LMFDB - Embedded newform 1152.2.r.d.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(193,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1152.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.r (of order $6$, 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:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			961.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.961
Dual form			1152.2.r.d.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72474 - 0.158919i) q^{3} +(2.94949 - 0.548188i) q^{9} +(3.27526 + 1.89097i) q^{11} +1.89898 q^{17} -2.51059i q^{19} +(-2.50000 + 4.33013i) q^{25} +(5.00000 - 1.41421i) q^{27} +(5.94949 + 2.74094i) q^{33} +(-6.39898 - 11.0834i) q^{41} +(11.1742 + 6.45145i) q^{43} +(3.50000 + 6.06218i) q^{49} +(3.27526 - 0.301783i) q^{51} +(-0.398979 - 4.33013i) q^{57} +(10.6237 - 6.13361i) q^{59} +(6.82577 - 3.94086i) q^{67} -13.6969 q^{73} +(-3.62372 + 7.86566i) q^{75} +(8.39898 - 3.23375i) q^{81} +(2.44949 + 1.41421i) q^{83} -18.0000 q^{89} +(-9.84847 + 17.0580i) q^{97} +(10.6969 + 3.78194i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 2 q^{9} + 18 q^{11} - 12 q^{17} - 10 q^{25} + 20 q^{27} + 14 q^{33} - 6 q^{41} + 30 q^{43} + 14 q^{49} + 18 q^{51} + 18 q^{57} + 18 q^{59} + 42 q^{67} + 4 q^{73} + 10 q^{75} + 14 q^{81}+ \cdots - 16 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 2 * q^9 + 18 * q^11 - 12 * q^17 - 10 * q^25 + 20 * q^27 + 14 * q^33 - 6 * q^41 + 30 * q^43 + 14 * q^49 + 18 * q^51 + 18 * q^57 + 18 * q^59 + 42 * q^67 + 4 * q^73 + 10 * q^75 + 14 * q^81 - 72 * q^89 - 10 * q^97 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$

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
$3$	1.72474	−	0.158919i	0.995782	−	0.0917517i
$3$	1.72474	−	0.158919i	0.995782	−	0.0917517i
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$	2.94949	−	0.548188i	0.983163	−	0.182729i
$10$		0			0
$11$	3.27526	+	1.89097i	0.987527	+	0.570149i	0.904534	−	0.426401i	$-0.140219\pi$
$11$	3.27526	+	1.89097i	0.987527	+	0.570149i	0.0829925	+	0.996550i	$0.473552\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.89898			0.460570			0.230285	−	0.973123i	$-0.426034\pi$
$17$	1.89898			0.460570			0.230285	+	0.973123i	$0.426034\pi$
$18$		0			0
$19$		−	2.51059i		−	0.575969i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$		−	2.51059i		−	0.575969i	0.957635	−	0.287984i	$-0.0929851\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$	5.94949	+	2.74094i	1.03567	+	0.477137i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.39898	−	11.0834i	−0.999353	−	1.73093i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	−6.39898	−	11.0834i	−0.999353	−	1.73093i	−0.468521	−	0.883452i	$-0.655213\pi$
$42$		0			0
$43$	11.1742	+	6.45145i	1.70405	+	0.983836i	0.941562	+	0.336840i	$0.109358\pi$
$43$	11.1742	+	6.45145i	1.70405	+	0.983836i	0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$	3.27526	−	0.301783i	0.458627	−	0.0422581i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.398979	−	4.33013i	−0.0528461	−	0.573539i
$58$		0			0
$59$	10.6237	−	6.13361i	1.38309	−	0.798528i	0.390567	−	0.920575i	$-0.372279\pi$
$59$	10.6237	−	6.13361i	1.38309	−	0.798528i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.82577	−	3.94086i	0.833900	−	0.481452i	−0.0212861	−	0.999773i	$-0.506776\pi$
$67$	6.82577	−	3.94086i	0.833900	−	0.481452i	0.855186	+	0.518321i	$0.173443\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−13.6969			−1.60311			−0.801553	−	0.597924i	$-0.795992\pi$
