LMFDB - Embedded newform 484.3.f.b.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [484,3,Mod(161,484)] 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("484.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(484, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 484 = 2^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	484.f (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-5,0,-1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.1880447950$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.18530015625.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 9x^{6} - 17x^{5} + 89x^{4} + 136x^{3} + 576x^{2} + 512x + 4096 $ x^8 - x^7 + 9x^6 - 17x^5 + 89x^4 + 136x^3 + 576x^2 + 512x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			161.2
Root			$-1.04209 + 3.20723i$ of defining polynomial
Character	$\chi$	$=$	484.161
Dual form			484.3.f.b.481.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66013 - 5.10934i) q^{3} +(6.56666 - 4.77096i) q^{5} +(-16.0682 - 11.6742i) q^{9} +(-13.4750 - 41.4717i) q^{15} +8.35053 q^{23} +(12.6336 - 38.8822i) q^{25} +(-47.2066 + 34.2976i) q^{27} +(19.8891 + 14.4502i) q^{31} +(22.5020 + 69.2539i) q^{37} -161.212 q^{45} +(15.4508 - 47.5528i) q^{47} +(-39.6418 + 28.8015i) q^{49} +(56.6312 + 41.1450i) q^{53} +(-29.8462 - 91.8571i) q^{59} -129.519 q^{67} +(13.8629 - 42.6657i) q^{69} +(-18.9438 + 13.7635i) q^{71} +(-177.689 - 129.099i) q^{75} +(41.6316 + 128.129i) q^{81} +177.753 q^{89} +(106.850 - 77.6308i) q^{93} +(156.938 + 114.022i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - q^{5} - 11 q^{9} + 47 q^{15} - 140 q^{23} - 99 q^{25} - 65 q^{27} - 37 q^{31} - 25 q^{37} - 968 q^{45} - 100 q^{47} - 98 q^{49} + 140 q^{53} + 107 q^{59} - 140 q^{67} - 61 q^{69} - 133 q^{71}+ \cdots + 95 q^{97}+O(q^{100}) $ 8 * q - 5 * q^3 - q^5 - 11 * q^9 + 47 * q^15 - 140 * q^23 - 99 * q^25 - 65 * q^27 - 37 * q^31 - 25 * q^37 - 968 * q^45 - 100 * q^47 - 98 * q^49 + 140 * q^53 + 107 * q^59 - 140 * q^67 - 61 * q^69 - 133 * q^71 - 198 * q^75 - 212 * q^81 + 388 * q^89 + 155 * q^93 + 95 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/484\mathbb{Z}\right)^\times$.

$n$	$243$	$365$
$\chi(n)$	$1$	$e\left(\frac{7}{10}\right)$

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
$3$	1.66013	−	5.10934i	0.553375	−	1.70311i	−0.146820	−	0.989163i	$-0.546904\pi$
$3$	1.66013	−	5.10934i	0.553375	−	1.70311i	0.700195	−	0.713951i	$-0.253096\pi$
$4$		0			0
$5$	6.56666	−	4.77096i	1.31333	−	0.954192i	0.313343	−	0.949640i	$-0.398551\pi$
$5$	6.56666	−	4.77096i	1.31333	−	0.954192i	0.999990	−	0.00455232i	$-0.00144905\pi$
$6$		0			0
$7$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$7$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$8$		0			0
$9$	−16.0682	−	11.6742i	−1.78536	−	1.29714i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$13$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$14$		0			0
$15$	−13.4750	−	41.4717i	−0.898332	−	2.76478i
$16$		0			0
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$		0			0
$19$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$19$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.35053			0.363067			0.181533	−	0.983385i	$-0.441894\pi$
$23$	8.35053			0.363067			0.181533	+	0.983385i	$0.441894\pi$
$24$		0			0
$25$	12.6336	−	38.8822i	0.505344	−	1.55529i
$26$		0			0
$27$	−47.2066	+	34.2976i	−1.74839	+	1.27028i
$28$		0			0
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$	19.8891	+	14.4502i	0.641582	+	0.466137i	0.860393	−	0.509630i	$-0.170218\pi$
$31$	19.8891	+	14.4502i	0.641582	+	0.466137i	−0.218811	+	0.975767i	$0.570218\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	22.5020	+	69.2539i	0.608161	+	1.87173i	0.473393	+	0.880851i	$0.343029\pi$
$37$	22.5020	+	69.2539i	0.608161	+	1.87173i	0.134768	+	0.990877i	$0.456971\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−161.212			−3.58249
$46$		0			0
$47$	15.4508	−	47.5528i	0.328741	−	1.01176i	−0.640982	−	0.767556i	$-0.721473\pi$
$47$	15.4508	−	47.5528i	0.328741	−	1.01176i	0.969723	−	0.244206i	$-0.0785274\pi$
$48$		0			0
$49$	−39.6418	+	28.8015i	−0.809017	+	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	56.6312	+	41.1450i	1.06851	+	0.776320i	0.975644	−	0.219358i	$-0.0703963\pi$
