LMFDB - Embedded newform 5776.2.a.bt.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.17557$ 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)

$f(q)$	$=$	$q+2.52015 q^{3} -2.45965 q^{5} +2.79360 q^{7} +3.35114 q^{9} +O(q^{10})$	$q+2.52015 q^{3} -2.45965 q^{5} +2.79360 q^{7} +3.35114 q^{9} +1.67853 q^{11} +6.34458 q^{13} -6.19868 q^{15} +4.96917 q^{17} +7.04029 q^{21} +2.49890 q^{23} +1.04988 q^{25} +0.884927 q^{27} +5.93179 q^{29} -7.28408 q^{31} +4.23015 q^{33} -6.87129 q^{35} +0.550972 q^{37} +15.9893 q^{39} -2.60741 q^{41} -2.87129 q^{43} -8.24263 q^{45} -0.745593 q^{47} +0.804226 q^{49} +12.5231 q^{51} -1.47027 q^{53} -4.12860 q^{55} +4.96261 q^{59} +9.33686 q^{61} +9.36176 q^{63} -15.6054 q^{65} -11.5604 q^{67} +6.29761 q^{69} +6.99698 q^{71} -6.18619 q^{73} +2.64584 q^{75} +4.68915 q^{77} +5.91158 q^{79} -7.82328 q^{81} -15.1773 q^{83} -12.2224 q^{85} +14.9490 q^{87} +6.90502 q^{89} +17.7242 q^{91} -18.3570 q^{93} +14.3874 q^{97} +5.62500 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9	$ 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 - 2 * q^11 + 18 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^21 + 10 * q^23 + 6 * q^25 + 4 * q^27 - 2 * q^29 - 26 * q^31 + 16 * q^33 - 6 * q^35 + 4 * q^37 + 6 * q^39 - 12 * q^41 + 10 * q^43 - 22 * q^45 + 12 * q^47 - 12 * q^49 + 2 * q^51 + 8 * q^53 + 26 * q^55 + 8 * q^59 + 22 * q^63 - 4 * q^65 - 10 * q^67 - 20 * q^69 - 14 * q^73 - 8 * q^75 + 4 * q^77 - 22 * q^79 - 4 * q^81 + 12 * q^83 - 18 * q^85 + 26 * q^87 - 16 * q^89 + 4 * q^91 + 8 * q^93 + 28 * 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
$3$	2.52015	1.45501	0.727504	−	0.686104i	$-0.240680\pi$
$3$	2.52015	1.45501	0.727504	+	0.686104i	$0.240680\pi$
$4$	0	0
$5$	−2.45965	−1.09999	−0.549994	−	0.835168i	$-0.685370\pi$
$5$	−2.45965	−1.09999	−0.549994	+	0.835168i	$0.685370\pi$
$6$	0	0
$7$	2.79360	1.05588	0.527942	−	0.849281i	$-0.322964\pi$
$7$	2.79360	1.05588	0.527942	+	0.849281i	$0.322964\pi$
$8$	0	0
$9$	3.35114	1.11705
$10$	0	0
$11$	1.67853	0.506096	0.253048	−	0.967454i	$-0.418567\pi$
$11$	1.67853	0.506096	0.253048	+	0.967454i	$0.418567\pi$
$12$	0	0
$13$	6.34458	1.75967	0.879834	−	0.475280i	$-0.157653\pi$
$13$	6.34458	1.75967	0.879834	+	0.475280i	$0.157653\pi$
$14$	0	0
$15$	−6.19868	−1.60049
$16$	0	0
$17$	4.96917	1.20520	0.602601	−	0.798043i	$-0.294131\pi$
$17$	4.96917	1.20520	0.602601	+	0.798043i	$0.294131\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	7.04029	1.53632
$22$	0	0
$23$	2.49890	0.521057	0.260529	−	0.965466i	$-0.416103\pi$
$23$	2.49890	0.521057	0.260529	+	0.965466i	$0.416103\pi$
$24$	0	0
$25$	1.04988	0.209975
$26$	0	0
$27$	0.884927	0.170304
$28$	0	0
$29$	5.93179	1.10150	0.550752	−	0.834669i	$-0.314341\pi$
$29$	5.93179	1.10150	0.550752	+	0.834669i	$0.314341\pi$
$30$	0	0
$31$	−7.28408	−1.30826	−0.654130	−	0.756382i	$-0.726965\pi$
$31$	−7.28408	−1.30826	−0.654130	+	0.756382i	$0.726965\pi$
$32$	0	0
$33$	4.23015	0.736374
$34$	0	0
$35$	−6.87129	−1.16146
$36$	0	0
$37$	0.550972	0.0905792	0.0452896	−	0.998974i	$-0.485579\pi$
$37$	0.550972	0.0905792	0.0452896	+	0.998974i	$0.485579\pi$
$38$	0	0
$39$	15.9893	2.56033
$40$	0	0
$41$	−2.60741	−0.407209	−0.203605	−	0.979053i	$-0.565266\pi$
$41$	−2.60741	−0.407209	−0.203605	+	0.979053i	$0.565266\pi$
$42$	0	0
$43$	−2.87129	−0.437867	−0.218934	−	0.975740i	$-0.570258\pi$
$43$	−2.87129	−0.437867	−0.218934	+	0.975740i	$0.570258\pi$
$44$	0	0
$45$	−8.24263	−1.22874
$46$	0	0
$47$	−0.745593	−0.108756	−0.0543780	−	0.998520i	$-0.517318\pi$
$47$	−0.745593	−0.108756	−0.0543780	+	0.998520i	$0.517318\pi$
$48$	0	0
$49$	0.804226	0.114889
$50$	0	0
$51$	12.5231	1.75358
$52$	0	0
$53$	−1.47027	−0.201957	−0.100979	−	0.994889i	$-0.532197\pi$
$53$	−1.47027	−0.201957	−0.100979	+	0.994889i	$0.532197\pi$
$54$	0	0
$55$	−4.12860	−0.556700
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.96261	0.646077	0.323038	−	0.946386i	$-0.395296\pi$
$59$	4.96261	0.646077	0.323038	+	0.946386i	$0.395296\pi$
$60$	0	0
$61$	9.33686	1.19546	0.597731	−	0.801697i	$-0.296069\pi$
$61$	9.33686	1.19546	0.597731	+	0.801697i	$0.296069\pi$
$62$	0	0
$63$	9.36176	1.17947
$64$	0	0
$65$	−15.6054	−1.93562
$66$	0	0
$67$	−11.5604	−1.41233	−0.706166	−	0.708046i	$-0.749577\pi$
$67$	−11.5604	−1.41233	−0.706166	+	0.708046i	$0.749577\pi$
$68$	0	0
$69$	6.29761	0.758143
$70$	0	0
$71$	6.99698	0.830389	0.415195	−	0.909733i	$-0.363714\pi$
$71$	6.99698	0.830389	0.415195	+	0.909733i	$0.363714\pi$
$72$	0	0
$73$	−6.18619	−0.724039	−0.362020	−	0.932171i	$-0.617913\pi$
$73$	−6.18619	−0.724039	−0.362020	+	0.932171i	$0.617913\pi$
$74$	0	0
$75$	2.64584	0.305515
$76$	0	0
$77$	4.68915	0.534379
$78$	0	0
$79$	5.91158	0.665105	0.332552	−	0.943085i	$-0.392090\pi$
