LMFDB - Embedded newform 368.3.k.a.229.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,3,Mod(45,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("368.45");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	368.k (of order $4$, degree $2$, 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:	$10.0272737285$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.322241908269256704.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7x^{6} + 64 $ x^12 - 7*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			229.2
Root			$-0.467979 + 1.33454i$ of defining polynomial
Character	$\chi$	$=$	368.229
Dual form			368.3.k.a.45.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.24907 - 1.56199i) q^{2} +(-3.67762 + 3.67762i) q^{3} +(-0.879635 + 3.90208i) q^{4} +(10.3380 + 1.15080i) q^{6} +(7.19375 - 3.50000i) q^{8} -18.0498i q^{9} +(-11.1154 - 17.5854i) q^{12} +(14.0322 - 14.0322i) q^{13} +(-14.4525 - 6.86482i) q^{16} +(-28.1937 + 22.5456i) q^{18} +23.0000i q^{23} +(-13.5842 + 39.3276i) q^{24} +25.0000i q^{25} +(-39.4455 - 4.39095i) q^{26} +(33.2819 + 33.2819i) q^{27} +(34.7897 - 34.7897i) q^{29} -3.44765 q^{31} +(7.32941 + 31.1493i) q^{32} +(70.4320 + 15.8773i) q^{36} +103.211i q^{39} +66.2117i q^{41} +(35.9258 - 28.7287i) q^{46} +50.9864 q^{47} +(78.3970 - 27.9046i) q^{48} -49.0000 q^{49} +(39.0498 - 31.2268i) q^{50} +(42.4117 + 67.0982i) q^{52} +(10.4146 - 93.5577i) q^{54} +(-97.7960 - 10.8864i) q^{58} +(44.5500 + 44.5500i) q^{59} +(4.30637 + 5.38520i) q^{62} +(39.5000 - 50.3562i) q^{64} +(-84.5854 - 84.5854i) q^{69} +130.792i q^{71} +(-63.1745 - 129.846i) q^{72} -88.0471i q^{73} +(-91.9406 - 91.9406i) q^{75} +(161.214 - 128.917i) q^{78} -82.3484 q^{81} +(103.422 - 82.7033i) q^{82} +255.887i q^{87} +(-89.7479 - 20.2316i) q^{92} +(12.6792 - 12.6792i) q^{93} +(-63.6857 - 79.6403i) q^{94} +(-141.510 - 87.6007i) q^{96} +(61.2046 + 76.5376i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 66 q^{6} - 6 q^{12} - 228 q^{27} + 294 q^{36} - 588 q^{49} - 546 q^{58} - 156 q^{59} + 606 q^{62} + 474 q^{64} - 378 q^{72} + 798 q^{78} - 972 q^{81} + 258 q^{82} + 1092 q^{93}+O(q^{100}) $ 12 * q + 66 * q^6 - 6 * q^12 - 228 * q^27 + 294 * q^36 - 588 * q^49 - 546 * q^58 - 156 * q^59 + 606 * q^62 + 474 * q^64 - 378 * q^72 + 798 * q^78 - 972 * q^81 + 258 * q^82 + 1092 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/368\mathbb{Z}\right)^\times$.

$n$	$47$	$97$	$277$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.24907	−	1.56199i	−0.624536	−	0.780996i
$2$	−1.24907	−	1.56199i	−0.624536	−	0.780996i
$3$	−3.67762	+	3.67762i	−1.22587	+	1.22587i	−0.260365	+	0.965510i	$0.583843\pi$
$3$	−3.67762	+	3.67762i	−1.22587	+	1.22587i	−0.965510	+	0.260365i	$0.916157\pi$
$4$	−0.879635	+	3.90208i	−0.219909	+	0.975520i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$	10.3380	+	1.15080i	1.72301	+	0.191800i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	7.19375	−	3.50000i	0.899218	−	0.437500i
$9$		−	18.0498i		−	2.00554i
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$	−11.1154	−	17.5854i	−0.926285	−	1.46545i
$13$	14.0322	−	14.0322i	1.07940	−	1.07940i	0.0828396	−	0.996563i	$-0.473601\pi$
$13$	14.0322	−	14.0322i	1.07940	−	1.07940i	0.996563	−	0.0828396i	$-0.0263989\pi$
$14$		0			0
$15$		0			0
$16$	−14.4525	−	6.86482i	−0.903280	−	0.429051i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−28.1937	+	22.5456i	−1.56632	+	1.25253i
$19$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			23.0000i			1.00000i
$23$			23.0000i			1.00000i
$24$	−13.5842	+	39.3276i	−0.566009	+	1.63865i
$25$			25.0000i			1.00000i
$26$	−39.4455	−	4.39095i	−1.51713	−	0.168883i
$27$	33.2819	+	33.2819i	1.23266	+	1.23266i
$28$		0			0
$29$	34.7897	−	34.7897i	1.19964	−	1.19964i	0.225371	−	0.974273i	$-0.427641\pi$
$29$	34.7897	−	34.7897i	1.19964	−	1.19964i	0.974273	−	0.225371i	$-0.0723595\pi$
$30$		0			0
$31$	−3.44765			−0.111215			−0.0556073	−	0.998453i	$-0.517709\pi$
$31$	−3.44765			−0.111215			−0.0556073	+	0.998453i	$0.517709\pi$
$32$	7.32941	+	31.1493i	0.229044	+	0.973416i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	70.4320	+	15.8773i	1.95644	+	0.441036i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$			103.211i			2.64642i
$40$		0			0
$41$			66.2117i			1.61492i	0.589922	+	0.807460i	$0.299158\pi$
$41$			66.2117i			1.61492i	−0.589922	+	0.807460i	$0.700842\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	35.9258	−	28.7287i	0.780996	−	0.624536i
$47$	50.9864			1.08482			0.542408	−	0.840115i	$-0.317513\pi$
$47$	50.9864			1.08482			0.542408	+	0.840115i	$0.317513\pi$
$48$	78.3970	−	27.9046i	1.63327	−	0.581345i
