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Properties

Label 896.2.bh.a
Level $896$
Weight $2$
Character orbit 896.bh
Analytic conductor $7.155$
Analytic rank $0$
Dimension $240$
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(81,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([0, 9, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.bh (of order \(24\), degree \(8\), 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: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{24})\)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{3} - 4 q^{5} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{3} - 4 q^{5} + 8 q^{7} - 4 q^{9} + 4 q^{11} - 16 q^{13} + 4 q^{19} - 8 q^{21} + 12 q^{23} - 4 q^{25} + 16 q^{27} - 16 q^{29} + 56 q^{31} - 8 q^{33} + 32 q^{35} - 4 q^{37} + 4 q^{39} - 16 q^{41} + 8 q^{45} + 28 q^{51} - 20 q^{53} + 16 q^{55} - 16 q^{57} + 36 q^{59} - 4 q^{61} + 16 q^{63} - 8 q^{65} - 36 q^{67} - 16 q^{69} - 48 q^{71} - 4 q^{73} - 16 q^{75} - 8 q^{77} + 96 q^{83} - 56 q^{85} + 4 q^{87} - 4 q^{89} + 56 q^{91} + 20 q^{93} + 8 q^{95} - 32 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.53727 + 1.94692i 0 0.157528 − 0.205294i 0 −0.893273 − 2.49039i 0 1.87081 − 6.98194i 0
81.2 0 −2.44576 + 1.87670i 0 0.0746500 − 0.0972858i 0 −1.07291 + 2.41844i 0 1.68329 − 6.28212i 0
81.3 0 −2.34780 + 1.80153i 0 −2.10755 + 2.74661i 0 2.53918 − 0.743361i 0 1.49020 − 5.56151i 0
81.4 0 −1.87086 + 1.43556i 0 −0.351264 + 0.457777i 0 −2.57326 + 0.615072i 0 0.662818 − 2.47367i 0
81.5 0 −1.80239 + 1.38302i 0 −0.194347 + 0.253278i 0 1.82678 + 1.91386i 0 0.559406 − 2.08773i 0
81.6 0 −1.59186 + 1.22147i 0 1.50255 − 1.95816i 0 2.25138 − 1.38970i 0 0.265550 − 0.991047i 0
81.7 0 −1.56993 + 1.20465i 0 −1.98898 + 2.59208i 0 −1.59198 − 2.11320i 0 0.237043 − 0.884657i 0
81.8 0 −1.42891 + 1.09644i 0 1.65207 − 2.15302i 0 −0.399279 − 2.61545i 0 0.0631376 − 0.235633i 0
81.9 0 −1.32101 + 1.01365i 0 2.54115 − 3.31169i 0 1.39322 + 2.24921i 0 −0.0588677 + 0.219697i 0
81.10 0 −0.867605 + 0.665736i 0 −0.00636490 + 0.00829490i 0 0.529064 + 2.59231i 0 −0.466924 + 1.74259i 0
81.11 0 −0.678599 + 0.520708i 0 −2.43841 + 3.17779i 0 −1.24705 + 2.33343i 0 −0.587096 + 2.19107i 0
81.12 0 −0.653842 + 0.501711i 0 −1.21118 + 1.57844i 0 1.14530 − 2.38501i 0 −0.600661 + 2.24170i 0
81.13 0 −0.194135 + 0.148965i 0 2.33985 − 3.04935i 0 −2.62216 − 0.352546i 0 −0.760959 + 2.83994i 0
81.14 0 −0.163619 + 0.125549i 0 −0.947705 + 1.23507i 0 2.63443 − 0.244488i 0 −0.765449 + 2.85669i 0
81.15 0 −0.112695 + 0.0864738i 0 1.22200 − 1.59255i 0 1.93515 − 1.80422i 0 −0.771235 + 2.87829i 0
81.16 0 −0.102520 + 0.0786667i 0 1.38225 − 1.80138i 0 −2.39854 + 1.11670i 0 −0.772135 + 2.88165i 0
81.17 0 0.360007 − 0.276243i 0 0.0970846 − 0.126523i 0 −2.18584 − 1.49067i 0 −0.723162 + 2.69888i 0
81.18 0 0.440708 − 0.338167i 0 −1.41512 + 1.84422i 0 −0.974187 − 2.45987i 0 −0.696591 + 2.59971i 0
81.19 0 0.656890 − 0.504050i 0 1.17870 − 1.53611i 0 0.376163 + 2.61887i 0 −0.599018 + 2.23557i 0
81.20 0 1.11575 − 0.856146i 0 0.209020 − 0.272400i 0 −1.04743 + 2.42958i 0 −0.264542 + 0.987286i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
32.g even 8 1 inner
224.bd even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.bh.a 240
4.b odd 2 1 224.2.bd.a ✓ 240
7.c even 3 1 inner 896.2.bh.a 240
28.g odd 6 1 224.2.bd.a ✓ 240
32.g even 8 1 inner 896.2.bh.a 240
32.h odd 8 1 224.2.bd.a ✓ 240
224.bd even 24 1 inner 896.2.bh.a 240
224.bf odd 24 1 224.2.bd.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.bd.a ✓ 240 4.b odd 2 1
224.2.bd.a ✓ 240 28.g odd 6 1
224.2.bd.a ✓ 240 32.h odd 8 1
224.2.bd.a ✓ 240 224.bf odd 24 1
896.2.bh.a 240 1.a even 1 1 trivial
896.2.bh.a 240 7.c even 3 1 inner
896.2.bh.a 240 32.g even 8 1 inner
896.2.bh.a 240 224.bd even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(896, [\chi])\).