LMFDB - Embedded newform 363.3.h.a.323.1

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:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
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			323.1
Root			$0.809017 + 0.587785i$ of defining polynomial
Character	$\chi$	$=$	363.323
Dual form			363.3.h.a.245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.927051 + 2.85317i) q^{3} +(1.23607 + 3.80423i) q^{4} +(3.39919 + 10.4616i) q^{7} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})$	$q+(-0.927051 + 2.85317i) q^{3} +(1.23607 + 3.80423i) q^{4} +(3.39919 + 10.4616i) q^{7} +(-7.28115 - 5.29007i) q^{9} -12.0000 q^{12} +(17.7984 + 12.9313i) q^{13} +(-12.9443 + 9.40456i) q^{16} +(3.39919 - 10.4616i) q^{19} -33.0000 q^{21} +(7.72542 - 23.7764i) q^{25} +(21.8435 - 15.8702i) q^{27} +(-35.5967 + 25.8626i) q^{28} +(-47.7320 - 34.6793i) q^{31} +(11.1246 - 34.2380i) q^{36} +(14.5238 + 44.6997i) q^{37} +(-53.3951 + 38.7938i) q^{39} -22.0000 q^{43} +(-14.8328 - 45.6507i) q^{48} +(-58.2492 + 42.3205i) q^{49} +(-27.1935 + 83.6930i) q^{52} +(26.6976 + 19.3969i) q^{57} +(97.8911 - 71.1220i) q^{61} +(30.5927 - 94.1546i) q^{63} +(-51.7771 - 37.6183i) q^{64} -13.0000 q^{67} +(44.1894 + 136.001i) q^{73} +(60.6763 + 44.0839i) q^{75} +44.0000 q^{76} +(-8.89919 - 6.46564i) q^{79} +(25.0304 + 77.0356i) q^{81} +(-40.7902 - 125.539i) q^{84} +(-74.7821 + 230.156i) q^{91} +(143.196 - 104.038i) q^{93} +(136.724 + 99.3357i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9	$ 4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9} - 48 q^{12} + 22 q^{13} - 16 q^{16} - 11 q^{19} - 132 q^{21} - 25 q^{25} + 27 q^{27} - 44 q^{28} - 59 q^{31} - 36 q^{36} - 47 q^{37} - 66 q^{39} - 88 q^{43} + 48 q^{48} - 72 q^{49} + 88 q^{52} + 33 q^{57} + 121 q^{61} - 99 q^{63} - 64 q^{64} - 52 q^{67} - 143 q^{73} + 75 q^{75} + 176 q^{76} - 11 q^{79} - 81 q^{81} + 132 q^{84} + 242 q^{91} + 177 q^{93} + 169 q^{97}+O(q^{100}) $ 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9 - 48 * q^12 + 22 * q^13 - 16 * q^16 - 11 * q^19 - 132 * q^21 - 25 * q^25 + 27 * q^27 - 44 * q^28 - 59 * q^31 - 36 * q^36 - 47 * q^37 - 66 * q^39 - 88 * q^43 + 48 * q^48 - 72 * q^49 + 88 * q^52 + 33 * q^57 + 121 * q^61 - 99 * q^63 - 64 * q^64 - 52 * q^67 - 143 * q^73 + 75 * q^75 + 176 * q^76 - 11 * q^79 - 81 * q^81 + 132 * q^84 + 242 * q^91 + 177 * q^93 + 169 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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	$-0.800000\pi$
$2$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$3$	−0.927051	+	2.85317i	−0.309017	+	0.951057i
$3$	−0.927051	+	2.85317i	−0.309017	+	0.951057i
$4$	1.23607	+	3.80423i	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
$7$	3.39919	+	10.4616i	0.485598	+	1.49452i	0.831113	+	0.556104i	$0.187704\pi$
$7$	3.39919	+	10.4616i	0.485598	+	1.49452i	−0.345515	+	0.938413i	$0.612296\pi$
$8$		0			0
$9$	−7.28115	−	5.29007i	−0.809017	−	0.587785i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−12.0000			−1.00000
$13$	17.7984	+	12.9313i	1.36911	+	0.994714i	0.997806	+	0.0662003i	$0.0210876\pi$
$13$	17.7984	+	12.9313i	1.36911	+	0.994714i	0.371299	+	0.928513i	$0.378912\pi$
$14$		0			0
$15$		0			0
$16$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$		0			0
$19$	3.39919	−	10.4616i	0.178905	−	0.550612i	−0.820886	−	0.571093i	$-0.806520\pi$
$19$	3.39919	−	10.4616i	0.178905	−	0.550612i	0.999790	+	0.0204809i	$0.00651973\pi$
$20$		0			0
$21$	−33.0000			−1.57143
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	7.72542	−	23.7764i	0.309017	−	0.951057i
$26$		0			0
$27$	21.8435	−	15.8702i	0.809017	−	0.587785i
$28$	−35.5967	+	25.8626i	−1.27131	+	0.923663i
$29$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$29$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$30$		0			0
$31$	−47.7320	−	34.6793i	−1.53974	−	1.11869i	−0.950497	−	0.310734i	$-0.899425\pi$
$31$	−47.7320	−	34.6793i	−1.53974	−	1.11869i	−0.589245	−	0.807954i	$-0.700575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	11.1246	−	34.2380i	0.309017	−	0.951057i
$37$	14.5238	+	44.6997i	0.392535	+	1.20810i	0.930865	+	0.365364i	$0.119056\pi$
$37$	14.5238	+	44.6997i	0.392535	+	1.20810i	−0.538329	+	0.842734i	$0.680944\pi$
$38$		0			0
$39$	−53.3951	+	38.7938i	−1.36911	+	0.994714i
$40$		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
$43$	−22.0000			−0.511628			−0.255814	−	0.966726i	$-0.582343\pi$
$43$	−22.0000			−0.511628			−0.255814	+	0.966726i	$0.582343\pi$
$44$		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	$0.600000\pi$
$48$	−14.8328	−	45.6507i	−0.309017	−	0.951057i
