LMFDB - Embedded newform 2394.2.e.d.1063.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2394,2,Mod(1063,2394)] 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("2394.1063"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2394, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,-24,0,0,-8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1063.13
Character	$\chi$	$=$	2394.1063
Dual form			2394.2.e.d.1063.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.63913i q^{5} +(1.91223 - 1.82849i) q^{7} +1.00000i q^{8} +1.63913 q^{10} -2.23105 q^{11} -0.801180 q^{13} +(-1.82849 - 1.91223i) q^{14} +1.00000 q^{16} -2.71542i q^{17} +(-3.13440 + 3.02912i) q^{19} -1.63913i q^{20} +2.23105i q^{22} -6.49732 q^{23} +2.31324 q^{25} +0.801180i q^{26} +(-1.91223 + 1.82849i) q^{28} -3.82446i q^{29} -5.19251 q^{31} -1.00000i q^{32} -2.71542 q^{34} +(2.99714 + 3.13440i) q^{35} -7.19617i q^{37} +(3.02912 + 3.13440i) q^{38} -1.63913 q^{40} -6.99299 q^{41} +7.13770 q^{43} +2.23105 q^{44} +6.49732i q^{46} +1.91424i q^{47} +(0.313241 - 6.99299i) q^{49} -2.31324i q^{50} +0.801180 q^{52} +1.19798i q^{53} -3.65698i q^{55} +(1.82849 + 1.91223i) q^{56} -3.82446 q^{58} -4.59174 q^{59} -0.934757i q^{61} +5.19251i q^{62} -1.00000 q^{64} -1.31324i q^{65} -15.2257i q^{67} +2.71542i q^{68} +(3.13440 - 2.99714i) q^{70} +1.19798i q^{71} -6.99299i q^{73} -7.19617 q^{74} +(3.13440 - 3.02912i) q^{76} +(-4.26627 + 4.07945i) q^{77} -0.503034i q^{79} +1.63913i q^{80} +6.99299i q^{82} +11.1494i q^{83} +4.45094 q^{85} -7.13770i q^{86} -2.23105i q^{88} -13.9860 q^{89} +(-1.53204 + 1.46495i) q^{91} +6.49732 q^{92} +1.91424 q^{94} +(-4.96513 - 5.13770i) q^{95} -17.9811 q^{97} +(-6.99299 - 0.313241i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85}+O(q^{100}) $ 24 * q - 24 * q^4 - 8 * q^7 + 24 * q^16 + 24 * q^25 + 8 * q^28 + 32 * q^43 - 24 * q^49 + 16 * q^58 - 24 * q^64 - 64 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.63913i			0.733043i	0.930410	+	0.366521i	$0.119451\pi$
$5$			1.63913i			0.733043i	−0.930410	+	0.366521i	$0.880549\pi$
$6$		0			0
$7$	1.91223	−	1.82849i	0.722755	−	0.691105i
$7$	1.91223	−	1.82849i	0.722755	−	0.691105i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.63913			0.518340
$11$	−2.23105			−0.672686			−0.336343	−	0.941740i	$-0.609190\pi$
$11$	−2.23105			−0.672686			−0.336343	+	0.941740i	$0.609190\pi$
$12$		0			0
$13$	−0.801180			−0.222207			−0.111104	−	0.993809i	$-0.535439\pi$
$13$	−0.801180			−0.222207			−0.111104	+	0.993809i	$0.535439\pi$
$14$	−1.82849	−	1.91223i	−0.488685	−	0.511065i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.71542i		−	0.658587i	−0.944228	−	0.329293i	$-0.893190\pi$
$17$		−	2.71542i		−	0.658587i	0.944228	−	0.329293i	$-0.106810\pi$
$18$		0			0
$19$	−3.13440	+	3.02912i	−0.719080	+	0.694927i
$19$	−3.13440	+	3.02912i	−0.719080	+	0.694927i
$20$		−	1.63913i		−	0.366521i
$21$		0			0
$22$			2.23105i			0.475661i
$23$	−6.49732			−1.35478			−0.677392	−	0.735622i	$-0.736890\pi$
$23$	−6.49732			−1.35478			−0.677392	+	0.735622i	$0.736890\pi$
$24$		0			0
$25$	2.31324			0.462648
$26$			0.801180i			0.157124i
$27$		0			0
$28$	−1.91223	+	1.82849i	−0.361377	+	0.345552i
$29$		−	3.82446i		−	0.710184i	−0.934831	−	0.355092i	$-0.884450\pi$
$29$		−	3.82446i		−	0.710184i	0.934831	−	0.355092i	$-0.115550\pi$
$30$		0			0
$31$	−5.19251			−0.932602			−0.466301	−	0.884626i	$-0.654414\pi$
$31$	−5.19251			−0.932602			−0.466301	+	0.884626i	$0.654414\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−2.71542			−0.465691
$35$	2.99714	+	3.13440i	0.506609	+	0.529810i
$36$		0			0
$37$		−	7.19617i		−	1.18304i	−0.806289	−	0.591522i	$-0.798527\pi$
$37$		−	7.19617i		−	1.18304i	0.806289	−	0.591522i	$-0.201473\pi$
$38$	3.02912	+	3.13440i	0.491387	+	0.508467i
$39$		0			0
$40$	−1.63913			−0.259170
$41$	−6.99299			−1.09212			−0.546061	−	0.837746i	$-0.683873\pi$
$41$	−6.99299			−1.09212			−0.546061	+	0.837746i	$0.683873\pi$
$42$		0			0
$43$	7.13770			1.08849			0.544244	−	0.838927i	$-0.316816\pi$
$43$	7.13770			1.08849			0.544244	+	0.838927i	$0.316816\pi$
$44$	2.23105			0.336343
$45$		0			0
$46$			6.49732i			0.957977i
$47$			1.91424i			0.279221i	0.990207	+	0.139610i	$0.0445850\pi$
$47$			1.91424i			0.279221i	−0.990207	+	0.139610i	$0.955415\pi$
$48$		0			0
$49$	0.313241	−	6.99299i	0.0447487	−	0.998998i
$50$		−	2.31324i		−	0.327142i
$51$		0			0
$52$	0.801180			0.111104
$53$			1.19798i			0.164555i	0.996609	+	0.0822774i	$0.0262193\pi$
$53$			1.19798i			0.164555i	−0.996609	+	0.0822774i	$0.973781\pi$
$54$		0			0
$55$		−	3.65698i		−	0.493107i
$56$	1.82849	+	1.91223i	0.244342	+	0.255532i
$57$		0			0
$58$	−3.82446			−0.502176
$59$	−4.59174			−0.597793			−0.298897	−	0.954285i	$-0.596619\pi$
$59$	−4.59174			−0.597793			−0.298897	+	0.954285i	$0.596619\pi$
$60$		0			0
$61$		−	0.934757i		−	0.119683i	−0.998208	−	0.0598417i	$-0.980940\pi$
$61$		−	0.934757i		−	0.119683i	0.998208	−	0.0598417i	$-0.0190596\pi$
$62$			5.19251i			0.659449i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	1.31324i		−	0.162887i
$66$		0			0
$67$		−	15.2257i		−	1.86011i	−0.367417	−	0.930056i	$-0.619758\pi$
$67$		−	15.2257i		−	1.86011i	0.367417	−	0.930056i	$-0.380242\pi$
$68$			2.71542i			0.329293i
$69$		0			0
$70$	3.13440	−	2.99714i	0.374632	−	0.358227i
$71$			1.19798i			0.142174i	0.997470	+	0.0710868i	$0.0226467\pi$
$71$			1.19798i			0.142174i	−0.997470	+	0.0710868i	$0.977353\pi$
$72$		0			0
$73$		−	6.99299i		−	0.818467i	−0.912430	−	0.409234i	$-0.865796\pi$
$73$		−	6.99299i		−	0.818467i	0.912430	−	0.409234i	$-0.134204\pi$
$74$	−7.19617			−0.836538
$75$		0			0
$76$	3.13440	−	3.02912i	0.359540	−	0.347463i
$77$	−4.26627	+	4.07945i	−0.486187	+	0.464896i
$78$		0			0
$79$		−	0.503034i		−	0.0565957i	−0.999600	−	0.0282979i	$-0.990991\pi$
$79$		−	0.503034i		−	0.0565957i	0.999600	−	0.0282979i	$-0.00900869\pi$
$80$			1.63913i			0.183261i
$81$		0			0
