LMFDB - Embedded newform 900.4.j.b.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(557,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("900.557");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.j (of order $4$, degree $2$, 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:	$53.1017190052$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 40x^{10} + 607x^{8} - 3980x^{6} + 10171x^{4} + 2180x^{2} + 7225 $ x^12 - 40x^10 + 607x^8 - 3980x^6 + 10171x^4 + 2180*x^2 + 7225
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			557.2
Root			$0.539657 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	900.557
Dual form			900.4.j.b.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.36478 - 7.36478i) q^{7} +O(q^{10})$	$q+(-7.36478 - 7.36478i) q^{7} +0.441268i q^{11} +(29.5231 - 29.5231i) q^{13} +(55.1338 - 55.1338i) q^{17} +133.424i q^{19} +(50.4452 + 50.4452i) q^{23} -199.370 q^{29} -21.9035 q^{31} +(-119.788 - 119.788i) q^{37} -8.29304i q^{41} +(237.424 - 237.424i) q^{43} +(96.0331 - 96.0331i) q^{47} -234.520i q^{49} +(-376.667 - 376.667i) q^{53} -111.204 q^{59} +449.006 q^{61} +(686.587 + 686.587i) q^{67} -887.582i q^{71} +(-246.169 + 246.169i) q^{73} +(3.24984 - 3.24984i) q^{77} -1238.96i q^{79} +(-864.074 - 864.074i) q^{83} -177.299 q^{89} -434.862 q^{91} +(229.366 + 229.366i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 84 q^{13} + 144 q^{31} + 564 q^{37} + 960 q^{43} + 1920 q^{61} + 2256 q^{67} + 5076 q^{73} + 4944 q^{91} + 2916 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 84 * q^13 + 144 * q^31 + 564 * q^37 + 960 * q^43 + 1920 * q^61 + 2256 * q^67 + 5076 * q^73 + 4944 * q^91 + 2916 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$5$		0			0
$6$		0			0
$7$	−7.36478	−	7.36478i	−0.397661	−	0.397661i	0.479746	−	0.877407i	$-0.340729\pi$
$7$	−7.36478	−	7.36478i	−0.397661	−	0.397661i	−0.877407	+	0.479746i	$0.840729\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.441268i			0.0120952i	0.999982	+	0.00604760i	$0.00192502\pi$
$11$			0.441268i			0.0120952i	−0.999982	+	0.00604760i	$0.998075\pi$
$12$		0			0
$13$	29.5231	−	29.5231i	0.629863	−	0.629863i	−0.318170	−	0.948034i	$-0.603068\pi$
$13$	29.5231	−	29.5231i	0.629863	−	0.629863i	0.948034	+	0.318170i	$0.103068\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	55.1338	−	55.1338i	0.786583	−	0.786583i	−0.194349	−	0.980932i	$-0.562259\pi$
$17$	55.1338	−	55.1338i	0.786583	−	0.786583i	0.980932	+	0.194349i	$0.0622594\pi$
$18$		0			0
$19$			133.424i			1.61102i	0.592580	+	0.805512i	$0.298110\pi$
$19$			133.424i			1.61102i	−0.592580	+	0.805512i	$0.701890\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	50.4452	+	50.4452i	0.457329	+	0.457329i	0.897778	−	0.440449i	$-0.145181\pi$
$23$	50.4452	+	50.4452i	0.457329	+	0.457329i	−0.440449	+	0.897778i	$0.645181\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−199.370			−1.27662			−0.638310	−	0.769779i	$-0.720366\pi$
$29$	−199.370			−1.27662			−0.638310	+	0.769779i	$0.720366\pi$
$30$		0			0
$31$	−21.9035			−0.126903			−0.0634515	−	0.997985i	$-0.520211\pi$
$31$	−21.9035			−0.126903			−0.0634515	+	0.997985i	$0.520211\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−119.788	−	119.788i	−0.532245	−	0.532245i	0.388995	−	0.921240i	$-0.372822\pi$
$37$	−119.788	−	119.788i	−0.532245	−	0.532245i	−0.921240	+	0.388995i	$0.872822\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.29304i		−	0.0315892i	−0.999875	−	0.0157946i	$-0.994972\pi$
$41$		−	8.29304i		−	0.0315892i	0.999875	−	0.0157946i	$-0.00502778\pi$
$42$		0			0
$43$	237.424	−	237.424i	0.842017	−	0.842017i	−0.147104	−	0.989121i	$-0.546995\pi$
$43$	237.424	−	237.424i	0.842017	−	0.842017i	0.989121	+	0.147104i	$0.0469951\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	96.0331	−	96.0331i	0.298040	−	0.298040i	−0.542206	−	0.840246i	$-0.682411\pi$
$47$	96.0331	−	96.0331i	0.298040	−	0.298040i	0.840246	+	0.542206i	$0.182411\pi$
$48$		0			0
$49$		−	234.520i		−	0.683732i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−376.667	−	376.667i	−0.976211	−	0.976211i	0.0235129	−	0.999724i	$-0.492515\pi$
$53$	−376.667	−	376.667i	−0.976211	−	0.976211i	−0.999724	+	0.0235129i	$0.992515\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−111.204			−0.245381			−0.122690	−	0.992445i	$-0.539152\pi$
$59$	−111.204			−0.245381			−0.122690	+	0.992445i	$0.539152\pi$
$60$		0			0
$61$	449.006			0.942449			0.471224	−	0.882013i	$-0.343812\pi$
$61$	449.006			0.942449			0.471224	+	0.882013i	$0.343812\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	686.587	+	686.587i	1.25194	+	1.25194i	0.954849	+	0.297090i	$0.0960163\pi$
$67$	686.587	+	686.587i	1.25194	+	1.25194i	0.297090	+	0.954849i	$0.403984\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	887.582i		−	1.48361i	−0.670614	−	0.741807i	$-0.733969\pi$
$71$		−	887.582i		−	1.48361i	0.670614	−	0.741807i	$-0.266031\pi$
$72$		0			0
$73$	−246.169	+	246.169i	−0.394683	+	0.394683i	−0.876353	−	0.481670i	$-0.840031\pi$
$73$	−246.169	+	246.169i	−0.394683	+	0.394683i	0.481670	+	0.876353i	$0.340031\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.24984	−	3.24984i	0.00480979	−	0.00480979i
