LMFDB - Embedded newform 8100.2.a.ba.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(1,8100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8100.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,1,0,0,0,3,0,-2,0,0,0,-9,0,-4,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6788256372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 900)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75080$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.99574 q^{7} -3.99574 q^{11} -1.54316 q^{13} -6.99574 q^{17} -2.25667 q^{19} +7.79556 q^{23} -6.16607 q^{29} -0.543164 q^{31} +6.25240 q^{37} -0.195906 q^{41} +0.0863273 q^{43} -3.82966 q^{47} +17.9574 q^{49} +4.19164 q^{53} -7.02557 q^{59} -2.90941 q^{61} -8.96590 q^{67} -8.79130 q^{71} +2.28650 q^{73} -19.9616 q^{77} -12.6442 q^{79} -13.9616 q^{83} +10.3577 q^{89} -7.70924 q^{91} -9.33873 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7} + 3 q^{11} - 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} - 9 q^{29} + 2 q^{31} + q^{37} + 9 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} - 15 q^{59} - q^{61} - 11 q^{67} + 12 q^{71}+ \cdots - 5 q^{97}+O(q^{100}) $ 4 * q + q^7 + 3 * q^11 - 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 - 9 * q^29 + 2 * q^31 + q^37 + 9 * q^41 - 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 - 15 * q^59 - q^61 - 11 * q^67 + 12 * q^71 + 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 - 3 * q^89 - 11 * q^91 - 5 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.99574	1.88821	0.944105	−	0.329644i	$-0.106929\pi$
$7$	4.99574	1.88821	0.944105	+	0.329644i	$0.106929\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.99574	−1.20476	−0.602380	−	0.798210i	$-0.705781\pi$
$11$	−3.99574	−1.20476	−0.602380	+	0.798210i	$0.705781\pi$
$12$	0	0
$13$	−1.54316	−0.427997	−0.213998	−	0.976834i	$-0.568649\pi$
$13$	−1.54316	−0.427997	−0.213998	+	0.976834i	$0.568649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.99574	−1.69672	−0.848358	−	0.529424i	$-0.822408\pi$
$17$	−6.99574	−1.69672	−0.848358	+	0.529424i	$0.822408\pi$
$18$	0	0
$19$	−2.25667	−0.517715	−0.258857	−	0.965916i	$-0.583346\pi$
$19$	−2.25667	−0.517715	−0.258857	+	0.965916i	$0.583346\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.79556	1.62549	0.812744	−	0.582621i	$-0.197973\pi$
$23$	7.79556	1.62549	0.812744	+	0.582621i	$0.197973\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.16607	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$29$	−6.16607	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$30$	0	0
$31$	−0.543164	−0.0975551	−0.0487775	−	0.998810i	$-0.515533\pi$
$31$	−0.543164	−0.0975551	−0.0487775	+	0.998810i	$0.515533\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.25240	1.02789	0.513944	−	0.857824i	$-0.328184\pi$
$37$	6.25240	1.02789	0.513944	+	0.857824i	$0.328184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.195906	−0.0305954	−0.0152977	−	0.999883i	$-0.504870\pi$
$41$	−0.195906	−0.0305954	−0.0152977	+	0.999883i	$0.504870\pi$
$42$	0	0
$43$	0.0863273	0.0131648	0.00658239	−	0.999978i	$-0.497905\pi$
$43$	0.0863273	0.0131648	0.00658239	+	0.999978i	$0.497905\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.82966	−0.558614	−0.279307	−	0.960202i	$-0.590105\pi$
$47$	−3.82966	−0.558614	−0.279307	+	0.960202i	$0.590105\pi$
$48$	0	0
$49$	17.9574	2.56534
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.19164	0.575766	0.287883	−	0.957666i	$-0.407049\pi$
$53$	4.19164	0.575766	0.287883	+	0.957666i	$0.407049\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.02557	−0.914651	−0.457326	−	0.889299i	$-0.651193\pi$
$59$	−7.02557	−0.914651	−0.457326	+	0.889299i	$0.651193\pi$
$60$	0	0
$61$	−2.90941	−0.372512	−0.186256	−	0.982501i	$-0.559635\pi$
$61$	−2.90941	−0.372512	−0.186256	+	0.982501i	$0.559635\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.96590	−1.09536	−0.547680	−	0.836688i	$-0.684489\pi$
$67$	−8.96590	−1.09536	−0.547680	+	0.836688i	$0.684489\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.79130	−1.04334	−0.521668	−	0.853149i	$-0.674690\pi$
$71$	−8.79130	−1.04334	−0.521668	+	0.853149i	$0.674690\pi$
$72$	0	0
$73$	2.28650	0.267614	0.133807	−	0.991007i	$-0.457280\pi$
$73$	2.28650	0.267614	0.133807	+	0.991007i	$0.457280\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−19.9616	−2.27484
$78$	0	0
