LMFDB - Embedded newform 2736.2.k.o.2431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (of order $2$, degree $1$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.3
Root			$1.68614 - 0.396143i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.o.2431.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228 q^{5} -0.792287i q^{7} +O(q^{10})$	$q+3.37228 q^{5} -0.792287i q^{7} +0.792287i q^{11} +5.04868i q^{13} -5.37228 q^{17} +(4.00000 + 1.73205i) q^{19} -8.51278i q^{23} +6.37228 q^{25} +10.0974i q^{29} +8.74456 q^{31} -2.67181i q^{35} +5.04868i q^{37} +6.92820i q^{41} +9.30506i q^{43} -4.25639i q^{47} +6.37228 q^{49} -3.16915i q^{53} +2.67181i q^{55} +4.00000 q^{59} +5.37228 q^{61} +17.0256i q^{65} -9.48913 q^{67} +4.00000 q^{71} -4.11684 q^{73} +0.627719 q^{77} +4.74456 q^{79} -3.46410i q^{83} -18.1168 q^{85} -13.2665i q^{89} +4.00000 q^{91} +(13.4891 + 5.84096i) q^{95} -13.2665i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5}+O(q^{10}) $ 4 * q + 2 * q^5	$ 4 q + 2 q^{5} - 10 q^{17} + 16 q^{19} + 14 q^{25} + 12 q^{31} + 14 q^{49} + 16 q^{59} + 10 q^{61} + 8 q^{67} + 16 q^{71} + 18 q^{73} + 14 q^{77} - 4 q^{79} - 38 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) $ 4 * q + 2 * q^5 - 10 * q^17 + 16 * q^19 + 14 * q^25 + 12 * q^31 + 14 * q^49 + 16 * q^59 + 10 * q^61 + 8 * q^67 + 16 * q^71 + 18 * q^73 + 14 * q^77 - 4 * q^79 - 38 * q^85 + 16 * q^91 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	3.37228			1.50813			0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228			1.50813			0.754065	+	0.656800i	$0.228090\pi$
$6$		0			0
$7$		−	0.792287i		−	0.299456i	−0.988727	−	0.149728i	$-0.952160\pi$
$7$		−	0.792287i		−	0.299456i	0.988727	−	0.149728i	$-0.0478399\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.792287i			0.238884i	0.992841	+	0.119442i	$0.0381105\pi$
$11$			0.792287i			0.238884i	−0.992841	+	0.119442i	$0.961890\pi$
$12$		0			0
$13$			5.04868i			1.40025i	0.714020	+	0.700125i	$0.246873\pi$
$13$			5.04868i			1.40025i	−0.714020	+	0.700125i	$0.753127\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.37228			−1.30297			−0.651485	−	0.758662i	$-0.725854\pi$
$17$	−5.37228			−1.30297			−0.651485	+	0.758662i	$0.725854\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.51278i		−	1.77504i	−0.460772	−	0.887518i	$-0.652428\pi$
$23$		−	8.51278i		−	1.77504i	0.460772	−	0.887518i	$-0.347572\pi$
$24$		0			0
$25$	6.37228			1.27446
$26$		0			0
$27$		0			0
$28$		0			0
$29$			10.0974i			1.87503i	0.347943	+	0.937516i	$0.386880\pi$
$29$			10.0974i			1.87503i	−0.347943	+	0.937516i	$0.613120\pi$
$30$		0			0
$31$	8.74456			1.57057			0.785285	−	0.619135i	$-0.212516\pi$
$31$	8.74456			1.57057			0.785285	+	0.619135i	$0.212516\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.67181i		−	0.451619i
$36$		0			0
$37$			5.04868i			0.829997i	0.909822	+	0.414999i	$0.136218\pi$
$37$			5.04868i			0.829997i	−0.909822	+	0.414999i	$0.863782\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		0			0
$43$			9.30506i			1.41901i	0.704701	+	0.709504i	$0.251081\pi$
$43$			9.30506i			1.41901i	−0.704701	+	0.709504i	$0.748919\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.25639i		−	0.620858i	−0.950597	−	0.310429i	$-0.899527\pi$
$47$		−	4.25639i		−	0.620858i	0.950597	−	0.310429i	$-0.100473\pi$
$48$		0			0
$49$	6.37228			0.910326
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.16915i		−	0.435316i	−0.976025	−	0.217658i	$-0.930158\pi$
$53$		−	3.16915i		−	0.435316i	0.976025	−	0.217658i	$-0.0698417\pi$
$54$		0			0
$55$			2.67181i			0.360267i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	5.37228			0.687850			0.343925	−	0.938997i	$-0.388243\pi$
$61$	5.37228			0.687850			0.343925	+	0.938997i	$0.388243\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			17.0256i			2.11176i
$66$		0			0
$67$	−9.48913			−1.15928			−0.579641	−	0.814872i	$-0.696807\pi$
$67$	−9.48913			−1.15928			−0.579641	+	0.814872i	$0.696807\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	−4.11684			−0.481840			−0.240920	−	0.970545i	$-0.577449\pi$
$73$	−4.11684			−0.481840			−0.240920	+	0.970545i	$0.577449\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.627719			0.0715352
$78$		0			0
$79$	4.74456			0.533805			0.266903	−	0.963724i	$-0.414000\pi$
$79$	4.74456			0.533805			0.266903	+	0.963724i	$0.414000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.46410i		−	0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$		−	3.46410i		−	0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$		0			0
$85$	−18.1168			−1.96505
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	13.2665i		−	1.40625i	−0.711068	−	0.703123i	$-0.751788\pi$
$89$		−	13.2665i		−	1.40625i	0.711068	−	0.703123i	$-0.248212\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.4891	+	5.84096i	1.38396	+	0.599270i