$78$		0			0
$79$		−	1238.96i		−	1.76447i	−0.470806	−	0.882237i	$-0.656037\pi$
$79$		−	1238.96i		−	1.76447i	0.470806	−	0.882237i	$-0.343963\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−864.074	−	864.074i	−1.14270	−	1.14270i	−0.987954	−	0.154751i	$-0.950543\pi$
$83$	−864.074	−	864.074i	−1.14270	−	1.14270i	−0.154751	−	0.987954i	$-0.549457\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−177.299			−0.211165			−0.105582	−	0.994411i	$-0.533671\pi$
$89$	−177.299			−0.211165			−0.105582	+	0.994411i	$0.533671\pi$
$90$		0			0
$91$	−434.862			−0.500944
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	229.366	+	229.366i	0.240088	+	0.240088i	0.816887	−	0.576798i	$-0.195698\pi$
$97$	229.366	+	229.366i	0.240088	+	0.240088i	−0.576798	+	0.816887i	$0.695698\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1191.24i			1.17359i	0.809736	+	0.586795i	$0.199610\pi$
$101$			1191.24i			1.17359i	−0.809736	+	0.586795i	$0.800390\pi$
$102$		0			0
$103$	759.419	−	759.419i	0.726484	−	0.726484i	−0.243434	−	0.969918i	$-0.578274\pi$
$103$	759.419	−	759.419i	0.726484	−	0.726484i	0.969918	+	0.243434i	$0.0782739\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	434.546	−	434.546i	0.392608	−	0.392608i	−0.483008	−	0.875616i	$-0.660456\pi$
$107$	434.546	−	434.546i	0.392608	−	0.392608i	0.875616	+	0.483008i	$0.160456\pi$
$108$		0			0
$109$		−	877.361i		−	0.770972i	−0.922714	−	0.385486i	$-0.874034\pi$
$109$		−	877.361i		−	0.770972i	0.922714	−	0.385486i	$-0.125966\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−899.085	−	899.085i	−0.748485	−	0.748485i	0.225709	−	0.974195i	$-0.427530\pi$
$113$	−899.085	−	899.085i	−0.748485	−	0.748485i	−0.974195	+	0.225709i	$0.927530\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−812.097			−0.625587
$120$		0			0
$121$	1330.81			0.999854
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	738.535	+	738.535i	0.516018	+	0.516018i	0.916364	−	0.400346i	$-0.131110\pi$
$127$	738.535	+	738.535i	0.516018	+	0.516018i	−0.400346	+	0.916364i	$0.631110\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2711.79i		−	1.80862i	−0.426871	−	0.904312i	$-0.640384\pi$
$131$		−	2711.79i		−	1.80862i	0.426871	−	0.904312i	$-0.359616\pi$
$132$		0			0
$133$	982.635	−	982.635i	0.640641	−	0.640641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2190.05	−	2190.05i	1.36576	−	1.36576i	0.499368	−	0.866390i	$-0.333566\pi$
$137$	2190.05	−	2190.05i	1.36576	−	1.36576i	0.866390	−	0.499368i	$-0.166434\pi$
$138$		0			0
$139$			1734.52i			1.05842i	0.848492	+	0.529209i	$0.177511\pi$
$139$			1734.52i			1.05842i	−0.848492	+	0.529209i	$0.822489\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	13.0276	+	13.0276i	0.00761832	+	0.00761832i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2909.59			−1.59975			−0.799875	−	0.600167i	$-0.795101\pi$
$149$	−2909.59			−1.59975			−0.799875	+	0.600167i	$0.795101\pi$
$150$		0			0
$151$	127.396			0.0686580			0.0343290	−	0.999411i	$-0.489071\pi$
$151$	127.396			0.0686580			0.0343290	+	0.999411i	$0.489071\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1053.78	−	1053.78i	−0.535673	−	0.535673i	0.386582	−	0.922255i	$-0.373656\pi$
$157$	−1053.78	−	1053.78i	−0.535673	−	0.535673i	−0.922255	+	0.386582i	$0.873656\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	743.036i		−	0.363723i
$162$		0			0
$163$	2355.63	−	2355.63i	1.13195	−	1.13195i	0.142093	−	0.989853i	$-0.454617\pi$
$163$	2355.63	−	2355.63i	1.13195	−	1.13195i	0.989853	−	0.142093i	$-0.0453833\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	366.858	−	366.858i	0.169990	−	0.169990i	−0.616985	−	0.786975i	$-0.711646\pi$
$167$	366.858	−	366.858i	0.169990	−	0.169990i	0.786975	+	0.616985i	$0.211646\pi$
$168$		0			0
$169$			453.778i			0.206544i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2764.16	−	2764.16i	−1.21477	−	1.21477i	−0.969440	−	0.245328i	$-0.921104\pi$
$173$	−2764.16	−	2764.16i	−1.21477	−	1.21477i	−0.245328	−	0.969440i	$-0.578896\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3015.74			−1.25926			−0.629629	−	0.776896i	$-0.716793\pi$
$179$	−3015.74			−1.25926			−0.629629	+	0.776896i	$0.716793\pi$
$180$		0			0
$181$	51.9520			0.0213346			0.0106673	−	0.999943i	$-0.496604\pi$
$181$	51.9520			0.0213346			0.0106673	+	0.999943i	$0.496604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	24.3288	+	24.3288i	0.00951388	+	0.00951388i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2191.86i			0.830352i	0.909741	+	0.415176i	$0.136280\pi$
$191$			2191.86i			0.830352i	−0.909741	+	0.415176i	$0.863720\pi$
$192$		0			0
$193$	1350.53	−	1350.53i	0.503698	−	0.503698i	−0.408887	−	0.912585i	$-0.634083\pi$
$193$	1350.53	−	1350.53i	0.503698	−	0.503698i	0.912585	+	0.408887i	$0.134083\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1312.73	+	1312.73i	−0.474761	+	0.474761i	−0.903452	−	0.428690i	$-0.858975\pi$
$197$	−1312.73	+	1312.73i	−0.474761	+	0.474761i	0.428690	+	0.903452i	$0.358975\pi$
$198$		0			0
$199$		−	2973.82i		−	1.05934i	−0.848204	−	0.529669i	$-0.822316\pi$