$49$	−58.2492	+	42.3205i	−1.18876	+	0.863684i
$50$		0			0
$51$		0			0
$52$	−27.1935	+	83.6930i	−0.522952	+	1.60948i
$53$		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
$57$	26.6976	+	19.3969i	0.468378	+	0.340297i
$58$		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
$61$	97.8911	−	71.1220i	1.60477	−	1.16593i	0.727282	−	0.686339i	$-0.240783\pi$
$61$	97.8911	−	71.1220i	1.60477	−	1.16593i	0.877490	−	0.479596i	$-0.159217\pi$
$62$		0			0
$63$	30.5927	−	94.1546i	0.485598	−	1.49452i
$64$	−51.7771	−	37.6183i	−0.809017	−	0.587785i
$65$		0			0
$66$		0			0
$67$	−13.0000			−0.194030			−0.0970149	−	0.995283i	$-0.530929\pi$
$67$	−13.0000			−0.194030			−0.0970149	+	0.995283i	$0.530929\pi$
$68$		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
$73$	44.1894	+	136.001i	0.605335	+	1.86303i	0.494474	+	0.869193i	$0.335361\pi$
$73$	44.1894	+	136.001i	0.605335	+	1.86303i	0.110861	+	0.993836i	$0.464639\pi$
$74$		0			0
$75$	60.6763	+	44.0839i	0.809017	+	0.587785i
$76$	44.0000			0.578947
$77$		0			0
$78$		0			0
$79$	−8.89919	−	6.46564i	−0.112648	−	0.0818435i	0.530035	−	0.847976i	$-0.322179\pi$
$79$	−8.89919	−	6.46564i	−0.112648	−	0.0818435i	−0.642683	+	0.766132i	$0.722179\pi$
$80$		0			0
$81$	25.0304	+	77.0356i	0.309017	+	0.951057i
$82$		0			0
$83$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$83$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$84$	−40.7902	−	125.539i	−0.485598	−	1.49452i
$85$		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
$91$	−74.7821	+	230.156i	−0.821781	+	2.52918i
$92$		0			0
$93$	143.196	−	104.038i	1.53974	−	1.11869i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	136.724	+	99.3357i	1.40952	+	1.02408i	0.993391	+	0.114776i	$0.0366149\pi$
$97$	136.724	+	99.3357i	1.40952	+	1.02408i	0.416133	+	0.909304i	$0.363385\pi$
$98$		0			0
$99$		0			0
$100$	100.000			1.00000
$101$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$101$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$102$		0			0
$103$	−48.5157	−	149.316i	−0.471026	−	1.44967i	−0.851243	−	0.524771i	$-0.824151\pi$
$103$	−48.5157	−	149.316i	−0.471026	−	1.44967i	0.380217	−	0.924897i	$-0.375849\pi$
$104$		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	$0.600000\pi$
$108$	87.3738	+	63.4808i	0.809017	+	0.587785i
$109$	143.000			1.31193			0.655963	−	0.754793i	$-0.272263\pi$
$109$	143.000			1.31193			0.655963	+	0.754793i	$0.272263\pi$
$110$		0			0
$111$	−141.000			−1.27027
$112$	−142.387	−	103.450i	−1.27131	−	0.923663i
$113$		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
$117$	−61.1854	−	188.309i	−0.522952	−	1.60948i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	72.9280	−	224.449i	0.588129	−	1.81008i
$125$		0			0
$126$		0			0
$127$	204.681	−	148.710i	1.61166	−	1.17094i	0.753726	−	0.657189i	$-0.228255\pi$
$127$	204.681	−	148.710i	1.61166	−	1.17094i	0.857938	−	0.513753i	$-0.171745\pi$
$128$		0			0
$129$	20.3951	−	62.7697i	0.158102	−	0.486587i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	121.000			0.909774
$134$		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
$139$	−6.79837	−	20.9232i	−0.0489092	−	0.150527i	0.923619	−	0.383311i	$-0.125216\pi$
$139$	−6.79837	−	20.9232i	−0.0489092	−	0.150527i	−0.972528	+	0.232784i	$0.925216\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	144.000			1.00000
$145$		0			0
$146$		0			0
$147$	−66.7477	−	205.428i	−0.454066	−	1.39747i
$148$	−152.095	+	110.504i	−1.02767	+	0.746646i
$149$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$149$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$	−88.3789	+	272.002i	−0.585290	+	1.80134i	0.0128103	+	0.999918i	$0.495922\pi$
$151$	−88.3789	+	272.002i	−0.585290	+	1.80134i	−0.598101	+	0.801421i	$0.704078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−213.580	−	155.175i	−1.36911	−	0.994714i
$157$	−59.6403	+	183.554i	−0.379874	+	1.16913i	0.560256	+	0.828319i	$0.310703\pi$
$157$	−59.6403	+	183.554i	−0.379874	+	1.16913i	−0.940131	+	0.340814i	$0.889297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	29.9336	+	21.7481i	0.183642	+	0.133424i	0.675808	−	0.737078i	$-0.263795\pi$
$163$	29.9336	+	21.7481i	0.183642	+	0.133424i	−0.492166	+	0.870501i	$0.663795\pi$
$164$		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
$169$	97.3404	+	299.583i	0.575978	+	1.77268i
$170$		0			0
$171$	−80.0927	+	58.1907i	−0.468378	+	0.340297i
$172$	−27.1935	−	83.6930i	−0.158102	−	0.486587i
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$	275.000			1.57143
$176$		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
$181$	0.809017	−	0.587785i	0.00446971	−	0.00324743i	−0.585548	−	0.810638i	$-0.699121\pi$