$73$	−13.6969			−1.60311			−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$	−3.62372	+	7.86566i	−0.418432	+	0.908248i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$	8.39898	−	3.23375i	0.933220	−	0.359306i
$82$		0			0
$83$	2.44949	+	1.41421i	0.268866	+	0.155230i	0.628372	−	0.777913i	$-0.283721\pi$
$83$	2.44949	+	1.41421i	0.268866	+	0.155230i	−0.359506	+	0.933143i	$0.617055\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.84847	+	17.0580i	−0.999961	+	1.73198i	−0.492287	+	0.870433i	$0.663839\pi$
$97$	−9.84847	+	17.0580i	−0.999961	+	1.73198i	−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	10.6969	+	3.78194i	1.07508	+	0.380099i
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.70334i		−	0.454689i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$		−	4.70334i		−	0.454689i	0.973814	−	0.227345i	$-0.0730044\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.00000	−	15.5885i	−0.846649	−	1.46644i	−0.884182	−	0.467143i	$-0.845283\pi$
$113$	−9.00000	−	15.5885i	−0.846649	−	1.46644i	0.0375328	−	0.999295i	$-0.488050\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.65153	+	2.86054i	0.150139	+	0.260049i
$122$		0			0
$123$	−12.7980	−	18.0990i	−1.15395	−	1.63194i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	20.2980	+	9.35131i	1.78714	+	0.823337i
$130$		0			0
$131$	−12.2474	+	7.07107i	−1.07006	+	0.617802i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−12.2474	+	7.07107i	−1.07006	+	0.617802i	−0.141865	+	0.989886i	$0.545310\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.29796	−	14.3725i	0.708942	−	1.22792i	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	8.29796	−	14.3725i	0.708942	−	1.22792i	0.965250	−	0.261329i	$-0.0841608\pi$
$138$		0			0
$139$	−20.1742	+	11.6476i	−1.71116	+	0.987937i	−0.778148	+	0.628080i	$0.783841\pi$
$139$	−20.1742	+	11.6476i	−1.71116	+	0.987937i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	+	9.89949i	0.577350	+	0.816497i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$		0			0
$153$	5.60102	−	1.04100i	0.452816	−	0.0841597i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	25.4558i		−	1.99386i	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$		−	25.4558i		−	1.99386i	0.0783260	−	0.996928i	$-0.475042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$	−1.37628	−	7.40496i	−0.105246	−	0.566271i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	17.3485	−	12.2672i	1.30399	−	0.922061i
$178$		0			0
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	6.21964	+	3.59091i	0.454825	+	0.262594i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	12.8485	+	22.2542i	0.924853	+	1.60189i	0.791797	+	0.610784i	$0.209146\pi$
$193$	12.8485	+	22.2542i	0.924853	+	1.60189i	0.133056	+	0.991109i	$0.457521\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	11.1464	−	7.88171i	0.786208	−	0.555933i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.74745	−	8.22282i	0.328388	−	0.568785i
$210$		0			0
$211$	−22.0454	+	12.7279i	−1.51767	+	0.876226i	−0.517884	+	0.855451i	$0.673280\pi$
$211$	−22.0454	+	12.7279i	−1.51767	+	0.876226i	−0.999784	+	0.0207756i	$0.993386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−23.6237	+	2.17670i	−1.59634	+	0.147088i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	−5.00000	+	14.1421i	−0.333333	+	0.942809i
$226$		0			0
$227$	−23.7247	−	13.6975i	−1.57467	−	0.909134i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$	−23.7247	−	13.6975i	−1.57467	−	0.909134i	−0.579082	−	0.815270i	$-0.696589\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.8990			−1.30363			−0.651813	−	0.758380i	$-0.725991\pi$