$199$		−	2973.82i		−	1.05934i	0.848204	−	0.529669i	$-0.177684\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1468.31	+	1468.31i	0.507662	+	0.507662i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−58.8755			−0.0194856
$210$		0			0
$211$	2039.33			0.665369			0.332685	−	0.943038i	$-0.392046\pi$
$211$	2039.33			0.665369			0.332685	+	0.943038i	$0.392046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	161.315	+	161.315i	0.0504644	+	0.0504644i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3255.44i		−	0.990880i
$222$		0			0
$223$	−3487.46	+	3487.46i	−1.04725	+	1.04725i	−0.0484262	+	0.998827i	$0.515421\pi$
$223$	−3487.46	+	3487.46i	−1.04725	+	1.04725i	−0.998827	+	0.0484262i	$0.984579\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3360.72	−	3360.72i	0.982637	−	0.982637i	−0.0172150	−	0.999852i	$-0.505480\pi$
$227$	3360.72	−	3360.72i	0.982637	−	0.982637i	0.999852	+	0.0172150i	$0.00547999\pi$
$228$		0			0
$229$		−	2901.92i		−	0.837400i	−0.908125	−	0.418700i	$-0.862486\pi$
$229$		−	2901.92i		−	0.837400i	0.908125	−	0.418700i	$-0.137514\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−665.578	−	665.578i	−0.187139	−	0.187139i	0.607319	−	0.794458i	$-0.292245\pi$
$233$	−665.578	−	665.578i	−0.187139	−	0.187139i	−0.794458	+	0.607319i	$0.792245\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5188.22			1.40418			0.702088	−	0.712090i	$-0.252251\pi$
$239$	5188.22			1.40418			0.702088	+	0.712090i	$0.252251\pi$
$240$		0			0
$241$	−2405.45			−0.642941			−0.321471	−	0.946920i	$-0.604177\pi$
$241$	−2405.45			−0.642941			−0.321471	+	0.946920i	$0.604177\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3939.07	+	3939.07i	1.01472	+	1.01472i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	711.869i		−	0.179015i	−0.995986	−	0.0895076i	$-0.971471\pi$
$251$		−	711.869i		−	0.179015i	0.995986	−	0.0895076i	$-0.0285293\pi$
$252$		0			0
$253$	−22.2598	+	22.2598i	−0.00553148	+	0.00553148i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5181.11	+	5181.11i	−1.25754	+	1.25754i	−0.305282	+	0.952262i	$0.598751\pi$
$257$	−5181.11	+	5181.11i	−1.25754	+	1.25754i	−0.952262	+	0.305282i	$0.901249\pi$
$258$		0			0
$259$			1764.43i			0.423306i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4003.72	−	4003.72i	−0.938708	−	0.938708i	0.0595191	−	0.998227i	$-0.481043\pi$
$263$	−4003.72	−	4003.72i	−0.938708	−	0.938708i	−0.998227	+	0.0595191i	$0.981043\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3154.16			0.714916			0.357458	−	0.933929i	$-0.383644\pi$
$269$	3154.16			0.714916			0.357458	+	0.933929i	$0.383644\pi$
$270$		0			0
$271$	626.964			0.140536			0.0702681	−	0.997528i	$-0.477615\pi$
$271$	626.964			0.140536			0.0702681	+	0.997528i	$0.477615\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2262.59	+	2262.59i	0.490780	+	0.490780i	0.908552	−	0.417772i	$-0.137189\pi$
$277$	2262.59	+	2262.59i	0.490780	+	0.490780i	−0.417772	+	0.908552i	$0.637189\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6428.97i			1.36484i	0.730960	+	0.682420i	$0.239072\pi$
$281$			6428.97i			1.36484i	−0.730960	+	0.682420i	$0.760928\pi$
$282$		0			0
$283$	−3527.10	+	3527.10i	−0.740863	+	0.740863i	−0.972744	−	0.231881i	$-0.925512\pi$
$283$	−3527.10	+	3527.10i	−0.740863	+	0.740863i	0.231881	+	0.972744i	$0.425512\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−61.0764	+	61.0764i	−0.0125618	+	0.0125618i
$288$		0			0
$289$		−	1166.48i		−	0.237427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3638.57	+	3638.57i	0.725486	+	0.725486i	0.969717	−	0.244231i	$-0.0785354\pi$
$293$	3638.57	+	3638.57i	0.725486	+	0.725486i	−0.244231	+	0.969717i	$0.578535\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2978.60			0.576109
$300$		0			0
$301$	−3497.14			−0.669675
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−6392.50	−	6392.50i	−1.18840	−	1.18840i	−0.977509	−	0.210892i	$-0.932363\pi$
$307$	−6392.50	−	6392.50i	−1.18840	−	1.18840i	−0.210892	−	0.977509i	$-0.567637\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	1082.07i		−	0.197294i	−0.995122	−	0.0986470i	$-0.968549\pi$
$311$		−	1082.07i		−	0.197294i	0.995122	−	0.0986470i	$-0.0314515\pi$
$312$		0			0
$313$	1117.70	−	1117.70i	0.201840	−	0.201840i	−0.598948	−	0.800788i	$-0.704414\pi$
$313$	1117.70	−	1117.70i	0.201840	−	0.201840i	0.800788	+	0.598948i	$0.204414\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1769.07	+	1769.07i	−0.313442	+	0.313442i	−0.846241	−	0.532800i	$-0.821140\pi$
$317$	−1769.07	+	1769.07i	−0.313442	+	0.313442i	0.532800	+	0.846241i	$0.321140\pi$
$318$		0			0
$319$		−	87.9753i		−	0.0154410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7356.15	+	7356.15i	1.26720	+	1.26720i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1414.53			−0.237037
$330$		0			0
$331$	5104.40			0.847623			0.423811	−	0.905751i	$-0.360692\pi$
$331$	5104.40			0.847623			0.423811	+	0.905751i	$0.360692\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3279.33	−	3279.33i	−0.530079	−	0.530079i	0.390517	−	0.920596i	$-0.372296\pi$
$337$	−3279.33	−	3279.33i	−0.530079	−	0.530079i	−0.920596	+	0.390517i	$0.872296\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	9.66532i		−	0.00153492i
$342$		0			0