$79$	−12.6442	−1.42259	−0.711293	−	0.702896i	$-0.751890\pi$
$79$	−12.6442	−1.42259	−0.711293	+	0.702896i	$0.751890\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.9616	−1.53249	−0.766244	−	0.642549i	$-0.777877\pi$
$83$	−13.9616	−1.53249	−0.766244	+	0.642549i	$0.777877\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.3577	1.09792	0.548958	−	0.835850i	$-0.315025\pi$
$89$	10.3577	1.09792	0.548958	+	0.835850i	$0.315025\pi$
$90$	0	0
$91$	−7.70924	−0.808148
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.33873	−0.948204	−0.474102	−	0.880470i	$-0.657227\pi$
$97$	−9.33873	−0.948204	−0.474102	+	0.880470i	$0.657227\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.96164	−0.792213	−0.396106	−	0.918205i	$-0.629639\pi$
$101$	−7.96164	−0.792213	−0.396106	+	0.918205i	$0.629639\pi$
$102$	0	0
$103$	16.4142	1.61734	0.808670	−	0.588262i	$-0.200188\pi$
$103$	16.4142	1.61734	0.808670	+	0.588262i	$0.200188\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3662	−1.09882	−0.549408	−	0.835554i	$-0.685147\pi$
$107$	−11.3662	−1.09882	−0.549408	+	0.835554i	$0.685147\pi$
$108$	0	0
$109$	−2.22683	−0.213292	−0.106646	−	0.994297i	$-0.534011\pi$
$109$	−2.22683	−0.213292	−0.106646	+	0.994297i	$0.534011\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.83393	−0.266593	−0.133297	−	0.991076i	$-0.542556\pi$
$113$	−2.83393	−0.266593	−0.133297	+	0.991076i	$0.542556\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−34.9488	−3.20376
$120$	0	0
$121$	4.96590	0.451446
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.02130	−0.623040	−0.311520	−	0.950240i	$-0.600838\pi$
$127$	−7.02130	−0.623040	−0.311520	+	0.950240i	$0.600838\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.36624	0.730962	0.365481	−	0.930819i	$-0.380904\pi$
$131$	8.36624	0.730962	0.365481	+	0.930819i	$0.380904\pi$
$132$	0	0
$133$	−11.2737	−0.977554
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.79130	0.238477	0.119238	−	0.992866i	$-0.461955\pi$
$137$	2.79130	0.238477	0.119238	+	0.992866i	$0.461955\pi$
$138$	0	0
$139$	−3.91599	−0.332150	−0.166075	−	0.986113i	$-0.553109\pi$
$139$	−3.91599	−0.332150	−0.166075	+	0.986113i	$0.553109\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.16607	0.515633
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.7572	1.04511	0.522555	−	0.852605i	$-0.324979\pi$
$149$	12.7572	1.04511	0.522555	+	0.852605i	$0.324979\pi$
$150$	0	0
$151$	−0.377090	−0.0306871	−0.0153436	−	0.999882i	$-0.504884\pi$
$151$	−0.377090	−0.0306871	−0.0153436	+	0.999882i	$0.504884\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.0968	−0.965429	−0.482714	−	0.875778i	$-0.660349\pi$
$157$	−12.0968	−0.965429	−0.482714	+	0.875778i	$0.660349\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.9446	3.06926
$162$	0	0
$163$	−10.8563	−0.850333	−0.425166	−	0.905115i	$-0.639784\pi$
$163$	−10.8563	−0.850333	−0.425166	+	0.905115i	$0.639784\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.18738	0.401411	0.200706	−	0.979652i	$-0.435677\pi$
$167$	5.18738	0.401411	0.200706	+	0.979652i	$0.435677\pi$
$168$	0	0
$169$	−10.6186	−0.816819
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.37477	−0.256579	−0.128290	−	0.991737i	$-0.540949\pi$
$173$	−3.37477	−0.256579	−0.128290	+	0.991737i	$0.540949\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.5997	−1.46495	−0.732474	−	0.680795i	$-0.761634\pi$
$179$	−19.5997	−1.46495	−0.732474	+	0.680795i	$0.761634\pi$
$180$	0	0
$181$	5.84865	0.434727	0.217363	−	0.976091i	$-0.430254\pi$
$181$	5.84865	0.434727	0.217363	+	0.976091i	$0.430254\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	27.9531	2.04413
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.5911	1.77935	0.889676	−	0.456593i	$-0.150930\pi$
$191$	24.5911	1.77935	0.889676	+	0.456593i	$0.150930\pi$
$192$	0	0
$193$	10.3430	0.744505	0.372252	−	0.928132i	$-0.378586\pi$
$193$	10.3430	0.744505	0.372252	+	0.928132i	$0.378586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.0256	−1.14177	−0.570887	−	0.821028i	$-0.693401\pi$
$197$	−16.0256	−1.14177	−0.570887	+	0.821028i	$0.693401\pi$
$198$	0	0
$199$	−22.2353	−1.57622	−0.788111	−	0.615533i	$-0.788941\pi$
$199$	−22.2353	−1.57622	−0.788111	+	0.615533i	$0.788941\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−30.8041	−2.16202
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.01704	0.623722