$79$	5.91158	0.665105	0.332552	+	0.943085i	$0.392090\pi$
$80$	0	0
$81$	−7.82328	−0.869253
$82$	0	0
$83$	−15.1773	−1.66593	−0.832964	−	0.553328i	$-0.813358\pi$
$83$	−15.1773	−1.66593	−0.832964	+	0.553328i	$0.813358\pi$
$84$	0	0
$85$	−12.2224	−1.32571
$86$	0	0
$87$	14.9490	1.60270
$88$	0	0
$89$	6.90502	0.731930	0.365965	−	0.930629i	$-0.380739\pi$
$89$	6.90502	0.731930	0.365965	+	0.930629i	$0.380739\pi$
$90$	0	0
$91$	17.7242	1.85800
$92$	0	0
$93$	−18.3570	−1.90353
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3874	1.46082	0.730408	−	0.683011i	$-0.239330\pi$
$97$	14.3874	1.46082	0.730408	+	0.683011i	$0.239330\pi$
$98$	0	0
$99$	5.62500	0.565333
$100$	0	0
$101$	11.2829	1.12269	0.561347	−	0.827581i	$-0.310283\pi$
$101$	11.2829	1.12269	0.561347	+	0.827581i	$0.310283\pi$
$102$	0	0
$103$	−14.8280	−1.46104	−0.730522	−	0.682889i	$-0.760723\pi$
$103$	−14.8280	−1.46104	−0.730522	+	0.682889i	$0.760723\pi$
$104$	0	0
$105$	−17.3167	−1.68993
$106$	0	0
$107$	2.82849	0.273440	0.136720	−	0.990610i	$-0.456344\pi$
$107$	2.82849	0.273440	0.136720	+	0.990610i	$0.456344\pi$
$108$	0	0
$109$	1.72539	0.165262	0.0826312	−	0.996580i	$-0.473668\pi$
$109$	1.72539	0.165262	0.0826312	+	0.996580i	$0.473668\pi$
$110$	0	0
$111$	1.38853	0.131793
$112$	0	0
$113$	−0.427785	−0.0402426	−0.0201213	−	0.999798i	$-0.506405\pi$
$113$	−0.427785	−0.0402426	−0.0201213	+	0.999798i	$0.506405\pi$
$114$	0	0
$115$	−6.14643	−0.573157
$116$	0	0
$117$	21.2616	1.96563
$118$	0	0
$119$	13.8819	1.27255
$120$	0	0
$121$	−8.18253	−0.743867
$122$	0	0
$123$	−6.57106	−0.592493
$124$	0	0
$125$	9.71592	0.869018
$126$	0	0
$127$	1.13632	0.100832	0.0504159	−	0.998728i	$-0.483945\pi$
$127$	1.13632	0.100832	0.0504159	+	0.998728i	$0.483945\pi$
$128$	0	0
$129$	−7.23607	−0.637100
$130$	0	0
$131$	17.8415	1.55882	0.779410	−	0.626515i	$-0.215519\pi$
$131$	17.8415	1.55882	0.779410	+	0.626515i	$0.215519\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.17661	−0.187333
$136$	0	0
$137$	−15.9156	−1.35976	−0.679882	−	0.733321i	$-0.737969\pi$
$137$	−15.9156	−1.35976	−0.679882	+	0.733321i	$0.737969\pi$
$138$	0	0
$139$	−10.1202	−0.858382	−0.429191	−	0.903214i	$-0.641201\pi$
$139$	−10.1202	−0.858382	−0.429191	+	0.903214i	$0.641201\pi$
$140$	0	0
$141$	−1.87901	−0.158241
$142$	0	0
$143$	10.6496	0.890562
$144$	0	0
$145$	−14.5901	−1.21164
$146$	0	0
$147$	2.02677	0.167165
$148$	0	0
$149$	9.39851	0.769956	0.384978	−	0.922926i	$-0.374209\pi$
$149$	9.39851	0.769956	0.384978	+	0.922926i	$0.374209\pi$
$150$	0	0
$151$	10.2632	0.835210	0.417605	−	0.908629i	$-0.362870\pi$
$151$	10.2632	0.835210	0.417605	+	0.908629i	$0.362870\pi$
$152$	0	0
$153$	16.6524	1.34627
$154$	0	0
$155$	17.9163	1.43907
$156$	0	0
$157$	2.97387	0.237341	0.118671	−	0.992934i	$-0.462137\pi$
$157$	2.97387	0.237341	0.118671	+	0.992934i	$0.462137\pi$
$158$	0	0
$159$	−3.70530	−0.293849
$160$	0	0
$161$	6.98095	0.550176
$162$	0	0
$163$	−5.59191	−0.437992	−0.218996	−	0.975726i	$-0.570278\pi$
$163$	−5.59191	−0.437992	−0.218996	+	0.975726i	$0.570278\pi$
$164$	0	0
$165$	−10.4047	−0.810003
$166$	0	0
$167$	7.08361	0.548146	0.274073	−	0.961709i	$-0.411629\pi$
$167$	7.08361	0.548146	0.274073	+	0.961709i	$0.411629\pi$
$168$	0	0
$169$	27.2537	2.09643
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−23.0208	−1.75024	−0.875120	−	0.483907i	$-0.839217\pi$
$173$	−23.0208	−1.75024	−0.875120	+	0.483907i	$0.839217\pi$
$174$	0	0
$175$	2.93294	0.221709
$176$	0	0
$177$	12.5065	0.940047
$178$	0	0
$179$	−8.21471	−0.613996	−0.306998	−	0.951710i	$-0.599325\pi$
$179$	−8.21471	−0.613996	−0.306998	+	0.951710i	$0.599325\pi$
$180$	0	0
$181$	−12.9999	−0.966274	−0.483137	−	0.875545i	$-0.660503\pi$
$181$	−12.9999	−0.966274	−0.483137	+	0.875545i	$0.660503\pi$
$182$	0	0
$183$	23.5303	1.73941
$184$	0	0
$185$	−1.35520	−0.0996361
$186$	0	0
$187$	8.34092	0.609948
$188$	0	0
$189$	2.47214	0.179821
$190$	0	0
$191$	−6.57479	−0.475735	−0.237868	−	0.971298i	$-0.576448\pi$
$191$	−6.57479	−0.475735	−0.237868	+	0.971298i	$0.576448\pi$
$192$	0	0
$193$	26.8826	1.93505	0.967527	−	0.252769i	$-0.0813412\pi$
$193$	26.8826	1.93505	0.967527	+	0.252769i	$0.0813412\pi$
$194$	0	0
$195$	−39.3280	−2.81634
$196$	0	0
$197$	−9.84940	−0.701741	−0.350870	−	0.936424i	$-0.614114\pi$
$197$	−9.84940	−0.701741	−0.350870	+	0.936424i	$0.614114\pi$
$198$	0	0
$199$	20.4661	1.45080	0.725402	−	0.688326i	$-0.241654\pi$
$199$	20.4661	1.45080	0.725402	+	0.688326i	$0.241654\pi$
$200$	0	0
$201$	−29.1340	−2.05495
$202$	0	0
$203$	16.5711	1.16306
$204$	0	0
$205$	6.41332	0.447926
$206$	0	0
$207$	8.37418	0.582046
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.11630	−0.558749	−0.279374	−	0.960182i	$-0.590127\pi$
$211$	−8.11630	−0.558749	−0.279374	+	0.960182i	$0.590127\pi$
$212$	0	0
