LMFDB - Embedded newform 1936.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,2,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1936.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:	$15.4590378313$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 242)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} -1.73205 q^{5} +4.73205 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} -1.73205 q^{5} +4.73205 q^{7} -2.46410 q^{9} -3.00000 q^{13} +1.26795 q^{15} +5.19615 q^{17} +1.26795 q^{19} -3.46410 q^{21} +1.26795 q^{23} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} -0.196152 q^{31} -8.19615 q^{35} -7.19615 q^{37} +2.19615 q^{39} +0.803848 q^{41} +4.26795 q^{45} +8.19615 q^{47} +15.3923 q^{49} -3.80385 q^{51} +11.1962 q^{53} -0.928203 q^{57} -9.46410 q^{59} -2.53590 q^{61} -11.6603 q^{63} +5.19615 q^{65} +10.1962 q^{67} -0.928203 q^{69} +9.46410 q^{71} +0.928203 q^{73} +1.46410 q^{75} -4.73205 q^{79} +4.46410 q^{81} +8.19615 q^{83} -9.00000 q^{85} -2.19615 q^{87} +6.46410 q^{89} -14.1962 q^{91} +0.143594 q^{93} -2.19615 q^{95} -1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9} - 6 q^{13} + 6 q^{15} + 6 q^{19} + 6 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{35} - 4 q^{37} - 6 q^{39} + 12 q^{41} + 12 q^{45} + 6 q^{47} + 10 q^{49} - 18 q^{51} + 12 q^{53} + 12 q^{57} - 12 q^{59} - 12 q^{61} - 6 q^{63} + 10 q^{67} + 12 q^{69} + 12 q^{71} - 12 q^{73} - 4 q^{75} - 6 q^{79} + 2 q^{81} + 6 q^{83} - 18 q^{85} + 6 q^{87} + 6 q^{89} - 18 q^{91} + 28 q^{93} + 6 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^7 + 2 * q^9 - 6 * q^13 + 6 * q^15 + 6 * q^19 + 6 * q^23 - 4 * q^25 + 8 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^35 - 4 * q^37 - 6 * q^39 + 12 * q^41 + 12 * q^45 + 6 * q^47 + 10 * q^49 - 18 * q^51 + 12 * q^53 + 12 * q^57 - 12 * q^59 - 12 * q^61 - 6 * q^63 + 10 * q^67 + 12 * q^69 + 12 * q^71 - 12 * q^73 - 4 * q^75 - 6 * q^79 + 2 * q^81 + 6 * q^83 - 18 * q^85 + 6 * q^87 + 6 * q^89 - 18 * q^91 + 28 * q^93 + 6 * q^95 - 2 * q^97

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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	1.26795	0.327383
$16$	0	0
$17$	5.19615	1.26025	0.630126	−	0.776493i	$-0.283003\pi$
$17$	5.19615	1.26025	0.630126	+	0.776493i	$0.283003\pi$
$18$	0	0
$19$	1.26795	0.290887	0.145444	−	0.989367i	$-0.453539\pi$
$19$	1.26795	0.290887	0.145444	+	0.989367i	$0.453539\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.19615	−1.38540
$36$	0	0
$37$	−7.19615	−1.18304	−0.591520	−	0.806290i	$-0.701472\pi$
$37$	−7.19615	−1.18304	−0.591520	+	0.806290i	$0.701472\pi$
$38$	0	0
$39$	2.19615	0.351666
$40$	0	0
$41$	0.803848	0.125540	0.0627700	−	0.998028i	$-0.480007\pi$
$41$	0.803848	0.125540	0.0627700	+	0.998028i	$0.480007\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	4.26795	0.636228
$46$	0	0
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	−3.80385	−0.532645
$52$	0	0
$53$	11.1962	1.53791	0.768955	−	0.639303i	$-0.220777\pi$
$53$	11.1962	1.53791	0.768955	+	0.639303i	$0.220777\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.928203	−0.122944
$58$	0	0
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	−2.53590	−0.324689	−0.162344	−	0.986734i	$-0.551906\pi$
$61$	−2.53590	−0.324689	−0.162344	+	0.986734i	$0.551906\pi$
$62$	0	0
$63$	−11.6603	−1.46905
$64$	0	0
$65$	5.19615	0.644503
$66$	0	0
$67$	10.1962	1.24566	0.622829	−	0.782358i	$-0.285983\pi$
$67$	10.1962	1.24566	0.622829	+	0.782358i	$0.285983\pi$
$68$	0	0
$69$	−0.928203	−0.111743
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	0.928203	0.108638	0.0543190	−	0.998524i	$-0.482701\pi$
$73$	0.928203	0.108638	0.0543190	+	0.998524i	$0.482701\pi$
$74$	0	0
$75$	1.46410	0.169060
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.73205	−0.532397	−0.266199	−	0.963918i	$-0.585768\pi$
$79$	−4.73205	−0.532397	−0.266199	+	0.963918i	$0.585768\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	8.19615	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	−2.19615	−0.235452
$88$	0	0
$89$	6.46410	0.685193	0.342597	−	0.939483i	$-0.388694\pi$
$89$	6.46410	0.685193	0.342597	+	0.939483i	$0.388694\pi$
$90$	0	0
$91$	−14.1962	−1.48816
$92$	0	0
$93$	0.143594	0.0148900
$94$	0	0
$95$	−2.19615	−0.225320