$233$	−19.8990			−1.30363			−0.651813	+	0.758380i	$0.725991\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	0.848469	−	1.46959i	0.0546547	−	0.0946647i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	0.848469	−	1.46959i	0.0546547	−	0.0946647i	0.892058	+	0.451920i	$0.149261\pi$
$242$		0			0
$243$	13.9722	−	6.91215i	0.896317	−	0.443415i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.44949	+	2.04989i	0.281975	+	0.129906i
$250$		0			0
$251$		−	20.7525i		−	1.30989i	−0.755678	−	0.654943i	$-0.772693\pi$
$251$		−	20.7525i		−	1.30989i	0.755678	−	0.654943i	$-0.227307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.60102	−	4.50510i	−0.162247	−	0.281020i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	−2.60102	−	4.50510i	−0.162247	−	0.281020i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−31.0454	+	2.86054i	−1.89995	+	0.175062i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.3763	+	9.45485i	−0.987527	+	0.570149i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	−0.462174	−	0.886789i	$-0.652930\pi$
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	0.999069	−	0.0431402i	$-0.0137362\pi$
$282$		0			0
$283$	−22.0454	+	12.7279i	−1.31046	+	0.756596i	−0.982173	−	0.187980i	$-0.939806\pi$
$283$	−22.0454	+	12.7279i	−1.31046	+	0.756596i	−0.328291	+	0.944577i	$0.606473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.3939			−0.787875
$290$		0			0
$291$	−14.2753	+	30.9859i	−0.836830	+	1.81642i
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	19.0505	+	4.82294i	1.10542	+	0.279855i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	33.6875i		−	1.92265i	−0.275421	−	0.961324i	$-0.588817\pi$
$307$		−	33.6875i		−	1.92265i	0.275421	−	0.961324i	$-0.411183\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−12.1969	+	21.1257i	−0.689412	+	1.19410i	0.282617	+	0.959233i	$0.408798\pi$
$313$	−12.1969	+	21.1257i	−0.689412	+	1.19410i	−0.972028	+	0.234863i	$0.924536\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.747449	−	8.11207i	−0.0417185	−	0.452771i
$322$		0			0
$323$		−	4.76756i		−	0.265274i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.0454	+	12.7279i	1.21173	+	0.699590i	0.963135	−	0.269019i	$-0.0866994\pi$
$331$	22.0454	+	12.7279i	1.21173	+	0.699590i	0.248590	+	0.968609i	$0.420033\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.609936	−	0.792451i	$-0.708805\pi$
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.381314	−	0.924445i	$-0.624528\pi$
$338$		0			0
$339$	−18.0000	−	25.4558i	−0.977626	−	1.38257i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.4217	−	6.59431i	0.613148	−	0.354001i	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	11.4217	−	6.59431i	0.613148	−	0.354001i	0.774197	+	0.632945i	$0.218154\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.2980	+	29.9609i	−0.920677	+	1.59466i	−0.122308	+	0.992492i	$0.539030\pi$
$353$	−17.2980	+	29.9609i	−0.920677	+	1.59466i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	12.6969			0.668260
$362$		0			0
$363$	3.30306	+	4.67123i	0.173366	+	0.245176i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	−24.9495	−	29.1824i	−1.29882	−	1.51918i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			28.6663i			1.47249i	0.676715	+	0.736245i	$0.263403\pi$
$379$			28.6663i			1.47249i	−0.676715	+	0.736245i	$0.736597\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	36.4949	+	12.9029i	1.85514	+	0.655891i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−20.0000	+	14.1421i	−1.00887	+	0.713376i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.6464	−	27.1004i	−0.781345	−	1.35333i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−15.6464	−	27.1004i	−0.781345	−	1.35333i	0.149813	−	0.988714i	$-0.452133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	12.0278	−	26.1076i	0.593288	−	1.28779i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−32.9444	+	23.2952i	−1.61329	+	1.14077i