$210$	0	0
$211$	24.3558	1.67672	0.838360	−	0.545117i	$-0.183515\pi$
$211$	24.3558	1.67672	0.838360	+	0.545117i	$0.183515\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.71350	−0.184205
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.7956	0.726188
$222$	0	0
$223$	−11.2481	−0.753231	−0.376615	−	0.926370i	$-0.622912\pi$
$223$	−11.2481	−0.753231	−0.376615	+	0.926370i	$0.622912\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.1490	0.673614	0.336807	−	0.941574i	$-0.390653\pi$
$227$	10.1490	0.673614	0.336807	+	0.941574i	$0.390653\pi$
$228$	0	0
$229$	−2.36856	−0.156519	−0.0782595	−	0.996933i	$-0.524936\pi$
$229$	−2.36856	−0.156519	−0.0782595	+	0.996933i	$0.524936\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.1831	−1.58429	−0.792144	−	0.610334i	$-0.791035\pi$
$233$	−24.1831	−1.58429	−0.792144	+	0.610334i	$0.791035\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.79556	0.116145	0.0580727	−	0.998312i	$-0.481504\pi$
$239$	1.79556	0.116145	0.0580727	+	0.998312i	$0.481504\pi$
$240$	0	0
$241$	−14.8668	−0.957654	−0.478827	−	0.877909i	$-0.658938\pi$
$241$	−14.8668	−0.957654	−0.478827	+	0.877909i	$0.658938\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.48240	0.221580
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.8297	0.999159	0.499580	−	0.866268i	$-0.333488\pi$
$251$	15.8297	0.999159	0.499580	+	0.866268i	$0.333488\pi$
$252$	0	0
$253$	−31.1490	−1.95832
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.9957	−1.37206	−0.686028	−	0.727575i	$-0.740647\pi$
$257$	−21.9957	−1.37206	−0.686028	+	0.727575i	$0.740647\pi$
$258$	0	0
$259$	31.2353	1.94087
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.1533	−0.934391	−0.467196	−	0.884154i	$-0.654736\pi$
$263$	−15.1533	−0.934391	−0.467196	+	0.884154i	$0.654736\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.99147	0.487249	0.243624	−	0.969870i	$-0.421664\pi$
$269$	7.99147	0.487249	0.243624	+	0.969870i	$0.421664\pi$
$270$	0	0
$271$	3.97248	0.241311	0.120656	−	0.992694i	$-0.461500\pi$
$271$	3.97248	0.241311	0.120656	+	0.992694i	$0.461500\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.2140	−1.09438	−0.547188	−	0.837010i	$-0.684302\pi$
$277$	−18.2140	−1.09438	−0.547188	+	0.837010i	$0.684302\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.2087	1.32486	0.662429	−	0.749124i	$-0.269525\pi$
$281$	22.2087	1.32486	0.662429	+	0.749124i	$0.269525\pi$
$282$	0	0
$283$	−11.6527	−0.692684	−0.346342	−	0.938108i	$-0.612576\pi$
$283$	−11.6527	−0.692684	−0.346342	+	0.938108i	$0.612576\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.978696	−0.0577706
$288$	0	0
$289$	31.9403	1.87884
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.18311	0.361221	0.180611	−	0.983555i	$-0.442193\pi$
$293$	6.18311	0.361221	0.180611	+	0.983555i	$0.442193\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.0298	−0.695703
$300$	0	0
$301$	0.431268	0.0248579
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.9872	0.741219	0.370610	−	0.928789i	$-0.379149\pi$
$307$	12.9872	0.741219	0.370610	+	0.928789i	$0.379149\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.1916	1.25837	0.629186	−	0.777255i	$-0.283388\pi$
$311$	22.1916	1.25837	0.629186	+	0.777255i	$0.283388\pi$
$312$	0	0
$313$	16.6922	0.943498	0.471749	−	0.881733i	$-0.343623\pi$
$313$	16.6922	0.943498	0.471749	+	0.881733i	$0.343623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.9574	−1.85107	−0.925535	−	0.378661i	$-0.876385\pi$
$317$	−32.9574	−1.85107	−0.925535	+	0.378661i	$0.876385\pi$
$318$	0	0
$319$	24.6380	1.37946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.7870	0.878414
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.1320	−1.05478
$330$	0	0
$331$	28.6666	1.57566	0.787830	−	0.615893i	$-0.211205\pi$
$331$	28.6666	1.57566	0.787830	+	0.615893i	$0.211205\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.8540	1.62625	0.813125	−	0.582089i	$-0.197764\pi$
$337$	29.8540	1.62625	0.813125	+	0.582089i	$0.197764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.17034	0.117530
$342$	0	0
$343$	54.7401	2.95569
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.83819	−0.259728	−0.129864	−	0.991532i	$-0.541454\pi$
$347$	−4.83819	−0.259728	−0.129864	+	0.991532i	$0.541454\pi$
$348$	0	0
$349$	−32.0650	−1.71640	−0.858200	−	0.513315i	$-0.828417\pi$
