LMFDB - Embedded newform 5776.2.a.ca.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5776,2,Mod(1,5776)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-6,0,-2,0,2,0,2,0,-12,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.1215922075$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.10564000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 24x^4 - 28x^3 - 21x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2888)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.94364$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.94364 q^{3} -0.422186 q^{5} -0.503690 q^{7} +0.777742 q^{9} +2.53017 q^{11} -0.681907 q^{13} -0.820579 q^{15} -5.42785 q^{17} -0.978993 q^{21} +3.01860 q^{23} -4.82176 q^{25} -4.31927 q^{27} +7.18948 q^{29} -10.4444 q^{31} +4.91775 q^{33} +0.212651 q^{35} -4.38572 q^{37} -1.32538 q^{39} -9.34554 q^{41} +4.16921 q^{43} -0.328352 q^{45} -10.2139 q^{47} -6.74630 q^{49} -10.5498 q^{51} +4.18583 q^{53} -1.06820 q^{55} +9.17729 q^{59} +0.515531 q^{61} -0.391741 q^{63} +0.287892 q^{65} -7.91082 q^{67} +5.86708 q^{69} +2.98275 q^{71} +10.7864 q^{73} -9.37177 q^{75} -1.27442 q^{77} +2.21860 q^{79} -10.7283 q^{81} -1.59618 q^{83} +2.29156 q^{85} +13.9738 q^{87} -12.8464 q^{89} +0.343470 q^{91} -20.3002 q^{93} -3.47329 q^{97} +1.96782 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 12 q^{11} + 6 q^{13} - 10 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 12 q^{27} + 18 q^{29} - 14 q^{31} + 40 q^{33} - 18 q^{35} + 16 q^{37} - 28 q^{39} + 12 q^{41}+ \cdots - 32 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 12 * q^11 + 6 * q^13 - 10 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 12 * q^27 + 18 * q^29 - 14 * q^31 + 40 * q^33 - 18 * q^35 + 16 * q^37 - 28 * q^39 + 12 * q^41 - 8 * q^43 - 2 * q^45 - 24 * q^47 + 12 * q^49 + 22 * q^51 + 12 * q^53 + 2 * q^55 + 16 * q^59 - 12 * q^63 + 12 * q^65 - 42 * q^67 - 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 - 12 * q^83 - 42 * q^85 + 12 * q^87 + 8 * q^89 - 16 * q^91 - 14 * q^93 + 28 * q^97 - 32 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.94364	1.12216	0.561081	−	0.827761i	$-0.310386\pi$
$3$	1.94364	1.12216	0.561081	+	0.827761i	$0.310386\pi$
$4$	0	0
$5$	−0.422186	−0.188807	−0.0944037	−	0.995534i	$-0.530094\pi$
$5$	−0.422186	−0.188807	−0.0944037	+	0.995534i	$0.530094\pi$
$6$	0	0
$7$	−0.503690	−0.190377	−0.0951885	−	0.995459i	$-0.530345\pi$
$7$	−0.503690	−0.190377	−0.0951885	+	0.995459i	$0.530345\pi$
$8$	0	0
$9$	0.777742	0.259247
$10$	0	0
$11$	2.53017	0.762876	0.381438	−	0.924394i	$-0.375429\pi$
$11$	2.53017	0.762876	0.381438	+	0.924394i	$0.375429\pi$
$12$	0	0
$13$	−0.681907	−0.189127	−0.0945636	−	0.995519i	$-0.530146\pi$
$13$	−0.681907	−0.189127	−0.0945636	+	0.995519i	$0.530146\pi$
$14$	0	0
$15$	−0.820579	−0.211872
$16$	0	0
$17$	−5.42785	−1.31645	−0.658224	−	0.752823i	$-0.728692\pi$
$17$	−5.42785	−1.31645	−0.658224	+	0.752823i	$0.728692\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.978993	−0.213634
$22$	0	0
$23$	3.01860	0.629423	0.314711	−	0.949187i	$-0.398092\pi$
$23$	3.01860	0.629423	0.314711	+	0.949187i	$0.398092\pi$
$24$	0	0
$25$	−4.82176	−0.964352
$26$	0	0
$27$	−4.31927	−0.831244
$28$	0	0
$29$	7.18948	1.33505	0.667527	−	0.744586i	$-0.267353\pi$
$29$	7.18948	1.33505	0.667527	+	0.744586i	$0.267353\pi$
$30$	0	0
$31$	−10.4444	−1.87587	−0.937935	−	0.346812i	$-0.887264\pi$
$31$	−10.4444	−1.87587	−0.937935	+	0.346812i	$0.887264\pi$
$32$	0	0
$33$	4.91775	0.856071
$34$	0	0
$35$	0.212651	0.0359446
$36$	0	0
$37$	−4.38572	−0.721009	−0.360504	−	0.932758i	$-0.617395\pi$
$37$	−4.38572	−0.721009	−0.360504	+	0.932758i	$0.617395\pi$
$38$	0	0
$39$	−1.32538	−0.212231
$40$	0	0
$41$	−9.34554	−1.45953	−0.729764	−	0.683699i	$-0.760370\pi$
$41$	−9.34554	−1.45953	−0.729764	+	0.683699i	$0.760370\pi$
$42$	0	0
$43$	4.16921	0.635798	0.317899	−	0.948125i	$-0.397023\pi$
$43$	4.16921	0.635798	0.317899	+	0.948125i	$0.397023\pi$
$44$	0	0
$45$	−0.328352	−0.0489478
$46$	0	0
$47$	−10.2139	−1.48984	−0.744922	−	0.667152i	$-0.767513\pi$
$47$	−10.2139	−1.48984	−0.744922	+	0.667152i	$0.767513\pi$
$48$	0	0
$49$	−6.74630	−0.963757
$50$	0	0
$51$	−10.5498	−1.47727
$52$	0	0
$53$	4.18583	0.574968	0.287484	−	0.957785i	$-0.407181\pi$
$53$	4.18583	0.574968	0.287484	+	0.957785i	$0.407181\pi$
$54$	0	0
$55$	−1.06820	−0.144037
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.17729	1.19478	0.597391	−	0.801950i	$-0.296204\pi$
$59$	9.17729	1.19478	0.597391	+	0.801950i	$0.296204\pi$
$60$	0	0
$61$	0.515531	0.0660070	0.0330035	−	0.999455i	$-0.489493\pi$
$61$	0.515531	0.0660070	0.0330035	+	0.999455i	$0.489493\pi$
$62$	0	0
$63$	−0.391741	−0.0493547
