LMFDB - Embedded newform 2592.2.i.bd.1729.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

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, 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("2592.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.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:	$20.6972242039$
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:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1729.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2592.1729
Dual form			2592.2.i.bd.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(2.44949 + 4.24264i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(2.44949 + 4.24264i) q^{7} +(-2.44949 - 4.24264i) q^{11} +(-1.50000 + 2.59808i) q^{13} -5.00000 q^{17} +4.89898 q^{19} +(-2.44949 + 4.24264i) q^{23} +(2.00000 + 3.46410i) q^{25} +(2.50000 + 4.33013i) q^{29} +4.89898 q^{35} -5.00000 q^{37} +(-1.00000 + 1.73205i) q^{41} +(-2.44949 - 4.24264i) q^{43} +(4.89898 + 8.48528i) q^{47} +(-8.50000 + 14.7224i) q^{49} -2.00000 q^{53} -4.89898 q^{55} +(4.89898 - 8.48528i) q^{59} +(6.50000 + 11.2583i) q^{61} +(1.50000 + 2.59808i) q^{65} +(-2.44949 + 4.24264i) q^{67} -4.89898 q^{71} +3.00000 q^{73} +(12.0000 - 20.7846i) q^{77} +(-7.34847 - 12.7279i) q^{79} +(4.89898 + 8.48528i) q^{83} +(-2.50000 + 4.33013i) q^{85} -13.0000 q^{89} -14.6969 q^{91} +(2.44949 - 4.24264i) q^{95} +(3.00000 + 5.19615i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5}+O(q^{10}) $ 4 * q + 2 * q^5	$ 4 q + 2 q^{5} - 6 q^{13} - 20 q^{17} + 8 q^{25} + 10 q^{29} - 20 q^{37} - 4 q^{41} - 34 q^{49} - 8 q^{53} + 26 q^{61} + 6 q^{65} + 12 q^{73} + 48 q^{77} - 10 q^{85} - 52 q^{89} + 12 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - 6 * q^13 - 20 * q^17 + 8 * q^25 + 10 * q^29 - 20 * q^37 - 4 * q^41 - 34 * q^49 - 8 * q^53 + 26 * q^61 + 6 * q^65 + 12 * q^73 + 48 * q^77 - 10 * q^85 - 52 * q^89 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$	2.44949	+	4.24264i	0.925820	+	1.60357i	0.790237	+	0.612801i	$0.209957\pi$
$7$	2.44949	+	4.24264i	0.925820	+	1.60357i	0.135583	+	0.990766i	$0.456709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.44949	−	4.24264i	−0.738549	−	1.27920i	−0.953149	−	0.302502i	$-0.902178\pi$
$11$	−2.44949	−	4.24264i	−0.738549	−	1.27920i	0.214600	−	0.976702i	$-0.431155\pi$
$12$		0			0
$13$	−1.50000	+	2.59808i	−0.416025	+	0.720577i	−0.995535	−	0.0943882i	$-0.969911\pi$
$13$	−1.50000	+	2.59808i	−0.416025	+	0.720577i	0.579510	+	0.814965i	$0.303244\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$		0			0
$19$	4.89898			1.12390			0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898			1.12390			0.561951	+	0.827170i	$0.310051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	0.489168	+	0.872189i	$0.337300\pi$
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	−0.999922	+	0.0124624i	$0.996033\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	0.999167	−	0.0408130i	$-0.0129948\pi$
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	−0.534928	+	0.844897i	$0.679661\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.89898			0.828079
$36$		0			0
$37$	−5.00000			−0.821995			−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000			−0.821995			−0.410997	+	0.911636i	$0.634819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	−0.933486	−	0.358614i	$-0.883249\pi$
$41$	−1.00000	+	1.73205i	−0.156174	+	0.270501i	0.777312	+	0.629115i	$0.216583\pi$
$42$		0			0
$43$	−2.44949	−	4.24264i	−0.373544	−	0.646997i	0.616564	−	0.787305i	$-0.288524\pi$
$43$	−2.44949	−	4.24264i	−0.373544	−	0.646997i	−0.990108	+	0.140308i	$0.955191\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	0.963118	+	0.269081i	$0.0867199\pi$
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	−0.248528	+	0.968625i	$0.579947\pi$
$48$		0			0
$49$	−8.50000	+	14.7224i	−1.21429	+	2.10320i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.89898	−	8.48528i	0.637793	−	1.10469i	−0.348123	−	0.937449i	$-0.613181\pi$
$59$	4.89898	−	8.48528i	0.637793	−	1.10469i	0.985916	−	0.167241i	$-0.0534857\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.50000	+	2.59808i	0.186052	+	0.322252i
$66$		0			0
$67$	−2.44949	+	4.24264i	−0.299253	+	0.518321i	−0.975965	−	0.217926i	$-0.930071\pi$
$67$	−2.44949	+	4.24264i	−0.299253	+	0.518321i	0.676712	+	0.736247i	$0.263404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.89898			−0.581402			−0.290701	−	0.956814i	$-0.593888\pi$
$71$	−4.89898			−0.581402			−0.290701	+	0.956814i	$0.593888\pi$
$72$		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
$77$	12.0000	−	20.7846i	1.36753	−	2.36863i
$78$		0			0
$79$	−7.34847	−	12.7279i	−0.826767	−	1.43200i	−0.900561	−	0.434730i	$-0.856844\pi$
$79$	−7.34847	−	12.7279i	−0.826767	−	1.43200i	0.0737937	−	0.997274i	$-0.476489\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.89898	+	8.48528i	0.537733	+	0.931381i	0.999026	+	0.0441327i	$0.0140524\pi$
$83$	4.89898	+	8.48528i	0.537733	+	0.931381i	−0.461293	+	0.887248i	$0.652614\pi$
$84$		0			0
$85$	−2.50000	+	4.33013i	−0.271163	+	0.469668i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.0000			−1.37800			−0.688999	−	0.724763i	$-0.741949\pi$
$89$	−13.0000			−1.37800			−0.688999	+	0.724763i	$0.741949\pi$
$90$		0			0
$91$	−14.6969			−1.54066
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.44949	−	4.24264i	0.251312	−	0.435286i
