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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(225,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 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("896.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.m (of order \(4\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.214798336.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_1 q^{5} + \beta_{5} q^{7} + ( - \beta_{7} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_1 q^{5} + \beta_{5} q^{7} + ( - \beta_{7} - \beta_{5}) q^{9} + (\beta_{7} - \beta_{3}) q^{11} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} - 1) q^{13} + (\beta_{6} + \beta_{3} - \beta_1 + 1) q^{15} + (\beta_{6} - \beta_1 + 3) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{19}+ \cdots + ( - \beta_{7} - 6 \beta_{5} - 4 \beta_{4} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + 8 q^{15} + 24 q^{17} - 12 q^{19} + 12 q^{27} + 16 q^{29} - 16 q^{31} - 24 q^{33} - 4 q^{35} - 16 q^{37} - 32 q^{43} + 8 q^{45} - 24 q^{47} - 8 q^{49} + 8 q^{51} + 8 q^{53} - 28 q^{59} - 28 q^{61} + 8 q^{63} - 48 q^{65} - 44 q^{69} + 28 q^{75} + 24 q^{79} + 40 q^{81} + 40 q^{85} - 16 q^{93} - 16 q^{95} - 32 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{4} + 7\nu^{3} + \nu^{2} + 2\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{5} - 2\nu^{4} + 5\nu^{3} + 6\nu^{2} - 2\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 2\nu^{5} + 2\nu^{4} - 7\nu^{3} - \nu^{2} + 4\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} - 18\nu^{4} + 25\nu^{3} + 18\nu^{2} + 16\nu - 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 2\nu^{6} - 4\nu^{5} - 18\nu^{4} + 21\nu^{3} + 12\nu^{2} + 20\nu - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 6\nu^{6} - 8\nu^{5} - 22\nu^{4} + 39\nu^{3} + 20\nu^{2} + 24\nu - 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{7} + 5\nu^{6} + 6\nu^{5} + 24\nu^{4} - 34\nu^{3} - 15\nu^{2} - 28\nu + 84 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 3\beta_{5} - 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{6} + 3\beta_{4} + \beta_{3} + 4\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} + 9\beta_{5} - 5\beta_{4} + \beta_{3} + 3\beta_{2} + 4\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 7\beta_{6} - 10\beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} - 3 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
−0.565036 − 1.29643i
1.41216 + 0.0762223i
1.23291 − 0.692769i
−1.08003 + 0.912978i
−0.565036 + 1.29643i
1.41216 − 0.0762223i
1.23291 + 0.692769i
−1.08003 − 0.912978i
0 −1.96506 + 1.96506i 0 0.627801 + 0.627801i 0 1.00000i 0 − 4.72294i 0
225.2 0 −0.715276 + 0.715276i 0 −0.867721 − 0.867721i 0 1.00000i 0 1.97676i 0
225.3 0 1.20825 − 1.20825i 0 2.59378 + 2.59378i 0 1.00000i 0 0.0802864i 0
225.4 0 1.47209 − 1.47209i 0 −0.353863 − 0.353863i 0 1.00000i 0 − 1.33411i 0
673.1 0 −1.96506 − 1.96506i 0 0.627801 − 0.627801i 0 − 1.00000i 0 4.72294i 0
673.2 0 −0.715276 − 0.715276i 0 −0.867721 + 0.867721i 0 − 1.00000i 0 − 1.97676i 0
673.3 0 1.20825 + 1.20825i 0 2.59378 − 2.59378i 0 − 1.00000i 0 − 0.0802864i 0
673.4 0 1.47209 + 1.47209i 0 −0.353863 + 0.353863i 0 − 1.00000i 0 1.33411i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.m.f 8
4.b odd 2 1 896.2.m.e 8
8.b even 2 1 448.2.m.c 8
8.d odd 2 1 112.2.m.c ✓ 8
16.e even 4 1 448.2.m.c 8
16.e even 4 1 inner 896.2.m.f 8
16.f odd 4 1 112.2.m.c ✓ 8
16.f odd 4 1 896.2.m.e 8
32.g even 8 2 7168.2.a.bd 8
32.h odd 8 2 7168.2.a.bc 8
56.e even 2 1 784.2.m.g 8
56.k odd 6 2 784.2.x.k 16
56.m even 6 2 784.2.x.j 16
112.j even 4 1 784.2.m.g 8
112.u odd 12 2 784.2.x.k 16
112.v even 12 2 784.2.x.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.c ✓ 8 8.d odd 2 1
112.2.m.c ✓ 8 16.f odd 4 1
448.2.m.c 8 8.b even 2 1
448.2.m.c 8 16.e even 4 1
784.2.m.g 8 56.e even 2 1
784.2.m.g 8 112.j even 4 1
784.2.x.j 16 56.m even 6 2
784.2.x.j 16 112.v even 12 2
784.2.x.k 16 56.k odd 6 2
784.2.x.k 16 112.u odd 12 2
896.2.m.e 8 4.b odd 2 1
896.2.m.e 8 16.f odd 4 1
896.2.m.f 8 1.a even 1 1 trivial
896.2.m.f 8 16.e even 4 1 inner
7168.2.a.bc 8 32.h odd 8 2
7168.2.a.bd 8 32.g even 8 2

Hecke kernels

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

\( T_{3}^{8} - 4T_{3}^{5} + 44T_{3}^{4} - 32T_{3}^{3} + 8T_{3}^{2} + 40T_{3} + 100 \) Copy content Toggle raw display
\( T_{11}^{8} + 32T_{11}^{5} + 544T_{11}^{4} + 768T_{11}^{3} + 512T_{11}^{2} - 512T_{11} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{5} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{5} + \cdots + 28900 \) Copy content Toggle raw display
$17$ \( (T^{4} - 12 T^{3} + 40 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 39204 \) Copy content Toggle raw display
$23$ \( T^{8} + 80 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + \cdots + 488)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + \cdots + 435600 \) Copy content Toggle raw display
$41$ \( T^{8} + 288 T^{6} + \cdots + 5198400 \) Copy content Toggle raw display
$43$ \( T^{8} + 32 T^{7} + \cdots + 10863616 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} + \cdots - 376)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$59$ \( T^{8} + 28 T^{7} + \cdots + 9771876 \) Copy content Toggle raw display
$61$ \( T^{8} + 28 T^{7} + \cdots + 161604 \) Copy content Toggle raw display
$67$ \( T^{8} - 1024 T^{5} + \cdots + 495616 \) Copy content Toggle raw display
$71$ \( T^{8} + 496 T^{6} + \cdots + 145926400 \) Copy content Toggle raw display
$73$ \( T^{8} + 272 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + \cdots - 2160)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{5} + \cdots + 100 \) Copy content Toggle raw display
$89$ \( T^{8} + 432 T^{6} + \cdots + 1290496 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 712)^{2} \) Copy content Toggle raw display
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