LMFDB - Embedded newform 576.4.k.b.433.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,4,Mod(145,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.145"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.7
Character	$\chi$	$=$	576.433
Dual form			576.4.k.b.145.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.644922 - 0.644922i) q^{5} -7.13926i q^{7} +(25.4455 + 25.4455i) q^{11} +(-14.6030 + 14.6030i) q^{13} -71.4024 q^{17} +(43.6238 - 43.6238i) q^{19} +211.845i q^{23} -124.168i q^{25} +(-5.84463 + 5.84463i) q^{29} +107.807 q^{31} +(-4.60426 + 4.60426i) q^{35} +(-184.865 - 184.865i) q^{37} +360.146i q^{41} +(312.475 + 312.475i) q^{43} +343.892 q^{47} +292.031 q^{49} +(249.900 + 249.900i) q^{53} -32.8207i q^{55} +(-152.755 - 152.755i) q^{59} +(-525.985 + 525.985i) q^{61} +18.8355 q^{65} +(35.3052 - 35.3052i) q^{67} +784.715i q^{71} +800.215i q^{73} +(181.662 - 181.662i) q^{77} -548.062 q^{79} +(464.431 - 464.431i) q^{83} +(46.0489 + 46.0489i) q^{85} +302.977i q^{89} +(104.254 + 104.254i) q^{91} -56.2679 q^{95} +1567.24 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79}+ \cdots - 7728 q^{95}+O(q^{100}) $ 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	−0.644922	−	0.644922i	−0.0576835	−	0.0576835i	0.677677	−	0.735360i	$-0.262987\pi$
$5$	−0.644922	−	0.644922i	−0.0576835	−	0.0576835i	−0.735360	+	0.677677i	$0.762987\pi$
$6$		0			0
$7$		−	7.13926i		−	0.385484i	−0.981249	−	0.192742i	$-0.938262\pi$
$7$		−	7.13926i		−	0.385484i	0.981249	−	0.192742i	$-0.0617380\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	25.4455	+	25.4455i	0.697464	+	0.697464i	0.963863	−	0.266399i	$-0.0858338\pi$
$11$	25.4455	+	25.4455i	0.697464	+	0.697464i	−0.266399	+	0.963863i	$0.585834\pi$
$12$		0			0
$13$	−14.6030	+	14.6030i	−0.311549	+	0.311549i	−0.845509	−	0.533961i	$-0.820703\pi$
$13$	−14.6030	+	14.6030i	−0.311549	+	0.311549i	0.533961	+	0.845509i	$0.320703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−71.4024			−1.01868			−0.509342	−	0.860564i	$-0.670111\pi$
$17$	−71.4024			−1.01868			−0.509342	+	0.860564i	$0.670111\pi$
$18$		0			0
$19$	43.6238	−	43.6238i	0.526736	−	0.526736i	−0.392861	−	0.919598i	$-0.628515\pi$
$19$	43.6238	−	43.6238i	0.526736	−	0.526736i	0.919598	+	0.392861i	$0.128515\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			211.845i			1.92056i	0.279045	+	0.960278i	$0.409982\pi$
$23$			211.845i			1.92056i	−0.279045	+	0.960278i	$0.590018\pi$
$24$		0			0
$25$		−	124.168i		−	0.993345i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.84463	+	5.84463i	−0.0374248	+	0.0374248i	−0.725572	−	0.688147i	$-0.758425\pi$
$29$	−5.84463	+	5.84463i	−0.0374248	+	0.0374248i	0.688147	+	0.725572i	$0.258425\pi$
$30$		0			0
$31$	107.807			0.624604			0.312302	−	0.949983i	$-0.398900\pi$
$31$	107.807			0.624604			0.312302	+	0.949983i	$0.398900\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.60426	+	4.60426i	−0.0222361	+	0.0222361i
$36$		0			0
$37$	−184.865	−	184.865i	−0.821395	−	0.821395i	0.164913	−	0.986308i	$-0.447266\pi$
$37$	−184.865	−	184.865i	−0.821395	−	0.821395i	−0.986308	+	0.164913i	$0.947266\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			360.146i			1.37184i	0.727679	+	0.685918i	$0.240599\pi$
$41$			360.146i			1.37184i	−0.727679	+	0.685918i	$0.759401\pi$
$42$		0			0
$43$	312.475	+	312.475i	1.10818	+	1.10818i	0.993389	+	0.114795i	$0.0366213\pi$
$43$	312.475	+	312.475i	1.10818	+	1.10818i	0.114795	+	0.993389i	$0.463379\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	343.892			1.06727			0.533636	−	0.845715i	$-0.320825\pi$
$47$	343.892			1.06727			0.533636	+	0.845715i	$0.320825\pi$
$48$		0			0
$49$	292.031			0.851402
$50$		0			0
$51$		0			0
$52$		0			0
$53$	249.900	+	249.900i	0.647667	+	0.647667i	0.952429	−	0.304762i	$-0.0985768\pi$
$53$	249.900	+	249.900i	0.647667	+	0.647667i	−0.304762	+	0.952429i	$0.598577\pi$
$54$		0			0
$55$		−	32.8207i		−	0.0804644i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−152.755	−	152.755i	−0.337067	−	0.337067i	0.518195	−	0.855262i	$-0.326604\pi$
$59$	−152.755	−	152.755i	−0.337067	−	0.337067i	−0.855262	+	0.518195i	$0.826604\pi$
$60$		0			0
$61$	−525.985	+	525.985i	−1.10402	+	1.10402i	−0.110104	+	0.993920i	$0.535118\pi$
$61$	−525.985	+	525.985i	−1.10402	+	1.10402i	−0.993920	+	0.110104i	$0.964882\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18.8355			0.0359425
$66$		0			0
$67$	35.3052	−	35.3052i	0.0643764	−	0.0643764i	−0.674186	−	0.738562i	$-0.735505\pi$
$67$	35.3052	−	35.3052i	0.0643764	−	0.0643764i	0.738562	+	0.674186i	$0.235505\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			784.715i			1.31167i	0.754905	+	0.655834i	$0.227683\pi$
$71$			784.715i			1.31167i	−0.754905	+	0.655834i	$0.772317\pi$
$72$		0			0
$73$			800.215i			1.28299i	0.767128	+	0.641494i	$0.221685\pi$
$73$			800.215i			1.28299i	−0.767128	+	0.641494i	$0.778315\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	181.662	−	181.662i	0.268861	−	0.268861i
$78$		0			0
$79$	−548.062			−0.780530			−0.390265	−	0.920703i	$-0.627617\pi$
$79$	−548.062			−0.780530			−0.390265	+	0.920703i	$0.627617\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	464.431	−	464.431i	0.614191	−	0.614191i	−0.329844	−	0.944035i	$-0.606996\pi$
$83$	464.431	−	464.431i	0.614191	−	0.614191i	0.944035	+	0.329844i	$0.106996\pi$
$84$		0			0
$85$	46.0489	+	46.0489i	0.0587613	+	0.0587613i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			302.977i			0.360849i	0.983589	+	0.180424i	$0.0577471\pi$