$82$			6.99299i			0.772246i
$83$			11.1494i			1.22381i	0.790932	+	0.611904i	$0.209596\pi$
$83$			11.1494i			1.22381i	−0.790932	+	0.611904i	$0.790404\pi$
$84$		0			0
$85$	4.45094			0.482772
$86$		−	7.13770i		−	0.769678i
$87$		0			0
$88$		−	2.23105i		−	0.237830i
$89$	−13.9860			−1.48251			−0.741255	−	0.671223i	$-0.765769\pi$
$89$	−13.9860			−1.48251			−0.741255	+	0.671223i	$0.765769\pi$
$90$		0			0
$91$	−1.53204	+	1.46495i	−0.160601	+	0.153569i
$92$	6.49732			0.677392
$93$		0			0
$94$	1.91424			0.197439
$95$	−4.96513	−	5.13770i	−0.509411	−	0.527117i
$96$		0			0
$97$	−17.9811			−1.82570			−0.912850	−	0.408294i	$-0.866124\pi$
$97$	−17.9811			−1.82570			−0.912850	+	0.408294i	$0.866124\pi$
$98$	−6.99299	−	0.313241i	−0.706398	−	0.0316421i
$99$		0			0
$100$	−2.31324			−0.231324
$101$		−	9.29610i		−	0.924997i	−0.886620	−	0.462498i	$-0.846953\pi$
$101$		−	9.29610i		−	0.924997i	0.886620	−	0.462498i	$-0.153047\pi$
$102$		0			0
$103$	12.0115			1.18353			0.591765	−	0.806110i	$-0.298431\pi$
$103$	12.0115			1.18353			0.591765	+	0.806110i	$0.298431\pi$
$104$		−	0.801180i		−	0.0785621i
$105$		0			0
$106$	1.19798			0.116358
$107$		−	2.62648i		−	0.253912i	−0.991908	−	0.126956i	$-0.959479\pi$
$107$		−	2.62648i		−	0.253912i	0.991908	−	0.126956i	$-0.0405206\pi$
$108$		0			0
$109$		−	4.26627i		−	0.408635i	−0.978905	−	0.204317i	$-0.934503\pi$
$109$		−	4.26627i		−	0.408635i	0.978905	−	0.204317i	$-0.0654974\pi$
$110$	−3.65698			−0.348680
$111$		0			0
$112$	1.91223	−	1.82849i	0.180689	−	0.172776i
$113$		−	11.5886i		−	1.09017i	−0.838382	−	0.545084i	$-0.816498\pi$
$113$		−	11.5886i		−	1.09017i	0.838382	−	0.545084i	$-0.183502\pi$
$114$		0			0
$115$		−	10.6500i		−	0.993115i
$116$			3.82446i			0.355092i
$117$		0			0
$118$			4.59174i			0.422704i
$119$	−4.96513	−	5.19251i	−0.455152	−	0.475997i
$120$		0			0
$121$	−6.02243			−0.547494
$122$	−0.934757			−0.0846289
$123$		0			0
$124$	5.19251			0.466301
$125$			11.9874i			1.07218i
$126$		0			0
$127$			0.503034i			0.0446370i	0.999751	+	0.0223185i	$0.00710480\pi$
$127$			0.503034i			0.0446370i	−0.999751	+	0.0223185i	$0.992895\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−1.31324			−0.115179
$131$			14.2135i			1.24184i	0.783874	+	0.620920i	$0.213241\pi$
$131$			14.2135i			1.24184i	−0.783874	+	0.620920i	$0.786759\pi$
$132$		0			0
$133$	−0.454979	+	11.5236i	−0.0394517	+	0.999221i
$134$	−15.2257			−1.31530
$135$		0			0
$136$	2.71542			0.232846
$137$	−1.53219			−0.130904			−0.0654520	−	0.997856i	$-0.520849\pi$
$137$	−1.53219			−0.130904			−0.0654520	+	0.997856i	$0.520849\pi$
$138$		0			0
$139$		−	3.33601i		−	0.282956i	−0.989941	−	0.141478i	$-0.954815\pi$
$139$		−	3.33601i		−	0.282956i	0.989941	−	0.141478i	$-0.0451855\pi$
$140$	−2.99714	−	3.13440i	−0.253305	−	0.264905i
$141$		0			0
$142$	1.19798			0.100532
$143$	1.78747			0.149476
$144$		0			0
$145$	6.26880			0.520595
$146$	−6.99299			−0.578744
$147$		0			0
$148$			7.19617i			0.591522i
$149$	−20.1908			−1.65410			−0.827048	−	0.562132i	$-0.809981\pi$
$149$	−20.1908			−1.65410			−0.827048	+	0.562132i	$0.809981\pi$
$150$		0			0
$151$			8.02951i			0.653432i	0.945123	+	0.326716i	$0.105942\pi$
$151$			8.02951i			0.653432i	−0.945123	+	0.326716i	$0.894058\pi$
$152$	−3.02912	−	3.13440i	−0.245694	−	0.254233i
$153$		0			0
$154$	4.07945	+	4.26627i	0.328731	+	0.343786i
$155$		−	8.51122i		−	0.683637i
$156$		0			0
$157$		−	3.33601i		−	0.266242i	−0.991100	−	0.133121i	$-0.957500\pi$
$157$		−	3.33601i		−	0.266242i	0.991100	−	0.133121i	$-0.0424999\pi$
$158$	−0.503034			−0.0400192
$159$		0			0
$160$	1.63913			0.129585
$161$	−12.4244	+	11.8803i	−0.979177	+	0.936298i
$162$		0			0
$163$	−2.62648			−0.205722			−0.102861	−	0.994696i	$-0.532800\pi$
$163$	−2.62648			−0.205722			−0.102861	+	0.994696i	$0.532800\pi$
$164$	6.99299			0.546061
$165$		0			0
$166$	11.1494			0.865363
$167$	−10.9709			−0.848957			−0.424479	−	0.905438i	$-0.639543\pi$
$167$	−10.9709			−0.848957			−0.424479	+	0.905438i	$0.639543\pi$
$168$		0			0
$169$	−12.3581			−0.950624
$170$		−	4.45094i		−	0.341372i
$171$		0			0
$172$	−7.13770			−0.544244
$173$	−20.3652			−1.54834			−0.774168	−	0.632980i	$-0.781832\pi$
$173$	−20.3652			−1.54834			−0.774168	+	0.632980i	$0.781832\pi$
$174$		0			0
$175$	4.42345	−	4.22974i	0.334381	−	0.319738i
$176$	−2.23105			−0.168171
$177$		0			0
$178$			13.9860i			1.04829i
$179$		−	18.7866i		−	1.40418i	−0.712089	−	0.702089i	$-0.752251\pi$
$179$		−	18.7866i		−	1.40418i	0.712089	−	0.702089i	$-0.247749\pi$
$180$		0			0
$181$	0.250962			0.0186539			0.00932694	−	0.999957i	$-0.497031\pi$
$181$	0.250962			0.0186539			0.00932694	+	0.999957i	$0.497031\pi$
$182$	1.46495	+	1.53204i	0.108589	+	0.113562i
$183$		0			0
$184$		−	6.49732i		−	0.478989i
$185$	11.7955			0.867221
$186$		0			0
$187$			6.05823i			0.443022i
$188$		−	1.91424i		−	0.139610i
$189$		0			0
$190$	−5.13770	+	4.96513i	−0.372728	+	0.360208i
$191$	23.9540			1.73325			0.866627	−	0.498957i	$-0.166283\pi$
$191$	23.9540			1.73325			0.866627	+	0.498957i	$0.166283\pi$
$192$		0			0
$193$			10.4564i			0.752666i	0.926484	+	0.376333i	$0.122815\pi$
$193$			10.4564i			0.752666i	−0.926484	+	0.376333i	$0.877185\pi$
$194$			17.9811i			1.29097i
$195$		0			0
$196$	−0.313241	+	6.99299i	−0.0223743	+	0.499499i
$197$	0.195821			0.0139516			0.00697582	−	0.999976i	$-0.497780\pi$
$197$	0.195821			0.0139516			0.00697582	+	0.999976i	$0.497780\pi$
$198$		0			0
$199$			13.9860i			0.991440i	0.868483	+	0.495720i	$0.165096\pi$
$199$			13.9860i			0.991440i	−0.868483	+	0.495720i	$0.834904\pi$
$200$			2.31324i			0.163571i
$201$		0			0
$202$	−9.29610			−0.654072
$203$	−6.99299	−	7.31324i	−0.490812	−	0.513289i
$204$		0			0
$205$		−	11.4624i		−	0.800572i
$206$		−	12.0115i		−	0.836883i
$207$		0			0
$208$	−0.801180			−0.0555518
$209$	6.99299	−	6.75810i	0.483715	−	0.467467i
$210$		0			0
$211$		−	6.69314i		−	0.460775i	−0.973099	−	0.230387i	$-0.926001\pi$