$96$		0			0
$97$	3.00000	+	5.19615i	0.304604	+	0.527589i	0.977173	−	0.212445i	$-0.0681426\pi$
$97$	3.00000	+	5.19615i	0.304604	+	0.527589i	−0.672569	+	0.740034i	$0.734809\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	0.999996	−	0.00286291i	$-0.000911295\pi$
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	−0.502477	+	0.864590i	$0.667578\pi$
$102$		0			0
$103$	4.89898	−	8.48528i	0.482711	−	0.836080i	−0.517092	−	0.855930i	$-0.672986\pi$
$103$	4.89898	−	8.48528i	0.482711	−	0.836080i	0.999803	+	0.0198501i	$0.00631890\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.5959			−1.89441			−0.947204	−	0.320630i	$-0.896105\pi$
$107$	−19.5959			−1.89441			−0.947204	+	0.320630i	$0.896105\pi$
$108$		0			0
$109$	−9.00000			−0.862044			−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000			−0.862044			−0.431022	+	0.902342i	$0.641847\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.500000	−	0.866025i	0.0470360	−	0.0814688i	−0.841549	−	0.540181i	$-0.818356\pi$
$113$	0.500000	−	0.866025i	0.0470360	−	0.0814688i	0.888585	+	0.458712i	$0.151689\pi$
$114$		0			0
$115$	2.44949	+	4.24264i	0.228416	+	0.395628i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.2474	−	21.2132i	−1.12272	−	1.94461i
$120$		0			0
$121$	−6.50000	+	11.2583i	−0.590909	+	1.02348i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	4.89898			0.434714			0.217357	−	0.976092i	$-0.430256\pi$
$127$	4.89898			0.434714			0.217357	+	0.976092i	$0.430256\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.34847	+	12.7279i	−0.642039	+	1.11204i	0.342938	+	0.939358i	$0.388578\pi$
$131$	−7.34847	+	12.7279i	−0.642039	+	1.11204i	−0.984977	+	0.172686i	$0.944755\pi$
$132$		0			0
$133$	12.0000	+	20.7846i	1.04053	+	1.80225i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.50000	+	14.7224i	0.726204	+	1.25782i	0.958477	+	0.285171i	$0.0920506\pi$
$137$	8.50000	+	14.7224i	0.726204	+	1.25782i	−0.232273	+	0.972651i	$0.574616\pi$
$138$		0			0
$139$	4.89898	−	8.48528i	0.415526	−	0.719712i	−0.579957	−	0.814647i	$-0.696931\pi$
$139$	4.89898	−	8.48528i	0.415526	−	0.719712i	0.995484	+	0.0949346i	$0.0302642\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.6969			1.22902
$144$		0			0
$145$	5.00000			0.415227
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.50000	−	4.33013i	0.204808	−	0.354738i	−0.745264	−	0.666770i	$-0.767676\pi$
$149$	2.50000	−	4.33013i	0.204808	−	0.354738i	0.950072	+	0.312032i	$0.101010\pi$
$150$		0			0
$151$	4.89898	+	8.48528i	0.398673	+	0.690522i	0.993562	−	0.113285i	$-0.0361374\pi$
$151$	4.89898	+	8.48528i	0.398673	+	0.690522i	−0.594889	+	0.803808i	$0.702804\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	−0.971219	−	0.238190i	$-0.923446\pi$
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	0.691888	+	0.722005i	$0.256779\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	9.79796			0.767435			0.383718	−	0.923450i	$-0.374644\pi$
$163$	9.79796			0.767435			0.383718	+	0.923450i	$0.374644\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.34847	−	12.7279i	0.568642	−	0.984916i	−0.428059	−	0.903751i	$-0.640802\pi$
$167$	7.34847	−	12.7279i	0.568642	−	0.984916i	0.996701	−	0.0811654i	$-0.0258642\pi$
$168$		0			0
$169$	2.00000	+	3.46410i	0.153846	+	0.266469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	0.884407	−	0.466717i	$-0.154563\pi$
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	−0.846392	+	0.532560i	$0.821230\pi$
$174$		0			0
$175$	−9.79796	+	16.9706i	−0.740656	+	1.28285i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	19.5959			1.46467			0.732334	−	0.680946i	$-0.238431\pi$
$179$	19.5959			1.46467			0.732334	+	0.680946i	$0.238431\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.50000	+	4.33013i	−0.183804	+	0.318357i
$186$		0			0
$187$	12.2474	+	21.2132i	0.895622	+	1.55126i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.44949	+	4.24264i	0.177239	+	0.306987i	0.940934	−	0.338591i	$-0.109950\pi$
$191$	2.44949	+	4.24264i	0.177239	+	0.306987i	−0.763695	+	0.645577i	$0.776617\pi$
$192$		0			0
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	0.289980	+	0.957033i	$0.406351\pi$
$193$	−9.50000	+	16.4545i	−0.683825	+	1.18442i	−0.973805	+	0.227387i	$0.926982\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	23.0000			1.63868			0.819341	−	0.573306i	$-0.194340\pi$
$197$	23.0000			1.63868			0.819341	+	0.573306i	$0.194340\pi$
$198$		0			0
$199$	19.5959			1.38912			0.694559	−	0.719436i	$-0.255600\pi$
$199$	19.5959			1.38912			0.694559	+	0.719436i	$0.255600\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.2474	+	21.2132i	−0.859602	+	1.48888i
$204$		0			0
$205$	1.00000	+	1.73205i	0.0698430	+	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.0000	−	20.7846i	−0.830057	−	1.43770i
$210$		0			0
$211$	−2.44949	+	4.24264i	−0.168630	+	0.292075i	−0.937938	−	0.346802i	$-0.887268\pi$
$211$	−2.44949	+	4.24264i	−0.168630	+	0.292075i	0.769309	+	0.638877i	$0.220601\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.89898			−0.334108
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.50000	−	12.9904i	0.504505	−	0.873828i
$222$		0			0
$223$	2.44949	+	4.24264i	0.164030	+	0.284108i	0.936310	−	0.351174i	$-0.114217\pi$
$223$	2.44949	+	4.24264i	0.164030	+	0.284108i	−0.772280	+	0.635282i	$0.780884\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.44949	+	4.24264i	0.162578	+	0.281594i	0.935793	−	0.352551i	$-0.114686\pi$