$89$			302.977i			0.360849i	−0.983589	+	0.180424i	$0.942253\pi$
$90$		0			0
$91$	104.254	+	104.254i	0.120097	+	0.120097i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−56.2679			−0.0607680
$96$		0			0
$97$	1567.24			1.64050			0.820252	−	0.572002i	$-0.193833\pi$
$97$	1567.24			1.64050			0.820252	+	0.572002i	$0.193833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−59.6129	−	59.6129i	−0.0587297	−	0.0587297i	0.677132	−	0.735862i	$-0.263223\pi$
$101$	−59.6129	−	59.6129i	−0.0587297	−	0.0587297i	−0.735862	+	0.677132i	$0.763223\pi$
$102$		0			0
$103$			1762.59i			1.68615i	0.537795	+	0.843076i	$0.319257\pi$
$103$			1762.59i			1.68615i	−0.537795	+	0.843076i	$0.680743\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	656.579	+	656.579i	0.593214	+	0.593214i	0.938498	−	0.345284i	$-0.112218\pi$
$107$	656.579	+	656.579i	0.593214	+	0.593214i	−0.345284	+	0.938498i	$0.612218\pi$
$108$		0			0
$109$	−327.776	+	327.776i	−0.288030	+	0.288030i	−0.836301	−	0.548271i	$-0.815286\pi$
$109$	−327.776	+	327.776i	−0.288030	+	0.288030i	0.548271	+	0.836301i	$0.315286\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1349.18			1.12319			0.561593	−	0.827414i	$-0.310189\pi$
$113$	1349.18			1.12319			0.561593	+	0.827414i	$0.310189\pi$
$114$		0			0
$115$	136.624	−	136.624i	0.110784	−	0.110784i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			509.761i			0.392686i
$120$		0			0
$121$		−	36.0534i		−	0.0270875i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−160.694	+	160.694i	−0.114983	+	0.114983i
$126$		0			0
$127$	−52.2111			−0.0364802			−0.0182401	−	0.999834i	$-0.505806\pi$
$127$	−52.2111			−0.0364802			−0.0182401	+	0.999834i	$0.505806\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−387.958	+	387.958i	−0.258749	+	0.258749i	−0.824545	−	0.565796i	$-0.808569\pi$
$131$	−387.958	+	387.958i	−0.258749	+	0.258749i	0.565796	+	0.824545i	$0.308569\pi$
$132$		0			0
$133$	−311.442	−	311.442i	−0.203048	−	0.203048i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			795.048i			0.495807i	0.968785	+	0.247903i	$0.0797415\pi$
$137$			795.048i			0.495807i	−0.968785	+	0.247903i	$0.920258\pi$
$138$		0			0
$139$	−1708.70	−	1708.70i	−1.04266	−	1.04266i	−0.999048	−	0.0436142i	$-0.986113\pi$
$139$	−1708.70	−	1708.70i	−1.04266	−	1.04266i	−0.0436142	−	0.999048i	$-0.513887\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−743.160			−0.434588
$144$		0			0
$145$	7.53865			0.00431759
$146$		0			0
$147$		0			0
$148$		0			0
$149$	69.6301	+	69.6301i	0.0382840	+	0.0382840i	0.725990	−	0.687706i	$-0.241382\pi$
$149$	69.6301	+	69.6301i	0.0382840	+	0.0382840i	−0.687706	+	0.725990i	$0.741382\pi$
$150$		0			0
$151$			1567.48i			0.844768i	0.906417	+	0.422384i	$0.138807\pi$
$151$			1567.48i			0.844768i	−0.906417	+	0.422384i	$0.861193\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−69.5271	−	69.5271i	−0.0360294	−	0.0360294i
$156$		0			0
$157$	1738.35	−	1738.35i	0.883667	−	0.883667i	−0.110238	−	0.993905i	$-0.535161\pi$
$157$	1738.35	−	1738.35i	0.883667	−	0.883667i	0.993905	+	0.110238i	$0.0351614\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1512.42			0.740344
$162$		0			0
$163$	2685.71	−	2685.71i	1.29056	−	1.29056i	0.356116	−	0.934442i	$-0.384101\pi$
$163$	2685.71	−	2685.71i	1.29056	−	1.29056i	0.934442	−	0.356116i	$-0.115899\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	27.5126i		−	0.0127484i	−0.999980	−	0.00637422i	$-0.997971\pi$
$167$		−	27.5126i		−	0.0127484i	0.999980	−	0.00637422i	$-0.00202899\pi$
$168$		0			0
$169$			1770.51i			0.805875i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3044.87	+	3044.87i	−1.33813	+	1.33813i	−0.440267	+	0.897867i	$0.645116\pi$
$173$	−3044.87	+	3044.87i	−1.33813	+	1.33813i	−0.897867	+	0.440267i	$0.854884\pi$
$174$		0			0
$175$	−886.469			−0.382919
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−365.808	+	365.808i	−0.152747	+	0.152747i	−0.779344	−	0.626597i	$-0.784447\pi$
$179$	−365.808	+	365.808i	−0.152747	+	0.152747i	0.626597	+	0.779344i	$0.284447\pi$
$180$		0			0
$181$	−1737.10	−	1737.10i	−0.713359	−	0.713359i	0.253877	−	0.967236i	$-0.418294\pi$
$181$	−1737.10	−	1737.10i	−0.713359	−	0.713359i	−0.967236	+	0.253877i	$0.918294\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			238.447i			0.0947620i
$186$		0			0
$187$	−1816.87	−	1816.87i	−0.710495	−	0.710495i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1709.44			−0.647595			−0.323798	−	0.946126i	$-0.604960\pi$
$191$	−1709.44			−0.647595			−0.323798	+	0.946126i	$0.604960\pi$
$192$		0			0
$193$	2404.54			0.896800			0.448400	−	0.893833i	$-0.351994\pi$
$193$	2404.54			0.896800			0.448400	+	0.893833i	$0.351994\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2772.54	−	2772.54i	−1.00272	−	1.00272i	−0.999996	−	0.00271983i	$-0.999134\pi$
$197$	−2772.54	−	2772.54i	−1.00272	−	1.00272i	−0.00271983	−	0.999996i	$-0.500866\pi$
$198$		0			0
$199$			1506.90i			0.536790i	0.963309	+	0.268395i	$0.0864932\pi$
$199$			1506.90i			0.536790i	−0.963309	+	0.268395i	$0.913507\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	41.7263	+	41.7263i	0.0144267	+	0.0144267i
$204$		0			0
$205$	232.266	−	232.266i	0.0791324	−	0.0791324i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2220.06			0.734760
$210$		0			0
$211$	1965.22	−	1965.22i	0.641190	−	0.641190i	−0.309658	−	0.950848i	$-0.600215\pi$
$211$	1965.22	−	1965.22i	0.641190	−	0.641190i	0.950848	+	0.309658i	$0.100215\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	403.043i		−	0.127848i
$216$		0			0
