LMFDB - Embedded newform 2592.2.i.bh.865.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(865,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,0,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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 $\Q(\zeta_{3})$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	2592.865
Dual form			2592.2.i.bh.1729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 - 2.29129i) q^{5} +(0.456850 - 0.791288i) q^{7} +(2.79129 - 4.83465i) q^{11} +(-1.29129 - 2.23658i) q^{13} -1.73205 q^{17} -7.84190 q^{19} +(-0.791288 - 1.37055i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(0.409175 - 0.708712i) q^{29} +(0.913701 + 1.58258i) q^{31} -2.41742 q^{35} -6.58258 q^{37} +(4.37780 + 7.58258i) q^{41} +(-3.92095 + 6.79129i) q^{43} +(4.00000 - 6.92820i) q^{47} +(3.08258 + 5.33918i) q^{49} +8.75560 q^{53} -14.7701 q^{55} +(4.00000 + 6.92820i) q^{59} +(5.29129 - 9.16478i) q^{61} +(-3.41643 + 5.91742i) q^{65} +(-3.92095 - 6.79129i) q^{67} -9.58258 q^{71} +12.1652 q^{73} +(-2.55040 - 4.41742i) q^{77} +(-7.38505 + 12.7913i) q^{79} +(-1.58258 + 2.74110i) q^{83} +(2.29129 + 3.96863i) q^{85} -8.85095 q^{89} -2.35970 q^{91} +(10.3739 + 17.9681i) q^{95} +(-6.58258 + 11.4014i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{11} + 8 q^{13} + 12 q^{23} - 8 q^{25} - 56 q^{35} - 16 q^{37} + 32 q^{47} - 12 q^{49} + 32 q^{59} + 24 q^{61} - 40 q^{71} + 24 q^{73} + 24 q^{83} + 28 q^{95} - 16 q^{97}+O(q^{100}) $ 8 * q + 4 * q^11 + 8 * q^13 + 12 * q^23 - 8 * q^25 - 56 * q^35 - 16 * q^37 + 32 * q^47 - 12 * q^49 + 32 * q^59 + 24 * q^61 - 40 * q^71 + 24 * q^73 + 24 * q^83 + 28 * q^95 - 16 * q^97

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
$5$	−1.32288	−	2.29129i	−0.591608	−	1.02470i	−0.994016	−	0.109235i	$-0.965160\pi$
$5$	−1.32288	−	2.29129i	−0.591608	−	1.02470i	0.402408	−	0.915460i	$-0.368173\pi$
$6$		0			0
$7$	0.456850	−	0.791288i	0.172673	−	0.299079i	−0.766680	−	0.642029i	$-0.778093\pi$
$7$	0.456850	−	0.791288i	0.172673	−	0.299079i	0.939354	+	0.342950i	$0.111426\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.79129	−	4.83465i	0.841605	−	1.45770i	−0.0469323	−	0.998898i	$-0.514945\pi$
$11$	2.79129	−	4.83465i	0.841605	−	1.45770i	0.888537	−	0.458804i	$-0.151722\pi$
$12$		0			0
$13$	−1.29129	−	2.23658i	−0.358139	−	0.620315i	0.629511	−	0.776991i	$-0.283255\pi$
$13$	−1.29129	−	2.23658i	−0.358139	−	0.620315i	−0.987650	+	0.156677i	$0.949922\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.73205			−0.420084			−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205			−0.420084			−0.210042	+	0.977692i	$0.567360\pi$
$18$		0			0
$19$	−7.84190			−1.79906			−0.899528	−	0.436863i	$-0.856089\pi$
$19$	−7.84190			−1.79906			−0.899528	+	0.436863i	$0.856089\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.791288	−	1.37055i	−0.164995	−	0.285780i	0.771659	−	0.636037i	$-0.219427\pi$
$23$	−0.791288	−	1.37055i	−0.164995	−	0.285780i	−0.936653	+	0.350257i	$0.886094\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.409175	−	0.708712i	0.0759819	−	0.131605i	−0.825531	−	0.564357i	$-0.809124\pi$
$29$	0.409175	−	0.708712i	0.0759819	−	0.131605i	0.901513	+	0.432752i	$0.142458\pi$
$30$		0			0
$31$	0.913701	+	1.58258i	0.164105	+	0.284239i	0.936337	−	0.351102i	$-0.114193\pi$
$31$	0.913701	+	1.58258i	0.164105	+	0.284239i	−0.772232	+	0.635341i	$0.780860\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.41742			−0.408619
$36$		0			0
$37$	−6.58258			−1.08217			−0.541084	−	0.840968i	$-0.681986\pi$
$37$	−6.58258			−1.08217			−0.541084	+	0.840968i	$0.681986\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.37780	+	7.58258i	0.683698	+	1.18420i	0.973844	+	0.227217i	$0.0729627\pi$
$41$	4.37780	+	7.58258i	0.683698	+	1.18420i	−0.290146	+	0.956982i	$0.593704\pi$
$42$		0			0
$43$	−3.92095	+	6.79129i	−0.597940	+	1.03566i	0.395185	+	0.918601i	$0.370680\pi$
$43$	−3.92095	+	6.79129i	−0.597940	+	1.03566i	−0.993125	+	0.117060i	$0.962653\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$	3.08258	+	5.33918i	0.440368	+	0.762740i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.75560			1.20267			0.601337	−	0.798995i	$-0.294635\pi$
$53$	8.75560			1.20267			0.601337	+	0.798995i	$0.294635\pi$
$54$		0			0
$55$	−14.7701			−1.99160
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	0.999709	+	0.0241347i	$0.00768307\pi$
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	−0.478953	+	0.877841i	$0.658984\pi$
$60$		0			0
$61$	5.29129	−	9.16478i	0.677480	−	1.17343i	−0.298257	−	0.954485i	$-0.596405\pi$
$61$	5.29129	−	9.16478i	0.677480	−	1.17343i	0.975737	−	0.218944i	$-0.0702613\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.41643	+	5.91742i	−0.423756	+	0.733966i
$66$		0			0
$67$	−3.92095	−	6.79129i	−0.479021	−	0.829688i	0.520690	−	0.853746i	$-0.325675\pi$
$67$	−3.92095	−	6.79129i	−0.479021	−	0.829688i	−0.999711	+	0.0240579i	$0.992341\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.58258			−1.13724			−0.568621	−	0.822599i	$-0.692523\pi$
$71$	−9.58258			−1.13724			−0.568621	+	0.822599i	$0.692523\pi$
$72$		0			0
$73$	12.1652			1.42382			0.711912	−	0.702269i	$-0.247830\pi$
$73$	12.1652			1.42382			0.711912	+	0.702269i	$0.247830\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.55040	−	4.41742i	−0.290645	−	0.503412i
$78$		0			0
$79$	−7.38505	+	12.7913i	−0.830883	+	1.43913i	0.0664551	+	0.997789i	$0.478831\pi$
$79$	−7.38505	+	12.7913i	−0.830883	+	1.43913i	−0.897339	+	0.441343i	$0.854502\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.58258	+	2.74110i	−0.173710	+	0.300875i	−0.939714	−	0.341961i	$-0.888909\pi$
