LMFDB - Embedded newform 2376.2.q.c.1585.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(793,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.793"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.9724555203$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1585.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2376.1585
Dual form			2376.2.q.c.793.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{11} +(-1.00000 - 1.73205i) q^{13} -6.00000 q^{17} +6.00000 q^{19} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +(5.00000 - 8.66025i) q^{29} +2.00000 q^{37} +(6.00000 + 10.3923i) q^{41} +(4.00000 - 6.92820i) q^{43} +(0.500000 - 0.866025i) q^{47} +(3.50000 + 6.06218i) q^{49} +9.00000 q^{53} -1.00000 q^{55} +(-4.00000 + 6.92820i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-6.50000 - 11.2583i) q^{67} -2.00000 q^{73} +(7.00000 - 12.1244i) q^{79} +(3.00000 - 5.19615i) q^{83} +(-3.00000 - 5.19615i) q^{85} +13.0000 q^{89} +(3.00000 + 5.19615i) q^{95} +(-5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - q^{11} - 2 q^{13} - 12 q^{17} + 12 q^{19} - 7 q^{23} + 4 q^{25} + 10 q^{29} + 4 q^{37} + 12 q^{41} + 8 q^{43} + q^{47} + 7 q^{49} + 18 q^{53} - 2 q^{55} - 8 q^{61} + 2 q^{65} - 13 q^{67}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q + q^5 - q^11 - 2 * q^13 - 12 * q^17 + 12 * q^19 - 7 * q^23 + 4 * q^25 + 10 * q^29 + 4 * q^37 + 12 * q^41 + 8 * q^43 + q^47 + 7 * q^49 + 18 * q^53 - 2 * q^55 - 8 * q^61 + 2 * q^65 - 13 * q^67 - 4 * q^73 + 14 * q^79 + 6 * q^83 - 6 * q^85 + 26 * q^89 + 6 * q^95 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2376\mathbb{Z}\right)^\times$.

$n$	$353$	$1189$	$1729$	$1783$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	−0.732294	+	0.680989i	$0.761550\pi$
$6$		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
$9$		0			0
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	−0.956967	−	0.290196i	$-0.906280\pi$
$23$	−3.50000	−	6.06218i	−0.729800	−	1.26405i	0.227167	−	0.973856i	$-0.427054\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.142605	−	0.989780i	$-0.454452\pi$
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.785872	−	0.618389i	$-0.212214\pi$
$30$		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
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.00000	+	10.3923i	0.937043	+	1.62301i	0.770950	+	0.636895i	$0.219782\pi$
$41$	6.00000	+	10.3923i	0.937043	+	1.62301i	0.166092	+	0.986110i	$0.446885\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	−0.827253	−	0.561830i	$-0.810098\pi$
$47$	0.500000	−	0.866025i	0.0729325	−	0.126323i	0.900185	+	0.435507i	$0.143431\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	−	1.73205i	0.124035	−	0.214834i
$66$		0			0
$67$	−6.50000	−	11.2583i	−0.794101	−	1.37542i	−0.923408	−	0.383819i	$-0.874609\pi$
$67$	−6.50000	−	11.2583i	−0.794101	−	1.37542i	0.129307	−	0.991605i	$-0.458725\pi$
$68$		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
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	−0.139895	−	0.990166i	$-0.544677\pi$
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	0.927457	−	0.373930i	$-0.121990\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.0000			1.37800			0.688999	−	0.724763i	$-0.258051\pi$
$89$	13.0000			1.37800			0.688999	+	0.724763i	$0.258051\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.00000	+	5.19615i	0.307794	+	0.533114i
$96$		0			0
$97$	−5.00000	+	8.66025i	−0.507673	+	0.879316i	0.492287	+	0.870433i	$0.336161\pi$
$97$	−5.00000	+	8.66025i	−0.507673	+	0.879316i	−0.999961	+	0.00888289i	$0.997172\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	0.500000	+	0.866025i	0.0492665	+	0.0853320i	0.889607	−	0.456727i	$-0.150978\pi$
$103$	0.500000	+	0.866025i	0.0492665	+	0.0853320i	−0.840341	+	0.542059i	$0.817645\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	3.50000	−	6.06218i	0.326377	−	0.565301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	1.00000	+	1.73205i	0.0848189	+	0.146911i	0.905314	−	0.424743i	$-0.139635\pi$
$139$	1.00000	+	1.73205i	0.0848189	+	0.146911i	−0.820495	+	0.571654i	$0.806302\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000			0.167248