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.3923	−1.63110	−0.815548	−	0.578690i	$-0.803564\pi$
$101$	−16.3923	−1.63110	−0.815548	+	0.578690i	$0.803564\pi$
$102$	0	0
$103$	8.39230	0.826918	0.413459	−	0.910523i	$-0.364320\pi$
$103$	8.39230	0.826918	0.413459	+	0.910523i	$0.364320\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	12.5885	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$107$	12.5885	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$108$	0	0
$109$	2.07180	0.198442	0.0992211	−	0.995065i	$-0.468365\pi$
$109$	2.07180	0.198442	0.0992211	+	0.995065i	$0.468365\pi$
$110$	0	0
$111$	5.26795	0.500012
$112$	0	0
$113$	10.8564	1.02128	0.510642	−	0.859793i	$-0.329408\pi$
$113$	10.8564	1.02128	0.510642	+	0.859793i	$0.329408\pi$
$114$	0	0
$115$	−2.19615	−0.204792
$116$	0	0
$117$	7.39230	0.683419
$118$	0	0
$119$	24.5885	2.25402
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.588457	−0.0530594
$124$	0	0
$125$	12.1244	1.08444
$126$	0	0
$127$	13.8564	1.22956	0.614779	−	0.788700i	$-0.289245\pi$
$127$	13.8564	1.22956	0.614779	+	0.788700i	$0.289245\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−6.92820	−0.596285
$136$	0	0
$137$	−9.46410	−0.808573	−0.404286	−	0.914632i	$-0.632480\pi$
$137$	−9.46410	−0.808573	−0.404286	+	0.914632i	$0.632480\pi$
$138$	0	0
$139$	20.1962	1.71302	0.856508	−	0.516134i	$-0.172629\pi$
$139$	20.1962	1.71302	0.856508	+	0.516134i	$0.172629\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.19615	−0.431517
$146$	0	0
$147$	−11.2679	−0.929365
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−4.39230	−0.357441	−0.178720	−	0.983900i	$-0.557196\pi$
$151$	−4.39230	−0.357441	−0.178720	+	0.983900i	$0.557196\pi$
$152$	0	0
$153$	−12.8038	−1.03513
$154$	0	0
$155$	0.339746	0.0272891
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	−8.19615	−0.649997
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−22.1962	−1.73854	−0.869268	−	0.494340i	$-0.835410\pi$
$163$	−22.1962	−1.73854	−0.869268	+	0.494340i	$0.835410\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.3923	1.26847	0.634237	−	0.773138i	$-0.281314\pi$
$167$	16.3923	1.26847	0.634237	+	0.773138i	$0.281314\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.12436	−0.238925
$172$	0	0
$173$	4.39230	0.333941	0.166970	−	0.985962i	$-0.446602\pi$
$173$	4.39230	0.333941	0.166970	+	0.985962i	$0.446602\pi$
$174$	0	0
$175$	−9.46410	−0.715419
$176$	0	0
$177$	6.92820	0.520756
$178$	0	0
$179$	13.8564	1.03568	0.517838	−	0.855479i	$-0.326737\pi$
$179$	13.8564	1.03568	0.517838	+	0.855479i	$0.326737\pi$
$180$	0	0
$181$	7.19615	0.534886	0.267443	−	0.963574i	$-0.413821\pi$
$181$	7.19615	0.534886	0.267443	+	0.963574i	$0.413821\pi$
$182$	0	0
$183$	1.85641	0.137230
$184$	0	0
$185$	12.4641	0.916379
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.9282	1.37682
$190$	0	0
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	0	0
$193$	−10.2679	−0.739103	−0.369552	−	0.929210i	$-0.620489\pi$
$193$	−10.2679	−0.739103	−0.369552	+	0.929210i	$0.620489\pi$
$194$	0	0
$195$	−3.80385	−0.272399
$196$	0	0
$197$	13.3923	0.954162	0.477081	−	0.878859i	$-0.341695\pi$
$197$	13.3923	0.954162	0.477081	+	0.878859i	$0.341695\pi$
$198$	0	0
$199$	0.392305	0.0278098	0.0139049	−	0.999903i	$-0.495574\pi$
$199$	0.392305	0.0278098	0.0139049	+	0.999903i	$0.495574\pi$
$200$	0	0
$201$	−7.46410	−0.526477
$202$	0	0
$203$	14.1962	0.996375
$204$	0	0
$205$	−1.39230	−0.0972428
$206$	0	0
$207$	−3.12436	−0.217158
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−25.8564	−1.78003	−0.890014	−	0.455933i	$-0.849306\pi$
$211$	−25.8564	−1.78003	−0.890014	+	0.455933i	$0.849306\pi$
$212$	0	0
$213$	−6.92820	−0.474713
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.928203	−0.0630105
$218$	0	0
$219$	−0.679492	−0.0459158
$220$	0	0
$221$	−15.5885	−1.04859
$222$	0	0
$223$	20.3923	1.36557	0.682785	−	0.730619i	$-0.260769\pi$
$223$	20.3923	1.36557	0.682785	+	0.730619i	$0.260769\pi$
$224$	0	0
$225$	4.92820	0.328547
$226$	0	0
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0	0
$229$	−13.1962	−0.872026	−0.436013	−	0.899940i	$-0.643610\pi$
$229$	−13.1962	−0.872026	−0.436013	+	0.899940i	$0.643610\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.80385	0.445735	0.222867	−	0.974849i	$-0.428458\pi$
$233$	6.80385	0.445735	0.222867	+	0.974849i	$0.428458\pi$
$234$	0	0