$49$	−49.0000			−1.00000
$50$	39.0498	−	31.2268i	0.780996	−	0.624536i
$51$		0			0
$52$	42.4117	+	67.0982i	0.815609	+	1.29035i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$	10.4146	−	93.5577i	0.192862	−	1.73255i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−97.7960	−	10.8864i	−1.68614	−	0.187696i
$59$	44.5500	+	44.5500i	0.755084	+	0.755084i	0.975423	−	0.220339i	$-0.0707164\pi$
$59$	44.5500	+	44.5500i	0.755084	+	0.755084i	−0.220339	+	0.975423i	$0.570716\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$	4.30637	+	5.38520i	0.0694575	+	0.0868581i
$63$		0			0
$64$	39.5000	−	50.3562i	0.617188	−	0.786816i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$	−84.5854	−	84.5854i	−1.22587	−	1.22587i
$70$		0			0
$71$			130.792i			1.84214i	0.389395	+	0.921071i	$0.372684\pi$
$71$			130.792i			1.84214i	−0.389395	+	0.921071i	$0.627316\pi$
$72$	−63.1745	−	129.846i	−0.877423	−	1.80342i
$73$		−	88.0471i		−	1.20612i	−0.797694	−	0.603062i	$-0.793947\pi$
$73$		−	88.0471i		−	1.20612i	0.797694	−	0.603062i	$-0.206053\pi$
$74$		0			0
$75$	−91.9406	−	91.9406i	−1.22587	−	1.22587i
$76$		0			0
$77$		0			0
$78$	161.214	−	128.917i	2.06685	−	1.65279i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−82.3484			−1.01665
$82$	103.422	−	82.7033i	1.26125	−	1.00858i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			255.887i			2.94123i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−89.7479	−	20.2316i	−0.975520	−	0.219909i
$93$	12.6792	−	12.6792i	0.136335	−	0.136335i
$94$	−63.6857	−	79.6403i	−0.677507	−	0.847237i
$95$		0			0
$96$	−141.510	−	87.6007i	−1.47407	−	0.912507i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	61.2046	+	76.5376i	0.624536	+	0.780996i
$99$		0			0
$100$	−97.5520	−	21.9909i	−0.975520	−	0.219909i
$101$	140.550	+	140.550i	1.39158	+	1.39158i	0.821782	+	0.569802i	$0.192980\pi$
$101$	140.550	+	140.550i	1.39158	+	1.39158i	0.569802	+	0.821782i	$0.307020\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	51.8315	−	150.057i	0.498380	−	1.44286i
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$	−159.145	+	100.593i	−1.47356	+	0.931416i
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	105.150	+	166.354i	0.906465	+	1.43409i
$117$	−253.280	−	253.280i	−2.16478	−	2.16478i
$118$	13.9405	−	125.233i	0.118140	−	1.06130i
$119$		0			0
$120$		0			0
$121$			121.000i			1.00000i
$122$		0			0
$123$	−243.502	−	243.502i	−1.97969	−	1.97969i
$124$	3.03268	−	13.4530i	0.0244571	−	0.108492i
$125$		0			0
$126$		0			0
$127$	183.410			1.44417			0.722086	−	0.691803i	$-0.243184\pi$
$127$	183.410			1.44417			0.722086	+	0.691803i	$0.243184\pi$
$128$	−127.994	+	1.19992i	−0.999956	+	0.00937437i
$129$		0			0
$130$		0			0
$131$	−159.010	+	159.010i	−1.21381	+	1.21381i	−0.244051	+	0.969762i	$0.578476\pi$
$131$	−159.010	+	159.010i	−1.21381	+	1.21381i	−0.969762	+	0.244051i	$0.921524\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−26.4684	+	237.775i	−0.191800	+	1.72301i
$139$	34.7625	+	34.7625i	0.250090	+	0.250090i	0.821007	−	0.570917i	$-0.193412\pi$
$139$	34.7625	+	34.7625i	0.250090	+	0.250090i	−0.570917	+	0.821007i	$0.693412\pi$
$140$		0			0
$141$	−187.509	+	187.509i	−1.32985	+	1.32985i
$142$	204.296	−	163.369i	1.43871	−	1.15048i
$143$		0			0
$144$	−123.909	+	260.865i	−0.860479	+	1.81156i
$145$		0			0
$146$	−137.529	+	109.977i	−0.941979	+	0.753269i
$147$	180.204	−	180.204i	1.22587	−	1.22587i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	−28.7700	+	258.451i	−0.191800	+	1.72301i
$151$			109.883i			0.727699i	0.931458	+	0.363849i	$0.118538\pi$
$151$			109.883i			0.727699i	−0.931458	+	0.363849i	$0.881462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−402.736	−	90.7877i	−2.58164	−	0.581972i
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	102.859	+	128.628i	0.634933	+	0.793997i
$163$	230.510	−	230.510i	1.41417	−	1.41417i	0.701774	−	0.712400i	$-0.252392\pi$
$163$	230.510	−	230.510i	1.41417	−	1.41417i	0.712400	−	0.701774i	$-0.247608\pi$
$164$	−258.364	−	58.2422i	−1.57539	−	0.355135i
$165$		0			0
$166$		0			0
$167$		−	230.200i		−	1.37844i	−0.724551	−	0.689221i	$-0.757953\pi$
$167$		−	230.200i		−	1.37844i	0.724551	−	0.689221i	$-0.242047\pi$
$168$		0			0
$169$		−	224.807i		−	1.33022i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	161.650	−	161.650i	0.934393	−	0.934393i	−0.0635838	−	0.997977i	$-0.520253\pi$
$173$	161.650	−	161.650i	0.934393	−	0.934393i	0.997977	+	0.0635838i	$0.0202530\pi$
$174$	399.693	−	319.621i	2.29709	−	1.83690i
$175$		0			0
$176$		0			0
$177$	−327.676			−1.85128
$178$		0			0
$179$	157.034	−	157.034i	0.877284	−	0.877284i	−0.115969	−	0.993253i	$-0.536997\pi$