$349$	−32.0650	−1.71640	−0.858200	+	0.513315i	$0.828417\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.85949	−0.524768	−0.262384	−	0.964964i	$-0.584509\pi$
$353$	−9.85949	−0.524768	−0.262384	+	0.964964i	$0.584509\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.93266	−0.524226	−0.262113	−	0.965037i	$-0.584419\pi$
$359$	−9.93266	−0.524226	−0.262113	+	0.965037i	$0.584419\pi$
$360$	0	0
$361$	−13.9075	−0.731972
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.0213	−0.992904	−0.496452	−	0.868064i	$-0.665364\pi$
$367$	−19.0213	−0.992904	−0.496452	+	0.868064i	$0.665364\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.9403	1.08717
$372$	0	0
$373$	1.87848	0.0972641	0.0486320	−	0.998817i	$-0.484514\pi$
$373$	1.87848	0.0972641	0.0486320	+	0.998817i	$0.484514\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.51526	0.490061
$378$	0	0
$379$	22.8435	1.17339	0.586697	−	0.809807i	$-0.300428\pi$
$379$	22.8435	1.17339	0.586697	+	0.809807i	$0.300428\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.70266	−0.240295	−0.120147	−	0.992756i	$-0.538337\pi$
$383$	−4.70266	−0.240295	−0.120147	+	0.992756i	$0.538337\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.7657	0.545844	0.272922	−	0.962036i	$-0.412010\pi$
$389$	10.7657	0.545844	0.272922	+	0.962036i	$0.412010\pi$
$390$	0	0
$391$	−54.5357	−2.75799
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.10958	0.407008	0.203504	−	0.979074i	$-0.434767\pi$
$397$	8.10958	0.407008	0.203504	+	0.979074i	$0.434767\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.4549	0.621967	0.310984	−	0.950415i	$-0.399342\pi$
$401$	12.4549	0.621967	0.310984	+	0.950415i	$0.399342\pi$
$402$	0	0
$403$	0.838190	0.0417532
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.9829	−1.23836
$408$	0	0
$409$	−31.4332	−1.55427	−0.777136	−	0.629333i	$-0.783328\pi$
$409$	−31.4332	−1.55427	−0.777136	+	0.629333i	$0.783328\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−35.0979	−1.72705
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.3697	−1.28824	−0.644121	−	0.764924i	$-0.722777\pi$
$419$	−26.3697	−1.28824	−0.644121	+	0.764924i	$0.722777\pi$
$420$	0	0
$421$	−13.2311	−0.644844	−0.322422	−	0.946596i	$-0.604497\pi$
$421$	−13.2311	−0.644844	−0.322422	+	0.946596i	$0.604497\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.5346	−0.703380
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.2895	−1.16999	−0.584993	−	0.811039i	$-0.698903\pi$
$431$	−24.2895	−1.16999	−0.584993	+	0.811039i	$0.698903\pi$
$432$	0	0
$433$	17.9840	0.864258	0.432129	−	0.901812i	$-0.357763\pi$
$433$	17.9840	0.864258	0.432129	+	0.901812i	$0.357763\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.5920	−0.841539
$438$	0	0
$439$	22.7154	1.08415	0.542074	−	0.840331i	$-0.317639\pi$
$439$	22.7154	1.08415	0.542074	+	0.840331i	$0.317639\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.9156	1.08875	0.544377	−	0.838841i	$-0.316766\pi$
$443$	22.9156	1.08875	0.544377	+	0.838841i	$0.316766\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.36966	0.253410	0.126705	−	0.991940i	$-0.459560\pi$
$449$	5.36966	0.253410	0.126705	+	0.991940i	$0.459560\pi$
$450$	0	0
$451$	0.782790	0.0368601
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.2462	0.900299	0.450149	−	0.892953i	$-0.351371\pi$
$457$	19.2462	0.900299	0.450149	+	0.892953i	$0.351371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−35.3739	−1.64753	−0.823763	−	0.566934i	$-0.808130\pi$
$461$	−35.3739	−1.64753	−0.823763	+	0.566934i	$0.808130\pi$
$462$	0	0
$463$	−17.1215	−0.795704	−0.397852	−	0.917450i	$-0.630244\pi$
$463$	−17.1215	−0.795704	−0.397852	+	0.917450i	$0.630244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.9318	−1.19998	−0.599990	−	0.800007i	$-0.704829\pi$
$467$	−25.9318	−1.19998	−0.599990	+	0.800007i	$0.704829\pi$
$468$	0	0
$469$	−44.7913	−2.06827
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.344941	−0.0158604
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.9147	0.955619	0.477810	−	0.878463i	$-0.341431\pi$
$479$	20.9147	0.955619	0.477810	+	0.878463i	$0.341431\pi$
$480$	0	0
$481$	−9.64848	−0.439933
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.2373	−0.917039	−0.458520	−	0.888684i	$-0.651620\pi$
$487$	−20.2373	−0.917039	−0.458520	+	0.888684i	$0.651620\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.2002	0.640845	0.320422	−	0.947275i	$-0.396175\pi$
