LMFDB - Embedded newform 8100.2.d.q.649.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8100,2,Mod(649,8100)] 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("8100.649"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4057180416.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 $ x^8 + 9x^6 + 22x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.8
Root			$-1.75080i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.q.649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.99574i q^{7} +3.99574 q^{11} +1.54316i q^{13} +6.99574i q^{17} +2.25667 q^{19} +7.79556i q^{23} -6.16607 q^{29} -0.543164 q^{31} +6.25240i q^{37} +0.195906 q^{41} -0.0863273i q^{43} +3.82966i q^{47} -17.9574 q^{49} +4.19164i q^{53} -7.02557 q^{59} -2.90941 q^{61} -8.96590i q^{67} +8.79130 q^{71} -2.28650i q^{73} +19.9616i q^{77} +12.6442 q^{79} -13.9616i q^{83} +10.3577 q^{89} -7.70924 q^{91} -9.33873i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) $ 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	4.99574i	1.88821i	0.329644	+	0.944105i	$0.393071\pi$
$7$	4.99574i	1.88821i	−0.329644	+	0.944105i	$0.606929\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.99574	1.20476	0.602380	−	0.798210i	$-0.294219\pi$
$11$	3.99574	1.20476	0.602380	+	0.798210i	$0.294219\pi$
$12$	0	0
$13$	1.54316i	0.427997i	0.976834	+	0.213998i	$0.0686487\pi$
$13$	1.54316i	0.427997i	−0.976834	+	0.213998i	$0.931351\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.99574i	1.69672i	0.529424	+	0.848358i	$0.322408\pi$
$17$	6.99574i	1.69672i	−0.529424	+	0.848358i	$0.677592\pi$
$18$	0	0
$19$	2.25667	0.517715	0.258857	−	0.965916i	$-0.416654\pi$
$19$	2.25667	0.517715	0.258857	+	0.965916i	$0.416654\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.79556i	1.62549i	0.582621	+	0.812744i	$0.302027\pi$
$23$	7.79556i	1.62549i	−0.582621	+	0.812744i	$0.697973\pi$
$24$	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
$37$	6.25240i	1.02789i	0.857824	+	0.513944i	$0.171816\pi$
$37$	6.25240i	1.02789i	−0.857824	+	0.513944i	$0.828184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.195906	0.0305954	0.0152977	−	0.999883i	$-0.495130\pi$
$41$	0.195906	0.0305954	0.0152977	+	0.999883i	$0.495130\pi$
$42$	0	0
$43$	− 0.0863273i	− 0.0131648i	−0.999978	−	0.00658239i	$-0.997905\pi$
$43$	− 0.0863273i	− 0.0131648i	0.999978	−	0.00658239i	$-0.00209526\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.82966i	0.558614i	0.960202	+	0.279307i	$0.0901046\pi$
$47$	3.82966i	0.558614i	−0.960202	+	0.279307i	$0.909895\pi$
$48$	0	0
$49$	−17.9574	−2.56534
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.19164i	0.575766i	0.957666	+	0.287883i	$0.0929515\pi$
$53$	4.19164i	0.575766i	−0.957666	+	0.287883i	$0.907049\pi$
$54$	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
$67$	− 8.96590i	− 1.09536i	−0.836688	−	0.547680i	$-0.815511\pi$
$67$	− 8.96590i	− 1.09536i	0.836688	−	0.547680i	$-0.184489\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.79130	1.04334	0.521668	−	0.853149i	$-0.325310\pi$
$71$	8.79130	1.04334	0.521668	+	0.853149i	$0.325310\pi$
$72$	0	0
$73$	− 2.28650i	− 0.267614i	−0.991007	−	0.133807i	$-0.957280\pi$
$73$	− 2.28650i	− 0.267614i	0.991007	−	0.133807i	$-0.0427203\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.9616i	2.27484i
$78$	0	0
