LMFDB - Embedded newform 5684.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5684,2,Mod(1,5684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5684, 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("5684.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5684 = 2^{2} \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5684.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:	$45.3869685089$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 10x^{4} + 18x^{3} + 38x^{2} - 12 $ x^6 - 3x^5 - 10x^4 + 18x^3 + 38x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 812)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.06652$ of defining polynomial
Character	$\chi$	$=$	5684.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.06652 q^{3} -2.61233 q^{5} +6.40352 q^{9} +O(q^{10})$	$q-3.06652 q^{3} -2.61233 q^{5} +6.40352 q^{9} -3.48251 q^{11} -0.0840835 q^{13} +8.01076 q^{15} +7.37042 q^{17} -6.02077 q^{19} -6.92421 q^{23} +1.82428 q^{25} -10.4369 q^{27} +1.00000 q^{29} +4.91912 q^{31} +10.6792 q^{33} -6.66128 q^{37} +0.257843 q^{39} +8.54902 q^{41} +5.19149 q^{43} -16.7281 q^{45} -11.7405 q^{47} -22.6015 q^{51} +1.28067 q^{53} +9.09747 q^{55} +18.4628 q^{57} +0.405400 q^{59} -2.29102 q^{61} +0.219654 q^{65} -6.42125 q^{67} +21.2332 q^{69} -14.2946 q^{71} -10.5743 q^{73} -5.59419 q^{75} +1.75488 q^{79} +12.7945 q^{81} -14.5599 q^{83} -19.2540 q^{85} -3.06652 q^{87} -4.04606 q^{89} -15.0846 q^{93} +15.7283 q^{95} -7.10017 q^{97} -22.3003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + q^{5} + 11 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 + q^5 + 11 * q^9	$ 6 q - 3 q^{3} + q^{5} + 11 q^{9} + q^{11} + 3 q^{13} + 13 q^{15} - 12 q^{19} + 17 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} - 17 q^{33} + 4 q^{37} + 13 q^{39} + 14 q^{41} + 29 q^{43} + 8 q^{45} - 31 q^{47} + 14 q^{51} - 7 q^{53} - 3 q^{55} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 5 q^{65} + 6 q^{67} + 48 q^{69} - 10 q^{73} - 22 q^{75} + 19 q^{79} - 18 q^{81} + 14 q^{83} - 12 q^{85} - 3 q^{87} + 16 q^{89} - 9 q^{93} - 34 q^{95} + 6 q^{97} - 22 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + q^5 + 11 * q^9 + q^11 + 3 * q^13 + 13 * q^15 - 12 * q^19 + 17 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 - 17 * q^33 + 4 * q^37 + 13 * q^39 + 14 * q^41 + 29 * q^43 + 8 * q^45 - 31 * q^47 + 14 * q^51 - 7 * q^53 - 3 * q^55 + 18 * q^57 + 2 * q^59 + 6 * q^61 + 5 * q^65 + 6 * q^67 + 48 * q^69 - 10 * q^73 - 22 * q^75 + 19 * q^79 - 18 * q^81 + 14 * q^83 - 12 * q^85 - 3 * q^87 + 16 * q^89 - 9 * q^93 - 34 * q^95 + 6 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.06652	−1.77045	−0.885227	−	0.465160i	$-0.845997\pi$
$3$	−3.06652	−1.77045	−0.885227	+	0.465160i	$0.845997\pi$
$4$	0	0
$5$	−2.61233	−1.16827	−0.584135	−	0.811656i	$-0.698566\pi$
$5$	−2.61233	−1.16827	−0.584135	+	0.811656i	$0.698566\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.40352	2.13451
$10$	0	0
$11$	−3.48251	−1.05002	−0.525008	−	0.851097i	$-0.675938\pi$
$11$	−3.48251	−1.05002	−0.525008	+	0.851097i	$0.675938\pi$
$12$	0	0
$13$	−0.0840835	−0.0233206	−0.0116603	−	0.999932i	$-0.503712\pi$
$13$	−0.0840835	−0.0233206	−0.0116603	+	0.999932i	$0.503712\pi$
$14$	0	0
$15$	8.01076	2.06837
$16$	0	0
$17$	7.37042	1.78759	0.893794	−	0.448477i	$-0.148033\pi$
$17$	7.37042	1.78759	0.893794	+	0.448477i	$0.148033\pi$
$18$	0	0
$19$	−6.02077	−1.38126	−0.690630	−	0.723208i	$-0.742667\pi$
$19$	−6.02077	−1.38126	−0.690630	+	0.723208i	$0.742667\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92421	−1.44380	−0.721899	−	0.691998i	$-0.756731\pi$
$23$	−6.92421	−1.44380	−0.721899	+	0.691998i	$0.756731\pi$
$24$	0	0
$25$	1.82428	0.364856
$26$	0	0
$27$	−10.4369	−2.00859
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.91912	0.883501	0.441750	−	0.897138i	$-0.354358\pi$
$31$	4.91912	0.883501	0.441750	+	0.897138i	$0.354358\pi$
$32$	0	0
$33$	10.6792	1.85900
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.66128	−1.09511	−0.547554	−	0.836770i	$-0.684441\pi$
$37$	−6.66128	−1.09511	−0.547554	+	0.836770i	$0.684441\pi$
$38$	0	0
$39$	0.257843	0.0412880
$40$	0	0
$41$	8.54902	1.33513	0.667567	−	0.744550i	$-0.267336\pi$
$41$	8.54902	1.33513	0.667567	+	0.744550i	$0.267336\pi$
$42$	0	0
$43$	5.19149	0.791695	0.395848	−	0.918316i	$-0.370451\pi$
$43$	5.19149	0.791695	0.395848	+	0.918316i	$0.370451\pi$
$44$	0	0
$45$	−16.7281	−2.49368
$46$	0	0
$47$	−11.7405	−1.71253	−0.856265	−	0.516537i	$-0.827221\pi$
$47$	−11.7405	−1.71253	−0.856265	+	0.516537i	$0.827221\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−22.6015	−3.16484
$52$	0	0
$53$	1.28067	0.175913	0.0879565	−	0.996124i	$-0.471966\pi$
$53$	1.28067	0.175913	0.0879565	+	0.996124i	$0.471966\pi$
$54$	0	0