$343$	−4253.31	+	4253.31i	−0.669554	+	0.669554i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2641.58	−	2641.58i	0.408666	−	0.408666i	−0.472607	−	0.881273i	$-0.656687\pi$
$347$	2641.58	−	2641.58i	0.408666	−	0.408666i	0.881273	+	0.472607i	$0.156687\pi$
$348$		0			0
$349$			7075.30i			1.08519i	0.839994	+	0.542596i	$0.182558\pi$
$349$			7075.30i			1.08519i	−0.839994	+	0.542596i	$0.817442\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3588.07	+	3588.07i	0.541003	+	0.541003i	0.923823	−	0.382820i	$-0.125047\pi$
$353$	3588.07	+	3588.07i	0.541003	+	0.541003i	−0.382820	+	0.923823i	$0.625047\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1477.52			0.217215			0.108608	−	0.994085i	$-0.465361\pi$
$359$	1477.52			0.217215			0.108608	+	0.994085i	$0.465361\pi$
$360$		0			0
$361$	−10942.8			−1.59540
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	602.472	+	602.472i	0.0856915	+	0.0856915i	0.748653	−	0.662962i	$-0.230701\pi$
$367$	602.472	+	602.472i	0.0856915	+	0.0856915i	−0.662962	+	0.748653i	$0.730701\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5548.14i			0.776402i
$372$		0			0
$373$	2823.61	−	2823.61i	0.391960	−	0.391960i	−0.483425	−	0.875386i	$-0.660608\pi$
$373$	2823.61	−	2823.61i	0.391960	−	0.391960i	0.875386	+	0.483425i	$0.160608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5886.00	+	5886.00i	−0.804096	+	0.804096i
$378$		0			0
$379$			14332.4i			1.94250i	0.238061	+	0.971250i	$0.423488\pi$
$379$			14332.4i			1.94250i	−0.238061	+	0.971250i	$0.576512\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4505.52	+	4505.52i	0.601099	+	0.601099i	0.940604	−	0.339505i	$-0.110259\pi$
$383$	4505.52	+	4505.52i	0.601099	+	0.601099i	−0.339505	+	0.940604i	$0.610259\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4986.81			−0.649978			−0.324989	−	0.945718i	$-0.605361\pi$
$389$	−4986.81			−0.649978			−0.324989	+	0.945718i	$0.605361\pi$
$390$		0			0
$391$	5562.48			0.719454
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8646.74	−	8646.74i	−1.09312	−	1.09312i	−0.995194	−	0.0979228i	$-0.968780\pi$
$397$	−8646.74	−	8646.74i	−1.09312	−	1.09312i	−0.0979228	−	0.995194i	$-0.531220\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2108.86i			0.262622i	0.991341	+	0.131311i	$0.0419186\pi$
$401$			2108.86i			0.262622i	−0.991341	+	0.131311i	$0.958081\pi$
$402$		0			0
$403$	−646.660	+	646.660i	−0.0799315	+	0.0799315i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	52.8587	−	52.8587i	0.00643761	−	0.00643761i
$408$		0			0
$409$			5640.34i			0.681899i	0.940082	+	0.340950i	$0.110749\pi$
$409$			5640.34i			0.681899i	−0.940082	+	0.340950i	$0.889251\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	818.990	+	818.990i	0.0975784	+	0.0975784i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1628.10			0.189828			0.0949139	−	0.995485i	$-0.469742\pi$
$419$	1628.10			0.189828			0.0949139	+	0.995485i	$0.469742\pi$
$420$		0			0
$421$	−2854.35			−0.330433			−0.165217	−	0.986257i	$-0.552832\pi$
$421$	−2854.35			−0.330433			−0.165217	+	0.986257i	$0.552832\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3306.83	−	3306.83i	−0.374775	−	0.374775i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	6570.13i		−	0.734274i	−0.930167	−	0.367137i	$-0.880338\pi$
$431$		−	6570.13i		−	0.734274i	0.930167	−	0.367137i	$-0.119662\pi$
$432$		0			0
$433$	581.325	−	581.325i	0.0645189	−	0.0645189i	−0.674111	−	0.738630i	$-0.735473\pi$
$433$	581.325	−	581.325i	0.0645189	−	0.0645189i	0.738630	+	0.674111i	$0.235473\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6730.58	+	6730.58i	−0.736767	+	0.736767i
$438$		0			0
$439$		−	2160.06i		−	0.234839i	−0.993082	−	0.117419i	$-0.962538\pi$
$439$		−	2160.06i		−	0.234839i	0.993082	−	0.117419i	$-0.0374622\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10199.2	+	10199.2i	1.09386	+	1.09386i	0.995113	+	0.0987457i	$0.0314830\pi$
$443$	10199.2	+	10199.2i	1.09386	+	1.09386i	0.0987457	+	0.995113i	$0.468517\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10978.3			1.15389			0.576946	−	0.816782i	$-0.304244\pi$
$449$	10978.3			1.15389			0.576946	+	0.816782i	$0.304244\pi$
$450$		0			0
$451$	3.65945			0.000382077
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	155.301	+	155.301i	0.0158965	+	0.0158965i	0.715010	−	0.699114i	$-0.246422\pi$
$457$	155.301	+	155.301i	0.0158965	+	0.0158965i	−0.699114	+	0.715010i	$0.746422\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			10032.7i			1.01360i	0.862064	+	0.506799i	$0.169172\pi$
$461$			10032.7i			1.01360i	−0.862064	+	0.506799i	$0.830828\pi$
$462$		0			0
$463$	−10019.6	+	10019.6i	−1.00572	+	1.00572i	−0.00574109	+	0.999984i	$0.501827\pi$
$463$	−10019.6	+	10019.6i	−1.00572	+	1.00572i	−0.999984	+	0.00574109i	$0.998173\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5914.26	−	5914.26i	0.586037	−	0.586037i	−0.350519	−	0.936556i	$-0.613995\pi$
$467$	5914.26	−	5914.26i	0.586037	−	0.586037i	0.936556	+	0.350519i	$0.113995\pi$
$468$		0			0
$469$		−	10113.1i		−	0.995695i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	104.767	+	104.767i	0.0101844	+	0.0101844i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19121.4			1.82396			0.911981	−	0.410232i	$-0.134552\pi$
