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Properties

Label 896.2.u.b
Level $896$
Weight $2$
Character orbit 896.u
Analytic conductor $7.155$
Analytic rank $0$
Dimension $40$
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(113,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(8))
 
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("896.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.u (of order \(8\), degree \(4\), 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: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 8 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} + 8 q^{5} + 8 q^{9} - 12 q^{11} - 4 q^{19} - 4 q^{21} - 16 q^{23} - 16 q^{25} + 32 q^{27} + 16 q^{29} - 24 q^{31} + 8 q^{33} + 16 q^{37} + 24 q^{39} + 8 q^{41} - 52 q^{43} - 64 q^{45} - 16 q^{51} + 8 q^{55} - 8 q^{57} - 20 q^{59} - 16 q^{61} + 24 q^{63} - 80 q^{65} + 4 q^{67} - 40 q^{69} - 24 q^{73} - 20 q^{75} + 4 q^{77} - 4 q^{83} - 64 q^{85} + 32 q^{87} + 32 q^{89} - 24 q^{93} + 80 q^{95} + 56 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −1.17309 − 2.83210i 0 0.671484 + 0.278138i 0 −0.707107 − 0.707107i 0 −4.52330 + 4.52330i 0
113.2 0 −0.965466 − 2.33084i 0 1.27468 + 0.527990i 0 −0.707107 − 0.707107i 0 −2.37937 + 2.37937i 0
113.3 0 −0.722871 − 1.74516i 0 −1.56259 − 0.647248i 0 −0.707107 − 0.707107i 0 −0.401738 + 0.401738i 0
113.4 0 −0.575733 − 1.38994i 0 1.63129 + 0.675703i 0 −0.707107 − 0.707107i 0 0.520847 − 0.520847i 0
113.5 0 0.129878 + 0.313552i 0 −1.65528 − 0.685640i 0 −0.707107 − 0.707107i 0 2.03987 − 2.03987i 0
113.6 0 0.149272 + 0.360376i 0 2.11490 + 0.876021i 0 −0.707107 − 0.707107i 0 2.01373 − 2.01373i 0
113.7 0 0.160595 + 0.387712i 0 −3.40274 − 1.40946i 0 −0.707107 − 0.707107i 0 1.99679 − 1.99679i 0
113.8 0 0.244479 + 0.590225i 0 −0.220493 − 0.0913310i 0 −0.707107 − 0.707107i 0 1.83272 − 1.83272i 0
113.9 0 0.745174 + 1.79901i 0 1.03098 + 0.427045i 0 −0.707107 − 0.707107i 0 −0.559827 + 0.559827i 0
113.10 0 1.00776 + 2.43296i 0 3.53199 + 1.46300i 0 −0.707107 − 0.707107i 0 −2.78237 + 2.78237i 0
337.1 0 −2.41040 − 0.998422i 0 0.0734640 + 0.177358i 0 0.707107 − 0.707107i 0 2.69188 + 2.69188i 0
337.2 0 −2.28471 − 0.946358i 0 0.446012 + 1.07677i 0 0.707107 − 0.707107i 0 2.20299 + 2.20299i 0
337.3 0 −2.04641 − 0.847650i 0 −1.31439 − 3.17322i 0 0.707107 − 0.707107i 0 1.34795 + 1.34795i 0
337.4 0 −1.29558 − 0.536647i 0 0.937867 + 2.26421i 0 0.707107 − 0.707107i 0 −0.730784 − 0.730784i 0
337.5 0 −0.350600 − 0.145223i 0 −1.27761 − 3.08441i 0 0.707107 − 0.707107i 0 −2.01949 − 2.01949i 0
337.6 0 −0.0422763 − 0.0175114i 0 1.39396 + 3.36532i 0 0.707107 − 0.707107i 0 −2.11984 − 2.11984i 0
337.7 0 0.910708 + 0.377228i 0 −0.227513 − 0.549265i 0 0.707107 − 0.707107i 0 −1.43423 − 1.43423i 0
337.8 0 1.66728 + 0.690611i 0 0.923458 + 2.22943i 0 0.707107 − 0.707107i 0 0.181564 + 0.181564i 0
337.9 0 1.81729 + 0.752744i 0 −0.920909 − 2.22327i 0 0.707107 − 0.707107i 0 0.614581 + 0.614581i 0
337.10 0 3.03470 + 1.25702i 0 0.551446 + 1.33131i 0 0.707107 − 0.707107i 0 5.50802 + 5.50802i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.u.b 40
4.b odd 2 1 224.2.u.b ✓ 40
32.g even 8 1 inner 896.2.u.b 40
32.h odd 8 1 224.2.u.b ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.b ✓ 40 4.b odd 2 1
224.2.u.b ✓ 40 32.h odd 8 1
896.2.u.b 40 1.a even 1 1 trivial
896.2.u.b 40 32.g even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 4 T_{3}^{39} + 4 T_{3}^{38} - 24 T_{3}^{37} - 88 T_{3}^{36} + 40 T_{3}^{35} + 872 T_{3}^{34} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display