LMFDB - Embedded newform 228.3.s.a.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,3,Mod(5,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 16]))

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

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

chi := DirichletCharacter("228.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	228.s (of order $18$, degree $6$, 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:	$6.21255002741$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			149.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	228.149
Dual form			228.3.s.a.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.81908 - 1.02606i) q^{3} +(3.30928 + 5.73184i) q^{7} +(6.89440 - 5.78509i) q^{9} +O(q^{10})$	$q+(2.81908 - 1.02606i) q^{3} +(3.30928 + 5.73184i) q^{7} +(6.89440 - 5.78509i) q^{9} +(16.4128 + 5.97378i) q^{13} +(5.50000 - 18.1865i) q^{19} +(15.2103 + 12.7630i) q^{21} +(-23.4923 - 8.55050i) q^{25} +(13.5000 - 23.3827i) q^{27} +(4.25893 + 7.37669i) q^{31} -24.6165 q^{37} +52.3985 q^{39} +(-12.2063 + 69.2254i) q^{43} +(2.59736 - 4.49876i) q^{49} +(-3.15555 - 56.9126i) q^{57} +(-14.3557 - 81.4153i) q^{61} +(55.9747 + 20.3731i) q^{63} +(-102.343 + 85.8759i) q^{67} +(-120.723 + 43.9397i) q^{73} -75.0000 q^{75} +(-103.122 + 37.5335i) q^{79} +(14.0655 - 79.7694i) q^{81} +(20.0739 + 113.844i) q^{91} +(19.5752 + 16.4255i) q^{93} +(-129.462 - 108.631i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) $ 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$e\left(\frac{2}{9}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.81908	−	1.02606i	0.939693	−	0.342020i
$3$	2.81908	−	1.02606i	0.939693	−	0.342020i
$4$		0			0
$5$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$5$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$6$		0			0
$7$	3.30928	+	5.73184i	0.472754	+	0.818834i	0.999514	−	0.0311803i	$-0.00992661\pi$
$7$	3.30928	+	5.73184i	0.472754	+	0.818834i	−0.526760	+	0.850014i	$0.676593\pi$
$8$		0			0
$9$	6.89440	−	5.78509i	0.766044	−	0.642788i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	16.4128	+	5.97378i	1.26252	+	0.459521i	0.884615	−	0.466321i	$-0.154421\pi$
$13$	16.4128	+	5.97378i	1.26252	+	0.459521i	0.377909	+	0.925843i	$0.376643\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$17$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$18$		0			0
$19$	5.50000	−	18.1865i	0.289474	−	0.957186i
$19$	5.50000	−	18.1865i	0.289474	−	0.957186i
$20$		0			0
$21$	15.2103	+	12.7630i	0.724301	+	0.607761i
$22$		0			0
$23$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$23$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$24$		0			0
$25$	−23.4923	−	8.55050i	−0.939693	−	0.342020i
$26$		0			0
$27$	13.5000	−	23.3827i	0.500000	−	0.866025i
$28$		0			0
$29$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$29$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$30$		0			0
$31$	4.25893	+	7.37669i	0.137385	+	0.237958i	0.926506	−	0.376280i	$-0.122797\pi$
$31$	4.25893	+	7.37669i	0.137385	+	0.237958i	−0.789121	+	0.614238i	$0.789464\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−24.6165			−0.665310			−0.332655	−	0.943049i	$-0.607944\pi$
$37$	−24.6165			−0.665310			−0.332655	+	0.943049i	$0.607944\pi$
$38$		0			0
$39$	52.3985			1.34355
$40$		0			0
$41$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$41$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$	−12.2063	+	69.2254i	−0.283868	+	1.60989i	0.425435	+	0.904989i	$0.360121\pi$
$43$	−12.2063	+	69.2254i	−0.283868	+	1.60989i	−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$47$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$48$		0			0
$49$	2.59736	−	4.49876i	0.0530074	−	0.0918115i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$53$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.15555	−	56.9126i	−0.0553606	−	0.998466i
$58$		0			0
$59$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$59$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$60$		0			0
$61$	−14.3557	−	81.4153i	−0.235340	−	1.33468i	−0.841897	−	0.539638i	$-0.818561\pi$
