LMFDB - Embedded newform 1152.3.m.e.415.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (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:	$31.3897264543$
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} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 $ x^16 - 6x^14 + 10x^12 + 88x^10 - 752x^8 + 1408x^6 + 2560x^4 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			415.4
Root			$-0.136762 + 1.99532i$ of defining polynomial
Character	$\chi$	$=$	1152.415
Dual form			1152.3.m.e.991.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +O(q^{10})$	$q+(-0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +(2.21045 - 2.21045i) q^{11} +(-5.08526 + 5.08526i) q^{13} -18.8341 q^{17} +(-11.7651 - 11.7651i) q^{19} -35.4354 q^{23} -24.8964i q^{25} +(21.2499 - 21.2499i) q^{29} +35.9691i q^{31} +(-0.888334 - 0.888334i) q^{35} +(34.4199 + 34.4199i) q^{37} -44.1055i q^{41} +(28.3018 - 28.3018i) q^{43} +32.8802i q^{47} -33.7729 q^{49} +(-42.1450 - 42.1450i) q^{53} -1.00642 q^{55} +(-66.9935 + 66.9935i) q^{59} +(-17.3728 + 17.3728i) q^{61} +2.31532 q^{65} +(-39.6756 - 39.6756i) q^{67} -63.0272 q^{71} -75.4549i q^{73} +(8.62561 - 8.62561i) q^{77} +59.1583i q^{79} +(-71.2155 - 71.2155i) q^{83} +(4.28757 + 4.28757i) q^{85} -150.671i q^{89} +(-19.8437 + 19.8437i) q^{91} +5.35665i q^{95} +51.5586 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{19} - 96 q^{37} + 32 q^{43} + 112 q^{49} - 256 q^{55} + 32 q^{61} + 256 q^{67} - 160 q^{85} - 288 q^{91}+O(q^{100}) $ 16 * q + 32 * q^19 - 96 * q^37 + 32 * q^43 + 112 * q^49 - 256 * q^55 + 32 * q^61 + 256 * q^67 - 160 * q^85 - 288 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−0.227650	−	0.227650i	−0.0455300	−	0.0455300i	0.683975	−	0.729505i	$-0.260250\pi$
$5$	−0.227650	−	0.227650i	−0.0455300	−	0.0455300i	−0.729505	+	0.683975i	$0.760250\pi$
$6$		0			0
$7$	3.90219			0.557456			0.278728	−	0.960370i	$-0.410087\pi$
$7$	3.90219			0.557456			0.278728	+	0.960370i	$0.410087\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.21045	−	2.21045i	0.200950	−	0.200950i	−0.599457	−	0.800407i	$-0.704617\pi$
$11$	2.21045	−	2.21045i	0.200950	−	0.200950i	0.800407	+	0.599457i	$0.204617\pi$
$12$		0			0
$13$	−5.08526	+	5.08526i	−0.391174	+	0.391174i	−0.875106	−	0.483932i	$-0.839208\pi$
$13$	−5.08526	+	5.08526i	−0.391174	+	0.391174i	0.483932	+	0.875106i	$0.339208\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−18.8341			−1.10789			−0.553943	−	0.832555i	$-0.686877\pi$
$17$	−18.8341			−1.10789			−0.553943	+	0.832555i	$0.686877\pi$
$18$		0			0
$19$	−11.7651	−	11.7651i	−0.619216	−	0.619216i	0.326114	−	0.945330i	$-0.394261\pi$
$19$	−11.7651	−	11.7651i	−0.619216	−	0.619216i	−0.945330	+	0.326114i	$0.894261\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−35.4354			−1.54067			−0.770335	−	0.637640i	$-0.779911\pi$
$23$	−35.4354			−1.54067			−0.770335	+	0.637640i	$0.779911\pi$
$24$		0			0
$25$		−	24.8964i		−	0.995854i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	21.2499	−	21.2499i	0.732755	−	0.732755i	−0.238410	−	0.971165i	$-0.576626\pi$
$29$	21.2499	−	21.2499i	0.732755	−	0.732755i	0.971165	+	0.238410i	$0.0766261\pi$
$30$		0			0
$31$			35.9691i			1.16029i	0.814512	+	0.580146i	$0.197005\pi$
$31$			35.9691i			1.16029i	−0.814512	+	0.580146i	$0.802995\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.888334	−	0.888334i	−0.0253810	−	0.0253810i
$36$		0			0
$37$	34.4199	+	34.4199i	0.930267	+	0.930267i	0.997722	−	0.0674555i	$-0.0214881\pi$
$37$	34.4199	+	34.4199i	0.930267	+	0.930267i	−0.0674555	+	0.997722i	$0.521488\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	44.1055i		−	1.07574i	−0.843026	−	0.537872i	$-0.819228\pi$
$41$		−	44.1055i		−	1.07574i	0.843026	−	0.537872i	$-0.180772\pi$
$42$		0			0
$43$	28.3018	−	28.3018i	0.658180	−	0.658180i	−0.296769	−	0.954949i	$-0.595909\pi$
$43$	28.3018	−	28.3018i	0.658180	−	0.658180i	0.954949	+	0.296769i	$0.0959091\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			32.8802i			0.699579i	0.936828	+	0.349790i	$0.113747\pi$
$47$			32.8802i			0.699579i	−0.936828	+	0.349790i	$0.886253\pi$
$48$		0			0
$49$	−33.7729			−0.689242
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−42.1450	−	42.1450i	−0.795189	−	0.795189i	0.187144	−	0.982333i	$-0.440077\pi$
$53$	−42.1450	−	42.1450i	−0.795189	−	0.795189i	−0.982333	+	0.187144i	$0.940077\pi$
$54$		0			0
$55$	−1.00642			−0.0182985
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−66.9935	+	66.9935i	−1.13548	+	1.13548i	−0.146233	+	0.989250i	$0.546715\pi$
$59$	−66.9935	+	66.9935i	−1.13548	+	1.13548i	−0.989250	+	0.146233i	$0.953285\pi$
$60$		0			0
$61$	−17.3728	+	17.3728i	−0.284800	+	0.284800i	−0.835020	−	0.550220i	$-0.814544\pi$
$61$	−17.3728	+	17.3728i	−0.284800	+	0.284800i	0.550220	+	0.835020i	$0.314544\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.31532			0.0356203
$66$		0			0
$67$	−39.6756	−	39.6756i	−0.592174	−	0.592174i	0.346044	−	0.938218i	$-0.387525\pi$
$67$	−39.6756	−	39.6756i	−0.592174	−	0.592174i	−0.938218	+	0.346044i	$0.887525\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−63.0272			−0.887707			−0.443853	−	0.896099i	$-0.646389\pi$
$71$	−63.0272			−0.887707			−0.443853	+	0.896099i	$0.646389\pi$
$72$		0			0
$73$		−	75.4549i		−	1.03363i	−0.856097	−	0.516815i	$-0.827118\pi$
$73$		−	75.4549i		−	1.03363i	0.856097	−	0.516815i	$-0.172882\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.62561	−	8.62561i	0.112021	−	0.112021i
$78$		0			0
$79$			59.1583i			0.748839i	0.927259	+	0.374419i	$0.122158\pi$