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.0000	+	20.7846i	0.983078	+	1.70274i	0.650183	+	0.759778i	$0.274692\pi$
$149$	12.0000	+	20.7846i	0.983078	+	1.70274i	0.332896	+	0.942964i	$0.391974\pi$
$150$		0			0
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	−0.511636	−	0.859202i	$-0.670960\pi$
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	0.999909	+	0.0134886i	$0.00429367\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.50000	−	11.2583i	−0.518756	−	0.898513i	−0.999762	−	0.0217953i	$-0.993062\pi$
$157$	−6.50000	−	11.2583i	−0.518756	−	0.898513i	0.481006	−	0.876717i	$-0.340272\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−15.0000			−1.17489			−0.587445	−	0.809264i	$-0.699866\pi$
$163$	−15.0000			−1.17489			−0.587445	+	0.809264i	$0.699866\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.00000	+	8.66025i	0.386912	+	0.670151i	0.992032	−	0.125983i	$-0.0402085\pi$
$167$	5.00000	+	8.66025i	0.386912	+	0.670151i	−0.605121	+	0.796134i	$0.706875\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	−0.729155	−	0.684349i	$-0.760087\pi$
$173$	3.00000	−	5.19615i	0.228086	−	0.395056i	0.957241	+	0.289292i	$0.0934200\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$	−21.0000			−1.56092			−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000			−1.56092			−0.780459	+	0.625207i	$0.785014\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	+	1.73205i	0.0735215	+	0.127343i
$186$		0			0
$187$	3.00000	−	5.19615i	0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.5000	−	21.6506i	0.904468	−	1.56658i	0.0828388	−	0.996563i	$-0.473601\pi$
$191$	12.5000	−	21.6506i	0.904468	−	1.56658i	0.821629	−	0.570022i	$-0.193065\pi$
$192$		0			0
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	−0.995948	−	0.0899262i	$-0.971337\pi$
$193$	−8.00000	−	13.8564i	−0.575853	−	0.997406i	0.420096	−	0.907480i	$-0.361996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−7.00000			−0.496217			−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000			−0.496217			−0.248108	+	0.968732i	$0.579809\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.00000	+	10.3923i	−0.419058	+	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.00000	+	5.19615i	−0.207514	+	0.359425i
$210$		0			0
$211$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$211$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	7.50000	−	12.9904i	0.502237	−	0.869900i	−0.497760	−	0.867315i	$-0.665844\pi$
$223$	7.50000	−	12.9904i	0.502237	−	0.869900i	0.999997	−	0.00258516i	$-0.000822884\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	−0.534577	−	0.845120i	$-0.679529\pi$
$227$	7.00000	−	12.1244i	0.464606	−	0.804722i	0.999184	+	0.0403978i	$0.0128625\pi$
$228$		0			0
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	0.996832	−	0.0795401i	$-0.0253452\pi$
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	−0.567300	+	0.823511i	$0.692012\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$	1.00000			0.0652328
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	0.707166	−	0.707048i	$-0.249973\pi$
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	−0.965904	+	0.258900i	$0.916640\pi$
$240$		0			0
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	0.0546547	+	0.998505i	$0.482594\pi$
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	−0.892058	+	0.451920i	$0.850739\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.50000	+	6.06218i	−0.223607	+	0.387298i
$246$		0			0
$247$	−6.00000	−	10.3923i	−0.381771	−	0.661247i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.00000			0.568075			0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000			0.568075			0.284037	+	0.958813i	$0.408326\pi$
$252$		0			0
$253$	7.00000			0.440086
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	4.50000	+	7.79423i	0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.0000			−1.40233			−0.701167	−	0.712997i	$-0.747337\pi$
$269$	−23.0000			−1.40233			−0.701167	+	0.712997i	$0.747337\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	−0.941966	−	0.335707i	$-0.891025\pi$
