Properties

Label 1472.2.h.a
Level $1472$
Weight $2$
Character orbit 1472.h
Analytic conductor $11.754$
Analytic rank $0$
Dimension $8$
CM discriminant -184
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1472,2,Mod(735,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.735");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.h (of order \(2\), degree \(1\), 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: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.73358639104.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 18x^{4} - 96x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} - 3 q^{9} - \beta_{4} q^{11} + \beta_{7} q^{19} + \beta_{6} q^{23} + (\beta_{5} + 5) q^{25} + 2 \beta_{6} q^{31} - \beta_1 q^{37} + \beta_{5} q^{41} + ( - \beta_{7} + 2 \beta_{4}) q^{43}+ \cdots + 3 \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 72 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 18x^{4} - 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} + 222\nu^{3} - 848\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 2\nu^{4} - 6\nu^{2} + 144 ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 22\nu^{5} - 10\nu^{3} + 464\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 2\nu^{5} - 2\nu^{3} + 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 2\nu^{2} + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 3\nu^{4} + 16\nu^{2} - 69 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 10\nu^{5} + 38\nu^{3} - 144\nu ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{4} + 3\beta_{3} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + \beta_{5} + 3\beta_{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 4\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - 21\beta_{4} + 25\beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{6} + \beta_{5} - 45\beta_{2} + 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19\beta_{7} - 23\beta_{4} + 27\beta_{3} - 23\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(833\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
735.1
−1.98720 0.225925i
−1.98720 + 0.225925i
1.24541 + 1.56491i
1.24541 1.56491i
−1.24541 + 1.56491i
−1.24541 1.56491i
1.98720 0.225925i
1.98720 + 0.225925i
0 0 0 −4.42625 0 0 0 −3.00000 0
735.2 0 0 0 −4.42625 0 0 0 −3.00000 0
735.3 0 0 0 −0.639012 0 0 0 −3.00000 0
735.4 0 0 0 −0.639012 0 0 0 −3.00000 0
735.5 0 0 0 0.639012 0 0 0 −3.00000 0
735.6 0 0 0 0.639012 0 0 0 −3.00000 0
735.7 0 0 0 4.42625 0 0 0 −3.00000 0
735.8 0 0 0 4.42625 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 735.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
184.e odd 2 1 CM by \(\Q(\sqrt{-46}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner
184.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1472.2.h.a 8
4.b odd 2 1 inner 1472.2.h.a 8
8.b even 2 1 inner 1472.2.h.a 8
8.d odd 2 1 inner 1472.2.h.a 8
23.b odd 2 1 inner 1472.2.h.a 8
92.b even 2 1 inner 1472.2.h.a 8
184.e odd 2 1 CM 1472.2.h.a 8
184.h even 2 1 inner 1472.2.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1472.2.h.a 8 1.a even 1 1 trivial
1472.2.h.a 8 4.b odd 2 1 inner
1472.2.h.a 8 8.b even 2 1 inner
1472.2.h.a 8 8.d odd 2 1 inner
1472.2.h.a 8 23.b odd 2 1 inner
1472.2.h.a 8 92.b even 2 1 inner
1472.2.h.a 8 184.e odd 2 1 CM
1472.2.h.a 8 184.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(1472, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 20 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 44 T^{2} + 392)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 76 T^{2} + 1352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + 92)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 148 T^{2} + 968)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 92)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 172 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 212 T^{2} + 6728)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 244 T^{2} + 14792)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 268 T^{2} + 13448)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 92)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 332 T^{2} + 968)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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