$79$			59.1583i			0.748839i	−0.927259	+	0.374419i	$0.877842\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−71.2155	−	71.2155i	−0.858018	−	0.858018i	0.133086	−	0.991104i	$-0.457511\pi$
$83$	−71.2155	−	71.2155i	−0.858018	−	0.858018i	−0.991104	+	0.133086i	$0.957511\pi$
$84$		0			0
$85$	4.28757	+	4.28757i	0.0504420	+	0.0504420i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	150.671i		−	1.69293i	−0.532443	−	0.846466i	$-0.678726\pi$
$89$		−	150.671i		−	1.69293i	0.532443	−	0.846466i	$-0.321274\pi$
$90$		0			0
$91$	−19.8437	+	19.8437i	−0.218062	+	0.218062i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.35665i			0.0563858i
$96$		0			0
$97$	51.5586			0.531532			0.265766	−	0.964038i	$-0.414375\pi$
$97$	51.5586			0.531532			0.265766	+	0.964038i	$0.414375\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−139.414	−	139.414i	−1.38034	−	1.38034i	−0.844010	−	0.536327i	$-0.819811\pi$
$101$	−139.414	−	139.414i	−1.38034	−	1.38034i	−0.536327	−	0.844010i	$-0.680189\pi$
$102$		0			0
$103$	67.9691			0.659894			0.329947	−	0.943999i	$-0.392969\pi$
$103$	67.9691			0.659894			0.329947	+	0.943999i	$0.392969\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−129.801	+	129.801i	−1.21309	+	1.21309i	−0.243090	+	0.970004i	$0.578161\pi$
$107$	−129.801	+	129.801i	−1.21309	+	1.21309i	−0.970004	+	0.243090i	$0.921839\pi$
$108$		0			0
$109$	−30.1323	+	30.1323i	−0.276443	+	0.276443i	−0.831687	−	0.555244i	$-0.812625\pi$
$109$	−30.1323	+	30.1323i	−0.276443	+	0.276443i	0.555244	+	0.831687i	$0.312625\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.11034			0.0452243			0.0226121	−	0.999744i	$-0.492802\pi$
$113$	5.11034			0.0452243			0.0226121	+	0.999744i	$0.492802\pi$
$114$		0			0
$115$	8.06687	+	8.06687i	0.0701467	+	0.0701467i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−73.4942			−0.617598
$120$		0			0
$121$			111.228i			0.919238i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.3589	+	11.3589i	−0.0908712	+	0.0908712i
$126$		0			0
$127$		−	123.225i		−	0.970277i	−0.874437	−	0.485138i	$-0.838769\pi$
$127$		−	123.225i		−	0.970277i	0.874437	−	0.485138i	$-0.161231\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−76.2899	−	76.2899i	−0.582365	−	0.582365i	0.353187	−	0.935553i	$-0.385098\pi$
$131$	−76.2899	−	76.2899i	−0.582365	−	0.582365i	−0.935553	+	0.353187i	$0.885098\pi$
$132$		0			0
$133$	−45.9098	−	45.9098i	−0.345186	−	0.345186i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	73.0194i		−	0.532988i	−0.963837	−	0.266494i	$-0.914135\pi$
$137$		−	73.0194i		−	0.532988i	0.963837	−	0.266494i	$-0.0858653\pi$
$138$		0			0
$139$	−16.5250	+	16.5250i	−0.118885	+	0.118885i	−0.764046	−	0.645162i	$-0.776790\pi$
$139$	−16.5250	+	16.5250i	−0.118885	+	0.118885i	0.645162	+	0.764046i	$0.276790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			22.4814i			0.157213i
$144$		0			0
$145$	−9.67507			−0.0667246
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−22.0334	−	22.0334i	−0.147875	−	0.147875i	0.629293	−	0.777168i	$-0.283344\pi$
$149$	−22.0334	−	22.0334i	−0.147875	−	0.147875i	−0.777168	+	0.629293i	$0.783344\pi$
$150$		0			0
$151$	−161.017			−1.06634			−0.533168	−	0.846010i	$-0.678999\pi$
$151$	−161.017			−1.06634			−0.533168	+	0.846010i	$0.678999\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.18836	−	8.18836i	0.0528281	−	0.0528281i
$156$		0			0
$157$	151.888	−	151.888i	0.967441	−	0.967441i	−0.0320455	−	0.999486i	$-0.510202\pi$
$157$	151.888	−	151.888i	0.967441	−	0.967441i	0.999486	+	0.0320455i	$0.0102022\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−138.276			−0.858856
$162$		0			0
$163$	186.334	+	186.334i	1.14315	+	1.14315i	0.987870	+	0.155283i	$0.0496290\pi$
$163$	186.334	+	186.334i	1.14315	+	1.14315i	0.155283	+	0.987870i	$0.450371\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−199.590			−1.19515			−0.597576	−	0.801813i	$-0.703869\pi$
$167$	−199.590			−1.19515			−0.597576	+	0.801813i	$0.703869\pi$
$168$		0			0
$169$			117.280i			0.693966i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	138.631	−	138.631i	0.801333	−	0.801333i	−0.181971	−	0.983304i	$-0.558248\pi$
$173$	138.631	−	138.631i	0.801333	−	0.801333i	0.983304	+	0.181971i	$0.0582477\pi$
$174$		0			0
$175$		−	97.1504i		−	0.555145i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	148.572	+	148.572i	0.830010	+	0.830010i	0.987518	−	0.157508i	$-0.0503458\pi$
$179$	148.572	+	148.572i	0.830010	+	0.830010i	−0.157508	+	0.987518i	$0.550346\pi$
$180$		0			0
$181$	−99.6006	−	99.6006i	−0.550280	−	0.550280i	0.376242	−	0.926522i	$-0.377216\pi$
$181$	−99.6006	−	99.6006i	−0.550280	−	0.550280i	−0.926522	+	0.376242i	$0.877216\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	15.6714i		−	0.0847101i
$186$		0			0
$187$	−41.6318	+	41.6318i	−0.222630	+	0.222630i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			149.517i			0.782812i	0.920218	+	0.391406i	$0.128011\pi$
$191$			149.517i			0.782812i	−0.920218	+	0.391406i	$0.871989\pi$
$192$		0			0
$193$	−52.8254			−0.273707			−0.136853	−	0.990591i	$-0.543699\pi$
$193$	−52.8254			−0.273707			−0.136853	+	0.990591i	$0.543699\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	108.072	+	108.072i	0.548590	+	0.548590i	0.926033	−	0.377443i	$-0.123197\pi$
$197$	108.072	+	108.072i	0.548590	+	0.548590i	−0.377443	+	0.926033i	$0.623197\pi$
$198$		0			0
$199$	−2.24707			−0.0112918			−0.00564590	−	0.999984i	$-0.501797\pi$
$199$	−2.24707			−0.0112918			−0.00564590	+	0.999984i	$0.501797\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	82.9212	−	82.9212i	0.408479	−	0.408479i
$204$		0			0
$205$	−10.0406	+	10.0406i	−0.0489787	+	0.0489787i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−52.0124			−0.248863
$210$		0			0
