LMFDB - Embedded newform 2592.2.r.p.2161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(433,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("2592.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.r (of order $6$, 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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2161.4
Root			$-1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	2592.2161
Dual form			2592.2.r.p.433.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q+(0.866025 - 0.500000i) q^{5} +(0.500000 - 0.866025i) q^{7} +(2.59808 + 1.50000i) q^{11} +(4.58258 - 2.64575i) q^{13} +5.29150 q^{17} +5.29150i q^{19} +(2.64575 + 4.58258i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-5.19615 - 3.00000i) q^{29} +(-3.50000 - 6.06218i) q^{31} -1.00000i q^{35} +5.29150i q^{37} +(2.64575 + 4.58258i) q^{41} +(-9.16515 - 5.29150i) q^{43} +(3.00000 + 5.19615i) q^{49} -9.00000i q^{53} +3.00000 q^{55} +(3.46410 - 2.00000i) q^{59} +(2.64575 - 4.58258i) q^{65} +15.8745 q^{71} +3.00000 q^{73} +(2.59808 - 1.50000i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-6.06218 - 3.50000i) q^{83} +(4.58258 - 2.64575i) q^{85} +10.5830 q^{89} -5.29150i q^{91} +(2.64575 + 4.58258i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} - 16 q^{25} - 28 q^{31} + 24 q^{49} + 24 q^{55} + 24 q^{73} + 16 q^{79} - 28 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 - 16 * q^25 - 28 * q^31 + 24 * q^49 + 24 * q^55 + 24 * q^73 + 16 * q^79 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i	−0.293691	−	0.955901i	$-0.594884\pi$
$5$	0.866025	−	0.500000i	0.387298	−	0.223607i	0.680989	+	0.732294i	$0.261550\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.59808	+	1.50000i	0.783349	+	0.452267i	0.837616	−	0.546259i	$-0.183949\pi$
$11$	2.59808	+	1.50000i	0.783349	+	0.452267i	−0.0542666	+	0.998526i	$0.517282\pi$
$12$		0			0
$13$	4.58258	−	2.64575i	1.27098	−	0.733799i	0.295806	−	0.955248i	$-0.404412\pi$
$13$	4.58258	−	2.64575i	1.27098	−	0.733799i	0.975172	+	0.221449i	$0.0710785\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.29150			1.28338			0.641689	−	0.766965i	$-0.278234\pi$
$17$	5.29150			1.28338			0.641689	+	0.766965i	$0.278234\pi$
$18$		0			0
$19$			5.29150i			1.21395i	0.794719	+	0.606977i	$0.207618\pi$
$19$			5.29150i			1.21395i	−0.794719	+	0.606977i	$0.792382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.64575	+	4.58258i	0.551677	+	0.955533i	0.998154	+	0.0607377i	$0.0193453\pi$
$23$	2.64575	+	4.58258i	0.551677	+	0.955533i	−0.446476	+	0.894795i	$0.647321\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.19615	−	3.00000i	−0.964901	−	0.557086i	−0.0672232	−	0.997738i	$-0.521414\pi$
$29$	−5.19615	−	3.00000i	−0.964901	−	0.557086i	−0.897678	+	0.440652i	$0.854747\pi$
$30$		0			0
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.00000i		−	0.169031i
$36$		0			0
$37$			5.29150i			0.869918i	0.900450	+	0.434959i	$0.143237\pi$
$37$			5.29150i			0.869918i	−0.900450	+	0.434959i	$0.856763\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.64575	+	4.58258i	0.413197	+	0.715678i	0.995237	−	0.0974818i	$-0.0310788\pi$
$41$	2.64575	+	4.58258i	0.413197	+	0.715678i	−0.582040	+	0.813160i	$0.697745\pi$
$42$		0			0
$43$	−9.16515	−	5.29150i	−1.39767	−	0.806947i	−0.403524	−	0.914969i	$-0.632215\pi$
$43$	−9.16515	−	5.29150i	−1.39767	−	0.806947i	−0.994148	+	0.108022i	$0.965548\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.46410	−	2.00000i	0.450988	−	0.260378i	−0.257260	−	0.966342i	$-0.582820\pi$
$59$	3.46410	−	2.00000i	0.450988	−	0.260378i	0.708247	+	0.705965i	$0.249486\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.64575	−	4.58258i	0.328165	−	0.568399i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.8745			1.88396			0.941979	−	0.335673i	$-0.108964\pi$
$71$	15.8745			1.88396			0.941979	+	0.335673i	$0.108964\pi$
$72$		0			0
$73$	3.00000			0.351123			0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000			0.351123			0.175562	+	0.984468i	$0.443826\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.59808	−	1.50000i	0.296078	−	0.170941i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.06218	−	3.50000i	−0.665410	−	0.384175i	0.128925	−	0.991654i	$-0.458847\pi$
$83$	−6.06218	−	3.50000i	−0.665410	−	0.384175i	−0.794335	+	0.607479i	$0.792181\pi$
$84$		0			0
$85$	4.58258	−	2.64575i	0.497050	−	0.286972i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.5830			1.12180			0.560898	−	0.827885i	$-0.310456\pi$
$89$	10.5830			1.12180			0.560898	+	0.827885i	$0.310456\pi$
$90$		0			0
$91$		−	5.29150i		−	0.554700i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.64575	+	4.58258i	0.271448	+	0.470162i
$96$		0			0
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	−0.987181	−	0.159602i	$-0.948979\pi$
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	0.631810	+	0.775123i	$0.282312\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.7224	+	8.50000i	1.46494	+	0.845782i	0.999233	−	0.0391591i	$-0.0124679\pi$
$101$	14.7224	+	8.50000i	1.46494	+	0.845782i	0.465704	+	0.884941i	$0.345801\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.00000i		−	0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$		−	3.00000i		−	0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$		0			0
