Properties

Label 138.4.d.a
Level $138$
Weight $4$
Character orbit 138.d
Analytic conductor $8.142$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} - 96 q^{4} + 8 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{3} - 96 q^{4} + 8 q^{6} + 36 q^{9} - 32 q^{12} + 96 q^{13} + 384 q^{16} + 128 q^{18} - 32 q^{24} + 144 q^{25} + 188 q^{27} + 72 q^{31} - 144 q^{36} - 660 q^{39} - 96 q^{46} + 128 q^{48} - 504 q^{49} - 384 q^{52} + 88 q^{54} - 672 q^{55} + 816 q^{58} - 1536 q^{64} + 352 q^{69} + 624 q^{70} - 512 q^{72} - 2688 q^{73} - 1072 q^{75} + 80 q^{78} - 2356 q^{81} + 1344 q^{82} + 4872 q^{85} + 3748 q^{87} - 2924 q^{93} - 1296 q^{94} + 128 q^{96}+O(q^{100}) \) 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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 2.00000i −4.96925 1.51872i −4.00000 −17.2734 −3.03745 + 9.93851i 11.5675i 8.00000i 22.3870 + 15.0938i 34.5468i
137.2 2.00000i −4.96925 1.51872i −4.00000 17.2734 −3.03745 + 9.93851i 11.5675i 8.00000i 22.3870 + 15.0938i 34.5468i
137.3 2.00000i −3.26555 + 4.04180i −4.00000 −6.82316 8.08360 + 6.53111i 3.09939i 8.00000i −5.67231 26.3974i 13.6463i
137.4 2.00000i −3.26555 + 4.04180i −4.00000 6.82316 8.08360 + 6.53111i 3.09939i 8.00000i −5.67231 26.3974i 13.6463i
137.5 2.00000i −1.34717 5.01848i −4.00000 −4.58818 −10.0370 + 2.69433i 31.2911i 8.00000i −23.3703 + 13.5215i 9.17636i
137.6 2.00000i −1.34717 5.01848i −4.00000 4.58818 −10.0370 + 2.69433i 31.2911i 8.00000i −23.3703 + 13.5215i 9.17636i
137.7 2.00000i 2.34341 + 4.63772i −4.00000 −10.8839 9.27544 4.68682i 21.8179i 8.00000i −16.0169 + 21.7361i 21.7678i
137.8 2.00000i 2.34341 + 4.63772i −4.00000 10.8839 9.27544 4.68682i 21.8179i 8.00000i −16.0169 + 21.7361i 21.7678i
137.9 2.00000i 4.33582 2.86368i −4.00000 −13.9937 −5.72736 8.67164i 0.843170i 8.00000i 10.5987 24.8328i 27.9874i
137.10 2.00000i 4.33582 2.86368i −4.00000 13.9937 −5.72736 8.67164i 0.843170i 8.00000i 10.5987 24.8328i 27.9874i
137.11 2.00000i 4.90275 + 1.72136i −4.00000 −10.2829 3.44272 9.80549i 24.1811i 8.00000i 21.0738 + 16.8788i 20.5659i
137.12 2.00000i 4.90275 + 1.72136i −4.00000 10.2829 3.44272 9.80549i 24.1811i 8.00000i 21.0738 + 16.8788i 20.5659i
137.13 2.00000i −4.96925 + 1.51872i −4.00000 −17.2734 −3.03745 9.93851i 11.5675i 8.00000i 22.3870 15.0938i 34.5468i
137.14 2.00000i −4.96925 + 1.51872i −4.00000 17.2734 −3.03745 9.93851i 11.5675i 8.00000i 22.3870 15.0938i 34.5468i
137.15 2.00000i −3.26555 4.04180i −4.00000 −6.82316 8.08360 6.53111i 3.09939i 8.00000i −5.67231 + 26.3974i 13.6463i
137.16 2.00000i −3.26555 4.04180i −4.00000 6.82316 8.08360 6.53111i 3.09939i 8.00000i −5.67231 + 26.3974i 13.6463i
137.17 2.00000i −1.34717 + 5.01848i −4.00000 −4.58818 −10.0370 2.69433i 31.2911i 8.00000i −23.3703 13.5215i 9.17636i
137.18 2.00000i −1.34717 + 5.01848i −4.00000 4.58818 −10.0370 2.69433i 31.2911i 8.00000i −23.3703 13.5215i 9.17636i
137.19 2.00000i 2.34341 4.63772i −4.00000 −10.8839 9.27544 + 4.68682i 21.8179i 8.00000i −16.0169 21.7361i 21.7678i
137.20 2.00000i 2.34341 4.63772i −4.00000 10.8839 9.27544 + 4.68682i 21.8179i 8.00000i −16.0169 21.7361i 21.7678i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.4.d.a 24
3.b odd 2 1 inner 138.4.d.a 24
23.b odd 2 1 inner 138.4.d.a 24
69.c even 2 1 inner 138.4.d.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.d.a 24 1.a even 1 1 trivial
138.4.d.a 24 3.b odd 2 1 inner
138.4.d.a 24 23.b odd 2 1 inner
138.4.d.a 24 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(138, [\chi])\).