$211$		−	6.69314i		−	0.460775i	0.973099	−	0.230387i	$-0.0739993\pi$
$212$		−	1.19798i		−	0.0822774i
$213$		0			0
$214$	−2.62648			−0.179543
$215$			11.6996i			0.797909i
$216$		0			0
$217$	−9.92927	+	9.49446i	−0.674043	+	0.644526i
$218$	−4.26627			−0.288948
$219$		0			0
$220$			3.65698i			0.246554i
$221$			2.17554i			0.146343i
$222$		0			0
$223$	3.03993			0.203569			0.101784	−	0.994806i	$-0.467545\pi$
$223$	3.03993			0.203569			0.101784	+	0.994806i	$0.467545\pi$
$224$	−1.82849	−	1.91223i	−0.122171	−	0.127766i
$225$		0			0
$226$	−11.5886			−0.770865
$227$	27.3582			1.81583			0.907913	−	0.419159i	$-0.137675\pi$
$227$	27.3582			1.81583			0.907913	+	0.419159i	$0.137675\pi$
$228$		0			0
$229$		−	5.84747i		−	0.386412i	−0.981158	−	0.193206i	$-0.938111\pi$
$229$		−	5.84747i		−	0.386412i	0.981158	−	0.193206i	$-0.0618885\pi$
$230$	−10.6500			−0.702238
$231$		0			0
$232$	3.82446			0.251088
$233$	3.06438			0.200754			0.100377	−	0.994949i	$-0.467995\pi$
$233$	3.06438			0.200754			0.100377	+	0.994949i	$0.467995\pi$
$234$		0			0
$235$	−3.13770			−0.204681
$236$	4.59174			0.298897
$237$		0			0
$238$	−5.19251	+	4.96513i	−0.336580	+	0.321841i
$239$	−15.4215			−0.997534			−0.498767	−	0.866736i	$-0.666214\pi$
$239$	−15.4215			−0.997534			−0.498767	+	0.866736i	$0.666214\pi$
$240$		0			0
$241$	−2.42769			−0.156381			−0.0781905	−	0.996938i	$-0.524914\pi$
$241$	−2.42769			−0.156381			−0.0781905	+	0.996938i	$0.524914\pi$
$242$			6.02243i			0.387137i
$243$		0			0
$244$			0.934757i			0.0598417i
$245$	11.4624	+	0.513444i	0.732309	+	0.0328027i
$246$		0			0
$247$	2.51122	−	2.42687i	0.159785	−	0.154418i
$248$		−	5.19251i		−	0.329725i
$249$		0			0
$250$	11.9874			0.758148
$251$		−	9.90835i		−	0.625409i	−0.949850	−	0.312705i	$-0.898765\pi$
$251$		−	9.90835i		−	0.625409i	0.949850	−	0.312705i	$-0.101235\pi$
$252$		0			0
$253$	14.4958			0.911344
$254$	0.503034			0.0315632
$255$		0			0
$256$	1.00000			0.0625000
$257$	−4.80250			−0.299572			−0.149786	−	0.988718i	$-0.547858\pi$
$257$	−4.80250			−0.299572			−0.149786	+	0.988718i	$0.547858\pi$
$258$		0			0
$259$	−13.1581	−	13.7607i	−0.817607	−	0.855050i
$260$			1.31324i			0.0814437i
$261$		0			0
$262$	14.2135			0.878113
$263$	7.89503			0.486828			0.243414	−	0.969922i	$-0.421733\pi$
$263$	7.89503			0.486828			0.243414	+	0.969922i	$0.421733\pi$
$264$		0			0
$265$	−1.96364			−0.120626
$266$	11.5236	+	0.454979i	0.706556	+	0.0278965i
$267$		0			0
$268$			15.2257i			0.930056i
$269$	8.78046			0.535354			0.267677	−	0.963509i	$-0.413744\pi$
$269$	8.78046			0.535354			0.267677	+	0.963509i	$0.413744\pi$
$270$		0			0
$271$			24.9569i			1.51603i	0.652240	+	0.758013i	$0.273829\pi$
$271$			24.9569i			1.51603i	−0.652240	+	0.758013i	$0.726171\pi$
$272$		−	2.71542i		−	0.164647i
$273$		0			0
$274$			1.53219i			0.0925630i
$275$	−5.16095			−0.311217
$276$		0			0
$277$	8.62648			0.518315			0.259158	−	0.965835i	$-0.416555\pi$
$277$	8.62648			0.518315			0.259158	+	0.965835i	$0.416555\pi$
$278$	−3.33601			−0.200080
$279$		0			0
$280$	−3.13440	+	2.99714i	−0.187316	+	0.179113i
$281$			26.8865i			1.60391i	0.597383	+	0.801956i	$0.296207\pi$
$281$			26.8865i			1.60391i	−0.597383	+	0.801956i	$0.703793\pi$
$282$		0			0
$283$		−	3.97796i		−	0.236465i	−0.992986	−	0.118233i	$-0.962277\pi$
$283$		−	3.97796i		−	0.236465i	0.992986	−	0.118233i	$-0.0377228\pi$
$284$		−	1.19798i		−	0.0710868i
$285$		0			0
$286$		−	1.78747i		−	0.105695i
$287$	−13.3722	+	12.7866i	−0.789336	+	0.754770i
$288$		0			0
$289$	9.62648			0.566264
$290$		−	6.26880i		−	0.368117i
$291$		0			0
$292$			6.99299i			0.409234i
$293$	3.97796			0.232395			0.116197	−	0.993226i	$-0.462929\pi$
$293$	3.97796			0.232395			0.116197	+	0.993226i	$0.462929\pi$
$294$		0			0
$295$		−	7.52647i		−	0.438208i
$296$	7.19617			0.418269
$297$		0			0
$298$			20.1908i			1.16962i
$299$	5.20552			0.301043
$300$		0			0
$301$	13.6489	−	13.0512i	0.786710	−	0.752260i
$302$	8.02951			0.462046
$303$		0			0
$304$	−3.13440	+	3.02912i	−0.179770	+	0.173732i
$305$	1.53219			0.0877330
$306$		0			0
$307$	19.8585			1.13339			0.566693	−	0.823929i	$-0.308222\pi$
$307$	19.8585			1.13339			0.566693	+	0.823929i	$0.308222\pi$
$308$	4.26627	−	4.07945i	0.243093	−	0.232448i
$309$		0			0
$310$	−8.51122			−0.483405
$311$		−	22.3230i		−	1.26582i	−0.774225	−	0.632911i	$-0.781860\pi$
$311$		−	22.3230i		−	1.26582i	0.774225	−	0.632911i	$-0.218140\pi$
$312$		0			0
$313$			5.12347i			0.289596i	0.989461	+	0.144798i	$0.0462532\pi$
$313$			5.12347i			0.289596i	−0.989461	+	0.144798i	$0.953747\pi$
$314$	−3.33601			−0.188262
$315$		0			0
$316$			0.503034i			0.0282979i
$317$			13.8794i			0.779547i	0.920911	+	0.389774i	$0.127447\pi$
$317$			13.8794i			0.779547i	−0.920911	+	0.389774i	$0.872553\pi$
$318$		0			0
$319$			8.53254i			0.477731i
$320$		−	1.63913i		−	0.0916304i
$321$		0			0
$322$	11.8803	+	12.4244i	0.662062	+	0.692382i
$323$	8.22533	+	8.51122i	0.457670	+	0.473577i
$324$		0			0
$325$	−1.85332			−0.102804
$326$			2.62648i			0.145467i
$327$		0			0
$328$		−	6.99299i		−	0.386123i
$329$	3.50018	+	3.66047i	0.192971	+	0.201808i
$330$		0			0
$331$			25.6821i			1.41161i	0.708405	+	0.705807i	$0.249415\pi$
$331$			25.6821i			1.41161i	−0.708405	+	0.705807i	$0.750585\pi$
$332$		−	11.1494i		−	0.611904i
$333$		0			0
$334$			10.9709i			0.600303i
$335$	24.9569			1.36354
$336$		0			0
$337$			1.92383i			0.104798i	0.998626	+	0.0523989i	$0.0166867\pi$
$337$			1.92383i			0.104798i	−0.998626	+	0.0523989i	$0.983313\pi$
$338$			12.3581i			0.672193i
$339$		0			0
$340$	−4.45094			−0.241386
$341$	11.5847			0.627348
$342$		0			0
$343$	−12.1876	−	13.9450i	−0.658070	−	0.752957i
$344$			7.13770i			0.384839i
$345$		0			0
$346$			20.3652i			1.09484i
$347$	34.0801			1.82952			0.914758	−	0.404002i	$-0.132381\pi$
$347$	34.0801			1.82952			0.914758	+	0.404002i	$0.132381\pi$
$348$		0			0
$349$		−	31.9499i		−	1.71024i	−0.518431	−	0.855120i	$-0.673484\pi$
