LMFDB - Embedded newform 1152.2.l.a.863.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(287,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("1152.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.6
Root			$-1.36166 - 0.381939i$ of defining polynomial
Character	$\chi$	$=$	1152.863
Dual form			1152.2.l.a.287.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +O(q^{10})$	$q+(0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +(1.95945 - 1.95945i) q^{11} +(-4.18757 - 4.18757i) q^{13} -4.03243i q^{17} +(4.26785 - 4.26785i) q^{19} +8.86408i q^{23} -3.83298i q^{25} +(1.23934 - 1.23934i) q^{29} -2.87835i q^{31} +(1.02070 + 1.02070i) q^{35} +(-0.434870 + 0.434870i) q^{37} +7.81179 q^{41} +(5.49678 + 5.49678i) q^{43} +3.20723 q^{47} -5.21456 q^{49} +(4.06777 + 4.06777i) q^{53} +2.99355 q^{55} +(-4.71811 + 4.71811i) q^{59} +(-3.26785 - 3.26785i) q^{61} -6.39758i q^{65} +(5.44348 - 5.44348i) q^{67} -3.76718i q^{71} -10.5357i q^{73} +(2.61822 - 2.61822i) q^{77} -11.1995i q^{79} +(9.73306 + 9.73306i) q^{83} +(3.08029 - 3.08029i) q^{85} +1.64130 q^{89} +(-5.59544 - 5.59544i) q^{91} +6.52023 q^{95} -5.70272 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 - 64 * q^55 + 32 * q^61 + 16 * q^67 + 32 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.763878	+	0.763878i	0.341617	+	0.341617i	0.856975	−	0.515358i	$-0.172341\pi$
$5$	0.763878	+	0.763878i	0.341617	+	0.341617i	−0.515358	+	0.856975i	$0.672341\pi$
$6$		0			0
$7$	1.33620			0.505038			0.252519	−	0.967592i	$-0.418741\pi$
$7$	1.33620			0.505038			0.252519	+	0.967592i	$0.418741\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.95945	−	1.95945i	0.590795	−	0.590795i	−0.347051	−	0.937846i	$-0.612817\pi$
$11$	1.95945	−	1.95945i	0.590795	−	0.590795i	0.937846	+	0.347051i	$0.112817\pi$
$12$		0			0
$13$	−4.18757	−	4.18757i	−1.16142	−	1.16142i	−0.984165	−	0.177257i	$-0.943278\pi$
$13$	−4.18757	−	4.18757i	−1.16142	−	1.16142i	−0.177257	−	0.984165i	$-0.556722\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.03243i		−	0.978009i	−0.872281	−	0.489004i	$-0.837360\pi$
$17$		−	4.03243i		−	0.978009i	0.872281	−	0.489004i	$-0.162640\pi$
$18$		0			0
$19$	4.26785	−	4.26785i	0.979112	−	0.979112i	−0.0206739	−	0.999786i	$-0.506581\pi$
$19$	4.26785	−	4.26785i	0.979112	−	0.979112i	0.999786	+	0.0206739i	$0.00658119\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.86408i			1.84829i	0.382044	+	0.924144i	$0.375220\pi$
$23$			8.86408i			1.84829i	−0.382044	+	0.924144i	$0.624780\pi$
$24$		0			0
$25$		−	3.83298i		−	0.766596i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.23934	−	1.23934i	0.230140	−	0.230140i	−0.582611	−	0.812751i	$-0.697969\pi$
$29$	1.23934	−	1.23934i	0.230140	−	0.230140i	0.812751	+	0.582611i	$0.197969\pi$
$30$		0			0
$31$		−	2.87835i		−	0.516968i	−0.966016	−	0.258484i	$-0.916777\pi$
$31$		−	2.87835i		−	0.516968i	0.966016	−	0.258484i	$-0.0832229\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.02070	+	1.02070i	0.172529	+	0.172529i
$36$		0			0
$37$	−0.434870	+	0.434870i	−0.0714922	+	0.0714922i	−0.741949	−	0.670457i	$-0.766098\pi$
$37$	−0.434870	+	0.434870i	−0.0714922	+	0.0714922i	0.670457	+	0.741949i	$0.266098\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.81179			1.22000			0.609998	−	0.792403i	$-0.291170\pi$
$41$	7.81179			1.22000			0.609998	+	0.792403i	$0.291170\pi$
$42$		0			0
$43$	5.49678	+	5.49678i	0.838251	+	0.838251i	0.988629	−	0.150378i	$-0.0480491\pi$
$43$	5.49678	+	5.49678i	0.838251	+	0.838251i	−0.150378	+	0.988629i	$0.548049\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.20723			0.467822			0.233911	−	0.972258i	$-0.424848\pi$
$47$	3.20723			0.467822			0.233911	+	0.972258i	$0.424848\pi$
$48$		0			0
$49$	−5.21456			−0.744937
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.06777	+	4.06777i	0.558751	+	0.558751i	0.928952	−	0.370201i	$-0.120711\pi$
$53$	4.06777	+	4.06777i	0.558751	+	0.558751i	−0.370201	+	0.928952i	$0.620711\pi$
$54$		0			0
$55$	2.99355			0.403651
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.71811	+	4.71811i	−0.614245	+	0.614245i	−0.944049	−	0.329804i	$-0.893017\pi$
$59$	−4.71811	+	4.71811i	−0.614245	+	0.614245i	0.329804	+	0.944049i	$0.393017\pi$
$60$		0			0
$61$	−3.26785	−	3.26785i	−0.418406	−	0.418406i	0.466248	−	0.884654i	$-0.345605\pi$
$61$	−3.26785	−	3.26785i	−0.418406	−	0.418406i	−0.884654	+	0.466248i	$0.845605\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.39758i		−	0.793522i
$66$		0			0
$67$	5.44348	−	5.44348i	0.665027	−	0.665027i	−0.291533	−	0.956561i	$-0.594165\pi$
$67$	5.44348	−	5.44348i	0.665027	−	0.665027i	0.956561	+	0.291533i	$0.0941654\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.76718i		−	0.447082i	−0.974695	−	0.223541i	$-0.928238\pi$
$71$		−	3.76718i		−	0.447082i	0.974695	−	0.223541i	$-0.0717616\pi$
$72$		0			0
$73$		−	10.5357i		−	1.23311i	−0.787312	−	0.616555i	$-0.788528\pi$
$73$		−	10.5357i		−	1.23311i	0.787312	−	0.616555i	$-0.211472\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.61822	−	2.61822i	0.298374	−	0.298374i
$78$		0			0
$79$		−	11.1995i		−	1.26004i	−0.776578	−	0.630021i	$-0.783046\pi$
$79$		−	11.1995i		−	1.26004i	0.776578	−	0.630021i	$-0.216954\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.73306	+	9.73306i	1.06834	+	1.06834i	0.997487	+	0.0708558i	$0.0225730\pi$
$83$	9.73306	+	9.73306i	1.06834	+	1.06834i	0.0708558	+	0.997487i	$0.477427\pi$
$84$		0			0
$85$	3.08029	−	3.08029i	0.334104	−	0.334104i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.64130			0.173977			0.0869886	−	0.996209i	$-0.472276\pi$