$217$		−	769.663i		−	0.240775i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1042.69	−	1042.69i	0.317370	−	0.317370i
$222$		0			0
$223$	−85.7869			−0.0257610			−0.0128805	−	0.999917i	$-0.504100\pi$
$223$	−85.7869			−0.0257610			−0.0128805	+	0.999917i	$0.504100\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2526.02	−	2526.02i	0.738582	−	0.738582i	−0.233722	−	0.972304i	$-0.575090\pi$
$227$	2526.02	−	2526.02i	0.738582	−	0.738582i	0.972304	+	0.233722i	$0.0750905\pi$
$228$		0			0
$229$	−1116.65	−	1116.65i	−0.322227	−	0.322227i	0.527394	−	0.849621i	$-0.323169\pi$
$229$	−1116.65	−	1116.65i	−0.322227	−	0.322227i	−0.849621	+	0.527394i	$0.823169\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5754.25i		−	1.61791i	−0.587869	−	0.808956i	$-0.700033\pi$
$233$		−	5754.25i		−	1.61791i	0.587869	−	0.808956i	$-0.299967\pi$
$234$		0			0
$235$	−221.783	−	221.783i	−0.0615640	−	0.0615640i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−641.784			−0.173697			−0.0868484	−	0.996222i	$-0.527680\pi$
$239$	−641.784			−0.173697			−0.0868484	+	0.996222i	$0.527680\pi$
$240$		0			0
$241$	−2887.45			−0.771771			−0.385885	−	0.922547i	$-0.626104\pi$
$241$	−2887.45			−0.771771			−0.385885	+	0.922547i	$0.626104\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−188.337	−	188.337i	−0.0491119	−	0.0491119i
$246$		0			0
$247$			1274.07i			0.328208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2367.55	−	2367.55i	−0.595372	−	0.595372i	0.343706	−	0.939077i	$-0.388318\pi$
$251$	−2367.55	−	2367.55i	−0.595372	−	0.595372i	−0.939077	+	0.343706i	$0.888318\pi$
$252$		0			0
$253$	−5390.51	+	5390.51i	−1.33952	+	1.33952i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5112.43			−1.24087			−0.620437	−	0.784256i	$-0.713045\pi$
$257$	−5112.43			−1.24087			−0.620437	+	0.784256i	$0.713045\pi$
$258$		0			0
$259$	−1319.80	+	1319.80i	−0.316635	+	0.316635i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5496.36i		−	1.28867i	−0.764744	−	0.644334i	$-0.777134\pi$
$263$		−	5496.36i		−	1.28867i	0.764744	−	0.644334i	$-0.222866\pi$
$264$		0			0
$265$		−	322.331i		−	0.0747194i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2474.35	−	2474.35i	0.560831	−	0.560831i	−0.368712	−	0.929544i	$-0.620201\pi$
$269$	2474.35	−	2474.35i	0.560831	−	0.560831i	0.929544	+	0.368712i	$0.120201\pi$
$270$		0			0
$271$	1718.58			0.385226			0.192613	−	0.981275i	$-0.438304\pi$
$271$	1718.58			0.385226			0.192613	+	0.981275i	$0.438304\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3159.52	−	3159.52i	0.692823	−	0.692823i
$276$		0			0
$277$	−4722.69	−	4722.69i	−1.02440	−	1.02440i	−0.999695	−	0.0247055i	$-0.992135\pi$
$277$	−4722.69	−	4722.69i	−1.02440	−	1.02440i	−0.0247055	−	0.999695i	$-0.507865\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2543.21i			0.539911i	0.962873	+	0.269955i	$0.0870090\pi$
$281$			2543.21i			0.539911i	−0.962873	+	0.269955i	$0.912991\pi$
$282$		0			0
$283$	3206.82	+	3206.82i	0.673589	+	0.673589i	0.958542	−	0.284952i	$-0.0919778\pi$
$283$	3206.82	+	3206.82i	0.673589	+	0.673589i	−0.284952	+	0.958542i	$0.591978\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2571.17			0.528821
$288$		0			0
$289$	185.302			0.0377167
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5911.40	−	5911.40i	−1.17866	−	1.17866i	−0.980086	−	0.198575i	$-0.936369\pi$
$293$	−5911.40	−	5911.40i	−1.17866	−	1.17866i	−0.198575	−	0.980086i	$-0.563631\pi$
$294$		0			0
$295$			197.030i			0.0388865i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3093.57	−	3093.57i	−0.598347	−	0.598347i
$300$		0			0
$301$	2230.84	−	2230.84i	0.427187	−	0.427187i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	678.438			0.127368
$306$		0			0
$307$	−1162.25	+	1162.25i	−0.216068	+	0.216068i	−0.806839	−	0.590771i	$-0.798824\pi$
$307$	−1162.25	+	1162.25i	−0.216068	+	0.216068i	0.590771	+	0.806839i	$0.298824\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2357.28i			0.429805i	0.976636	+	0.214902i	$0.0689433\pi$
$311$			2357.28i			0.429805i	−0.976636	+	0.214902i	$0.931057\pi$
$312$		0			0
$313$		−	3785.95i		−	0.683689i	−0.939757	−	0.341844i	$-0.888948\pi$
$313$		−	3785.95i		−	0.683689i	0.939757	−	0.341844i	$-0.111052\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3135.51	−	3135.51i	0.555545	−	0.555545i	−0.372491	−	0.928036i	$-0.621496\pi$
$317$	3135.51	−	3135.51i	0.555545	−	0.555545i	0.928036	+	0.372491i	$0.121496\pi$
$318$		0			0
$319$	−297.439			−0.0522050
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3114.85	+	3114.85i	−0.536578	+	0.536578i
$324$		0			0
$325$	1813.22	+	1813.22i	0.309476	+	0.309476i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2455.13i		−	0.411416i
$330$		0			0
$331$	1734.12	+	1734.12i	0.287963	+	0.287963i	0.836274	−	0.548312i	$-0.184729\pi$
$331$	1734.12	+	1734.12i	0.287963	+	0.287963i	−0.548312	+	0.836274i	$0.684729\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−45.5382			−0.00742691
$336$		0			0
$337$	5828.89			0.942196			0.471098	−	0.882081i	$-0.343858\pi$
$337$	5828.89			0.942196			0.471098	+	0.882081i	$0.343858\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2743.21	+	2743.21i	0.435639	+	0.435639i
$342$		0			0
$343$		−	4533.65i		−	0.713686i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−716.234	−	716.234i	−0.110805	−	0.110805i	0.649530	−	0.760336i	$-0.274966\pi$
$347$	−716.234	−	716.234i	−0.110805	−	0.110805i	−0.760336	+	0.649530i	$0.774966\pi$
$348$		0			0
$349$	−4388.39	+	4388.39i	−0.673081	+	0.673081i	−0.958425	−	0.285344i	$-0.907892\pi$