$79$	12.6442	1.42259	0.711293	−	0.702896i	$-0.248110\pi$
$79$	12.6442	1.42259	0.711293	+	0.702896i	$0.248110\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.9616i	− 1.53249i	−0.642549	−	0.766244i	$-0.722123\pi$
$83$	− 13.9616i	− 1.53249i	0.642549	−	0.766244i	$-0.277877\pi$
$84$	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
$97$	− 9.33873i	− 0.948204i	−0.880470	−	0.474102i	$-0.842773\pi$
$97$	− 9.33873i	− 0.948204i	0.880470	−	0.474102i	$-0.157227\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.96164	0.792213	0.396106	−	0.918205i	$-0.370361\pi$
$101$	7.96164	0.792213	0.396106	+	0.918205i	$0.370361\pi$
$102$	0	0
$103$	− 16.4142i	− 1.61734i	−0.588262	−	0.808670i	$-0.700188\pi$
$103$	− 16.4142i	− 1.61734i	0.588262	−	0.808670i	$-0.299812\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3662i	1.09882i	0.835554	+	0.549408i	$0.185147\pi$
$107$	11.3662i	1.09882i	−0.835554	+	0.549408i	$0.814853\pi$
$108$	0	0
$109$	2.22683	0.213292	0.106646	−	0.994297i	$-0.465989\pi$
$109$	2.22683	0.213292	0.106646	+	0.994297i	$0.465989\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.83393i	− 0.266593i	−0.991076	−	0.133297i	$-0.957444\pi$
$113$	− 2.83393i	− 0.266593i	0.991076	−	0.133297i	$-0.0425563\pi$
$114$	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
$127$	− 7.02130i	− 0.623040i	−0.950240	−	0.311520i	$-0.899162\pi$
$127$	− 7.02130i	− 0.623040i	0.950240	−	0.311520i	$-0.100838\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.36624	−0.730962	−0.365481	−	0.930819i	$-0.619096\pi$
$131$	−8.36624	−0.730962	−0.365481	+	0.930819i	$0.619096\pi$
$132$	0	0
$133$	11.2737i	0.977554i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.79130i	− 0.238477i	−0.992866	−	0.119238i	$-0.961955\pi$
$137$	− 2.79130i	− 0.238477i	0.992866	−	0.119238i	$-0.0380453\pi$
$138$	0	0
$139$	3.91599	0.332150	0.166075	−	0.986113i	$-0.446891\pi$
$139$	3.91599	0.332150	0.166075	+	0.986113i	$0.446891\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.16607i	0.515633i
$144$	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
$157$	− 12.0968i	− 0.965429i	−0.875778	−	0.482714i	$-0.839651\pi$
$157$	− 12.0968i	− 0.965429i	0.875778	−	0.482714i	$-0.160349\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−38.9446	−3.06926
$162$	0	0
$163$	10.8563i	0.850333i	0.905115	+	0.425166i	$0.139784\pi$
$163$	10.8563i	0.850333i	−0.905115	+	0.425166i	$0.860216\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.18738i	− 0.401411i	−0.979652	−	0.200706i	$-0.935677\pi$
$167$	− 5.18738i	− 0.401411i	0.979652	−	0.200706i	$-0.0643235\pi$
$168$	0	0
$169$	10.6186	0.816819
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.37477i	− 0.256579i	−0.991737	−	0.128290i	$-0.959051\pi$
$173$	− 3.37477i	− 0.256579i	0.991737	−	0.128290i	$-0.0409487\pi$
$174$	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
$187$	27.9531i	2.04413i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.5911	−1.77935	−0.889676	−	0.456593i	$-0.849070\pi$
$191$	−24.5911	−1.77935	−0.889676	+	0.456593i	$0.849070\pi$
$192$	0	0
$193$	− 10.3430i	− 0.744505i	−0.928132	−	0.372252i	$-0.878586\pi$
$193$	− 10.3430i	− 0.744505i	0.928132	−	0.372252i	$-0.121414\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.0256i	1.14177i	0.821028	+	0.570887i	$0.193401\pi$
$197$	16.0256i	1.14177i	−0.821028	+	0.570887i	$0.806599\pi$