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	0.761714	+	0.647913i	$0.224358\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.00000	+	3.46410i	−0.119310	+	0.206651i	−0.919494	−	0.393103i	$-0.871402\pi$
$281$	−2.00000	+	3.46410i	−0.119310	+	0.206651i	0.800184	+	0.599754i	$0.204735\pi$
$282$		0			0
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	0.762978	−	0.646425i	$-0.223737\pi$
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	−0.941309	+	0.337546i	$0.890403\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.0000	+	25.9808i	0.876309	+	1.51781i	0.855361	+	0.518032i	$0.173335\pi$
$293$	15.0000	+	25.9808i	0.876309	+	1.51781i	0.0209480	+	0.999781i	$0.493332\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.00000	+	12.1244i	−0.404820	+	0.701170i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000			−0.458079
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−16.5000	−	28.5788i	−0.935629	−	1.62056i	−0.773508	−	0.633786i	$-0.781500\pi$
$311$	−16.5000	−	28.5788i	−0.935629	−	1.62056i	−0.162121	−	0.986771i	$-0.551833\pi$
$312$		0			0
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	6.50000	−	11.2583i	0.367402	−	0.636358i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.50000	−	14.7224i	0.477408	−	0.826894i	−0.522257	−	0.852788i	$-0.674910\pi$
$317$	8.50000	−	14.7224i	0.477408	−	0.826894i	0.999665	+	0.0258939i	$0.00824321\pi$
$318$		0			0
$319$	5.00000	+	8.66025i	0.279946	+	0.484881i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.0000			−2.00309
$324$		0			0
$325$	−8.00000			−0.443760
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−17.5000	+	30.3109i	−0.961887	+	1.66604i	−0.244131	+	0.969742i	$0.578503\pi$
$331$	−17.5000	+	30.3109i	−0.961887	+	1.66604i	−0.717756	+	0.696295i	$0.754831\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.50000	−	11.2583i	0.355133	−	0.615108i
$336$		0			0
$337$	8.00000	+	13.8564i	0.435788	+	0.754807i	0.997360	−	0.0726214i	$-0.0231365\pi$
$337$	8.00000	+	13.8564i	0.435788	+	0.754807i	−0.561572	+	0.827428i	$0.689803\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	14.0000	−	24.2487i	0.749403	−	1.29800i	−0.198706	−	0.980059i	$-0.563674\pi$
$349$	14.0000	−	24.2487i	0.749403	−	1.29800i	0.948109	−	0.317945i	$-0.102993\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.500000	+	0.866025i	−0.0266123	+	0.0460939i	−0.879025	−	0.476776i	$-0.841805\pi$
$353$	−0.500000	+	0.866025i	−0.0266123	+	0.0460939i	0.852413	+	0.522870i	$0.175139\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.0000			−1.37223			−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000			−1.37223			−0.686114	+	0.727494i	$0.740685\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.00000	−	1.73205i	−0.0523424	−	0.0906597i
$366$		0			0
$367$	0.500000	−	0.866025i	0.0260998	−	0.0452062i	−0.852680	−	0.522433i	$-0.825025\pi$
$367$	0.500000	−	0.866025i	0.0260998	−	0.0452062i	0.878780	+	0.477227i	$0.158358\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	0.809591	−	0.586994i	$-0.199689\pi$
$373$	−2.00000	−	3.46410i	−0.103556	−	0.179364i	−0.913147	+	0.407630i	$0.866355\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0000			−1.03005
$378$		0			0
$379$	−19.0000			−0.975964			−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000			−0.975964			−0.487982	+	0.872854i	$0.662267\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.5000	+	19.9186i	0.587623	+	1.01779i	0.994543	+	0.104328i	$0.0332693\pi$
$383$	11.5000	+	19.9186i	0.587623	+	1.01779i	−0.406920	+	0.913464i	$0.633397\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	−0.998763	−	0.0497253i	$-0.984165\pi$
$389$	−9.00000	+	15.5885i	−0.456318	+	0.790366i	0.542445	+	0.840091i	$0.317499\pi$
$390$		0			0
$391$	21.0000	+	36.3731i	1.06202	+	1.83947i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.0000			0.704416
$396$		0			0
$397$	−11.0000			−0.552074			−0.276037	−	0.961147i	$-0.589021\pi$
$397$	−11.0000			−0.552074			−0.276037	+	0.961147i	$0.589021\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.50000	−	9.52628i	−0.274657	−	0.475720i	0.695392	−	0.718631i	$-0.255231\pi$
$401$	−5.50000	−	9.52628i	−0.274657	−	0.475720i	−0.970049	+	0.242911i	$0.921898\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.00000	+	1.73205i	−0.0495682	+	0.0858546i