$61$	−14.3557	−	81.4153i	−0.235340	−	1.33468i	0.606557	−	0.795040i	$-0.292550\pi$
$62$		0			0
$63$	55.9747	+	20.3731i	0.888487	+	0.323383i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−102.343	+	85.8759i	−1.52751	+	1.28173i	−0.714073	+	0.700071i	$0.753152\pi$
$67$	−102.343	+	85.8759i	−1.52751	+	1.28173i	−0.813433	+	0.581659i	$0.802404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$71$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$72$		0			0
$73$	−120.723	+	43.9397i	−1.65375	+	0.601914i	−0.989362	−	0.145478i	$-0.953528\pi$
$73$	−120.723	+	43.9397i	−1.65375	+	0.601914i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	−75.0000			−1.00000
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−103.122	+	37.5335i	−1.30535	+	0.475107i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−103.122	+	37.5335i	−1.30535	+	0.475107i	−0.406612	+	0.913601i	$0.633290\pi$
$80$		0			0
$81$	14.0655	−	79.7694i	0.173648	−	0.984808i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$89$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$90$		0			0
$91$	20.0739	+	113.844i	0.220592	+	1.25104i
$92$		0			0
$93$	19.5752	+	16.4255i	0.210486	+	0.176619i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−129.462	−	108.631i	−1.33465	−	1.11991i	−0.982965	−	0.183792i	$-0.941163\pi$
$97$	−129.462	−	108.631i	−1.33465	−	1.11991i	−0.351690	−	0.936117i	$-0.614393\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$101$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$102$		0			0
$103$	17.2709	−	29.9140i	0.167678	−	0.290427i	−0.769925	−	0.638135i	$-0.779706\pi$
$103$	17.2709	−	29.9140i	0.167678	−	0.290427i	0.937603	+	0.347707i	$0.113040\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	12.3290	−	69.9214i	0.113110	−	0.641480i	−0.874558	−	0.484921i	$-0.838849\pi$
$109$	12.3290	−	69.9214i	0.113110	−	0.641480i	0.987668	−	0.156560i	$-0.0500404\pi$
$110$		0			0
$111$	−69.3958	+	25.2580i	−0.625187	+	0.227549i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	147.715	−	53.7640i	1.26252	−	0.459521i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	237.742	+	86.5311i	1.87199	+	0.681347i	0.966313	+	0.257371i	$0.0828564\pi$
$127$	237.742	+	86.5311i	1.87199	+	0.681347i	0.905674	+	0.423976i	$0.139366\pi$
$128$		0			0
$129$	36.6189	+	207.676i	0.283868	+	1.60989i
$130$		0			0
$131$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$131$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$132$		0			0
$133$	122.443	−	28.6592i	0.920626	−	0.215483i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$137$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	260.049	+	94.6500i	1.87085	+	0.680936i	0.967977	+	0.251038i	$0.0807719\pi$
$139$	260.049	+	94.6500i	1.87085	+	0.680936i	0.902878	+	0.429898i	$0.141450\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.70616	−	15.3474i	0.0184093	−	0.104404i
$148$		0			0
$149$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$149$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$150$		0			0
$151$	227.000			1.50331			0.751656	−	0.659556i	$-0.229256\pi$
$151$	227.000			1.50331			0.751656	+	0.659556i	$0.229256\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	36.5367	−	207.210i	0.232718	−	1.31981i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	36.5367	−	207.210i	0.232718	−	1.31981i	0.847367	−	0.531007i	$-0.178186\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−72.7734	+	126.047i	−0.446462	+	0.773295i	−0.998153	−	0.0607535i	$-0.980650\pi$
$163$	−72.7734	+	126.047i	−0.446462	+	0.773295i	0.551690	+	0.834049i	$0.313983\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$167$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$168$		0			0
$169$	104.233	+	87.4619i	0.616764	+	0.517526i
$170$		0			0
$171$	−67.2915	−	157.203i	−0.393518	−	0.919317i
$172$		0			0
$173$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$173$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$174$		0			0
$175$	−28.7325	−	162.950i	−0.164186	−	0.931144i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	−239.772	+	201.193i	−1.32471	+	1.11156i	−0.339423	+	0.940634i	$0.610232\pi$
