LMFDB - Embedded newform 1568.2.q.f.1391.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(815,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.815");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			1391.1
Root			$-0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1568.1391
Dual form			1568.2.q.f.815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.92586 + 1.68925i) q^{3} +(4.20711 - 7.28692i) q^{9} +O(q^{10})$	$q+(-2.92586 + 1.68925i) q^{3} +(4.20711 - 7.28692i) q^{9} +(1.12132 + 1.94218i) q^{11} +(-2.92586 + 1.68925i) q^{17} +(-1.21193 - 0.699709i) q^{19} +(2.50000 + 4.33013i) q^{25} +18.2919i q^{27} +(-6.56165 - 3.78837i) q^{33} +8.15640i q^{41} -13.0711 q^{43} +(5.70711 - 9.88500i) q^{51} +4.72792 q^{57} +(9.48751 - 5.47762i) q^{59} +(-4.24264 - 7.34847i) q^{67} +(-12.9154 + 7.45669i) q^{73} +(-14.6293 - 8.44623i) q^{75} +(-18.2782 - 31.6587i) q^{81} -17.7122i q^{83} +(-10.4915 - 6.05728i) q^{89} -15.7331i q^{97} +18.8701 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 28 q^{9}+O(q^{10}) $ 8 * q + 28 * q^9	$ 8 q + 28 q^{9} - 8 q^{11} + 20 q^{25} - 48 q^{43} + 40 q^{51} - 64 q^{57} - 84 q^{81} - 64 q^{99}+O(q^{100}) $ 8 * q + 28 * q^9 - 8 * q^11 + 20 * q^25 - 48 * q^43 + 40 * q^51 - 64 * q^57 - 84 * q^81 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.92586	+	1.68925i	−1.68925	+	0.975287i	−0.734152	+	0.678985i	$0.762420\pi$
$3$	−2.92586	+	1.68925i	−1.68925	+	0.975287i	−0.955094	+	0.296302i	$0.904247\pi$
$4$		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
$7$		0			0
$8$		0			0
$9$	4.20711	−	7.28692i	1.40237	−	2.42897i
$10$		0			0
$11$	1.12132	+	1.94218i	0.338091	+	0.585590i	0.984074	−	0.177762i	$-0.0568856\pi$
$11$	1.12132	+	1.94218i	0.338091	+	0.585590i	−0.645983	+	0.763352i	$0.723552\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.92586	+	1.68925i	−0.709625	+	0.409702i	−0.810922	−	0.585154i	$-0.801034\pi$
$17$	−2.92586	+	1.68925i	−0.709625	+	0.409702i	0.101297	+	0.994856i	$0.467701\pi$
$18$		0			0
$19$	−1.21193	−	0.699709i	−0.278036	−	0.160524i	0.354498	−	0.935057i	$-0.384652\pi$
$19$	−1.21193	−	0.699709i	−0.278036	−	0.160524i	−0.632534	+	0.774533i	$0.717985\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$			18.2919i			3.52027i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		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
$33$	−6.56165	−	3.78837i	−1.14224	−	0.659471i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.15640i			1.27382i	0.770940	+	0.636908i	$0.219787\pi$
$41$			8.15640i			1.27382i	−0.770940	+	0.636908i	$0.780213\pi$
$42$		0			0
$43$	−13.0711			−1.99332			−0.996660	−	0.0816682i	$-0.973975\pi$
$43$	−13.0711			−1.99332			−0.996660	+	0.0816682i	$0.973975\pi$
$44$		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
$49$		0			0
$50$		0			0
$51$	5.70711	−	9.88500i	0.799155	−	1.38418i
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.72792			0.626229
$58$		0			0
$59$	9.48751	−	5.47762i	1.23517	−	0.713125i	0.267066	−	0.963678i	$-0.413946\pi$
$59$	9.48751	−	5.47762i	1.23517	−	0.713125i	0.968103	+	0.250553i	$0.0806124\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.24264	−	7.34847i	−0.518321	−	0.897758i	−0.999773	−	0.0212861i	$-0.993224\pi$