$349$	−4388.39	+	4388.39i	−0.673081	+	0.673081i	0.285344	+	0.958425i	$0.407892\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9348.76			−1.40959			−0.704794	−	0.709412i	$-0.748960\pi$
$353$	−9348.76			−1.40959			−0.704794	+	0.709412i	$0.748960\pi$
$354$		0			0
$355$	506.079	−	506.079i	0.0756617	−	0.0756617i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12531.1i			1.84224i	0.389275	+	0.921122i	$0.372726\pi$
$359$			12531.1i			1.84224i	−0.389275	+	0.921122i	$0.627274\pi$
$360$		0			0
$361$			3052.92i			0.445097i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	516.076	−	516.076i	0.0740073	−	0.0740073i
$366$		0			0
$367$	−9276.83			−1.31947			−0.659736	−	0.751497i	$-0.729332\pi$
$367$	−9276.83			−1.31947			−0.659736	+	0.751497i	$0.729332\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1784.10	−	1784.10i	0.249665	−	0.249665i
$372$		0			0
$373$	1765.32	+	1765.32i	0.245053	+	0.245053i	0.818937	−	0.573884i	$-0.194564\pi$
$373$	1765.32	+	1765.32i	0.245053	+	0.245053i	−0.573884	+	0.818937i	$0.694564\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	170.698i		−	0.0233193i
$378$		0			0
$379$	−2909.05	−	2909.05i	−0.394269	−	0.394269i	0.481937	−	0.876206i	$-0.339933\pi$
$379$	−2909.05	−	2909.05i	−0.394269	−	0.394269i	−0.876206	+	0.481937i	$0.839933\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1520.26			0.202824			0.101412	−	0.994845i	$-0.467664\pi$
$383$	1520.26			0.202824			0.101412	+	0.994845i	$0.467664\pi$
$384$		0			0
$385$	−234.316			−0.0310177
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8093.35	+	8093.35i	1.05488	+	1.05488i	0.998404	+	0.0564785i	$0.0179872\pi$
$389$	8093.35	+	8093.35i	1.05488	+	1.05488i	0.0564785	+	0.998404i	$0.482013\pi$
$390$		0			0
$391$		−	15126.3i		−	1.95644i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	353.457	+	353.457i	0.0450237	+	0.0450237i
$396$		0			0
$397$	8897.26	−	8897.26i	1.12479	−	1.12479i	0.133776	−	0.991012i	$-0.457290\pi$
$397$	8897.26	−	8897.26i	1.12479	−	1.12479i	0.991012	−	0.133776i	$-0.0427101\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4199.74			0.523004			0.261502	−	0.965203i	$-0.415782\pi$
$401$	4199.74			0.523004			0.261502	+	0.965203i	$0.415782\pi$
$402$		0			0
$403$	−1574.30	+	1574.30i	−0.194595	+	0.194595i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9407.97i		−	1.14579i
$408$		0			0
$409$		−	7881.87i		−	0.952894i	−0.879203	−	0.476447i	$-0.841924\pi$
$409$		−	7881.87i		−	0.952894i	0.879203	−	0.476447i	$-0.158076\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1090.56	+	1090.56i	−0.129934	+	0.129934i
$414$		0			0
$415$	−599.043			−0.0708575
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5502.69	+	5502.69i	−0.641585	+	0.641585i	−0.950945	−	0.309360i	$-0.899885\pi$
$419$	−5502.69	+	5502.69i	−0.641585	+	0.641585i	0.309360	+	0.950945i	$0.399885\pi$
$420$		0			0
$421$	11302.2	+	11302.2i	1.30840	+	1.30840i	0.922573	+	0.385823i	$0.126082\pi$
$421$	11302.2	+	11302.2i	1.30840	+	1.30840i	0.385823	+	0.922573i	$0.373918\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			8865.90i			1.01190i
$426$		0			0
$427$	3755.14	+	3755.14i	0.425584	+	0.425584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5428.20			−0.606652			−0.303326	−	0.952887i	$-0.598097\pi$
$431$	−5428.20			−0.606652			−0.303326	+	0.952887i	$0.598097\pi$
$432$		0			0
$433$	−5143.78			−0.570888			−0.285444	−	0.958395i	$-0.592141\pi$
$433$	−5143.78			−0.570888			−0.285444	+	0.958395i	$0.592141\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9241.50	+	9241.50i	1.01163	+	1.01163i
$438$		0			0
$439$		−	2183.50i		−	0.237386i	−0.992931	−	0.118693i	$-0.962130\pi$
$439$		−	2183.50i		−	0.237386i	0.992931	−	0.118693i	$-0.0378705\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	561.821	+	561.821i	0.0602549	+	0.0602549i	0.736592	−	0.676337i	$-0.236434\pi$
$443$	561.821	+	561.821i	0.0602549	+	0.0602549i	−0.676337	+	0.736592i	$0.736434\pi$
$444$		0			0
$445$	195.397	−	195.397i	0.0208150	−	0.0208150i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2639.02			0.277378			0.138689	−	0.990336i	$-0.455711\pi$
$449$	2639.02			0.277378			0.138689	+	0.990336i	$0.455711\pi$
$450$		0			0
$451$	−9164.08	+	9164.08i	−0.956807	+	0.956807i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	134.472i		−	0.0138552i
$456$		0			0
$457$			5546.69i			0.567753i	0.958861	+	0.283876i	$0.0916205\pi$
$457$			5546.69i			0.567753i	−0.958861	+	0.283876i	$0.908379\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8475.23	+	8475.23i	−0.856249	+	0.856249i	−0.990894	−	0.134645i	$-0.957011\pi$
$461$	−8475.23	+	8475.23i	−0.856249	+	0.856249i	0.134645	+	0.990894i	$0.457011\pi$
$462$		0			0
$463$	16410.6			1.64722			0.823612	−	0.567154i	$-0.191956\pi$
$463$	16410.6			1.64722			0.823612	+	0.567154i	$0.191956\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6689.53	+	6689.53i	−0.662858	+	0.662858i	−0.956053	−	0.293195i	$-0.905281\pi$
$467$	−6689.53	+	6689.53i	−0.662858	+	0.662858i	0.293195	+	0.956053i	$0.405281\pi$
$468$		0			0
$469$	−252.053	−	252.053i	−0.0248161	−	0.0248161i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			15902.1i			1.54584i
$474$		0			0
$475$	−5416.69	−	5416.69i	−0.523231	−	0.523231i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7188.55			0.685707			0.342853	−	0.939389i	$-0.388607\pi$
$479$	7188.55			0.685707			0.342853	+	0.939389i	$0.388607\pi$
$480$		0			0
$481$	5399.16			0.511810
