	0.991673i	$-0.541107\pi$
$787$	−7.22568	−0.257568	−0.128784	+	0.991673i	$0.541107\pi$
$788$	0	0
$789$	−12.9736	−0.461872
$790$	0	0
$791$	−7.69956	−0.273765
$792$	0	0
$793$	−18.7260	−0.664979
$794$	0	0
$795$	−45.3282	−1.60763
$796$	0	0
$797$	25.3558	0.898149	0.449075	−	0.893494i	$-0.351754\pi$
$797$	25.3558	0.898149	0.449075	+	0.893494i	$0.351754\pi$
$798$	0	0
$799$	−57.7126	−2.04173
$800$	0	0
$801$	50.2882	1.77684
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.15579	0.181718
$806$	0	0
$807$	57.0250	2.00737
$808$	0	0
$809$	36.1256	1.27011	0.635055	−	0.772467i	$-0.280978\pi$
$809$	36.1256	1.27011	0.635055	+	0.772467i	$0.280978\pi$
$810$	0	0
$811$	39.9298	1.40212	0.701062	−	0.713100i	$-0.252710\pi$
$811$	39.9298	1.40212	0.701062	+	0.713100i	$0.252710\pi$
$812$	0	0
$813$	53.0635	1.86102
$814$	0	0
$815$	25.3856	0.889220
$816$	0	0
$817$	2.90049	0.101475
$818$	0	0
$819$	47.8870	1.67331
$820$	0	0
$821$	42.2807	1.47561	0.737803	−	0.675017i	$-0.235864\pi$
$821$	42.2807	1.47561	0.737803	+	0.675017i	$0.235864\pi$
$822$	0	0
$823$	−38.9439	−1.35750	−0.678750	−	0.734369i	$-0.737478\pi$
$823$	−38.9439	−1.35750	−0.678750	+	0.734369i	$0.737478\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.1059	0.490510	0.245255	−	0.969459i	$-0.421128\pi$
$827$	14.1059	0.490510	0.245255	+	0.969459i	$0.421128\pi$
$828$	0	0
$829$	1.01184	0.0351425	0.0175713	−	0.999846i	$-0.494407\pi$
$829$	1.01184	0.0351425	0.0175713	+	0.999846i	$0.494407\pi$
$830$	0	0
$831$	−39.6003	−1.37372
$832$	0	0
$833$	−27.4367	−0.950624
$834$	0	0
$835$	−21.3573	−0.739100
$836$	0	0
$837$	1.25201	0.0432756
$838$	0	0
$839$	23.4592	0.809901	0.404951	−	0.914339i	$-0.367289\pi$
$839$	23.4592	0.809901	0.404951	+	0.914339i	$0.367289\pi$
$840$	0	0
$841$	−20.0554	−0.691567
$842$	0	0
$843$	−27.0749	−0.932508
$844$	0	0
$845$	22.3941	0.770381
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−46.9317	−1.61069
$850$	0	0
$851$	5.05416	0.173254
$852$	0	0
$853$	−51.6125	−1.76718	−0.883589	−	0.468262i	$-0.844880\pi$
$853$	−51.6125	−1.76718	−0.883589	+	0.468262i	$0.844880\pi$
$854$	0	0
$855$	−7.54163	−0.257918
$856$	0	0
$857$	−32.4436	−1.10825	−0.554126	−	0.832433i	$-0.686948\pi$
$857$	−32.4436	−1.10825	−0.554126	+	0.832433i	$0.686948\pi$
$858$	0	0
$859$	9.45118	0.322470	0.161235	−	0.986916i	$-0.448452\pi$
$859$	9.45118	0.322470	0.161235	+	0.986916i	$0.448452\pi$
$860$	0	0
$861$	50.9813	1.73744
$862$	0	0
$863$	18.7459	0.638118	0.319059	−	0.947735i	$-0.396633\pi$
$863$	18.7459	0.638118	0.319059	+	0.947735i	$0.396633\pi$
$864$	0	0
$865$	−23.2203	−0.789513
$866$	0	0
$867$	122.836	4.17174
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.37894	0.283910
$872$	0	0
$873$	−86.5220	−2.92832
$874$	0	0
$875$	−21.9190	−0.740998
$876$	0	0
$877$	0.740973	0.0250209	0.0125104	−	0.999922i	$-0.496018\pi$