$213$	17.6334	1.20822
$214$	0	0
$215$	7.06236	0.481649
$216$	0	0
$217$	−20.3488	−1.38137
$218$	0	0
$219$	−15.5901	−1.05348
$220$	0	0
$221$	31.5273	2.12076
$222$	0	0
$223$	−10.7540	−0.720143	−0.360071	−	0.932925i	$-0.617248\pi$
$223$	−10.7540	−0.720143	−0.360071	+	0.932925i	$0.617248\pi$
$224$	0	0
$225$	3.51828	0.234552
$226$	0	0
$227$	−27.0936	−1.79827	−0.899133	−	0.437676i	$-0.855802\pi$
$227$	−27.0936	−1.79827	−0.899133	+	0.437676i	$0.855802\pi$
$228$	0	0
$229$	9.46557	0.625503	0.312751	−	0.949835i	$-0.398749\pi$
$229$	9.46557	0.625503	0.312751	+	0.949835i	$0.398749\pi$
$230$	0	0
$231$	11.8174	0.777525
$232$	0	0
$233$	22.8621	1.49775	0.748873	−	0.662713i	$-0.230595\pi$
$233$	22.8621	1.49775	0.748873	+	0.662713i	$0.230595\pi$
$234$	0	0
$235$	1.83390	0.119630
$236$	0	0
$237$	14.8981	0.967733
$238$	0	0
$239$	−8.66611	−0.560564	−0.280282	−	0.959918i	$-0.590428\pi$
$239$	−8.66611	−0.560564	−0.280282	+	0.959918i	$0.590428\pi$
$240$	0	0
$241$	5.65826	0.364480	0.182240	−	0.983254i	$-0.441665\pi$
$241$	5.65826	0.364480	0.182240	+	0.983254i	$0.441665\pi$
$242$	0	0
$243$	−22.3706	−1.43507
$244$	0	0
$245$	−1.97811	−0.126377
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−38.2491	−2.42394
$250$	0	0
$251$	13.5552	0.855599	0.427799	−	0.903874i	$-0.359289\pi$
$251$	13.5552	0.855599	0.427799	+	0.903874i	$0.359289\pi$
$252$	0	0
$253$	4.19449	0.263705
$254$	0	0
$255$	−30.8023	−1.92892
$256$	0	0
$257$	10.0393	0.626231	0.313116	−	0.949715i	$-0.398627\pi$
$257$	10.0393	0.626231	0.313116	+	0.949715i	$0.398627\pi$
$258$	0	0
$259$	1.53920	0.0956411
$260$	0	0
$261$	19.8782	1.23043
$262$	0	0
$263$	18.8262	1.16087	0.580436	−	0.814306i	$-0.302882\pi$
$263$	18.8262	1.16087	0.580436	+	0.814306i	$0.302882\pi$
$264$	0	0
$265$	3.61635	0.222151
$266$	0	0
$267$	17.4017	1.06496
$268$	0	0
$269$	−16.3827	−0.998870	−0.499435	−	0.866351i	$-0.666459\pi$
$269$	−16.3827	−0.998870	−0.499435	+	0.866351i	$0.666459\pi$
$270$	0	0
$271$	2.76684	0.168073	0.0840367	−	0.996463i	$-0.473219\pi$
$271$	2.76684	0.168073	0.0840367	+	0.996463i	$0.473219\pi$
$272$	0	0
$273$	44.6677	2.70341
$274$	0	0
$275$	1.76225	0.106268
$276$	0	0
$277$	24.5321	1.47399	0.736996	−	0.675897i	$-0.236244\pi$
$277$	24.5321	1.47399	0.736996	+	0.675897i	$0.236244\pi$
$278$	0	0
$279$	−24.4100	−1.46139
$280$	0	0
$281$	10.1704	0.606713	0.303356	−	0.952877i	$-0.401893\pi$
$281$	10.1704	0.606713	0.303356	+	0.952877i	$0.401893\pi$
$282$	0	0
$283$	−27.6678	−1.64468	−0.822340	−	0.568997i	$-0.807332\pi$
$283$	−27.6678	−1.64468	−0.822340	+	0.568997i	$0.807332\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.28408	−0.429966
$288$	0	0
$289$	7.69270	0.452512
$290$	0	0
$291$	36.2583	2.12550
$292$	0	0
$293$	23.8386	1.39267	0.696333	−	0.717719i	$-0.254814\pi$
$293$	23.8386	1.39267	0.696333	+	0.717719i	$0.254814\pi$
$294$	0	0
$295$	−12.2063	−0.710677
$296$	0	0
$297$	1.48538	0.0861904
$298$	0	0
$299$	15.8545	0.916889
$300$	0	0
$301$	−8.02124	−0.462337
$302$	0	0
$303$	28.4346	1.63353
$304$	0	0
$305$	−22.9654	−1.31500
$306$	0	0
$307$	−25.9884	−1.48324	−0.741619	−	0.670821i	$-0.765942\pi$
$307$	−25.9884	−1.48324	−0.741619	+	0.670821i	$0.765942\pi$
$308$	0	0
$309$	−37.3687	−2.12583
$310$	0	0
$311$	23.1043	1.31013	0.655063	−	0.755574i	$-0.272642\pi$
$311$	23.1043	1.31013	0.655063	+	0.755574i	$0.272642\pi$
$312$	0	0
$313$	12.2647	0.693242	0.346621	−	0.938005i	$-0.387329\pi$
$313$	12.2647	0.693242	0.346621	+	0.938005i	$0.387329\pi$
$314$	0	0
$315$	−23.0267	−1.29741
$316$	0	0
$317$	−0.272305	−0.0152942	−0.00764708	−	0.999971i	$-0.502434\pi$
$317$	−0.272305	−0.0152942	−0.00764708	+	0.999971i	$0.502434\pi$
$318$	0	0
$319$	9.95669	0.557468
$320$	0	0
$321$	7.12820	0.397857
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.66102	0.369487
$326$	0	0
$327$	4.34824	0.240458
$328$	0	0
$329$	−2.08289	−0.114834
$330$	0	0
$331$	19.9930	1.09891	0.549457	−	0.835522i	$-0.314835\pi$
$331$	19.9930	1.09891	0.549457	+	0.835522i	$0.314835\pi$
$332$	0	0
$333$	1.84638	0.101181
$334$	0	0
$335$	28.4346	1.55355
$336$	0	0
$337$	−17.2824	−0.941433	−0.470717	−	0.882284i	$-0.656005\pi$
$337$	−17.2824	−0.941433	−0.470717	+	0.882284i	$0.656005\pi$
$338$	0	0
$339$	−1.07808	−0.0585533
$340$	0	0
$341$	−12.2266	−0.662105
$342$	0	0
$343$	−17.3085	−0.934573
$344$	0	0
$345$	−15.4899	−0.833948
$346$	0	0
$347$	18.5456	0.995579	0.497789	−	0.867298i	$-0.334145\pi$
$347$	18.5456	0.995579	0.497789	+	0.867298i	$0.334145\pi$
$348$	0	0
$349$	−20.1210	−1.07705	−0.538527	−	0.842609i	$-0.681019\pi$
$349$	−20.1210	−1.07705	−0.538527	+	0.842609i	$0.681019\pi$
$350$	0	0
$351$	5.61449	0.299679
$352$	0	0
$353$	24.0557	1.28035	0.640177	−	0.768227i	$-0.278861\pi$
$353$	24.0557	1.28035	0.640177	+	0.768227i	$0.278861\pi$