$109$		−	5.29150i		−	0.506834i	−0.967357	−	0.253417i	$-0.918446\pi$
$109$		−	5.29150i		−	0.506834i	0.967357	−	0.253417i	$-0.0815545\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.93725	+	13.7477i	0.746674	+	1.29328i	0.949409	+	0.314044i	$0.101684\pi$
$113$	7.93725	+	13.7477i	0.746674	+	1.29328i	−0.202735	+	0.979234i	$0.564983\pi$
$114$		0			0
$115$	4.58258	+	2.64575i	0.427327	+	0.246718i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.64575	−	4.58258i	0.242536	−	0.420084i
$120$		0			0
$121$	−1.00000	−	1.73205i	−0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.06218	−	3.50000i	0.529655	−	0.305796i	−0.211221	−	0.977438i	$-0.567744\pi$
$131$	6.06218	−	3.50000i	0.529655	−	0.305796i	0.740876	+	0.671642i	$0.234411\pi$
$132$		0			0
$133$	4.58258	+	2.64575i	0.397360	+	0.229416i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−9.16515	+	5.29150i	−0.777378	+	0.448819i	−0.835500	−	0.549490i	$-0.814822\pi$
$139$	−9.16515	+	5.29150i	−0.777378	+	0.448819i	0.0581222	+	0.998309i	$0.481489\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.8745			1.32749
$144$		0			0
$145$	−6.00000			−0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.59808	+	1.50000i	−0.212843	+	0.122885i	−0.602632	−	0.798019i	$-0.705881\pi$
$149$	−2.59808	+	1.50000i	−0.212843	+	0.122885i	0.389789	+	0.920904i	$0.372548\pi$
$150$		0			0
$151$	1.50000	−	2.59808i	0.122068	−	0.211428i	−0.798515	−	0.601975i	$-0.794381\pi$
$151$	1.50000	−	2.59808i	0.122068	−	0.211428i	0.920583	+	0.390547i	$0.127714\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.06218	−	3.50000i	−0.486926	−	0.281127i
$156$		0			0
$157$	−9.16515	+	5.29150i	−0.731459	+	0.422308i	−0.818956	−	0.573857i	$-0.805447\pi$
$157$	−9.16515	+	5.29150i	−0.731459	+	0.422308i	0.0874969	+	0.996165i	$0.472113\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.29150			0.417029
$162$		0			0
$163$		−	5.29150i		−	0.414462i	−0.978292	−	0.207231i	$-0.933555\pi$
$163$		−	5.29150i		−	0.414462i	0.978292	−	0.207231i	$-0.0664452\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.29150	+	9.16515i	0.409469	+	0.709221i	0.994830	−	0.101552i	$-0.0323807\pi$
$167$	5.29150	+	9.16515i	0.409469	+	0.709221i	−0.585361	+	0.810772i	$0.699047\pi$
$168$		0			0
$169$	7.50000	−	12.9904i	0.576923	−	0.999260i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.9904	−	7.50000i	−0.987640	−	0.570214i	−0.0830722	−	0.996544i	$-0.526473\pi$
$173$	−12.9904	−	7.50000i	−0.987640	−	0.570214i	−0.904568	+	0.426329i	$0.859807\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	23.0000i		−	1.71910i	−0.511051	−	0.859550i	$-0.670744\pi$
$179$		−	23.0000i		−	1.71910i	0.511051	−	0.859550i	$-0.329256\pi$
$180$		0			0
$181$		−	21.1660i		−	1.57326i	−0.617426	−	0.786629i	$-0.711825\pi$
$181$		−	21.1660i		−	1.57326i	0.617426	−	0.786629i	$-0.288175\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.64575	+	4.58258i	0.194520	+	0.336918i
$186$		0			0
$187$	13.7477	+	7.93725i	1.00533	+	0.580429i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.64575	−	4.58258i	0.191440	−	0.331584i	−0.754288	−	0.656544i	$-0.772018\pi$
$191$	2.64575	−	4.58258i	0.191440	−	0.331584i	0.945728	+	0.324960i	$0.105351\pi$
$192$		0			0
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	0.914949	−	0.403570i	$-0.132231\pi$
$193$	1.50000	+	2.59808i	0.107972	+	0.187014i	−0.806976	+	0.590584i	$0.798898\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.0000i		−	0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$		−	13.0000i		−	0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$		0			0
$199$	17.0000			1.20510			0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000			1.20510			0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.19615	+	3.00000i	−0.364698	+	0.210559i
$204$		0			0
$205$	4.58258	+	2.64575i	0.320061	+	0.184787i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.93725	+	13.7477i	−0.549031	+	0.950950i
$210$		0			0
$211$	−4.58258	+	2.64575i	−0.315478	+	0.182141i	−0.649375	−	0.760468i	$-0.724969\pi$
$211$	−4.58258	+	2.64575i	−0.315478	+	0.182141i	0.333897	+	0.942609i	$0.391636\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.5830			−0.721755
$216$		0			0
$217$	−7.00000			−0.475191
$218$		0			0
$219$		0			0
$220$		0			0
$221$	24.2487	−	14.0000i	1.63114	−	0.941742i
$222$		0			0
$223$	−8.00000	+	13.8564i	−0.535720	+	0.927894i	0.463409	+	0.886145i	$0.346626\pi$
$223$	−8.00000	+	13.8564i	−0.535720	+	0.927894i	−0.999128	+	0.0417488i	$0.986707\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	0.610535	−	0.791989i	$-0.290954\pi$
$227$	3.46410	+	2.00000i	0.229920	+	0.132745i	−0.380615	+	0.924734i	$0.624288\pi$
$228$		0			0
$229$	−18.3303	+	10.5830i	−1.21130	+	0.699345i	−0.963043	−	0.269349i	$-0.913191\pi$
$229$	−18.3303	+	10.5830i	−1.21130	+	0.699345i	−0.248258	+	0.968694i	$0.579858\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.5830			0.693316			0.346658	−	0.937992i	$-0.387316\pi$