$179$	157.034	−	157.034i	0.877284	−	0.877284i	0.993253	+	0.115969i	$0.0369974\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	80.5000	+	165.456i	0.437500	+	0.899218i
$185$		0			0
$186$	−35.6419	−	3.96755i	−0.191623	−	0.0213309i
$187$		0			0
$188$	−44.8494	+	198.953i	−0.238561	+	1.05826i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	39.9251	+	330.458i	0.207943	+	1.72113i
$193$	350.889			1.81808			0.909038	−	0.416713i	$-0.136818\pi$
$193$	350.889			1.81808			0.909038	+	0.416713i	$0.136818\pi$
$194$		0			0
$195$		0			0
$196$	43.1021	−	191.202i	0.219909	−	0.975520i
$197$	209.780	+	209.780i	1.06487	+	1.06487i	0.997744	+	0.0671289i	$0.0213839\pi$
$197$	209.780	+	209.780i	1.06487	+	1.06487i	0.0671289	+	0.997744i	$0.478616\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	87.5000	+	179.844i	0.437500	+	0.899218i
$201$		0			0
$202$	43.9808	−	395.095i	0.217727	−	1.95592i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	415.147			2.00554
$208$	−299.129	+	106.472i	−1.43812	+	0.511884i
$209$		0			0
$210$		0			0
$211$	260.550	−	260.550i	1.23483	−	1.23483i	0.272749	−	0.962085i	$-0.412067\pi$
$211$	260.550	−	260.550i	1.23483	−	1.23483i	0.962085	−	0.272749i	$-0.0879328\pi$
$212$		0			0
$213$	−481.004	−	481.004i	−2.25824	−	2.25824i
$214$		0			0
$215$		0			0
$216$	355.909	+	122.935i	1.64773	+	0.569144i
$217$		0			0
$218$		0			0
$219$	323.804	+	323.804i	1.47856	+	1.47856i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−230.200			−1.03229			−0.516143	−	0.856502i	$-0.672633\pi$
$223$	−230.200			−1.03229			−0.516143	+	0.856502i	$0.672633\pi$
$224$		0			0
$225$	451.246			2.00554
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$229$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$230$		0			0
$231$		0			0
$232$	128.504	−	372.032i	0.553898	−	1.60359i
$233$			273.867i			1.17539i	0.809081	+	0.587697i	$0.199965\pi$
$233$			273.867i			1.17539i	−0.809081	+	0.587697i	$0.800035\pi$
$234$	−79.2560	+	711.985i	−0.338701	+	3.04267i
$235$		0			0
$236$	−213.025	+	134.650i	−0.902650	+	0.570550i
$237$		0			0
$238$		0			0
$239$	−401.211			−1.67871			−0.839354	−	0.543585i	$-0.817066\pi$
$239$	−401.211			−1.67871			−0.839354	+	0.543585i	$0.817066\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	189.001	−	151.138i	0.780996	−	0.624536i
$243$	3.30901	−	3.30901i	0.0136173	−	0.0136173i
$244$		0			0
$245$		0			0
$246$	−76.1964	+	684.499i	−0.309741	+	2.78252i
$247$		0			0
$248$	−24.8015	+	12.0668i	−0.100006	+	0.0486564i
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$	−229.092	−	286.485i	−0.901938	−	1.12789i
$255$		0			0
$256$	161.749	+	198.427i	0.631830	+	0.775107i
$257$	497.958			1.93758			0.968791	−	0.247880i	$-0.0797338\pi$
$257$	497.958			1.93758			0.968791	+	0.247880i	$0.0797338\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−627.948	−	627.948i	−2.40593	−	2.40593i
$262$	446.986	+	49.7571i	1.70605	+	0.189913i
$263$		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
$268$		0			0
$269$	−288.615	+	288.615i	−1.07292	+	1.07292i	−0.0757948	+	0.997123i	$0.524149\pi$
$269$	−288.615	+	288.615i	−1.07292	+	1.07292i	−0.997123	+	0.0757948i	$0.975851\pi$
$270$		0			0
$271$	−286.000			−1.05535			−0.527675	−	0.849446i	$-0.676936\pi$
$271$	−286.000			−1.05535			−0.527675	+	0.849446i	$0.676936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	404.463	−	255.655i	1.46545	−	0.926285i
$277$	−50.0092	−	50.0092i	−0.180539	−	0.180539i	0.611052	−	0.791591i	$-0.290747\pi$
$277$	−50.0092	−	50.0092i	−0.180539	−	0.180539i	−0.791591	+	0.611052i	$0.790747\pi$
$278$	10.8778	−	97.7196i	0.0391290	−	0.351509i
$279$			62.2296i			0.223045i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	527.099	+	58.6751i	1.86915	+	0.208068i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$	−510.361	−	115.049i	−1.79705	−	0.405103i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	562.240	−	132.295i	1.95222	−	0.459357i
$289$	289.000			1.00000
$290$		0			0
$291$		0			0
$292$	343.567	+	77.4494i	1.17660	+	0.265238i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$	−506.564	−	56.3891i	−1.72301	−	0.191800i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	322.741	+	322.741i	1.07940	+	1.07940i
$300$	439.634	−	277.886i	1.46545	−	0.926285i
$301$		0			0
$302$	171.636	−	137.251i	0.568330	−	0.454474i
$303$	−1033.78			−3.41182
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−394.750	+	394.750i	−1.28583	+	1.28583i	−0.348534	+	0.937296i	$0.613320\pi$
$307$	−394.750	+	394.750i	−1.28583	+	1.28583i	−0.937296	+	0.348534i	$0.886680\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			578.675i			1.86069i	0.366684	+	0.930346i	$0.380493\pi$
