LMFDB - Embedded newform 6498.2.a.bx.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
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	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} -1.00000 q^{8} -2.45965 q^{10} +1.67853 q^{11} +6.34458 q^{13} +2.79360 q^{14} +1.00000 q^{16} -4.96917 q^{17} +2.45965 q^{20} -1.67853 q^{22} +2.49890 q^{23} +1.04988 q^{25} -6.34458 q^{26} -2.79360 q^{28} -5.93179 q^{29} +7.28408 q^{31} -1.00000 q^{32} +4.96917 q^{34} -6.87129 q^{35} +0.550972 q^{37} -2.45965 q^{40} +2.60741 q^{41} +2.87129 q^{43} +1.67853 q^{44} -2.49890 q^{46} -0.745593 q^{47} +0.804226 q^{49} -1.04988 q^{50} +6.34458 q^{52} +1.47027 q^{53} +4.12860 q^{55} +2.79360 q^{56} +5.93179 q^{58} +4.96261 q^{59} +9.33686 q^{61} -7.28408 q^{62} +1.00000 q^{64} +15.6054 q^{65} +11.5604 q^{67} -4.96917 q^{68} +6.87129 q^{70} +6.99698 q^{71} -6.18619 q^{73} -0.550972 q^{74} -4.68915 q^{77} -5.91158 q^{79} +2.45965 q^{80} -2.60741 q^{82} -15.1773 q^{83} -12.2224 q^{85} -2.87129 q^{86} -1.67853 q^{88} -6.90502 q^{89} -17.7242 q^{91} +2.49890 q^{92} +0.745593 q^{94} +14.3874 q^{97} -0.804226 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8} - 2 q^{10} - 2 q^{11} + 18 q^{13} + 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{20} + 2 q^{22} + 10 q^{23} + 6 q^{25} - 18 q^{26} - 2 q^{28} + 2 q^{29} + 26 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{37} - 2 q^{40} + 12 q^{41} - 10 q^{43} - 2 q^{44} - 10 q^{46} + 12 q^{47} - 12 q^{49} - 6 q^{50} + 18 q^{52} - 8 q^{53} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 26 q^{62} + 4 q^{64} + 4 q^{65} + 10 q^{67} - 6 q^{68} + 6 q^{70} - 14 q^{73} - 4 q^{74} - 4 q^{77} + 22 q^{79} + 2 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{85} + 10 q^{86} + 2 q^{88} + 16 q^{89} - 4 q^{91} + 10 q^{92} - 12 q^{94} + 28 q^{97} + 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 - 4 * q^8 - 2 * q^10 - 2 * q^11 + 18 * q^13 + 2 * q^14 + 4 * q^16 - 6 * q^17 + 2 * q^20 + 2 * q^22 + 10 * q^23 + 6 * q^25 - 18 * q^26 - 2 * q^28 + 2 * q^29 + 26 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^37 - 2 * q^40 + 12 * q^41 - 10 * q^43 - 2 * q^44 - 10 * q^46 + 12 * q^47 - 12 * q^49 - 6 * q^50 + 18 * q^52 - 8 * q^53 - 26 * q^55 + 2 * q^56 - 2 * q^58 + 8 * q^59 - 26 * q^62 + 4 * q^64 + 4 * q^65 + 10 * q^67 - 6 * q^68 + 6 * q^70 - 14 * q^73 - 4 * q^74 - 4 * q^77 + 22 * q^79 + 2 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^85 + 10 * q^86 + 2 * q^88 + 16 * q^89 - 4 * q^91 + 10 * q^92 - 12 * q^94 + 28 * q^97 + 12 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.45965	1.09999	0.549994	−	0.835168i	$-0.314630\pi$
$5$	2.45965	1.09999	0.549994	+	0.835168i	$0.314630\pi$
$6$	0	0
$7$	−2.79360	−1.05588	−0.527942	−	0.849281i	$-0.677036\pi$
$7$	−2.79360	−1.05588	−0.527942	+	0.849281i	$0.677036\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.45965	−0.777809
$11$	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	$0.157653\pi$
$14$	2.79360	0.746622
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.96917	−1.20520	−0.602601	−	0.798043i	$-0.705869\pi$
$17$	−4.96917	−1.20520	−0.602601	+	0.798043i	$0.705869\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	2.45965	0.549994
$21$	0	0
$22$	−1.67853	−0.357864
$23$	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
$26$	−6.34458	−1.24427
$27$	0	0
$28$	−2.79360	−0.527942
$29$	−5.93179	−1.10150	−0.550752	−	0.834669i	$-0.685659\pi$
$29$	−5.93179	−1.10150	−0.550752	+	0.834669i	$0.685659\pi$
$30$	0	0
$31$	7.28408	1.30826	0.654130	−	0.756382i	$-0.273035\pi$
$31$	7.28408	1.30826	0.654130	+	0.756382i	$0.273035\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.96917	0.852206
$35$	−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
$39$	0	0
$40$	−2.45965	−0.388905
$41$	2.60741	0.407209	0.203605	−	0.979053i	$-0.434734\pi$
$41$	2.60741	0.407209	0.203605	+	0.979053i	$0.434734\pi$
$42$	0	0
$43$	2.87129	0.437867	0.218934	−	0.975740i	$-0.429742\pi$
$43$	2.87129	0.437867	0.218934	+	0.975740i	$0.429742\pi$
$44$	1.67853	0.253048
$45$	0	0
$46$	−2.49890	−0.368443
$47$	−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
$50$	−1.04988	−0.148475
$51$	0	0
$52$	6.34458	0.879834
$53$	1.47027	0.201957	0.100979	−	0.994889i	$-0.467803\pi$
$53$	1.47027	0.201957	0.100979	+	0.994889i	$0.467803\pi$
$54$	0	0