$181$	−239.772	+	201.193i	−1.32471	+	1.11156i	−0.985284	+	0.170927i	$0.945324\pi$
$182$		0			0
$183$	−124.007	−	214.786i	−0.677634	−	1.17370i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	178.701			0.945508
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	308.460	−	112.270i	1.59824	−	0.581710i	0.619171	−	0.785256i	$-0.287469\pi$
$193$	308.460	−	112.270i	1.59824	−	0.581710i	0.979065	+	0.203546i	$0.0652466\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	178.753	−	149.992i	0.898259	−	0.753728i	−0.0715907	−	0.997434i	$-0.522808\pi$
$199$	178.753	−	149.992i	0.898259	−	0.753728i	0.969849	+	0.243706i	$0.0783631\pi$
$200$		0			0
$201$	−200.399	+	347.101i	−0.997009	+	1.72687i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−288.455	−	242.043i	−1.36709	−	1.14712i	−0.973722	−	0.227740i	$-0.926866\pi$
$211$	−288.455	−	242.043i	−1.36709	−	1.14712i	−0.393365	−	0.919382i	$-0.628689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−28.1880	+	48.8230i	−0.129899	+	0.224991i
$218$		0			0
$219$	−295.244	+	247.739i	−1.34815	+	1.13123i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−37.8715	+	214.780i	−0.169828	+	0.963140i	0.774119	+	0.633041i	$0.218193\pi$
$223$	−37.8715	+	214.780i	−0.169828	+	0.963140i	−0.943946	+	0.330099i	$0.892918\pi$
$224$		0			0
$225$	−211.431	+	76.9545i	−0.939693	+	0.342020i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	313.839			1.37048			0.685238	−	0.728319i	$-0.259698\pi$
$229$	313.839			1.37048			0.685238	+	0.728319i	$0.259698\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$233$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−252.198	+	211.620i	−1.06413	+	0.892909i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−127.850	−	46.5337i	−0.530499	−	0.193086i	0.0628623	−	0.998022i	$-0.479977\pi$
$241$	−127.850	−	46.5337i	−0.530499	−	0.193086i	−0.593361	+	0.804936i	$0.702199\pi$
$242$		0			0
$243$	−42.1965	−	239.308i	−0.173648	−	0.984808i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	198.913	−	265.636i	0.805315	−	1.07545i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$251$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$257$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$258$		0			0
$259$	−81.4628	−	141.098i	−0.314528	−	0.544779i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$263$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$269$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$270$		0			0
$271$	51.9208	−	294.458i	0.191590	−	1.08656i	−0.725603	−	0.688114i	$-0.758439\pi$
$271$	51.9208	−	294.458i	0.191590	−	1.08656i	0.917192	−	0.398445i	$-0.130450\pi$
$272$		0			0
$273$	173.401	+	300.339i	0.635169	+	1.10014i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	0.220217	+	0.975451i	$0.429324\pi$
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$	72.0376	+	26.2195i	0.258199	+	0.0939768i
$280$		0			0
$281$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$281$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$282$		0			0
$283$	45.1966	+	37.9245i	0.159705	+	0.134009i	0.719138	−	0.694867i	$-0.244537\pi$
$283$	45.1966	+	37.9245i	0.159705	+	0.134009i	−0.559433	+	0.828876i	$0.688981\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	50.1843	+	284.609i	0.173648	+	0.984808i
$290$		0			0
$291$	−476.424	−	173.404i	−1.63720	−	0.595891i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−437.183	+	159.122i	−1.45243	+	0.528643i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−574.152	+	208.974i	−1.87020	+	0.680698i	−0.901333	+	0.433127i	$0.857410\pi$
$307$	−574.152	+	208.974i	−1.87020	+	0.680698i	−0.968870	+	0.247571i	$0.920368\pi$
$308$		0			0
$309$	17.9943	−	102.051i	0.0582341	−	0.330262i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	458.861	−	385.030i	1.46601	−	1.23013i	0.546262	−	0.837615i	$-0.316050\pi$
$313$	458.861	−	385.030i	1.46601	−	1.23013i	0.919747	−	0.392513i	$-0.128394\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$317$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−334.496	−	280.676i	−1.02922	−	0.863618i
$326$		0			0