$89$	1.64130			0.173977			0.0869886	+	0.996209i	$0.472276\pi$
$90$		0			0
$91$	−5.59544	−	5.59544i	−0.586562	−	0.586562i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.52023			0.668962
$96$		0			0
$97$	−5.70272			−0.579024			−0.289512	−	0.957174i	$-0.593493\pi$
$97$	−5.70272			−0.579024			−0.289512	+	0.957174i	$0.593493\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.68599	−	6.68599i	−0.665281	−	0.665281i	0.291339	−	0.956620i	$-0.405899\pi$
$101$	−6.68599	−	6.68599i	−0.665281	−	0.665281i	−0.956620	+	0.291339i	$0.905899\pi$
$102$		0			0
$103$	10.8784			1.07188			0.535938	−	0.844257i	$-0.319958\pi$
$103$	10.8784			1.07188			0.535938	+	0.844257i	$0.319958\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.31755	+	1.31755i	−0.127372	+	0.127372i	−0.767919	−	0.640547i	$-0.778708\pi$
$107$	−1.31755	+	1.31755i	−0.127372	+	0.127372i	0.640547	+	0.767919i	$0.278708\pi$
$108$		0			0
$109$	3.51516	+	3.51516i	0.336691	+	0.336691i	0.855120	−	0.518429i	$-0.173483\pi$
$109$	3.51516	+	3.51516i	0.336691	+	0.336691i	−0.518429	+	0.855120i	$0.673483\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.3139i			1.53469i	0.641236	+	0.767344i	$0.278422\pi$
$113$			16.3139i			1.53469i	−0.641236	+	0.767344i	$0.721578\pi$
$114$		0			0
$115$	−6.77107	+	6.77107i	−0.631406	+	0.631406i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.38815i		−	0.493931i
$120$		0			0
$121$			3.32115i			0.301922i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.74732	−	6.74732i	0.603499	−	0.603499i
$126$		0			0
$127$			20.7416i			1.84052i	0.391303	+	0.920262i	$0.372024\pi$
$127$			20.7416i			1.84052i	−0.391303	+	0.920262i	$0.627976\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.43621	−	9.43621i	−0.824446	−	0.824446i	0.162296	−	0.986742i	$-0.448110\pi$
$131$	−9.43621	−	9.43621i	−0.824446	−	0.824446i	−0.986742	+	0.162296i	$0.948110\pi$
$132$		0			0
$133$	5.70272	−	5.70272i	0.494489	−	0.494489i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.8211			−1.09538			−0.547692	−	0.836680i	$-0.684494\pi$
$137$	−12.8211			−1.09538			−0.547692	+	0.836680i	$0.684494\pi$
$138$		0			0
$139$	−1.44348	−	1.44348i	−0.122435	−	0.122435i	0.643235	−	0.765669i	$-0.277592\pi$
$139$	−1.44348	−	1.44348i	−0.122435	−	0.122435i	−0.765669	+	0.643235i	$0.777592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.4106			−1.37232
$144$		0			0
$145$	1.89341			0.157239
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.42073	−	6.42073i	−0.526007	−	0.526007i	0.393372	−	0.919379i	$-0.371308\pi$
$149$	−6.42073	−	6.42073i	−0.526007	−	0.526007i	−0.919379	+	0.393372i	$0.871308\pi$
$150$		0			0
$151$	0.205945			0.0167596			0.00837978	−	0.999965i	$-0.497333\pi$
$151$	0.205945			0.0167596			0.00837978	+	0.999965i	$0.497333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.19871	−	2.19871i	0.176605	−	0.176605i
$156$		0			0
$157$	1.26785	+	1.26785i	0.101186	+	0.101186i	0.755887	−	0.654702i	$-0.227206\pi$
$157$	1.26785	+	1.26785i	0.101186	+	0.101186i	−0.654702	+	0.755887i	$0.727206\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			11.8442i			0.933456i
$162$		0			0
$163$	−0.169186	+	0.169186i	−0.0132517	+	0.0132517i	−0.713702	−	0.700450i	$-0.752983\pi$
$163$	−0.169186	+	0.169186i	−0.0132517	+	0.0132517i	0.700450	+	0.713702i	$0.252983\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.4503i		−	0.808671i	−0.914611	−	0.404335i	$-0.867503\pi$
$167$		−	10.4503i		−	0.808671i	0.914611	−	0.404335i	$-0.132497\pi$
$168$		0			0
$169$			22.0714i			1.69780i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.974085	−	0.974085i	0.0740583	−	0.0740583i	−0.669107	−	0.743166i	$-0.733323\pi$
$173$	0.974085	−	0.974085i	0.0740583	−	0.0740583i	0.743166	+	0.669107i	$0.233323\pi$
$174$		0			0
$175$		−	5.12165i		−	0.387160i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.59560	+	6.59560i	0.492979	+	0.492979i	0.909243	−	0.416265i	$-0.136661\pi$
$179$	6.59560	+	6.59560i	0.492979	+	0.492979i	−0.416265	+	0.909243i	$0.636661\pi$
$180$		0			0
$181$	−10.1876	+	10.1876i	−0.757236	+	0.757236i	−0.975818	−	0.218583i	$-0.929857\pi$
$181$	−10.1876	+	10.1876i	−0.757236	+	0.757236i	0.218583	+	0.975818i	$0.429857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.664376			−0.0488459
$186$		0			0
$187$	−7.90133	−	7.90133i	−0.577803	−	0.577803i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.2297			1.02962			0.514812	−	0.857303i	$-0.327862\pi$
$191$	14.2297			1.02962			0.514812	+	0.857303i	$0.327862\pi$
$192$		0			0
$193$	6.53570			0.470450			0.235225	−	0.971941i	$-0.424417\pi$
$193$	6.53570			0.470450			0.235225	+	0.971941i	$0.424417\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.07713	+	9.07713i	0.646718	+	0.646718i	0.952198	−	0.305480i	$-0.0988169\pi$
$197$	9.07713	+	9.07713i	0.646718	+	0.646718i	−0.305480	+	0.952198i	$0.598817\pi$
$198$		0			0
$199$	−10.0865			−0.715011			−0.357505	−	0.933911i	$-0.616373\pi$
$199$	−10.0865			−0.715011			−0.357505	+	0.933911i	$0.616373\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.65601	−	1.65601i	0.116229	−	0.116229i
$204$		0			0
$205$	5.96725	+	5.96725i	0.416771	+	0.416771i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	16.7252i		−	1.15691i
$210$		0			0
$211$	−17.9792	+	17.9792i	−1.23774	+	1.23774i	−0.276815	+	0.960923i	$0.589279\pi$
$211$	−17.9792	+	17.9792i	−1.23774	+	1.23774i	−0.960923	+	0.276815i	$0.910721\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.39773i			0.572721i
$216$		0			0