$227$	2.44949	+	4.24264i	0.162578	+	0.281594i	−0.773214	+	0.634145i	$0.781352\pi$
$228$		0			0
$229$	−1.50000	+	2.59808i	−0.0991228	+	0.171686i	−0.911322	−	0.411695i	$-0.864937\pi$
$229$	−1.50000	+	2.59808i	−0.0991228	+	0.171686i	0.812199	+	0.583380i	$0.198270\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−29.0000			−1.89985			−0.949927	−	0.312473i	$-0.898843\pi$
$233$	−29.0000			−1.89985			−0.949927	+	0.312473i	$0.898843\pi$
$234$		0			0
$235$	9.79796			0.639148
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.89898	+	8.48528i	−0.316889	+	0.548867i	−0.979837	−	0.199798i	$-0.935971\pi$
$239$	−4.89898	+	8.48528i	−0.316889	+	0.548867i	0.662949	+	0.748665i	$0.269305\pi$
$240$		0			0
$241$	−13.5000	−	23.3827i	−0.869611	−	1.50621i	−0.862394	−	0.506237i	$-0.831036\pi$
$241$	−13.5000	−	23.3827i	−0.869611	−	1.50621i	−0.00721719	−	0.999974i	$-0.502297\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.50000	+	14.7224i	0.543045	+	0.940582i
$246$		0			0
$247$	−7.34847	+	12.7279i	−0.467572	+	0.809858i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.6969			0.927663			0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969			0.927663			0.463831	+	0.885924i	$0.346474\pi$
$252$		0			0
$253$	24.0000			1.50887
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.50000	+	6.06218i	−0.218324	+	0.378148i	−0.954296	−	0.298864i	$-0.903392\pi$
$257$	−3.50000	+	6.06218i	−0.218324	+	0.378148i	0.735972	+	0.677012i	$0.236726\pi$
$258$		0			0
$259$	−12.2474	−	21.2132i	−0.761019	−	1.31812i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.89898	+	8.48528i	0.302084	+	0.523225i	0.976608	−	0.215028i	$-0.0689843\pi$
$263$	4.89898	+	8.48528i	0.302084	+	0.523225i	−0.674524	+	0.738253i	$0.735651\pi$
$264$		0			0
$265$	−1.00000	+	1.73205i	−0.0614295	+	0.106399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.00000			−0.0609711			−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000			−0.0609711			−0.0304855	+	0.999535i	$0.509705\pi$
$270$		0			0
$271$	−4.89898			−0.297592			−0.148796	−	0.988868i	$-0.547540\pi$
$271$	−4.89898			−0.297592			−0.148796	+	0.988868i	$0.547540\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.79796	−	16.9706i	0.590839	−	1.02336i
$276$		0			0
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	0.998861	+	0.0477206i	$0.0151957\pi$
$277$	9.00000	+	15.5885i	0.540758	+	0.936620i	−0.458103	+	0.888899i	$0.651471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.50000	+	11.2583i	0.387757	+	0.671616i	0.992148	−	0.125073i	$-0.0399165\pi$
$281$	6.50000	+	11.2583i	0.387757	+	0.671616i	−0.604390	+	0.796689i	$0.706583\pi$
$282$		0			0
$283$	9.79796	−	16.9706i	0.582428	−	1.00880i	−0.412762	−	0.910839i	$-0.635436\pi$
$283$	9.79796	−	16.9706i	0.582428	−	1.00880i	0.995191	−	0.0979565i	$-0.0312306\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.79796			−0.578355
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.50000	−	4.33013i	0.146052	−	0.252969i	−0.783713	−	0.621123i	$-0.786677\pi$
$293$	2.50000	−	4.33013i	0.146052	−	0.252969i	0.929765	+	0.368154i	$0.120010\pi$
$294$		0			0
$295$	−4.89898	−	8.48528i	−0.285230	−	0.494032i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.34847	−	12.7279i	−0.424973	−	0.736075i
$300$		0			0
$301$	12.0000	−	20.7846i	0.691669	−	1.19800i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.0000			0.744378
$306$		0			0
$307$	−19.5959			−1.11840			−0.559199	−	0.829033i	$-0.688891\pi$
$307$	−19.5959			−1.11840			−0.559199	+	0.829033i	$0.688891\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	−0.984092	−	0.177660i	$-0.943147\pi$
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	0.338188	−	0.941079i	$-0.390186\pi$
$318$		0			0
$319$	12.2474	−	21.2132i	0.685725	−	1.18771i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.4949			−1.36293
$324$		0			0
$325$	−12.0000			−0.665640
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.0000	+	41.5692i	−1.32316	+	2.29179i
$330$		0			0
$331$	−12.2474	−	21.2132i	−0.673181	−	1.16598i	−0.976997	−	0.213253i	$-0.931594\pi$
$331$	−12.2474	−	21.2132i	−0.673181	−	1.16598i	0.303816	−	0.952731i	$-0.401739\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.44949	+	4.24264i	0.133830	+	0.231800i
$336$		0			0
$337$	3.00000	−	5.19615i	0.163420	−	0.283052i	−0.772673	−	0.634804i	$-0.781081\pi$
$337$	3.00000	−	5.19615i	0.163420	−	0.283052i	0.936093	+	0.351752i	$0.114414\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−48.9898			−2.64520
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.1464	−	29.6985i	0.920468	−	1.59430i	0.121777	−	0.992557i	$-0.461141\pi$
$347$	17.1464	−	29.6985i	0.920468	−	1.59430i	0.798692	−	0.601741i	$-0.205526\pi$
$348$		0			0
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.00000	−	1.73205i	−0.0532246	−	0.0921878i	0.838186	−	0.545385i	$-0.183617\pi$
$353$	−1.00000	−	1.73205i	−0.0532246	−	0.0921878i	−0.891410	+	0.453197i	$0.850283\pi$
$354$		0			0
$355$	−2.44949	+	4.24264i	−0.130005	+	0.225176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.6969			−0.775675			−0.387837	−	0.921728i	$-0.626778\pi$
$359$	−14.6969			−0.775675			−0.387837	+	0.921728i	$0.626778\pi$
$360$		0			0
$361$	5.00000			0.263158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.50000	−	2.59808i	0.0785136	−	0.135990i