$181$	0.809017	−	0.587785i	0.00446971	−	0.00324743i	0.590018	+	0.807390i	$0.299121\pi$
$182$		0			0
$183$	112.173	+	345.234i	0.612968	+	1.88652i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	240.278	+	174.572i	1.27131	+	0.923663i
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$	155.331	−	112.855i	0.809017	−	0.587785i
$193$	−115.689	+	84.0533i	−0.599427	+	0.435509i	−0.845675	−	0.533697i	$-0.820802\pi$
$193$	−115.689	+	84.0533i	−0.599427	+	0.435509i	0.246248	+	0.969207i	$0.420802\pi$
$194$		0			0
$195$		0			0
$196$	−232.997	−	169.282i	−1.18876	−	0.863684i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−277.000			−1.39196			−0.695980	−	0.718061i	$-0.745030\pi$
$199$	−277.000			−1.39196			−0.695980	+	0.718061i	$0.745030\pi$
$200$		0			0
$201$	12.0517	−	37.0912i	0.0599585	−	0.184533i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−352.000			−1.69231
$209$		0			0
$210$		0			0
$211$	204.681	+	148.710i	0.970054	+	0.704785i	0.955464	−	0.295108i	$-0.0953558\pi$
$211$	204.681	+	148.710i	0.970054	+	0.704785i	0.0145898	+	0.999894i	$0.495356\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	200.552	−	617.236i	0.924203	−	2.84440i
$218$		0			0
$219$	−429.000			−1.95890
$220$		0			0
$221$		0			0
$222$		0			0
$223$	25.6484	−	78.9377i	0.115015	−	0.353981i	−0.876935	−	0.480609i	$-0.840416\pi$
$223$	25.6484	−	78.9377i	0.115015	−	0.353981i	0.991950	+	0.126628i	$0.0404156\pi$
$224$		0			0
$225$	−182.029	+	132.252i	−0.809017	+	0.587785i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$	−40.7902	+	125.539i	−0.178905	+	0.550612i
$229$	−21.0344	−	15.2824i	−0.0918535	−	0.0667354i	0.540911	−	0.841080i	$-0.318080\pi$
$229$	−21.0344	−	15.2824i	−0.0918535	−	0.0667354i	−0.632764	+	0.774345i	$0.718080\pi$
$230$		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
$235$		0			0
$236$		0			0
$237$	26.6976	−	19.3969i	0.112648	−	0.0818435i
$238$		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
$241$	−286.000			−1.18672			−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−286.000			−1.18672			−0.593361	+	0.804936i	$0.702199\pi$
$242$		0			0
$243$	−243.000			−1.00000
$244$	391.564	+	284.488i	1.60477	+	1.16593i
$245$		0			0
$246$		0			0
$247$	195.782	−	142.244i	0.792640	−	0.575887i
$248$		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	$0.200000\pi$
$252$	396.000			1.57143
$253$		0			0
$254$		0			0
$255$		0			0
$256$	79.1084	−	243.470i	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
$259$	−418.262	+	303.885i	−1.61491	+	1.17330i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−16.0689	−	49.4549i	−0.0599585	−	0.184533i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$	74.7821	+	230.156i	0.275949	+	0.849283i	0.988967	+	0.148137i	$0.0473277\pi$
$271$	74.7821	+	230.156i	0.275949	+	0.849283i	−0.713018	+	0.701146i	$0.752672\pi$
$272$		0			0
$273$	−587.346	−	426.732i	−2.15145	−	1.56312i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−329.270	−	239.229i	−1.18870	−	0.863641i	−0.195574	−	0.980689i	$-0.562657\pi$
$277$	−329.270	−	239.229i	−1.18870	−	0.863641i	−0.993126	+	0.117048i	$0.962657\pi$
$278$		0			0
$279$	164.088	+	505.011i	0.588129	+	1.81008i
$280$		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
$283$	−159.762	+	491.696i	−0.564529	+	1.73744i	0.104817	+	0.994492i	$0.466574\pi$
$283$	−159.762	+	491.696i	−0.564529	+	1.73744i	−0.669346	+	0.742951i	$0.733426\pi$
$284$		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
$291$	−410.172	+	298.007i	−1.40952	+	1.02408i
$292$	−462.758	+	336.213i	−1.58479	+	1.15141i
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−92.7051	+	285.317i	−0.309017	+	0.951057i
$301$	−74.7821	−	230.156i	−0.248446	−	0.764637i
$302$		0			0
$303$		0			0
$304$	54.3870	+	167.386i	0.178905	+	0.550612i
$305$		0			0
$306$		0			0
$307$	−253.000			−0.824104			−0.412052	−	0.911160i	$-0.635188\pi$
$307$	−253.000			−0.824104			−0.412052	+	0.911160i	$0.635188\pi$
$308$		0			0
$309$	471.000			1.52427
$310$		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
$313$	114.880	−	83.4655i	0.367030	−	0.266663i	−0.388948	−	0.921260i	$-0.627162\pi$
$313$	114.880	−	83.4655i	0.367030	−	0.266663i	0.755978	+	0.654597i	$0.227162\pi$
$314$		0			0
$315$		0			0
$316$	13.5967	−	41.8465i	0.0430277	−	0.132426i
$317$		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
$324$	−262.122	+	190.442i	−0.809017	+	0.587785i
$325$	444.959	−	323.282i	1.36911	−	0.994714i
$326$		0			0
$327$	−132.568	+	408.003i	−0.405408	+	1.24772i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	299.000			0.903323			0.451662	−	0.892189i	$-0.350831\pi$