$211$	−50.4041	−	50.4041i	−0.238882	−	0.238882i	0.577505	−	0.816387i	$-0.304026\pi$
$211$	−50.4041	−	50.4041i	−0.238882	−	0.238882i	−0.816387	+	0.577505i	$0.804026\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.8858			−0.0599339
$216$		0			0
$217$			140.358i			0.646812i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	95.7760	−	95.7760i	0.433376	−	0.433376i
$222$		0			0
$223$		−	230.340i		−	1.03291i	−0.856313	−	0.516457i	$-0.827251\pi$
$223$		−	230.340i		−	1.03291i	0.856313	−	0.516457i	$-0.172749\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.1395	−	12.1395i	−0.0534779	−	0.0534779i	0.679862	−	0.733340i	$-0.262040\pi$
$227$	−12.1395	−	12.1395i	−0.0534779	−	0.0534779i	−0.733340	+	0.679862i	$0.762040\pi$
$228$		0			0
$229$	−39.1259	−	39.1259i	−0.170855	−	0.170855i	0.616500	−	0.787355i	$-0.288550\pi$
$229$	−39.1259	−	39.1259i	−0.170855	−	0.170855i	−0.787355	+	0.616500i	$0.788550\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			36.6784i			0.157418i	0.996898	+	0.0787090i	$0.0250798\pi$
$233$			36.6784i			0.157418i	−0.996898	+	0.0787090i	$0.974920\pi$
$234$		0			0
$235$	7.48518	−	7.48518i	0.0318518	−	0.0318518i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	147.098i		−	0.615474i	−0.951471	−	0.307737i	$-0.900428\pi$
$239$		−	147.098i		−	0.615474i	0.951471	−	0.307737i	$-0.0995717\pi$
$240$		0			0
$241$	205.488			0.852649			0.426324	−	0.904570i	$-0.359808\pi$
$241$	205.488			0.852649			0.426324	+	0.904570i	$0.359808\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.68839	+	7.68839i	0.0313812	+	0.0313812i
$246$		0			0
$247$	119.657			0.484442
$248$		0			0
$249$		0			0
$250$		0			0
$251$	55.4867	−	55.4867i	0.221063	−	0.221063i	−0.587883	−	0.808946i	$-0.700039\pi$
$251$	55.4867	−	55.4867i	0.221063	−	0.221063i	0.808946	+	0.587883i	$0.200039\pi$
$252$		0			0
$253$	−78.3282	+	78.3282i	−0.309598	+	0.309598i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	142.878			0.555944			0.277972	−	0.960589i	$-0.410338\pi$
$257$	142.878			0.555944			0.277972	+	0.960589i	$0.410338\pi$
$258$		0			0
$259$	134.313	+	134.313i	0.518583	+	0.518583i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	242.691			0.922780			0.461390	−	0.887197i	$-0.347351\pi$
$263$	242.691			0.922780			0.461390	+	0.887197i	$0.347351\pi$
$264$		0			0
$265$			19.1886i			0.0724099i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−292.665	+	292.665i	−1.08797	+	1.08797i	−0.0922353	+	0.995737i	$0.529401\pi$
$269$	−292.665	+	292.665i	−1.08797	+	1.08797i	−0.995737	+	0.0922353i	$0.970599\pi$
$270$		0			0
$271$			258.180i			0.952694i	0.879257	+	0.476347i	$0.158039\pi$
$271$			258.180i			0.952694i	−0.879257	+	0.476347i	$0.841961\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−55.0322	−	55.0322i	−0.200117	−	0.200117i
$276$		0			0
$277$	−63.2023	−	63.2023i	−0.228167	−	0.228167i	0.583759	−	0.811927i	$-0.301581\pi$
$277$	−63.2023	−	63.2023i	−0.228167	−	0.228167i	−0.811927	+	0.583759i	$0.801581\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			421.406i			1.49966i	0.661628	+	0.749832i	$0.269866\pi$
$281$			421.406i			1.49966i	−0.661628	+	0.749832i	$0.730134\pi$
$282$		0			0
$283$	−275.707	+	275.707i	−0.974229	+	0.974229i	−0.999676	−	0.0254475i	$-0.991899\pi$
$283$	−275.707	+	275.707i	−0.974229	+	0.974229i	0.0254475	+	0.999676i	$0.491899\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	172.108i		−	0.599681i
$288$		0			0
$289$	65.7217			0.227411
$290$		0			0
$291$		0			0
$292$		0			0
$293$	181.330	+	181.330i	0.618873	+	0.618873i	0.945242	−	0.326369i	$-0.105825\pi$
$293$	181.330	+	181.330i	0.618873	+	0.618873i	−0.326369	+	0.945242i	$0.605825\pi$
$294$		0			0
$295$	30.5021			0.103397
$296$		0			0
$297$		0			0
$298$		0			0
$299$	180.198	−	180.198i	0.602669	−	0.602669i
$300$		0			0
$301$	110.439	−	110.439i	0.366907	−	0.366907i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.90985			0.0259339
$306$		0			0
$307$	−279.027	−	279.027i	−0.908883	−	0.908883i	0.0872996	−	0.996182i	$-0.472176\pi$
$307$	−279.027	−	279.027i	−0.908883	−	0.908883i	−0.996182	+	0.0872996i	$0.972176\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	414.511			1.33283			0.666417	−	0.745579i	$-0.267827\pi$
$311$	414.511			1.33283			0.666417	+	0.745579i	$0.267827\pi$
$312$		0			0
$313$		−	331.008i		−	1.05753i	−0.848767	−	0.528767i	$-0.822654\pi$
$313$		−	331.008i		−	1.05753i	0.848767	−	0.528767i	$-0.177346\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−194.563	+	194.563i	−0.613764	+	0.613764i	−0.943925	−	0.330161i	$-0.892897\pi$
$317$	−194.563	+	194.563i	−0.613764	+	0.613764i	0.330161	+	0.943925i	$0.392897\pi$
$318$		0			0
$319$		−	93.9437i		−	0.294494i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	221.585	+	221.585i	0.686021	+	0.686021i
$324$		0			0
$325$	126.604	+	126.604i	0.389552	+	0.389552i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			128.305i			0.389985i
$330$		0			0
$331$	462.054	−	462.054i	1.39593	−	1.39593i	0.584641	−	0.811292i	$-0.301235\pi$
$331$	462.054	−	462.054i	1.39593	−	1.39593i	0.811292	−	0.584641i	$-0.198765\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			18.0643i			0.0539233i
$336$		0			0
$337$	−472.439			−1.40190			−0.700948	−	0.713212i	$-0.747240\pi$
$337$	−472.439			−1.40190			−0.700948	+	0.713212i	$0.747240\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	79.5078	+	79.5078i	0.233161	+	0.233161i
$342$		0			0
$343$	−322.996			−0.941679
$344$		0			0
$345$		0			0
$346$		0			0
$347$	92.8976	−	92.8976i	0.267716	−	0.267716i	−0.560463	−	0.828179i	$-0.689377\pi$
$347$	92.8976	−	92.8976i	0.267716	−	0.267716i	0.828179	+	0.560463i	$0.189377\pi$
