LMFDB - Embedded newform 363.3.h.f.245.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(245,363)]

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, 8]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("363.245");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	363.h (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:	$9.89103359628$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.228765625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 $ x^8 - x^7 - 2x^6 + 5x^5 + x^4 + 15x^3 - 18x^2 - 27*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			245.2
Root			$-1.73166 + 0.0369185i$ of defining polynomial
Character	$\chi$	$=$	363.245
Dual form			363.3.h.f.323.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.34969 + 1.86519i) q^{3} +(1.23607 - 3.80423i) q^{4} +(5.84839 - 8.04962i) q^{5} +(2.04210 + 8.76526i) q^{9} +O(q^{10})$	$q+(2.34969 + 1.86519i) q^{3} +(1.23607 - 3.80423i) q^{4} +(5.84839 - 8.04962i) q^{5} +(2.04210 + 8.76526i) q^{9} +(10.0000 - 6.63325i) q^{12} +(28.7560 - 8.00573i) q^{15} +(-12.9443 - 9.40456i) q^{16} +(-23.3936 - 32.1985i) q^{20} +29.8496i q^{23} +(-22.8673 - 70.3782i) q^{25} +(-11.5506 + 24.4046i) q^{27} +(-29.9336 + 21.7481i) q^{31} +(35.8692 + 3.06586i) q^{36} +(7.72542 - 23.7764i) q^{37} +(82.5000 + 34.8246i) q^{45} +(75.7031 - 24.5974i) q^{47} +(-12.8737 - 46.2414i) q^{48} +(39.6418 + 28.8015i) q^{49} +(46.7871 + 64.3969i) q^{53} +(47.3145 + 15.3734i) q^{59} +(5.08877 - 119.290i) q^{60} +(-51.7771 + 37.6183i) q^{64} -35.0000 q^{67} +(-55.6754 + 70.1374i) q^{69} +(-29.2419 + 40.2481i) q^{71} +(77.5380 - 208.019i) q^{75} +(-151.406 + 49.1949i) q^{80} +(-72.6597 + 35.7991i) q^{81} -149.248i q^{89} +(113.555 + 36.8962i) q^{92} +(-110.899 - 4.73083i) q^{93} +(-76.8566 + 55.8396i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9	$ 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9} + 80 q^{12} - 33 q^{15} - 32 q^{16} + 148 q^{25} + 10 q^{27} - 74 q^{31} - 28 q^{36} - 50 q^{37} + 660 q^{45} - 80 q^{48} + 98 q^{49} - 132 q^{60} - 128 q^{64} - 280 q^{67} - 99 q^{69} + 370 q^{75} + 113 q^{81} - 185 q^{93} - 190 q^{97}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9 + 80 * q^12 - 33 * q^15 - 32 * q^16 + 148 * q^25 + 10 * q^27 - 74 * q^31 - 28 * q^36 - 50 * q^37 + 660 * q^45 - 80 * q^48 + 98 * q^49 - 132 * q^60 - 128 * q^64 - 280 * q^67 - 99 * q^69 + 370 * q^75 + 113 * q^81 - 185 * q^93 - 190 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$122$	$244$
$\chi(n)$	$-1$	$e\left(\frac{4}{5}\right)$

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		0.809017	−	0.587785i	$-0.200000\pi$
$2$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$3$	2.34969	+	1.86519i	0.783230	+	0.621732i
$3$	2.34969	+	1.86519i	0.783230	+	0.621732i
$4$	1.23607	−	3.80423i	0.309017	−	0.951057i
$5$	5.84839	−	8.04962i	1.16968	−	1.60992i	0.503937	−	0.863740i	$-0.331884\pi$
$5$	5.84839	−	8.04962i	1.16968	−	1.60992i	0.665741	−	0.746183i	$-0.268116\pi$
$6$		0			0
$7$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$7$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$8$		0			0
$9$	2.04210	+	8.76526i	0.226900	+	0.973918i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	10.0000	−	6.63325i	0.833333	−	0.552771i
$13$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$13$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$14$		0			0
$15$	28.7560	−	8.00573i	1.91707	−	0.533715i
$16$	−12.9443	−	9.40456i	−0.809017	−	0.587785i
$17$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$17$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	−23.3936	−	32.1985i	−1.16968	−	1.60992i
$21$		0			0
$22$		0			0
$23$			29.8496i			1.29781i	0.760870	+	0.648905i	$0.224773\pi$
$23$			29.8496i			1.29781i	−0.760870	+	0.648905i	$0.775227\pi$
$24$		0			0
$25$	−22.8673	−	70.3782i	−0.914690	−	2.81513i
$26$		0			0
$27$	−11.5506	+	24.4046i	−0.427801	+	0.903873i
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$	−29.9336	+	21.7481i	−0.965601	+	0.701550i	−0.954445	−	0.298388i	$-0.903551\pi$
$31$	−29.9336	+	21.7481i	−0.965601	+	0.701550i	−0.0111562	+	0.999938i	$0.503551\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	35.8692	+	3.06586i	0.996367	+	0.0851627i
$37$	7.72542	−	23.7764i	0.208795	−	0.642606i	−0.790741	−	0.612151i	$-0.790304\pi$
$37$	7.72542	−	23.7764i	0.208795	−	0.642606i	0.999536	−	0.0304547i	$-0.00969555\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$41$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	82.5000	+	34.8246i	1.83333	+	0.773879i
$46$		0			0
$47$	75.7031	−	24.5974i	1.61071	−	0.523350i	0.640982	−	0.767556i	$-0.278527\pi$
$47$	75.7031	−	24.5974i	1.61071	−	0.523350i	0.969723	+	0.244206i	$0.0785274\pi$
$48$	−12.8737	−	46.2414i	−0.268202	−	0.963363i