$479$	19121.4			1.82396			0.911981	+	0.410232i	$0.134552\pi$
$480$		0			0
$481$	−7073.03			−0.670484
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	10324.3	+	10324.3i	0.960655	+	0.960655i	0.999255	−	0.0385994i	$-0.0122896\pi$
$487$	10324.3	+	10324.3i	0.960655	+	0.960655i	−0.0385994	+	0.999255i	$0.512290\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3340.23i			0.307011i	0.988148	+	0.153506i	$0.0490563\pi$
$491$			3340.23i			0.307011i	−0.988148	+	0.153506i	$0.950944\pi$
$492$		0			0
$493$	−10992.0	+	10992.0i	−1.00417	+	1.00417i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6536.85	+	6536.85i	−0.589975	+	0.589975i
$498$		0			0
$499$			7322.23i			0.656890i	0.944523	+	0.328445i	$0.106525\pi$
$499$			7322.23i			0.656890i	−0.944523	+	0.328445i	$0.893475\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1856.85	−	1856.85i	−0.164598	−	0.164598i	0.620002	−	0.784600i	$-0.287132\pi$
$503$	−1856.85	−	1856.85i	−0.164598	−	0.164598i	−0.784600	+	0.620002i	$0.787132\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3169.85			0.276033			0.138017	−	0.990430i	$-0.455927\pi$
$509$	3169.85			0.276033			0.138017	+	0.990430i	$0.455927\pi$
$510$		0			0
$511$	3625.96			0.313900
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	42.3763	+	42.3763i	0.00360485	+	0.00360485i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15862.5i			1.33388i	0.745113	+	0.666939i	$0.232396\pi$
$521$			15862.5i			1.33388i	−0.745113	+	0.666939i	$0.767604\pi$
$522$		0			0
$523$	3954.58	−	3954.58i	0.330634	−	0.330634i	−0.522193	−	0.852827i	$-0.674886\pi$
$523$	3954.58	−	3954.58i	0.330634	−	0.330634i	0.852827	+	0.522193i	$0.174886\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1207.63	+	1207.63i	−0.0998198	+	0.0998198i
$528$		0			0
$529$		−	7077.56i		−	0.581701i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−244.836	−	244.836i	−0.0198969	−	0.0198969i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	103.486			0.00826987
$540$		0			0
$541$	−4469.46			−0.355188			−0.177594	−	0.984104i	$-0.556831\pi$
$541$	−4469.46			−0.355188			−0.177594	+	0.984104i	$0.556831\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8307.94	+	8307.94i	0.649400	+	0.649400i	0.952848	−	0.303448i	$-0.0981378\pi$
$547$	8307.94	+	8307.94i	0.649400	+	0.649400i	−0.303448	+	0.952848i	$0.598138\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	26600.6i		−	2.05667i
$552$		0			0
$553$	−9124.64	+	9124.64i	−0.701662	+	0.701662i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10304.4	+	10304.4i	−0.783861	+	0.783861i	−0.980480	−	0.196619i	$-0.937004\pi$
$557$	−10304.4	+	10304.4i	−0.783861	+	0.783861i	0.196619	+	0.980480i	$0.437004\pi$
$558$		0			0
$559$		−	14018.9i		−	1.06071i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	741.796	+	741.796i	0.0555293	+	0.0555293i	0.734326	−	0.678797i	$-0.237498\pi$
$563$	741.796	+	741.796i	0.0555293	+	0.0555293i	−0.678797	+	0.734326i	$0.737498\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10605.2			−0.781357			−0.390679	−	0.920527i	$-0.627760\pi$
$569$	−10605.2			−0.781357			−0.390679	+	0.920527i	$0.627760\pi$
$570$		0			0
$571$	−11883.2			−0.870920			−0.435460	−	0.900208i	$-0.643414\pi$
$571$	−11883.2			−0.870920			−0.435460	+	0.900208i	$0.643414\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−14849.3	−	14849.3i	−1.07138	−	1.07138i	−0.997249	−	0.0741303i	$-0.976382\pi$
$577$	−14849.3	−	14849.3i	−1.07138	−	1.07138i	−0.0741303	−	0.997249i	$-0.523618\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			12727.4i			0.908818i
$582$		0			0
$583$	166.211	−	166.211i	0.0118075	−	0.0118075i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3401.02	+	3401.02i	−0.239140	+	0.239140i	−0.816494	−	0.577354i	$-0.804085\pi$
$587$	−3401.02	+	3401.02i	−0.239140	+	0.239140i	0.577354	+	0.816494i	$0.304085\pi$
$588$		0			0
$589$		−	2922.45i		−	0.204444i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12345.8	−	12345.8i	−0.854946	−	0.854946i	0.135791	−	0.990737i	$-0.456642\pi$
$593$	−12345.8	−	12345.8i	−0.854946	−	0.854946i	−0.990737	+	0.135791i	$0.956642\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7552.82			0.515191			0.257596	−	0.966253i	$-0.417070\pi$
$599$	7552.82			0.515191			0.257596	+	0.966253i	$0.417070\pi$
$600$		0			0
$601$	26434.0			1.79412			0.897061	−	0.441907i	$-0.145698\pi$
$601$	26434.0			1.79412			0.897061	+	0.441907i	$0.145698\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	14610.2	+	14610.2i	0.976950	+	0.976950i	0.999740	−	0.0227906i	$-0.00725509\pi$
$607$	14610.2	+	14610.2i	0.976950	+	0.976950i	−0.0227906	+	0.999740i	$0.507255\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	5670.38i		−	0.375449i
$612$		0			0
$613$	19365.1	−	19365.1i	1.27593	−	1.27593i	0.333010	−	0.942923i	$-0.391936\pi$
$613$	19365.1	−	19365.1i	1.27593	−	1.27593i	0.942923	−	0.333010i	$-0.108064\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13145.5	−	13145.5i	0.857727	−	0.857727i	−0.133343	−	0.991070i	$-0.542571\pi$
$617$	13145.5	−	13145.5i	0.857727	−	0.857727i	0.991070	+	0.133343i	$0.0425712\pi$