$418$		0			0
$419$	−31.8434	+	18.3848i	−1.55565	+	0.898155i	−0.557986	+	0.829851i	$0.688426\pi$
$419$	−31.8434	+	18.3848i	−1.55565	+	0.898155i	−0.997665	+	0.0683046i	$0.978241\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.74745	+	8.22282i	−0.230285	+	0.398865i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−4.30306			−0.206792			−0.103396	−	0.994640i	$-0.532971\pi$
$433$	−4.30306			−0.206792			−0.103396	+	0.994640i	$0.532971\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	13.6464	+	15.9617i	0.649830	+	0.760080i
$442$		0			0
$443$	30.2753	+	17.4794i	1.43842	+	0.830473i	0.997740	−	0.0671913i	$-0.0214038\pi$
$443$	30.2753	+	17.4794i	1.43842	+	0.830473i	0.440681	+	0.897664i	$0.354737\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.1010			0.759854			0.379927	−	0.925016i	$-0.375949\pi$
$449$	16.1010			0.759854			0.379927	+	0.925016i	$0.375949\pi$
$450$		0			0
$451$		−	48.4011i		−	2.27912i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	21.1969	−	36.7142i	0.991551	−	1.71742i	0.383437	−	0.923567i	$-0.374740\pi$
$457$	21.1969	−	36.7142i	0.991551	−	1.71742i	0.608114	−	0.793849i	$-0.291926\pi$
$458$		0			0
$459$	9.49490	−	2.68556i	0.443184	−	0.125351i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$463$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.4244i			0.482384i	0.970477	+	0.241192i	$0.0775384\pi$
$467$			10.4244i			0.482384i	−0.970477	+	0.241192i	$0.922462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	24.3990	+	42.2603i	1.12187	+	1.94313i
$474$		0			0
$475$	10.8712	+	6.27647i	0.498804	+	0.287984i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	−4.04541	−	43.9048i	−0.182940	−	1.98545i
$490$		0			0
$491$	37.6237	−	21.7221i	1.69793	−	0.980303i	0.750218	−	0.661190i	$-0.229948\pi$
$491$	37.6237	−	21.7221i	1.69793	−	0.980303i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−28.8712	+	16.6688i	−1.29245	+	0.746197i	−0.979088	−	0.203436i	$-0.934789\pi$
$499$	−28.8712	+	16.6688i	−1.29245	+	0.746197i	−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$	−9.42168	+	20.4507i	−0.418432	+	0.908248i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.55051	−	12.5529i	−0.156759	−	0.554226i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.1918			1.84846			0.924229	−	0.381839i	$-0.124709\pi$
$521$	42.1918			1.84846			0.924229	+	0.381839i	$0.124709\pi$
$522$		0			0
$523$		−	25.4558i		−	1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$		−	25.4558i		−	1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$	27.9722	−	23.9148i	1.21389	−	1.03781i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.14643	+	34.1482i	0.135778	+	1.47360i
$538$		0			0
$539$			26.4736i			1.14030i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	38.1742	+	22.0399i	1.63221	+	0.942358i	0.983409	+	0.181402i	$0.0580636\pi$
$547$	38.1742	+	22.0399i	1.63221	+	0.942358i	0.648803	+	0.760956i	$0.275270\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	11.2980	+	5.20499i	0.477000	+	0.219755i
$562$		0			0
$563$	38.4217	−	22.1828i	1.61928	−	0.934892i	0.632175	−	0.774826i	$-0.282163\pi$