$67$	−4.24264	−	7.34847i	−0.518321	−	0.897758i	0.481452	−	0.876472i	$-0.340109\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−12.9154	+	7.45669i	−1.51163	+	0.872740i	−0.511722	+	0.859151i	$0.670992\pi$
$73$	−12.9154	+	7.45669i	−1.51163	+	0.872740i	−0.999908	+	0.0135893i	$0.995674\pi$
$74$		0			0
$75$	−14.6293	−	8.44623i	−1.68925	−	0.975287i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−18.2782	−	31.6587i	−2.03091	−	3.51764i
$82$		0			0
$83$		−	17.7122i		−	1.94417i	−0.234631	−	0.972085i	$-0.575388\pi$
$83$		−	17.7122i		−	1.94417i	0.234631	−	0.972085i	$-0.424612\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.4915	−	6.05728i	−1.11210	−	0.642070i	−0.172726	−	0.984970i	$-0.555258\pi$
$89$	−10.4915	−	6.05728i	−1.11210	−	0.642070i	−0.939372	+	0.342900i	$0.888591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	15.7331i		−	1.59746i	−0.601690	−	0.798730i	$-0.705506\pi$
$97$		−	15.7331i		−	1.59746i	0.601690	−	0.798730i	$-0.294494\pi$
$98$		0			0
$99$	18.8701			1.89651
$100$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	0.683793	+	0.729676i	$0.260329\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.7279			−1.94992			−0.974959	−	0.222383i	$-0.928617\pi$
$113$	−20.7279			−1.94992			−0.974959	+	0.222383i	$0.928617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.98528	−	5.17066i	0.271389	−	0.470060i
$122$		0			0
$123$	−13.7782	−	23.8645i	−1.24234	−	2.15179i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	38.2441	−	22.0803i	3.36721	−	1.94406i
$130$		0			0
$131$	−5.34972	−	3.08866i	−0.467407	−	0.269858i	0.247746	−	0.968825i	$-0.420310\pi$
$131$	−5.34972	−	3.08866i	−0.467407	−	0.269858i	−0.715154	+	0.698967i	$0.753643\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.87868	+	10.1822i	0.502249	+	0.869922i	0.999997	+	0.00259945i	$0.000827431\pi$
$137$	5.87868	+	10.1822i	0.502249	+	0.869922i	−0.497747	+	0.867322i	$0.665839\pi$
$138$		0			0
$139$		−	0.579658i		−	0.0491659i	−0.999698	−	0.0245830i	$-0.992174\pi$
$139$		−	0.579658i		−	0.0491659i	0.999698	−	0.0245830i	$-0.00782579\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		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
$153$			28.4274i			2.29822i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−9.70711	+	16.8132i	−0.760319	+	1.31691i	0.182367	+	0.983231i	$0.441624\pi$
$163$	−9.70711	+	16.8132i	−0.760319	+	1.31691i	−0.942686	+	0.333681i	$0.891709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−10.1974	+	5.88750i	−0.779818	+	0.450228i
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−18.5061	+	32.0535i	−1.39100	+	2.40929i
$178$		0			0
$179$	9.00000	+	15.5885i	0.672692	+	1.16514i	0.977138	+	0.212607i	$0.0681952\pi$
$179$	9.00000	+	15.5885i	0.672692	+	1.16514i	−0.304446	+	0.952529i	$0.598471\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.56165	−	3.78837i	−0.479836	−	0.277033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	1.77817	+	3.07989i	0.127996	+	0.221695i	0.922900	−	0.385040i	$-0.125812\pi$
$193$	1.77817	+	3.07989i	0.127996	+	0.221695i	−0.794904	+	0.606735i	$0.792479\pi$
$194$		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		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$	24.8268	+	14.3337i	1.75114	+	1.01102i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.13839i		−	0.217087i