$618$		0			0
$619$			13542.6i			0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$619$			13542.6i			0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1305.77	+	1305.77i	0.0839719	+	0.0839719i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−13208.8			−0.837311
$630$		0			0
$631$	−3149.60			−0.198706			−0.0993531	−	0.995052i	$-0.531677\pi$
$631$	−3149.60			−0.198706			−0.0993531	+	0.995052i	$0.531677\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6923.75	−	6923.75i	−0.430658	−	0.430658i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17886.1i		−	1.10212i	−0.834466	−	0.551059i	$-0.814224\pi$
$641$		−	17886.1i		−	1.10212i	0.834466	−	0.551059i	$-0.185776\pi$
$642$		0			0
$643$	4072.99	−	4072.99i	0.249802	−	0.249802i	−0.571087	−	0.820889i	$-0.693478\pi$
$643$	4072.99	−	4072.99i	0.249802	−	0.249802i	0.820889	+	0.571087i	$0.193478\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20784.3	−	20784.3i	1.26293	−	1.26293i	0.313261	−	0.949667i	$-0.398578\pi$
$647$	20784.3	−	20784.3i	1.26293	−	1.26293i	0.949667	−	0.313261i	$-0.101422\pi$
$648$		0			0
$649$		−	49.0705i		−	0.00296793i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15886.2	−	15886.2i	−0.952032	−	0.952032i	0.0468692	−	0.998901i	$-0.485076\pi$
$653$	−15886.2	−	15886.2i	−0.952032	−	0.952032i	−0.998901	+	0.0468692i	$0.985076\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9906.30			−0.585576			−0.292788	−	0.956177i	$-0.594583\pi$
$659$	−9906.30			−0.585576			−0.292788	+	0.956177i	$0.594583\pi$
$660$		0			0
$661$	12288.2			0.723082			0.361541	−	0.932356i	$-0.382251\pi$
$661$	12288.2			0.723082			0.361541	+	0.932356i	$0.382251\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−10057.2	−	10057.2i	−0.583835	−	0.583835i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			198.132i			0.0113991i
$672$		0			0
$673$	8626.20	−	8626.20i	0.494080	−	0.494080i	−0.415509	−	0.909589i	$-0.636397\pi$
$673$	8626.20	−	8626.20i	0.494080	−	0.494080i	0.909589	+	0.415509i	$0.136397\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2851.84	+	2851.84i	−0.161898	+	0.161898i	−0.783407	−	0.621509i	$-0.786520\pi$
$677$	−2851.84	+	2851.84i	−0.161898	+	0.161898i	0.621509	+	0.783407i	$0.286520\pi$
$678$		0			0
$679$		−	3378.46i		−	0.190948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	19698.9	+	19698.9i	1.10360	+	1.10360i	0.993973	+	0.109626i	$0.0349653\pi$
$683$	19698.9	+	19698.9i	1.10360	+	1.10360i	0.109626	+	0.993973i	$0.465035\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22240.7			−1.22976
$690$		0			0
$691$	−5246.03			−0.288811			−0.144405	−	0.989519i	$-0.546127\pi$
$691$	−5246.03			−0.288811			−0.144405	+	0.989519i	$0.546127\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−457.227	−	457.227i	−0.0248475	−	0.0248475i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13161.9i			0.709158i	0.935026	+	0.354579i	$0.115376\pi$
$701$			13161.9i			0.709158i	−0.935026	+	0.354579i	$0.884624\pi$
$702$		0			0
$703$	15982.6	−	15982.6i	0.857460	−	0.857460i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8773.20	−	8773.20i	0.466691	−	0.466691i
$708$		0			0
$709$			14372.0i			0.761288i	0.924722	+	0.380644i	$0.124298\pi$
$709$			14372.0i			0.761288i	−0.924722	+	0.380644i	$0.875702\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1104.93	−	1104.93i	−0.0580364	−	0.0580364i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−32834.6			−1.70309			−0.851547	−	0.524278i	$-0.824335\pi$
$719$	−32834.6			−1.70309			−0.851547	+	0.524278i	$0.824335\pi$
$720$		0			0
$721$	−11185.9			−0.577788
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	12524.6	+	12524.6i	0.638941	+	0.638941i	0.950294	−	0.311353i	$-0.100782\pi$
$727$	12524.6	+	12524.6i	0.638941	+	0.638941i	−0.311353	+	0.950294i	$0.600782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	26180.1i		−	1.32463i
$732$		0			0
$733$	3594.77	−	3594.77i	0.181141	−	0.181141i	−0.610712	−	0.791853i	$-0.709117\pi$
$733$	3594.77	−	3594.77i	0.181141	−	0.181141i	0.791853	+	0.610712i	$0.209117\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−302.969	+	302.969i	−0.0151425	+	0.0151425i
$738$		0			0
$739$			5227.58i			0.260216i	0.991500	+	0.130108i	$0.0415324\pi$
$739$			5227.58i			0.260216i	−0.991500	+	0.130108i	$0.958468\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17434.7	−	17434.7i	−0.860858	−	0.860858i	0.130580	−	0.991438i	$-0.458316\pi$
$743$	−17434.7	−	17434.7i	−0.860858	−	0.860858i	−0.991438	+	0.130580i	$0.958316\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6400.67			−0.312250
$750$		0			0
$751$	9871.85			0.479666			0.239833	−	0.970814i	$-0.422907\pi$
$751$	9871.85			0.479666			0.239833	+	0.970814i	$0.422907\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10863.6	−	10863.6i	−0.521589	−	0.521589i	0.396462	−	0.918051i	$-0.370238\pi$
$757$	−10863.6	−	10863.6i	−0.521589	−	0.521589i	−0.918051	+	0.396462i	$0.870238\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			12446.7i			0.592896i	0.955049	+	0.296448i	$0.0958022\pi$
$761$			12446.7i			0.592896i	−0.955049	+	0.296448i	$0.904198\pi$
$762$		0			0