$331$	299.000			0.903323			0.451662	+	0.892189i	$0.350831\pi$
$332$		0			0
$333$	130.714	−	402.297i	0.392535	−	1.20810i
$334$		0			0
$335$		0			0
$336$	427.161	−	310.351i	1.27131	−	0.923663i
$337$	−200.552	−	617.236i	−0.595110	−	1.83156i	−0.554178	−	0.832398i	$-0.686967\pi$
$337$	−200.552	−	617.236i	−0.595110	−	1.83156i	−0.0409315	−	0.999162i	$-0.513033\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−204.681	−	148.710i	−0.596738	−	0.433556i
$344$		0			0
$345$		0			0
$346$		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
$349$	207.350	−	638.159i	0.594127	−	1.82854i	0.0351039	−	0.999384i	$-0.488824\pi$
$349$	207.350	−	638.159i	0.594127	−	1.82854i	0.559023	−	0.829152i	$-0.311176\pi$
$350$		0			0
$351$	594.000			1.69231
$352$		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		−0.309017	−	0.951057i	$-0.600000\pi$
$359$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$360$		0			0
$361$	194.164	+	141.068i	0.537851	+	0.390771i
$362$		0			0
$363$		0			0
$364$	−968.000			−2.65934
$365$		0			0
$366$		0			0
$367$	−221.874	−	682.859i	−0.604562	−	1.86065i	−0.499775	−	0.866155i	$-0.666584\pi$
$367$	−221.874	−	682.859i	−0.604562	−	1.86065i	−0.104787	−	0.994495i	$-0.533416\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	572.784	+	416.152i	1.53974	+	1.11869i
$373$	−121.000			−0.324397			−0.162198	−	0.986758i	$-0.551858\pi$
$373$	−121.000			−0.324397			−0.162198	+	0.986758i	$0.551858\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	561.458	−	407.923i	1.48142	−	1.07631i	0.504323	−	0.863515i	$-0.331742\pi$
$379$	561.458	−	407.923i	1.48142	−	1.07631i	0.977096	−	0.212799i	$-0.0682579\pi$
$380$		0			0
$381$	234.544	+	721.852i	0.615601	+	1.89462i
$382$		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
$387$	160.185	+	116.381i	0.413916	+	0.300727i
$388$	−208.895	+	642.914i	−0.538390	+	1.65700i
$389$		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
$397$	431.000			1.08564			0.542821	−	0.839848i	$-0.317356\pi$
$397$	431.000			1.08564			0.542821	+	0.839848i	$0.317356\pi$
$398$		0			0
$399$	−112.173	+	345.234i	−0.281136	+	0.865247i
$400$	123.607	+	380.423i	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
$403$	−401.104	−	1234.47i	−0.995295	−	3.06320i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−115.689	−	84.0533i	−0.282859	−	0.205509i	0.437304	−	0.899314i	$-0.355933\pi$
$409$	−115.689	−	84.0533i	−0.282859	−	0.205509i	−0.720164	+	0.693804i	$0.755933\pi$
$410$		0			0
$411$		0			0
$412$	508.063	−	369.129i	1.23316	−	0.895945i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	66.0000			0.158273
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−110.628	+	340.478i	−0.262775	+	0.808737i	0.729423	+	0.684063i	$0.239789\pi$
$421$	−110.628	+	340.478i	−0.262775	+	0.808737i	−0.992198	+	0.124674i	$0.960211\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1076.80	+	782.342i	2.52178	+	1.83218i
$428$		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	$0.200000\pi$
$432$	−133.495	+	410.856i	−0.309017	+	0.951057i
$433$	155.436	+	478.381i	0.358974	+	1.10481i	0.953669	+	0.300856i	$0.0972724\pi$
$433$	155.436	+	478.381i	0.358974	+	1.10481i	−0.594696	+	0.803951i	$0.702728\pi$
$434$		0			0
$435$		0			0
$436$	176.758	+	544.004i	0.405408	+	1.24772i
$437$		0			0
$438$		0			0
$439$	803.000			1.82916			0.914579	−	0.404408i	$-0.132522\pi$
$439$	803.000			1.82916			0.914579	+	0.404408i	$0.132522\pi$
$440$		0			0
$441$	648.000			1.46939
$442$		0			0
$443$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$	−174.286	−	536.396i	−0.392535	−	1.20810i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	217.548	−	669.544i	0.485598	−	1.49452i
$449$		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
$453$	−694.137	−	504.320i	−1.53231	−	1.11329i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	658.540	−	478.457i	1.44101	−	1.04695i	0.453175	−	0.891421i	$-0.350291\pi$
$457$	658.540	−	478.457i	1.44101	−	1.04695i	0.987831	−	0.155531i	$-0.0497088\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−526.000			−1.13607			−0.568035	−	0.823005i	$-0.692296\pi$
$463$	−526.000			−1.13607			−0.568035	+	0.823005i	$0.692296\pi$
$464$		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	$0.800000\pi$
$468$	640.741	−	465.526i	1.36911	−	0.994714i
$469$	−44.1894	−	136.001i	−0.0942205	−	0.289981i
$470$		0			0
$471$	−468.421	−	340.328i	−0.994524	−	0.722564i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−222.480	−	161.641i	−0.468378	−	0.340297i
$476$		0			0
$477$		0			0
$478$		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