$64$	0	0
$65$	0.287892	0.0357086
$66$	0	0
$67$	−7.91082	−0.966460	−0.483230	−	0.875493i	$-0.660537\pi$
$67$	−7.91082	−0.966460	−0.483230	+	0.875493i	$0.660537\pi$
$68$	0	0
$69$	5.86708	0.706314
$70$	0	0
$71$	2.98275	0.353987	0.176994	−	0.984212i	$-0.443363\pi$
$71$	2.98275	0.353987	0.176994	+	0.984212i	$0.443363\pi$
$72$	0	0
$73$	10.7864	1.26245	0.631224	−	0.775601i	$-0.282553\pi$
$73$	10.7864	1.26245	0.631224	+	0.775601i	$0.282553\pi$
$74$	0	0
$75$	−9.37177	−1.08216
$76$	0	0
$77$	−1.27442	−0.145234
$78$	0	0
$79$	2.21860	0.249612	0.124806	−	0.992181i	$-0.460169\pi$
$79$	2.21860	0.249612	0.124806	+	0.992181i	$0.460169\pi$
$80$	0	0
$81$	−10.7283	−1.19204
$82$	0	0
$83$	−1.59618	−0.175204	−0.0876018	−	0.996156i	$-0.527920\pi$
$83$	−1.59618	−0.175204	−0.0876018	+	0.996156i	$0.527920\pi$
$84$	0	0
$85$	2.29156	0.248555
$86$	0	0
$87$	13.9738	1.49815
$88$	0	0
$89$	−12.8464	−1.36171	−0.680857	−	0.732416i	$-0.738393\pi$
$89$	−12.8464	−1.36171	−0.680857	+	0.732416i	$0.738393\pi$
$90$	0	0
$91$	0.343470	0.0360055
$92$	0	0
$93$	−20.3002	−2.10503
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.47329	−0.352659	−0.176329	−	0.984331i	$-0.556422\pi$
$97$	−3.47329	−0.352659	−0.176329	+	0.984331i	$0.556422\pi$
$98$	0	0
$99$	1.96782	0.197774
$100$	0	0
$101$	−3.36644	−0.334973	−0.167487	−	0.985874i	$-0.553565\pi$
$101$	−3.36644	−0.334973	−0.167487	+	0.985874i	$0.553565\pi$
$102$	0	0
$103$	−4.62521	−0.455736	−0.227868	−	0.973692i	$-0.573175\pi$
$103$	−4.62521	−0.455736	−0.227868	+	0.973692i	$0.573175\pi$
$104$	0	0
$105$	0.413318	0.0403357
$106$	0	0
$107$	−5.95957	−0.576134	−0.288067	−	0.957610i	$-0.593013\pi$
$107$	−5.95957	−0.576134	−0.288067	+	0.957610i	$0.593013\pi$
$108$	0	0
$109$	18.3703	1.75956	0.879780	−	0.475382i	$-0.157690\pi$
$109$	18.3703	1.75956	0.879780	+	0.475382i	$0.157690\pi$
$110$	0	0
$111$	−8.52428	−0.809088
$112$	0	0
$113$	6.75443	0.635403	0.317702	−	0.948191i	$-0.397089\pi$
$113$	6.75443	0.635403	0.317702	+	0.948191i	$0.397089\pi$
$114$	0	0
$115$	−1.27441	−0.118840
$116$	0	0
$117$	−0.530348	−0.0490307
$118$	0	0
$119$	2.73396	0.250621
$120$	0	0
$121$	−4.59822	−0.418020
$122$	0	0
$123$	−18.1644	−1.63783
$124$	0	0
$125$	4.14661	0.370884
$126$	0	0
$127$	−15.9078	−1.41159	−0.705796	−	0.708415i	$-0.749410\pi$
$127$	−15.9078	−1.41159	−0.705796	+	0.708415i	$0.749410\pi$
$128$	0	0
$129$	8.10344	0.713468
$130$	0	0
$131$	−9.95021	−0.869354	−0.434677	−	0.900586i	$-0.643137\pi$
$131$	−9.95021	−0.869354	−0.434677	+	0.900586i	$0.643137\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.82354	0.156945
$136$	0	0
$137$	14.6304	1.24996	0.624981	−	0.780640i	$-0.285107\pi$
$137$	14.6304	1.24996	0.624981	+	0.780640i	$0.285107\pi$
$138$	0	0
$139$	−1.05200	−0.0892292	−0.0446146	−	0.999004i	$-0.514206\pi$
$139$	−1.05200	−0.0892292	−0.0446146	+	0.999004i	$0.514206\pi$
$140$	0	0
$141$	−19.8521	−1.67185
$142$	0	0
$143$	−1.72534	−0.144281
$144$	0	0
$145$	−3.03530	−0.252068
$146$	0	0
$147$	−13.1124	−1.08149
$148$	0	0
$149$	19.0971	1.56449	0.782246	−	0.622970i	$-0.214074\pi$
$149$	19.0971	1.56449	0.782246	+	0.622970i	$0.214074\pi$
$150$	0	0
$151$	−23.7911	−1.93609	−0.968045	−	0.250778i	$-0.919314\pi$
$151$	−23.7911	−1.93609	−0.968045	+	0.250778i	$0.919314\pi$
$152$	0	0
$153$	−4.22147	−0.341285
$154$	0	0
$155$	4.40948	0.354178
$156$	0	0
$157$	8.46925	0.675920	0.337960	−	0.941160i	$-0.390263\pi$
$157$	8.46925	0.675920	0.337960	+	0.941160i	$0.390263\pi$
$158$	0	0
$159$	8.13575	0.645207
$160$	0	0
$161$	−1.52044	−0.119828
$162$	0	0
$163$	−16.9968	−1.33129	−0.665647	−	0.746267i	$-0.731844\pi$
$163$	−16.9968	−1.33129	−0.665647	+	0.746267i	$0.731844\pi$
$164$	0	0
$165$	−2.07621	−0.161632
$166$	0	0
$167$	16.7819	1.29862	0.649311	−	0.760523i	$-0.275058\pi$
$167$	16.7819	1.29862	0.649311	+	0.760523i	$0.275058\pi$
$168$	0	0
$169$	−12.5350	−0.964231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.00240	−0.0762112	−0.0381056	−	0.999274i	$-0.512132\pi$
$173$	−1.00240	−0.0762112	−0.0381056	+	0.999274i	$0.512132\pi$
$174$	0	0
$175$	2.42867	0.183590
$176$	0	0
$177$	17.8374	1.34074
$178$	0	0
$179$	−1.72867	−0.129207	−0.0646033	−	0.997911i	$-0.520578\pi$
$179$	−1.72867	−0.129207	−0.0646033	+	0.997911i	$0.520578\pi$
$180$	0	0
$181$	24.3966	1.81339	0.906693	−	0.421791i	$-0.138598\pi$
$181$	24.3966	1.81339	0.906693	+	0.421791i	$0.138598\pi$
$182$	0	0
$183$	1.00201	0.0740705
$184$	0	0
$185$	1.85159	0.136132
$186$	0	0
$187$	−13.7334	−1.00429
$188$	0	0
$189$	2.17558	0.158250
$190$	0	0
$191$	3.77858	0.273408	0.136704	−	0.990612i	$-0.456349\pi$
$191$	3.77858	0.273408	0.136704	+	0.990612i	$0.456349\pi$
$192$	0	0
$193$	15.7012	1.13020	0.565099	−	0.825023i	$-0.308838\pi$