$233$	10.5830			0.693316			0.346658	+	0.937992i	$0.387316\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.93725	+	13.7477i	0.513418	+	0.889267i	0.999879	+	0.0155640i	$0.00495438\pi$
$239$	7.93725	+	13.7477i	0.513418	+	0.889267i	−0.486461	+	0.873703i	$0.661712\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.19615	+	3.00000i	0.331970	+	0.191663i
$246$		0			0
$247$	14.0000	+	24.2487i	0.890799	+	1.54291i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$		0			0
$253$			15.8745i			0.998022i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.93725	−	13.7477i	−0.495112	−	0.857560i	0.504872	−	0.863194i	$-0.331540\pi$
$257$	−7.93725	−	13.7477i	−0.495112	−	0.857560i	−0.999984	+	0.00563467i	$0.998206\pi$
$258$		0			0
$259$	4.58258	+	2.64575i	0.284747	+	0.164399i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.64575	+	4.58258i	−0.163144	+	0.282574i	−0.935995	−	0.352014i	$-0.885497\pi$
$263$	−2.64575	+	4.58258i	−0.163144	+	0.282574i	0.772851	+	0.634588i	$0.218830\pi$
$264$		0			0
$265$	−4.50000	−	7.79423i	−0.276433	−	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$	−15.0000			−0.911185			−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000			−0.911185			−0.455593	+	0.890188i	$0.650573\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.3923	+	6.00000i	−0.626680	+	0.361814i
$276$		0			0
$277$	−9.16515	−	5.29150i	−0.550681	−	0.317936i	0.198716	−	0.980057i	$-0.436323\pi$
$277$	−9.16515	−	5.29150i	−0.550681	−	0.317936i	−0.749396	+	0.662122i	$0.769656\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.93725	−	13.7477i	0.473497	−	0.820121i	−0.526043	−	0.850458i	$-0.676325\pi$
$281$	7.93725	−	13.7477i	0.473497	−	0.820121i	0.999540	+	0.0303374i	$0.00965819\pi$
$282$		0			0
$283$	22.9129	−	13.2288i	1.36203	−	0.786368i	0.372135	−	0.928178i	$-0.378626\pi$
$283$	22.9129	−	13.2288i	1.36203	−	0.786368i	0.989894	+	0.141810i	$0.0452924\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.29150			0.312348
$288$		0			0
$289$	11.0000			0.647059
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−25.9808	+	15.0000i	−1.51781	+	0.876309i	−0.518032	+	0.855361i	$0.673335\pi$
$293$	−25.9808	+	15.0000i	−1.51781	+	0.876309i	−0.999781	+	0.0209480i	$0.993332\pi$
$294$		0			0
$295$	2.00000	−	3.46410i	0.116445	−	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.2487	+	14.0000i	1.40234	+	0.809641i
$300$		0			0
$301$	−9.16515	+	5.29150i	−0.528271	+	0.304997i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	15.8745i		−	0.906006i	−0.891509	−	0.453003i	$-0.850353\pi$
$307$		−	15.8745i		−	0.906006i	0.891509	−	0.453003i	$-0.149647\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.2288	−	22.9129i	−0.750134	−	1.29927i	−0.947757	−	0.318992i	$-0.896656\pi$
$311$	−13.2288	−	22.9129i	−0.750134	−	1.29927i	0.197623	−	0.980278i	$-0.436678\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.2583	−	6.50000i	−0.632331	−	0.365076i	0.149323	−	0.988788i	$-0.452290\pi$
$317$	−11.2583	−	6.50000i	−0.632331	−	0.365076i	−0.781654	+	0.623712i	$0.785624\pi$
$318$		0			0
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			28.0000i			1.55796i
$324$		0			0
$325$			21.1660i			1.17408i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.3303	+	10.5830i	1.00752	+	0.581695i	0.910466	−	0.413584i	$-0.135723\pi$
$331$	18.3303	+	10.5830i	1.00752	+	0.581695i	0.0970586	+	0.995279i	$0.469057\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−17.0000	−	29.4449i	−0.926049	−	1.60396i	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−17.0000	−	29.4449i	−0.926049	−	1.60396i	−0.136184	−	0.990684i	$-0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	21.0000i		−	1.13721i
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.9186	−	11.5000i	1.06929	−	0.617352i	0.141299	−	0.989967i	$-0.454872\pi$
$347$	19.9186	−	11.5000i	1.06929	−	0.617352i	0.927986	+	0.372615i	$0.121539\pi$
$348$		0			0
$349$	−4.58258	−	2.64575i	−0.245300	−	0.141624i	0.372310	−	0.928108i	$-0.378566\pi$
$349$	−4.58258	−	2.64575i	−0.245300	−	0.141624i	−0.617610	+	0.786484i	$0.711899\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.93725	−	13.7477i	0.422457	−	0.731718i	−0.573722	−	0.819050i	$-0.694501\pi$
$353$	7.93725	−	13.7477i	0.422457	−	0.731718i	0.996179	+	0.0873325i	$0.0278343\pi$
$354$		0			0
$355$	13.7477	−	7.93725i	0.729654	−	0.421266i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.1660			−1.11710			−0.558550	−	0.829471i	$-0.688642\pi$
$359$	−21.1660			−1.11710			−0.558550	+	0.829471i	$0.688642\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.59808	−	1.50000i	0.135990	−	0.0785136i
$366$		0			0
$367$	−1.50000	+	2.59808i	−0.0782994	+	0.135618i	−0.902516	−	0.430656i	$-0.858282\pi$
$367$	−1.50000	+	2.59808i	−0.0782994	+	0.135618i	0.824217	+	0.566274i	$0.191616\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.79423	−	4.50000i	−0.404656	−	0.233628i
$372$		0			0
$373$	22.9129	−	13.2288i	1.18638	−	0.684959i	0.228901	−	0.973450i	$-0.426487\pi$
$373$	22.9129	−	13.2288i	1.18638	−	0.684959i	0.957483	+	0.288491i	$0.0931534\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−31.7490			−1.63516
$378$		0			0
$379$			21.1660i			1.08722i	0.839336	+	0.543612i	$0.182944\pi$
