LMFDB - Embedded newform 864.2.i.e.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(289,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("864.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.i (of order $3$, 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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.289
Dual form			864.2.i.e.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(1.72474 - 2.98735i) q^{11} +(1.94949 + 3.37662i) q^{13} +4.89898 q^{17} -4.00000 q^{19} +(0.275255 + 0.476756i) q^{23} +(2.00000 - 3.46410i) q^{25} +(4.94949 - 8.57277i) q^{29} +(3.72474 + 6.45145i) q^{31} +1.44949 q^{35} +8.89898 q^{37} +(-1.05051 - 1.81954i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-4.17423 + 7.22999i) q^{47} +(2.44949 + 4.24264i) q^{49} +0.898979 q^{53} -3.44949 q^{55} +(0.174235 + 0.301783i) q^{59} +(-0.949490 + 1.64456i) q^{61} +(1.94949 - 3.37662i) q^{65} +(-1.17423 - 2.03383i) q^{67} -11.7980 q^{71} +4.89898 q^{73} +(2.50000 + 4.33013i) q^{77} +(4.27526 - 7.40496i) q^{79} +(2.72474 - 4.71940i) q^{83} +(-2.44949 - 4.24264i) q^{85} +3.10102 q^{89} -5.65153 q^{91} +(2.00000 + 3.46410i) q^{95} +(-2.94949 + 5.10867i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 2 * q^7	$ 4 q - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{13} - 16 q^{19} + 6 q^{23} + 8 q^{25} + 10 q^{29} + 10 q^{31} - 4 q^{35} + 16 q^{37} - 14 q^{41} + 10 q^{43} - 2 q^{47} - 16 q^{53} - 4 q^{55} - 14 q^{59} + 6 q^{61} - 2 q^{65} + 10 q^{67} - 8 q^{71} + 10 q^{77} + 22 q^{79} + 6 q^{83} + 32 q^{89} - 52 q^{91} + 8 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 - 2 * q^13 - 16 * q^19 + 6 * q^23 + 8 * q^25 + 10 * q^29 + 10 * q^31 - 4 * q^35 + 16 * q^37 - 14 * q^41 + 10 * q^43 - 2 * q^47 - 16 * q^53 - 4 * q^55 - 14 * q^59 + 6 * q^61 - 2 * q^65 + 10 * q^67 - 8 * q^71 + 10 * q^77 + 22 * q^79 + 6 * q^83 + 32 * q^89 - 52 * q^91 + 8 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$	−0.724745	+	1.25529i	−0.273928	+	0.474457i	−0.969864	−	0.243647i	$-0.921656\pi$
$7$	−0.724745	+	1.25529i	−0.273928	+	0.474457i	0.695936	+	0.718104i	$0.254990\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.72474	−	2.98735i	0.520030	−	0.900719i	−0.479699	−	0.877433i	$-0.659254\pi$
$11$	1.72474	−	2.98735i	0.520030	−	0.900719i	0.999729	−	0.0232854i	$-0.00741263\pi$
$12$		0			0
$13$	1.94949	+	3.37662i	0.540691	+	0.936505i	0.998864	+	0.0476417i	$0.0151706\pi$
$13$	1.94949	+	3.37662i	0.540691	+	0.936505i	−0.458173	+	0.888863i	$0.651496\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.89898			1.18818			0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898			1.18818			0.594089	+	0.804400i	$0.297513\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.275255	+	0.476756i	0.0573947	+	0.0994105i	0.893295	−	0.449471i	$-0.148387\pi$
$23$	0.275255	+	0.476756i	0.0573947	+	0.0994105i	−0.835900	+	0.548881i	$0.815054\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.94949	−	8.57277i	0.919097	−	1.59192i	0.118308	−	0.992977i	$-0.462253\pi$
$29$	4.94949	−	8.57277i	0.919097	−	1.59192i	0.800789	−	0.598946i	$-0.204414\pi$
$30$		0			0
$31$	3.72474	+	6.45145i	0.668984	+	1.15871i	0.978189	+	0.207719i	$0.0666038\pi$
$31$	3.72474	+	6.45145i	0.668984	+	1.15871i	−0.309205	+	0.950996i	$0.600063\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.44949			0.245008
$36$		0			0
$37$	8.89898			1.46298			0.731492	−	0.681850i	$-0.238825\pi$
$37$	8.89898			1.46298			0.731492	+	0.681850i	$0.238825\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.05051	−	1.81954i	−0.164062	−	0.284164i	0.772260	−	0.635307i	$-0.219126\pi$
$41$	−1.05051	−	1.81954i	−0.164062	−	0.284164i	−0.936322	+	0.351143i	$0.885793\pi$
$42$		0			0
$43$	6.17423	−	10.6941i	0.941562	−	1.63083i	0.179069	−	0.983836i	$-0.442691\pi$
$43$	6.17423	−	10.6941i	0.941562	−	1.63083i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.17423	+	7.22999i	−0.608875	+	1.05460i	0.382552	+	0.923934i	$0.375045\pi$
$47$	−4.17423	+	7.22999i	−0.608875	+	1.05460i	−0.991426	+	0.130668i	$0.958288\pi$
$48$		0			0
$49$	2.44949	+	4.24264i	0.349927	+	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.898979			0.123484			0.0617422	−	0.998092i	$-0.480334\pi$
$53$	0.898979			0.123484			0.0617422	+	0.998092i	$0.480334\pi$
$54$		0			0
$55$	−3.44949			−0.465129
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.174235	+	0.301783i	0.0226834	+	0.0392888i	0.877144	−	0.480227i	$-0.159446\pi$
$59$	0.174235	+	0.301783i	0.0226834	+	0.0392888i	−0.854461	+	0.519516i	$0.826112\pi$
$60$		0			0
$61$	−0.949490	+	1.64456i	−0.121570	+	0.210565i	−0.920387	−	0.391009i	$-0.872126\pi$
$61$	−0.949490	+	1.64456i	−0.121570	+	0.210565i	0.798817	+	0.601574i	$0.205459\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.94949	−	3.37662i	0.241804	−	0.418818i
$66$		0			0
$67$	−1.17423	−	2.03383i	−0.143456	−	0.248472i	0.785340	−	0.619065i	$-0.212488\pi$
$67$	−1.17423	−	2.03383i	−0.143456	−	0.248472i	−0.928796	+	0.370592i	$0.879155\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.7980			−1.40016			−0.700080	−	0.714064i	$-0.746852\pi$
$71$	−11.7980			−1.40016			−0.700080	+	0.714064i	$0.746852\pi$
$72$		0			0
$73$	4.89898			0.573382			0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898			0.573382			0.286691	+	0.958023i	$0.407445\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.50000	+	4.33013i	0.284901	+	0.493464i
$78$		0			0
$79$	4.27526	−	7.40496i	0.481004	−	0.833123i	−0.518759	−	0.854921i	$-0.673606\pi$
$79$	4.27526	−	7.40496i	0.481004	−	0.833123i	0.999762	+	0.0217978i	$0.00693899\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.72474	−	4.71940i	0.299080	−	0.518021i	−0.676846	−	0.736125i	$-0.736654\pi$