$349$		−	31.9499i		−	1.71024i	0.518431	−	0.855120i	$-0.326516\pi$
$350$	−4.22974	−	4.42345i	−0.226089	−	0.236443i
$351$		0			0
$352$			2.23105i			0.118915i
$353$		−	19.2957i		−	1.02701i	−0.858088	−	0.513503i	$-0.828348\pi$
$353$		−	19.2957i		−	1.02701i	0.858088	−	0.513503i	$-0.171652\pi$
$354$		0			0
$355$	−1.96364			−0.104219
$356$	13.9860			0.741255
$357$		0			0
$358$	−18.7866			−0.992904
$359$	−9.56170			−0.504647			−0.252324	−	0.967643i	$-0.581195\pi$
$359$	−9.56170			−0.504647			−0.252324	+	0.967643i	$0.581195\pi$
$360$		0			0
$361$	0.648917	−	18.9889i	0.0341535	−	0.999417i
$362$		−	0.250962i		−	0.0131903i
$363$		0			0
$364$	1.53204	−	1.46495i	0.0803007	−	0.0767843i
$365$	11.4624			0.599972
$366$		0			0
$367$			10.9709i			0.572679i	0.958128	+	0.286339i	$0.0924385\pi$
$367$			10.9709i			0.572679i	−0.958128	+	0.286339i	$0.907562\pi$
$368$	−6.49732			−0.338696
$369$		0			0
$370$		−	11.7955i		−	0.613218i
$371$	2.19049	+	2.29081i	0.113725	+	0.118933i
$372$		0			0
$373$		−	33.7116i		−	1.74552i	−0.488151	−	0.872759i	$-0.662328\pi$
$373$		−	33.7116i		−	1.74552i	0.488151	−	0.872759i	$-0.337672\pi$
$374$	6.05823			0.313264
$375$		0			0
$376$	−1.91424			−0.0987195
$377$			3.06408i			0.157808i
$378$		0			0
$379$			28.6120i			1.46970i	0.678231	+	0.734849i	$0.262747\pi$
$379$			28.6120i			1.46970i	−0.678231	+	0.734849i	$0.737253\pi$
$380$	4.96513	+	5.13770i	0.254706	+	0.263558i
$381$		0			0
$382$		−	23.9540i		−	1.22560i
$383$	−18.7885			−0.960046			−0.480023	−	0.877256i	$-0.659372\pi$
$383$	−18.7885			−0.960046			−0.480023	+	0.877256i	$0.659372\pi$
$384$		0			0
$385$	−6.68676	−	6.99299i	−0.340789	−	0.356396i
$386$	10.4564			0.532215
$387$		0			0
$388$	17.9811			0.912850
$389$	17.1264			0.868344			0.434172	−	0.900830i	$-0.357041\pi$
$389$	17.1264			0.868344			0.434172	+	0.900830i	$0.357041\pi$
$390$		0			0
$391$			17.6430i			0.892243i
$392$	6.99299	+	0.313241i	0.353199	+	0.0158211i
$393$		0			0
$394$		−	0.195821i		−	0.00986531i
$395$	0.824540			0.0414871
$396$		0			0
$397$		−	1.57671i		−	0.0791328i	−0.999217	−	0.0395664i	$-0.987402\pi$
$397$		−	1.57671i		−	0.0791328i	0.999217	−	0.0395664i	$-0.0125977\pi$
$398$	13.9860			0.701054
$399$		0			0
$400$	2.31324			0.115662
$401$			18.2600i			0.911860i	0.890016	+	0.455930i	$0.150693\pi$
$401$			18.2600i			0.911860i	−0.890016	+	0.455930i	$0.849307\pi$
$402$		0			0
$403$	4.16013			0.207231
$404$			9.29610i			0.462498i
$405$		0			0
$406$	−7.31324	+	6.99299i	−0.362950	+	0.347056i
$407$			16.0550i			0.795816i
$408$		0			0
$409$	−27.3139			−1.35059			−0.675294	−	0.737549i	$-0.735983\pi$
$409$	−27.3139			−1.35059			−0.675294	+	0.737549i	$0.735983\pi$
$410$	−11.4624			−0.566090
$411$		0			0
$412$	−12.0115			−0.591765
$413$	−8.78046	+	8.39595i	−0.432058	+	0.413138i
$414$		0			0
$415$	−18.2754			−0.897104
$416$			0.801180i			0.0392811i
$417$		0			0
$418$	−6.75810	−	6.99299i	−0.330549	−	0.342038i
$419$			2.72805i			0.133274i	0.997777	+	0.0666370i	$0.0212269\pi$
$419$			2.72805i			0.133274i	−0.997777	+	0.0666370i	$0.978773\pi$
$420$		0			0
$421$		−	10.1261i		−	0.493515i	−0.969077	−	0.246757i	$-0.920635\pi$
$421$		−	10.1261i		−	0.493515i	0.969077	−	0.246757i	$-0.0793650\pi$
$422$	−6.69314			−0.325817
$423$		0			0
$424$	−1.19798			−0.0581789
$425$		−	6.28143i		−	0.304694i
$426$		0			0
$427$	−1.70919	−	1.78747i	−0.0827137	−	0.0865017i
$428$			2.62648i			0.126956i
$429$		0			0
$430$	11.6996			0.564207
$431$		−	5.02243i		−	0.241922i	−0.992657	−	0.120961i	$-0.961402\pi$
$431$		−	5.02243i		−	0.241922i	0.992657	−	0.120961i	$-0.0385976\pi$
$432$		0			0
$433$	18.5313			0.890557			0.445278	−	0.895392i	$-0.353105\pi$
$433$	18.5313			0.890557			0.445278	+	0.895392i	$0.353105\pi$
$434$	9.49446	+	9.92927i	0.455749	+	0.476620i
$435$		0			0
$436$			4.26627i			0.204317i
$437$	20.3652	−	19.6811i	0.974199	−	0.941476i
$438$		0			0
$439$	26.6534			1.27210			0.636049	−	0.771649i	$-0.280568\pi$
$439$	26.6534			1.27210			0.636049	+	0.771649i	$0.280568\pi$
$440$	3.65698			0.174340
$441$		0			0
$442$	2.17554			0.103480
$443$	0.698855			0.0332036			0.0166018	−	0.999862i	$-0.494715\pi$
$443$	0.698855			0.0332036			0.0166018	+	0.999862i	$0.494715\pi$
$444$		0			0
$445$		−	22.9249i		−	1.08674i
$446$		−	3.03993i		−	0.143945i
$447$		0			0
$448$	−1.91223	+	1.82849i	−0.0903443	+	0.0863881i
$449$		−	26.3203i		−	1.24213i	−0.783759	−	0.621065i	$-0.786700\pi$
$449$		−	26.3203i		−	1.24213i	0.783759	−	0.621065i	$-0.213300\pi$
$450$		0			0
$451$	15.6017			0.734654
$452$			11.5886i			0.545084i
$453$		0			0
$454$		−	27.3582i		−	1.28398i
$455$	−2.40125	−	2.51122i	−0.112572	−	0.117728i
$456$		0			0
$457$	2.97757			0.139285			0.0696423	−	0.997572i	$-0.477814\pi$
$457$	2.97757			0.139285			0.0696423	+	0.997572i	$0.477814\pi$
$458$	−5.84747			−0.273234
$459$		0			0
$460$			10.6500i			0.496557i
$461$		−	38.0779i		−	1.77347i	−0.462281	−	0.886733i	$-0.652969\pi$
$461$		−	38.0779i		−	1.77347i	0.462281	−	0.886733i	$-0.347031\pi$
$462$		0			0
$463$	−8.72634			−0.405547			−0.202774	−	0.979226i	$-0.564996\pi$
$463$	−8.72634			−0.405547			−0.202774	+	0.979226i	$0.564996\pi$
$464$		−	3.82446i		−	0.177546i
$465$		0			0
$466$		−	3.06438i		−	0.141955i
$467$		−	3.63955i		−	0.168418i	−0.996448	−	0.0842092i	$-0.973164\pi$
$467$		−	3.63955i		−	0.168418i	0.996448	−	0.0842092i	$-0.0268364\pi$
$468$		0			0
$469$	−27.8400	−	29.1150i	−1.28553	−	1.34441i
$470$			3.13770i			0.144731i
$471$		0			0
$472$		−	4.59174i		−	0.211352i
$473$	−15.9245			−0.732211
$474$		0			0
$475$	−7.25062	+	7.00707i	−0.332681	+	0.321507i
$476$	4.96513	+	5.19251i	0.227576	+	0.237998i
$477$		0			0
$478$			15.4215i			0.705363i
$479$		−	25.5277i		−	1.16639i	−0.812332	−	0.583196i	$-0.801802\pi$
$479$		−	25.5277i		−	1.16639i	0.812332	−	0.583196i	$-0.198198\pi$
$480$		0			0
$481$			5.76543i			0.262881i
$482$			2.42769i			0.110578i
$483$		0			0
$484$	6.02243			0.273747