$481$	−319.524	+	983.392i	−0.664290	+	2.04447i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	297.274	−	914.916i	0.610420	−	1.87868i	0.156380	−	0.987697i	$-0.450018\pi$
$487$	297.274	−	914.916i	0.610420	−	1.87868i	0.454040	−	0.890981i	$-0.349982\pi$
$488$		0			0
$489$	−89.8009	+	65.2442i	−0.183642	+	0.133424i
$490$		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
$496$	944.000			1.90323
$497$		0			0
$498$		0			0
$499$	−271.008	−	834.077i	−0.543102	−	1.67150i	−0.725460	−	0.688264i	$-0.758373\pi$
$499$	−271.008	−	834.077i	−0.543102	−	1.67150i	0.182358	−	0.983232i	$-0.441627\pi$
$500$		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
$507$	−945.000			−1.86391
$508$	818.725	+	594.839i	1.61166	+	1.17094i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$	−1272.58	+	924.586i	−2.49038	+	1.80937i
$512$		0			0
$513$	−91.7780	−	282.464i	−0.178905	−	0.550612i
$514$		0			0
$515$		0			0
$516$	264.000			0.511628
$517$		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
$523$	−649.641	+	471.992i	−1.24214	+	0.902470i	−0.997739	−	0.0672041i	$-0.978592\pi$
$523$	−649.641	+	471.992i	−1.24214	+	0.902470i	−0.244403	+	0.969674i	$0.578592\pi$
$524$		0			0
$525$	−254.939	+	784.622i	−0.485598	+	1.49452i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	529.000			1.00000
$530$		0			0
$531$		0			0
$532$	149.564	+	460.311i	0.281136	+	0.865247i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−836.524	−	607.770i	−1.54625	−	1.12342i	−0.946256	−	0.323418i	$-0.895168\pi$
$541$	−836.524	−	607.770i	−1.54625	−	1.12342i	−0.599998	−	0.800001i	$-0.704832\pi$
$542$		0			0
$543$	0.927051	+	2.85317i	0.00170728	+	0.00525446i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	156.363	−	481.235i	0.285855	−	0.879771i	−0.700286	−	0.713862i	$-0.746944\pi$
$547$	156.363	−	481.235i	0.285855	−	0.879771i	0.986141	−	0.165909i	$-0.0530557\pi$
$548$		0			0
$549$	−1089.00			−1.98361
$550$		0			0
$551$		0			0
$552$		0			0
$553$	37.3911	−	115.078i	0.0676149	−	0.208097i
$554$		0			0
$555$		0			0
$556$	71.1935	−	51.7251i	0.128046	−	0.0930308i
$557$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$557$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$	−391.564	−	284.488i	−0.700473	−	0.508923i
$560$		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
$565$		0			0
$566$		0			0
$567$	−720.834	+	523.717i	−1.27131	+	0.923663i
$568$		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
$571$	1067.00			1.86865			0.934326	−	0.356420i	$-0.116003\pi$
$571$	1067.00			1.86865			0.934326	+	0.356420i	$0.116003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	177.994	+	547.809i	0.309017	+	0.951057i
$577$	835.715	−	607.182i	1.44838	−	1.05231i	0.462172	−	0.886790i	$-0.347070\pi$
$577$	835.715	−	607.182i	1.44838	−	1.05231i	0.986207	−	0.165519i	$-0.0529298\pi$
$578$		0			0
$579$	−132.568	−	408.003i	−0.228961	−	0.704669i
$580$		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	$0.400000\pi$
$588$	698.991	−	507.846i	1.18876	−	0.863684i
$589$	−525.052	+	381.473i	−0.891430	+	0.647662i
$590$		0			0
$591$		0			0
$592$	−608.381	−	442.015i	−1.02767	−	0.746646i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	256.793	−	790.328i	0.430139	−	1.32383i
$598$		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
$601$	370.511	+	1140.32i	0.616491	+	1.89737i	0.375398	+	0.926864i	$0.377506\pi$
$601$	370.511	+	1140.32i	0.616491	+	1.89737i	0.241094	+	0.970502i	$0.422494\pi$
$602$		0			0
$603$	94.6550	+	68.7709i	0.156973	+	0.114048i
$604$	−1144.00			−1.89404
$605$		0			0
$606$		0			0
$607$	658.540	+	478.457i	1.08491	+	0.788233i	0.978532	−	0.206093i	$-0.0660750\pi$
$607$	658.540	+	478.457i	1.08491	+	0.788233i	0.106377	+	0.994326i	$0.466075\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	44.1894	−	136.001i	0.0720872	−	0.221861i	−0.908521	−	0.417838i	$-0.862788\pi$
$613$	44.1894	−	136.001i	0.0720872	−	0.221861i	0.980608	+	0.195977i	$0.0627878\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−66.1296	+	203.526i	−0.106833	+	0.328798i	−0.990156	−	0.139966i	$-0.955301\pi$
$619$	−66.1296	+	203.526i	−0.106833	+	0.328798i	0.883323	+	0.468764i	$0.155301\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	326.322	−	1004.32i	0.522952	−	1.60948i
$625$	−505.636	−	367.366i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$	−772.000			−1.22930
$629$		0			0
$630$		0			0
$631$	208.277	+	641.012i	0.330075	+	1.01587i	0.969097	+	0.246679i	$0.0793392\pi$
$631$	208.277	+	641.012i	0.330075	+	1.01587i	−0.639022	+	0.769188i	$0.720661\pi$
$632$		0			0
$633$	−614.044	+	446.129i	−0.970054	+	0.704785i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1584.00			−2.48666