$55$	4.12860	0.556700
$56$	2.79360	0.373311
$57$	0	0
$58$	5.93179	0.778882
$59$	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	$0.296069\pi$
$62$	−7.28408	−0.925079
$63$	0	0
$64$	1.00000	0.125000
$65$	15.6054	1.93562
$66$	0	0
$67$	11.5604	1.41233	0.706166	−	0.708046i	$-0.250423\pi$
$67$	11.5604	1.41233	0.706166	+	0.708046i	$0.250423\pi$
$68$	−4.96917	−0.602601
$69$	0	0
$70$	6.87129	0.821276
$71$	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	$0.617913\pi$
$74$	−0.550972	−0.0640492
$75$	0	0
$76$	0	0
$77$	−4.68915	−0.534379
$78$	0	0
$79$	−5.91158	−0.665105	−0.332552	−	0.943085i	$-0.607910\pi$
$79$	−5.91158	−0.665105	−0.332552	+	0.943085i	$0.607910\pi$
$80$	2.45965	0.274997
$81$	0	0
$82$	−2.60741	−0.287941
$83$	−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
$86$	−2.87129	−0.309619
$87$	0	0
$88$	−1.67853	−0.178932
$89$	−6.90502	−0.731930	−0.365965	−	0.930629i	$-0.619261\pi$
$89$	−6.90502	−0.731930	−0.365965	+	0.930629i	$0.619261\pi$
$90$	0	0
$91$	−17.7242	−1.85800
$92$	2.49890	0.260529
$93$	0	0
$94$	0.745593	0.0769021
$95$	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	$0.239330\pi$
$98$	−0.804226	−0.0812391
$99$	0	0
$100$	1.04988	0.104988
$101$	−11.2829	−1.12269	−0.561347	−	0.827581i	$-0.689717\pi$
$101$	−11.2829	−1.12269	−0.561347	+	0.827581i	$0.689717\pi$
$102$	0	0
$103$	14.8280	1.46104	0.730522	−	0.682889i	$-0.239277\pi$
$103$	14.8280	1.46104	0.730522	+	0.682889i	$0.239277\pi$
$104$	−6.34458	−0.622137
$105$	0	0
$106$	−1.47027	−0.142805
$107$	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	$0.473668\pi$
$110$	−4.12860	−0.393646
$111$	0	0
$112$	−2.79360	−0.263971
$113$	0.427785	0.0402426	0.0201213	−	0.999798i	$-0.493595\pi$
$113$	0.427785	0.0402426	0.0201213	+	0.999798i	$0.493595\pi$
$114$	0	0
$115$	6.14643	0.573157
$116$	−5.93179	−0.550752
$117$	0	0
$118$	−4.96261	−0.456845
$119$	13.8819	1.27255
$120$	0	0
$121$	−8.18253	−0.743867
$122$	−9.33686	−0.845320
$123$	0	0
$124$	7.28408	0.654130
$125$	−9.71592	−0.869018
$126$	0	0
$127$	−1.13632	−0.100832	−0.0504159	−	0.998728i	$-0.516055\pi$
$127$	−1.13632	−0.100832	−0.0504159	+	0.998728i	$0.516055\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−15.6054	−1.36869
$131$	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
$134$	−11.5604	−0.998670
$135$	0	0
$136$	4.96917	0.426103
$137$	15.9156	1.35976	0.679882	−	0.733321i	$-0.262031\pi$
$137$	15.9156	1.35976	0.679882	+	0.733321i	$0.262031\pi$
$138$	0	0
$139$	10.1202	0.858382	0.429191	−	0.903214i	$-0.358799\pi$
$139$	10.1202	0.858382	0.429191	+	0.903214i	$0.358799\pi$
$140$	−6.87129	−0.580730
$141$	0	0
$142$	−6.99698	−0.587174
$143$	10.6496	0.890562
$144$	0	0
$145$	−14.5901	−1.21164
$146$	6.18619	0.511973
$147$	0	0
$148$	0.550972	0.0452896
$149$	−9.39851	−0.769956	−0.384978	−	0.922926i	$-0.625791\pi$
$149$	−9.39851	−0.769956	−0.384978	+	0.922926i	$0.625791\pi$
$150$	0	0
$151$	−10.2632	−0.835210	−0.417605	−	0.908629i	$-0.637130\pi$
$151$	−10.2632	−0.835210	−0.417605	+	0.908629i	$0.637130\pi$
$152$	0	0
$153$	0	0
$154$	4.68915	0.377863
$155$	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	$0.462137\pi$
$158$	5.91158	0.470300
$159$	0	0
$160$	−2.45965	−0.194452
$161$	−6.98095	−0.550176
$162$	0	0
$163$	5.59191	0.437992	0.218996	−	0.975726i	$-0.429722\pi$
$163$	5.59191	0.437992	0.218996	+	0.975726i	$0.429722\pi$
$164$	2.60741	0.203605
$165$	0	0
$166$	15.1773	1.17799
$167$	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
$170$	12.2224	0.937418
$171$	0	0
$172$	2.87129	0.218934
$173$	23.0208	1.75024	0.875120	−	0.483907i	$-0.160783\pi$
$173$	23.0208	1.75024	0.875120	+	0.483907i	$0.160783\pi$
$174$	0	0
$175$	−2.93294	−0.221709
$176$	1.67853	0.126524
$177$	0	0
$178$	6.90502	0.517553
$179$	−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	$0.660503\pi$
$182$	17.7242	1.31381
$183$	0	0
$184$	−2.49890	−0.184222
$185$	1.35520	0.0996361
$186$	0	0
$187$	−8.34092	−0.609948
$188$	−0.745593	−0.0543780
$189$	0	0
$190$	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	$0.0813412\pi$
$194$	−14.3874	−1.03295
$195$	0	0
$196$	0.804226	0.0574447
$197$	9.84940	0.701741	0.350870	−	0.936424i	$-0.385886\pi$
$197$	9.84940	0.701741	0.350870	+	0.936424i	$0.385886\pi$
$198$	0	0
$199$	−20.4661	−1.45080	−0.725402	−	0.688326i	$-0.758346\pi$
$199$	−20.4661	−1.45080	−0.725402	+	0.688326i	$0.758346\pi$
