LMFDB - Embedded newform 363.2.f.g.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(161,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 7]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("363.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.f (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			161.4
Root			$-1.13551 + 1.56290i$ of defining polynomial
Character	$\chi$	$=$	363.161
Dual form			363.2.f.g.239.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.535233 - 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(4.52729 - 1.47101i) q^{7} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})$	$q+(0.535233 - 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(4.52729 - 1.47101i) q^{7} +(-2.42705 - 1.76336i) q^{9} -3.46410 q^{12} +(-2.04829 + 2.81923i) q^{13} +(-3.23607 + 2.35114i) q^{16} +(1.21308 + 0.394155i) q^{19} -8.24504i q^{21} +(1.54508 - 4.75528i) q^{25} +(-4.20378 + 3.05422i) q^{27} +(-5.59604 - 7.70229i) q^{28} +(1.40126 + 1.01807i) q^{31} +(-1.85410 + 5.70634i) q^{36} +(-1.60570 - 4.94183i) q^{37} +(3.54775 + 4.88306i) q^{39} +13.0053i q^{43} +(2.14093 + 6.58911i) q^{48} +(12.6694 - 9.20487i) q^{49} +(6.62842 + 2.15370i) q^{52} +(1.29857 - 1.78732i) q^{57} +(6.89461 + 9.48962i) q^{61} +(-13.5819 - 4.41302i) q^{63} +(6.47214 + 4.70228i) q^{64} +12.1244 q^{67} +(-10.2677 + 3.33617i) q^{73} +(-7.00629 - 5.09037i) q^{75} -2.55103i q^{76} +(10.4424 - 14.3727i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-15.6830 + 5.09572i) q^{84} +(-5.12612 + 15.7766i) q^{91} +(2.42705 - 1.76336i) q^{93} +(4.04508 + 2.93893i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 16 * q + 8 * q^4 - 12 * q^9	$ 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) $ 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$

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		0.587785	−	0.809017i	$-0.300000\pi$
$2$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$3$	0.535233	−	1.64728i	0.309017	−	0.951057i
$3$	0.535233	−	1.64728i	0.309017	−	0.951057i
$4$	−0.618034	−	1.90211i	−0.309017	−	0.951057i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$		0			0
$7$	4.52729	−	1.47101i	1.71116	−	0.555988i	0.720631	−	0.693319i	$-0.243852\pi$
$7$	4.52729	−	1.47101i	1.71116	−	0.555988i	0.990525	+	0.137330i	$0.0438522\pi$
$8$		0			0
$9$	−2.42705	−	1.76336i	−0.809017	−	0.587785i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−3.46410			−1.00000
$13$	−2.04829	+	2.81923i	−0.568095	+	0.781915i	−0.992327	−	0.123638i	$-0.960544\pi$
$13$	−2.04829	+	2.81923i	−0.568095	+	0.781915i	0.424233	+	0.905553i	$0.360544\pi$
$14$		0			0
$15$		0			0
$16$	−3.23607	+	2.35114i	−0.809017	+	0.587785i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$		0			0
$19$	1.21308	+	0.394155i	0.278301	+	0.0904254i	0.444842	−	0.895609i	$-0.353260\pi$
$19$	1.21308	+	0.394155i	0.278301	+	0.0904254i	−0.166541	+	0.986035i	$0.553260\pi$
$20$		0			0
$21$		−	8.24504i		−	1.79922i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$	−4.20378	+	3.05422i	−0.809017	+	0.587785i
$28$	−5.59604	−	7.70229i	−1.05755	−	1.45560i
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	1.40126	+	1.01807i	0.251673	+	0.182851i	0.706468	−	0.707745i	$-0.250287\pi$
$31$	1.40126	+	1.01807i	0.251673	+	0.182851i	−0.454795	+	0.890596i	$0.650287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.85410	+	5.70634i	−0.309017	+	0.951057i
$37$	−1.60570	−	4.94183i	−0.263975	−	0.812433i	−0.991928	−	0.126805i	$-0.959528\pi$
$37$	−1.60570	−	4.94183i	−0.263975	−	0.812433i	0.727952	−	0.685628i	$-0.240472\pi$
$38$		0			0
$39$	3.54775	+	4.88306i	0.568095	+	0.781915i
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$			13.0053i			1.98329i	0.128984	+	0.991647i	$0.458828\pi$
$43$			13.0053i			1.98329i	−0.128984	+	0.991647i	$0.541172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	2.14093	+	6.58911i	0.309017	+	0.951057i
$49$	12.6694	−	9.20487i	1.80992	−	1.31498i
$50$		0			0
$51$		0			0
$52$	6.62842	+	2.15370i	0.919196	+	0.298665i
$53$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$53$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.29857	−	1.78732i	0.171999	−	0.236737i
$58$		0			0
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$	6.89461	+	9.48962i	0.882764	+	1.21502i	0.975648	+	0.219344i	$0.0703916\pi$
$61$	6.89461	+	9.48962i	0.882764	+	1.21502i	−0.0928833	+	0.995677i	$0.529608\pi$
$62$		0			0
$63$	−13.5819	−	4.41302i	−1.71116	−	0.555988i
