LMFDB - Embedded newform 2376.2.q.f.1585.5

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)

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")

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);

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 = [12,0,0,0,4] 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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 7x^{10} - 2x^{9} + 39x^{8} - 9x^{7} + 67x^{6} - 18x^{5} + 88x^{4} - 16x^{3} + 24x^{2} + 4x + 4 $ x^12 + 7x^10 - 2x^9 + 39x^8 - 9x^7 + 67x^6 - 18x^5 + 88x^4 - 16x^3 + 24x^2 + 4x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 792)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1585.5
Root			$0.322589 - 0.558741i$ of defining polynomial
Character	$\chi$	$=$	2376.1585
Dual form			2376.2.q.f.793.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.29187 + 2.23759i) q^{5} +(-0.251298 + 0.435260i) q^{7} +(0.500000 - 0.866025i) q^{11} +(-1.11008 - 1.92272i) q^{13} +2.46350 q^{17} +3.08115 q^{19} +(3.34183 + 5.78822i) q^{23} +(-0.837868 + 1.45123i) q^{25} +(4.14591 - 7.18093i) q^{29} +(-1.35878 - 2.35348i) q^{31} -1.29858 q^{35} +2.51492 q^{37} +(0.578267 + 1.00159i) q^{41} +(-1.58657 + 2.74802i) q^{43} +(-5.10003 + 8.83351i) q^{47} +(3.37370 + 5.84342i) q^{49} +12.4202 q^{53} +2.58374 q^{55} +(3.45322 + 5.98116i) q^{59} +(0.631684 - 1.09411i) q^{61} +(2.86817 - 4.96781i) q^{65} +(2.74005 + 4.74591i) q^{67} -8.68115 q^{71} +5.92397 q^{73} +(0.251298 + 0.435260i) q^{77} +(2.96672 - 5.13851i) q^{79} +(-5.37161 + 9.30390i) q^{83} +(3.18252 + 5.51229i) q^{85} -9.82062 q^{89} +1.11584 q^{91} +(3.98045 + 6.89434i) q^{95} +(-0.971719 + 1.68307i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{5} - 5 q^{7} + 6 q^{11} - 3 q^{13} - 14 q^{17} + 10 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} - 16 q^{35} + 6 q^{37} - 5 q^{43} + 17 q^{47} - 3 q^{49} - 28 q^{53} + 8 q^{55} + 4 q^{59}+ \cdots + 16 q^{97}+O(q^{100}) $ 12 * q + 4 * q^5 - 5 * q^7 + 6 * q^11 - 3 * q^13 - 14 * q^17 + 10 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 - 16 * q^35 + 6 * q^37 - 5 * q^43 + 17 * q^47 - 3 * q^49 - 28 * q^53 + 8 * q^55 + 4 * q^59 - q^61 + 15 * q^65 + 4 * q^67 - 14 * q^71 - 24 * q^73 + 5 * q^77 - q^79 + 22 * q^83 + 3 * q^85 - 44 * q^89 + 22 * q^91 + 24 * q^95 + 16 * 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
$5$	1.29187	+	2.23759i	0.577743	+	1.00068i	0.995738	+	0.0922306i	$0.0293997\pi$
$5$	1.29187	+	2.23759i	0.577743	+	1.00068i	−0.417995	+	0.908449i	$0.637267\pi$
$6$		0			0
$7$	−0.251298	+	0.435260i	−0.0949816	+	0.164513i	−0.909601	−	0.415483i	$-0.863613\pi$
$7$	−0.251298	+	0.435260i	−0.0949816	+	0.164513i	0.814619	+	0.579996i	$0.196946\pi$
$8$		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
$13$	−1.11008	−	1.92272i	−0.307881	−	0.533266i	0.670017	−	0.742345i	$-0.266287\pi$
$13$	−1.11008	−	1.92272i	−0.307881	−	0.533266i	−0.977899	+	0.209079i	$0.932953\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.46350			0.597486			0.298743	−	0.954334i	$-0.403433\pi$
$17$	2.46350			0.597486			0.298743	+	0.954334i	$0.403433\pi$
$18$		0			0
$19$	3.08115			0.706864			0.353432	−	0.935460i	$-0.385015\pi$
$19$	3.08115			0.706864			0.353432	+	0.935460i	$0.385015\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.34183	+	5.78822i	0.696820	+	1.20693i	0.969563	+	0.244841i	$0.0787356\pi$
$23$	3.34183	+	5.78822i	0.696820	+	1.20693i	−0.272743	+	0.962087i	$0.587931\pi$
$24$		0			0
$25$	−0.837868	+	1.45123i	−0.167574	+	0.290246i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.14591	−	7.18093i	0.769877	−	1.33347i	−0.167753	−	0.985829i	$-0.553651\pi$
$29$	4.14591	−	7.18093i	0.769877	−	1.33347i	0.937629	−	0.347636i	$-0.113016\pi$
$30$		0			0
$31$	−1.35878	−	2.35348i	−0.244045	−	0.422698i	0.717818	−	0.696231i	$-0.245141\pi$
$31$	−1.35878	−	2.35348i	−0.244045	−	0.422698i	−0.961863	+	0.273533i	$0.911808\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.29858			−0.219500
$36$		0			0
$37$	2.51492			0.413451			0.206725	−	0.978399i	$-0.433719\pi$
$37$	2.51492			0.413451			0.206725	+	0.978399i	$0.433719\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.578267	+	1.00159i	0.0903101	+	0.156422i	0.907642	−	0.419746i	$-0.137881\pi$
$41$	0.578267	+	1.00159i	0.0903101	+	0.156422i	−0.817332	+	0.576168i	$0.804548\pi$
$42$		0			0
$43$	−1.58657	+	2.74802i	−0.241950	+	0.419069i	−0.961270	−	0.275610i	$-0.911120\pi$
$43$	−1.58657	+	2.74802i	−0.241950	+	0.419069i	0.719320	+	0.694679i	$0.244454\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.10003	+	8.83351i	−0.743915	+	1.28850i	0.206785	+	0.978387i	$0.433700\pi$
$47$	−5.10003	+	8.83351i	−0.743915	+	1.28850i	−0.950700	+	0.310113i	$0.899633\pi$
$48$		0			0
$49$	3.37370	+	5.84342i	0.481957	+	0.834774i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.4202			1.70605			0.853024	−	0.521872i	$-0.174766\pi$
$53$	12.4202			1.70605			0.853024	+	0.521872i	$0.174766\pi$
$54$		0			0
$55$	2.58374			0.348392
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.45322	+	5.98116i	0.449571	+	0.778680i	0.998358	−	0.0572820i	$-0.0182434\pi$
$59$	3.45322	+	5.98116i	0.449571	+	0.778680i	−0.548787	+	0.835962i	$0.684910\pi$
$60$		0			0
$61$	0.631684	−	1.09411i	0.0808788	−	0.140086i	−0.822749	−	0.568405i	$-0.807561\pi$
$61$	0.631684	−	1.09411i	0.0808788	−	0.140086i	0.903628	+	0.428319i	$0.140894\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.86817	−	4.96781i	0.355752	−	0.616181i
$66$		0			0
$67$	2.74005	+	4.74591i	0.334751	+	0.579805i	0.983437	−	0.181251i	$-0.0580146\pi$
$67$	2.74005	+	4.74591i	0.334751	+	0.579805i	−0.648686	+	0.761056i	$0.724681\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.68115			−1.03026			−0.515131	−	0.857111i	$-0.672257\pi$
$71$	−8.68115			−1.03026			−0.515131	+	0.857111i	$0.672257\pi$