$49$	39.6418	+	28.8015i	0.809017	+	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	46.7871	+	64.3969i	0.882776	+	1.21504i	0.975644	+	0.219358i	$0.0703963\pi$
$53$	46.7871	+	64.3969i	0.882776	+	1.21504i	−0.0928686	+	0.995678i	$0.529604\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	47.3145	+	15.3734i	0.801940	+	0.260566i	0.681180	−	0.732116i	$-0.261467\pi$
$59$	47.3145	+	15.3734i	0.801940	+	0.260566i	0.120760	+	0.992682i	$0.461467\pi$
$60$	5.08877	−	119.290i	0.0848129	−	1.98817i
$61$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$61$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$62$		0			0
$63$		0			0
$64$	−51.7771	+	37.6183i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−35.0000			−0.522388			−0.261194	−	0.965286i	$-0.584116\pi$
$67$	−35.0000			−0.522388			−0.261194	+	0.965286i	$0.584116\pi$
$68$		0			0
$69$	−55.6754	+	70.1374i	−0.806889	+	1.01648i
$70$		0			0
$71$	−29.2419	+	40.2481i	−0.411858	+	0.566874i	−0.963670	−	0.267094i	$-0.913937\pi$
$71$	−29.2419	+	40.2481i	−0.411858	+	0.566874i	0.551812	+	0.833968i	$0.313937\pi$
$72$		0			0
$73$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$73$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$74$		0			0
$75$	77.5380	−	208.019i	1.03384	−	2.77359i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$79$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$80$	−151.406	+	49.1949i	−1.89258	+	0.614936i
$81$	−72.6597	+	35.7991i	−0.897033	+	0.441964i
$82$		0			0
$83$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$83$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	149.248i		−	1.67695i	−0.544944	−	0.838473i	$-0.683449\pi$
$89$		−	149.248i		−	1.67695i	0.544944	−	0.838473i	$-0.316551\pi$
$90$		0			0
$91$		0			0
$92$	113.555	+	36.8962i	1.23429	+	0.401045i
$93$	−110.899	−	4.73083i	−1.19246	−	0.0508691i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−76.8566	+	55.8396i	−0.792336	+	0.575666i	−0.908656	−	0.417546i	$-0.862890\pi$
$97$	−76.8566	+	55.8396i	−0.792336	+	0.575666i	0.116320	+	0.993212i	$0.462890\pi$
$98$		0			0
$99$		0			0
$100$	−296.000			−2.96000
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	−58.7132	+	180.701i	−0.570031	+	1.75438i	0.0824753	+	0.996593i	$0.473717\pi$
$103$	−58.7132	+	180.701i	−0.570031	+	1.75438i	−0.652507	+	0.757783i	$0.726283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$107$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$108$	78.5632	+	74.1069i	0.727437	+	0.686175i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	62.5000	−	41.4578i	0.563063	−	0.373494i
$112$		0			0
$113$	66.2402	−	21.5228i	0.586197	−	0.190467i	−0.000877893	−	1.00000i	$-0.500279\pi$
$113$	66.2402	−	21.5228i	0.586197	−	0.190467i	0.587075	+	0.809533i	$0.300279\pi$
$114$		0			0
$115$	240.278	+	174.572i	2.08937	+	1.51802i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	45.7345	+	140.756i	0.368827	+	1.13513i
$125$	−463.682	−	150.659i	−3.70945	−	1.20527i
$126$		0			0
$127$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$127$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$128$		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
$135$	128.895	+	235.706i	0.954777	+	1.74597i
$136$		0			0
$137$	40.9387	−	56.3473i	0.298823	−	0.411294i	−0.633032	−	0.774126i	$-0.718190\pi$
$137$	40.9387	−	56.3473i	0.298823	−	0.411294i	0.931855	+	0.362831i	$0.118190\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$	223.758	+	83.4047i	1.58694	+	0.591523i
$142$		0			0
$143$		0			0
$144$	56.0000	−	132.665i	0.388889	−	0.921285i
$145$		0			0
$146$		0			0
$147$	39.4257	+	141.614i	0.268202	+	0.963363i
$148$	−80.9017	−	58.7785i	−0.546633	−	0.397152i
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$		0			0
$151$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$151$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			368.145i			2.37513i
$156$		0			0
$157$	−66.4387	−	204.477i	−0.423176	−	1.30240i	−0.904730	−	0.425986i	$-0.859927\pi$
$157$	−66.4387	−	204.477i	−0.423176	−	1.30240i	0.481554	−	0.876417i	$-0.340073\pi$
$158$		0			0
$159$	−10.1775	+	238.580i	−0.0640097	+	1.50050i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	56.6312	−	41.1450i	0.347431	−	0.252423i	−0.400360	−	0.916358i	$-0.631115\pi$
$163$	56.6312	−	41.1450i	0.347431	−	0.252423i	0.747790	+	0.663935i	$0.231115\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		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	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	82.5000	+	124.373i	0.466102	+	0.702675i
$178$		0			0
$179$	−331.201	+	107.614i	−1.85029	+	0.601194i	−0.853500	+	0.521094i	$0.825524\pi$
$179$	−331.201	+	107.614i	−1.85029	+	0.601194i	−0.996787	+	0.0801009i	$0.974476\pi$
$180$	234.456	−	270.803i	1.30253	−	1.50446i