$200$	−1.04988	−0.0742374
$201$	0	0
$202$	11.2829	0.793864
$203$	16.5711	1.16306
$204$	0	0
$205$	6.41332	0.447926
$206$	−14.8280	−1.03311
$207$	0	0
$208$	6.34458	0.439917
$209$	0	0
$210$	0	0
$211$	8.11630	0.558749	0.279374	−	0.960182i	$-0.409873\pi$
$211$	8.11630	0.558749	0.279374	+	0.960182i	$0.409873\pi$
$212$	1.47027	0.100979
$213$	0	0
$214$	−2.82849	−0.193351
$215$	7.06236	0.481649
$216$	0	0
$217$	−20.3488	−1.38137
$218$	−1.72539	−0.116858
$219$	0	0
$220$	4.12860	0.278350
$221$	−31.5273	−2.12076
$222$	0	0
$223$	10.7540	0.720143	0.360071	−	0.932925i	$-0.382752\pi$
$223$	10.7540	0.720143	0.360071	+	0.932925i	$0.382752\pi$
$224$	2.79360	0.186656
$225$	0	0
$226$	−0.427785	−0.0284558
$227$	−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	$0.398749\pi$
$230$	−6.14643	−0.405283
$231$	0	0
$232$	5.93179	0.389441
$233$	−22.8621	−1.49775	−0.748873	−	0.662713i	$-0.769405\pi$
$233$	−22.8621	−1.49775	−0.748873	+	0.662713i	$0.769405\pi$
$234$	0	0
$235$	−1.83390	−0.119630
$236$	4.96261	0.323038
$237$	0	0
$238$	−13.8819	−0.899831
$239$	−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	$0.441665\pi$
$242$	8.18253	0.525993
$243$	0	0
$244$	9.33686	0.597731
$245$	1.97811	0.126377
$246$	0	0
$247$	0	0
$248$	−7.28408	−0.462539
$249$	0	0
$250$	9.71592	0.614489
$251$	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
$254$	1.13632	0.0712988
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0393	−0.626231	−0.313116	−	0.949715i	$-0.601373\pi$
$257$	−10.0393	−0.626231	−0.313116	+	0.949715i	$0.601373\pi$
$258$	0	0
$259$	−1.53920	−0.0956411
$260$	15.6054	0.967808
$261$	0	0
$262$	−17.8415	−1.10225
$263$	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
$267$	0	0
$268$	11.5604	0.706166
$269$	16.3827	0.998870	0.499435	−	0.866351i	$-0.333541\pi$
$269$	16.3827	0.998870	0.499435	+	0.866351i	$0.333541\pi$
$270$	0	0
$271$	−2.76684	−0.168073	−0.0840367	−	0.996463i	$-0.526781\pi$
$271$	−2.76684	−0.168073	−0.0840367	+	0.996463i	$0.526781\pi$
$272$	−4.96917	−0.301300
$273$	0	0
$274$	−15.9156	−0.961499
$275$	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	$0.236244\pi$
$278$	−10.1202	−0.606967
$279$	0	0
$280$	6.87129	0.410638
$281$	−10.1704	−0.606713	−0.303356	−	0.952877i	$-0.598107\pi$
$281$	−10.1704	−0.606713	−0.303356	+	0.952877i	$0.598107\pi$
$282$	0	0
$283$	27.6678	1.64468	0.822340	−	0.568997i	$-0.192668\pi$
$283$	27.6678	1.64468	0.822340	+	0.568997i	$0.192668\pi$
$284$	6.99698	0.415195
$285$	0	0
$286$	−10.6496	−0.629722
$287$	−7.28408	−0.429966
$288$	0	0
$289$	7.69270	0.452512
$290$	14.5901	0.856761
$291$	0	0
$292$	−6.18619	−0.362020
$293$	−23.8386	−1.39267	−0.696333	−	0.717719i	$-0.745186\pi$
$293$	−23.8386	−1.39267	−0.696333	+	0.717719i	$0.745186\pi$
$294$	0	0
$295$	12.2063	0.710677
$296$	−0.550972	−0.0320246
$297$	0	0
$298$	9.39851	0.544441
$299$	15.8545	0.916889
$300$	0	0
$301$	−8.02124	−0.462337
$302$	10.2632	0.590583
$303$	0	0
$304$	0	0
$305$	22.9654	1.31500
$306$	0	0
$307$	25.9884	1.48324	0.741619	−	0.670821i	$-0.234058\pi$
$307$	25.9884	1.48324	0.741619	+	0.670821i	$0.234058\pi$
$308$	−4.68915	−0.267189
$309$	0	0
$310$	−17.9163	−1.01758
$311$	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	$0.387329\pi$
$314$	−2.97387	−0.167825
$315$	0	0
$316$	−5.91158	−0.332552
$317$	0.272305	0.0152942	0.00764708	−	0.999971i	$-0.497566\pi$
$317$	0.272305	0.0152942	0.00764708	+	0.999971i	$0.497566\pi$
$318$	0	0
$319$	−9.95669	−0.557468
$320$	2.45965	0.137499
$321$	0	0
$322$	6.98095	0.389033
$323$	0	0
$324$	0	0
$325$	6.66102	0.369487
$326$	−5.59191	−0.309707
$327$	0	0
$328$	−2.60741	−0.143970
$329$	2.08289	0.114834
$330$	0	0
$331$	−19.9930	−1.09891	−0.549457	−	0.835522i	$-0.685165\pi$
$331$	−19.9930	−1.09891	−0.549457	+	0.835522i	$0.685165\pi$
$332$	−15.1773	−0.832964
$333$	0	0
$334$	−7.08361	−0.387598
$335$	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	$0.656005\pi$
$338$	−27.2537	−1.48240
$339$	0	0
$340$	−12.2224	−0.662854
$341$	12.2266	0.662105
$342$	0	0
$343$	17.3085	0.934573
$344$	−2.87129	−0.154809
$345$	0	0
$346$	−23.0208	−1.23761
$347$	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	$0.681019\pi$
$350$	2.93294	0.156772
$351$	0	0
$352$	−1.67853	−0.0894660
$353$	−24.0557	−1.28035	−0.640177	−	0.768227i	$-0.721139\pi$
$353$	−24.0557	−1.28035	−0.640177	+	0.768227i	$0.721139\pi$