$348$		0			0
$349$	−264.708	+	264.708i	−0.758475	+	0.758475i	−0.976045	−	0.217570i	$-0.930187\pi$
$349$	−264.708	+	264.708i	−0.758475	+	0.758475i	0.217570	+	0.976045i	$0.430187\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	413.360			1.17099			0.585495	−	0.810676i	$-0.300900\pi$
$353$	413.360			1.17099			0.585495	+	0.810676i	$0.300900\pi$
$354$		0			0
$355$	14.3481	+	14.3481i	0.0404173	+	0.0404173i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	625.198			1.74150			0.870749	−	0.491728i	$-0.163635\pi$
$359$	625.198			1.74150			0.870749	+	0.491728i	$0.163635\pi$
$360$		0			0
$361$		−	84.1643i		−	0.233142i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17.1773	+	17.1773i	−0.0470611	+	0.0470611i
$366$		0			0
$367$			601.904i			1.64007i	0.572316	+	0.820033i	$0.306045\pi$
$367$			601.904i			1.64007i	−0.572316	+	0.820033i	$0.693955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−164.458	−	164.458i	−0.443283	−	0.443283i
$372$		0			0
$373$	493.026	+	493.026i	1.32179	+	1.32179i	0.912330	+	0.409455i	$0.134281\pi$
$373$	493.026	+	493.026i	1.32179	+	1.32179i	0.409455	+	0.912330i	$0.365719\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			216.122i			0.573269i
$378$		0			0
$379$	144.970	−	144.970i	0.382506	−	0.382506i	−0.489498	−	0.872004i	$-0.662820\pi$
$379$	144.970	−	144.970i	0.382506	−	0.382506i	0.872004	+	0.489498i	$0.162820\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	417.764i		−	1.09077i	−0.838187	−	0.545384i	$-0.816384\pi$
$383$		−	417.764i		−	1.09077i	0.838187	−	0.545384i	$-0.183616\pi$
$384$		0			0
$385$	−3.92724			−0.0102006
$386$		0			0
$387$		0			0
$388$		0			0
$389$	144.751	+	144.751i	0.372109	+	0.372109i	0.868245	−	0.496136i	$-0.165248\pi$
$389$	144.751	+	144.751i	0.372109	+	0.372109i	−0.496136	+	0.868245i	$0.665248\pi$
$390$		0			0
$391$	667.392			1.70689
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.4674	−	13.4674i	0.0340946	−	0.0340946i
$396$		0			0
$397$	105.531	−	105.531i	0.265822	−	0.265822i	−0.561592	−	0.827414i	$-0.689811\pi$
$397$	105.531	−	105.531i	0.265822	−	0.265822i	0.827414	+	0.561592i	$0.189811\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	529.383			1.32016			0.660078	−	0.751197i	$-0.270523\pi$
$401$	529.383			1.32016			0.660078	+	0.751197i	$0.270523\pi$
$402$		0			0
$403$	−182.912	−	182.912i	−0.453876	−	0.453876i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	152.167			0.373874
$408$		0			0
$409$		−	371.204i		−	0.907590i	−0.891106	−	0.453795i	$-0.850070\pi$
$409$		−	371.204i		−	0.907590i	0.891106	−	0.453795i	$-0.149930\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−261.422	+	261.422i	−0.632982	+	0.632982i
$414$		0			0
$415$			32.4244i			0.0781311i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	124.486	+	124.486i	0.297103	+	0.297103i	0.839878	−	0.542775i	$-0.182626\pi$
$419$	124.486	+	124.486i	0.297103	+	0.297103i	−0.542775	+	0.839878i	$0.682626\pi$
$420$		0			0
$421$	−51.7267	−	51.7267i	−0.122866	−	0.122866i	0.643000	−	0.765866i	$-0.277690\pi$
$421$	−51.7267	−	51.7267i	−0.122866	−	0.122866i	−0.765866	+	0.643000i	$0.777690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			468.899i			1.10329i
$426$		0			0
$427$	−67.7922	+	67.7922i	−0.158764	+	0.158764i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			424.978i			0.986029i	0.870021	+	0.493014i	$0.164105\pi$
$431$			424.978i			0.986029i	−0.870021	+	0.493014i	$0.835895\pi$
$432$		0			0
$433$	283.151			0.653927			0.326964	−	0.945037i	$-0.393975\pi$
$433$	283.151			0.653927			0.326964	+	0.945037i	$0.393975\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	416.901	+	416.901i	0.954008	+	0.954008i
$438$		0			0
$439$	−701.459			−1.59786			−0.798928	−	0.601427i	$-0.794599\pi$
$439$	−701.459			−1.59786			−0.798928	+	0.601427i	$0.794599\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	272.651	−	272.651i	0.615464	−	0.615464i	−0.328900	−	0.944365i	$-0.606678\pi$
$443$	272.651	−	272.651i	0.615464	−	0.615464i	0.944365	+	0.328900i	$0.106678\pi$
$444$		0			0
$445$	−34.3002	+	34.3002i	−0.0770792	+	0.0770792i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−614.711			−1.36907			−0.684534	−	0.728981i	$-0.739994\pi$
$449$	−614.711			−1.36907			−0.684534	+	0.728981i	$0.739994\pi$
$450$		0			0
$451$	−97.4931	−	97.4931i	−0.216171	−	0.216171i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.03482			0.0198567
$456$		0			0
$457$		−	117.859i		−	0.257898i	−0.991651	−	0.128949i	$-0.958840\pi$
$457$		−	117.859i		−	0.257898i	0.991651	−	0.128949i	$-0.0411603\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	262.235	−	262.235i	0.568839	−	0.568839i	−0.362964	−	0.931803i	$-0.618235\pi$
$461$	262.235	−	262.235i	0.568839	−	0.568839i	0.931803	+	0.362964i	$0.118235\pi$
$462$		0			0
$463$			192.234i			0.415192i	0.978215	+	0.207596i	$0.0665640\pi$
$463$			192.234i			0.415192i	−0.978215	+	0.207596i	$0.933436\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	529.169	+	529.169i	1.13312	+	1.13312i	0.989655	+	0.143470i	$0.0458259\pi$
$467$	529.169	+	529.169i	1.13312	+	1.13312i	0.143470	+	0.989655i	$0.454174\pi$
$468$		0			0
$469$	−154.822	−	154.822i	−0.330111	−	0.330111i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	125.119i		−	0.264523i
$474$		0			0
$475$	−292.908	+	292.908i	−0.616649	+	0.616649i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			1.43750i			0.00300104i	0.999999	+	0.00150052i	$0.000477630\pi$
$479$			1.43750i			0.00300104i	−0.999999	+	0.00150052i	$0.999522\pi$
$480$		0			0
$481$	−350.068			−0.727792
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.7373	−	11.7373i	−0.0242006	−	0.0242006i