$181$	−212.771	−	154.588i	−1.17553	−	0.854075i	−0.183872	−	0.982950i	$-0.558863\pi$
$181$	−212.771	−	154.588i	−1.17553	−	0.854075i	−0.991661	+	0.128875i	$0.958863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−146.210	−	201.240i	−0.790323	−	1.08779i
$186$		0			0
$187$		0			0
$188$		−	318.396i		−	1.69360i
$189$		0			0
$190$		0			0
$191$	331.201	+	107.614i	1.73404	+	0.563423i	0.994023	−	0.109168i	$-0.0348186\pi$
$191$	331.201	+	107.614i	1.73404	+	0.563423i	0.740015	+	0.672591i	$0.234819\pi$
$192$	−191.826	−	8.18305i	−0.999091	−	0.0426201i
$193$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$193$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$194$		0			0
$195$		0			0
$196$	158.567	−	115.206i	0.809017	−	0.587785i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−2.00000			−0.0100503			−0.00502513	−	0.999987i	$-0.501600\pi$
$199$	−2.00000			−0.0100503			−0.00502513	+	0.999987i	$0.501600\pi$
$200$		0			0
$201$	−82.2392	−	65.2818i	−0.409150	−	0.324785i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−261.640	+	60.9559i	−1.26396	+	0.294473i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$211$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$212$	302.813	−	98.3898i	1.42836	−	0.464103i
$213$	−143.780	+	40.0287i	−0.675024	+	0.187928i
$214$		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
$223$	−137.513	−	423.220i	−0.616648	−	1.89785i	−0.371976	−	0.928242i	$-0.621320\pi$
$223$	−137.513	−	423.220i	−0.616648	−	1.89785i	−0.244672	−	0.969606i	$-0.578680\pi$
$224$		0			0
$225$	570.186	−	344.157i	2.53416	−	1.52959i
$226$		0			0
$227$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$227$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$228$		0			0
$229$	−350.304	+	254.511i	−1.52971	+	1.11140i	−0.573316	+	0.819335i	$0.694343\pi$
$229$	−350.304	+	254.511i	−1.52971	+	1.11140i	−0.956398	+	0.292067i	$0.905657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$233$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$234$		0			0
$235$	244.741	−	753.237i	1.04145	−	3.20526i
$236$	116.968	−	160.992i	0.495626	−	0.682171i
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$	−447.516	−	166.809i	−1.86465	−	0.695039i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−237.500	−	51.4077i	−0.977366	−	0.211554i
$244$		0			0
$245$	463.682	−	150.659i	1.89258	−	0.614936i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	263.178	+	362.233i	1.04852	+	1.44316i	0.890088	+	0.455788i	$0.150642\pi$
$251$	263.178	+	362.233i	1.04852	+	1.44316i	0.158428	+	0.987371i	$0.449358\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	79.1084	+	243.470i	0.309017	+	0.951057i
$257$	454.219	+	147.585i	1.76739	+	0.574259i	0.997922	−	0.0644282i	$-0.0205223\pi$
$257$	454.219	+	147.585i	1.76739	+	0.574259i	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
$265$	792.000			2.98868
$266$		0			0
$267$	278.377	−	350.687i	1.04261	−	1.31343i
$268$	−43.2624	+	133.148i	−0.161427	+	0.496821i
$269$	−233.936	+	321.985i	−0.869649	+	1.19697i	0.109533	+	0.993983i	$0.465065\pi$
$269$	−233.936	+	321.985i	−0.869649	+	1.19697i	−0.979182	+	0.202986i	$0.934935\pi$
$270$		0			0
$271$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$271$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	198.000	+	298.496i	0.717391	+	1.08151i
$277$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$277$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$278$		0			0
$279$	−251.755	−	217.964i	−0.902347	−	0.781235i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$283$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$284$	116.968	+	160.992i	0.411858	+	0.566874i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	89.3059	+	274.855i	0.309017	+	0.951057i
$290$		0			0
$291$	−284.741	−	12.1467i	−0.978492	−	0.0417413i
$292$		0			0
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$		0			0
$295$	400.463	−	290.954i	1.35750	−	0.986284i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−695.509	−	552.098i	−2.31836	−	1.84033i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−475.000	+	315.079i	−1.53722	+	1.01967i
$310$		0			0
$311$	−378.516	+	122.987i	−1.21709	+	0.395457i	−0.846022	−	0.533148i	$-0.821009\pi$
$311$	−378.516	+	122.987i	−1.21709	+	0.395457i	−0.371070	+	0.928605i	$0.621009\pi$
$312$		0			0
$313$	−214.390	−	155.763i	−0.684950	−	0.497646i	0.190046	−	0.981775i	$-0.439136\pi$
$313$	−214.390	−	155.763i	−0.684950	−	0.497646i	−0.874996	+	0.484130i	$0.839136\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−368.449	−	507.126i	−1.16230	−	1.59977i	−0.702374	−	0.711808i	$-0.747877\pi$
$317$	−368.449	−	507.126i	−1.16230	−	1.59977i	−0.459924	−	0.887958i	$-0.652123\pi$
$318$		0			0
$319$		0			0
