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Properties

Label 399.2.j.d
Level $399$
Weight $2$
Character orbit 399.j
Analytic conductor $3.186$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(58,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("399.58");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.j (of order \(3\), 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: \(3.18603104065\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{6} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{7}+ \cdots + ( - 2 \beta_{6} + \beta_{3} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{5} + 2 q^{7} - 6 q^{8} - 4 q^{9} + 3 q^{10} + 2 q^{11} + 12 q^{13} + 2 q^{14} - 8 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{19} + 24 q^{20} + q^{21} - 12 q^{22} - 5 q^{23} - 3 q^{24} + 4 q^{25}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{7} + 64\nu^{6} - 256\nu^{5} + 338\nu^{4} - 1088\nu^{3} + 1216\nu^{2} - 1385\nu + 256 ) / 289 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\nu^{7} + 16\nu^{6} + 225\nu^{5} - 60\nu^{4} + 884\nu^{3} + 15\nu^{2} + 304\nu + 64 ) / 289 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63\nu^{7} - 41\nu^{6} + 164\nu^{5} - 352\nu^{4} + 697\nu^{3} - 779\nu^{2} - 490\nu - 164 ) / 289 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -64\nu^{7} + 60\nu^{6} - 240\nu^{5} + 353\nu^{4} - 1020\nu^{3} + 1140\nu^{2} - 305\nu + 240 ) / 289 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -164\nu^{7} - 63\nu^{6} - 615\nu^{5} + 164\nu^{4} - 2108\nu^{3} - 41\nu^{2} - 41\nu + 37 ) / 289 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 180\nu^{7} + 48\nu^{6} + 675\nu^{5} - 180\nu^{4} + 2363\nu^{3} + 45\nu^{2} + 45\nu + 481 ) / 289 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 3\beta_{3} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 4\beta_{4} + 4\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 11\beta_{3} - 5\beta_{2} - 5\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{7} - 15\beta_{6} + 7\beta_{3} - 7\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -30\beta_{5} - 8\beta_{4} + 23\beta_{2} + 65\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(134\) \(211\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
0.882007 − 1.52768i
0.346911 − 0.600868i
−0.198169 + 0.343239i
−1.03075 + 1.78531i
0.882007 + 1.52768i
0.346911 + 0.600868i
−0.198169 − 0.343239i
−1.03075 − 1.78531i
−0.882007 + 1.52768i 0.500000 + 0.866025i −0.555874 − 0.962803i −1.05587 + 1.82883i −1.76401 −0.893022 + 2.49048i −1.56689 −0.500000 + 0.866025i −1.86258 − 3.22608i
58.2 −0.346911 + 0.600868i 0.500000 + 0.866025i 0.759305 + 1.31516i 0.259305 − 0.449130i −0.693822 −2.54751 − 0.714287i −2.44129 −0.500000 + 0.866025i 0.179912 + 0.311616i
58.3 0.198169 − 0.343239i 0.500000 + 0.866025i 0.921458 + 1.59601i 0.421458 − 0.729986i 0.396339 1.79981 − 1.93925i 1.52310 −0.500000 + 0.866025i −0.167040 − 0.289322i
58.4 1.03075 − 1.78531i 0.500000 + 0.866025i −1.12489 − 1.94836i −1.62489 + 2.81439i 2.06150 2.64072 + 0.163054i −0.514916 −0.500000 + 0.866025i 3.34971 + 5.80186i
172.1 −0.882007 − 1.52768i 0.500000 − 0.866025i −0.555874 + 0.962803i −1.05587 − 1.82883i −1.76401 −0.893022 − 2.49048i −1.56689 −0.500000 − 0.866025i −1.86258 + 3.22608i
172.2 −0.346911 − 0.600868i 0.500000 − 0.866025i 0.759305 − 1.31516i 0.259305 + 0.449130i −0.693822 −2.54751 + 0.714287i −2.44129 −0.500000 − 0.866025i 0.179912 − 0.311616i
172.3 0.198169 + 0.343239i 0.500000 − 0.866025i 0.921458 − 1.59601i 0.421458 + 0.729986i 0.396339 1.79981 + 1.93925i 1.52310 −0.500000 − 0.866025i −0.167040 + 0.289322i
172.4 1.03075 + 1.78531i 0.500000 − 0.866025i −1.12489 + 1.94836i −1.62489 − 2.81439i 2.06150 2.64072 − 0.163054i −0.514916 −0.500000 − 0.866025i 3.34971 − 5.80186i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.2.j.d ✓ 8
3.b odd 2 1 1197.2.j.k 8
7.c even 3 1 inner 399.2.j.d ✓ 8
7.c even 3 1 2793.2.a.bc 4
7.d odd 6 1 2793.2.a.bd 4
21.g even 6 1 8379.2.a.bt 4
21.h odd 6 1 1197.2.j.k 8
21.h odd 6 1 8379.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.j.d ✓ 8 1.a even 1 1 trivial
399.2.j.d ✓ 8 7.c even 3 1 inner
1197.2.j.k 8 3.b odd 2 1
1197.2.j.k 8 21.h odd 6 1
2793.2.a.bc 4 7.c even 3 1
2793.2.a.bd 4 7.d odd 6 1
8379.2.a.br 4 21.h odd 6 1
8379.2.a.bt 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{6} + 2T_{2}^{5} + 15T_{2}^{4} + 4T_{2}^{3} + 5T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + \cdots - 141)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 5 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 303)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 17 T^{7} + \cdots + 974169 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( (T^{4} - 7 T^{3} - 47 T^{2} + \cdots - 43)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 15 T^{3} + \cdots - 301)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + \cdots + 12439729 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 423801 \) Copy content Toggle raw display
$59$ \( T^{8} + 17 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$61$ \( T^{8} - 9 T^{7} + \cdots + 237169 \) Copy content Toggle raw display
$67$ \( T^{8} - 5 T^{7} + \cdots + 2211169 \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} + 43 T - 47)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 15 T^{7} + \cdots + 6135529 \) Copy content Toggle raw display
$79$ \( T^{8} + 5 T^{7} + \cdots + 321520761 \) Copy content Toggle raw display
$83$ \( (T^{4} - 34 T^{3} + \cdots + 2733)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 3 T^{7} + \cdots + 328805689 \) Copy content Toggle raw display
$97$ \( (T^{4} - 11 T^{3} + \cdots + 1451)^{2} \) Copy content Toggle raw display
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