$486$		0			0
$487$	701.238			1.43991			0.719957	−	0.694019i	$-0.244162\pi$
$487$	701.238			1.43991			0.719957	+	0.694019i	$0.244162\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	519.668	−	519.668i	1.05839	−	1.05839i	0.0602007	−	0.998186i	$-0.480826\pi$
$491$	519.668	−	519.668i	1.05839	−	1.05839i	0.998186	−	0.0602007i	$-0.0191741\pi$
$492$		0			0
$493$	−400.222	+	400.222i	−0.811809	+	0.811809i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−245.944			−0.494858
$498$		0			0
$499$	−359.284	−	359.284i	−0.720009	−	0.720009i	0.248598	−	0.968607i	$-0.420030\pi$
$499$	−359.284	−	359.284i	−0.720009	−	0.720009i	−0.968607	+	0.248598i	$0.920030\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−174.376			−0.346671			−0.173335	−	0.984863i	$-0.555454\pi$
$503$	−174.376			−0.346671			−0.173335	+	0.984863i	$0.555454\pi$
$504$		0			0
$505$			63.4752i			0.125693i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	314.247	−	314.247i	0.617381	−	0.617381i	−0.327478	−	0.944859i	$-0.606199\pi$
$509$	314.247	−	314.247i	0.617381	−	0.617381i	0.944859	+	0.327478i	$0.106199\pi$
$510$		0			0
$511$		−	294.440i		−	0.576203i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.4732	−	15.4732i	−0.0300450	−	0.0300450i
$516$		0			0
$517$	72.6801	+	72.6801i	0.140580	+	0.140580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			468.678i			0.899575i	0.893136	+	0.449787i	$0.148500\pi$
$521$			468.678i			0.899575i	−0.893136	+	0.449787i	$0.851500\pi$
$522$		0			0
$523$	417.239	−	417.239i	0.797780	−	0.797780i	−0.184965	−	0.982745i	$-0.559217\pi$
$523$	417.239	−	417.239i	0.797780	−	0.797780i	0.982745	+	0.184965i	$0.0592173\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	677.443i		−	1.28547i
$528$		0			0
$529$	726.667			1.37366
$530$		0			0
$531$		0			0
$532$		0			0
$533$	224.288	+	224.288i	0.420803	+	0.420803i
$534$		0			0
$535$	59.0984			0.110464
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−74.6533	+	74.6533i	−0.138503	+	0.138503i
$540$		0			0
$541$	163.152	−	163.152i	0.301574	−	0.301574i	−0.540055	−	0.841629i	$-0.681597\pi$
$541$	163.152	−	163.152i	0.301574	−	0.301574i	0.841629	+	0.540055i	$0.181597\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.7192			0.0251729
$546$		0			0
$547$	595.267	+	595.267i	1.08824	+	1.08824i	0.995710	+	0.0925300i	$0.0294954\pi$
$547$	595.267	+	595.267i	1.08824	+	1.08824i	0.0925300	+	0.995710i	$0.470505\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−500.015			−0.907468
$552$		0			0
$553$			230.847i			0.417445i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−370.177	+	370.177i	−0.664590	+	0.664590i	−0.956459	−	0.291868i	$-0.905723\pi$
$557$	−370.177	+	370.177i	−0.664590	+	0.664590i	0.291868	+	0.956459i	$0.405723\pi$
$558$		0			0
$559$			287.843i			0.514926i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−426.543	−	426.543i	−0.757625	−	0.757625i	0.218264	−	0.975890i	$-0.429960\pi$
$563$	−426.543	−	426.543i	−0.757625	−	0.757625i	−0.975890	+	0.218264i	$0.929960\pi$
$564$		0			0
$565$	−1.16337	−	1.16337i	−0.00205906	−	0.00205906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	291.547i		−	0.512385i	−0.966626	−	0.256192i	$-0.917532\pi$
$569$		−	291.547i		−	0.512385i	0.966626	−	0.256192i	$-0.0824680\pi$
$570$		0			0
$571$	−569.328	+	569.328i	−0.997071	+	0.997071i	−0.999996	−	0.00292477i	$-0.999069\pi$
$571$	−569.328	+	569.328i	−0.997071	+	0.997071i	0.00292477	+	0.999996i	$0.499069\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			882.212i			1.53428i
$576$		0			0
$577$	−360.847			−0.625385			−0.312692	−	0.949854i	$-0.601231\pi$
$577$	−360.847			−0.625385			−0.312692	+	0.949854i	$0.601231\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−277.897	−	277.897i	−0.478308	−	0.478308i
$582$		0			0
$583$	−186.319			−0.319586
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−373.439	+	373.439i	−0.636183	+	0.636183i	−0.949612	−	0.313429i	$-0.898522\pi$
$587$	−373.439	+	373.439i	−0.636183	+	0.636183i	0.313429	+	0.949612i	$0.398522\pi$
$588$		0			0
$589$	423.180	−	423.180i	0.718472	−	0.718472i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−608.053			−1.02538			−0.512692	−	0.858572i	$-0.671352\pi$
$593$	−608.053			−1.02538			−0.512692	+	0.858572i	$0.671352\pi$
$594$		0			0
$595$	16.7309	+	16.7309i	0.0281192	+	0.0281192i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−211.563			−0.353194			−0.176597	−	0.984283i	$-0.556509\pi$
$599$	−211.563			−0.353194			−0.176597	+	0.984283i	$0.556509\pi$
$600$		0			0
$601$			343.381i			0.571349i	0.958327	+	0.285674i	$0.0922176\pi$
$601$			343.381i			0.571349i	−0.958327	+	0.285674i	$0.907782\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	25.3210	−	25.3210i	0.0418529	−	0.0418529i
$606$		0			0
$607$		−	1006.23i		−	1.65771i	−0.559461	−	0.828856i	$-0.688992\pi$
$607$		−	1006.23i		−	1.65771i	0.559461	−	0.828856i	$-0.311008\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−167.204	−	167.204i	−0.273657	−	0.273657i
$612$		0			0
$613$	−380.705	−	380.705i	−0.621052	−	0.621052i	0.324748	−	0.945801i	$-0.394720\pi$
$613$	−380.705	−	380.705i	−0.621052	−	0.621052i	−0.945801	+	0.324748i	$0.894720\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	693.808i		−	1.12449i	−0.826972	−	0.562243i	$-0.809939\pi$
$617$		−	693.808i		−	1.12449i	0.826972	−	0.562243i	$-0.190061\pi$
$618$		0			0
$619$	619.279	−	619.279i	1.00045	−	1.00045i	0.000450619	−	1.00000i	$-0.499857\pi$
$619$	619.279	−	619.279i	1.00045	−	1.00045i	1.00000		0.000450619i	$-0.000143436\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	587.947i		−	0.943736i
$624$		0			0