$193$	15.7012	1.13020	0.565099	+	0.825023i	$0.308838\pi$
$194$	0	0
$195$	0.559559	0.0400708
$196$	0	0
$197$	−14.1468	−1.00792	−0.503958	−	0.863728i	$-0.668123\pi$
$197$	−14.1468	−1.00792	−0.503958	+	0.863728i	$0.668123\pi$
$198$	0	0
$199$	−5.16067	−0.365830	−0.182915	−	0.983129i	$-0.558553\pi$
$199$	−5.16067	−0.365830	−0.182915	+	0.983129i	$0.558553\pi$
$200$	0	0
$201$	−15.3758	−1.08452
$202$	0	0
$203$	−3.62127	−0.254164
$204$	0	0
$205$	3.94556	0.275570
$206$	0	0
$207$	2.34770	0.163176
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.06846	0.417770	0.208885	−	0.977940i	$-0.433017\pi$
$211$	6.06846	0.417770	0.208885	+	0.977940i	$0.433017\pi$
$212$	0	0
$213$	5.79740	0.397231
$214$	0	0
$215$	−1.76018	−0.120043
$216$	0	0
$217$	5.26074	0.357122
$218$	0	0
$219$	20.9648	1.41667
$220$	0	0
$221$	3.70129	0.248976
$222$	0	0
$223$	5.95773	0.398959	0.199480	−	0.979902i	$-0.436075\pi$
$223$	5.95773	0.398959	0.199480	+	0.979902i	$0.436075\pi$
$224$	0	0
$225$	−3.75008	−0.250006
$226$	0	0
$227$	−24.4566	−1.62324	−0.811619	−	0.584187i	$-0.801414\pi$
$227$	−24.4566	−1.62324	−0.811619	+	0.584187i	$0.801414\pi$
$228$	0	0
$229$	−20.6998	−1.36788	−0.683941	−	0.729538i	$-0.739735\pi$
$229$	−20.6998	−1.36788	−0.683941	+	0.729538i	$0.739735\pi$
$230$	0	0
$231$	−2.47702	−0.162976
$232$	0	0
$233$	−29.1566	−1.91011	−0.955057	−	0.296423i	$-0.904206\pi$
$233$	−29.1566	−1.91011	−0.955057	+	0.296423i	$0.904206\pi$
$234$	0	0
$235$	4.31215	0.281293
$236$	0	0
$237$	4.31216	0.280105
$238$	0	0
$239$	0.754852	0.0488273	0.0244136	−	0.999702i	$-0.492228\pi$
$239$	0.754852	0.0488273	0.0244136	+	0.999702i	$0.492228\pi$
$240$	0	0
$241$	−13.3471	−0.859764	−0.429882	−	0.902885i	$-0.641445\pi$
$241$	−13.3471	−0.859764	−0.429882	+	0.902885i	$0.641445\pi$
$242$	0	0
$243$	−7.89424	−0.506415
$244$	0	0
$245$	2.84819	0.181964
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.10240	−0.196607
$250$	0	0
$251$	−21.1508	−1.33503	−0.667514	−	0.744597i	$-0.732642\pi$
$251$	−21.1508	−1.33503	−0.667514	+	0.744597i	$0.732642\pi$
$252$	0	0
$253$	7.63760	0.480172
$254$	0	0
$255$	4.45398	0.278919
$256$	0	0
$257$	−12.8564	−0.801958	−0.400979	−	0.916087i	$-0.631330\pi$
$257$	−12.8564	−0.801958	−0.400979	+	0.916087i	$0.631330\pi$
$258$	0	0
$259$	2.20905	0.137264
$260$	0	0
$261$	5.59156	0.346109
$262$	0	0
$263$	−10.4154	−0.642242	−0.321121	−	0.947038i	$-0.604060\pi$
$263$	−10.4154	−0.642242	−0.321121	+	0.947038i	$0.604060\pi$
$264$	0	0
$265$	−1.76720	−0.108558
$266$	0	0
$267$	−24.9688	−1.52806
$268$	0	0
$269$	17.9585	1.09495	0.547473	−	0.836823i	$-0.315590\pi$
$269$	17.9585	1.09495	0.547473	+	0.836823i	$0.315590\pi$
$270$	0	0
$271$	−21.5676	−1.31014	−0.655069	−	0.755569i	$-0.727361\pi$
$271$	−21.5676	−1.31014	−0.655069	+	0.755569i	$0.727361\pi$
$272$	0	0
$273$	0.667583	0.0404040
$274$	0	0
$275$	−12.1999	−0.735681
$276$	0	0
$277$	19.4391	1.16798	0.583992	−	0.811759i	$-0.301490\pi$
$277$	19.4391	1.16798	0.583992	+	0.811759i	$0.301490\pi$
$278$	0	0
$279$	−8.12305	−0.486314
$280$	0	0
$281$	1.28133	0.0764377	0.0382188	−	0.999269i	$-0.487832\pi$
$281$	1.28133	0.0764377	0.0382188	+	0.999269i	$0.487832\pi$
$282$	0	0
$283$	−18.8498	−1.12050	−0.560251	−	0.828323i	$-0.689295\pi$
$283$	−18.8498	−1.12050	−0.560251	+	0.828323i	$0.689295\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.70726	0.277861
$288$	0	0
$289$	12.4616	0.733033
$290$	0	0
$291$	−6.75083	−0.395740
$292$	0	0
$293$	13.5171	0.789676	0.394838	−	0.918751i	$-0.370801\pi$
$293$	13.5171	0.789676	0.394838	+	0.918751i	$0.370801\pi$
$294$	0	0
$295$	−3.87453	−0.225584
$296$	0	0
$297$	−10.9285	−0.634137
$298$	0	0
$299$	−2.05841	−0.119041
$300$	0	0
$301$	−2.09999	−0.121041
$302$	0	0
$303$	−6.54315	−0.375894
$304$	0	0
$305$	−0.217650	−0.0124626
$306$	0	0
$307$	19.6472	1.12133	0.560663	−	0.828044i	$-0.310546\pi$
$307$	19.6472	1.12133	0.560663	+	0.828044i	$0.310546\pi$
$308$	0	0
$309$	−8.98976	−0.511410
$310$	0	0
$311$	−20.0407	−1.13640	−0.568202	−	0.822889i	$-0.692361\pi$
$311$	−20.0407	−1.13640	−0.568202	+	0.822889i	$0.692361\pi$
$312$	0	0
$313$	−10.9810	−0.620683	−0.310341	−	0.950625i	$-0.600443\pi$
$313$	−10.9810	−0.620683	−0.310341	+	0.950625i	$0.600443\pi$
$314$	0	0
$315$	0.165388	0.00931854
$316$	0	0
$317$	12.5017	0.702167	0.351083	−	0.936344i	$-0.385813\pi$
$317$	12.5017	0.702167	0.351083	+	0.936344i	$0.385813\pi$
$318$	0	0
$319$	18.1906	1.01848
$320$	0	0
$321$	−11.5833	−0.646515
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.28799	0.182385
$326$	0	0
$327$	35.7054	1.97451
$328$	0	0
$329$	5.14462	0.283632
$330$	0	0
$331$	17.4321	0.958152	0.479076	−	0.877773i	$-0.340972\pi$
$331$	17.4321	0.958152	0.479076	+	0.877773i	$0.340972\pi$