$72$		0			0
$73$	5.92397			0.693348			0.346674	−	0.937986i	$-0.387311\pi$
$73$	5.92397			0.693348			0.346674	+	0.937986i	$0.387311\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.251298	+	0.435260i	0.0286380	+	0.0496025i
$78$		0			0
$79$	2.96672	−	5.13851i	0.333782	−	0.578127i	−0.649468	−	0.760389i	$-0.725008\pi$
$79$	2.96672	−	5.13851i	0.333782	−	0.578127i	0.983250	+	0.182262i	$0.0583418\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.37161	+	9.30390i	−0.589610	+	1.02124i	0.404673	+	0.914462i	$0.367385\pi$
$83$	−5.37161	+	9.30390i	−0.589610	+	1.02124i	−0.994283	+	0.106774i	$0.965948\pi$
$84$		0			0
$85$	3.18252	+	5.51229i	0.345193	+	0.597892i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.82062			−1.04098			−0.520492	−	0.853867i	$-0.674251\pi$
$89$	−9.82062			−1.04098			−0.520492	+	0.853867i	$0.674251\pi$
$90$		0			0
$91$	1.11584			0.116972
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.98045	+	6.89434i	0.408386	+	0.707345i
$96$		0			0
$97$	−0.971719	+	1.68307i	−0.0986631	+	0.170890i	−0.911132	−	0.412116i	$-0.864790\pi$
$97$	−0.971719	+	1.68307i	−0.0986631	+	0.170890i	0.812468	+	0.583005i	$0.198123\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.74863	+	11.6890i	−0.671514	+	1.16310i	0.305961	+	0.952044i	$0.401022\pi$
$101$	−6.74863	+	11.6890i	−0.671514	+	1.16310i	−0.977475	+	0.211052i	$0.932311\pi$
$102$		0			0
$103$	−8.54821	−	14.8059i	−0.842280	−	1.45887i	−0.887963	−	0.459915i	$-0.847880\pi$
$103$	−8.54821	−	14.8059i	−0.842280	−	1.45887i	0.0456829	−	0.998956i	$-0.485454\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.5517			−1.50344			−0.751721	−	0.659482i	$-0.770776\pi$
$107$	−15.5517			−1.50344			−0.751721	+	0.659482i	$0.770776\pi$
$108$		0			0
$109$	−7.40551			−0.709319			−0.354660	−	0.934996i	$-0.615403\pi$
$109$	−7.40551			−0.709319			−0.354660	+	0.934996i	$0.615403\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.12659	+	12.3436i	0.670414	+	1.16119i	0.977787	+	0.209601i	$0.0672166\pi$
$113$	7.12659	+	12.3436i	0.670414	+	1.16119i	−0.307373	+	0.951589i	$0.599450\pi$
$114$		0			0
$115$	−8.63444	+	14.9553i	−0.805165	+	1.39459i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.619071	+	1.07226i	−0.0567502	+	0.0982942i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.58905			0.768228
$126$		0			0
$127$	12.3963			1.09999			0.549995	−	0.835168i	$-0.314630\pi$
$127$	12.3963			1.09999			0.549995	+	0.835168i	$0.314630\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.87297	+	3.24409i	0.163642	+	0.283437i	0.936172	−	0.351541i	$-0.114342\pi$
$131$	1.87297	+	3.24409i	0.163642	+	0.283437i	−0.772530	+	0.634978i	$0.781009\pi$
$132$		0			0
$133$	−0.774286	+	1.34110i	−0.0671391	+	0.116288i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.0727359	−	0.125982i	0.00621425	−	0.0107634i	−0.862902	−	0.505372i	$-0.831355\pi$
$137$	0.0727359	−	0.125982i	0.00621425	−	0.0107634i	0.869116	+	0.494609i	$0.164689\pi$
$138$		0			0
$139$	0.970286	+	1.68058i	0.0822986	+	0.142545i	0.904237	−	0.427031i	$-0.140441\pi$
$139$	0.970286	+	1.68058i	0.0822986	+	0.142545i	−0.821938	+	0.569577i	$0.807107\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.22016			−0.185659
$144$		0			0
$145$	21.4240			1.77916
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.69848	−	6.40595i	−0.302991	−	0.524796i	0.673821	−	0.738895i	$-0.264652\pi$
$149$	−3.69848	−	6.40595i	−0.302991	−	0.524796i	−0.976812	+	0.214099i	$0.931319\pi$
$150$		0			0
$151$	6.58322	−	11.4025i	0.535735	−	0.927920i	−0.463392	−	0.886153i	$-0.653368\pi$
$151$	6.58322	−	11.4025i	0.535735	−	0.927920i	0.999127	−	0.0417670i	$-0.0132987\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.51075	−	6.08080i	0.281990	−	0.488422i
$156$		0			0
$157$	1.31627	+	2.27985i	0.105050	+	0.181952i	0.913759	−	0.406257i	$-0.133166\pi$
$157$	1.31627	+	2.27985i	0.105050	+	0.181952i	−0.808709	+	0.588210i	$0.799833\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.35918			−0.264740
$162$		0			0
$163$	6.72257			0.526552			0.263276	−	0.964721i	$-0.415197\pi$
$163$	6.72257			0.526552			0.263276	+	0.964721i	$0.415197\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.52713	+	7.84122i	0.350320	+	0.606772i	0.986305	−	0.164929i	$-0.0527395\pi$
$167$	4.52713	+	7.84122i	0.350320	+	0.606772i	−0.635985	+	0.771701i	$0.719406\pi$
$168$		0			0
$169$	4.03544	−	6.98958i	0.310418	−	0.537660i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.18467	−	7.24807i	0.318155	−	0.551060i	−0.661948	−	0.749550i	$-0.730270\pi$
$173$	4.18467	−	7.24807i	0.318155	−	0.551060i	0.980103	+	0.198489i	$0.0636035\pi$
$174$		0			0
$175$	−0.421109	−	0.729382i	−0.0318328	−	0.0551361i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.0418			−1.12428			−0.562138	−	0.827043i	$-0.690021\pi$
$179$	−15.0418			−1.12428			−0.562138	+	0.827043i	$0.690021\pi$
$180$		0			0
$181$	7.53854			0.560336			0.280168	−	0.959951i	$-0.409610\pi$
$181$	7.53854			0.560336			0.280168	+	0.959951i	$0.409610\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.24896	+	5.62736i	0.238868	+	0.413732i
$186$		0			0
$187$	1.23175	−	2.13345i	0.0900744	−	0.156013i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.14165	−	3.70945i	0.154964	−	0.268406i	−0.778082	−	0.628163i	$-0.783807\pi$
$191$	2.14165	−	3.70945i	0.154964	−	0.268406i	0.933046	+	0.359757i	$0.117140\pi$
$192$		0			0
$193$	11.2417	+	19.4712i	0.809197	+	1.40157i	0.913421	+	0.407016i	$0.133431\pi$
$193$	11.2417	+	19.4712i	0.809197	+	1.40157i	−0.104224	+	0.994554i	$0.533236\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.0796			1.14562			0.572811	−	0.819688i	$-0.305853\pi$
$197$	16.0796			1.14562			0.572811	+	0.819688i	$0.305853\pi$
$198$		0			0
$199$	−2.71681			−0.192590			−0.0962948	−	0.995353i	$-0.530699\pi$