$563$	38.4217	−	22.1828i	1.61928	−	0.934892i	0.987106	−	0.160066i	$-0.0511708\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.702041	−	1.21597i	0.0294311	−	0.0509761i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	0.702041	−	1.21597i	0.0294311	−	0.0509761i	0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	−1.87117	+	1.08032i	−0.0783062	+	0.0452101i	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	−1.87117	+	1.08032i	−0.0783062	+	0.0452101i	0.460336	+	0.887745i	$0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	12.3939			0.515964			0.257982	−	0.966150i	$-0.416942\pi$
$577$	12.3939			0.515964			0.257982	+	0.966150i	$0.416942\pi$
$578$		0			0
$579$	25.6969	+	36.3410i	1.06793	+	1.51028i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.3207	−	14.6189i	−1.04510	−	0.603386i	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−25.3207	−	14.6189i	−1.04510	−	0.603386i	−0.921272	+	0.388918i	$0.872849\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	18.8485	−	32.6465i	0.768845	−	1.33168i	−0.169344	−	0.985557i	$-0.554165\pi$
$601$	18.8485	−	32.6465i	0.768845	−	1.33168i	0.938190	−	0.346122i	$-0.112502\pi$
$602$		0			0
$603$	17.9722	−	15.3653i	0.731884	−	0.625724i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.6464	+	42.6889i	0.992228	+	1.71859i	0.603877	+	0.797077i	$0.293622\pi$
$617$	24.6464	+	42.6889i	0.992228	+	1.71859i	0.388351	+	0.921512i	$0.373045\pi$
$618$		0			0
$619$	37.8712	+	21.8649i	1.52217	+	0.878826i	0.999657	+	0.0261952i	$0.00833914\pi$
$619$	37.8712	+	21.8649i	1.52217	+	0.878826i	0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	6.88138	−	14.9367i	0.274816	−	0.596515i
$628$		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
$633$	−36.0000	+	25.4558i	−1.43087	+	1.01178i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.7474	+	39.3997i	−0.898470	+	1.55620i	−0.0690201	+	0.997615i	$0.521987\pi$
$641$	−22.7474	+	39.3997i	−0.898470	+	1.55620i	−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	33.8258	−	19.5293i	1.33396	−	0.770161i	0.348054	−	0.937474i	$-0.386843\pi$
$643$	33.8258	−	19.5293i	1.33396	−	0.770161i	0.985904	+	0.167313i	$0.0535092\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	46.3939			1.82112
$650$		0			0
$651$		0			0
$652$		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
$657$	−40.3990	+	7.50850i	−1.57611	+	0.292934i
$658$		0			0
$659$	41.6413	+	24.0416i	1.62212	+	0.936529i	0.986353	+	0.164644i	$0.0526477\pi$
$659$	41.6413	+	24.0416i	1.62212	+	0.936529i	0.635763	+	0.771885i	$0.280686\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	5.00000	−	8.66025i	0.192736	−	0.333828i	−0.753420	−	0.657539i	$-0.771597\pi$
$673$	5.00000	−	8.66025i	0.192736	−	0.333828i	0.946156	+	0.323711i	$0.104931\pi$
$674$		0			0
$675$	−6.37628	+	25.1862i	−0.245423	+	0.969416i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−43.0959	−	19.8544i	−1.65144	−	0.760821i
$682$		0			0
$683$		−	51.9294i		−	1.98702i	−0.113728	−	0.993512i	$-0.536279\pi$
$683$		−	51.9294i		−	1.98702i	0.113728	−	0.993512i	$-0.463721\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	22.0454	+	12.7279i	0.838647	+	0.484193i	0.856804	−	0.515642i	$-0.172447\pi$
$691$	22.0454	+	12.7279i	0.838647	+	0.484193i	−0.0181572	+	0.999835i	$0.505780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−12.1515	−	21.0471i	−0.460272	−	0.797215i
$698$		0			0
$699$	−34.3207	+	3.16232i	−1.29813	+	0.119610i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	1.22985	−	2.66951i	0.0457385	−	0.0992801i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$	21.2196	+	12.2512i	0.784837	+	0.453126i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.8082			1.09800
$738$		0			0