$96$		0			0
$97$		−	13.2665i		−	1.34701i	−0.739183	−	0.673504i	$-0.764788\pi$
$97$		−	13.2665i		−	1.34701i	0.739183	−	0.673504i	$-0.235212\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.7446			1.46714			0.733569	−	0.679615i	$-0.237853\pi$
$101$	14.7446			1.46714			0.733569	+	0.679615i	$0.237853\pi$
$102$		0			0
$103$	−12.7446			−1.25576			−0.627880	−	0.778311i	$-0.716077\pi$
$103$	−12.7446			−1.25576			−0.627880	+	0.778311i	$0.716077\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.48913			−0.917348			−0.458674	−	0.888605i	$-0.651675\pi$
$107$	−9.48913			−0.917348			−0.458674	+	0.888605i	$0.651675\pi$
$108$		0			0
$109$		−	8.21782i		−	0.787125i	−0.919298	−	0.393562i	$-0.871243\pi$
$109$		−	8.21782i		−	0.787125i	0.919298	−	0.393562i	$-0.128757\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.75906i			0.353622i	0.984245	+	0.176811i	$0.0565782\pi$
$113$			3.75906i			0.353622i	−0.984245	+	0.176811i	$0.943422\pi$
$114$		0			0
$115$		−	28.7075i		−	2.67699i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.25639i			0.390183i
$120$		0			0
$121$	10.3723			0.942935
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.62772			0.413916
$126$		0			0
$127$	12.7446			1.13090			0.565449	−	0.824784i	$-0.308703\pi$
$127$	12.7446			1.13090			0.565449	+	0.824784i	$0.308703\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.37686i		−	0.207667i	−0.994595	−	0.103834i	$-0.966889\pi$
$131$		−	2.37686i		−	0.207667i	0.994595	−	0.103834i	$-0.0331110\pi$
$132$		0			0
$133$	1.37228	−	3.16915i	0.118992	−	0.274800i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.11684			0.351726			0.175863	−	0.984415i	$-0.443728\pi$
$137$	4.11684			0.351726			0.175863	+	0.984415i	$0.443728\pi$
$138$		0			0
$139$			5.54601i			0.470406i	0.971946	+	0.235203i	$0.0755756\pi$
$139$			5.54601i			0.470406i	−0.971946	+	0.235203i	$0.924424\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.00000			−0.334497
$144$		0			0
$145$			34.0511i			2.82779i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.11684			−0.173419			−0.0867093	−	0.996234i	$-0.527635\pi$
$149$	−2.11684			−0.173419			−0.0867093	+	0.996234i	$0.527635\pi$
$150$		0			0
$151$	−8.74456			−0.711622			−0.355811	−	0.934558i	$-0.615795\pi$
$151$	−8.74456			−0.711622			−0.355811	+	0.934558i	$0.615795\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	29.4891			2.36862
$156$		0			0
$157$	7.48913			0.597697			0.298849	−	0.954301i	$-0.403397\pi$
$157$	7.48913			0.597697			0.298849	+	0.954301i	$0.403397\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.74456			−0.531546
$162$		0			0
$163$			19.8997i			1.55867i	0.626608	+	0.779334i	$0.284443\pi$
$163$			19.8997i			1.55867i	−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.4891			−1.35335			−0.676675	−	0.736282i	$-0.736580\pi$
$167$	−17.4891			−1.35335			−0.676675	+	0.736282i	$0.736580\pi$
$168$		0			0
$169$	−12.4891			−0.960702
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.1947i			1.53537i	0.640825	+	0.767687i	$0.278593\pi$
$173$			20.1947i			1.53537i	−0.640825	+	0.767687i	$0.721407\pi$
$174$		0			0
$175$		−	5.04868i		−	0.381644i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.48913			−0.111302			−0.0556512	−	0.998450i	$-0.517723\pi$
$179$	−1.48913			−0.111302			−0.0556512	+	0.998450i	$0.517723\pi$
$180$		0			0
$181$			5.63858i			0.419113i	0.977797	+	0.209556i	$0.0672019\pi$
$181$			5.63858i			0.419113i	−0.977797	+	0.209556i	$0.932798\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			17.0256i			1.25174i
$186$		0			0
$187$		−	4.25639i		−	0.311258i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.66648i		−	0.265297i	−0.991163	−	0.132649i	$-0.957652\pi$
$191$		−	3.66648i		−	0.265297i	0.991163	−	0.132649i	$-0.0423482\pi$
$192$		0			0
$193$		−	13.8564i		−	0.997406i	−0.866773	−	0.498703i	$-0.833810\pi$
$193$		−	13.8564i		−	0.997406i	0.866773	−	0.498703i	$-0.166190\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.25544			0.374434			0.187217	−	0.982319i	$-0.440053\pi$
$197$	5.25544			0.374434			0.187217	+	0.982319i	$0.440053\pi$
$198$		0			0
$199$		−	21.5769i		−	1.52955i	−0.644300	−	0.764773i	$-0.722851\pi$
$199$		−	21.5769i		−	1.52955i	0.644300	−	0.764773i	$-0.277149\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.00000			0.561490
$204$		0			0
$205$			23.3639i			1.63180i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.37228	+	3.16915i	−0.0949227	+	0.219215i
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			31.3793i			2.14005i
$216$		0			0
$217$		−	6.92820i		−	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	27.1229i		−	1.82448i
$222$		0			0
$223$	0.744563			0.0498596			0.0249298	−	0.999689i	$-0.492064\pi$