$354$	0	0
$355$	17.2101	0.913419
$356$	−6.90502	−0.365965
$357$	0	0
$358$	8.21471	0.434161
$359$	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
$362$	12.9999	0.683259
$363$	0	0
$364$	−17.7242	−0.929002
$365$	−15.2159	−0.796435
$366$	0	0
$367$	−32.6121	−1.70234	−0.851168	−	0.524893i	$-0.824105\pi$
$367$	−32.6121	−1.70234	−0.851168	+	0.524893i	$0.824105\pi$
$368$	2.49890	0.130264
$369$	0	0
$370$	−1.35520	−0.0704534
$371$	−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	$0.543831\pi$
$374$	8.34092	0.431299
$375$	0	0
$376$	0.745593	0.0384510
$377$	−37.6347	−1.93828
$378$	0	0
$379$	24.6656	1.26699	0.633494	−	0.773748i	$-0.281620\pi$
$379$	24.6656	1.26699	0.633494	+	0.773748i	$0.281620\pi$
$380$	0	0
$381$	0	0
$382$	6.57479	0.336396
$383$	−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
$386$	−26.8826	−1.36829
$387$	0	0
$388$	14.3874	0.730408
$389$	−2.79353	−0.141638	−0.0708189	−	0.997489i	$-0.522561\pi$
$389$	−2.79353	−0.141638	−0.0708189	+	0.997489i	$0.522561\pi$
$390$	0	0
$391$	−12.4175	−0.627979
$392$	−0.804226	−0.0406196
$393$	0	0
$394$	−9.84940	−0.496206
$395$	−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	$0.958415\pi$
$398$	20.4661	1.02587
$399$	0	0
$400$	1.04988	0.0524938
$401$	18.6215	0.929912	0.464956	−	0.885334i	$-0.346070\pi$
$401$	18.6215	0.929912	0.464956	+	0.885334i	$0.346070\pi$
$402$	0	0
$403$	46.2144	2.30210
$404$	−11.2829	−0.561347
$405$	0	0
$406$	−16.5711	−0.822408
$407$	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	$0.639318\pi$
$410$	−6.41332	−0.316731
$411$	0	0
$412$	14.8280	0.730522
$413$	−13.8636	−0.682182
$414$	0	0
$415$	−37.3309	−1.83250
$416$	−6.34458	−0.311068
$417$	0	0
$418$	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	$0.629172\pi$
$422$	−8.11630	−0.395095
$423$	0	0
$424$	−1.47027	−0.0714027
$425$	−5.21702	−0.253062
$426$	0	0
$427$	−26.0835	−1.26227
$428$	2.82849	0.136720
$429$	0	0
$430$	−7.06236	−0.340577
$431$	−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	$0.388880\pi$
$434$	20.3488	0.976775
$435$	0	0
$436$	1.72539	0.0826312
$437$	0	0
$438$	0	0
$439$	28.1900	1.34544	0.672718	−	0.739899i	$-0.265127\pi$
$439$	28.1900	1.34544	0.672718	+	0.739899i	$0.265127\pi$
$440$	−4.12860	−0.196823
$441$	0	0
$442$	31.5273	1.49960
$443$	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
$446$	−10.7540	−0.509218
$447$	0	0
$448$	−2.79360	−0.131985
$449$	17.1975	0.811601	0.405801	−	0.913962i	$-0.366993\pi$
$449$	17.1975	0.811601	0.405801	+	0.913962i	$0.366993\pi$
$450$	0	0
$451$	4.37662	0.206087
$452$	0.427785	0.0201213
$453$	0	0
$454$	27.0936	1.27157
$455$	−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	$0.759832\pi$
$458$	−9.46557	−0.442297
$459$	0	0
$460$	6.14643	0.286579
$461$	−15.5410	−0.723816	−0.361908	−	0.932214i	$-0.617875\pi$
$461$	−15.5410	−0.723816	−0.361908	+	0.932214i	$0.617875\pi$
$462$	0	0
$463$	20.6648	0.960374	0.480187	−	0.877166i	$-0.340569\pi$
$463$	20.6648	0.960374	0.480187	+	0.877166i	$0.340569\pi$
$464$	−5.93179	−0.275376
$465$	0	0
$466$	22.8621	1.05907
$467$	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
$470$	1.83390	0.0845914
$471$	0	0
$472$	−4.96261	−0.228423
$473$	4.81955	0.221603
$474$	0	0
$475$	0	0
$476$	13.8819	0.636276
$477$	0	0
$478$	8.66611	0.396379
$479$	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
$482$	−5.65826	−0.257727
$483$	0	0
$484$	−8.18253	−0.371933
$485$	35.3879	1.60688
$486$	0	0
$487$	2.29894	0.104175	0.0520875	−	0.998643i	$-0.483413\pi$
$487$	2.29894	0.104175	0.0520875	+	0.998643i	$0.483413\pi$
$488$	−9.33686	−0.422660
$489$	0	0
$490$	−1.97811	−0.0893621
$491$	−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
$495$	0	0
$496$	7.28408	0.327065
$497$	−19.5468	−0.876794
$498$	0	0
$499$	15.1348	0.677528	0.338764	−	0.940871i	$-0.389991\pi$
$499$	15.1348	0.677528	0.338764	+	0.940871i	$0.389991\pi$
$500$	−9.71592	−0.434509
$501$	0	0
$502$	−13.5552	−0.605000
$503$	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
$506$	−4.19449	−0.186468
$507$	0	0
$508$	−1.13632	−0.0504159
$509$	26.2163	1.16202	0.581008	−	0.813898i	$-0.302659\pi$
$509$	26.2163	1.16202	0.581008	+	0.813898i	$0.302659\pi$
$510$	0	0
$511$	17.2818	0.764501
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.0393	0.442813
$515$	36.4716	1.60713
$516$	0	0
$517$	−1.25150	−0.0550410