$83$	−1.58258	+	2.74110i	−0.173710	+	0.300875i	0.766004	+	0.642836i	$0.222242\pi$
$84$		0			0
$85$	2.29129	+	3.96863i	0.248525	+	0.430458i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.85095			−0.938199			−0.469100	−	0.883145i	$-0.655421\pi$
$89$	−8.85095			−0.938199			−0.469100	+	0.883145i	$0.655421\pi$
$90$		0			0
$91$	−2.35970			−0.247364
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.3739	+	17.9681i	1.06434	+	1.84348i
$96$		0			0
$97$	−6.58258	+	11.4014i	−0.668359	+	1.15763i	0.310003	+	0.950735i	$0.399670\pi$
$97$	−6.58258	+	11.4014i	−0.668359	+	1.15763i	−0.978363	+	0.206897i	$0.933664\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.92820	−	12.0000i	0.689382	−	1.19404i	−0.282656	−	0.959221i	$-0.591216\pi$
$101$	6.92820	−	12.0000i	0.689382	−	1.19404i	0.972038	−	0.234823i	$-0.0754512\pi$
$102$		0			0
$103$	−9.66930	−	16.7477i	−0.952745	−	1.65020i	−0.739446	−	0.673216i	$-0.764913\pi$
$103$	−9.66930	−	16.7477i	−0.952745	−	1.65020i	−0.213299	−	0.976987i	$-0.568421\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.1652			−1.07938			−0.539688	−	0.841865i	$-0.681458\pi$
$107$	−11.1652			−1.07938			−0.539688	+	0.841865i	$0.681458\pi$
$108$		0			0
$109$	−5.41742			−0.518895			−0.259448	−	0.965757i	$-0.583540\pi$
$109$	−5.41742			−0.518895			−0.259448	+	0.965757i	$0.583540\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.961376	−	1.66515i	−0.0904386	−	0.156644i	0.817257	−	0.576273i	$-0.195494\pi$
$113$	−0.961376	−	1.66515i	−0.0904386	−	0.156644i	−0.907696	+	0.419629i	$0.862160\pi$
$114$		0			0
$115$	−2.09355	+	3.62614i	−0.195225	+	0.338139i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.791288	+	1.37055i	−0.0725372	+	0.125638i
$120$		0			0
$121$	−10.0826	−	17.4635i	−0.916598	−	1.58759i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.93725			−0.709930
$126$		0			0
$127$	−12.9427			−1.14848			−0.574240	−	0.818687i	$-0.694702\pi$
$127$	−12.9427			−1.14848			−0.574240	+	0.818687i	$0.694702\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.79129	+	4.83465i	0.243876	+	0.422406i	0.961815	−	0.273700i	$-0.0882477\pi$
$131$	2.79129	+	4.83465i	0.243876	+	0.422406i	−0.717939	+	0.696106i	$0.754914\pi$
$132$		0			0
$133$	−3.58258	+	6.20520i	−0.310649	+	0.538059i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.33013	+	7.50000i	−0.369948	+	0.640768i	−0.989557	−	0.144142i	$-0.953958\pi$
$137$	−4.33013	+	7.50000i	−0.369948	+	0.640768i	0.619609	+	0.784910i	$0.287291\pi$
$138$		0			0
$139$	0.913701	+	1.58258i	0.0774991	+	0.134232i	0.902170	−	0.431380i	$-0.141973\pi$
$139$	0.913701	+	1.58258i	0.0774991	+	0.134232i	−0.824671	+	0.565612i	$0.808640\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.4174			−1.20565
$144$		0			0
$145$	−2.16515			−0.179806
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.43273	−	12.8739i	−0.608913	−	1.05467i	−0.991420	−	0.130715i	$-0.958273\pi$
$149$	−7.43273	−	12.8739i	−0.608913	−	1.05467i	0.382507	−	0.923953i	$-0.375061\pi$
$150$		0			0
$151$	−9.66930	+	16.7477i	−0.786877	+	1.36291i	0.140995	+	0.990010i	$0.454970\pi$
$151$	−9.66930	+	16.7477i	−0.786877	+	1.36291i	−0.927871	+	0.372900i	$0.878363\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.41742	−	4.18710i	0.194172	−	0.336316i
$156$		0			0
$157$	5.70871	+	9.88778i	0.455605	+	0.789131i	0.998723	−	0.0505257i	$-0.0160897\pi$
$157$	5.70871	+	9.88778i	0.455605	+	0.789131i	−0.543118	+	0.839656i	$0.682756\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.44600			−0.113961
$162$		0			0
$163$	−17.5112			−1.37158			−0.685792	−	0.727798i	$-0.740544\pi$
$163$	−17.5112			−1.37158			−0.685792	+	0.727798i	$0.740544\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.20871	+	5.55765i	0.248298	+	0.430064i	0.963054	−	0.269310i	$-0.0867956\pi$
$167$	3.20871	+	5.55765i	0.248298	+	0.430064i	−0.714756	+	0.699374i	$0.753462\pi$
$168$		0			0
$169$	3.16515	−	5.48220i	0.243473	−	0.421708i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.16478	−	15.8739i	0.696785	−	1.20687i	−0.272790	−	0.962074i	$-0.587946\pi$
$173$	9.16478	−	15.8739i	0.696785	−	1.20687i	0.969575	−	0.244794i	$-0.0787203\pi$
$174$		0			0
$175$	0.913701	+	1.58258i	0.0690693	+	0.119631i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.70793	+	15.0826i	0.640220	+	1.10889i
$186$		0			0
$187$	−4.83465	+	8.37386i	−0.353545	+	0.612358i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.791288	−	1.37055i	0.0572556	−	0.0991696i	−0.835977	−	0.548765i	$-0.815098\pi$
$191$	0.791288	−	1.37055i	0.0572556	−	0.0991696i	0.893233	+	0.449595i	$0.148432\pi$
$192$		0			0
$193$	−3.08258	−	5.33918i	−0.221889	−	0.384322i	0.733493	−	0.679697i	$-0.237889\pi$
$193$	−3.08258	−	5.33918i	−0.221889	−	0.384322i	−0.955381	+	0.295375i	$0.904555\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	19.9663			1.42254			0.711269	−	0.702920i	$-0.248121\pi$
$197$	19.9663			1.42254			0.711269	+	0.702920i	$0.248121\pi$
$198$		0			0
$199$	3.65480			0.259082			0.129541	−	0.991574i	$-0.458650\pi$
$199$	3.65480			0.259082			0.129541	+	0.991574i	$0.458650\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.373864	−	0.647551i	−0.0262401	−	0.0454491i
$204$		0			0
$205$	11.5826	−	20.0616i	0.808962	−	1.40116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.8890	+	37.9129i	−1.51409	+	2.62249i
$210$		0			0
$211$	−3.00725	−	5.20871i	−0.207028	−	0.358583i	0.743749	−	0.668459i	$-0.233046\pi$
$211$	−3.00725	−	5.20871i	−0.207028	−	0.358583i	−0.950777	+	0.309876i	$0.899712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	20.7477			1.41498
$216$		0			0
$217$	1.66970			0.113346