$235$	−14.1962	−0.926055
$236$	0	0
$237$	3.46410	0.225018
$238$	0	0
$239$	−14.1962	−0.918273	−0.459136	−	0.888366i	$-0.651841\pi$
$239$	−14.1962	−0.918273	−0.459136	+	0.888366i	$0.651841\pi$
$240$	0	0
$241$	−26.7846	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$241$	−26.7846	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−26.6603	−1.70326
$246$	0	0
$247$	−3.80385	−0.242033
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−9.12436	−0.575924	−0.287962	−	0.957642i	$-0.592978\pi$
$251$	−9.12436	−0.575924	−0.287962	+	0.957642i	$0.592978\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.58846	0.412585
$256$	0	0
$257$	−19.3923	−1.20966	−0.604829	−	0.796355i	$-0.706759\pi$
$257$	−19.3923	−1.20966	−0.604829	+	0.796355i	$0.706759\pi$
$258$	0	0
$259$	−34.0526	−2.11592
$260$	0	0
$261$	−7.39230	−0.457572
$262$	0	0
$263$	9.80385	0.604531	0.302266	−	0.953224i	$-0.402257\pi$
$263$	9.80385	0.604531	0.302266	+	0.953224i	$0.402257\pi$
$264$	0	0
$265$	−19.3923	−1.19126
$266$	0	0
$267$	−4.73205	−0.289597
$268$	0	0
$269$	9.58846	0.584619	0.292309	−	0.956324i	$-0.405576\pi$
$269$	9.58846	0.584619	0.292309	+	0.956324i	$0.405576\pi$
$270$	0	0
$271$	−16.3923	−0.995762	−0.497881	−	0.867245i	$-0.665888\pi$
$271$	−16.3923	−0.995762	−0.497881	+	0.867245i	$0.665888\pi$
$272$	0	0
$273$	10.3923	0.628971
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.32051	0.499931	0.249965	−	0.968255i	$-0.419581\pi$
$277$	8.32051	0.499931	0.249965	+	0.968255i	$0.419581\pi$
$278$	0	0
$279$	0.483340	0.0289368
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	6.92820	0.411839	0.205919	−	0.978569i	$-0.433982\pi$
$283$	6.92820	0.411839	0.205919	+	0.978569i	$0.433982\pi$
$284$	0	0
$285$	1.60770	0.0952316
$286$	0	0
$287$	3.80385	0.224534
$288$	0	0
$289$	10.0000	0.588235
$290$	0	0
$291$	0.732051	0.0429136
$292$	0	0
$293$	−23.7846	−1.38951	−0.694756	−	0.719246i	$-0.744488\pi$
$293$	−23.7846	−1.38951	−0.694756	+	0.719246i	$0.744488\pi$
$294$	0	0
$295$	16.3923	0.954397
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.80385	−0.219982
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	0	0
$305$	4.39230	0.251503
$306$	0	0
$307$	25.2679	1.44212	0.721059	−	0.692874i	$-0.243656\pi$
$307$	25.2679	1.44212	0.721059	+	0.692874i	$0.243656\pi$
$308$	0	0
$309$	−6.14359	−0.349497
$310$	0	0
$311$	−8.78461	−0.498130	−0.249065	−	0.968487i	$-0.580123\pi$
$311$	−8.78461	−0.498130	−0.249065	+	0.968487i	$0.580123\pi$
$312$	0	0
$313$	−31.7846	−1.79657	−0.898286	−	0.439411i	$-0.855187\pi$
$313$	−31.7846	−1.79657	−0.898286	+	0.439411i	$0.855187\pi$
$314$	0	0
$315$	20.1962	1.13792
$316$	0	0
$317$	−7.85641	−0.441260	−0.220630	−	0.975358i	$-0.570811\pi$
$317$	−7.85641	−0.441260	−0.220630	+	0.975358i	$0.570811\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.21539	−0.514353
$322$	0	0
$323$	6.58846	0.366592
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−1.51666	−0.0838715
$328$	0	0
$329$	38.7846	2.13826
$330$	0	0
$331$	−28.7846	−1.58215	−0.791073	−	0.611722i	$-0.790477\pi$
$331$	−28.7846	−1.58215	−0.791073	+	0.611722i	$0.790477\pi$
$332$	0	0
$333$	17.7321	0.971710
$334$	0	0
$335$	−17.6603	−0.964883
$336$	0	0
$337$	−2.66025	−0.144913	−0.0724566	−	0.997372i	$-0.523084\pi$
$337$	−2.66025	−0.144913	−0.0724566	+	0.997372i	$0.523084\pi$
$338$	0	0
$339$	−7.94744	−0.431646
$340$	0	0
$341$	0	0
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	1.60770	0.0865554
$346$	0	0
$347$	−28.3923	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$347$	−28.3923	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$348$	0	0
$349$	21.9282	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$349$	21.9282	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$350$	0	0
$351$	−12.0000	−0.640513
$352$	0	0
$353$	21.9282	1.16712	0.583560	−	0.812070i	$-0.301659\pi$
$353$	21.9282	1.16712	0.583560	+	0.812070i	$0.301659\pi$
$354$	0	0
$355$	−16.3923	−0.870013
$356$	0	0
$357$	−18.0000	−0.952661
$358$	0	0
$359$	−6.58846	−0.347725	−0.173863	−	0.984770i	$-0.555625\pi$
$359$	−6.58846	−0.347725	−0.173863	+	0.984770i	$0.555625\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.60770	−0.0841506
$366$	0	0
$367$	−8.58846	−0.448314	−0.224157	−	0.974553i	$-0.571963\pi$
$367$	−8.58846	−0.448314	−0.224157	+	0.974553i	$0.571963\pi$
$368$	0	0
