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Properties

Label 1104.3.g.d
Level $1104$
Weight $3$
Character orbit 1104.g
Analytic conductor $30.082$
Analytic rank $0$
Dimension $44$
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] = [1104,3,Mod(737,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.737");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.g (of order \(2\), degree \(1\), 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: \(30.0818211854\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 552)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} - 24 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} - 24 q^{7} + 12 q^{9} - 40 q^{13} + 8 q^{19} - 180 q^{25} + 4 q^{27} - 72 q^{31} - 8 q^{37} + 96 q^{39} + 344 q^{43} - 32 q^{45} + 420 q^{49} - 256 q^{51} - 112 q^{55} - 104 q^{57} + 184 q^{61} + 176 q^{63} - 232 q^{67} - 360 q^{73} - 260 q^{75} + 152 q^{79} + 148 q^{81} - 400 q^{85} - 160 q^{87} - 224 q^{91} + 64 q^{93} + 504 q^{97} + 616 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1 0 −2.95090 − 0.540569i 0 0.499498i 0 3.08204 0 8.41557 + 3.19032i 0
737.2 0 −2.95090 + 0.540569i 0 − 0.499498i 0 3.08204 0 8.41557 − 3.19032i 0
737.3 0 −2.86400 − 0.893038i 0 − 3.63890i 0 12.6596 0 7.40497 + 5.11532i 0
737.4 0 −2.86400 + 0.893038i 0 3.63890i 0 12.6596 0 7.40497 − 5.11532i 0
737.5 0 −2.83547 − 0.979858i 0 − 3.20437i 0 −7.83276 0 7.07976 + 5.55671i 0
737.6 0 −2.83547 + 0.979858i 0 3.20437i 0 −7.83276 0 7.07976 − 5.55671i 0
737.7 0 −2.68471 − 1.33879i 0 − 9.07133i 0 −11.7074 0 5.41530 + 7.18850i 0
737.8 0 −2.68471 + 1.33879i 0 9.07133i 0 −11.7074 0 5.41530 − 7.18850i 0
737.9 0 −2.33636 − 1.88187i 0 6.58362i 0 −0.477243 0 1.91716 + 8.79344i 0
737.10 0 −2.33636 + 1.88187i 0 − 6.58362i 0 −0.477243 0 1.91716 − 8.79344i 0
737.11 0 −2.31152 − 1.91229i 0 7.18138i 0 2.18881 0 1.68626 + 8.84062i 0
737.12 0 −2.31152 + 1.91229i 0 − 7.18138i 0 2.18881 0 1.68626 − 8.84062i 0
737.13 0 −1.66505 − 2.49552i 0 − 4.55723i 0 1.13857 0 −3.45522 + 8.31032i 0
737.14 0 −1.66505 + 2.49552i 0 4.55723i 0 1.13857 0 −3.45522 − 8.31032i 0
737.15 0 −1.31323 − 2.69730i 0 − 2.01018i 0 9.63197 0 −5.55086 + 7.08434i 0
737.16 0 −1.31323 + 2.69730i 0 2.01018i 0 9.63197 0 −5.55086 − 7.08434i 0
737.17 0 −1.09749 − 2.79204i 0 3.78975i 0 −8.32649 0 −6.59103 + 6.12849i 0
737.18 0 −1.09749 + 2.79204i 0 − 3.78975i 0 −8.32649 0 −6.59103 − 6.12849i 0
737.19 0 −0.188624 − 2.99406i 0 1.86524i 0 −12.4728 0 −8.92884 + 1.12951i 0
737.20 0 −0.188624 + 2.99406i 0 − 1.86524i 0 −12.4728 0 −8.92884 − 1.12951i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.3.g.d 44
3.b odd 2 1 inner 1104.3.g.d 44
4.b odd 2 1 552.3.g.a ✓ 44
12.b even 2 1 552.3.g.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.3.g.a ✓ 44 4.b odd 2 1
552.3.g.a ✓ 44 12.b even 2 1
1104.3.g.d 44 1.a even 1 1 trivial
1104.3.g.d 44 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{44} + 640 T_{5}^{42} + 186836 T_{5}^{40} + 32998064 T_{5}^{38} + 3943966980 T_{5}^{36} + \cdots + 97\!\cdots\!16 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display