$366$		0			0
$367$	9.79796	+	16.9706i	0.511449	+	0.885856i	0.999912	+	0.0132714i	$0.00422453\pi$
$367$	9.79796	+	16.9706i	0.511449	+	0.885856i	−0.488463	+	0.872585i	$0.662442\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.89898	−	8.48528i	−0.254342	−	0.440534i
$372$		0			0
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.0291379	−	0.999575i	$-0.490724\pi$
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.851089	−	0.525022i	$-0.175943\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.0000			−0.772539
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.34847	+	12.7279i	−0.375489	+	0.650366i	−0.990400	−	0.138230i	$-0.955859\pi$
$383$	−7.34847	+	12.7279i	−0.375489	+	0.650366i	0.614911	+	0.788597i	$0.289192\pi$
$384$		0			0
$385$	−12.0000	−	20.7846i	−0.611577	−	1.05928i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.00000	+	1.73205i	0.0507020	+	0.0878185i	0.890263	−	0.455448i	$-0.150521\pi$
$389$	1.00000	+	1.73205i	0.0507020	+	0.0878185i	−0.839561	+	0.543266i	$0.817187\pi$
$390$		0			0
$391$	12.2474	−	21.2132i	0.619380	−	1.07280i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−14.6969			−0.739483
$396$		0			0
$397$	−13.0000			−0.652451			−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000			−0.652451			−0.326226	+	0.945292i	$0.605777\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.5000	−	25.1147i	0.724095	−	1.25417i	−0.235250	−	0.971935i	$-0.575591\pi$
$401$	14.5000	−	25.1147i	0.724095	−	1.25417i	0.959345	−	0.282235i	$-0.0910758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.2474	+	21.2132i	0.607083	+	1.05150i
$408$		0			0
$409$	4.50000	−	7.79423i	0.222511	−	0.385400i	−0.733059	−	0.680165i	$-0.761908\pi$
$409$	4.50000	−	7.79423i	0.222511	−	0.385400i	0.955570	+	0.294765i	$0.0952414\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	48.0000			2.36193
$414$		0			0
$415$	9.79796			0.480963
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.79796	−	16.9706i	0.478662	−	0.829066i	−0.521039	−	0.853533i	$-0.674455\pi$
$419$	9.79796	−	16.9706i	0.478662	−	0.829066i	0.999701	+	0.0244666i	$0.00778874\pi$
$420$		0			0
$421$	−7.50000	−	12.9904i	−0.365528	−	0.633112i	0.623333	−	0.781956i	$-0.285778\pi$
$421$	−7.50000	−	12.9904i	−0.365528	−	0.633112i	−0.988861	+	0.148844i	$0.952445\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	−	17.3205i	−0.485071	−	0.840168i
$426$		0			0
$427$	−31.8434	+	55.1543i	−1.54101	+	2.66911i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.79796			0.471951			0.235976	−	0.971759i	$-0.424171\pi$
$431$	9.79796			0.471951			0.235976	+	0.971759i	$0.424171\pi$
$432$		0			0
$433$	19.0000			0.913082			0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000			0.913082			0.456541	+	0.889702i	$0.349088\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	20.7846i	−0.574038	+	0.994263i
$438$		0			0
$439$	−14.6969	−	25.4558i	−0.701447	−	1.21494i	−0.967959	−	0.251110i	$-0.919205\pi$
$439$	−14.6969	−	25.4558i	−0.701447	−	1.21494i	0.266512	−	0.963832i	$-0.414129\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.89898	+	8.48528i	0.232758	+	0.403148i	0.958619	−	0.284693i	$-0.0918918\pi$
$443$	4.89898	+	8.48528i	0.232758	+	0.403148i	−0.725861	+	0.687841i	$0.758558\pi$
$444$		0			0
$445$	−6.50000	+	11.2583i	−0.308130	+	0.533696i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	9.79796			0.461368
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.34847	+	12.7279i	−0.344502	+	0.596694i
$456$		0			0
$457$	−7.50000	−	12.9904i	−0.350835	−	0.607664i	0.635561	−	0.772051i	$-0.280769\pi$
$457$	−7.50000	−	12.9904i	−0.350835	−	0.607664i	−0.986396	+	0.164386i	$0.947436\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.0000	−	19.0526i	−0.512321	−	0.887366i	−0.999898	−	0.0142861i	$-0.995452\pi$
$461$	−11.0000	−	19.0526i	−0.512321	−	0.887366i	0.487577	−	0.873080i	$-0.337881\pi$
$462$		0			0
$463$	4.89898	−	8.48528i	0.227675	−	0.394344i	−0.729444	−	0.684041i	$-0.760221\pi$
$463$	4.89898	−	8.48528i	0.227675	−	0.394344i	0.957119	+	0.289696i	$0.0935543\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.79796			−0.453395			−0.226698	−	0.973965i	$-0.572793\pi$
$467$	−9.79796			−0.453395			−0.226698	+	0.973965i	$0.572793\pi$
$468$		0			0
$469$	−24.0000			−1.10822
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.0000	+	20.7846i	−0.551761	+	0.955677i
$474$		0			0
$475$	9.79796	+	16.9706i	0.449561	+	0.778663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.2474	−	21.2132i	−0.559600	−	0.969256i	−0.997530	−	0.0702467i	$-0.977621\pi$
$479$	−12.2474	−	21.2132i	−0.559600	−	0.969256i	0.437929	−	0.899009i	$-0.355712\pi$
$480$		0			0
$481$	7.50000	−	12.9904i	0.341971	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000			0.272446
$486$		0			0
$487$	19.5959			0.887976			0.443988	−	0.896033i	$-0.353563\pi$
$487$	19.5959			0.887976			0.443988	+	0.896033i	$0.353563\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.1464	+	29.6985i	−0.773807	+	1.34027i	0.161655	+	0.986847i	$0.448317\pi$
$491$	−17.1464	+	29.6985i	−0.773807	+	1.34027i	−0.935462	+	0.353426i	$0.885016\pi$
$492$		0			0
$493$	−12.5000	−	21.6506i	−0.562972	−	0.975096i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	12.2474	−	21.2132i	0.548271	−	0.949633i	−0.450122	−	0.892967i	$-0.648620\pi$