$625$	−617.237			−0.987579
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−648.266	−	648.266i	−1.03063	−	1.03063i
$630$		0			0
$631$	−1007.91			−1.59732			−0.798661	−	0.601781i	$-0.794458\pi$
$631$	−1007.91			−1.59732			−0.798661	+	0.601781i	$0.794458\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−28.0522	+	28.0522i	−0.0441767	+	0.0441767i
$636$		0			0
$637$	171.744	−	171.744i	0.269613	−	0.269613i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	511.481			0.797943			0.398971	−	0.916963i	$-0.369367\pi$
$641$	511.481			0.797943			0.398971	+	0.916963i	$0.369367\pi$
$642$		0			0
$643$	497.727	+	497.727i	0.774071	+	0.774071i	0.978815	−	0.204745i	$-0.0656365\pi$
$643$	497.727	+	497.727i	0.774071	+	0.774071i	−0.204745	+	0.978815i	$0.565636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−59.8694			−0.0925339			−0.0462669	−	0.998929i	$-0.514732\pi$
$647$	−59.8694			−0.0925339			−0.0462669	+	0.998929i	$0.514732\pi$
$648$		0			0
$649$			296.172i			0.456351i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	489.952	−	489.952i	0.750310	−	0.750310i	−0.224227	−	0.974537i	$-0.571986\pi$
$653$	489.952	−	489.952i	0.750310	−	0.750310i	0.974537	+	0.224227i	$0.0719858\pi$
$654$		0			0
$655$			34.7348i			0.0530302i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	234.984	+	234.984i	0.356577	+	0.356577i	0.862549	−	0.505973i	$-0.168866\pi$
$659$	234.984	+	234.984i	0.356577	+	0.356577i	−0.505973	+	0.862549i	$0.668866\pi$
$660$		0			0
$661$	−305.060	−	305.060i	−0.461512	−	0.461512i	0.437639	−	0.899151i	$-0.355815\pi$
$661$	−305.060	−	305.060i	−0.461512	−	0.461512i	−0.899151	+	0.437639i	$0.855815\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			20.9027i			0.0314326i
$666$		0			0
$667$	−752.998	+	752.998i	−1.12893	+	1.12893i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			76.8036i			0.114461i
$672$		0			0
$673$	458.738			0.681632			0.340816	−	0.940130i	$-0.389297\pi$
$673$	458.738			0.681632			0.340816	+	0.940130i	$0.389297\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	171.344	+	171.344i	0.253094	+	0.253094i	0.822238	−	0.569144i	$-0.192725\pi$
$677$	171.344	+	171.344i	0.253094	+	0.253094i	−0.569144	+	0.822238i	$0.692725\pi$
$678$		0			0
$679$	201.192			0.296306
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−769.447	+	769.447i	−1.12657	+	1.12657i	−0.135838	+	0.990731i	$0.543373\pi$
$683$	−769.447	+	769.447i	−1.12657	+	1.12657i	−0.990731	+	0.135838i	$0.956627\pi$
$684$		0			0
$685$	−16.6229	+	16.6229i	−0.0242670	+	0.0242670i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	428.636			0.622114
$690$		0			0
$691$	−44.7951	−	44.7951i	−0.0648264	−	0.0648264i	0.673950	−	0.738777i	$-0.264596\pi$
$691$	−44.7951	−	44.7951i	−0.0648264	−	0.0648264i	−0.738777	+	0.673950i	$0.764596\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.52381			0.0108256
$696$		0			0
$697$			830.686i			1.19180i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.7012	−	31.7012i	0.0452229	−	0.0452229i	−0.684134	−	0.729357i	$-0.739820\pi$
$701$	31.7012	−	31.7012i	0.0452229	−	0.0452229i	0.729357	+	0.684134i	$0.239820\pi$
$702$		0			0
$703$		−	809.907i		−	1.15207i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−544.021	−	544.021i	−0.769478	−	0.769478i
$708$		0			0
$709$	−256.706	−	256.706i	−0.362068	−	0.362068i	0.502506	−	0.864574i	$-0.332411\pi$
$709$	−256.706	−	256.706i	−0.362068	−	0.362068i	−0.864574	+	0.502506i	$0.832411\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1274.58i		−	1.78763i
$714$		0			0
$715$	5.11789	−	5.11789i	0.00715789	−	0.00715789i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1368.29i			1.90305i	0.307574	+	0.951524i	$0.400483\pi$
$719$			1368.29i			1.90305i	−0.307574	+	0.951524i	$0.599517\pi$
$720$		0			0
$721$	265.229			0.367862
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−529.045	−	529.045i	−0.729717	−	0.729717i
$726$		0			0
$727$	37.5336			0.0516281			0.0258141	−	0.999667i	$-0.491782\pi$
$727$	37.5336			0.0516281			0.0258141	+	0.999667i	$0.491782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−533.037	+	533.037i	−0.729189	+	0.729189i
$732$		0			0
$733$	385.216	−	385.216i	0.525534	−	0.525534i	−0.393704	−	0.919237i	$-0.628806\pi$
$733$	385.216	−	385.216i	0.525534	−	0.525534i	0.919237	+	0.393704i	$0.128806\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−175.402			−0.237995
$738$		0			0
$739$	889.039	+	889.039i	1.20303	+	1.20303i	0.973240	+	0.229790i	$0.0738040\pi$
$739$	889.039	+	889.039i	1.20303	+	1.20303i	0.229790	+	0.973240i	$0.426196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−506.252			−0.681362			−0.340681	−	0.940179i	$-0.610658\pi$
$743$	−506.252			−0.681362			−0.340681	+	0.940179i	$0.610658\pi$
$744$		0			0
$745$			10.0318i			0.0134655i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−506.509	+	506.509i	−0.676247	+	0.676247i
$750$		0			0
$751$			456.844i			0.608314i	0.952622	+	0.304157i	$0.0983748\pi$
$751$			456.844i			0.608314i	−0.952622	+	0.304157i	$0.901625\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	36.6554	+	36.6554i	0.0485502	+	0.0485502i
$756$		0			0
$757$	221.708	+	221.708i	0.292877	+	0.292877i	0.838216	−	0.545339i	$-0.183599\pi$
$757$	221.708	+	221.708i	0.292877	+	0.292877i	−0.545339	+	0.838216i	$0.683599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	205.900i		−	0.270566i	−0.990807	−	0.135283i	$-0.956806\pi$
$761$		−	205.900i		−	0.270566i	0.990807	−	0.135283i	$-0.0431943\pi$
$762$		0			0
$763$	−117.582	+	117.582i	−0.154105	+	0.154105i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	681.359i		−	0.888342i
$768$		0			0