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1010.75	−	1010.75i	−0.0946301	−	0.0946301i
$486$		0			0
$487$			11181.6i			1.04042i	0.854038	+	0.520210i	$0.174146\pi$
$487$			11181.6i			1.04042i	−0.854038	+	0.520210i	$0.825854\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11679.8	+	11679.8i	1.07353	+	1.07353i	0.997073	+	0.0764529i	$0.0243595\pi$
$491$	11679.8	+	11679.8i	1.07353	+	1.07353i	0.0764529	+	0.997073i	$0.475641\pi$
$492$		0			0
$493$	417.321	−	417.321i	0.0381241	−	0.0381241i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5602.28			0.505627
$498$		0			0
$499$	2094.26	−	2094.26i	0.187880	−	0.187880i	−0.606899	−	0.794779i	$-0.707587\pi$
$499$	2094.26	−	2094.26i	0.187880	−	0.187880i	0.794779	+	0.606899i	$0.207587\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4293.63i		−	0.380604i	−0.981726	−	0.190302i	$-0.939053\pi$
$503$		−	4293.63i		−	0.380604i	0.981726	−	0.190302i	$-0.0609466\pi$
$504$		0			0
$505$			76.8912i			0.00677548i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9115.13	−	9115.13i	0.793755	−	0.793755i	−0.188348	−	0.982102i	$-0.560313\pi$
$509$	9115.13	−	9115.13i	0.793755	−	0.793755i	0.982102	+	0.188348i	$0.0603132\pi$
$510$		0			0
$511$	5712.95			0.494571
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1136.73	−	1136.73i	0.0972631	−	0.0972631i
$516$		0			0
$517$	8750.49	+	8750.49i	0.744383	+	0.744383i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	23150.3i		−	1.94671i	−0.229309	−	0.973354i	$-0.573647\pi$
$521$		−	23150.3i		−	1.94671i	0.229309	−	0.973354i	$-0.426353\pi$
$522$		0			0
$523$	8510.52	+	8510.52i	0.711547	+	0.711547i	0.966859	−	0.255312i	$-0.0821781\pi$
$523$	8510.52	+	8510.52i	0.711547	+	0.711547i	−0.255312	+	0.966859i	$0.582178\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7697.69			−0.636274
$528$		0			0
$529$	−32711.4			−2.68854
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5259.19	−	5259.19i	−0.427394	−	0.427394i
$534$		0			0
$535$		−	846.884i		−	0.0684373i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7430.87	+	7430.87i	0.593822	+	0.593822i
$540$		0			0
$541$	3403.84	−	3403.84i	0.270504	−	0.270504i	−0.558799	−	0.829303i	$-0.688738\pi$
$541$	3403.84	−	3403.84i	0.270504	−	0.270504i	0.829303	+	0.558799i	$0.188738\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	422.780			0.0332292
$546$		0			0
$547$	−15672.4	+	15672.4i	−1.22505	+	1.22505i	−0.259237	+	0.965814i	$0.583471\pi$
$547$	−15672.4	+	15672.4i	−1.22505	+	1.22505i	−0.965814	+	0.259237i	$0.916529\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			509.930i			0.0394261i
$552$		0			0
$553$			3912.76i			0.300882i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6285.58	+	6285.58i	−0.478149	+	0.478149i	−0.904539	−	0.426391i	$-0.859785\pi$
$557$	−6285.58	+	6285.58i	−0.478149	+	0.478149i	0.426391	+	0.904539i	$0.359785\pi$
$558$		0			0
$559$	−9126.11			−0.690507
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1763.14	+	1763.14i	−0.131985	+	0.131985i	−0.770013	−	0.638028i	$-0.779750\pi$
$563$	−1763.14	+	1763.14i	−0.131985	+	0.131985i	0.638028	+	0.770013i	$0.279750\pi$
$564$		0			0
$565$	−870.114	−	870.114i	−0.0647894	−	0.0647894i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1150.05i			0.0847323i	0.999102	+	0.0423662i	$0.0134896\pi$
$569$			1150.05i			0.0847323i	−0.999102	+	0.0423662i	$0.986510\pi$
$570$		0			0
$571$	4853.30	+	4853.30i	0.355699	+	0.355699i	0.862225	−	0.506526i	$-0.169070\pi$
$571$	4853.30	+	4853.30i	0.355699	+	0.355699i	−0.506526	+	0.862225i	$0.669070\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26304.4			1.90778
$576$		0			0
$577$	−5152.84			−0.371777			−0.185889	−	0.982571i	$-0.559516\pi$
$577$	−5152.84			−0.371777			−0.185889	+	0.982571i	$0.559516\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3315.69	−	3315.69i	−0.236761	−	0.236761i
$582$		0			0
$583$			12717.6i			0.903449i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12549.2	+	12549.2i	0.882387	+	0.882387i	0.993777	−	0.111390i	$-0.0355302\pi$
$587$	12549.2	+	12549.2i	0.882387	+	0.882387i	−0.111390	+	0.993777i	$0.535530\pi$
$588$		0			0
$589$	4702.96	−	4702.96i	0.329002	−	0.329002i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1116.75			0.0773347			0.0386674	−	0.999252i	$-0.487689\pi$
$593$	1116.75			0.0773347			0.0386674	+	0.999252i	$0.487689\pi$
$594$		0			0
$595$	328.756	−	328.756i	0.0226515	−	0.0226515i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	15827.4i		−	1.07962i	−0.841788	−	0.539808i	$-0.818497\pi$
$599$		−	15827.4i		−	1.07962i	0.841788	−	0.539808i	$-0.181503\pi$
$600$		0			0
$601$			12216.9i			0.829180i	0.910008	+	0.414590i	$0.136075\pi$
$601$			12216.9i			0.829180i	−0.910008	+	0.414590i	$0.863925\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.2516	+	23.2516i	−0.00156250	+	0.00156250i
$606$		0			0
$607$	24175.5			1.61656			0.808282	−	0.588795i	$-0.200398\pi$
$607$	24175.5			1.61656			0.808282	+	0.588795i	$0.200398\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5021.84	+	5021.84i	−0.332507	+	0.332507i
$612$		0			0
$613$	12893.7	+	12893.7i	0.849544	+	0.849544i	0.990076	−	0.140532i	$-0.0448812\pi$
$613$	12893.7	+	12893.7i	0.849544	+	0.849544i	−0.140532	+	0.990076i	$0.544881\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	13762.7i		−	0.897998i	−0.893532	−	0.448999i	$-0.851781\pi$
$617$		−	13762.7i		−	0.897998i	0.893532	−	0.448999i	$-0.148219\pi$
$618$		0			0
$619$	326.797	+	326.797i	0.0212199	+	0.0212199i	0.717637	−	0.696417i	$-0.245224\pi$