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.23658	+	3.87386i	0.150448	+	0.260584i
$222$		0			0
$223$	10.1262	−	17.5390i	0.678097	−	1.17450i	−0.297456	−	0.954736i	$-0.596138\pi$
$223$	10.1262	−	17.5390i	0.678097	−	1.17450i	0.975553	−	0.219764i	$-0.0705286\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.37386	−	14.5040i	0.555793	−	0.962661i	−0.442049	−	0.896991i	$-0.645748\pi$
$227$	8.37386	−	14.5040i	0.555793	−	0.962661i	0.997841	−	0.0656703i	$-0.0209186\pi$
$228$		0			0
$229$	11.8739	+	20.5661i	0.784647	+	1.35905i	0.929210	+	0.369553i	$0.120489\pi$
$229$	11.8739	+	20.5661i	0.784647	+	1.35905i	−0.144563	+	0.989496i	$0.546178\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.0707			1.38038			0.690192	−	0.723626i	$-0.257526\pi$
$233$	21.0707			1.38038			0.690192	+	0.723626i	$0.257526\pi$
$234$		0			0
$235$	−21.1660			−1.38072
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.5826	−	23.5257i	−0.878584	−	1.52175i	−0.852895	−	0.522082i	$-0.825156\pi$
$239$	−13.5826	−	23.5257i	−0.878584	−	1.52175i	−0.0256883	−	0.999670i	$-0.508178\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.15573	−	14.1261i	0.521050	−	0.902486i
$246$		0			0
$247$	10.1262	+	17.5390i	0.644312	+	1.11598i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.5826			−0.857325			−0.428662	−	0.903465i	$-0.641015\pi$
$251$	−13.5826			−0.857325			−0.428662	+	0.903465i	$0.641015\pi$
$252$		0			0
$253$	−8.83485			−0.555442
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.51178	+	6.08258i	0.219059	+	0.379421i	0.954520	−	0.298146i	$-0.0963681\pi$
$257$	3.51178	+	6.08258i	0.219059	+	0.379421i	−0.735462	+	0.677566i	$0.763035\pi$
$258$		0			0
$259$	−3.00725	+	5.20871i	−0.186862	+	0.323654i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	15.1652	−	26.2668i	0.935123	−	1.61968i	0.160709	−	0.987002i	$-0.448622\pi$
$263$	15.1652	−	26.2668i	0.935123	−	1.61968i	0.774414	−	0.632679i	$-0.218045\pi$
$264$		0			0
$265$	−11.5826	−	20.0616i	−0.711512	−	1.23237i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.7937			−1.32878			−0.664391	−	0.747385i	$-0.731309\pi$
$269$	−21.7937			−1.32878			−0.664391	+	0.747385i	$0.731309\pi$
$270$		0			0
$271$	−2.74110			−0.166510			−0.0832550	−	0.996528i	$-0.526532\pi$
$271$	−2.74110			−0.166510			−0.0832550	+	0.996528i	$0.526532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.58258	+	9.66930i	0.336642	+	0.583081i
$276$		0			0
$277$	−8.16515	+	14.1425i	−0.490596	+	0.849738i	−0.999941	−	0.0108245i	$-0.996554\pi$
$277$	−8.16515	+	14.1425i	−0.490596	+	0.849738i	0.509345	+	0.860562i	$0.329888\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.60713	−	6.24773i	0.215183	−	0.372708i	−0.738146	−	0.674641i	$-0.764298\pi$
$281$	3.60713	−	6.24773i	0.215183	−	0.372708i	0.953329	+	0.301933i	$0.0976318\pi$
$282$		0			0
$283$	−1.82740	−	3.16515i	−0.108628	−	0.188149i	0.806587	−	0.591116i	$-0.201312\pi$
$283$	−1.82740	−	3.16515i	−0.108628	−	0.188149i	−0.915215	+	0.402967i	$0.867979\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.00000			0.472225
$288$		0			0
$289$	−14.0000			−0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.504525	+	0.873864i	0.0294747	+	0.0510517i	0.880386	−	0.474257i	$-0.157283\pi$
$293$	0.504525	+	0.873864i	0.0294747	+	0.0510517i	−0.850912	+	0.525309i	$0.823950\pi$
$294$		0			0
$295$	10.5830	−	18.3303i	0.616166	−	1.06723i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.04356	+	3.53955i	−0.118182	+	0.204698i
$300$		0			0
$301$	3.58258	+	6.20520i	0.206496	+	0.357662i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−27.9989			−1.60321
$306$		0			0
$307$	19.3386			1.10371			0.551856	−	0.833939i	$-0.313920\pi$
$307$	19.3386			1.10371			0.551856	+	0.833939i	$0.313920\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.58258	−	2.74110i	−0.0897396	−	0.155434i	0.817661	−	0.575699i	$-0.195270\pi$
$311$	−1.58258	−	2.74110i	−0.0897396	−	0.155434i	−0.907401	+	0.420266i	$0.861937\pi$
$312$		0			0
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	0.462093	+	0.886831i	$0.347098\pi$
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	−0.999065	+	0.0432311i	$0.986235\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.1738	−	17.6216i	0.571419	−	0.989727i	−0.425001	−	0.905193i	$-0.639726\pi$
$317$	10.1738	−	17.6216i	0.571419	−	0.989727i	0.996421	−	0.0845344i	$-0.0269403\pi$
$318$		0			0
$319$	−2.28425	−	3.95644i	−0.127894	−	0.221518i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.5826			0.755755
$324$		0			0
$325$	5.16515			0.286511
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.65480	−	6.33030i	−0.201496	−	0.349001i
$330$		0			0
$331$	12.6766	−	21.9564i	0.696767	−	1.20684i	−0.272815	−	0.962067i	$-0.587955\pi$
$331$	12.6766	−	21.9564i	0.696767	−	1.20684i	0.969582	−	0.244769i	$-0.0787120\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.3739	+	17.9681i	−0.566785	+	0.981700i
$336$		0			0
$337$	0.582576	+	1.00905i	0.0317349	+	0.0549665i	0.881457	−	0.472265i	$-0.156563\pi$
$337$	0.582576	+	1.00905i	0.0317349	+	0.0549665i	−0.849722	+	0.527231i	$0.823230\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	10.2016			0.552448
$342$		0			0
$343$	12.0290			0.649505
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.2087	−	22.8782i	−0.709081	−	1.22816i	−0.965198	−	0.261518i	$-0.915777\pi$
$347$	−13.2087	−	22.8782i	−0.709081	−	1.22816i	0.256118	−	0.966646i	$-0.417557\pi$
$348$		0			0
$349$	8.58258	−	14.8655i	0.459415	−	0.795730i	−0.539515	−	0.841976i	$-0.681393\pi$
$349$	8.58258	−	14.8655i	0.459415	−	0.795730i	0.998930	+	0.0462461i	$0.0147258\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5.10080	+	8.83485i	−0.271488	+	0.470232i	−0.969243	−	0.246105i	$-0.920849\pi$