$379$			21.1660i			1.08722i	−0.839336	+	0.543612i	$0.817056\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.93725	−	13.7477i	−0.405575	−	0.702476i	0.588813	−	0.808269i	$-0.299595\pi$
$383$	−7.93725	−	13.7477i	−0.405575	−	0.702476i	−0.994388	+	0.105793i	$0.966262\pi$
$384$		0			0
$385$	1.50000	−	2.59808i	0.0764471	−	0.132410i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.33013	−	2.50000i	−0.219546	−	0.126755i	0.386194	−	0.922418i	$-0.373790\pi$
$389$	−4.33013	−	2.50000i	−0.219546	−	0.126755i	−0.605740	+	0.795663i	$0.707123\pi$
$390$		0			0
$391$	14.0000	+	24.2487i	0.708010	+	1.22631i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	4.00000i		−	0.201262i
$396$		0			0
$397$		−	5.29150i		−	0.265573i	−0.991145	−	0.132786i	$-0.957608\pi$
$397$		−	5.29150i		−	0.265573i	0.991145	−	0.132786i	$-0.0423924\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−32.0780	−	18.5203i	−1.59792	−	0.922560i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.93725	+	13.7477i	−0.393435	+	0.681450i
$408$		0			0
$409$	5.50000	+	9.52628i	0.271957	+	0.471044i	0.969363	−	0.245633i	$-0.0789957\pi$
$409$	5.50000	+	9.52628i	0.271957	+	0.471044i	−0.697406	+	0.716677i	$0.745662\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	−7.00000			−0.343616
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−9.16515	−	5.29150i	−0.446682	−	0.257892i	0.259746	−	0.965677i	$-0.416361\pi$
$421$	−9.16515	−	5.29150i	−0.446682	−	0.257892i	−0.706428	+	0.707785i	$0.749695\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.5830	+	18.3303i	−0.513351	+	0.889150i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−31.7490			−1.52930			−0.764648	−	0.644448i	$-0.777087\pi$
$431$	−31.7490			−1.52930			−0.764648	+	0.644448i	$0.777087\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.2487	+	14.0000i	−1.15997	+	0.669711i
$438$		0			0
$439$	−10.5000	+	18.1865i	−0.501138	+	0.867996i	0.498861	+	0.866682i	$0.333752\pi$
$439$	−10.5000	+	18.1865i	−0.501138	+	0.867996i	−0.999999	+	0.00131415i	$0.999582\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.7846	−	12.0000i	−0.987507	−	0.570137i	−0.0829786	−	0.996551i	$-0.526443\pi$
$443$	−20.7846	−	12.0000i	−0.987507	−	0.570137i	−0.904528	+	0.426414i	$0.859777\pi$
$444$		0			0
$445$	9.16515	−	5.29150i	0.434470	−	0.250841i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.8745			−0.749164			−0.374582	−	0.927194i	$-0.622214\pi$
$449$	−15.8745			−0.749164			−0.374582	+	0.927194i	$0.622214\pi$
$450$		0			0
$451$			15.8745i			0.747501i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.64575	−	4.58258i	−0.124035	−	0.214834i
$456$		0			0
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$	−8.50000	+	14.7224i	−0.397613	+	0.688686i	0.595818	+	0.803120i	$0.296828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.79423	+	4.50000i	0.363013	+	0.209586i	0.670402	−	0.741998i	$-0.266122\pi$
$461$	7.79423	+	4.50000i	0.363013	+	0.209586i	−0.307388	+	0.951584i	$0.599455\pi$
$462$		0			0
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.213205	−	0.977007i	$-0.568390\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.00000i		−	0.0462745i	−0.999732	−	0.0231372i	$-0.992635\pi$
$467$		−	1.00000i		−	0.0462745i	0.999732	−	0.0231372i	$-0.00736547\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.8745	−	27.4955i	−0.729911	−	1.26424i
$474$		0			0
$475$	−18.3303	−	10.5830i	−0.841052	−	0.485582i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.5830	+	18.3303i	−0.483550	+	0.837533i	−0.999822	−	0.0188920i	$-0.993986\pi$
$479$	−10.5830	+	18.3303i	−0.483550	+	0.837533i	0.516272	+	0.856425i	$0.327319\pi$
$480$		0			0
$481$	14.0000	+	24.2487i	0.638345	+	1.10565i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			7.00000i			0.317854i
$486$		0			0
$487$	40.0000			1.81257			0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000			1.81257			0.906287	+	0.422664i	$0.138905\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−25.1147	+	14.5000i	−1.13341	+	0.654376i	−0.944791	−	0.327674i	$-0.893735\pi$
$491$	−25.1147	+	14.5000i	−1.13341	+	0.654376i	−0.188621	+	0.982050i	$0.560402\pi$
$492$		0			0
$493$	−27.4955	−	15.8745i	−1.23833	−	0.714952i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.93725	−	13.7477i	0.356034	−	0.616670i
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.8745			0.707809			0.353905	−	0.935282i	$-0.384854\pi$
$503$	15.8745			0.707809			0.353905	+	0.935282i	$0.384854\pi$
$504$		0			0
$505$	17.0000			0.756490
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.1865	−	10.5000i	0.806104	−	0.465404i	−0.0394971	−	0.999220i	$-0.512576\pi$
$509$	18.1865	−	10.5000i	0.806104	−	0.465404i	0.845601	+	0.533815i	$0.179242\pi$
$510$		0			0
$511$	1.50000	−	2.59808i	0.0663561	−	0.114932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.8745			−0.695475			−0.347737	−	0.937592i	$-0.613050\pi$
$521$	−15.8745			−0.695475			−0.347737	+	0.937592i	$0.613050\pi$
$522$		0			0
$523$			15.8745i			0.694144i	0.937839	+	0.347072i	$0.112824\pi$
$523$			15.8745i			0.694144i	−0.937839	+	0.347072i	$0.887176\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.5203	−	32.0780i	−0.806755	−	1.39734i
