LMFDB - Embedded newform 1152.2.p.c.959.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(191,1152)]

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([3, 3, 1]))

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

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

chi := DirichletCharacter("1152.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.p (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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			959.2
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1152.959
Dual form			1152.2.p.c.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.158919 + 1.72474i) q^{3} +(-2.94949 - 0.548188i) q^{9} +O(q^{10})$	$q+(-0.158919 + 1.72474i) q^{3} +(-2.94949 - 0.548188i) q^{9} +(4.71940 - 2.72474i) q^{11} +8.02458i q^{17} +2.51059 q^{19} +(2.50000 + 4.33013i) q^{25} +(1.41421 - 5.00000i) q^{27} +(3.94949 + 8.57277i) q^{33} +(-0.398979 - 0.230351i) q^{41} +(6.45145 + 11.1742i) q^{43} +(-3.50000 + 6.06218i) q^{49} +(-13.8404 - 1.27526i) q^{51} +(-0.398979 + 4.33013i) q^{57} +(-8.00853 - 4.62372i) q^{59} +(-3.94086 + 6.82577i) q^{67} -13.6969 q^{73} +(-7.86566 + 3.62372i) q^{75} +(8.39898 + 3.23375i) q^{81} +(15.5885 - 9.00000i) q^{83} -5.65685i q^{89} +(9.84847 + 17.0580i) q^{97} +(-15.4135 + 5.44949i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 20 q^{25} + 12 q^{33} + 36 q^{41} - 28 q^{49} + 36 q^{57} + 8 q^{73} + 28 q^{81} + 20 q^{97}+O(q^{100}) $ 8 * q - 4 * q^9 + 20 * q^25 + 12 * q^33 + 36 * q^41 - 28 * q^49 + 36 * q^57 + 8 * q^73 + 28 * q^81 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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
$3$	−0.158919	+	1.72474i	−0.0917517	+	0.995782i
$3$	−0.158919	+	1.72474i	−0.0917517	+	0.995782i
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$		0			0
$9$	−2.94949	−	0.548188i	−0.983163	−	0.182729i
$10$		0			0
$11$	4.71940	−	2.72474i	1.42295	−	0.821541i	0.426401	−	0.904534i	$-0.359781\pi$
$11$	4.71940	−	2.72474i	1.42295	−	0.821541i	0.996550	+	0.0829925i	$0.0264478\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			8.02458i			1.94625i	0.230285	+	0.973123i	$0.426034\pi$
$17$			8.02458i			1.94625i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	2.51059			0.575969			0.287984	−	0.957635i	$-0.407015\pi$
$19$	2.51059			0.575969			0.287984	+	0.957635i	$0.407015\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	1.41421	−	5.00000i	0.272166	−	0.962250i
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$	3.94949	+	8.57277i	0.687518	+	1.49233i
$34$		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
$41$	−0.398979	−	0.230351i	−0.0623101	−	0.0359748i	0.468521	−	0.883452i	$-0.344787\pi$
$41$	−0.398979	−	0.230351i	−0.0623101	−	0.0359748i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	6.45145	+	11.1742i	0.983836	+	1.70405i	0.646997	+	0.762493i	$0.276025\pi$
$43$	6.45145	+	11.1742i	0.983836	+	1.70405i	0.336840	+	0.941562i	$0.390642\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$		0			0
$51$	−13.8404	−	1.27526i	−1.93804	−	0.178571i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.398979	+	4.33013i	−0.0528461	+	0.573539i
$58$		0			0
$59$	−8.00853	−	4.62372i	−1.04262	−	0.601958i	−0.122047	−	0.992524i	$-0.538946\pi$
$59$	−8.00853	−	4.62372i	−1.04262	−	0.601958i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.94086	+	6.82577i	−0.481452	+	0.833900i	−0.999773	−	0.0212861i	$-0.993224\pi$
$67$	−3.94086	+	6.82577i	−0.481452	+	0.833900i	0.518321	+	0.855186i	$0.326557\pi$
$68$		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
$75$	−7.86566	+	3.62372i	−0.908248	+	0.418432i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	8.39898	+	3.23375i	0.933220	+	0.359306i