$353$	−5.10080	+	8.83485i	−0.271488	+	0.470232i	0.697755	+	0.716337i	$0.254182\pi$
$354$		0			0
$355$	12.6766	+	21.9564i	0.672802	+	1.16533i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−23.9129			−1.26207			−0.631037	−	0.775753i	$-0.717370\pi$
$359$	−23.9129			−1.26207			−0.631037	+	0.775753i	$0.717370\pi$
$360$		0			0
$361$	42.4955			2.23660
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.0930	−	27.8739i	−0.842345	−	1.45898i
$366$		0			0
$367$	−2.74110	+	4.74773i	−0.143084	+	0.247829i	−0.928657	−	0.370941i	$-0.879035\pi$
$367$	−2.74110	+	4.74773i	−0.143084	+	0.247829i	0.785572	+	0.618770i	$0.212369\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.00000	−	6.92820i	0.207670	−	0.359694i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.11345			−0.108848
$378$		0			0
$379$	15.6838			0.805623			0.402812	−	0.915283i	$-0.368033\pi$
$379$	15.6838			0.805623			0.402812	+	0.915283i	$0.368033\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.20871	+	5.55765i	0.163958	+	0.283983i	0.936285	−	0.351242i	$-0.114241\pi$
$383$	3.20871	+	5.55765i	0.163958	+	0.283983i	−0.772327	+	0.635225i	$0.780907\pi$
$384$		0			0
$385$	−6.74773	+	11.6874i	−0.343896	+	0.595645i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.723000	−	1.25227i	0.0366576	−	0.0634928i	−0.847115	−	0.531410i	$-0.821662\pi$
$389$	0.723000	−	1.25227i	0.0366576	−	0.0634928i	0.883772	+	0.467918i	$0.154996\pi$
$390$		0			0
$391$	1.37055	+	2.37386i	0.0693117	+	0.120051i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	39.0780			1.96623
$396$		0			0
$397$	−9.74773			−0.489224			−0.244612	−	0.969621i	$-0.578661\pi$
$397$	−9.74773			−0.489224			−0.244612	+	0.969621i	$0.578661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.24383	−	9.08258i	−0.261864	−	0.453562i	0.704873	−	0.709333i	$-0.251004\pi$
$401$	−5.24383	−	9.08258i	−0.261864	−	0.453562i	−0.966737	+	0.255771i	$0.917671\pi$
$402$		0			0
$403$	2.35970	−	4.08712i	0.117545	−	0.203594i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−18.3739	+	31.8245i	−0.910759	+	1.57748i
$408$		0			0
$409$	1.08258	+	1.87508i	0.0535299	+	0.0927165i	0.891549	−	0.452925i	$-0.149619\pi$
$409$	1.08258	+	1.87508i	0.0535299	+	0.0927165i	−0.838019	+	0.545641i	$0.816286\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.30960			0.359682
$414$		0			0
$415$	8.37420			0.411073
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.58258	−	9.66930i	−0.272727	−	0.472376i	0.696832	−	0.717234i	$-0.254592\pi$
$419$	−5.58258	−	9.66930i	−0.272727	−	0.472376i	−0.969559	+	0.244858i	$0.921259\pi$
$420$		0			0
$421$	4.29129	−	7.43273i	0.209145	−	0.362249i	−0.742301	−	0.670067i	$-0.766265\pi$
$421$	4.29129	−	7.43273i	0.209145	−	0.362249i	0.951445	+	0.307818i	$0.0995987\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.73205	−	3.00000i	0.0840168	−	0.145521i
$426$		0			0
$427$	−4.83465	−	8.37386i	−0.233965	−	0.405240i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.8348			1.00358			0.501790	−	0.864990i	$-0.332675\pi$
$431$	20.8348			1.00358			0.501790	+	0.864990i	$0.332675\pi$
$432$		0			0
$433$	13.3303			0.640613			0.320307	−	0.947314i	$-0.396214\pi$
$433$	13.3303			0.640613			0.320307	+	0.947314i	$0.396214\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.20520	+	10.7477i	0.296835	+	0.514134i
$438$		0			0
$439$	−10.5830	+	18.3303i	−0.505099	+	0.874858i	0.494883	+	0.868959i	$0.335211\pi$
$439$	−10.5830	+	18.3303i	−0.505099	+	0.874858i	−0.999983	+	0.00589819i	$0.998123\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.3303	+	31.7490i	−0.870899	+	1.50844i	−0.00983083	+	0.999952i	$0.503129\pi$
$443$	−18.3303	+	31.7490i	−0.870899	+	1.50844i	−0.861068	+	0.508490i	$0.830204\pi$
$444$		0			0
$445$	11.7087	+	20.2801i	0.555046	+	0.961368i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.9572			−0.894646			−0.447323	−	0.894372i	$-0.647623\pi$
$449$	−18.9572			−0.894646			−0.447323	+	0.894372i	$0.647623\pi$
$450$		0			0
$451$	48.8788			2.30161
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.12159	+	5.40675i	0.146342	+	0.253473i
$456$		0			0
$457$	5.08258	−	8.80328i	0.237753	−	0.411800i	−0.722316	−	0.691563i	$-0.756922\pi$
$457$	5.08258	−	8.80328i	0.237753	−	0.411800i	0.960069	+	0.279763i	$0.0902558\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.1334	+	22.7477i	−0.611684	+	1.05947i	0.379273	+	0.925285i	$0.376174\pi$
$461$	−13.1334	+	22.7477i	−0.611684	+	1.05947i	−0.990957	+	0.134182i	$0.957159\pi$
$462$		0			0
$463$	−8.75560	−	15.1652i	−0.406907	−	0.704784i	0.587634	−	0.809127i	$-0.300060\pi$
$463$	−8.75560	−	15.1652i	−0.406907	−	0.704784i	−0.994541	+	0.104343i	$0.966726\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.1652			−0.886857			−0.443429	−	0.896310i	$-0.646238\pi$
$467$	−19.1652			−0.886857			−0.443429	+	0.896310i	$0.646238\pi$
$468$		0			0
$469$	−7.16515			−0.330856
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.8890	+	37.9129i	1.00646	+	1.74324i
$474$		0			0
$475$	7.84190	−	13.5826i	0.359811	−	0.623211i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.37386	+	4.11165i	−0.108465	+	0.187866i	−0.915148	−	0.403117i	$-0.867927\pi$
$479$	−2.37386	+	4.11165i	−0.108465	+	0.187866i	0.806684	+	0.590983i	$0.201260\pi$
$480$		0			0
$481$	8.50000	+	14.7224i	0.387567	+	0.671285i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	34.8317			1.58163
$486$		0			0
$487$	29.5402			1.33859			0.669297	−	0.742995i	$-0.266595\pi$
$487$	29.5402			1.33859			0.669297	+	0.742995i	$0.266595\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.79129	+	4.83465i	0.125969	+	0.218185i	0.922111	−	0.386925i	$-0.126463\pi$