$217$		−	3.84607i		−	0.261088i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.8861	+	16.8861i	−1.13588	+	1.13588i
$222$		0			0
$223$			4.00861i			0.268437i	0.990952	+	0.134218i	$0.0428523\pi$
$223$			4.00861i			0.268437i	−0.990952	+	0.134218i	$0.957148\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.7915	−	13.7915i	−0.915373	−	0.915373i	0.0813152	−	0.996688i	$-0.474088\pi$
$227$	−13.7915	−	13.7915i	−0.915373	−	0.915373i	−0.996688	+	0.0813152i	$0.974088\pi$
$228$		0			0
$229$	−3.47840	+	3.47840i	−0.229859	+	0.229859i	−0.812634	−	0.582775i	$-0.801967\pi$
$229$	−3.47840	+	3.47840i	−0.229859	+	0.229859i	0.582775	+	0.812634i	$0.301967\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.23973			−0.212241			−0.106121	−	0.994353i	$-0.533843\pi$
$233$	−3.23973			−0.212241			−0.106121	+	0.994353i	$0.533843\pi$
$234$		0			0
$235$	2.44993	+	2.44993i	0.159816	+	0.159816i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.2484			−1.56850			−0.784249	−	0.620446i	$-0.786952\pi$
$239$	−24.2484			−1.56850			−0.784249	+	0.620446i	$0.786952\pi$
$240$		0			0
$241$	16.5596			1.06670			0.533348	−	0.845896i	$-0.320934\pi$
$241$	16.5596			1.06670			0.533348	+	0.845896i	$0.320934\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.98329	−	3.98329i	−0.254483	−	0.254483i
$246$		0			0
$247$	−35.7438			−2.27432
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.3957	+	11.3957i	−0.719287	+	0.719287i	−0.968459	−	0.249172i	$-0.919842\pi$
$251$	−11.3957	+	11.3957i	−0.719287	+	0.719287i	0.249172	+	0.968459i	$0.419842\pi$
$252$		0			0
$253$	17.3687	+	17.3687i	1.09196	+	1.09196i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.54316i			0.221016i	0.993875	+	0.110508i	$0.0352478\pi$
$257$			3.54316i			0.221016i	−0.993875	+	0.110508i	$0.964752\pi$
$258$		0			0
$259$	−0.581076	+	0.581076i	−0.0361063	+	0.0361063i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	21.0534i		−	1.29821i	−0.760701	−	0.649103i	$-0.775145\pi$
$263$		−	21.0534i		−	1.29821i	0.760701	−	0.649103i	$-0.224855\pi$
$264$		0			0
$265$			6.21456i			0.381757i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.4077	+	20.4077i	−1.24428	+	1.24428i	−0.286072	+	0.958208i	$0.592350\pi$
$269$	−20.4077	+	20.4077i	−1.24428	+	1.24428i	−0.958208	+	0.286072i	$0.907650\pi$
$270$		0			0
$271$			5.06279i			0.307543i	0.988106	+	0.153771i	$0.0491419\pi$
$271$			5.06279i			0.307543i	−0.988106	+	0.153771i	$0.950858\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.51052	−	7.51052i	−0.452901	−	0.452901i
$276$		0			0
$277$	5.86642	−	5.86642i	0.352479	−	0.352479i	−0.508552	−	0.861031i	$-0.669819\pi$
$277$	5.86642	−	5.86642i	0.352479	−	0.352479i	0.861031	+	0.508552i	$0.169819\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.77316			−0.284743			−0.142371	−	0.989813i	$-0.545473\pi$
$281$	−4.77316			−0.284743			−0.142371	+	0.989813i	$0.545473\pi$
$282$		0			0
$283$	10.0779	+	10.0779i	0.599066	+	0.599066i	0.940064	−	0.340998i	$-0.110765\pi$
$283$	10.0779	+	10.0779i	0.599066	+	0.599066i	−0.340998	+	0.940064i	$0.610765\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.4381			0.616144
$288$		0			0
$289$	0.739481			0.0434989
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.7829	+	11.7829i	0.688364	+	0.688364i	0.961870	−	0.273506i	$-0.0881834\pi$
$293$	11.7829	+	11.7829i	0.688364	+	0.688364i	−0.273506	+	0.961870i	$0.588183\pi$
$294$		0			0
$295$	−7.20811			−0.419673
$296$		0			0
$297$		0			0
$298$		0			0
$299$	37.1189	−	37.1189i	2.14664	−	2.14664i
$300$		0			0
$301$	7.34482	+	7.34482i	0.423348	+	0.423348i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4.99248i		−	0.285869i
$306$		0			0
$307$	4.31322	−	4.31322i	0.246169	−	0.246169i	−0.573228	−	0.819396i	$-0.694309\pi$
$307$	4.31322	−	4.31322i	0.246169	−	0.246169i	0.819396	+	0.573228i	$0.194309\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.4434i			1.55617i	0.628156	+	0.778087i	$0.283810\pi$
$311$			27.4434i			1.55617i	−0.628156	+	0.778087i	$0.716190\pi$
$312$		0			0
$313$			18.8568i			1.06585i	0.846162	+	0.532926i	$0.178908\pi$
$313$			18.8568i			1.06585i	−0.846162	+	0.532926i	$0.821092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.154552	+	0.154552i	−0.00868053	+	0.00868053i	−0.711434	−	0.702753i	$-0.751954\pi$
$317$	−0.154552	+	0.154552i	−0.00868053	+	0.00868053i	0.702753	+	0.711434i	$0.251954\pi$
$318$		0			0
$319$		−	4.85685i		−	0.271931i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−17.2098	−	17.2098i	−0.957580	−	0.957580i
$324$		0			0
$325$	−16.0509	+	16.0509i	−0.890342	+	0.890342i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.28551			0.236268
$330$		0			0
$331$	16.3132	+	16.3132i	0.896656	+	0.896656i	0.995139	−	0.0984829i	$-0.0313990\pi$
$331$	16.3132	+	16.3132i	0.896656	+	0.896656i	−0.0984829	+	0.995139i	$0.531399\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.31631			0.454369
$336$		0			0
$337$	1.89341			0.103141			0.0515704	−	0.998669i	$-0.483577\pi$
$337$	1.89341			0.103141			0.0515704	+	0.998669i	$0.483577\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.63998	−	5.63998i	−0.305422	−	0.305422i
$342$		0			0
$343$	−16.3211			−0.881259
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.7630	+	19.7630i	−1.06093	+	1.06093i	−0.0629131	+	0.998019i	$0.520039\pi$
$347$	−19.7630	+	19.7630i	−1.06093	+	1.06093i	−0.998019	+	0.0629131i	$0.979961\pi$
$348$		0			0
$349$	−18.1008	−	18.1008i	−0.968915	−	0.968915i	0.0306158	−	0.999531i	$-0.490253\pi$
$349$	−18.1008	−	18.1008i	−0.968915	−	0.968915i	−0.999531	+	0.0306158i	$0.990253\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	19.3695i		−	1.03093i	−0.856910	−	0.515466i	$-0.827619\pi$