$769$	900.133			1.17052			0.585262	−	0.810844i	$-0.300992\pi$
$769$	900.133			1.17052			0.585262	+	0.810844i	$0.300992\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−870.407	−	870.407i	−1.12601	−	1.12601i	−0.990819	−	0.135193i	$-0.956835\pi$
$773$	−870.407	−	870.407i	−1.12601	−	1.12601i	−0.135193	−	0.990819i	$-0.543165\pi$
$774$		0			0
$775$	895.498			1.15548
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−518.907	+	518.907i	−0.666119	+	0.666119i
$780$		0			0
$781$	−139.318	+	139.318i	−0.178385	+	0.178385i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−69.1547			−0.0880951
$786$		0			0
$787$	−1005.24	−	1005.24i	−1.27730	−	1.27730i	−0.942171	−	0.335133i	$-0.891219\pi$
$787$	−1005.24	−	1005.24i	−1.27730	−	1.27730i	−0.335133	−	0.942171i	$-0.608781\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	19.9415			0.0252106
$792$		0			0
$793$		−	176.691i		−	0.222813i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	434.779	−	434.779i	0.545520	−	0.545520i	−0.379622	−	0.925142i	$-0.623946\pi$
$797$	434.779	−	434.779i	0.545520	−	0.545520i	0.925142	+	0.379622i	$0.123946\pi$
$798$		0			0
$799$		−	619.268i		−	0.775054i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−166.789	−	166.789i	−0.207708	−	0.207708i
$804$		0			0
$805$	31.4785	+	31.4785i	0.0391037	+	0.0391037i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	50.0002i		−	0.0618049i	−0.999522	−	0.0309024i	$-0.990162\pi$
$809$		−	50.0002i		−	0.0618049i	0.999522	−	0.0309024i	$-0.00983812\pi$
$810$		0			0
$811$	36.9957	−	36.9957i	0.0456173	−	0.0456173i	−0.683930	−	0.729547i	$-0.739731\pi$
$811$	36.9957	−	36.9957i	0.0456173	−	0.0456173i	0.729547	+	0.683930i	$0.239731\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	84.8378i		−	0.104096i
$816$		0			0
$817$	−665.947			−0.815112
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−297.373	−	297.373i	−0.362208	−	0.362208i	0.502417	−	0.864625i	$-0.332444\pi$
$821$	−297.373	−	297.373i	−0.362208	−	0.362208i	−0.864625	+	0.502417i	$0.832444\pi$
$822$		0			0
$823$	−345.052			−0.419262			−0.209631	−	0.977781i	$-0.567226\pi$
$823$	−345.052			−0.419262			−0.209631	+	0.977781i	$0.567226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−426.416	+	426.416i	−0.515618	+	0.515618i	−0.916242	−	0.400624i	$-0.868793\pi$
$827$	−426.416	+	426.416i	−0.515618	+	0.515618i	0.400624	+	0.916242i	$0.368793\pi$
$828$		0			0
$829$	−897.159	+	897.159i	−1.08222	+	1.08222i	−0.0859157	+	0.996302i	$0.527382\pi$
$829$	−897.159	+	897.159i	−1.08222	+	1.08222i	−0.996302	+	0.0859157i	$0.972618\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	636.080			0.763602
$834$		0			0
$835$	45.4367	+	45.4367i	0.0544152	+	0.0544152i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−708.037			−0.843906			−0.421953	−	0.906618i	$-0.638655\pi$
$839$	−708.037			−0.843906			−0.421953	+	0.906618i	$0.638655\pi$
$840$		0			0
$841$		−	62.1159i		−	0.0738595i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.6989	−	26.6989i	0.0315963	−	0.0315963i
$846$		0			0
$847$			434.033i			0.512435i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1219.68	−	1219.68i	−1.43323	−	1.43323i
$852$		0			0
$853$	−795.484	−	795.484i	−0.932572	−	0.932572i	0.0652942	−	0.997866i	$-0.479201\pi$
$853$	−795.484	−	795.484i	−0.932572	−	0.932572i	−0.997866	+	0.0652942i	$0.979201\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1447.28i		−	1.68877i	−0.535735	−	0.844386i	$-0.679966\pi$
$857$		−	1447.28i		−	1.68877i	0.535735	−	0.844386i	$-0.320034\pi$
$858$		0			0
$859$	99.6061	−	99.6061i	0.115956	−	0.115956i	−0.646748	−	0.762704i	$-0.723871\pi$
$859$	99.6061	−	99.6061i	0.115956	−	0.115956i	0.762704	+	0.646748i	$0.223871\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			549.773i			0.637049i	0.947915	+	0.318525i	$0.103187\pi$
$863$			549.773i			0.637049i	−0.947915	+	0.318525i	$0.896813\pi$
$864$		0			0
$865$	−63.1185			−0.0729694
$866$		0			0
$867$		0			0
$868$		0			0
$869$	130.766	+	130.766i	0.150479	+	0.150479i
$870$		0			0
$871$	403.522			0.463286
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−44.3246	+	44.3246i	−0.0506567	+	0.0506567i
$876$		0			0
$877$	165.469	−	165.469i	0.188677	−	0.188677i	−0.606447	−	0.795124i	$-0.707406\pi$
$877$	165.469	−	165.469i	0.188677	−	0.188677i	0.795124	+	0.606447i	$0.207406\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1046.60			−1.18796			−0.593982	−	0.804479i	$-0.702445\pi$
$881$	−1046.60			−1.18796			−0.593982	+	0.804479i	$0.702445\pi$
$882$		0			0
$883$	−338.581	−	338.581i	−0.383444	−	0.383444i	0.488897	−	0.872341i	$-0.337399\pi$
$883$	−338.581	−	338.581i	−0.383444	−	0.383444i	−0.872341	+	0.488897i	$0.837399\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1486.13			−1.67546			−0.837728	−	0.546088i	$-0.816117\pi$
$887$	−1486.13			−1.67546			−0.837728	+	0.546088i	$0.816117\pi$
$888$		0			0
$889$		−	480.848i		−	0.540887i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	386.840	−	386.840i	0.433191	−	0.433191i
$894$		0			0
$895$		−	67.6447i		−	0.0755807i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	764.339	+	764.339i	0.850210	+	0.850210i
$900$		0			0
$901$	793.761	+	793.761i	0.880978	+	0.880978i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			45.3482i			0.0501085i
$906$		0			0
$907$	66.8336	−	66.8336i	0.0736864	−	0.0736864i	−0.669303	−	0.742989i	$-0.733407\pi$
$907$	66.8336	−	66.8336i	0.0736864	−	0.0736864i	0.742989	+	0.669303i	$0.233407\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	227.208i		−	0.249406i	−0.992194	−	0.124703i	$-0.960202\pi$
$911$		−	227.208i		−	0.249406i	0.992194	−	0.124703i	$-0.0397977\pi$