$491$	2.79129	+	4.83465i	0.125969	+	0.218185i	−0.796142	+	0.605109i	$0.793129\pi$
$492$		0			0
$493$	−0.708712	+	1.22753i	−0.0319188	+	0.0552850i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.37780	+	7.58258i	−0.196371	+	0.340125i
$498$		0			0
$499$	9.02175	+	15.6261i	0.403869	+	0.699522i	0.994189	−	0.107647i	$-0.0343318\pi$
$499$	9.02175	+	15.6261i	0.403869	+	0.699522i	−0.590320	+	0.807169i	$0.700998\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.16515			−0.141127			−0.0705636	−	0.997507i	$-0.522480\pi$
$503$	−3.16515			−0.141127			−0.0705636	+	0.997507i	$0.522480\pi$
$504$		0			0
$505$	−36.6606			−1.63138
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.723000	+	1.25227i	0.0320464	+	0.0555060i	0.881604	−	0.471990i	$-0.156464\pi$
$509$	0.723000	+	1.25227i	0.0320464	+	0.0555060i	−0.849557	+	0.527496i	$0.823131\pi$
$510$		0			0
$511$	5.55765	−	9.62614i	0.245856	−	0.425835i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−25.5826	+	44.3103i	−1.12730	+	1.95255i
$516$		0			0
$517$	−22.3303	−	38.6772i	−0.982086	−	1.70102i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	26.2668			1.15077			0.575385	−	0.817883i	$-0.304852\pi$
$521$	26.2668			1.15077			0.575385	+	0.817883i	$0.304852\pi$
$522$		0			0
$523$	37.3821			1.63461			0.817303	−	0.576208i	$-0.195468\pi$
$523$	37.3821			1.63461			0.817303	+	0.576208i	$0.195468\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.58258	−	2.74110i	−0.0689381	−	0.119404i
$528$		0			0
$529$	10.2477	−	17.7496i	0.445553	−	0.771721i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.3060	−	19.5826i	0.489717	−	0.848216i
$534$		0			0
$535$	14.7701	+	25.5826i	0.638567	+	1.10603i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	34.4174			1.48246
$540$		0			0
$541$	−42.9129			−1.84497			−0.922484	−	0.386034i	$-0.873845\pi$
$541$	−42.9129			−1.84497			−0.922484	+	0.386034i	$0.873845\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.16658	+	12.4129i	0.306983	+	0.531709i
$546$		0			0
$547$	6.92820	−	12.0000i	0.296229	−	0.513083i	−0.679041	−	0.734100i	$-0.737604\pi$
$547$	6.92820	−	12.0000i	0.296229	−	0.513083i	0.975270	+	0.221017i	$0.0709377\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.20871	+	5.55765i	−0.136696	+	0.236764i
$552$		0			0
$553$	6.74773	+	11.6874i	0.286943	+	0.496999i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.5312			−1.20890			−0.604452	−	0.796641i	$-0.706608\pi$
$557$	−28.5312			−1.20890			−0.604452	+	0.796641i	$0.706608\pi$
$558$		0			0
$559$	20.2523			0.856581
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	−0.999978	−	0.00664037i	$-0.997886\pi$
$563$	−12.0000	−	20.7846i	−0.505740	−	0.875967i	0.494238	−	0.869326i	$-0.335447\pi$
$564$		0			0
$565$	−2.54356	+	4.40558i	−0.107008	+	0.185344i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.6361	−	27.0826i	0.655501	−	1.13536i	−0.326267	−	0.945278i	$-0.605791\pi$
$569$	15.6361	−	27.0826i	0.655501	−	1.13536i	0.981768	−	0.190083i	$-0.0608757\pi$
$570$		0			0
$571$	−10.5830	−	18.3303i	−0.442885	−	0.767099i	0.555017	−	0.831839i	$-0.312712\pi$
$571$	−10.5830	−	18.3303i	−0.442885	−	0.767099i	−0.997902	+	0.0647396i	$0.979378\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.16515			0.131996
$576$		0			0
$577$	−0.165151			−0.00687534			−0.00343767	−	0.999994i	$-0.501094\pi$
$577$	−0.165151			−0.00687534			−0.00343767	+	0.999994i	$0.501094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.44600	+	2.50455i	0.0599902	+	0.103906i
$582$		0			0
$583$	24.4394	−	42.3303i	1.01218	−	1.75314i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.79129	−	11.7629i	0.280306	−	0.485505i	−0.691154	−	0.722708i	$-0.742897\pi$
$587$	6.79129	−	11.7629i	0.280306	−	0.485505i	0.971460	+	0.237203i	$0.0762306\pi$
$588$		0			0
$589$	−7.16515	−	12.4104i	−0.295235	−	0.511362i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.1244			−0.497888			−0.248944	−	0.968518i	$-0.580083\pi$
$593$	−12.1244			−0.497888			−0.248944	+	0.968518i	$0.580083\pi$
$594$		0			0
$595$	4.18710			0.171654
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.41742	−	11.1153i	−0.262209	−	0.454159i	0.704620	−	0.709585i	$-0.251118\pi$
$599$	−6.41742	−	11.1153i	−0.262209	−	0.454159i	−0.966829	+	0.255426i	$0.917784\pi$
$600$		0			0
$601$	−20.2477	+	35.0701i	−0.825922	+	1.43054i	0.0752906	+	0.997162i	$0.476012\pi$
$601$	−20.2477	+	35.0701i	−0.825922	+	1.43054i	−0.901213	+	0.433377i	$0.857322\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−26.6760	+	46.2042i	−1.08453	+	1.87847i
$606$		0			0
$607$	−3.19795	−	5.53901i	−0.129801	−	0.224822i	0.793798	−	0.608181i	$-0.208100\pi$
$607$	−3.19795	−	5.53901i	−0.129801	−	0.224822i	−0.923599	+	0.383359i	$0.874767\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−20.6606			−0.835839
$612$		0			0
$613$	30.0000			1.21169			0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000			1.21169			0.605844	+	0.795583i	$0.292835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.5353	−	18.2477i	−0.424136	−	0.734626i	0.572203	−	0.820112i	$-0.306089\pi$
$617$	−10.5353	−	18.2477i	−0.424136	−	0.734626i	−0.996339	+	0.0854862i	$0.972756\pi$
$618$		0			0
$619$	−4.18710	+	7.25227i	−0.168294	+	0.291493i	−0.937820	−	0.347122i	$-0.887159\pi$
$619$	−4.18710	+	7.25227i	−0.168294	+	0.291493i	0.769526	+	0.638615i	$0.220492\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.04356	+	7.00365i	−0.162002	+	0.280595i
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	11.4014			0.454602
$630$		0			0
$631$	12.0290			0.478867			0.239434	−	0.970913i	$-0.423038\pi$
