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Properties

Label 368.2.i.a
Level $368$
Weight $2$
Character orbit 368.i
Analytic conductor $2.938$
Analytic rank $0$
Dimension $12$
CM discriminant -23
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,2,Mod(91,368)]
 
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([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.i (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: \(2.93849479438\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{9} - \beta_{7}) q^{3} + \beta_{2} q^{4} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{4}) q^{6}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{10} + \beta_{9} - \beta_{7}) q^{3} + \beta_{2} q^{4} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{4}) q^{6}+ \cdots + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{6} - 30 q^{12} - 24 q^{27} - 66 q^{36} + 96 q^{39} + 84 q^{49} - 30 q^{58} - 72 q^{59} + 90 q^{62} + 42 q^{64} + 54 q^{72} - 102 q^{78} - 108 q^{81} - 66 q^{82} + 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + \nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 2\nu^{7} + \nu^{2} - 10\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} + 23\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 4\nu^{8} + 23\nu^{4} + 4\nu^{2} ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - \nu^{4} ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} + 8\nu^{8} + 21\nu^{5} + 8\nu^{2} ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + \nu^{5} ) / 24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{11} - 8\nu^{8} + 21\nu^{5} - 8\nu^{2} ) / 96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{8} + 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} + 3\beta_{10} + 2\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{11} - 3\beta_{9} + 6\beta_{5} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6\beta_{11} + 6\beta_{9} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -\beta_{6} + 23\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -\beta_{11} + \beta_{9} - 23\beta_{8} - 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2\beta_{11} + 21\beta_{10} - 2\beta_{9} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−1.38973 − 0.261988i
−0.921756 − 1.07255i
−0.467979 + 1.33454i
0.467979 + 1.33454i
0.921756 − 1.07255i
1.38973 − 0.261988i
−1.38973 + 0.261988i
−0.921756 + 1.07255i
−0.467979 − 1.33454i
0.467979 − 1.33454i
0.921756 + 1.07255i
1.38973 + 0.261988i
−1.38973 − 0.261988i −2.26228 − 2.26228i 1.86272 + 0.728188i 0 2.55128 + 3.73666i 0 −2.39792 − 1.50000i 7.23585i 0
91.2 −0.921756 − 1.07255i 2.42879 + 2.42879i −0.300733 + 1.97726i 0 0.366251 − 4.84375i 0 2.39792 − 1.50000i 8.79803i 0
91.3 −0.467979 + 1.33454i −1.48960 − 1.48960i −1.56199 − 1.24907i 0 2.68503 − 1.29083i 0 2.39792 − 1.50000i 1.43781i 0
91.4 0.467979 + 1.33454i 1.94450 + 1.94450i −1.56199 + 1.24907i 0 −1.68503 + 3.50500i 0 −2.39792 − 1.50000i 4.56219i 0
91.5 0.921756 − 1.07255i 0.317779 + 0.317779i −0.300733 − 1.97726i 0 0.633749 − 0.0479197i 0 −2.39792 − 1.50000i − 2.79803i 0
91.6 1.38973 − 0.261988i −0.939190 − 0.939190i 1.86272 − 0.728188i 0 −1.55128 − 1.05917i 0 2.39792 − 1.50000i − 1.23585i 0
275.1 −1.38973 + 0.261988i −2.26228 + 2.26228i 1.86272 − 0.728188i 0 2.55128 − 3.73666i 0 −2.39792 + 1.50000i − 7.23585i 0
275.2 −0.921756 + 1.07255i 2.42879 − 2.42879i −0.300733 − 1.97726i 0 0.366251 + 4.84375i 0 2.39792 + 1.50000i − 8.79803i 0
275.3 −0.467979 − 1.33454i −1.48960 + 1.48960i −1.56199 + 1.24907i 0 2.68503 + 1.29083i 0 2.39792 + 1.50000i − 1.43781i 0
275.4 0.467979 − 1.33454i 1.94450 − 1.94450i −1.56199 − 1.24907i 0 −1.68503 − 3.50500i 0 −2.39792 + 1.50000i − 4.56219i 0
275.5 0.921756 + 1.07255i 0.317779 − 0.317779i −0.300733 + 1.97726i 0 0.633749 + 0.0479197i 0 −2.39792 + 1.50000i 2.79803i 0
275.6 1.38973 + 0.261988i −0.939190 + 0.939190i 1.86272 + 0.728188i 0 −1.55128 + 1.05917i 0 2.39792 + 1.50000i 1.23585i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.6
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.f odd 4 1 inner
368.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.i.a ✓ 12
4.b odd 2 1 1472.2.i.a 12
16.e even 4 1 1472.2.i.a 12
16.f odd 4 1 inner 368.2.i.a ✓ 12
23.b odd 2 1 CM 368.2.i.a ✓ 12
92.b even 2 1 1472.2.i.a 12
368.i even 4 1 inner 368.2.i.a ✓ 12
368.k odd 4 1 1472.2.i.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.i.a ✓ 12 1.a even 1 1 trivial
368.2.i.a ✓ 12 16.f odd 4 1 inner
368.2.i.a ✓ 12 23.b odd 2 1 CM
368.2.i.a ✓ 12 368.i even 4 1 inner
1472.2.i.a 12 4.b odd 2 1
1472.2.i.a 12 16.e even 4 1
1472.2.i.a 12 92.b even 2 1
1472.2.i.a 12 368.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 8 T_{3}^{9} + 162 T_{3}^{8} + 72 T_{3}^{7} + 32 T_{3}^{6} + 648 T_{3}^{5} + 5193 T_{3}^{4} + \cdots + 1444 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7T^{6} + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{9} + \cdots + 1444 \) 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
$13$ \( T^{12} - 148 T^{9} + \cdots + 1170724 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{2} - 23)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 945316516 \) Copy content Toggle raw display
$31$ \( (T^{6} + 186 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( (T^{6} + 246 T^{4} + \cdots + 181476)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( (T^{6} + 282 T^{4} + \cdots + 412988)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{3} + \cdots + 676)^{3} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{3} - 213 T - 1176)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 438 T^{4} + \cdots + 1503076)^{2} \) 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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