$638$		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
$643$	−746.723	+	542.526i	−1.16131	+	0.843742i	−0.989943	−	0.141466i	$-0.954819\pi$
$643$	−746.723	+	542.526i	−1.16131	+	0.843742i	−0.171367	+	0.985207i	$0.554819\pi$
$644$		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
$651$	1575.16	+	1144.42i	2.41959	+	1.75794i
$652$	−45.7345	+	140.756i	−0.0701450	+	0.215884i
$653$		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
$657$	397.705	−	1224.01i	0.605335	−	1.86303i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1201.00			−1.81694			−0.908472	−	0.417946i	$-0.862750\pi$
$661$	−1201.00			−1.81694			−0.908472	+	0.417946i	$0.862750\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	201.445	+	146.359i	0.301114	+	0.218772i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	952.213	+	691.823i	1.41488	+	1.02797i	0.992591	+	0.121507i	$0.0387728\pi$
$673$	952.213	+	691.823i	1.41488	+	1.02797i	0.422288	+	0.906462i	$0.361227\pi$
$674$		0			0
$675$	−208.586	−	641.963i	−0.309017	−	0.951057i
$676$	−1019.36	+	740.609i	−1.50793	+	1.09558i
$677$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$677$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$	−574.463	+	1768.01i	−0.846042	+	2.60385i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−320.371	−	232.763i	−0.468378	−	0.340297i
$685$		0			0
$686$		0			0
$687$	63.1033	−	45.8472i	0.0918535	−	0.0667354i
$688$	284.774	−	206.900i	0.413916	−	0.300727i
$689$		0			0
$690$		0			0
$691$	−824.388	−	598.953i	−1.19304	−	0.866792i	−0.199455	−	0.979907i	$-0.563917\pi$
$691$	−824.388	−	598.953i	−1.19304	−	0.866792i	−0.993582	+	0.113115i	$0.963917\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	339.919	+	1046.16i	0.485598	+	1.49452i
$701$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$	517.000			0.735420
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	755.622	−	548.991i	1.06576	−	0.774318i	0.0906124	−	0.995886i	$-0.471118\pi$
$709$	755.622	−	548.991i	1.06576	−	0.774318i	0.975145	+	0.221568i	$0.0711176\pi$
$710$		0			0
$711$	30.5927	+	94.1546i	0.0430277	+	0.132426i
$712$		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
$721$	1397.17	−	1015.11i	1.93783	−	1.40791i
$722$		0			0
$723$	265.137	−	816.006i	0.366717	−	1.12864i
$724$	3.23607	+	2.35114i	0.00446971	+	0.00324743i
$725$		0			0
$726$		0			0
$727$	482.000			0.662999			0.331499	−	0.943455i	$-0.392446\pi$
$727$	482.000			0.662999			0.331499	+	0.943455i	$0.392446\pi$
$728$		0			0
$729$	225.273	−	693.320i	0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$	−1174.69	+	853.464i	−1.60477	+	1.16593i
$733$	319.524	+	983.392i	0.435912	+	1.34160i	0.892149	+	0.451741i	$0.149197\pi$
$733$	319.524	+	983.392i	0.435912	+	1.34160i	−0.456237	+	0.889858i	$0.650803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1076.80	−	782.342i	−1.45711	−	1.05865i	−0.984105	−	0.177588i	$-0.943171\pi$
$739$	−1076.80	−	782.342i	−1.45711	−	1.05865i	−0.473001	−	0.881062i	$-0.656829\pi$
$740$		0			0
$741$	224.346	+	690.467i	0.302762	+	0.931804i
$742$		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
$751$	−426.752	+	1313.41i	−0.568246	+	1.74888i	0.0898598	+	0.995954i	$0.471358\pi$
$751$	−426.752	+	1313.41i	−0.568246	+	1.74888i	−0.658105	+	0.752926i	$0.728642\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−367.112	+	1129.86i	−0.485598	+	1.49452i
$757$	−1222.42	−	888.144i	−1.61483	−	1.17324i	−0.844398	−	0.535716i	$-0.820042\pi$
$757$	−1222.42	−	888.144i	−1.61483	−	1.17324i	−0.770430	−	0.637525i	$-0.779958\pi$
$758$		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
$763$	486.084	+	1496.01i	0.637069	+	1.96070i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	621.325	+	451.419i	0.809017	+	0.587785i
$769$	671.000			0.872562			0.436281	−	0.899811i	$-0.356295\pi$
$769$	671.000			0.872562			0.436281	+	0.899811i	$0.356295\pi$
$770$		0			0
$771$		0			0
$772$	−462.758	−	336.213i	−0.599427	−	0.435509i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$		0			0
$775$	−1193.30	+	866.983i	−1.53974	+	1.11869i
$776$		0			0
$777$	−479.285	−	1475.09i	−0.616841	−	1.89844i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	355.988	−	1095.62i	0.454066	−	1.39747i
$785$		0			0
$786$		0			0
$787$	−1263.68	+	918.121i	−1.60570	+	1.16661i	−0.730407	+	0.683012i	$0.760669\pi$
$787$	−1263.68	+	918.121i	−1.60570	+	1.16661i	−0.875292	+	0.483596i	$0.839331\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2662.00			3.35687
$794$		0			0
$795$		0			0
$796$	−342.391	−	1053.77i	−0.430139	−	1.32383i
$797$		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