$353$		−	19.3695i		−	1.03093i	0.856910	−	0.515466i	$-0.172381\pi$
$354$		0			0
$355$	2.87766	−	2.87766i	0.152730	−	0.152730i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	7.09236i		−	0.374321i	−0.982329	−	0.187160i	$-0.940072\pi$
$359$		−	7.09236i		−	0.374321i	0.982329	−	0.187160i	$-0.0599284\pi$
$360$		0			0
$361$		−	17.4291i		−	0.917322i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.04799	−	8.04799i	0.421251	−	0.421251i
$366$		0			0
$367$			1.65735i			0.0865130i	0.999064	+	0.0432565i	$0.0137733\pi$
$367$			1.65735i			0.0865130i	−0.999064	+	0.0432565i	$0.986227\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.43537	+	5.43537i	0.282190	+	0.282190i
$372$		0			0
$373$	3.48241	−	3.48241i	0.180312	−	0.180312i	−0.611180	−	0.791492i	$-0.709305\pi$
$373$	3.48241	−	3.48241i	0.180312	−	0.180312i	0.791492	+	0.611180i	$0.209305\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.3797			−0.534579
$378$		0			0
$379$	−19.1548	−	19.1548i	−0.983917	−	0.983917i	0.0159558	−	0.999873i	$-0.494921\pi$
$379$	−19.1548	−	19.1548i	−0.983917	−	0.983917i	−0.999873	+	0.0159558i	$0.994921\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.2907			−0.781319			−0.390660	−	0.920535i	$-0.627753\pi$
$383$	−15.2907			−0.781319			−0.390660	+	0.920535i	$0.627753\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.7144	−	22.7144i	−1.15166	−	1.15166i	−0.986219	−	0.165445i	$-0.947094\pi$
$389$	−22.7144	−	22.7144i	−1.15166	−	1.15166i	−0.165445	−	0.986219i	$-0.552906\pi$
$390$		0			0
$391$	35.7438			1.80764
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.55505	−	8.55505i	0.430451	−	0.430451i
$396$		0			0
$397$	−7.45854	−	7.45854i	−0.374334	−	0.374334i	0.494719	−	0.869053i	$-0.335271\pi$
$397$	−7.45854	−	7.45854i	−0.374334	−	0.374334i	−0.869053	+	0.494719i	$0.835271\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			11.5911i			0.578834i	0.957203	+	0.289417i	$0.0934615\pi$
$401$			11.5911i			0.578834i	−0.957203	+	0.289417i	$0.906539\pi$
$402$		0			0
$403$	−12.0533	+	12.0533i	−0.600417	+	0.600417i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.70421i			0.0844745i
$408$		0			0
$409$			12.4659i			0.616398i	0.951322	+	0.308199i	$0.0997262\pi$
$409$			12.4659i			0.616398i	−0.951322	+	0.308199i	$0.900274\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.30435	+	6.30435i	−0.310217	+	0.310217i
$414$		0			0
$415$			14.8697i			0.729927i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.4096	+	14.4096i	0.703953	+	0.703953i	0.965257	−	0.261304i	$-0.0841525\pi$
$419$	14.4096	+	14.4096i	0.703953	+	0.703953i	−0.261304	+	0.965257i	$0.584152\pi$
$420$		0			0
$421$	5.83630	−	5.83630i	0.284444	−	0.284444i	−0.550434	−	0.834878i	$-0.685538\pi$
$421$	5.83630	−	5.83630i	0.284444	−	0.284444i	0.834878	+	0.550434i	$0.185538\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.4562			−0.749738
$426$		0			0
$427$	−4.36652	−	4.36652i	−0.211311	−	0.211311i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.4510			−0.888755			−0.444377	−	0.895840i	$-0.646575\pi$
$431$	−18.4510			−0.888755			−0.444377	+	0.895840i	$0.646575\pi$
$432$		0			0
$433$	6.58166			0.316295			0.158147	−	0.987416i	$-0.449448\pi$
$433$	6.58166			0.316295			0.158147	+	0.987416i	$0.449448\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	37.8306	+	37.8306i	1.80968	+	1.80968i
$438$		0			0
$439$	3.68747			0.175993			0.0879966	−	0.996121i	$-0.471954\pi$
$439$	3.68747			0.175993			0.0879966	+	0.996121i	$0.471954\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.31861	+	3.31861i	−0.157672	+	0.157672i	−0.781534	−	0.623862i	$-0.785563\pi$
$443$	−3.31861	+	3.31861i	−0.157672	+	0.157672i	0.623862	+	0.781534i	$0.285563\pi$
$444$		0			0
$445$	1.25375	+	1.25375i	0.0594335	+	0.0594335i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.3555i			0.960635i	0.877095	+	0.480318i	$0.159479\pi$
$449$			20.3555i			0.960635i	−0.877095	+	0.480318i	$0.840521\pi$
$450$		0			0
$451$	15.3068	−	15.3068i	0.720768	−	0.720768i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	8.54847i		−	0.400758i
$456$		0			0
$457$		−	10.0239i		−	0.468897i	−0.972129	−	0.234448i	$-0.924672\pi$
$457$		−	10.0239i		−	0.468897i	0.972129	−	0.234448i	$-0.0753284\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.4043	−	16.4043i	0.764026	−	0.764026i	−0.213021	−	0.977048i	$-0.568330\pi$
$461$	16.4043	−	16.4043i	0.764026	−	0.764026i	0.977048	+	0.213021i	$0.0683304\pi$
$462$		0			0
$463$		−	0.997833i		−	0.0463732i	−0.999731	−	0.0231866i	$-0.992619\pi$
$463$		−	0.997833i		−	0.0463732i	0.999731	−	0.0231866i	$-0.00738119\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.73306	+	9.73306i	0.450392	+	0.450392i	0.895485	−	0.445092i	$-0.146829\pi$
$467$	9.73306	+	9.73306i	0.450392	+	0.450392i	−0.445092	+	0.895485i	$0.646829\pi$
$468$		0			0
$469$	7.27361	−	7.27361i	0.335864	−	0.335864i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.5413			0.990469
$474$		0			0
$475$	−16.3586	−	16.3586i	−0.750584	−	0.750584i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	37.0669			1.69363			0.846816	−	0.531886i	$-0.178517\pi$
$479$	37.0669			1.69363			0.846816	+	0.531886i	$0.178517\pi$
$480$		0			0
$481$	3.64210			0.166065
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.35618	−	4.35618i	−0.197804	−	0.197804i
$486$		0			0
$487$	33.1866			1.50383			0.751914	−	0.659261i	$-0.229131\pi$