$320$			636.792i			1.98997i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	46.3754	+	320.664i	0.143134	+	0.989703i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	563.000			1.70091			0.850453	−	0.526051i	$-0.176328\pi$
$331$	563.000			1.70091			0.850453	+	0.526051i	$0.176328\pi$
$332$		0			0
$333$	224.183	+	19.1616i	0.673221	+	0.0575424i
$334$		0			0
$335$	−204.694	+	281.737i	−0.611026	+	0.841005i
$336$		0			0
$337$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$337$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$338$		0			0
$339$	195.788	+	72.9791i	0.577547	+	0.215278i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	238.968	+	858.356i	0.692661	+	2.48799i
$346$		0			0
$347$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$347$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	328.346i		−	0.930158i	−0.885269	−	0.465079i	$-0.846026\pi$
$353$		−	328.346i		−	0.930158i	0.885269	−	0.465079i	$-0.153974\pi$
$354$		0			0
$355$	152.963	+	470.773i	0.430883	+	1.32612i
$356$	−567.774	−	184.481i	−1.59487	−	0.518205i
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		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
$367$	196.226	−	603.921i	0.534675	−	1.64556i	−0.209675	−	0.977771i	$-0.567241\pi$
$367$	196.226	−	603.921i	0.534675	−	1.64556i	0.744350	−	0.667790i	$-0.232759\pi$
$368$	280.723	−	386.382i	0.762833	−	1.04995i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−155.076	+	416.038i	−0.416871	+	1.11838i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−808.500	−	1218.86i	−2.15600	−	3.25029i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	107.599	+	78.1754i	0.283903	+	0.206268i	0.720618	−	0.693332i	$-0.243858\pi$
$379$	107.599	+	78.1754i	0.283903	+	0.206268i	−0.436715	+	0.899600i	$0.643858\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	76.0291	+	104.645i	0.198509	+	0.273225i	0.896654	−	0.442732i	$-0.145991\pi$
$383$	76.0291	+	104.645i	0.198509	+	0.273225i	−0.698145	+	0.715957i	$0.745991\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	117.426	+	361.401i	0.302646	+	0.931447i
$389$	−520.459	−	169.107i	−1.33794	−	0.434723i	−0.449322	−	0.893370i	$-0.648334\pi$
$389$	−520.459	−	169.107i	−1.33794	−	0.434723i	−0.888619	+	0.458646i	$0.848334\pi$
$390$		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
$400$	−365.876	+	1126.05i	−0.914690	+	2.81513i
$401$	467.871	−	643.969i	1.16676	−	1.60591i	0.484523	−	0.874778i	$-0.338993\pi$
$401$	467.871	−	643.969i	1.16676	−	1.60591i	0.682238	−	0.731130i	$-0.261007\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−136.773	+	794.249i	−0.337712	+	1.96111i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$	201.292	−	56.0401i	0.489762	−	0.136351i
$412$	614.853	+	446.717i	1.49236	+	1.08426i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	795.990i		−	1.89974i	−0.312649	−	0.949869i	$-0.601216\pi$
$419$		−	795.990i		−	1.89974i	0.312649	−	0.949869i	$-0.398784\pi$
$420$		0			0
$421$	229.291	+	705.684i	0.544633	+	1.67621i	0.721860	+	0.692039i	$0.243287\pi$
$421$	229.291	+	705.684i	0.544633	+	1.67621i	−0.177227	+	0.984170i	$0.556713\pi$
$422$		0			0
$423$	370.196	+	613.328i	0.875169	+	1.44995i
$424$		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	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	379.029	−	207.271i	0.877381	−	0.479794i
$433$	−7.72542	+	23.7764i	−0.0178416	+	0.0549109i	−0.959581	−	0.281433i	$-0.909190\pi$
$433$	−7.72542	+	23.7764i	−0.0178416	+	0.0549109i	0.941739	+	0.336344i	$0.109190\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−171.500	+	406.287i	−0.388889	+	0.921285i
$442$		0			0
$443$	−728.643	+	236.750i	−1.64479	+	0.534425i	−0.977602	−	0.210462i	$-0.932503\pi$
$443$	−728.643	+	236.750i	−1.64479	+	0.534425i	−0.667190	+	0.744888i	$0.732503\pi$
$444$	−80.4606	−	289.009i	−0.181218	−	0.650921i
$445$	−1201.39	−	872.861i	−2.69975	−	1.96149i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−380.145	−	523.225i	−0.846649	−	1.16531i	−0.984591	−	0.174872i	$-0.944049\pi$
$449$	−380.145	−	523.225i	−0.846649	−	1.16531i	0.137942	−	0.990440i	$-0.455951\pi$
$450$		0			0
$451$		0			0
$452$		−	278.596i		−	0.616364i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$457$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$458$		0			0
$459$		0			0
$460$	961.112	−	698.289i	2.08937	−	1.51802i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	35.0000			0.0755940			0.0377970	−	0.999285i	$-0.487966\pi$
$463$	35.0000			0.0755940			0.0377970	+	0.999285i	$0.487966\pi$
$464$		0			0
$465$	−686.663	+	865.028i	−1.47669	+	1.86028i
$466$		0			0
$467$	76.0291	−	104.645i	0.162803	−	0.224079i	−0.719820	−	0.694161i	$-0.755776\pi$
$467$	76.0291	−	104.645i	0.162803	−	0.224079i	0.882623	+	0.470082i	$0.155776\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	225.279	−	604.379i	0.478300	−	1.28318i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−468.912	+	541.606i	−0.983044	+	1.13544i