$199$	−2.71681			−0.192590			−0.0962948	+	0.995353i	$0.530699\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.08372	+	3.60910i	0.146248	+	0.253309i
$204$		0			0
$205$	−1.49409	+	2.58785i	−0.104352	+	0.180743i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.54057	−	2.66835i	0.106564	−	0.184574i
$210$		0			0
$211$	10.1201	+	17.5286i	0.696699	+	1.20672i	0.969605	+	0.244677i	$0.0786820\pi$
$211$	10.1201	+	17.5286i	0.696699	+	1.20672i	−0.272906	+	0.962041i	$0.587985\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.19858			−0.559139
$216$		0			0
$217$	1.36584			0.0927191
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.73468	−	4.73661i	−0.183955	−	0.318619i
$222$		0			0
$223$	3.07520	−	5.32640i	0.205930	−	0.356682i	−0.744498	−	0.667624i	$-0.767311\pi$
$223$	3.07520	−	5.32640i	0.205930	−	0.356682i	0.950429	+	0.310942i	$0.100645\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.82077	−	10.0819i	0.386338	−	0.669157i	−0.605616	−	0.795757i	$-0.707073\pi$
$227$	5.82077	−	10.0819i	0.386338	−	0.669157i	0.991954	+	0.126600i	$0.0404065\pi$
$228$		0			0
$229$	9.85030	+	17.0612i	0.650926	+	1.12744i	0.982899	+	0.184148i	$0.0589525\pi$
$229$	9.85030	+	17.0612i	0.650926	+	1.12744i	−0.331973	+	0.943289i	$0.607714\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.3270			−1.39718			−0.698588	−	0.715524i	$-0.746188\pi$
$233$	−21.3270			−1.39718			−0.698588	+	0.715524i	$0.746188\pi$
$234$		0			0
$235$	−26.3543			−1.71917
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.33890	+	7.51520i	0.280660	+	0.486118i	0.971548	−	0.236845i	$-0.0761132\pi$
$239$	4.33890	+	7.51520i	0.280660	+	0.486118i	−0.690887	+	0.722963i	$0.742780\pi$
$240$		0			0
$241$	12.9052	−	22.3524i	0.831295	−	1.43984i	−0.0657173	−	0.997838i	$-0.520934\pi$
$241$	12.9052	−	22.3524i	0.831295	−	1.43984i	0.897012	−	0.442006i	$-0.145733\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.71678	+	15.0979i	−0.556894	+	0.964569i
$246$		0			0
$247$	−3.42033	−	5.92418i	−0.217630	−	0.376947i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.10916			0.259368			0.129684	−	0.991555i	$-0.458604\pi$
$251$	4.10916			0.259368			0.129684	+	0.991555i	$0.458604\pi$
$252$		0			0
$253$	6.68366			0.420198
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.42959	−	16.3325i	−0.588201	−	1.01879i	−0.994468	−	0.105040i	$-0.966503\pi$
$257$	−9.42959	−	16.3325i	−0.588201	−	1.01879i	0.406267	−	0.913755i	$-0.366830\pi$
$258$		0			0
$259$	−0.631995	+	1.09465i	−0.0392702	+	0.0680180i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	8.91584	−	15.4427i	0.549774	−	0.952237i	−0.448516	−	0.893775i	$-0.648047\pi$
$263$	8.91584	−	15.4427i	0.549774	−	0.952237i	0.998290	−	0.0584616i	$-0.0186195\pi$
$264$		0			0
$265$	16.0453	+	27.7913i	0.985657	+	1.70721i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.1462			−0.740564			−0.370282	−	0.928919i	$-0.620739\pi$
$269$	−12.1462			−0.740564			−0.370282	+	0.928919i	$0.620739\pi$
$270$		0			0
$271$	−0.860803			−0.0522900			−0.0261450	−	0.999658i	$-0.508323\pi$
$271$	−0.860803			−0.0522900			−0.0261450	+	0.999658i	$0.508323\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.837868	+	1.45123i	0.0505253	+	0.0875125i
$276$		0			0
$277$	8.17937	−	14.1671i	0.491451	−	0.851217i	−0.508501	−	0.861061i	$-0.669800\pi$
$277$	8.17937	−	14.1671i	0.491451	−	0.851217i	0.999952	+	0.00984399i	$0.00313349\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.8107	−	18.7247i	0.644913	−	1.11702i	−0.339408	−	0.940639i	$-0.610227\pi$
$281$	10.8107	−	18.7247i	0.644913	−	1.11702i	0.984321	−	0.176384i	$-0.0564399\pi$
$282$		0			0
$283$	−14.0655	−	24.3622i	−0.836109	−	1.44818i	−0.893124	−	0.449810i	$-0.851492\pi$
$283$	−14.0655	−	24.3622i	−0.836109	−	1.44818i	0.0570148	−	0.998373i	$-0.481842\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.581269			−0.0343112
$288$		0			0
$289$	−10.9312			−0.643011
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.34267	+	14.4499i	0.487384	+	0.844174i	0.999895	−	0.0145067i	$-0.00461778\pi$
$293$	8.34267	+	14.4499i	0.487384	+	0.844174i	−0.512511	+	0.858681i	$0.671284\pi$
$294$		0			0
$295$	−8.92225	+	15.4538i	−0.519473	+	0.899754i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.41941	−	12.8508i	0.429076	−	0.743181i
$300$		0			0
$301$	−0.797403	−	1.38114i	−0.0459616	−	0.0796077i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.26422			0.186909
$306$		0			0
$307$	2.19917			0.125513			0.0627567	−	0.998029i	$-0.480011\pi$
$307$	2.19917			0.125513			0.0627567	+	0.998029i	$0.480011\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.07869	−	8.79655i	−0.287986	−	0.498807i	0.685343	−	0.728221i	$-0.259652\pi$
$311$	−5.07869	−	8.79655i	−0.287986	−	0.498807i	−0.973329	+	0.229414i	$0.926319\pi$
$312$		0			0
$313$	−5.53771	+	9.59160i	−0.313010	+	0.542149i	−0.979012	−	0.203800i	$-0.934671\pi$
$313$	−5.53771	+	9.59160i	−0.313010	+	0.542149i	0.666003	+	0.745949i	$0.268004\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.7318	−	18.5881i	0.602760	−	1.04401i	−0.389641	−	0.920967i	$-0.627401\pi$
$317$	10.7318	−	18.5881i	0.602760	−	1.04401i	0.992401	−	0.123044i	$-0.0392657\pi$
$318$		0			0
$319$	−4.14591	−	7.18093i	−0.232127	−	0.402055i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.59040			0.422341
$324$		0			0
$325$	3.72041			0.206371
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.56325	−	4.43968i	−0.141317	−	0.244767i
$330$		0			0
$331$	−12.4516	+	21.5668i	−0.684401	+	1.18542i	0.289224	+	0.957261i	$0.406603\pi$
$331$	−12.4516	+	21.5668i	−0.684401	+	1.18542i	−0.973625	+	0.228155i	$0.926731\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.07960	+	12.2622i	−0.386800	+	0.669957i
$336$		0			0
$337$	0.399568	+	0.692072i	0.0217659	+	0.0376996i	0.876703	−	0.481032i	$-0.159738\pi$