$55$	9.09747	1.22670
$56$	0	0
$57$	18.4628	2.44546
$58$	0	0
$59$	0.405400	0.0527785	0.0263893	−	0.999652i	$-0.491599\pi$
$59$	0.405400	0.0527785	0.0263893	+	0.999652i	$0.491599\pi$
$60$	0	0
$61$	−2.29102	−0.293335	−0.146667	−	0.989186i	$-0.546855\pi$
$61$	−2.29102	−0.293335	−0.146667	+	0.989186i	$0.546855\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.219654	0.0272447
$66$	0	0
$67$	−6.42125	−0.784480	−0.392240	−	0.919863i	$-0.628300\pi$
$67$	−6.42125	−0.784480	−0.392240	+	0.919863i	$0.628300\pi$
$68$	0	0
$69$	21.2332	2.55618
$70$	0	0
$71$	−14.2946	−1.69646	−0.848230	−	0.529628i	$-0.822332\pi$
$71$	−14.2946	−1.69646	−0.848230	+	0.529628i	$0.822332\pi$
$72$	0	0
$73$	−10.5743	−1.23763	−0.618815	−	0.785537i	$-0.712387\pi$
$73$	−10.5743	−1.23763	−0.618815	+	0.785537i	$0.712387\pi$
$74$	0	0
$75$	−5.59419	−0.645961
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.75488	0.197439	0.0987196	−	0.995115i	$-0.468525\pi$
$79$	1.75488	0.197439	0.0987196	+	0.995115i	$0.468525\pi$
$80$	0	0
$81$	12.7945	1.42161
$82$	0	0
$83$	−14.5599	−1.59816	−0.799081	−	0.601224i	$-0.794680\pi$
$83$	−14.5599	−1.59816	−0.799081	+	0.601224i	$0.794680\pi$
$84$	0	0
$85$	−19.2540	−2.08839
$86$	0	0
$87$	−3.06652	−0.328765
$88$	0	0
$89$	−4.04606	−0.428881	−0.214441	−	0.976737i	$-0.568793\pi$
$89$	−4.04606	−0.428881	−0.214441	+	0.976737i	$0.568793\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−15.0846	−1.56420
$94$	0	0
$95$	15.7283	1.61369
$96$	0	0
$97$	−7.10017	−0.720914	−0.360457	−	0.932776i	$-0.617379\pi$
$97$	−7.10017	−0.720914	−0.360457	+	0.932776i	$0.617379\pi$
$98$	0	0
$99$	−22.3003	−2.24126
$100$	0	0
$101$	−17.4891	−1.74023	−0.870114	−	0.492850i	$-0.835955\pi$
$101$	−17.4891	−1.74023	−0.870114	+	0.492850i	$0.835955\pi$
$102$	0	0
$103$	5.66143	0.557838	0.278919	−	0.960315i	$-0.410024\pi$
$103$	5.66143	0.557838	0.278919	+	0.960315i	$0.410024\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.9369	1.34733	0.673667	−	0.739035i	$-0.264718\pi$
$107$	13.9369	1.34733	0.673667	+	0.739035i	$0.264718\pi$
$108$	0	0
$109$	−1.03325	−0.0989676	−0.0494838	−	0.998775i	$-0.515758\pi$
$109$	−1.03325	−0.0989676	−0.0494838	+	0.998775i	$0.515758\pi$
$110$	0	0
$111$	20.4269	1.93884
$112$	0	0
$113$	−6.49311	−0.610821	−0.305410	−	0.952221i	$-0.598794\pi$
$113$	−6.49311	−0.610821	−0.305410	+	0.952221i	$0.598794\pi$
$114$	0	0
$115$	18.0883	1.68675
$116$	0	0
$117$	−0.538430	−0.0497779
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.12787	0.102533
$122$	0	0
$123$	−26.2157	−2.36379
$124$	0	0
$125$	8.29603	0.742020
$126$	0	0
$127$	9.37042	0.831490	0.415745	−	0.909481i	$-0.363521\pi$
$127$	9.37042	0.831490	0.415745	+	0.909481i	$0.363521\pi$
$128$	0	0
$129$	−15.9198	−1.40166
$130$	0	0
$131$	−14.9523	−1.30639	−0.653194	−	0.757191i	$-0.726571\pi$
$131$	−14.9523	−1.30639	−0.653194	+	0.757191i	$0.726571\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	27.2647	2.34657
$136$	0	0
$137$	−12.7984	−1.09344	−0.546720	−	0.837316i	$-0.684124\pi$
$137$	−12.7984	−1.09344	−0.546720	+	0.837316i	$0.684124\pi$
$138$	0	0
$139$	−18.9967	−1.61128	−0.805640	−	0.592405i	$-0.798178\pi$
$139$	−18.9967	−1.61128	−0.805640	+	0.592405i	$0.798178\pi$
$140$	0	0
$141$	36.0025	3.03195
$142$	0	0
$143$	0.292821	0.0244870
$144$	0	0
$145$	−2.61233	−0.216942
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.08673	−0.0890283	−0.0445142	−	0.999009i	$-0.514174\pi$
$149$	−1.08673	−0.0890283	−0.0445142	+	0.999009i	$0.514174\pi$
$150$	0	0
$151$	2.65561	0.216111	0.108055	−	0.994145i	$-0.465538\pi$
$151$	2.65561	0.216111	0.108055	+	0.994145i	$0.465538\pi$
$152$	0	0
$153$	47.1966	3.81562
$154$	0	0
$155$	−12.8504	−1.03217
$156$	0	0
$157$	−9.64707	−0.769920	−0.384960	−	0.922933i	$-0.625785\pi$
$157$	−9.64707	−0.769920	−0.384960	+	0.922933i	$0.625785\pi$
$158$	0	0
$159$	−3.92718	−0.311446
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.8777	0.930335	0.465168	−	0.885223i	$-0.345994\pi$
$163$	11.8777	0.930335	0.465168	+	0.885223i	$0.345994\pi$
$164$	0	0
$165$	−27.8975	−2.17182
$166$	0	0
$167$	9.46190	0.732184	0.366092	−	0.930579i	$-0.380696\pi$
$167$	9.46190	0.732184	0.366092	+	0.930579i	$0.380696\pi$
$168$	0	0
$169$	−12.9929	−0.999456
$170$	0	0
$171$	−38.5541	−2.94831
$172$	0	0
$173$	23.7924	1.80890	0.904452	−	0.426576i	$-0.140280\pi$
$173$	23.7924	1.80890	0.904452	+	0.426576i	$0.140280\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.24316	−0.0934419
$178$	0	0
$179$	−2.26219	−0.169084	−0.0845419	−	0.996420i	$-0.526943\pi$
$179$	−2.26219	−0.169084	−0.0845419	+	0.996420i	$0.526943\pi$
$180$	0	0
$181$	20.0929	1.49349	0.746746	−	0.665109i	$-0.231615\pi$
$181$	20.0929	1.49349	0.746746	+	0.665109i	$0.231615\pi$
$182$	0	0
$183$	7.02544	0.519335
$184$	0	0
$185$	17.4015	1.27938