$619$	326.797	+	326.797i	0.0212199	+	0.0212199i	−0.696417	+	0.717637i	$0.745224\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2163.04			0.139101
$624$		0			0
$625$	−15313.7			−0.980080
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13199.8	+	13199.8i	0.836742	+	0.836742i
$630$		0			0
$631$		−	12260.2i		−	0.773487i	−0.922187	−	0.386744i	$-0.873600\pi$
$631$		−	12260.2i		−	0.773487i	0.922187	−	0.386744i	$-0.126400\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	33.6721	+	33.6721i	0.00210431	+	0.00210431i
$636$		0			0
$637$	−4264.52	+	4264.52i	−0.265253	+	0.265253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10329.4			−0.636485			−0.318242	−	0.948009i	$-0.603093\pi$
$641$	−10329.4			−0.636485			−0.318242	+	0.948009i	$0.603093\pi$
$642$		0			0
$643$	19299.9	−	19299.9i	1.18369	−	1.18369i	0.204911	−	0.978781i	$-0.434309\pi$
$643$	19299.9	−	19299.9i	1.18369	−	1.18369i	0.978781	−	0.204911i	$-0.0656905\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8657.07i			0.526035i	0.964791	+	0.263017i	$0.0847177\pi$
$647$			8657.07i			0.526035i	−0.964791	+	0.263017i	$0.915282\pi$
$648$		0			0
$649$		−	7773.84i		−	0.470185i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5966.31	+	5966.31i	−0.357549	+	0.357549i	−0.862909	−	0.505360i	$-0.831360\pi$
$653$	−5966.31	+	5966.31i	−0.357549	+	0.357549i	0.505360	+	0.862909i	$0.331360\pi$
$654$		0			0
$655$	500.405			0.0298511
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7127.52	−	7127.52i	0.421318	−	0.421318i	−0.464339	−	0.885657i	$-0.653708\pi$
$659$	7127.52	−	7127.52i	0.421318	−	0.421318i	0.885657	+	0.464339i	$0.153708\pi$
$660$		0			0
$661$	2386.19	+	2386.19i	0.140412	+	0.140412i	0.773819	−	0.633407i	$-0.218344\pi$
$661$	2386.19	+	2386.19i	0.140412	+	0.140412i	−0.633407	+	0.773819i	$0.718344\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			401.711i			0.0234251i
$666$		0			0
$667$	−1238.16	−	1238.16i	−0.0718765	−	0.0718765i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−26767.9			−1.54003
$672$		0			0
$673$	−3709.39			−0.212461			−0.106231	−	0.994342i	$-0.533878\pi$
$673$	−3709.39			−0.212461			−0.106231	+	0.994342i	$0.533878\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1602.20	+	1602.20i	0.0909564	+	0.0909564i	0.751121	−	0.660165i	$-0.229513\pi$
$677$	1602.20	+	1602.20i	0.0909564	+	0.0909564i	−0.660165	+	0.751121i	$0.729513\pi$
$678$		0			0
$679$		−	11188.9i		−	0.632388i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15495.9	−	15495.9i	−0.868130	−	0.868130i	0.124135	−	0.992265i	$-0.460384\pi$
$683$	−15495.9	−	15495.9i	−0.868130	−	0.868130i	−0.992265	+	0.124135i	$0.960384\pi$
$684$		0			0
$685$	512.743	−	512.743i	0.0285999	−	0.0285999i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7298.55			−0.403560
$690$		0			0
$691$	−6337.43	+	6337.43i	−0.348896	+	0.348896i	−0.859698	−	0.510802i	$-0.829348\pi$
$691$	−6337.43	+	6337.43i	−0.348896	+	0.348896i	0.510802	+	0.859698i	$0.329348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2203.96i			0.120289i
$696$		0			0
$697$		−	25715.3i		−	1.39747i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	952.656	−	952.656i	0.0513285	−	0.0513285i	−0.680977	−	0.732305i	$-0.738444\pi$
$701$	952.656	−	952.656i	0.0513285	−	0.0513285i	0.732305	+	0.680977i	$0.238444\pi$
$702$		0			0
$703$	−16129.0			−0.865318
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−425.592	+	425.592i	−0.0226394	+	0.0226394i
$708$		0			0
$709$	−20781.1	−	20781.1i	−1.10078	−	1.10078i	−0.994317	−	0.106461i	$-0.966048\pi$
$709$	−20781.1	−	20781.1i	−1.10078	−	1.10078i	−0.106461	−	0.994317i	$-0.533952\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			22838.4i			1.19959i
$714$		0			0
$715$	479.280	+	479.280i	0.0250686	+	0.0250686i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1927.96			−0.100001			−0.0500006	−	0.998749i	$-0.515922\pi$
$719$	−1927.96			−0.100001			−0.0500006	+	0.998749i	$0.515922\pi$
$720$		0			0
$721$	12583.6			0.649984
$722$		0			0
$723$		0			0
$724$		0			0
$725$	725.717	+	725.717i	0.0371758	+	0.0371758i
$726$		0			0
$727$		−	5482.27i		−	0.279678i	−0.990174	−	0.139839i	$-0.955341\pi$
$727$		−	5482.27i		−	0.279678i	0.990174	−	0.139839i	$-0.0446585\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−22311.4	−	22311.4i	−1.12889	−	1.12889i
$732$		0			0
$733$	15048.8	−	15048.8i	0.758311	−	0.758311i	−0.217704	−	0.976015i	$-0.569857\pi$
$733$	15048.8	−	15048.8i	0.758311	−	0.758311i	0.976015	+	0.217704i	$0.0698568\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1796.72			0.0898004
$738$		0			0
$739$	−12622.3	+	12622.3i	−0.628306	+	0.628306i	−0.947642	−	0.319335i	$-0.896540\pi$
$739$	−12622.3	+	12622.3i	−0.628306	+	0.628306i	0.319335	+	0.947642i	$0.396540\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2335.45i		−	0.115315i	−0.998336	−	0.0576576i	$-0.981637\pi$
$743$		−	2335.45i		−	0.115315i	0.998336	−	0.0576576i	$-0.0183632\pi$
$744$		0			0
$745$		−	89.8119i		−	0.00441672i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4687.49	−	4687.49i	0.228674	−	0.228674i
$750$		0			0
$751$	3513.73			0.170729			0.0853647	−	0.996350i	$-0.472794\pi$
$751$	3513.73			0.170729			0.0853647	+	0.996350i	$0.472794\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1010.90	−	1010.90i	0.0487292	−	0.0487292i
$756$		0			0
$757$	−10416.0	−	10416.0i	−0.500102	−	0.500102i	0.411368	−	0.911470i	$-0.365051\pi$
$757$	−10416.0	−	10416.0i	−0.500102	−	0.500102i	−0.911470	+	0.411368i	$0.865051\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			17937.9i			0.854464i	0.904142	+	0.427232i	$0.140511\pi$