$327$	−36.9871	−	209.764i	−0.113110	−	0.641480i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−316.789	+	548.694i	−0.957065	+	1.65769i	−0.227495	+	0.973779i	$0.573054\pi$
$331$	−316.789	+	548.694i	−0.957065	+	1.65769i	−0.729570	+	0.683906i	$0.760280\pi$
$332$		0			0
$333$	−169.716	+	142.408i	−0.509657	+	0.427653i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	95.1000	−	539.339i	0.282196	−	1.60041i	−0.432938	−	0.901424i	$-0.642523\pi$
$337$	95.1000	−	539.339i	0.282196	−	1.60041i	0.715134	−	0.698988i	$-0.246366\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	358.691			1.04575
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$347$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$348$		0			0
$349$	348.145	+	603.005i	0.997550	+	1.72781i	0.559362	+	0.828923i	$0.311046\pi$
$349$	348.145	+	603.005i	0.997550	+	1.72781i	0.438187	+	0.898884i	$0.355621\pi$
$350$		0			0
$351$	361.256	−	303.130i	1.02922	−	0.863618i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$359$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$360$		0			0
$361$	−300.500	−	200.052i	−0.832410	−	0.554160i
$362$		0			0
$363$	−278.074	−	233.332i	−0.766044	−	0.642788i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	608.915	+	221.627i	1.65917	+	0.603888i	0.990232	−	0.139431i	$-0.0445272\pi$
$367$	608.915	+	221.627i	1.65917	+	0.603888i	0.668937	+	0.743319i	$0.266749\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.162198	−	0.986758i	$-0.551858\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	179.698			0.474137			0.237068	−	0.971493i	$-0.423813\pi$
$379$	179.698			0.474137			0.237068	+	0.971493i	$0.423813\pi$
$380$		0			0
$381$	759.000			1.99213
$382$		0			0
$383$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$383$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	316.320	+	547.882i	0.817364	+	1.41572i
$388$		0			0
$389$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$389$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	581.276	+	487.748i	1.46417	+	1.22858i	0.921349	+	0.388736i	$0.127088\pi$
$397$	581.276	+	487.748i	1.46417	+	1.22858i	0.542821	+	0.839848i	$0.317356\pi$
$398$		0			0
$399$	315.771	−	206.427i	0.791406	−	0.517360i
$400$		0			0
$401$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$401$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$402$		0			0
$403$	25.8344	+	146.514i	0.0641052	+	0.363559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	479.544	−	402.385i	1.17248	−	0.983827i	0.172479	−	0.985013i	$-0.444822\pi$
$409$	479.544	−	402.385i	1.17248	−	0.983827i	0.999999	+	0.00118660i	$0.000377706\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	830.215			1.99092
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−788.402	+	286.955i	−1.87269	+	0.681603i	−0.907499	+	0.420055i	$0.862011\pi$
$421$	−788.402	+	286.955i	−1.87269	+	0.681603i	−0.965190	+	0.261548i	$0.915767\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	419.152	−	351.711i	0.981622	−	0.823678i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$431$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$432$		0			0
$433$	135.720	+	769.706i	0.313441	+	1.77761i	0.580831	+	0.814024i	$0.302728\pi$
$433$	135.720	+	769.706i	0.313441	+	1.77761i	−0.267390	+	0.963588i	$0.586161\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−671.066	−	563.091i	−1.52862	−	1.28267i	−0.807517	−	0.589844i	$-0.799189\pi$
$439$	−671.066	−	563.091i	−1.52862	−	1.28267i	−0.721107	−	0.692824i	$-0.756366\pi$
$440$		0			0
$441$	−8.11849	−	46.0422i	−0.0184093	−	0.104404i
$442$		0			0
$443$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$443$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	639.931	−	232.916i	1.41265	−	0.514163i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	622.735			1.36266			0.681329	−	0.731977i	$-0.261402\pi$
$457$	622.735			1.36266			0.681329	+	0.731977i	$0.261402\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$461$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$	420.932	+	729.075i	0.909140	+	1.57468i	0.815262	+	0.579092i	$0.196593\pi$