$487$	33.1866			1.50383			0.751914	+	0.659261i	$0.229131\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.2580	+	10.2580i	−0.462936	+	0.462936i	−0.899617	−	0.436681i	$-0.856154\pi$
$491$	−10.2580	+	10.2580i	−0.462936	+	0.462936i	0.436681	+	0.899617i	$0.356154\pi$
$492$		0			0
$493$	−4.99757	−	4.99757i	−0.225079	−	0.225079i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5.03372i		−	0.225793i
$498$		0			0
$499$	−4.19733	+	4.19733i	−0.187898	+	0.187898i	−0.794787	−	0.606889i	$-0.792417\pi$
$499$	−4.19733	+	4.19733i	−0.187898	+	0.187898i	0.606889	+	0.794787i	$0.292417\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.0655i			0.939266i	0.882862	+	0.469633i	$0.155614\pi$
$503$			21.0655i			0.939266i	−0.882862	+	0.469633i	$0.844386\pi$
$504$		0			0
$505$		−	10.2146i		−	0.454542i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.6424	−	22.6424i	1.00361	−	1.00361i	0.00361481	−	0.999993i	$-0.498849\pi$
$509$	22.6424	−	22.6424i	1.00361	−	1.00361i	0.999993	−	0.00361481i	$-0.00115063\pi$
$510$		0			0
$511$		−	14.0779i		−	0.622768i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.30973	+	8.30973i	0.366171	+	0.366171i
$516$		0			0
$517$	6.28439	−	6.28439i	0.276387	−	0.276387i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.3351			1.67949			0.839746	−	0.542980i	$-0.182704\pi$
$521$	38.3351			1.67949			0.839746	+	0.542980i	$0.182704\pi$
$522$		0			0
$523$	5.39811	+	5.39811i	0.236043	+	0.236043i	0.815209	−	0.579166i	$-0.196622\pi$
$523$	5.39811	+	5.39811i	0.236043	+	0.236043i	−0.579166	+	0.815209i	$0.696622\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.6068			−0.505599
$528$		0			0
$529$	−55.5719			−2.41617
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−32.7124	−	32.7124i	−1.41693	−	1.41693i
$534$		0			0
$535$	−2.01289			−0.0870249
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10.2176	+	10.2176i	−0.440105	+	0.440105i
$540$		0			0
$541$	−5.51516	−	5.51516i	−0.237115	−	0.237115i	0.578539	−	0.815654i	$-0.303623\pi$
$541$	−5.51516	−	5.51516i	−0.237115	−	0.237115i	−0.815654	+	0.578539i	$0.803623\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			5.37030i			0.230038i
$546$		0			0
$547$	23.2535	−	23.2535i	0.994247	−	0.994247i	−0.00573636	−	0.999984i	$-0.501826\pi$
$547$	23.2535	−	23.2535i	0.994247	−	0.994247i	0.999984	+	0.00573636i	$0.00182595\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.5787i		−	0.450666i
$552$		0			0
$553$		−	14.9648i		−	0.636369i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.9066	−	26.9066i	1.14007	−	1.14007i	0.151635	−	0.988437i	$-0.451546\pi$
$557$	26.9066	−	26.9066i	1.14007	−	1.14007i	0.988437	−	0.151635i	$-0.0484537\pi$
$558$		0			0
$559$		−	46.0362i		−	1.94712i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.4144	+	21.4144i	0.902508	+	0.902508i	0.995653	−	0.0931446i	$-0.0296919\pi$
$563$	21.4144	+	21.4144i	0.902508	+	0.902508i	−0.0931446	+	0.995653i	$0.529692\pi$
$564$		0			0
$565$	−12.4619	+	12.4619i	−0.524275	+	0.524275i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.54807			0.0648986			0.0324493	−	0.999473i	$-0.489669\pi$
$569$	1.54807			0.0648986			0.0324493	+	0.999473i	$0.489669\pi$
$570$		0			0
$571$	12.3384	+	12.3384i	0.516345	+	0.516345i	0.916463	−	0.400119i	$-0.131031\pi$
$571$	12.3384	+	12.3384i	0.516345	+	0.516345i	−0.400119	+	0.916463i	$0.631031\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.9759			1.41689
$576$		0			0
$577$	−24.9648			−1.03930			−0.519650	−	0.854380i	$-0.673937\pi$
$577$	−24.9648			−1.03930			−0.519650	+	0.854380i	$0.673937\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.0054	+	13.0054i	0.539553	+	0.539553i
$582$		0			0
$583$	15.9411			0.660215
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.99426	+	3.99426i	−0.164861	+	0.164861i	−0.784716	−	0.619855i	$-0.787191\pi$
$587$	−3.99426	+	3.99426i	−0.164861	+	0.164861i	0.619855	+	0.784716i	$0.287191\pi$
$588$		0			0
$589$	−12.2844	−	12.2844i	−0.506169	−	0.506169i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			21.0442i			0.864183i	0.901830	+	0.432092i	$0.142224\pi$
$593$			21.0442i			0.864183i	−0.901830	+	0.432092i	$0.857776\pi$
$594$		0			0
$595$	4.11589	−	4.11589i	0.168735	−	0.168735i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	12.7718i		−	0.521840i	−0.965360	−	0.260920i	$-0.915974\pi$
$599$		−	12.7718i		−	0.521840i	0.965360	−	0.260920i	$-0.0840259\pi$
$600$		0			0
$601$		−	44.2967i		−	1.80690i	−0.428691	−	0.903451i	$-0.641025\pi$
$601$		−	44.2967i		−	1.80690i	0.428691	−	0.903451i	$-0.358975\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.53695	+	2.53695i	−0.103142	+	0.103142i
$606$		0			0
$607$			26.8784i			1.09096i	0.838124	+	0.545479i	$0.183652\pi$
$607$			26.8784i			1.09096i	−0.838124	+	0.545479i	$0.816348\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.4305	−	13.4305i	−0.543339	−	0.543339i
$612$		0			0
$613$	27.6602	−	27.6602i	1.11719	−	1.11719i	0.125033	−	0.992153i	$-0.460096\pi$
$613$	27.6602	−	27.6602i	1.11719	−	1.11719i	0.992153	−	0.125033i	$-0.0399036\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.6148			−0.789661			−0.394831	−	0.918754i	$-0.629197\pi$
$617$	−19.6148			−0.789661			−0.394831	+	0.918754i	$0.629197\pi$
$618$		0			0
$619$	11.7854	+	11.7854i	0.473697	+	0.473697i	0.903109	−	0.429412i	$-0.141279\pi$
$619$	11.7854	+	11.7854i	0.473697	+	0.473697i	−0.429412	+	0.903109i	$0.641279\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.19311			0.0878651
$624$		0			0
$625$	−8.85665			−0.354266
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.75359	+	1.75359i	0.0699200	+	0.0699200i
$630$		0			0