$210$		0			0
$211$	−25.4558			−1.75245			−0.876226	−	0.481900i	$-0.839947\pi$
$211$	−25.4558			−1.75245			−0.876226	+	0.481900i	$0.839947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	25.1924	−	43.6345i	1.70234	−	2.94855i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	42.0711			2.80474
$226$		0			0
$227$	−12.2054	+	7.04681i	−0.810104	+	0.467714i	−0.846992	−	0.531606i	$-0.821589\pi$
$227$	−12.2054	+	7.04681i	−0.810104	+	0.467714i	0.0368883	+	0.999319i	$0.488255\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6066	−	21.8353i	0.825886	−	1.43048i	−0.0753544	−	0.997157i	$-0.524009\pi$
$233$	12.6066	−	21.8353i	0.825886	−	1.43048i	0.901240	−	0.433320i	$-0.142658\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−22.1950	+	12.8143i	−1.42970	+	0.825439i	−0.997097	−	0.0761440i	$-0.975739\pi$
$241$	−22.1950	+	12.8143i	−1.42970	+	0.825439i	−0.432606	+	0.901583i	$0.642406\pi$
$242$		0			0
$243$	59.4351	+	34.3149i	3.81276	+	2.20130i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	29.9203	+	51.8235i	1.89612	+	3.28418i
$250$		0			0
$251$		−	26.4483i		−	1.66940i	−0.550704	−	0.834700i	$-0.685641\pi$
$251$		−	26.4483i		−	1.66940i	0.550704	−	0.834700i	$-0.314359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.7526	−	16.0230i	−1.73116	−	0.999486i	−0.881611	−	0.471976i	$-0.843541\pi$
$257$	−27.7526	−	16.0230i	−1.73116	−	0.999486i	−0.849549	−	0.527510i	$-0.823126\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	40.9289			2.50481
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.60660	+	9.71092i	−0.338091	+	0.585590i
$276$		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
$281$	7.27208			0.433816			0.216908	−	0.976192i	$-0.430403\pi$
$281$	7.27208			0.433816			0.216908	+	0.976192i	$0.430403\pi$
$282$		0			0
$283$	−26.0387	+	15.0334i	−1.54784	+	0.893645i	−0.549532	+	0.835472i	$0.685194\pi$
$283$	−26.0387	+	15.0334i	−1.54784	+	0.893645i	−0.998307	+	0.0581728i	$0.981473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.79289	+	4.83743i	−0.164288	+	0.284555i
$290$		0			0
$291$	26.5772	+	46.0330i	1.55798	+	2.69850i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−35.5262	+	20.5111i	−2.06144	+	1.19017i
$298$		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
$307$			20.8506i			1.19001i	0.803723	+	0.595004i	$0.202849\pi$
$307$			20.8506i			1.19001i	−0.803723	+	0.595004i	$0.797151\pi$
$308$		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
$313$	30.4705	+	17.5922i	1.72230	+	0.994368i	0.914138	+	0.405404i	$0.132869\pi$
$313$	30.4705	+	17.5922i	1.72230	+	0.994368i	0.808159	+	0.588964i	$0.200464\pi$
$314$		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
$321$		−	20.2710i		−	1.13141i
$322$		0			0
$323$	4.72792			0.263069
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.192388	−	0.333226i	0.0105746	−	0.0183158i	−0.860690	−	0.509130i	$-0.829967\pi$
$331$	0.192388	−	0.333226i	0.0105746	−	0.0183158i	0.871264	+	0.490814i	$0.163301\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−33.9411			−1.84889			−0.924445	−	0.381314i	$-0.875472\pi$
$337$	−33.9411			−1.84889			−0.924445	+	0.381314i	$0.875472\pi$