$631$	12.0290			0.478867			0.239434	+	0.970913i	$0.423038\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.1216	+	29.6555i	0.679450	+	1.17684i
$636$		0			0
$637$	7.96099	−	13.7888i	0.315426	−	0.546333i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.961376	−	1.66515i	0.0379721	−	0.0657695i	−0.846415	−	0.532524i	$-0.821244\pi$
$641$	0.961376	−	1.66515i	0.0379721	−	0.0657695i	0.884387	+	0.466755i	$0.154577\pi$
$642$		0			0
$643$	2.74110	+	4.74773i	0.108098	+	0.187232i	0.915000	−	0.403454i	$-0.132190\pi$
$643$	2.74110	+	4.74773i	0.108098	+	0.187232i	−0.806901	+	0.590686i	$0.798857\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.9129			1.25462			0.627312	−	0.778768i	$-0.284155\pi$
$647$	31.9129			1.25462			0.627312	+	0.778768i	$0.284155\pi$
$648$		0			0
$649$	44.6606			1.75308
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.2342	−	31.5826i	−0.713560	−	1.23592i	−0.963512	−	0.267664i	$-0.913748\pi$
$653$	−18.2342	−	31.5826i	−0.713560	−	1.23592i	0.249953	−	0.968258i	$-0.419585\pi$
$654$		0			0
$655$	7.38505	−	12.7913i	0.288558	−	0.499797i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.62614	+	6.28065i	−0.141254	+	0.244659i	−0.927969	−	0.372657i	$-0.878447\pi$
$659$	−3.62614	+	6.28065i	−0.141254	+	0.244659i	0.786715	+	0.617316i	$0.211780\pi$
$660$		0			0
$661$	−7.87386	−	13.6379i	−0.306258	−	0.530454i	0.671283	−	0.741201i	$-0.265744\pi$
$661$	−7.87386	−	13.6379i	−0.306258	−	0.530454i	−0.977541	+	0.210747i	$0.932410\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.9572			0.735129
$666$		0			0
$667$	−1.29510			−0.0501465
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−29.5390	−	51.1631i	−1.14034	−	1.97513i
$672$		0			0
$673$	20.0826	−	34.7840i	0.774126	−	1.34083i	−0.161158	−	0.986929i	$-0.551523\pi$
$673$	20.0826	−	34.7840i	0.774126	−	1.34083i	0.935284	−	0.353898i	$-0.115144\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.03260	−	13.9129i	0.308718	−	0.534715i	−0.669364	−	0.742934i	$-0.733433\pi$
$677$	8.03260	−	13.9129i	0.308718	−	0.534715i	0.978082	+	0.208219i	$0.0667667\pi$
$678$		0			0
$679$	6.01450	+	10.4174i	0.230815	+	0.399784i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.0000			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	22.9129			0.875456
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.3060	−	19.5826i	−0.430725	−	0.746037i
$690$		0			0
$691$	10.8492	−	18.7913i	0.412721	−	0.714854i	−0.582465	−	0.812856i	$-0.697912\pi$
$691$	10.8492	−	18.7913i	0.412721	−	0.714854i	0.995186	+	0.0980015i	$0.0312450\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.41742	−	4.18710i	0.0916981	−	0.158826i
$696$		0			0
$697$	−7.58258	−	13.1334i	−0.287211	−	0.497463i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.3477			−0.768521			−0.384260	−	0.923225i	$-0.625543\pi$
$701$	−20.3477			−0.768521			−0.384260	+	0.923225i	$0.625543\pi$
$702$		0			0
$703$	51.6199			1.94688
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.33030	−	10.9644i	−0.238076	−	0.412359i
$708$		0			0
$709$	−4.12614	+	7.14668i	−0.154960	+	0.268399i	−0.933045	−	0.359761i	$-0.882858\pi$
$709$	−4.12614	+	7.14668i	−0.154960	+	0.268399i	0.778084	+	0.628160i	$0.216192\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.44600	−	2.50455i	0.0541531	−	0.0937960i
$714$		0			0
$715$	19.0725	+	33.0345i	0.713270	+	1.23542i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	47.9129			1.78685			0.893424	−	0.449214i	$-0.148296\pi$
$719$	47.9129			1.78685			0.893424	+	0.449214i	$0.148296\pi$
$720$		0			0
$721$	−17.6697			−0.658054
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.818350	+	1.41742i	0.0303928	+	0.0526418i
$726$		0			0
$727$	8.29875	−	14.3739i	0.307784	−	0.533097i	−0.670093	−	0.742277i	$-0.733746\pi$
$727$	8.29875	−	14.3739i	0.307784	−	0.533097i	0.977877	+	0.209180i	$0.0670793\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.79129	−	11.7629i	0.251185	−	0.435065i
$732$		0			0
$733$	2.16515	+	3.75015i	0.0799717	+	0.138515i	0.903237	−	0.429141i	$-0.141184\pi$
$733$	2.16515	+	3.75015i	0.0799717	+	0.138515i	−0.823266	+	0.567656i	$0.807850\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−43.7780			−1.61258
$738$		0			0
$739$	−42.3320			−1.55721			−0.778604	−	0.627515i	$-0.784072\pi$
$739$	−42.3320			−1.55721			−0.778604	+	0.627515i	$0.784072\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.16515	−	5.48220i	−0.116118	−	0.201123i	0.802108	−	0.597179i	$-0.203712\pi$
$743$	−3.16515	−	5.48220i	−0.116118	−	0.201123i	−0.918226	+	0.396056i	$0.870378\pi$
$744$		0			0
$745$	−19.6652	+	34.0610i	−0.720475	+	1.24790i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.10080	+	8.83485i	−0.186379	+	0.322818i
$750$		0			0
$751$	18.8818	+	32.7042i	0.689005	+	1.19339i	0.972160	+	0.234318i	$0.0752856\pi$
$751$	18.8818	+	32.7042i	0.689005	+	1.19339i	−0.283155	+	0.959074i	$0.591381\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	51.1652			1.86209
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.2583	−	19.5000i	−0.408114	−	0.706874i	0.586564	−	0.809903i	$-0.300480\pi$
$761$	−11.2583	−	19.5000i	−0.408114	−	0.706874i	−0.994678	+	0.103028i	$0.967147\pi$
$762$		0			0
$763$	−2.47495	+	4.28674i	−0.0895993	+	0.155190i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.3303	−	17.8926i	0.373006	−	0.646065i
$768$		0			0
$769$	−11.0826	−	19.1956i	−0.399648	−	0.692210i	0.594034	−	0.804440i	$-0.297534\pi$
$769$	−11.0826	−	19.1956i	−0.399648	−	0.692210i	−0.993682	+	0.112229i	$0.964201\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.95536			−0.358069			−0.179035	−	0.983843i	$-0.557297\pi$