$223$	0.744563			0.0498596			0.0249298	+	0.999689i	$0.492064\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	−24.1168			−1.59369			−0.796843	−	0.604186i	$-0.793498\pi$
$229$	−24.1168			−1.59369			−0.796843	+	0.604186i	$0.793498\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.11684			0.269703			0.134852	−	0.990866i	$-0.456944\pi$
$233$	4.11684			0.269703			0.134852	+	0.990866i	$0.456944\pi$
$234$		0			0
$235$		−	14.3537i		−	0.936335i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			29.7947i			1.92726i	0.267240	+	0.963630i	$0.413888\pi$
$239$			29.7947i			1.92726i	−0.267240	+	0.963630i	$0.586112\pi$
$240$		0			0
$241$			17.0256i			1.09671i	0.836245	+	0.548356i	$0.184746\pi$
$241$			17.0256i			1.09671i	−0.836245	+	0.548356i	$0.815254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	21.4891			1.37289
$246$		0			0
$247$	−8.74456	+	20.1947i	−0.556403	+	1.28496i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	6.13592i		−	0.387296i	−0.981071	−	0.193648i	$-0.937968\pi$
$251$		−	6.13592i		−	0.387296i	0.981071	−	0.193648i	$-0.0620319\pi$
$252$		0			0
$253$	6.74456			0.424027
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	3.75906i		−	0.234483i	−0.993103	−	0.117242i	$-0.962595\pi$
$257$		−	3.75906i		−	0.234483i	0.993103	−	0.117242i	$-0.0374052\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.1128i		−	1.11688i	−0.829544	−	0.558441i	$-0.811400\pi$
$263$		−	18.1128i		−	1.11688i	0.829544	−	0.558441i	$-0.188600\pi$
$264$		0			0
$265$		−	10.6873i		−	0.656513i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.6873i			0.651614i	0.945436	+	0.325807i	$0.105636\pi$
$269$			10.6873i			0.651614i	−0.945436	+	0.325807i	$0.894364\pi$
$270$		0			0
$271$		−	13.5615i		−	0.823800i	−0.911229	−	0.411900i	$-0.864865\pi$
$271$		−	13.5615i		−	0.823800i	0.911229	−	0.411900i	$-0.135135\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			5.04868i			0.304447i
$276$		0			0
$277$	−2.62772			−0.157884			−0.0789422	−	0.996879i	$-0.525154\pi$
$277$	−2.62772			−0.157884			−0.0789422	+	0.996879i	$0.525154\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.0974i			0.602357i	0.953568	+	0.301179i	$0.0973801\pi$
$281$			10.0974i			0.602357i	−0.953568	+	0.301179i	$0.902620\pi$
$282$		0			0
$283$		−	27.3253i		−	1.62432i	−0.583435	−	0.812160i	$-0.698292\pi$
$283$		−	27.3253i		−	1.62432i	0.583435	−	0.812160i	$-0.301708\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.48913			0.324013
$288$		0			0
$289$	11.8614			0.697730
$290$		0			0
$291$		0			0
$292$		0			0
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$	13.4891			0.785367
$296$		0			0
$297$		0			0
$298$		0			0
$299$	42.9783			2.48550
$300$		0			0
$301$	7.37228			0.424931
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.1168			1.03737
$306$		0			0
$307$	−17.4891			−0.998157			−0.499079	−	0.866557i	$-0.666328\pi$
$307$	−17.4891			−0.998157			−0.499079	+	0.866557i	$0.666328\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.3537i		−	0.813926i	−0.913445	−	0.406963i	$-0.866588\pi$
$311$		−	14.3537i		−	0.813926i	0.913445	−	0.406963i	$-0.133412\pi$
$312$		0			0
$313$	19.4891			1.10159			0.550795	−	0.834640i	$-0.314325\pi$
$313$	19.4891			1.10159			0.550795	+	0.834640i	$0.314325\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10.6873i		−	0.600256i	−0.953899	−	0.300128i	$-0.902971\pi$
$317$		−	10.6873i		−	0.600256i	0.953899	−	0.300128i	$-0.0970294\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.4891	−	9.30506i	−1.19569	−	0.517748i
$324$		0			0
$325$			32.1716i			1.78456i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.37228			−0.185920
$330$		0			0
$331$	9.48913			0.521569			0.260785	−	0.965397i	$-0.416019\pi$
$331$	9.48913			0.521569			0.260785	+	0.965397i	$0.416019\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−32.0000			−1.74835
$336$		0			0
$337$		−	20.7846i		−	1.13221i	−0.824333	−	0.566105i	$-0.808450\pi$
$337$		−	20.7846i		−	1.13221i	0.824333	−	0.566105i	$-0.191550\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.92820i			0.375183i
$342$		0			0
$343$		−	10.5947i		−	0.572059i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	13.0641i		−	0.701319i	−0.936503	−	0.350659i	$-0.885958\pi$
$347$		−	13.0641i		−	0.701319i	0.936503	−	0.350659i	$-0.114042\pi$
$348$		0			0
$349$	−25.6060			−1.37066			−0.685328	−	0.728234i	$-0.740341\pi$
$349$	−25.6060			−1.37066			−0.685328	+	0.728234i	$0.740341\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	23.4891			1.25020			0.625100	−	0.780545i	$-0.285058\pi$
$353$	23.4891			1.25020			0.625100	+	0.780545i	$0.285058\pi$
$354$		0			0
$355$	13.4891			0.715928
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.7638i		−	0.726427i	−0.931706	−	0.363214i	$-0.881680\pi$
$359$		−	13.7638i		−	0.726427i	0.931706	−	0.363214i	$-0.118320\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.8832			−0.726678