$491$	14.2002	0.640845	0.320422	+	0.947275i	$0.396175\pi$
$492$	0	0
$493$	43.1362	1.94276
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−43.9190	−1.97004
$498$	0	0
$499$	−28.2767	−1.26584	−0.632920	−	0.774217i	$-0.718144\pi$
$499$	−28.2767	−1.26584	−0.632920	+	0.774217i	$0.718144\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.4165	−0.776565	−0.388282	−	0.921540i	$-0.626931\pi$
$503$	−17.4165	−0.776565	−0.388282	+	0.921540i	$0.626931\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.2121	−0.984534	−0.492267	−	0.870444i	$-0.663832\pi$
$509$	−22.2121	−0.984534	−0.492267	+	0.870444i	$0.663832\pi$
$510$	0	0
$511$	11.4227	0.505312
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.3023	0.672995
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.48898	−0.240477	−0.120238	−	0.992745i	$-0.538366\pi$
$521$	−5.48898	−0.240477	−0.120238	+	0.992745i	$0.538366\pi$
$522$	0	0
$523$	−26.0991	−1.14123	−0.570617	−	0.821216i	$-0.693296\pi$
$523$	−26.0991	−1.14123	−0.570617	+	0.821216i	$0.693296\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.79983	0.165523
$528$	0	0
$529$	37.7708	1.64221
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.302316	0.0130947
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−71.7529	−3.09062
$540$	0	0
$541$	18.7828	0.807534	0.403767	−	0.914862i	$-0.367701\pi$
$541$	18.7828	0.807534	0.403767	+	0.914862i	$0.367701\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.05649	0.387228	0.193614	−	0.981078i	$-0.437979\pi$
$547$	9.05649	0.387228	0.193614	+	0.981078i	$0.437979\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.9148	0.592789
$552$	0	0
$553$	−63.1672	−2.68614
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.8467	−1.05279	−0.526394	−	0.850241i	$-0.676456\pi$
$557$	−24.8467	−1.05279	−0.526394	+	0.850241i	$0.676456\pi$
$558$	0	0
$559$	−0.133217	−0.00563448
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.3535	0.984231	0.492115	−	0.870530i	$-0.336224\pi$
$563$	23.3535	0.984231	0.492115	+	0.870530i	$0.336224\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0902	0.632614	0.316307	−	0.948657i	$-0.397557\pi$
$569$	15.0902	0.632614	0.316307	+	0.948657i	$0.397557\pi$
$570$	0	0
$571$	−6.53573	−0.273512	−0.136756	−	0.990605i	$-0.543668\pi$
$571$	−6.53573	−0.273512	−0.136756	+	0.990605i	$0.543668\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.12662	0.338316	0.169158	−	0.985589i	$-0.445895\pi$
$577$	8.12662	0.338316	0.169158	+	0.985589i	$0.445895\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−69.7487	−2.89366
$582$	0	0
$583$	−16.7487	−0.693660
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.0987	−0.540643	−0.270321	−	0.962770i	$-0.587130\pi$
$587$	−13.0987	−0.540643	−0.270321	+	0.962770i	$0.587130\pi$
$588$	0	0
$589$	1.22574	0.0505057
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.60478	−0.353356	−0.176678	−	0.984269i	$-0.556535\pi$
$593$	−8.60478	−0.353356	−0.176678	+	0.984269i	$0.556535\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.3654	−0.995543	−0.497771	−	0.867308i	$-0.665848\pi$
$599$	−24.3654	−0.995543	−0.497771	+	0.867308i	$0.665848\pi$
$600$	0	0
$601$	−24.1799	−0.986320	−0.493160	−	0.869938i	$-0.664158\pi$
$601$	−24.1799	−0.986320	−0.493160	+	0.869938i	$0.664158\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.42700	0.0579203	0.0289602	−	0.999581i	$-0.490780\pi$
$607$	1.42700	0.0579203	0.0289602	+	0.999581i	$0.490780\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.90979	0.239085
$612$	0	0
$613$	6.92878	0.279851	0.139925	−	0.990162i	$-0.455314\pi$
$613$	6.92878	0.279851	0.139925	+	0.990162i	$0.455314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.4931	1.06657	0.533286	−	0.845935i	$-0.320957\pi$
$617$	26.4931	1.06657	0.533286	+	0.845935i	$0.320957\pi$
$618$	0	0
$619$	−26.9597	−1.08360	−0.541801	−	0.840507i	$-0.682257\pi$
$619$	−26.9597	−1.08360	−0.541801	+	0.840507i	$0.682257\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	51.7444	2.07310
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−43.7401	−1.74403
$630$	0	0
$631$	−15.9350	−0.634361	−0.317181	−	0.948365i	$-0.602736\pi$
$631$	−15.9350	−0.634361	−0.317181	+	0.948365i	$0.602736\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−27.7112	−1.09796
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.16181	0.282874	0.141437	−	0.989947i	$-0.454828\pi$