$53$	56.6312	+	41.1450i	1.06851	+	0.776320i	0.0928686	+	0.995678i	$0.470396\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−29.8462	−	91.8571i	−0.505867	−	1.55690i	−0.799308	−	0.600922i	$-0.794800\pi$
$59$	−29.8462	−	91.8571i	−0.505867	−	1.55690i	0.293441	−	0.955977i	$-0.405200\pi$
$60$		0			0
$61$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$61$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−129.519			−1.93312			−0.966559	−	0.256442i	$-0.917450\pi$
$67$	−129.519			−1.93312			−0.966559	+	0.256442i	$0.917450\pi$
$68$		0			0
$69$	13.8629	−	42.6657i	0.200912	−	0.618344i
$70$		0			0
$71$	−18.9438	+	13.7635i	−0.266814	+	0.193851i	−0.713145	−	0.701016i	$-0.752730\pi$
$71$	−18.9438	+	13.7635i	−0.266814	+	0.193851i	0.446332	+	0.894868i	$0.352730\pi$
$72$		0			0
$73$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$73$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$74$		0			0
$75$	−177.689	−	129.099i	−2.36919	−	1.72132i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$79$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$80$		0			0
$81$	41.6316	+	128.129i	0.513971	+	1.58184i
$82$		0			0
$83$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$83$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	177.753			1.99722			0.998610	−	0.0526989i	$-0.0167824\pi$
$89$	177.753			1.99722			0.998610	+	0.0526989i	$0.0167824\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	106.850	−	77.6308i	1.14892	−	0.834739i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	156.938	+	114.022i	1.61792	+	1.17549i	0.815933	+	0.578146i	$0.196224\pi$
$97$	156.938	+	114.022i	1.61792	+	1.17549i	0.801987	+	0.597342i	$0.203776\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$101$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$102$		0			0
$103$	−58.7132	−	180.701i	−0.570031	−	1.75438i	−0.652507	−	0.757783i	$-0.726283\pi$
$103$	−58.7132	−	180.701i	−0.570031	−	1.75438i	0.0824753	−	0.996593i	$-0.473717\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$107$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	391.198			3.52431
$112$		0			0
$113$	−14.5801	+	44.8728i	−0.129027	+	0.397105i	−0.994613	−	0.103655i	$-0.966946\pi$
$113$	−14.5801	+	44.8728i	−0.129027	+	0.397105i	0.865586	+	0.500760i	$0.166946\pi$
$114$		0			0
$115$	54.8351	−	39.8401i	0.476827	−	0.346435i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−39.8388	−	122.611i	−0.318710	−	0.980890i
$126$		0			0
$127$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$127$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−146.357	+	450.441i	−1.08413	+	3.33660i
$136$		0			0
$137$	−155.993	+	113.336i	−1.13863	+	0.827267i	−0.986928	−	0.161159i	$-0.948477\pi$
$137$	−155.993	+	113.336i	−1.13863	+	0.827267i	−0.151706	+	0.988426i	$0.548477\pi$
$138$		0			0
$139$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$139$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$140$		0			0
$141$	−217.313	−	157.887i	−1.54123	−	1.11977i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	81.3462	+	250.358i	0.553375	+	1.70311i
$148$		0			0
$149$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$149$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$151$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	199.546			1.28740
$156$		0			0
$157$	28.0240	−	86.2489i	0.178497	−	0.549356i	−0.821279	−	0.570526i	$-0.806739\pi$
$157$	28.0240	−	86.2489i	0.178497	−	0.549356i	0.999776	+	0.0211705i	$0.00673928\pi$
$158$		0			0
$159$	304.239	−	221.042i	1.91345	−	1.39020i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	56.6312	+	41.1450i	0.347431	+	0.252423i	0.747790	−	0.663935i	$-0.231115\pi$
$163$	56.6312	+	41.1450i	0.347431	+	0.252423i	−0.400360	+	0.916358i	$0.631115\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$167$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$168$		0			0
$169$	52.2239	+	160.729i	0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−518.878			−2.93151
$178$		0			0
$179$	80.3721	−	247.360i	0.449006	−	1.38190i	−0.429024	−	0.903293i	$-0.641142\pi$
$179$	80.3721	−	247.360i	0.449006	−	1.38190i	0.878030	−	0.478606i	$-0.158858\pi$
$180$		0			0