$198$	0	0
$199$	22.2353	1.57622	0.788111	−	0.615533i	$-0.211059\pi$
$199$	22.2353	1.57622	0.788111	+	0.615533i	$0.211059\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 30.8041i	− 2.16202i
$204$	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
$217$	− 2.71350i	− 0.184205i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.7956	−0.726188
$222$	0	0
$223$	11.2481i	0.753231i	0.926370	+	0.376615i	$0.122912\pi$
$223$	11.2481i	0.753231i	−0.926370	+	0.376615i	$0.877088\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.1490i	− 0.673614i	−0.941574	−	0.336807i	$-0.890653\pi$
$227$	− 10.1490i	− 0.673614i	0.941574	−	0.336807i	$-0.109347\pi$
$228$	0	0
$229$	2.36856	0.156519	0.0782595	−	0.996933i	$-0.475064\pi$
$229$	2.36856	0.156519	0.0782595	+	0.996933i	$0.475064\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.1831i	− 1.58429i	−0.610334	−	0.792144i	$-0.708965\pi$
$233$	− 24.1831i	− 1.58429i	0.610334	−	0.792144i	$-0.291035\pi$
$234$	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
$247$	3.48240i	0.221580i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.8297	−0.999159	−0.499580	−	0.866268i	$-0.666512\pi$
$251$	−15.8297	−0.999159	−0.499580	+	0.866268i	$0.666512\pi$
$252$	0	0
$253$	31.1490i	1.95832i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.9957i	1.37206i	0.727575	+	0.686028i	$0.240647\pi$
$257$	21.9957i	1.37206i	−0.727575	+	0.686028i	$0.759353\pi$
$258$	0	0
$259$	−31.2353	−1.94087
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 15.1533i	− 0.934391i	−0.884154	−	0.467196i	$-0.845264\pi$
$263$	− 15.1533i	− 0.934391i	0.884154	−	0.467196i	$-0.154736\pi$
$264$	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
$277$	− 18.2140i	− 1.09438i	−0.837010	−	0.547188i	$-0.815698\pi$
$277$	− 18.2140i	− 1.09438i	0.837010	−	0.547188i	$-0.184302\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.2087	−1.32486	−0.662429	−	0.749124i	$-0.730475\pi$
$281$	−22.2087	−1.32486	−0.662429	+	0.749124i	$0.730475\pi$
$282$	0	0
$283$	11.6527i	0.692684i	0.938108	+	0.346342i	$0.112576\pi$
$283$	11.6527i	0.692684i	−0.938108	+	0.346342i	$0.887424\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.978696i	0.0577706i
$288$	0	0
$289$	−31.9403	−1.87884
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.18311i	0.361221i	0.983555	+	0.180611i	$0.0578074\pi$
$293$	6.18311i	0.361221i	−0.983555	+	0.180611i	$0.942193\pi$
$294$	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
$307$	12.9872i	0.741219i	0.928789	+	0.370610i	$0.120851\pi$
$307$	12.9872i	0.741219i	−0.928789	+	0.370610i	$0.879149\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.1916	−1.25837	−0.629186	−	0.777255i	$-0.716612\pi$
$311$	−22.1916	−1.25837	−0.629186	+	0.777255i	$0.716612\pi$
$312$	0	0
$313$	− 16.6922i	− 0.943498i	−0.881733	−	0.471749i	$-0.843623\pi$
$313$	− 16.6922i	− 0.943498i	0.881733	−	0.471749i	$-0.156377\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.9574i	1.85107i	0.378661	+	0.925535i	$0.376385\pi$
$317$	32.9574i	1.85107i	−0.378661	+	0.925535i	$0.623615\pi$
$318$	0	0
$319$	−24.6380	−1.37946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.7870i	0.878414i
$324$	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
$337$	29.8540i	1.62625i	0.582089	+	0.813125i	$0.302236\pi$