$485$		−	29.4734i		−	1.33832i
$486$		0			0
$487$			31.9605i			1.44827i	0.689660	+	0.724133i	$0.257760\pi$
$487$			31.9605i			1.44827i	−0.689660	+	0.724133i	$0.742240\pi$
$488$	0.934757			0.0423144
$489$		0			0
$490$	0.513444	−	11.4624i	0.0231950	−	0.517820i
$491$	0.307213			0.0138643			0.00693217	−	0.999976i	$-0.497793\pi$
$491$	0.307213			0.0138643			0.00693217	+	0.999976i	$0.497793\pi$
$492$		0			0
$493$	−10.3850			−0.467718
$494$	−2.42687	−	2.51122i	−0.109190	−	0.112985i
$495$		0			0
$496$	−5.19251			−0.233151
$497$	2.19049	+	2.29081i	0.0982569	+	0.102757i
$498$		0			0
$499$	14.4355			0.646223			0.323112	−	0.946361i	$-0.395271\pi$
$499$	14.4355			0.646223			0.323112	+	0.946361i	$0.395271\pi$
$500$		−	11.9874i		−	0.536092i
$501$		0			0
$502$	−9.90835			−0.442231
$503$			10.5751i			0.471519i	0.971811	+	0.235759i	$0.0757577\pi$
$503$			10.5751i			0.471519i	−0.971811	+	0.235759i	$0.924242\pi$
$504$		0			0
$505$	15.2376			0.678062
$506$		−	14.4958i		−	0.644417i
$507$		0			0
$508$		−	0.503034i		−	0.0223185i
$509$	−30.7223			−1.36174			−0.680872	−	0.732402i	$-0.738399\pi$
$509$	−30.7223			−1.36174			−0.680872	+	0.732402i	$0.738399\pi$
$510$		0			0
$511$	−12.7866	−	13.3722i	−0.565647	−	0.591551i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			4.80250i			0.211829i
$515$			19.6885i			0.867579i
$516$		0			0
$517$		−	4.27076i		−	0.187828i
$518$	−13.7607	+	13.1581i	−0.604612	+	0.578135i
$519$		0			0
$520$	1.31324			0.0575894
$521$	40.7304			1.78443			0.892215	−	0.451611i	$-0.149151\pi$
$521$	40.7304			1.78443			0.892215	+	0.451611i	$0.149151\pi$
$522$		0			0
$523$	−8.42138			−0.368241			−0.184121	−	0.982904i	$-0.558944\pi$
$523$	−8.42138			−0.368241			−0.184121	+	0.982904i	$0.558944\pi$
$524$		−	14.2135i		−	0.620920i
$525$		0			0
$526$		−	7.89503i		−	0.344239i
$527$			14.0999i			0.614199i
$528$		0			0
$529$	19.2151			0.835440
$530$			1.96364i			0.0852952i
$531$		0			0
$532$	0.454979	−	11.5236i	0.0197258	−	0.499611i
$533$	5.60264			0.242677
$534$		0			0
$535$	4.30515			0.186128
$536$	15.2257			0.657649
$537$		0			0
$538$		−	8.78046i		−	0.378552i
$539$	−0.698855	+	15.6017i	−0.0301018	+	0.672012i
$540$		0			0
$541$	11.8245			0.508373			0.254187	−	0.967155i	$-0.418192\pi$
$541$	11.8245			0.508373			0.254187	+	0.967155i	$0.418192\pi$
$542$	24.9569			1.07199
$543$		0			0
$544$	−2.71542			−0.116423
$545$	6.99299			0.299547
$546$		0			0
$547$		−	10.6291i		−	0.454468i	−0.973840	−	0.227234i	$-0.927032\pi$
$547$		−	10.6291i		−	0.454468i	0.973840	−	0.227234i	$-0.0729682\pi$
$548$	1.53219			0.0654520
$549$		0			0
$550$			5.16095i			0.220063i
$551$	11.5847	+	11.9874i	0.493526	+	0.510680i
$552$		0			0
$553$	−0.919793	−	0.961916i	−0.0391136	−	0.0409048i
$554$		−	8.62648i		−	0.366504i
$555$		0			0
$556$			3.33601i			0.141478i
$557$	−15.1143			−0.640413			−0.320206	−	0.947348i	$-0.603752\pi$
$557$	−15.1143			−0.640413			−0.320206	+	0.947348i	$0.603752\pi$
$558$		0			0
$559$	−5.71858			−0.241870
$560$	2.99714	+	3.13440i	0.126652	+	0.132453i
$561$		0			0
$562$	26.8865			1.13414
$563$	36.1386			1.52306			0.761531	−	0.648129i	$-0.224448\pi$
$563$	36.1386			1.52306			0.761531	+	0.648129i	$0.224448\pi$
$564$		0			0
$565$	18.9953			0.799139
$566$	−3.97796			−0.167206
$567$		0			0
$568$	−1.19798			−0.0502660
$569$			17.2530i			0.723282i	0.932317	+	0.361641i	$0.117783\pi$
$569$			17.2530i			0.723282i	−0.932317	+	0.361641i	$0.882217\pi$
$570$		0			0
$571$	−39.4580			−1.65126			−0.825632	−	0.564209i	$-0.809181\pi$
$571$	−39.4580			−1.65126			−0.825632	+	0.564209i	$0.809181\pi$
$572$	−1.78747			−0.0747378
$573$		0			0
$574$	12.7866	+	13.3722i	0.533703	+	0.558145i
$575$	−15.0299			−0.626788
$576$		0			0
$577$		−	28.6139i		−	1.19121i	−0.803277	−	0.595606i	$-0.796912\pi$
$577$		−	28.6139i		−	1.19121i	0.803277	−	0.595606i	$-0.203088\pi$
$578$		−	9.62648i		−	0.400409i
$579$		0			0
$580$	−6.26880			−0.260298
$581$	20.3866	+	21.3203i	0.845780	+	0.884513i
$582$		0			0
$583$		−	2.67274i		−	0.110694i
$584$	6.99299			0.289372
$585$		0			0
$586$		−	3.97796i		−	0.164328i
$587$		−	42.3045i		−	1.74609i	−0.487637	−	0.873046i	$-0.662141\pi$
$587$		−	42.3045i		−	1.74609i	0.487637	−	0.873046i	$-0.337859\pi$
$588$		0			0
$589$	16.2754	−	15.7287i	0.670616	−	0.648090i
$590$	−7.52647			−0.309860
$591$		0			0
$592$		−	7.19617i		−	0.295761i
$593$			40.4270i			1.66014i	0.557660	+	0.830069i	$0.311699\pi$
$593$			40.4270i			1.66014i	−0.557660	+	0.830069i	$0.688301\pi$
$594$		0			0
$595$	8.51122	−	8.13850i	0.348926	−	0.333646i
$596$	20.1908			0.827048
$597$		0			0
$598$		−	5.20552i		−	0.212870i
$599$			1.72460i			0.0704653i	0.999379	+	0.0352327i	$0.0112172\pi$
$599$			1.72460i			0.0704653i	−0.999379	+	0.0352327i	$0.988783\pi$
$600$		0			0
$601$	−23.3384			−0.951992			−0.475996	−	0.879448i	$-0.657912\pi$
$601$	−23.3384			−0.951992			−0.475996	+	0.879448i	$0.657912\pi$
$602$	−13.0512	−	13.6489i	−0.531928	−	0.556288i
$603$		0			0
$604$		−	8.02951i		−	0.326716i
$605$		−	9.87158i		−	0.401337i
$606$		0			0
$607$	15.9388			0.646936			0.323468	−	0.946239i	$-0.395151\pi$
$607$	15.9388			0.646936			0.323468	+	0.946239i	$0.395151\pi$
$608$	3.02912	+	3.13440i	0.122847	+	0.127117i
$609$		0			0
$610$		−	1.53219i		−	0.0620366i
$611$		−	1.53365i		−	0.0620449i
$612$		0			0
$613$	23.9243			0.966294			0.483147	−	0.875539i	$-0.339494\pi$
$613$	23.9243			0.966294			0.483147	+	0.875539i	$0.339494\pi$
$614$		−	19.8585i		−	0.801425i
$615$		0			0
$616$	−4.07945	−	4.26627i	−0.164366	−	0.171893i
$617$	31.8491			1.28220			0.641098	−	0.767459i	$-0.278479\pi$
$617$	31.8491			1.28220			0.641098	+	0.767459i	$0.278479\pi$
$618$		0			0
$619$		−	6.67201i		−	0.268171i	−0.990970	−	0.134085i	$-0.957190\pi$
$619$		−	6.67201i		−	0.268171i	0.990970	−	0.134085i	$-0.0428096\pi$
$620$			8.51122i			0.341819i
$621$		0			0
$622$	−22.3230			−0.895071
$623$	−26.7444	+	25.5732i	−1.07149	+	1.02457i
$624$		0			0