$773$	−9.95536			−0.358069			−0.179035	+	0.983843i	$0.557297\pi$
$774$		0			0
$775$	−3.65480			−0.131284
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−34.3303	−	59.4618i	−1.23001	−	2.13044i
$780$		0			0
$781$	−26.7477	+	46.3284i	−0.957109	+	1.65776i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.1038	−	26.1606i	0.539079	−	0.933712i
$786$		0			0
$787$	−7.57575	−	13.1216i	−0.270046	−	0.467734i	0.698827	−	0.715291i	$-0.253706\pi$
$787$	−7.57575	−	13.1216i	−0.270046	−	0.467734i	−0.968873	+	0.247557i	$0.920372\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.75682			−0.0624653
$792$		0			0
$793$	−27.3303			−0.970528
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.2982	+	38.6216i	0.789842	+	1.36805i	0.926063	+	0.377368i	$0.123171\pi$
$797$	22.2982	+	38.6216i	0.789842	+	1.36805i	−0.136221	+	0.990678i	$0.543496\pi$
$798$		0			0
$799$	−6.92820	+	12.0000i	−0.245102	+	0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.9564	−	58.8143i	1.19830	−	2.07551i
$804$		0			0
$805$	1.91288	+	3.31320i	0.0674201	+	0.116775i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.1287			1.58664			0.793320	−	0.608805i	$-0.208351\pi$
$809$	45.1287			1.58664			0.793320	+	0.608805i	$0.208351\pi$
$810$		0			0
$811$	−3.65480			−0.128337			−0.0641687	−	0.997939i	$-0.520440\pi$
$811$	−3.65480			−0.128337			−0.0641687	+	0.997939i	$0.520440\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.1652	+	40.1232i	0.811440	+	1.40546i
$816$		0			0
$817$	30.7477	−	53.2566i	1.07573	−	1.86321i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.88233	−	8.45644i	0.170394	−	0.295132i	−0.768163	−	0.640254i	$-0.778829\pi$
$821$	4.88233	−	8.45644i	0.170394	−	0.295132i	0.938558	+	0.345122i	$0.112162\pi$
$822$		0			0
$823$	−12.0290	−	20.8348i	−0.419305	−	0.726257i	0.576565	−	0.817051i	$-0.304393\pi$
$823$	−12.0290	−	20.8348i	−0.419305	−	0.726257i	−0.995870	+	0.0907942i	$0.971059\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.74773			−0.304188			−0.152094	−	0.988366i	$-0.548602\pi$
$827$	−8.74773			−0.304188			−0.152094	+	0.988366i	$0.548602\pi$
$828$		0			0
$829$	8.33030			0.289323			0.144662	−	0.989481i	$-0.453791\pi$
$829$	8.33030			0.289323			0.144662	+	0.989481i	$0.453791\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.33918	−	9.24773i	−0.184992	−	0.320415i
$834$		0			0
$835$	8.48945	−	14.7042i	0.293790	−	0.508859i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.3739	+	38.7527i	−0.772432	+	1.33789i	0.163795	+	0.986494i	$0.447626\pi$
$839$	−22.3739	+	38.7527i	−0.772432	+	1.33789i	−0.936227	+	0.351396i	$0.885707\pi$
$840$		0			0
$841$	14.1652	+	24.5348i	0.488453	+	0.846026i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16.7484			−0.576163
$846$		0			0
$847$	−18.4249			−0.633087
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.20871	+	9.02175i	0.178552	+	0.309262i
$852$		0			0
$853$	0.165151	−	0.286051i	0.00565468	−	0.00979419i	−0.863184	−	0.504889i	$-0.831533\pi$
$853$	0.165151	−	0.286051i	0.00565468	−	0.00979419i	0.868839	+	0.495095i	$0.164867\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.0767	−	20.9174i	0.412532	−	0.714526i	−0.582634	−	0.812734i	$-0.697978\pi$
$857$	12.0767	−	20.9174i	0.412532	−	0.714526i	0.995166	+	0.0982089i	$0.0313113\pi$
$858$		0			0
$859$	24.4394	+	42.3303i	0.833862	+	1.44429i	0.894954	+	0.446159i	$0.147208\pi$
$859$	24.4394	+	42.3303i	0.833862	+	1.44429i	−0.0610917	+	0.998132i	$0.519458\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−49.4083			−1.68188			−0.840940	−	0.541129i	$-0.817997\pi$
$863$	−49.4083			−1.68188			−0.840940	+	0.541129i	$0.817997\pi$
$864$		0			0
$865$	−48.4955			−1.64889
$866$		0			0
$867$		0			0
$868$		0			0
$869$	41.2276	+	71.4083i	1.39855	+	2.42236i
$870$		0			0
$871$	−10.1262	+	17.5390i	−0.343112	+	0.594287i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.62614	+	6.28065i	−0.122586	+	0.212325i
$876$		0			0
$877$	20.0390	+	34.7086i	0.676669	+	1.17203i	0.975978	+	0.217869i	$0.0699107\pi$
$877$	20.0390	+	34.7086i	0.676669	+	1.17203i	−0.299308	+	0.954156i	$0.596756\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.44600			−0.0487170			−0.0243585	−	0.999703i	$-0.507754\pi$
$881$	−1.44600			−0.0487170			−0.0243585	+	0.999703i	$0.507754\pi$
$882$		0			0
$883$	9.13701			0.307485			0.153742	−	0.988111i	$-0.450867\pi$
$883$	9.13701			0.307485			0.153742	+	0.988111i	$0.450867\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.7913	−	22.1552i	−0.429489	−	0.743897i	0.567338	−	0.823485i	$-0.307973\pi$
$887$	−12.7913	−	22.1552i	−0.429489	−	0.743897i	−0.996828	+	0.0795872i	$0.974640\pi$
$888$		0			0
$889$	−5.91288	+	10.2414i	−0.198312	+	0.343486i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−31.3676	+	54.3303i	−1.04968	+	1.81809i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.49545			0.0498762
$900$		0			0
$901$	−15.1652			−0.505224
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.64575	−	4.58258i	−0.0879477	−	0.152330i
$906$		0			0
$907$	11.4967	−	19.9129i	0.381742	−	0.661196i	−0.609569	−	0.792733i	$-0.708658\pi$
$907$	11.4967	−	19.9129i	0.381742	−	0.661196i	0.991311	+	0.131536i	$0.0419910\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.2087	−	19.4141i	0.371361	−	0.643216i	−0.618414	−	0.785852i	$-0.712224\pi$
$911$	11.2087	−	19.4141i	0.371361	−	0.643216i	0.989775	+	0.142636i	$0.0455578\pi$
$912$		0			0
$913$	8.83485	+	15.3024i	0.292391	+	0.506436i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.10080			0.168443
$918$		0			0
$919$	−30.4539			−1.00458			−0.502291	−	0.864699i	$-0.667509\pi$