$369$	−1.98076	−0.103114
$370$	0	0
$371$	52.9808	2.75062
$372$	0	0
$373$	11.3205	0.586154	0.293077	−	0.956089i	$-0.405321\pi$
$373$	11.3205	0.586154	0.293077	+	0.956089i	$0.405321\pi$
$374$	0	0
$375$	−8.87564	−0.458336
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−3.60770	−0.185315	−0.0926574	−	0.995698i	$-0.529536\pi$
$379$	−3.60770	−0.185315	−0.0926574	+	0.995698i	$0.529536\pi$
$380$	0	0
$381$	−10.1436	−0.519672
$382$	0	0
$383$	0.679492	0.0347204	0.0173602	−	0.999849i	$-0.494474\pi$
$383$	0.679492	0.0347204	0.0173602	+	0.999849i	$0.494474\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.6603	1.65594	0.827970	−	0.560772i	$-0.189496\pi$
$389$	32.6603	1.65594	0.827970	+	0.560772i	$0.189496\pi$
$390$	0	0
$391$	6.58846	0.333193
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.19615	0.412393
$396$	0	0
$397$	5.58846	0.280477	0.140238	−	0.990118i	$-0.455213\pi$
$397$	5.58846	0.280477	0.140238	+	0.990118i	$0.455213\pi$
$398$	0	0
$399$	−4.39230	−0.219890
$400$	0	0
$401$	−7.39230	−0.369154	−0.184577	−	0.982818i	$-0.559092\pi$
$401$	−7.39230	−0.369154	−0.184577	+	0.982818i	$0.559092\pi$
$402$	0	0
$403$	0.588457	0.0293131
$404$	0	0
$405$	−7.73205	−0.384209
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.66025	0.131541	0.0657705	−	0.997835i	$-0.479049\pi$
$409$	2.66025	0.131541	0.0657705	+	0.997835i	$0.479049\pi$
$410$	0	0
$411$	6.92820	0.341743
$412$	0	0
$413$	−44.7846	−2.20371
$414$	0	0
$415$	−14.1962	−0.696862
$416$	0	0
$417$	−14.7846	−0.724005
$418$	0	0
$419$	39.3731	1.92350	0.961750	−	0.273928i	$-0.0883231\pi$
$419$	39.3731	1.92350	0.961750	+	0.273928i	$0.0883231\pi$
$420$	0	0
$421$	7.58846	0.369839	0.184919	−	0.982754i	$-0.440798\pi$
$421$	7.58846	0.369839	0.184919	+	0.982754i	$0.440798\pi$
$422$	0	0
$423$	−20.1962	−0.981971
$424$	0	0
$425$	−10.3923	−0.504101
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.3923	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$431$	16.3923	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$432$	0	0
$433$	31.7846	1.52747	0.763735	−	0.645529i	$-0.223363\pi$
$433$	31.7846	1.52747	0.763735	+	0.645529i	$0.223363\pi$
$434$	0	0
$435$	3.80385	0.182381
$436$	0	0
$437$	1.60770	0.0769065
$438$	0	0
$439$	−21.1244	−1.00821	−0.504105	−	0.863642i	$-0.668178\pi$
$439$	−21.1244	−1.00821	−0.504105	+	0.863642i	$0.668178\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	0	0
$443$	−17.0718	−0.811106	−0.405553	−	0.914072i	$-0.632921\pi$
$443$	−17.0718	−0.811106	−0.405553	+	0.914072i	$0.632921\pi$
$444$	0	0
$445$	−11.1962	−0.530749
$446$	0	0
$447$	−10.9808	−0.519372
$448$	0	0
$449$	−9.92820	−0.468541	−0.234270	−	0.972171i	$-0.575270\pi$
$449$	−9.92820	−0.468541	−0.234270	+	0.972171i	$0.575270\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.21539	0.151072
$454$	0	0
$455$	24.5885	1.15272
$456$	0	0
$457$	5.19615	0.243066	0.121533	−	0.992587i	$-0.461219\pi$
$457$	5.19615	0.243066	0.121533	+	0.992587i	$0.461219\pi$
$458$	0	0
$459$	20.7846	0.970143
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	−28.7846	−1.33773	−0.668867	−	0.743382i	$-0.733220\pi$
$463$	−28.7846	−1.33773	−0.668867	+	0.743382i	$0.733220\pi$
$464$	0	0
$465$	−0.248711	−0.0115337
$466$	0	0
$467$	−21.4641	−0.993240	−0.496620	−	0.867968i	$-0.665426\pi$
$467$	−21.4641	−0.993240	−0.496620	+	0.867968i	$0.665426\pi$
$468$	0	0
$469$	48.2487	2.22792
$470$	0	0
$471$	−2.92820	−0.134924
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.53590	−0.116355
$476$	0	0
$477$	−27.5885	−1.26319
$478$	0	0
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	0	0
$481$	21.5885	0.984349
$482$	0	0
$483$	−4.39230	−0.199857
$484$	0	0
$485$	1.73205	0.0786484
$486$	0	0
$487$	12.9808	0.588214	0.294107	−	0.955772i	$-0.404978\pi$
$487$	12.9808	0.588214	0.294107	+	0.955772i	$0.404978\pi$
$488$	0	0
$489$	16.2487	0.734792
$490$	0	0
$491$	−33.3731	−1.50611	−0.753053	−	0.657960i	$-0.771419\pi$
$491$	−33.3731	−1.50611	−0.753053	+	0.657960i	$0.771419\pi$
$492$	0	0
$493$	15.5885	0.702069
$494$	0	0
$495$	0	0
$496$	0	0
$497$	44.7846	2.00886
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−14.1962	−0.632975	−0.316488	−	0.948597i	$-0.602504\pi$
$503$	−14.1962	−0.632975	−0.316488	+	0.948597i	$0.602504\pi$
$504$	0	0
$505$	28.3923	1.26344
$506$	0	0
$507$	2.92820	0.130046
$508$	0	0