$338$		0			0
$339$	60.6470	−	35.0146i	3.29389	−	1.90173i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.1213	+	26.1909i	0.811755	+	1.40600i	0.911635	+	0.411001i	$0.134821\pi$
$347$	15.1213	+	26.1909i	0.811755	+	1.40600i	−0.0998797	+	0.995000i	$0.531846\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.0467	−	16.1928i	1.49277	−	0.861853i	0.492808	−	0.870138i	$-0.335970\pi$
$353$	28.0467	−	16.1928i	1.49277	−	0.861853i	0.999966	+	0.00828457i	$0.00263709\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−8.52082	−	14.7585i	−0.448464	−	0.776762i
$362$		0			0
$363$			20.1715i			1.05873i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$	59.4351	+	34.3149i	3.09407	+	1.78636i
$370$		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
$379$	−32.8701			−1.68842			−0.844211	−	0.536011i	$-0.819930\pi$
$379$	−32.8701			−1.68842			−0.844211	+	0.536011i	$0.819930\pi$
$380$		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
$387$	−54.9914	+	95.2479i	−2.79537	+	4.84172i
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	20.8701			1.05276
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.35372	−	3.66832i	0.314171	−	0.181387i	−0.334620	−	0.942353i	$-0.608608\pi$
$409$	6.35372	−	3.66832i	0.314171	−	0.181387i	0.648792	+	0.760966i	$0.275275\pi$
$410$		0			0
$411$	−34.4004	−	19.8611i	−1.69685	−	0.979675i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.979185	+	1.69600i	0.0479509	+	0.0830534i
$418$		0			0
$419$		−	2.55873i		−	0.125002i	−0.998045	−	0.0625011i	$-0.980092\pi$
$419$		−	2.55873i		−	0.125002i	0.998045	−	0.0625011i	$-0.0199077\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.6293	−	8.44623i	−0.709625	−	0.409702i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		−	41.6018i		−	1.99925i	−0.0273152	−	0.999627i	$-0.508696\pi$
$433$		−	41.6018i		−	1.99925i	0.0273152	−	0.999627i	$-0.491304\pi$
$434$		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
$441$		0			0
$442$		0			0
$443$	1.41421	−	2.44949i	0.0671913	−	0.116379i	−0.830473	−	0.557059i	$-0.811930\pi$
$443$	1.41421	−	2.44949i	0.0671913	−	0.116379i	0.897664	+	0.440681i	$0.145263\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.65685			0.266963			0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685			0.266963			0.133482	+	0.991051i	$0.457384\pi$
$450$		0			0
$451$	−15.8412	+	9.14594i	−0.745935	+	0.430666i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−21.1924	+	36.7063i	−0.991338	+	1.71705i	−0.381930	+	0.924191i	$0.624740\pi$
$457$	−21.1924	+	36.7063i	−0.991338	+	1.71705i	−0.609408	+	0.792857i	$0.708593\pi$
$458$		0			0
$459$	−30.8995	−	53.5195i	−1.44226	−	2.49808i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.3143	+	19.8114i	1.58787	+	0.916760i	0.993657	+	0.112456i	$0.0358717\pi$
$467$	34.3143	+	19.8114i	1.58787	+	0.916760i	0.594218	+	0.804304i	$0.297462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.6569	−	25.3864i	−0.673923	−	1.16727i
$474$		0			0
$475$		−	6.99709i		−	0.321048i
$476$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		−	65.5908i		−	2.96612i
$490$		0			0
$491$	14.1421			0.638226			0.319113	−	0.947717i	$-0.396615\pi$