$408$		0			0
$409$	4.00000	+	6.92820i	0.197787	+	0.342578i	0.947811	−	0.318834i	$-0.103291\pi$
$409$	4.00000	+	6.92820i	0.197787	+	0.342578i	−0.750023	+	0.661411i	$0.769958\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.50000	−	16.4545i	−0.464105	−	0.803854i	0.535055	−	0.844817i	$-0.320291\pi$
$419$	−9.50000	−	16.4545i	−0.464105	−	0.803854i	−0.999161	+	0.0409630i	$0.986957\pi$
$420$		0			0
$421$	5.50000	−	9.52628i	0.268054	−	0.464282i	−0.700306	−	0.713843i	$-0.746953\pi$
$421$	5.50000	−	9.52628i	0.268054	−	0.464282i	0.968359	+	0.249561i	$0.0802862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000	+	20.7846i	−0.582086	+	1.00820i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.0000			−0.963366			−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000			−0.963366			−0.481683	+	0.876346i	$0.659974\pi$
$432$		0			0
$433$	1.00000			0.0480569			0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000			0.0480569			0.0240285	+	0.999711i	$0.492351\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0000	−	36.3731i	−1.00457	−	1.73996i
$438$		0			0
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	−0.721686	−	0.692220i	$-0.756633\pi$
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	0.960323	+	0.278889i	$0.0899661\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.0000	+	17.3205i	−0.475114	+	0.822922i	−0.999594	−	0.0285009i	$-0.990927\pi$
$443$	−10.0000	+	17.3205i	−0.475114	+	0.822922i	0.524479	+	0.851423i	$0.324260\pi$
$444$		0			0
$445$	6.50000	+	11.2583i	0.308130	+	0.533696i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	0.127469	+	0.991843i	$0.459315\pi$
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	−0.922695	+	0.385530i	$0.874019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	−0.787668	−	0.616100i	$-0.788712\pi$
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	0.927392	+	0.374091i	$0.122045\pi$
$462$		0			0
$463$	20.5000	+	35.5070i	0.952716	+	1.65015i	0.739511	+	0.673145i	$0.235057\pi$
$463$	20.5000	+	35.5070i	0.952716	+	1.65015i	0.213205	+	0.977007i	$0.431610\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.00000			−0.0462745			−0.0231372	−	0.999732i	$-0.507365\pi$
$467$	−1.00000			−0.0462745			−0.0231372	+	0.999732i	$0.507365\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	12.0000	−	20.7846i	0.550598	−	0.953663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	0.399704	+	0.916644i	$0.369113\pi$
$479$	−13.0000	+	22.5167i	−0.593985	+	1.02881i	−0.993689	+	0.112168i	$0.964220\pi$
$480$		0			0
$481$	−2.00000	−	3.46410i	−0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.0000			−0.454077
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.00000	+	8.66025i	0.225647	+	0.390832i	0.956513	−	0.291689i	$-0.0942171\pi$
$491$	5.00000	+	8.66025i	0.225647	+	0.390832i	−0.730866	+	0.682520i	$0.760884\pi$
$492$		0			0
$493$	−30.0000	+	51.9615i	−1.35113	+	2.34023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	0.877000	−	0.480490i	$-0.159541\pi$
$499$	0.500000	+	0.866025i	0.0223831	+	0.0387686i	−0.854617	+	0.519259i	$0.826208\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.00000			−0.267527			−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000			−0.267527			−0.133763	+	0.991013i	$0.542706\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.5000	+	23.3827i	0.598377	+	1.03642i	0.993061	+	0.117602i	$0.0375208\pi$
$509$	13.5000	+	23.3827i	0.598377	+	1.03642i	−0.394684	+	0.918817i	$0.629146\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.500000	+	0.866025i	−0.0220326	+	0.0381616i
$516$		0			0
$517$	0.500000	+	0.866025i	0.0219900	+	0.0380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$		0			0
$523$	−38.0000			−1.66162			−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000			−1.66162			−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.0000	+	22.5167i	−0.565217	+	0.978985i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.0000	−	20.7846i	0.519778	−	0.900281i
$534$		0			0
$535$	−4.00000	−	6.92820i	−0.172935	−	0.299532i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.00000			−0.301511