$332$	0	0
$333$	−3.41096	−0.186920
$334$	0	0
$335$	3.33984	0.182475
$336$	0	0
$337$	−11.7431	−0.639687	−0.319843	−	0.947470i	$-0.603630\pi$
$337$	−11.7431	−0.639687	−0.319843	+	0.947470i	$0.603630\pi$
$338$	0	0
$339$	13.1282	0.713025
$340$	0	0
$341$	−26.4261	−1.43106
$342$	0	0
$343$	6.92388	0.373854
$344$	0	0
$345$	−2.47700	−0.133357
$346$	0	0
$347$	−20.8857	−1.12120	−0.560602	−	0.828085i	$-0.689430\pi$
$347$	−20.8857	−1.12120	−0.560602	+	0.828085i	$0.689430\pi$
$348$	0	0
$349$	−20.1827	−1.08036	−0.540178	−	0.841551i	$-0.681643\pi$
$349$	−20.1827	−1.08036	−0.540178	+	0.841551i	$0.681643\pi$
$350$	0	0
$351$	2.94534	0.157211
$352$	0	0
$353$	−16.7155	−0.889675	−0.444838	−	0.895611i	$-0.646739\pi$
$353$	−16.7155	−0.889675	−0.444838	+	0.895611i	$0.646739\pi$
$354$	0	0
$355$	−1.25928	−0.0668354
$356$	0	0
$357$	5.31383	0.281238
$358$	0	0
$359$	22.9404	1.21075	0.605373	−	0.795942i	$-0.293024\pi$
$359$	22.9404	1.21075	0.605373	+	0.795942i	$0.293024\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−8.93729	−0.469086
$364$	0	0
$365$	−4.55385	−0.238360
$366$	0	0
$367$	36.0951	1.88415	0.942074	−	0.335406i	$-0.108874\pi$
$367$	36.0951	1.88415	0.942074	+	0.335406i	$0.108874\pi$
$368$	0	0
$369$	−7.26842	−0.378379
$370$	0	0
$371$	−2.10836	−0.109461
$372$	0	0
$373$	29.4376	1.52422	0.762109	−	0.647448i	$-0.224164\pi$
$373$	29.4376	1.52422	0.762109	+	0.647448i	$0.224164\pi$
$374$	0	0
$375$	8.05953	0.416192
$376$	0	0
$377$	−4.90256	−0.252495
$378$	0	0
$379$	34.7516	1.78507	0.892534	−	0.450981i	$-0.148926\pi$
$379$	34.7516	1.78507	0.892534	+	0.450981i	$0.148926\pi$
$380$	0	0
$381$	−30.9191	−1.58403
$382$	0	0
$383$	−1.24480	−0.0636065	−0.0318033	−	0.999494i	$-0.510125\pi$
$383$	−1.24480	−0.0636065	−0.0318033	+	0.999494i	$0.510125\pi$
$384$	0	0
$385$	0.538044	0.0274213
$386$	0	0
$387$	3.24257	0.164829
$388$	0	0
$389$	−30.7442	−1.55879	−0.779397	−	0.626531i	$-0.784474\pi$
$389$	−30.7442	−1.55879	−0.779397	+	0.626531i	$0.784474\pi$
$390$	0	0
$391$	−16.3845	−0.828602
$392$	0	0
$393$	−19.3396	−0.975556
$394$	0	0
$395$	−0.936662	−0.0471286
$396$	0	0
$397$	25.9477	1.30228	0.651138	−	0.758959i	$-0.274292\pi$
$397$	25.9477	1.30228	0.651138	+	0.758959i	$0.274292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.49940	−0.174752	−0.0873758	−	0.996175i	$-0.527848\pi$
$401$	−3.49940	−0.174752	−0.0873758	+	0.996175i	$0.527848\pi$
$402$	0	0
$403$	7.12211	0.354778
$404$	0	0
$405$	4.52936	0.225066
$406$	0	0
$407$	−11.0966	−0.550040
$408$	0	0
$409$	23.7449	1.17411	0.587055	−	0.809547i	$-0.300287\pi$
$409$	23.7449	1.17411	0.587055	+	0.809547i	$0.300287\pi$
$410$	0	0
$411$	28.4363	1.40266
$412$	0	0
$413$	−4.62251	−0.227459
$414$	0	0
$415$	0.673886	0.0330797
$416$	0	0
$417$	−2.04470	−0.100130
$418$	0	0
$419$	−38.7765	−1.89436	−0.947178	−	0.320708i	$-0.896079\pi$
$419$	−38.7765	−1.89436	−0.947178	+	0.320708i	$0.896079\pi$
$420$	0	0
$421$	6.12507	0.298518	0.149259	−	0.988798i	$-0.452311\pi$
$421$	6.12507	0.298518	0.149259	+	0.988798i	$0.452311\pi$
$422$	0	0
$423$	−7.94374	−0.386238
$424$	0	0
$425$	26.1718	1.26952
$426$	0	0
$427$	−0.259668	−0.0125662
$428$	0	0
$429$	−3.35345	−0.161906
$430$	0	0
$431$	21.8752	1.05369	0.526845	−	0.849961i	$-0.323375\pi$
$431$	21.8752	1.05369	0.526845	+	0.849961i	$0.323375\pi$
$432$	0	0
$433$	11.4485	0.550177	0.275089	−	0.961419i	$-0.411293\pi$
$433$	11.4485	0.550177	0.275089	+	0.961419i	$0.411293\pi$
$434$	0	0
$435$	−5.89954	−0.282861
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−27.7756	−1.32566	−0.662828	−	0.748772i	$-0.730644\pi$
$439$	−27.7756	−1.32566	−0.662828	+	0.748772i	$0.730644\pi$
$440$	0	0
$441$	−5.24688	−0.249851
$442$	0	0
$443$	25.9628	1.23353	0.616764	−	0.787148i	$-0.288443\pi$
$443$	25.9628	1.23353	0.616764	+	0.787148i	$0.288443\pi$
$444$	0	0
$445$	5.42357	0.257102
$446$	0	0
$447$	37.1178	1.75561
$448$	0	0
$449$	−13.3509	−0.630066	−0.315033	−	0.949081i	$-0.602016\pi$
$449$	−13.3509	−0.630066	−0.315033	+	0.949081i	$0.602016\pi$
$450$	0	0
$451$	−23.6458	−1.11344
$452$	0	0
$453$	−46.2413	−2.17261
$454$	0	0
$455$	−0.145008	−0.00679810
$456$	0	0
$457$	30.5280	1.42804	0.714020	−	0.700125i	$-0.246873\pi$
$457$	30.5280	1.42804	0.714020	+	0.700125i	$0.246873\pi$
$458$	0	0
$459$	23.4444	1.09429
$460$	0	0
$461$	−4.80002	−0.223559	−0.111780	−	0.993733i	$-0.535655\pi$
$461$	−4.80002	−0.223559	−0.111780	+	0.993733i	$0.535655\pi$
$462$	0	0
$463$	−7.92519	−0.368315	−0.184157	−	0.982897i	$-0.558956\pi$
$463$	−7.92519	−0.368315	−0.184157	+	0.982897i	$0.558956\pi$
$464$	0	0
$465$	8.57045	0.397445
$466$	0	0
$467$	−5.47639	−0.253417	−0.126708	−	0.991940i	$-0.540441\pi$
