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Properties

Label 164.4.g.a
Level $164$
Weight $4$
Character orbit 164.g
Analytic conductor $9.676$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [164,4,Mod(37,164)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("164.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{3} + 270 q^{9} + 83 q^{11} - 94 q^{13} - 48 q^{15} - 144 q^{17} - 179 q^{19} - 176 q^{21} + 180 q^{23} - 542 q^{25} - 236 q^{27} + 68 q^{29} - 298 q^{31} + 823 q^{33} + 178 q^{35} - 253 q^{37}+ \cdots + 3203 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0 −8.21967 0 2.52208 − 7.76217i 0 −12.5165 − 9.09375i 0 40.5630 0
37.2 0 −8.16150 0 −5.04262 + 15.5196i 0 12.7699 + 9.27787i 0 39.6101 0
37.3 0 −4.80812 0 −0.365636 + 1.12531i 0 −3.13921 − 2.28077i 0 −3.88195 0
37.4 0 −2.76331 0 6.39959 − 19.6959i 0 14.0004 + 10.1719i 0 −19.3641 0
37.5 0 −0.00279865 0 −5.78452 + 17.8029i 0 −6.36253 − 4.62265i 0 −27.0000 0
37.6 0 0.660929 0 −0.853567 + 2.62701i 0 −8.94503 − 6.49895i 0 −26.5632 0
37.7 0 1.42190 0 −1.32265 + 4.07070i 0 26.1745 + 19.0169i 0 −24.9782 0
37.8 0 3.93993 0 1.95847 − 6.02755i 0 −25.1761 − 18.2915i 0 −11.4769 0
37.9 0 7.79300 0 2.83592 − 8.72806i 0 7.21035 + 5.23862i 0 33.7308 0
37.10 0 8.52161 0 −5.93723 + 18.2729i 0 −4.01572 − 2.91759i 0 45.6179 0
57.1 0 −9.50420 0 10.5878 + 7.69246i 0 −0.866952 − 2.66820i 0 63.3299 0
57.2 0 −7.26044 0 −10.9541 − 7.95862i 0 −3.86362 − 11.8910i 0 25.7140 0
57.3 0 −4.92844 0 −5.60949 − 4.07554i 0 9.35499 + 28.7917i 0 −2.71043 0
57.4 0 −3.22589 0 5.85485 + 4.25380i 0 −3.98024 − 12.2499i 0 −16.5936 0
57.5 0 −1.34246 0 12.1086 + 8.79743i 0 3.71815 + 11.4433i 0 −25.1978 0
57.6 0 2.14664 0 −2.20052 − 1.59877i 0 −7.08323 − 21.7999i 0 −22.3919 0
57.7 0 2.15973 0 −11.6302 − 8.44982i 0 4.32972 + 13.3255i 0 −22.3356 0
57.8 0 7.29617 0 12.1060 + 8.79552i 0 −7.93674 − 24.4268i 0 26.2340 0
57.9 0 7.60991 0 −11.2392 − 8.16579i 0 −0.776601 − 2.39013i 0 30.9107 0
57.10 0 7.66703 0 6.56647 + 4.77082i 0 7.10452 + 21.8655i 0 31.7833 0
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.4.g.a ✓ 40
41.d even 5 1 inner 164.4.g.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.4.g.a ✓ 40 1.a even 1 1 trivial
164.4.g.a ✓ 40 41.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(164, [\chi])\).