$186$	0	0
$187$	−25.6675	−1.87700
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.93078	0.356779	0.178389	−	0.983960i	$-0.442911\pi$
$191$	4.93078	0.356779	0.178389	+	0.983960i	$0.442911\pi$
$192$	0	0
$193$	15.2662	1.09889	0.549443	−	0.835531i	$-0.314840\pi$
$193$	15.2662	1.09889	0.549443	+	0.835531i	$0.314840\pi$
$194$	0	0
$195$	−0.673572	−0.0482355
$196$	0	0
$197$	−7.43364	−0.529625	−0.264812	−	0.964300i	$-0.585310\pi$
$197$	−7.43364	−0.529625	−0.264812	+	0.964300i	$0.585310\pi$
$198$	0	0
$199$	11.5472	0.818561	0.409281	−	0.912409i	$-0.365780\pi$
$199$	11.5472	0.818561	0.409281	+	0.912409i	$0.365780\pi$
$200$	0	0
$201$	19.6909	1.38889
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−22.3329	−1.55980
$206$	0	0
$207$	−44.3393	−3.08179
$208$	0	0
$209$	20.9674	1.45035
$210$	0	0
$211$	20.8178	1.43316	0.716578	−	0.697507i	$-0.245707\pi$
$211$	20.8178	1.43316	0.716578	+	0.697507i	$0.245707\pi$
$212$	0	0
$213$	43.8347	3.00350
$214$	0	0
$215$	−13.5619	−0.924914
$216$	0	0
$217$	0	0
$218$	0	0
$219$	32.4263	2.19116
$220$	0	0
$221$	−0.619730	−0.0416876
$222$	0	0
$223$	21.8096	1.46048	0.730238	−	0.683193i	$-0.239409\pi$
$223$	21.8096	1.46048	0.730238	+	0.683193i	$0.239409\pi$
$224$	0	0
$225$	11.6818	0.778788
$226$	0	0
$227$	7.68385	0.509995	0.254997	−	0.966942i	$-0.417925\pi$
$227$	7.68385	0.509995	0.254997	+	0.966942i	$0.417925\pi$
$228$	0	0
$229$	0.178454	0.0117925	0.00589627	−	0.999983i	$-0.498123\pi$
$229$	0.178454	0.0117925	0.00589627	+	0.999983i	$0.498123\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.1062	−1.51374	−0.756868	−	0.653567i	$-0.773272\pi$
$233$	−23.1062	−1.51374	−0.756868	+	0.653567i	$0.773272\pi$
$234$	0	0
$235$	30.6701	2.00070
$236$	0	0
$237$	−5.38136	−0.349557
$238$	0	0
$239$	−0.313432	−0.0202742	−0.0101371	−	0.999949i	$-0.503227\pi$
$239$	−0.313432	−0.0202742	−0.0101371	+	0.999949i	$0.503227\pi$
$240$	0	0
$241$	9.74552	0.627764	0.313882	−	0.949462i	$-0.398370\pi$
$241$	9.74552	0.627764	0.313882	+	0.949462i	$0.398370\pi$
$242$	0	0
$243$	−7.92362	−0.508301
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.506248	0.0322118
$248$	0	0
$249$	44.6483	2.82947
$250$	0	0
$251$	3.97896	0.251149	0.125575	−	0.992084i	$-0.459923\pi$
$251$	3.97896	0.251149	0.125575	+	0.992084i	$0.459923\pi$
$252$	0	0
$253$	24.1136	1.51601
$254$	0	0
$255$	59.0426	3.69739
$256$	0	0
$257$	3.10952	0.193967	0.0969833	−	0.995286i	$-0.469081\pi$
$257$	3.10952	0.193967	0.0969833	+	0.995286i	$0.469081\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.40352	0.396368
$262$	0	0
$263$	2.14024	0.131973	0.0659865	−	0.997821i	$-0.478981\pi$
$263$	2.14024	0.131973	0.0659865	+	0.997821i	$0.478981\pi$
$264$	0	0
$265$	−3.34553	−0.205514
$266$	0	0
$267$	12.4073	0.759314
$268$	0	0
$269$	30.6031	1.86591	0.932953	−	0.359998i	$-0.117223\pi$
$269$	30.6031	1.86591	0.932953	+	0.359998i	$0.117223\pi$
$270$	0	0
$271$	11.8257	0.718359	0.359180	−	0.933268i	$-0.383057\pi$
$271$	11.8257	0.718359	0.359180	+	0.933268i	$0.383057\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.35308	−0.383105
$276$	0	0
$277$	−17.0221	−1.02276	−0.511380	−	0.859355i	$-0.670865\pi$
$277$	−17.0221	−1.02276	−0.511380	+	0.859355i	$0.670865\pi$
$278$	0	0
$279$	31.4997	1.88584
$280$	0	0
$281$	1.21663	0.0725782	0.0362891	−	0.999341i	$-0.488446\pi$
$281$	1.21663	0.0725782	0.0362891	+	0.999341i	$0.488446\pi$
$282$	0	0
$283$	−25.7761	−1.53223	−0.766116	−	0.642703i	$-0.777813\pi$
$283$	−25.7761	−1.53223	−0.766116	+	0.642703i	$0.777813\pi$
$284$	0	0
$285$	−48.2310	−2.85696
$286$	0	0
$287$	0	0
$288$	0	0
$289$	37.3230	2.19547
$290$	0	0
$291$	21.7728	1.27634
$292$	0	0
$293$	−8.14649	−0.475923	−0.237962	−	0.971275i	$-0.576479\pi$
$293$	−8.14649	−0.475923	−0.237962	+	0.971275i	$0.576479\pi$
$294$	0	0
$295$	−1.05904	−0.0616596
$296$	0	0
$297$	36.3467	2.10905
$298$	0	0
$299$	0.582212	0.0336702
$300$	0	0
$301$	0	0
$302$	0	0
$303$	53.6305	3.08099
$304$	0	0
$305$	5.98490	0.342694
$306$	0	0
$307$	−16.1847	−0.923710	−0.461855	−	0.886956i	$-0.652816\pi$
$307$	−16.1847	−0.923710	−0.461855	+	0.886956i	$0.652816\pi$
$308$	0	0
$309$	−17.3609	−0.987625
$310$	0	0
$311$	−5.20012	−0.294872	−0.147436	−	0.989072i	$-0.547102\pi$
$311$	−5.20012	−0.294872	−0.147436	+	0.989072i	$0.547102\pi$
$312$	0	0
$313$	−5.09442	−0.287954	−0.143977	−	0.989581i	$-0.545989\pi$
$313$	−5.09442	−0.287954	−0.143977	+	0.989581i	$0.545989\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2160	0.854614	0.427307	−	0.904107i	$-0.359462\pi$
$317$	15.2160	0.854614	0.427307	+	0.904107i	$0.359462\pi$
$318$	0	0
$319$	−3.48251	−0.194983
$320$	0	0
$321$	−42.7378	−2.38539
$322$	0	0
$323$	−44.3756	−2.46913
$324$	0	0
$325$	−0.153392	−0.00850866
$326$	0	0
$327$	3.16848	0.175218
