Properties

Label 138.6.d.a
Level $138$
Weight $6$
Character orbit 138.d
Analytic conductor $22.133$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 138.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.1329671342\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{3} - 640 q^{4} + 80 q^{6} + 504 q^{9} + 64 q^{12} - 1048 q^{13} + 10240 q^{16} + 1280 q^{18} - 1280 q^{24} + 30480 q^{25} + 1700 q^{27} - 22576 q^{31} - 8064 q^{36} + 55608 q^{39} + 1088 q^{46} - 1024 q^{48} - 23224 q^{49} + 16768 q^{52} + 25456 q^{54} + 210400 q^{55} - 83168 q^{58} - 163840 q^{64} + 99076 q^{69} + 167520 q^{70} - 20480 q^{72} + 241160 q^{73} - 255604 q^{75} - 233440 q^{78} + 78512 q^{81} - 8832 q^{82} - 460296 q^{85} - 4136 q^{87} + 500704 q^{93} - 138272 q^{94} + 20480 q^{96}+O(q^{100}) \) 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 4.00000i −11.7474 + 10.2469i −16.0000 −107.494 40.9875 + 46.9896i 192.413i 64.0000i 33.0032 240.748i 429.977i
137.2 4.00000i −11.7474 + 10.2469i −16.0000 107.494 40.9875 + 46.9896i 192.413i 64.0000i 33.0032 240.748i 429.977i
137.3 4.00000i −11.7474 10.2469i −16.0000 −107.494 40.9875 46.9896i 192.413i 64.0000i 33.0032 + 240.748i 429.977i
137.4 4.00000i −11.7474 10.2469i −16.0000 107.494 40.9875 46.9896i 192.413i 64.0000i 33.0032 + 240.748i 429.977i
137.5 4.00000i −1.79544 + 15.4847i −16.0000 −69.6024 61.9389 + 7.18174i 161.717i 64.0000i −236.553 55.6036i 278.410i
137.6 4.00000i −1.79544 + 15.4847i −16.0000 69.6024 61.9389 + 7.18174i 161.717i 64.0000i −236.553 55.6036i 278.410i
137.7 4.00000i −1.79544 15.4847i −16.0000 −69.6024 61.9389 7.18174i 161.717i 64.0000i −236.553 + 55.6036i 278.410i
137.8 4.00000i −1.79544 15.4847i −16.0000 69.6024 61.9389 7.18174i 161.717i 64.0000i −236.553 + 55.6036i 278.410i
137.9 4.00000i −15.1862 + 3.51856i −16.0000 −0.0328033 14.0742 + 60.7447i 156.501i 64.0000i 218.240 106.867i 0.131213i
137.10 4.00000i −15.1862 + 3.51856i −16.0000 0.0328033 14.0742 + 60.7447i 156.501i 64.0000i 218.240 106.867i 0.131213i
137.11 4.00000i −15.1862 3.51856i −16.0000 −0.0328033 14.0742 60.7447i 156.501i 64.0000i 218.240 + 106.867i 0.131213i
137.12 4.00000i −15.1862 3.51856i −16.0000 0.0328033 14.0742 60.7447i 156.501i 64.0000i 218.240 + 106.867i 0.131213i
137.13 4.00000i 15.5676 + 0.805444i −16.0000 −10.2990 3.22177 62.2705i 139.754i 64.0000i 241.703 + 25.0777i 41.1960i
137.14 4.00000i 15.5676 + 0.805444i −16.0000 10.2990 3.22177 62.2705i 139.754i 64.0000i 241.703 + 25.0777i 41.1960i
137.15 4.00000i 15.5676 0.805444i −16.0000 −10.2990 3.22177 + 62.2705i 139.754i 64.0000i 241.703 25.0777i 41.1960i
137.16 4.00000i 15.5676 0.805444i −16.0000 10.2990 3.22177 + 62.2705i 139.754i 64.0000i 241.703 25.0777i 41.1960i
137.17 4.00000i −14.2654 6.28478i −16.0000 −46.4212 −25.1391 + 57.0616i 104.982i 64.0000i 164.003 + 179.310i 185.685i
137.18 4.00000i −14.2654 6.28478i −16.0000 46.4212 −25.1391 + 57.0616i 104.982i 64.0000i 164.003 + 179.310i 185.685i
137.19 4.00000i −14.2654 + 6.28478i −16.0000 −46.4212 −25.1391 57.0616i 104.982i 64.0000i 164.003 179.310i 185.685i
137.20 4.00000i −14.2654 + 6.28478i −16.0000 46.4212 −25.1391 57.0616i 104.982i 64.0000i 164.003 179.310i 185.685i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.40
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 138.6.d.a 40
3.b odd 2 1 inner 138.6.d.a 40
23.b odd 2 1 inner 138.6.d.a 40
69.c even 2 1 inner 138.6.d.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.d.a 40 1.a even 1 1 trivial
138.6.d.a 40 3.b odd 2 1 inner
138.6.d.a 40 23.b odd 2 1 inner
138.6.d.a 40 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(138, [\chi])\).