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$			945.238i			1.94894i
$486$		0			0
$487$	−270.390	−	832.174i	−0.555215	−	1.70878i	−0.695374	−	0.718648i	$-0.744761\pi$
$487$	−270.390	−	832.174i	−0.555215	−	1.70878i	0.140159	−	0.990129i	$-0.455239\pi$
$488$		0			0
$489$	209.809	+	8.95022i	0.429058	+	0.0183031i
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	592.000			1.19355
$497$		0			0
$498$		0			0
$499$	−186.028	+	572.536i	−0.372802	+	1.14737i	0.572147	+	0.820151i	$0.306110\pi$
$499$	−186.028	+	572.536i	−0.372802	+	1.14737i	−0.944950	+	0.327216i	$0.893890\pi$
$500$	−1146.28	+	1577.73i	−2.29257	+	3.15545i
$501$		0			0
$502$		0			0
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−422.500	+	280.255i	−0.833333	+	0.552771i
$508$		0			0
$509$	−141.943	+	46.1202i	−0.278867	+	0.0906094i	−0.445111	−	0.895475i	$-0.646836\pi$
$509$	−141.943	+	46.1202i	−0.278867	+	0.0906094i	0.166244	+	0.986085i	$0.446836\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1111.19	+	1529.43i	2.15766	+	2.96976i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	615.088	+	199.854i	1.18059	+	0.383597i	0.832586	−	0.553895i	$-0.186859\pi$
$521$	615.088	+	199.854i	1.18059	+	0.383597i	0.348005	+	0.937493i	$0.386859\pi$
$522$		0			0
$523$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$523$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−362.000			−0.684310
$530$		0			0
$531$	−38.1311	+	446.118i	−0.0718100	+	0.840146i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−978.941	−	364.896i	−1.82298	−	0.679508i
$538$		0			0
$539$		0			0
$540$	1056.00	−	198.997i	1.95556	−	0.368514i
$541$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$541$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$542$		0			0
$543$	−211.611	−	760.093i	−0.389708	−	1.39980i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$547$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$548$	−163.755	−	225.389i	−0.298823	−	0.411294i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	31.8048	−	745.563i	0.0573060	−	1.34336i
$556$		0			0
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$563$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$564$	593.870	−	748.132i	1.05296	−	1.32648i
$565$	214.149	−	659.082i	0.379024	−	1.16652i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$569$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	577.500	+	870.614i	1.00785	+	1.51940i
$574$		0			0
$575$	2100.76	−	682.579i	3.65350	−	1.18709i
$576$	−435.468	−	377.020i	−0.756021	−	0.654548i
$577$	853.513	+	620.113i	1.47923	+	1.07472i	0.977805	+	0.209516i	$0.0671888\pi$
$577$	853.513	+	620.113i	1.47923	+	1.07472i	0.501420	+	0.865204i	$0.332811\pi$
$578$		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
$587$	−302.813	−	98.3898i	−0.515865	−	0.167615i	0.0395031	−	0.999219i	$-0.487423\pi$
$587$	−302.813	−	98.3898i	−0.515865	−	0.167615i	−0.555368	+	0.831605i	$0.687423\pi$
$588$	587.466	+	25.0606i	0.999091	+	0.0426201i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−323.607	+	235.114i	−0.546633	+	0.397152i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.69938	−	3.73039i	−0.00787166	−	0.00624856i
$598$		0			0
$599$	701.807	−	965.954i	1.17163	−	1.61261i	0.519635	−	0.854388i	$-0.326068\pi$
$599$	701.807	−	965.954i	1.17163	−	1.61261i	0.651995	−	0.758223i	$-0.273932\pi$
$600$		0			0
$601$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$601$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$602$		0			0
$603$	−71.4734	−	306.784i	−0.118530	−	0.508763i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$607$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$613$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1114.39i		−	1.80614i	−0.429498	−	0.903068i	$-0.641309\pi$
$617$		−	1114.39i		−	1.80614i	0.429498	−	0.903068i	$-0.358691\pi$
$618$		0			0
$619$	−382.254	−	1176.46i	−0.617535	−	1.90058i	−0.346986	−	0.937870i	$-0.612795\pi$
$619$	−382.254	−	1176.46i	−0.617535	−	1.90058i	−0.270549	−	0.962706i	$-0.587205\pi$
$620$	1400.51	+	455.053i	2.25888	+	0.733956i
$621$	−728.467	−	344.782i	−1.17306	−	0.555204i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2427.86	+	1763.94i	−3.88458	+	2.82231i
$626$		0			0
$627$		0			0
$628$	−860.000			−1.36943
$629$		0			0
$630$		0			0
$631$	374.838	−	1153.63i	0.594037	−	1.82826i	0.0345772	−	0.999402i	$-0.488992\pi$
$631$	374.838	−	1153.63i	0.594037	−	1.82826i	0.559460	−	0.828857i	$-0.311008\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	895.032	+	333.619i	1.40728	+	0.524558i
$637$		0			0
$638$		0			0
$639$	−412.500	−	174.123i	−0.645540	−	0.272493i
$640$		0			0