$631$	18.8195			0.749193			0.374596	−	0.927188i	$-0.377781\pi$
$631$	18.8195			0.749193			0.374596	+	0.927188i	$0.377781\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.8441	+	15.8441i	−0.628753	+	0.628753i
$636$		0			0
$637$	21.8363	+	21.8363i	0.865186	+	0.865186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17.9718i		−	0.709845i	−0.934896	−	0.354923i	$-0.884507\pi$
$641$		−	17.9718i		−	0.709845i	0.934896	−	0.354923i	$-0.115493\pi$
$642$		0			0
$643$	9.92589	−	9.92589i	0.391439	−	0.391439i	−0.483761	−	0.875200i	$-0.660730\pi$
$643$	9.92589	−	9.92589i	0.391439	−	0.391439i	0.875200	+	0.483761i	$0.160730\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.6801i			0.537820i	0.963165	+	0.268910i	$0.0866635\pi$
$647$			13.6801i			0.537820i	−0.963165	+	0.268910i	$0.913337\pi$
$648$		0			0
$649$			18.4897i			0.725786i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.1940	+	31.1940i	−1.22071	+	1.22071i	−0.253336	+	0.967378i	$0.581528\pi$
$653$	−31.1940	+	31.1940i	−1.22071	+	1.22071i	−0.967378	+	0.253336i	$0.918472\pi$
$654$		0			0
$655$		−	14.4162i		−	0.563288i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27.2397	+	27.2397i	1.06111	+	1.06111i	0.998007	+	0.0631026i	$0.0200995\pi$
$659$	27.2397	+	27.2397i	1.06111	+	1.06111i	0.0631026	+	0.998007i	$0.479900\pi$
$660$		0			0
$661$	12.0770	−	12.0770i	0.469740	−	0.469740i	−0.432091	−	0.901830i	$-0.642224\pi$
$661$	12.0770	−	12.0770i	0.469740	−	0.469740i	0.901830	+	0.432091i	$0.142224\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.71237			0.337851
$666$		0			0
$667$	10.9856	+	10.9856i	0.425365	+	0.425365i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12.8064			−0.494384
$672$		0			0
$673$	11.0108			0.424434			0.212217	−	0.977223i	$-0.431932\pi$
$673$	11.0108			0.424434			0.212217	+	0.977223i	$0.431932\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.26708	−	5.26708i	−0.202430	−	0.202430i	0.598610	−	0.801040i	$-0.295720\pi$
$677$	−5.26708	−	5.26708i	−0.202430	−	0.202430i	−0.801040	+	0.598610i	$0.795720\pi$
$678$		0			0
$679$	−7.62000			−0.292429
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−16.0080	+	16.0080i	−0.612528	+	0.612528i	−0.943604	−	0.331076i	$-0.892588\pi$
$683$	−16.0080	+	16.0080i	−0.612528	+	0.612528i	0.331076	+	0.943604i	$0.392588\pi$
$684$		0			0
$685$	−9.79379	−	9.79379i	−0.374201	−	0.374201i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	34.0681i		−	1.29789i
$690$		0			0
$691$	−7.63348	+	7.63348i	−0.290391	+	0.290391i	−0.837235	−	0.546843i	$-0.815829\pi$
$691$	−7.63348	+	7.63348i	−0.290391	+	0.290391i	0.546843	+	0.837235i	$0.315829\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	2.20529i		−	0.0836514i
$696$		0			0
$697$		−	31.5005i		−	1.19317i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.3494	−	18.3494i	0.693049	−	0.693049i	−0.269853	−	0.962902i	$-0.586975\pi$
$701$	18.3494	−	18.3494i	0.693049	−	0.693049i	0.962902	+	0.269853i	$0.0869750\pi$
$702$		0			0
$703$			3.71192i			0.139998i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.93385	−	8.93385i	−0.335992	−	0.335992i
$708$		0			0
$709$	−19.8774	+	19.8774i	−0.746511	+	0.746511i	−0.973822	−	0.227311i	$-0.927007\pi$
$709$	−19.8774	+	19.8774i	−0.746511	+	0.746511i	0.227311	+	0.973822i	$0.427007\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	25.5140			0.955505
$714$		0			0
$715$	−12.5357	−	12.5357i	−0.468809	−	0.468809i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.1176			−0.601083			−0.300542	−	0.953769i	$-0.597167\pi$
$719$	−16.1176			−0.601083			−0.300542	+	0.953769i	$0.597167\pi$
$720$		0			0
$721$	14.5357			0.541338
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.75038	−	4.75038i	−0.176425	−	0.176425i
$726$		0			0
$727$	16.4536			0.610229			0.305115	−	0.952316i	$-0.401305\pi$
$727$	16.4536			0.610229			0.305115	+	0.952316i	$0.401305\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.1654	−	22.1654i	0.819816	−	0.819816i
$732$		0			0
$733$	−24.5995	−	24.5995i	−0.908602	−	0.908602i	0.0875578	−	0.996159i	$-0.472094\pi$
$733$	−24.5995	−	24.5995i	−0.908602	−	0.908602i	−0.996159	+	0.0875578i	$0.972094\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	21.3324i		−	0.785790i
$738$		0			0
$739$	−3.06707	+	3.06707i	−0.112824	+	0.112824i	−0.761265	−	0.648441i	$-0.775421\pi$
$739$	−3.06707	+	3.06707i	−0.112824	+	0.112824i	0.648441	+	0.761265i	$0.275421\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.16681i			0.0428061i	0.999771	+	0.0214031i	$0.00681333\pi$
$743$			1.16681i			0.0428061i	−0.999771	+	0.0214031i	$0.993187\pi$
$744$		0			0
$745$		−	9.80931i		−	0.359385i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.76051	+	1.76051i	−0.0643278	+	0.0643278i
$750$		0			0
$751$		−	2.12809i		−	0.0776552i	−0.999246	−	0.0388276i	$-0.987638\pi$
$751$		−	2.12809i		−	0.0776552i	0.999246	−	0.0388276i	$-0.0123623\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.157317	+	0.157317i	0.00572534	+	0.00572534i
$756$		0			0
$757$	−19.1573	+	19.1573i	−0.696282	+	0.696282i	−0.963607	−	0.267324i	$-0.913860\pi$
$757$	−19.1573	+	19.1573i	−0.696282	+	0.696282i	0.267324	+	0.963607i	$0.413860\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.2848			−0.372824			−0.186412	−	0.982472i	$-0.559686\pi$
$761$	−10.2848			−0.372824			−0.186412	+	0.982472i	$0.559686\pi$
$762$		0			0
$763$	4.69697	+	4.69697i	0.170042	+	0.170042i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	39.5147			1.42679
$768$		0			0
$769$	−15.2860			−0.551226			−0.275613	−	0.961269i	$-0.588881\pi$