$83$	2.72474	−	4.71940i	0.299080	−	0.518021i	0.975926	+	0.218104i	$0.0699871\pi$
$84$		0			0
$85$	−2.44949	−	4.24264i	−0.265684	−	0.460179i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.10102			0.328708			0.164354	−	0.986401i	$-0.447446\pi$
$89$	3.10102			0.328708			0.164354	+	0.986401i	$0.447446\pi$
$90$		0			0
$91$	−5.65153			−0.592441
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	−2.94949	+	5.10867i	−0.299475	+	0.518706i	−0.976016	−	0.217699i	$-0.930145\pi$
$97$	−2.94949	+	5.10867i	−0.299475	+	0.518706i	0.676541	+	0.736405i	$0.263478\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.39898	+	5.88721i	−0.338211	+	0.585799i	−0.984096	−	0.177636i	$-0.943155\pi$
$101$	−3.39898	+	5.88721i	−0.338211	+	0.585799i	0.645885	+	0.763435i	$0.276488\pi$
$102$		0			0
$103$	8.72474	+	15.1117i	0.859675	+	1.48900i	0.872239	+	0.489079i	$0.162667\pi$
$103$	8.72474	+	15.1117i	0.859675	+	1.48900i	−0.0125648	+	0.999921i	$0.504000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.7980			1.33390			0.666950	−	0.745103i	$-0.267600\pi$
$107$	13.7980			1.33390			0.666950	+	0.745103i	$0.267600\pi$
$108$		0			0
$109$	−8.89898			−0.852368			−0.426184	−	0.904637i	$-0.640142\pi$
$109$	−8.89898			−0.852368			−0.426184	+	0.904637i	$0.640142\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.39898	−	9.35131i	−0.507893	−	0.879697i	−0.999958	−	0.00913847i	$-0.997091\pi$
$113$	−5.39898	−	9.35131i	−0.507893	−	0.879697i	0.492065	−	0.870558i	$-0.336242\pi$
$114$		0			0
$115$	0.275255	−	0.476756i	0.0256677	−	0.0444577i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.55051	+	6.14966i	−0.325475	+	0.563739i
$120$		0			0
$121$	−0.449490	−	0.778539i	−0.0408627	−	0.0707763i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.62372	−	9.74058i	−0.491347	−	0.851038i	0.508603	−	0.861001i	$-0.330162\pi$
$131$	−5.62372	−	9.74058i	−0.491347	−	0.851038i	−0.999950	+	0.00996288i	$0.996829\pi$
$132$		0			0
$133$	2.89898	−	5.02118i	0.251373	−	0.435392i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.94949	+	17.2330i	−0.850042	+	1.47232i	0.0311270	+	0.999515i	$0.490090\pi$
$137$	−9.94949	+	17.2330i	−0.850042	+	1.47232i	−0.881169	+	0.472801i	$0.843243\pi$
$138$		0			0
$139$	0.724745	+	1.25529i	0.0614721	+	0.106473i	0.895124	−	0.445818i	$-0.147087\pi$
$139$	0.724745	+	1.25529i	0.0614721	+	0.106473i	−0.833652	+	0.552291i	$0.813754\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	13.4495			1.12470
$144$		0			0
$145$	−9.89898			−0.822066
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.94949	+	12.0369i	0.569324	+	0.986099i	0.996633	+	0.0819929i	$0.0261285\pi$
$149$	6.94949	+	12.0369i	0.569324	+	0.986099i	−0.427309	+	0.904106i	$0.640538\pi$
$150$		0			0
$151$	−4.62372	+	8.00853i	−0.376273	+	0.651725i	−0.990517	−	0.137392i	$-0.956128\pi$
$151$	−4.62372	+	8.00853i	−0.376273	+	0.651725i	0.614243	+	0.789117i	$0.289461\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.72474	−	6.45145i	0.299179	−	0.518193i
$156$		0			0
$157$	−4.39898	−	7.61926i	−0.351077	−	0.608083i	0.635362	−	0.772215i	$-0.280851\pi$
$157$	−4.39898	−	7.61926i	−0.351077	−	0.608083i	−0.986438	+	0.164132i	$0.947518\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.797959			−0.0628880
$162$		0			0
$163$	−13.7980			−1.08074			−0.540370	−	0.841428i	$-0.681716\pi$
$163$	−13.7980			−1.08074			−0.540370	+	0.841428i	$0.681716\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.72474	−	8.18350i	−0.365612	−	0.633258i	0.623262	−	0.782013i	$-0.285807\pi$
$167$	−4.72474	−	8.18350i	−0.365612	−	0.633258i	−0.988874	+	0.148755i	$0.952474\pi$
$168$		0			0
$169$	−1.10102	+	1.90702i	−0.0846939	+	0.146694i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.50000	−	2.59808i	0.114043	−	0.197528i	−0.803354	−	0.595502i	$-0.796953\pi$
$173$	1.50000	−	2.59808i	0.114043	−	0.197528i	0.917397	+	0.397974i	$0.130287\pi$
$174$		0			0
$175$	2.89898	+	5.02118i	0.219142	+	0.379566i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−18.6969			−1.38973			−0.694866	−	0.719139i	$-0.744536\pi$
$181$	−18.6969			−1.38973			−0.694866	+	0.719139i	$0.744536\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.44949	−	7.70674i	−0.327133	−	0.566611i
$186$		0			0
$187$	8.44949	−	14.6349i	0.617888	−	1.07021i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.72474	−	15.1117i	0.631300	−	1.09344i	−0.355986	−	0.934491i	$-0.615855\pi$
$191$	8.72474	−	15.1117i	0.631300	−	1.09344i	0.987286	−	0.158953i	$-0.0508118\pi$
$192$		0			0
$193$	−6.94949	−	12.0369i	−0.500235	−	0.866433i	−1.00000		0.000271627i	$-0.999914\pi$
$193$	−6.94949	−	12.0369i	−0.500235	−	0.866433i	0.499765	−	0.866161i	$-0.333420\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−21.5959			−1.53865			−0.769323	−	0.638860i	$-0.779406\pi$
$197$	−21.5959			−1.53865			−0.769323	+	0.638860i	$0.779406\pi$
$198$		0			0
$199$	−11.7980			−0.836335			−0.418168	−	0.908370i	$-0.637328\pi$
$199$	−11.7980			−0.836335			−0.418168	+	0.908370i	$0.637328\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.17423	+	12.4261i	0.503533	+	0.872144i
$204$		0			0
$205$	−1.05051	+	1.81954i	−0.0733708	+	0.127082i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.89898	+	11.9494i	−0.477212	+	0.826556i
$210$		0			0
$211$	7.72474	+	13.3797i	0.531793	+	0.921093i	0.999311	+	0.0371095i	$0.0118150\pi$
$211$	7.72474	+	13.3797i	0.531793	+	0.921093i	−0.467518	+	0.883984i	$0.654852\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.3485			−0.842159
$216$		0			0
$217$	−10.7980			−0.733013
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.55051	+	16.5420i	0.642437	+	1.11273i
$222$		0			0
$223$	9.07321	−	15.7153i	0.607587	−	1.05237i	−0.384049	−	0.923313i	$-0.625471\pi$