$641$	7.16181	0.282874	0.141437	+	0.989947i	$0.454828\pi$
$642$	0	0
$643$	7.06308	0.278541	0.139270	−	0.990254i	$-0.455524\pi$
$643$	7.06308	0.278541	0.139270	+	0.990254i	$0.455524\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.4037	−1.03804	−0.519019	−	0.854763i	$-0.673703\pi$
$647$	−26.4037	−1.03804	−0.519019	+	0.854763i	$0.673703\pi$
$648$	0	0
$649$	28.0723	1.10193
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.71970	0.302095	0.151048	−	0.988527i	$-0.451735\pi$
$653$	7.71970	0.302095	0.151048	+	0.988527i	$0.451735\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.2853	0.868110	0.434055	−	0.900886i	$-0.357082\pi$
$659$	22.2853	0.868110	0.434055	+	0.900886i	$0.357082\pi$
$660$	0	0
$661$	32.1087	1.24888	0.624442	−	0.781071i	$-0.285326\pi$
$661$	32.1087	1.24888	0.624442	+	0.781071i	$0.285326\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−48.0680	−1.86120
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.6252	0.448787
$672$	0	0
$673$	7.83161	0.301886	0.150943	−	0.988542i	$-0.451769\pi$
$673$	7.83161	0.301886	0.150943	+	0.988542i	$0.451769\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0034	−0.691927	−0.345964	−	0.938248i	$-0.612448\pi$
$677$	−18.0034	−0.691927	−0.345964	+	0.938248i	$0.612448\pi$
$678$	0	0
$679$	−46.6538	−1.79041
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.3356	1.16076	0.580379	−	0.814347i	$-0.302904\pi$
$683$	30.3356	1.16076	0.580379	+	0.814347i	$0.302904\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.46839	−0.246426
$690$	0	0
$691$	28.1149	1.06954	0.534771	−	0.844997i	$-0.320398\pi$
$691$	28.1149	1.06954	0.534771	+	0.844997i	$0.320398\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.37051	0.0519117
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.4378	−1.07408	−0.537041	−	0.843556i	$-0.680458\pi$
$701$	−28.4378	−1.07408	−0.537041	+	0.843556i	$0.680458\pi$
$702$	0	0
$703$	−14.1096	−0.532153
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.7742	−1.49586
$708$	0	0
$709$	30.9000	1.16047	0.580237	−	0.814447i	$-0.302960\pi$
$709$	30.9000	1.16047	0.580237	+	0.814447i	$0.302960\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.23427	−0.158575
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.2462	0.792349	0.396175	−	0.918175i	$-0.370338\pi$
$719$	21.2462	0.792349	0.396175	+	0.918175i	$0.370338\pi$
$720$	0	0
$721$	82.0011	3.05388
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−24.9821	−0.926534	−0.463267	−	0.886219i	$-0.653323\pi$
$727$	−24.9821	−0.926534	−0.463267	+	0.886219i	$0.653323\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.603923	−0.0223369
$732$	0	0
$733$	10.4677	0.386633	0.193316	−	0.981136i	$-0.438076\pi$
$733$	10.4677	0.386633	0.193316	+	0.981136i	$0.438076\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.8254	1.31964
$738$	0	0
$739$	35.6244	1.31046	0.655232	−	0.755428i	$-0.272571\pi$
$739$	35.6244	1.31046	0.655232	+	0.755428i	$0.272571\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.0554	0.368897	0.184448	−	0.982842i	$-0.440950\pi$
$743$	10.0554	0.368897	0.184448	+	0.982842i	$0.440950\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−56.7828	−2.07480
$750$	0	0
$751$	17.7348	0.647152	0.323576	−	0.946202i	$-0.395115\pi$
$751$	17.7348	0.647152	0.323576	+	0.946202i	$0.395115\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.42042	0.233354	0.116677	−	0.993170i	$-0.462776\pi$
$757$	6.42042	0.233354	0.116677	+	0.993170i	$0.462776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.3910	−0.920422	−0.460211	−	0.887810i	$-0.652226\pi$
$761$	−25.3910	−0.920422	−0.460211	+	0.887810i	$0.652226\pi$
$762$	0	0
$763$	−11.1247	−0.402740
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.8416	0.391468
$768$	0	0
$769$	−14.8018	−0.533765	−0.266883	−	0.963729i	$-0.585994\pi$
$769$	−14.8018	−0.533765	−0.266883	+	0.963729i	$0.585994\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.48046	0.269053	0.134527	−	0.990910i	$-0.457049\pi$
$773$	7.48046	0.269053	0.134527	+	0.990910i	$0.457049\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.442095	0.0158397
$780$	0	0
$781$	35.1277	1.25697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.3174	1.86491	0.932457	−	0.361281i	$-0.117660\pi$
$787$	52.3174	1.86491	0.932457	+	0.361281i	$0.117660\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.1575	−0.503384