$311$			578.675i			1.86069i	−0.366684	+	0.930346i	$0.619507\pi$
$312$	361.237	+	742.471i	1.15781	+	2.37971i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	154.750	−	154.750i	0.488170	−	0.488170i	−0.419558	−	0.907728i	$-0.637815\pi$
$317$	154.750	−	154.750i	0.488170	−	0.488170i	0.907728	+	0.419558i	$0.137815\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	72.4366	−	321.330i	0.223570	−	0.991760i
$325$	350.806	+	350.806i	1.07940	+	1.07940i
$326$	−647.979	−	72.1311i	−1.98767	−	0.221261i
$327$		0			0
$328$	231.741	+	476.310i	0.706528	+	1.45217i
$329$		0			0
$330$		0			0
$331$	−452.347	−	452.347i	−1.36661	−	1.36661i	−0.865228	−	0.501379i	$-0.832826\pi$
$331$	−452.347	−	452.347i	−1.36661	−	1.36661i	−0.501379	−	0.865228i	$-0.667174\pi$
$332$		0			0
$333$		0			0
$334$	−359.570	+	287.536i	−1.07656	+	0.860887i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−351.147	+	280.800i	−1.03890	+	0.830770i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−454.408	−	50.5833i	−1.31332	−	0.146195i
$347$	473.650	+	473.650i	1.36499	+	1.36499i	0.867435	+	0.497550i	$0.165767\pi$
$347$	473.650	+	473.650i	1.36499	+	1.36499i	0.497550	+	0.867435i	$0.334233\pi$
$348$	−998.491	−	225.087i	−2.86923	−	0.646802i
$349$	−481.178	+	481.178i	−1.37873	+	1.37873i	−0.531970	+	0.846763i	$0.678548\pi$
$349$	−481.178	+	481.178i	−1.37873	+	1.37873i	−0.846763	+	0.531970i	$0.821452\pi$
$350$		0			0
$351$	934.040			2.66108
$352$		0			0
$353$	−241.712			−0.684736			−0.342368	−	0.939566i	$-0.611229\pi$
$353$	−241.712			−0.684736			−0.342368	+	0.939566i	$0.611229\pi$
$354$	409.291	+	511.827i	1.15619	+	1.44584i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−441.432	−	49.1389i	−1.23305	−	0.137259i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$		−	361.000i		−	1.00000i
$362$		0			0
$363$	−444.993	−	444.993i	−1.22587	−	1.22587i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	157.891	−	332.407i	0.429051	−	0.903280i
$369$	1195.11			3.23879
$370$		0			0
$371$		0			0
$372$	38.3221	+	60.6282i	0.103016	+	0.162979i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	366.783	−	178.452i	0.975487	−	0.474607i
$377$		−	976.354i		−	2.58980i
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$	−674.513	+	674.513i	−1.77037	+	1.77037i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	466.302	−	475.128i	1.21433	−	1.23731i
$385$		0			0
$386$	−438.286	−	548.085i	−1.13545	−	1.41991i
$387$		0			0
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	−352.494	+	171.500i	−0.899218	+	0.437500i
$393$		−	1169.55i		−	2.97597i
$394$	65.6442	−	589.705i	0.166610	−	1.49671i
$395$		0			0
$396$		0			0
$397$	308.116	−	308.116i	0.776111	−	0.776111i	−0.203056	−	0.979167i	$-0.565087\pi$
$397$	308.116	−	308.116i	0.776111	−	0.776111i	0.979167	+	0.203056i	$0.0650875\pi$
$398$		0			0
$399$		0			0
$400$	171.620	−	361.312i	0.429051	−	0.903280i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−48.3782	+	48.3782i	−0.120045	+	0.120045i
$404$	−672.070	+	424.805i	−1.66354	+	1.05150i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			610.436i			1.49251i	0.665661	+	0.746254i	$0.268150\pi$
$409$			610.436i			1.49251i	−0.665661	+	0.746254i	$0.731850\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−518.548	−	648.455i	−1.25253	−	1.56632i
$415$		0			0
$416$	539.942	+	334.246i	1.29794	+	0.803477i
$417$	−255.687			−0.613158
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	−732.423	−	81.5310i	−1.73560	−	0.193202i
$423$		−	920.296i		−	2.17564i
$424$		0			0
$425$		0			0
$426$	−150.515	+	1352.13i	−0.353322	+	3.17402i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−252.532	−	709.481i	−0.584565	−	1.64232i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	101.325	−	910.235i	0.231335	−	2.07816i
$439$		−	235.511i		−	0.536471i	−0.963353	−	0.268236i	$-0.913559\pi$
$439$		−	235.511i		−	0.536471i	0.963353	−	0.268236i	$-0.0864406\pi$
$440$		0			0
$441$			884.443i			2.00554i
$442$		0			0
$443$	−625.389	−	625.389i	−1.41171	−	1.41171i	−0.747893	−	0.663820i	$-0.768934\pi$
$443$	−625.389	−	625.389i	−1.41171	−	1.41171i	−0.663820	−	0.747893i	$-0.731066\pi$
$444$		0			0
$445$		0			0
$446$	287.536	+	359.570i	0.644700	+	0.806211i
$447$		0			0
$448$		0			0
$449$	690.600			1.53808			0.769042	−	0.639198i	$-0.220734\pi$
$449$	690.600			1.53808			0.769042	+	0.639198i	$0.220734\pi$
$450$	−563.639	−	704.843i	−1.25253	−	1.56632i
$451$		0			0
$452$		0			0
$453$	−404.107	−	404.107i	−0.892068	−	0.892068i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−608.417	+	608.417i	−1.31978	+	1.31978i	−0.405827	+	0.913950i	$0.633016\pi$