$354$	0	0
$355$	−17.2101	−0.913419
$356$	0	0
$357$	34.9845	1.85157
$358$	0	0
$359$	10.8535	0.572824	0.286412	−	0.958107i	$-0.407537\pi$
$359$	10.8535	0.572824	0.286412	+	0.958107i	$0.407537\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−20.6212	−1.08233
$364$	0	0
$365$	15.2159	0.796435
$366$	0	0
$367$	32.6121	1.70234	0.851168	−	0.524893i	$-0.175895\pi$
$367$	32.6121	1.70234	0.851168	+	0.524893i	$0.175895\pi$
$368$	0	0
$369$	−8.73781	−0.454872
$370$	0	0
$371$	−4.10736	−0.213243
$372$	0	0
$373$	−5.30198	−0.274526	−0.137263	−	0.990535i	$-0.543831\pi$
$373$	−5.30198	−0.274526	−0.137263	+	0.990535i	$0.543831\pi$
$374$	0	0
$375$	24.4855	1.26443
$376$	0	0
$377$	37.6347	1.93828
$378$	0	0
$379$	−24.6656	−1.26699	−0.633494	−	0.773748i	$-0.718380\pi$
$379$	−24.6656	−1.26699	−0.633494	+	0.773748i	$0.718380\pi$
$380$	0	0
$381$	2.86368	0.146711
$382$	0	0
$383$	−3.64416	−0.186208	−0.0931039	−	0.995656i	$-0.529679\pi$
$383$	−3.64416	−0.186208	−0.0931039	+	0.995656i	$0.529679\pi$
$384$	0	0
$385$	−11.5337	−0.587810
$386$	0	0
$387$	−9.62209	−0.489118
$388$	0	0
$389$	2.79353	0.141638	0.0708189	−	0.997489i	$-0.477439\pi$
$389$	2.79353	0.141638	0.0708189	+	0.997489i	$0.477439\pi$
$390$	0	0
$391$	12.4175	0.627979
$392$	0	0
$393$	44.9632	2.26809
$394$	0	0
$395$	−14.5404	−0.731608
$396$	0	0
$397$	−39.5101	−1.98296	−0.991478	−	0.130273i	$-0.958415\pi$
$397$	−39.5101	−1.98296	−0.991478	+	0.130273i	$0.958415\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.6215	−0.929912	−0.464956	−	0.885334i	$-0.653930\pi$
$401$	−18.6215	−0.929912	−0.464956	+	0.885334i	$0.653930\pi$
$402$	0	0
$403$	−46.2144	−2.30210
$404$	0	0
$405$	19.2425	0.956168
$406$	0	0
$407$	0.924824	0.0458418
$408$	0	0
$409$	−17.1433	−0.847681	−0.423840	−	0.905737i	$-0.639318\pi$
$409$	−17.1433	−0.847681	−0.423840	+	0.905737i	$0.639318\pi$
$410$	0	0
$411$	−40.1098	−1.97847
$412$	0	0
$413$	13.8636	0.682182
$414$	0	0
$415$	37.3309	1.83250
$416$	0	0
$417$	−25.5043	−1.24895
$418$	0	0
$419$	18.5042	0.903989	0.451995	−	0.892021i	$-0.350713\pi$
$419$	18.5042	0.903989	0.451995	+	0.892021i	$0.350713\pi$
$420$	0	0
$421$	−16.1996	−0.789520	−0.394760	−	0.918784i	$-0.629172\pi$
$421$	−16.1996	−0.789520	−0.394760	+	0.918784i	$0.629172\pi$
$422$	0	0
$423$	−2.49859	−0.121486
$424$	0	0
$425$	5.21702	0.253062
$426$	0	0
$427$	26.0835	1.26227
$428$	0	0
$429$	26.8385	1.29577
$430$	0	0
$431$	−12.6997	−0.611721	−0.305861	−	0.952076i	$-0.598944\pi$
$431$	−12.6997	−0.611721	−0.305861	+	0.952076i	$0.598944\pi$
$432$	0	0
$433$	14.2350	0.684092	0.342046	−	0.939683i	$-0.388880\pi$
$433$	14.2350	0.684092	0.342046	+	0.939683i	$0.388880\pi$
$434$	0	0
$435$	−36.7692	−1.76295
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−28.1900	−1.34544	−0.672718	−	0.739899i	$-0.734873\pi$
$439$	−28.1900	−1.34544	−0.672718	+	0.739899i	$0.734873\pi$
$440$	0	0
$441$	2.69507	0.128337
$442$	0	0
$443$	32.9944	1.56761	0.783805	−	0.621007i	$-0.213276\pi$
$443$	32.9944	1.56761	0.783805	+	0.621007i	$0.213276\pi$
$444$	0	0
$445$	−16.9839	−0.805115
$446$	0	0
$447$	23.6856	1.12029
$448$	0	0
$449$	−17.1975	−0.811601	−0.405801	−	0.913962i	$-0.633007\pi$
$449$	−17.1975	−0.811601	−0.405801	+	0.913962i	$0.633007\pi$
$450$	0	0
$451$	−4.37662	−0.206087
$452$	0	0
$453$	25.8649	1.21524
$454$	0	0
$455$	−43.5954	−2.04378
$456$	0	0
$457$	−31.1517	−1.45722	−0.728608	−	0.684931i	$-0.759832\pi$
$457$	−31.1517	−1.45722	−0.728608	+	0.684931i	$0.759832\pi$
$458$	0	0
$459$	4.39736	0.205251
$460$	0	0
$461$	15.5410	0.723816	0.361908	−	0.932214i	$-0.382125\pi$
$461$	15.5410	0.723816	0.361908	+	0.932214i	$0.382125\pi$
$462$	0	0
$463$	−20.6648	−0.960374	−0.480187	−	0.877166i	$-0.659431\pi$
$463$	−20.6648	−0.960374	−0.480187	+	0.877166i	$0.659431\pi$
$464$	0	0
$465$	45.1517	2.09386
$466$	0	0
$467$	28.4830	1.31803	0.659017	−	0.752128i	$-0.270972\pi$
$467$	28.4830	1.31803	0.659017	+	0.752128i	$0.270972\pi$
$468$	0	0
$469$	−32.2953	−1.49126
$470$	0	0
$471$	7.49460	0.345333
$472$	0	0
$473$	−4.81955	−0.221603
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.92709	−0.225596
$478$	0	0
$479$	10.4726	0.478507	0.239254	−	0.970957i	$-0.423097\pi$
$479$	10.4726	0.478507	0.239254	+	0.970957i	$0.423097\pi$
$480$	0	0
$481$	3.49568	0.159389
$482$	0	0
$483$	17.5930	0.800510
$484$	0	0
$485$	−35.3879	−1.60688
$486$	0	0
$487$	−2.29894	−0.104175	−0.0520875	−	0.998643i	$-0.516587\pi$
$487$	−2.29894	−0.104175	−0.0520875	+	0.998643i	$0.516587\pi$
$488$	0	0
$489$	−14.0924	−0.637282
$490$	0	0
$491$	−12.1612	−0.548826	−0.274413	−	0.961612i	$-0.588484\pi$
$491$	−12.1612	−0.548826	−0.274413	+	0.961612i	$0.588484\pi$
$492$	0	0
$493$	29.4761	1.32754
$494$	0	0
$495$	−13.8355	−0.621860
$496$	0	0
$497$	19.5468	0.876794
$498$	0	0
