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Properties

Label 405.3.g.d
Level $405$
Weight $3$
Character orbit 405.g
Analytic conductor $11.035$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,3,Mod(82,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.g (of order \(4\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{2}) q^{2} + ( - \beta_{6} - \beta_{5} - 2 \beta_1) q^{4} + (\beta_{7} + \beta_{4} + \cdots + 3 \beta_{2}) q^{5} + (\beta_{6} + 7 \beta_1 - 7) q^{7} + (2 \beta_{4} + 2 \beta_{2}) q^{8}+ \cdots + (26 \beta_{4} + 85 \beta_{3} + 26 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 52 q^{7} + 44 q^{10} + 56 q^{13} - 16 q^{16} - 92 q^{22} - 64 q^{25} + 136 q^{28} - 296 q^{31} + 20 q^{37} + 96 q^{40} - 4 q^{43} + 232 q^{46} + 224 q^{52} + 52 q^{55} + 132 q^{58} - 224 q^{61} + 152 q^{67}+ \cdots + 548 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{5} + \nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 2\nu^{6} - 15\nu^{5} + 10\nu^{4} - 25\nu^{3} + 30\nu^{2} - 20\nu + 16 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} - 6\nu^{4} + 5\nu^{3} - 2\nu^{2} + 20\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 6\nu^{4} + 5\nu^{3} + 2\nu^{2} + 20\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + \nu^{6} - 5\nu^{5} - 5\nu^{4} - 15\nu^{3} - 15\nu^{2} - 30\nu - 8 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} - 3\beta_{5} - 4\beta_{4} + 3\beta_{3} - 3\beta_{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 11\beta_{2} - 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} + 5\beta_{4} - 5\beta_{3} + 5\beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 7\beta_{6} - 7\beta_{5} - 7\beta_{3} + 13\beta_{2} - 20\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.09445 − 0.895644i
0.228425 + 1.39564i
−0.228425 + 1.39564i
−1.09445 − 0.895644i
1.09445 + 0.895644i
0.228425 − 1.39564i
−0.228425 − 1.39564i
−1.09445 + 0.895644i
−2.18890 − 2.18890i 0 5.58258i 1.27520 + 4.83465i 0 −8.79129 − 8.79129i 3.46410 − 3.46410i 0 7.79129 − 13.3739i
82.2 −0.456850 − 0.456850i 0 − 3.58258i −3.92095 + 3.10260i 0 −4.20871 − 4.20871i −3.46410 + 3.46410i 0 3.20871 + 0.373864i
82.3 0.456850 + 0.456850i 0 − 3.58258i 3.92095 − 3.10260i 0 −4.20871 − 4.20871i 3.46410 − 3.46410i 0 3.20871 + 0.373864i
82.4 2.18890 + 2.18890i 0 5.58258i −1.27520 − 4.83465i 0 −8.79129 − 8.79129i −3.46410 + 3.46410i 0 7.79129 − 13.3739i
163.1 −2.18890 + 2.18890i 0 − 5.58258i 1.27520 − 4.83465i 0 −8.79129 + 8.79129i 3.46410 + 3.46410i 0 7.79129 + 13.3739i
163.2 −0.456850 + 0.456850i 0 3.58258i −3.92095 − 3.10260i 0 −4.20871 + 4.20871i −3.46410 − 3.46410i 0 3.20871 − 0.373864i
163.3 0.456850 − 0.456850i 0 3.58258i 3.92095 + 3.10260i 0 −4.20871 + 4.20871i 3.46410 + 3.46410i 0 3.20871 − 0.373864i
163.4 2.18890 − 2.18890i 0 − 5.58258i −1.27520 + 4.83465i 0 −8.79129 + 8.79129i −3.46410 − 3.46410i 0 7.79129 + 13.3739i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.g.d ✓ 8
3.b odd 2 1 inner 405.3.g.d ✓ 8
5.c odd 4 1 inner 405.3.g.d ✓ 8
9.c even 3 1 405.3.l.e 8
9.c even 3 1 405.3.l.i 8
9.d odd 6 1 405.3.l.e 8
9.d odd 6 1 405.3.l.i 8
15.e even 4 1 inner 405.3.g.d ✓ 8
45.k odd 12 1 405.3.l.e 8
45.k odd 12 1 405.3.l.i 8
45.l even 12 1 405.3.l.e 8
45.l even 12 1 405.3.l.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.g.d ✓ 8 1.a even 1 1 trivial
405.3.g.d ✓ 8 3.b odd 2 1 inner
405.3.g.d ✓ 8 5.c odd 4 1 inner
405.3.g.d ✓ 8 15.e even 4 1 inner
405.3.l.e 8 9.c even 3 1
405.3.l.e 8 9.d odd 6 1
405.3.l.e 8 45.k odd 12 1
405.3.l.e 8 45.l even 12 1
405.3.l.i 8 9.c even 3 1
405.3.l.i 8 9.d odd 6 1
405.3.l.i 8 45.k odd 12 1
405.3.l.i 8 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 92T_{2}^{4} + 16 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 92T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 32 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 26 T^{3} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 110 T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 28 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 172412 T^{4} + 1336336 \) Copy content Toggle raw display
$19$ \( (T^{4} + 666 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 152072 T^{4} + 78074896 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1734 T^{2} + 71289)^{2} \) Copy content Toggle raw display
$31$ \( (T + 37)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots + 21325924)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 1550 T^{2} + 7921)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + \cdots + 1612900)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 121173610000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 12750 T^{2} + 27468081)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 56 T + 28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 76 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 5270 T^{2} + 5851561)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 130 T^{3} + \cdots + 4072324)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16620 T^{2} + 46949904)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9630 T^{2} + 18584721)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 274 T^{3} + \cdots + 87871876)^{2} \) Copy content Toggle raw display
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