$82$		0			0
$83$	15.5885	−	9.00000i	1.71106	−	0.987878i	0.777913	−	0.628372i	$-0.216279\pi$
$83$	15.5885	−	9.00000i	1.71106	−	0.987878i	0.933143	−	0.359506i	$-0.117055\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		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	0.507673	+	0.861550i	$0.330506\pi$
$97$	9.84847	+	17.0580i	0.999961	+	1.73198i	0.492287	+	0.870433i	$0.336161\pi$
$98$		0			0
$99$	−15.4135	+	5.44949i	−1.54911	+	0.547694i
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	20.1464i		−	1.94763i	−0.227345	−	0.973814i	$-0.573004\pi$
$107$		−	20.1464i		−	1.94763i	0.227345	−	0.973814i	$-0.426996\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.79796	+	5.65685i	0.921714	+	0.532152i	0.884182	−	0.467143i	$-0.154717\pi$
$113$	9.79796	+	5.65685i	0.921714	+	0.532152i	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
$121$	9.34847	−	16.1920i	0.849861	−	1.47200i
$122$		0			0
$123$	0.460702	−	0.651531i	0.0415401	−	0.0587466i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−20.2980	+	9.35131i	−1.78714	+	0.823337i
$130$		0			0
$131$	15.5885	+	9.00000i	1.36197	+	0.786334i	0.989886	−	0.141865i	$-0.0453100\pi$
$131$	15.5885	+	9.00000i	1.36197	+	0.786334i	0.372084	+	0.928199i	$0.378643\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.2980	+	8.25493i	−1.22156	+	0.705266i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−14.2980	+	8.25493i	−1.22156	+	0.705266i	−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	−11.6476	+	20.1742i	−0.987937	+	1.71116i	−0.359856	+	0.933008i	$0.617174\pi$
$139$	−11.6476	+	20.1742i	−0.987937	+	1.71116i	−0.628080	+	0.778148i	$0.716159\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−9.89949	−	7.00000i	−0.816497	−	0.577350i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	4.39898	−	23.6684i	0.355636	−	1.91348i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	25.4558			1.99386			0.996928	−	0.0783260i	$-0.0249575\pi$
$163$	25.4558			1.99386			0.996928	+	0.0783260i	$0.0249575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$	−7.40496	−	1.37628i	−0.566271	−	0.105246i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	9.24745	−	13.0779i	0.695081	−	0.982993i
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		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
$187$	21.8649	+	37.8712i	1.59892	+	2.76941i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	12.8485	−	22.2542i	0.924853	−	1.60189i	0.133056	−	0.991109i	$-0.457521\pi$
$193$	12.8485	−	22.2542i	0.924853	−	1.60189i	0.791797	−	0.610784i	$-0.209146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		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
$209$	11.8485	−	6.84072i	0.819576	−	0.473182i
$210$		0			0
$211$	12.7279	−	22.0454i	0.876226	−	1.51767i	0.0207756	−	0.999784i	$-0.493386\pi$
$211$	12.7279	−	22.0454i	0.876226	−	1.51767i	0.855451	−	0.517884i	$-0.173280\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.17670	−	23.6237i	0.147088	−	1.59634i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$	−5.00000	−	14.1421i	−0.333333	−	0.942809i
$226$		0			0
$227$	10.8691	−	6.27526i	0.721405	−	0.416503i	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	10.8691	−	6.27526i	0.721405	−	0.416503i	0.815270	+	0.579082i	$0.196589\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	23.1523i		−	1.51676i	−0.651813	−	0.758380i	$-0.725991\pi$
$233$		−	23.1523i		−	1.51676i	0.651813	−	0.758380i	$-0.274009\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	0.848469	+	1.46959i	0.0546547	+	0.0946647i	0.892058	−	0.451920i	$-0.149261\pi$