$540$		0			0
$541$	26.0000			1.11783			0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000			1.11783			0.558914	+	0.829226i	$0.311218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.00000	+	1.73205i	0.0428353	+	0.0741929i
$546$		0			0
$547$	−18.0000	+	31.1769i	−0.769624	+	1.33303i	0.168142	+	0.985763i	$0.446223\pi$
$547$	−18.0000	+	31.1769i	−0.769624	+	1.33303i	−0.937767	+	0.347266i	$0.887110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	30.0000	−	51.9615i	1.27804	−	2.21364i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.0000			0.593199			0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000			0.593199			0.296600	+	0.955002i	$0.404147\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	0.999978	+	0.00664037i	$0.00211371\pi$
$563$	12.0000	+	20.7846i	0.505740	+	0.875967i	−0.494238	+	0.869326i	$0.664553\pi$
$564$		0			0
$565$	3.00000	−	5.19615i	0.126211	−	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.00000	+	12.1244i	−0.293455	+	0.508279i	−0.974624	−	0.223847i	$-0.928139\pi$
$569$	−7.00000	+	12.1244i	−0.293455	+	0.508279i	0.681169	+	0.732126i	$0.261472\pi$
$570$		0			0
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	0.963827	−	0.266529i	$-0.0858769\pi$
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	−0.712735	+	0.701434i	$0.752544\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.0000			−1.16768
$576$		0			0
$577$	25.0000			1.04076			0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000			1.04076			0.520382	+	0.853934i	$0.325790\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.50000	+	7.79423i	−0.186371	+	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.50000	−	2.59808i	0.0619116	−	0.107234i	−0.833408	−	0.552658i	$-0.813614\pi$
$587$	1.50000	−	2.59808i	0.0619116	−	0.107234i	0.895320	+	0.445424i	$0.146947\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.00000			−0.164260			−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000			−0.164260			−0.0821302	+	0.996622i	$0.526172\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	8.00000	−	13.8564i	0.326327	−	0.565215i	−0.655453	−	0.755236i	$-0.727522\pi$
$601$	8.00000	−	13.8564i	0.326327	−	0.565215i	0.981780	+	0.190021i	$0.0608557\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.500000	−	0.866025i	0.0203279	−	0.0352089i
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.00000			−0.0809113
$612$		0			0
$613$	−8.00000			−0.323117			−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000			−0.323117			−0.161558	+	0.986863i	$0.551652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.50000	+	14.7224i	0.342197	+	0.592703i	0.984840	−	0.173463i	$-0.0554956\pi$
$617$	8.50000	+	14.7224i	0.342197	+	0.592703i	−0.642643	+	0.766165i	$0.722162\pi$
$618$		0			0
$619$	18.0000	−	31.1769i	0.723481	−	1.25311i	−0.236115	−	0.971725i	$-0.575874\pi$
$619$	18.0000	−	31.1769i	0.723481	−	1.25311i	0.959596	−	0.281381i	$-0.0907924\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	35.0000			1.39333			0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000			1.39333			0.696664	+	0.717398i	$0.254667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.00000	+	10.3923i	0.238103	+	0.412406i
$636$		0			0
$637$	7.00000	−	12.1244i	0.277350	−	0.480384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.5000	−	19.9186i	0.454223	−	0.786737i	−0.544420	−	0.838812i	$-0.683250\pi$
$641$	11.5000	−	19.9186i	0.454223	−	0.786737i	0.998643	+	0.0520757i	$0.0165837\pi$
$642$		0			0
$643$	12.5000	+	21.6506i	0.492952	+	0.853818i	0.999967	−	0.00811944i	$-0.00258453\pi$
$643$	12.5000	+	21.6506i	0.492952	+	0.853818i	−0.507015	+	0.861937i	$0.669251\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000			0.353827			0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000			0.353827			0.176913	+	0.984226i	$0.443389\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	0.947118	−	0.320884i	$-0.103980\pi$
$653$	5.00000	+	8.66025i	0.195665	+	0.338902i	−0.751453	+	0.659786i	$0.770647\pi$
$654$		0			0
$655$	−3.00000	+	5.19615i	−0.117220	+	0.203030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	−23.0000	−	39.8372i	−0.894596	−	1.54949i	−0.834303	−	0.551306i	$-0.814130\pi$