$641$	−993.604	+	322.841i	−1.55008	+	0.503653i	−0.954137	−	0.299371i	$-0.903223\pi$
$641$	−993.604	+	322.841i	−1.55008	+	0.503653i	−0.595947	+	0.803024i	$0.703223\pi$
$642$		0			0
$643$	−319.562	−	232.175i	−0.496986	−	0.361081i	0.310879	−	0.950450i	$-0.399377\pi$
$643$	−319.562	−	232.175i	−0.496986	−	0.361081i	−0.807864	+	0.589368i	$0.799377\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−684.262	−	941.805i	−1.05759	−	1.45565i	−0.882037	−	0.471180i	$-0.843828\pi$
$647$	−684.262	−	941.805i	−1.05759	−	1.45565i	−0.175554	−	0.984470i	$-0.556172\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−86.5248	−	266.296i	−0.132707	−	0.408429i
$653$	−160.869	−	52.2696i	−0.246354	−	0.0800453i	0.183237	−	0.983069i	$-0.441342\pi$
$653$	−160.869	−	52.2696i	−0.246354	−	0.0800453i	−0.429591	+	0.903023i	$0.641342\pi$
$654$		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
$661$	1153.00			1.74433			0.872163	−	0.489215i	$-0.162717\pi$
$661$	1153.00			1.74433			0.872163	+	0.489215i	$0.162717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	466.276	−	1250.92i	0.696974	−	1.86984i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$673$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$674$		0			0
$675$	1981.68	+	254.846i	2.93582	+	0.377550i
$676$	546.895	+	397.343i	0.809017	+	0.587785i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1114.39i			1.63160i	0.578331	+	0.815802i	$0.303704\pi$
$683$			1114.39i			1.63160i	−0.578331	+	0.815802i	$0.696296\pi$
$684$		0			0
$685$	−214.149	−	659.082i	−0.312626	−	0.962164i
$686$		0			0
$687$	−1297.82	−	55.3635i	−1.88911	−	0.0805873i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	884.256	−	642.449i	1.27968	−	0.929738i	0.280132	−	0.959961i	$-0.409622\pi$
$691$	884.256	−	642.449i	1.27968	−	0.929738i	0.999543	+	0.0302230i	$0.00962175\pi$
$692$		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	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	1980.00	−	1313.38i	2.80851	−	1.86296i
$706$		0			0
$707$		0			0
$708$	575.120	−	160.115i	0.812317	−	0.226151i
$709$	−855.131	−	621.289i	−1.20611	−	0.876289i	−0.211237	−	0.977435i	$-0.567749\pi$
$709$	−855.131	−	621.289i	−1.20611	−	0.876289i	−0.994872	+	0.101146i	$0.967749\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−649.171	−	893.508i	−0.910479	−	1.25317i
$714$		0			0
$715$		0			0
$716$			1392.98i			1.94551i
$717$		0			0
$718$		0			0
$719$	804.346	+	261.348i	1.11870	+	0.363488i	0.809272	−	0.587435i	$-0.199862\pi$
$719$	804.346	+	261.348i	1.11870	+	0.363488i	0.309429	+	0.950922i	$0.399862\pi$
$720$	−740.393	−	1226.66i	−1.02832	−	1.70369i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−851.086	+	618.350i	−1.17553	+	0.854075i
$725$		0			0
$726$		0			0
$727$	−1355.00			−1.86382			−0.931912	−	0.362685i	$-0.881860\pi$
$727$	−1355.00			−1.86382			−0.931912	+	0.362685i	$0.881860\pi$
$728$		0			0
$729$	−462.166	−	563.776i	−0.633973	−	0.773355i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$733$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$734$		0			0
$735$	1370.52	+	510.854i	1.86465	+	0.695039i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$	−946.289	+	307.468i	−1.27877	+	0.415497i
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	300.674	+	925.378i	0.400364	+	1.23219i	0.924705	+	0.380686i	$0.124312\pi$
$751$	300.674	+	925.378i	0.400364	+	1.23219i	−0.524340	+	0.851509i	$0.675688\pi$
$752$	−1211.25	−	393.559i	−1.61071	−	0.523350i
$753$	−57.2487	+	1342.01i	−0.0760275	+	1.78222i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−56.6312	+	41.1450i	−0.0748100	+	0.0543527i	−0.624562	−	0.780975i	$-0.714722\pi$
$757$	−56.6312	+	41.1450i	−0.0748100	+	0.0543527i	0.549752	+	0.835328i	$0.314722\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$761$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$		0			0
$764$	818.775	−	1126.95i	1.07169	−	1.47506i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−268.240	+	719.633i	−0.349270	+	0.937022i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	792.000	+	1193.98i	1.02724	+	1.54862i
$772$		0			0
$773$	681.328	−	221.377i	0.881408	−	0.286387i	0.166866	−	0.985980i	$-0.446635\pi$
$773$	681.328	−	221.377i	0.881408	−	0.286387i	0.714542	+	0.699593i	$0.246635\pi$
$774$		0			0
$775$	2215.09	+	1609.36i	2.85818	+	2.07659i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−242.269	−	745.628i	−0.309017	−	0.951057i
$785$	−2034.52	−	661.056i	−2.59175	−	0.842110i
$786$		0			0
$787$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$787$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1860.96	+	1477.23i	2.34082	+	1.85816i
$796$	−2.47214	+	7.60845i	−0.00310570	+	0.00955836i
$797$	−99.4226	+	136.843i	−0.124746	+	0.171698i	−0.866822	−	0.498617i	$-0.833841\pi$