$241$	0.848469	+	1.46959i	0.0546547	+	0.0946647i	−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	−6.91215	+	13.9722i	−0.443415	+	0.896317i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	13.0454	+	28.3164i	0.826719	+	1.79448i
$250$		0			0
$251$		−	23.9444i		−	1.51136i	−0.654943	−	0.755678i	$-0.727307\pi$
$251$		−	23.9444i		−	1.51136i	0.654943	−	0.755678i	$-0.272693\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.3990	−	15.8188i	−1.70910	−	0.986750i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−27.3990	−	15.8188i	−1.70910	−	0.986750i	−0.773427	−	0.633885i	$-0.781459\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	9.75663	+	0.898979i	0.597096	+	0.0550167i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	23.5970	+	13.6237i	1.42295	+	0.821541i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.4949	−	14.1421i	1.46124	−	0.843649i	0.462174	−	0.886789i	$-0.347070\pi$
$281$	24.4949	−	14.1421i	1.46124	−	0.843649i	0.999069	+	0.0431402i	$0.0137362\pi$
$282$		0			0
$283$	−12.7279	+	22.0454i	−0.756596	+	1.31046i	0.187980	+	0.982173i	$0.439806\pi$
$283$	−12.7279	+	22.0454i	−0.756596	+	1.31046i	−0.944577	+	0.328291i	$0.893527\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−47.3939			−2.78788
$290$		0			0
$291$	−30.9859	+	14.2753i	−1.81642	+	0.836830i
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.94949	−	27.4504i	−0.403250	−	1.59283i
$298$		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
$307$	−33.6875			−1.92265			−0.961324	−	0.275421i	$-0.911183\pi$
$307$	−33.6875			−1.92265			−0.961324	+	0.275421i	$0.911183\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	12.1969	+	21.1257i	0.689412	+	1.19410i	0.972028	+	0.234863i	$0.0754642\pi$
$313$	12.1969	+	21.1257i	0.689412	+	1.19410i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	34.7474	+	3.20164i	1.93941	+	0.178698i
$322$		0			0
$323$			20.1464i			1.12098i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.7279	−	22.0454i	−0.699590	−	1.21173i	−0.968609	−	0.248590i	$-0.920033\pi$
$331$	−12.7279	−	22.0454i	−0.699590	−	1.21173i	0.269019	−	0.963135i	$-0.413301\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.1969	−	31.5180i	0.991250	−	1.71690i	0.381314	−	0.924445i	$-0.375472\pi$
$337$	18.1969	−	31.5180i	0.991250	−	1.71690i	0.609936	−	0.792451i	$-0.291195\pi$
$338$		0			0
$339$	−11.3137	+	16.0000i	−0.614476	+	0.869001i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.1752	−	17.4217i	−1.61989	−	0.935245i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	−30.1752	−	17.4217i	−1.61989	−	0.935245i	−0.632945	−	0.774197i	$-0.718154\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.7020	−	7.33353i	0.676061	−	0.390324i	−0.122308	−	0.992492i	$-0.539030\pi$
$353$	12.7020	−	7.33353i	0.676061	−	0.390324i	0.798369	+	0.602168i	$0.205696\pi$
$354$		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
$363$	26.4415	+	18.6969i	1.38782	+	0.981335i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	1.05051	+	0.898133i	0.0546874	+	0.0467550i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	28.6663			1.47249			0.736245	−	0.676715i	$-0.236597\pi$
$379$	28.6663			1.47249			0.736245	+	0.676715i	$0.236597\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−12.9029	−	36.4949i	−0.655891	−	1.85514i
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−18.0000	+	25.4558i	−0.907980	+	1.28408i
$394$		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
$401$	−21.6464	−	12.4976i	−1.08097	−	0.624099i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−21.6464	−	12.4976i	−1.08097	−	0.624099i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		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	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	9.19694	−	15.9296i	0.454759	−	0.787666i	0.998674	+	0.0514740i	$0.0163919\pi$
