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Properties

Label 1350.2.j.f
Level $1350$
Weight $2$
Character orbit 1350.j
Analytic conductor $10.780$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,2,Mod(199,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{4} - \beta_{7} q^{7} + (\beta_{5} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{4} - \beta_{7} q^{7} + (\beta_{5} + \beta_1) q^{8} + ( - 2 \beta_{3} + \beta_{2} - 2) q^{11} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{13}+ \cdots + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 6 q^{11} + 2 q^{14} - 4 q^{16} - 4 q^{19} - 8 q^{26} + 6 q^{29} + 4 q^{31} + 18 q^{34} - 12 q^{41} - 12 q^{44} + 12 q^{46} + 6 q^{49} - 2 q^{56} - 6 q^{59} - 2 q^{61} - 8 q^{64} + 48 q^{71} - 16 q^{74} - 2 q^{76} + 8 q^{79} + 34 q^{86} + 60 q^{89} - 136 q^{91} + 18 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.396143 + 1.68614i
1.26217 − 1.18614i
−1.26217 + 1.18614i
0.396143 − 1.68614i
−0.396143 − 1.68614i
1.26217 + 1.18614i
−1.26217 − 1.18614i
0.396143 + 1.68614i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 −2.92048 + 1.68614i 1.00000i 0 0
199.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 2.05446 − 1.18614i 1.00000i 0 0
199.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 −2.05446 + 1.18614i − 1.00000i 0 0
199.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 2.92048 − 1.68614i − 1.00000i 0 0
1099.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 −2.92048 − 1.68614i − 1.00000i 0 0
1099.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 2.05446 + 1.18614i − 1.00000i 0 0
1099.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −2.05446 − 1.18614i 1.00000i 0 0
1099.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 2.92048 + 1.68614i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.j.f 8
3.b odd 2 1 450.2.j.g 8
5.b even 2 1 inner 1350.2.j.f 8
5.c odd 4 1 270.2.e.c 4
5.c odd 4 1 1350.2.e.l 4
9.c even 3 1 inner 1350.2.j.f 8
9.c even 3 1 4050.2.c.ba 4
9.d odd 6 1 450.2.j.g 8
9.d odd 6 1 4050.2.c.v 4
15.d odd 2 1 450.2.j.g 8
15.e even 4 1 90.2.e.c ✓ 4
15.e even 4 1 450.2.e.j 4
20.e even 4 1 2160.2.q.f 4
45.h odd 6 1 450.2.j.g 8
45.h odd 6 1 4050.2.c.v 4
45.j even 6 1 inner 1350.2.j.f 8
45.j even 6 1 4050.2.c.ba 4
45.k odd 12 1 270.2.e.c 4
45.k odd 12 1 810.2.a.k 2
45.k odd 12 1 1350.2.e.l 4
45.k odd 12 1 4050.2.a.bo 2
45.l even 12 1 90.2.e.c ✓ 4
45.l even 12 1 450.2.e.j 4
45.l even 12 1 810.2.a.i 2
45.l even 12 1 4050.2.a.bw 2
60.l odd 4 1 720.2.q.f 4
180.v odd 12 1 720.2.q.f 4
180.v odd 12 1 6480.2.a.be 2
180.x even 12 1 2160.2.q.f 4
180.x even 12 1 6480.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c ✓ 4 15.e even 4 1
90.2.e.c ✓ 4 45.l even 12 1
270.2.e.c 4 5.c odd 4 1
270.2.e.c 4 45.k odd 12 1
450.2.e.j 4 15.e even 4 1
450.2.e.j 4 45.l even 12 1
450.2.j.g 8 3.b odd 2 1
450.2.j.g 8 9.d odd 6 1
450.2.j.g 8 15.d odd 2 1
450.2.j.g 8 45.h odd 6 1
720.2.q.f 4 60.l odd 4 1
720.2.q.f 4 180.v odd 12 1
810.2.a.i 2 45.l even 12 1
810.2.a.k 2 45.k odd 12 1
1350.2.e.l 4 5.c odd 4 1
1350.2.e.l 4 45.k odd 12 1
1350.2.j.f 8 1.a even 1 1 trivial
1350.2.j.f 8 5.b even 2 1 inner
1350.2.j.f 8 9.c even 3 1 inner
1350.2.j.f 8 45.j even 6 1 inner
2160.2.q.f 4 20.e even 4 1
2160.2.q.f 4 180.x even 12 1
4050.2.a.bo 2 45.k odd 12 1
4050.2.a.bw 2 45.l even 12 1
4050.2.c.v 4 9.d odd 6 1
4050.2.c.v 4 45.h odd 6 1
4050.2.c.ba 4 9.c even 3 1
4050.2.c.ba 4 45.j even 6 1
6480.2.a.be 2 180.v odd 12 1
6480.2.a.bn 2 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{8} - 17T_{7}^{6} + 225T_{7}^{4} - 1088T_{7}^{2} + 4096 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} + 15T_{11}^{2} - 18T_{11} + 36 \) Copy content Toggle raw display
\( T_{19}^{2} + T_{19} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 17 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{4} + 3 T^{3} + 15 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 68 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$17$ \( (T^{4} + 57 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 8)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 21 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + 15 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - 161 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$47$ \( T^{8} - 57 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( (T^{2} + 132)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3 T^{3} + 15 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} + 75 T^{2} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 209 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} - 57 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T - 18)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 77 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
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