$518$	1.53920	0.0676285
$519$	0	0
$520$	−15.6054	−0.684344
$521$	12.9637	0.567948	0.283974	−	0.958832i	$-0.408347\pi$
$521$	12.9637	0.567948	0.283974	+	0.958832i	$0.408347\pi$
$522$	0	0
$523$	17.7793	0.777432	0.388716	−	0.921358i	$-0.372919\pi$
$523$	17.7793	0.777432	0.388716	+	0.921358i	$0.372919\pi$
$524$	17.8415	0.779410
$525$	0	0
$526$	−18.8262	−0.820861
$527$	−36.1959	−1.57672
$528$	0	0
$529$	−16.7555	−0.728499
$530$	−3.61635	−0.157084
$531$	0	0
$532$	0	0
$533$	16.5429	0.716554
$534$	0	0
$535$	6.95709	0.300781
$536$	−11.5604	−0.499335
$537$	0	0
$538$	−16.3827	−0.706307
$539$	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	$0.536649\pi$
$542$	2.76684	0.118846
$543$	0	0
$544$	4.96917	0.213052
$545$	4.24385	0.181787
$546$	0	0
$547$	5.10112	0.218108	0.109054	−	0.994036i	$-0.465218\pi$
$547$	5.10112	0.218108	0.109054	+	0.994036i	$0.465218\pi$
$548$	15.9156	0.679882
$549$	0	0
$550$	−1.76225	−0.0751426
$551$	0	0
$552$	0	0
$553$	16.5146	0.702273
$554$	−24.5321	−1.04227
$555$	0	0
$556$	10.1202	0.429191
$557$	10.7235	0.454370	0.227185	−	0.973852i	$-0.427048\pi$
$557$	10.7235	0.454370	0.227185	+	0.973852i	$0.427048\pi$
$558$	0	0
$559$	18.2171	0.770502
$560$	−6.87129	−0.290365
$561$	0	0
$562$	10.1704	0.429011
$563$	−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
$566$	−27.6678	−1.16296
$567$	0	0
$568$	−6.99698	−0.293587
$569$	8.84194	0.370674	0.185337	−	0.982675i	$-0.440662\pi$
$569$	8.84194	0.370674	0.185337	+	0.982675i	$0.440662\pi$
$570$	0	0
$571$	22.8411	0.955871	0.477935	−	0.878395i	$-0.341385\pi$
$571$	22.8411	0.955871	0.477935	+	0.878395i	$0.341385\pi$
$572$	10.6496	0.445281
$573$	0	0
$574$	7.28408	0.304032
$575$	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	$0.594247\pi$
$578$	−7.69270	−0.319974
$579$	0	0
$580$	−14.5901	−0.605821
$581$	42.3994	1.75903
$582$	0	0
$583$	2.46790	0.102210
$584$	6.18619	0.255986
$585$	0	0
$586$	23.8386	0.984763
$587$	−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
$590$	−12.2063	−0.502525
$591$	0	0
$592$	0.550972	0.0226448
$593$	34.9193	1.43396	0.716981	−	0.697093i	$-0.245523\pi$
$593$	34.9193	1.43396	0.716981	+	0.697093i	$0.245523\pi$
$594$	0	0
$595$	34.1446	1.39979
$596$	−9.39851	−0.384978
$597$	0	0
$598$	−15.8545	−0.648338
$599$	−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	$0.594489\pi$
$602$	8.02124	0.326921
$603$	0	0
$604$	−10.2632	−0.417605
$605$	−20.1262	−0.818245
$606$	0	0
$607$	−10.3488	−0.420046	−0.210023	−	0.977696i	$-0.567354\pi$
$607$	−10.3488	−0.420046	−0.210023	+	0.977696i	$0.567354\pi$
$608$	0	0
$609$	0	0
$610$	−22.9654	−0.929842
$611$	−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	$0.370689\pi$
$614$	−25.9884	−1.04881
$615$	0	0
$616$	4.68915	0.188931
$617$	5.92522	0.238540	0.119270	−	0.992862i	$-0.461945\pi$
$617$	5.92522	0.238540	0.119270	+	0.992862i	$0.461945\pi$
$618$	0	0
$619$	8.03958	0.323138	0.161569	−	0.986861i	$-0.448345\pi$
$619$	8.03958	0.323138	0.161569	+	0.986861i	$0.448345\pi$
$620$	17.9163	0.719535
$621$	0	0
$622$	−23.1043	−0.926400
$623$	19.2899	0.772833
$624$	0	0
$625$	−29.1471	−1.16589
$626$	−12.2647	−0.490196
$627$	0	0
$628$	2.97387	0.118671
$629$	−2.73788	−0.109166
$630$	0	0
$631$	−12.9006	−0.513567	−0.256783	−	0.966469i	$-0.582663\pi$
$631$	−12.9006	−0.513567	−0.256783	+	0.966469i	$0.582663\pi$
$632$	5.91158	0.235150
$633$	0	0
$634$	−0.272305	−0.0108146
$635$	−2.79494	−0.110914
$636$	0	0
$637$	5.10247	0.202167
$638$	9.95669	0.394189
$639$	0	0
$640$	−2.45965	−0.0972262
$641$	28.8639	1.14006	0.570028	−	0.821625i	$-0.306932\pi$
$641$	28.8639	1.14006	0.570028	+	0.821625i	$0.306932\pi$
$642$	0	0
$643$	26.3394	1.03873	0.519363	−	0.854554i	$-0.326169\pi$
$643$	26.3394	1.03873	0.519363	+	0.854554i	$0.326169\pi$
$644$	−6.98095	−0.275088
$645$	0	0
$646$	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
$650$	−6.66102	−0.261267
$651$	0	0
$652$	5.59191	0.218996
$653$	15.1973	0.594717	0.297359	−	0.954766i	$-0.403894\pi$
$653$	15.1973	0.594717	0.297359	+	0.954766i	$0.403894\pi$
$654$	0	0
$655$	43.8838	1.71468
$656$	2.60741	0.101802
$657$	0	0
$658$	−2.08289	−0.0811996
$659$	−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	$0.236644\pi$
$662$	19.9930	0.777050
$663$	0	0
$664$	15.1773	0.588994
$665$	0	0
$666$	0	0
$667$	−14.8230	−0.573947
$668$	7.08361	0.274073
$669$	0	0