$410$		0			0
$411$	−11.9654	−	25.9722i	−0.590211	−	1.28111i
$412$		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
$419$	−15.5885	−	9.00000i	−0.761546	−	0.439679i	0.0683046	−	0.997665i	$-0.478241\pi$
$419$	−15.5885	−	9.00000i	−0.761546	−	0.439679i	−0.829851	+	0.557986i	$0.811574\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.7474	+	20.0614i	−1.68550	+	0.973123i
$426$		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	$-0.467029\pi$
$433$	4.30306			0.206792			0.103396	+	0.994640i	$0.467029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	13.6464	−	15.9617i	0.649830	−	0.760080i
$442$		0			0
$443$	−20.3079	+	11.7247i	−0.964855	+	0.557059i	−0.897664	−	0.440681i	$-0.854737\pi$
$443$	−20.3079	+	11.7247i	−0.964855	+	0.557059i	−0.0671913	+	0.997740i	$0.521404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			39.2015i			1.85003i	0.379927	+	0.925016i	$0.375949\pi$
$449$			39.2015i			1.85003i	−0.379927	+	0.925016i	$0.624051\pi$
$450$		0			0
$451$	−2.51059			−0.118219
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	21.1969	+	36.7142i	0.991551	+	1.71742i	0.608114	+	0.793849i	$0.291926\pi$
$457$	21.1969	+	36.7142i	0.991551	+	1.71742i	0.383437	+	0.923567i	$0.374740\pi$
$458$		0			0
$459$	40.1229	+	11.3485i	1.87278	+	0.529701i
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			41.9444i			1.94095i	0.241192	+	0.970477i	$0.422462\pi$
$467$			41.9444i			1.94095i	−0.241192	+	0.970477i	$0.577538\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	60.8939	+	35.1571i	2.79990	+	1.61653i
$474$		0			0
$475$	6.27647	+	10.8712i	0.287984	+	0.498804i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−4.04541	+	43.9048i	−0.182940	+	1.98545i
$490$		0			0
$491$	−7.57993	−	4.37628i	−0.342078	−	0.197499i	0.319113	−	0.947717i	$-0.396615\pi$
$491$	−7.57993	−	4.37628i	−0.342078	−	0.197499i	−0.661190	+	0.750218i	$0.729948\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.6688	−	28.8712i	0.746197	−	1.29245i	−0.203436	−	0.979088i	$-0.565211\pi$
$499$	16.6688	−	28.8712i	0.746197	−	1.29245i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		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
$507$	20.4507	−	9.42168i	0.908248	−	0.418432i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		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
$521$			17.4313i			0.763678i	0.924229	+	0.381839i	$0.124709\pi$
$521$			17.4313i			0.763678i	−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$	−25.4558			−1.11311			−0.556553	−	0.830812i	$-0.687876\pi$
$523$	−25.4558			−1.11311			−0.556553	+	0.830812i	$0.687876\pi$
$524$		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
$531$	21.0864	+	18.0278i	0.915072	+	0.782340i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	31.0454	+	2.86054i	1.33971	+	0.123441i
$538$		0			0
$539$			38.1464i			1.64308i
$540$		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
$547$	−22.0399	−	38.1742i	−0.942358	−	1.63221i	−0.760956	−	0.648803i	$-0.775270\pi$
$547$	−22.0399	−	38.1742i	−0.942358	−	1.63221i	−0.181402	−	0.983409i	$-0.558064\pi$
$548$		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			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−68.7929	+	31.6930i	−2.90444	+	1.33808i
$562$		0			0
$563$	14.5868	+	8.42168i	0.614760	+	0.354932i	0.774826	−	0.632175i	$-0.217837\pi$
$563$	14.5868	+	8.42168i	0.614760	+	0.354932i	−0.160066	+	0.987106i	$0.551171\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−41.2980	+	23.8434i	−1.73130	+	0.999567i	−0.850935	+	0.525271i	$0.823964\pi$
$569$	−41.2980	+	23.8434i	−1.73130	+	0.999567i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−1.08032	+	1.87117i	−0.0452101	+	0.0783062i	−0.887745	−	0.460336i	$-0.847729\pi$
