Properties

Label 1920.2.b.e
Level $1920$
Weight $2$
Character orbit 1920.b
Analytic conductor $15.331$
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] = [1920,2,Mod(191,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.b (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: \(15.3312771881\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} - \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} - \beta_{3} q^{7} + \beta_{2} q^{9} + (\beta_{6} + \beta_{5}) q^{13} - \beta_1 q^{15} - \beta_{5} q^{17} + (\beta_{4} + \beta_1) q^{19} + ( - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{21} + ( - \beta_{7} - 2 \beta_1) q^{23} + q^{25} + ( - \beta_{4} - \beta_{3}) q^{27} + (\beta_{6} + 2 \beta_{2} + 4) q^{29} + (\beta_{7} + \beta_{4} + \cdots - 3 \beta_1) q^{31}+ \cdots + (2 \beta_{6} + 4 \beta_{2} - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 8 q^{21} + 8 q^{25} + 32 q^{29} - 24 q^{49} - 16 q^{53} - 32 q^{57} + 40 q^{69} - 16 q^{73} + 40 q^{81} - 64 q^{93} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 17\nu^{3} - 63\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 9\nu^{5} + 37\nu^{3} + 63\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 9\nu^{4} - \nu^{2} - 45 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + \nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 9\nu^{5} + 23\nu^{3} - 63\nu ) / 54 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 2\beta_{5} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 2\beta_{4} + 3\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{7} + \beta_{4} + 10\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.68014 + 0.420861i
1.68014 0.420861i
0.420861 + 1.68014i
0.420861 1.68014i
−0.420861 + 1.68014i
−0.420861 1.68014i
−1.68014 + 0.420861i
−1.68014 0.420861i
0 −1.68014 0.420861i 0 1.00000 0 3.91044i 0 2.64575 + 1.41421i 0
191.2 0 −1.68014 + 0.420861i 0 1.00000 0 3.91044i 0 2.64575 1.41421i 0
191.3 0 −0.420861 1.68014i 0 1.00000 0 2.16991i 0 −2.64575 + 1.41421i 0
191.4 0 −0.420861 + 1.68014i 0 1.00000 0 2.16991i 0 −2.64575 1.41421i 0
191.5 0 0.420861 1.68014i 0 1.00000 0 2.16991i 0 −2.64575 1.41421i 0
191.6 0 0.420861 + 1.68014i 0 1.00000 0 2.16991i 0 −2.64575 + 1.41421i 0
191.7 0 1.68014 0.420861i 0 1.00000 0 3.91044i 0 2.64575 1.41421i 0
191.8 0 1.68014 + 0.420861i 0 1.00000 0 3.91044i 0 2.64575 + 1.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.2.b.e yes 8
3.b odd 2 1 1920.2.b.c 8
4.b odd 2 1 inner 1920.2.b.e yes 8
8.b even 2 1 1920.2.b.c 8
8.d odd 2 1 1920.2.b.c 8
12.b even 2 1 1920.2.b.c 8
24.f even 2 1 inner 1920.2.b.e yes 8
24.h odd 2 1 inner 1920.2.b.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.b.c 8 3.b odd 2 1
1920.2.b.c 8 8.b even 2 1
1920.2.b.c 8 8.d odd 2 1
1920.2.b.c 8 12.b even 2 1
1920.2.b.e yes 8 1.a even 1 1 trivial
1920.2.b.e yes 8 4.b odd 2 1 inner
1920.2.b.e yes 8 24.f even 2 1 inner
1920.2.b.e yes 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1920, [\chi])\):

\( T_{7}^{4} + 20T_{7}^{2} + 72 \) Copy content Toggle raw display
\( T_{19}^{4} - 40T_{19}^{2} + 288 \) Copy content Toggle raw display
\( T_{23}^{4} - 52T_{23}^{2} + 648 \) Copy content Toggle raw display
\( T_{29}^{2} - 8T_{29} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 10T^{4} + 81 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T - 12)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 104 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 180 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 108)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 208 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 216 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 140 T^{2} + 3528)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 256 T^{2} + 256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T - 76)^{4} \) Copy content Toggle raw display
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