$797$	−99.4226	+	136.843i	−0.124746	+	0.171698i	0.742076	+	0.670316i	$0.233841\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1308.20	−	304.779i	1.63321	−	0.380499i
$802$		0			0
$803$		0			0
$804$	−350.000	+	232.164i	−0.435323	+	0.288761i
$805$		0			0
$806$		0			0
$807$	−1150.24	+	320.229i	−1.42533	+	0.396814i
$808$		0			0
$809$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$809$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$810$		0			0
$811$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$811$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	696.491i		−	0.854590i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$	−610.808	+	443.778i	−0.742172	+	0.539220i	−0.893391	−	0.449281i	$-0.851680\pi$
$823$	−610.808	+	443.778i	−0.742172	+	0.539220i	0.151218	+	0.988500i	$0.451680\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$827$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$828$	−91.5147	+	1070.68i	−0.110525	+	1.29309i
$829$	−252.467	+	777.013i	−0.304544	+	0.937290i	0.675303	+	0.737540i	$0.264013\pi$
$829$	−252.467	+	777.013i	−0.304544	+	0.937290i	−0.979847	+	0.199749i	$0.935987\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−185.000	−	981.721i	−0.221027	−	1.17290i
$838$		0			0
$839$	1561.38	−	507.322i	1.86100	−	0.604675i	0.866596	−	0.499010i	$-0.166303\pi$
$839$	1561.38	−	507.322i	1.86100	−	0.604675i	0.994402	−	0.105665i	$-0.0336971\pi$
$840$		0			0
$841$	−680.383	−	494.327i	−0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	988.378	+	1360.39i	1.16968	+	1.60992i
$846$		0			0
$847$		0			0
$848$		−	1273.58i		−	1.50187i
$849$		0			0
$850$		0			0
$851$	709.717	+	230.601i	0.833980	+	0.270977i
$852$	−25.4439	+	596.450i	−0.0298637	+	0.700059i
$853$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$853$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−757.000			−0.881257			−0.440629	−	0.897689i	$-0.645244\pi$
$859$	−757.000			−0.881257			−0.440629	+	0.897689i	$0.645244\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−982.529	+	1352.34i	−1.13850	+	1.56702i	−0.367701	+	0.929944i	$0.619855\pi$
$863$	−982.529	+	1352.34i	−1.13850	+	1.56702i	−0.770803	+	0.637073i	$0.780145\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−302.817	+	812.398i	−0.349270	+	0.937022i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−646.398	−	559.638i	−0.740432	−	0.641052i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$877$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			945.238i			1.07291i	0.843927	+	0.536457i	$0.180238\pi$
$881$			945.238i			1.07291i	−0.843927	+	0.536457i	$0.819762\pi$
$882$		0			0
$883$	−423.353	−	1302.95i	−0.479449	−	1.47559i	−0.839863	−	0.542799i	$-0.817364\pi$
$883$	−423.353	−	1302.95i	−0.479449	−	1.47559i	0.360414	−	0.932793i	$-0.382636\pi$
$884$		0			0
$885$	1483.65	+	63.2908i	1.67644	+	0.0715150i
$886$		0			0
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1780.00			−1.99552
$893$		0			0
$894$		0			0
$895$	−1070.74	+	3295.41i	−1.19636	+	3.68202i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−604.461	−	2594.52i	−0.671623	−	2.88280i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2488.74	+	808.641i	−2.74999	+	0.893526i
$906$		0			0
$907$	1415.78	+	1028.62i	1.56095	+	1.13410i	0.935227	+	0.354048i	$0.115195\pi$
$907$	1415.78	+	1028.62i	1.56095	+	1.13410i	0.625721	+	0.780047i	$0.284805\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	233.936	+	321.985i	0.256790	+	0.353441i	0.917875	−	0.396870i	$-0.129904\pi$
$911$	233.936	+	321.985i	0.256790	+	0.353441i	−0.661085	+	0.750311i	$0.729904\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	535.217	+	1647.23i	0.584299	+	1.79829i
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1850.00			−2.00000
$926$		0			0
$927$	−1703.79	−	145.628i	−1.83796	−	0.157096i
$928$		0			0
$929$	−935.742	+	1287.94i	−1.00726	+	1.38637i	−0.0864930	+	0.996252i	$0.527566\pi$
$929$	−935.742	+	1287.94i	−1.00726	+	1.38637i	−0.920765	+	0.390119i	$0.872434\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1118.79	−	417.023i	−1.19913	−	0.446971i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$937$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$938$		0			0
$939$	−213.221	−	765.873i	−0.227072	−	0.815627i
$940$	−2562.97	−	1862.10i	−2.72656	−	1.98096i
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	−467.871	−	643.969i	−0.495626	−	0.682171i
$945$		0			0
$946$		0			0
$947$		−	1323.33i		−	1.39740i	−0.715417	−	0.698698i	$-0.753763\pi$
$947$		−	1323.33i		−	1.39740i	0.715417	−	0.698698i	$-0.246237\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	80.1482	−	1878.82i	0.0842778	−	1.97562i
$952$		0			0
$953$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$953$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$954$		0			0