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.36268	−0.129865	−0.0649323	−	0.997890i	$-0.520683\pi$
$331$	−2.36268	−0.129865	−0.0649323	+	0.997890i	$0.520683\pi$
$332$	0	0
$333$	−42.6556	−2.33751
$334$	0	0
$335$	16.7744	0.916485
$336$	0	0
$337$	−1.58878	−0.0865463	−0.0432731	−	0.999063i	$-0.513779\pi$
$337$	−1.58878	−0.0865463	−0.0432731	+	0.999063i	$0.513779\pi$
$338$	0	0
$339$	19.9112	1.08143
$340$	0	0
$341$	−17.1309	−0.927690
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−55.4682	−2.98631
$346$	0	0
$347$	−1.83214	−0.0983544	−0.0491772	−	0.998790i	$-0.515660\pi$
$347$	−1.83214	−0.0983544	−0.0491772	+	0.998790i	$0.515660\pi$
$348$	0	0
$349$	10.5433	0.564368	0.282184	−	0.959360i	$-0.408941\pi$
$349$	10.5433	0.564368	0.282184	+	0.959360i	$0.408941\pi$
$350$	0	0
$351$	0.877574	0.0468414
$352$	0	0
$353$	−16.5927	−0.883140	−0.441570	−	0.897227i	$-0.645578\pi$
$353$	−16.5927	−0.883140	−0.441570	+	0.897227i	$0.645578\pi$
$354$	0	0
$355$	37.3423	1.98192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.5123	−1.61038	−0.805189	−	0.593018i	$-0.797936\pi$
$359$	−30.5123	−1.61038	−0.805189	+	0.593018i	$0.797936\pi$
$360$	0	0
$361$	17.2497	0.907880
$362$	0	0
$363$	−3.45862	−0.181530
$364$	0	0
$365$	27.6236	1.44589
$366$	0	0
$367$	6.81270	0.355620	0.177810	−	0.984065i	$-0.443099\pi$
$367$	6.81270	0.355620	0.177810	+	0.984065i	$0.443099\pi$
$368$	0	0
$369$	54.7438	2.84985
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.35555	−0.432634	−0.216317	−	0.976323i	$-0.569404\pi$
$373$	−8.35555	−0.432634	−0.216317	+	0.976323i	$0.569404\pi$
$374$	0	0
$375$	−25.4399	−1.31371
$376$	0	0
$377$	−0.0840835	−0.00433052
$378$	0	0
$379$	4.95637	0.254592	0.127296	−	0.991865i	$-0.459370\pi$
$379$	4.95637	0.254592	0.127296	+	0.991865i	$0.459370\pi$
$380$	0	0
$381$	−28.7345	−1.47211
$382$	0	0
$383$	−35.8677	−1.83275	−0.916377	−	0.400316i	$-0.868900\pi$
$383$	−35.8677	−1.83275	−0.916377	+	0.400316i	$0.868900\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	33.2438	1.68988
$388$	0	0
$389$	12.6604	0.641907	0.320953	−	0.947095i	$-0.395997\pi$
$389$	12.6604	0.641907	0.320953	+	0.947095i	$0.395997\pi$
$390$	0	0
$391$	−51.0343	−2.58092
$392$	0	0
$393$	45.8514	2.31290
$394$	0	0
$395$	−4.58432	−0.230662
$396$	0	0
$397$	−21.1147	−1.05972	−0.529859	−	0.848086i	$-0.677755\pi$
$397$	−21.1147	−1.05972	−0.529859	+	0.848086i	$0.677755\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.0545	0.751784	0.375892	−	0.926664i	$-0.377336\pi$
$401$	15.0545	0.751784	0.375892	+	0.926664i	$0.377336\pi$
$402$	0	0
$403$	−0.413617	−0.0206037
$404$	0	0
$405$	−33.4234	−1.66082
$406$	0	0
$407$	23.1980	1.14988
$408$	0	0
$409$	5.38255	0.266150	0.133075	−	0.991106i	$-0.457515\pi$
$409$	5.38255	0.266150	0.133075	+	0.991106i	$0.457515\pi$
$410$	0	0
$411$	39.2464	1.93588
$412$	0	0
$413$	0	0
$414$	0	0
$415$	38.0354	1.86709
$416$	0	0
$417$	58.2537	2.85270
$418$	0	0
$419$	−18.6204	−0.909664	−0.454832	−	0.890577i	$-0.650301\pi$
$419$	−18.6204	−0.909664	−0.454832	+	0.890577i	$0.650301\pi$
$420$	0	0
$421$	27.5009	1.34031	0.670156	−	0.742220i	$-0.266227\pi$
$421$	27.5009	1.34031	0.670156	+	0.742220i	$0.266227\pi$
$422$	0	0
$423$	−75.1806	−3.65540
$424$	0	0
$425$	13.4457	0.652213
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.897941	−0.0433530
$430$	0	0
$431$	6.08608	0.293156	0.146578	−	0.989199i	$-0.453174\pi$
$431$	6.08608	0.293156	0.146578	+	0.989199i	$0.453174\pi$
$432$	0	0
$433$	26.1860	1.25842	0.629209	−	0.777236i	$-0.283379\pi$
$433$	26.1860	1.25842	0.629209	+	0.777236i	$0.283379\pi$
$434$	0	0
$435$	8.01076	0.384086
$436$	0	0
$437$	41.6891	1.99426
$438$	0	0
$439$	−0.424226	−0.0202472	−0.0101236	−	0.999949i	$-0.503222\pi$
$439$	−0.424226	−0.0202472	−0.0101236	+	0.999949i	$0.503222\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.9991	1.47281	0.736406	−	0.676540i	$-0.236521\pi$
$443$	30.9991	1.47281	0.736406	+	0.676540i	$0.236521\pi$
$444$	0	0
$445$	10.5696	0.501049
$446$	0	0
$447$	3.33247	0.157621
$448$	0	0
$449$	2.25301	0.106326	0.0531631	−	0.998586i	$-0.483070\pi$
$449$	2.25301	0.106326	0.0531631	+	0.998586i	$0.483070\pi$
$450$	0	0
$451$	−29.7720	−1.40191
$452$	0	0
$453$	−8.14348	−0.382614
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.3668	0.625273	0.312636	−	0.949873i	$-0.398788\pi$
$457$	13.3668	0.625273	0.312636	+	0.949873i	$0.398788\pi$
$458$	0	0
$459$	−76.9245	−3.59053
$460$	0	0
$461$	0.0603755	0.00281197	0.00140598	−	0.999999i	$-0.499552\pi$
$461$	0.0603755	0.00281197	0.00140598	+	0.999999i	$0.499552\pi$
$462$	0	0
$463$	33.6735	1.56494	0.782470	−	0.622688i	$-0.213959\pi$
$463$	33.6735	1.56494	0.782470	+	0.622688i	$0.213959\pi$
$464$	0	0
$465$	39.4059	1.82740
$466$	0	0
$467$	−37.9126	−1.75438	−0.877192	−	0.480140i	$-0.840586\pi$