$499$	−15.1348	−0.677528	−0.338764	−	0.940871i	$-0.610009\pi$
$499$	−15.1348	−0.677528	−0.338764	+	0.940871i	$0.610009\pi$
$500$	0	0
$501$	17.8517	0.797556
$502$	0	0
$503$	0.573683	0.0255793	0.0127896	−	0.999918i	$-0.495929\pi$
$503$	0.573683	0.0255793	0.0127896	+	0.999918i	$0.495929\pi$
$504$	0	0
$505$	−27.7520	−1.23495
$506$	0	0
$507$	68.6832	3.05033
$508$	0	0
$509$	−26.2163	−1.16202	−0.581008	−	0.813898i	$-0.697341\pi$
$509$	−26.2163	−1.16202	−0.581008	+	0.813898i	$0.697341\pi$
$510$	0	0
$511$	−17.2818	−0.764501
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.4716	1.60713
$516$	0	0
$517$	−1.25150	−0.0550410
$518$	0	0
$519$	−58.0158	−2.54661
$520$	0	0
$521$	−12.9637	−0.567948	−0.283974	−	0.958832i	$-0.591653\pi$
$521$	−12.9637	−0.567948	−0.283974	+	0.958832i	$0.591653\pi$
$522$	0	0
$523$	−17.7793	−0.777432	−0.388716	−	0.921358i	$-0.627081\pi$
$523$	−17.7793	−0.777432	−0.388716	+	0.921358i	$0.627081\pi$
$524$	0	0
$525$	7.39144	0.322589
$526$	0	0
$527$	−36.1959	−1.57672
$528$	0	0
$529$	−16.7555	−0.728499
$530$	0	0
$531$	16.6304	0.721698
$532$	0	0
$533$	−16.5429	−0.716554
$534$	0	0
$535$	−6.95709	−0.300781
$536$	0	0
$537$	−20.7023	−0.893369
$538$	0	0
$539$	1.34992	0.0581451
$540$	0	0
$541$	−5.34418	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$541$	−5.34418	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$542$	0	0
$543$	−32.7616	−1.40594
$544$	0	0
$545$	−4.24385	−0.181787
$546$	0	0
$547$	−5.10112	−0.218108	−0.109054	−	0.994036i	$-0.534782\pi$
$547$	−5.10112	−0.218108	−0.109054	+	0.994036i	$0.534782\pi$
$548$	0	0
$549$	31.2891	1.33539
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.5146	0.702273
$554$	0	0
$555$	−3.41530	−0.144971
$556$	0	0
$557$	−10.7235	−0.454370	−0.227185	−	0.973852i	$-0.572952\pi$
$557$	−10.7235	−0.454370	−0.227185	+	0.973852i	$0.572952\pi$
$558$	0	0
$559$	−18.2171	−0.770502
$560$	0	0
$561$	21.0203	0.887479
$562$	0	0
$563$	−9.76285	−0.411455	−0.205728	−	0.978609i	$-0.565956\pi$
$563$	−9.76285	−0.411455	−0.205728	+	0.978609i	$0.565956\pi$
$564$	0	0
$565$	1.05220	0.0442664
$566$	0	0
$567$	−21.8551	−0.917830
$568$	0	0
$569$	−8.84194	−0.370674	−0.185337	−	0.982675i	$-0.559338\pi$
$569$	−8.84194	−0.370674	−0.185337	+	0.982675i	$0.559338\pi$
$570$	0	0
$571$	−22.8411	−0.955871	−0.477935	−	0.878395i	$-0.658615\pi$
$571$	−22.8411	−0.955871	−0.477935	+	0.878395i	$0.658615\pi$
$572$	0	0
$573$	−16.5694	−0.692198
$574$	0	0
$575$	2.62354	0.109409
$576$	0	0
$577$	−14.0176	−0.583560	−0.291780	−	0.956486i	$-0.594247\pi$
$577$	−14.0176	−0.583560	−0.291780	+	0.956486i	$0.594247\pi$
$578$	0	0
$579$	67.7482	2.81552
$580$	0	0
$581$	−42.3994	−1.75903
$582$	0	0
$583$	−2.46790	−0.102210
$584$	0	0
$585$	−52.2960	−2.16217
$586$	0	0
$587$	−21.0356	−0.868232	−0.434116	−	0.900857i	$-0.642939\pi$
$587$	−21.0356	−0.868232	−0.434116	+	0.900857i	$0.642939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−24.8219	−1.02104
$592$	0	0
$593$	−34.9193	−1.43396	−0.716981	−	0.697093i	$-0.754477\pi$
$593$	−34.9193	−1.43396	−0.716981	+	0.697093i	$0.754477\pi$
$594$	0	0
$595$	−34.1446	−1.39979
$596$	0	0
$597$	51.5776	2.11093
$598$	0	0
$599$	−21.5106	−0.878901	−0.439450	−	0.898267i	$-0.644827\pi$
$599$	−21.5106	−0.878901	−0.439450	+	0.898267i	$0.644827\pi$
$600$	0	0
$601$	−14.3417	−0.585012	−0.292506	−	0.956264i	$-0.594489\pi$
$601$	−14.3417	−0.585012	−0.292506	+	0.956264i	$0.594489\pi$
$602$	0	0
$603$	−38.7407	−1.57764
$604$	0	0
$605$	20.1262	0.818245
$606$	0	0
$607$	10.3488	0.420046	0.210023	−	0.977696i	$-0.432646\pi$
$607$	10.3488	0.420046	0.210023	+	0.977696i	$0.432646\pi$
$608$	0	0
$609$	41.7615	1.69226
$610$	0	0
$611$	−4.73047	−0.191375
$612$	0	0
$613$	19.5674	0.790320	0.395160	−	0.918612i	$-0.370689\pi$
$613$	19.5674	0.790320	0.395160	+	0.918612i	$0.370689\pi$
$614$	0	0
$615$	16.1625	0.651735
$616$	0	0
$617$	−5.92522	−0.238540	−0.119270	−	0.992862i	$-0.538055\pi$
$617$	−5.92522	−0.238540	−0.119270	+	0.992862i	$0.538055\pi$
$618$	0	0
$619$	−8.03958	−0.323138	−0.161569	−	0.986861i	$-0.551655\pi$
$619$	−8.03958	−0.323138	−0.161569	+	0.986861i	$0.551655\pi$
$620$	0	0
$621$	2.21135	0.0887383
$622$	0	0
$623$	19.2899	0.772833
$624$	0	0
$625$	−29.1471	−1.16589
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.73788	0.109166
$630$	0	0
$631$	12.9006	0.513567	0.256783	−	0.966469i	$-0.417337\pi$
$631$	12.9006	0.513567	0.256783	+	0.966469i	$0.417337\pi$
$632$	0	0
$633$	−20.4543	−0.812984
$634$	0	0
$635$	−2.79494	−0.110914
$636$	0	0
$637$	5.10247	0.202167
$638$	0	0
$639$	23.4479	0.927584
$640$	0	0
$641$	−28.8639	−1.14006	−0.570028	−	0.821625i	$-0.693068\pi$
$641$	−28.8639	−1.14006	−0.570028	+	0.821625i	$0.693068\pi$
$642$	0	0
$643$	−26.3394	−1.03873	−0.519363	−	0.854554i	$-0.673831\pi$