$661$	−23.0000	−	39.8372i	−0.894596	−	1.54949i	−0.0602929	−	0.998181i	$-0.519203\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−70.0000			−2.71041
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.00000	−	6.92820i	−0.154418	−	0.267460i
$672$		0			0
$673$	−25.0000	+	43.3013i	−0.963679	+	1.66914i	−0.250557	+	0.968102i	$0.580614\pi$
$673$	−25.0000	+	43.3013i	−0.963679	+	1.66914i	−0.713123	+	0.701039i	$0.752720\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	0.375467	+	0.926836i	$0.377482\pi$
$677$	−16.0000	+	27.7128i	−0.614930	+	1.06509i	−0.990397	+	0.138254i	$0.955851\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.00000			−0.0382639			−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000			−0.0382639			−0.0191320	+	0.999817i	$0.506090\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.00000	−	15.5885i	−0.342873	−	0.593873i
$690$		0			0
$691$	−16.5000	+	28.5788i	−0.627690	+	1.08719i	0.360325	+	0.932827i	$0.382666\pi$
$691$	−16.5000	+	28.5788i	−0.627690	+	1.08719i	−0.988014	+	0.154363i	$0.950667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.00000	+	1.73205i	−0.0379322	+	0.0657004i
$696$		0			0
$697$	−36.0000	−	62.3538i	−1.36360	−	2.36182i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$		0			0
$703$	12.0000			0.452589
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.50000	+	9.52628i	−0.206557	+	0.357767i	−0.950628	−	0.310334i	$-0.899559\pi$
$709$	−5.50000	+	9.52628i	−0.206557	+	0.357767i	0.744071	+	0.668101i	$0.232892\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	1.00000	+	1.73205i	0.0373979	+	0.0647750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.00000			0.0372937			0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000			0.0372937			0.0186469	+	0.999826i	$0.494064\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−20.0000	−	34.6410i	−0.742781	−	1.28654i
$726$		0			0
$727$	6.00000	−	10.3923i	0.222528	−	0.385429i	−0.733047	−	0.680178i	$-0.761903\pi$
$727$	6.00000	−	10.3923i	0.222528	−	0.385429i	0.955575	+	0.294749i	$0.0952359\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−24.0000	+	41.5692i	−0.887672	+	1.53749i
$732$		0			0
$733$	16.0000	+	27.7128i	0.590973	+	1.02360i	0.994102	+	0.108453i	$0.0345896\pi$
$733$	16.0000	+	27.7128i	0.590973	+	1.02360i	−0.403128	+	0.915144i	$0.632077\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.0000			0.478861
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	0.988797	+	0.149270i	$0.0476922\pi$
$743$	17.0000	+	29.4449i	0.623670	+	1.08023i	−0.365127	+	0.930958i	$0.618974\pi$
$744$		0			0
$745$	−12.0000	+	20.7846i	−0.439646	+	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.855938	−	0.517079i	$-0.827019\pi$
$751$	−24.0000	−	41.5692i	−0.875772	−	1.51688i	−0.0198348	−	0.999803i	$-0.506314\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−31.0000			−1.12671			−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000			−1.12671			−0.563357	+	0.826214i	$0.690490\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.0000	−	17.3205i	−0.362500	−	0.627868i	0.625872	−	0.779926i	$-0.284743\pi$
$761$	−10.0000	−	17.3205i	−0.362500	−	0.627868i	−0.988372	+	0.152058i	$0.951410\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	8.00000	+	13.8564i	0.288487	+	0.499675i	0.973449	−	0.228904i	$-0.0735143\pi$
$769$	8.00000	+	13.8564i	0.288487	+	0.499675i	−0.684962	+	0.728579i	$0.740181\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36.0000	+	62.3538i	1.28983	+	2.23406i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.50000	−	11.2583i	0.231995	−	0.401827i
$786$		0			0
$787$	24.0000	+	41.5692i	0.855508	+	1.48178i	0.876173	+	0.481996i	$0.160088\pi$
$787$	24.0000	+	41.5692i	0.855508	+	1.48178i	−0.0206657	+	0.999786i	$0.506579\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	16.0000			0.568177
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	0.661441	−	0.749997i	$-0.269945\pi$
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	−0.980237	+	0.197826i	$0.936612\pi$
$798$		0			0