$366$		0			0
$367$			26.8280i			1.40041i	0.713943	+	0.700204i	$0.246908\pi$
$367$			26.8280i			1.40041i	−0.713943	+	0.700204i	$0.753092\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.51087			−0.130358
$372$		0			0
$373$			1.87953i			0.0973183i	0.998815	+	0.0486591i	$0.0154948\pi$
$373$			1.87953i			0.0973183i	−0.998815	+	0.0486591i	$0.984505\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−50.9783			−2.62551
$378$		0			0
$379$	26.9783			1.38578			0.692890	−	0.721043i	$-0.256337\pi$
$379$	26.9783			1.38578			0.692890	+	0.721043i	$0.256337\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−30.9783			−1.58291			−0.791457	−	0.611224i	$-0.790677\pi$
$383$	−30.9783			−1.58291			−0.791457	+	0.611224i	$0.790677\pi$
$384$		0			0
$385$	2.11684			0.107884
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.11684			−0.310136			−0.155068	−	0.987904i	$-0.549560\pi$
$389$	−6.11684			−0.310136			−0.155068	+	0.987904i	$0.549560\pi$
$390$		0			0
$391$			45.7330i			2.31282i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	11.8832			0.596399			0.298199	−	0.954504i	$-0.403614\pi$
$397$	11.8832			0.596399			0.298199	+	0.954504i	$0.403614\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	23.9538i		−	1.19619i	−0.801424	−	0.598097i	$-0.795924\pi$
$401$		−	23.9538i		−	1.19619i	0.801424	−	0.598097i	$-0.204076\pi$
$402$		0			0
$403$			44.1485i			2.19919i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.00000			−0.198273
$408$		0			0
$409$			6.92820i			0.342578i	0.985221	+	0.171289i	$0.0547931\pi$
$409$			6.92820i			0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	3.16915i		−	0.155944i
$414$		0			0
$415$		−	11.6819i		−	0.573443i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			0.294954i			0.0144094i	0.999974	+	0.00720471i	$0.00229335\pi$
$419$			0.294954i			0.0144094i	−0.999974	+	0.00720471i	$0.997707\pi$
$420$		0			0
$421$		−	32.7615i		−	1.59670i	−0.602196	−	0.798349i	$-0.705707\pi$
$421$		−	32.7615i		−	1.59670i	0.602196	−	0.798349i	$-0.294293\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.2337			−1.66058
$426$		0			0
$427$		−	4.25639i		−	0.205981i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.5109			−0.891637			−0.445819	−	0.895123i	$-0.647087\pi$
$431$	−18.5109			−0.891637			−0.445819	+	0.895123i	$0.647087\pi$
$432$		0			0
$433$		−	16.4356i		−	0.789847i	−0.918714	−	0.394923i	$-0.870771\pi$
$433$		−	16.4356i		−	0.789847i	0.918714	−	0.394923i	$-0.129229\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.7446	−	34.0511i	0.705328	−	1.62889i
$438$		0			0
$439$	−39.7228			−1.89587			−0.947933	−	0.318469i	$-0.896831\pi$
$439$	−39.7228			−1.89587			−0.947933	+	0.318469i	$0.896831\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	19.4024i		−	0.921837i	−0.887443	−	0.460918i	$-0.847520\pi$
$443$		−	19.4024i		−	0.921837i	0.887443	−	0.460918i	$-0.152480\pi$
$444$		0			0
$445$		−	44.7384i		−	2.12080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	17.6155i		−	0.831325i	−0.909519	−	0.415663i	$-0.863550\pi$
$449$		−	17.6155i		−	0.831325i	0.909519	−	0.415663i	$-0.136450\pi$
$450$		0			0
$451$	−5.48913			−0.258473
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.4891			0.632380
$456$		0			0
$457$	−8.11684			−0.379690			−0.189845	−	0.981814i	$-0.560799\pi$
$457$	−8.11684			−0.379690			−0.189845	+	0.981814i	$0.560799\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−32.6277			−1.51962			−0.759812	−	0.650143i	$-0.774709\pi$
$461$	−32.6277			−1.51962			−0.759812	+	0.650143i	$0.774709\pi$
$462$		0			0
$463$		−	0.202380i		−	0.00940538i	−0.999989	−	0.00470269i	$-0.998503\pi$
$463$		−	0.202380i		−	0.00940538i	0.999989	−	0.00470269i	$-0.00149692\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.0843i			1.43841i	0.694797	+	0.719206i	$0.255494\pi$
$467$			31.0843i			1.43841i	−0.694797	+	0.719206i	$0.744506\pi$
$468$		0			0
$469$			7.51811i			0.347154i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.37228			−0.338978
$474$		0			0
$475$	25.4891	+	11.0371i	1.16952	+	0.506418i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			21.7793i			0.995121i	0.867429	+	0.497560i	$0.165771\pi$
$479$			21.7793i			0.995121i	−0.867429	+	0.497560i	$0.834229\pi$
$480$		0			0
$481$	−25.4891			−1.16220
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	44.7384i		−	2.03146i
$486$		0			0
$487$	5.76631			0.261297			0.130648	−	0.991429i	$-0.458294\pi$
$487$	5.76631			0.261297			0.130648	+	0.991429i	$0.458294\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.0796i		−	0.951307i	−0.879633	−	0.475654i	$-0.842212\pi$
$491$		−	21.0796i		−	0.951307i	0.879633	−	0.475654i	$-0.157788\pi$
$492$		0			0
$493$		−	54.2458i		−	2.44311i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3.16915i		−	0.142156i
$498$		0			0