$643$	−26.3394	−1.03873	−0.519363	+	0.854554i	$0.673831\pi$
$644$	0	0
$645$	17.7982	0.700803
$646$	0	0
$647$	3.04601	0.119751	0.0598756	−	0.998206i	$-0.480930\pi$
$647$	3.04601	0.119751	0.0598756	+	0.998206i	$0.480930\pi$
$648$	0	0
$649$	8.32990	0.326977
$650$	0	0
$651$	−51.2821	−2.00990
$652$	0	0
$653$	−15.1973	−0.594717	−0.297359	−	0.954766i	$-0.596106\pi$
$653$	−15.1973	−0.594717	−0.297359	+	0.954766i	$0.596106\pi$
$654$	0	0
$655$	−43.8838	−1.71468
$656$	0	0
$657$	−20.7308	−0.808786
$658$	0	0
$659$	−17.8741	−0.696275	−0.348138	−	0.937443i	$-0.613186\pi$
$659$	−17.8741	−0.696275	−0.348138	+	0.937443i	$0.613186\pi$
$660$	0	0
$661$	37.8525	1.47229	0.736146	−	0.676823i	$-0.236644\pi$
$661$	37.8525	1.47229	0.736146	+	0.676823i	$0.236644\pi$
$662$	0	0
$663$	79.4535	3.08572
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.8230	0.573947
$668$	0	0
$669$	−27.1017	−1.04781
$670$	0	0
$671$	15.6722	0.605019
$672$	0	0
$673$	−8.28461	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$673$	−8.28461	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$674$	0	0
$675$	0.929063	0.0357597
$676$	0	0
$677$	−18.3233	−0.704223	−0.352111	−	0.935958i	$-0.614536\pi$
$677$	−18.3233	−0.704223	−0.352111	+	0.935958i	$0.614536\pi$
$678$	0	0
$679$	40.1926	1.54245
$680$	0	0
$681$	−68.2799	−2.61649
$682$	0	0
$683$	49.2217	1.88342	0.941709	−	0.336429i	$-0.109219\pi$
$683$	49.2217	1.88342	0.941709	+	0.336429i	$0.109219\pi$
$684$	0	0
$685$	39.1469	1.49573
$686$	0	0
$687$	23.8546	0.910111
$688$	0	0
$689$	−9.32825	−0.355378
$690$	0	0
$691$	−16.6779	−0.634458	−0.317229	−	0.948349i	$-0.602752\pi$
$691$	−16.6779	−0.634458	−0.317229	+	0.948349i	$0.602752\pi$
$692$	0	0
$693$	15.7140	0.596926
$694$	0	0
$695$	24.8921	0.944210
$696$	0	0
$697$	−12.9567	−0.490770
$698$	0	0
$699$	57.6159	2.17923
$700$	0	0
$701$	−30.9066	−1.16733	−0.583663	−	0.811996i	$-0.698381\pi$
$701$	−30.9066	−1.16733	−0.583663	+	0.811996i	$0.698381\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.62169	0.174063
$706$	0	0
$707$	31.5200	1.18543
$708$	0	0
$709$	40.1056	1.50620	0.753098	−	0.657908i	$-0.228558\pi$
$709$	40.1056	1.50620	0.753098	+	0.657908i	$0.228558\pi$
$710$	0	0
$711$	19.8105	0.742953
$712$	0	0
$713$	−18.2022	−0.681678
$714$	0	0
$715$	−26.1942	−0.979608
$716$	0	0
$717$	−21.8399	−0.815625
$718$	0	0
$719$	45.6529	1.70256	0.851282	−	0.524708i	$-0.175825\pi$
$719$	45.6529	1.70256	0.851282	+	0.524708i	$0.175825\pi$
$720$	0	0
$721$	−41.4235	−1.54269
$722$	0	0
$723$	14.2596	0.530322
$724$	0	0
$725$	6.22764	0.231289
$726$	0	0
$727$	15.1811	0.563034	0.281517	−	0.959556i	$-0.409162\pi$
$727$	15.1811	0.563034	0.281517	+	0.959556i	$0.409162\pi$
$728$	0	0
$729$	−32.9073	−1.21879
$730$	0	0
$731$	−14.2679	−0.527719
$732$	0	0
$733$	−26.9986	−0.997217	−0.498608	−	0.866827i	$-0.666155\pi$
$733$	−26.9986	−0.997217	−0.498608	+	0.866827i	$0.666155\pi$
$734$	0	0
$735$	−4.98514	−0.183880
$736$	0	0
$737$	−19.4046	−0.714776
$738$	0	0
$739$	5.20782	0.191573	0.0957864	−	0.995402i	$-0.469463\pi$
$739$	5.20782	0.191573	0.0957864	+	0.995402i	$0.469463\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.1079	−0.811059	−0.405529	−	0.914082i	$-0.632913\pi$
$743$	−22.1079	−0.811059	−0.405529	+	0.914082i	$0.632913\pi$
$744$	0	0
$745$	−23.1170	−0.846943
$746$	0	0
$747$	−50.8613	−1.86092
$748$	0	0
$749$	7.90167	0.288721
$750$	0	0
$751$	12.4960	0.455984	0.227992	−	0.973663i	$-0.426784\pi$
$751$	12.4960	0.455984	0.227992	+	0.973663i	$0.426784\pi$
$752$	0	0
$753$	34.1612	1.24490
$754$	0	0
$755$	−25.2440	−0.918722
$756$	0	0
$757$	−30.9988	−1.12667	−0.563336	−	0.826228i	$-0.690482\pi$
$757$	−30.9988	−1.12667	−0.563336	+	0.826228i	$0.690482\pi$
$758$	0	0
$759$	10.5707	0.383693
$760$	0	0
$761$	1.96398	0.0711941	0.0355971	−	0.999366i	$-0.488667\pi$
$761$	1.96398	0.0711941	0.0355971	+	0.999366i	$0.488667\pi$
$762$	0	0
$763$	4.82006	0.174498
$764$	0	0
$765$	−40.9591	−1.48088
$766$	0	0
$767$	31.4857	1.13688
$768$	0	0
$769$	−22.3322	−0.805318	−0.402659	−	0.915350i	$-0.631914\pi$
$769$	−22.3322	−0.805318	−0.402659	+	0.915350i	$0.631914\pi$
$770$	0	0
$771$	25.3004	0.911172
$772$	0	0
$773$	27.3553	0.983901	0.491951	−	0.870623i	$-0.336284\pi$
$773$	27.3553	0.983901	0.491951	+	0.870623i	$0.336284\pi$
$774$	0	0
$775$	−7.64738	−0.274702
$776$	0	0
$777$	3.87901	0.139159
$778$	0	0
$779$	0	0
$780$	0	0
$781$	11.7447	0.420257
$782$	0	0
$783$	5.24920	0.187591
$784$	0	0
$785$	−7.31469	−0.261072
$786$	0	0
$787$	−7.78962	−0.277670	−0.138835	−	0.990316i	$-0.544336\pi$
$787$	−7.78962	−0.277670	−0.138835	+	0.990316i	$0.544336\pi$
$788$	0	0
$789$	47.4447	1.68908
$790$	0	0
$791$	−1.19506	−0.0424915
$792$	0	0
$793$	59.2384	2.10362
$794$	0	0
$795$	9.11374	0.323231