$491$	14.1421			0.638226			0.319113	+	0.947717i	$0.396615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	21.2132	−	36.7423i	0.949633	−	1.64481i	0.203436	−	0.979088i	$-0.434789\pi$
$499$	21.2132	−	36.7423i	0.949633	−	1.64481i	0.746197	−	0.665725i	$-0.231878\pi$
$500$		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
$507$	38.0362	−	21.9602i	1.68925	−	0.975287i
$508$		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
$513$	12.7990	−	22.1685i	0.565089	−	0.978763i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.1974	−	5.88750i	0.446758	−	0.257936i	−0.259702	−	0.965689i	$-0.583624\pi$
$521$	10.1974	−	5.88750i	0.446758	−	0.257936i	0.706460	+	0.707753i	$0.250291\pi$
$522$		0			0
$523$	7.77359	+	4.48808i	0.339915	+	0.196250i	0.660235	−	0.751060i	$-0.270457\pi$
$523$	7.77359	+	4.48808i	0.339915	+	0.196250i	−0.320319	+	0.947310i	$0.603790\pi$
$524$		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
$531$		−	92.1797i		−	4.00026i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−52.6655	−	30.4064i	−2.27268	−	1.31213i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	26.5269			1.13421			0.567104	−	0.823646i	$-0.308064\pi$
$547$	26.5269			1.13421			0.567104	+	0.823646i	$0.308064\pi$
$548$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	25.5980			1.08075
$562$		0			0
$563$	19.4770	−	11.2451i	0.820859	−	0.473923i	−0.0298537	−	0.999554i	$-0.509504\pi$
$563$	19.4770	−	11.2451i	0.820859	−	0.473923i	0.850713	+	0.525631i	$0.176171\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.3137	+	19.5959i	−0.474295	+	0.821504i	−0.999567	−	0.0294311i	$-0.990630\pi$
$569$	−11.3137	+	19.5959i	−0.474295	+	0.821504i	0.525271	+	0.850935i	$0.323964\pi$
$570$		0			0
$571$	−22.7782	−	39.4530i	−0.953237	−	1.65105i	−0.738352	−	0.674415i	$-0.764396\pi$
$571$	−22.7782	−	39.4530i	−0.953237	−	1.65105i	−0.214885	−	0.976639i	$-0.568938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−38.4521	+	22.2003i	−1.60078	+	0.924211i	−0.609449	+	0.792826i	$0.708609\pi$
$577$	−38.4521	+	22.2003i	−1.60078	+	0.924211i	−0.991331	+	0.131385i	$0.958058\pi$
$578$		0			0
$579$	−10.4054	−	6.00755i	−0.432433	−	0.249665i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			46.7192i			1.92831i	0.265341	+	0.964155i	$0.414516\pi$
$587$			46.7192i			1.92831i	−0.265341	+	0.964155i	$0.585484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−42.1740	−	24.3492i	−1.73188	−	0.999900i	−0.873007	−	0.487708i	$-0.837833\pi$
$593$	−42.1740	−	24.3492i	−1.73188	−	0.999900i	−0.858871	−	0.512192i	$-0.828834\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$			36.0041i			1.46864i	0.678804	+	0.734319i	$0.262498\pi$
$601$			36.0041i			1.46864i	−0.678804	+	0.734319i	$0.737502\pi$
$602$		0			0
$603$	−71.3970			−2.90751
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	−25.3287	+	14.6236i	−1.01805	+	0.587770i	−0.913538	−	0.406753i	$-0.866661\pi$
$619$	−25.3287	+	14.6236i	−1.01805	+	0.587770i	−0.104510	+	0.994524i	$0.533328\pi$
$620$		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
$627$	5.30152	+	9.18249i	0.211722	+	0.366713i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	74.4803	−	43.0012i	2.96032	−	1.70914i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.1421	+	24.4949i	0.558581	+	0.967490i	0.997615	+	0.0690201i	$0.0219873\pi$