$499$		−	4.55134i		−	0.203746i	−0.994797	−	0.101873i	$-0.967516\pi$
$499$		−	4.55134i		−	0.203746i	0.994797	−	0.101873i	$-0.0324835\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	16.0309i		−	0.714782i	−0.933955	−	0.357391i	$-0.883666\pi$
$503$		−	16.0309i		−	0.714782i	0.933955	−	0.357391i	$-0.116334\pi$
$504$		0			0
$505$	49.7228			2.21264
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.2665i			0.588027i	0.955801	+	0.294014i	$0.0949911\pi$
$509$			13.2665i			0.588027i	−0.955801	+	0.294014i	$0.905009\pi$
$510$		0			0
$511$			3.26172i			0.144290i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−42.9783			−1.89385
$516$		0			0
$517$	3.37228			0.148313
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.75906i			0.164687i	0.996604	+	0.0823436i	$0.0262405\pi$
$521$			3.75906i			0.164687i	−0.996604	+	0.0823436i	$0.973760\pi$
$522$		0			0
$523$	32.0000			1.39926			0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000			1.39926			0.699631	+	0.714504i	$0.253348\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−46.9783			−2.04640
$528$		0			0
$529$	−49.4674			−2.15076
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.9783			−1.51508
$534$		0			0
$535$	−32.0000			−1.38348
$536$		0			0
$537$		0			0
$538$		0			0
$539$			5.04868i			0.217462i
$540$		0			0
$541$	−4.11684			−0.176997			−0.0884985	−	0.996076i	$-0.528207\pi$
$541$	−4.11684			−0.176997			−0.0884985	+	0.996076i	$0.528207\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	27.7128i		−	1.18709i
$546$		0			0
$547$	−29.4891			−1.26086			−0.630432	−	0.776245i	$-0.717122\pi$
$547$	−29.4891			−1.26086			−0.630432	+	0.776245i	$0.717122\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−17.4891	+	40.3894i	−0.745062	+	1.72065i
$552$		0			0
$553$		−	3.75906i		−	0.159851i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.1386			−0.471957			−0.235979	−	0.971758i	$-0.575830\pi$
$557$	−11.1386			−0.471957			−0.235979	+	0.971758i	$0.575830\pi$
$558$		0			0
$559$	−46.9783			−1.98697
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.9783			0.968418			0.484209	−	0.874952i	$-0.339107\pi$
$563$	22.9783			0.968418			0.484209	+	0.874952i	$0.339107\pi$
$564$		0			0
$565$			12.6766i			0.533308i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			27.1229i			1.13705i	0.822666	+	0.568526i	$0.192486\pi$
$569$			27.1229i			1.13705i	−0.822666	+	0.568526i	$0.807514\pi$
$570$		0			0
$571$		−	2.87419i		−	0.120281i	−0.998190	−	0.0601406i	$-0.980845\pi$
$571$		−	2.87419i		−	0.120281i	0.998190	−	0.0601406i	$-0.0191549\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	54.2458i		−	2.26221i
$576$		0			0
$577$	−37.6060			−1.56556			−0.782778	−	0.622300i	$-0.786198\pi$
$577$	−37.6060			−1.56556			−0.782778	+	0.622300i	$0.786198\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.74456			−0.113864
$582$		0			0
$583$	2.51087			0.103990
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.38219i			0.0570493i	0.999593	+	0.0285246i	$0.00908090\pi$
$587$			1.38219i			0.0570493i	−0.999593	+	0.0285246i	$0.990919\pi$
$588$		0			0
$589$	34.9783	+	15.1460i	1.44125	+	0.624081i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23.4891			0.964583			0.482291	−	0.876011i	$-0.339805\pi$
$593$	23.4891			0.964583			0.482291	+	0.876011i	$0.339805\pi$
$594$		0			0
$595$			14.3537i			0.588446i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−44.4674			−1.81689			−0.908444	−	0.418007i	$-0.862729\pi$
$599$	−44.4674			−1.81689			−0.908444	+	0.418007i	$0.862729\pi$
$600$		0			0
$601$		−	19.6048i		−	0.799696i	−0.916581	−	0.399848i	$-0.869063\pi$
$601$		−	19.6048i		−	0.799696i	0.916581	−	0.399848i	$-0.130937\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	34.9783			1.42207
$606$		0			0
$607$	−40.7446			−1.65377			−0.826885	−	0.562370i	$-0.809889\pi$
$607$	−40.7446			−1.65377			−0.826885	+	0.562370i	$0.809889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21.4891			0.869357
$612$		0			0
$613$	5.37228			0.216984			0.108492	−	0.994097i	$-0.465398\pi$
$613$	5.37228			0.216984			0.108492	+	0.994097i	$0.465398\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.0951			0.768740			0.384370	−	0.923179i	$-0.374419\pi$
$617$	19.0951			0.768740			0.384370	+	0.923179i	$0.374419\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.5109			−0.421109
$624$		0			0
$625$	−16.2554			−0.650217
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	27.1229i		−	1.08146i
$630$		0			0
$631$			42.7663i			1.70250i	0.524762	+	0.851249i	$0.324154\pi$
$631$			42.7663i			1.70250i	−0.524762	+	0.851249i	$0.675846\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	42.9783			1.70554
$636$		0			0
$637$			32.1716i			1.27468i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13.2665i		−	0.523995i	−0.965069	−	0.261998i	$-0.915619\pi$