$467$	−37.9126	−1.75438	−0.877192	+	0.480140i	$0.840586\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	29.5829	1.36311
$472$	0	0
$473$	−18.0794	−0.831292
$474$	0	0
$475$	−10.9836	−0.503962
$476$	0	0
$477$	8.20076	0.375487
$478$	0	0
$479$	9.37636	0.428417	0.214208	−	0.976788i	$-0.431283\pi$
$479$	9.37636	0.428417	0.214208	+	0.976788i	$0.431283\pi$
$480$	0	0
$481$	0.560104	0.0255385
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.5480	0.842222
$486$	0	0
$487$	16.0310	0.726432	0.363216	−	0.931705i	$-0.381679\pi$
$487$	16.0310	0.726432	0.363216	+	0.931705i	$0.381679\pi$
$488$	0	0
$489$	−36.4232	−1.64712
$490$	0	0
$491$	−16.3808	−0.739257	−0.369629	−	0.929180i	$-0.620515\pi$
$491$	−16.3808	−0.739257	−0.369629	+	0.929180i	$0.620515\pi$
$492$	0	0
$493$	7.37042	0.331947
$494$	0	0
$495$	58.2558	2.61840
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.1045	1.61626	0.808130	−	0.589004i	$-0.200480\pi$
$499$	36.1045	1.61626	0.808130	+	0.589004i	$0.200480\pi$
$500$	0	0
$501$	−29.0151	−1.29630
$502$	0	0
$503$	33.8345	1.50861	0.754304	−	0.656526i	$-0.227975\pi$
$503$	33.8345	1.50861	0.754304	+	0.656526i	$0.227975\pi$
$504$	0	0
$505$	45.6873	2.03306
$506$	0	0
$507$	39.8430	1.76949
$508$	0	0
$509$	21.8561	0.968756	0.484378	−	0.874859i	$-0.339046\pi$
$509$	21.8561	0.968756	0.484378	+	0.874859i	$0.339046\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	62.8384	2.77438
$514$	0	0
$515$	−14.7895	−0.651705
$516$	0	0
$517$	40.8864	1.79818
$518$	0	0
$519$	−72.9598	−3.20258
$520$	0	0
$521$	2.93612	0.128634	0.0643170	−	0.997930i	$-0.479513\pi$
$521$	2.93612	0.128634	0.0643170	+	0.997930i	$0.479513\pi$
$522$	0	0
$523$	38.8056	1.69685	0.848425	−	0.529315i	$-0.177551\pi$
$523$	38.8056	1.69685	0.848425	+	0.529315i	$0.177551\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.2560	1.57934
$528$	0	0
$529$	24.9447	1.08455
$530$	0	0
$531$	2.59598	0.112656
$532$	0	0
$533$	−0.718832	−0.0311361
$534$	0	0
$535$	−36.4079	−1.57405
$536$	0	0
$537$	6.93703	0.299355
$538$	0	0
$539$	0	0
$540$	0	0
$541$	27.3743	1.17691	0.588456	−	0.808529i	$-0.299736\pi$
$541$	27.3743	1.17691	0.588456	+	0.808529i	$0.299736\pi$
$542$	0	0
$543$	−61.6151	−2.64416
$544$	0	0
$545$	2.69920	0.115621
$546$	0	0
$547$	−45.2353	−1.93412	−0.967061	−	0.254543i	$-0.918075\pi$
$547$	−45.2353	−1.93412	−0.967061	+	0.254543i	$0.918075\pi$
$548$	0	0
$549$	−14.6706	−0.626124
$550$	0	0
$551$	−6.02077	−0.256494
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−53.3619	−2.26509
$556$	0	0
$557$	−38.5885	−1.63505	−0.817524	−	0.575895i	$-0.804654\pi$
$557$	−38.5885	−1.63505	−0.817524	+	0.575895i	$0.804654\pi$
$558$	0	0
$559$	−0.436519	−0.0184628
$560$	0	0
$561$	78.7099	3.32313
$562$	0	0
$563$	36.7297	1.54797	0.773986	−	0.633203i	$-0.218260\pi$
$563$	36.7297	1.54797	0.773986	+	0.633203i	$0.218260\pi$
$564$	0	0
$565$	16.9622	0.713604
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.69436	−0.280642	−0.140321	−	0.990106i	$-0.544813\pi$
$569$	−6.69436	−0.280642	−0.140321	+	0.990106i	$0.544813\pi$
$570$	0	0
$571$	24.7697	1.03658	0.518290	−	0.855205i	$-0.326569\pi$
$571$	24.7697	1.03658	0.518290	+	0.855205i	$0.326569\pi$
$572$	0	0
$573$	−15.1203	−0.631660
$574$	0	0
$575$	−12.6317	−0.526779
$576$	0	0
$577$	−36.3032	−1.51132	−0.755660	−	0.654964i	$-0.772684\pi$
$577$	−36.3032	−1.51132	−0.755660	+	0.654964i	$0.772684\pi$
$578$	0	0
$579$	−46.8141	−1.94553
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.45993	−0.184711
$584$	0	0
$585$	1.40656	0.0581540
$586$	0	0
$587$	−20.1537	−0.831832	−0.415916	−	0.909403i	$-0.636539\pi$
$587$	−20.1537	−0.831832	−0.415916	+	0.909403i	$0.636539\pi$
$588$	0	0
$589$	−29.6169	−1.22034
$590$	0	0
$591$	22.7954	0.937676
$592$	0	0
$593$	−44.8161	−1.84038	−0.920188	−	0.391478i	$-0.871964\pi$
$593$	−44.8161	−1.84038	−0.920188	+	0.391478i	$0.871964\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−35.4098	−1.44922
$598$	0	0
$599$	−12.6685	−0.517621	−0.258810	−	0.965928i	$-0.583330\pi$
$599$	−12.6685	−0.517621	−0.258810	+	0.965928i	$0.583330\pi$
$600$	0	0
$601$	31.7548	1.29531	0.647653	−	0.761935i	$-0.275750\pi$
$601$	31.7548	1.29531	0.647653	+	0.761935i	$0.275750\pi$
$602$	0	0
$603$	−41.1186	−1.67448
$604$	0	0
$605$	−2.94636	−0.119787
$606$	0	0
$607$	−17.5456	−0.712156	−0.356078	−	0.934456i	$-0.615886\pi$
$607$	−17.5456	−0.712156	−0.356078	+	0.934456i	$0.615886\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.987184	0.0399372
$612$	0	0
$613$	7.22245	0.291712	0.145856	−	0.989306i	$-0.453406\pi$
$613$	7.22245	0.291712	0.145856	+	0.989306i	$0.453406\pi$
$614$	0	0
$615$	68.4842	2.76155
$616$	0	0
$617$	−40.8964	−1.64643	−0.823213	−	0.567732i	$-0.807821\pi$
$617$	−40.8964	−1.64643	−0.823213	+	0.567732i	$0.807821\pi$
$618$	0	0