$761$			17937.9i			0.854464i	−0.904142	+	0.427232i	$0.859489\pi$
$762$		0			0
$763$	2340.08	+	2340.08i	0.111031	+	0.111031i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4461.34			0.210026
$768$		0			0
$769$	32224.0			1.51109			0.755544	−	0.655097i	$-0.227372\pi$
$769$	32224.0			1.51109			0.755544	+	0.655097i	$0.227372\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13719.5	−	13719.5i	−0.638365	−	0.638365i	0.311787	−	0.950152i	$-0.399072\pi$
$773$	−13719.5	−	13719.5i	−0.638365	−	0.638365i	−0.950152	+	0.311787i	$0.899072\pi$
$774$		0			0
$775$		−	13386.2i		−	0.620448i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	15710.9	+	15710.9i	0.722596	+	0.722596i
$780$		0			0
$781$	−19967.5	+	19967.5i	−0.914842	+	0.914842i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2242.20			−0.101946
$786$		0			0
$787$	9919.72	−	9919.72i	0.449301	−	0.449301i	−0.445821	−	0.895122i	$-0.647088\pi$
$787$	9919.72	−	9919.72i	0.449301	−	0.449301i	0.895122	+	0.445821i	$0.147088\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	9632.14i		−	0.432970i
$792$		0			0
$793$		−	15361.9i		−	0.687915i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20018.0	−	20018.0i	0.889679	−	0.889679i	−0.104813	−	0.994492i	$-0.533424\pi$
$797$	20018.0	−	20018.0i	0.889679	−	0.889679i	0.994492	+	0.104813i	$0.0334243\pi$
$798$		0			0
$799$	−24554.7			−1.08721
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20361.9	+	20361.9i	−0.894838	+	0.894838i
$804$		0			0
$805$	−975.392	−	975.392i	−0.0427056	−	0.0427056i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	9967.99i		−	0.433196i	−0.976261	−	0.216598i	$-0.930504\pi$
$809$		−	9967.99i		−	0.433196i	0.976261	−	0.216598i	$-0.0694961\pi$
$810$		0			0
$811$	−16080.2	−	16080.2i	−0.696243	−	0.696243i	0.267355	−	0.963598i	$-0.413850\pi$
$811$	−16080.2	−	16080.2i	−0.696243	−	0.696243i	−0.963598	+	0.267355i	$0.913850\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3464.14			−0.148888
$816$		0			0
$817$	27262.7			1.16744
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8235.64	−	8235.64i	−0.350092	−	0.350092i	0.510051	−	0.860144i	$-0.329626\pi$
$821$	−8235.64	−	8235.64i	−0.350092	−	0.350092i	−0.860144	+	0.510051i	$0.829626\pi$
$822$		0			0
$823$		−	21762.8i		−	0.921752i	−0.887464	−	0.460876i	$-0.847535\pi$
$823$		−	21762.8i		−	0.921752i	0.887464	−	0.460876i	$-0.152465\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30211.4	−	30211.4i	−1.27032	−	1.27032i	−0.945921	−	0.324396i	$-0.894839\pi$
$827$	−30211.4	−	30211.4i	−1.27032	−	1.27032i	−0.324396	−	0.945921i	$-0.605161\pi$
$828$		0			0
$829$	−16821.1	+	16821.1i	−0.704729	+	0.704729i	−0.965422	−	0.260692i	$-0.916049\pi$
$829$	−16821.1	+	16821.1i	−0.704729	+	0.704729i	0.260692	+	0.965422i	$0.416049\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20851.7			−0.867310
$834$		0			0
$835$	−17.7435	+	17.7435i	−0.000735375	+	0.000735375i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			21686.9i			0.892389i	0.894936	+	0.446194i	$0.147221\pi$
$839$			21686.9i			0.892389i	−0.894936	+	0.446194i	$0.852779\pi$
$840$		0			0
$841$			24320.7i			0.997199i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1141.84	−	1141.84i	0.0464857	−	0.0464857i
$846$		0			0
$847$	−257.395			−0.0104418
$848$		0			0
$849$		0			0
$850$		0			0
$851$	39162.8	−	39162.8i	1.57754	−	1.57754i
$852$		0			0
$853$	12612.5	+	12612.5i	0.506266	+	0.506266i	0.913378	−	0.407112i	$-0.133464\pi$
$853$	12612.5	+	12612.5i	0.506266	+	0.506266i	−0.407112	+	0.913378i	$0.633464\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	46080.3i		−	1.83672i	−0.395741	−	0.918362i	$-0.629512\pi$
$857$		−	46080.3i		−	1.83672i	0.395741	−	0.918362i	$-0.370488\pi$
$858$		0			0
$859$	−8935.85	−	8935.85i	−0.354933	−	0.354933i	0.507008	−	0.861941i	$-0.330751\pi$
$859$	−8935.85	−	8935.85i	−0.354933	−	0.354933i	−0.861941	+	0.507008i	$0.830751\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19360.8			0.763674			0.381837	−	0.924230i	$-0.375292\pi$
$863$	19360.8			0.763674			0.381837	+	0.924230i	$0.375292\pi$
$864$		0			0
$865$	3927.40			0.154377
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13945.7	−	13945.7i	−0.544391	−	0.544391i
$870$		0			0
$871$			1031.12i			0.0401128i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1147.24	+	1147.24i	0.0443242	+	0.0443242i
$876$		0			0
$877$	19139.4	−	19139.4i	0.736933	−	0.736933i	−0.235050	−	0.971983i	$-0.575525\pi$
$877$	19139.4	−	19139.4i	0.736933	−	0.736933i	0.971983	+	0.235050i	$0.0755254\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4836.79			−0.184967			−0.0924833	−	0.995714i	$-0.529480\pi$
$881$	−4836.79			−0.184967			−0.0924833	+	0.995714i	$0.529480\pi$
$882$		0			0
$883$	7730.89	−	7730.89i	0.294638	−	0.294638i	−0.544271	−	0.838909i	$-0.683194\pi$
$883$	7730.89	−	7730.89i	0.294638	−	0.294638i	0.838909	+	0.544271i	$0.183194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			12163.5i			0.460442i	0.973138	+	0.230221i	$0.0739448\pi$
$887$			12163.5i			0.460442i	−0.973138	+	0.230221i	$0.926055\pi$
$888$		0			0
$889$			372.749i			0.0140625i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15001.9	−	15001.9i	0.562171	−	0.562171i
$894$		0			0
$895$	471.835			0.0176220
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−630.093	+	630.093i	−0.0233757	+	0.0233757i
$900$		0			0
$901$	−17843.4	−	17843.4i	−0.659768	−	0.659768i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2240.59i			0.0822981i