$64$	6.47214	+	4.70228i	0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	12.1244			1.48123			0.740613	−	0.671932i	$-0.234535\pi$
$67$	12.1244			1.48123			0.740613	+	0.671932i	$0.234535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$		0			0
$73$	−10.2677	+	3.33617i	−1.20174	+	0.390469i	−0.840401	−	0.541965i	$-0.817680\pi$
$73$	−10.2677	+	3.33617i	−1.20174	+	0.390469i	−0.361339	+	0.932434i	$0.617680\pi$
$74$		0			0
$75$	−7.00629	−	5.09037i	−0.809017	−	0.587785i
$76$		−	2.55103i		−	0.292623i
$77$		0			0
$78$		0			0
$79$	10.4424	−	14.3727i	1.17486	−	1.61705i	0.559250	−	0.828999i	$-0.311089\pi$
$79$	10.4424	−	14.3727i	1.17486	−	1.61705i	0.615608	−	0.788053i	$-0.288911\pi$
$80$		0			0
$81$	2.78115	+	8.55951i	0.309017	+	0.951057i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$	−15.6830	+	5.09572i	−1.71116	+	0.555988i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−5.12612	+	15.7766i	−0.537363	+	1.65383i
$92$		0			0
$93$	2.42705	−	1.76336i	0.251673	−	0.182851i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.04508	+	2.93893i	0.410716	+	0.298403i	0.773892	−	0.633318i	$-0.218307\pi$
$97$	4.04508	+	2.93893i	0.410716	+	0.298403i	−0.363176	+	0.931721i	$0.618307\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$	2.16312	+	6.65740i	0.213138	+	0.655973i	0.999281	+	0.0379269i	$0.0120754\pi$
$103$	2.16312	+	6.65740i	0.213138	+	0.655973i	−0.786142	+	0.618046i	$0.787925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$	8.40755	+	6.10844i	0.809017	+	0.587785i
$109$			7.31130i			0.700296i	0.936694	+	0.350148i	$0.113869\pi$
$109$			7.31130i			0.700296i	−0.936694	+	0.350148i	$0.886131\pi$
$110$		0			0
$111$	−9.00000			−0.854242
$112$	−11.1921	+	15.4046i	−1.05755	+	1.45560i
$113$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$113$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.94263	−	3.23056i	0.919196	−	0.298665i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	1.07047	−	3.29456i	0.0961307	−	0.295860i
$125$		0			0
$126$		0			0
$127$	−8.94290	−	12.3089i	−0.793554	−	1.09223i	−0.993656	−	0.112459i	$-0.964127\pi$
$127$	−8.94290	−	12.3089i	−0.793554	−	1.09223i	0.200102	−	0.979775i	$-0.435873\pi$
$128$		0			0
$129$	21.4234	+	6.96088i	1.88622	+	0.612871i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	6.07180			0.526492
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$137$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$	−0.888040	+	0.288542i	−0.0753225	+	0.0244738i	−0.346436	−	0.938074i	$-0.612608\pi$
$139$	−0.888040	+	0.288542i	−0.0753225	+	0.0244738i	0.271113	+	0.962547i	$0.412608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$	−8.38189	−	25.7968i	−0.691326	−	2.12768i
$148$	−8.40755	+	6.10844i	−0.691096	+	0.502111i
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$		0			0
$151$	−18.9972	−	6.17257i	−1.54597	−	0.502317i	−0.592955	−	0.805235i	$-0.702039\pi$
$151$	−18.9972	−	6.17257i	−1.54597	−	0.502317i	−0.953016	+	0.302919i	$0.902039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	7.09550	−	9.76612i	0.568095	−	0.781915i
$157$	6.95803	−	21.4146i	0.555311	−	1.70907i	−0.139808	−	0.990179i	$-0.544649\pi$
$157$	6.95803	−	21.4146i	0.555311	−	1.70907i	0.695120	−	0.718894i	$-0.255351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−13.7533	−	9.99235i	−1.07724	−	0.782661i	−0.100041	−	0.994983i	$-0.531897\pi$
$163$	−13.7533	−	9.99235i	−1.07724	−	0.782661i	−0.977200	+	0.212322i	$0.931897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	0.264644	+	0.814491i	0.0203572	+	0.0626531i
$170$		0			0
$171$	−2.24918	−	3.09573i	−0.171999	−	0.236737i
$172$	24.7376	−	8.03773i	1.88622	−	0.612871i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		−	23.8014i		−	1.79922i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$179$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$180$		0			0
$181$	−15.3713	+	11.1679i	−1.14254	+	0.830105i	−0.987471	−	0.157799i	$-0.949560\pi$
$181$	−15.3713	+	11.1679i	−1.14254	+	0.830105i	−0.155070	+	0.987903i	$0.549560\pi$
$182$		0			0
$183$	19.3223	−	6.27818i	1.42834	−	0.464097i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−14.5389	+	20.0111i	−1.05755	+	1.45560i
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$	11.2101	−	8.14459i	0.809017	−	0.587785i
$193$	16.0384	+	22.0750i	1.15447	+	1.58899i	0.729831	+	0.683628i	$0.239599\pi$
$193$	16.0384	+	22.0750i	1.15447	+	1.58899i	0.424639	+	0.905363i	$0.360401\pi$