$181$	280.665	−	203.915i	1.55064	−	1.12660i	0.607440	−	0.794366i	$-0.292197\pi$
$181$	280.665	−	203.915i	1.55064	−	1.12660i	0.943196	−	0.332237i	$-0.107803\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	478.171	+	347.411i	2.58471	+	1.87790i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	117.454	+	361.487i	0.614943	+	1.89260i	0.402470	+	0.915433i	$0.368152\pi$
$191$	117.454	+	361.487i	0.614943	+	1.89260i	0.212473	+	0.977167i	$0.431848\pi$
$192$		0			0
$193$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$193$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	2.00000			0.0100503			0.00502513	−	0.999987i	$-0.498400\pi$
$199$	2.00000			0.0100503			0.00502513	+	0.999987i	$0.498400\pi$
$200$		0			0
$201$	−215.018	+	661.757i	−1.06974	+	3.29232i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−134.178	−	97.4861i	−0.648204	−	0.470947i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$		0			0
$213$	38.8732	+	119.639i	0.182503	+	0.561687i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	76.7445	−	236.195i	0.344146	−	1.05917i	−0.617894	−	0.786262i	$-0.712014\pi$
$223$	76.7445	−	236.195i	0.344146	−	1.05917i	0.962039	−	0.272910i	$-0.0879863\pi$
$224$		0			0
$225$	−656.919	+	477.280i	−2.91964	+	2.12124i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$	−279.720	−	203.228i	−1.22148	−	0.887460i	−0.225261	−	0.974298i	$-0.572324\pi$
$229$	−279.720	−	203.228i	−1.22148	−	0.887460i	−0.996222	+	0.0868385i	$0.972324\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$		0			0
$235$	−125.412	−	385.979i	−0.533669	−	1.64246i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$239$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	198.614			0.817342
$244$		0			0
$245$	−122.904	+	378.259i	−0.501649	+	1.54392i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	405.526	+	294.632i	1.61564	+	1.17383i	0.839768	+	0.542945i	$0.182691\pi$
$251$	405.526	+	294.632i	1.61564	+	1.17383i	0.775874	+	0.630888i	$0.217309\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−58.7132	−	180.701i	−0.228456	−	0.703116i	−0.997922	−	0.0644282i	$-0.979478\pi$
$257$	−58.7132	−	180.701i	−0.228456	−	0.703116i	0.769466	−	0.638688i	$-0.220522\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	568.179			2.14407
$266$		0			0
$267$	295.092	−	908.199i	1.10521	−	3.40150i
$268$		0			0
$269$	−292.864	+	212.778i	−1.08871	+	0.790997i	−0.979182	−	0.202986i	$-0.934935\pi$
$269$	−292.864	+	212.778i	−1.08871	+	0.790997i	−0.109533	+	0.993983i	$0.534935\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$277$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$278$		0			0
$279$	−150.886	−	464.379i	−0.540810	−	1.66444i
$280$		0			0
$281$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$281$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$283$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	89.3059	−	274.855i	0.309017	−	0.951057i
$290$		0			0
$291$	843.116	−	612.560i	2.89731	−	2.10502i
$292$		0			0
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$	−634.236	−	460.800i	−2.14995	−	1.56203i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−1020.73			−3.30334
$310$		0			0
$311$	−147.710	+	454.605i	−0.474952	+	1.46175i	0.371070	+	0.928605i	$0.378991\pi$
$311$	−147.710	+	454.605i	−0.474952	+	1.46175i	−0.846022	+	0.533148i	$0.821009\pi$
$312$		0			0
$313$	290.162	−	210.815i	0.927036	−	0.673531i	−0.0182297	−	0.999834i	$-0.505803\pi$
$313$	290.162	−	210.815i	0.927036	−	0.673531i	0.945265	+	0.326303i	$0.105803\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−400.756	−	291.166i	−1.26421	−	0.918505i	−0.265257	−	0.964178i	$-0.585457\pi$
$317$	−400.756	−	291.166i	−1.26421	−	0.918505i	−0.998956	+	0.0456731i	$0.985457\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.0895			0.0606935			0.0303467	−	0.999539i	$-0.490339\pi$
$331$	20.0895			0.0606935			0.0303467	+	0.999539i	$0.490339\pi$
$332$		0			0
$333$	446.921	−	1375.48i	1.34210	−	4.13057i
$334$		0			0
$335$	−850.508	+	617.930i	−2.53883	+	1.84457i
$336$		0			0
$337$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$337$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$338$		0			0
