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Properties

Label 896.2.f.c
Level $896$
Weight $2$
Character orbit 896.f
Analytic conductor $7.155$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(895,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.895");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.f (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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{7} + ( - \beta_{3} + 2) q^{9} + (\beta_{7} - \beta_{4}) q^{11} - \beta_{6} q^{13} + (\beta_{7} - 3 \beta_{4}) q^{15} + ( - \beta_{6} - \beta_1) q^{17} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{19} + (\beta_{3} - \beta_1 - 3) q^{21} - \beta_{7} q^{23} + \beta_{3} q^{25} + ( - 2 \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{27} + 6 q^{29} + (2 \beta_{5} + \beta_{4}) q^{31} + ( - \beta_{6} + 5 \beta_1) q^{33} + ( - \beta_{7} + 2 \beta_{4} + \beta_{2}) q^{35} + 2 q^{37} + (\beta_{7} - \beta_{4}) q^{39} - 2 \beta_1 q^{41} + ( - \beta_{7} + \beta_{4}) q^{43} + ( - \beta_{6} + 6 \beta_1) q^{45} + (2 \beta_{5} + \beta_{4}) q^{47} + ( - \beta_{6} + \beta_1 + 5) q^{49} + 2 \beta_{4} q^{51} - 2 \beta_{3} q^{53} + (2 \beta_{5} + \beta_{4} - 6 \beta_{2}) q^{55} + (\beta_{3} - 9) q^{57} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{59} + ( - 2 \beta_{6} + \beta_1) q^{61} + ( - \beta_{7} - \beta_{5} + \cdots - 5 \beta_{2}) q^{63}+ \cdots + (3 \beta_{7} - 13 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} - 24 q^{21} + 48 q^{29} + 16 q^{37} + 40 q^{49} - 72 q^{57} - 8 q^{65} + 8 q^{77} + 80 q^{81} + 32 q^{85} + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 24\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 23 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} - 48\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + \nu^{6} + \nu^{5} - 115\nu^{3} + 24\nu^{2} + 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} + 2\nu^{5} + 206\nu^{3} + 43\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{6} - 134\nu^{2} ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} + \beta_{4} - 10\beta_{2} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{3} - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{6} + 10\beta_{5} + 5\beta_{4} - 48\beta_{2} - 43\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -24\beta_{7} + 67\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12\beta_{6} - 24\beta_{5} - 12\beta_{4} + 115\beta_{2} - 103\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
−1.54779 − 1.54779i
−1.54779 + 1.54779i
−0.323042 − 0.323042i
−0.323042 + 0.323042i
0.323042 − 0.323042i
0.323042 + 0.323042i
1.54779 − 1.54779i
1.54779 + 1.54779i
0 −3.09557 0 − 3.09557i 0 2.44949 − 1.00000i 0 6.58258 0
895.2 0 −3.09557 0 3.09557i 0 2.44949 + 1.00000i 0 6.58258 0
895.3 0 −0.646084 0 − 0.646084i 0 −2.44949 − 1.00000i 0 −2.58258 0
895.4 0 −0.646084 0 0.646084i 0 −2.44949 + 1.00000i 0 −2.58258 0
895.5 0 0.646084 0 − 0.646084i 0 2.44949 + 1.00000i 0 −2.58258 0
895.6 0 0.646084 0 0.646084i 0 2.44949 − 1.00000i 0 −2.58258 0
895.7 0 3.09557 0 − 3.09557i 0 −2.44949 + 1.00000i 0 6.58258 0
895.8 0 3.09557 0 3.09557i 0 −2.44949 − 1.00000i 0 6.58258 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.f.c ✓ 8
4.b odd 2 1 inner 896.2.f.c ✓ 8
7.b odd 2 1 inner 896.2.f.c ✓ 8
8.b even 2 1 896.2.f.d yes 8
8.d odd 2 1 896.2.f.d yes 8
16.e even 4 1 1792.2.e.d 8
16.e even 4 1 1792.2.e.e 8
16.f odd 4 1 1792.2.e.d 8
16.f odd 4 1 1792.2.e.e 8
28.d even 2 1 inner 896.2.f.c ✓ 8
56.e even 2 1 896.2.f.d yes 8
56.h odd 2 1 896.2.f.d yes 8
112.j even 4 1 1792.2.e.d 8
112.j even 4 1 1792.2.e.e 8
112.l odd 4 1 1792.2.e.d 8
112.l odd 4 1 1792.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.f.c ✓ 8 1.a even 1 1 trivial
896.2.f.c ✓ 8 4.b odd 2 1 inner
896.2.f.c ✓ 8 7.b odd 2 1 inner
896.2.f.c ✓ 8 28.d even 2 1 inner
896.2.f.d yes 8 8.b even 2 1
896.2.f.d yes 8 8.d odd 2 1
896.2.f.d yes 8 56.e even 2 1
896.2.f.d yes 8 56.h odd 2 1
1792.2.e.d 8 16.e even 4 1
1792.2.e.d 8 16.f odd 4 1
1792.2.e.d 8 112.j even 4 1
1792.2.e.d 8 112.l odd 4 1
1792.2.e.e 8 16.e even 4 1
1792.2.e.e 8 16.f odd 4 1
1792.2.e.e 8 112.j even 4 1
1792.2.e.e 8 112.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 66 T^{2} + 900)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 40 T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 84)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 154 T^{2} + 4900)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 204 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 56)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 360 T^{2} + 5184)^{2} \) Copy content Toggle raw display
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