$670$	−28.4346	−1.09853
$671$	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	$0.551044\pi$
$674$	17.2824	0.665694
$675$	0	0
$676$	27.2537	1.04822
$677$	18.3233	0.704223	0.352111	−	0.935958i	$-0.385464\pi$
$677$	18.3233	0.704223	0.352111	+	0.935958i	$0.385464\pi$
$678$	0	0
$679$	−40.1926	−1.54245
$680$	12.2224	0.468709
$681$	0	0
$682$	−12.2266	−0.468179
$683$	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
$686$	−17.3085	−0.660843
$687$	0	0
$688$	2.87129	0.109467
$689$	9.32825	0.355378
$690$	0	0
$691$	16.6779	0.634458	0.317229	−	0.948349i	$-0.397248\pi$
$691$	16.6779	0.634458	0.317229	+	0.948349i	$0.397248\pi$
$692$	23.0208	0.875120
$693$	0	0
$694$	−18.5456	−0.703981
$695$	24.8921	0.944210
$696$	0	0
$697$	−12.9567	−0.490770
$698$	20.1210	0.761592
$699$	0	0
$700$	−2.93294	−0.110855
$701$	30.9066	1.16733	0.583663	−	0.811996i	$-0.301619\pi$
$701$	30.9066	1.16733	0.583663	+	0.811996i	$0.301619\pi$
$702$	0	0
$703$	0	0
$704$	1.67853	0.0632620
$705$	0	0
$706$	24.0557	0.905348
$707$	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	$0.228558\pi$
$710$	−17.2101	−0.645884
$711$	0	0
$712$	6.90502	0.258776
$713$	18.2022	0.681678
$714$	0	0
$715$	26.1942	0.979608
$716$	−8.21471	−0.306998
$717$	0	0
$718$	−10.8535	−0.405048
$719$	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
$723$	0	0
$724$	−12.9999	−0.483137
$725$	−6.22764	−0.231289
$726$	0	0
$727$	−15.1811	−0.563034	−0.281517	−	0.959556i	$-0.590838\pi$
$727$	−15.1811	−0.563034	−0.281517	+	0.959556i	$0.590838\pi$
$728$	17.7242	0.656904
$729$	0	0
$730$	15.2159	0.563164
$731$	−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	$0.666155\pi$
$734$	32.6121	1.20373
$735$	0	0
$736$	−2.49890	−0.0921108
$737$	19.4046	0.714776
$738$	0	0
$739$	−5.20782	−0.191573	−0.0957864	−	0.995402i	$-0.530537\pi$
$739$	−5.20782	−0.191573	−0.0957864	+	0.995402i	$0.530537\pi$
$740$	1.35520	0.0498181
$741$	0	0
$742$	4.10736	0.150786
$743$	−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
$746$	5.30198	0.194119
$747$	0	0
$748$	−8.34092	−0.304974
$749$	−7.90167	−0.288721
$750$	0	0
$751$	−12.4960	−0.455984	−0.227992	−	0.973663i	$-0.573216\pi$
$751$	−12.4960	−0.455984	−0.227992	+	0.973663i	$0.573216\pi$
$752$	−0.745593	−0.0271890
$753$	0	0
$754$	37.6347	1.37057
$755$	−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	$0.690482\pi$
$758$	−24.6656	−0.895895
$759$	0	0
$760$	0	0
$761$	−1.96398	−0.0711941	−0.0355971	−	0.999366i	$-0.511333\pi$
$761$	−1.96398	−0.0711941	−0.0355971	+	0.999366i	$0.511333\pi$
$762$	0	0
$763$	−4.82006	−0.174498
$764$	−6.57479	−0.237868
$765$	0	0
$766$	3.64416	0.131669
$767$	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	$0.631914\pi$
$770$	11.5337	0.415645
$771$	0	0
$772$	26.8826	0.967527
$773$	−27.3553	−0.983901	−0.491951	−	0.870623i	$-0.663716\pi$
$773$	−27.3553	−0.983901	−0.491951	+	0.870623i	$0.663716\pi$
$774$	0	0
$775$	7.64738	0.274702
$776$	−14.3874	−0.516477
$777$	0	0
$778$	2.79353	0.100153
$779$	0	0
$780$	0	0
$781$	11.7447	0.420257
$782$	12.4175	0.444049
$783$	0	0
$784$	0.804226	0.0287224
$785$	7.31469	0.261072
$786$	0	0
$787$	7.78962	0.277670	0.138835	−	0.990316i	$-0.455664\pi$
$787$	7.78962	0.277670	0.138835	+	0.990316i	$0.455664\pi$
$788$	9.84940	0.350870
$789$	0	0
$790$	14.5404	0.517325
$791$	−1.19506	−0.0424915
$792$	0	0
$793$	59.2384	2.10362
$794$	39.5101	1.40216
$795$	0	0
$796$	−20.4661	−0.725402
$797$	6.09655	0.215951	0.107975	−	0.994154i	$-0.465563\pi$
$797$	6.09655	0.215951	0.107975	+	0.994154i	$0.465563\pi$
$798$	0	0
$799$	3.70498	0.131073
$800$	−1.04988	−0.0371187
$801$	0	0
$802$	−18.6215	−0.657547
$803$	−10.3837	−0.366433
$804$	0	0
$805$	−17.1707	−0.605187
$806$	−46.2144	−1.62783
$807$	0	0
$808$	11.2829	0.396932
$809$	−56.8455	−1.99858	−0.999291	−	0.0376413i	$-0.988016\pi$
$809$	−56.8455	−1.99858	−0.999291	+	0.0376413i	$0.988016\pi$
$810$	0	0
$811$	−20.9209	−0.734631	−0.367316	−	0.930096i	$-0.619723\pi$
$811$	−20.9209	−0.734631	−0.367316	+	0.930096i	$0.619723\pi$
$812$	16.5711	0.581530
$813$	0	0
$814$	−0.924824	−0.0324151
$815$	13.7541	0.481786
$816$	0	0
$817$	0	0
$818$	17.1433	0.599401
$819$	0	0
$820$	6.41332	0.223963
$821$	−16.2582	−0.567416	−0.283708	−	0.958911i	$-0.591565\pi$
$821$	−16.2582	−0.567416	−0.283708	+	0.958911i	$0.591565\pi$
$822$	0	0
$823$	−17.5328	−0.611157	−0.305578	−	0.952167i	$-0.598850\pi$