$625$	−8.08271			−0.323308
$626$	5.12347			0.204775
$627$		0			0
$628$			3.33601i			0.133121i
$629$	−19.5406			−0.779136
$630$		0			0
$631$	−21.7488			−0.865805			−0.432903	−	0.901441i	$-0.642511\pi$
$631$	−21.7488			−0.865805			−0.432903	+	0.901441i	$0.642511\pi$
$632$	0.503034			0.0200096
$633$		0			0
$634$	13.8794			0.551223
$635$	−0.824540			−0.0327209
$636$		0			0
$637$	−0.250962	+	5.60264i	−0.00994349	+	0.221985i
$638$	8.53254			0.337807
$639$		0			0
$640$	−1.63913			−0.0647924
$641$			18.9019i			0.746579i	0.927715	+	0.373290i	$0.121770\pi$
$641$			18.9019i			0.746579i	−0.927715	+	0.373290i	$0.878230\pi$
$642$		0			0
$643$			23.3802i			0.922026i	0.887393	+	0.461013i	$0.152514\pi$
$643$			23.3802i			0.922026i	−0.887393	+	0.461013i	$0.847486\pi$
$644$	12.4244	−	11.8803i	0.489588	−	0.468149i
$645$		0			0
$646$	8.51122	−	8.22533i	0.334869	−	0.323621i
$647$		−	45.2456i		−	1.77879i	−0.457141	−	0.889394i	$-0.651126\pi$
$647$		−	45.2456i		−	1.77879i	0.457141	−	0.889394i	$-0.348874\pi$
$648$		0			0
$649$	10.2444			0.402127
$650$			1.85332i			0.0726933i
$651$		0			0
$652$	2.62648			0.102861
$653$	12.1844			0.476812			0.238406	−	0.971166i	$-0.423375\pi$
$653$	12.1844			0.476812			0.238406	+	0.971166i	$0.423375\pi$
$654$		0			0
$655$	−23.2978			−0.910322
$656$	−6.99299			−0.273030
$657$		0			0
$658$	3.66047	−	3.50018i	0.142700	−	0.136451i
$659$			11.8091i			0.460015i	0.973189	+	0.230008i	$0.0738751\pi$
$659$			11.8091i			0.460015i	−0.973189	+	0.230008i	$0.926125\pi$
$660$		0			0
$661$	−29.0811			−1.13112			−0.565562	−	0.824706i	$-0.691341\pi$
$661$	−29.0811			−1.13112			−0.565562	+	0.824706i	$0.691341\pi$
$662$	25.6821			0.998161
$663$		0			0
$664$	−11.1494			−0.432682
$665$	−18.8887	−	0.745771i	−0.732472	−	0.0289198i
$666$		0			0
$667$			24.8487i			0.962146i
$668$	10.9709			0.424479
$669$		0			0
$670$		−	24.9569i		−	0.964170i
$671$			2.08548i			0.0805092i
$672$		0			0
$673$			10.4564i			0.403064i	0.979482	+	0.201532i	$0.0645919\pi$
$673$			10.4564i			0.403064i	−0.979482	+	0.201532i	$0.935408\pi$
$674$	1.92383			0.0741033
$675$		0			0
$676$	12.3581			0.475312
$677$	24.7462			0.951072			0.475536	−	0.879696i	$-0.342254\pi$
$677$	24.7462			0.951072			0.475536	+	0.879696i	$0.342254\pi$
$678$		0			0
$679$	−34.3839	+	32.8782i	−1.31953	+	1.26175i
$680$			4.45094i			0.170686i
$681$		0			0
$682$		−	11.5847i		−	0.443602i
$683$		−	8.74175i		−	0.334494i	−0.985915	−	0.167247i	$-0.946512\pi$
$683$		−	8.74175i		−	0.334494i	0.985915	−	0.167247i	$-0.0534877\pi$
$684$		0			0
$685$		−	2.51147i		−	0.0959582i
$686$	−13.9450	+	12.1876i	−0.532421	+	0.465326i
$687$		0			0
$688$	7.13770			0.272122
$689$		−	0.959795i		−	0.0365653i
$690$		0			0
$691$		−	22.5557i		−	0.858058i	−0.903291	−	0.429029i	$-0.858856\pi$
$691$		−	22.5557i		−	0.858058i	0.903291	−	0.429029i	$-0.141144\pi$
$692$	20.3652			0.774168
$693$		0			0
$694$		−	34.0801i		−	1.29366i
$695$	5.46816			0.207419
$696$		0			0
$697$			18.9889i			0.719256i
$698$	−31.9499			−1.20932
$699$		0			0
$700$	−4.42345	+	4.22974i	−0.167191	+	0.159869i
$701$	−5.79846			−0.219005			−0.109502	−	0.993987i	$-0.534926\pi$
$701$	−5.79846			−0.219005			−0.109502	+	0.993987i	$0.534926\pi$
$702$		0			0
$703$	21.7980	+	22.5557i	0.822128	+	0.850703i
$704$	2.23105			0.0840857
$705$		0			0
$706$	−19.2957			−0.726203
$707$	−16.9978	−	17.7763i	−0.639270	−	0.668546i
$708$		0			0
$709$	41.0466			1.54154			0.770769	−	0.637115i	$-0.219872\pi$
$709$	41.0466			1.54154			0.770769	+	0.637115i	$0.219872\pi$
$710$			1.96364i			0.0736942i
$711$		0			0
$712$		−	13.9860i		−	0.524147i
$713$	33.7374			1.26347
$714$		0			0
$715$			2.92990i			0.109572i
$716$			18.7866i			0.702089i
$717$		0			0
$718$			9.56170i			0.356839i
$719$			14.1894i			0.529174i	0.964362	+	0.264587i	$0.0852356\pi$
$719$			14.1894i			0.529174i	−0.964362	+	0.264587i	$0.914764\pi$
$720$		0			0
$721$	22.9688	−	21.9630i	0.855402	−	0.817944i
$722$	−18.9889	−	0.648917i	−0.706694	−	0.0241502i
$723$		0			0
$724$	−0.250962			−0.00932694
$725$		−	8.84689i		−	0.328565i
$726$		0			0
$727$			38.9429i			1.44431i	0.691730	+	0.722156i	$0.256849\pi$
$727$			38.9429i			1.44431i	−0.691730	+	0.722156i	$0.743151\pi$
$728$	−1.46495	−	1.53204i	−0.0542947	−	0.0567812i
$729$		0			0
$730$		−	11.4624i		−	0.424244i
$731$		−	19.3819i		−	0.716864i
$732$		0			0
$733$			36.7524i			1.35748i	0.734378	+	0.678741i	$0.237474\pi$
$733$			36.7524i			1.35748i	−0.734378	+	0.678741i	$0.762526\pi$
$734$	10.9709			0.404945
$735$		0			0
$736$			6.49732i			0.239494i
$737$			33.9692i			1.25127i
$738$		0			0
$739$	−3.29783			−0.121313			−0.0606564	−	0.998159i	$-0.519319\pi$
$739$	−3.29783			−0.121313			−0.0606564	+	0.998159i	$0.519319\pi$
$740$	−11.7955			−0.433611
$741$		0			0
$742$	2.29081	−	2.19049i	0.0840981	−	0.0804154i
$743$			37.8693i			1.38929i	0.719352	+	0.694645i	$0.244439\pi$
$743$			37.8693i			1.38929i	−0.719352	+	0.694645i	$0.755561\pi$
$744$		0			0
$745$		−	33.0954i		−	1.21252i
$746$	−33.7116			−1.23427
$747$		0			0
$748$		−	6.05823i		−	0.221511i
$749$	−4.80250	−	5.02243i	−0.175479	−	0.183516i
$750$		0			0
$751$			45.3467i			1.65473i	0.561668	+	0.827363i	$0.310160\pi$
$751$			45.3467i			1.65473i	−0.561668	+	0.827363i	$0.689840\pi$
$752$			1.91424i			0.0698052i
$753$		0			0
$754$	3.06408			0.111587
$755$	−13.1614			−0.478994
$756$		0			0
$757$	−21.2978			−0.774083			−0.387041	−	0.922062i	$-0.626503\pi$
$757$	−21.2978			−0.774083			−0.387041	+	0.922062i	$0.626503\pi$
$758$	28.6120			1.03923
$759$		0			0
$760$	5.13770	−	4.96513i	0.186364	−	0.180104i
$761$			7.57079i			0.274441i	0.990541	+	0.137221i	$0.0438169\pi$
$761$			7.57079i			0.274441i	−0.990541	+	0.137221i	$0.956183\pi$
$762$		0			0
$763$	−7.80084	−	8.15809i	−0.282409	−	0.295343i
$764$	−23.9540			−0.866627
$765$		0			0
$766$			18.7885i			0.678855i
$767$	3.67881			0.132834
$768$		0			0