$467$	−5.47639	−0.253417	−0.126708	+	0.991940i	$0.540441\pi$
$468$	0	0
$469$	3.98460	0.183992
$470$	0	0
$471$	16.4612	0.758492
$472$	0	0
$473$	10.5488	0.485035
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.25550	0.149059
$478$	0	0
$479$	0.802184	0.0366527	0.0183264	−	0.999832i	$-0.494166\pi$
$479$	0.802184	0.0366527	0.0183264	+	0.999832i	$0.494166\pi$
$480$	0	0
$481$	2.99066	0.136362
$482$	0	0
$483$	−2.95519	−0.134466
$484$	0	0
$485$	1.46637	0.0665846
$486$	0	0
$487$	−15.3683	−0.696404	−0.348202	−	0.937419i	$-0.613208\pi$
$487$	−15.3683	−0.696404	−0.348202	+	0.937419i	$0.613208\pi$
$488$	0	0
$489$	−33.0357	−1.49393
$490$	0	0
$491$	31.7186	1.43144	0.715720	−	0.698387i	$-0.246099\pi$
$491$	31.7186	1.43144	0.715720	+	0.698387i	$0.246099\pi$
$492$	0	0
$493$	−39.0234	−1.75753
$494$	0	0
$495$	−0.830788	−0.0373411
$496$	0	0
$497$	−1.50238	−0.0673911
$498$	0	0
$499$	6.20215	0.277646	0.138823	−	0.990317i	$-0.455668\pi$
$499$	6.20215	0.277646	0.138823	+	0.990317i	$0.455668\pi$
$500$	0	0
$501$	32.6180	1.45726
$502$	0	0
$503$	26.9000	1.19941	0.599706	−	0.800220i	$-0.295284\pi$
$503$	26.9000	1.19941	0.599706	+	0.800220i	$0.295284\pi$
$504$	0	0
$505$	1.42126	0.0632454
$506$	0	0
$507$	−24.3635	−1.08202
$508$	0	0
$509$	15.4859	0.686402	0.343201	−	0.939262i	$-0.388489\pi$
$509$	15.4859	0.686402	0.343201	+	0.939262i	$0.388489\pi$
$510$	0	0
$511$	−5.43299	−0.240341
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.95270	0.0860463
$516$	0	0
$517$	−25.8428	−1.13657
$518$	0	0
$519$	−1.94831	−0.0855214
$520$	0	0
$521$	34.2598	1.50095	0.750475	−	0.660899i	$-0.229825\pi$
$521$	34.2598	1.50095	0.750475	+	0.660899i	$0.229825\pi$
$522$	0	0
$523$	−8.19316	−0.358262	−0.179131	−	0.983825i	$-0.557329\pi$
$523$	−8.19316	−0.358262	−0.179131	+	0.983825i	$0.557329\pi$
$524$	0	0
$525$	4.72047	0.206018
$526$	0	0
$527$	56.6906	2.46948
$528$	0	0
$529$	−13.8880	−0.603827
$530$	0	0
$531$	7.13757	0.309744
$532$	0	0
$533$	6.37279	0.276036
$534$	0	0
$535$	2.51605	0.108778
$536$	0	0
$537$	−3.35991	−0.144991
$538$	0	0
$539$	−17.0693	−0.735227
$540$	0	0
$541$	−15.8082	−0.679646	−0.339823	−	0.940489i	$-0.610367\pi$
$541$	−15.8082	−0.679646	−0.339823	+	0.940489i	$0.610367\pi$
$542$	0	0
$543$	47.4183	2.03491
$544$	0	0
$545$	−7.75570	−0.332218
$546$	0	0
$547$	1.24357	0.0531712	0.0265856	−	0.999647i	$-0.491537\pi$
$547$	1.24357	0.0531712	0.0265856	+	0.999647i	$0.491537\pi$
$548$	0	0
$549$	0.400950	0.0171121
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1.11749	−0.0475204
$554$	0	0
$555$	3.59883	0.152762
$556$	0	0
$557$	−3.88239	−0.164502	−0.0822510	−	0.996612i	$-0.526211\pi$
$557$	−3.88239	−0.164502	−0.0822510	+	0.996612i	$0.526211\pi$
$558$	0	0
$559$	−2.84301	−0.120247
$560$	0	0
$561$	−26.6928	−1.12697
$562$	0	0
$563$	27.5748	1.16214	0.581070	−	0.813853i	$-0.302634\pi$
$563$	27.5748	1.16214	0.581070	+	0.813853i	$0.302634\pi$
$564$	0	0
$565$	−2.85163	−0.119969
$566$	0	0
$567$	5.40376	0.226937
$568$	0	0
$569$	−20.8746	−0.875110	−0.437555	−	0.899192i	$-0.644155\pi$
$569$	−20.8746	−0.875110	−0.437555	+	0.899192i	$0.644155\pi$
$570$	0	0
$571$	35.8475	1.50017	0.750086	−	0.661340i	$-0.230012\pi$
$571$	35.8475	1.50017	0.750086	+	0.661340i	$0.230012\pi$
$572$	0	0
$573$	7.34420	0.306808
$574$	0	0
$575$	−14.5550	−0.606985
$576$	0	0
$577$	3.68540	0.153425	0.0767127	−	0.997053i	$-0.475558\pi$
$577$	3.68540	0.153425	0.0767127	+	0.997053i	$0.475558\pi$
$578$	0	0
$579$	30.5175	1.26827
$580$	0	0
$581$	0.803981	0.0333547
$582$	0	0
$583$	10.5909	0.438629
$584$	0	0
$585$	0.223906	0.00925736
$586$	0	0
$587$	−25.0321	−1.03319	−0.516593	−	0.856231i	$-0.672800\pi$
$587$	−25.0321	−1.03319	−0.516593	+	0.856231i	$0.672800\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−27.4962	−1.13104
$592$	0	0
$593$	7.69797	0.316118	0.158059	−	0.987430i	$-0.449476\pi$
$593$	7.69797	0.316118	0.158059	+	0.987430i	$0.449476\pi$
$594$	0	0
$595$	−1.15424	−0.0473192
$596$	0	0
$597$	−10.0305	−0.410521
$598$	0	0
$599$	−17.2487	−0.704761	−0.352381	−	0.935857i	$-0.614628\pi$
$599$	−17.2487	−0.704761	−0.352381	+	0.935857i	$0.614628\pi$
$600$	0	0
$601$	17.0334	0.694808	0.347404	−	0.937715i	$-0.387063\pi$
$601$	17.0334	0.694808	0.347404	+	0.937715i	$0.387063\pi$
$602$	0	0
$603$	−6.15257	−0.250552
$604$	0	0
$605$	1.94130	0.0789252
$606$	0	0
$607$	19.6440	0.797327	0.398664	−	0.917097i	$-0.369474\pi$
$607$	19.6440	0.797327	0.398664	+	0.917097i	$0.369474\pi$
$608$	0	0
$609$	−7.03846	−0.285213
$610$	0	0
$611$	6.96490	0.281770
$612$	0	0
$613$	39.2667	1.58597	0.792983	−	0.609244i	$-0.208527\pi$
$613$	39.2667	1.58597	0.792983	+	0.609244i	$0.208527\pi$
$614$	0	0
$615$	7.66875	0.309234
$616$	0	0