$337$	29.8540i	1.62625i	−0.582089	+	0.813125i	$0.697764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.17034	−0.117530
$342$	0	0
$343$	− 54.7401i	− 2.95569i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.83819i	0.259728i	0.991532	+	0.129864i	$0.0414540\pi$
$347$	4.83819i	0.259728i	−0.991532	+	0.129864i	$0.958546\pi$
$348$	0	0
$349$	32.0650	1.71640	0.858200	−	0.513315i	$-0.171583\pi$
$349$	32.0650	1.71640	0.858200	+	0.513315i	$0.171583\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.85949i	− 0.524768i	−0.964964	−	0.262384i	$-0.915491\pi$
$353$	− 9.85949i	− 0.524768i	0.964964	−	0.262384i	$-0.0845087\pi$
$354$	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
$367$	− 19.0213i	− 0.992904i	−0.868064	−	0.496452i	$-0.834636\pi$
$367$	− 19.0213i	− 0.992904i	0.868064	−	0.496452i	$-0.165364\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.9403	−1.08717
$372$	0	0
$373$	− 1.87848i	− 0.0972641i	−0.998817	−	0.0486320i	$-0.984514\pi$
$373$	− 1.87848i	− 0.0972641i	0.998817	−	0.0486320i	$-0.0154862\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.51526i	− 0.490061i
$378$	0	0
$379$	−22.8435	−1.17339	−0.586697	−	0.809807i	$-0.699572\pi$
$379$	−22.8435	−1.17339	−0.586697	+	0.809807i	$0.699572\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 4.70266i	− 0.240295i	−0.992756	−	0.120147i	$-0.961663\pi$
$383$	− 4.70266i	− 0.240295i	0.992756	−	0.120147i	$-0.0383367\pi$
$384$	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
$397$	8.10958i	0.407008i	0.979074	+	0.203504i	$0.0652330\pi$
$397$	8.10958i	0.407008i	−0.979074	+	0.203504i	$0.934767\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.4549	−0.621967	−0.310984	−	0.950415i	$-0.600658\pi$
$401$	−12.4549	−0.621967	−0.310984	+	0.950415i	$0.600658\pi$
$402$	0	0
$403$	− 0.838190i	− 0.0417532i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.9829i	1.23836i
$408$	0	0
$409$	31.4332	1.55427	0.777136	−	0.629333i	$-0.216672\pi$
$409$	31.4332	1.55427	0.777136	+	0.629333i	$0.216672\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 35.0979i	− 1.72705i
$414$	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
$427$	− 14.5346i	− 0.703380i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2895	1.16999	0.584993	−	0.811039i	$-0.301097\pi$
$431$	24.2895	1.16999	0.584993	+	0.811039i	$0.301097\pi$
$432$	0	0
$433$	− 17.9840i	− 0.864258i	−0.901812	−	0.432129i	$-0.857763\pi$
$433$	− 17.9840i	− 0.864258i	0.901812	−	0.432129i	$-0.142237\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.5920i	0.841539i
$438$	0	0
$439$	−22.7154	−1.08415	−0.542074	−	0.840331i	$-0.682361\pi$
$439$	−22.7154	−1.08415	−0.542074	+	0.840331i	$0.682361\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.9156i	1.08875i	0.838841	+	0.544377i	$0.183234\pi$
$443$	22.9156i	1.08875i	−0.838841	+	0.544377i	$0.816766\pi$
$444$	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
$457$	19.2462i	0.900299i	0.892953	+	0.450149i	$0.148629\pi$
$457$	19.2462i	0.900299i	−0.892953	+	0.450149i	$0.851371\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.3739	1.64753	0.823763	−	0.566934i	$-0.191870\pi$
$461$	35.3739	1.64753	0.823763	+	0.566934i	$0.191870\pi$
$462$	0	0
$463$	17.1215i	0.795704i	0.917450	+	0.397852i	$0.130244\pi$
$463$	17.1215i	0.795704i	−0.917450	+	0.397852i	$0.869756\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.9318i	1.19998i	0.800007	+	0.599990i	$0.204829\pi$