$194$		0			0
$195$		0			0
$196$	−25.3388	−	18.4097i	−1.80992	−	1.31498i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−25.9808			−1.84173			−0.920864	−	0.389885i	$-0.872515\pi$
$199$	−25.9808			−1.84173			−0.920864	+	0.389885i	$0.872515\pi$
$200$		0			0
$201$	6.48936	−	19.9722i	0.457724	−	1.40873i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	13.9391i		−	0.966500i
$209$		0			0
$210$		0			0
$211$	−5.39515	+	7.42579i	−0.371418	+	0.511213i	−0.953286	−	0.302071i	$-0.902322\pi$
$211$	−5.39515	+	7.42579i	−0.371418	+	0.511213i	0.581868	+	0.813283i	$0.302322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.84150	+	2.54786i	0.532316	+	0.172960i
$218$		0			0
$219$			18.6993i			1.26358i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−7.10739	+	21.8743i	−0.475946	+	1.46481i	0.368731	+	0.929536i	$0.379792\pi$
$223$	−7.10739	+	21.8743i	−0.475946	+	1.46481i	−0.844678	+	0.535275i	$0.820208\pi$
$224$		0			0
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$		0			0
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$	−4.20225	−	1.36539i	−0.278301	−	0.0904254i
$229$	16.8151	+	12.2169i	1.11117	+	0.807315i	0.982848	−	0.184418i	$-0.0590399\pi$
$229$	16.8151	+	12.2169i	1.11117	+	0.807315i	0.128325	+	0.991732i	$0.459040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$233$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−18.0867	−	24.8942i	−1.17486	−	1.61705i
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$		−	29.4954i		−	1.89997i	−0.312301	−	0.949983i	$-0.601100\pi$
$241$		−	29.4954i		−	1.89997i	0.312301	−	0.949983i	$-0.398900\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$	13.7892	−	18.9792i	0.882764	−	1.21502i
$245$		0			0
$246$		0			0
$247$	−3.59597	+	2.61263i	−0.228806	+	0.166237i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$			28.5617i			1.79922i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$257$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$258$		0			0
$259$	−14.5389	−	20.0111i	−0.903406	−	1.24343i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−7.49326	−	23.0619i	−0.457724	−	1.40873i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$	−30.4780	+	9.90289i	−1.85141	+	0.601558i	−0.854826	+	0.518915i	$0.826336\pi$
$271$	−30.4780	+	9.90289i	−1.85141	+	0.601558i	−0.996579	+	0.0826430i	$0.973664\pi$
$272$		0			0
$273$	23.2447	+	16.8883i	1.40683	+	1.02212i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.84523	+	10.7980i	−0.471374	+	0.648791i	−0.976819	−	0.214068i	$-0.931328\pi$
$277$	−7.84523	+	10.7980i	−0.471374	+	0.648791i	0.505445	+	0.862859i	$0.331328\pi$
$278$		0			0
$279$	−1.60570	−	4.94183i	−0.0961307	−	0.295860i
$280$		0			0
$281$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$281$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$282$		0			0
$283$	26.8387	+	8.72043i	1.59540	+	0.518376i	0.965963	−	0.258679i	$-0.0832872\pi$
$283$	26.8387	+	8.72043i	1.59540	+	0.518376i	0.629433	+	0.777055i	$0.283287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.25329	+	16.1680i	−0.309017	+	0.951057i
$290$		0			0
$291$	7.00629	−	5.09037i	0.410716	−	0.298403i
$292$	12.6915	+	17.4684i	0.742716	+	1.02226i
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−5.35233	+	16.4728i	−0.309017	+	0.951057i
$301$	19.1309	+	58.8789i	1.10269	+	3.39372i
$302$		0			0
$303$		0			0
$304$	−4.85234	+	1.57662i	−0.278301	+	0.0904254i
$305$		0			0
$306$		0			0
$307$			34.2557i			1.95508i	0.210760	+	0.977538i	$0.432406\pi$
$307$			34.2557i			1.95508i	−0.210760	+	0.977538i	$0.567594\pi$
$308$		0			0
$309$	12.1244			0.689730
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	22.4201	−	16.2892i	1.26726	−	0.920719i	0.268171	−	0.963371i	$-0.413581\pi$
$313$	22.4201	−	16.2892i	1.26726	−	0.920719i	0.999090	+	0.0426523i	$0.0135808\pi$
$314$		0			0
$315$		0			0
$316$	−33.7922	−	10.9797i	−1.90096	−	0.617659i
$317$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$317$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	14.5623	−	10.5801i	0.809017	−	0.587785i
$325$	10.2415	+	14.0962i	0.568095	+	0.781915i
$326$		0			0
$327$	12.0438	+	3.91325i	0.666021	+	0.216403i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−31.0000			−1.70391			−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000			−1.70391			−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	−4.81710	+	14.8255i	−0.263975	+	0.812433i
$334$		0			0
$335$		0			0