$617$	−11.5240	−0.463937	−0.231968	−	0.972723i	$-0.574517\pi$
$617$	−11.5240	−0.463937	−0.231968	+	0.972723i	$0.574517\pi$
$618$	0	0
$619$	−46.2942	−1.86072	−0.930360	−	0.366648i	$-0.880505\pi$
$619$	−46.2942	−1.86072	−0.930360	+	0.366648i	$0.880505\pi$
$620$	0	0
$621$	−13.0382	−0.523204
$622$	0	0
$623$	6.47060	0.259239
$624$	0	0
$625$	22.3582	0.894326
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.8051	0.949170
$630$	0	0
$631$	21.9295	0.873000	0.436500	−	0.899704i	$-0.356218\pi$
$631$	21.9295	0.873000	0.436500	+	0.899704i	$0.356218\pi$
$632$	0	0
$633$	11.7949	0.468806
$634$	0	0
$635$	6.71607	0.266519
$636$	0	0
$637$	4.60035	0.182272
$638$	0	0
$639$	2.31981	0.0917703
$640$	0	0
$641$	−8.73616	−0.345058	−0.172529	−	0.985004i	$-0.555194\pi$
$641$	−8.73616	−0.345058	−0.172529	+	0.985004i	$0.555194\pi$
$642$	0	0
$643$	37.4284	1.47603	0.738016	−	0.674783i	$-0.235763\pi$
$643$	37.4284	1.47603	0.738016	+	0.674783i	$0.235763\pi$
$644$	0	0
$645$	−3.42116	−0.134708
$646$	0	0
$647$	−14.6325	−0.575261	−0.287631	−	0.957741i	$-0.592868\pi$
$647$	−14.6325	−0.575261	−0.287631	+	0.957741i	$0.592868\pi$
$648$	0	0
$649$	23.2201	0.911471
$650$	0	0
$651$	10.2250	0.400749
$652$	0	0
$653$	−35.1120	−1.37404	−0.687020	−	0.726639i	$-0.741081\pi$
$653$	−35.1120	−1.37404	−0.687020	+	0.726639i	$0.741081\pi$
$654$	0	0
$655$	4.20084	0.164140
$656$	0	0
$657$	8.38901	0.327286
$658$	0	0
$659$	22.9883	0.895495	0.447748	−	0.894160i	$-0.352226\pi$
$659$	22.9883	0.895495	0.447748	+	0.894160i	$0.352226\pi$
$660$	0	0
$661$	−7.06550	−0.274816	−0.137408	−	0.990515i	$-0.543877\pi$
$661$	−7.06550	−0.274816	−0.137408	+	0.990515i	$0.543877\pi$
$662$	0	0
$663$	7.19398	0.279391
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.7022	0.840313
$668$	0	0
$669$	11.5797	0.447697
$670$	0	0
$671$	1.30438	0.0503552
$672$	0	0
$673$	−16.0233	−0.617654	−0.308827	−	0.951118i	$-0.599936\pi$
$673$	−16.0233	−0.617654	−0.308827	+	0.951118i	$0.599936\pi$
$674$	0	0
$675$	20.8265	0.801612
$676$	0	0
$677$	−26.8510	−1.03197	−0.515984	−	0.856598i	$-0.672574\pi$
$677$	−26.8510	−1.03197	−0.515984	+	0.856598i	$0.672574\pi$
$678$	0	0
$679$	1.74946	0.0671382
$680$	0	0
$681$	−47.5348	−1.82154
$682$	0	0
$683$	20.2128	0.773421	0.386710	−	0.922201i	$-0.373611\pi$
$683$	20.2128	0.773421	0.386710	+	0.922201i	$0.373611\pi$
$684$	0	0
$685$	−6.17677	−0.236002
$686$	0	0
$687$	−40.2330	−1.53498
$688$	0	0
$689$	−2.85435	−0.108742
$690$	0	0
$691$	−13.3849	−0.509185	−0.254593	−	0.967048i	$-0.581941\pi$
$691$	−13.3849	−0.509185	−0.254593	+	0.967048i	$0.581941\pi$
$692$	0	0
$693$	−0.991173	−0.0376516
$694$	0	0
$695$	0.444139	0.0168471
$696$	0	0
$697$	50.7262	1.92139
$698$	0	0
$699$	−56.6700	−2.14346
$700$	0	0
$701$	21.0604	0.795440	0.397720	−	0.917507i	$-0.369802\pi$
$701$	21.0604	0.795440	0.397720	+	0.917507i	$0.369802\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	8.38127	0.315657
$706$	0	0
$707$	1.69564	0.0637712
$708$	0	0
$709$	−31.7794	−1.19350	−0.596750	−	0.802427i	$-0.703542\pi$
$709$	−31.7794	−1.19350	−0.596750	+	0.802427i	$0.703542\pi$
$710$	0	0
$711$	1.72550	0.0647112
$712$	0	0
$713$	−31.5275	−1.18071
$714$	0	0
$715$	0.728417	0.0272412
$716$	0	0
$717$	1.46716	0.0547921
$718$	0	0
$719$	−16.1164	−0.601040	−0.300520	−	0.953775i	$-0.597160\pi$
$719$	−16.1164	−0.601040	−0.300520	+	0.953775i	$0.597160\pi$
$720$	0	0
$721$	2.32968	0.0867617
$722$	0	0
$723$	−25.9420	−0.964794
$724$	0	0
$725$	−34.6660	−1.28746
$726$	0	0
$727$	−45.0557	−1.67103	−0.835513	−	0.549471i	$-0.814829\pi$
$727$	−45.0557	−1.67103	−0.835513	+	0.549471i	$0.814829\pi$
$728$	0	0
$729$	16.8415	0.623758
$730$	0	0
$731$	−22.6298	−0.836995
$732$	0	0
$733$	39.6974	1.46626	0.733129	−	0.680090i	$-0.238059\pi$
$733$	39.6974	1.46626	0.733129	+	0.680090i	$0.238059\pi$
$734$	0	0
$735$	5.53587	0.204194
$736$	0	0
$737$	−20.0157	−0.737289
$738$	0	0
$739$	1.33212	0.0490027	0.0245013	−	0.999700i	$-0.492200\pi$
$739$	1.33212	0.0490027	0.0245013	+	0.999700i	$0.492200\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.68197	−0.318511	−0.159255	−	0.987237i	$-0.550909\pi$
$743$	−8.68197	−0.318511	−0.159255	+	0.987237i	$0.550909\pi$
$744$	0	0
$745$	−8.06252	−0.295388
$746$	0	0
$747$	−1.24142	−0.0454211
$748$	0	0
$749$	3.00178	0.109683
$750$	0	0
$751$	4.51054	0.164592	0.0822960	−	0.996608i	$-0.473775\pi$
$751$	4.51054	0.164592	0.0822960	+	0.996608i	$0.473775\pi$
$752$	0	0
$753$	−41.1096	−1.49812
$754$	0	0
$755$	10.0443	0.365548
$756$	0	0
$757$	20.3309	0.738938	0.369469	−	0.929243i	$-0.379540\pi$
$757$	20.3309	0.738938	0.369469	+	0.929243i	$0.379540\pi$
$758$	0	0
$759$	14.8447	0.538830
$760$	0	0
$761$	−2.76315	−0.100164	−0.0500820	−	0.998745i	$-0.515948\pi$