$528$		0			0
$529$	−2.50000	+	4.33013i	−0.108696	+	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.2487	+	14.0000i	1.05033	+	0.606407i
$534$		0			0
$535$	−1.50000	−	2.59808i	−0.0648507	−	0.112325i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			18.0000i			0.775315i
$540$		0			0
$541$		−	10.5830i		−	0.454999i	−0.973778	−	0.227499i	$-0.926945\pi$
$541$		−	10.5830i		−	0.454999i	0.973778	−	0.227499i	$-0.0730550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.64575	−	4.58258i	−0.113332	−	0.196296i
$546$		0			0
$547$	−4.58258	−	2.64575i	−0.195937	−	0.113124i	0.398822	−	0.917028i	$-0.369419\pi$
$547$	−4.58258	−	2.64575i	−0.195937	−	0.113124i	−0.594759	+	0.803904i	$0.702752\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.8745	−	27.4955i	0.676277	−	1.17135i
$552$		0			0
$553$	−2.00000	−	3.46410i	−0.0850487	−	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			39.0000i			1.65248i	0.563316	+	0.826242i	$0.309525\pi$
$557$			39.0000i			1.65248i	−0.563316	+	0.826242i	$0.690475\pi$
$558$		0			0
$559$	−56.0000			−2.36855
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.59808	+	1.50000i	−0.109496	+	0.0632175i	−0.553748	−	0.832684i	$-0.686803\pi$
$563$	−2.59808	+	1.50000i	−0.109496	+	0.0632175i	0.444252	+	0.895902i	$0.353470\pi$
$564$		0			0
$565$	13.7477	+	7.93725i	0.578371	+	0.333923i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.8745	+	27.4955i	−0.665494	+	1.15267i	0.313657	+	0.949536i	$0.398446\pi$
$569$	−15.8745	+	27.4955i	−0.665494	+	1.15267i	−0.979151	+	0.203133i	$0.934888\pi$
$570$		0			0
$571$	−22.9129	+	13.2288i	−0.958874	+	0.553606i	−0.895826	−	0.444404i	$-0.853415\pi$
$571$	−22.9129	+	13.2288i	−0.958874	+	0.553606i	−0.0630478	+	0.998011i	$0.520082\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.1660			−0.882684
$576$		0			0
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.06218	+	3.50000i	−0.251502	+	0.145204i
$582$		0			0
$583$	13.5000	−	23.3827i	0.559113	−	0.968412i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.9186	+	11.5000i	0.822128	+	0.474656i	0.851150	−	0.524923i	$-0.175906\pi$
$587$	19.9186	+	11.5000i	0.822128	+	0.474656i	−0.0290218	+	0.999579i	$0.509239\pi$
$588$		0			0
$589$	32.0780	−	18.5203i	1.32175	−	0.763114i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.7490			−1.30378			−0.651888	−	0.758315i	$-0.726023\pi$
$593$	−31.7490			−1.30378			−0.651888	+	0.758315i	$0.726023\pi$
$594$		0			0
$595$		−	5.29150i		−	0.216930i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.5830	+	18.3303i	0.432410	+	0.748956i	0.997080	−	0.0763606i	$-0.0243300\pi$
$599$	10.5830	+	18.3303i	0.432410	+	0.748956i	−0.564670	+	0.825317i	$0.690997\pi$
$600$		0			0
$601$	−2.50000	+	4.33013i	−0.101977	+	0.176630i	−0.912499	−	0.409079i	$-0.865850\pi$
$601$	−2.50000	+	4.33013i	−0.101977	+	0.176630i	0.810522	+	0.585708i	$0.199184\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.73205	−	1.00000i	−0.0704179	−	0.0406558i
$606$		0			0
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	0.773358	−	0.633970i	$-0.218576\pi$
$607$	−4.00000	−	6.92820i	−0.162355	−	0.281207i	−0.935713	+	0.352763i	$0.885242\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			10.5830i			0.427444i	0.976895	+	0.213722i	$0.0685586\pi$
$613$			10.5830i			0.427444i	−0.976895	+	0.213722i	$0.931441\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.64575	−	4.58258i	−0.106514	−	0.184488i	0.807842	−	0.589399i	$-0.200636\pi$
$617$	−2.64575	−	4.58258i	−0.106514	−	0.184488i	−0.914356	+	0.404912i	$0.867302\pi$
$618$		0			0
$619$	−13.7477	−	7.93725i	−0.552568	−	0.319025i	0.197589	−	0.980285i	$-0.436689\pi$
$619$	−13.7477	−	7.93725i	−0.552568	−	0.319025i	−0.750157	+	0.661260i	$0.770022\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5.29150	−	9.16515i	0.212000	−	0.367194i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			28.0000i			1.11643i
$630$		0			0
$631$	23.0000			0.915616			0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000			0.915616			0.457808	+	0.889051i	$0.348635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.2583	+	6.50000i	−0.446773	+	0.257945i
$636$		0			0
$637$	27.4955	+	15.8745i	1.08941	+	0.628971i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.5203	+	32.0780i	−0.731506	+	1.26701i	0.224733	+	0.974420i	$0.427849\pi$
$641$	−18.5203	+	32.0780i	−0.731506	+	1.26701i	−0.956239	+	0.292586i	$0.905484\pi$
$642$		0			0
$643$	−18.3303	+	10.5830i	−0.722877	+	0.417353i	−0.815811	−	0.578319i	$-0.803709\pi$
$643$	−18.3303	+	10.5830i	−0.722877	+	0.417353i	0.0929339	+	0.995672i	$0.470375\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−31.7490			−1.24818			−0.624091	−	0.781351i	$-0.714531\pi$
$647$	−31.7490			−1.24818			−0.624091	+	0.781351i	$0.714531\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.59808	−	1.50000i	0.101671	−	0.0586995i	−0.448303	−	0.893882i	$-0.647971\pi$
$653$	2.59808	−	1.50000i	0.101671	−	0.0586995i	0.549973	+	0.835182i	$0.314638\pi$
$654$		0			0
$655$	3.50000	−	6.06218i	0.136756	−	0.236869i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.9904	+	7.50000i	0.506033	+	0.292159i	0.731202	−	0.682161i	$-0.238960\pi$