$641$	14.1421	+	24.4949i	0.558581	+	0.967490i	−0.439034	+	0.898470i	$0.644679\pi$
$642$		0			0
$643$		−	11.2948i		−	0.445423i	−0.974884	−	0.222712i	$-0.928509\pi$
$643$		−	11.2948i		−	0.445423i	0.974884	−	0.222712i	$-0.0714908\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	21.2771	+	12.2843i	0.835199	+	0.482202i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			125.484i			4.89561i
$658$		0			0
$659$	−21.2721			−0.828643			−0.414321	−	0.910131i	$-0.635981\pi$
$659$	−21.2721			−0.828643			−0.414321	+	0.910131i	$0.635981\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$		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
$673$	−28.9289			−1.11513			−0.557564	−	0.830134i	$-0.688264\pi$
$673$	−28.9289			−1.11513			−0.557564	+	0.830134i	$0.688264\pi$
$674$		0			0
$675$	−79.2062	+	45.7297i	−3.04865	+	1.76014i
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	23.8076	−	41.2360i	0.912310	−	1.58017i
$682$		0			0
$683$	15.5563	+	26.9444i	0.595247	+	1.03100i	0.993512	+	0.113728i	$0.0362792\pi$
$683$	15.5563	+	26.9444i	0.595247	+	1.03100i	−0.398265	+	0.917270i	$0.630387\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	35.6123	+	20.5608i	1.35476	+	0.782169i	0.988911	−	0.148507i	$-0.0474466\pi$
$691$	35.6123	+	20.5608i	1.35476	+	0.782169i	0.365845	+	0.930676i	$0.380780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−13.7782	−	23.8645i	−0.521886	−	0.903932i
$698$		0			0
$699$			85.1826i			3.22190i
$700$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	43.2929	−	74.9855i	1.61008	−	2.78874i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−122.196			−4.52578
$730$		0			0
$731$	38.2441	−	22.0803i	1.41451	−	0.816668i
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.51472	−	16.4800i	0.350479	−	0.607048i
$738$		0			0
$739$	−2.97918	−	5.16010i	−0.109591	−	0.189817i	0.806014	−	0.591897i	$-0.201621\pi$
$739$	−2.97918	−	5.16010i	−0.109591	−	0.189817i	−0.915605	+	0.402080i	$0.868287\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−129.068	−	74.5172i	−4.72234	−	2.72644i
$748$		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
$753$	44.6777	+	77.3840i	1.62814	+	2.82003i
$754$		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
$761$	39.4561	+	22.7800i	1.43028	+	0.825773i	0.997142	−	0.0755541i	$-0.0240725\pi$
$761$	39.4561	+	22.7800i	1.43028	+	0.825773i	0.433139	+	0.901327i	$0.357406\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			55.4553i			1.99977i	0.0151869	+	0.999885i	$0.495166\pi$
$769$			55.4553i			1.99977i	−0.0151869	+	0.999885i	$0.504834\pi$
$770$		0			0
$771$	108.267			3.89914
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.70711	−	9.88500i	0.204478	−	0.354167i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−48.4416	+	27.9678i	−1.72676	+	0.996943i	−0.824310	+	0.566139i	$0.808437\pi$
$787$	−48.4416	+	27.9678i	−1.72676	+	0.996943i	−0.902446	+	0.430804i	$0.858230\pi$
$788$		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			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−88.2778	+	50.9672i	−3.11914	+	1.80084i
$802$		0			0
$803$	−28.9645	−	16.7227i	−1.02214	−	0.590131i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.1213	−	38.3153i	−0.777744	−	1.34709i	−0.933239	−	0.359256i	$-0.883031\pi$