$339$	205.066	+	148.989i	0.604914	+	0.439496i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−112.523	−	346.311i	−0.326154	−	1.00380i
$346$		0			0
$347$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$347$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$349$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.1441			0.0797285			0.0398642	−	0.999205i	$-0.487307\pi$
$353$	28.1441			0.0797285			0.0398642	+	0.999205i	$0.487307\pi$
$354$		0			0
$355$	−58.7325	+	180.760i	−0.165444	+	0.509183i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$		0			0
$361$	−292.055	−	212.190i	−0.809017	−	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−196.635	−	605.179i	−0.535789	−	1.64899i	−0.741937	−	0.670469i	$-0.766093\pi$
$367$	−196.635	−	605.179i	−0.535789	−	1.64899i	0.206148	−	0.978521i	$-0.433907\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−692.600			−1.84693
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−576.638	+	418.952i	−1.52147	+	1.10541i	−0.560719	+	0.828006i	$0.689475\pi$
$379$	−576.638	+	418.952i	−1.52147	+	1.10541i	−0.960752	+	0.277407i	$0.910525\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	396.029	+	287.732i	1.03402	+	0.751259i	0.969109	−	0.246633i	$-0.0793241\pi$
$383$	396.029	+	287.732i	1.03402	+	0.751259i	0.0649097	+	0.997891i	$0.479324\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	231.895	+	713.698i	0.596130	+	1.83470i	0.549020	+	0.835809i	$0.315001\pi$
$389$	231.895	+	713.698i	0.596130	+	1.83470i	0.0471101	+	0.998890i	$0.484999\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−790.000			−1.98992			−0.994962	−	0.100251i	$-0.968036\pi$
$397$	−790.000			−1.98992			−0.994962	+	0.100251i	$0.968036\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−79.2837	+	57.6030i	−0.197715	+	0.143648i	−0.682238	−	0.731130i	$-0.738993\pi$
$401$	−79.2837	+	57.6030i	−0.197715	+	0.143648i	0.484523	+	0.874778i	$0.338993\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	884.680	+	642.757i	2.18439	+	1.58706i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$409$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$410$		0			0
$411$	320.102	+	985.173i	0.778837	+	2.39701i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−262.000			−0.625298			−0.312649	−	0.949869i	$-0.601216\pi$
$419$	−262.000			−0.625298			−0.312649	+	0.949869i	$0.601216\pi$
$420$		0			0
$421$	−229.291	+	705.684i	−0.544633	+	1.67621i	0.177227	+	0.984170i	$0.443287\pi$
$421$	−229.291	+	705.684i	−0.544633	+	1.67621i	−0.721860	+	0.692039i	$0.756713\pi$
$422$		0			0
$423$	−803.411	+	583.712i	−1.89932	+	1.37993i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$431$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$432$		0			0
$433$	235.522	+	724.862i	0.543931	+	1.67405i	0.723518	+	0.690305i	$0.242524\pi$
$433$	235.522	+	724.862i	0.543931	+	1.67405i	−0.179588	+	0.983742i	$0.557476\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	973.209			2.20682
$442$		0			0
$443$	−136.276	+	419.413i	−0.307620	+	0.946756i	0.671067	+	0.741397i	$0.265836\pi$
$443$	−136.276	+	419.413i	−0.307620	+	0.946756i	−0.978687	+	0.205359i	$0.934164\pi$
$444$		0			0
$445$	1167.24	−	848.051i	2.62302	−	1.90573i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	705.135	+	512.311i	1.57046	+	1.14100i	0.926718	+	0.375759i	$0.122618\pi$
$449$	705.135	+	512.311i	1.57046	+	1.14100i	0.643739	+	0.765245i	$0.277382\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$457$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−818.866			−1.76861			−0.884305	−	0.466910i	$-0.845367\pi$
$463$	−818.866			−1.76861			−0.884305	+	0.466910i	$0.845367\pi$
$464$		0			0
$465$	331.272	−	1019.55i	0.712413	−	2.19258i
$466$		0			0
$467$	−283.545	+	206.008i	−0.607163	+	0.441130i	−0.848414	−	0.529333i	$-0.822442\pi$
$467$	−283.545	+	206.008i	−0.607163	+	0.441130i	0.241251	+	0.970463i	$0.422442\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−394.152	−	286.368i	−0.836840	−	0.608000i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−429.626	−	1322.25i	−0.900683	−	2.77202i
$478$		0			0
$479$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$479$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1574.56			3.24651
$486$		0			0