$823$	−17.5328	−0.611157	−0.305578	+	0.952167i	$0.598850\pi$
$824$	−14.8280	−0.516557
$825$	0	0
$826$	13.8636	0.482375
$827$	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	$0.531904\pi$
$830$	37.3309	1.29577
$831$	0	0
$832$	6.34458	0.219959
$833$	−3.99634	−0.138465
$834$	0	0
$835$	17.4232	0.602954
$836$	0	0
$837$	0	0
$838$	−18.5042	−0.639217
$839$	−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
$842$	16.1996	0.558275
$843$	0	0
$844$	8.11630	0.279374
$845$	67.0344	2.30605
$846$	0	0
$847$	22.8588	0.785436
$848$	1.47027	0.0504893
$849$	0	0
$850$	5.21702	0.178942
$851$	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	$0.484795\pi$
$854$	26.0835	0.892559
$855$	0	0
$856$	−2.82849	−0.0966757
$857$	−16.3945	−0.560025	−0.280012	−	0.959996i	$-0.590339\pi$
$857$	−16.3945	−0.560025	−0.280012	+	0.959996i	$0.590339\pi$
$858$	0	0
$859$	9.52505	0.324990	0.162495	−	0.986709i	$-0.448046\pi$
$859$	9.52505	0.324990	0.162495	+	0.986709i	$0.448046\pi$
$860$	7.06236	0.240825
$861$	0	0
$862$	12.6997	0.432552
$863$	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
$866$	−14.2350	−0.483726
$867$	0	0
$868$	−20.3488	−0.690684
$869$	−9.92278	−0.336607
$870$	0	0
$871$	73.3461	2.48524
$872$	−1.72539	−0.0584291
$873$	0	0
$874$	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	$0.394587\pi$
$878$	−28.1900	−0.951367
$879$	0	0
$880$	4.12860	0.139175
$881$	−19.1076	−0.643750	−0.321875	−	0.946782i	$-0.604313\pi$
$881$	−19.1076	−0.643750	−0.321875	+	0.946782i	$0.604313\pi$
$882$	0	0
$883$	49.9980	1.68257	0.841283	−	0.540595i	$-0.181801\pi$
$883$	49.9980	1.68257	0.841283	+	0.540595i	$0.181801\pi$
$884$	−31.5273	−1.06038
$885$	0	0
$886$	−32.9944	−1.10847
$887$	−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
$890$	16.9839	0.569302
$891$	0	0
$892$	10.7540	0.360071
$893$	0	0
$894$	0	0
$895$	−20.2053	−0.675389
$896$	2.79360	0.0933278
$897$	0	0
$898$	−17.1975	−0.573889
$899$	−43.2076	−1.44105
$900$	0	0
$901$	−7.30603	−0.243399
$902$	−4.37662	−0.145726
$903$	0	0
$904$	−0.427785	−0.0142279
$905$	−31.9752	−1.06289
$906$	0	0
$907$	55.2545	1.83470	0.917348	−	0.398087i	$-0.130326\pi$
$907$	55.2545	1.83470	0.917348	+	0.398087i	$0.130326\pi$
$908$	−27.0936	−0.899133
$909$	0	0
$910$	43.5954	1.44517
$911$	−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
$914$	31.1517	1.03041
$915$	0	0
$916$	9.46557	0.312751
$917$	−49.8421	−1.64593
$918$	0	0
$919$	27.2500	0.898894	0.449447	−	0.893307i	$-0.351621\pi$
$919$	27.2500	0.898894	0.449447	+	0.893307i	$0.351621\pi$
$920$	−6.14643	−0.202642
$921$	0	0
$922$	15.5410	0.511815
$923$	44.3929	1.46121
$924$	0	0
$925$	0.578452	0.0190194
$926$	−20.6648	−0.679087
$927$	0	0
$928$	5.93179	0.194720
$929$	−44.4410	−1.45806	−0.729031	−	0.684481i	$-0.760029\pi$
$929$	−44.4410	−1.45806	−0.729031	+	0.684481i	$0.760029\pi$
$930$	0	0
$931$	0	0
$932$	−22.8621	−0.748873
$933$	0	0
$934$	−28.4830	−0.931991
$935$	−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	$0.596134\pi$
$938$	32.2953	1.05448
$939$	0	0
$940$	−1.83390	−0.0598152
$941$	40.2680	1.31270	0.656350	−	0.754457i	$-0.272099\pi$
$941$	40.2680	1.31270	0.656350	+	0.754457i	$0.272099\pi$
$942$	0	0
$943$	6.51567	0.212180
$944$	4.96261	0.161519
$945$	0	0
$946$	−4.81955	−0.156697
$947$	−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	0
$952$	−13.8819	−0.449915
$953$	32.6944	1.05907	0.529537	−	0.848287i	$-0.322366\pi$
$953$	32.6944	1.05907	0.529537	+	0.848287i	$0.322366\pi$
$954$	0	0
$955$	−16.1717	−0.523303
$956$	−8.66611	−0.280282
$957$	0	0
$958$	−10.4726	−0.338356
$959$	−44.4620	−1.43575
$960$	0	0
$961$	22.0578	0.711542
$962$	−3.49568	−0.112705
$963$	0	0
$964$	5.65826	0.182240
$965$	66.1218	2.12854
$966$	0	0
$967$	8.55936	0.275250	0.137625	−	0.990484i	$-0.456053\pi$
$967$	8.55936	0.275250	0.137625	+	0.990484i	$0.456053\pi$
$968$	8.18253	0.262997
$969$	0	0
$970$	−35.3879	−1.13624
$971$	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
$974$	−2.29894	−0.0736628
$975$	0	0
$976$	9.33686	0.298866
$977$	6.18021	0.197722	0.0988612	−	0.995101i	$-0.468480\pi$
$977$	6.18021	0.197722	0.0988612	+	0.995101i	$0.468480\pi$
$978$	0	0
$979$	−11.5903	−0.370427
$980$	1.97811	0.0631885
$981$	0	0
$982$	12.1612	0.388079
$983$	−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
$986$	−29.4761	−0.938710
$987$	0	0
$988$	0	0