$809$	−22.1213	−	38.3153i	−0.777744	−	1.34709i	0.155495	−	0.987837i	$-0.450303\pi$
$810$		0			0
$811$			18.8715i			0.662669i	0.943513	+	0.331335i	$0.107499\pi$
$811$			18.8715i			0.662669i	−0.943513	+	0.331335i	$0.892501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	15.8412	+	9.14594i	0.554215	+	0.319976i
$818$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		−	37.8837i		−	1.31894i
$826$		0			0
$827$	54.0000			1.87776			0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000			1.87776			0.938882	+	0.344239i	$0.111863\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−21.2771	+	12.2843i	−0.732822	+	0.423095i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	50.7904	−	87.9715i	1.74312	−	3.01917i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.207935	−	0.120051i	0.00710291	−	0.00410087i	−0.496444	−	0.868069i	$-0.665361\pi$
$857$	0.207935	−	0.120051i	0.00710291	−	0.00410087i	0.503547	+	0.863968i	$0.332028\pi$
$858$		0			0
$859$	43.5938	+	25.1689i	1.48740	+	0.858752i	0.999897	−	0.0143672i	$-0.00457339\pi$
$859$	43.5938	+	25.1689i	1.48740	+	0.858752i	0.487506	+	0.873120i	$0.337907\pi$
$860$		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
$867$		−	18.8715i		−	0.640911i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−114.646	−	66.1910i	−3.88019	−	2.24023i
$874$		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
$881$		−	5.01801i		−	0.169061i	−0.996421	−	0.0845306i	$-0.973061\pi$
$881$		−	5.01801i		−	0.169061i	0.996421	−	0.0845306i	$-0.0269391\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		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
$891$	40.9914	−	70.9991i	1.37326	−	2.37856i
$892$		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
$907$	−5.00000	−	8.66025i	−0.166022	−	0.287559i	0.770996	−	0.636841i	$-0.219759\pi$
$907$	−5.00000	−	8.66025i	−0.166022	−	0.287559i	−0.937018	+	0.349281i	$0.886426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	34.4004	−	19.8611i	1.13849	−	0.657306i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−35.2218	−	61.0060i	−1.16060	−	2.01022i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	52.5794	+	30.3567i	1.72507	+	0.995971i	0.907373	+	0.420327i	$0.138085\pi$
$929$	52.5794	+	30.3567i	1.72507	+	0.995971i	0.817700	+	0.575644i	$0.195249\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			34.0250i			1.11155i	0.831333	+	0.555775i	$0.187578\pi$
$937$			34.0250i			1.11155i	−0.831333	+	0.555775i	$0.812422\pi$
$938$		0			0
$939$	−118.870			−3.87918
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.39340	−	14.5378i	0.272749	−	0.472415i	−0.696816	−	0.717250i	$-0.745401\pi$
$947$	8.39340	−	14.5378i	0.272749	−	0.472415i	0.969565	+	0.244835i	$0.0787339\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.2548			1.46595			0.732974	−	0.680257i	$-0.238132\pi$
$953$	45.2548			1.46595			0.732974	+	0.680257i	$0.238132\pi$
$954$		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
$963$	25.2426	+	43.7215i	0.813433	+	1.40891i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−13.8332	+	7.98663i	−0.444388	+	0.256567i
$970$		0			0
$971$	−32.8944	−	18.9916i	−1.05563	−	0.609469i	−0.131411	−	0.991328i	$-0.541951\pi$