$223$	9.07321	−	15.7153i	0.607587	−	1.05237i	0.991637	−	0.129060i	$-0.0411959\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.1742	+	19.3543i	−0.741660	+	1.28459i	0.210079	+	0.977684i	$0.432628\pi$
$227$	−11.1742	+	19.3543i	−0.741660	+	1.28459i	−0.951739	+	0.306908i	$0.900705\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	0.882073	−	0.471113i	$-0.156147\pi$
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	−0.849032	+	0.528341i	$0.822814\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	8.34847			0.544594
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.0732	+	17.4473i	0.651582	+	1.12857i	0.982739	+	0.184998i	$0.0592277\pi$
$239$	10.0732	+	17.4473i	0.651582	+	1.12857i	−0.331157	+	0.943576i	$0.607439\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.44949	−	4.24264i	0.156492	−	0.271052i
$246$		0			0
$247$	−7.79796	−	13.5065i	−0.496172	−	0.859396i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.20204			0.138992			0.0694958	−	0.997582i	$-0.477861\pi$
$251$	2.20204			0.138992			0.0694958	+	0.997582i	$0.477861\pi$
$252$		0			0
$253$	1.89898			0.119388
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.39898	+	7.61926i	0.274401	+	0.475276i	0.969984	−	0.243170i	$-0.0781872\pi$
$257$	4.39898	+	7.61926i	0.274401	+	0.475276i	−0.695583	+	0.718446i	$0.744854\pi$
$258$		0			0
$259$	−6.44949	+	11.1708i	−0.400752	+	0.694122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.275255	+	0.476756i	−0.0169730	+	0.0293980i	−0.874387	−	0.485229i	$-0.838736\pi$
$263$	−0.275255	+	0.476756i	−0.0169730	+	0.0293980i	0.857414	+	0.514627i	$0.172070\pi$
$264$		0			0
$265$	−0.449490	−	0.778539i	−0.0276119	−	0.0478253i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−16.8990			−1.03035			−0.515174	−	0.857085i	$-0.672273\pi$
$269$	−16.8990			−1.03035			−0.515174	+	0.857085i	$0.672273\pi$
$270$		0			0
$271$	29.3939			1.78555			0.892775	−	0.450502i	$-0.148755\pi$
$271$	29.3939			1.78555			0.892775	+	0.450502i	$0.148755\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.89898	−	11.9494i	−0.416024	−	0.720575i
$276$		0			0
$277$	−2.39898	+	4.15515i	−0.144141	+	0.249659i	−0.929052	−	0.369949i	$-0.879375\pi$
$277$	−2.39898	+	4.15515i	−0.144141	+	0.249659i	0.784911	+	0.619608i	$0.212708\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.94949	+	17.2330i	−0.593537	+	1.02804i	0.400215	+	0.916421i	$0.368936\pi$
$281$	−9.94949	+	17.2330i	−0.593537	+	1.02804i	−0.993752	+	0.111615i	$0.964398\pi$
$282$		0			0
$283$	0.724745	+	1.25529i	0.0430816	+	0.0746195i	0.886762	−	0.462226i	$-0.152949\pi$
$283$	0.724745	+	1.25529i	0.0430816	+	0.0746195i	−0.843681	+	0.536846i	$0.819616\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.04541			0.179765
$288$		0			0
$289$	7.00000			0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.39898	−	16.2795i	−0.549094	−	0.951059i	−0.998337	−	0.0576493i	$-0.981639\pi$
$293$	−9.39898	−	16.2795i	−0.549094	−	0.951059i	0.449243	−	0.893410i	$-0.351694\pi$
$294$		0			0
$295$	0.174235	−	0.301783i	0.0101443	−	0.0175705i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.07321	+	1.85886i	−0.0620656	+	0.107501i
$300$		0			0
$301$	8.94949	+	15.5010i	0.515840	+	0.893461i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.89898			0.108735
$306$		0			0
$307$	2.20204			0.125677			0.0628386	−	0.998024i	$-0.479985\pi$
$307$	2.20204			0.125677			0.0628386	+	0.998024i	$0.479985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.62372	−	8.00853i	−0.262187	−	0.454122i	0.704636	−	0.709569i	$-0.251111\pi$
$311$	−4.62372	−	8.00853i	−0.262187	−	0.454122i	−0.966823	+	0.255448i	$0.917777\pi$
$312$		0			0
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	−0.519300	−	0.854592i	$-0.673807\pi$
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	0.999748	+	0.0224310i	$0.00714060\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.05051	−	13.9439i	0.452162	−	0.783167i	−0.546358	−	0.837552i	$-0.683986\pi$
$317$	8.05051	−	13.9439i	0.452162	−	0.783167i	0.998520	+	0.0543845i	$0.0173197\pi$
$318$		0			0
$319$	−17.0732	−	29.5717i	−0.955916	−	1.65570i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−19.5959			−1.09035
$324$		0			0
$325$	15.5959			0.865106
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.05051	−	10.4798i	−0.333575	−	0.577770i
$330$		0			0
$331$	−6.62372	+	11.4726i	−0.364073	+	0.630593i	−0.988627	−	0.150389i	$-0.951947\pi$
$331$	−6.62372	+	11.4726i	−0.364073	+	0.630593i	0.624554	+	0.780982i	$0.285281\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.17423	+	2.03383i	−0.0641553	+	0.111120i
$336$		0			0
$337$	−4.39898	−	7.61926i	−0.239628	−	0.415047i	0.720980	−	0.692956i	$-0.243692\pi$
$337$	−4.39898	−	7.61926i	−0.239628	−	0.415047i	−0.960607	+	0.277909i	$0.910359\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	25.6969			1.39157
$342$		0			0
$343$	−17.2474			−0.931275
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0732	+	24.3755i	0.755490	+	1.30855i	0.945130	+	0.326693i	$0.105934\pi$
$347$	14.0732	+	24.3755i	0.755490	+	1.30855i	−0.189641	+	0.981854i	$0.560732\pi$
$348$		0			0
$349$	−2.39898	+	4.15515i	−0.128414	+	0.222420i	−0.923062	−	0.384650i	$-0.874322\pi$
$349$	−2.39898	+	4.15515i	−0.128414	+	0.222420i	0.794648	+	0.607070i	$0.207655\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.84847	−	13.5939i	0.417732	−	0.723533i	−0.577979	−	0.816052i	$-0.696158\pi$
$353$	7.84847	−	13.5939i	0.417732	−	0.723533i	0.995711	+	0.0925188i	$0.0294918\pi$
$354$		0			0
$355$	5.89898	+	10.2173i	0.313085	+	0.542280i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	17.7980			0.939340			0.469670	−	0.882842i	$-0.344373\pi$