$336$	19.3853	+	26.6815i	1.05755	+	1.45560i
$337$	−21.0983	+	6.85527i	−1.14930	+	0.373430i	−0.820879	−	0.571101i	$-0.806516\pi$
$337$	−21.0983	+	6.85527i	−1.14930	+	0.373430i	−0.328420	+	0.944532i	$0.606516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	24.2316	−	33.3519i	1.30838	−	1.80083i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$347$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$348$		0			0
$349$	−28.3768	−	9.22020i	−1.51898	−	0.493546i	−0.573493	−	0.819211i	$-0.694412\pi$
$349$	−28.3768	−	9.22020i	−1.51898	−	0.493546i	−0.945485	+	0.325665i	$0.894412\pi$
$350$		0			0
$351$		−	18.1074i		−	0.966500i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	−14.0551	−	10.2116i	−0.739742	−	0.537454i
$362$		0			0
$363$		0			0
$364$	33.1769			1.73894
$365$		0			0
$366$		0			0
$367$	−1.23607	−	3.80423i	−0.0645222	−	0.198579i	0.913598	−	0.406618i	$-0.133292\pi$
$367$	−1.23607	−	3.80423i	−0.0645222	−	0.198579i	−0.978121	+	0.208039i	$0.933292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−4.85410	−	3.52671i	−0.251673	−	0.182851i
$373$			3.14299i			0.162738i	0.996684	+	0.0813690i	$0.0259292\pi$
$373$			3.14299i			0.162738i	−0.996684	+	0.0813690i	$0.974071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.47214	+	4.70228i	−0.332451	+	0.241540i	−0.741470	−	0.670986i	$-0.765871\pi$
$379$	−6.47214	+	4.70228i	−0.332451	+	0.241540i	0.409019	+	0.912526i	$0.365871\pi$
$380$		0			0
$381$	−25.0626	+	8.14335i	−1.28400	+	0.417196i
$382$		0			0
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.9330	−	31.5646i	1.16575	−	1.60452i
$388$	3.09017	−	9.51057i	0.156880	−	0.482826i
$389$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$389$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	35.0000			1.75660			0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000			1.75660			0.878300	+	0.478110i	$0.158678\pi$
$398$		0			0
$399$	3.24983	−	10.0019i	0.162695	−	0.500723i
$400$	6.18034	+	19.0211i	0.309017	+	0.951057i
$401$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$401$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$	−5.74038	+	1.86516i	−0.285949	+	0.0929103i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−23.6827	+	32.5965i	−1.17104	+	1.61179i	−0.514529	+	0.857473i	$0.672033\pi$
$409$	−23.6827	+	32.5965i	−1.17104	+	1.61179i	−0.656507	+	0.754320i	$0.727967\pi$
$410$		0			0
$411$		0			0
$412$	11.3262	−	8.22899i	0.558004	−	0.405413i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			1.61729i			0.0791988i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−10.7047	+	32.9456i	−0.521713	+	1.60567i	0.249012	+	0.968501i	$0.419894\pi$
$421$	−10.7047	+	32.9456i	−0.521713	+	1.60567i	−0.770725	+	0.637168i	$0.780106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	45.1732	+	32.8203i	2.18609	+	1.58828i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	6.42280	−	19.7673i	0.309017	−	0.951057i
$433$	−11.4336	−	35.1891i	−0.549465	−	1.69108i	−0.710130	−	0.704071i	$-0.751364\pi$
$433$	−11.4336	−	35.1891i	−0.549465	−	1.69108i	0.160665	−	0.987009i	$-0.448636\pi$
$434$		0			0
$435$		0			0
$436$	13.9069	−	4.51863i	0.666021	−	0.216403i
$437$		0			0
$438$		0			0
$439$			22.8677i			1.09141i	0.837976	+	0.545707i	$0.183739\pi$
$439$			22.8677i			1.09141i	−0.837976	+	0.545707i	$0.816261\pi$
$440$		0			0
$441$	−46.9808			−2.23718
$442$		0			0
$443$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$	5.56231	+	17.1190i	0.263975	+	0.812433i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	36.2184	+	11.7681i	1.71116	+	0.555988i
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−20.3359	+	27.9899i	−0.955463	+	1.31508i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−21.4336	−	29.5008i	−1.00262	−	1.37999i	−0.923704	−	0.383106i	$-0.874854\pi$
$457$	−21.4336	−	29.5008i	−1.00262	−	1.37999i	−0.0789151	−	0.996881i	$-0.525146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$	−12.2898	−	16.9154i	−0.568095	−	0.781915i
$469$	54.8905	−	17.8350i	2.53461	−	0.823544i
$470$		0			0
$471$	−31.5517	−	22.9236i	−1.45382	−	1.05626i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3.74864	−	5.15956i	0.171999	−	0.236737i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$	17.2211	+	5.59549i	0.785216	+	0.255132i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	1.07047	−	3.29456i	0.0485075	−	0.149291i	−0.923869	−	0.382709i	$-0.874991\pi$
$487$	1.07047	−	3.29456i	0.0485075	−	0.149291i	0.972376	+	0.233418i	$0.0749912\pi$