$619$	22.4389	0.901894	0.450947	−	0.892551i	$-0.351086\pi$
$619$	22.4389	0.901894	0.450947	+	0.892551i	$0.351086\pi$
$620$	0	0
$621$	72.2675	2.90000
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.7934	−1.23174
$626$	0	0
$627$	−64.2968	−2.56777
$628$	0	0
$629$	−49.0964	−1.95760
$630$	0	0
$631$	−13.3863	−0.532899	−0.266449	−	0.963849i	$-0.585851\pi$
$631$	−13.3863	−0.532899	−0.266449	+	0.963849i	$0.585851\pi$
$632$	0	0
$633$	−63.8381	−2.53734
$634$	0	0
$635$	−24.4786	−0.971405
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−91.5359	−3.62110
$640$	0	0
$641$	−15.9475	−0.629888	−0.314944	−	0.949110i	$-0.601986\pi$
$641$	−15.9475	−0.629888	−0.314944	+	0.949110i	$0.601986\pi$
$642$	0	0
$643$	29.8136	1.17573	0.587867	−	0.808958i	$-0.299968\pi$
$643$	29.8136	1.17573	0.587867	+	0.808958i	$0.299968\pi$
$644$	0	0
$645$	41.5878	1.63752
$646$	0	0
$647$	−29.3625	−1.15436	−0.577180	−	0.816617i	$-0.695847\pi$
$647$	−29.3625	−1.15436	−0.577180	+	0.816617i	$0.695847\pi$
$648$	0	0
$649$	−1.41181	−0.0554183
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.6252	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$653$	38.6252	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$654$	0	0
$655$	39.0604	1.52621
$656$	0	0
$657$	−67.7127	−2.64173
$658$	0	0
$659$	−18.4737	−0.719634	−0.359817	−	0.933023i	$-0.617161\pi$
$659$	−18.4737	−0.719634	−0.359817	+	0.933023i	$0.617161\pi$
$660$	0	0
$661$	−11.4773	−0.446414	−0.223207	−	0.974771i	$-0.571653\pi$
$661$	−11.4773	−0.446414	−0.223207	+	0.974771i	$0.571653\pi$
$662$	0	0
$663$	1.90041	0.0738059
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.92421	−0.268107
$668$	0	0
$669$	−66.8794	−2.58571
$670$	0	0
$671$	7.97848	0.308006
$672$	0	0
$673$	−16.0746	−0.619632	−0.309816	−	0.950797i	$-0.600267\pi$
$673$	−16.0746	−0.619632	−0.309816	+	0.950797i	$0.600267\pi$
$674$	0	0
$675$	−19.0399	−0.732846
$676$	0	0
$677$	28.3402	1.08920	0.544601	−	0.838695i	$-0.316681\pi$
$677$	28.3402	1.08920	0.544601	+	0.838695i	$0.316681\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−23.5626	−0.902922
$682$	0	0
$683$	−13.8358	−0.529410	−0.264705	−	0.964329i	$-0.585275\pi$
$683$	−13.8358	−0.529410	−0.264705	+	0.964329i	$0.585275\pi$
$684$	0	0
$685$	33.4336	1.27743
$686$	0	0
$687$	−0.547230	−0.0208781
$688$	0	0
$689$	−0.107683	−0.00410239
$690$	0	0
$691$	20.8670	0.793817	0.396908	−	0.917858i	$-0.370083\pi$
$691$	20.8670	0.793817	0.396908	+	0.917858i	$0.370083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	49.6257	1.88241
$696$	0	0
$697$	63.0099	2.38667
$698$	0	0
$699$	70.8555	2.68000
$700$	0	0
$701$	18.7398	0.707794	0.353897	−	0.935284i	$-0.384856\pi$
$701$	18.7398	0.707794	0.353897	+	0.935284i	$0.384856\pi$
$702$	0	0
$703$	40.1061	1.51263
$704$	0	0
$705$	−94.0504	−3.54214
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.9135	−0.484978	−0.242489	−	0.970154i	$-0.577964\pi$
$709$	−12.9135	−0.484978	−0.242489	+	0.970154i	$0.577964\pi$
$710$	0	0
$711$	11.2374	0.421435
$712$	0	0
$713$	−34.0611	−1.27560
$714$	0	0
$715$	−0.764947	−0.0286074
$716$	0	0
$717$	0.961145	0.0358946
$718$	0	0
$719$	20.5663	0.766992	0.383496	−	0.923543i	$-0.374720\pi$
$719$	20.5663	0.766992	0.383496	+	0.923543i	$0.374720\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−29.8848	−1.11143
$724$	0	0
$725$	1.82428	0.0677521
$726$	0	0
$727$	−36.2357	−1.34391	−0.671955	−	0.740592i	$-0.734545\pi$
$727$	−36.2357	−1.34391	−0.671955	+	0.740592i	$0.734545\pi$
$728$	0	0
$729$	−14.0855	−0.521685
$730$	0	0
$731$	38.2635	1.41523
$732$	0	0
$733$	52.5895	1.94244	0.971219	−	0.238188i	$-0.0765535\pi$
$733$	52.5895	1.94244	0.971219	+	0.238188i	$0.0765535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.3620	0.823717
$738$	0	0
$739$	21.2675	0.782336	0.391168	−	0.920319i	$-0.372071\pi$
$739$	21.2675	0.782336	0.391168	+	0.920319i	$0.372071\pi$
$740$	0	0
$741$	−1.55242	−0.0570294
$742$	0	0
$743$	2.68575	0.0985306	0.0492653	−	0.998786i	$-0.484312\pi$
$743$	2.68575	0.0985306	0.0492653	+	0.998786i	$0.484312\pi$
$744$	0	0
$745$	2.83890	0.104009
$746$	0	0
$747$	−93.2348	−3.41128
$748$	0	0
$749$	0	0
$750$	0	0
$751$	45.5675	1.66278	0.831390	−	0.555689i	$-0.187545\pi$
$751$	45.5675	1.66278	0.831390	+	0.555689i	$0.187545\pi$
$752$	0	0
$753$	−12.2015	−0.444648
$754$	0	0
$755$	−6.93734	−0.252476
$756$	0	0
$757$	−3.67082	−0.133418	−0.0667091	−	0.997772i	$-0.521250\pi$
$757$	−3.67082	−0.133418	−0.0667091	+	0.997772i	$0.521250\pi$
$758$	0	0
$759$	−73.9448	−2.68403
$760$	0	0
$761$	−29.2624	−1.06076	−0.530381	−	0.847759i	$-0.677951\pi$
$761$	−29.2624	−1.06076	−0.530381	+	0.847759i	$0.677951\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−123.293	−4.45767
$766$	0	0
$767$	−0.0340874	−0.00123082
$768$	0	0
$769$	33.9850	1.22553	0.612764	−	0.790266i	$-0.290057\pi$