$971$	−32.8944	−	18.9916i	−1.05563	−	0.609469i	−0.924221	+	0.381859i	$0.875284\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.8787	+	34.4309i	0.635975	+	1.10154i	0.986308	+	0.164916i	$0.0527353\pi$
$977$	19.8787	+	34.4309i	0.635975	+	1.10154i	−0.350332	+	0.936625i	$0.613931\pi$
$978$		0			0
$979$		−	27.1686i		−	0.868312i
$980$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$			1.29996i			0.0412531i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.q.f.1391.1		8
4.3	odd	2		392.2.m.d.19.2		8
7.2	even	3		1568.2.e.a.783.1		4
7.3	odd	6	inner	1568.2.q.f.815.1		8
7.4	even	3	inner	1568.2.q.f.815.4		8
7.5	odd	6		1568.2.e.a.783.4		4
7.6	odd	2	inner	1568.2.q.f.1391.4		8
8.3	odd	2	CM	1568.2.q.f.1391.1		8
8.5	even	2		392.2.m.d.19.2		8
28.3	even	6		392.2.m.d.227.2		8
28.11	odd	6		392.2.m.d.227.1		8
28.19	even	6		392.2.e.a.195.3	✓	4
28.23	odd	6		392.2.e.a.195.4	yes	4
28.27	even	2		392.2.m.d.19.1		8
56.3	even	6	inner	1568.2.q.f.815.1		8
56.5	odd	6		392.2.e.a.195.3	✓	4
56.11	odd	6	inner	1568.2.q.f.815.4		8
56.13	odd	2		392.2.m.d.19.1		8
56.19	even	6		1568.2.e.a.783.4		4
56.27	even	2	inner	1568.2.q.f.1391.4		8
56.37	even	6		392.2.e.a.195.4	yes	4
56.45	odd	6		392.2.m.d.227.2		8
56.51	odd	6		1568.2.e.a.783.1		4
56.53	even	6		392.2.m.d.227.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.e.a.195.3	✓	4	28.19	even	6
392.2.e.a.195.3	✓	4	56.5	odd	6
392.2.e.a.195.4	yes	4	28.23	odd	6
392.2.e.a.195.4	yes	4	56.37	even	6
392.2.m.d.19.1		8	28.27	even	2
392.2.m.d.19.1		8	56.13	odd	2
392.2.m.d.19.2		8	4.3	odd	2
392.2.m.d.19.2		8	8.5	even	2
392.2.m.d.227.1		8	28.11	odd	6
392.2.m.d.227.1		8	56.53	even	6
392.2.m.d.227.2		8	28.3	even	6
392.2.m.d.227.2		8	56.45	odd	6
1568.2.e.a.783.1		4	7.2	even	3
1568.2.e.a.783.1		4	56.51	odd	6
1568.2.e.a.783.4		4	7.5	odd	6
1568.2.e.a.783.4		4	56.19	even	6
1568.2.q.f.815.1		8	7.3	odd	6	inner
1568.2.q.f.815.1		8	56.3	even	6	inner
1568.2.q.f.815.4		8	7.4	even	3	inner
1568.2.q.f.815.4		8	56.11	odd	6	inner
1568.2.q.f.1391.1		8	1.1	even	1	trivial
1568.2.q.f.1391.1		8	8.3	odd	2	CM
1568.2.q.f.1391.4		8	7.6	odd	2	inner
1568.2.q.f.1391.4		8	56.27	even	2	inner

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.92586 + 1.68925i) q^{3} +(4.20711 - 7.28692i) q^{9} +O(q^{10})\)	\(q+(-2.92586 + 1.68925i) q^{3} +(4.20711 - 7.28692i) q^{9} +(1.12132 + 1.94218i) q^{11} +(-2.92586 + 1.68925i) q^{17} +(-1.21193 - 0.699709i) q^{19} +(2.50000 + 4.33013i) q^{25} +18.2919i q^{27} +(-6.56165 - 3.78837i) q^{33} +8.15640i q^{41} -13.0711 q^{43} +(5.70711 - 9.88500i) q^{51} +4.72792 q^{57} +(9.48751 - 5.47762i) q^{59} +(-4.24264 - 7.34847i) q^{67} +(-12.9154 + 7.45669i) q^{73} +(-14.6293 - 8.44623i) q^{75} +(-18.2782 - 31.6587i) q^{81} -17.7122i q^{83} +(-10.4915 - 6.05728i) q^{89} -15.7331i q^{97} +18.8701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 28 q^{9}+O(q^{10}) \) 8 * q + 28 * q^9	\( 8 q + 28 q^{9} - 8 q^{11} + 20 q^{25} - 48 q^{43} + 40 q^{51} - 64 q^{57} - 84 q^{81} - 64 q^{99}+O(q^{100}) \) 8 * q + 28 * q^9 - 8 * q^11 + 20 * q^25 - 48 * q^43 + 40 * q^51 - 64 * q^57 - 84 * q^81 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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