$912$		0			0
$913$	−314.837			−0.344838
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−297.698	−	297.698i	−0.324643	−	0.324643i
$918$		0			0
$919$	−1375.00			−1.49619			−0.748097	−	0.663589i	$-0.769032\pi$
$919$	−1375.00			−1.49619			−0.748097	+	0.663589i	$0.769032\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	320.509	−	320.509i	0.347247	−	0.347247i
$924$		0			0
$925$	856.929	−	856.929i	0.926410	−	0.926410i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1788.04			−1.92470			−0.962348	−	0.271822i	$-0.912374\pi$
$929$	−1788.04			−1.92470			−0.962348	+	0.271822i	$0.912374\pi$
$930$		0			0
$931$	397.342	+	397.342i	0.426790	+	0.426790i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.9549			0.0202727
$936$		0			0
$937$		−	768.834i		−	0.820527i	−0.911967	−	0.410263i	$-0.865437\pi$
$937$		−	768.834i		−	0.820527i	0.911967	−	0.410263i	$-0.134563\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1016.85	−	1016.85i	1.08061	−	1.08061i	0.0841571	−	0.996453i	$-0.473180\pi$
$941$	1016.85	−	1016.85i	1.08061	−	1.08061i	0.996453	−	0.0841571i	$-0.0268197\pi$
$942$		0			0
$943$			1562.90i			1.65737i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	476.431	+	476.431i	0.503095	+	0.503095i	0.912398	−	0.409304i	$-0.134228\pi$
$947$	476.431	+	476.431i	0.503095	+	0.503095i	−0.409304	+	0.912398i	$0.634228\pi$
$948$		0			0
$949$	383.708	+	383.708i	0.404329	+	0.404329i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			577.529i			0.606011i	0.952989	+	0.303006i	$0.0979901\pi$
$953$			577.529i			0.606011i	−0.952989	+	0.303006i	$0.902010\pi$
$954$		0			0
$955$	34.0375	−	34.0375i	0.0356414	−	0.0356414i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	284.936i		−	0.297118i
$960$		0			0
$961$	−332.773			−0.346278
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.0257	+	12.0257i	0.0124619	+	0.0124619i
$966$		0			0
$967$	864.047			0.893534			0.446767	−	0.894650i	$-0.352575\pi$
$967$	864.047			0.893534			0.446767	+	0.894650i	$0.352575\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	898.515	−	898.515i	0.925350	−	0.925350i	−0.0720505	−	0.997401i	$-0.522954\pi$
$971$	898.515	−	898.515i	0.925350	−	0.925350i	0.997401	+	0.0720505i	$0.0229543\pi$
$972$		0			0
$973$	−64.4836	+	64.4836i	−0.0662730	+	0.0662730i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	538.131			0.550799			0.275400	−	0.961330i	$-0.411190\pi$
$977$	538.131			0.550799			0.275400	+	0.961330i	$0.411190\pi$
$978$		0			0
$979$	−333.051	−	333.051i	−0.340195	−	0.340195i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−759.452			−0.772586			−0.386293	−	0.922376i	$-0.626245\pi$
$983$	−759.452			−0.772586			−0.386293	+	0.922376i	$0.626245\pi$
$984$		0			0
$985$		−	49.2052i		−	0.0499546i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1002.88	+	1002.88i	−1.01404	+	1.01404i
$990$		0			0
$991$		−	514.565i		−	0.519238i	−0.965711	−	0.259619i	$-0.916403\pi$
$991$		−	514.565i		−	0.519238i	0.965711	−	0.259619i	$-0.0835969\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.511545	+	0.511545i	0.000514115	+	0.000514115i
$996$		0			0
$997$	1249.59	+	1249.59i	1.25335	+	1.25335i	0.954208	+	0.299145i	$0.0967013\pi$
$997$	1249.59	+	1249.59i	1.25335	+	1.25335i	0.299145	+	0.954208i	$0.403299\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.e.415.4		16
3.2	odd	2	inner	1152.3.m.e.415.5		16
4.3	odd	2		1152.3.m.d.415.4		16
8.3	odd	2		576.3.m.b.271.5		16
8.5	even	2		144.3.m.b.91.8	yes	16
12.11	even	2		1152.3.m.d.415.5		16
16.3	odd	4	inner	1152.3.m.e.991.4		16
16.5	even	4		576.3.m.b.559.5		16
16.11	odd	4		144.3.m.b.19.8	yes	16
16.13	even	4		1152.3.m.d.991.4		16
24.5	odd	2		144.3.m.b.91.1	yes	16
24.11	even	2		576.3.m.b.271.4		16
48.5	odd	4		576.3.m.b.559.4		16
48.11	even	4		144.3.m.b.19.1	✓	16
48.29	odd	4		1152.3.m.d.991.5		16
48.35	even	4	inner	1152.3.m.e.991.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.m.b.19.1	✓	16	48.11	even	4
144.3.m.b.19.8	yes	16	16.11	odd	4
144.3.m.b.91.1	yes	16	24.5	odd	2
144.3.m.b.91.8	yes	16	8.5	even	2
576.3.m.b.271.4		16	24.11	even	2
576.3.m.b.271.5		16	8.3	odd	2
576.3.m.b.559.4		16	48.5	odd	4
576.3.m.b.559.5		16	16.5	even	4
1152.3.m.d.415.4		16	4.3	odd	2
1152.3.m.d.415.5		16	12.11	even	2
1152.3.m.d.991.4		16	16.13	even	4
1152.3.m.d.991.5		16	48.29	odd	4
1152.3.m.e.415.4		16	1.1	even	1	trivial
1152.3.m.e.415.5		16	3.2	odd	2	inner
1152.3.m.e.991.4		16	16.3	odd	4	inner
1152.3.m.e.991.5		16	48.35	even	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +O(q^{10})\)	\(q+(-0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +(2.21045 - 2.21045i) q^{11} +(-5.08526 + 5.08526i) q^{13} -18.8341 q^{17} +(-11.7651 - 11.7651i) q^{19} -35.4354 q^{23} -24.8964i q^{25} +(21.2499 - 21.2499i) q^{29} +35.9691i q^{31} +(-0.888334 - 0.888334i) q^{35} +(34.4199 + 34.4199i) q^{37} -44.1055i q^{41} +(28.3018 - 28.3018i) q^{43} +32.8802i q^{47} -33.7729 q^{49} +(-42.1450 - 42.1450i) q^{53} -1.00642 q^{55} +(-66.9935 + 66.9935i) q^{59} +(-17.3728 + 17.3728i) q^{61} +2.31532 q^{65} +(-39.6756 - 39.6756i) q^{67} -63.0272 q^{71} -75.4549i q^{73} +(8.62561 - 8.62561i) q^{77} +59.1583i q^{79} +(-71.2155 - 71.2155i) q^{83} +(4.28757 + 4.28757i) q^{85} -150.671i q^{89} +(-19.8437 + 19.8437i) q^{91} +5.35665i q^{95} +51.5586 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{19} - 96 q^{37} + 32 q^{43} + 112 q^{49} - 256 q^{55} + 32 q^{61} + 256 q^{67} - 160 q^{85} - 288 q^{91}+O(q^{100}) \) 16 * q + 32 * q^19 - 96 * q^37 + 32 * q^43 + 112 * q^49 - 256 * q^55 + 32 * q^61 + 256 * q^67 - 160 * q^85 - 288 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more