$337$	0.399568	+	0.692072i	0.0217659	+	0.0376996i	−0.854937	+	0.518731i	$0.826404\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.71757			−0.147165
$342$		0			0
$343$	−6.90938			−0.373071
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.2648	+	29.9035i	0.926822	+	1.60530i	0.788604	+	0.614901i	$0.210804\pi$
$347$	17.2648	+	29.9035i	0.926822	+	1.60530i	0.138218	+	0.990402i	$0.455863\pi$
$348$		0			0
$349$	−2.94069	+	5.09343i	−0.157412	+	0.272645i	−0.933935	−	0.357444i	$-0.883648\pi$
$349$	−2.94069	+	5.09343i	−0.157412	+	0.272645i	0.776523	+	0.630089i	$0.216982\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.0132	−	19.0754i	0.586174	−	1.01528i	−0.408555	−	0.912734i	$-0.633967\pi$
$353$	11.0132	−	19.0754i	0.586174	−	1.01528i	0.994728	−	0.102548i	$-0.0326996\pi$
$354$		0			0
$355$	−11.2149	−	19.4248i	−0.595227	−	1.03096i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.76577			0.357084			0.178542	−	0.983932i	$-0.442862\pi$
$359$	6.76577			0.357084			0.178542	+	0.983932i	$0.442862\pi$
$360$		0			0
$361$	−9.50652			−0.500343
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.65301	+	13.2554i	0.400577	+	0.693819i
$366$		0			0
$367$	11.1999	−	19.3989i	0.584632	−	1.01261i	−0.410289	−	0.911956i	$-0.634572\pi$
$367$	11.1999	−	19.3989i	0.584632	−	1.01261i	0.994921	−	0.100657i	$-0.0320946\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.12117	+	5.40603i	−0.162043	+	0.280667i
$372$		0			0
$373$	−13.9329	−	24.1324i	−0.721416	−	1.24953i	−0.960432	−	0.278514i	$-0.910158\pi$
$373$	−13.9329	−	24.1324i	−0.721416	−	1.24953i	0.239016	−	0.971016i	$-0.423175\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.4092			−0.948123
$378$		0			0
$379$	−26.9794			−1.38584			−0.692920	−	0.721014i	$-0.743676\pi$
$379$	−26.9794			−1.38584			−0.692920	+	0.721014i	$0.743676\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.9356	−	31.0653i	−0.916466	−	1.58737i	−0.804741	−	0.593626i	$-0.797696\pi$
$383$	−17.9356	−	31.0653i	−0.916466	−	1.58737i	−0.111725	−	0.993739i	$-0.535637\pi$
$384$		0			0
$385$	−0.649289	+	1.12460i	−0.0330908	+	0.0573150i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.63112	+	8.02133i	−0.234807	+	0.406697i	−0.959217	−	0.282672i	$-0.908779\pi$
$389$	−4.63112	+	8.02133i	−0.234807	+	0.406697i	0.724410	+	0.689370i	$0.242112\pi$
$390$		0			0
$391$	8.23259	+	14.2593i	0.416340	+	0.721122i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.3305			0.771360
$396$		0			0
$397$	−28.2180			−1.41622			−0.708110	−	0.706102i	$-0.750452\pi$
$397$	−28.2180			−1.41622			−0.708110	+	0.706102i	$0.750452\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.23371	+	15.9933i	0.461110	+	0.798665i	0.999017	−	0.0443388i	$-0.0141181\pi$
$401$	9.23371	+	15.9933i	0.461110	+	0.798665i	−0.537907	+	0.843004i	$0.680785\pi$
$402$		0			0
$403$	−3.01672	+	5.22512i	−0.150274	+	0.260282i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.25746	−	2.17799i	0.0623301	−	0.107959i
$408$		0			0
$409$	−12.2043	−	21.1384i	−0.603462	−	1.04523i	−0.992293	−	0.123918i	$-0.960454\pi$
$409$	−12.2043	−	21.1384i	−0.603462	−	1.04523i	0.388830	−	0.921309i	$-0.372879\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.47115			−0.170804
$414$		0			0
$415$	−27.7577			−1.36257
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.30710	+	3.99601i	0.112709	+	0.195218i	0.916862	−	0.399205i	$-0.130714\pi$
$419$	2.30710	+	3.99601i	0.112709	+	0.195218i	−0.804153	+	0.594423i	$0.797381\pi$
$420$		0			0
$421$	8.16163	−	14.1364i	0.397774	−	0.688964i	−0.595677	−	0.803224i	$-0.703116\pi$
$421$	8.16163	−	14.1364i	0.397774	−	0.688964i	0.993451	+	0.114260i	$0.0364496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.06409	+	3.57510i	−0.100123	+	0.173418i
$426$		0			0
$427$	0.317482	+	0.549894i	0.0153640	+	0.0266112i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−25.6444			−1.23525			−0.617623	−	0.786475i	$-0.711904\pi$
$431$	−25.6444			−1.23525			−0.617623	+	0.786475i	$0.711904\pi$
$432$		0			0
$433$	−3.62673			−0.174290			−0.0871448	−	0.996196i	$-0.527774\pi$
$433$	−3.62673			−0.174290			−0.0871448	+	0.996196i	$0.527774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.2967	+	17.8344i	0.492557	+	0.853134i
$438$		0			0
$439$	−18.9834	+	32.8802i	−0.906029	+	1.56929i	−0.0864986	+	0.996252i	$0.527568\pi$
$439$	−18.9834	+	32.8802i	−0.906029	+	1.56929i	−0.819530	+	0.573036i	$0.805766\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.6397	+	30.5528i	−0.838087	+	1.45161i	0.0534047	+	0.998573i	$0.482993\pi$
$443$	−17.6397	+	30.5528i	−0.838087	+	1.45161i	−0.891492	+	0.453037i	$0.850341\pi$
$444$		0			0
$445$	−12.6870	−	21.9745i	−0.601421	−	1.04169i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	25.6451			1.21027			0.605134	−	0.796123i	$-0.293119\pi$
$449$	25.6451			1.21027			0.605134	+	0.796123i	$0.293119\pi$
$450$		0			0
$451$	1.15653			0.0544590
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.44153	+	2.49680i	0.0675799	+	0.117052i
$456$		0			0
$457$	20.2624	−	35.0956i	0.947837	−	1.64170i	0.197869	−	0.980228i	$-0.436598\pi$
$457$	20.2624	−	35.0956i	0.947837	−	1.64170i	0.749968	−	0.661474i	$-0.230069\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.5005	−	25.1157i	0.675358	−	1.16975i	−0.301006	−	0.953622i	$-0.597323\pi$
$461$	14.5005	−	25.1157i	0.675358	−	1.16975i	0.976364	−	0.216132i	$-0.0693441\pi$
$462$		0			0
$463$	−17.1853	−	29.7658i	−0.798669	−	1.38334i	−0.920483	−	0.390783i	$-0.872204\pi$
$463$	−17.1853	−	29.7658i	−0.798669	−	1.38334i	0.121813	−	0.992553i	$-0.461129\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.16966			0.146674			0.0733371	−	0.997307i	$-0.476635\pi$
$467$	3.16966			0.146674			0.0733371	+	0.997307i	$0.476635\pi$
$468$		0			0
$469$	−2.75428			−0.127181