$487$	−20.6966	+	63.6977i	−0.0424982	+	0.130796i	−0.970054	−	0.242888i	$-0.921905\pi$
$487$	−20.6966	+	63.6977i	−0.0424982	+	0.130796i	0.927556	+	0.373684i	$0.121905\pi$
$488$		0			0
$489$	304.239	−	221.042i	0.622165	−	0.452029i
$490$		0			0
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	186.028	+	572.536i	0.372802	+	1.14737i	0.944950	+	0.327216i	$0.106110\pi$
$499$	186.028	+	572.536i	0.372802	+	1.14737i	−0.572147	+	0.820151i	$0.693890\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	907.916			1.79076
$508$		0			0
$509$	−115.649	+	355.930i	−0.227208	+	0.699274i	0.770852	+	0.637014i	$0.219831\pi$
$509$	−115.649	+	355.930i	−0.227208	+	0.699274i	−0.998060	+	0.0622597i	$0.980169\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1247.67	−	906.483i	−2.42265	−	1.76016i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−46.8454	−	144.175i	−0.0899144	−	0.276728i	0.895981	−	0.444093i	$-0.146474\pi$
$521$	−46.8454	−	144.175i	−0.0899144	−	0.276728i	−0.985895	+	0.167365i	$0.946474\pi$
$522$		0			0
$523$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$523$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−459.269			−0.868183
$530$		0			0
$531$	−592.787	+	1824.41i	−1.11636	+	3.43580i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1130.42	−	821.297i	−2.10506	−	1.52942i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$541$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$542$		0			0
$543$	−575.933	−	1772.54i	−1.06065	−	3.26434i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$547$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	2568.87	−	1866.39i	4.62859	−	3.36287i
$556$		0			0
$557$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$557$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$563$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$564$		0			0
$565$	118.344	+	364.226i	0.209459	+	0.644647i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$569$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	2041.95			3.56361
$574$		0			0
$575$	105.497	−	324.687i	0.183473	−	0.564673i
$576$		0			0
$577$	99.1113	−	72.0086i	0.171770	−	0.124798i	−0.498579	−	0.866844i	$-0.666145\pi$
$577$	99.1113	−	72.0086i	0.171770	−	0.124798i	0.670349	+	0.742046i	$0.266145\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	349.189	+	1074.69i	0.594871	+	1.83082i	0.555368	+	0.831605i	$0.312577\pi$
$587$	349.189	+	1074.69i	0.594871	+	1.83082i	0.0395031	+	0.999219i	$0.487423\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.32025	−	10.2187i	0.00556156	−	0.0171167i
$598$		0			0
$599$	−79.2837	+	57.6030i	−0.132360	+	0.0961652i	−0.651995	−	0.758223i	$-0.726068\pi$
$599$	−79.2837	+	57.6030i	−0.132360	+	0.0961652i	0.519635	+	0.854388i	$0.326068\pi$
$600$		0			0
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	2081.14	+	1512.04i	3.45131	+	2.50752i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$607$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	530.000			0.858995			0.429498	−	0.903068i	$-0.358691\pi$
$617$	530.000			0.858995			0.429498	+	0.903068i	$0.358691\pi$
$618$		0			0
$619$	177.813	−	547.253i	0.287259	−	0.884092i	−0.698454	−	0.715655i	$-0.746128\pi$
$619$	177.813	−	547.253i	0.287259	−	0.884092i	0.985712	−	0.168437i	$-0.0538719\pi$
$620$		0			0
$621$	−394.200	+	286.403i	−0.634783	+	0.461197i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.7024	−	14.3147i	−0.0315239	−	0.0229034i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	280.615	+	863.644i	0.444715	+	1.36869i	0.882796	+	0.469756i	$0.155658\pi$
$631$	280.615	+	863.644i	0.444715	+	1.36869i	−0.438081	+	0.898935i	$0.644342\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	465.070			0.727810
$640$		0			0
$641$	−394.389	+	1213.80i	−0.615271	+	1.89361i	−0.217806	+	0.975992i	$0.569890\pi$
$641$	−394.389	+	1213.80i	−0.615271	+	1.89361i	−0.397465	+	0.917617i	$0.630110\pi$
$642$		0			0
$643$	−697.673	+	506.889i	−1.08503	+	0.788319i	−0.978553	−	0.205996i	$-0.933957\pi$
$643$	−697.673	+	506.889i	−1.08503	+	0.788319i	−0.106476	+	0.994315i	$0.533957\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	587.080	+	426.539i	0.907388	+	0.659256i	0.940353	−	0.340201i	$-0.110495\pi$