$509$	12.9282	0.573033	0.286516	−	0.958075i	$-0.407503\pi$
$509$	12.9282	0.573033	0.286516	+	0.958075i	$0.407503\pi$
$510$	0	0
$511$	4.39230	0.194304
$512$	0	0
$513$	5.07180	0.223925
$514$	0	0
$515$	−14.5359	−0.640528
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.21539	−0.141140
$520$	0	0
$521$	9.46410	0.414630	0.207315	−	0.978274i	$-0.433528\pi$
$521$	9.46410	0.414630	0.207315	+	0.978274i	$0.433528\pi$
$522$	0	0
$523$	2.53590	0.110887	0.0554435	−	0.998462i	$-0.482343\pi$
$523$	2.53590	0.110887	0.0554435	+	0.998462i	$0.482343\pi$
$524$	0	0
$525$	6.92820	0.302372
$526$	0	0
$527$	−1.01924	−0.0443987
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	23.3205	1.01202
$532$	0	0
$533$	−2.41154	−0.104456
$534$	0	0
$535$	−21.8038	−0.942663
$536$	0	0
$537$	−10.1436	−0.437728
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−5.26795	−0.226069
$544$	0	0
$545$	−3.58846	−0.153713
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	0	0
$549$	6.24871	0.266688
$550$	0	0
$551$	3.80385	0.162049
$552$	0	0
$553$	−22.3923	−0.952218
$554$	0	0
$555$	−9.12436	−0.387307
$556$	0	0
$557$	−37.1769	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$557$	−37.1769	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.588457	−0.0248005	−0.0124003	−	0.999923i	$-0.503947\pi$
$563$	−0.588457	−0.0248005	−0.0124003	+	0.999923i	$0.503947\pi$
$564$	0	0
$565$	−18.8038	−0.791084
$566$	0	0
$567$	21.1244	0.887140
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	5.66025	0.236874	0.118437	−	0.992962i	$-0.462212\pi$
$571$	5.66025	0.236874	0.118437	+	0.992962i	$0.462212\pi$
$572$	0	0
$573$	−15.7128	−0.656412
$574$	0	0
$575$	−2.53590	−0.105754
$576$	0	0
$577$	14.6077	0.608126	0.304063	−	0.952652i	$-0.401657\pi$
$577$	14.6077	0.608126	0.304063	+	0.952652i	$0.401657\pi$
$578$	0	0
$579$	7.51666	0.312382
$580$	0	0
$581$	38.7846	1.60906
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12.8038	−0.529374
$586$	0	0
$587$	−30.5885	−1.26252	−0.631260	−	0.775571i	$-0.717462\pi$
$587$	−30.5885	−1.26252	−0.631260	+	0.775571i	$0.717462\pi$
$588$	0	0
$589$	−0.248711	−0.0102480
$590$	0	0
$591$	−9.80385	−0.403276
$592$	0	0
$593$	23.1962	0.952552	0.476276	−	0.879296i	$-0.341986\pi$
$593$	23.1962	0.952552	0.476276	+	0.879296i	$0.341986\pi$
$594$	0	0
$595$	−42.5885	−1.74596
$596$	0	0
$597$	−0.287187	−0.0117538
$598$	0	0
$599$	3.80385	0.155421	0.0777105	−	0.996976i	$-0.475239\pi$
$599$	3.80385	0.155421	0.0777105	+	0.996976i	$0.475239\pi$
$600$	0	0
$601$	8.66025	0.353259	0.176630	−	0.984277i	$-0.443481\pi$
$601$	8.66025	0.353259	0.176630	+	0.984277i	$0.443481\pi$
$602$	0	0
$603$	−25.1244	−1.02314
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.1962	1.06327	0.531635	−	0.846974i	$-0.321578\pi$
$607$	26.1962	1.06327	0.531635	+	0.846974i	$0.321578\pi$
$608$	0	0
$609$	−10.3923	−0.421117
$610$	0	0
$611$	−24.5885	−0.994743
$612$	0	0
$613$	−20.3205	−0.820738	−0.410369	−	0.911920i	$-0.634600\pi$
$613$	−20.3205	−0.820738	−0.410369	+	0.911920i	$0.634600\pi$
$614$	0	0
$615$	1.01924	0.0410996
$616$	0	0
$617$	10.6077	0.427050	0.213525	−	0.976938i	$-0.431506\pi$
$617$	10.6077	0.427050	0.213525	+	0.976938i	$0.431506\pi$
$618$	0	0
$619$	30.1962	1.21369	0.606843	−	0.794822i	$-0.292436\pi$
$619$	30.1962	1.21369	0.606843	+	0.794822i	$0.292436\pi$
$620$	0	0
$621$	5.07180	0.203524
$622$	0	0
$623$	30.5885	1.22550
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.3923	−1.49093
$630$	0	0
$631$	20.5885	0.819614	0.409807	−	0.912172i	$-0.365596\pi$
$631$	20.5885	0.819614	0.409807	+	0.912172i	$0.365596\pi$
$632$	0	0
$633$	18.9282	0.752329
$634$	0	0
$635$	−24.0000	−0.952411
$636$	0	0
$637$	−46.1769	−1.82960
$638$	0	0
$639$	−23.3205	−0.922545
$640$	0	0
$641$	3.67949	0.145331	0.0726656	−	0.997356i	$-0.476849\pi$
$641$	3.67949	0.145331	0.0726656	+	0.997356i	$0.476849\pi$
$642$	0	0
$643$	25.8038	1.01760	0.508802	−	0.860883i	$-0.330088\pi$
$643$	25.8038	1.01760	0.508802	+	0.860883i	$0.330088\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.679492	0.0266314
$652$	0	0
$653$	−1.85641	−0.0726468	−0.0363234	−	0.999340i	$-0.511565\pi$
$653$	−1.85641	−0.0726468	−0.0363234	+	0.999340i	$0.511565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.28719	−0.0892317
$658$	0	0
$659$	16.9808	0.661477	0.330738	−	0.943723i	$-0.392702\pi$
