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Properties

Label 637.2.i.b
Level $637$
Weight $2$
Character orbit 637.i
Analytic conductor $5.086$
Analytic rank $0$
Dimension $56$
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] = [637,2,Mod(489,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.489");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.i (of order \(4\), degree \(2\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{9} - 16 q^{11} + 48 q^{15} - 56 q^{16} - 32 q^{18} - 48 q^{29} - 32 q^{39} - 64 q^{44} + 32 q^{46} - 40 q^{50} - 16 q^{53} - 96 q^{57} + 72 q^{58} - 64 q^{60} + 32 q^{65} + 32 q^{71} + 208 q^{72} + 208 q^{74} + 160 q^{78} - 80 q^{79} + 152 q^{81} + 24 q^{85} + 64 q^{86} - 128 q^{92} + 64 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
489.1 −1.87195 − 1.87195i − 0.0697458i 5.00838i 1.60950 − 1.60950i −0.130560 + 0.130560i 0 5.63154 − 5.63154i 2.99514 −6.02579
489.2 −1.87195 − 1.87195i 0.0697458i 5.00838i −1.60950 + 1.60950i 0.130560 − 0.130560i 0 5.63154 − 5.63154i 2.99514 6.02579
489.3 −1.67405 − 1.67405i − 1.42381i 3.60486i 1.67159 − 1.67159i −2.38352 + 2.38352i 0 2.68660 − 2.68660i 0.972778 −5.59662
489.4 −1.67405 − 1.67405i 1.42381i 3.60486i −1.67159 + 1.67159i 2.38352 − 2.38352i 0 2.68660 − 2.68660i 0.972778 5.59662
489.5 −1.31218 − 1.31218i − 2.95980i 1.44363i −0.400884 + 0.400884i −3.88379 + 3.88379i 0 −0.730060 + 0.730060i −5.76043 1.05206
489.6 −1.31218 − 1.31218i 2.95980i 1.44363i 0.400884 − 0.400884i 3.88379 − 3.88379i 0 −0.730060 + 0.730060i −5.76043 −1.05206
489.7 −1.16293 − 1.16293i − 2.79366i 0.704814i −1.90208 + 1.90208i −3.24884 + 3.24884i 0 −1.50621 + 1.50621i −4.80456 4.42397
489.8 −1.16293 − 1.16293i 2.79366i 0.704814i 1.90208 − 1.90208i 3.24884 − 3.24884i 0 −1.50621 + 1.50621i −4.80456 −4.42397
489.9 −1.04867 − 1.04867i − 0.884495i 0.199431i −2.77428 + 2.77428i −0.927546 + 0.927546i 0 −1.88821 + 1.88821i 2.21767 5.81862
489.10 −1.04867 − 1.04867i 0.884495i 0.199431i 2.77428 − 2.77428i 0.927546 − 0.927546i 0 −1.88821 + 1.88821i 2.21767 −5.81862
489.11 −0.313676 − 0.313676i − 2.15152i − 1.80322i −1.37124 + 1.37124i −0.674879 + 0.674879i 0 −1.19298 + 1.19298i −1.62904 0.860251
489.12 −0.313676 − 0.313676i 2.15152i − 1.80322i 1.37124 − 1.37124i 0.674879 − 0.674879i 0 −1.19298 + 1.19298i −1.62904 −0.860251
489.13 −0.0315097 − 0.0315097i − 0.476410i − 1.99801i −2.33607 + 2.33607i −0.0150115 + 0.0150115i 0 −0.125976 + 0.125976i 2.77303 0.147218
489.14 −0.0315097 − 0.0315097i 0.476410i − 1.99801i 2.33607 − 2.33607i 0.0150115 − 0.0150115i 0 −0.125976 + 0.125976i 2.77303 −0.147218
489.15 0.319250 + 0.319250i − 3.18289i − 1.79616i 0.0250354 − 0.0250354i 1.01614 − 1.01614i 0 1.21192 − 1.21192i −7.13080 0.0159851
489.16 0.319250 + 0.319250i 3.18289i − 1.79616i −0.0250354 + 0.0250354i −1.01614 + 1.01614i 0 1.21192 − 1.21192i −7.13080 −0.0159851
489.17 0.356966 + 0.356966i − 1.89794i − 1.74515i 0.631571 − 0.631571i 0.677500 − 0.677500i 0 1.33689 − 1.33689i −0.602176 0.450899
489.18 0.356966 + 0.356966i 1.89794i − 1.74515i −0.631571 + 0.631571i −0.677500 + 0.677500i 0 1.33689 − 1.33689i −0.602176 −0.450899
489.19 0.477302 + 0.477302i − 0.696451i − 1.54437i 2.64777 − 2.64777i 0.332417 − 0.332417i 0 1.69173 − 1.69173i 2.51496 2.52757
489.20 0.477302 + 0.477302i 0.696451i − 1.54437i −2.64777 + 2.64777i −0.332417 + 0.332417i 0 1.69173 − 1.69173i 2.51496 −2.52757
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 489.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.i.b ✓ 56
7.b odd 2 1 inner 637.2.i.b ✓ 56
7.c even 3 2 637.2.bc.c 112
7.d odd 6 2 637.2.bc.c 112
13.d odd 4 1 inner 637.2.i.b ✓ 56
91.i even 4 1 inner 637.2.i.b ✓ 56
91.z odd 12 2 637.2.bc.c 112
91.bb even 12 2 637.2.bc.c 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.i.b ✓ 56 1.a even 1 1 trivial
637.2.i.b ✓ 56 7.b odd 2 1 inner
637.2.i.b ✓ 56 13.d odd 4 1 inner
637.2.i.b ✓ 56 91.i even 4 1 inner
637.2.bc.c 112 7.c even 3 2
637.2.bc.c 112 7.d odd 6 2
637.2.bc.c 112 91.z odd 12 2
637.2.bc.c 112 91.bb even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 105 T_{2}^{24} + 16 T_{2}^{21} + 3788 T_{2}^{20} + 192 T_{2}^{19} + 24 T_{2}^{17} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display