$792$	0	0
$793$	4.48969	0.159434
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−53.7359	−1.90342	−0.951711	−	0.306994i	$-0.900677\pi$
$797$	−53.7359	−1.90342	−0.951711	+	0.306994i	$0.900677\pi$
$798$	0	0
$799$	26.7913	0.947808
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.13624	−0.322411
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.1908	0.920819	0.460410	−	0.887707i	$-0.347703\pi$
$809$	26.1908	0.920819	0.460410	+	0.887707i	$0.347703\pi$
$810$	0	0
$811$	−52.3506	−1.83828	−0.919140	−	0.393931i	$-0.871115\pi$
$811$	−52.3506	−1.83828	−0.919140	+	0.393931i	$0.871115\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.194812	−0.00681560
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.85864	0.169568	0.0847839	−	0.996399i	$-0.472980\pi$
$821$	4.85864	0.169568	0.0847839	+	0.996399i	$0.472980\pi$
$822$	0	0
$823$	−1.17777	−0.0410546	−0.0205273	−	0.999789i	$-0.506534\pi$
$823$	−1.17777	−0.0410546	−0.0205273	+	0.999789i	$0.506534\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.7188	1.13775	0.568873	−	0.822426i	$-0.307380\pi$
$827$	32.7188	1.13775	0.568873	+	0.822426i	$0.307380\pi$
$828$	0	0
$829$	−5.99342	−0.208160	−0.104080	−	0.994569i	$-0.533190\pi$
$829$	−5.99342	−0.208160	−0.104080	+	0.994569i	$0.533190\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−125.625	−4.35265
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.3867	1.32526	0.662628	−	0.748949i	$-0.269441\pi$
$839$	38.3867	1.32526	0.662628	+	0.748949i	$0.269441\pi$
$840$	0	0
$841$	9.02047	0.311051
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24.8083	0.852425
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.7410	1.67082
$852$	0	0
$853$	−21.7190	−0.743644	−0.371822	−	0.928304i	$-0.621267\pi$
$853$	−21.7190	−0.743644	−0.371822	+	0.928304i	$0.621267\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.59966	−0.157121	−0.0785607	−	0.996909i	$-0.525032\pi$
$857$	−4.59966	−0.157121	−0.0785607	+	0.996909i	$0.525032\pi$
$858$	0	0
$859$	14.9179	0.508993	0.254496	−	0.967074i	$-0.418090\pi$
$859$	14.9179	0.508993	0.254496	+	0.967074i	$0.418090\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.4089	0.388362	0.194181	−	0.980966i	$-0.437795\pi$
$863$	11.4089	0.388362	0.194181	+	0.980966i	$0.437795\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	50.5229	1.71387
$870$	0	0
$871$	13.8359	0.468810
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.9284	−0.943075	−0.471537	−	0.881846i	$-0.656301\pi$
$877$	−27.9284	−0.943075	−0.471537	+	0.881846i	$0.656301\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.3270	1.42603	0.713017	−	0.701147i	$-0.247328\pi$
$881$	42.3270	1.42603	0.713017	+	0.701147i	$0.247328\pi$
$882$	0	0
$883$	20.3895	0.686161	0.343081	−	0.939306i	$-0.388530\pi$
$883$	20.3895	0.686161	0.343081	+	0.939306i	$0.388530\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.5451	1.32779	0.663897	−	0.747824i	$-0.268901\pi$
$887$	39.5451	1.32779	0.663897	+	0.747824i	$0.268901\pi$
$888$	0	0
$889$	−35.0766	−1.17643
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.64227	0.289202
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.34919	0.111702
$900$	0	0
$901$	−29.3236	−0.976911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.9597	−1.22723	−0.613613	−	0.789607i	$-0.710285\pi$
$907$	−36.9597	−1.22723	−0.613613	+	0.789607i	$0.710285\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.84670	0.326236	0.163118	−	0.986607i	$-0.447845\pi$
$911$	9.84670	0.326236	0.163118	+	0.986607i	$0.447845\pi$
$912$	0	0
$913$	55.7870	1.84628
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.7955	1.38021
$918$	0	0
$919$	−8.42079	−0.277776	−0.138888	−	0.990308i	$-0.544353\pi$
$919$	−8.42079	−0.277776	−0.138888	+	0.990308i	$0.544353\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.5664	0.446544
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.3875	0.931365	0.465683	−	0.884952i	$-0.345809\pi$
$929$	28.3875	0.931365	0.465683	+	0.884952i	$0.345809\pi$
$930$	0	0
$931$	−40.5238	−1.32811
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.69342	0.153327	0.0766637	−	0.997057i	$-0.475573\pi$
$937$	4.69342	0.153327	0.0766637	+	0.997057i	$0.475573\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−19.0256	−0.620216	−0.310108	−	0.950701i	$-0.600365\pi$
$941$	−19.0256	−0.620216	−0.310108	+	0.950701i	$0.600365\pi$