$467$	25.9318i	1.19998i	−0.800007	+	0.599990i	$0.795171\pi$
$468$	0	0
$469$	44.7913	2.06827
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 0.344941i	− 0.0158604i
$474$	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
$487$	− 20.2373i	− 0.917039i	−0.888684	−	0.458520i	$-0.848380\pi$
$487$	− 20.2373i	− 0.917039i	0.888684	−	0.458520i	$-0.151620\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.2002	−0.640845	−0.320422	−	0.947275i	$-0.603825\pi$
$491$	−14.2002	−0.640845	−0.320422	+	0.947275i	$0.603825\pi$
$492$	0	0
$493$	− 43.1362i	− 1.94276i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.9190i	1.97004i
$498$	0	0
$499$	28.2767	1.26584	0.632920	−	0.774217i	$-0.281856\pi$
$499$	28.2767	1.26584	0.632920	+	0.774217i	$0.281856\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 17.4165i	− 0.776565i	−0.921540	−	0.388282i	$-0.873069\pi$
$503$	− 17.4165i	− 0.776565i	0.921540	−	0.388282i	$-0.126931\pi$
$504$	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
$517$	15.3023i	0.672995i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5.48898	0.240477	0.120238	−	0.992745i	$-0.461634\pi$
$521$	5.48898	0.240477	0.120238	+	0.992745i	$0.461634\pi$
$522$	0	0
$523$	26.0991i	1.14123i	0.821216	+	0.570617i	$0.193296\pi$
$523$	26.0991i	1.14123i	−0.821216	+	0.570617i	$0.806704\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.79983i	− 0.165523i
$528$	0	0
$529$	−37.7708	−1.64221
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.302316i	0.0130947i
$534$	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
$547$	9.05649i	0.387228i	0.981078	+	0.193614i	$0.0620209\pi$
$547$	9.05649i	0.387228i	−0.981078	+	0.193614i	$0.937979\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13.9148	−0.592789
$552$	0	0
$553$	63.1672i	2.68614i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.8467i	1.05279i	0.850241	+	0.526394i	$0.176456\pi$
$557$	24.8467i	1.05279i	−0.850241	+	0.526394i	$0.823544\pi$
$558$	0	0
$559$	0.133217	0.00563448
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.3535i	0.984231i	0.870530	+	0.492115i	$0.163776\pi$
$563$	23.3535i	0.984231i	−0.870530	+	0.492115i	$0.836224\pi$
$564$	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
$577$	8.12662i	0.338316i	0.985589	+	0.169158i	$0.0541047\pi$
$577$	8.12662i	0.338316i	−0.985589	+	0.169158i	$0.945895\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	69.7487	2.89366
$582$	0	0
$583$	16.7487i	0.693660i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.0987i	0.540643i	0.962770	+	0.270321i	$0.0871299\pi$
$587$	13.0987i	0.540643i	−0.962770	+	0.270321i	$0.912870\pi$
$588$	0	0
$589$	−1.22574	−0.0505057
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 8.60478i	− 0.353356i	−0.984269	−	0.176678i	$-0.943465\pi$
$593$	− 8.60478i	− 0.353356i	0.984269	−	0.176678i	$-0.0565351\pi$
$594$	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
$607$	1.42700i	0.0579203i	0.999581	+	0.0289602i	$0.00921959\pi$
$607$	1.42700i	0.0579203i	−0.999581	+	0.0289602i	$0.990780\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.90979	−0.239085
$612$	0	0
$613$	− 6.92878i	− 0.279851i	−0.990162	−	0.139925i	$-0.955314\pi$
$613$	− 6.92878i	− 0.279851i	0.990162	−	0.139925i	$-0.0446863\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.4931i	− 1.06657i	−0.845935	−	0.533286i	$-0.820957\pi$