$659$	16.9808	0.661477	0.330738	+	0.943723i	$0.392702\pi$
$660$	0	0
$661$	−16.4115	−0.638335	−0.319168	−	0.947698i	$-0.603403\pi$
$661$	−16.4115	−0.638335	−0.319168	+	0.947698i	$0.603403\pi$
$662$	0	0
$663$	11.4115	0.443188
$664$	0	0
$665$	−10.3923	−0.402996
$666$	0	0
$667$	3.80385	0.147286
$668$	0	0
$669$	−14.9282	−0.577158
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−0.928203	−0.0357796	−0.0178898	−	0.999840i	$-0.505695\pi$
$673$	−0.928203	−0.0357796	−0.0178898	+	0.999840i	$0.505695\pi$
$674$	0	0
$675$	−8.00000	−0.307920
$676$	0	0
$677$	−4.60770	−0.177088	−0.0885441	−	0.996072i	$-0.528221\pi$
$677$	−4.60770	−0.177088	−0.0885441	+	0.996072i	$0.528221\pi$
$678$	0	0
$679$	−4.73205	−0.181599
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	5.41154	0.207067	0.103533	−	0.994626i	$-0.466985\pi$
$683$	5.41154	0.207067	0.103533	+	0.994626i	$0.466985\pi$
$684$	0	0
$685$	16.3923	0.626318
$686$	0	0
$687$	9.66025	0.368562
$688$	0	0
$689$	−33.5885	−1.27962
$690$	0	0
$691$	41.1769	1.56644	0.783222	−	0.621742i	$-0.213575\pi$
$691$	41.1769	1.56644	0.783222	+	0.621742i	$0.213575\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−34.9808	−1.32690
$696$	0	0
$697$	4.17691	0.158212
$698$	0	0
$699$	−4.98076	−0.188390
$700$	0	0
$701$	17.7846	0.671715	0.335858	−	0.941913i	$-0.390974\pi$
$701$	17.7846	0.671715	0.335858	+	0.941913i	$0.390974\pi$
$702$	0	0
$703$	−9.12436	−0.344132
$704$	0	0
$705$	10.3923	0.391397
$706$	0	0
$707$	−77.5692	−2.91729
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	11.6603	0.437294
$712$	0	0
$713$	−0.248711	−0.00931431
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.3923	0.388108
$718$	0	0
$719$	46.7321	1.74281	0.871406	−	0.490563i	$-0.163209\pi$
$719$	46.7321	1.74281	0.871406	+	0.490563i	$0.163209\pi$
$720$	0	0
$721$	39.7128	1.47898
$722$	0	0
$723$	19.6077	0.729218
$724$	0	0
$725$	−6.00000	−0.222834
$726$	0	0
$727$	36.9808	1.37154	0.685770	−	0.727818i	$-0.259465\pi$
$727$	36.9808	1.37154	0.685770	+	0.727818i	$0.259465\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−16.6077	−0.613419	−0.306710	−	0.951803i	$-0.599228\pi$
$733$	−16.6077	−0.613419	−0.306710	+	0.951803i	$0.599228\pi$
$734$	0	0
$735$	19.5167	0.719883
$736$	0	0
$737$	0	0
$738$	0	0
$739$	10.7321	0.394785	0.197392	−	0.980325i	$-0.436753\pi$
$739$	10.7321	0.394785	0.197392	+	0.980325i	$0.436753\pi$
$740$	0	0
$741$	2.78461	0.102295
$742$	0	0
$743$	−38.1962	−1.40128	−0.700640	−	0.713514i	$-0.747102\pi$
$743$	−38.1962	−1.40128	−0.700640	+	0.713514i	$0.747102\pi$
$744$	0	0
$745$	−25.9808	−0.951861
$746$	0	0
$747$	−20.1962	−0.738939
$748$	0	0
$749$	59.5692	2.17661
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	6.67949	0.243414
$754$	0	0
$755$	7.60770	0.276872
$756$	0	0
$757$	−13.1962	−0.479622	−0.239811	−	0.970820i	$-0.577086\pi$
$757$	−13.1962	−0.479622	−0.239811	+	0.970820i	$0.577086\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.9808	−0.724302	−0.362151	−	0.932119i	$-0.617958\pi$
$761$	−19.9808	−0.724302	−0.362151	+	0.932119i	$0.617958\pi$
$762$	0	0
$763$	9.80385	0.354923
$764$	0	0
$765$	22.1769	0.801808
$766$	0	0
$767$	28.3923	1.02519
$768$	0	0
$769$	−34.5167	−1.24470	−0.622351	−	0.782738i	$-0.713822\pi$
$769$	−34.5167	−1.24470	−0.622351	+	0.782738i	$0.713822\pi$
$770$	0	0
$771$	14.1962	0.511262
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	0.392305	0.0140920
$776$	0	0
$777$	24.9282	0.894294
$778$	0	0
$779$	1.01924	0.0365180
$780$	0	0
$781$	0	0
$782$	0	0
$783$	12.0000	0.428845
$784$	0	0
$785$	−6.92820	−0.247278
$786$	0	0
$787$	44.7846	1.59640	0.798199	−	0.602393i	$-0.205786\pi$
$787$	44.7846	1.59640	0.798199	+	0.602393i	$0.205786\pi$
$788$	0	0
$789$	−7.17691	−0.255505
$790$	0	0
$791$	51.3731	1.82662
$792$	0	0
$793$	7.60770	0.270157
$794$	0	0
$795$	14.1962	0.503486
$796$	0	0
$797$	36.9282	1.30806	0.654032	−	0.756467i	$-0.273076\pi$
$797$	36.9282	1.30806	0.654032	+	0.756467i	$0.273076\pi$
$798$	0	0
$799$	42.5885	1.50667
$800$	0	0
$801$	−15.9282	−0.562795
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−10.3923	−0.366281
$806$	0	0
$807$	−7.01924	−0.247089
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	6.92820	0.243282	0.121641	−	0.992574i	$-0.461184\pi$
$811$	6.92820	0.243282	0.121641	+	0.992574i	$0.461184\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	0	0