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.58657	+	2.74802i	0.0729506	+	0.126354i
$474$		0			0
$475$	−2.58160	+	4.47146i	−0.118452	+	0.205164i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.82843	−	10.0951i	0.266308	−	0.461259i	−0.701598	−	0.712573i	$-0.747530\pi$
$479$	5.82843	−	10.0951i	0.266308	−	0.461259i	0.967905	+	0.251315i	$0.0808629\pi$
$480$		0			0
$481$	−2.79177	−	4.83549i	−0.127294	−	0.220479i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.02135			−0.228008
$486$		0			0
$487$	−2.06578			−0.0936095			−0.0468048	−	0.998904i	$-0.514904\pi$
$487$	−2.06578			−0.0936095			−0.0468048	+	0.998904i	$0.514904\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−18.5423	−	32.1162i	−0.836803	−	1.44939i	−0.892554	−	0.450941i	$-0.851089\pi$
$491$	−18.5423	−	32.1162i	−0.836803	−	1.44939i	0.0557510	−	0.998445i	$-0.482245\pi$
$492$		0			0
$493$	10.2134	−	17.6902i	0.459991	−	0.796727i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.18155	−	3.77856i	0.0978560	−	0.169492i
$498$		0			0
$499$	8.61373	+	14.9194i	0.385604	+	0.667885i	0.991853	−	0.127390i	$-0.0406599\pi$
$499$	8.61373	+	14.9194i	0.385604	+	0.667885i	−0.606249	+	0.795275i	$0.707327\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−41.1144			−1.83320			−0.916600	−	0.399805i	$-0.869078\pi$
$503$	−41.1144			−1.83320			−0.916600	+	0.399805i	$0.869078\pi$
$504$		0			0
$505$	−34.8735			−1.55185
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.73267	−	16.8575i	−0.431393	−	0.747194i	0.565601	−	0.824679i	$-0.308644\pi$
$509$	−9.73267	−	16.8575i	−0.431393	−	0.747194i	−0.996994	+	0.0774848i	$0.975311\pi$
$510$		0			0
$511$	−1.48868	+	2.57847i	−0.0658553	+	0.114065i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	22.0864	−	38.2547i	0.973242	−	1.68571i
$516$		0			0
$517$	5.10003	+	8.83351i	0.224299	+	0.388497i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.9681			−0.699576			−0.349788	−	0.936829i	$-0.613746\pi$
$521$	−15.9681			−0.699576			−0.349788	+	0.936829i	$0.613746\pi$
$522$		0			0
$523$	36.6045			1.60060			0.800302	−	0.599597i	$-0.204672\pi$
$523$	36.6045			1.60060			0.800302	+	0.599597i	$0.204672\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.34736	−	5.79780i	−0.145813	−	0.252556i
$528$		0			0
$529$	−10.8357	+	18.7679i	−0.471116	+	0.815997i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.28385	−	2.22369i	0.0556096	−	0.0963186i
$534$		0			0
$535$	−20.0908	−	34.7983i	−0.868602	−	1.50446i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.74740			0.290631
$540$		0			0
$541$	−14.0034			−0.602055			−0.301028	−	0.953615i	$-0.597330\pi$
$541$	−14.0034			−0.602055			−0.301028	+	0.953615i	$0.597330\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.56697	−	16.5705i	−0.409804	−	0.709801i
$546$		0			0
$547$	−12.0437	+	20.8604i	−0.514953	+	0.891925i	0.484896	+	0.874572i	$0.338857\pi$
$547$	−12.0437	+	20.8604i	−0.514953	+	0.891925i	−0.999849	+	0.0173533i	$0.994476\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.7742	−	22.1255i	0.544198	−	0.942579i
$552$		0			0
$553$	1.49106	+	2.58259i	0.0634063	+	0.109823i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.8881			0.885057			0.442529	−	0.896754i	$-0.354082\pi$
$557$	20.8881			0.885057			0.442529	+	0.896754i	$0.354082\pi$
$558$		0			0
$559$	7.04489			0.297967
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.06276	−	1.84076i	−0.0447901	−	0.0775787i	0.842761	−	0.538288i	$-0.180929\pi$
$563$	−1.06276	−	1.84076i	−0.0447901	−	0.0775787i	−0.887551	+	0.460709i	$0.847595\pi$
$564$		0			0
$565$	−18.4133	+	31.8928i	−0.774653	+	1.34174i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.0627	−	24.3574i	0.589540	−	1.02111i	−0.404752	−	0.914426i	$-0.632642\pi$
$569$	14.0627	−	24.3574i	0.589540	−	1.02111i	0.994293	−	0.106687i	$-0.0340244\pi$
$570$		0			0
$571$	4.72244	+	8.17950i	0.197628	+	0.342301i	0.947759	−	0.318988i	$-0.103343\pi$
$571$	4.72244	+	8.17950i	0.197628	+	0.342301i	−0.750131	+	0.661289i	$0.770010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.2001			−0.467074
$576$		0			0
$577$	−11.3534			−0.472648			−0.236324	−	0.971674i	$-0.575943\pi$
$577$	−11.3534			−0.472648			−0.236324	+	0.971674i	$0.575943\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.69975	−	4.67610i	−0.112004	−	0.193997i
$582$		0			0
$583$	6.21011	−	10.7562i	0.257196	−	0.445477i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.3372	+	23.1007i	−0.550486	+	0.953469i	0.447754	+	0.894157i	$0.352224\pi$
$587$	−13.3372	+	23.1007i	−0.550486	+	0.953469i	−0.998239	+	0.0593125i	$0.981109\pi$
$588$		0			0
$589$	−4.18662	−	7.25143i	−0.172507	−	0.298790i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−42.0073			−1.72504			−0.862518	−	0.506027i	$-0.831114\pi$
$593$	−42.0073			−1.72504			−0.862518	+	0.506027i	$0.831114\pi$
$594$		0			0
$595$	−3.19905			−0.131148
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.26821	−	12.5889i	−0.296971	−	0.514369i	0.678471	−	0.734628i	$-0.262643\pi$
$599$	−7.26821	−	12.5889i	−0.296971	−	0.514369i	−0.975441	+	0.220259i	$0.929310\pi$
$600$		0			0
$601$	−4.04850	+	7.01220i	−0.165142	+	0.286034i	−0.936706	−	0.350118i	$-0.886141\pi$
$601$	−4.04850	+	7.01220i	−0.165142	+	0.286034i	0.771564	+	0.636152i	$0.219475\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.29187	−	2.23759i	0.0525221	−	0.0909709i
$606$		0			0
$607$	8.82417	+	15.2839i	0.358162	+	0.620354i	0.987654	−	0.156653i	$-0.0500703\pi$
$607$	8.82417	+	15.2839i	0.358162	+	0.620354i	−0.629492	+	0.777007i	$0.716737\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.6458			0.916151
$612$		0			0
$613$	−40.8291			−1.64907			−0.824537	−	0.565808i	$-0.808564\pi$
$613$	−40.8291			−1.64907			−0.824537	+	0.565808i	$0.808564\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.97609	+	17.2791i	0.401622	+	0.695630i	0.993922	−	0.110087i	$-0.0351131\pi$