$359$	17.7980			0.939340			0.469670	+	0.882842i	$0.344373\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.44949	−	4.24264i	−0.128212	−	0.222070i
$366$		0			0
$367$	5.17423	−	8.96204i	0.270093	−	0.467815i	−0.698793	−	0.715324i	$-0.746279\pi$
$367$	5.17423	−	8.96204i	0.270093	−	0.467815i	0.968885	+	0.247510i	$0.0796123\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.651531	+	1.12848i	−0.0338258	+	0.0585880i
$372$		0			0
$373$	−1.15153	−	1.99451i	−0.0596240	−	0.103272i	0.834673	−	0.550746i	$-0.185657\pi$
$373$	−1.15153	−	1.99451i	−0.0596240	−	0.103272i	−0.894297	+	0.447475i	$0.852324\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	38.5959			1.98779
$378$		0			0
$379$	26.0000			1.33553			0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000			1.33553			0.667765	+	0.744372i	$0.267251\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.7247	−	28.9681i	−0.854595	−	1.48020i	−0.877021	−	0.480453i	$-0.840472\pi$
$383$	−16.7247	−	28.9681i	−0.854595	−	1.48020i	0.0224261	−	0.999749i	$-0.492861\pi$
$384$		0			0
$385$	2.50000	−	4.33013i	0.127412	−	0.220684i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.39898	−	11.0834i	0.324441	−	0.561949i	−0.656958	−	0.753927i	$-0.728157\pi$
$389$	6.39898	−	11.0834i	0.324441	−	0.561949i	0.981399	+	0.191979i	$0.0614904\pi$
$390$		0			0
$391$	1.34847	+	2.33562i	0.0681950	+	0.118117i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.55051			−0.430223
$396$		0			0
$397$	18.6969			0.938372			0.469186	−	0.883099i	$-0.344547\pi$
$397$	18.6969			0.938372			0.469186	+	0.883099i	$0.344547\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.84847	+	13.5939i	0.391934	+	0.678849i	0.992705	−	0.120572i	$-0.0384729\pi$
$401$	7.84847	+	13.5939i	0.391934	+	0.678849i	−0.600771	+	0.799421i	$0.705140\pi$
$402$		0			0
$403$	−14.5227	+	25.1541i	−0.723427	+	1.25301i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15.3485	−	26.5843i	0.760795	−	1.31774i
$408$		0			0
$409$	6.29796	+	10.9084i	0.311414	+	0.539385i	0.978669	−	0.205445i	$-0.0658641\pi$
$409$	6.29796	+	10.9084i	0.311414	+	0.539385i	−0.667255	+	0.744830i	$0.732531\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.505103			−0.0248545
$414$		0			0
$415$	−5.44949			−0.267505
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.37628	+	5.84788i	0.164942	+	0.285688i	0.936635	−	0.350308i	$-0.113923\pi$
$419$	3.37628	+	5.84788i	0.164942	+	0.285688i	−0.771693	+	0.635995i	$0.780590\pi$
$420$		0			0
$421$	−4.94949	+	8.57277i	−0.241223	+	0.417811i	−0.961063	−	0.276329i	$-0.910882\pi$
$421$	−4.94949	+	8.57277i	−0.241223	+	0.417811i	0.719840	+	0.694140i	$0.244215\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.79796	−	16.9706i	0.475271	−	0.823193i
$426$		0			0
$427$	−1.37628	−	2.38378i	−0.0666026	−	0.115359i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	33.7980			1.62799			0.813995	−	0.580872i	$-0.197288\pi$
$431$	33.7980			1.62799			0.813995	+	0.580872i	$0.197288\pi$
$432$		0			0
$433$	40.4949			1.94606			0.973030	−	0.230677i	$-0.0740942\pi$
$433$	40.4949			1.94606			0.973030	+	0.230677i	$0.0740942\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.10102	−	1.90702i	−0.0526690	−	0.0912253i
$438$		0			0
$439$	−10.8258	+	18.7508i	−0.516686	+	0.894926i	0.483127	+	0.875550i	$0.339501\pi$
$439$	−10.8258	+	18.7508i	−0.516686	+	0.894926i	−0.999812	+	0.0193752i	$0.993832\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.27526	+	16.0652i	−0.440681	+	0.763281i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−9.27526	+	16.0652i	−0.440681	+	0.763281i	0.557059	+	0.830473i	$0.311930\pi$
$444$		0			0
$445$	−1.55051	−	2.68556i	−0.0735012	−	0.127308i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.8990			−0.986284			−0.493142	−	0.869949i	$-0.664152\pi$
$449$	−20.8990			−0.986284			−0.493142	+	0.869949i	$0.664152\pi$
$450$		0			0
$451$	−7.24745			−0.341269
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.82577	+	4.89437i	0.132474	+	0.229452i
$456$		0			0
$457$	−8.74745	+	15.1510i	−0.409188	+	0.708735i	−0.994799	−	0.101857i	$-0.967521\pi$
$457$	−8.74745	+	15.1510i	−0.409188	+	0.708735i	0.585611	+	0.810593i	$0.300855\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.15153	−	5.45861i	0.146781	−	0.254233i	−0.783255	−	0.621701i	$-0.786442\pi$
$461$	3.15153	−	5.45861i	0.146781	−	0.254233i	0.930036	+	0.367468i	$0.119775\pi$
$462$		0			0
$463$	−3.37628	−	5.84788i	−0.156909	−	0.271774i	0.776844	−	0.629694i	$-0.216820\pi$
$463$	−3.37628	−	5.84788i	−0.156909	−	0.271774i	−0.933752	+	0.357920i	$0.883486\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.404082			0.0186987			0.00934934	−	0.999956i	$-0.497024\pi$
$467$	0.404082			0.0186987			0.00934934	+	0.999956i	$0.497024\pi$
$468$		0			0
$469$	3.40408			0.157186
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.2980	−	36.8891i	−0.979281	−	1.69616i
$474$		0			0
$475$	−8.00000	+	13.8564i	−0.367065	+	0.635776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.52270	−	16.4938i	0.435103	−	0.753621i	−0.562201	−	0.827001i	$-0.690045\pi$
$479$	9.52270	−	16.4938i	0.435103	−	0.753621i	0.997304	+	0.0733796i	$0.0233785\pi$
$480$		0			0
$481$	17.3485	+	30.0484i	0.791022	+	1.37009i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.89898			0.267859
$486$		0			0
$487$	−1.79796			−0.0814733			−0.0407366	−	0.999170i	$-0.512970\pi$
$487$	−1.79796			−0.0814733			−0.0407366	+	0.999170i	$0.512970\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.6237	−	30.5252i	−0.795348	−	1.37758i	−0.922618	−	0.385714i	$-0.873955\pi$
$491$	−17.6237	−	30.5252i	−0.795348	−	1.37758i	0.127271	−	0.991868i	$-0.459378\pi$
$492$		0			0
$493$	24.2474	−	41.9978i	1.09205	−	1.89149i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.55051	−	14.8099i	0.383543	−	0.664316i