$761$	−2.76315	−0.100164	−0.0500820	+	0.998745i	$0.515948\pi$
$762$	0	0
$763$	−9.25296	−0.334980
$764$	0	0
$765$	1.78225	0.0644372
$766$	0	0
$767$	−6.25806	−0.225966
$768$	0	0
$769$	−26.7310	−0.963944	−0.481972	−	0.876187i	$-0.660079\pi$
$769$	−26.7310	−0.963944	−0.481972	+	0.876187i	$0.660079\pi$
$770$	0	0
$771$	−24.9882	−0.899926
$772$	0	0
$773$	−15.9543	−0.573836	−0.286918	−	0.957955i	$-0.592631\pi$
$773$	−15.9543	−0.573836	−0.286918	+	0.957955i	$0.592631\pi$
$774$	0	0
$775$	50.3604	1.80900
$776$	0	0
$777$	4.29360	0.154032
$778$	0	0
$779$	0	0
$780$	0	0
$781$	7.54688	0.270049
$782$	0	0
$783$	−31.0533	−1.10976
$784$	0	0
$785$	−3.57560	−0.127619
$786$	0	0
$787$	−22.0591	−0.786323	−0.393162	−	0.919469i	$-0.628619\pi$
$787$	−22.0591	−0.786323	−0.393162	+	0.919469i	$0.628619\pi$
$788$	0	0
$789$	−20.2438	−0.720700
$790$	0	0
$791$	−3.40214	−0.120966
$792$	0	0
$793$	−0.351544	−0.0124837
$794$	0	0
$795$	−3.43480	−0.121820
$796$	0	0
$797$	50.5254	1.78970	0.894851	−	0.446365i	$-0.147282\pi$
$797$	50.5254	1.78970	0.894851	+	0.446365i	$0.147282\pi$
$798$	0	0
$799$	55.4393	1.96130
$800$	0	0
$801$	−9.99118	−0.353021
$802$	0	0
$803$	27.2914	0.963092
$804$	0	0
$805$	0.641910	0.0226243
$806$	0	0
$807$	34.9048	1.22871
$808$	0	0
$809$	−19.2675	−0.677409	−0.338704	−	0.940893i	$-0.609989\pi$
$809$	−19.2675	−0.677409	−0.338704	+	0.940893i	$0.609989\pi$
$810$	0	0
$811$	6.46450	0.226999	0.113500	−	0.993538i	$-0.463794\pi$
$811$	6.46450	0.226999	0.113500	+	0.993538i	$0.463794\pi$
$812$	0	0
$813$	−41.9197	−1.47019
$814$	0	0
$815$	7.17583	0.251358
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.267131	0.00933432
$820$	0	0
$821$	−5.45017	−0.190212	−0.0951061	−	0.995467i	$-0.530319\pi$
$821$	−5.45017	−0.190212	−0.0951061	+	0.995467i	$0.530319\pi$
$822$	0	0
$823$	−38.7321	−1.35012	−0.675059	−	0.737764i	$-0.735882\pi$
$823$	−38.7321	−1.35012	−0.675059	+	0.737764i	$0.735882\pi$
$824$	0	0
$825$	−23.7122	−0.825553
$826$	0	0
$827$	21.3171	0.741269	0.370635	−	0.928779i	$-0.379140\pi$
$827$	21.3171	0.741269	0.370635	+	0.928779i	$0.379140\pi$
$828$	0	0
$829$	−35.7240	−1.24075	−0.620373	−	0.784307i	$-0.713019\pi$
$829$	−35.7240	−1.24075	−0.620373	+	0.784307i	$0.713019\pi$
$830$	0	0
$831$	37.7827	1.31067
$832$	0	0
$833$	36.6179	1.26873
$834$	0	0
$835$	−7.08508	−0.245189
$836$	0	0
$837$	45.1122	1.55931
$838$	0	0
$839$	−52.3427	−1.80707	−0.903535	−	0.428514i	$-0.859037\pi$
$839$	−52.3427	−1.80707	−0.903535	+	0.428514i	$0.859037\pi$
$840$	0	0
$841$	22.6887	0.782369
$842$	0	0
$843$	2.49044	0.0857755
$844$	0	0
$845$	5.29211	0.182054
$846$	0	0
$847$	2.31608	0.0795814
$848$	0	0
$849$	−36.6372	−1.25739
$850$	0	0
$851$	−13.2388	−0.453819
$852$	0	0
$853$	−26.5703	−0.909751	−0.454875	−	0.890555i	$-0.650316\pi$
$853$	−26.5703	−0.909751	−0.454875	+	0.890555i	$0.650316\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.4735	0.972636	0.486318	−	0.873782i	$-0.338340\pi$
$857$	28.4735	0.972636	0.486318	+	0.873782i	$0.338340\pi$
$858$	0	0
$859$	−39.4433	−1.34579	−0.672893	−	0.739740i	$-0.734949\pi$
$859$	−39.4433	−1.34579	−0.672893	+	0.739740i	$0.734949\pi$
$860$	0	0
$861$	9.14922	0.311805
$862$	0	0
$863$	34.4270	1.17191	0.585954	−	0.810344i	$-0.300720\pi$
$863$	34.4270	1.17191	0.585954	+	0.810344i	$0.300720\pi$
$864$	0	0
$865$	0.423201	0.0143892
$866$	0	0
$867$	24.2208	0.822582
$868$	0	0
$869$	5.61344	0.190423
$870$	0	0
$871$	5.39445	0.182784
$872$	0	0
$873$	−2.70132	−0.0914259
$874$	0	0
$875$	−2.08861	−0.0706078
$876$	0	0
$877$	29.3645	0.991568	0.495784	−	0.868446i	$-0.334881\pi$
$877$	29.3645	0.991568	0.495784	+	0.868446i	$0.334881\pi$
$878$	0	0
$879$	26.2724	0.886145
$880$	0	0
$881$	−22.4439	−0.756154	−0.378077	−	0.925774i	$-0.623415\pi$
$881$	−22.4439	−0.756154	−0.378077	+	0.925774i	$0.623415\pi$
$882$	0	0
$883$	37.8540	1.27389	0.636945	−	0.770910i	$-0.280198\pi$
$883$	37.8540	1.27389	0.636945	+	0.770910i	$0.280198\pi$
$884$	0	0
$885$	−7.53069	−0.253141
$886$	0	0
$887$	15.4888	0.520064	0.260032	−	0.965600i	$-0.416267\pi$
$887$	15.4888	0.520064	0.260032	+	0.965600i	$0.416267\pi$
$888$	0	0
$889$	8.01262	0.268735
$890$	0	0
$891$	−27.1446	−0.909378
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0.729819	0.0243952
$896$	0	0
$897$	−4.00081	−0.133583
$898$	0	0
$899$	−75.0898	−2.50439
$900$	0	0
$901$	−22.7201	−0.756915
$902$	0	0
$903$	−4.08163	−0.135828
$904$	0	0
$905$	−10.2999	−0.342381
$906$	0	0
$907$	−40.5616	−1.34683	−0.673413	−	0.739267i	$-0.735172\pi$
$907$	−40.5616	−1.34683	−0.673413	+	0.739267i	$0.735172\pi$
$908$	0	0
$909$	−2.61822	−0.0868409
$910$	0	0
$911$	−14.5475	−0.481981	−0.240991	−	0.970527i	$-0.577472\pi$
$911$	−14.5475	−0.481981	−0.240991	+	0.970527i	$0.577472\pi$