$799$	−3.00000	+	5.19615i	−0.106132	+	0.183827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.00000	−	1.73205i	0.0352892	−	0.0611227i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.50000	−	12.9904i	−0.262714	−	0.455033i
$816$		0			0
$817$	24.0000	−	41.5692i	0.839654	−	1.45432i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.00000	−	13.8564i	0.279202	−	0.483592i	−0.691985	−	0.721912i	$-0.743263\pi$
$821$	8.00000	−	13.8564i	0.279202	−	0.483592i	0.971187	+	0.238320i	$0.0765968\pi$
$822$		0			0
$823$	25.5000	+	44.1673i	0.888874	+	1.53958i	0.841208	+	0.540712i	$0.181845\pi$
$823$	25.5000	+	44.1673i	0.888874	+	1.53958i	0.0476662	+	0.998863i	$0.484822\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.00000			−0.208640			−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000			−0.208640			−0.104320	+	0.994544i	$0.533267\pi$
$828$		0			0
$829$	21.0000			0.729360			0.364680	−	0.931133i	$-0.381178\pi$
$829$	21.0000			0.729360			0.364680	+	0.931133i	$0.381178\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.0000	−	36.3731i	−0.727607	−	1.26025i
$834$		0			0
$835$	−5.00000	+	8.66025i	−0.173032	+	0.299700i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.5000	−	18.1865i	0.362500	−	0.627869i	−0.625871	−	0.779926i	$-0.715257\pi$
$839$	10.5000	−	18.1865i	0.362500	−	0.627869i	0.988372	+	0.152057i	$0.0485899\pi$
$840$		0			0
$841$	−35.5000	−	61.4878i	−1.22414	−	2.12027i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.00000			0.309609
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.00000	−	12.1244i	−0.239957	−	0.415618i
$852$		0			0
$853$	8.00000	−	13.8564i	0.273915	−	0.474434i	−0.695946	−	0.718094i	$-0.745015\pi$
$853$	8.00000	−	13.8564i	0.273915	−	0.474434i	0.969861	+	0.243660i	$0.0783480\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	−0.584967	−	0.811057i	$-0.698893\pi$
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	0.994880	+	0.101068i	$0.0322260\pi$
$858$		0			0
$859$	−14.5000	−	25.1147i	−0.494734	−	0.856904i	0.505248	−	0.862974i	$-0.331401\pi$
$859$	−14.5000	−	25.1147i	−0.494734	−	0.856904i	−0.999982	+	0.00607046i	$0.998068\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	52.0000			1.77010			0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000			1.77010			0.885050	+	0.465495i	$0.154124\pi$
$864$		0			0
$865$	6.00000			0.204006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.00000	+	12.1244i	0.237459	+	0.411291i
$870$		0			0
$871$	−13.0000	+	22.5167i	−0.440488	+	0.762948i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	16.0000	+	27.7128i	0.540282	+	0.935795i	0.998888	+	0.0471555i	$0.0150156\pi$
$877$	16.0000	+	27.7128i	0.540282	+	0.935795i	−0.458606	+	0.888640i	$0.651651\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	31.0000			1.04442			0.522208	−	0.852818i	$-0.325108\pi$
$881$	31.0000			1.04442			0.522208	+	0.852818i	$0.325108\pi$
$882$		0			0
$883$	−29.0000			−0.975928			−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000			−0.975928			−0.487964	+	0.872864i	$0.662260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.00000	−	1.73205i	−0.0335767	−	0.0581566i	0.848749	−	0.528796i	$-0.177356\pi$
$887$	−1.00000	−	1.73205i	−0.0335767	−	0.0581566i	−0.882325	+	0.470640i	$0.844023\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.00000	−	5.19615i	0.100391	−	0.173883i
$894$		0			0
$895$	−4.50000	−	7.79423i	−0.150418	−	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−54.0000			−1.79900
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.5000	−	18.1865i	−0.349032	−	0.604541i
$906$		0			0
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	−0.226173	−	0.974087i	$-0.572621\pi$
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	0.956671	−	0.291172i	$-0.0940453\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14.5000	+	25.1147i	−0.480406	+	0.832088i	−0.999747	−	0.0224788i	$-0.992844\pi$
$911$	−14.5000	+	25.1147i	−0.480406	+	0.832088i	0.519341	+	0.854567i	$0.326178\pi$
$912$		0			0
$913$	3.00000	+	5.19615i	0.0992855	+	0.171968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.00000	−	6.92820i	0.131519	−	0.227798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.5000	+	44.1673i	−0.836628	+	1.44908i	0.0560703	+	0.998427i	$0.482143\pi$