$617$	9.97609	+	17.2791i	0.401622	+	0.695630i	−0.592300	+	0.805718i	$0.701780\pi$
$618$		0			0
$619$	−5.32594	+	9.22480i	−0.214068	+	0.370776i	−0.952984	−	0.303021i	$-0.902005\pi$
$619$	−5.32594	+	9.22480i	−0.214068	+	0.370776i	0.738916	+	0.673797i	$0.235338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.46790	−	4.27453i	0.0988743	−	0.171255i
$624$		0			0
$625$	15.2853	+	26.4749i	0.611412	+	1.05900i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.19551			0.247031
$630$		0			0
$631$	3.16199			0.125877			0.0629385	−	0.998017i	$-0.479953\pi$
$631$	3.16199			0.125877			0.0629385	+	0.998017i	$0.479953\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.0144	+	27.7377i	0.635511	+	1.10074i
$636$		0			0
$637$	7.49016	−	12.9733i	0.296771	−	0.514023i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.5188	+	32.0756i	−0.731450	+	1.26691i	0.224813	+	0.974402i	$0.427823\pi$
$641$	−18.5188	+	32.0756i	−0.731450	+	1.26691i	−0.956263	+	0.292507i	$0.905510\pi$
$642$		0			0
$643$	1.99681	+	3.45858i	0.0787465	+	0.136393i	0.902709	−	0.430251i	$-0.141575\pi$
$643$	1.99681	+	3.45858i	0.0787465	+	0.136393i	−0.823963	+	0.566644i	$0.808242\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.79267			−0.109791			−0.0548955	−	0.998492i	$-0.517483\pi$
$647$	−2.79267			−0.109791			−0.0548955	+	0.998492i	$0.517483\pi$
$648$		0			0
$649$	6.90645			0.271102
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.9182	−	20.6429i	−0.466395	−	0.807820i	0.532868	−	0.846198i	$-0.321114\pi$
$653$	−11.9182	−	20.6429i	−0.466395	−	0.807820i	−0.999263	+	0.0383785i	$0.987781\pi$
$654$		0			0
$655$	−4.83928	+	8.38189i	−0.189087	+	0.327507i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.655300	−	1.13501i	0.0255269	−	0.0442138i	−0.852980	−	0.521944i	$-0.825207\pi$
$659$	0.655300	−	1.13501i	0.0255269	−	0.0442138i	0.878507	+	0.477730i	$0.158540\pi$
$660$		0			0
$661$	0.725184	+	1.25606i	0.0282064	+	0.0488549i	0.879784	−	0.475374i	$-0.157687\pi$
$661$	0.725184	+	1.25606i	0.0282064	+	0.0488549i	−0.851578	+	0.524229i	$0.824354\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.00111			−0.155157
$666$		0			0
$667$	55.4198			2.14586
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.631684	−	1.09411i	−0.0243859	−	0.0422376i
$672$		0			0
$673$	22.2698	−	38.5723i	0.858436	−	1.48685i	−0.0149843	−	0.999888i	$-0.504770\pi$
$673$	22.2698	−	38.5723i	0.858436	−	1.48685i	0.873420	−	0.486967i	$-0.161897\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	24.7487	−	42.8660i	0.951169	−	1.64747i	0.208268	−	0.978072i	$-0.433217\pi$
$677$	24.7487	−	42.8660i	0.951169	−	1.64747i	0.742901	−	0.669401i	$-0.233449\pi$
$678$		0			0
$679$	−0.488382	−	0.845902i	−0.0187424	−	0.0324627i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	40.8454			1.56291			0.781454	−	0.623963i	$-0.214479\pi$
$683$	40.8454			1.56291			0.781454	+	0.623963i	$0.214479\pi$
$684$		0			0
$685$	0.375862			0.0143610
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.7875	−	23.8806i	−0.525260	−	0.909778i
$690$		0			0
$691$	7.35816	−	12.7447i	0.279918	−	0.484831i	−0.691446	−	0.722428i	$-0.743026\pi$
$691$	7.35816	−	12.7447i	0.279918	−	0.484831i	0.971364	+	0.237596i	$0.0763595\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.50697	+	4.34220i	−0.0950949	+	0.164709i
$696$		0			0
$697$	1.42456	+	2.46741i	0.0539590	+	0.0934598i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−18.5732			−0.701501			−0.350751	−	0.936469i	$-0.614074\pi$
$701$	−18.5732			−0.701501			−0.350751	+	0.936469i	$0.614074\pi$
$702$		0			0
$703$	7.74885			0.292254
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.39183	−	5.87482i	−0.127563	−	0.220945i
$708$		0			0
$709$	−20.4373	+	35.3985i	−0.767539	+	1.32942i	0.171354	+	0.985210i	$0.445186\pi$
$709$	−20.4373	+	35.3985i	−0.767539	+	1.32942i	−0.938893	+	0.344208i	$0.888148\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.08165	−	15.7299i	0.340111	−	0.589089i
$714$		0			0
$715$	−2.86817	−	4.96781i	−0.107263	−	0.185786i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−30.8115			−1.14908			−0.574538	−	0.818478i	$-0.694818\pi$
$719$	−30.8115			−1.14908			−0.574538	+	0.818478i	$0.694818\pi$
$720$		0			0
$721$	8.59258			0.320004
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.94746	+	12.0333i	0.258022	+	0.446907i
$726$		0			0
$727$	18.0554	−	31.2729i	0.669638	−	1.15985i	−0.308368	−	0.951267i	$-0.599783\pi$
$727$	18.0554	−	31.2729i	0.669638	−	1.15985i	0.978006	−	0.208579i	$-0.0668839\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.90851	+	6.76974i	−0.144562	+	0.250388i
$732$		0			0
$733$	16.8793	+	29.2358i	0.623452	+	1.07985i	0.988838	+	0.148994i	$0.0476035\pi$
$733$	16.8793	+	29.2358i	0.623452	+	1.07985i	−0.365386	+	0.930856i	$0.619063\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.48010			0.201862
$738$		0			0
$739$	13.7582			0.506104			0.253052	−	0.967453i	$-0.418566\pi$
$739$	13.7582			0.506104			0.253052	+	0.967453i	$0.418566\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.7807	−	29.0650i	−0.615623	−	1.06629i	−0.990275	−	0.139124i	$-0.955571\pi$
$743$	−16.7807	−	29.0650i	−0.615623	−	1.06629i	0.374652	−	0.927165i	$-0.377762\pi$
$744$		0			0
$745$	9.55592	−	16.5513i	0.350102	−	0.606394i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.90811	−	6.76905i	0.142799	−	0.247336i
$750$		0			0
$751$	−24.5938	−	42.5977i	−0.897441	−	1.55441i	−0.830755	−	0.556639i	$-0.812091\pi$
$751$	−24.5938	−	42.5977i	−0.897441	−	1.55441i	−0.0666861	−	0.997774i	$-0.521243\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	34.0187			1.23807
$756$		0			0
$757$	34.9762			1.27123			0.635617	−	0.772005i	$-0.280746\pi$