$617$	− 26.4931i	− 1.06657i	0.845935	−	0.533286i	$-0.179043\pi$
$618$	0	0
$619$	26.9597	1.08360	0.541801	−	0.840507i	$-0.317743\pi$
$619$	26.9597	1.08360	0.541801	+	0.840507i	$0.317743\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	51.7444i	2.07310i
$624$	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
$637$	− 27.7112i	− 1.09796i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.16181	−0.282874	−0.141437	−	0.989947i	$-0.545172\pi$
$641$	−7.16181	−0.282874	−0.141437	+	0.989947i	$0.545172\pi$
$642$	0	0
$643$	− 7.06308i	− 0.278541i	−0.990254	−	0.139270i	$-0.955524\pi$
$643$	− 7.06308i	− 0.278541i	0.990254	−	0.139270i	$-0.0444757\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.4037i	1.03804i	0.854763	+	0.519019i	$0.173703\pi$
$647$	26.4037i	1.03804i	−0.854763	+	0.519019i	$0.826297\pi$
$648$	0	0
$649$	−28.0723	−1.10193
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.71970i	0.302095i	0.988527	+	0.151048i	$0.0482647\pi$
$653$	7.71970i	0.302095i	−0.988527	+	0.151048i	$0.951735\pi$
$654$	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
$667$	− 48.0680i	− 1.86120i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.6252	−0.448787
$672$	0	0
$673$	− 7.83161i	− 0.301886i	−0.988542	−	0.150943i	$-0.951769\pi$
$673$	− 7.83161i	− 0.301886i	0.988542	−	0.150943i	$-0.0482311\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0034i	0.691927i	0.938248	+	0.345964i	$0.112448\pi$
$677$	18.0034i	0.691927i	−0.938248	+	0.345964i	$0.887552\pi$
$678$	0	0
$679$	46.6538	1.79041
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.3356i	1.16076i	0.814347	+	0.580379i	$0.197096\pi$
$683$	30.3356i	1.16076i	−0.814347	+	0.580379i	$0.802904\pi$
$684$	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
$697$	1.37051i	0.0519117i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.4378	1.07408	0.537041	−	0.843556i	$-0.319542\pi$
$701$	28.4378	1.07408	0.537041	+	0.843556i	$0.319542\pi$
$702$	0	0
$703$	14.1096i	0.532153i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	39.7742i	1.49586i
$708$	0	0
$709$	−30.9000	−1.16047	−0.580237	−	0.814447i	$-0.697040\pi$
$709$	−30.9000	−1.16047	−0.580237	+	0.814447i	$0.697040\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4.23427i	− 0.158575i
$714$	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
$727$	− 24.9821i	− 0.926534i	−0.886219	−	0.463267i	$-0.846677\pi$
$727$	− 24.9821i	− 0.926534i	0.886219	−	0.463267i	$-0.153323\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.603923	0.0223369
$732$	0	0
$733$	− 10.4677i	− 0.386633i	−0.981136	−	0.193316i	$-0.938076\pi$
$733$	− 10.4677i	− 0.386633i	0.981136	−	0.193316i	$-0.0619244\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 35.8254i	− 1.31964i
$738$	0	0
$739$	−35.6244	−1.31046	−0.655232	−	0.755428i	$-0.727429\pi$
$739$	−35.6244	−1.31046	−0.655232	+	0.755428i	$0.727429\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.0554i	0.368897i	0.982842	+	0.184448i	$0.0590499\pi$
$743$	10.0554i	0.368897i	−0.982842	+	0.184448i	$0.940950\pi$
$744$	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
$757$	6.42042i	0.233354i	0.993170	+	0.116677i	$0.0372243\pi$
$757$	6.42042i	0.233354i	−0.993170	+	0.116677i	$0.962776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.3910	0.920422	0.460211	−	0.887810i	$-0.347774\pi$