$659$	12.9904	+	7.50000i	0.506033	+	0.292159i	−0.225168	+	0.974320i	$0.572293\pi$
$660$		0			0
$661$	−22.9129	+	13.2288i	−0.891208	+	0.514539i	−0.874337	−	0.485319i	$-0.838704\pi$
$661$	−22.9129	+	13.2288i	−0.891208	+	0.514539i	−0.0168704	+	0.999858i	$0.505370\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.29150			0.205196
$666$		0			0
$667$		−	31.7490i		−	1.22933i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.5000	+	25.1147i	−0.558934	+	0.968102i	0.438652	+	0.898657i	$0.355456\pi$
$673$	−14.5000	+	25.1147i	−0.558934	+	0.968102i	−0.997586	+	0.0694449i	$0.977877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.19615	−	3.00000i	−0.199704	−	0.115299i	0.396813	−	0.917899i	$-0.370116\pi$
$677$	−5.19615	−	3.00000i	−0.199704	−	0.115299i	−0.596518	+	0.802600i	$0.703449\pi$
$678$		0			0
$679$	3.50000	+	6.06218i	0.134318	+	0.232645i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.0000i		−	1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$		−	44.0000i		−	1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.8118	−	41.2432i	−0.907156	−	1.57124i
$690$		0			0
$691$	36.6606	+	21.1660i	1.39464	+	0.805193i	0.993824	−	0.110968i	$-0.0353950\pi$
$691$	36.6606	+	21.1660i	1.39464	+	0.805193i	0.400811	+	0.916161i	$0.368728\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.29150	+	9.16515i	−0.200718	+	0.347654i
$696$		0			0
$697$	14.0000	+	24.2487i	0.530288	+	0.918485i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.00000i			0.0377695i	0.999822	+	0.0188847i	$0.00601156\pi$
$701$			1.00000i			0.0377695i	−0.999822	+	0.0188847i	$0.993988\pi$
$702$		0			0
$703$	−28.0000			−1.05604
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.7224	−	8.50000i	0.553694	−	0.319675i
$708$		0			0
$709$	18.3303	+	10.5830i	0.688409	+	0.397453i	0.803016	−	0.595958i	$-0.203227\pi$
$709$	18.3303	+	10.5830i	0.688409	+	0.397453i	−0.114607	+	0.993411i	$0.536561\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.5203	−	32.0780i	0.693589	−	1.20133i
$714$		0			0
$715$	13.7477	−	7.93725i	0.514136	−	0.296836i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.1660			−0.789359			−0.394679	−	0.918819i	$-0.629144\pi$
$719$	−21.1660			−0.789359			−0.394679	+	0.918819i	$0.629144\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.7846	−	12.0000i	0.771921	−	0.445669i
$726$		0			0
$727$	16.5000	−	28.5788i	0.611951	−	1.05993i	−0.378960	−	0.925413i	$-0.623718\pi$
$727$	16.5000	−	28.5788i	0.611951	−	1.05993i	0.990911	−	0.134517i	$-0.0429484\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−48.4974	−	28.0000i	−1.79374	−	1.03562i
$732$		0			0
$733$	4.58258	−	2.64575i	0.169261	−	0.0977231i	−0.412976	−	0.910742i	$-0.635511\pi$
$733$	4.58258	−	2.64575i	0.169261	−	0.0977231i	0.582237	+	0.813019i	$0.302177\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	5.29150i		−	0.194651i	−0.995253	−	0.0973255i	$-0.968971\pi$
$739$		−	5.29150i		−	0.194651i	0.995253	−	0.0973255i	$-0.0310288\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.5203	−	32.0780i	−0.679442	−	1.17683i	−0.975149	−	0.221550i	$-0.928888\pi$
$743$	−18.5203	−	32.0780i	−0.679442	−	1.17683i	0.295707	−	0.955279i	$-0.404445\pi$
$744$		0			0
$745$	−1.50000	+	2.59808i	−0.0549557	+	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.59808	−	1.50000i	−0.0949316	−	0.0548088i
$750$		0			0
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	−0.996993	−	0.0774878i	$-0.975310\pi$
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	0.431390	−	0.902165i	$-0.358023\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	3.00000i		−	0.109181i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.29150	+	9.16515i	0.191817	+	0.332236i	0.945852	−	0.324597i	$-0.105229\pi$
$761$	5.29150	+	9.16515i	0.191817	+	0.332236i	−0.754036	+	0.656834i	$0.771895\pi$
$762$		0			0
$763$	−4.58258	−	2.64575i	−0.165900	−	0.0957826i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.5830	−	18.3303i	0.382130	−	0.661869i
$768$		0			0
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	0.987334	+	0.158655i	$0.0507157\pi$
$769$	17.5000	+	30.3109i	0.631066	+	1.09304i	−0.356268	+	0.934384i	$0.615951\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	14.0000i		−	0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$		−	14.0000i		−	0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$		0			0
$775$	28.0000			1.00579
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.2487	+	14.0000i	−0.868800	+	0.501602i
$780$		0			0
$781$	41.2432	+	23.8118i	1.47580	+	0.852052i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.29150	+	9.16515i	−0.188862	+	0.327118i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.8745			0.564433
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−42.4352	+	24.5000i	−1.50313	+	0.867835i	−0.503140	+	0.864205i	$0.667822\pi$
$797$	−42.4352	+	24.5000i	−1.50313	+	0.867835i	−0.999993	+	0.00362965i	$0.998845\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.79423	+	4.50000i	0.275052	+	0.158802i
$804$		0			0
$805$	4.58258	−	2.64575i	0.161515	−	0.0932505i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−31.7490			−1.11624			−0.558118	−	0.829762i	$-0.688476\pi$