$739$			8.23166i			0.302807i	0.988472	+	0.151403i	$0.0483792\pi$
$739$			8.23166i			0.302807i	−0.988472	+	0.151403i	$0.951621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	8.00000	+	2.82843i	0.292705	+	0.103487i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	−3.29796	−	35.7928i	−0.120184	−	1.30436i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0000	+	46.7654i	0.978749	+	1.69524i	0.666962	+	0.745091i	$0.267594\pi$
$761$	27.0000	+	46.7654i	0.978749	+	1.69524i	0.311787	+	0.950152i	$0.399073\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−11.0000	−	19.0526i	−0.396670	−	0.687053i	0.596643	−	0.802507i	$-0.296501\pi$
$769$	−11.0000	−	19.0526i	−0.396670	−	0.687053i	−0.993313	+	0.115454i	$0.963168\pi$
$770$		0			0
$771$	−5.20204	−	7.35680i	−0.187347	−	0.264949i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.8258	+	16.0652i	−0.996962	+	0.575596i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−22.0454	+	12.7279i	−0.785834	+	0.453701i	−0.838494	−	0.544911i	$-0.816563\pi$
$787$	−22.0454	+	12.7279i	−0.785834	+	0.453701i	0.0526599	+	0.998613i	$0.483230\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−53.0908	+	9.86739i	−1.87587	+	0.348647i
$802$		0			0
$803$	−44.8610	−	25.9005i	−1.58311	−	0.914009i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−45.9898			−1.61692			−0.808458	−	0.588555i	$-0.799697\pi$
$809$	−45.9898			−1.61692			−0.808458	+	0.588555i	$0.799697\pi$
$810$		0			0
$811$		−	54.1222i		−	1.90049i	−0.311509	−	0.950243i	$-0.600834\pi$
$811$		−	54.1222i		−	1.90049i	0.311509	−	0.950243i	$-0.399166\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	16.1969	−	28.0539i	0.566659	−	0.981482i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$	−26.7423	+	18.9097i	−0.931049	+	0.658351i
$826$		0			0
$827$			19.7990i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$827$			19.7990i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.64643	+	11.5120i	0.230285	+	0.398865i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$	13.0454	−	28.3164i	0.449308	−	0.975268i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−36.0000	+	25.4558i	−1.23552	+	0.873642i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.0000	−	46.7654i	0.922302	−	1.59747i	0.126459	−	0.991972i	$-0.459639\pi$
$857$	27.0000	−	46.7654i	0.922302	−	1.59747i	0.795843	−	0.605503i	$-0.207028\pi$
$858$		0			0
$859$	−47.1742	+	27.2361i	−1.60956	+	0.929282i	−0.620097	+	0.784525i	$0.712907\pi$
$859$	−47.1742	+	27.2361i	−1.60956	+	0.929282i	−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−23.1010	+	2.12854i	−0.784552	+	0.0722889i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−19.6969	+	55.7114i	−0.666640	+	1.88554i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$			27.9664i			0.941145i	0.882361	+	0.470573i	$0.155953\pi$
$883$			27.9664i			0.941145i	−0.882361	+	0.470573i	$0.844047\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	33.6237	+	5.29085i	1.12644	+	0.177250i
$892$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−33.2196	−	19.1794i	−1.10304	−	0.636841i	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−33.2196	−	19.1794i	−1.10304	−	0.636841i	−0.937018	+	0.349281i	$0.886426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$	5.34847	+	9.26382i	0.177008	+	0.306588i
$914$		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
$921$	−5.35357	−	58.1024i	−0.176406	−	1.91454i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.0410949	+	0.999155i	$0.513085\pi$
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.844746	+	0.535167i	$0.820249\pi$