$919$	−30.4539			−1.00458			−0.502291	+	0.864699i	$0.667509\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.3739	+	21.4322i	0.407291	+	0.705448i
$924$		0			0
$925$	6.58258	−	11.4014i	0.216434	−	0.374874i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.79423	−	13.5000i	0.255720	−	0.442921i	−0.709371	−	0.704836i	$-0.751021\pi$
$929$	7.79423	−	13.5000i	0.255720	−	0.442921i	0.965091	+	0.261915i	$0.0843540\pi$
$930$		0			0
$931$	−24.1733	−	41.8693i	−0.792247	−	1.37221i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	25.5826			0.836640
$936$		0			0
$937$	2.16515			0.0707324			0.0353662	−	0.999374i	$-0.488740\pi$
$937$	2.16515			0.0707324			0.0353662	+	0.999374i	$0.488740\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.06398	+	7.03901i	0.132482	+	0.229465i	0.924633	−	0.380860i	$-0.124372\pi$
$941$	4.06398	+	7.03901i	0.132482	+	0.229465i	−0.792151	+	0.610325i	$0.791039\pi$
$942$		0			0
$943$	6.92820	−	12.0000i	0.225613	−	0.390774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.5390	+	40.7708i	−0.764915	+	1.32487i	0.175376	+	0.984501i	$0.443886\pi$
$947$	−23.5390	+	40.7708i	−0.764915	+	1.32487i	−0.940291	+	0.340371i	$0.889448\pi$
$948$		0			0
$949$	−15.7087	−	27.2083i	−0.509926	−	0.883218i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.7835			1.58025			0.790126	−	0.612945i	$-0.210015\pi$
$953$	48.7835			1.58025			0.790126	+	0.612945i	$0.210015\pi$
$954$		0			0
$955$	−4.18710			−0.135491
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.95644	+	6.85275i	0.127760	+	0.221287i
$960$		0			0
$961$	13.8303	−	23.9548i	0.446139	−	0.772735i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.15573	+	14.1261i	−0.262542	+	0.454736i
$966$		0			0
$967$	−15.2270	−	26.3739i	−0.489666	−	0.848126i	0.510263	−	0.860018i	$-0.329548\pi$
$967$	−15.2270	−	26.3739i	−0.489666	−	0.848126i	−0.999929	+	0.0118919i	$0.996215\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.747727			0.0239957			0.0119979	−	0.999928i	$-0.496181\pi$
$971$	0.747727			0.0239957			0.0119979	+	0.999928i	$0.496181\pi$
$972$		0			0
$973$	1.66970			0.0535280
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.47860	−	16.4174i	−0.303247	−	0.525240i	0.673622	−	0.739076i	$-0.264738\pi$
$977$	−9.47860	−	16.4174i	−0.303247	−	0.525240i	−0.976870	+	0.213836i	$0.931404\pi$
$978$		0			0
$979$	−24.7056	+	42.7913i	−0.789593	+	1.36762i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.7477	−	49.7925i	0.916910	−	1.58813i	0.112830	−	0.993614i	$-0.464009\pi$
$983$	28.7477	−	49.7925i	0.916910	−	1.58813i	0.804080	−	0.594521i	$-0.202658\pi$
$984$		0			0
$985$	−26.4129	−	45.7484i	−0.841584	−	1.45767i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.4104			0.394628
$990$		0			0
$991$	−12.9427			−0.411139			−0.205569	−	0.978643i	$-0.565905\pi$
$991$	−12.9427			−0.411139			−0.205569	+	0.978643i	$0.565905\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.83485	−	8.37420i	−0.153275	−	0.265480i
$996$		0			0
$997$	−5.12614	+	8.87873i	−0.162346	+	0.281192i	−0.935710	−	0.352771i	$-0.885240\pi$
$997$	−5.12614	+	8.87873i	−0.162346	+	0.281192i	0.773363	+	0.633963i	$0.218573\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.bh.865.1		8
3.2	odd	2		2592.2.i.bg.865.3		8
4.3	odd	2		2592.2.i.bg.865.2		8
9.2	odd	6		2592.2.a.w.1.2	yes	4
9.4	even	3	inner	2592.2.i.bh.1729.1		8
9.5	odd	6		2592.2.i.bg.1729.3		8
9.7	even	3		2592.2.a.v.1.4	yes	4
12.11	even	2	inner	2592.2.i.bh.865.4		8
36.7	odd	6		2592.2.a.w.1.3	yes	4
36.11	even	6		2592.2.a.v.1.1	✓	4
36.23	even	6	inner	2592.2.i.bh.1729.4		8
36.31	odd	6		2592.2.i.bg.1729.2		8
72.11	even	6		5184.2.a.ce.1.3		4
72.29	odd	6		5184.2.a.cd.1.4		4
72.43	odd	6		5184.2.a.cd.1.1		4
72.61	even	6		5184.2.a.ce.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.v.1.1	✓	4	36.11	even	6
2592.2.a.v.1.4	yes	4	9.7	even	3
2592.2.a.w.1.2	yes	4	9.2	odd	6
2592.2.a.w.1.3	yes	4	36.7	odd	6
2592.2.i.bg.865.2		8	4.3	odd	2
2592.2.i.bg.865.3		8	3.2	odd	2
2592.2.i.bg.1729.2		8	36.31	odd	6
2592.2.i.bg.1729.3		8	9.5	odd	6
2592.2.i.bh.865.1		8	1.1	even	1	trivial
2592.2.i.bh.865.4		8	12.11	even	2	inner
2592.2.i.bh.1729.1		8	9.4	even	3	inner
2592.2.i.bh.1729.4		8	36.23	even	6	inner
5184.2.a.cd.1.1		4	72.43	odd	6
5184.2.a.cd.1.4		4	72.29	odd	6
5184.2.a.ce.1.2		4	72.61	even	6
5184.2.a.ce.1.3		4	72.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.32288 - 2.29129i) q^{5} +(0.456850 - 0.791288i) q^{7} +(2.79129 - 4.83465i) q^{11} +(-1.29129 - 2.23658i) q^{13} -1.73205 q^{17} -7.84190 q^{19} +(-0.791288 - 1.37055i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(0.409175 - 0.708712i) q^{29} +(0.913701 + 1.58258i) q^{31} -2.41742 q^{35} -6.58258 q^{37} +(4.37780 + 7.58258i) q^{41} +(-3.92095 + 6.79129i) q^{43} +(4.00000 - 6.92820i) q^{47} +(3.08258 + 5.33918i) q^{49} +8.75560 q^{53} -14.7701 q^{55} +(4.00000 + 6.92820i) q^{59} +(5.29129 - 9.16478i) q^{61} +(-3.41643 + 5.91742i) q^{65} +(-3.92095 - 6.79129i) q^{67} -9.58258 q^{71} +12.1652 q^{73} +(-2.55040 - 4.41742i) q^{77} +(-7.38505 + 12.7913i) q^{79} +(-1.58258 + 2.74110i) q^{83} +(2.29129 + 3.96863i) q^{85} -8.85095 q^{89} -2.35970 q^{91} +(10.3739 + 17.9681i) q^{95} +(-6.58258 + 11.4014i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{11} + 8 q^{13} + 12 q^{23} - 8 q^{25} - 56 q^{35} - 16 q^{37} + 32 q^{47} - 12 q^{49} + 32 q^{59} + 24 q^{61} - 40 q^{71} + 24 q^{73} + 24 q^{83} + 28 q^{95} - 16 q^{97}+O(q^{100}) \) 8 * q + 4 * q^11 + 8 * q^13 + 12 * q^23 - 8 * q^25 - 56 * q^35 - 16 * q^37 + 32 * q^47 - 12 * q^49 + 32 * q^59 + 24 * q^61 - 40 * q^71 + 24 * q^73 + 24 * q^83 + 28 * q^95 - 16 * q^97

Introduction

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