$647$	587.080	+	426.539i	0.907388	+	0.659256i	−0.0329650	+	0.999457i	$0.510495\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−245.355	−	755.126i	−0.375735	−	1.15639i	−0.942981	−	0.332846i	$-0.891991\pi$
$653$	−245.355	−	755.126i	−0.375735	−	1.15639i	0.567246	−	0.823549i	$-0.308009\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	16.4051			0.0248187			0.0124093	−	0.999923i	$-0.496050\pi$
$661$	16.4051			0.0248187			0.0124093	+	0.999923i	$0.496050\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1079.40	−	784.228i	−1.61345	−	1.17224i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$673$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$674$		0			0
$675$	737.177	+	2268.80i	1.09211	+	3.36118i
$676$		0			0
$677$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$677$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−790.000			−1.15666			−0.578331	−	0.815802i	$-0.696296\pi$
$683$	−790.000			−1.15666			−0.578331	+	0.815802i	$0.696296\pi$
$684$		0			0
$685$	−483.634	+	1488.47i	−0.706035	+	2.17295i
$686$		0			0
$687$	−1502.73	+	1091.80i	−2.18738	+	1.58923i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	150.422	+	109.288i	0.217687	+	0.158159i	0.691285	−	0.722582i	$-0.257045\pi$
$691$	150.422	+	109.288i	0.217687	+	0.158159i	−0.473598	+	0.880741i	$0.657045\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−2180.30			−3.09262
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1089.83	+	791.806i	−1.53713	+	1.11679i	−0.585031	+	0.811011i	$0.698918\pi$
$709$	−1089.83	+	791.806i	−1.53713	+	1.11679i	−0.952102	+	0.305781i	$0.901082\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	166.084	+	120.667i	0.232937	+	0.169239i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−406.027	−	1249.62i	−0.564711	−	1.73800i	−0.668808	−	0.743435i	$-0.733195\pi$
$719$	−406.027	−	1249.62i	−0.564711	−	1.73800i	0.104097	−	0.994567i	$-0.466805\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1134.19			−1.56010			−0.780050	−	0.625717i	$-0.784806\pi$
$727$	−1134.19			−1.56010			−0.780050	+	0.625717i	$0.784806\pi$
$728$		0			0
$729$	−44.9603	+	138.374i	−0.0616740	+	0.189813i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$733$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$734$		0			0
$735$	1728.62	+	1255.92i	2.35186	+	1.70873i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$739$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$743$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	456.553	−	1405.13i	0.607927	−	1.87101i	0.132669	−	0.991160i	$-0.457645\pi$
$751$	456.553	−	1405.13i	0.607927	−	1.87101i	0.475258	−	0.879846i	$-0.342355\pi$
$752$		0			0
$753$	2178.60	−	1582.85i	2.89323	−	2.10205i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	56.6312	+	41.1450i	0.0748100	+	0.0543527i	0.624562	−	0.780975i	$-0.285278\pi$
$757$	56.6312	+	41.1450i	0.0748100	+	0.0543527i	−0.549752	+	0.835328i	$0.685278\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$761$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−1020.73			−1.32391
$772$		0			0
$773$	423.353	−	1302.95i	0.547676	−	1.68557i	−0.166866	−	0.985980i	$-0.553365\pi$
$773$	423.353	−	1302.95i	0.547676	−	1.68557i	0.714542	−	0.699593i	$-0.246635\pi$
$774$		0			0
$775$	813.127	−	590.771i	1.04920	−	0.762286i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−227.466	−	700.068i	−0.289766	−	0.891807i
$786$		0			0
$787$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$787$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	943.249	−	2903.02i	1.18648	−	3.65160i
$796$		0			0
$797$	−522.636	+	379.717i	−0.655754	+	0.476433i	−0.865226	−	0.501381i	$-0.832825\pi$
$797$	−522.636	+	379.717i	−0.655754	+	0.476433i	0.209472	+	0.977815i	$0.432825\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−2856.17	−	2075.13i	−3.56575	−	2.59067i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	600.966	+	1849.58i	0.744691	+	2.29192i
$808$		0			0
$809$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$809$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$810$		0			0
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	568.179			0.697152
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$821$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$822$		0			0