$804$	156.000			0.194030
$805$		0			0
$806$		0			0
$807$		0			0
$808$		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
$811$	−78.1813	+	240.617i	−0.0964011	+	0.296692i	−0.987616	−	0.156889i	$-0.949854\pi$
$811$	−78.1813	+	240.617i	−0.0964011	+	0.296692i	0.891215	+	0.453581i	$0.149854\pi$
$812$		0			0
$813$	−726.000			−0.892989
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−74.7821	+	230.156i	−0.0915326	+	0.281708i
$818$		0			0
$819$	1762.04	−	1280.20i	2.15145	−	1.56312i
$820$		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
$823$	1311.42	+	952.800i	1.59346	+	1.15772i	0.898784	+	0.438391i	$0.144452\pi$
$823$	1311.42	+	952.800i	1.59346	+	1.15772i	0.694675	+	0.719324i	$0.255548\pi$
$824$		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
$829$	−497.208	−	1530.25i	−0.599769	−	1.84590i	−0.529391	−	0.848378i	$-0.677579\pi$
$829$	−497.208	−	1530.25i	−0.599769	−	1.84590i	−0.0703782	−	0.997520i	$-0.522421\pi$
$830$		0			0
$831$	987.810	−	717.686i	1.18870	−	0.863641i
$832$	−435.096	−	1339.09i	−0.522952	−	1.60948i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1593.00			−1.90323
$838$		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
$841$	−680.383	+	494.327i	−0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$	−312.725	+	962.469i	−0.370527	+	1.14037i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1254.79	−	911.655i	−1.47796	−	1.07380i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	952.213	−	691.823i	1.11631	−	0.811047i	0.132665	−	0.991161i	$-0.457647\pi$
$853$	952.213	−	691.823i	1.11631	−	0.811047i	0.983646	+	0.180114i	$0.0576466\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1549.00			−1.80326			−0.901630	−	0.432509i	$-0.857629\pi$
$859$	−1549.00			−1.80326			−0.901630	+	0.432509i	$0.857629\pi$
$860$		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
$867$	701.418	+	509.610i	0.809017	+	0.587785i
$868$	2596.00			2.99078
$869$		0			0
$870$		0			0
$871$	−231.379	−	168.107i	−0.265647	−	0.193004i
$872$		0			0
$873$	−470.015	−	1446.56i	−0.538390	−	1.65700i
$874$		0			0
$875$		0			0
$876$	−530.273	−	1632.01i	−0.605335	−	1.86303i
$877$	533.672	−	1642.47i	0.608520	−	1.87283i	0.138030	−	0.990428i	$-0.455923\pi$
$877$	533.672	−	1642.47i	0.608520	−	1.87283i	0.470491	−	0.882405i	$-0.344077\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	389.052	−	1197.38i	0.440603	−	1.35604i	−0.446632	−	0.894718i	$-0.647377\pi$
$883$	389.052	−	1197.38i	0.440603	−	1.35604i	0.887235	−	0.461318i	$-0.152623\pi$
$884$		0			0
$885$		0			0
$886$		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
$889$	2251.49	+	1635.81i	2.53261	+	1.84005i
$890$		0			0
$891$		0			0
$892$	332.000			0.372197
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−728.115	−	529.007i	−0.809017	−	0.587785i
$901$		0			0
$902$		0			0
$903$	726.000			0.803987
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1175.50	−	854.052i	1.29603	−	0.941623i	0.296124	−	0.955149i	$-0.404306\pi$
$907$	1175.50	−	854.052i	1.29603	−	0.941623i	0.999909	+	0.0135265i	$0.00430576\pi$
$908$		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	$0.200000\pi$
$912$	−528.000			−0.578947
$913$		0			0
$914$		0			0
$915$		0			0
$916$	32.1378	−	98.9099i	0.0350849	−	0.107980i
$917$		0			0
$918$		0			0
$919$	1486.16	−	1079.76i	1.61715	−	1.17493i	0.789190	−	0.614149i	$-0.210501\pi$
$919$	1486.16	−	1079.76i	1.61715	−	1.17493i	0.827963	−	0.560782i	$-0.189499\pi$
$920$		0			0
$921$	234.544	−	721.852i	0.254662	−	0.783770i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1175.00			1.27027
$926$		0			0
$927$	−436.641	+	1343.84i	−0.471026	+	1.44967i
$928$		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
$931$	244.741	+	753.237i	0.262880	+	0.809062i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	525.052	+	381.473i	0.560354	+	0.407121i	0.831589	−	0.555392i	$-0.187432\pi$
$937$	525.052	+	381.473i	0.560354	+	0.407121i	−0.271234	+	0.962513i	$0.587432\pi$
$938$		0			0
$939$	131.641	+	405.150i	0.140193	+	0.431470i
$940$		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
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	106.790	+	77.5877i	0.112648	+	0.0818435i
$949$	−972.167	+	2992.02i	−1.02441	+	3.15282i
$950$		0			0
$951$		0			0
$952$		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
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	778.723	+	2396.66i	0.810326	+	2.49393i
$962$		0			0
$963$		0			0
$964$	−353.515	−	1088.01i	−0.366717	−	1.12864i
$965$		0			0
$966$		0			0
$967$	−253.000			−0.261634			−0.130817	−	0.991407i	$-0.541760\pi$