$929$	−25.5000	+	44.1673i	−0.836628	+	1.44908i	−0.892698	+	0.450655i	$0.851190\pi$
$930$		0			0
$931$	21.0000	+	36.3731i	0.688247	+	1.19208i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.00000			0.196221
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	0.989599	+	0.143856i	$0.0459502\pi$
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	−0.370216	+	0.928946i	$0.620716\pi$
$942$		0			0
$943$	42.0000	−	72.7461i	1.36771	−	2.36894i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.50000	+	9.52628i	−0.178726	+	0.309562i	−0.941444	−	0.337168i	$-0.890531\pi$
$947$	−5.50000	+	9.52628i	−0.178726	+	0.309562i	0.762718	+	0.646731i	$0.223864\pi$
$948$		0			0
$949$	2.00000	+	3.46410i	0.0649227	+	0.112449i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$	25.0000			0.808981
$956$		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
$963$		0			0
$964$		0			0
$965$	8.00000	−	13.8564i	0.257529	−	0.446054i
$966$		0			0
$967$	−2.00000	−	3.46410i	−0.0643157	−	0.111398i	0.832075	−	0.554664i	$-0.187153\pi$
$967$	−2.00000	−	3.46410i	−0.0643157	−	0.111398i	−0.896390	+	0.443266i	$0.853820\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.0000			−0.673922			−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000			−0.673922			−0.336961	+	0.941519i	$0.609399\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.5000	+	38.9711i	0.719839	+	1.24680i	0.961063	+	0.276328i	$0.0891176\pi$
$977$	22.5000	+	38.9711i	0.719839	+	1.24680i	−0.241225	+	0.970469i	$0.577549\pi$
$978$		0			0
$979$	−6.50000	+	11.2583i	−0.207741	+	0.359818i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.0000	−	27.7128i	0.510321	−	0.883901i	−0.489608	−	0.871943i	$-0.662860\pi$
$983$	16.0000	−	27.7128i	0.510321	−	0.883901i	0.999928	−	0.0119587i	$-0.00380665\pi$
$984$		0			0
$985$	3.00000	+	5.19615i	0.0955879	+	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−56.0000			−1.78070
$990$		0			0
$991$	−56.0000			−1.77890			−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000			−1.77890			−0.889449	+	0.457034i	$0.848912\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.50000	−	6.06218i	−0.110957	−	0.192184i
$996$		0			0
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	−0.637587	−	0.770378i	$-0.720067\pi$
$997$	11.0000	−	19.0526i	0.348373	−	0.603401i	0.985961	+	0.166978i	$0.0534008\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.q.c.1585.1		2
3.2	odd	2		792.2.q.a.529.1	yes	2
9.2	odd	6		7128.2.a.f.1.1		1
9.4	even	3	inner	2376.2.q.c.793.1		2
9.5	odd	6		792.2.q.a.265.1	✓	2
9.7	even	3		7128.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.q.a.265.1	✓	2	9.5	odd	6
792.2.q.a.529.1	yes	2	3.2	odd	2
2376.2.q.c.793.1		2	9.4	even	3	inner
2376.2.q.c.1585.1		2	1.1	even	1	trivial
7128.2.a.d.1.1		1	9.7	even	3
7128.2.a.f.1.1		1	9.2	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{11} +(-1.00000 - 1.73205i) q^{13} -6.00000 q^{17} +6.00000 q^{19} +(-3.50000 - 6.06218i) q^{23} +(2.00000 - 3.46410i) q^{25} +(5.00000 - 8.66025i) q^{29} +2.00000 q^{37} +(6.00000 + 10.3923i) q^{41} +(4.00000 - 6.92820i) q^{43} +(0.500000 - 0.866025i) q^{47} +(3.50000 + 6.06218i) q^{49} +9.00000 q^{53} -1.00000 q^{55} +(-4.00000 + 6.92820i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-6.50000 - 11.2583i) q^{67} -2.00000 q^{73} +(7.00000 - 12.1244i) q^{79} +(3.00000 - 5.19615i) q^{83} +(-3.00000 - 5.19615i) q^{85} +13.0000 q^{89} +(3.00000 + 5.19615i) q^{95} +(-5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - q^{11} - 2 q^{13} - 12 q^{17} + 12 q^{19} - 7 q^{23} + 4 q^{25} + 10 q^{29} + 4 q^{37} + 12 q^{41} + 8 q^{43} + q^{47} + 7 q^{49} + 18 q^{53} - 2 q^{55} - 8 q^{61} + 2 q^{65} - 13 q^{67}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q + q^5 - q^11 - 2 * q^13 - 12 * q^17 + 12 * q^19 - 7 * q^23 + 4 * q^25 + 10 * q^29 + 4 * q^37 + 12 * q^41 + 8 * q^43 + q^47 + 7 * q^49 + 18 * q^53 - 2 * q^55 - 8 * q^61 + 2 * q^65 - 13 * q^67 - 4 * q^73 + 14 * q^79 + 6 * q^83 - 6 * q^85 + 26 * q^89 + 6 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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