$571$	−1.08032	+	1.87117i	−0.0452101	+	0.0783062i	0.842535	+	0.538642i	$0.181062\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−12.3939			−0.515964			−0.257982	−	0.966150i	$-0.583058\pi$
$577$	−12.3939			−0.515964			−0.257982	+	0.966150i	$0.583058\pi$
$578$		0			0
$579$	36.3410	+	25.6969i	1.51028	+	1.06793i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.4644	−	19.3207i	1.38122	−	0.797449i	0.388918	−	0.921272i	$-0.372849\pi$
$587$	33.4644	−	19.3207i	1.38122	−	0.797449i	0.992304	+	0.123823i	$0.0395156\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	45.2548i		−	1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$		−	45.2548i		−	1.85839i	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	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	18.8485	+	32.6465i	0.768845	+	1.33168i	0.938190	+	0.346122i	$0.112502\pi$
$601$	18.8485	+	32.6465i	0.768845	+	1.33168i	−0.169344	+	0.985557i	$0.554165\pi$
$602$		0			0
$603$	15.3653	−	17.9722i	0.625724	−	0.731884i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		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
$617$	5.35357	+	3.09089i	0.215527	+	0.124434i	0.603877	−	0.797077i	$-0.293622\pi$
$617$	5.35357	+	3.09089i	0.215527	+	0.124434i	−0.388351	+	0.921512i	$0.626955\pi$
$618$		0			0
$619$	−21.8649	−	37.8712i	−0.878826	−	1.52217i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	−21.8649	−	37.8712i	−0.878826	−	1.52217i	−0.0261952	−	0.999657i	$-0.508339\pi$
$620$		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
$627$	9.91555	+	21.5227i	0.395989	+	0.859534i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		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
$641$	19.2526	−	11.1155i	0.760430	−	0.439034i	−0.0690201	−	0.997615i	$-0.521987\pi$
$641$	19.2526	−	11.1155i	0.760430	−	0.439034i	0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$	19.5293	−	33.8258i	0.770161	−	1.33396i	−0.167313	−	0.985904i	$-0.553509\pi$
$643$	19.5293	−	33.8258i	0.770161	−	1.33396i	0.937474	−	0.348054i	$-0.113157\pi$
$644$		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
$649$	−50.3939			−1.97813
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	40.3990	+	7.50850i	1.57611	+	0.292934i
$658$		0			0
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	−0.771885	−	0.635763i	$-0.780686\pi$
$659$	−15.5885	+	9.00000i	−0.607240	+	0.350590i	0.164644	+	0.986353i	$0.447352\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		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	$-0.228403\pi$
$673$	−5.00000	−	8.66025i	−0.192736	−	0.333828i	−0.946156	+	0.323711i	$0.895069\pi$
$674$		0			0
$675$	25.1862	−	6.37628i	0.969416	−	0.245423i
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	9.09592	+	19.7436i	0.348556	+	0.756577i
$682$		0			0
$683$			5.94439i			0.227456i	0.993512	+	0.113728i	$0.0362792\pi$
$683$			5.94439i			0.227456i	−0.993512	+	0.113728i	$0.963721\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.7279	−	22.0454i	−0.484193	−	0.838647i	0.515642	−	0.856804i	$-0.327553\pi$
$691$	−12.7279	−	22.0454i	−0.484193	−	0.838647i	−0.999835	+	0.0181572i	$0.994220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.84847	−	3.20164i	0.0700158	−	0.121271i
$698$		0			0
$699$	39.9319	+	3.67934i	1.51036	+	0.139165i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		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	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		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
$723$	−2.66951	+	1.22985i	−0.0992801	+	0.0457385i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$730$		0			0