$499$	12.2474	−	21.2132i	0.548271	−	0.949633i	0.998393	−	0.0566664i	$-0.0180471\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	39.1918			1.74748			0.873739	−	0.486395i	$-0.161689\pi$
$503$	39.1918			1.74748			0.873739	+	0.486395i	$0.161689\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.0000	−	22.5167i	0.576215	−	0.998033i	−0.419694	−	0.907666i	$-0.637862\pi$
$509$	13.0000	−	22.5167i	0.576215	−	0.998033i	0.995908	−	0.0903676i	$-0.0288042\pi$
$510$		0			0
$511$	7.34847	+	12.7279i	0.325077	+	0.563050i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.89898	−	8.48528i	−0.215875	−	0.373906i
$516$		0			0
$517$	24.0000	−	41.5692i	1.05552	−	1.82821i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.0000			1.48957			0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000			1.48957			0.744784	+	0.667306i	$0.232553\pi$
$522$		0			0
$523$	44.0908			1.92796			0.963978	−	0.265981i	$-0.0856957\pi$
$523$	44.0908			1.92796			0.963978	+	0.265981i	$0.0856957\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.0217391	−	0.0376533i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.00000	−	5.19615i	−0.129944	−	0.225070i
$534$		0			0
$535$	−9.79796	+	16.9706i	−0.423603	+	0.733701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	83.2827			3.58724
$540$		0			0
$541$	27.0000			1.16082			0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000			1.16082			0.580410	+	0.814324i	$0.302892\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.50000	+	7.79423i	−0.192759	+	0.333868i
$546$		0			0
$547$	14.6969	+	25.4558i	0.628396	+	1.08841i	0.987874	+	0.155260i	$0.0496214\pi$
$547$	14.6969	+	25.4558i	0.628396	+	1.08841i	−0.359478	+	0.933154i	$0.617045\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.2474	+	21.2132i	0.521759	+	0.903713i
$552$		0			0
$553$	36.0000	−	62.3538i	1.53088	−	2.65155i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−17.0000			−0.720313			−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000			−0.720313			−0.360157	+	0.932892i	$0.617277\pi$
$558$		0			0
$559$	14.6969			0.621614
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.89898	+	8.48528i	−0.206467	+	0.357612i	−0.950599	−	0.310421i	$-0.899530\pi$
$563$	−4.89898	+	8.48528i	−0.206467	+	0.357612i	0.744132	+	0.668033i	$0.232863\pi$
$564$		0			0
$565$	−0.500000	−	0.866025i	−0.0210352	−	0.0364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.50000	+	14.7224i	0.356339	+	0.617196i	0.987346	−	0.158580i	$-0.0506917\pi$
$569$	8.50000	+	14.7224i	0.356339	+	0.617196i	−0.631008	+	0.775777i	$0.717358\pi$
$570$		0			0
$571$	4.89898	−	8.48528i	0.205016	−	0.355098i	−0.745122	−	0.666928i	$-0.767609\pi$
$571$	4.89898	−	8.48528i	0.205016	−	0.355098i	0.950138	+	0.311830i	$0.100942\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.5959			−0.817206
$576$		0			0
$577$	−1.00000			−0.0416305			−0.0208153	−	0.999783i	$-0.506626\pi$
$577$	−1.00000			−0.0416305			−0.0208153	+	0.999783i	$0.506626\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	+	41.5692i	−0.995688	+	1.72458i
$582$		0			0
$583$	4.89898	+	8.48528i	0.202895	+	0.351424i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.34847	+	12.7279i	0.303304	+	0.525338i	0.976882	−	0.213778i	$-0.0685770\pi$
$587$	7.34847	+	12.7279i	0.303304	+	0.525338i	−0.673578	+	0.739116i	$0.735244\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.0000			−0.698106			−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000			−0.698106			−0.349053	+	0.937103i	$0.613497\pi$
$594$		0			0
$595$	−24.4949			−1.00419
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.79796	+	16.9706i	−0.400334	+	0.693398i	−0.993766	−	0.111486i	$-0.964439\pi$
$599$	−9.79796	+	16.9706i	−0.400334	+	0.693398i	0.593432	+	0.804884i	$0.297772\pi$
$600$		0			0
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	0.912499	−	0.409079i	$-0.134150\pi$
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	−0.810522	+	0.585708i	$0.800816\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.50000	+	11.2583i	0.264263	+	0.457716i
$606$		0			0
$607$	−12.2474	+	21.2132i	−0.497109	+	0.861017i	−0.999994	−	0.00333548i	$-0.998938\pi$
$607$	−12.2474	+	21.2132i	−0.497109	+	0.861017i	0.502886	+	0.864353i	$0.332272\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−29.3939			−1.18915
$612$		0			0
$613$	14.0000			0.565455			0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000			0.565455			0.282727	+	0.959200i	$0.408761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.50000	+	9.52628i	−0.221422	+	0.383514i	−0.955240	−	0.295832i	$-0.904403\pi$
$617$	−5.50000	+	9.52628i	−0.221422	+	0.383514i	0.733818	+	0.679346i	$0.237736\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−31.8434	−	55.1543i	−1.27578	−	2.20971i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25.0000			0.996815
$630$		0			0
$631$	29.3939			1.17015			0.585076	−	0.810979i	$-0.301065\pi$
$631$	29.3939			1.17015			0.585076	+	0.810979i	$0.301065\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.44949	−	4.24264i	0.0972050	−	0.168364i
$636$		0			0
$637$	−25.5000	−	44.1673i	−1.01035	−	1.74997i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−21.5000	−	37.2391i	−0.849199	−	1.47086i	−0.881924	−	0.471391i	$-0.843752\pi$
$641$	−21.5000	−	37.2391i	−0.849199	−	1.47086i	0.0327252	−	0.999464i	$-0.489581\pi$
$642$		0			0
$643$	14.6969	−	25.4558i	0.579591	−	1.00388i	−0.415935	−	0.909394i	$-0.636546\pi$