$877$	0.740973	0.0250209	0.0125104	+	0.999922i	$0.496018\pi$
$878$	0	0
$879$	−32.9180	−1.11030
$880$	0	0
$881$	−25.4342	−0.856899	−0.428450	−	0.903566i	$-0.640940\pi$
$881$	−25.4342	−0.856899	−0.428450	+	0.903566i	$0.640940\pi$
$882$	0	0
$883$	−5.05448	−0.170097	−0.0850485	−	0.996377i	$-0.527105\pi$
$883$	−5.05448	−0.170097	−0.0850485	+	0.996377i	$0.527105\pi$
$884$	0	0
$885$	2.40164	0.0807303
$886$	0	0
$887$	7.21513	0.242260	0.121130	−	0.992637i	$-0.461348\pi$
$887$	7.21513	0.242260	0.121130	+	0.992637i	$0.461348\pi$
$888$	0	0
$889$	−18.9358	−0.635087
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.40622	0.247840
$894$	0	0
$895$	−37.3613	−1.24885
$896$	0	0
$897$	26.4368	0.882698
$898$	0	0
$899$	−0.704017	−0.0234803
$900$	0	0
$901$	−81.5716	−2.71754
$902$	0	0
$903$	−15.1994	−0.505805
$904$	0	0
$905$	−31.7251	−1.05458
$906$	0	0
$907$	−56.5190	−1.87668	−0.938341	−	0.345712i	$-0.887637\pi$
$907$	−56.5190	−1.87668	−0.938341	+	0.345712i	$0.887637\pi$
$908$	0	0
$909$	30.3024	1.00507
$910$	0	0
$911$	−53.5531	−1.77429	−0.887147	−	0.461487i	$-0.847316\pi$
$911$	−53.5531	−1.77429	−0.887147	+	0.461487i	$0.847316\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−15.4542	−0.510902
$916$	0	0
$917$	−12.2987	−0.406139
$918$	0	0
$919$	16.2472	0.535946	0.267973	−	0.963426i	$-0.413646\pi$
$919$	16.2472	0.535946	0.267973	+	0.963426i	$0.413646\pi$
$920$	0	0
$921$	39.1987	1.29164
$922$	0	0
$923$	−25.2151	−0.829964
$924$	0	0
$925$	−7.39619	−0.243185
$926$	0	0
$927$	57.0931	1.87518
$928$	0	0
$929$	−25.5616	−0.838650	−0.419325	−	0.907836i	$-0.637733\pi$
$929$	−25.5616	−0.838650	−0.419325	+	0.907836i	$0.637733\pi$
$930$	0	0
$931$	3.52093	0.115394
$932$	0	0
$933$	58.4304	1.91293
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.2756	1.21774	0.608871	−	0.793269i	$-0.291623\pi$
$937$	37.2756	1.21774	0.608871	+	0.793269i	$0.291623\pi$
$938$	0	0
$939$	−37.2793	−1.21657
$940$	0	0
$941$	14.0711	0.458706	0.229353	−	0.973343i	$-0.426339\pi$
$941$	14.0711	0.458706	0.229353	+	0.973343i	$0.426339\pi$
$942$	0	0
$943$	17.4477	0.568176
$944$	0	0
$945$	15.2902	0.497391
$946$	0	0
$947$	−42.8042	−1.39095	−0.695475	−	0.718550i	$-0.744806\pi$
$947$	−42.8042	−1.39095	−0.695475	+	0.718550i	$0.744806\pi$
$948$	0	0
$949$	−26.8947	−0.873040
$950$	0	0
$951$	−24.3238	−0.788754
$952$	0	0
$953$	54.3682	1.76116	0.880579	−	0.473899i	$-0.157154\pi$
$953$	54.3682	1.76116	0.880579	+	0.473899i	$0.157154\pi$
$954$	0	0
$955$	−5.07012	−0.164065
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.7851	−1.05868
$960$	0	0
$961$	−30.9446	−0.998213
$962$	0	0
$963$	12.2603	0.395082
$964$	0	0
$965$	−33.8767	−1.09053
$966$	0	0
$967$	−33.3309	−1.07185	−0.535925	−	0.844266i	$-0.680037\pi$
$967$	−33.3309	−1.07185	−0.535925	+	0.844266i	$0.680037\pi$
$968$	0	0
$969$	−21.8927	−0.703293
$970$	0	0
