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Properties

Label 930.2.s.b
Level $930$
Weight $2$
Character orbit 930.s
Analytic conductor $7.426$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,2,Mod(439,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("930.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.s (of order \(6\), degree \(2\), 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: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_{2} q^{3} - q^{4} + (\beta_{4} + 2 \beta_{3}) q^{5} + ( - \beta_{3} - 1) q^{6} - \beta_{6} q^{7} + (\beta_{4} + \beta_{2}) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{2}) q^{2} - \beta_{2} q^{3} - q^{4} + (\beta_{4} + 2 \beta_{3}) q^{5} + ( - \beta_{3} - 1) q^{6} - \beta_{6} q^{7} + (\beta_{4} + \beta_{2}) q^{8} - \beta_{3} q^{9} + (2 \beta_{4} - \beta_{3}) q^{10} + ( - \beta_{5} + 2 \beta_{3}) q^{11} + \beta_{2} q^{12} + 2 \beta_1 q^{13} + \beta_{7} q^{14} + (2 \beta_{4} + 2 \beta_{2} + 1) q^{15} + q^{16} + 6 \beta_{2} q^{17} - \beta_{4} q^{18} - 2 \beta_{7} q^{19} + ( - \beta_{4} - 2 \beta_{3}) q^{20} + \beta_{5} q^{21} + (2 \beta_{4} + \beta_1) q^{22} + (2 \beta_{6} - 4 \beta_{4} + \cdots + 2 \beta_1) q^{23}+ \cdots + ( - \beta_{7} + 2 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{5} - 4 q^{6} + 4 q^{9} + 4 q^{10} - 8 q^{11} + 8 q^{15} + 8 q^{16} + 8 q^{20} + 4 q^{24} - 12 q^{25} + 16 q^{30} - 12 q^{31} + 24 q^{34} - 4 q^{36} - 4 q^{40} + 8 q^{41} + 8 q^{44} + 8 q^{45} - 32 q^{46} + 16 q^{50} - 24 q^{51} - 8 q^{54} - 16 q^{55} + 8 q^{59} - 8 q^{60} + 80 q^{61} - 8 q^{64} + 16 q^{66} - 16 q^{69} + 16 q^{71} - 16 q^{74} - 16 q^{75} + 16 q^{79} - 8 q^{80} - 4 q^{81} - 48 q^{85} + 24 q^{86} - 16 q^{89} - 4 q^{90} + 112 q^{91} - 32 q^{94} - 4 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 7\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 3\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 11\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 5\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
1.09445 + 0.895644i
−0.228425 − 1.39564i
−1.09445 − 0.895644i
0.228425 + 1.39564i
−1.09445 + 0.895644i
0.228425 − 1.39564i
1.09445 − 0.895644i
−0.228425 + 1.39564i
− 1.00000i −0.866025 − 0.500000i −1.00000 −1.86603 − 1.23205i −0.500000 + 0.866025i −2.29129 − 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.2 − 1.00000i −0.866025 − 0.500000i −1.00000 −1.86603 − 1.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.3 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 − 2.23205i −0.500000 + 0.866025i −2.29129 − 1.32288i − 1.00000i 0.500000 + 0.866025i 2.23205 − 0.133975i
439.4 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 − 2.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i − 1.00000i 0.500000 + 0.866025i 2.23205 − 0.133975i
769.1 − 1.00000i 0.866025 − 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 − 0.866025i −2.29129 + 1.32288i 1.00000i 0.500000 − 0.866025i 2.23205 + 0.133975i
769.2 − 1.00000i 0.866025 − 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 − 0.866025i 2.29129 − 1.32288i 1.00000i 0.500000 − 0.866025i 2.23205 + 0.133975i
769.3 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 − 0.866025i −2.29129 + 1.32288i − 1.00000i 0.500000 − 0.866025i −1.23205 − 1.86603i
769.4 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 − 0.866025i 2.29129 − 1.32288i − 1.00000i 0.500000 − 0.866025i −1.23205 − 1.86603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.b ✓ 8
5.b even 2 1 inner 930.2.s.b ✓ 8
31.c even 3 1 inner 930.2.s.b ✓ 8
155.j even 6 1 inner 930.2.s.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.b ✓ 8 1.a even 1 1 trivial
930.2.s.b ✓ 8 5.b even 2 1 inner
930.2.s.b ✓ 8 31.c even 3 1 inner
930.2.s.b ✓ 8 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 7T_{7}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{3} + 11 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + 19 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{2} + 784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 88 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 128 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( (T^{4} + 88 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 74 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 19 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 128 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 256 T^{6} + \cdots + 84934656 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 226 T^{6} + \cdots + 151807041 \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 24)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 358 T^{2} + 29241)^{2} \) Copy content Toggle raw display
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