$498$		0			0
$499$	−8.17423	−	14.1582i	−0.365929	−	0.633808i	0.622996	−	0.782225i	$-0.285915\pi$
$499$	−8.17423	−	14.1582i	−0.365929	−	0.633808i	−0.988925	+	0.148418i	$0.952582\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.2020			−0.900764			−0.450382	−	0.892836i	$-0.648712\pi$
$503$	−20.2020			−0.900764			−0.450382	+	0.892836i	$0.648712\pi$
$504$		0			0
$505$	6.79796			0.302505
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.7474	+	29.0074i	0.742318	+	1.28573i	0.951438	+	0.307842i	$0.0996068\pi$
$509$	16.7474	+	29.0074i	0.742318	+	1.28573i	−0.209120	+	0.977890i	$0.567060\pi$
$510$		0			0
$511$	−3.55051	+	6.14966i	−0.157065	+	0.272045i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.72474	−	15.1117i	0.384458	−	0.665901i
$516$		0			0
$517$	14.3990	+	24.9398i	0.633266	+	1.09685i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	25.5959			1.11923			0.559616	−	0.828752i	$-0.310949\pi$
$523$	25.5959			1.11923			0.559616	+	0.828752i	$0.310949\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.2474	+	31.6055i	0.794871	+	1.37676i
$528$		0			0
$529$	11.3485	−	19.6561i	0.493412	−	0.854614i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.09592	−	7.09434i	0.177414	−	0.307290i
$534$		0			0
$535$	−6.89898	−	11.9494i	−0.298269	−	0.516617i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.8990			0.727891
$540$		0			0
$541$	−9.59592			−0.412561			−0.206280	−	0.978493i	$-0.566136\pi$
$541$	−9.59592			−0.412561			−0.206280	+	0.978493i	$0.566136\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.44949	+	7.70674i	0.190595	+	0.330121i
$546$		0			0
$547$	−5.72474	+	9.91555i	−0.244772	+	0.423958i	−0.962068	−	0.272811i	$-0.912047\pi$
$547$	−5.72474	+	9.91555i	−0.244772	+	0.423958i	0.717295	+	0.696769i	$0.245380\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−19.7980	+	34.2911i	−0.843421	+	1.46085i
$552$		0			0
$553$	6.19694	+	10.7334i	0.263521	+	0.456431i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.6969			−0.453244			−0.226622	−	0.973983i	$-0.572768\pi$
$557$	−10.6969			−0.453244			−0.226622	+	0.973983i	$0.572768\pi$
$558$		0			0
$559$	48.1464			2.03638
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.17423	+	8.96204i	0.218068	+	0.377705i	0.954217	−	0.299114i	$-0.0966912\pi$
$563$	5.17423	+	8.96204i	0.218068	+	0.377705i	−0.736149	+	0.676819i	$0.763358\pi$
$564$		0			0
$565$	−5.39898	+	9.35131i	−0.227137	+	0.393412i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	−9.17423	−	15.8902i	−0.383930	−	0.664986i	0.607690	−	0.794174i	$-0.292096\pi$
$571$	−9.17423	−	15.8902i	−0.383930	−	0.664986i	−0.991620	+	0.129188i	$0.958763\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.20204			0.0918315
$576$		0			0
$577$	−28.8990			−1.20308			−0.601540	−	0.798843i	$-0.705446\pi$
$577$	−28.8990			−1.20308			−0.601540	+	0.798843i	$0.705446\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.94949	+	6.84072i	0.163852	+	0.283801i
$582$		0			0
$583$	1.55051	−	2.68556i	0.0642156	−	0.111225i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.9722	+	20.7364i	−0.494145	+	0.855885i	−0.999977	−	0.00674727i	$-0.997852\pi$
$587$	−11.9722	+	20.7364i	−0.494145	+	0.855885i	0.505832	+	0.862632i	$0.331186\pi$
$588$		0			0
$589$	−14.8990	−	25.8058i	−0.613902	−	1.06331i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.1010			−0.784385			−0.392192	−	0.919883i	$-0.628283\pi$
$593$	−19.1010			−0.784385			−0.392192	+	0.919883i	$0.628283\pi$
$594$		0			0
$595$	7.10102			0.291113
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.7247	−	28.9681i	−0.683355	−	1.18360i	−0.973951	−	0.226759i	$-0.927187\pi$
$599$	−16.7247	−	28.9681i	−0.683355	−	1.18360i	0.290596	−	0.956846i	$-0.406146\pi$
$600$		0			0
$601$	−19.8485	+	34.3786i	−0.809636	+	1.40233i	0.103480	+	0.994631i	$0.467002\pi$
$601$	−19.8485	+	34.3786i	−0.809636	+	1.40233i	−0.913116	+	0.407699i	$0.866331\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.449490	+	0.778539i	−0.0182744	+	0.0316521i
$606$		0			0
$607$	−13.9722	−	24.2005i	−0.567114	−	0.982270i	−0.996850	−	0.0793153i	$-0.974727\pi$
$607$	−13.9722	−	24.2005i	−0.567114	−	0.982270i	0.429736	−	0.902955i	$-0.358607\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.5505			−1.31685
$612$		0			0
$613$	2.69694			0.108928			0.0544642	−	0.998516i	$-0.482655\pi$
$613$	2.69694			0.108928			0.0544642	+	0.998516i	$0.482655\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.84847	−	4.93369i	−0.114675	−	0.198623i	0.802975	−	0.596013i	$-0.203249\pi$
$617$	−2.84847	−	4.93369i	−0.114675	−	0.198623i	−0.917650	+	0.397390i	$0.869916\pi$
$618$		0			0
$619$	4.07321	−	7.05501i	0.163716	−	0.283565i	−0.772482	−	0.635036i	$-0.780985\pi$
$619$	4.07321	−	7.05501i	0.163716	−	0.283565i	0.936199	+	0.351471i	$0.114319\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.24745	+	3.89270i	−0.0900421	+	0.155958i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	43.5959			1.73828
$630$		0			0
$631$	25.7980			1.02700			0.513500	−	0.858089i	$-0.328349\pi$
$631$	25.7980			1.02700			0.513500	+	0.858089i	$0.328349\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	−9.55051	+	16.5420i	−0.378405	+	0.655417i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.29796	+	10.9084i	−0.248754	+	0.430855i	−0.963180	−	0.268856i	$-0.913355\pi$
$641$	−6.29796	+	10.9084i	−0.248754	+	0.430855i	0.714426	+	0.699711i	$0.246688\pi$
$642$		0			0
$643$	−5.17423	−	8.96204i	−0.204052	−	0.353428i	0.745778	−	0.666194i	$-0.232078\pi$
$643$	−5.17423	−	8.96204i	−0.204052	−	0.353428i	−0.949830	+	0.312766i	$0.898744\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.79796			−0.385198			−0.192599	−	0.981278i	$-0.561692\pi$