$930$		0			0
$931$	15.2196	−	8.78706i	0.498804	−	0.287984i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−17.6793	+	38.3748i	−0.576943	+	1.25231i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	45.7702	+	26.4254i	1.48733	+	0.858710i	0.999896	−	0.0144491i	$-0.00459946\pi$
$947$	45.7702	+	26.4254i	1.48733	+	0.858710i	0.487435	+	0.873160i	$0.337933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−60.1918			−1.94980			−0.974902	−	0.222633i	$-0.928535\pi$
$953$	−60.1918			−1.94980			−0.974902	+	0.222633i	$0.928535\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$	−2.57832	−	13.8725i	−0.0830851	−	0.447034i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$		0			0
$969$	−0.757654	−	8.22282i	−0.0243393	−	0.264155i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.4444	+	49.2671i	0.910017	+	1.57619i	0.814038	+	0.580812i	$0.197265\pi$
$977$	28.4444	+	49.2671i	0.910017	+	1.57619i	0.0959785	+	0.995383i	$0.469402\pi$
$978$		0			0
$979$	−58.9546	−	34.0374i	−1.88420	−	1.08784i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	40.0454	+	18.4490i	1.27080	+	0.585461i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.r.d.961.2	yes	4
3.2	odd	2		3456.2.r.b.2881.2		4
4.3	odd	2		1152.2.r.a.961.1	yes	4
8.3	odd	2	CM	1152.2.r.d.961.2	yes	4
8.5	even	2		1152.2.r.a.961.1	yes	4
9.4	even	3		1152.2.r.a.193.1	✓	4
9.5	odd	6		3456.2.r.c.577.1		4
12.11	even	2		3456.2.r.c.2881.1		4
24.5	odd	2		3456.2.r.c.2881.1		4
24.11	even	2		3456.2.r.b.2881.2		4
36.23	even	6		3456.2.r.b.577.2		4
36.31	odd	6	inner	1152.2.r.d.193.2	yes	4
72.5	odd	6		3456.2.r.b.577.2		4
72.13	even	6	inner	1152.2.r.d.193.2	yes	4
72.59	even	6		3456.2.r.c.577.1		4
72.67	odd	6		1152.2.r.a.193.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.r.a.193.1	✓	4	9.4	even	3
1152.2.r.a.193.1	✓	4	72.67	odd	6
1152.2.r.a.961.1	yes	4	4.3	odd	2
1152.2.r.a.961.1	yes	4	8.5	even	2
1152.2.r.d.193.2	yes	4	36.31	odd	6	inner
1152.2.r.d.193.2	yes	4	72.13	even	6	inner
1152.2.r.d.961.2	yes	4	1.1	even	1	trivial
1152.2.r.d.961.2	yes	4	8.3	odd	2	CM
3456.2.r.b.577.2		4	36.23	even	6
3456.2.r.b.577.2		4	72.5	odd	6
3456.2.r.b.2881.2		4	3.2	odd	2
3456.2.r.b.2881.2		4	24.11	even	2
3456.2.r.c.577.1		4	9.5	odd	6
3456.2.r.c.577.1		4	72.59	even	6
3456.2.r.c.2881.1		4	12.11	even	2
3456.2.r.c.2881.1		4	24.5	odd	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.72474 - 0.158919i) q^{3} +(2.94949 - 0.548188i) q^{9} +(3.27526 + 1.89097i) q^{11} +1.89898 q^{17} -2.51059i q^{19} +(-2.50000 + 4.33013i) q^{25} +(5.00000 - 1.41421i) q^{27} +(5.94949 + 2.74094i) q^{33} +(-6.39898 - 11.0834i) q^{41} +(11.1742 + 6.45145i) q^{43} +(3.50000 + 6.06218i) q^{49} +(3.27526 - 0.301783i) q^{51} +(-0.398979 - 4.33013i) q^{57} +(10.6237 - 6.13361i) q^{59} +(6.82577 - 3.94086i) q^{67} -13.6969 q^{73} +(-3.62372 + 7.86566i) q^{75} +(8.39898 - 3.23375i) q^{81} +(2.44949 + 1.41421i) q^{83} -18.0000 q^{89} +(-9.84847 + 17.0580i) q^{97} +(10.6969 + 3.78194i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{9} + 18 q^{11} - 12 q^{17} - 10 q^{25} + 20 q^{27} + 14 q^{33} - 6 q^{41} + 30 q^{43} + 14 q^{49} + 18 q^{51} + 18 q^{57} + 18 q^{59} + 42 q^{67} + 4 q^{73} + 10 q^{75} + 14 q^{81}+ \cdots - 16 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 2 * q^9 + 18 * q^11 - 12 * q^17 - 10 * q^25 + 20 * q^27 + 14 * q^33 - 6 * q^41 + 30 * q^43 + 14 * q^49 + 18 * q^51 + 18 * q^57 + 18 * q^59 + 42 * q^67 + 4 * q^73 + 10 * q^75 + 14 * q^81 - 72 * q^89 - 10 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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