$488$		0			0
$489$	−23.8214	+	17.3073i	−1.07724	+	0.782661i
$490$		0			0
$491$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$491$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−6.92820			−0.311086
$497$		0			0
$498$		0			0
$499$	−13.3808	−	41.1820i	−0.599008	−	1.84356i	−0.533667	−	0.845694i	$-0.679187\pi$
$499$	−13.3808	−	41.1820i	−0.599008	−	1.84356i	−0.0653408	−	0.997863i	$-0.520813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.48334			0.0658774
$508$	−17.8858	+	24.6177i	−0.793554	+	1.09223i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$	−41.5772	+	30.2076i	−1.83927	+	1.33631i
$512$		0			0
$513$	−6.30337	+	2.04809i	−0.278301	+	0.0904254i
$514$		0			0
$515$		0			0
$516$		−	45.0518i		−	1.98329i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	−11.3930	−	15.6811i	−0.498180	−	0.685686i	0.483690	−	0.875239i	$-0.339296\pi$
$523$	−11.3930	−	15.6811i	−0.498180	−	0.685686i	−0.981870	+	0.189553i	$0.939296\pi$
$524$		0			0
$525$	−39.2075	−	12.7393i	−1.71116	−	0.555988i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	−3.75258	−	11.5492i	−0.162695	−	0.500723i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	16.2393	−	22.3515i	0.698182	−	0.960965i	−0.301790	−	0.953375i	$-0.597584\pi$
$541$	16.2393	−	22.3515i	0.698182	−	0.960965i	0.999971	−	0.00759004i	$-0.00241601\pi$
$542$		0			0
$543$	10.1694	+	31.2983i	0.436412	+	1.34314i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.5927	+	3.44178i	0.452912	+	0.147160i	0.526585	−	0.850122i	$-0.323472\pi$
$547$	10.5927	+	3.44178i	0.452912	+	0.147160i	−0.0736732	+	0.997282i	$0.523472\pi$
$548$		0			0
$549$		−	35.1894i		−	1.50185i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	26.1333	−	80.4301i	1.11130	−	3.42024i
$554$		0			0
$555$		0			0
$556$	1.09768	+	1.51082i	0.0465519	+	0.0640732i
$557$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$557$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$558$		0			0
$559$	−36.6651	−	26.6387i	−1.55077	−	1.12670i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	25.1822	+	34.6603i	1.05755	+	1.45560i
$568$		0			0
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		0			0
$571$		−	39.3577i		−	1.64707i	−0.567265	−	0.823535i	$-0.691999\pi$
$571$		−	39.3577i		−	1.64707i	0.567265	−	0.823535i	$-0.308001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−7.41641	−	22.8254i	−0.309017	−	0.951057i
$577$	37.8340	−	27.4880i	1.57505	−	1.14434i	0.652941	−	0.757409i	$-0.273535\pi$
$577$	37.8340	−	27.4880i	1.57505	−	1.14434i	0.922109	−	0.386931i	$-0.126465\pi$
$578$		0			0
$579$	44.9479	−	14.6045i	1.86797	−	0.606941i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$	−43.8881	+	31.8866i	−1.80992	+	1.31498i
$589$	1.29857	+	1.78732i	0.0535065	+	0.0736454i
$590$		0			0
$591$		0			0
$592$	16.8151	+	12.2169i	0.691096	+	0.502111i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.9058	+	42.7975i	−0.569125	+	1.75159i
$598$		0			0
$599$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$599$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$600$		0			0
$601$	34.1172	−	11.0854i	1.39167	−	0.452181i	0.485183	−	0.874413i	$-0.338753\pi$
$601$	34.1172	−	11.0854i	1.39167	−	0.452181i	0.906488	+	0.422232i	$0.138753\pi$
$602$		0			0
$603$	−29.4264	−	21.3796i	−1.19834	−	0.870643i
$604$			39.9497i			1.62553i
$605$		0			0
$606$		0			0
$607$	−27.0296	+	37.2031i	−1.09710	+	1.51003i	−0.257921	+	0.966166i	$0.583038\pi$
$607$	−27.0296	+	37.2031i	−1.09710	+	1.51003i	−0.839176	+	0.543859i	$0.816962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.75121	−	0.893924i	−0.111121	−	0.0361052i	0.252929	−	0.967485i	$-0.418606\pi$
$613$	−2.75121	−	0.893924i	−0.111121	−	0.0361052i	−0.364049	+	0.931380i	$0.618606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	9.88854	−	30.4338i	0.397454	−	1.22324i	−0.529580	−	0.848260i	$-0.677650\pi$
$619$	9.88854	−	30.4338i	0.397454	−	1.22324i	0.927034	−	0.374978i	$-0.122350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−22.9615	−	7.46065i	−0.919196	−	0.298665i
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$	−45.0333			−1.79703
$629$		0			0
$630$		0			0
$631$	−7.49326	−	23.0619i	−0.298302	−	0.918080i	−0.982092	−	0.188401i	$-0.939670\pi$
$631$	−7.49326	−	23.0619i	−0.298302	−	0.918080i	0.683790	−	0.729679i	$-0.260330\pi$
$632$		0			0