$647$	−9.79796			−0.385198			−0.192599	+	0.981278i	$0.561692\pi$
$648$		0			0
$649$	1.20204			0.0471842
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.50000	−	14.7224i	−0.332631	−	0.576133i	0.650396	−	0.759595i	$-0.274603\pi$
$653$	−8.50000	−	14.7224i	−0.332631	−	0.576133i	−0.983027	+	0.183462i	$0.941270\pi$
$654$		0			0
$655$	−5.62372	+	9.74058i	−0.219737	+	0.380596i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.1742	+	19.3543i	−0.435286	+	0.753938i	−0.997319	−	0.0731770i	$-0.976686\pi$
$659$	−11.1742	+	19.3543i	−0.435286	+	0.753938i	0.562033	+	0.827115i	$0.310020\pi$
$660$		0			0
$661$	−14.1969	−	24.5898i	−0.552197	−	0.956433i	−0.998116	−	0.0613597i	$-0.980456\pi$
$661$	−14.1969	−	24.5898i	−0.552197	−	0.956433i	0.445919	−	0.895073i	$-0.352877\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.79796			−0.224835
$666$		0			0
$667$	5.44949			0.211005
$668$		0			0
$669$		0			0
$670$		0			0
$671$	3.27526	+	5.67291i	0.126440	+	0.219000i
$672$		0			0
$673$	20.6464	−	35.7607i	0.795861	−	1.37847i	−0.126429	−	0.991976i	$-0.540352\pi$
$673$	20.6464	−	35.7607i	0.795861	−	1.37847i	0.922291	−	0.386497i	$-0.126315\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−17.1969	+	29.7860i	−0.660932	+	1.14477i	0.319439	+	0.947607i	$0.396506\pi$
$677$	−17.1969	+	29.7860i	−0.660932	+	1.14477i	−0.980371	+	0.197161i	$0.936828\pi$
$678$		0			0
$679$	−4.27526	−	7.40496i	−0.164069	−	0.284176i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	41.3939			1.58389			0.791946	−	0.610591i	$-0.209068\pi$
$683$	41.3939			1.58389			0.791946	+	0.610591i	$0.209068\pi$
$684$		0			0
$685$	19.8990			0.760301
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.75255	+	3.03551i	0.0667669	+	0.115644i
$690$		0			0
$691$	7.97219	−	13.8082i	0.303277	−	0.525290i	−0.673600	−	0.739096i	$-0.735253\pi$
$691$	7.97219	−	13.8082i	0.303277	−	0.525290i	0.976876	+	0.213806i	$0.0685861\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.724745	−	1.25529i	0.0274911	−	0.0476161i
$696$		0			0
$697$	−5.14643	−	8.91388i	−0.194935	−	0.337637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.4949			−1.37839			−0.689197	−	0.724574i	$-0.742036\pi$
$701$	−36.4949			−1.37839			−0.689197	+	0.724574i	$0.742036\pi$
$702$		0			0
$703$	−35.5959			−1.34253
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.92679	−	8.53344i	−0.185291	−	0.320933i
$708$		0			0
$709$	2.50000	−	4.33013i	0.0938895	−	0.162621i	−0.815255	−	0.579102i	$-0.803403\pi$
$709$	2.50000	−	4.33013i	0.0938895	−	0.162621i	0.909145	+	0.416481i	$0.136737\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.05051	+	3.55159i	−0.0767922	+	0.133008i
$714$		0			0
$715$	−6.72474	−	11.6476i	−0.251491	−	0.435596i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−47.1918			−1.75996			−0.879979	−	0.475012i	$-0.842444\pi$
$719$	−47.1918			−1.75996			−0.879979	+	0.475012i	$0.842444\pi$
$720$		0			0
$721$	−25.2929			−0.941955
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−19.7980	−	34.2911i	−0.735278	−	1.27354i
$726$		0			0
$727$	−11.7247	+	20.3079i	−0.434847	+	0.753177i	−0.997283	−	0.0736639i	$-0.976531\pi$
$727$	−11.7247	+	20.3079i	−0.434847	+	0.753177i	0.562436	+	0.826841i	$0.309864\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	30.2474	−	52.3901i	1.11874	−	1.93772i
$732$		0			0
$733$	1.05051	+	1.81954i	0.0388015	+	0.0672061i	0.884774	−	0.466020i	$-0.154313\pi$
$733$	1.05051	+	1.81954i	0.0388015	+	0.0672061i	−0.845972	+	0.533227i	$0.820979\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.10102			−0.298405
$738$		0			0
$739$	14.0000			0.514998			0.257499	−	0.966279i	$-0.417102\pi$
$739$	14.0000			0.514998			0.257499	+	0.966279i	$0.417102\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.2753	+	26.4575i	0.560395	+	0.970632i	0.997462	+	0.0712033i	$0.0226839\pi$
$743$	15.2753	+	26.4575i	0.560395	+	0.970632i	−0.437067	+	0.899429i	$0.643983\pi$
$744$		0			0
$745$	6.94949	−	12.0369i	0.254610	−	0.440997i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.0000	+	17.3205i	−0.365392	+	0.632878i
$750$		0			0
$751$	20.3207	+	35.1964i	0.741512	+	1.28434i	0.951807	+	0.306698i	$0.0992241\pi$
$751$	20.3207	+	35.1964i	0.741512	+	1.28434i	−0.210295	+	0.977638i	$0.567443\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.24745			0.336549
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.6464	+	37.4927i	0.784682	+	1.35911i	0.929189	+	0.369606i	$0.120507\pi$
$761$	21.6464	+	37.4927i	0.784682	+	1.35911i	−0.144506	+	0.989504i	$0.546159\pi$
$762$		0			0
$763$	6.44949	−	11.1708i	0.233487	−	0.404412i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.679337	+	1.17665i	−0.0245294	+	0.0424862i
$768$		0			0
$769$	−1.29796	−	2.24813i	−0.0468056	−	0.0810697i	0.841673	−	0.539987i	$-0.181571\pi$
$769$	−1.29796	−	2.24813i	−0.0468056	−	0.0810697i	−0.888479	+	0.458917i	$0.848237\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.4949			−0.449410			−0.224705	−	0.974427i	$-0.572142\pi$
$773$	−12.4949			−0.449410			−0.224705	+	0.974427i	$0.572142\pi$
$774$		0			0
$775$	29.7980			1.07037
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.20204	+	7.27815i	0.150554	+	0.260767i
$780$		0			0
$781$	−20.3485	+	35.2446i	−0.728125	+	1.26115i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.39898	+	7.61926i	−0.157006	+	0.271943i
$786$		0			0
$787$	1.62372	+	2.81237i	0.0578795	+	0.100250i	0.893513	−	0.449037i	$-0.148233\pi$
$787$	1.62372	+	2.81237i	0.0578795	+	0.100250i	−0.835634	+	0.549287i	$0.814899\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.6515			0.556504
$792$		0			0
$793$	−7.40408			−0.262927