$761$	25.3910	0.920422	0.460211	+	0.887810i	$0.347774\pi$
$762$	0	0
$763$	11.1247i	0.402740i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 10.8416i	− 0.391468i
$768$	0	0
$769$	14.8018	0.533765	0.266883	−	0.963729i	$-0.414006\pi$
$769$	14.8018	0.533765	0.266883	+	0.963729i	$0.414006\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.48046i	0.269053i	0.990910	+	0.134527i	$0.0429514\pi$
$773$	7.48046i	0.269053i	−0.990910	+	0.134527i	$0.957049\pi$
$774$	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
$787$	52.3174i	1.86491i	0.361281	+	0.932457i	$0.382340\pi$
$787$	52.3174i	1.86491i	−0.361281	+	0.932457i	$0.617660\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.1575	0.503384
$792$	0	0
$793$	− 4.48969i	− 0.159434i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.7359i	1.90342i	0.306994	+	0.951711i	$0.400677\pi$
$797$	53.7359i	1.90342i	−0.306994	+	0.951711i	$0.599323\pi$
$798$	0	0
$799$	−26.7913	−0.947808
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 9.13624i	− 0.322411i
$804$	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
$817$	− 0.194812i	− 0.00681560i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.85864	−0.169568	−0.0847839	−	0.996399i	$-0.527020\pi$
$821$	−4.85864	−0.169568	−0.0847839	+	0.996399i	$0.527020\pi$
$822$	0	0
$823$	1.17777i	0.0410546i	0.999789	+	0.0205273i	$0.00653450\pi$
$823$	1.17777i	0.0410546i	−0.999789	+	0.0205273i	$0.993466\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32.7188i	− 1.13775i	−0.822426	−	0.568873i	$-0.807380\pi$
$827$	− 32.7188i	− 1.13775i	0.822426	−	0.568873i	$-0.192620\pi$
$828$	0	0
$829$	5.99342	0.208160	0.104080	−	0.994569i	$-0.466810\pi$
$829$	5.99342	0.208160	0.104080	+	0.994569i	$0.466810\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 125.625i	− 4.35265i
$834$	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
$847$	24.8083i	0.852425i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−48.7410	−1.67082
$852$	0	0
$853$	21.7190i	0.743644i	0.928304	+	0.371822i	$0.121267\pi$
$853$	21.7190i	0.743644i	−0.928304	+	0.371822i	$0.878733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.59966i	0.157121i	0.996909	+	0.0785607i	$0.0250324\pi$
$857$	4.59966i	0.157121i	−0.996909	+	0.0785607i	$0.974968\pi$
$858$	0	0
$859$	−14.9179	−0.508993	−0.254496	−	0.967074i	$-0.581910\pi$
$859$	−14.9179	−0.508993	−0.254496	+	0.967074i	$0.581910\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.4089i	0.388362i	0.980966	+	0.194181i	$0.0622050\pi$
$863$	11.4089i	0.388362i	−0.980966	+	0.194181i	$0.937795\pi$
$864$	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
$877$	− 27.9284i	− 0.943075i	−0.881846	−	0.471537i	$-0.843699\pi$
$877$	− 27.9284i	− 0.943075i	0.881846	−	0.471537i	$-0.156301\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.3270	−1.42603	−0.713017	−	0.701147i	$-0.752672\pi$
$881$	−42.3270	−1.42603	−0.713017	+	0.701147i	$0.752672\pi$
$882$	0	0
$883$	− 20.3895i	− 0.686161i	−0.939306	−	0.343081i	$-0.888530\pi$
$883$	− 20.3895i	− 0.686161i	0.939306	−	0.343081i	$-0.111470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 39.5451i	− 1.32779i	−0.747824	−	0.663897i	$-0.768901\pi$
$887$	− 39.5451i	− 1.32779i	0.747824	−	0.663897i	$-0.231099\pi$
$888$	0	0
$889$	35.0766	1.17643