$463$	420.932	+	729.075i	0.909140	+	1.57468i	0.0938775	+	0.995584i	$0.470074\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−830.908	−	302.426i	−1.77166	−	0.644831i
$470$		0			0
$471$	−109.610	−	621.629i	−0.232718	−	1.31981i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−284.712	+	380.216i	−0.599393	+	0.800455i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$479$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$480$		0			0
$481$	−404.026	−	147.053i	−0.839970	−	0.305724i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	174.500	+	302.243i	0.358316	+	0.620622i	0.987680	−	0.156489i	$-0.0500176\pi$
$487$	174.500	+	302.243i	0.358316	+	0.620622i	−0.629363	+	0.777111i	$0.716684\pi$
$488$		0			0
$489$	−75.8218	+	430.007i	−0.155055	+	0.879359i
$490$		0			0
$491$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$491$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	171.394	−	972.024i	0.343475	−	1.94794i	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	171.394	−	972.024i	0.343475	−	1.94794i	0.317423	−	0.948284i	$-0.397182\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$503$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	383.582	+	139.613i	0.756573	+	0.275370i
$508$		0			0
$509$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$509$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$510$		0			0
$511$	−651.363	−	546.558i	−1.27468	−	1.06959i
$512$		0			0
$513$	−351.000	−	374.123i	−0.684211	−	0.729285i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−753.658	+	632.394i	−1.44103	+	1.20917i	−0.502214	+	0.864743i	$0.667481\pi$
$523$	−753.658	+	632.394i	−1.44103	+	1.20917i	−0.938815	+	0.344423i	$0.888075\pi$
$524$		0			0
$525$	−248.196	−	429.888i	−0.472754	−	0.818834i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−497.097	+	180.929i	−0.939693	+	0.342020i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−827.821	+	694.624i	−1.53017	+	1.28396i	−0.732902	+	0.680334i	$0.761835\pi$
$541$	−827.821	+	694.624i	−1.53017	+	1.28396i	−0.797265	+	0.603629i	$0.793721\pi$
$542$		0			0
$543$	−469.500	+	813.198i	−0.864641	+	1.49760i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−24.9606	−	141.559i	−0.0456319	−	0.258791i	0.953454	−	0.301538i	$-0.0975001\pi$
$547$	−24.9606	−	141.559i	−0.0456319	−	0.258791i	−0.999086	+	0.0427471i	$0.986389\pi$
$548$		0			0
$549$	−569.969	−	478.261i	−1.03820	−	0.871149i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−556.396	−	466.872i	−1.00614	−	0.844253i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$557$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$558$		0			0
$559$	−613.877	+	1063.27i	−1.09817	+	1.90209i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	503.772	−	183.358i	0.888487	−	0.323383i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−215.566			−0.377523			−0.188761	−	0.982023i	$-0.560447\pi$
$571$	−215.566			−0.377523			−0.188761	+	0.982023i	$0.560447\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.833622	+	0.552335i	$0.186263\pi$
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.0615251	+	0.998106i	$0.480404\pi$
$578$		0			0
$579$	754.376	−	632.996i	1.30289	−	1.09326i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$587$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$588$		0			0
$589$	157.581	−	36.8834i	0.267539	−	0.0626204i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$593$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	350.019	−	606.251i	0.586297	−	1.01550i
$598$		0			0
$599$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$599$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$600$		0			0
$601$	−486.520	−	842.678i	−0.809518	−	1.40213i	−0.913198	−	0.407516i	$-0.866395\pi$
$601$	−486.520	−	842.678i	−0.809518	−	1.40213i	0.103680	−	0.994611i	$-0.466938\pi$
$602$		0			0
$603$	−208.793	+	1184.13i	−0.346258	+	1.96372i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1072.96			1.76764			0.883819	−	0.467830i	$-0.154964\pi$
$607$	1072.96			1.76764			0.883819	+	0.467830i	$0.154964\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	170.696	−	968.066i	0.278460	−	1.57923i	−0.449291	−	0.893386i	$-0.648323\pi$