$769$		−	9.50445i		−	0.342739i	−0.985207	−	0.171370i	$-0.945181\pi$
$769$		−	9.50445i		−	0.342739i	0.985207	−	0.171370i	$-0.0548193\pi$
$770$	−6.99299	+	6.68676i	−0.252010	+	0.240974i
$771$		0			0
$772$		−	10.4564i		−	0.376333i
$773$	−0.403021			−0.0144956			−0.00724782	−	0.999974i	$-0.502307\pi$
$773$	−0.403021			−0.0144956			−0.00724782	+	0.999974i	$0.502307\pi$
$774$		0			0
$775$	−12.0115			−0.431467
$776$		−	17.9811i		−	0.645483i
$777$		0			0
$778$		−	17.1264i		−	0.614012i
$779$	21.9188	−	21.1826i	0.785323	−	0.758944i
$780$		0			0
$781$		−	2.67274i		−	0.0956382i
$782$	17.6430			0.630911
$783$		0			0
$784$	0.313241	−	6.99299i	0.0111872	−	0.249750i
$785$	5.46816			0.195167
$786$		0			0
$787$	34.3115			1.22307			0.611536	−	0.791216i	$-0.290552\pi$
$787$	34.3115			1.22307			0.611536	+	0.791216i	$0.290552\pi$
$788$	−0.195821			−0.00697582
$789$		0			0
$790$		−	0.824540i		−	0.0293358i
$791$	−21.1897	−	22.1601i	−0.753420	−	0.787924i
$792$		0			0
$793$			0.748908i			0.0265945i
$794$	−1.57671			−0.0559553
$795$		0			0
$796$		−	13.9860i		−	0.495720i
$797$	6.37921			0.225963			0.112982	−	0.993597i	$-0.463960\pi$
$797$	6.37921			0.225963			0.112982	+	0.993597i	$0.463960\pi$
$798$		0			0
$799$	5.19798			0.183891
$800$		−	2.31324i		−	0.0817854i
$801$		0			0
$802$	18.2600			0.644783
$803$			15.6017i			0.550571i
$804$		0			0
$805$	−19.4734	−	20.3652i	−0.686346	−	0.717778i
$806$		−	4.16013i		−	0.146534i
$807$		0			0
$808$	9.29610			0.327036
$809$	22.4449			0.789122			0.394561	−	0.918870i	$-0.370897\pi$
$809$	22.4449			0.789122			0.394561	+	0.918870i	$0.370897\pi$
$810$		0			0
$811$	10.3850			0.364667			0.182334	−	0.983237i	$-0.441635\pi$
$811$	10.3850			0.364667			0.182334	+	0.983237i	$0.441635\pi$
$812$	6.99299	+	7.31324i	0.245406	+	0.256644i
$813$		0			0
$814$	16.0550			0.562727
$815$		−	4.30515i		−	0.150803i
$816$		0			0
$817$	−22.3724	+	21.6209i	−0.782711	+	0.756420i
$818$			27.3139i			0.955010i
$819$		0			0
$820$			11.4624i			0.400286i
$821$	29.2495			1.02081			0.510407	−	0.859933i	$-0.329495\pi$
$821$	29.2495			1.02081			0.510407	+	0.859933i	$0.329495\pi$
$822$		0			0
$823$	21.8143			0.760400			0.380200	−	0.924904i	$-0.375855\pi$
$823$	21.8143			0.760400			0.380200	+	0.924904i	$0.375855\pi$
$824$			12.0115i			0.418441i
$825$		0			0
$826$	8.39595	+	8.78046i	0.292133	+	0.305511i
$827$		−	13.5337i		−	0.470611i	−0.971921	−	0.235306i	$-0.924391\pi$
$827$		−	13.5337i		−	0.470611i	0.971921	−	0.235306i	$-0.0756091\pi$
$828$		0			0
$829$	−26.5989			−0.923819			−0.461910	−	0.886927i	$-0.652836\pi$
$829$	−26.5989			−0.923819			−0.461910	+	0.886927i	$0.652836\pi$
$830$			18.2754i			0.634348i
$831$		0			0
$832$	0.801180			0.0277759
$833$	−18.9889	−	0.850581i	−0.657927	−	0.0294709i
$834$		0			0
$835$		−	17.9828i		−	0.622322i
$836$	−6.99299	+	6.75810i	−0.241858	+	0.233734i
$837$		0			0
$838$	2.72805			0.0942389
$839$	51.7013			1.78493			0.892464	−	0.451119i	$-0.148975\pi$
$839$	51.7013			1.78493			0.892464	+	0.451119i	$0.148975\pi$
$840$		0			0
$841$	14.3735			0.495639
$842$	−10.1261			−0.348967
$843$		0			0
$844$			6.69314i			0.230387i
$845$		−	20.2566i		−	0.696848i
$846$		0			0
$847$	−11.5163	+	11.0120i	−0.395704	+	0.378376i
$848$			1.19798i			0.0411387i
$849$		0			0
$850$	−6.28143			−0.215451
$851$			46.7558i			1.60277i
$852$		0			0
$853$			12.4093i			0.424885i	0.977174	+	0.212443i	$0.0681419\pi$
$853$			12.4093i			0.424885i	−0.977174	+	0.212443i	$0.931858\pi$
$854$	−1.78747	+	1.70919i	−0.0611659	+	0.0584874i
$855$		0			0
$856$	2.62648			0.0897713
$857$	6.03006			0.205983			0.102991	−	0.994682i	$-0.467159\pi$
$857$	6.03006			0.205983			0.102991	+	0.994682i	$0.467159\pi$
$858$		0			0
$859$			21.2999i			0.726745i	0.931644	+	0.363372i	$0.118375\pi$
$859$			21.2999i			0.726745i	−0.931644	+	0.363372i	$0.881625\pi$
$860$		−	11.6996i		−	0.398955i
$861$		0			0
$862$	−5.02243			−0.171065
$863$		−	44.9261i		−	1.52930i	−0.644445	−	0.764650i	$-0.722912\pi$
$863$		−	44.9261i		−	1.52930i	0.644445	−	0.764650i	$-0.277088\pi$
$864$		0			0
$865$		−	33.3813i		−	1.13500i
$866$		−	18.5313i		−	0.629719i
$867$		0			0
$868$	9.92927	−	9.49446i	0.337021	−	0.322263i
$869$			1.12229i			0.0380711i
$870$		0			0
$871$			12.1985i			0.413331i
$872$	4.26627			0.144474
$873$		0			0
$874$	−19.6811	−	20.3652i	−0.665724	−	0.688863i
$875$	21.9188	+	22.9226i	0.740991	+	0.774926i
$876$		0			0
$877$			42.5013i			1.43517i	0.696473	+	0.717583i	$0.254752\pi$
$877$			42.5013i			1.43517i	−0.696473	+	0.717583i	$0.745248\pi$
$878$		−	26.6534i		−	0.899509i
$879$		0			0
$880$		−	3.65698i		−	0.123277i
$881$		−	28.6286i		−	0.964521i	−0.876028	−	0.482261i	$-0.839816\pi$
$881$		−	28.6286i		−	0.964521i	0.876028	−	0.482261i	$-0.160184\pi$
$882$		0			0
$883$	−36.0396			−1.21283			−0.606414	−	0.795149i	$-0.707393\pi$
$883$	−36.0396			−1.21283			−0.606414	+	0.795149i	$0.707393\pi$
$884$		−	2.17554i		−	0.0731714i
$885$		0			0
$886$		−	0.698855i		−	0.0234785i
$887$	−17.5609			−0.589638			−0.294819	−	0.955553i	$-0.595259\pi$
$887$	−17.5609			−0.589638			−0.294819	+	0.955553i	$0.595259\pi$
$888$		0			0
$889$	0.919793	+	0.961916i	0.0308489	+	0.0322616i
$890$	−22.9249			−0.768444
$891$		0			0
$892$	−3.03993			−0.101784
$893$	−5.79846	−	6.00000i	−0.194038	−	0.200782i
$894$		0			0
$895$	30.7938			1.02932
$896$	1.82849	+	1.91223i	0.0610856	+	0.0638831i
$897$		0			0
$898$	−26.3203			−0.878319
$899$			19.8585i			0.662319i
$900$		0			0
$901$	3.25301			0.108374
$902$		−	15.6017i		−	0.519479i
$903$		0			0
$904$	11.5886			0.385432
$905$			0.411361i			0.0136741i
$906$		0			0
$907$		−	28.6120i		−	0.950044i	−0.879974	−	0.475022i	$-0.842440\pi$
$907$		−	28.6120i		−	0.950044i	0.879974	−	0.475022i	$-0.157560\pi$
$908$	−27.3582			−0.907913
$909$		0			0
$910$	−2.51122	+	2.40125i	−0.0832460	+	0.0796006i
$911$			0.230528i			0.00763775i	0.999993	+	0.00381887i	$0.00121559\pi$
$911$			0.230528i			0.00763775i	−0.999993	+	0.00381887i	$0.998784\pi$