$955$	2803.24	−	2036.68i	2.93533	−	2.13264i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1187.74	+	1496.26i	−1.23723	+	1.55861i
$961$	126.079	−	388.031i	0.131196	−	0.403778i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1845.26	+	599.563i	−1.90037	+	0.617469i	−0.936980	+	0.349382i	$0.886391\pi$
$971$	−1845.26	+	599.563i	−1.90037	+	0.617469i	−0.963395	+	0.268087i	$0.913609\pi$
$972$	−489.133	+	839.960i	−0.503223	+	0.864157i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	894.804	+	1231.59i	0.915869	+	1.26058i	0.965123	+	0.261799i	$0.0843156\pi$
$977$	894.804	+	1231.59i	0.915869	+	1.26058i	−0.0492540	+	0.998786i	$0.515684\pi$
$978$		0			0
$979$		0			0
$980$		−	1950.18i		−	1.98997i
$981$		0			0
$982$		0			0
$983$	1769.56	+	574.965i	1.80016	+	0.584909i	0.999892	−	0.0147282i	$-0.00468831\pi$
$983$	1769.56	+	574.965i	1.80016	+	0.584909i	0.800272	+	0.599637i	$0.204688\pi$
$984$		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	$-0.205794\pi$
$991$	1582.00			1.59637			0.798184	+	0.602414i	$0.205794\pi$
$992$		0			0
$993$	1322.88	+	1050.10i	1.33220	+	1.05751i
$994$		0			0
$995$	−11.6968	+	16.0992i	−0.0117556	+	0.0161801i
$996$		0			0
$997$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$997$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$998$		0			0
$999$	491.020	+	463.168i	0.491511	+	0.463632i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.f.245.2		8
3.2	odd	2	inner	363.3.h.f.245.1		8
11.2	odd	10		363.3.b.c.122.1	✓	2
11.3	even	5	inner	363.3.h.f.269.1		8
11.4	even	5	inner	363.3.h.f.323.1		8
11.5	even	5	inner	363.3.h.f.251.2		8
11.6	odd	10	inner	363.3.h.f.251.2		8
11.7	odd	10	inner	363.3.h.f.323.1		8
11.8	odd	10	inner	363.3.h.f.269.1		8
11.9	even	5		363.3.b.c.122.1	✓	2
11.10	odd	2	CM	363.3.h.f.245.2		8
33.2	even	10		363.3.b.c.122.2	yes	2
33.5	odd	10	inner	363.3.h.f.251.1		8
33.8	even	10	inner	363.3.h.f.269.2		8
33.14	odd	10	inner	363.3.h.f.269.2		8
33.17	even	10	inner	363.3.h.f.251.1		8
33.20	odd	10		363.3.b.c.122.2	yes	2
33.26	odd	10	inner	363.3.h.f.323.2		8
33.29	even	10	inner	363.3.h.f.323.2		8
33.32	even	2	inner	363.3.h.f.245.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.b.c.122.1	✓	2	11.2	odd	10
363.3.b.c.122.1	✓	2	11.9	even	5
363.3.b.c.122.2	yes	2	33.2	even	10
363.3.b.c.122.2	yes	2	33.20	odd	10
363.3.h.f.245.1		8	3.2	odd	2	inner
363.3.h.f.245.1		8	33.32	even	2	inner
363.3.h.f.245.2		8	1.1	even	1	trivial
363.3.h.f.245.2		8	11.10	odd	2	CM
363.3.h.f.251.1		8	33.5	odd	10	inner
363.3.h.f.251.1		8	33.17	even	10	inner
363.3.h.f.251.2		8	11.5	even	5	inner
363.3.h.f.251.2		8	11.6	odd	10	inner
363.3.h.f.269.1		8	11.3	even	5	inner
363.3.h.f.269.1		8	11.8	odd	10	inner
363.3.h.f.269.2		8	33.8	even	10	inner
363.3.h.f.269.2		8	33.14	odd	10	inner
363.3.h.f.323.1		8	11.4	even	5	inner
363.3.h.f.323.1		8	11.7	odd	10	inner
363.3.h.f.323.2		8	33.26	odd	10	inner
363.3.h.f.323.2		8	33.29	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.34969 + 1.86519i) q^{3} +(1.23607 - 3.80423i) q^{4} +(5.84839 - 8.04962i) q^{5} +(2.04210 + 8.76526i) q^{9} +O(q^{10})\)	\(q+(2.34969 + 1.86519i) q^{3} +(1.23607 - 3.80423i) q^{4} +(5.84839 - 8.04962i) q^{5} +(2.04210 + 8.76526i) q^{9} +(10.0000 - 6.63325i) q^{12} +(28.7560 - 8.00573i) q^{15} +(-12.9443 - 9.40456i) q^{16} +(-23.3936 - 32.1985i) q^{20} +29.8496i q^{23} +(-22.8673 - 70.3782i) q^{25} +(-11.5506 + 24.4046i) q^{27} +(-29.9336 + 21.7481i) q^{31} +(35.8692 + 3.06586i) q^{36} +(7.72542 - 23.7764i) q^{37} +(82.5000 + 34.8246i) q^{45} +(75.7031 - 24.5974i) q^{47} +(-12.8737 - 46.2414i) q^{48} +(39.6418 + 28.8015i) q^{49} +(46.7871 + 64.3969i) q^{53} +(47.3145 + 15.3734i) q^{59} +(5.08877 - 119.290i) q^{60} +(-51.7771 + 37.6183i) q^{64} -35.0000 q^{67} +(-55.6754 + 70.1374i) q^{69} +(-29.2419 + 40.2481i) q^{71} +(77.5380 - 208.019i) q^{75} +(-151.406 + 49.1949i) q^{80} +(-72.6597 + 35.7991i) q^{81} -149.248i q^{89} +(113.555 + 36.8962i) q^{92} +(-110.899 - 4.73083i) q^{93} +(-76.8566 + 55.8396i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9	\( 8 q - 5 q^{3} - 8 q^{4} - 7 q^{9} + 80 q^{12} - 33 q^{15} - 32 q^{16} + 148 q^{25} + 10 q^{27} - 74 q^{31} - 28 q^{36} - 50 q^{37} + 660 q^{45} - 80 q^{48} + 98 q^{49} - 132 q^{60} - 128 q^{64} - 280 q^{67} - 99 q^{69} + 370 q^{75} + 113 q^{81} - 185 q^{93} - 190 q^{97}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^4 - 7 * q^9 + 80 * q^12 - 33 * q^15 - 32 * q^16 + 148 * q^25 + 10 * q^27 - 74 * q^31 - 28 * q^36 - 50 * q^37 + 660 * q^45 - 80 * q^48 + 98 * q^49 - 132 * q^60 - 128 * q^64 - 280 * q^67 - 99 * q^69 + 370 * q^75 + 113 * q^81 - 185 * q^93 - 190 * q^97

Introduction

L-functions

Modular forms

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Database

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