$613$	170.696	−	968.066i	0.278460	−	1.57923i	0.727751	−	0.685841i	$-0.240566\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$617$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$618$		0			0
$619$	−309.071	+	535.326i	−0.499306	+	0.864824i	−1.00000		0.000800838i	$-0.999745\pi$
$619$	−309.071	+	535.326i	−0.499306	+	0.864824i	0.500693	+	0.865625i	$0.333078\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	478.778	+	401.742i	0.766044	+	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29.4341	−	166.929i	−0.0466467	−	0.264546i	0.952561	−	0.304348i	$-0.0984386\pi$
$631$	−29.4341	−	166.929i	−0.0466467	−	0.264546i	−0.999208	+	0.0398015i	$0.987327\pi$
$632$		0			0
$633$	−1061.53	−	386.365i	−1.67698	−	0.610371i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	69.5047	−	58.3213i	0.109112	−	0.0915562i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$641$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$642$		0			0
$643$	−123.534	+	44.9626i	−0.192121	+	0.0699263i	−0.436289	−	0.899807i	$-0.643707\pi$
$643$	−123.534	+	44.9626i	−0.192121	+	0.0699263i	0.244168	+	0.969733i	$0.421485\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−29.3687	+	166.558i	−0.0451133	+	0.255850i
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−578.120	+	1001.33i	−0.879939	+	1.52410i
$658$		0			0
$659$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$659$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$660$		0			0
$661$	21.1851	+	120.147i	0.0320500	+	0.181765i	0.996630	−	0.0820248i	$-0.0261387\pi$
$661$	21.1851	+	120.147i	0.0320500	+	0.181765i	−0.964580	+	0.263790i	$0.915028\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	113.615	+	644.341i	0.169828	+	0.963140i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	423.723	−	733.910i	0.629604	−	1.09051i	−0.358027	−	0.933711i	$-0.616551\pi$
$673$	423.723	−	733.910i	0.629604	−	1.09051i	0.987631	−	0.156795i	$-0.0501161\pi$
$674$		0			0
$675$	−517.080	+	433.882i	−0.766044	+	0.642788i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	194.232	−	1101.54i	0.286056	−	1.62230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	884.737	−	322.018i	1.28783	−	0.468731i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	0.216353	−	0.976315i	$-0.430584\pi$
$692$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$701$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$	−135.391	+	447.688i	−0.192590	+	0.636826i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1139.77	+	414.844i	1.60758	+	0.585111i	0.980959	−	0.194214i	$-0.0622158\pi$
$709$	1139.77	+	414.844i	1.60758	+	0.585111i	0.626620	+	0.779325i	$0.284438\pi$
$710$		0			0
$711$	−493.832	+	855.343i	−0.694560	+	1.20301i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$719$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$720$		0			0
$721$	228.616			0.317082
$722$		0			0
$723$	−408.166			−0.564545
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−220.055	+	1247.99i	−0.302689	+	1.71663i	0.331499	+	0.943455i	$0.392446\pi$
$727$	−220.055	+	1247.99i	−0.302689	+	1.71663i	−0.634188	+	0.773179i	$0.718666\pi$
$728$		0			0
$729$	−364.500	−	631.333i	−0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.261255	−	0.965270i	$-0.415864\pi$
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.705321	−	0.708888i	$-0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1097.48	+	920.893i	1.48508	+	1.24613i	0.900541	+	0.434771i	$0.143171\pi$
$739$	1097.48	+	920.893i	1.48508	+	1.24613i	0.584543	+	0.811363i	$0.301274\pi$
$740$		0			0
$741$	288.192	−	952.947i	0.388923	−	1.28603i
$742$		0			0
$743$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$743$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	839.361	−	704.308i	1.11766	−	0.937827i	0.119174	−	0.992873i	$-0.461975\pi$
$751$	839.361	−	704.308i	1.11766	−	0.937827i	0.998484	+	0.0550467i	$0.0175308\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−158.401	+	57.6533i	−0.209248	+	0.0761602i	−0.444518	−	0.895770i	$-0.646625\pi$