$763$	−6461.57	+	6461.57i	−0.306585	+	0.306585i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3283.07	+	3283.07i	−0.154556	+	0.154556i
$768$		0			0
$769$		−	10910.6i		−	0.511631i	−0.966726	−	0.255816i	$-0.917656\pi$
$769$		−	10910.6i		−	0.511631i	0.966726	−	0.255816i	$-0.0823440\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27307.6	−	27307.6i	−1.27062	−	1.27062i	−0.945764	−	0.324854i	$-0.894685\pi$
$773$	−27307.6	−	27307.6i	−1.27062	−	1.27062i	−0.324854	−	0.945764i	$-0.605315\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1106.49			0.0508909
$780$		0			0
$781$	391.661			0.0179446
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	11121.6	+	11121.6i	0.503740	+	0.503740i	0.912598	−	0.408858i	$-0.134073\pi$
$787$	11121.6	+	11121.6i	0.503740	+	0.503740i	−0.408858	+	0.912598i	$0.634073\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			13243.1i			0.595287i
$792$		0			0
$793$	13256.0	−	13256.0i	0.593614	−	0.593614i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24507.3	−	24507.3i	1.08920	−	1.08920i	0.0935904	−	0.995611i	$-0.470166\pi$
$797$	24507.3	−	24507.3i	1.08920	−	1.08920i	0.995611	−	0.0935904i	$-0.0298344\pi$
$798$		0			0
$799$		−	10589.3i		−	0.468866i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−108.626	−	108.626i	−0.00477377	−	0.00477377i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23667.2			1.02855			0.514274	−	0.857626i	$-0.328062\pi$
$809$	23667.2			1.02855			0.514274	+	0.857626i	$0.328062\pi$
$810$		0			0
$811$	−319.087			−0.0138158			−0.00690792	−	0.999976i	$-0.502199\pi$
$811$	−319.087			−0.0138158			−0.00690792	+	0.999976i	$0.502199\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	31677.9	+	31677.9i	1.35651	+	1.35651i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	32126.2i		−	1.36567i	−0.730574	−	0.682834i	$-0.760747\pi$
$821$		−	32126.2i		−	1.36567i	0.730574	−	0.682834i	$-0.239253\pi$
$822$		0			0
$823$	14595.7	−	14595.7i	0.618193	−	0.618193i	−0.326874	−	0.945068i	$-0.605995\pi$
$823$	14595.7	−	14595.7i	0.618193	−	0.618193i	0.945068	+	0.326874i	$0.105995\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10162.3	−	10162.3i	0.427300	−	0.427300i	−0.460408	−	0.887707i	$-0.652297\pi$
$827$	10162.3	−	10162.3i	0.427300	−	0.427300i	0.887707	+	0.460408i	$0.152297\pi$
$828$		0			0
$829$		−	37045.6i		−	1.55204i	−0.630705	−	0.776022i	$-0.717234\pi$
$829$		−	37045.6i		−	1.55204i	0.630705	−	0.776022i	$-0.282766\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12930.0	−	12930.0i	−0.537812	−	0.537812i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11456.4			0.471417			0.235708	−	0.971824i	$-0.424259\pi$
$839$	11456.4			0.471417			0.235708	+	0.971824i	$0.424259\pi$
$840$		0			0
$841$	15359.2			0.629760
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9801.09	−	9801.09i	−0.397603	−	0.397603i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	12085.5i		−	0.486822i
$852$		0			0
$853$	−10404.9	+	10404.9i	−0.417652	+	0.417652i	−0.884394	−	0.466741i	$-0.845428\pi$
$853$	−10404.9	+	10404.9i	−0.417652	+	0.417652i	0.466741	+	0.884394i	$0.345428\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4276.86	+	4276.86i	−0.170472	+	0.170472i	−0.787187	−	0.616714i	$-0.788463\pi$
$857$	−4276.86	+	4276.86i	−0.170472	+	0.170472i	0.616714	+	0.787187i	$0.288463\pi$
$858$		0			0
$859$		−	15693.6i		−	0.623353i	−0.950188	−	0.311677i	$-0.899109\pi$
$859$		−	15693.6i		−	0.623353i	0.950188	−	0.311677i	$-0.100891\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17125.8	−	17125.8i	−0.675514	−	0.675514i	0.283468	−	0.958982i	$-0.408515\pi$
$863$	−17125.8	−	17125.8i	−0.675514	−	0.675514i	−0.958982	+	0.283468i	$0.908515\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	546.711			0.0213417
$870$		0			0
$871$	40540.3			1.57710
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	17154.1	+	17154.1i	0.660495	+	0.660495i	0.955497	−	0.295002i	$-0.0953203\pi$
$877$	17154.1	+	17154.1i	0.660495	+	0.660495i	−0.295002	+	0.955497i	$0.595320\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	18377.7i		−	0.702795i	−0.936226	−	0.351397i	$-0.885707\pi$
$881$		−	18377.7i		−	0.702795i	0.936226	−	0.351397i	$-0.114293\pi$
$882$		0			0
$883$	−15827.1	+	15827.1i	−0.603198	+	0.603198i	−0.941160	−	0.337962i	$-0.890263\pi$
$883$	−15827.1	+	15827.1i	−0.603198	+	0.603198i	0.337962	+	0.941160i	$0.390263\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3329.22	−	3329.22i	0.126025	−	0.126025i	−0.641281	−	0.767306i	$-0.721597\pi$
$887$	3329.22	−	3329.22i	0.126025	−	0.126025i	0.767306	+	0.641281i	$0.221597\pi$
$888$		0			0
$889$		−	10878.3i		−	0.410401i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12813.1	+	12813.1i	0.480149	+	0.480149i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4366.90			0.162007
$900$		0			0
$901$	−41534.2			−1.53574
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	26187.0	+	26187.0i	0.958681	+	0.958681i	0.999180	−	0.0404987i	$-0.0128947\pi$
$907$	26187.0	+	26187.0i	0.958681	+	0.958681i	−0.0404987	+	0.999180i	$0.512895\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			2763.32i			0.100497i	0.998737	+	0.0502486i	$0.0160014\pi$