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.15153	+	15.8509i	0.324164	+	0.561468i	0.981343	−	0.192266i	$-0.0615838\pi$
$797$	9.15153	+	15.8509i	0.324164	+	0.561468i	−0.657179	+	0.753735i	$0.728250\pi$
$798$		0			0
$799$	−20.4495	+	35.4196i	−0.723451	+	1.25305i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.44949	−	14.6349i	0.298176	−	0.516456i
$804$		0			0
$805$	0.398979	+	0.691053i	0.0140622	+	0.0243564i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.4949			−0.579930			−0.289965	−	0.957037i	$-0.593644\pi$
$809$	−16.4949			−0.579930			−0.289965	+	0.957037i	$0.593644\pi$
$810$		0			0
$811$	−47.5959			−1.67132			−0.835659	−	0.549248i	$-0.814914\pi$
$811$	−47.5959			−1.67132			−0.835659	+	0.549248i	$0.814914\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.89898	+	11.9494i	0.241661	+	0.418569i
$816$		0			0
$817$	−24.6969	+	42.7764i	−0.864037	+	1.49656i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.6464	+	46.1530i	−0.929967	+	1.61075i	−0.146595	+	0.989197i	$0.546831\pi$
$821$	−26.6464	+	46.1530i	−0.929967	+	1.61075i	−0.783372	+	0.621553i	$0.786502\pi$
$822$		0			0
$823$	7.72474	+	13.3797i	0.269268	+	0.466385i	0.968673	−	0.248340i	$-0.0798850\pi$
$823$	7.72474	+	13.3797i	0.269268	+	0.466385i	−0.699405	+	0.714725i	$0.746552\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−28.0000			−0.973655			−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000			−0.973655			−0.486828	+	0.873498i	$0.661846\pi$
$828$		0			0
$829$	−32.8990			−1.14263			−0.571314	−	0.820731i	$-0.693566\pi$
$829$	−32.8990			−1.14263			−0.571314	+	0.820731i	$0.693566\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0000	+	20.7846i	0.415775	+	0.720144i
$834$		0			0
$835$	−4.72474	+	8.18350i	−0.163507	+	0.283202i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.82577	−	13.5546i	0.270175	−	0.467958i	−0.698731	−	0.715384i	$-0.746252\pi$
$839$	7.82577	−	13.5546i	0.270175	−	0.467958i	0.968907	+	0.247427i	$0.0795850\pi$
$840$		0			0
$841$	−34.4949	−	59.7469i	−1.18948	−	2.06024i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.20204			0.0757525
$846$		0			0
$847$	1.30306			0.0447737
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.44949	+	4.24264i	0.0839674	+	0.145436i
$852$		0			0
$853$	−0.949490	+	1.64456i	−0.0325099	+	0.0563088i	−0.881823	−	0.471581i	$-0.843683\pi$
$853$	−0.949490	+	1.64456i	−0.0325099	+	0.0563088i	0.849313	+	0.527890i	$0.177017\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.05051	+	15.6759i	−0.309160	+	0.535480i	−0.978179	−	0.207765i	$-0.933381\pi$
$857$	−9.05051	+	15.6759i	−0.309160	+	0.535480i	0.669019	+	0.743245i	$0.266714\pi$
$858$		0			0
$859$	−8.27526	−	14.3332i	−0.282348	−	0.489041i	0.689615	−	0.724177i	$-0.257780\pi$
$859$	−8.27526	−	14.3332i	−0.282348	−	0.489041i	−0.971963	+	0.235135i	$0.924447\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.79796			0.0612032			0.0306016	−	0.999532i	$-0.490258\pi$
$863$	1.79796			0.0612032			0.0306016	+	0.999532i	$0.490258\pi$
$864$		0			0
$865$	−3.00000			−0.102003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−14.7474	−	25.5433i	−0.500273	−	0.866498i
$870$		0			0
$871$	4.57832	−	7.92988i	0.155130	−	0.268694i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.52270	−	11.2977i	0.220508	−	0.381930i
$876$		0			0
$877$	9.05051	+	15.6759i	0.305614	+	0.529339i	0.977398	−	0.211408i	$-0.0678050\pi$
$877$	9.05051	+	15.6759i	0.305614	+	0.529339i	−0.671784	+	0.740747i	$0.734472\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.4949			−0.825254			−0.412627	−	0.910900i	$-0.635389\pi$
$881$	−24.4949			−0.825254			−0.412627	+	0.910900i	$0.635389\pi$
$882$		0			0
$883$	−0.404082			−0.0135984			−0.00679922	−	0.999977i	$-0.502164\pi$
$883$	−0.404082			−0.0135984			−0.00679922	+	0.999977i	$0.502164\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.4217	−	24.9791i	−0.484233	−	0.838716i	0.515603	−	0.856827i	$-0.327568\pi$
$887$	−14.4217	−	24.9791i	−0.484233	−	0.838716i	−0.999836	+	0.0181118i	$0.994235\pi$
$888$		0			0
$889$	5.79796	−	10.0424i	0.194457	−	0.336810i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.6969	−	28.9199i	0.558742	−	0.967769i
$894$		0			0
$895$	6.00000	+	10.3923i	0.200558	+	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	73.7423			2.45944
$900$		0			0
$901$	4.40408			0.146721
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.34847	+	16.1920i	0.310754	+	0.538241i
$906$		0			0
$907$	0.376276	−	0.651729i	0.0124940	−	0.0216403i	−0.859711	−	0.510781i	$-0.829356\pi$
$907$	0.376276	−	0.651729i	0.0124940	−	0.0216403i	0.872205	+	0.489141i	$0.162690\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.7702	+	42.9032i	−0.820672	+	1.42145i	0.0845109	+	0.996423i	$0.473067\pi$
$911$	−24.7702	+	42.9032i	−0.820672	+	1.42145i	−0.905183	+	0.425023i	$0.860266\pi$
$912$		0			0
$913$	−9.39898	−	16.2795i	−0.311061	−	0.538773i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.3031			0.538375
$918$		0			0
$919$	−9.79796			−0.323205			−0.161602	−	0.986856i	$-0.551666\pi$
$919$	−9.79796			−0.323205			−0.161602	+	0.986856i	$0.551666\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−23.0000	−	39.8372i	−0.757054	−	1.31126i
$924$		0			0
$925$	17.7980	−	30.8270i	0.585193	−	1.01358i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.15153	−	15.8509i	0.300252	−	0.520052i	−0.675941	−	0.736956i	$-0.736263\pi$
$929$	9.15153	−	15.8509i	0.300252	−	0.520052i	0.976193	+	0.216904i	$0.0695959\pi$
$930$		0			0
$931$	−9.79796	−	16.9706i	−0.321115	−	0.556188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.8990			−0.552656
$936$		0			0
$937$	7.50510			0.245181			0.122591	−	0.992457i	$-0.460880\pi$