$757$	−158.401	+	57.6533i	−0.209248	+	0.0761602i	0.235269	+	0.971930i	$0.424403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	441.578	−	160.721i	0.578739	−	0.210644i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−120.429	+	101.052i	−0.156605	+	0.131407i	−0.717723	−	0.696328i	$-0.754816\pi$
$769$	−120.429	+	101.052i	−0.156605	+	0.131407i	0.561118	+	0.827736i	$0.310371\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$773$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$774$		0			0
$775$	−36.9778	−	209.711i	−0.0477133	−	0.270595i
$776$		0			0
$777$	−374.425	−	314.180i	−0.481885	−	0.404349i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−231.141	+	400.347i	−0.293698	+	0.508700i	−0.974681	−	0.223599i	$-0.928219\pi$
$787$	−231.141	+	400.347i	−0.293698	+	0.508700i	0.680983	+	0.732299i	$0.261553\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	250.739	−	1422.01i	0.316191	−	1.79321i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	237.742	+	86.5311i	0.293147	+	0.106697i	0.484407	−	0.874843i	$-0.339035\pi$
$811$	237.742	+	86.5311i	0.293147	+	0.106697i	−0.191260	+	0.981539i	$0.561257\pi$
$812$		0			0
$813$	−155.762	−	883.373i	−0.191590	−	1.08656i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1191.84	+	602.730i	1.45880	+	0.737736i
$818$		0			0
$819$	796.998	+	668.760i	0.973135	+	0.816557i
$820$		0			0
$821$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$821$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$822$		0			0
$823$	1523.24	+	554.415i	1.85084	+	0.673651i	0.984804	+	0.173670i	$0.0555627\pi$
$823$	1523.24	+	554.415i	1.85084	+	0.673651i	0.866037	+	0.499980i	$0.166659\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$827$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$828$		0			0
$829$	487.692	+	844.707i	0.588289	+	1.01895i	0.994457	+	0.105148i	$0.0335317\pi$
$829$	487.692	+	844.707i	0.588289	+	1.01895i	−0.406167	+	0.913799i	$0.633135\pi$
$830$		0			0
$831$	−212.024	+	1202.45i	−0.255144	+	1.44699i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	229.982			0.274770
$838$		0			0
$839$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$839$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$840$		0			0
$841$	146.038	−	828.223i	0.173648	−	0.984808i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	400.423	−	693.552i	0.472754	−	0.818834i
$848$		0			0
$849$	166.326	+	60.5376i	0.195908	+	0.0713046i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1170.82	+	982.434i	1.37259	+	1.15174i	0.971864	+	0.235543i	$0.0756867\pi$
$853$	1170.82	+	982.434i	1.37259	+	1.15174i	0.400727	+	0.916198i	$0.368758\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$857$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$858$		0			0
$859$	−173.777	−	985.539i	−0.202302	−	1.14731i	−0.901630	−	0.432509i	$-0.857629\pi$
$859$	−173.777	−	985.539i	−0.202302	−	1.14731i	0.699328	−	0.714801i	$-0.253483\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	433.500	+	750.844i	0.500000	+	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2192.74	+	798.092i	−2.51750	+	0.916294i
$872$		0			0
$873$	−1521.00			−1.74227
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.90293	−	0.692610i	0.00216982	−	0.000789750i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	1.90293	−	0.692610i	0.00216982	−	0.000789750i	0.343105	+	0.939297i	$0.388521\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	1348.61	−	1131.62i	1.52731	−	1.28156i	0.712911	−	0.701255i	$-0.247377\pi$
$883$	1348.61	−	1131.62i	1.52731	−	1.28156i	0.814397	−	0.580308i	$-0.197068\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$887$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$888$		0			0
$889$	290.773	+	1649.06i	0.327079	+	1.85496i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−1069.18	+	897.152i	−1.18404	+	0.993524i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	289.472	−	1641.67i	0.319153	−	1.81000i	−0.228775	−	0.973479i	$-0.573472\pi$
$907$	289.472	−	1641.67i	0.319153	−	1.81000i	0.547928	−	0.836526i	$-0.315417\pi$
$908$		0			0
$909$		0			0
$910$		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
$919$	−852.741	−	1476.99i	−0.927901	−	1.60717i	−0.786828	−	0.617172i	$-0.788278\pi$