$461$	−608.417	+	608.417i	−1.31978	+	1.31978i	−0.913950	+	0.405827i	$0.866984\pi$
$462$		0			0
$463$	−98.0000			−0.211663			−0.105832	−	0.994384i	$-0.533750\pi$
$463$	−98.0000			−0.211663			−0.105832	+	0.994384i	$0.533750\pi$
$464$	−741.622	+	263.972i	−1.59832	+	0.568906i
$465$		0			0
$466$	427.778	−	342.079i	0.917978	−	0.734076i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	1211.11	−	765.524i	2.58785	−	1.63574i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	476.406	+	164.556i	1.00934	+	0.348636i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	501.142	+	626.689i	1.04841	+	1.31106i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−472.152	−	106.436i	−0.975520	−	0.219909i
$485$		0			0
$486$	−9.30183	−	1.03545i	−0.0191396	−	0.00213056i
$487$		−	678.526i		−	1.39328i	−0.717423	−	0.696638i	$-0.754678\pi$
$487$		−	678.526i		−	1.39328i	0.717423	−	0.696638i	$-0.245322\pi$
$488$		0			0
$489$			1695.46i			3.46720i
$490$		0			0
$491$	−583.051	−	583.051i	−1.18748	−	1.18748i	−0.977763	−	0.209715i	$-0.932747\pi$
$491$	−583.051	−	583.051i	−1.18748	−	1.18748i	−0.209715	−	0.977763i	$-0.567253\pi$
$492$	1164.36	−	735.971i	2.36658	−	1.49588i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	49.8271	+	23.6675i	0.100458	+	0.0477167i
$497$		0			0
$498$		0			0
$499$	681.461	−	681.461i	1.36565	−	1.36565i	0.499119	−	0.866533i	$-0.333657\pi$
$499$	681.461	−	681.461i	1.36565	−	1.36565i	0.866533	−	0.499119i	$-0.166343\pi$
$500$		0			0
$501$	846.589	+	846.589i	1.68980	+	1.68980i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	826.756	+	826.756i	1.63068	+	1.63068i
$508$	−161.334	+	715.680i	−0.317586	+	1.40882i
$509$	719.595	−	719.595i	1.41374	−	1.41374i	0.688639	−	0.725104i	$-0.258208\pi$
$509$	719.595	−	719.595i	1.41374	−	1.41374i	0.725104	−	0.688639i	$-0.241792\pi$
$510$		0			0
$511$		0			0
$512$	107.906	−	500.500i	0.210754	−	0.977539i
$513$		0			0
$514$	−621.986	−	777.807i	−1.21009	−	1.51324i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			1188.98i			2.29090i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−196.497	+	1765.20i	−0.376431	+	3.38162i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$	−480.598	−	760.339i	−0.917171	−	1.45103i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−529.000			−1.00000
$530$		0			0
$531$	804.120	−	804.120i	1.51435	−	1.51435i
$532$		0			0
$533$	929.098	+	929.098i	1.74315	+	1.74315i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			1155.02i			2.15088i
$538$	811.315	+	90.3131i	1.50802	+	0.167868i
$539$		0			0
$540$		0			0
$541$	554.221	−	554.221i	1.02444	−	1.02444i	0.0247449	−	0.999694i	$-0.492123\pi$
$541$	554.221	−	554.221i	1.02444	−	1.02444i	0.999694	−	0.0247449i	$-0.00787734\pi$
$542$	357.235	+	446.730i	0.659105	+	0.824224i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−297.970	+	297.970i	−0.544734	+	0.544734i	−0.924913	−	0.380179i	$-0.875862\pi$
$547$	−297.970	+	297.970i	−0.544734	+	0.544734i	0.380179	+	0.924913i	$0.375862\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−904.535	−	312.437i	−1.63865	−	0.566009i
$553$		0			0
$554$	−15.6488	+	140.579i	−0.0282470	+	0.253753i
$555$		0			0
$556$	−166.224	+	105.068i	−0.298965	+	0.188971i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	97.2021	−	77.7292i	0.174197	−	0.139300i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$	−566.735	−	896.614i	−1.00485	−	1.58974i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	457.772	+	940.885i	0.805937	+	1.65649i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−575.000			−1.00000
$576$	−908.922	−	712.969i	−1.57799	−	1.23779i
$577$	−1150.41			−1.99378			−0.996890	−	0.0788074i	$-0.974889\pi$
$577$	−1150.41			−1.99378			−0.996890	+	0.0788074i	$0.974889\pi$
$578$	−360.982	−	451.416i	−0.624536	−	0.780996i
$579$	−1290.44	+	1290.44i	−2.22873	+	2.22873i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−308.165	−	633.389i	−0.527680	−	1.08457i
$585$		0			0
$586$		0			0
$587$	−790.016	−	790.016i	−1.34585	−	1.34585i	−0.890112	−	0.455742i	$-0.849374\pi$
$587$	−790.016	−	790.016i	−1.34585	−	1.34585i	−0.455742	−	0.890112i	$-0.650626\pi$
$588$	544.656	+	861.683i	0.926285	+	1.46545i
$589$		0			0
$590$		0			0
$591$	−1542.98			−2.61080
$592$		0			0
$593$	1151.00			1.94098			0.970489	−	0.241147i	$-0.0775235\pi$
$593$	1151.00			1.94098			0.970489	+	0.241147i	$0.0775235\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	100.992	−	907.247i	0.168883	−	1.51713i
$599$			1106.00i			1.84641i	0.384307	+	0.923205i	$0.374440\pi$
$599$			1106.00i			1.84641i	−0.384307	+	0.923205i	$0.625560\pi$
$600$	−983.190	−	339.605i	−1.63865	−	0.566009i
$601$			593.306i			0.987198i	0.869690	+	0.493599i	$0.164319\pi$