$633$	9.34468	+	12.8619i	0.371418	+	0.511213i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			54.5723i			2.16223i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$641$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$642$		0			0
$643$	38.0238	−	27.6259i	1.49951	−	1.08946i	0.528937	−	0.848661i	$-0.322591\pi$
$643$	38.0238	−	27.6259i	1.49951	−	1.08946i	0.970575	−	0.240798i	$-0.0774093\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8.39406	−	11.5534i	0.328989	−	0.452815i
$652$	−10.5066	+	32.3359i	−0.411469	+	1.26637i
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	30.8030	+	10.0085i	1.20174	+	0.390469i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	50.2295			1.95370			0.976850	−	0.213925i	$-0.0686249\pi$
$661$	50.2295			1.95370			0.976850	+	0.213925i	$0.0686249\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	32.2289	+	23.4157i	1.24604	+	0.905303i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.2799	−	36.1711i	1.01301	−	1.39430i	0.0960273	−	0.995379i	$-0.469386\pi$
$673$	26.2799	−	36.1711i	1.01301	−	1.39430i	0.916987	−	0.398916i	$-0.130614\pi$
$674$		0			0
$675$	8.02850	+	24.7092i	0.309017	+	0.951057i
$676$	1.38569	−	1.00677i	0.0532960	−	0.0387218i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$	22.6365	+	7.35504i	0.868708	+	0.282260i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−4.49837	+	6.19147i	−0.171999	+	0.236737i
$685$		0			0
$686$		0			0
$687$	29.1246	−	21.1603i	1.11117	−	0.807315i
$688$	−30.5773	−	42.0861i	−1.16575	−	1.60452i
$689$		0			0
$690$		0			0
$691$	39.6418	+	28.8015i	1.50805	+	1.09566i	0.967035	+	0.254645i	$0.0819585\pi$
$691$	39.6418	+	28.8015i	1.50805	+	1.09566i	0.541012	+	0.841015i	$0.318041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−45.2729	+	14.7101i	−1.71116	+	0.555988i
$701$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$701$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$702$		0			0
$703$		−	6.62776i		−	0.249971i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−39.2352	+	28.5061i	−1.47351	+	1.07057i	−0.493933	+	0.869500i	$0.664441\pi$
$709$	−39.2352	+	28.5061i	−1.47351	+	1.07057i	−0.979577	+	0.201068i	$0.935559\pi$
$710$		0			0
$711$	−50.6883	+	16.4696i	−1.90096	+	0.617659i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$	19.5862	+	26.9580i	0.729426	+	1.00397i
$722$		0			0
$723$	−48.5872	−	15.7869i	−1.80698	−	0.587122i
$724$	30.7426	+	22.3358i	1.14254	+	0.830105i
$725$		0			0
$726$		0			0
$727$	31.1769			1.15629			0.578144	−	0.815935i	$-0.303777\pi$
$727$	31.1769			1.15629			0.578144	+	0.815935i	$0.303777\pi$
$728$		0			0
$729$	8.34346	−	25.6785i	0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$	−23.8836	−	32.8730i	−0.882764	−	1.21502i
$733$	−19.6473	+	6.38380i	−0.725690	+	0.235791i	−0.648488	−	0.761225i	$-0.724598\pi$
$733$	−19.6473	+	6.38380i	−0.725690	+	0.235791i	−0.0772015	+	0.997016i	$0.524598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−16.9890	+	23.3834i	−0.624951	+	0.860171i	−0.997702	−	0.0677601i	$-0.978415\pi$
$739$	−16.9890	+	23.3834i	−0.624951	+	0.860171i	0.372750	+	0.927932i	$0.378415\pi$
$740$		0			0
$741$	2.37904	+	7.32193i	0.0873962	+	0.268978i
$742$		0			0
$743$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$743$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.5922	+	51.0656i	−0.605459	+	1.86341i	−0.111854	+	0.993725i	$0.535679\pi$
$751$	−16.5922	+	51.0656i	−0.605459	+	1.86341i	−0.493604	+	0.869687i	$0.664321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	47.0490	+	15.2872i	1.71116	+	0.555988i
$757$	1.40126	+	1.01807i	0.0509296	+	0.0370025i	0.612959	−	0.790115i	$-0.289979\pi$
$757$	1.40126	+	1.01807i	0.0509296	+	0.0370025i	−0.562029	+	0.827117i	$0.689979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$	10.7550	+	33.1004i	0.389356	+	1.19832i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−22.4201	−	16.2892i	−0.809017	−	0.587785i
$769$		−	12.4134i		−	0.447637i	−0.974631	−	0.223819i	$-0.928148\pi$
$769$		−	12.4134i		−	0.447637i	0.974631	−	0.223819i	$-0.0718523\pi$
$770$		0			0
$771$		0			0
$772$	32.0768	−	44.1499i	1.15447	−	1.58899i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$		0			0
$775$	7.00629	−	5.09037i	0.251673	−	0.182851i
$776$		0			0
$777$	−40.7456	+	13.2391i	−1.46174	+	0.474949i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−19.3571	+	59.5751i	−0.691326	+	2.12768i
$785$		0			0
$786$		0			0
$787$	21.8353	+	30.0538i	0.778346	+	1.07130i	0.995462	+	0.0951552i	$0.0303347\pi$