$919$	−852.741	−	1476.99i	−0.927901	−	1.60717i	−0.141072	−	0.989999i	$-0.545055\pi$
$920$		0			0
$921$	−1404.16	+	1178.23i	−1.52460	+	1.27929i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	578.298	+	210.483i	0.625187	+	0.227549i
$926$		0			0
$927$	−53.9830	−	306.153i	−0.0582341	−	0.330262i
$928$		0			0
$929$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$929$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$930$		0			0
$931$	−67.5314	−	71.9802i	−0.0725364	−	0.0773150i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−594.711	−	216.457i	−0.634697	−	0.231011i	0.00457754	−	0.999990i	$-0.498543\pi$
$937$	−594.711	−	216.457i	−0.634697	−	0.231011i	−0.639274	+	0.768979i	$0.720765\pi$
$938$		0			0
$939$	898.500	−	1556.25i	0.956869	−	1.65735i
$940$		0			0
$941$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$941$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$947$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$948$		0			0
$949$	−2243.90			−2.36449
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$953$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	444.223	−	769.417i	0.462251	−	0.800642i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1480.14	−	1241.98i	−1.53065	−	1.28437i	−0.793175	−	0.608994i	$-0.791573\pi$
$967$	−1480.14	−	1241.98i	−1.53065	−	1.28437i	−0.737475	−	0.675374i	$-0.763982\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$971$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$972$		0			0
$973$	318.055	+	1803.78i	0.326881	+	1.85383i
$974$		0			0
$975$	−1230.96	−	448.033i	−1.26252	−	0.459521i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−319.500	−	553.390i	−0.325688	−	0.564108i
$982$		0			0
$983$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$983$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1826.18	+	664.674i	−1.84276	+	0.670710i	−0.854188	+	0.519965i	$0.825945\pi$
$991$	−1826.18	+	664.674i	−1.84276	+	0.670710i	−0.988573	+	0.150745i	$0.951833\pi$
$992$		0			0
$993$	−330.059	+	1871.86i	−0.332385	+	1.88505i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1418.48	+	1190.24i	−1.42275	+	1.19383i	−0.472896	+	0.881118i	$0.656791\pi$
$997$	−1418.48	+	1190.24i	−1.42275	+	1.19383i	−0.949850	+	0.312707i	$0.898764\pi$
$998$		0			0
$999$	−332.322	+	575.599i	−0.332655	+	0.576176i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.3.s.a.149.1	yes	6
3.2	odd	2	CM	228.3.s.a.149.1	yes	6
19.6	even	9	inner	228.3.s.a.101.1	✓	6
57.44	odd	18	inner	228.3.s.a.101.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.s.a.101.1	✓	6	19.6	even	9	inner
228.3.s.a.101.1	✓	6	57.44	odd	18	inner
228.3.s.a.149.1	yes	6	1.1	even	1	trivial
228.3.s.a.149.1	yes	6	3.2	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:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	228.s (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(2.81908 - 1.02606i) q^{3} +(3.30928 + 5.73184i) q^{7} +(6.89440 - 5.78509i) q^{9} +O(q^{10})\)	\(q+(2.81908 - 1.02606i) q^{3} +(3.30928 + 5.73184i) q^{7} +(6.89440 - 5.78509i) q^{9} +(16.4128 + 5.97378i) q^{13} +(5.50000 - 18.1865i) q^{19} +(15.2103 + 12.7630i) q^{21} +(-23.4923 - 8.55050i) q^{25} +(13.5000 - 23.3827i) q^{27} +(4.25893 + 7.37669i) q^{31} -24.6165 q^{37} +52.3985 q^{39} +(-12.2063 + 69.2254i) q^{43} +(2.59736 - 4.49876i) q^{49} +(-3.15555 - 56.9126i) q^{57} +(-14.3557 - 81.4153i) q^{61} +(55.9747 + 20.3731i) q^{63} +(-102.343 + 85.8759i) q^{67} +(-120.723 + 43.9397i) q^{73} -75.0000 q^{75} +(-103.122 + 37.5335i) q^{79} +(14.0655 - 79.7694i) q^{81} +(20.0739 + 113.844i) q^{91} +(19.5752 + 16.4255i) q^{93} +(-129.462 - 108.631i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 69 q^{13} + 33 q^{19} + 117 q^{21} + 81 q^{27} - 183 q^{43} - 147 q^{49} + 222 q^{61} + 54 q^{63} - 327 q^{67} - 291 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) \) 6 * q + 69 * q^13 + 33 * q^19 + 117 * q^21 + 81 * q^27 - 183 * q^43 - 147 * q^49 + 222 * q^61 + 54 * q^63 - 327 * q^67 - 291 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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