$796$	0	0
$797$	−6.09655	−0.215951	−0.107975	−	0.994154i	$-0.534437\pi$
$797$	−6.09655	−0.215951	−0.107975	+	0.994154i	$0.534437\pi$
$798$	0	0
$799$	−3.70498	−0.131073
$800$	0	0
$801$	23.1397	0.817601
$802$	0	0
$803$	−10.3837	−0.366433
$804$	0	0
$805$	−17.1707	−0.605187
$806$	0	0
$807$	−41.2868	−1.45336
$808$	0	0
$809$	56.8455	1.99858	0.999291	−	0.0376413i	$-0.0119844\pi$
$809$	56.8455	1.99858	0.999291	+	0.0376413i	$0.0119844\pi$
$810$	0	0
$811$	20.9209	0.734631	0.367316	−	0.930096i	$-0.380277\pi$
$811$	20.9209	0.734631	0.367316	+	0.930096i	$0.380277\pi$
$812$	0	0
$813$	6.97283	0.244548
$814$	0	0
$815$	13.7541	0.481786
$816$	0	0
$817$	0	0
$818$	0	0
$819$	59.3964	2.07548
$820$	0	0
$821$	16.2582	0.567416	0.283708	−	0.958911i	$-0.408435\pi$
$821$	16.2582	0.567416	0.283708	+	0.958911i	$0.408435\pi$
$822$	0	0
$823$	17.5328	0.611157	0.305578	−	0.952167i	$-0.401150\pi$
$823$	17.5328	0.611157	0.305578	+	0.952167i	$0.401150\pi$
$824$	0	0
$825$	4.44113	0.154620
$826$	0	0
$827$	17.0254	0.592030	0.296015	−	0.955183i	$-0.404342\pi$
$827$	17.0254	0.592030	0.296015	+	0.955183i	$0.404342\pi$
$828$	0	0
$829$	−5.76207	−0.200125	−0.100062	−	0.994981i	$-0.531904\pi$
$829$	−5.76207	−0.200125	−0.100062	+	0.994981i	$0.531904\pi$
$830$	0	0
$831$	61.8246	2.14467
$832$	0	0
$833$	3.99634	0.138465
$834$	0	0
$835$	−17.4232	−0.602954
$836$	0	0
$837$	−6.44588	−0.222802
$838$	0	0
$839$	−8.89954	−0.307246	−0.153623	−	0.988130i	$-0.549094\pi$
$839$	−8.89954	−0.307246	−0.153623	+	0.988130i	$0.549094\pi$
$840$	0	0
$841$	6.18608	0.213313
$842$	0	0
$843$	25.6308	0.882772
$844$	0	0
$845$	−67.0344	−2.30605
$846$	0	0
$847$	−22.8588	−0.785436
$848$	0	0
$849$	−69.7269	−2.39302
$850$	0	0
$851$	1.37683	0.0471970
$852$	0	0
$853$	2.78915	0.0954987	0.0477493	−	0.998859i	$-0.484795\pi$
$853$	2.78915	0.0954987	0.0477493	+	0.998859i	$0.484795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.3945	0.560025	0.280012	−	0.959996i	$-0.409661\pi$
$857$	16.3945	0.560025	0.280012	+	0.959996i	$0.409661\pi$
$858$	0	0
$859$	−9.52505	−0.324990	−0.162495	−	0.986709i	$-0.551954\pi$
$859$	−9.52505	−0.324990	−0.162495	+	0.986709i	$0.551954\pi$
$860$	0	0
$861$	−18.3570	−0.625603
$862$	0	0
$863$	50.6764	1.72505	0.862523	−	0.506018i	$-0.168883\pi$
$863$	50.6764	1.72505	0.862523	+	0.506018i	$0.168883\pi$
$864$	0	0
$865$	56.6231	1.92524
$866$	0	0
$867$	19.3867	0.658408
$868$	0	0
$869$	9.92278	0.336607
$870$	0	0
$871$	−73.3461	−2.48524
$872$	0	0
$873$	48.2141	1.63180
$874$	0	0
$875$	27.1424	0.917582
$876$	0	0
$877$	19.2578	0.650291	0.325146	−	0.945664i	$-0.394587\pi$
$877$	19.2578	0.650291	0.325146	+	0.945664i	$0.394587\pi$
$878$	0	0
$879$	60.0768	2.02634
$880$	0	0
$881$	19.1076	0.643750	0.321875	−	0.946782i	$-0.395687\pi$
$881$	19.1076	0.643750	0.321875	+	0.946782i	$0.395687\pi$
$882$	0	0
$883$	−49.9980	−1.68257	−0.841283	−	0.540595i	$-0.818199\pi$
$883$	−49.9980	−1.68257	−0.841283	+	0.540595i	$0.818199\pi$
$884$	0	0
$885$	−30.7616	−1.03404
$886$	0	0
$887$	−39.6195	−1.33029	−0.665146	−	0.746713i	$-0.731631\pi$
$887$	−39.6195	−1.33029	−0.665146	+	0.746713i	$0.731631\pi$
$888$	0	0
$889$	3.17442	0.106467
$890$	0	0
$891$	−13.1316	−0.439926
$892$	0	0
$893$	0	0
$894$	0	0
$895$	20.2053	0.675389
$896$	0	0
$897$	39.9556	1.33408
$898$	0	0
$899$	−43.2076	−1.44105
$900$	0	0
$901$	−7.30603	−0.243399
$902$	0	0
$903$	−20.2147	−0.672703
$904$	0	0
$905$	31.9752	1.06289
$906$	0	0
$907$	−55.2545	−1.83470	−0.917348	−	0.398087i	$-0.869674\pi$
$907$	−55.2545	−1.83470	−0.917348	+	0.398087i	$0.869674\pi$
$908$	0	0
$909$	37.8107	1.25410
$910$	0	0
$911$	−27.7087	−0.918031	−0.459015	−	0.888428i	$-0.651798\pi$
$911$	−27.7087	−0.918031	−0.459015	+	0.888428i	$0.651798\pi$
$912$	0	0
$913$	−25.4756	−0.843120
$914$	0	0
$915$	−57.8762	−1.91333
$916$	0	0
$917$	49.8421	1.64593
$918$	0	0
$919$	−27.2500	−0.898894	−0.449447	−	0.893307i	$-0.648379\pi$
$919$	−27.2500	−0.898894	−0.449447	+	0.893307i	$0.648379\pi$
$920$	0	0
$921$	−65.4947	−2.15812
$922$	0	0
$923$	44.3929	1.46121
$924$	0	0
$925$	0.578452	0.0190194
$926$	0	0
$927$	−49.6906	−1.63205
$928$	0	0
$929$	44.4410	1.45806	0.729031	−	0.684481i	$-0.239971\pi$
$929$	44.4410	1.45806	0.729031	+	0.684481i	$0.239971\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	58.2263	1.90624
$934$	0	0
$935$	−20.5157	−0.670936
$936$	0	0
$937$	−18.2097	−0.594885	−0.297442	−	0.954740i	$-0.596134\pi$
$937$	−18.2097	−0.594885	−0.297442	+	0.954740i	$0.596134\pi$
$938$	0	0
$939$	30.9088	1.00867
$940$	0	0
$941$	−40.2680	−1.31270	−0.656350	−	0.754457i	$-0.727901\pi$
$941$	−40.2680	−1.31270	−0.656350	+	0.754457i	$0.727901\pi$
$942$	0	0
$943$	−6.51567	−0.212180
$944$	0	0
$945$	−6.08059	−0.197802
$946$	0	0
$947$	−6.36561	−0.206854	−0.103427	−	0.994637i	$-0.532981\pi$