$823$	−719.359	−	522.645i	−0.874069	−	0.635048i	0.0576069	−	0.998339i	$-0.481653\pi$
$823$	−719.359	−	522.645i	−0.874069	−	0.635048i	−0.931676	+	0.363291i	$0.881653\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$827$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$828$		0			0
$829$	−259.865	−	799.784i	−0.313469	−	0.964757i	−0.976380	−	0.216060i	$-0.930679\pi$
$829$	−259.865	−	799.784i	−0.313469	−	0.964757i	0.662912	−	0.748698i	$-0.269321\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1434.50			−1.71386
$838$		0			0
$839$	−492.968	+	1517.20i	−0.587567	+	1.80834i	0.00114272	+	0.999999i	$0.499636\pi$
$839$	−492.968	+	1517.20i	−0.587567	+	1.80834i	−0.588709	+	0.808345i	$0.700364\pi$
$840$		0			0
$841$	−680.383	+	494.327i	−0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1109.77	+	806.292i	1.31333	+	0.954192i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	187.903	+	578.307i	0.220803	+	0.679562i
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	1714.11			1.99547			0.997736	−	0.0672491i	$-0.0214222\pi$
$859$	1714.11			1.99547			0.997736	+	0.0672491i	$0.0214222\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	347.877	−	252.748i	0.403102	−	0.292871i	−0.367701	−	0.929944i	$-0.619855\pi$
$863$	347.877	−	252.748i	0.403102	−	0.292871i	0.770803	+	0.637073i	$0.219855\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1256.07	−	912.589i	−1.44876	−	1.05258i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1190.59	−	3664.27i	−1.36380	−	4.19733i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$877$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	75.1002			0.0852442			0.0426221	−	0.999091i	$-0.486429\pi$
$881$	75.1002			0.0852442			0.0426221	+	0.999091i	$0.486429\pi$
$882$		0			0
$883$	423.353	−	1302.95i	0.479449	−	1.47559i	−0.360414	−	0.932793i	$-0.617364\pi$
$883$	423.353	−	1302.95i	0.479449	−	1.47559i	0.839863	−	0.542799i	$-0.182636\pi$
$884$		0			0
$885$	−3407.30	+	2475.54i	−3.85005	+	2.79723i
$886$		0			0
$887$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$887$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−652.368	−	2007.78i	−0.728902	−	2.24333i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	870.162	−	2678.08i	0.961505	−	2.95921i
$906$		0			0
$907$	1415.78	−	1028.62i	1.56095	−	1.13410i	0.625721	−	0.780047i	$-0.284805\pi$
$907$	1415.78	−	1028.62i	1.56095	−	1.13410i	0.935227	−	0.354048i	$-0.115195\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1438.43	−	1045.08i	−1.57896	−	1.14718i	−0.917875	−	0.396870i	$-0.870096\pi$
$911$	−1438.43	−	1045.08i	−1.57896	−	1.14718i	−0.661085	−	0.750311i	$-0.729904\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$919$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2977.03			3.21841
$926$		0			0
$927$	−1166.13	+	3588.97i	−1.25796	+	3.87160i
$928$		0			0
$929$	775.038	−	563.098i	0.834272	−	0.606134i	−0.0864930	−	0.996252i	$-0.527566\pi$
$929$	775.038	−	563.098i	0.834272	−	0.606134i	0.920765	+	0.390119i	$0.127566\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	2077.52	+	1509.40i	2.22670	+	1.61780i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$937$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$	−595.421	−	1832.52i	−0.634101	−	1.95156i
$940$		0			0
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1823.54			−1.92560			−0.962798	−	0.270221i	$-0.912903\pi$
$947$	−1823.54			−1.92560			−0.962798	+	0.270221i	$0.912903\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2152.97	+	1564.23i	−2.26390	+	1.64482i
$952$		0			0
$953$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$953$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$954$		0			0
$955$	2495.92	+	1813.39i	2.61353	+	1.89884i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−110.200	−	339.162i	−0.114673	−	0.352926i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	506.412	−	1558.58i	0.521537	−	1.60513i	−0.249527	−	0.968368i	$-0.580275\pi$
$971$	506.412	−	1558.58i	0.521537	−	1.60513i	0.771064	−	0.636758i	$-0.219725\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	571.067	+	414.904i	0.584510	+	0.424672i	0.840347	−	0.542048i	$-0.182351\pi$