$601$			593.306i			0.987198i	−0.869690	+	0.493599i	$0.835681\pi$
$602$		0			0
$603$		0			0
$604$	−428.771	−	96.6566i	−0.709885	−	0.160027i
$605$		0			0
$606$	1291.27	+	1614.76i	2.13080	+	2.66461i
$607$	−1151.00			−1.89621			−0.948105	−	0.317957i	$-0.897003\pi$
$607$	−1151.00			−1.89621			−0.948105	+	0.317957i	$0.897003\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	715.453	−	715.453i	1.17095	−	1.17095i
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$	1109.67	+	123.525i	1.80728	+	0.201180i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$	−765.485	+	765.485i	−1.23266	+	1.23266i
$622$	903.885	−	722.807i	1.45319	−	1.16207i
$623$		0			0
$624$	708.522	−	1491.65i	1.13545	−	2.39046i
$625$	−625.000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$			1916.41i			3.02750i
$634$	−435.012	−	48.4242i	−0.686139	−	0.0763788i
$635$		0			0
$636$		0			0
$637$	−687.579	+	687.579i	−1.07940	+	1.07940i
$638$		0			0
$639$	2360.78			3.69449
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	987.754i		−	1.52667i	−0.646004	−	0.763334i	$-0.723561\pi$
$647$		−	987.754i		−	1.52667i	0.646004	−	0.763334i	$-0.276439\pi$
$648$	−592.394	+	288.219i	−0.914188	+	0.444783i
$649$		0			0
$650$	109.774	−	986.138i	0.168883	−	1.51713i
$651$		0			0
$652$	696.705	+	1102.24i	1.06857	+	1.69054i
$653$	277.195	−	277.195i	0.424494	−	0.424494i	−0.462254	−	0.886748i	$-0.652959\pi$
$653$	277.195	−	277.195i	0.424494	−	0.424494i	0.886748	+	0.462254i	$0.152959\pi$
$654$		0			0
$655$		0			0
$656$	454.532	−	956.924i	0.692883	−	1.45873i
$657$	−1589.24			−2.41893
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$661$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$662$	−141.548	+	1271.58i	−0.213819	+	1.92081i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	800.163	+	800.163i	1.19964	+	1.19964i
$668$	898.259	+	202.492i	1.34470	+	0.303132i
$669$	846.589	−	846.589i	1.26545	−	1.26545i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1022.20			−1.51888			−0.759438	−	0.650580i	$-0.774526\pi$
$673$	−1022.20			−1.51888			−0.759438	+	0.650580i	$0.774526\pi$
$674$		0			0
$675$	−832.049	+	832.049i	−1.23266	+	1.23266i
$676$	877.216	+	197.748i	1.29766	+	0.292527i
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−334.162	−	334.162i	−0.489256	−	0.489256i	0.418816	−	0.908071i	$-0.362445\pi$
$683$	−334.162	−	334.162i	−0.489256	−	0.489256i	−0.908071	+	0.418816i	$0.862445\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	910.050	−	910.050i	1.31700	−	1.31700i	0.400868	−	0.916136i	$-0.368708\pi$
$691$	910.050	−	910.050i	1.31700	−	1.31700i	0.916136	−	0.400868i	$-0.131292\pi$
$692$	488.578	+	772.964i	0.706038	+	1.11700i
$693$		0			0
$694$	148.214	−	1331.46i	0.213565	−	1.91853i
$695$		0			0
$696$	895.604	+	1840.78i	1.28679	+	2.64481i
$697$		0			0
$698$	1352.62	+	150.570i	1.93785	+	0.215716i
$699$	−1007.18	−	1007.18i	−1.44089	−	1.44089i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$	−1166.68	−	1458.96i	−1.66194	−	2.07829i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	301.916	+	377.552i	0.427642	+	0.534776i
$707$		0			0
$708$	288.236	−	1278.62i	0.407112	−	1.80596i
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	79.2960i		−	0.111215i
$714$		0			0
$715$		0			0
$716$	474.626	+	750.891i	0.662886	+	1.04873i
$717$	1475.50	−	1475.50i	2.05789	−	2.05789i
$718$		0			0
$719$	1151.00			1.60083			0.800417	−	0.599444i	$-0.204611\pi$
$719$	1151.00			1.60083			0.800417	+	0.599444i	$0.204611\pi$
$720$		0			0
$721$		0			0
$722$	−563.879	+	450.915i	−0.780996	+	0.624536i
$723$		0			0
$724$		0			0
$725$	869.742	+	869.742i	1.19964	+	1.19964i
$726$	−139.247	+	1250.90i	−0.191800	+	1.72301i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		−	716.797i		−	0.983261i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	−716.434	+	168.576i	−0.973416	+	0.229044i
$737$		0			0
$738$	−1492.78	−	1866.75i	−2.02274	−	2.52948i
$739$	1022.24	−	1022.24i	1.38328	−	1.38328i	0.544550	−	0.838729i	$-0.316701\pi$
$739$	1022.24	−	1022.24i	1.38328	−	1.38328i	0.838729	−	0.544550i	$-0.183299\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	46.8336	−	135.588i	0.0629484	−	0.182242i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−736.880	−	350.012i	−0.979893	−	0.465442i
$753$		0			0
$754$	−1525.06	+	1219.54i	−2.02262	+	1.61742i
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1464.35i		−	1.92425i	−0.272608	−	0.962125i	$-0.587886\pi$
$761$		−	1464.35i		−	1.92425i	0.272608	−	0.962125i	$-0.412114\pi$
$762$	1896.10	+	211.068i	2.48832	+	0.276992i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1250.27			1.63008
