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Properties

Label 368.3.k.a
Level $368$
Weight $3$
Character orbit 368.k
Analytic conductor $10.027$
Analytic rank $0$
Dimension $12$
CM discriminant -23
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,3,Mod(45,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.45");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.k (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: \(10.0272737285\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.322241908269256704.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{11} - 2 \beta_{10} + \cdots + \beta_{4}) q^{3} + (\beta_{9} + \beta_{8} + \cdots + \beta_{2}) q^{4} + ( - 3 \beta_{10} + \beta_{9} + \cdots + 5) q^{6} + (3 \beta_{7} + 2 \beta_{3}) q^{8}+ \cdots + 49 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 66 q^{6} - 6 q^{12} - 228 q^{27} + 294 q^{36} - 588 q^{49} - 546 q^{58} - 156 q^{59} + 606 q^{62} + 474 q^{64} - 378 q^{72} + 798 q^{78} - 972 q^{81} + 258 q^{82} + 1092 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{6} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + \nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + \nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 25\nu^{5} + 96\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 11\nu^{2} + 12\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} + 23\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - \nu^{4} + 24\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} + \nu^{4} + 24\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} + 7\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} + 7\nu^{4} ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{6} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{11} + 3\beta_{9} - 3\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{10} - 3\beta_{9} - 3\beta_{8} + 6\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{5} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 12\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{9} + \beta_{8} - 2\beta_{6} + 22\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -\beta_{7} + 23\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -4\beta_{11} + 21\beta_{9} - 21\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -50\beta_{10} - 21\beta_{9} - 21\beta_{8} + 42\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.921756 + 1.07255i
−0.467979 − 1.33454i
1.38973 + 0.261988i
−1.38973 + 0.261988i
0.467979 − 1.33454i
−0.921756 + 1.07255i
0.921756 − 1.07255i
−0.467979 + 1.33454i
1.38973 − 0.261988i
−1.38973 − 0.261988i
0.467979 + 1.33454i
−0.921756 − 1.07255i
−1.97726 + 0.300733i −2.12720 − 2.12720i 3.81912 − 1.18925i 0 4.84574 + 3.56631i 0 −7.19375 + 3.50000i 0.0499413i 0
45.2 −1.24907 + 1.56199i −3.67762 − 3.67762i −0.879635 − 3.90208i 0 10.3380 − 1.15080i 0 7.19375 + 3.50000i 18.0498i 0
45.3 −0.728188 − 1.86272i 0.00678988 + 0.00678988i −2.93948 + 2.71283i 0 0.00770336 − 0.0175920i 0 7.19375 + 3.50000i − 8.99991i 0
45.4 0.728188 − 1.86272i 4.24264 + 4.24264i −2.93948 − 2.71283i 0 10.9923 − 4.81342i 0 −7.19375 + 3.50000i 26.9999i 0
45.5 1.24907 + 1.56199i −2.11544 − 2.11544i −0.879635 + 3.90208i 0 0.661961 − 5.94663i 0 −7.19375 + 3.50000i − 0.0498490i 0
45.6 1.97726 + 0.300733i 3.67083 + 3.67083i 3.81912 + 1.18925i 0 6.15426 + 8.36214i 0 7.19375 + 3.50000i 17.9501i 0
229.1 −1.97726 − 0.300733i −2.12720 + 2.12720i 3.81912 + 1.18925i 0 4.84574 − 3.56631i 0 −7.19375 − 3.50000i − 0.0499413i 0
229.2 −1.24907 − 1.56199i −3.67762 + 3.67762i −0.879635 + 3.90208i 0 10.3380 + 1.15080i 0 7.19375 − 3.50000i − 18.0498i 0
229.3 −0.728188 + 1.86272i 0.00678988 − 0.00678988i −2.93948 − 2.71283i 0 0.00770336 + 0.0175920i 0 7.19375 − 3.50000i 8.99991i 0
229.4 0.728188 + 1.86272i 4.24264 − 4.24264i −2.93948 + 2.71283i 0 10.9923 + 4.81342i 0 −7.19375 − 3.50000i − 26.9999i 0
229.5 1.24907 − 1.56199i −2.11544 + 2.11544i −0.879635 − 3.90208i 0 0.661961 + 5.94663i 0 −7.19375 − 3.50000i 0.0498490i 0
229.6 1.97726 − 0.300733i 3.67083 − 3.67083i 3.81912 − 1.18925i 0 6.15426 − 8.36214i 0 7.19375 − 3.50000i − 17.9501i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
16.e even 4 1 inner
368.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.k.a ✓ 12
16.e even 4 1 inner 368.3.k.a ✓ 12
23.b odd 2 1 CM 368.3.k.a ✓ 12
368.k odd 4 1 inner 368.3.k.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.3.k.a ✓ 12 1.a even 1 1 trivial
368.3.k.a ✓ 12 16.e even 4 1 inner
368.3.k.a ✓ 12 23.b odd 2 1 CM
368.3.k.a ✓ 12 368.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 76 T_{3}^{9} + 1458 T_{3}^{8} + 2052 T_{3}^{7} + 2888 T_{3}^{6} + 55404 T_{3}^{5} + \cdots + 196 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 79T^{6} + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} + 76 T^{9} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 71959490615236 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{6} - 5766 T^{4} + \cdots - 97982208)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( (T^{6} + 10086 T^{4} + \cdots + 1903925956)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( (T^{3} - 6627 T + 205342)^{4} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{3} + \cdots + 39513796)^{3} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{6} + 30246 T^{4} + \cdots + 67306675968)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 31974 T^{4} + \cdots + 525685901764)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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