$809$	−31.7490			−1.11624			−0.558118	+	0.829762i	$0.688476\pi$
$810$		0			0
$811$		−	37.0405i		−	1.30067i	−0.759648	−	0.650334i	$-0.774629\pi$
$811$		−	37.0405i		−	1.30067i	0.759648	−	0.650334i	$-0.225371\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.64575	−	4.58258i	−0.0926766	−	0.160521i
$816$		0			0
$817$	28.0000	−	48.4974i	0.979596	−	1.69671i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.66025	−	5.00000i	−0.302245	−	0.174501i	0.341206	−	0.939989i	$-0.389165\pi$
$821$	−8.66025	−	5.00000i	−0.302245	−	0.174501i	−0.643451	+	0.765487i	$0.722498\pi$
$822$		0			0
$823$	11.5000	+	19.9186i	0.400865	+	0.694318i	0.993831	−	0.110910i	$-0.0353764\pi$
$823$	11.5000	+	19.9186i	0.400865	+	0.694318i	−0.592966	+	0.805228i	$0.702043\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$		−	21.1660i		−	0.735126i	−0.929999	−	0.367563i	$-0.880192\pi$
$829$		−	21.1660i		−	0.735126i	0.929999	−	0.367563i	$-0.119808\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.8745	+	27.4955i	0.550019	+	0.952661i
$834$		0			0
$835$	9.16515	+	5.29150i	0.317173	+	0.183120i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−18.5203	+	32.0780i	−0.639390	+	1.10746i	0.346176	+	0.938169i	$0.387480\pi$
$839$	−18.5203	+	32.0780i	−0.639390	+	1.10746i	−0.985567	+	0.169287i	$0.945853\pi$
$840$		0			0
$841$	3.50000	+	6.06218i	0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	15.0000i		−	0.516016i
$846$		0			0
$847$	−2.00000			−0.0687208
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.2487	+	14.0000i	−0.831235	+	0.479914i
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.93725	−	13.7477i	0.271131	−	0.469613i	−0.698020	−	0.716078i	$-0.745936\pi$
$857$	7.93725	−	13.7477i	0.271131	−	0.469613i	0.969152	+	0.246464i	$0.0792689\pi$
$858$		0			0
$859$	−45.8258	+	26.4575i	−1.56355	+	0.902719i	−0.566662	+	0.823951i	$0.691765\pi$
$859$	−45.8258	+	26.4575i	−1.56355	+	0.902719i	−0.996893	+	0.0787681i	$0.974901\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−15.0000			−0.510015
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.3923	−	6.00000i	0.352535	−	0.203536i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.79423	+	4.50000i	0.263493	+	0.152128i
$876$		0			0
$877$	−45.8258	+	26.4575i	−1.54743	+	0.893407i	−0.549089	+	0.835764i	$0.685025\pi$
$877$	−45.8258	+	26.4575i	−1.54743	+	0.893407i	−0.998337	+	0.0576426i	$0.981642\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.4575			0.891376			0.445688	−	0.895188i	$-0.352959\pi$
$881$	26.4575			0.891376			0.445688	+	0.895188i	$0.352959\pi$
$882$		0			0
$883$			58.2065i			1.95881i	0.201916	+	0.979403i	$0.435283\pi$
$883$			58.2065i			1.95881i	−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.93725	+	13.7477i	0.266507	+	0.461603i	0.967957	−	0.251115i	$-0.0807972\pi$
$887$	7.93725	+	13.7477i	0.266507	+	0.461603i	−0.701450	+	0.712718i	$0.747464\pi$
$888$		0			0
$889$	−6.50000	+	11.2583i	−0.218003	+	0.377592i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−11.5000	−	19.9186i	−0.384403	−	0.665805i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			42.0000i			1.40078i
$900$		0			0
$901$		−	47.6235i		−	1.58657i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.5830	−	18.3303i	−0.351791	−	0.609320i
$906$		0			0
$907$	−4.58258	−	2.64575i	−0.152162	−	0.0878507i	0.421986	−	0.906602i	$-0.361333\pi$
$907$	−4.58258	−	2.64575i	−0.152162	−	0.0878507i	−0.574148	+	0.818752i	$0.694667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.8745	+	27.4955i	−0.525946	+	0.910965i	0.473597	+	0.880742i	$0.342955\pi$
$911$	−15.8745	+	27.4955i	−0.525946	+	0.910965i	−0.999543	+	0.0302235i	$0.990378\pi$
$912$		0			0
$913$	−10.5000	−	18.1865i	−0.347499	−	0.601886i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	7.00000i		−	0.231160i
$918$		0			0
$919$	−3.00000			−0.0989609			−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000			−0.0989609			−0.0494804	+	0.998775i	$0.515757\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	72.7461	−	42.0000i	2.39447	−	1.38245i
$924$		0			0
$925$	−18.3303	−	10.5830i	−0.602697	−	0.347967i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.29150	−	9.16515i	0.173609	−	0.300699i	−0.766070	−	0.642757i	$-0.777791\pi$
$929$	5.29150	−	9.16515i	0.173609	−	0.300699i	0.939679	+	0.342058i	$0.111124\pi$
$930$		0			0
$931$	−27.4955	+	15.8745i	−0.901127	+	0.520266i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15.8745			0.519152
$936$		0			0
$937$	−19.0000			−0.620703			−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000			−0.620703			−0.310351	+	0.950622i	$0.600447\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18.1865	+	10.5000i	−0.592864	+	0.342290i	−0.766229	−	0.642567i	$-0.777869\pi$
$941$	−18.1865	+	10.5000i	−0.592864	+	0.342290i	0.173365	+	0.984858i	$0.444536\pi$
$942$		0			0
$943$	−14.0000	+	24.2487i	−0.455903	+	0.789647i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.7750	+	19.5000i	1.09754	+	0.633665i	0.935574	−	0.353131i	$-0.114883\pi$
$947$	33.7750	+	19.5000i	1.09754	+	0.633665i	0.161966	+	0.986796i	$0.448217\pi$
$948$		0			0