$769$	−15.2860			−0.551226			−0.275613	+	0.961269i	$0.588881\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.6903	+	23.6903i	0.852081	+	0.852081i	0.990389	−	0.138308i	$-0.0441664\pi$
$773$	23.6903	+	23.6903i	0.852081	+	0.852081i	−0.138308	+	0.990389i	$0.544166\pi$
$774$		0			0
$775$	−11.0327			−0.396305
$776$		0			0
$777$		0			0
$778$		0			0
$779$	33.3396	−	33.3396i	1.19451	−	1.19451i
$780$		0			0
$781$	−7.38158	−	7.38158i	−0.264134	−	0.264134i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.93697i			0.0691333i
$786$		0			0
$787$	29.3220	−	29.3220i	1.04522	−	1.04522i	0.0462895	−	0.998928i	$-0.485260\pi$
$787$	29.3220	−	29.3220i	1.04522	−	1.04522i	0.998928	−	0.0462895i	$-0.0147397\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			21.7988i			0.775075i
$792$		0			0
$793$			27.3687i			0.971890i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.41153	+	3.41153i	−0.120843	+	0.120843i	−0.764942	−	0.644099i	$-0.777232\pi$
$797$	−3.41153	+	3.41153i	−0.120843	+	0.120843i	0.644099	+	0.764942i	$0.277232\pi$
$798$		0			0
$799$		−	12.9329i		−	0.457534i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.6441	−	20.6441i	−0.728516	−	0.728516i
$804$		0			0
$805$	−9.04754	+	9.04754i	−0.318884	+	0.318884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.15877			0.322005			0.161003	−	0.986954i	$-0.448527\pi$
$809$	9.15877			0.322005			0.161003	+	0.986954i	$0.448527\pi$
$810$		0			0
$811$	8.91014	+	8.91014i	0.312877	+	0.312877i	0.846023	−	0.533146i	$-0.178990\pi$
$811$	8.91014	+	8.91014i	0.312877	+	0.312877i	−0.533146	+	0.846023i	$0.678990\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.258474			−0.00905397
$816$		0			0
$817$	46.9189			1.64148
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.0618	+	27.0618i	0.944464	+	0.944464i	0.998537	−	0.0540734i	$-0.0172205\pi$
$821$	27.0618	+	27.0618i	0.944464	+	0.944464i	−0.0540734	+	0.998537i	$0.517221\pi$
$822$		0			0
$823$	11.6286			0.405348			0.202674	−	0.979246i	$-0.435037\pi$
$823$	11.6286			0.405348			0.202674	+	0.979246i	$0.435037\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.57283	−	8.57283i	0.298107	−	0.298107i	−0.542165	−	0.840272i	$-0.682395\pi$
$827$	8.57283	−	8.57283i	0.298107	−	0.298107i	0.840272	+	0.542165i	$0.182395\pi$
$828$		0			0
$829$	−19.5629	−	19.5629i	−0.679447	−	0.679447i	0.280428	−	0.959875i	$-0.409524\pi$
$829$	−19.5629	−	19.5629i	−0.679447	−	0.679447i	−0.959875	+	0.280428i	$0.909524\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			21.0274i			0.728555i
$834$		0			0
$835$	7.98277	−	7.98277i	0.276255	−	0.276255i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			14.8018i			0.511015i	0.966807	+	0.255508i	$0.0822426\pi$
$839$			14.8018i			0.511015i	−0.966807	+	0.255508i	$0.917757\pi$
$840$		0			0
$841$			25.9281i			0.894071i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16.8599	+	16.8599i	−0.579997	+	0.579997i
$846$		0			0
$847$			4.43773i			0.152482i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.85473	−	3.85473i	−0.132138	−	0.132138i
$852$		0			0
$853$	27.2550	−	27.2550i	0.933192	−	0.933192i	−0.0647119	−	0.997904i	$-0.520613\pi$
$853$	27.2550	−	27.2550i	0.933192	−	0.933192i	0.997904	+	0.0647119i	$0.0206129\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28.7462			−0.981951			−0.490976	−	0.871173i	$-0.663360\pi$
$857$	−28.7462			−0.981951			−0.490976	+	0.871173i	$0.663360\pi$
$858$		0			0
$859$	−29.2578	−	29.2578i	−0.998264	−	0.998264i	0.00173461	−	0.999998i	$-0.499448\pi$
$859$	−29.2578	−	29.2578i	−0.998264	−	0.998264i	−0.999998	+	0.00173461i	$0.999448\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−40.6816			−1.38482			−0.692408	−	0.721506i	$-0.743450\pi$
$863$	−40.6816			−1.38482			−0.692408	+	0.721506i	$0.743450\pi$
$864$		0			0
$865$	1.48816			0.0505991
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.9448	−	21.9448i	−0.744427	−	0.744427i
$870$		0			0
$871$	−45.5899			−1.54475
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.01580	−	9.01580i	0.304790	−	0.304790i
$876$		0			0
$877$	28.9577	+	28.9577i	0.977831	+	0.977831i	0.999760	−	0.0219281i	$-0.00698049\pi$
$877$	28.9577	+	28.9577i	0.977831	+	0.977831i	−0.0219281	+	0.999760i	$0.506980\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.2311i			0.580531i	0.956946	+	0.290266i	$0.0937437\pi$
$881$			17.2311i			0.580531i	−0.956946	+	0.290266i	$0.906256\pi$
$882$		0			0
$883$	−24.8129	+	24.8129i	−0.835019	+	0.835019i	−0.988198	−	0.153179i	$-0.951049\pi$
$883$	−24.8129	+	24.8129i	−0.835019	+	0.835019i	0.153179	+	0.988198i	$0.451049\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			39.0606i			1.31153i	0.754967	+	0.655763i	$0.227653\pi$
$887$			39.0606i			1.31153i	−0.754967	+	0.655763i	$0.772347\pi$
$888$		0			0
$889$			27.7151i			0.929534i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.6880	−	13.6880i	0.458050	−	0.458050i
$894$		0			0
$895$			10.0765i			0.336819i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.56727	−	3.56727i	−0.118975	−	0.118975i
$900$		0			0
$901$	16.4030	−	16.4030i	0.546463	−	0.546463i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.5641			−0.517369
$906$		0			0
$907$	−0.366518	−	0.366518i	−0.0121700	−	0.0121700i	0.700996	−	0.713166i	$-0.252739\pi$
$907$	−0.366518	−	0.366518i	−0.0121700	−	0.0121700i	−0.713166	+	0.700996i	$0.752739\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.8950			−0.626018			−0.313009	−	0.949750i	$-0.601337\pi$
$911$	−18.8950			−0.626018			−0.313009	+	0.949750i	$0.601337\pi$
$912$		0			0
$913$	38.1428			1.26234