$815$	38.4449	1.34666
$816$	0	0
$817$	0	0
$818$	0	0
$819$	34.9808	1.22233
$820$	0	0
$821$	16.3923	0.572095	0.286048	−	0.958215i	$-0.407658\pi$
$821$	16.3923	0.572095	0.286048	+	0.958215i	$0.407658\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.1962	−0.702289	−0.351145	−	0.936321i	$-0.614207\pi$
$827$	−20.1962	−0.702289	−0.351145	+	0.936321i	$0.614207\pi$
$828$	0	0
$829$	14.8038	0.514159	0.257079	−	0.966390i	$-0.417240\pi$
$829$	14.8038	0.514159	0.257079	+	0.966390i	$0.417240\pi$
$830$	0	0
$831$	−6.09103	−0.211296
$832$	0	0
$833$	79.9808	2.77117
$834$	0	0
$835$	−28.3923	−0.982556
$836$	0	0
$837$	−0.784610	−0.0271201
$838$	0	0
$839$	50.4449	1.74155	0.870775	−	0.491682i	$-0.163618\pi$
$839$	50.4449	1.74155	0.870775	+	0.491682i	$0.163618\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	13.1769	0.453837
$844$	0	0
$845$	6.92820	0.238337
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.07180	−0.174064
$850$	0	0
$851$	−9.12436	−0.312779
$852$	0	0
$853$	10.8564	0.371716	0.185858	−	0.982577i	$-0.440494\pi$
$853$	10.8564	0.371716	0.185858	+	0.982577i	$0.440494\pi$
$854$	0	0
$855$	5.41154	0.185071
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	24.7846	0.845640	0.422820	−	0.906214i	$-0.361040\pi$
$859$	24.7846	0.845640	0.422820	+	0.906214i	$0.361040\pi$
$860$	0	0
$861$	−2.78461	−0.0948992
$862$	0	0
$863$	3.80385	0.129484	0.0647422	−	0.997902i	$-0.479377\pi$
$863$	3.80385	0.129484	0.0647422	+	0.997902i	$0.479377\pi$
$864$	0	0
$865$	−7.60770	−0.258669
$866$	0	0
$867$	−7.32051	−0.248617
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−30.5885	−1.03645
$872$	0	0
$873$	2.46410	0.0833972
$874$	0	0
$875$	57.3731	1.93956
$876$	0	0
$877$	6.46410	0.218277	0.109139	−	0.994027i	$-0.465191\pi$
$877$	6.46410	0.218277	0.109139	+	0.994027i	$0.465191\pi$
$878$	0	0
$879$	17.4115	0.587277
$880$	0	0
$881$	41.5359	1.39938	0.699690	−	0.714447i	$-0.253321\pi$
$881$	41.5359	1.39938	0.699690	+	0.714447i	$0.253321\pi$
$882$	0	0
$883$	15.6077	0.525241	0.262620	−	0.964899i	$-0.415413\pi$
$883$	15.6077	0.525241	0.262620	+	0.964899i	$0.415413\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−45.8038	−1.53794	−0.768971	−	0.639283i	$-0.779231\pi$
$887$	−45.8038	−1.53794	−0.768971	+	0.639283i	$0.779231\pi$
$888$	0	0
$889$	65.5692	2.19912
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.3923	0.347765
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	2.78461	0.0929754
$898$	0	0
$899$	−0.588457	−0.0196261
$900$	0	0
$901$	58.1769	1.93815
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.4641	−0.414321
$906$	0	0
$907$	−37.5692	−1.24747	−0.623733	−	0.781638i	$-0.714385\pi$
$907$	−37.5692	−1.24747	−0.623733	+	0.781638i	$0.714385\pi$
$908$	0	0
$909$	40.3923	1.33973
$910$	0	0
$911$	31.6077	1.04721	0.523605	−	0.851961i	$-0.324587\pi$
$911$	31.6077	1.04721	0.523605	+	0.851961i	$0.324587\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−3.21539	−0.106298
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.1436	0.730450	0.365225	−	0.930919i	$-0.380992\pi$
$919$	22.1436	0.730450	0.365225	+	0.930919i	$0.380992\pi$
$920$	0	0
$921$	−18.4974	−0.609511
$922$	0	0
$923$	−28.3923	−0.934544
$924$	0	0
$925$	14.3923	0.473216
$926$	0	0
$927$	−20.6795	−0.679204
$928$	0	0
$929$	39.0000	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$930$	0	0
$931$	19.5167	0.639633
$932$	0	0
$933$	6.43078	0.210534
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.4449	−1.15793	−0.578967	−	0.815351i	$-0.696544\pi$
$937$	−35.4449	−1.15793	−0.578967	+	0.815351i	$0.696544\pi$
$938$	0	0
$939$	23.2679	0.759321
$940$	0	0
$941$	34.6077	1.12818	0.564089	−	0.825714i	$-0.309227\pi$
$941$	34.6077	1.12818	0.564089	+	0.825714i	$0.309227\pi$
$942$	0	0
$943$	1.01924	0.0331910
$944$	0	0
$945$	−32.7846	−1.06648
$946$	0	0
$947$	−5.90897	−0.192016	−0.0960078	−	0.995381i	$-0.530607\pi$
$947$	−5.90897	−0.192016	−0.0960078	+	0.995381i	$0.530607\pi$
$948$	0	0
$949$	−2.78461	−0.0903923
$950$	0	0
$951$	5.75129	0.186498
$952$	0	0
$953$	39.5885	1.28240	0.641198	−	0.767376i	$-0.278438\pi$
$953$	39.5885	1.28240	0.641198	+	0.767376i	$0.278438\pi$
$954$	0	0
$955$	−37.1769	−1.20302
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−44.7846	−1.44617
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	−31.0192	−0.999581