$769$	33.9850	1.22553	0.612764	+	0.790266i	$0.290057\pi$
$770$	0	0
$771$	−9.53539	−0.343409
$772$	0	0
$773$	−23.5558	−0.847242	−0.423621	−	0.905839i	$-0.639241\pi$
$773$	−23.5558	−0.847242	−0.423621	+	0.905839i	$0.639241\pi$
$774$	0	0
$775$	8.97387	0.322351
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−51.4717	−1.84417
$780$	0	0
$781$	49.7812	1.78131
$782$	0	0
$783$	−10.4369	−0.372986
$784$	0	0
$785$	25.2014	0.899475
$786$	0	0
$787$	−38.8676	−1.38548	−0.692739	−	0.721188i	$-0.743596\pi$
$787$	−38.8676	−1.38548	−0.692739	+	0.721188i	$0.743596\pi$
$788$	0	0
$789$	−6.56309	−0.233652
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.192637	0.00684073
$794$	0	0
$795$	10.2591	0.363853
$796$	0	0
$797$	−37.3256	−1.32214	−0.661070	−	0.750324i	$-0.729897\pi$
$797$	−37.3256	−1.32214	−0.661070	+	0.750324i	$0.729897\pi$
$798$	0	0
$799$	−86.5325	−3.06130
$800$	0	0
$801$	−25.9090	−0.915449
$802$	0	0
$803$	36.8251	1.29953
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−93.8450	−3.30350
$808$	0	0
$809$	17.9109	0.629714	0.314857	−	0.949139i	$-0.398043\pi$
$809$	17.9109	0.629714	0.314857	+	0.949139i	$0.398043\pi$
$810$	0	0
$811$	−32.0190	−1.12434	−0.562169	−	0.827022i	$-0.690033\pi$
$811$	−32.0190	−1.12434	−0.562169	+	0.827022i	$0.690033\pi$
$812$	0	0
$813$	−36.2637	−1.27182
$814$	0	0
$815$	−31.0286	−1.08688
$816$	0	0
$817$	−31.2568	−1.09354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.18218	0.0412582	0.0206291	−	0.999787i	$-0.493433\pi$
$821$	1.18218	0.0412582	0.0206291	+	0.999787i	$0.493433\pi$
$822$	0	0
$823$	40.4843	1.41119	0.705596	−	0.708614i	$-0.250679\pi$
$823$	40.4843	1.41119	0.705596	+	0.708614i	$0.250679\pi$
$824$	0	0
$825$	19.4818	0.678269
$826$	0	0
$827$	18.2982	0.636290	0.318145	−	0.948042i	$-0.396940\pi$
$827$	18.2982	0.636290	0.318145	+	0.948042i	$0.396940\pi$
$828$	0	0
$829$	−7.69970	−0.267422	−0.133711	−	0.991020i	$-0.542689\pi$
$829$	−7.69970	−0.267422	−0.133711	+	0.991020i	$0.542689\pi$
$830$	0	0
$831$	52.1985	1.81075
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.7176	−0.855389
$836$	0	0
$837$	−51.3405	−1.77459
$838$	0	0
$839$	20.3577	0.702826	0.351413	−	0.936221i	$-0.385701\pi$
$839$	20.3577	0.702826	0.351413	+	0.936221i	$0.385701\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.73082	−0.128496
$844$	0	0
$845$	33.9419	1.16764
$846$	0	0
$847$	0	0
$848$	0	0
$849$	79.0429	2.71274
$850$	0	0
$851$	46.1241	1.58111
$852$	0	0
$853$	−5.40584	−0.185092	−0.0925462	−	0.995708i	$-0.529501\pi$
$853$	−5.40584	−0.185092	−0.0925462	+	0.995708i	$0.529501\pi$
$854$	0	0
$855$	100.716	3.44442
$856$	0	0
$857$	33.7805	1.15392	0.576959	−	0.816773i	$-0.304239\pi$
$857$	33.7805	1.15392	0.576959	+	0.816773i	$0.304239\pi$
$858$	0	0
$859$	29.0338	0.990619	0.495310	−	0.868716i	$-0.335055\pi$
$859$	29.0338	0.990619	0.495310	+	0.868716i	$0.335055\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.0543	−1.29538	−0.647691	−	0.761903i	$-0.724265\pi$
$863$	−38.0543	−1.29538	−0.647691	+	0.761903i	$0.724265\pi$
$864$	0	0
$865$	−62.1537	−2.11329
$866$	0	0
$867$	−114.452	−3.88698
$868$	0	0
$869$	−6.11138	−0.207314
$870$	0	0
$871$	0.539921	0.0182945
$872$	0	0
$873$	−45.4661	−1.53879
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.26539	−0.279102	−0.139551	−	0.990215i	$-0.544566\pi$
$877$	−8.26539	−0.279102	−0.139551	+	0.990215i	$0.544566\pi$
$878$	0	0
$879$	24.9813	0.842600
$880$	0	0
$881$	50.7616	1.71020	0.855100	−	0.518463i	$-0.173496\pi$
$881$	50.7616	1.71020	0.855100	+	0.518463i	$0.173496\pi$
$882$	0	0
$883$	−12.9714	−0.436523	−0.218262	−	0.975890i	$-0.570039\pi$
$883$	−12.9714	−0.436523	−0.218262	+	0.975890i	$0.570039\pi$
$884$	0	0
$885$	3.24756	0.109165
$886$	0	0
$887$	−1.18996	−0.0399549	−0.0199774	−	0.999800i	$-0.506359\pi$
$887$	−1.18996	−0.0399549	−0.0199774	+	0.999800i	$0.506359\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−44.5568	−1.49271
$892$	0	0
$893$	70.6870	2.36545
$894$	0	0
$895$	5.90959	0.197536
$896$	0	0
$897$	−1.78536	−0.0596115
$898$	0	0
$899$	4.91912	0.164062
$900$	0	0
$901$	9.43904	0.314460
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−52.4893	−1.74480
$906$	0	0
$907$	−39.9992	−1.32815	−0.664076	−	0.747665i	$-0.731175\pi$
$907$	−39.9992	−1.32815	−0.664076	+	0.747665i	$0.731175\pi$
$908$	0	0
$909$	−111.992	−3.71453
$910$	0	0
$911$	−37.6415	−1.24712	−0.623559	−	0.781777i	$-0.714314\pi$
$911$	−37.6415	−1.24712	−0.623559	+	0.781777i	$0.714314\pi$
$912$	0	0
$913$	50.7051	1.67809
$914$	0	0
$915$	−18.3528	−0.606724
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−20.3214	−0.670343	−0.335171	−	0.942157i	$-0.608794\pi$
$919$	−20.3214	−0.670343	−0.335171	+	0.942157i	$0.608794\pi$
$920$	0	0
$921$	49.6306	1.63539
$922$	0	0
$923$	1.20194	0.0395624