$937$	7.50510			0.245181			0.122591	+	0.992457i	$0.460880\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.50000	+	6.06218i	0.114097	+	0.197621i	0.917418	−	0.397924i	$-0.130269\pi$
$941$	3.50000	+	6.06218i	0.114097	+	0.197621i	−0.803322	+	0.595545i	$0.796936\pi$
$942$		0			0
$943$	0.578317	−	1.00167i	0.0188326	−	0.0326190i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.6237	−	39.1854i	0.735172	−	1.27336i	−0.219475	−	0.975618i	$-0.570435\pi$
$947$	22.6237	−	39.1854i	0.735172	−	1.27336i	0.954648	−	0.297738i	$-0.0962321\pi$
$948$		0			0
$949$	9.55051	+	16.5420i	0.310023	+	0.536975i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−52.8990			−1.71357			−0.856783	−	0.515677i	$-0.827540\pi$
$953$	−52.8990			−1.71357			−0.856783	+	0.515677i	$0.827540\pi$
$954$		0			0
$955$	−17.4495			−0.564652
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.4217	−	24.9791i	−0.465700	−	0.806617i
$960$		0			0
$961$	−12.2474	+	21.2132i	−0.395079	+	0.684297i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.94949	+	12.0369i	−0.223712	+	0.387481i
$966$		0			0
$967$	4.62372	+	8.00853i	0.148689	+	0.257537i	0.930743	−	0.365674i	$-0.119161\pi$
$967$	4.62372	+	8.00853i	0.148689	+	0.257537i	−0.782054	+	0.623210i	$0.785828\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−10.0000			−0.320915			−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000			−0.320915			−0.160458	+	0.987043i	$0.551297\pi$
$972$		0			0
$973$	−2.10102			−0.0673556
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.3990	+	28.4039i	0.524650	+	0.908720i	0.999588	+	0.0287010i	$0.00913707\pi$
$977$	16.3990	+	28.4039i	0.524650	+	0.908720i	−0.474938	+	0.880019i	$0.657530\pi$
$978$		0			0
$979$	5.34847	−	9.26382i	0.170938	−	0.296073i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.1742	+	29.7466i	−0.547773	+	0.948771i	0.450654	+	0.892699i	$0.351191\pi$
$983$	−17.1742	+	29.7466i	−0.547773	+	0.948771i	−0.998427	+	0.0560718i	$0.982142\pi$
$984$		0			0
$985$	10.7980	+	18.7026i	0.344052	+	0.595915i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.79796			0.216163
$990$		0			0
$991$	−37.3939			−1.18786			−0.593928	−	0.804518i	$-0.702424\pi$
$991$	−37.3939			−1.18786			−0.593928	+	0.804518i	$0.702424\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.89898	+	10.2173i	0.187010	+	0.323911i
$996$		0			0
$997$	13.1969	−	22.8578i	0.417951	−	0.723913i	−0.577782	−	0.816191i	$-0.696082\pi$
$997$	13.1969	−	22.8578i	0.417951	−	0.723913i	0.995733	+	0.0922783i	$0.0294150\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.i.e.289.1		4
3.2	odd	2		288.2.i.c.97.2	✓	4
4.3	odd	2		864.2.i.c.289.2		4
8.3	odd	2		1728.2.i.k.1153.2		4
8.5	even	2		1728.2.i.m.1153.1		4
9.2	odd	6		2592.2.a.j.1.2		2
9.4	even	3	inner	864.2.i.e.577.1		4
9.5	odd	6		288.2.i.c.193.1	yes	4
9.7	even	3		2592.2.a.o.1.2		2
12.11	even	2		288.2.i.e.97.1	yes	4
24.5	odd	2		576.2.i.m.385.1		4
24.11	even	2		576.2.i.i.385.2		4
36.7	odd	6		2592.2.a.s.1.1		2
36.11	even	6		2592.2.a.n.1.1		2
36.23	even	6		288.2.i.e.193.2	yes	4
36.31	odd	6		864.2.i.c.577.2		4
72.5	odd	6		576.2.i.m.193.2		4
72.11	even	6		5184.2.a.by.1.1		2
72.13	even	6		1728.2.i.m.577.1		4
72.29	odd	6		5184.2.a.bu.1.2		2
72.43	odd	6		5184.2.a.bn.1.1		2
72.59	even	6		576.2.i.i.193.1		4
72.61	even	6		5184.2.a.bj.1.2		2
72.67	odd	6		1728.2.i.k.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.c.97.2	✓	4	3.2	odd	2
288.2.i.c.193.1	yes	4	9.5	odd	6
288.2.i.e.97.1	yes	4	12.11	even	2
288.2.i.e.193.2	yes	4	36.23	even	6
576.2.i.i.193.1		4	72.59	even	6
576.2.i.i.385.2		4	24.11	even	2
576.2.i.m.193.2		4	72.5	odd	6
576.2.i.m.385.1		4	24.5	odd	2
864.2.i.c.289.2		4	4.3	odd	2
864.2.i.c.577.2		4	36.31	odd	6
864.2.i.e.289.1		4	1.1	even	1	trivial
864.2.i.e.577.1		4	9.4	even	3	inner
1728.2.i.k.577.2		4	72.67	odd	6
1728.2.i.k.1153.2		4	8.3	odd	2
1728.2.i.m.577.1		4	72.13	even	6
1728.2.i.m.1153.1		4	8.5	even	2
2592.2.a.j.1.2		2	9.2	odd	6
2592.2.a.n.1.1		2	36.11	even	6
2592.2.a.o.1.2		2	9.7	even	3
2592.2.a.s.1.1		2	36.7	odd	6
5184.2.a.bj.1.2		2	72.61	even	6
5184.2.a.bn.1.1		2	72.43	odd	6
5184.2.a.bu.1.2		2	72.29	odd	6
5184.2.a.by.1.1		2	72.11	even	6

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(1.72474 - 2.98735i) q^{11} +(1.94949 + 3.37662i) q^{13} +4.89898 q^{17} -4.00000 q^{19} +(0.275255 + 0.476756i) q^{23} +(2.00000 - 3.46410i) q^{25} +(4.94949 - 8.57277i) q^{29} +(3.72474 + 6.45145i) q^{31} +1.44949 q^{35} +8.89898 q^{37} +(-1.05051 - 1.81954i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-4.17423 + 7.22999i) q^{47} +(2.44949 + 4.24264i) q^{49} +0.898979 q^{53} -3.44949 q^{55} +(0.174235 + 0.301783i) q^{59} +(-0.949490 + 1.64456i) q^{61} +(1.94949 - 3.37662i) q^{65} +(-1.17423 - 2.03383i) q^{67} -11.7980 q^{71} +4.89898 q^{73} +(2.50000 + 4.33013i) q^{77} +(4.27526 - 7.40496i) q^{79} +(2.72474 - 4.71940i) q^{83} +(-2.44949 - 4.24264i) q^{85} +3.10102 q^{89} -5.65153 q^{91} +(2.00000 + 3.46410i) q^{95} +(-2.94949 + 5.10867i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 2 * q^7	\( 4 q - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{13} - 16 q^{19} + 6 q^{23} + 8 q^{25} + 10 q^{29} + 10 q^{31} - 4 q^{35} + 16 q^{37} - 14 q^{41} + 10 q^{43} - 2 q^{47} - 16 q^{53} - 4 q^{55} - 14 q^{59} + 6 q^{61} - 2 q^{65} + 10 q^{67} - 8 q^{71} + 10 q^{77} + 22 q^{79} + 6 q^{83} + 32 q^{89} - 52 q^{91} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 - 2 * q^13 - 16 * q^19 + 6 * q^23 + 8 * q^25 + 10 * q^29 + 10 * q^31 - 4 * q^35 + 16 * q^37 - 14 * q^41 + 10 * q^43 - 2 * q^47 - 16 * q^53 - 4 * q^55 - 14 * q^59 + 6 * q^61 - 2 * q^65 + 10 * q^67 - 8 * q^71 + 10 * q^77 + 22 * q^79 + 6 * q^83 + 32 * q^89 - 52 * q^91 + 8 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more