$906$		0			0
$907$	8264.83	+	8264.83i	0.302568	+	0.302568i	0.842018	−	0.539450i	$-0.181368\pi$
$907$	8264.83	+	8264.83i	0.302568	+	0.302568i	−0.539450	+	0.842018i	$0.681368\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−41364.8			−1.50437			−0.752183	−	0.658955i	$-0.770999\pi$
$911$	−41364.8			−1.50437			−0.752183	+	0.658955i	$0.770999\pi$
$912$		0			0
$913$	23635.3			0.856753
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2769.74	+	2769.74i	0.0997434	+	0.0997434i
$918$		0			0
$919$			36966.7i			1.32690i	0.748221	+	0.663449i	$0.230908\pi$
$919$			36966.7i			1.32690i	−0.748221	+	0.663449i	$0.769092\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11459.2	−	11459.2i	−0.408649	−	0.408649i
$924$		0			0
$925$	−22954.4	+	22954.4i	−0.815929	+	0.815929i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18882.5			0.666862			0.333431	−	0.942774i	$-0.391794\pi$
$929$	18882.5			0.666862			0.333431	+	0.942774i	$0.391794\pi$
$930$		0			0
$931$	12739.5	−	12739.5i	0.448464	−	0.448464i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2343.48i			0.0819678i
$936$		0			0
$937$			8581.23i			0.299185i	0.988748	+	0.149593i	$0.0477962\pi$
$937$			8581.23i			0.299185i	−0.988748	+	0.149593i	$0.952204\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5618.83	−	5618.83i	0.194653	−	0.194653i	−0.603050	−	0.797703i	$-0.706048\pi$
$941$	5618.83	−	5618.83i	0.194653	−	0.194653i	0.797703	+	0.603050i	$0.206048\pi$
$942$		0			0
$943$	−76295.1			−2.63469
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18722.7	−	18722.7i	0.642456	−	0.642456i	−0.308702	−	0.951159i	$-0.599895\pi$
$947$	18722.7	−	18722.7i	0.642456	−	0.642456i	0.951159	+	0.308702i	$0.0998947\pi$
$948$		0			0
$949$	−11685.5	−	11685.5i	−0.399713	−	0.399713i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	17197.8i		−	0.584565i	−0.956332	−	0.292282i	$-0.905585\pi$
$953$		−	17197.8i		−	0.584565i	0.956332	−	0.292282i	$-0.0944147\pi$
$954$		0			0
$955$	1102.45	+	1102.45i	0.0373556	+	0.0373556i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5676.06			0.191126
$960$		0			0
$961$	−18168.6			−0.609870
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1550.74	−	1550.74i	−0.0517306	−	0.0517306i
$966$		0			0
$967$			58259.3i			1.93743i	0.248182	+	0.968713i	$0.420167\pi$
$967$			58259.3i			1.93743i	−0.248182	+	0.968713i	$0.579833\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19203.8	−	19203.8i	−0.634685	−	0.634685i	0.314555	−	0.949239i	$-0.398145\pi$
$971$	−19203.8	−	19203.8i	−0.634685	−	0.634685i	−0.949239	+	0.314555i	$0.898145\pi$
$972$		0			0
$973$	−12198.9	+	12198.9i	−0.401930	+	0.401930i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16691.4			0.546578			0.273289	−	0.961932i	$-0.411888\pi$
$977$	16691.4			0.546578			0.273289	+	0.961932i	$0.411888\pi$
$978$		0			0
$979$	−7709.41	+	7709.41i	−0.251679	+	0.251679i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	36043.6i		−	1.16949i	−0.811216	−	0.584747i	$-0.801194\pi$
$983$		−	36043.6i		−	1.16949i	0.811216	−	0.584747i	$-0.198806\pi$
$984$		0			0
$985$			3576.14i			0.115680i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−66196.3	+	66196.3i	−2.12833	+	2.12833i
$990$		0			0
$991$	48948.9			1.56904			0.784518	−	0.620107i	$-0.212911\pi$
$991$	48948.9			1.56904			0.784518	+	0.620107i	$0.212911\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	971.831	−	971.831i	0.0309639	−	0.0309639i
$996$		0			0
$997$	−22542.1	−	22542.1i	−0.716062	−	0.716062i	0.251734	−	0.967796i	$-0.418999\pi$
$997$	−22542.1	−	22542.1i	−0.716062	−	0.716062i	−0.967796	+	0.251734i	$0.918999\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.b.433.7		24
3.2	odd	2		192.4.j.a.49.3		24
4.3	odd	2		144.4.k.b.37.3		24
12.11	even	2		48.4.j.a.37.10	yes	24
16.3	odd	4		144.4.k.b.109.3		24
16.13	even	4	inner	576.4.k.b.145.7		24
24.5	odd	2		384.4.j.a.97.9		24
24.11	even	2		384.4.j.b.97.4		24
48.5	odd	4		384.4.j.a.289.9		24
48.11	even	4		384.4.j.b.289.4		24
48.29	odd	4		192.4.j.a.145.3		24
48.35	even	4		48.4.j.a.13.10	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.10	✓	24	48.35	even	4
48.4.j.a.37.10	yes	24	12.11	even	2
144.4.k.b.37.3		24	4.3	odd	2
144.4.k.b.109.3		24	16.3	odd	4
192.4.j.a.49.3		24	3.2	odd	2
192.4.j.a.145.3		24	48.29	odd	4
384.4.j.a.97.9		24	24.5	odd	2
384.4.j.a.289.9		24	48.5	odd	4
384.4.j.b.97.4		24	24.11	even	2
384.4.j.b.289.4		24	48.11	even	4
576.4.k.b.145.7		24	16.13	even	4	inner
576.4.k.b.433.7		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.644922 - 0.644922i) q^{5} -7.13926i q^{7} +(25.4455 + 25.4455i) q^{11} +(-14.6030 + 14.6030i) q^{13} -71.4024 q^{17} +(43.6238 - 43.6238i) q^{19} +211.845i q^{23} -124.168i q^{25} +(-5.84463 + 5.84463i) q^{29} +107.807 q^{31} +(-4.60426 + 4.60426i) q^{35} +(-184.865 - 184.865i) q^{37} +360.146i q^{41} +(312.475 + 312.475i) q^{43} +343.892 q^{47} +292.031 q^{49} +(249.900 + 249.900i) q^{53} -32.8207i q^{55} +(-152.755 - 152.755i) q^{59} +(-525.985 + 525.985i) q^{61} +18.8355 q^{65} +(35.3052 - 35.3052i) q^{67} +784.715i q^{71} +800.215i q^{73} +(181.662 - 181.662i) q^{77} -548.062 q^{79} +(464.431 - 464.431i) q^{83} +(46.0489 + 46.0489i) q^{85} +302.977i q^{89} +(104.254 + 104.254i) q^{91} -56.2679 q^{95} +1567.24 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79}+ \cdots - 7728 q^{95}+O(q^{100}) \) 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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