↑

Properties

Label 370.2.n.e
Level $370$
Weight $2$
Character orbit 370.n
Analytic conductor $2.954$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [370,2,Mod(269,370)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("370.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0,-16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 2 \beta_{5} q^{3} - \beta_{4} q^{4} + (\beta_{5} + \beta_1) q^{5} - 2 q^{6} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{7} + ( - \beta_{5} - \beta_{3}) q^{8} + (\beta_{4} + 1) q^{9}+ \cdots + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 16 q^{6} + 4 q^{9} + 12 q^{10} + 8 q^{11} - 8 q^{14} - 12 q^{15} - 4 q^{16} + 4 q^{19} + 8 q^{21} - 8 q^{24} - 2 q^{25} - 32 q^{26} - 8 q^{31} - 8 q^{34} - 6 q^{35} + 8 q^{36} + 32 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−1.26217 − 1.18614i
0.396143 + 1.68614i
−0.396143 − 1.68614i
1.26217 + 1.18614i
−1.26217 + 1.18614i
0.396143 − 1.68614i
−0.396143 + 1.68614i
1.26217 − 1.18614i
−0.866025 − 0.500000i 1.73205 − 1.00000i 0.500000 + 0.866025i −2.12819 − 0.686141i −2.00000 3.73831 − 2.15831i − 1.00000i 0.500000 − 0.866025i 1.50000 + 1.65831i
269.2 −0.866025 − 0.500000i 1.73205 − 1.00000i 0.500000 + 0.866025i −0.469882 + 2.18614i −2.00000 −2.00626 + 1.15831i − 1.00000i 0.500000 − 0.866025i 1.50000 − 1.65831i
269.3 0.866025 + 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0.469882 − 2.18614i −2.00000 −3.73831 + 2.15831i 1.00000i 0.500000 − 0.866025i 1.50000 − 1.65831i
269.4 0.866025 + 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 2.12819 + 0.686141i −2.00000 2.00626 − 1.15831i 1.00000i 0.500000 − 0.866025i 1.50000 + 1.65831i
359.1 −0.866025 + 0.500000i 1.73205 + 1.00000i 0.500000 − 0.866025i −2.12819 + 0.686141i −2.00000 3.73831 + 2.15831i 1.00000i 0.500000 + 0.866025i 1.50000 − 1.65831i
359.2 −0.866025 + 0.500000i 1.73205 + 1.00000i 0.500000 − 0.866025i −0.469882 − 2.18614i −2.00000 −2.00626 − 1.15831i 1.00000i 0.500000 + 0.866025i 1.50000 + 1.65831i
359.3 0.866025 − 0.500000i −1.73205 − 1.00000i 0.500000 − 0.866025i 0.469882 + 2.18614i −2.00000 −3.73831 − 2.15831i − 1.00000i 0.500000 + 0.866025i 1.50000 + 1.65831i
359.4 0.866025 − 0.500000i −1.73205 − 1.00000i 0.500000 − 0.866025i 2.12819 − 0.686141i −2.00000 2.00626 + 1.15831i − 1.00000i 0.500000 + 0.866025i 1.50000 − 1.65831i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.n.e ✓ 8
5.b even 2 1 inner 370.2.n.e ✓ 8
37.c even 3 1 inner 370.2.n.e ✓ 8
185.n even 6 1 inner 370.2.n.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.n.e ✓ 8 1.a even 1 1 trivial
370.2.n.e ✓ 8 5.b even 2 1 inner
370.2.n.e ✓ 8 37.c even 3 1 inner
370.2.n.e ✓ 8 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 24T_{7}^{6} + 476T_{7}^{4} - 2400T_{7}^{2} + 10000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 24 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 10)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 30 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 10)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 47 T^{2} + 1369)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 200 T^{2} + 9604)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 24 T^{2} + 100)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 96 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 72 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$71$ \( (T^{4} - 22 T^{3} + \cdots + 12100)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 120 T^{2} + 784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 44 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 99)^{4} \) Copy content Toggle raw display
show more
show less