$757$	34.9762			1.27123			0.635617	+	0.772005i	$0.280746\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.426628	+	0.738942i	0.0154653	+	0.0267866i	0.873654	−	0.486547i	$-0.161744\pi$
$761$	0.426628	+	0.738942i	0.0154653	+	0.0267866i	−0.858189	+	0.513334i	$0.828410\pi$
$762$		0			0
$763$	1.86099	−	3.22333i	0.0673723	−	0.116692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.66672	−	13.2792i	0.276829	−	0.479482i
$768$		0			0
$769$	4.62455	+	8.00996i	0.166766	+	0.288846i	0.937281	−	0.348575i	$-0.113334\pi$
$769$	4.62455	+	8.00996i	0.166766	+	0.288846i	−0.770515	+	0.637422i	$0.780001\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−38.9134			−1.39962			−0.699809	−	0.714330i	$-0.746731\pi$
$773$	−38.9134			−1.39962			−0.699809	+	0.714330i	$0.746731\pi$
$774$		0			0
$775$	4.55393			0.163582
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.78173	+	3.08604i	0.0638370	+	0.110569i
$780$		0			0
$781$	−4.34057	+	7.51809i	−0.155318	+	0.269019i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.40092	+	5.89056i	−0.121384	+	0.210243i
$786$		0			0
$787$	3.54778	+	6.14493i	0.126465	+	0.219043i	0.922305	−	0.386464i	$-0.126304\pi$
$787$	3.54778	+	6.14493i	0.126465	+	0.219043i	−0.795840	+	0.605507i	$0.792970\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.16359			−0.254708
$792$		0			0
$793$	−2.80488			−0.0996043
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.96247	−	13.7914i	−0.282045	−	0.488516i	0.689843	−	0.723959i	$-0.257679\pi$
$797$	−7.96247	−	13.7914i	−0.282045	−	0.488516i	−0.971888	+	0.235442i	$0.924346\pi$
$798$		0			0
$799$	−12.5639	+	21.7613i	−0.444479	+	0.769860i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.96198	−	5.13030i	0.104526	−	0.181045i
$804$		0			0
$805$	−4.33963	−	7.51646i	−0.152952	−	0.264920i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.5647			0.828492			0.414246	−	0.910165i	$-0.364045\pi$
$809$	23.5647			0.828492			0.414246	+	0.910165i	$0.364045\pi$
$810$		0			0
$811$	51.3678			1.80377			0.901884	−	0.431978i	$-0.142184\pi$
$811$	51.3678			1.80377			0.901884	+	0.431978i	$0.142184\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.68470	+	15.0423i	0.304212	+	0.526910i
$816$		0			0
$817$	−4.88846	+	8.46706i	−0.171026	+	0.296225i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.0358	+	45.0954i	−0.908656	+	1.57384i	−0.0927238	+	0.995692i	$0.529557\pi$
$821$	−26.0358	+	45.0954i	−0.908656	+	1.57384i	−0.815933	+	0.578147i	$0.803776\pi$
$822$		0			0
$823$	15.5424	+	26.9203i	0.541776	+	0.938383i	0.998802	+	0.0489299i	$0.0155811\pi$
$823$	15.5424	+	26.9203i	0.541776	+	0.938383i	−0.457027	+	0.889453i	$0.651086\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.4394			0.571655			0.285828	−	0.958281i	$-0.407732\pi$
$827$	16.4394			0.571655			0.285828	+	0.958281i	$0.407732\pi$
$828$		0			0
$829$	−21.0593			−0.731421			−0.365711	−	0.930729i	$-0.619174\pi$
$829$	−21.0593			−0.731421			−0.365711	+	0.930729i	$0.619174\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.31110	+	14.3952i	0.287963	+	0.498766i
$834$		0			0
$835$	−11.6970	+	20.2597i	−0.404790	+	0.701117i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.73325	+	6.46618i	−0.128886	+	0.223237i	−0.923245	−	0.384211i	$-0.874473\pi$
$839$	−3.73325	+	6.46618i	−0.128886	+	0.223237i	0.794359	+	0.607448i	$0.207807\pi$
$840$		0			0
$841$	−19.8772	−	34.4283i	−0.685420	−	1.18718i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.8531			0.717368
$846$		0			0
$847$	0.502596			0.0172694
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8.40445	+	14.5569i	0.288101	+	0.499005i
$852$		0			0
$853$	−13.6648	+	23.6681i	−0.467873	+	0.810380i	−0.999326	−	0.0367077i	$-0.988313\pi$
$853$	−13.6648	+	23.6681i	−0.467873	+	0.810380i	0.531453	+	0.847088i	$0.321646\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.8199	+	34.3290i	−0.677034	+	1.17266i	0.298836	+	0.954305i	$0.403402\pi$
$857$	−19.8199	+	34.3290i	−0.677034	+	1.17266i	−0.975870	+	0.218353i	$0.929932\pi$
$858$		0			0
$859$	−4.61839	−	7.99928i	−0.157577	−	0.272932i	0.776417	−	0.630219i	$-0.217035\pi$
$859$	−4.61839	−	7.99928i	−0.157577	−	0.272932i	−0.933994	+	0.357287i	$0.883702\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.8545			−0.743938			−0.371969	−	0.928245i	$-0.621317\pi$
$863$	−21.8545			−0.743938			−0.371969	+	0.928245i	$0.621317\pi$
$864$		0			0
$865$	21.6243			0.735247
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.96672	−	5.13851i	−0.100639	−	0.174312i
$870$		0			0
$871$	6.08336	−	10.5367i	0.206127	−	0.357022i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.15841	+	3.73847i	−0.0729675	+	0.126383i
$876$		0			0
$877$	−10.5679	−	18.3042i	−0.356854	−	0.618089i	0.630579	−	0.776125i	$-0.282817\pi$
$877$	−10.5679	−	18.3042i	−0.356854	−	0.618089i	−0.987433	+	0.158035i	$0.949484\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−46.1967			−1.55641			−0.778203	−	0.628013i	$-0.783869\pi$
$881$	−46.1967			−1.55641			−0.778203	+	0.628013i	$0.783869\pi$
$882$		0			0
$883$	4.03099			0.135654			0.0678269	−	0.997697i	$-0.478393\pi$
$883$	4.03099			0.135654			0.0678269	+	0.997697i	$0.478393\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.9508	−	44.9482i	−0.871344	−	1.50921i	−0.860607	−	0.509269i	$-0.829916\pi$
$887$	−25.9508	−	44.9482i	−0.871344	−	1.50921i	−0.0107365	−	0.999942i	$-0.503418\pi$
$888$		0			0
$889$	−3.11515	+	5.39560i	−0.104479	+	0.180963i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.7139	+	27.2173i	−0.525847	+	0.910794i
$894$		0			0
$895$	−19.4321	−	33.6573i	−0.649543	−	1.12504i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.5336			−0.751538
$900$		0			0
$901$	30.5972			1.01934
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.73884	+	16.8682i	0.323730	+	0.560717i
$906$		0			0
$907$	9.40612	−	16.2919i	0.312325	−	0.540963i	−0.666540	−	0.745469i	$-0.732226\pi$