$911$			2763.32i			0.100497i	−0.998737	+	0.0502486i	$0.983999\pi$
$912$		0			0
$913$	381.288	−	381.288i	0.0138212	−	0.0138212i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−19971.7	+	19971.7i	−0.719219	+	0.719219i
$918$		0			0
$919$		−	34747.3i		−	1.24723i	−0.781730	−	0.623617i	$-0.785663\pi$
$919$		−	34747.3i		−	1.24723i	0.781730	−	0.623617i	$-0.214337\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−26204.1	−	26204.1i	−0.934474	−	0.934474i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−45786.1			−1.61700			−0.808500	−	0.588497i	$-0.799720\pi$
$929$	−45786.1			−1.61700			−0.808500	+	0.588497i	$0.799720\pi$
$930$		0			0
$931$	31290.5			1.10151
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−27887.2	−	27887.2i	−0.972291	−	0.972291i	0.0273358	−	0.999626i	$-0.491298\pi$
$937$	−27887.2	−	27887.2i	−0.972291	−	0.972291i	−0.999626	+	0.0273358i	$0.991298\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			29770.9i			1.03135i	0.856783	+	0.515677i	$0.172460\pi$
$941$			29770.9i			1.03135i	−0.856783	+	0.515677i	$0.827540\pi$
$942$		0			0
$943$	418.344	−	418.344i	0.0144466	−	0.0144466i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21380.8	+	21380.8i	−0.733666	+	0.733666i	−0.971344	−	0.237678i	$-0.923614\pi$
$947$	−21380.8	+	21380.8i	−0.733666	+	0.733666i	0.237678	+	0.971344i	$0.423614\pi$
$948$		0			0
$949$			14535.3i			0.497193i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13008.2	−	13008.2i	−0.442159	−	0.442159i	0.450578	−	0.892737i	$-0.351218\pi$
$953$	−13008.2	−	13008.2i	−0.442159	−	0.442159i	−0.892737	+	0.450578i	$0.851218\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−32258.5			−1.08622
$960$		0			0
$961$	−29311.2			−0.983896
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−7717.38	−	7717.38i	−0.256643	−	0.256643i	0.567044	−	0.823687i	$-0.308087\pi$
$967$	−7717.38	−	7717.38i	−0.256643	−	0.256643i	−0.823687	+	0.567044i	$0.808087\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			2258.41i			0.0746404i	0.999303	+	0.0373202i	$0.0118822\pi$
$971$			2258.41i			0.0746404i	−0.999303	+	0.0373202i	$0.988118\pi$
$972$		0			0
$973$	12774.4	−	12774.4i	0.420891	−	0.420891i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23378.2	+	23378.2i	−0.765543	+	0.765543i	−0.977318	−	0.211775i	$-0.932076\pi$
$977$	−23378.2	+	23378.2i	−0.765543	+	0.765543i	0.211775	+	0.977318i	$0.432076\pi$
$978$		0			0
$979$		−	78.2362i		−	0.00255408i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14219.9	−	14219.9i	−0.461387	−	0.461387i	0.437723	−	0.899110i	$-0.355785\pi$
$983$	−14219.9	−	14219.9i	−0.461387	−	0.461387i	−0.899110	+	0.437723i	$0.855785\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	23953.8			0.770157
$990$		0			0
$991$	−50784.7			−1.62788			−0.813940	−	0.580949i	$-0.802682\pi$
$991$	−50784.7			−1.62788			−0.813940	+	0.580949i	$0.802682\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20268.5	+	20268.5i	0.643843	+	0.643843i	0.951498	−	0.307655i	$-0.0995443\pi$
$997$	20268.5	+	20268.5i	0.643843	+	0.643843i	−0.307655	+	0.951498i	$0.599544\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.j.b.557.2		12
3.2	odd	2	inner	900.4.j.b.557.1		12
5.2	odd	4		180.4.j.a.53.5	yes	12
5.3	odd	4	inner	900.4.j.b.593.2		12
5.4	even	2		180.4.j.a.17.2	✓	12
15.2	even	4		180.4.j.a.53.2	yes	12
15.8	even	4	inner	900.4.j.b.593.1		12
15.14	odd	2		180.4.j.a.17.5	yes	12
20.7	even	4		720.4.w.e.593.5		12
20.19	odd	2		720.4.w.e.17.2		12
60.47	odd	4		720.4.w.e.593.2		12
60.59	even	2		720.4.w.e.17.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.j.a.17.2	✓	12	5.4	even	2
180.4.j.a.17.5	yes	12	15.14	odd	2
180.4.j.a.53.2	yes	12	15.2	even	4
180.4.j.a.53.5	yes	12	5.2	odd	4
720.4.w.e.17.2		12	20.19	odd	2
720.4.w.e.17.5		12	60.59	even	2
720.4.w.e.593.2		12	60.47	odd	4
720.4.w.e.593.5		12	20.7	even	4
900.4.j.b.557.1		12	3.2	odd	2	inner
900.4.j.b.557.2		12	1.1	even	1	trivial
900.4.j.b.593.1		12	15.8	even	4	inner
900.4.j.b.593.2		12	5.3	odd	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.36478 - 7.36478i) q^{7} +O(q^{10})\)	\(q+(-7.36478 - 7.36478i) q^{7} +0.441268i q^{11} +(29.5231 - 29.5231i) q^{13} +(55.1338 - 55.1338i) q^{17} +133.424i q^{19} +(50.4452 + 50.4452i) q^{23} -199.370 q^{29} -21.9035 q^{31} +(-119.788 - 119.788i) q^{37} -8.29304i q^{41} +(237.424 - 237.424i) q^{43} +(96.0331 - 96.0331i) q^{47} -234.520i q^{49} +(-376.667 - 376.667i) q^{53} -111.204 q^{59} +449.006 q^{61} +(686.587 + 686.587i) q^{67} -887.582i q^{71} +(-246.169 + 246.169i) q^{73} +(3.24984 - 3.24984i) q^{77} -1238.96i q^{79} +(-864.074 - 864.074i) q^{83} -177.299 q^{89} -434.862 q^{91} +(229.366 + 229.366i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 84 q^{13} + 144 q^{31} + 564 q^{37} + 960 q^{43} + 1920 q^{61} + 2256 q^{67} + 5076 q^{73} + 4944 q^{91} + 2916 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 84 * q^13 + 144 * q^31 + 564 * q^37 + 960 * q^43 + 1920 * q^61 + 2256 * q^67 + 5076 * q^73 + 4944 * q^91 + 2916 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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