$912$		0			0
$913$		−	24.8749i		−	0.823238i
$914$		−	2.97757i		−	0.0984891i
$915$		0			0
$916$			5.84747i			0.193206i
$917$	25.9893	+	27.1795i	0.858241	+	0.897545i
$918$		0			0
$919$	−37.8038			−1.24703			−0.623515	−	0.781811i	$-0.714296\pi$
$919$	−37.8038			−1.24703			−0.623515	+	0.781811i	$0.714296\pi$
$920$	10.6500			0.351119
$921$		0			0
$922$	−38.0779			−1.25403
$923$		−	0.959795i		−	0.0315920i
$924$		0			0
$925$		−	16.6465i		−	0.547333i
$926$			8.72634i			0.286765i
$927$		0			0
$928$	−3.82446			−0.125544
$929$			32.3352i			1.06088i	0.847721	+	0.530442i	$0.177974\pi$
$929$			32.3352i			1.06088i	−0.847721	+	0.530442i	$0.822026\pi$
$930$		0			0
$931$	20.2007	+	22.8677i	0.662053	+	0.749457i
$932$	−3.06438			−0.100377
$933$		0			0
$934$	−3.63955			−0.119090
$935$	−9.93025			−0.324754
$936$		0			0
$937$			41.6933i			1.36206i	0.732255	+	0.681030i	$0.238468\pi$
$937$			41.6933i			1.36206i	−0.732255	+	0.681030i	$0.761532\pi$
$938$	−29.1150	+	27.8400i	−0.950638	+	0.909009i
$939$		0			0
$940$	3.13770			0.102340
$941$	36.7524			1.19809			0.599047	−	0.800714i	$-0.295546\pi$
$941$	36.7524			1.19809			0.599047	+	0.800714i	$0.295546\pi$
$942$		0			0
$943$	45.4357			1.47959
$944$	−4.59174			−0.149448
$945$		0			0
$946$			15.9245i			0.517751i
$947$	21.6999			0.705152			0.352576	−	0.935783i	$-0.385306\pi$
$947$	21.6999			0.705152			0.352576	+	0.935783i	$0.385306\pi$
$948$		0			0
$949$			5.60264i			0.181869i
$950$	7.00707	+	7.25062i	0.227340	+	0.235241i
$951$		0			0
$952$	5.19251	−	4.96513i	0.168290	−	0.160921i
$953$		−	59.4373i		−	1.92536i	−0.270639	−	0.962681i	$-0.587235\pi$
$953$		−	59.4373i		−	1.92536i	0.270639	−	0.962681i	$-0.412765\pi$
$954$		0			0
$955$			39.2639i			1.27055i
$956$	15.4215			0.498767
$957$		0			0
$958$	−25.5277			−0.824763
$959$	−2.92990	+	2.80160i	−0.0946114	+	0.0904683i
$960$		0			0
$961$	−4.03784			−0.130253
$962$	5.76543			0.185885
$963$		0			0
$964$	2.42769			0.0781905
$965$	−17.1394			−0.551737
$966$		0			0
$967$	−43.0567			−1.38461			−0.692305	−	0.721605i	$-0.743405\pi$
$967$	−43.0567			−1.38461			−0.692305	+	0.721605i	$0.743405\pi$
$968$		−	6.02243i		−	0.193568i
$969$		0			0
$970$	−29.4734			−0.946333
$971$	−44.4976			−1.42799			−0.713997	−	0.700148i	$-0.753117\pi$
$971$	−44.4976			−1.42799			−0.713997	+	0.700148i	$0.753117\pi$
$972$		0			0
$973$	−6.09986	−	6.37921i	−0.195552	−	0.204508i
$974$	31.9605			1.02408
$975$		0			0
$976$		−	0.934757i		−	0.0299208i
$977$		−	37.0070i		−	1.18396i	−0.805953	−	0.591980i	$-0.798346\pi$
$977$		−	37.0070i		−	1.18396i	0.805953	−	0.591980i	$-0.201654\pi$
$978$		0			0
$979$	31.2034			0.997263
$980$	−11.4624	−	0.513444i	−0.366154	−	0.0164014i
$981$		0			0
$982$		−	0.307213i		−	0.00980357i
$983$	−34.1404			−1.08891			−0.544455	−	0.838790i	$-0.683263\pi$
$983$	−34.1404			−1.08891			−0.544455	+	0.838790i	$0.683263\pi$
$984$		0			0
$985$			0.320976i			0.0102272i
$986$			10.3850i			0.330726i
$987$		0			0
$988$	−2.51122	+	2.42687i	−0.0798925	+	0.0772089i
$989$	−46.3759			−1.47467
$990$		0			0
$991$			36.8142i			1.16944i	0.811235	+	0.584721i	$0.198796\pi$
$991$			36.8142i			1.16944i	−0.811235	+	0.584721i	$0.801204\pi$
$992$			5.19251i			0.164862i
$993$		0			0
$994$	2.29081	−	2.19049i	0.0726600	−	0.0694781i
$995$	−22.9249			−0.726768
$996$		0			0
$997$		−	42.1969i		−	1.33639i	−0.743987	−	0.668194i	$-0.767068\pi$
$997$		−	42.1969i		−	1.33639i	0.743987	−	0.668194i	$-0.232932\pi$
$998$		−	14.4355i		−	0.456949i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.e.d.1063.13	yes	24
3.2	odd	2	inner	2394.2.e.d.1063.8	yes	24
7.6	odd	2	inner	2394.2.e.d.1063.5	✓	24
19.18	odd	2	inner	2394.2.e.d.1063.7	yes	24
21.20	even	2	inner	2394.2.e.d.1063.16	yes	24
57.56	even	2	inner	2394.2.e.d.1063.14	yes	24
133.132	even	2	inner	2394.2.e.d.1063.15	yes	24
399.398	odd	2	inner	2394.2.e.d.1063.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.e.d.1063.5	✓	24	7.6	odd	2	inner
2394.2.e.d.1063.6	yes	24	399.398	odd	2	inner
2394.2.e.d.1063.7	yes	24	19.18	odd	2	inner
2394.2.e.d.1063.8	yes	24	3.2	odd	2	inner
2394.2.e.d.1063.13	yes	24	1.1	even	1	trivial
2394.2.e.d.1063.14	yes	24	57.56	even	2	inner
2394.2.e.d.1063.15	yes	24	133.132	even	2	inner
2394.2.e.d.1063.16	yes	24	21.20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.63913i q^{5} +(1.91223 - 1.82849i) q^{7} +1.00000i q^{8} +1.63913 q^{10} -2.23105 q^{11} -0.801180 q^{13} +(-1.82849 - 1.91223i) q^{14} +1.00000 q^{16} -2.71542i q^{17} +(-3.13440 + 3.02912i) q^{19} -1.63913i q^{20} +2.23105i q^{22} -6.49732 q^{23} +2.31324 q^{25} +0.801180i q^{26} +(-1.91223 + 1.82849i) q^{28} -3.82446i q^{29} -5.19251 q^{31} -1.00000i q^{32} -2.71542 q^{34} +(2.99714 + 3.13440i) q^{35} -7.19617i q^{37} +(3.02912 + 3.13440i) q^{38} -1.63913 q^{40} -6.99299 q^{41} +7.13770 q^{43} +2.23105 q^{44} +6.49732i q^{46} +1.91424i q^{47} +(0.313241 - 6.99299i) q^{49} -2.31324i q^{50} +0.801180 q^{52} +1.19798i q^{53} -3.65698i q^{55} +(1.82849 + 1.91223i) q^{56} -3.82446 q^{58} -4.59174 q^{59} -0.934757i q^{61} +5.19251i q^{62} -1.00000 q^{64} -1.31324i q^{65} -15.2257i q^{67} +2.71542i q^{68} +(3.13440 - 2.99714i) q^{70} +1.19798i q^{71} -6.99299i q^{73} -7.19617 q^{74} +(3.13440 - 3.02912i) q^{76} +(-4.26627 + 4.07945i) q^{77} -0.503034i q^{79} +1.63913i q^{80} +6.99299i q^{82} +11.1494i q^{83} +4.45094 q^{85} -7.13770i q^{86} -2.23105i q^{88} -13.9860 q^{89} +(-1.53204 + 1.46495i) q^{91} +6.49732 q^{92} +1.91424 q^{94} +(-4.96513 - 5.13770i) q^{95} -17.9811 q^{97} +(-6.99299 - 0.313241i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} - 8 q^{7} + 24 q^{16} + 24 q^{25} + 8 q^{28} + 32 q^{43} - 24 q^{49} + 16 q^{58} - 24 q^{64} - 64 q^{85}+O(q^{100}) \) 24 * q - 24 * q^4 - 8 * q^7 + 24 * q^16 + 24 * q^25 + 8 * q^28 + 32 * q^43 - 24 * q^49 + 16 * q^58 - 24 * q^64 - 64 * q^85

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