LMFDB - Newform orbit 936.2.r.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,2,Mod(601,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("936.601"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.r (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{9} + ( - 4 \zeta_{6} + 3) q^{13} + (\zeta_{6} + 1) q^{15} + 5 \zeta_{6} q^{17} + \zeta_{6} q^{19} + (3 \zeta_{6} + 3) q^{21}+ \cdots + (3 \zeta_{6} - 3) q^{97}+O(q^{100}) $ q + (-2z + 1) q^3 + (z - 1) * q^5 + (3z - 3) q^7 - 3 * q^9 + (-4z + 3) q^13 + (z + 1) * q^15 + 5z q^17 + z * q^19 + (3z + 3) q^21 + 3z q^23 + 4z q^25 + (6z - 3) q^27 + 6 * q^29 + (3z - 3) q^31 - 3z q^35 + (z - 1) * q^37 + (-2z - 5) q^39 + 9z q^41 + (9z - 9) q^43 + (-3z + 3) q^45 - z * q^47 - 2z q^49 + (-5z + 10) q^51 + 6 * q^53 + (-z + 2) * q^57 - 4 * q^59 + (-11z + 11) q^61 + (-9z + 9) q^63 + (3z + 1) q^65 - 7z q^67 + (-3z + 6) q^69 + 3z q^71 - 2 * q^73 + (-4z + 8) q^75 - 9z q^79 + 9 * q^81 + 5z q^83 - 5 * q^85 + (-12z + 6) q^87 + (15z - 15) q^89 + (9z + 3) q^91 + (3z + 3) q^93 - q^95 + (3z - 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 3 q^{7} - 6 q^{9} + 2 q^{13} + 3 q^{15} + 5 q^{17} + q^{19} + 9 q^{21} + 3 q^{23} + 4 q^{25} + 12 q^{29} - 3 q^{31} - 3 q^{35} - q^{37} - 12 q^{39} + 9 q^{41} - 9 q^{43} + 3 q^{45} - q^{47}+ \cdots - 3 q^{97}+O(q^{100}) $ 2 * q - q^5 - 3 * q^7 - 6 * q^9 + 2 * q^13 + 3 * q^15 + 5 * q^17 + q^19 + 9 * q^21 + 3 * q^23 + 4 * q^25 + 12 * q^29 - 3 * q^31 - 3 * q^35 - q^37 - 12 * q^39 + 9 * q^41 - 9 * q^43 + 3 * q^45 - q^47 - 2 * q^49 + 15 * q^51 + 12 * q^53 + 3 * q^57 - 8 * q^59 + 11 * q^61 + 9 * q^63 + 5 * q^65 - 7 * q^67 + 9 * q^69 + 3 * q^71 - 4 * q^73 + 12 * q^75 - 9 * q^79 + 18 * q^81 + 5 * q^83 - 10 * q^85 - 15 * q^89 + 15 * q^91 + 9 * q^93 - 2 * q^95 - 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-\zeta_{6}$	$-1 + \zeta_{6}$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

601.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.73205i

−0.500000

−

0.866025i

−1.50000

−

2.59808i

−3.00000

841.1

−

1.73205i

−0.500000

0.866025i

−1.50000

2.59808i

−3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	936.2.r.c	✓	2
3.b	odd	2	1		2808.2.r.b		2
9.c	even	3	1		936.2.s.c	yes	2
9.d	odd	6	1		2808.2.s.b		2
13.c	even	3	1		936.2.s.c	yes	2
39.i	odd	6	1		2808.2.s.b		2
117.h	even	3	1	inner	936.2.r.c	✓	2
117.k	odd	6	1		2808.2.r.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
936.2.r.c	✓	2	1.a	even	1	1	trivial
936.2.r.c	✓	2	117.h	even	3	1	inner
936.2.s.c	yes	2	9.c	even	3	1
936.2.s.c	yes	2	13.c	even	3	1
2808.2.r.b		2	3.b	odd	2	1
2808.2.r.b		2	117.k	odd	6	1
2808.2.s.b		2	9.d	odd	6	1
2808.2.s.b		2	39.i	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(936, [\chi])$:

$ T_{5}^{2} + T_{5} + 1 $

T5^2 + T5 + 1

$ T_{7}^{2} + 3T_{7} + 9 $

T7^2 + 3*T7 + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 3 $ T^2 + 3
$5$	$ T^{2} + T + 1 $ T^2 + T + 1
$7$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
$11$	$ T^{2} $ T^2
$13$	$ T^{2} - 2T + 13 $ T^2 - 2*T + 13
$17$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$19$	$ T^{2} - T + 1 $ T^2 - T + 1
$23$	$ T^{2} - 3T + 9 $ T^2 - 3*T + 9
$29$	$ (T - 6)^{2} $ (T - 6)^2
$31$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
$37$	$ T^{2} + T + 1 $ T^2 + T + 1
$41$	$ T^{2} - 9T + 81 $ T^2 - 9*T + 81
$43$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$47$	$ T^{2} + T + 1 $ T^2 + T + 1
$53$	$ (T - 6)^{2} $ (T - 6)^2
$59$	$ (T + 4)^{2} $ (T + 4)^2
$61$	$ T^{2} - 11T + 121 $ T^2 - 11*T + 121
$67$	$ T^{2} + 7T + 49 $ T^2 + 7*T + 49
$71$	$ T^{2} - 3T + 9 $ T^2 - 3*T + 9
$73$	$ (T + 2)^{2} $ (T + 2)^2
$79$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$83$	$ T^{2} - 5T + 25 $ T^2 - 5*T + 25
$89$	$ T^{2} + 15T + 225 $ T^2 + 15*T + 225
$97$	$ T^{2} + 3T + 9 $ T^2 + 3*T + 9
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Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.r (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{5} + (3 \zeta_{6} - 3) q^{7} - 3 q^{9} + ( - 4 \zeta_{6} + 3) q^{13} + (\zeta_{6} + 1) q^{15} + 5 \zeta_{6} q^{17} + \zeta_{6} q^{19} + (3 \zeta_{6} + 3) q^{21}+ \cdots + (3 \zeta_{6} - 3) q^{97}+O(q^{100}) \) q + (-2z + 1) q^3 + (z - 1) * q^5 + (3z - 3) q^7 - 3 * q^9 + (-4z + 3) q^13 + (z + 1) * q^15 + 5z q^17 + z * q^19 + (3z + 3) q^21 + 3z q^23 + 4z q^25 + (6z - 3) q^27 + 6 * q^29 + (3z - 3) q^31 - 3z q^35 + (z - 1) * q^37 + (-2z - 5) q^39 + 9z q^41 + (9z - 9) q^43 + (-3z + 3) q^45 - z * q^47 - 2z q^49 + (-5z + 10) q^51 + 6 * q^53 + (-z + 2) * q^57 - 4 * q^59 + (-11z + 11) q^61 + (-9z + 9) q^63 + (3z + 1) q^65 - 7z q^67 + (-3z + 6) q^69 + 3z q^71 - 2 * q^73 + (-4z + 8) q^75 - 9z q^79 + 9 * q^81 + 5z q^83 - 5 * q^85 + (-12z + 6) q^87 + (15z - 15) q^89 + (9z + 3) q^91 + (3z + 3) q^93 - q^95 + (3z - 3) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 3 q^{7} - 6 q^{9} + 2 q^{13} + 3 q^{15} + 5 q^{17} + q^{19} + 9 q^{21} + 3 q^{23} + 4 q^{25} + 12 q^{29} - 3 q^{31} - 3 q^{35} - q^{37} - 12 q^{39} + 9 q^{41} - 9 q^{43} + 3 q^{45} - q^{47}+ \cdots - 3 q^{97}+O(q^{100}) \) 2 * q - q^5 - 3 * q^7 - 6 * q^9 + 2 * q^13 + 3 * q^15 + 5 * q^17 + q^19 + 9 * q^21 + 3 * q^23 + 4 * q^25 + 12 * q^29 - 3 * q^31 - 3 * q^35 - q^37 - 12 * q^39 + 9 * q^41 - 9 * q^43 + 3 * q^45 - q^47 - 2 * q^49 + 15 * q^51 + 12 * q^53 + 3 * q^57 - 8 * q^59 + 11 * q^61 + 9 * q^63 + 5 * q^65 - 7 * q^67 + 9 * q^69 + 3 * q^71 - 4 * q^73 + 12 * q^75 - 9 * q^79 + 18 * q^81 + 5 * q^83 - 10 * q^85 - 15 * q^89 + 15 * q^91 + 9 * q^93 - 2 * q^95 - 3 * q^97

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