$912$	0	0
$913$	−4.03862	−0.133659
$914$	0	0
$915$	−0.423034	−0.0139851
$916$	0	0
$917$	5.01183	0.165505
$918$	0	0
$919$	55.1050	1.81775	0.908873	−	0.417074i	$-0.136944\pi$
$919$	55.1050	1.81775	0.908873	+	0.417074i	$0.136944\pi$
$920$	0	0
$921$	38.1872	1.25831
$922$	0	0
$923$	−2.03396	−0.0669486
$924$	0	0
$925$	21.1469	0.695306
$926$	0	0
$927$	−3.59722	−0.118148
$928$	0	0
$929$	13.6049	0.446362	0.223181	−	0.974777i	$-0.428356\pi$
$929$	13.6049	0.446362	0.223181	+	0.974777i	$0.428356\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−38.9519	−1.27523
$934$	0	0
$935$	5.79806	0.189617
$936$	0	0
$937$	−28.6593	−0.936257	−0.468129	−	0.883660i	$-0.655072\pi$
$937$	−28.6593	−0.936257	−0.468129	+	0.883660i	$0.655072\pi$
$938$	0	0
$939$	−21.3431	−0.696507
$940$	0	0
$941$	−33.4354	−1.08996	−0.544981	−	0.838448i	$-0.683463\pi$
$941$	−33.4354	−1.08996	−0.544981	+	0.838448i	$0.683463\pi$
$942$	0	0
$943$	−28.2105	−0.918660
$944$	0	0
$945$	−0.918498	−0.0298787
$946$	0	0
$947$	21.0257	0.683244	0.341622	−	0.939837i	$-0.389024\pi$
$947$	21.0257	0.683244	0.341622	+	0.939837i	$0.389024\pi$
$948$	0	0
$949$	−7.35530	−0.238763
$950$	0	0
$951$	24.2989	0.787945
$952$	0	0
$953$	−22.6405	−0.733399	−0.366700	−	0.930339i	$-0.619512\pi$
$953$	−22.6405	−0.733399	−0.366700	+	0.930339i	$0.619512\pi$
$954$	0	0
$955$	−1.59526	−0.0516215
$956$	0	0
$957$	35.3561	1.14290
$958$	0	0
$959$	−7.36921	−0.237964
$960$	0	0
$961$	78.0854	2.51888
$962$	0	0
$963$	−4.63501	−0.149361
$964$	0	0
$965$	−6.62884	−0.213390
$966$	0	0
$967$	−30.3802	−0.976962	−0.488481	−	0.872574i	$-0.662449\pi$
$967$	−30.3802	−0.976962	−0.488481	+	0.872574i	$0.662449\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.7731	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$971$	37.7731	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$972$	0	0
$973$	0.529881	0.0169872
$974$	0	0
$975$	6.39068	0.204666
$976$	0	0
$977$	22.5754	0.722251	0.361126	−	0.932517i	$-0.382393\pi$
$977$	22.5754	0.722251	0.361126	+	0.932517i	$0.382393\pi$
$978$	0	0
$979$	−32.5036	−1.03882
$980$	0	0
$981$	14.2874	0.456161
$982$	0	0
$983$	4.22342	0.134706	0.0673531	−	0.997729i	$-0.478545\pi$
$983$	4.22342	0.134706	0.0673531	+	0.997729i	$0.478545\pi$
$984$	0	0
$985$	5.97257	0.190302
$986$	0	0
$987$	9.99929	0.318281
$988$	0	0
$989$	12.5852	0.400186
$990$	0	0
$991$	−22.1713	−0.704295	−0.352148	−	0.935944i	$-0.614549\pi$
$991$	−22.1713	−0.704295	−0.352148	+	0.935944i	$0.614549\pi$
$992$	0	0
$993$	33.8817	1.07520
$994$	0	0
$995$	2.17876	0.0690715
$996$	0	0
$997$	44.3442	1.40440	0.702198	−	0.711982i	$-0.252202\pi$
$997$	44.3442	1.40440	0.702198	+	0.711982i	$0.252202\pi$
$998$	0	0
$999$	18.9431	0.599334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.ca.1.8		8
4.3	odd	2		2888.2.a.w.1.1	yes	8
19.18	odd	2		5776.2.a.cc.1.1		8
76.75	even	2		2888.2.a.v.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.8	✓	8	76.75	even	2
2888.2.a.w.1.1	yes	8	4.3	odd	2
5776.2.a.ca.1.8		8	1.1	even	1	trivial
5776.2.a.cc.1.1		8	19.18	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+1.94364 q^{3} -0.422186 q^{5} -0.503690 q^{7} +0.777742 q^{9} +2.53017 q^{11} -0.681907 q^{13} -0.820579 q^{15} -5.42785 q^{17} -0.978993 q^{21} +3.01860 q^{23} -4.82176 q^{25} -4.31927 q^{27} +7.18948 q^{29} -10.4444 q^{31} +4.91775 q^{33} +0.212651 q^{35} -4.38572 q^{37} -1.32538 q^{39} -9.34554 q^{41} +4.16921 q^{43} -0.328352 q^{45} -10.2139 q^{47} -6.74630 q^{49} -10.5498 q^{51} +4.18583 q^{53} -1.06820 q^{55} +9.17729 q^{59} +0.515531 q^{61} -0.391741 q^{63} +0.287892 q^{65} -7.91082 q^{67} +5.86708 q^{69} +2.98275 q^{71} +10.7864 q^{73} -9.37177 q^{75} -1.27442 q^{77} +2.21860 q^{79} -10.7283 q^{81} -1.59618 q^{83} +2.29156 q^{85} +13.9738 q^{87} -12.8464 q^{89} +0.343470 q^{91} -20.3002 q^{93} -3.47329 q^{97} +1.96782 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 12 q^{11} + 6 q^{13} - 10 q^{17} - 4 q^{21} + 12 q^{23} + 2 q^{25} - 12 q^{27} + 18 q^{29} - 14 q^{31} + 40 q^{33} - 18 q^{35} + 16 q^{37} - 28 q^{39} + 12 q^{41}+ \cdots - 32 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 12 * q^11 + 6 * q^13 - 10 * q^17 - 4 * q^21 + 12 * q^23 + 2 * q^25 - 12 * q^27 + 18 * q^29 - 14 * q^31 + 40 * q^33 - 18 * q^35 + 16 * q^37 - 28 * q^39 + 12 * q^41 - 8 * q^43 - 2 * q^45 - 24 * q^47 + 12 * q^49 + 22 * q^51 + 12 * q^53 + 2 * q^55 + 16 * q^59 - 12 * q^63 + 12 * q^65 - 42 * q^67 - 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 - 12 * q^83 - 42 * q^85 + 12 * q^87 + 8 * q^89 - 16 * q^91 - 14 * q^93 + 28 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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