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.6087	−	12.6087i	−0.416376	−	0.416376i
$918$		0			0
$919$	−54.5805			−1.80045			−0.900223	−	0.435429i	$-0.856597\pi$
$919$	−54.5805			−1.80045			−0.900223	+	0.435429i	$0.856597\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15.7753	+	15.7753i	−0.519250	+	0.519250i
$924$		0			0
$925$	1.66685	+	1.66685i	0.0548057	+	0.0548057i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	21.2118i		−	0.695938i	−0.937506	−	0.347969i	$-0.886871\pi$
$929$		−	21.2118i		−	0.695938i	0.937506	−	0.347969i	$-0.113129\pi$
$930$		0			0
$931$	−22.2550	+	22.2550i	−0.729377	+	0.729377i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	12.0713i		−	0.394774i
$936$		0			0
$937$			15.7395i			0.514186i	0.966387	+	0.257093i	$0.0827647\pi$
$937$			15.7395i			0.514186i	−0.966387	+	0.257093i	$0.917235\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.43393	+	1.43393i	−0.0467447	+	0.0467447i	−0.730093	−	0.683348i	$-0.760523\pi$
$941$	−1.43393	+	1.43393i	−0.0467447	+	0.0467447i	0.683348	+	0.730093i	$0.260523\pi$
$942$		0			0
$943$			69.2443i			2.25491i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.5461	−	28.5461i	−0.927623	−	0.927623i	0.0699288	−	0.997552i	$-0.477723\pi$
$947$	−28.5461	−	28.5461i	−0.927623	−	0.927623i	−0.997552	+	0.0699288i	$0.977723\pi$
$948$		0			0
$949$	−44.1189	+	44.1189i	−1.43216	+	1.43216i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.6986			0.476136			0.238068	−	0.971249i	$-0.423486\pi$
$953$	14.6986			0.476136			0.238068	+	0.971249i	$0.423486\pi$
$954$		0			0
$955$	10.8697	+	10.8697i	0.351737	+	0.351737i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−17.1317			−0.553210
$960$		0			0
$961$	22.7151			0.732745
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.99248	+	4.99248i	0.160714	+	0.160714i
$966$		0			0
$967$	−7.34054			−0.236056			−0.118028	−	0.993010i	$-0.537657\pi$
$967$	−7.34054			−0.236056			−0.118028	+	0.993010i	$0.537657\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.48251	−	3.48251i	0.111759	−	0.111759i	−0.649016	−	0.760775i	$-0.724819\pi$
$971$	3.48251	−	3.48251i	0.111759	−	0.111759i	0.760775	+	0.649016i	$0.224819\pi$
$972$		0			0
$973$	−1.92879	−	1.92879i	−0.0618341	−	0.0618341i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			9.95013i			0.318333i	0.987252	+	0.159166i	$0.0508806\pi$
$977$			9.95013i			0.318333i	−0.987252	+	0.159166i	$0.949119\pi$
$978$		0			0
$979$	3.21603	−	3.21603i	0.102785	−	0.102785i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			18.7648i			0.598505i	0.954174	+	0.299253i	$0.0967373\pi$
$983$			18.7648i			0.598505i	−0.954174	+	0.299253i	$0.903263\pi$
$984$		0			0
$985$			13.8676i			0.441859i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−48.7239	+	48.7239i	−1.54933	+	1.54933i
$990$		0			0
$991$		−	3.18227i		−	0.101088i	−0.998722	−	0.0505441i	$-0.983904\pi$
$991$		−	3.18227i		−	0.101088i	0.998722	−	0.0505441i	$-0.0160955\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.70483	−	7.70483i	−0.244259	−	0.244259i
$996$		0			0
$997$	−26.9944	+	26.9944i	−0.854923	+	0.854923i	−0.990735	−	0.135812i	$-0.956636\pi$
$997$	−26.9944	+	26.9944i	−0.854923	+	0.854923i	0.135812	+	0.990735i	$0.456636\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.l.a.863.6		16
3.2	odd	2	inner	1152.2.l.a.863.3		16
4.3	odd	2		1152.2.l.b.863.6		16
8.3	odd	2		576.2.l.a.431.3		16
8.5	even	2		144.2.l.a.35.7	yes	16
12.11	even	2		1152.2.l.b.863.3		16
16.3	odd	4		144.2.l.a.107.2	yes	16
16.5	even	4		1152.2.l.b.287.3		16
16.11	odd	4	inner	1152.2.l.a.287.3		16
16.13	even	4		576.2.l.a.143.6		16
24.5	odd	2		144.2.l.a.35.2	✓	16
24.11	even	2		576.2.l.a.431.6		16
48.5	odd	4		1152.2.l.b.287.6		16
48.11	even	4	inner	1152.2.l.a.287.6		16
48.29	odd	4		576.2.l.a.143.3		16
48.35	even	4		144.2.l.a.107.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.l.a.35.2	✓	16	24.5	odd	2
144.2.l.a.35.7	yes	16	8.5	even	2
144.2.l.a.107.2	yes	16	16.3	odd	4
144.2.l.a.107.7	yes	16	48.35	even	4
576.2.l.a.143.3		16	48.29	odd	4
576.2.l.a.143.6		16	16.13	even	4
576.2.l.a.431.3		16	8.3	odd	2
576.2.l.a.431.6		16	24.11	even	2
1152.2.l.a.287.3		16	16.11	odd	4	inner
1152.2.l.a.287.6		16	48.11	even	4	inner
1152.2.l.a.863.3		16	3.2	odd	2	inner
1152.2.l.a.863.6		16	1.1	even	1	trivial
1152.2.l.b.287.3		16	16.5	even	4
1152.2.l.b.287.6		16	48.5	odd	4
1152.2.l.b.863.3		16	12.11	even	2
1152.2.l.b.863.6		16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +O(q^{10})\)	\(q+(0.763878 + 0.763878i) q^{5} +1.33620 q^{7} +(1.95945 - 1.95945i) q^{11} +(-4.18757 - 4.18757i) q^{13} -4.03243i q^{17} +(4.26785 - 4.26785i) q^{19} +8.86408i q^{23} -3.83298i q^{25} +(1.23934 - 1.23934i) q^{29} -2.87835i q^{31} +(1.02070 + 1.02070i) q^{35} +(-0.434870 + 0.434870i) q^{37} +7.81179 q^{41} +(5.49678 + 5.49678i) q^{43} +3.20723 q^{47} -5.21456 q^{49} +(4.06777 + 4.06777i) q^{53} +2.99355 q^{55} +(-4.71811 + 4.71811i) q^{59} +(-3.26785 - 3.26785i) q^{61} -6.39758i q^{65} +(5.44348 - 5.44348i) q^{67} -3.76718i q^{71} -10.5357i q^{73} +(2.61822 - 2.61822i) q^{77} -11.1995i q^{79} +(9.73306 + 9.73306i) q^{83} +(3.08029 - 3.08029i) q^{85} +1.64130 q^{89} +(-5.59544 - 5.59544i) q^{91} +6.52023 q^{95} -5.70272 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} - 64 q^{55} + 32 q^{61} + 16 q^{67} + 32 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 - 64 * q^55 + 32 * q^61 + 16 * q^67 + 32 * q^85 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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