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Properties

Label 3078.2.a.x
Level $3078$
Weight $2$
Character orbit 3078.a
Self dual yes
Analytic conductor $24.578$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3078,2,Mod(1,3078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3078, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3078.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3078 = 2 \cdot 3^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3078.a (trivial)

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: yes
Analytic conductor: \(24.5779537422\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.37354176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} + 9x^{2} - 2x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 342)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_1 q^{5} + ( - \beta_{5} + 1) q^{7} + q^{8} - \beta_1 q^{10} + (\beta_{4} - \beta_{3} + 1) q^{11} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{13} + ( - \beta_{5} + 1) q^{14} + q^{16} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{17} - q^{19} - \beta_1 q^{20} + (\beta_{4} - \beta_{3} + 1) q^{22} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 2) q^{23}+ \cdots + ( - \beta_{5} - \beta_{2} + 2 \beta_1 + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 6 q^{16} - 6 q^{19} + 6 q^{22} + 12 q^{23} + 30 q^{25} + 6 q^{26} + 6 q^{28} - 6 q^{29} + 6 q^{31} + 6 q^{32} + 12 q^{35} + 6 q^{37} - 6 q^{38} - 6 q^{41} + 12 q^{43} + 6 q^{44} + 12 q^{46} + 6 q^{47} + 18 q^{49} + 30 q^{50} + 6 q^{52} - 18 q^{53} + 24 q^{55} + 6 q^{56} - 6 q^{58} - 12 q^{59} + 12 q^{61} + 6 q^{62} + 6 q^{64} + 6 q^{65} + 18 q^{67} + 12 q^{70} + 18 q^{71} + 24 q^{73} + 6 q^{74} - 6 q^{76} + 6 q^{77} + 18 q^{79} - 6 q^{82} - 12 q^{83} + 36 q^{85} + 12 q^{86} + 6 q^{88} + 6 q^{89} - 12 q^{91} + 12 q^{92} + 6 q^{94} - 18 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} + 9x^{2} - 2x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{5} - 8\nu^{3} - \nu^{2} + 9\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{5} - \nu^{4} + 15\nu^{3} + 7\nu^{2} - 9\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{5} - \nu^{4} + 15\nu^{3} + 7\nu^{2} - 12\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{5} - \nu^{4} + 16\nu^{3} + 7\nu^{2} - 17\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} + 2\nu^{4} - 15\nu^{3} - 15\nu^{2} + 11\nu + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} - 2\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{4} - 8\beta_{3} + 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} - 8\beta_{4} - 6\beta_{3} + \beta_{2} - 16\beta _1 + 46 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 23\beta_{4} - 56\beta_{3} + 31\beta_{2} + \beta _1 + 5 ) / 3 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.579865
0.850971
−1.24208
−0.244728
2.61017
−2.55420
1.00000 0 1.00000 −4.38829 0 −3.76746 1.00000 0 −4.38829
1.2 1.00000 0 1.00000 −3.45098 0 3.80379 1.00000 0 −3.45098
1.3 1.00000 0 1.00000 −0.652132 0 4.21313 1.00000 0 −0.652132
1.4 1.00000 0 1.00000 1.14606 0 −1.63489 1.00000 0 1.14606
1.5 1.00000 0 1.00000 3.42989 0 0.0828222 1.00000 0 3.42989
1.6 1.00000 0 1.00000 3.91545 0 3.30261 1.00000 0 3.91545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3078.2.a.x 6
3.b odd 2 1 3078.2.a.v 6
9.c even 3 2 342.2.e.c ✓ 12
9.d odd 6 2 1026.2.e.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.e.c ✓ 12 9.c even 3 2
1026.2.e.d 12 9.d odd 6 2
3078.2.a.v 6 3.b odd 2 1
3078.2.a.x 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3078))\):

\( T_{5}^{6} - 30T_{5}^{4} + 8T_{5}^{3} + 228T_{5}^{2} - 96T_{5} - 152 \) Copy content Toggle raw display
\( T_{7}^{6} - 6T_{7}^{5} - 12T_{7}^{4} + 108T_{7}^{3} - 33T_{7}^{2} - 324T_{7} + 27 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} - 24T_{11}^{4} + 116T_{11}^{3} + 276T_{11}^{2} - 432T_{11} - 872 \) Copy content Toggle raw display
\( T_{23}^{6} - 12T_{23}^{5} + 6T_{23}^{4} + 398T_{23}^{3} - 1701T_{23}^{2} + 2172T_{23} - 257 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 30 T^{4} + \cdots - 152 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$11$ \( T^{6} - 6 T^{5} + \cdots - 872 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + \cdots + 4363 \) Copy content Toggle raw display
$17$ \( T^{6} - 66 T^{4} + \cdots + 405 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + \cdots - 257 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} + \cdots - 269 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots - 14216 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots - 90329 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 40 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + \cdots + 8536 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 54343 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + \cdots + 1927 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + \cdots + 4509 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots - 2120 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} + \cdots + 524421 \) Copy content Toggle raw display
$71$ \( T^{6} - 18 T^{5} + \cdots - 50840 \) Copy content Toggle raw display
$73$ \( T^{6} - 24 T^{5} + \cdots - 59967 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + \cdots - 33560 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + \cdots + 3240 \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots - 5400 \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + \cdots + 2415352 \) Copy content Toggle raw display
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