$787$	21.8353	+	30.0538i	0.778346	+	1.07130i	−0.217117	+	0.976146i	$0.569665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−40.8756			−1.45154
$794$		0			0
$795$		0			0
$796$	16.0570	+	49.4183i	0.569125	+	1.75159i
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−42.0000			−1.48123
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$809$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$810$		0			0
$811$	−54.0025	−	17.5465i	−1.89628	−	0.616140i	−0.972316	−	0.233669i	$-0.924927\pi$
$811$	−54.0025	−	17.5465i	−1.89628	−	0.616140i	−0.923967	−	0.382471i	$-0.875073\pi$
$812$		0			0
$813$			55.5061i			1.94668i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−5.12612	+	15.7766i	−0.179340	+	0.551952i
$818$		0			0
$819$	40.2610	−	29.2514i	1.40683	−	1.02212i
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	4.04508	+	2.93893i	0.141003	+	0.102445i	0.656051	−	0.754717i	$-0.272226\pi$
$823$	4.04508	+	2.93893i	0.141003	+	0.102445i	−0.515048	+	0.857161i	$0.672226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$827$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$828$		0			0
$829$	2.16312	+	6.65740i	0.0751282	+	0.231221i	0.981568	−	0.191115i	$-0.0612103\pi$
$829$	2.16312	+	6.65740i	0.0751282	+	0.231221i	−0.906439	+	0.422336i	$0.861210\pi$
$830$		0			0
$831$	13.5883	+	18.7027i	0.471374	+	0.648791i
$832$	−26.5137	+	8.61482i	−0.919196	+	0.298665i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.00000			−0.311086
$838$		0			0
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	17.4591	+	5.67280i	0.600967	+	0.195266i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	28.7299	−	39.5434i	0.986009	−	1.35713i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	31.8759	+	43.8734i	1.09141	+	1.50220i	0.846303	+	0.532702i	$0.178823\pi$
$853$	31.8759	+	43.8734i	1.09141	+	1.50220i	0.245108	+	0.969496i	$0.421177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	13.0000			0.443554			0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000			0.443554			0.221777	+	0.975097i	$0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$863$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.8214	+	17.3073i	0.809017	+	0.587785i
$868$		−	16.4901i		−	0.559710i
$869$		0			0
$870$		0			0
$871$	−24.8342	+	34.1814i	−0.841476	+	1.15819i
$872$		0			0
$873$	−4.63525	−	14.2658i	−0.156880	−	0.482826i
$874$		0			0
$875$		0			0
$876$	35.5683	−	11.5568i	1.20174	−	0.390469i
$877$	−43.1718	−	14.0274i	−1.45781	−	0.473671i	−0.530409	−	0.847742i	$-0.677962\pi$
$877$	−43.1718	−	14.0274i	−1.45781	−	0.473671i	−0.927400	+	0.374071i	$0.877962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	6.95803	−	21.4146i	0.234156	−	0.720659i	−0.763076	−	0.646309i	$-0.776312\pi$
$883$	6.95803	−	21.4146i	0.234156	−	0.720659i	0.997232	−	0.0743502i	$-0.0236883\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$	−58.5936	−	42.5707i	−1.96517	−	1.42778i
$890$		0			0
$891$		0			0
$892$	46.0000			1.54019
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	24.2705	+	17.6336i	0.809017	+	0.587785i
$901$		0			0
$902$		0			0
$903$	107.229			3.56837
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−15.3713	+	11.1679i	−0.510396	+	0.370825i	−0.812974	−	0.582300i	$-0.802153\pi$
$907$	−15.3713	+	11.1679i	−0.510396	+	0.370825i	0.302578	+	0.953125i	$0.402153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$			8.83701i			0.292623i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	12.8456	−	39.5347i	0.424430	−	1.30626i
$917$		0			0
$918$		0			0
$919$	−24.7804	−	34.1073i	−0.817431	−	1.12510i	−0.990134	−	0.140123i	$-0.955250\pi$
$919$	−24.7804	−	34.1073i	−0.817431	−	1.12510i	0.172704	−	0.984974i	$-0.444750\pi$
$920$		0			0
$921$	56.4287	+	18.3348i	1.85939	+	0.604152i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−25.9808			−0.854242
$926$		0			0
$927$	6.48936	−	19.9722i	0.213138	−	0.655973i
$928$		0			0
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	18.9972	−	6.17257i	0.622609	−	0.202298i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.4237	−	48.7565i	1.15724	−	1.59281i	0.436313	−	0.899795i	$-0.356284\pi$
$937$	35.4237	−	48.7565i	1.15724	−	1.59281i	0.720928	−	0.693010i	$-0.243716\pi$
$938$		0			0
$939$	−14.8328	−	45.6507i	−0.484051	−	1.48975i
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	−36.1734	+	49.7884i	−1.17486	+	1.61705i