$641$		−	13.2665i		−	0.523995i	0.965069	−	0.261998i	$-0.0843813\pi$
$642$		0			0
$643$		−	0.792287i		−	0.0312447i	−0.999878	−	0.0156224i	$-0.995027\pi$
$643$		−	0.792287i		−	0.0312447i	0.999878	−	0.0156224i	$-0.00497296\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.7229i			1.44373i	0.692035	+	0.721864i	$0.256714\pi$
$647$			36.7229i			1.44373i	−0.692035	+	0.721864i	$0.743286\pi$
$648$		0			0
$649$			3.16915i			0.124400i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.1168			−0.865499			−0.432749	−	0.901514i	$-0.642457\pi$
$653$	−22.1168			−0.865499			−0.432749	+	0.901514i	$0.642457\pi$
$654$		0			0
$655$		−	8.01544i		−	0.313189i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.48913			−0.369644			−0.184822	−	0.982772i	$-0.559171\pi$
$659$	−9.48913			−0.369644			−0.184822	+	0.982772i	$0.559171\pi$
$660$		0			0
$661$		−	46.0280i		−	1.79028i	−0.445784	−	0.895141i	$-0.647075\pi$
$661$		−	46.0280i		−	1.79028i	0.445784	−	0.895141i	$-0.352925\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.62772	−	10.6873i	0.179455	−	0.414434i
$666$		0			0
$667$	85.9565			3.32825
$668$		0			0
$669$		0			0
$670$		0			0
$671$			4.25639i			0.164316i
$672$		0			0
$673$		−	14.4463i		−	0.556864i	−0.960456	−	0.278432i	$-0.910185\pi$
$673$		−	14.4463i		−	0.556864i	0.960456	−	0.278432i	$-0.0898148\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	21.3745i		−	0.821489i	−0.911750	−	0.410745i	$-0.865269\pi$
$677$		−	21.3745i		−	0.821489i	0.911750	−	0.410745i	$-0.134731\pi$
$678$		0			0
$679$	−10.5109			−0.403370
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.97825			−0.267015			−0.133508	−	0.991048i	$-0.542624\pi$
$683$	−6.97825			−0.267015			−0.133508	+	0.991048i	$0.542624\pi$
$684$		0			0
$685$	13.8832			0.530448
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.0000			0.609551
$690$		0			0
$691$		−	27.9152i		−	1.06194i	−0.847389	−	0.530972i	$-0.821827\pi$
$691$		−	27.9152i		−	1.06194i	0.847389	−	0.530972i	$-0.178173\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			18.7027i			0.709434i
$696$		0			0
$697$		−	37.2203i		−	1.40982i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−37.7228			−1.42477			−0.712385	−	0.701788i	$-0.752385\pi$
$701$	−37.7228			−1.42477			−0.712385	+	0.701788i	$0.752385\pi$
$702$		0			0
$703$	−8.74456	+	20.1947i	−0.329807	+	0.761658i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	11.6819i		−	0.439344i
$708$		0			0
$709$	43.9565			1.65082			0.825411	−	0.564533i	$-0.190944\pi$
$709$	43.9565			1.65082			0.825411	+	0.564533i	$0.190944\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	74.4405i		−	2.78782i
$714$		0			0
$715$	−13.4891			−0.504465
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	13.7638i		−	0.513304i	−0.966504	−	0.256652i	$-0.917381\pi$
$719$		−	13.7638i		−	0.513304i	0.966504	−	0.256652i	$-0.0826195\pi$
$720$		0			0
$721$			10.0974i			0.376045i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			64.3432i			2.38965i
$726$		0			0
$727$		−	24.7460i		−	0.917780i	−0.888493	−	0.458890i	$-0.848247\pi$
$727$		−	24.7460i		−	0.917780i	0.888493	−	0.458890i	$-0.151753\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	49.9894i		−	1.84893i
$732$		0			0
$733$	−20.9783			−0.774849			−0.387425	−	0.921901i	$-0.626635\pi$
$733$	−20.9783			−0.774849			−0.387425	+	0.921901i	$0.626635\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	7.51811i		−	0.276933i
$738$		0			0
$739$		−	38.0125i		−	1.39831i	−0.714968	−	0.699157i	$-0.753559\pi$
$739$		−	38.0125i		−	1.39831i	0.714968	−	0.699157i	$-0.246441\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	34.9783			1.28323			0.641614	−	0.767028i	$-0.278265\pi$
$743$	34.9783			1.28323			0.641614	+	0.767028i	$0.278265\pi$
$744$		0			0
$745$	−7.13859			−0.261538
$746$		0			0
$747$		0			0
$748$		0			0
$749$			7.51811i			0.274706i
$750$		0			0
$751$	15.7228			0.573734			0.286867	−	0.957970i	$-0.407386\pi$
$751$	15.7228			0.573734			0.286867	+	0.957970i	$0.407386\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.4891			−1.07322
$756$		0			0
$757$	10.8614			0.394765			0.197382	−	0.980327i	$-0.436756\pi$
$757$	10.8614			0.394765			0.197382	+	0.980327i	$0.436756\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.86141			−0.248726			−0.124363	−	0.992237i	$-0.539689\pi$
$761$	−6.86141			−0.248726			−0.124363	+	0.992237i	$0.539689\pi$
$762$		0			0
$763$	−6.51087			−0.235709
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20.1947i			0.729188i
$768$		0			0
$769$	−9.13859			−0.329546			−0.164773	−	0.986332i	$-0.552689\pi$
$769$	−9.13859			−0.329546			−0.164773	+	0.986332i	$0.552689\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			39.7995i			1.43149i	0.698363	+	0.715744i	$0.253912\pi$
$773$			39.7995i			1.43149i	−0.698363	+	0.715744i	$0.746088\pi$