$924$	0	0
$925$	−12.1521	−0.399557
$926$	0	0
$927$	36.2531	1.19071
$928$	0	0
$929$	−45.4942	−1.49262	−0.746308	−	0.665601i	$-0.768175\pi$
$929$	−45.4942	−1.49262	−0.746308	+	0.665601i	$0.768175\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.9463	0.522057
$934$	0	0
$935$	67.0521	2.19284
$936$	0	0
$937$	37.1711	1.21433	0.607164	−	0.794577i	$-0.292307\pi$
$937$	37.1711	1.21433	0.607164	+	0.794577i	$0.292307\pi$
$938$	0	0
$939$	15.6221	0.509808
$940$	0	0
$941$	−16.3159	−0.531883	−0.265942	−	0.963989i	$-0.585683\pi$
$941$	−16.3159	−0.531883	−0.265942	+	0.963989i	$0.585683\pi$
$942$	0	0
$943$	−59.1953	−1.92766
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.3063	−1.43976	−0.719880	−	0.694098i	$-0.755803\pi$
$947$	−44.3063	−1.43976	−0.719880	+	0.694098i	$0.755803\pi$
$948$	0	0
$949$	0.889125	0.0288622
$950$	0	0
$951$	−46.6600	−1.51305
$952$	0	0
$953$	29.4540	0.954108	0.477054	−	0.878874i	$-0.341705\pi$
$953$	29.4540	0.954108	0.477054	+	0.878874i	$0.341705\pi$
$954$	0	0
$955$	−12.8808	−0.416814
$956$	0	0
$957$	10.6792	0.345208
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.80223	−0.219427
$962$	0	0
$963$	89.2453	2.87589
$964$	0	0
$965$	−39.8804	−1.28380
$966$	0	0
$967$	8.62218	0.277271	0.138635	−	0.990344i	$-0.455728\pi$
$967$	8.62218	0.277271	0.138635	+	0.990344i	$0.455728\pi$
$968$	0	0
$969$	136.079	4.37147
$970$	0	0
$971$	35.5862	1.14202	0.571008	−	0.820944i	$-0.306552\pi$
$971$	35.5862	1.14202	0.571008	+	0.820944i	$0.306552\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.470379	0.0150642
$976$	0	0
$977$	−19.6859	−0.629808	−0.314904	−	0.949124i	$-0.601972\pi$
$977$	−19.6859	−0.629808	−0.314904	+	0.949124i	$0.601972\pi$
$978$	0	0
$979$	14.0904	0.450332
$980$	0	0
$981$	−6.61645	−0.211247
$982$	0	0
$983$	12.2452	0.390561	0.195280	−	0.980747i	$-0.437438\pi$
$983$	12.2452	0.390561	0.195280	+	0.980747i	$0.437438\pi$
$984$	0	0
$985$	19.4191	0.618745
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35.9470	−1.14305
$990$	0	0
$991$	−22.6196	−0.718534	−0.359267	−	0.933235i	$-0.616973\pi$
$991$	−22.6196	−0.718534	−0.359267	+	0.933235i	$0.616973\pi$
$992$	0	0
$993$	7.24520	0.229919
$994$	0	0
$995$	−30.1652	−0.956301
$996$	0	0
$997$	26.4392	0.837339	0.418669	−	0.908139i	$-0.362497\pi$
$997$	26.4392	0.837339	0.418669	+	0.908139i	$0.362497\pi$
$998$	0	0
$999$	69.5233	2.19962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5684.2.a.r.1.1		6
7.6	odd	2		812.2.a.e.1.6	✓	6
21.20	even	2		7308.2.a.o.1.2		6
28.27	even	2		3248.2.a.bc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
812.2.a.e.1.6	✓	6	7.6	odd	2
3248.2.a.bc.1.1		6	28.27	even	2
5684.2.a.r.1.1		6	1.1	even	1	trivial
7308.2.a.o.1.2		6	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5684 = 2^{2} \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5684.a (trivial)

\(f(q)\)	\(=\)	\(q-3.06652 q^{3} -2.61233 q^{5} +6.40352 q^{9} +O(q^{10})\)	\(q-3.06652 q^{3} -2.61233 q^{5} +6.40352 q^{9} -3.48251 q^{11} -0.0840835 q^{13} +8.01076 q^{15} +7.37042 q^{17} -6.02077 q^{19} -6.92421 q^{23} +1.82428 q^{25} -10.4369 q^{27} +1.00000 q^{29} +4.91912 q^{31} +10.6792 q^{33} -6.66128 q^{37} +0.257843 q^{39} +8.54902 q^{41} +5.19149 q^{43} -16.7281 q^{45} -11.7405 q^{47} -22.6015 q^{51} +1.28067 q^{53} +9.09747 q^{55} +18.4628 q^{57} +0.405400 q^{59} -2.29102 q^{61} +0.219654 q^{65} -6.42125 q^{67} +21.2332 q^{69} -14.2946 q^{71} -10.5743 q^{73} -5.59419 q^{75} +1.75488 q^{79} +12.7945 q^{81} -14.5599 q^{83} -19.2540 q^{85} -3.06652 q^{87} -4.04606 q^{89} -15.0846 q^{93} +15.7283 q^{95} -7.10017 q^{97} -22.3003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + q^{5} + 11 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 + q^5 + 11 * q^9	\( 6 q - 3 q^{3} + q^{5} + 11 q^{9} + q^{11} + 3 q^{13} + 13 q^{15} - 12 q^{19} + 17 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} - 17 q^{33} + 4 q^{37} + 13 q^{39} + 14 q^{41} + 29 q^{43} + 8 q^{45} - 31 q^{47} + 14 q^{51} - 7 q^{53} - 3 q^{55} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 5 q^{65} + 6 q^{67} + 48 q^{69} - 10 q^{73} - 22 q^{75} + 19 q^{79} - 18 q^{81} + 14 q^{83} - 12 q^{85} - 3 q^{87} + 16 q^{89} - 9 q^{93} - 34 q^{95} + 6 q^{97} - 22 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + q^5 + 11 * q^9 + q^11 + 3 * q^13 + 13 * q^15 - 12 * q^19 + 17 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 - 17 * q^33 + 4 * q^37 + 13 * q^39 + 14 * q^41 + 29 * q^43 + 8 * q^45 - 31 * q^47 + 14 * q^51 - 7 * q^53 - 3 * q^55 + 18 * q^57 + 2 * q^59 + 6 * q^61 + 5 * q^65 + 6 * q^67 + 48 * q^69 - 10 * q^73 - 22 * q^75 + 19 * q^79 - 18 * q^81 + 14 * q^83 - 12 * q^85 - 3 * q^87 + 16 * q^89 - 9 * q^93 - 34 * q^95 + 6 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more