$949$	11.6258	−	35.7804i	0.377388	−	1.16148i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$953$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8.65248	−	26.6296i	−0.279112	−	0.859019i
$962$		0			0
$963$		0			0
$964$	−56.1036	+	18.2292i	−1.80698	+	0.587122i
$965$		0			0
$966$		0			0
$967$		−	4.07673i		−	0.131099i	−0.997849	−	0.0655495i	$-0.979120\pi$
$967$		−	4.07673i		−	0.131099i	0.997849	−	0.0655495i	$-0.0208800\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$971$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$972$	−9.63420	−	29.6510i	−0.309017	−	0.951057i
$973$	−3.59597	+	2.61263i	−0.115282	+	0.0837569i
$974$		0			0
$975$	28.7019	−	9.32581i	0.919196	−	0.298665i
$976$	−44.6229	−	14.4988i	−1.42834	−	0.464097i
$977$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$977$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	12.8924	−	17.7449i	0.411624	−	0.566551i
$982$		0			0
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	7.19194	+	5.22525i	0.228806	+	0.166237i
$989$		0			0
$990$		0			0
$991$	−45.0333			−1.43053			−0.715265	−	0.698853i	$-0.753694\pi$
$991$	−45.0333			−1.43053			−0.715265	+	0.698853i	$0.753694\pi$
$992$		0			0
$993$	−16.5922	+	51.0656i	−0.526538	+	1.62052i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	59.7429	−	19.4116i	1.89208	−	0.614773i	0.914333	−	0.404962i	$-0.132715\pi$
$997$	59.7429	−	19.4116i	1.89208	−	0.614773i	0.977742	−	0.209810i	$-0.0672846\pi$
$998$		0			0
$999$	21.8435	+	15.8702i	0.691096	+	0.502111i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.f.g.161.4		16
3.2	odd	2	CM	363.2.f.g.161.4		16
11.2	odd	10	inner	363.2.f.g.215.1		16
11.3	even	5	inner	363.2.f.g.239.3		16
11.4	even	5	inner	363.2.f.g.233.1		16
11.5	even	5		363.2.d.d.362.3	✓	4
11.6	odd	10		363.2.d.d.362.4	yes	4
11.7	odd	10	inner	363.2.f.g.233.2		16
11.8	odd	10	inner	363.2.f.g.239.4		16
11.9	even	5	inner	363.2.f.g.215.2		16
11.10	odd	2	inner	363.2.f.g.161.3		16
33.2	even	10	inner	363.2.f.g.215.1		16
33.5	odd	10		363.2.d.d.362.3	✓	4
33.8	even	10	inner	363.2.f.g.239.4		16
33.14	odd	10	inner	363.2.f.g.239.3		16
33.17	even	10		363.2.d.d.362.4	yes	4
33.20	odd	10	inner	363.2.f.g.215.2		16
33.26	odd	10	inner	363.2.f.g.233.1		16
33.29	even	10	inner	363.2.f.g.233.2		16
33.32	even	2	inner	363.2.f.g.161.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.d.d.362.3	✓	4	11.5	even	5
363.2.d.d.362.3	✓	4	33.5	odd	10
363.2.d.d.362.4	yes	4	11.6	odd	10
363.2.d.d.362.4	yes	4	33.17	even	10
363.2.f.g.161.3		16	11.10	odd	2	inner
363.2.f.g.161.3		16	33.32	even	2	inner
363.2.f.g.161.4		16	1.1	even	1	trivial
363.2.f.g.161.4		16	3.2	odd	2	CM
363.2.f.g.215.1		16	11.2	odd	10	inner
363.2.f.g.215.1		16	33.2	even	10	inner
363.2.f.g.215.2		16	11.9	even	5	inner
363.2.f.g.215.2		16	33.20	odd	10	inner
363.2.f.g.233.1		16	11.4	even	5	inner
363.2.f.g.233.1		16	33.26	odd	10	inner
363.2.f.g.233.2		16	11.7	odd	10	inner
363.2.f.g.233.2		16	33.29	even	10	inner
363.2.f.g.239.3		16	11.3	even	5	inner
363.2.f.g.239.3		16	33.14	odd	10	inner
363.2.f.g.239.4		16	11.8	odd	10	inner
363.2.f.g.239.4		16	33.8	even	10	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.f (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.535233 - 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(4.52729 - 1.47101i) q^{7} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)	\(q+(0.535233 - 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(4.52729 - 1.47101i) q^{7} +(-2.42705 - 1.76336i) q^{9} -3.46410 q^{12} +(-2.04829 + 2.81923i) q^{13} +(-3.23607 + 2.35114i) q^{16} +(1.21308 + 0.394155i) q^{19} -8.24504i q^{21} +(1.54508 - 4.75528i) q^{25} +(-4.20378 + 3.05422i) q^{27} +(-5.59604 - 7.70229i) q^{28} +(1.40126 + 1.01807i) q^{31} +(-1.85410 + 5.70634i) q^{36} +(-1.60570 - 4.94183i) q^{37} +(3.54775 + 4.88306i) q^{39} +13.0053i q^{43} +(2.14093 + 6.58911i) q^{48} +(12.6694 - 9.20487i) q^{49} +(6.62842 + 2.15370i) q^{52} +(1.29857 - 1.78732i) q^{57} +(6.89461 + 9.48962i) q^{61} +(-13.5819 - 4.41302i) q^{63} +(6.47214 + 4.70228i) q^{64} +12.1244 q^{67} +(-10.2677 + 3.33617i) q^{73} +(-7.00629 - 5.09037i) q^{75} -2.55103i q^{76} +(10.4424 - 14.3727i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-15.6830 + 5.09572i) q^{84} +(-5.12612 + 15.7766i) q^{91} +(2.42705 - 1.76336i) q^{93} +(4.04508 + 2.93893i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 16 * q + 8 * q^4 - 12 * q^9	\( 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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