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Properties

Label 2790.2.d.j
Level $2790$
Weight $2$
Character orbit 2790.d
Analytic conductor $22.278$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2790,2,Mod(559,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2790.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.d (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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.11669056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - q^{4} + ( - \beta_1 + 1) q^{5} + 2 \beta_{3} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - q^{4} + ( - \beta_1 + 1) q^{5} + 2 \beta_{3} q^{7} + \beta_{3} q^{8} + (\beta_{5} - \beta_{3}) q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + (\beta_{5} + \beta_{4} - 4 \beta_{3}) q^{13} + 2 q^{14} + q^{16} + (2 \beta_{5} + 2 \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots - 3 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 4 q^{5} - 16 q^{11} + 12 q^{14} + 6 q^{16} + 4 q^{19} - 4 q^{20} - 8 q^{25} - 20 q^{26} - 4 q^{29} - 6 q^{31} - 4 q^{34} - 12 q^{41} + 16 q^{44} + 4 q^{46} + 18 q^{49} + 8 q^{50} + 6 q^{55} - 12 q^{56} - 4 q^{59} + 28 q^{61} - 6 q^{64} + 8 q^{65} + 8 q^{70} + 16 q^{71} - 12 q^{74} - 4 q^{76} + 12 q^{79} + 4 q^{80} - 22 q^{85} - 4 q^{89} + 40 q^{91} + 4 q^{94} + 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 6\nu^{3} + 14\nu^{2} + 13\nu + 42 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 14\nu^{4} + 49\nu^{3} - 29\nu^{2} + 27\nu + 213 ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 16\nu^{4} - 61\nu^{3} + 26\nu^{2} - 53\nu - 272 ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 21\nu^{4} - 61\nu^{3} + 31\nu^{2} - 3\nu - 232 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{4} + 4\beta_{3} + 2\beta_{2} - 3\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{5} - 6\beta_{4} - \beta_{3} - 2\beta_{2} - 10\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 26\beta_{4} - 39\beta_{3} - 17\beta_{2} - 5\beta _1 + 13 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times\).

\(n\) \(1117\) \(1801\) \(2171\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
1.60509 + 2.15264i
0.630356 − 2.20530i
−1.23545 + 0.0526623i
1.60509 − 2.15264i
0.630356 + 2.20530i
−1.23545 − 0.0526623i
− 1.00000i 0 −1.00000 −0.605092 − 2.15264i 0 2.00000i 1.00000i 0 −2.15264 + 0.605092i
559.2 − 1.00000i 0 −1.00000 0.369644 + 2.20530i 0 2.00000i 1.00000i 0 2.20530 − 0.369644i
559.3 − 1.00000i 0 −1.00000 2.23545 − 0.0526623i 0 2.00000i 1.00000i 0 −0.0526623 − 2.23545i
559.4 1.00000i 0 −1.00000 −0.605092 + 2.15264i 0 − 2.00000i − 1.00000i 0 −2.15264 − 0.605092i
559.5 1.00000i 0 −1.00000 0.369644 − 2.20530i 0 − 2.00000i − 1.00000i 0 2.20530 + 0.369644i
559.6 1.00000i 0 −1.00000 2.23545 + 0.0526623i 0 − 2.00000i − 1.00000i 0 −0.0526623 + 2.23545i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2790.2.d.j 6
3.b odd 2 1 930.2.d.i ✓ 6
5.b even 2 1 inner 2790.2.d.j 6
15.d odd 2 1 930.2.d.i ✓ 6
15.e even 4 1 4650.2.a.ci 3
15.e even 4 1 4650.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.i ✓ 6 3.b odd 2 1
930.2.d.i ✓ 6 15.d odd 2 1
2790.2.d.j 6 1.a even 1 1 trivial
2790.2.d.j 6 5.b even 2 1 inner
4650.2.a.ci 3 15.e even 4 1
4650.2.a.cp 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2790, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 8T_{11}^{2} + 13T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 4 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 8 T^{2} + 13 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 50 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$17$ \( T^{6} + 74 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 35 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 68 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 32 T + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + 164 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T + 2)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 152 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{6} + 74 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{6} + 188 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} + \cdots - 280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 286 T^{4} + \cdots + 425104 \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 79 T - 20)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 268 T^{4} + \cdots + 153664 \) Copy content Toggle raw display
$79$ \( (T^{3} - 6 T^{2} - 7 T + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 370 T^{4} + \cdots + 59536 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} + \cdots - 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 150 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
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