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.64227i	0.289202i
$894$	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
$907$	− 36.9597i	− 1.22723i	−0.789607	−	0.613613i	$-0.789715\pi$
$907$	− 36.9597i	− 1.22723i	0.789607	−	0.613613i	$-0.210285\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.84670	−0.326236	−0.163118	−	0.986607i	$-0.552155\pi$
$911$	−9.84670	−0.326236	−0.163118	+	0.986607i	$0.552155\pi$
$912$	0	0
$913$	− 55.7870i	− 1.84628i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 41.7955i	− 1.38021i
$918$	0	0
$919$	8.42079	0.277776	0.138888	−	0.990308i	$-0.455647\pi$
$919$	8.42079	0.277776	0.138888	+	0.990308i	$0.455647\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.5664i	0.446544i
$924$	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
$937$	4.69342i	0.153327i	0.997057	+	0.0766637i	$0.0244268\pi$
$937$	4.69342i	0.153327i	−0.997057	+	0.0766637i	$0.975573\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.0256	0.620216	0.310108	−	0.950701i	$-0.399635\pi$
$941$	19.0256	0.620216	0.310108	+	0.950701i	$0.399635\pi$
$942$	0	0
$943$	1.52720i	0.0497325i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.8169i	0.416492i	0.978076	+	0.208246i	$0.0667755\pi$
$947$	12.8169i	0.416492i	−0.978076	+	0.208246i	$0.933224\pi$
$948$	0	0
$949$	3.52844	0.114538
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 45.9701i	− 1.48912i	−0.667556	−	0.744560i	$-0.732660\pi$
$953$	− 45.9701i	− 1.48912i	0.667556	−	0.744560i	$-0.267340\pi$
$954$	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
$967$	− 5.14709i	− 0.165519i	−0.996570	−	0.0827596i	$-0.973627\pi$
$967$	− 5.14709i	− 0.165519i	0.996570	−	0.0827596i	$-0.0263734\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.3875	−0.333352	−0.166676	−	0.986012i	$-0.553303\pi$
$971$	−10.3875	−0.333352	−0.166676	+	0.986012i	$0.553303\pi$
$972$	0	0
$973$	19.5632i	0.627169i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.5621i	− 1.26570i	−0.774272	−	0.632852i	$-0.781884\pi$
$977$	− 39.5621i	− 1.26570i	0.774272	−	0.632852i	$-0.218116\pi$
$978$	0	0
$979$	41.3867	1.32272
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.4719i	0.652954i	0.945205	+	0.326477i	$0.105862\pi$
$983$	20.4719i	0.652954i	−0.945205	+	0.326477i	$0.894138\pi$
$984$	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
$997$	23.2385i	0.735971i	0.929832	+	0.367986i	$0.119952\pi$
$997$	23.2385i	0.735971i	−0.929832	+	0.367986i	$0.880048\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.99574i q^{7} +3.99574 q^{11} +1.54316i q^{13} +6.99574i q^{17} +2.25667 q^{19} +7.79556i q^{23} -6.16607 q^{29} -0.543164 q^{31} +6.25240i q^{37} +0.195906 q^{41} -0.0863273i q^{43} +3.82966i q^{47} -17.9574 q^{49} +4.19164i q^{53} -7.02557 q^{59} -2.90941 q^{61} -8.96590i q^{67} +8.79130 q^{71} -2.28650i q^{73} +19.9616i q^{77} +12.6442 q^{79} -13.9616i q^{83} +10.3577 q^{89} -7.70924 q^{91} -9.33873i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{11} + 8 q^{19} - 18 q^{29} + 4 q^{31} - 18 q^{41} - 18 q^{49} - 30 q^{59} - 2 q^{61} - 24 q^{71} + 14 q^{79} - 6 q^{89} - 22 q^{91}+O(q^{100}) \) 8 * q - 6 * q^11 + 8 * q^19 - 18 * q^29 + 4 * q^31 - 18 * q^41 - 18 * q^49 - 30 * q^59 - 2 * q^61 - 24 * q^71 + 14 * q^79 - 6 * q^89 - 22 * q^91

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