$977$	571.067	+	414.904i	0.584510	+	0.424672i	−0.255837	+	0.966720i	$0.582351\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	399.822	+	1230.52i	0.406736	+	1.25180i	0.919437	+	0.393238i	$0.128645\pi$
$983$	399.822	+	1230.52i	0.406736	+	1.25180i	−0.512701	+	0.858567i	$0.671355\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1582.00			−1.59637			−0.798184	−	0.602414i	$-0.794206\pi$
$991$	−1582.00			−1.59637			−0.798184	+	0.602414i	$0.794206\pi$
$992$		0			0
$993$	33.3512	−	102.644i	0.0335863	−	0.103368i
$994$		0			0
$995$	13.1333	−	9.54192i	0.0131993	−	0.00958987i
$996$		0			0
$997$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$997$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$998$		0			0
$999$	−3437.48	−	2497.48i	−3.44092	−	2.49998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	484.3.f.b.161.2		8
11.2	odd	10	inner	484.3.f.b.457.1		8
11.3	even	5	inner	484.3.f.b.481.2		8
11.4	even	5	inner	484.3.f.b.233.1		8
11.5	even	5		44.3.d.a.21.2	✓	2
11.6	odd	10		44.3.d.a.21.2	✓	2
11.7	odd	10	inner	484.3.f.b.233.1		8
11.8	odd	10	inner	484.3.f.b.481.2		8
11.9	even	5	inner	484.3.f.b.457.1		8
11.10	odd	2	CM	484.3.f.b.161.2		8
33.5	odd	10		396.3.f.a.109.2		2
33.17	even	10		396.3.f.a.109.2		2
44.27	odd	10		176.3.h.b.65.1		2
44.39	even	10		176.3.h.b.65.1		2
55.17	even	20		1100.3.e.a.549.1		4
55.27	odd	20		1100.3.e.a.549.1		4
55.28	even	20		1100.3.e.a.549.4		4
55.38	odd	20		1100.3.e.a.549.4		4
55.39	odd	10		1100.3.f.a.901.1		2
55.49	even	10		1100.3.f.a.901.1		2
77.6	even	10		2156.3.h.a.197.1		2
77.27	odd	10		2156.3.h.a.197.1		2
88.5	even	10		704.3.h.c.65.1		2
88.27	odd	10		704.3.h.f.65.2		2
88.61	odd	10		704.3.h.c.65.1		2
88.83	even	10		704.3.h.f.65.2		2
132.71	even	10		1584.3.j.c.1297.2		2
132.83	odd	10		1584.3.j.c.1297.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.3.d.a.21.2	✓	2	11.5	even	5
44.3.d.a.21.2	✓	2	11.6	odd	10
176.3.h.b.65.1		2	44.27	odd	10
176.3.h.b.65.1		2	44.39	even	10
396.3.f.a.109.2		2	33.5	odd	10
396.3.f.a.109.2		2	33.17	even	10
484.3.f.b.161.2		8	1.1	even	1	trivial
484.3.f.b.161.2		8	11.10	odd	2	CM
484.3.f.b.233.1		8	11.4	even	5	inner
484.3.f.b.233.1		8	11.7	odd	10	inner
484.3.f.b.457.1		8	11.2	odd	10	inner
484.3.f.b.457.1		8	11.9	even	5	inner
484.3.f.b.481.2		8	11.3	even	5	inner
484.3.f.b.481.2		8	11.8	odd	10	inner
704.3.h.c.65.1		2	88.5	even	10
704.3.h.c.65.1		2	88.61	odd	10
704.3.h.f.65.2		2	88.27	odd	10
704.3.h.f.65.2		2	88.83	even	10
1100.3.e.a.549.1		4	55.17	even	20
1100.3.e.a.549.1		4	55.27	odd	20
1100.3.e.a.549.4		4	55.28	even	20
1100.3.e.a.549.4		4	55.38	odd	20
1100.3.f.a.901.1		2	55.39	odd	10
1100.3.f.a.901.1		2	55.49	even	10
1584.3.j.c.1297.2		2	132.71	even	10
1584.3.j.c.1297.2		2	132.83	odd	10
2156.3.h.a.197.1		2	77.6	even	10
2156.3.h.a.197.1		2	77.27	odd	10

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:	\( N \)	\(=\)	\( 484 = 2^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	484.f (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.66013 - 5.10934i) q^{3} +(6.56666 - 4.77096i) q^{5} +(-16.0682 - 11.6742i) q^{9} +(-13.4750 - 41.4717i) q^{15} +8.35053 q^{23} +(12.6336 - 38.8822i) q^{25} +(-47.2066 + 34.2976i) q^{27} +(19.8891 + 14.4502i) q^{31} +(22.5020 + 69.2539i) q^{37} -161.212 q^{45} +(15.4508 - 47.5528i) q^{47} +(-39.6418 + 28.8015i) q^{49} +(56.6312 + 41.1450i) q^{53} +(-29.8462 - 91.8571i) q^{59} -129.519 q^{67} +(13.8629 - 42.6657i) q^{69} +(-18.9438 + 13.7635i) q^{71} +(-177.689 - 129.099i) q^{75} +(41.6316 + 128.129i) q^{81} +177.753 q^{89} +(106.850 - 77.6308i) q^{93} +(156.938 + 114.022i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - q^{5} - 11 q^{9} + 47 q^{15} - 140 q^{23} - 99 q^{25} - 65 q^{27} - 37 q^{31} - 25 q^{37} - 968 q^{45} - 100 q^{47} - 98 q^{49} + 140 q^{53} + 107 q^{59} - 140 q^{67} - 61 q^{69} - 133 q^{71}+ \cdots + 95 q^{97}+O(q^{100}) \) 8 * q - 5 * q^3 - q^5 - 11 * q^9 + 47 * q^15 - 140 * q^23 - 99 * q^25 - 65 * q^27 - 37 * q^31 - 25 * q^37 - 968 * q^45 - 100 * q^47 - 98 * q^49 + 140 * q^53 + 107 * q^59 - 140 * q^67 - 61 * q^69 - 133 * q^71 - 198 * q^75 - 212 * q^81 + 388 * q^89 + 155 * q^93 + 95 * q^97

Introduction

L-functions

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