$768$	−1324.59	−	134.891i	−1.72473	−	0.175639i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−1831.30	+	1831.30i	−2.37523	+	2.37523i
$772$	−308.654	+	1369.20i	−0.399811	+	1.77357i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		−	86.1913i		−	0.111215i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	2315.74			2.95752
$784$	708.172	+	336.376i	0.903280	+	0.429051i
$785$		0			0
$786$	−1826.83	+	1460.86i	−2.32422	+	1.85860i
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−1003.11	+	634.049i	−1.27298	+	0.804631i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−866.134	−	96.4153i	−1.09085	−	0.121430i
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	−778.733	+	183.235i	−0.973416	+	0.229044i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	135.994	+	15.1385i	0.168727	+	0.0187822i
$807$		−	2122.84i		−	2.63053i
$808$	1503.01	+	519.156i	1.86016	+	0.642520i
$809$		−	1611.40i		−	1.99184i	−0.0902349	−	0.995921i	$-0.528762\pi$
$809$		−	1611.40i		−	1.99184i	0.0902349	−	0.995921i	$-0.471238\pi$
$810$		0			0
$811$	350.310	+	350.310i	0.431949	+	0.431949i	0.889291	−	0.457342i	$-0.151199\pi$
$811$	350.310	+	350.310i	0.431949	+	0.431949i	−0.457342	+	0.889291i	$0.651199\pi$
$812$		0			0
$813$	1051.80	−	1051.80i	1.29373	−	1.29373i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	953.496	−	762.479i	1.16564	−	0.932126i
$819$		0			0
$820$		0			0
$821$	119.050	+	119.050i	0.145006	+	0.145006i	0.775883	−	0.630877i	$-0.217305\pi$
$821$	119.050	+	119.050i	0.145006	+	0.145006i	−0.630877	+	0.775883i	$0.717305\pi$
$822$		0			0
$823$			1155.23i			1.40369i	0.712332	+	0.701843i	$0.247639\pi$
$823$			1155.23i			1.40369i	−0.712332	+	0.701843i	$0.752361\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	−365.178	+	1619.94i	−0.441036	+	1.95644i
$829$	−884.550	+	884.550i	−1.06701	+	1.06701i	−0.0694210	+	0.997587i	$0.522115\pi$
$829$	−884.550	+	884.550i	−1.06701	+	1.06701i	−0.997587	+	0.0694210i	$0.977885\pi$
$830$		0			0
$831$	367.830			0.442635
$832$	−152.337	−	1260.88i	−0.183098	−	1.51548i
$833$		0			0
$834$	319.371	+	399.381i	0.382939	+	0.478874i
$835$		0			0
$836$		0			0
$837$	−114.745	−	114.745i	−0.137090	−	0.137090i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		−	1579.64i		−	1.87829i
$842$		0			0
$843$		0			0
$844$	787.498	+	1245.88i	0.933055	+	1.47616i
$845$		0			0
$846$	−1437.50	+	1149.52i	−1.69917	+	1.35877i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	2300.03	−	1453.81i	2.69956	−	1.70635i
$853$	1090.75	+	1090.75i	1.27872	+	1.27872i	0.941383	+	0.337339i	$0.109527\pi$
$853$	1090.75	+	1090.75i	1.27872	+	1.27872i	0.337339	+	0.941383i	$0.390473\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1697.28i		−	1.98049i	−0.139332	−	0.990246i	$-0.544496\pi$
$857$		−	1697.28i		−	1.98049i	0.139332	−	0.990246i	$-0.455504\pi$
$858$		0			0
$859$	−304.121	−	304.121i	−0.354041	−	0.354041i	0.507570	−	0.861611i	$-0.330544\pi$
$859$	−304.121	−	304.121i	−0.354041	−	0.354041i	−0.861611	+	0.507570i	$0.830544\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−700.501			−0.811704			−0.405852	−	0.913939i	$-0.633025\pi$
$863$	−700.501			−0.811704			−0.405852	+	0.913939i	$0.633025\pi$
$864$	−792.773	+	1280.65i	−0.917561	+	1.48223i
$865$		0			0
$866$		0			0
$867$	−1062.83	+	1062.83i	−1.22587	+	1.22587i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−1548.34	+	978.681i	−1.76751	+	1.11722i
$877$	376.150	−	376.150i	0.428906	−	0.428906i	−0.459350	−	0.888255i	$-0.651918\pi$
$877$	376.150	−	376.150i	0.428906	−	0.428906i	0.888255	+	0.459350i	$0.151918\pi$
$878$	−367.866	+	294.170i	−0.418982	+	0.335046i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1381.49	−	1104.73i	1.56632	−	1.25253i
$883$	−279.150	+	279.150i	−0.316138	+	0.316138i	−0.847282	−	0.531144i	$-0.821762\pi$
$883$	−279.150	+	279.150i	−0.316138	+	0.316138i	0.531144	+	0.847282i	$0.321762\pi$
$884$		0			0
$885$		0			0
$886$	−195.696	+	1758.01i	−0.220876	+	1.98421i
$887$			1509.61i			1.70192i	0.525228	+	0.850961i	$0.323980\pi$
$887$			1509.61i			1.70192i	−0.525228	+	0.850961i	$0.676020\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	202.492	−	898.259i	0.227009	−	1.00702i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2373.84			−2.64642
$898$	−862.609	−	1078.71i	−0.960589	−	1.20124i
$899$	−119.943	+	119.943i	−0.133418	+	0.133418i
$900$	−396.932	+	1760.80i	−0.441036	+	1.95644i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−126.453	+	1135.97i	−0.139572	+	1.25383i
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$	2536.91	−	2536.91i	2.79088	−	2.79088i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		−	2903.48i		−	3.15253i