$942$	0	0
$943$	−1.52720	−0.0497325
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.8169	−0.416492	−0.208246	−	0.978076i	$-0.566776\pi$
$947$	−12.8169	−0.416492	−0.208246	+	0.978076i	$0.566776\pi$
$948$	0	0
$949$	−3.52844	−0.114538
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.9701	−1.48912	−0.744560	−	0.667556i	$-0.767340\pi$
$953$	−45.9701	−1.48912	−0.744560	+	0.667556i	$0.767340\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.9446	0.450295
$960$	0	0
$961$	−30.7050	−0.990483
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.14709	−0.165519	−0.0827596	−	0.996570i	$-0.526373\pi$
$967$	−5.14709	−0.165519	−0.0827596	+	0.996570i	$0.526373\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.3875	0.333352	0.166676	−	0.986012i	$-0.446697\pi$
$971$	10.3875	0.333352	0.166676	+	0.986012i	$0.446697\pi$
$972$	0	0
$973$	−19.5632	−0.627169
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.5621	1.26570	0.632852	−	0.774272i	$-0.281884\pi$
$977$	39.5621	1.26570	0.632852	+	0.774272i	$0.281884\pi$
$978$	0	0
$979$	−41.3867	−1.32272
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.4719	0.652954	0.326477	−	0.945205i	$-0.394138\pi$
$983$	20.4719	0.652954	0.326477	+	0.945205i	$0.394138\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.672970	0.0213992
$990$	0	0
$991$	−19.2415	−0.611228	−0.305614	−	0.952156i	$-0.598862\pi$
$991$	−19.2415	−0.611228	−0.305614	+	0.952156i	$0.598862\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.2385	0.735971	0.367986	−	0.929832i	$-0.380048\pi$
$997$	23.2385	0.735971	0.367986	+	0.929832i	$0.380048\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.ba.1.4		4
3.2	odd	2		8100.2.a.z.1.4		4
5.2	odd	4		8100.2.d.s.649.8		8
5.3	odd	4		8100.2.d.s.649.1		8
5.4	even	2		8100.2.a.y.1.1		4
9.2	odd	6		900.2.i.e.301.3	yes	8
9.4	even	3		2700.2.i.d.1801.1		8
9.5	odd	6		900.2.i.e.601.3	yes	8
9.7	even	3		2700.2.i.d.901.1		8
15.2	even	4		8100.2.d.q.649.8		8
15.8	even	4		8100.2.d.q.649.1		8
15.14	odd	2		8100.2.a.x.1.1		4
45.2	even	12		900.2.s.d.49.7		16
45.4	even	6		2700.2.i.e.1801.4		8
45.7	odd	12		2700.2.s.d.1549.8		16
45.13	odd	12		2700.2.s.d.2449.8		16
45.14	odd	6		900.2.i.d.601.2	yes	8
45.22	odd	12		2700.2.s.d.2449.1		16
45.23	even	12		900.2.s.d.349.7		16
45.29	odd	6		900.2.i.d.301.2	✓	8
45.32	even	12		900.2.s.d.349.2		16
45.34	even	6		2700.2.i.e.901.4		8
45.38	even	12		900.2.s.d.49.2		16
45.43	odd	12		2700.2.s.d.1549.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.2	✓	8	45.29	odd	6
900.2.i.d.601.2	yes	8	45.14	odd	6
900.2.i.e.301.3	yes	8	9.2	odd	6
900.2.i.e.601.3	yes	8	9.5	odd	6
900.2.s.d.49.2		16	45.38	even	12
900.2.s.d.49.7		16	45.2	even	12
900.2.s.d.349.2		16	45.32	even	12
900.2.s.d.349.7		16	45.23	even	12
2700.2.i.d.901.1		8	9.7	even	3
2700.2.i.d.1801.1		8	9.4	even	3
2700.2.i.e.901.4		8	45.34	even	6
2700.2.i.e.1801.4		8	45.4	even	6
2700.2.s.d.1549.1		16	45.43	odd	12
2700.2.s.d.1549.8		16	45.7	odd	12
2700.2.s.d.2449.1		16	45.22	odd	12
2700.2.s.d.2449.8		16	45.13	odd	12
8100.2.a.x.1.1		4	15.14	odd	2
8100.2.a.y.1.1		4	5.4	even	2
8100.2.a.z.1.4		4	3.2	odd	2
8100.2.a.ba.1.4		4	1.1	even	1	trivial
8100.2.d.q.649.1		8	15.8	even	4
8100.2.d.q.649.8		8	15.2	even	4
8100.2.d.s.649.1		8	5.3	odd	4
8100.2.d.s.649.8		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q+4.99574 q^{7} -3.99574 q^{11} -1.54316 q^{13} -6.99574 q^{17} -2.25667 q^{19} +7.79556 q^{23} -6.16607 q^{29} -0.543164 q^{31} +6.25240 q^{37} -0.195906 q^{41} +0.0863273 q^{43} -3.82966 q^{47} +17.9574 q^{49} +4.19164 q^{53} -7.02557 q^{59} -2.90941 q^{61} -8.96590 q^{67} -8.79130 q^{71} +2.28650 q^{73} -19.9616 q^{77} -12.6442 q^{79} -13.9616 q^{83} +10.3577 q^{89} -7.70924 q^{91} -9.33873 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7} + 3 q^{11} - 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} - 9 q^{29} + 2 q^{31} + q^{37} + 9 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} - 15 q^{59} - q^{61} - 11 q^{67} + 12 q^{71}+ \cdots - 5 q^{97}+O(q^{100}) \) 4 * q + q^7 + 3 * q^11 - 2 * q^13 - 9 * q^17 - 4 * q^19 + 3 * q^23 - 9 * q^29 + 2 * q^31 + q^37 + 9 * q^41 - 8 * q^43 - 12 * q^47 + 9 * q^49 - 12 * q^53 - 15 * q^59 - q^61 - 11 * q^67 + 12 * q^71 + 10 * q^73 - 36 * q^77 - 7 * q^79 - 12 * q^83 - 3 * q^89 - 11 * q^91 - 5 * q^97

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