$774$		0			0
$775$	55.7228			2.00162
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	+	27.7128i	−0.429945	+	0.992915i
$780$		0			0
$781$			3.16915i			0.113401i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25.2554			0.901405
$786$		0			0
$787$	10.9783			0.391332			0.195666	−	0.980671i	$-0.437313\pi$
$787$	10.9783			0.391332			0.195666	+	0.980671i	$0.437313\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.97825			0.105894
$792$		0			0
$793$			27.1229i			0.963163i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			10.0974i			0.357667i	0.983879	+	0.178833i	$0.0572323\pi$
$797$			10.0974i			0.357667i	−0.983879	+	0.178833i	$0.942768\pi$
$798$		0			0
$799$			22.8665i			0.808959i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	3.26172i		−	0.115104i
$804$		0			0
$805$	−22.7446			−0.801640
$806$		0			0
$807$		0			0
$808$		0			0
$809$	32.1168			1.12917			0.564584	−	0.825376i	$-0.309037\pi$
$809$	32.1168			1.12917			0.564584	+	0.825376i	$0.309037\pi$
$810$		0			0
$811$	−32.4674			−1.14008			−0.570042	−	0.821616i	$-0.693073\pi$
$811$	−32.4674			−1.14008			−0.570042	+	0.821616i	$0.693073\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			67.1076i			2.35067i
$816$		0			0
$817$	−16.1168	+	37.2203i	−0.563857	+	1.30217i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16.6277			−0.580311			−0.290156	−	0.956979i	$-0.593707\pi$
$821$	−16.6277			−0.580311			−0.290156	+	0.956979i	$0.593707\pi$
$822$		0			0
$823$			16.2333i			0.565856i	0.959141	+	0.282928i	$0.0913058\pi$
$823$			16.2333i			0.565856i	−0.959141	+	0.282928i	$0.908694\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.48913			0.0517820			0.0258910	−	0.999665i	$-0.491758\pi$
$827$	1.48913			0.0517820			0.0258910	+	0.999665i	$0.491758\pi$
$828$		0			0
$829$			43.4487i			1.50904i	0.656279	+	0.754519i	$0.272130\pi$
$829$			43.4487i			1.50904i	−0.656279	+	0.754519i	$0.727870\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.2337			−1.18613
$834$		0			0
$835$	−58.9783			−2.04103
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−72.9565			−2.51574
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−42.1168			−1.44886
$846$		0			0
$847$		−	8.21782i		−	0.282368i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	42.9783			1.47328
$852$		0			0
$853$	22.0000			0.753266			0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000			0.753266			0.376633	+	0.926363i	$0.377082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	47.3176i		−	1.61634i	−0.588950	−	0.808169i	$-0.700459\pi$
$857$		−	47.3176i		−	1.61634i	0.588950	−	0.808169i	$-0.299541\pi$
$858$		0			0
$859$		−	52.4589i		−	1.78987i	−0.446193	−	0.894937i	$-0.647221\pi$
$859$		−	52.4589i		−	1.78987i	0.446193	−	0.894937i	$-0.352779\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	56.4674			1.92217			0.961086	−	0.276250i	$-0.0890917\pi$
$863$	56.4674			1.92217			0.961086	+	0.276250i	$0.0890917\pi$
$864$		0			0
$865$			68.1022i			2.31554i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			3.75906i			0.127517i
$870$		0			0
$871$		−	47.9075i		−	1.62328i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	3.66648i		−	0.123950i
$876$		0			0
$877$		−	33.3514i		−	1.12620i	−0.826390	−	0.563098i	$-0.809609\pi$
$877$		−	33.3514i		−	1.12620i	0.826390	−	0.563098i	$-0.190391\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.6060			1.13221			0.566107	−	0.824332i	$-0.308449\pi$
$881$	33.6060			1.13221			0.566107	+	0.824332i	$0.308449\pi$
$882$		0			0
$883$			6.13592i			0.206490i	0.994656	+	0.103245i	$0.0329226\pi$
$883$			6.13592i			0.206490i	−0.994656	+	0.103245i	$0.967077\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000			−0.940148			−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000			−0.940148			−0.470074	+	0.882627i	$0.655773\pi$
$888$		0			0
$889$		−	10.0974i		−	0.338654i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.37228	−	17.0256i	0.246704	−	0.569738i
$894$		0			0
$895$	−5.02175			−0.167859
$896$		0			0
$897$		0			0
$898$		0			0
$899$			88.2969i			2.94487i
$900$		0			0
$901$			17.0256i			0.567203i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			19.0149i			0.632076i
$906$		0			0
$907$	−48.4674			−1.60933			−0.804666	−	0.593727i	$-0.797656\pi$
$907$	−48.4674			−1.60933			−0.804666	+	0.593727i	$0.797656\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.4891			−0.844492			−0.422246	−	0.906481i	$-0.638758\pi$
$911$	−25.4891			−0.844492			−0.422246	+	0.906481i	$0.638758\pi$
$912$		0			0
$913$	2.74456			0.0908318
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.88316			−0.0621873
$918$		0			0
$919$		−	51.3716i		−	1.69459i	−0.531120	−	0.847297i	$-0.678229\pi$
$919$		−	51.3716i		−	1.69459i	0.531120	−	0.847297i	$-0.321771\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			20.1947i			0.664717i