$907$	9.40612	−	16.2919i	0.312325	−	0.540963i	0.978865	+	0.204506i	$0.0655589\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.535073	+	0.926774i	−0.0177278	+	0.0307054i	−0.874753	−	0.484569i	$-0.838977\pi$
$911$	−0.535073	+	0.926774i	−0.0177278	+	0.0307054i	0.857025	+	0.515274i	$0.172310\pi$
$912$		0			0
$913$	5.37161	+	9.30390i	0.177774	+	0.307914i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.88270			−0.0621721
$918$		0			0
$919$	9.97597			0.329077			0.164538	−	0.986371i	$-0.447387\pi$
$919$	9.97597			0.329077			0.164538	+	0.986371i	$0.447387\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.63678	+	16.6914i	0.317199	+	0.549404i
$924$		0			0
$925$	−2.10717	+	3.64973i	−0.0692835	+	0.120002i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.8413	−	32.6341i	0.618162	−	1.07069i	−0.371658	−	0.928370i	$-0.621211\pi$
$929$	18.8413	−	32.6341i	0.618162	−	1.07069i	0.989821	−	0.142319i	$-0.0454559\pi$
$930$		0			0
$931$	10.3949	+	18.0044i	0.340678	+	0.590072i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.36505			0.208159
$936$		0			0
$937$	−35.8799			−1.17214			−0.586072	−	0.810259i	$-0.699326\pi$
$937$	−35.8799			−1.17214			−0.586072	+	0.810259i	$0.699326\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.62812	−	14.9443i	−0.281269	−	0.487172i	0.690429	−	0.723400i	$-0.257422\pi$
$941$	−8.62812	−	14.9443i	−0.281269	−	0.487172i	−0.971697	+	0.236229i	$0.924089\pi$
$942$		0			0
$943$	−3.86494	+	6.69427i	−0.125860	+	0.217995i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.9864	−	36.3496i	0.681968	−	1.18120i	−0.292412	−	0.956293i	$-0.594458\pi$
$947$	20.9864	−	36.3496i	0.681968	−	1.18120i	0.974379	−	0.224910i	$-0.0722090\pi$
$948$		0			0
$949$	−6.57609	−	11.3901i	−0.213469	−	0.369739i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17.6110			0.570477			0.285239	−	0.958457i	$-0.407927\pi$
$953$	17.6110			0.570477			0.285239	+	0.958457i	$0.407927\pi$
$954$		0			0
$955$	11.0670			0.358118
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.0365568	+	0.0633182i	0.00118048	+	0.00204465i
$960$		0			0
$961$	11.8074	−	20.4510i	0.380884	−	0.659711i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−29.0457	+	50.3087i	−0.935015	+	1.61949i
$966$		0			0
$967$	6.86559	+	11.8916i	0.220783	+	0.382407i	0.955046	−	0.296458i	$-0.0958056\pi$
$967$	6.86559	+	11.8916i	0.220783	+	0.382407i	−0.734263	+	0.678865i	$0.762472\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	39.0717			1.25387			0.626935	−	0.779072i	$-0.284309\pi$
$971$	39.0717			1.25387			0.626935	+	0.779072i	$0.284309\pi$
$972$		0			0
$973$	−0.975323			−0.0312674
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.8006	+	37.7597i	0.697462	+	1.20804i	0.969344	+	0.245708i	$0.0790205\pi$
$977$	21.8006	+	37.7597i	0.697462	+	1.20804i	−0.271882	+	0.962331i	$0.587646\pi$
$978$		0			0
$979$	−4.91031	+	8.50491i	−0.156934	+	0.271818i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27.1583	+	47.0396i	−0.866217	+	1.50033i	−0.000382692		1.00000i	$0.500122\pi$
$983$	−27.1583	+	47.0396i	−0.866217	+	1.50033i	−0.865834	+	0.500331i	$0.833212\pi$
$984$		0			0
$985$	20.7727	+	35.9794i	0.661875	+	1.14640i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21.2082			−0.674381
$990$		0			0
$991$	−13.7473			−0.436697			−0.218348	−	0.975871i	$-0.570067\pi$
$991$	−13.7473			−0.436697			−0.218348	+	0.975871i	$0.570067\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.50977	−	6.07911i	−0.111267	−	0.192721i
$996$		0			0
$997$	−0.263061	+	0.455634i	−0.00833121	+	0.0144301i	−0.870161	−	0.492768i	$-0.835985\pi$
$997$	−0.263061	+	0.455634i	−0.00833121	+	0.0144301i	0.861830	+	0.507198i	$0.169319\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.q.f.1585.5		12
3.2	odd	2		792.2.q.f.529.4	yes	12
9.2	odd	6		7128.2.a.ba.1.5		6
9.4	even	3	inner	2376.2.q.f.793.5		12
9.5	odd	6		792.2.q.f.265.4	✓	12
9.7	even	3		7128.2.a.w.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.q.f.265.4	✓	12	9.5	odd	6
792.2.q.f.529.4	yes	12	3.2	odd	2
2376.2.q.f.793.5		12	9.4	even	3	inner
2376.2.q.f.1585.5		12	1.1	even	1	trivial
7128.2.a.w.1.2		6	9.7	even	3
7128.2.a.ba.1.5		6	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)

\(f(q)\)	\(=\)	\(q+(1.29187 + 2.23759i) q^{5} +(-0.251298 + 0.435260i) q^{7} +(0.500000 - 0.866025i) q^{11} +(-1.11008 - 1.92272i) q^{13} +2.46350 q^{17} +3.08115 q^{19} +(3.34183 + 5.78822i) q^{23} +(-0.837868 + 1.45123i) q^{25} +(4.14591 - 7.18093i) q^{29} +(-1.35878 - 2.35348i) q^{31} -1.29858 q^{35} +2.51492 q^{37} +(0.578267 + 1.00159i) q^{41} +(-1.58657 + 2.74802i) q^{43} +(-5.10003 + 8.83351i) q^{47} +(3.37370 + 5.84342i) q^{49} +12.4202 q^{53} +2.58374 q^{55} +(3.45322 + 5.98116i) q^{59} +(0.631684 - 1.09411i) q^{61} +(2.86817 - 4.96781i) q^{65} +(2.74005 + 4.74591i) q^{67} -8.68115 q^{71} +5.92397 q^{73} +(0.251298 + 0.435260i) q^{77} +(2.96672 - 5.13851i) q^{79} +(-5.37161 + 9.30390i) q^{83} +(3.18252 + 5.51229i) q^{85} -9.82062 q^{89} +1.11584 q^{91} +(3.98045 + 6.89434i) q^{95} +(-0.971719 + 1.68307i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{5} - 5 q^{7} + 6 q^{11} - 3 q^{13} - 14 q^{17} + 10 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{29} - 4 q^{31} - 16 q^{35} + 6 q^{37} - 5 q^{43} + 17 q^{47} - 3 q^{49} - 28 q^{53} + 8 q^{55} + 4 q^{59}+ \cdots + 16 q^{97}+O(q^{100}) \) 12 * q + 4 * q^5 - 5 * q^7 + 6 * q^11 - 3 * q^13 - 14 * q^17 + 10 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^29 - 4 * q^31 - 16 * q^35 + 6 * q^37 - 5 * q^43 + 17 * q^47 - 3 * q^49 - 28 * q^53 + 8 * q^55 + 4 * q^59 - q^61 + 15 * q^65 + 4 * q^67 - 14 * q^71 - 24 * q^73 + 5 * q^77 - q^79 + 22 * q^83 + 3 * q^85 - 44 * q^89 + 22 * q^91 + 24 * q^95 + 16 * q^97

Introduction

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