$967$	−253.000			−0.261634			−0.130817	+	0.991407i	$0.541760\pi$
$968$		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	$0.600000\pi$
$972$	−300.365	−	924.427i	−0.309017	−	0.951057i
$973$	195.782	−	142.244i	0.201215	−	0.146191i
$974$		0			0
$975$	509.878	+	1569.24i	0.522952	+	1.60948i
$976$	−598.257	+	1841.25i	−0.612968	+	1.88652i
$977$		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
$981$	−1041.20	−	756.480i	−1.06137	−	0.771131i
$982$		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
$988$	783.128	+	568.976i	0.792640	+	0.575887i
$989$		0			0
$990$		0			0
$991$	−46.0000			−0.0464178			−0.0232089	−	0.999731i	$-0.507388\pi$
$991$	−46.0000			−0.0464178			−0.0232089	+	0.999731i	$0.507388\pi$
$992$		0			0
$993$	−277.188	+	853.098i	−0.279142	+	0.859111i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	125.770	+	387.080i	0.126148	+	0.388245i	0.994109	−	0.108389i	$-0.0345692\pi$
$997$	125.770	+	387.080i	0.126148	+	0.388245i	−0.867960	+	0.496634i	$0.834569\pi$
$998$		0			0
$999$	1026.64	+	745.899i	1.02767	+	0.746646i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.a.323.1		4
3.2	odd	2	CM	363.3.h.a.323.1		4
11.2	odd	10		363.3.h.b.269.1		4
11.3	even	5	inner	363.3.h.a.245.1		4
11.4	even	5	inner	363.3.h.a.251.1		4
11.5	even	5		363.3.b.b.122.1	yes	1
11.6	odd	10		363.3.b.a.122.1	✓	1
11.7	odd	10		363.3.h.b.251.1		4
11.8	odd	10		363.3.h.b.245.1		4
11.9	even	5	inner	363.3.h.a.269.1		4
11.10	odd	2		363.3.h.b.323.1		4
33.2	even	10		363.3.h.b.269.1		4
33.5	odd	10		363.3.b.b.122.1	yes	1
33.8	even	10		363.3.h.b.245.1		4
33.14	odd	10	inner	363.3.h.a.245.1		4
33.17	even	10		363.3.b.a.122.1	✓	1
33.20	odd	10	inner	363.3.h.a.269.1		4
33.26	odd	10	inner	363.3.h.a.251.1		4
33.29	even	10		363.3.h.b.251.1		4
33.32	even	2		363.3.h.b.323.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.b.a.122.1	✓	1	11.6	odd	10
363.3.b.a.122.1	✓	1	33.17	even	10
363.3.b.b.122.1	yes	1	11.5	even	5
363.3.b.b.122.1	yes	1	33.5	odd	10
363.3.h.a.245.1		4	11.3	even	5	inner
363.3.h.a.245.1		4	33.14	odd	10	inner
363.3.h.a.251.1		4	11.4	even	5	inner
363.3.h.a.251.1		4	33.26	odd	10	inner
363.3.h.a.269.1		4	11.9	even	5	inner
363.3.h.a.269.1		4	33.20	odd	10	inner
363.3.h.a.323.1		4	1.1	even	1	trivial
363.3.h.a.323.1		4	3.2	odd	2	CM
363.3.h.b.245.1		4	11.8	odd	10
363.3.h.b.245.1		4	33.8	even	10
363.3.h.b.251.1		4	11.7	odd	10
363.3.h.b.251.1		4	33.29	even	10
363.3.h.b.269.1		4	11.2	odd	10
363.3.h.b.269.1		4	33.2	even	10
363.3.h.b.323.1		4	11.10	odd	2
363.3.h.b.323.1		4	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.927051 + 2.85317i) q^{3} +(1.23607 + 3.80423i) q^{4} +(3.39919 + 10.4616i) q^{7} +(-7.28115 - 5.29007i) q^{9} +O(q^{10})\)	\(q+(-0.927051 + 2.85317i) q^{3} +(1.23607 + 3.80423i) q^{4} +(3.39919 + 10.4616i) q^{7} +(-7.28115 - 5.29007i) q^{9} -12.0000 q^{12} +(17.7984 + 12.9313i) q^{13} +(-12.9443 + 9.40456i) q^{16} +(3.39919 - 10.4616i) q^{19} -33.0000 q^{21} +(7.72542 - 23.7764i) q^{25} +(21.8435 - 15.8702i) q^{27} +(-35.5967 + 25.8626i) q^{28} +(-47.7320 - 34.6793i) q^{31} +(11.1246 - 34.2380i) q^{36} +(14.5238 + 44.6997i) q^{37} +(-53.3951 + 38.7938i) q^{39} -22.0000 q^{43} +(-14.8328 - 45.6507i) q^{48} +(-58.2492 + 42.3205i) q^{49} +(-27.1935 + 83.6930i) q^{52} +(26.6976 + 19.3969i) q^{57} +(97.8911 - 71.1220i) q^{61} +(30.5927 - 94.1546i) q^{63} +(-51.7771 - 37.6183i) q^{64} -13.0000 q^{67} +(44.1894 + 136.001i) q^{73} +(60.6763 + 44.0839i) q^{75} +44.0000 q^{76} +(-8.89919 - 6.46564i) q^{79} +(25.0304 + 77.0356i) q^{81} +(-40.7902 - 125.539i) q^{84} +(-74.7821 + 230.156i) q^{91} +(143.196 - 104.038i) q^{93} +(136.724 + 99.3357i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9	\( 4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9} - 48 q^{12} + 22 q^{13} - 16 q^{16} - 11 q^{19} - 132 q^{21} - 25 q^{25} + 27 q^{27} - 44 q^{28} - 59 q^{31} - 36 q^{36} - 47 q^{37} - 66 q^{39} - 88 q^{43} + 48 q^{48} - 72 q^{49} + 88 q^{52} + 33 q^{57} + 121 q^{61} - 99 q^{63} - 64 q^{64} - 52 q^{67} - 143 q^{73} + 75 q^{75} + 176 q^{76} - 11 q^{79} - 81 q^{81} + 132 q^{84} + 242 q^{91} + 177 q^{93} + 169 q^{97}+O(q^{100}) \) 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9 - 48 * q^12 + 22 * q^13 - 16 * q^16 - 11 * q^19 - 132 * q^21 - 25 * q^25 + 27 * q^27 - 44 * q^28 - 59 * q^31 - 36 * q^36 - 47 * q^37 - 66 * q^39 - 88 * q^43 + 48 * q^48 - 72 * q^49 + 88 * q^52 + 33 * q^57 + 121 * q^61 - 99 * q^63 - 64 * q^64 - 52 * q^67 - 143 * q^73 + 75 * q^75 + 176 * q^76 - 11 * q^79 - 81 * q^81 + 132 * q^84 + 242 * q^91 + 177 * q^93 + 169 * q^97

Introduction

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