$989$	7.17507	0.228154
$990$	0	0
$991$	−46.1746	−1.46679	−0.733393	−	0.679805i	$-0.762064\pi$
$991$	−46.1746	−1.46679	−0.733393	+	0.679805i	$0.762064\pi$
$992$	−7.28408	−0.231270
$993$	0	0
$994$	19.5468	0.619987
$995$	−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	$0.504129\pi$
$998$	−15.1348	−0.479085
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bx.1.4		4
3.2	odd	2		722.2.a.n.1.1	yes	4
12.11	even	2		5776.2.a.bt.1.4		4
19.18	odd	2		6498.2.a.ca.1.4		4
57.2	even	18		722.2.e.s.99.4		24
57.5	odd	18		722.2.e.r.595.4		24
57.8	even	6		722.2.c.n.653.1		8
57.11	odd	6		722.2.c.m.653.4		8
57.14	even	18		722.2.e.s.595.1		24
57.17	odd	18		722.2.e.r.99.1		24
57.23	odd	18		722.2.e.r.415.4		24
57.26	odd	6		722.2.c.m.429.4		8
57.29	even	18		722.2.e.s.423.4		24
57.32	even	18		722.2.e.s.245.4		24
57.35	odd	18		722.2.e.r.389.1		24
57.41	even	18		722.2.e.s.389.4		24
57.44	odd	18		722.2.e.r.245.1		24
57.47	odd	18		722.2.e.r.423.1		24
57.50	even	6		722.2.c.n.429.1		8
57.53	even	18		722.2.e.s.415.1		24
57.56	even	2		722.2.a.m.1.4	✓	4
228.227	odd	2		5776.2.a.bv.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.m.1.4	✓	4	57.56	even	2
722.2.a.n.1.1	yes	4	3.2	odd	2
722.2.c.m.429.4		8	57.26	odd	6
722.2.c.m.653.4		8	57.11	odd	6
722.2.c.n.429.1		8	57.50	even	6
722.2.c.n.653.1		8	57.8	even	6
722.2.e.r.99.1		24	57.17	odd	18
722.2.e.r.245.1		24	57.44	odd	18
722.2.e.r.389.1		24	57.35	odd	18
722.2.e.r.415.4		24	57.23	odd	18
722.2.e.r.423.1		24	57.47	odd	18
722.2.e.r.595.4		24	57.5	odd	18
722.2.e.s.99.4		24	57.2	even	18
722.2.e.s.245.4		24	57.32	even	18
722.2.e.s.389.4		24	57.41	even	18
722.2.e.s.415.1		24	57.53	even	18
722.2.e.s.423.4		24	57.29	even	18
722.2.e.s.595.1		24	57.14	even	18
5776.2.a.bt.1.4		4	12.11	even	2
5776.2.a.bv.1.1		4	228.227	odd	2
6498.2.a.bx.1.4		4	1.1	even	1	trivial
6498.2.a.ca.1.4		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.45965 q^{5} -2.79360 q^{7} -1.00000 q^{8} -2.45965 q^{10} +1.67853 q^{11} +6.34458 q^{13} +2.79360 q^{14} +1.00000 q^{16} -4.96917 q^{17} +2.45965 q^{20} -1.67853 q^{22} +2.49890 q^{23} +1.04988 q^{25} -6.34458 q^{26} -2.79360 q^{28} -5.93179 q^{29} +7.28408 q^{31} -1.00000 q^{32} +4.96917 q^{34} -6.87129 q^{35} +0.550972 q^{37} -2.45965 q^{40} +2.60741 q^{41} +2.87129 q^{43} +1.67853 q^{44} -2.49890 q^{46} -0.745593 q^{47} +0.804226 q^{49} -1.04988 q^{50} +6.34458 q^{52} +1.47027 q^{53} +4.12860 q^{55} +2.79360 q^{56} +5.93179 q^{58} +4.96261 q^{59} +9.33686 q^{61} -7.28408 q^{62} +1.00000 q^{64} +15.6054 q^{65} +11.5604 q^{67} -4.96917 q^{68} +6.87129 q^{70} +6.99698 q^{71} -6.18619 q^{73} -0.550972 q^{74} -4.68915 q^{77} -5.91158 q^{79} +2.45965 q^{80} -2.60741 q^{82} -15.1773 q^{83} -12.2224 q^{85} -2.87129 q^{86} -1.67853 q^{88} -6.90502 q^{89} -17.7242 q^{91} +2.49890 q^{92} +0.745593 q^{94} +14.3874 q^{97} -0.804226 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8} - 2 q^{10} - 2 q^{11} + 18 q^{13} + 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{20} + 2 q^{22} + 10 q^{23} + 6 q^{25} - 18 q^{26} - 2 q^{28} + 2 q^{29} + 26 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{37} - 2 q^{40} + 12 q^{41} - 10 q^{43} - 2 q^{44} - 10 q^{46} + 12 q^{47} - 12 q^{49} - 6 q^{50} + 18 q^{52} - 8 q^{53} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 26 q^{62} + 4 q^{64} + 4 q^{65} + 10 q^{67} - 6 q^{68} + 6 q^{70} - 14 q^{73} - 4 q^{74} - 4 q^{77} + 22 q^{79} + 2 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{85} + 10 q^{86} + 2 q^{88} + 16 q^{89} - 4 q^{91} + 10 q^{92} - 12 q^{94} + 28 q^{97} + 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 - 4 * q^8 - 2 * q^10 - 2 * q^11 + 18 * q^13 + 2 * q^14 + 4 * q^16 - 6 * q^17 + 2 * q^20 + 2 * q^22 + 10 * q^23 + 6 * q^25 - 18 * q^26 - 2 * q^28 + 2 * q^29 + 26 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^37 - 2 * q^40 + 12 * q^41 - 10 * q^43 - 2 * q^44 - 10 * q^46 + 12 * q^47 - 12 * q^49 - 6 * q^50 + 18 * q^52 - 8 * q^53 - 26 * q^55 + 2 * q^56 - 2 * q^58 + 8 * q^59 - 26 * q^62 + 4 * q^64 + 4 * q^65 + 10 * q^67 - 6 * q^68 + 6 * q^70 - 14 * q^73 - 4 * q^74 - 4 * q^77 + 22 * q^79 + 2 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^85 + 10 * q^86 + 2 * q^88 + 16 * q^89 - 4 * q^91 + 10 * q^92 - 12 * q^94 + 28 * q^97 + 12 * q^98

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