$971$	−61.1816	−1.96341	−0.981706	−	0.190404i	$-0.939020\pi$
$971$	−61.1816	−1.96341	−0.981706	+	0.190404i	$0.939020\pi$
$972$	0	0
$973$	38.3966	1.23094
$974$	0	0
$975$	−38.6873	−1.23898
$976$	0	0
$977$	16.5055	0.528057	0.264029	−	0.964515i	$-0.414949\pi$
$977$	16.5055	0.528057	0.264029	+	0.964515i	$0.414949\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−39.9146	−1.27438
$982$	0	0
$983$	−12.1534	−0.387635	−0.193817	−	0.981038i	$-0.562087\pi$
$983$	−12.1534	−0.387635	−0.193817	+	0.981038i	$0.562087\pi$
$984$	0	0
$985$	13.1583	0.419257
$986$	0	0
$987$	−38.8108	−1.23536
$988$	0	0
$989$	−5.20182	−0.165408
$990$	0	0
$991$	52.1321	1.65603	0.828016	−	0.560705i	$-0.189470\pi$
$991$	52.1321	1.65603	0.828016	+	0.560705i	$0.189470\pi$
$992$	0	0
$993$	4.94154	0.156815
$994$	0	0
$995$	29.9503	0.949487
$996$	0	0
$997$	−10.2550	−0.324780	−0.162390	−	0.986727i	$-0.551920\pi$
$997$	−10.2550	−0.324780	−0.162390	+	0.986727i	$0.551920\pi$
$998$	0	0
$999$	14.9888	0.474226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.v.1.13	yes	14
11.10	odd	2		9196.2.a.u.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.u.1.13	✓	14	11.10	odd	2
9196.2.a.v.1.13	yes	14	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.80947 q^{3} +1.54127 q^{5} +1.86523 q^{7} +4.89312 q^{9} +5.24687 q^{13} +4.33016 q^{15} +7.79245 q^{17} -1.00000 q^{19} +5.24030 q^{21} +1.79343 q^{23} -2.62448 q^{25} +5.31867 q^{27} -2.99075 q^{29} +0.235398 q^{31} +2.87482 q^{35} +2.81816 q^{37} +14.7409 q^{39} +9.72869 q^{41} -2.90049 q^{43} +7.54163 q^{45} -7.40622 q^{47} -3.52093 q^{49} +21.8927 q^{51} -10.4680 q^{53} -2.80947 q^{57} +0.554631 q^{59} -3.56898 q^{61} +9.12678 q^{63} +8.08685 q^{65} +1.59694 q^{67} +5.03858 q^{69} -4.80573 q^{71} -5.12586 q^{73} -7.37340 q^{75} +7.05506 q^{79} +0.263273 q^{81} -12.5690 q^{83} +12.0103 q^{85} -8.40241 q^{87} +10.2773 q^{89} +9.78660 q^{91} +0.661345 q^{93} -1.54127 q^{95} -17.6824 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 6 q^{5} + 4 q^{7} + 12 q^{9} + 4 q^{13} + 2 q^{15} + 8 q^{17} - 14 q^{19} + 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} - 4 q^{29} - 2 q^{31} + 24 q^{35} - 10 q^{37} + 24 q^{39} + 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) \) 14 * q - 2 * q^3 - 6 * q^5 + 4 * q^7 + 12 * q^9 + 4 * q^13 + 2 * q^15 + 8 * q^17 - 14 * q^19 + 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 - 4 * q^29 - 2 * q^31 + 24 * q^35 - 10 * q^37 + 24 * q^39 + 4 * q^41 + 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 + 22 * q^51 + 6 * q^53 + 2 * q^57 - 6 * q^59 - 4 * q^61 - 16 * q^63 + 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 - 24 * q^73 - 10 * q^75 + 48 * q^79 + 2 * q^81 - 40 * q^85 + 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 + 6 * q^95 - 20 * q^97

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