$731$	−89.6685	+	51.7702i	−3.31651	+	1.91479i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			42.9513i			1.58213i
$738$		0			0
$739$	8.23166			0.302807			0.151403	−	0.988472i	$-0.451621\pi$
$739$	8.23166			0.302807			0.151403	+	0.988472i	$0.451621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−50.9117	+	18.0000i	−1.86276	+	0.658586i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	41.2980	+	3.80521i	1.50498	+	0.138670i
$754$		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
$761$	9.79796	+	5.65685i	0.355176	+	0.205061i	0.666962	−	0.745091i	$-0.267594\pi$
$761$	9.79796	+	5.65685i	0.355176	+	0.205061i	−0.311787	+	0.950152i	$0.600927\pi$
$762$		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	−0.993313	−	0.115454i	$-0.963168\pi$
$769$	−11.0000	+	19.0526i	−0.396670	+	0.687053i	0.596643	+	0.802507i	$0.296501\pi$
$770$		0			0
$771$	31.6376	−	44.7423i	1.13940	−	1.61136i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.00167	−	0.578317i	−0.0358887	−	0.0207203i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.7279	+	22.0454i	−0.453701	+	0.785834i	−0.998613	−	0.0526599i	$-0.983230\pi$
$787$	−12.7279	+	22.0454i	−0.453701	+	0.785834i	0.544911	+	0.838494i	$0.316563\pi$
$788$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−3.10102	+	16.6848i	−0.109569	+	0.589530i
$802$		0			0
$803$	−64.6413	+	37.3207i	−2.28114	+	1.31702i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	33.4804i		−	1.17711i	−0.808458	−	0.588555i	$-0.799697\pi$
$809$		−	33.4804i		−	1.17711i	0.808458	−	0.588555i	$-0.200303\pi$
$810$		0			0
$811$	−54.1222			−1.90049			−0.950243	−	0.311509i	$-0.899166\pi$
$811$	−54.1222			−1.90049			−0.950243	+	0.311509i	$0.899166\pi$
$812$		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		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	−27.2474	+	38.5337i	−0.948634	+	1.34157i
$826$		0			0
$827$			54.0000i			1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$			54.0000i			1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−48.6464	−	28.0860i	−1.68550	−	0.973123i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$	20.4989	+	44.4949i	0.706019	+	1.53249i
$844$		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	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.5959	−	11.3137i	0.669384	−	0.386469i	−0.126459	−	0.991972i	$-0.540361\pi$
$857$	19.5959	−	11.3137i	0.669384	−	0.386469i	0.795843	+	0.605503i	$0.207028\pi$
$858$		0			0
$859$	27.2361	−	47.1742i	0.929282	−	1.60956i	0.144757	−	0.989467i	$-0.453760\pi$
$859$	27.2361	−	47.1742i	0.929282	−	1.60956i	0.784525	−	0.620097i	$-0.212907\pi$
$860$		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
$867$	7.53177	−	81.7423i	0.255792	−	2.77612i
$868$		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	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			56.5685i			1.90584i	0.303218	+	0.952921i	$0.401939\pi$
$881$			56.5685i			1.90584i	−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	−27.9664			−0.941145			−0.470573	−	0.882361i	$-0.655953\pi$
$883$	−27.9664			−0.941145			−0.470573	+	0.882361i	$0.655953\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	48.4493	−	7.62372i	1.62311	−	0.255404i
$892$		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
$907$	−19.1794	−	33.2196i	−0.636841	−	1.10304i	−0.986122	−	0.166022i	$-0.946908\pi$
$907$	−19.1794	−	33.2196i	−0.636841	−	1.10304i	0.349281	−	0.937018i	$-0.386426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	49.0454	−	84.9491i	1.62317	−	2.81141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		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
$929$	24.4949	−	14.1421i	0.803652	−	0.463988i	−0.0410949	−	0.999155i	$-0.513085\pi$
$929$	24.4949	−	14.1421i	0.803652	−	0.463988i	0.844746	+	0.535167i	$0.179751\pi$