$924$		0			0
$925$			32.1716i			1.05780i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	55.4891			1.82054			0.910270	−	0.414016i	$-0.135874\pi$
$929$	55.4891			1.82054			0.910270	+	0.414016i	$0.135874\pi$
$930$		0			0
$931$	25.4891	+	11.0371i	0.835372	+	0.361727i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	14.3537i		−	0.469417i
$936$		0			0
$937$	17.3723			0.567528			0.283764	−	0.958894i	$-0.408417\pi$
$937$	17.3723			0.567528			0.283764	+	0.958894i	$0.408417\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	3.75906i		−	0.122542i	−0.998121	−	0.0612708i	$-0.980485\pi$
$941$		−	3.75906i		−	0.122542i	0.998121	−	0.0612708i	$-0.0195153\pi$
$942$		0			0
$943$	58.9783			1.92060
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.9715i			0.421519i	0.977538	+	0.210759i	$0.0675936\pi$
$947$			12.9715i			0.421519i	−0.977538	+	0.210759i	$0.932406\pi$
$948$		0			0
$949$		−	20.7846i		−	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			37.8102i			1.22479i	0.790551	+	0.612396i	$0.209794\pi$
$953$			37.8102i			1.22479i	−0.790551	+	0.612396i	$0.790206\pi$
$954$		0			0
$955$		−	12.3644i		−	0.400103i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	3.26172i		−	0.105326i
$960$		0			0
$961$	45.4674			1.46669
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	46.7277i		−	1.50422i
$966$		0			0
$967$		−	16.1407i		−	0.519050i	−0.965736	−	0.259525i	$-0.916434\pi$
$967$		−	16.1407i		−	0.519050i	0.965736	−	0.259525i	$-0.0835659\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−44.4674			−1.42703			−0.713513	−	0.700642i	$-0.752897\pi$
$971$	−44.4674			−1.42703			−0.713513	+	0.700642i	$0.752897\pi$
$972$		0			0
$973$	4.39403			0.140866
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	19.6048i		−	0.627213i	−0.949553	−	0.313606i	$-0.898463\pi$
$977$		−	19.6048i		−	0.627213i	0.949553	−	0.313606i	$-0.101537\pi$
$978$		0			0
$979$	10.5109			0.335929
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.02175			0.160169			0.0800845	−	0.996788i	$-0.474481\pi$
$983$	5.02175			0.160169			0.0800845	+	0.996788i	$0.474481\pi$
$984$		0			0
$985$	17.7228			0.564696
$986$		0			0
$987$		0			0
$988$		0			0
$989$	79.2119			2.51879
$990$		0			0
$991$	8.74456			0.277780			0.138890	−	0.990308i	$-0.455647\pi$
$991$	8.74456			0.277780			0.138890	+	0.990308i	$0.455647\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	72.7634i		−	2.30675i
$996$		0			0
$997$	−53.6060			−1.69772			−0.848859	−	0.528619i	$-0.822710\pi$
$997$	−53.6060			−1.69772			−0.848859	+	0.528619i	$0.822710\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.k.o.2431.3		4
3.2	odd	2		912.2.k.h.607.1	yes	4
4.3	odd	2		2736.2.k.n.2431.4		4
12.11	even	2		912.2.k.g.607.2	yes	4
19.18	odd	2		2736.2.k.n.2431.3		4
24.5	odd	2		3648.2.k.g.2431.3		4
24.11	even	2		3648.2.k.h.2431.4		4
57.56	even	2		912.2.k.g.607.1	✓	4
76.75	even	2	inner	2736.2.k.o.2431.4		4
228.227	odd	2		912.2.k.h.607.2	yes	4
456.227	odd	2		3648.2.k.g.2431.4		4
456.341	even	2		3648.2.k.h.2431.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.k.g.607.1	✓	4	57.56	even	2
912.2.k.g.607.2	yes	4	12.11	even	2
912.2.k.h.607.1	yes	4	3.2	odd	2
912.2.k.h.607.2	yes	4	228.227	odd	2
2736.2.k.n.2431.3		4	19.18	odd	2
2736.2.k.n.2431.4		4	4.3	odd	2
2736.2.k.o.2431.3		4	1.1	even	1	trivial
2736.2.k.o.2431.4		4	76.75	even	2	inner
3648.2.k.g.2431.3		4	24.5	odd	2
3648.2.k.g.2431.4		4	456.227	odd	2
3648.2.k.h.2431.3		4	456.341	even	2
3648.2.k.h.2431.4		4	24.11	even	2

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.37228 q^{5} -0.792287i q^{7} +O(q^{10})\)	\(q+3.37228 q^{5} -0.792287i q^{7} +0.792287i q^{11} +5.04868i q^{13} -5.37228 q^{17} +(4.00000 + 1.73205i) q^{19} -8.51278i q^{23} +6.37228 q^{25} +10.0974i q^{29} +8.74456 q^{31} -2.67181i q^{35} +5.04868i q^{37} +6.92820i q^{41} +9.30506i q^{43} -4.25639i q^{47} +6.37228 q^{49} -3.16915i q^{53} +2.67181i q^{55} +4.00000 q^{59} +5.37228 q^{61} +17.0256i q^{65} -9.48913 q^{67} +4.00000 q^{71} -4.11684 q^{73} +0.627719 q^{77} +4.74456 q^{79} -3.46410i q^{83} -18.1168 q^{85} -13.2665i q^{89} +4.00000 q^{91} +(13.4891 + 5.84096i) q^{95} -13.2665i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5}+O(q^{10}) \) 4 * q + 2 * q^5	\( 4 q + 2 q^{5} - 10 q^{17} + 16 q^{19} + 14 q^{25} + 12 q^{31} + 14 q^{49} + 16 q^{59} + 10 q^{61} + 8 q^{67} + 16 q^{71} + 18 q^{73} + 14 q^{77} - 4 q^{79} - 38 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) \) 4 * q + 2 * q^5 - 10 * q^17 + 16 * q^19 + 14 * q^25 + 12 * q^31 + 14 * q^49 + 16 * q^59 + 10 * q^61 + 8 * q^67 + 16 * q^71 + 18 * q^73 + 14 * q^77 - 4 * q^79 - 38 * q^85 + 16 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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