$643$	14.6969	−	25.4558i	0.579591	−	1.00388i	0.995526	−	0.0944864i	$-0.0301209\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.89898			0.192599			0.0962994	−	0.995352i	$-0.469299\pi$
$647$	4.89898			0.192599			0.0962994	+	0.995352i	$0.469299\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.00000	−	8.66025i	0.195665	−	0.338902i	−0.751453	−	0.659786i	$-0.770647\pi$
$653$	5.00000	−	8.66025i	0.195665	−	0.338902i	0.947118	+	0.320884i	$0.103980\pi$
$654$		0			0
$655$	7.34847	+	12.7279i	0.287128	+	0.497321i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.44949	−	4.24264i	−0.0954186	−	0.165270i	0.814365	−	0.580354i	$-0.197086\pi$
$659$	−2.44949	−	4.24264i	−0.0954186	−	0.165270i	−0.909783	+	0.415084i	$0.863752\pi$
$660$		0			0
$661$	14.5000	−	25.1147i	0.563985	−	0.976850i	−0.433159	−	0.901318i	$-0.642601\pi$
$661$	14.5000	−	25.1147i	0.563985	−	0.976850i	0.997143	−	0.0755324i	$-0.0240656\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.0000			0.930680
$666$		0			0
$667$	−24.4949			−0.948446
$668$		0			0
$669$		0			0
$670$		0			0
$671$	31.8434	−	55.1543i	1.22930	−	2.12921i
$672$		0			0
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	0.740334	−	0.672239i	$-0.234667\pi$
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	−0.952343	+	0.305028i	$0.901334\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.0000	+	29.4449i	0.653363	+	1.13166i	0.982301	+	0.187307i	$0.0599758\pi$
$677$	17.0000	+	29.4449i	0.653363	+	1.13166i	−0.328938	+	0.944351i	$0.606691\pi$
$678$		0			0
$679$	−14.6969	+	25.4558i	−0.564017	+	0.976906i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−48.9898			−1.87454			−0.937271	−	0.348601i	$-0.886657\pi$
$683$	−48.9898			−1.87454			−0.937271	+	0.348601i	$0.886657\pi$
$684$		0			0
$685$	17.0000			0.649537
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.00000	−	5.19615i	0.114291	−	0.197958i
$690$		0			0
$691$	2.44949	+	4.24264i	0.0931830	+	0.161398i	0.908849	−	0.417126i	$-0.136963\pi$
$691$	2.44949	+	4.24264i	0.0931830	+	0.161398i	−0.815666	+	0.578523i	$0.803629\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.89898	−	8.48528i	−0.185829	−	0.321865i
$696$		0			0
$697$	5.00000	−	8.66025i	0.189389	−	0.328031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17.0000			−0.642081			−0.321041	−	0.947065i	$-0.604033\pi$
$701$	−17.0000			−0.642081			−0.321041	+	0.947065i	$0.604033\pi$
$702$		0			0
$703$	−24.4949			−0.923843
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.4949	+	42.4264i	−0.921225	+	1.59561i
$708$		0			0
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	−0.999967	−	0.00810550i	$-0.997420\pi$
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	0.492964	−	0.870050i	$-0.335913\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	7.34847	−	12.7279i	0.274817	−	0.475997i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−44.0908			−1.64431			−0.822155	−	0.569264i	$-0.807228\pi$
$719$	−44.0908			−1.64431			−0.822155	+	0.569264i	$0.807228\pi$
$720$		0			0
$721$	48.0000			1.78761
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10.0000	+	17.3205i	−0.371391	+	0.643268i
$726$		0			0
$727$	−2.44949	−	4.24264i	−0.0908465	−	0.157351i	0.817021	−	0.576608i	$-0.195624\pi$
$727$	−2.44949	−	4.24264i	−0.0908465	−	0.157351i	−0.907868	+	0.419257i	$0.862291\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.2474	+	21.2132i	0.452988	+	0.784599i
$732$		0			0
$733$	−3.00000	+	5.19615i	−0.110808	+	0.191924i	−0.916096	−	0.400959i	$-0.868677\pi$
$733$	−3.00000	+	5.19615i	−0.110808	+	0.191924i	0.805289	+	0.592883i	$0.202010\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$	−39.1918			−1.44169			−0.720847	−	0.693094i	$-0.756247\pi$
$739$	−39.1918			−1.44169			−0.720847	+	0.693094i	$0.756247\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.5959	−	33.9411i	0.718905	−	1.24518i	−0.242530	−	0.970144i	$-0.577977\pi$
$743$	19.5959	−	33.9411i	0.718905	−	1.24518i	0.961434	−	0.275035i	$-0.0886895\pi$
$744$		0			0
$745$	−2.50000	−	4.33013i	−0.0915929	−	0.158644i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−48.0000	−	83.1384i	−1.75388	−	3.03781i
$750$		0			0
$751$	7.34847	−	12.7279i	0.268149	−	0.464448i	−0.700235	−	0.713913i	$-0.746921\pi$
$751$	7.34847	−	12.7279i	0.268149	−	0.464448i	0.968384	+	0.249464i	$0.0802546\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.79796			0.356584
$756$		0			0
$757$	30.0000			1.09037			0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000			1.09037			0.545184	+	0.838316i	$0.316460\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.5000	+	19.9186i	−0.416875	+	0.722048i	−0.995623	−	0.0934579i	$-0.970208\pi$
$761$	−11.5000	+	19.9186i	−0.416875	+	0.722048i	0.578749	+	0.815506i	$0.303541\pi$
$762$		0			0
$763$	−22.0454	−	38.1838i	−0.798097	−	1.38235i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14.6969	+	25.4558i	0.530676	+	0.919157i
$768$		0			0
$769$	−5.50000	+	9.52628i	−0.198335	+	0.343526i	−0.947989	−	0.318304i	$-0.896887\pi$
$769$	−5.50000	+	9.52628i	−0.198335	+	0.343526i	0.749654	+	0.661830i	$0.230220\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−41.0000			−1.47467			−0.737334	−	0.675529i	$-0.763915\pi$
$773$	−41.0000			−1.47467			−0.737334	+	0.675529i	$0.763915\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.89898	+	8.48528i	−0.175524	+	0.304017i
$780$		0			0