$947$	−6.36561	−0.206854	−0.103427	+	0.994637i	$0.532981\pi$
$948$	0	0
$949$	−39.2488	−1.27407
$950$	0	0
$951$	−0.686248	−0.0222531
$952$	0	0
$953$	−32.6944	−1.05907	−0.529537	−	0.848287i	$-0.677634\pi$
$953$	−32.6944	−1.05907	−0.529537	+	0.848287i	$0.677634\pi$
$954$	0	0
$955$	16.1717	0.523303
$956$	0	0
$957$	25.0923	0.811119
$958$	0	0
$959$	−44.4620	−1.43575
$960$	0	0
$961$	22.0578	0.711542
$962$	0	0
$963$	9.47866	0.305445
$964$	0	0
$965$	−66.1218	−2.12854
$966$	0	0
$967$	−8.55936	−0.275250	−0.137625	−	0.990484i	$-0.543947\pi$
$967$	−8.55936	−0.275250	−0.137625	+	0.990484i	$0.543947\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.8789	1.34396	0.671978	−	0.740571i	$-0.265445\pi$
$971$	41.8789	1.34396	0.671978	+	0.740571i	$0.265445\pi$
$972$	0	0
$973$	−28.2718	−0.906351
$974$	0	0
$975$	16.7867	0.537606
$976$	0	0
$977$	−6.18021	−0.197722	−0.0988612	−	0.995101i	$-0.531520\pi$
$977$	−6.18021	−0.197722	−0.0988612	+	0.995101i	$0.531520\pi$
$978$	0	0
$979$	11.5903	0.370427
$980$	0	0
$981$	5.78203	0.184606
$982$	0	0
$983$	−33.5742	−1.07085	−0.535425	−	0.844583i	$-0.679848\pi$
$983$	−33.5742	−1.07085	−0.535425	+	0.844583i	$0.679848\pi$
$984$	0	0
$985$	24.2261	0.771907
$986$	0	0
$987$	−5.24920	−0.167084
$988$	0	0
$989$	−7.17507	−0.228154
$990$	0	0
$991$	46.1746	1.46679	0.733393	−	0.679805i	$-0.237936\pi$
$991$	46.1746	1.46679	0.733393	+	0.679805i	$0.237936\pi$
$992$	0	0
$993$	50.3853	1.59893
$994$	0	0
$995$	−50.3394	−1.59587
$996$	0	0
$997$	−0.819130	−0.0259421	−0.0129711	−	0.999916i	$-0.504129\pi$
$997$	−0.819130	−0.0259421	−0.0129711	+	0.999916i	$0.504129\pi$
$998$	0	0
$999$	0.487570	0.0154260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.bt.1.4		4
4.3	odd	2		722.2.a.n.1.1	yes	4
12.11	even	2		6498.2.a.bx.1.4		4
19.18	odd	2		5776.2.a.bv.1.1		4
76.3	even	18		722.2.e.s.389.4		24
76.7	odd	6		722.2.c.m.429.4		8
76.11	odd	6		722.2.c.m.653.4		8
76.15	even	18		722.2.e.s.415.1		24
76.23	odd	18		722.2.e.r.415.4		24
76.27	even	6		722.2.c.n.653.1		8
76.31	even	6		722.2.c.n.429.1		8
76.35	odd	18		722.2.e.r.389.1		24
76.43	odd	18		722.2.e.r.595.4		24
76.47	odd	18		722.2.e.r.423.1		24
76.51	even	18		722.2.e.s.245.4		24
76.55	odd	18		722.2.e.r.99.1		24
76.59	even	18		722.2.e.s.99.4		24
76.63	odd	18		722.2.e.r.245.1		24
76.67	even	18		722.2.e.s.423.4		24
76.71	even	18		722.2.e.s.595.1		24
76.75	even	2		722.2.a.m.1.4	✓	4
228.227	odd	2		6498.2.a.ca.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.m.1.4	✓	4	76.75	even	2
722.2.a.n.1.1	yes	4	4.3	odd	2
722.2.c.m.429.4		8	76.7	odd	6
722.2.c.m.653.4		8	76.11	odd	6
722.2.c.n.429.1		8	76.31	even	6
722.2.c.n.653.1		8	76.27	even	6
722.2.e.r.99.1		24	76.55	odd	18
722.2.e.r.245.1		24	76.63	odd	18
722.2.e.r.389.1		24	76.35	odd	18
722.2.e.r.415.4		24	76.23	odd	18
722.2.e.r.423.1		24	76.47	odd	18
722.2.e.r.595.4		24	76.43	odd	18
722.2.e.s.99.4		24	76.59	even	18
722.2.e.s.245.4		24	76.51	even	18
722.2.e.s.389.4		24	76.3	even	18
722.2.e.s.415.1		24	76.15	even	18
722.2.e.s.423.4		24	76.67	even	18
722.2.e.s.595.1		24	76.71	even	18
5776.2.a.bt.1.4		4	1.1	even	1	trivial
5776.2.a.bv.1.1		4	19.18	odd	2
6498.2.a.bx.1.4		4	12.11	even	2
6498.2.a.ca.1.4		4	228.227	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.52015 q^{3} -2.45965 q^{5} +2.79360 q^{7} +3.35114 q^{9} +O(q^{10})\)	\(q+2.52015 q^{3} -2.45965 q^{5} +2.79360 q^{7} +3.35114 q^{9} +1.67853 q^{11} +6.34458 q^{13} -6.19868 q^{15} +4.96917 q^{17} +7.04029 q^{21} +2.49890 q^{23} +1.04988 q^{25} +0.884927 q^{27} +5.93179 q^{29} -7.28408 q^{31} +4.23015 q^{33} -6.87129 q^{35} +0.550972 q^{37} +15.9893 q^{39} -2.60741 q^{41} -2.87129 q^{43} -8.24263 q^{45} -0.745593 q^{47} +0.804226 q^{49} +12.5231 q^{51} -1.47027 q^{53} -4.12860 q^{55} +4.96261 q^{59} +9.33686 q^{61} +9.36176 q^{63} -15.6054 q^{65} -11.5604 q^{67} +6.29761 q^{69} +6.99698 q^{71} -6.18619 q^{73} +2.64584 q^{75} +4.68915 q^{77} +5.91158 q^{79} -7.82328 q^{81} -15.1773 q^{83} -12.2224 q^{85} +14.9490 q^{87} +6.90502 q^{89} +17.7242 q^{91} -18.3570 q^{93} +14.3874 q^{97} +5.62500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9	\( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 - 2 * q^11 + 18 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^21 + 10 * q^23 + 6 * q^25 + 4 * q^27 - 2 * q^29 - 26 * q^31 + 16 * q^33 - 6 * q^35 + 4 * q^37 + 6 * q^39 - 12 * q^41 + 10 * q^43 - 22 * q^45 + 12 * q^47 - 12 * q^49 + 2 * q^51 + 8 * q^53 + 26 * q^55 + 8 * q^59 + 22 * q^63 - 4 * q^65 - 10 * q^67 - 20 * q^69 - 14 * q^73 - 8 * q^75 + 4 * q^77 - 22 * q^79 - 4 * q^81 + 12 * q^83 - 18 * q^85 + 26 * q^87 - 16 * q^89 + 4 * q^91 + 8 * q^93 + 28 * q^97 - 22 * q^99

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