$964$	0	0
$965$	17.7846	0.572507
$966$	0	0
$967$	−32.4449	−1.04336	−0.521678	−	0.853142i	$-0.674694\pi$
$967$	−32.4449	−1.04336	−0.521678	+	0.853142i	$0.674694\pi$
$968$	0	0
$969$	−4.82309	−0.154940
$970$	0	0
$971$	4.05256	0.130053	0.0650264	−	0.997884i	$-0.479287\pi$
$971$	4.05256	0.130053	0.0650264	+	0.997884i	$0.479287\pi$
$972$	0	0
$973$	95.5692	3.06381
$974$	0	0
$975$	−4.39230	−0.140666
$976$	0	0
$977$	32.3205	1.03402	0.517012	−	0.855978i	$-0.327044\pi$
$977$	32.3205	1.03402	0.517012	+	0.855978i	$0.327044\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5.10512	−0.162994
$982$	0	0
$983$	−32.1051	−1.02399	−0.511997	−	0.858987i	$-0.671094\pi$
$983$	−32.1051	−1.02399	−0.511997	+	0.858987i	$0.671094\pi$
$984$	0	0
$985$	−23.1962	−0.739091
$986$	0	0
$987$	−28.3923	−0.903737
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	21.0718	0.668693
$994$	0	0
$995$	−0.679492	−0.0215413
$996$	0	0
$997$	−12.4641	−0.394742	−0.197371	−	0.980329i	$-0.563240\pi$
$997$	−12.4641	−0.394742	−0.197371	+	0.980329i	$0.563240\pi$
$998$	0	0
$999$	−28.7846	−0.910705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.2.a.y.1.1		2
4.3	odd	2		242.2.a.c.1.2	✓	2
8.3	odd	2		7744.2.a.cs.1.1		2
8.5	even	2		7744.2.a.bt.1.2		2
11.10	odd	2		1936.2.a.v.1.1		2
12.11	even	2		2178.2.a.y.1.2		2
20.19	odd	2		6050.2.a.cv.1.1		2
44.3	odd	10		242.2.c.g.9.1		8
44.7	even	10		242.2.c.f.27.1		8
44.15	odd	10		242.2.c.g.27.1		8
44.19	even	10		242.2.c.f.9.1		8
44.27	odd	10		242.2.c.g.3.2		8
44.31	odd	10		242.2.c.g.81.2		8
44.35	even	10		242.2.c.f.81.2		8
44.39	even	10		242.2.c.f.3.2		8
44.43	even	2		242.2.a.e.1.2	yes	2
88.21	odd	2		7744.2.a.bq.1.2		2
88.43	even	2		7744.2.a.cv.1.1		2
132.131	odd	2		2178.2.a.s.1.2		2
220.219	even	2		6050.2.a.cc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
242.2.a.c.1.2	✓	2	4.3	odd	2
242.2.a.e.1.2	yes	2	44.43	even	2
242.2.c.f.3.2		8	44.39	even	10
242.2.c.f.9.1		8	44.19	even	10
242.2.c.f.27.1		8	44.7	even	10
242.2.c.f.81.2		8	44.35	even	10
242.2.c.g.3.2		8	44.27	odd	10
242.2.c.g.9.1		8	44.3	odd	10
242.2.c.g.27.1		8	44.15	odd	10
242.2.c.g.81.2		8	44.31	odd	10
1936.2.a.v.1.1		2	11.10	odd	2
1936.2.a.y.1.1		2	1.1	even	1	trivial
2178.2.a.s.1.2		2	132.131	odd	2
2178.2.a.y.1.2		2	12.11	even	2
6050.2.a.cc.1.1		2	220.219	even	2
6050.2.a.cv.1.1		2	20.19	odd	2
7744.2.a.bq.1.2		2	88.21	odd	2
7744.2.a.bt.1.2		2	8.5	even	2
7744.2.a.cs.1.1		2	8.3	odd	2
7744.2.a.cv.1.1		2	88.43	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} -1.73205 q^{5} +4.73205 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} -1.73205 q^{5} +4.73205 q^{7} -2.46410 q^{9} -3.00000 q^{13} +1.26795 q^{15} +5.19615 q^{17} +1.26795 q^{19} -3.46410 q^{21} +1.26795 q^{23} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} -0.196152 q^{31} -8.19615 q^{35} -7.19615 q^{37} +2.19615 q^{39} +0.803848 q^{41} +4.26795 q^{45} +8.19615 q^{47} +15.3923 q^{49} -3.80385 q^{51} +11.1962 q^{53} -0.928203 q^{57} -9.46410 q^{59} -2.53590 q^{61} -11.6603 q^{63} +5.19615 q^{65} +10.1962 q^{67} -0.928203 q^{69} +9.46410 q^{71} +0.928203 q^{73} +1.46410 q^{75} -4.73205 q^{79} +4.46410 q^{81} +8.19615 q^{83} -9.00000 q^{85} -2.19615 q^{87} +6.46410 q^{89} -14.1962 q^{91} +0.143594 q^{93} -2.19615 q^{95} -1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 6 q^{7} + 2 q^{9} - 6 q^{13} + 6 q^{15} + 6 q^{19} + 6 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{35} - 4 q^{37} - 6 q^{39} + 12 q^{41} + 12 q^{45} + 6 q^{47} + 10 q^{49} - 18 q^{51} + 12 q^{53} + 12 q^{57} - 12 q^{59} - 12 q^{61} - 6 q^{63} + 10 q^{67} + 12 q^{69} + 12 q^{71} - 12 q^{73} - 4 q^{75} - 6 q^{79} + 2 q^{81} + 6 q^{83} - 18 q^{85} + 6 q^{87} + 6 q^{89} - 18 q^{91} + 28 q^{93} + 6 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^7 + 2 * q^9 - 6 * q^13 + 6 * q^15 + 6 * q^19 + 6 * q^23 - 4 * q^25 + 8 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^35 - 4 * q^37 - 6 * q^39 + 12 * q^41 + 12 * q^45 + 6 * q^47 + 10 * q^49 - 18 * q^51 + 12 * q^53 + 12 * q^57 - 12 * q^59 - 12 * q^61 - 6 * q^63 + 10 * q^67 + 12 * q^69 + 12 * q^71 - 12 * q^73 - 4 * q^75 - 6 * q^79 + 2 * q^81 + 6 * q^83 - 18 * q^85 + 6 * q^87 + 6 * q^89 - 18 * q^91 + 28 * q^93 + 6 * q^95 - 2 * q^97

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