$781$	12.0000	+	20.7846i	0.429394	+	0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.50000	+	6.06218i	0.124920	+	0.216368i
$786$		0			0
$787$	7.34847	−	12.7279i	0.261945	−	0.453701i	−0.704814	−	0.709392i	$-0.748970\pi$
$787$	7.34847	−	12.7279i	0.261945	−	0.453701i	0.966759	+	0.255691i	$0.0823029\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.89898			0.174188
$792$		0			0
$793$	−39.0000			−1.38493
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.5000	+	19.9186i	−0.407351	+	0.705552i	−0.994592	−	0.103860i	$-0.966881\pi$
$797$	−11.5000	+	19.9186i	−0.407351	+	0.705552i	0.587241	+	0.809412i	$0.300214\pi$
$798$		0			0
$799$	−24.4949	−	42.4264i	−0.866567	−	1.50094i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.34847	−	12.7279i	−0.259322	−	0.449159i
$804$		0			0
$805$	−12.0000	+	20.7846i	−0.422944	+	0.732561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.0000			0.808637			0.404318	−	0.914618i	$-0.367509\pi$
$809$	23.0000			0.808637			0.404318	+	0.914618i	$0.367509\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.89898	−	8.48528i	0.171604	−	0.297226i
$816$		0			0
$817$	−12.0000	−	20.7846i	−0.419827	−	0.727161i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.50000	−	9.52628i	−0.191951	−	0.332469i	0.753946	−	0.656937i	$-0.228148\pi$
$821$	−5.50000	−	9.52628i	−0.191951	−	0.332469i	−0.945897	+	0.324468i	$0.894815\pi$
$822$		0			0
$823$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$823$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.89898			0.170354			0.0851771	−	0.996366i	$-0.472854\pi$
$827$	4.89898			0.170354			0.0851771	+	0.996366i	$0.472854\pi$
$828$		0			0
$829$	−42.0000			−1.45872			−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000			−1.45872			−0.729360	+	0.684130i	$0.760182\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	42.5000	−	73.6122i	1.47254	−	2.55051i
$834$		0			0
$835$	−7.34847	−	12.7279i	−0.254304	−	0.440468i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.34847	+	12.7279i	0.253697	+	0.439417i	0.964541	−	0.263933i	$-0.0850199\pi$
$839$	7.34847	+	12.7279i	0.253697	+	0.439417i	−0.710844	+	0.703350i	$0.751687\pi$
$840$		0			0
$841$	2.00000	−	3.46410i	0.0689655	−	0.119452i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.00000			0.137604
$846$		0			0
$847$	−63.6867			−2.18830
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.2474	−	21.2132i	0.419837	−	0.727179i
$852$		0			0
$853$	1.00000	+	1.73205i	0.0342393	+	0.0593043i	0.882637	−	0.470055i	$-0.155766\pi$
$853$	1.00000	+	1.73205i	0.0342393	+	0.0593043i	−0.848398	+	0.529359i	$0.822432\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.5000	+	21.6506i	0.426992	+	0.739572i	0.996604	−	0.0823419i	$-0.0262400\pi$
$857$	12.5000	+	21.6506i	0.426992	+	0.739572i	−0.569612	+	0.821914i	$0.692907\pi$
$858$		0			0
$859$	14.6969	−	25.4558i	0.501453	−	0.868542i	−0.498546	−	0.866864i	$-0.666132\pi$
$859$	14.6969	−	25.4558i	0.501453	−	0.868542i	0.999999	−	0.00167867i	$-0.000534338\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.4949			0.833816			0.416908	−	0.908949i	$-0.363114\pi$
$863$	24.4949			0.833816			0.416908	+	0.908949i	$0.363114\pi$
$864$		0			0
$865$	1.00000			0.0340010
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−36.0000	+	62.3538i	−1.22122	+	2.11521i
$870$		0			0
$871$	−7.34847	−	12.7279i	−0.248993	−	0.431269i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.0454	+	38.1838i	0.745271	+	1.29085i
$876$		0			0
$877$	−17.5000	+	30.3109i	−0.590933	+	1.02353i	0.403174	+	0.915123i	$0.367907\pi$
$877$	−17.5000	+	30.3109i	−0.590933	+	1.02353i	−0.994107	+	0.108403i	$0.965426\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.0000			0.336909			0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		0			0
$883$	29.3939			0.989183			0.494591	−	0.869126i	$-0.335318\pi$
$883$	29.3939			0.989183			0.494591	+	0.869126i	$0.335318\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7.34847	−	12.7279i	0.246737	−	0.427362i	−0.715881	−	0.698222i	$-0.753975\pi$
$887$	7.34847	−	12.7279i	0.246737	−	0.427362i	0.962619	+	0.270860i	$0.0873081\pi$
$888$		0			0
$889$	12.0000	+	20.7846i	0.402467	+	0.697093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.0000	+	41.5692i	0.803129	+	1.39106i
$894$		0			0
$895$	9.79796	−	16.9706i	0.327510	−	0.567263i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	10.0000			0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.00000	+	15.5885i	−0.299170	+	0.518178i
$906$		0			0
$907$	−9.79796	−	16.9706i	−0.325336	−	0.563498i	0.656244	−	0.754548i	$-0.272144\pi$
$907$	−9.79796	−	16.9706i	−0.325336	−	0.563498i	−0.981580	+	0.191050i	$0.938811\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.44949	+	4.24264i	0.0811552	+	0.140565i	0.903746	−	0.428068i	$-0.140806\pi$
$911$	2.44949	+	4.24264i	0.0811552	+	0.140565i	−0.822591	+	0.568633i	$0.807472\pi$
$912$		0			0
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−72.0000			−2.37765
$918$		0			0
$919$	34.2929			1.13122			0.565608	−	0.824674i	$-0.308641\pi$
$919$	34.2929			1.13122			0.565608	+	0.824674i	$0.308641\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.34847	−	12.7279i	0.241878	−	0.418945i
$924$		0			0
$925$	−10.0000	−	17.3205i	−0.328798	−	0.569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.50000	+	4.33013i	0.0820223	+	0.142067i	0.904118	−	0.427282i	$-0.140529\pi$