$949$	13.7477	−	7.93725i	0.446270	−	0.257654i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	31.7490			1.02845			0.514226	−	0.857655i	$-0.328079\pi$
$953$	31.7490			1.02845			0.514226	+	0.857655i	$0.328079\pi$
$954$		0			0
$955$		−	5.29150i		−	0.171229i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.59808	+	1.50000i	0.0836350	+	0.0482867i
$966$		0			0
$967$	20.5000	+	35.5070i	0.659236	+	1.14183i	0.980814	+	0.194946i	$0.0624533\pi$
$967$	20.5000	+	35.5070i	0.659236	+	1.14183i	−0.321578	+	0.946883i	$0.604213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	15.0000i		−	0.481373i	−0.970603	−	0.240686i	$-0.922627\pi$
$971$		−	15.0000i		−	0.481373i	0.970603	−	0.240686i	$-0.0773725\pi$
$972$		0			0
$973$			10.5830i			0.339276i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.29150	+	9.16515i	0.169290	+	0.293219i	0.938170	−	0.346174i	$-0.112519\pi$
$977$	5.29150	+	9.16515i	0.169290	+	0.293219i	−0.768880	+	0.639393i	$0.779186\pi$
$978$		0			0
$979$	27.4955	+	15.8745i	0.878759	+	0.507351i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.1660	−	36.6606i	0.675091	−	1.16929i	−0.301351	−	0.953513i	$-0.597438\pi$
$983$	21.1660	−	36.6606i	0.675091	−	1.16929i	0.976442	−	0.215779i	$-0.0692289\pi$
$984$		0			0
$985$	−6.50000	−	11.2583i	−0.207107	−	0.358720i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	56.0000i		−	1.78070i
$990$		0			0
$991$	−47.0000			−1.49300			−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000			−1.49300			−0.746502	+	0.665383i	$0.768268\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.7224	−	8.50000i	0.466732	−	0.269468i
$996$		0			0
$997$	−13.7477	−	7.93725i	−0.435395	−	0.251375i	0.266247	−	0.963905i	$-0.414216\pi$
$997$	−13.7477	−	7.93725i	−0.435395	−	0.251375i	−0.701642	+	0.712529i	$0.747550\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.r.p.2161.4		8
3.2	odd	2	inner	2592.2.r.p.2161.2		8
4.3	odd	2		648.2.n.n.541.4		8
8.3	odd	2		648.2.n.n.541.3		8
8.5	even	2	inner	2592.2.r.p.2161.1		8
9.2	odd	6		864.2.d.a.433.2		4
9.4	even	3	inner	2592.2.r.p.433.1		8
9.5	odd	6	inner	2592.2.r.p.433.3		8
9.7	even	3		864.2.d.a.433.4		4
12.11	even	2		648.2.n.n.541.1		8
24.5	odd	2	inner	2592.2.r.p.2161.3		8
24.11	even	2		648.2.n.n.541.2		8
36.7	odd	6		216.2.d.b.109.1	✓	4
36.11	even	6		216.2.d.b.109.4	yes	4
36.23	even	6		648.2.n.n.109.2		8
36.31	odd	6		648.2.n.n.109.3		8
72.5	odd	6	inner	2592.2.r.p.433.2		8
72.11	even	6		216.2.d.b.109.3	yes	4
72.13	even	6	inner	2592.2.r.p.433.4		8
72.29	odd	6		864.2.d.a.433.3		4
72.43	odd	6		216.2.d.b.109.2	yes	4
72.59	even	6		648.2.n.n.109.1		8
72.61	even	6		864.2.d.a.433.1		4
72.67	odd	6		648.2.n.n.109.4		8
144.11	even	12		6912.2.a.bu.1.2		2
144.29	odd	12		6912.2.a.bd.1.1		2
144.43	odd	12		6912.2.a.bc.1.2		2
144.61	even	12		6912.2.a.bv.1.1		2
144.83	even	12		6912.2.a.bc.1.1		2
144.101	odd	12		6912.2.a.bv.1.2		2
144.115	odd	12		6912.2.a.bu.1.1		2
144.133	even	12		6912.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.b.109.1	✓	4	36.7	odd	6
216.2.d.b.109.2	yes	4	72.43	odd	6
216.2.d.b.109.3	yes	4	72.11	even	6
216.2.d.b.109.4	yes	4	36.11	even	6
648.2.n.n.109.1		8	72.59	even	6
648.2.n.n.109.2		8	36.23	even	6
648.2.n.n.109.3		8	36.31	odd	6
648.2.n.n.109.4		8	72.67	odd	6
648.2.n.n.541.1		8	12.11	even	2
648.2.n.n.541.2		8	24.11	even	2
648.2.n.n.541.3		8	8.3	odd	2
648.2.n.n.541.4		8	4.3	odd	2
864.2.d.a.433.1		4	72.61	even	6
864.2.d.a.433.2		4	9.2	odd	6
864.2.d.a.433.3		4	72.29	odd	6
864.2.d.a.433.4		4	9.7	even	3
2592.2.r.p.433.1		8	9.4	even	3	inner
2592.2.r.p.433.2		8	72.5	odd	6	inner
2592.2.r.p.433.3		8	9.5	odd	6	inner
2592.2.r.p.433.4		8	72.13	even	6	inner
2592.2.r.p.2161.1		8	8.5	even	2	inner
2592.2.r.p.2161.2		8	3.2	odd	2	inner
2592.2.r.p.2161.3		8	24.5	odd	2	inner
2592.2.r.p.2161.4		8	1.1	even	1	trivial
6912.2.a.bc.1.1		2	144.83	even	12
6912.2.a.bc.1.2		2	144.43	odd	12
6912.2.a.bd.1.1		2	144.29	odd	12
6912.2.a.bd.1.2		2	144.133	even	12
6912.2.a.bu.1.1		2	144.115	odd	12
6912.2.a.bu.1.2		2	144.11	even	12
6912.2.a.bv.1.1		2	144.61	even	12
6912.2.a.bv.1.2		2	144.101	odd	12

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{5} +(0.500000 - 0.866025i) q^{7} +(2.59808 + 1.50000i) q^{11} +(4.58258 - 2.64575i) q^{13} +5.29150 q^{17} +5.29150i q^{19} +(2.64575 + 4.58258i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-5.19615 - 3.00000i) q^{29} +(-3.50000 - 6.06218i) q^{31} -1.00000i q^{35} +5.29150i q^{37} +(2.64575 + 4.58258i) q^{41} +(-9.16515 - 5.29150i) q^{43} +(3.00000 + 5.19615i) q^{49} -9.00000i q^{53} +3.00000 q^{55} +(3.46410 - 2.00000i) q^{59} +(2.64575 - 4.58258i) q^{65} +15.8745 q^{71} +3.00000 q^{73} +(2.59808 - 1.50000i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-6.06218 - 3.50000i) q^{83} +(4.58258 - 2.64575i) q^{85} +10.5830 q^{89} -5.29150i q^{91} +(2.64575 + 4.58258i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} - 16 q^{25} - 28 q^{31} + 24 q^{49} + 24 q^{55} + 24 q^{73} + 16 q^{79} - 28 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 - 16 * q^25 - 28 * q^31 + 24 * q^49 + 24 * q^55 + 24 * q^73 + 16 * q^79 - 28 * q^97

Introduction

L-functions

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