$922$	1710.30	+	190.385i	1.85499	+	0.206492i
$923$	1835.30	+	1835.30i	1.98841	+	1.98841i
$924$		0			0
$925$		0			0
$926$	122.409	+	153.075i	0.132191	+	0.165308i
$927$		0			0
$928$	1338.66	+	828.687i	1.44252	+	0.892981i
$929$	618.195			0.665441			0.332720	−	0.943026i	$-0.392033\pi$
$929$	618.195			0.665441			0.332720	+	0.943026i	$0.392033\pi$
$930$		0			0
$931$		0			0
$932$	−1068.65	−	240.903i	−1.14662	−	0.258480i
$933$	−2128.15	−	2128.15i	−2.28097	−	2.28097i
$934$		0			0
$935$		0			0
$936$	−2708.51	−	935.551i	−2.89371	−	0.999520i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$	−1522.87			−1.61492
$944$	−338.030	−	949.685i	−0.358083	−	1.00602i
$945$		0			0
$946$		0			0
$947$	−625.246	+	625.246i	−0.660238	+	0.660238i	−0.955436	−	0.295198i	$-0.904614\pi$
$947$	−625.246	+	625.246i	−0.660238	+	0.660238i	0.295198	+	0.955436i	$0.404614\pi$
$948$		0			0
$949$	−1235.50	−	1235.50i	−1.30189	−	1.30189i
$950$		0			0
$951$			1138.22i			1.19687i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	352.920	−	1565.56i	0.369163	−	1.63761i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−949.114			−0.987631
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1925.34i		−	1.99104i	−0.0945328	−	0.995522i	$-0.530136\pi$
$967$		−	1925.34i		−	1.99104i	0.0945328	−	0.995522i	$-0.469864\pi$
$968$	423.500	+	870.443i	0.437500	+	0.899218i
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$	10.0013	+	15.8227i	0.0102894	+	0.0162785i
$973$		0			0
$974$	−1059.85	+	847.528i	−1.08814	+	0.870152i
$975$	−2580.26			−2.64642
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$	2648.30	−	2117.75i	2.70787	−	2.16539i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−182.448	+	1638.99i	−0.185792	+	1.66904i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−2603.95	−	899.434i	−2.64629	−	0.914059i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1611.40			1.62603			0.813017	−	0.582240i	$-0.197824\pi$
$991$	1611.40			1.62603			0.813017	+	0.582240i	$0.197824\pi$
$992$	−25.2692	−	107.392i	−0.0254730	−	0.108258i
$993$	3327.12			3.35058
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1407.15	−	1407.15i	−1.41138	−	1.41138i	−0.750401	−	0.660983i	$-0.770140\pi$
$997$	−1407.15	−	1407.15i	−1.41138	−	1.41138i	−0.660983	−	0.750401i	$-0.729860\pi$
$998$	−1915.63	−	213.242i	−1.91947	−	0.213669i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.3.k.a.229.2	yes	12
16.13	even	4	inner	368.3.k.a.45.2	✓	12
23.22	odd	2	CM	368.3.k.a.229.2	yes	12
368.45	odd	4	inner	368.3.k.a.45.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.3.k.a.45.2	✓	12	16.13	even	4	inner
368.3.k.a.45.2	✓	12	368.45	odd	4	inner
368.3.k.a.229.2	yes	12	1.1	even	1	trivial
368.3.k.a.229.2	yes	12	23.22	odd	2	CM

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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	368.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.24907 - 1.56199i) q^{2} +(-3.67762 + 3.67762i) q^{3} +(-0.879635 + 3.90208i) q^{4} +(10.3380 + 1.15080i) q^{6} +(7.19375 - 3.50000i) q^{8} -18.0498i q^{9} +(-11.1154 - 17.5854i) q^{12} +(14.0322 - 14.0322i) q^{13} +(-14.4525 - 6.86482i) q^{16} +(-28.1937 + 22.5456i) q^{18} +23.0000i q^{23} +(-13.5842 + 39.3276i) q^{24} +25.0000i q^{25} +(-39.4455 - 4.39095i) q^{26} +(33.2819 + 33.2819i) q^{27} +(34.7897 - 34.7897i) q^{29} -3.44765 q^{31} +(7.32941 + 31.1493i) q^{32} +(70.4320 + 15.8773i) q^{36} +103.211i q^{39} +66.2117i q^{41} +(35.9258 - 28.7287i) q^{46} +50.9864 q^{47} +(78.3970 - 27.9046i) q^{48} -49.0000 q^{49} +(39.0498 - 31.2268i) q^{50} +(42.4117 + 67.0982i) q^{52} +(10.4146 - 93.5577i) q^{54} +(-97.7960 - 10.8864i) q^{58} +(44.5500 + 44.5500i) q^{59} +(4.30637 + 5.38520i) q^{62} +(39.5000 - 50.3562i) q^{64} +(-84.5854 - 84.5854i) q^{69} +130.792i q^{71} +(-63.1745 - 129.846i) q^{72} -88.0471i q^{73} +(-91.9406 - 91.9406i) q^{75} +(161.214 - 128.917i) q^{78} -82.3484 q^{81} +(103.422 - 82.7033i) q^{82} +255.887i q^{87} +(-89.7479 - 20.2316i) q^{92} +(12.6792 - 12.6792i) q^{93} +(-63.6857 - 79.6403i) q^{94} +(-141.510 - 87.6007i) q^{96} +(61.2046 + 76.5376i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 66 q^{6} - 6 q^{12} - 228 q^{27} + 294 q^{36} - 588 q^{49} - 546 q^{58} - 156 q^{59} + 606 q^{62} + 474 q^{64} - 378 q^{72} + 798 q^{78} - 972 q^{81} + 258 q^{82} + 1092 q^{93}+O(q^{100}) \) 12 * q + 66 * q^6 - 6 * q^12 - 228 * q^27 + 294 * q^36 - 588 * q^49 - 546 * q^58 - 156 * q^59 + 606 * q^62 + 474 * q^64 - 378 * q^72 + 798 * q^78 - 972 * q^81 + 258 * q^82 + 1092 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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