$929$	2.50000	+	4.33013i	0.0820223	+	0.142067i	−0.822096	+	0.569349i	$0.807195\pi$
$930$		0			0
$931$	−41.6413	+	72.1249i	−1.36474	+	2.36380i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.4949			0.801069
$936$		0			0
$937$	19.0000			0.620703			0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000			0.620703			0.310351	+	0.950622i	$0.399553\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.5000	−	35.5070i	0.668281	−	1.15750i	−0.310104	−	0.950703i	$-0.600364\pi$
$941$	20.5000	−	35.5070i	0.668281	−	1.15750i	0.978385	−	0.206794i	$-0.0663029\pi$
$942$		0			0
$943$	−4.89898	−	8.48528i	−0.159533	−	0.276319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.34847	−	12.7279i	−0.238793	−	0.413602i	0.721575	−	0.692336i	$-0.243418\pi$
$947$	−7.34847	−	12.7279i	−0.238793	−	0.413602i	−0.960368	+	0.278734i	$0.910085\pi$
$948$		0			0
$949$	−4.50000	+	7.79423i	−0.146076	+	0.253011i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.00000			0.226752			0.113376	−	0.993552i	$-0.463833\pi$
$953$	7.00000			0.226752			0.113376	+	0.993552i	$0.463833\pi$
$954$		0			0
$955$	4.89898			0.158527
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−41.6413	+	72.1249i	−1.34467	+	2.32903i
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.50000	+	16.4545i	0.305816	+	0.529689i
$966$		0			0
$967$	−2.44949	+	4.24264i	−0.0787703	+	0.136434i	−0.902720	−	0.430229i	$-0.858433\pi$
$967$	−2.44949	+	4.24264i	−0.0787703	+	0.136434i	0.823949	+	0.566663i	$0.191766\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.89898			−0.157216			−0.0786079	−	0.996906i	$-0.525048\pi$
$971$	−4.89898			−0.157216			−0.0786079	+	0.996906i	$0.525048\pi$
$972$		0			0
$973$	48.0000			1.53881
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.0000	−	19.0526i	0.351921	−	0.609545i	−0.634665	−	0.772787i	$-0.718862\pi$
$977$	11.0000	−	19.0526i	0.351921	−	0.609545i	0.986586	+	0.163242i	$0.0521952\pi$
$978$		0			0
$979$	31.8434	+	55.1543i	1.01772	+	1.76274i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.79796	+	16.9706i	0.312506	+	0.541277i	0.978904	−	0.204319i	$-0.0654981\pi$
$983$	9.79796	+	16.9706i	0.312506	+	0.541277i	−0.666398	+	0.745596i	$0.732165\pi$
$984$		0			0
$985$	11.5000	−	19.9186i	0.366420	−	0.634659i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	34.2929			1.08935			0.544674	−	0.838648i	$-0.316653\pi$
$991$	34.2929			1.08935			0.544674	+	0.838648i	$0.316653\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.79796	−	16.9706i	0.310616	−	0.538003i
$996$		0			0
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	0.993213	−	0.116310i	$-0.0371066\pi$
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	−0.597334	+	0.801993i	$0.703773\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.bd.1729.2		4
3.2	odd	2		2592.2.i.z.1729.2		4
4.3	odd	2	inner	2592.2.i.bd.1729.1		4
9.2	odd	6		2592.2.i.z.865.2		4
9.4	even	3		2592.2.a.m.1.1	✓	2
9.5	odd	6		2592.2.a.r.1.1	yes	2
9.7	even	3	inner	2592.2.i.bd.865.2		4
12.11	even	2		2592.2.i.z.1729.1		4
36.7	odd	6	inner	2592.2.i.bd.865.1		4
36.11	even	6		2592.2.i.z.865.1		4
36.23	even	6		2592.2.a.r.1.2	yes	2
36.31	odd	6		2592.2.a.m.1.2	yes	2
72.5	odd	6		5184.2.a.bk.1.1		2
72.13	even	6		5184.2.a.bv.1.1		2
72.59	even	6		5184.2.a.bk.1.2		2
72.67	odd	6		5184.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.m.1.1	✓	2	9.4	even	3
2592.2.a.m.1.2	yes	2	36.31	odd	6
2592.2.a.r.1.1	yes	2	9.5	odd	6
2592.2.a.r.1.2	yes	2	36.23	even	6
2592.2.i.z.865.1		4	36.11	even	6
2592.2.i.z.865.2		4	9.2	odd	6
2592.2.i.z.1729.1		4	12.11	even	2
2592.2.i.z.1729.2		4	3.2	odd	2
2592.2.i.bd.865.1		4	36.7	odd	6	inner
2592.2.i.bd.865.2		4	9.7	even	3	inner
2592.2.i.bd.1729.1		4	4.3	odd	2	inner
2592.2.i.bd.1729.2		4	1.1	even	1	trivial
5184.2.a.bk.1.1		2	72.5	odd	6
5184.2.a.bk.1.2		2	72.59	even	6
5184.2.a.bv.1.1		2	72.13	even	6
5184.2.a.bv.1.2		2	72.67	odd	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.44949 + 4.24264i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.44949 + 4.24264i) q^{7} +(-2.44949 - 4.24264i) q^{11} +(-1.50000 + 2.59808i) q^{13} -5.00000 q^{17} +4.89898 q^{19} +(-2.44949 + 4.24264i) q^{23} +(2.00000 + 3.46410i) q^{25} +(2.50000 + 4.33013i) q^{29} +4.89898 q^{35} -5.00000 q^{37} +(-1.00000 + 1.73205i) q^{41} +(-2.44949 - 4.24264i) q^{43} +(4.89898 + 8.48528i) q^{47} +(-8.50000 + 14.7224i) q^{49} -2.00000 q^{53} -4.89898 q^{55} +(4.89898 - 8.48528i) q^{59} +(6.50000 + 11.2583i) q^{61} +(1.50000 + 2.59808i) q^{65} +(-2.44949 + 4.24264i) q^{67} -4.89898 q^{71} +3.00000 q^{73} +(12.0000 - 20.7846i) q^{77} +(-7.34847 - 12.7279i) q^{79} +(4.89898 + 8.48528i) q^{83} +(-2.50000 + 4.33013i) q^{85} -13.0000 q^{89} -14.6969 q^{91} +(2.44949 - 4.24264i) q^{95} +(3.00000 + 5.19615i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5}+O(q^{10}) \) 4 * q + 2 * q^5	\( 4 q + 2 q^{5} - 6 q^{13} - 20 q^{17} + 8 q^{25} + 10 q^{29} - 20 q^{37} - 4 q^{41} - 34 q^{49} - 8 q^{53} + 26 q^{61} + 6 q^{65} + 12 q^{73} + 48 q^{77} - 10 q^{85} - 52 q^{89} + 12 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - 6 * q^13 - 20 * q^17 + 8 * q^25 + 10 * q^29 - 20 * q^37 - 4 * q^41 - 34 * q^49 - 8 * q^53 + 26 * q^61 + 6 * q^65 + 12 * q^73 + 48 * q^77 - 10 * q^85 - 52 * q^89 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

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