$930$		0			0
$931$	−8.78706	+	15.2196i	−0.287984	+	0.498804i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$	−38.3748	+	17.6793i	−1.25231	+	0.576943i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.3147	−	15.7702i	0.887609	−	0.512461i	0.0144491	−	0.999896i	$-0.495401\pi$
$947$	27.3147	−	15.7702i	0.887609	−	0.512461i	0.873160	+	0.487435i	$0.162067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	13.7456i		−	0.445265i	−0.974902	−	0.222633i	$-0.928535\pi$
$953$		−	13.7456i		−	0.445265i	0.974902	−	0.222633i	$-0.0714650\pi$
$954$		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
$963$	−11.0440	+	59.4217i	−0.355889	+	1.91484i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$	−34.7474	−	3.20164i	−1.11625	−	0.102852i
$970$		0			0
$971$		−	54.0000i		−	1.73294i	−0.499227	−	0.866471i	$-0.666383\pi$
$971$		−	54.0000i		−	1.73294i	0.499227	−	0.866471i	$-0.333617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.4444	+	12.9583i	0.718060	+	0.414572i	0.814038	−	0.580812i	$-0.197265\pi$
$977$	22.4444	+	12.9583i	0.718060	+	0.414572i	−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$	−15.4135	−	26.6969i	−0.492617	−	0.853238i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		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	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.p.c.959.2	yes	8
3.2	odd	2		3456.2.p.c.2879.1		8
4.3	odd	2	inner	1152.2.p.c.959.3	yes	8
8.3	odd	2	CM	1152.2.p.c.959.2	yes	8
8.5	even	2	inner	1152.2.p.c.959.3	yes	8
9.2	odd	6	inner	1152.2.p.c.191.2	✓	8
9.7	even	3		3456.2.p.c.575.1		8
12.11	even	2		3456.2.p.c.2879.4		8
24.5	odd	2		3456.2.p.c.2879.4		8
24.11	even	2		3456.2.p.c.2879.1		8
36.7	odd	6		3456.2.p.c.575.4		8
36.11	even	6	inner	1152.2.p.c.191.3	yes	8
72.11	even	6	inner	1152.2.p.c.191.2	✓	8
72.29	odd	6	inner	1152.2.p.c.191.3	yes	8
72.43	odd	6		3456.2.p.c.575.1		8
72.61	even	6		3456.2.p.c.575.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.p.c.191.2	✓	8	9.2	odd	6	inner
1152.2.p.c.191.2	✓	8	72.11	even	6	inner
1152.2.p.c.191.3	yes	8	36.11	even	6	inner
1152.2.p.c.191.3	yes	8	72.29	odd	6	inner
1152.2.p.c.959.2	yes	8	1.1	even	1	trivial
1152.2.p.c.959.2	yes	8	8.3	odd	2	CM
1152.2.p.c.959.3	yes	8	4.3	odd	2	inner
1152.2.p.c.959.3	yes	8	8.5	even	2	inner
3456.2.p.c.575.1		8	9.7	even	3
3456.2.p.c.575.1		8	72.43	odd	6
3456.2.p.c.575.4		8	36.7	odd	6
3456.2.p.c.575.4		8	72.61	even	6
3456.2.p.c.2879.1		8	3.2	odd	2
3456.2.p.c.2879.1		8	24.11	even	2
3456.2.p.c.2879.4		8	12.11	even	2
3456.2.p.c.2879.4		8	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.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.158919 + 1.72474i) q^{3} +(-2.94949 - 0.548188i) q^{9} +O(q^{10})\)	\(q+(-0.158919 + 1.72474i) q^{3} +(-2.94949 - 0.548188i) q^{9} +(4.71940 - 2.72474i) q^{11} +8.02458i q^{17} +2.51059 q^{19} +(2.50000 + 4.33013i) q^{25} +(1.41421 - 5.00000i) q^{27} +(3.94949 + 8.57277i) q^{33} +(-0.398979 - 0.230351i) q^{41} +(6.45145 + 11.1742i) q^{43} +(-3.50000 + 6.06218i) q^{49} +(-13.8404 - 1.27526i) q^{51} +(-0.398979 + 4.33013i) q^{57} +(-8.00853 - 4.62372i) q^{59} +(-3.94086 + 6.82577i) q^{67} -13.6969 q^{73} +(-7.86566 + 3.62372i) q^{75} +(8.39898 + 3.23375i) q^{81} +(15.5885 - 9.00000i) q^{83} -5.65685i q^{89} +(9.84847 + 17.0580i) q^{97} +(-15.4135 + 5.44949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 20 q^{25} + 12 q^{33} + 36 q^{41} - 28 q^{49} + 36 q^{57} + 8 q^{73} + 28 q^{81} + 20 q^{97}+O(q^{100}) \) 8 * q - 4 * q^9 + 20 * q^25 + 12 * q^33 + 36 * q^41 - 28 * q^49 + 36 * q^57 + 8 * q^73 + 28 * q^81 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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