Properties

Label 40.6.f.a
Level 40
Weight 6
Character orbit 40.f
Analytic conductor 6.415
Analytic rank 0
Dimension 28
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41535279252\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 12q^{4} - 156q^{6} + 1940q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 12q^{4} - 156q^{6} + 1940q^{9} - 16q^{10} - 692q^{14} - 488q^{15} + 1560q^{16} + 2732q^{20} - 2224q^{24} + 1556q^{25} - 9976q^{26} - 15012q^{30} + 4368q^{31} + 13016q^{34} - 34116q^{36} + 23360q^{39} - 22496q^{40} - 2480q^{41} + 10712q^{44} + 58372q^{46} - 38420q^{49} + 45624q^{50} - 3568q^{54} - 48776q^{55} + 110944q^{56} + 111688q^{60} - 46944q^{64} + 37200q^{65} - 136120q^{66} - 112852q^{70} - 69232q^{71} + 34176q^{74} - 13944q^{76} - 35984q^{79} - 47064q^{80} + 122596q^{81} - 165688q^{84} - 73676q^{86} - 178744q^{89} + 51496q^{90} + 314740q^{94} + 89416q^{95} - 236176q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −5.53936 1.14695i −16.0077 29.3690 + 12.7067i 19.1184 + 52.5308i 88.6724 + 18.3600i 20.5525i −148.112 104.072i 13.2465 −45.6540 312.915i
29.2 −5.53936 + 1.14695i −16.0077 29.3690 12.7067i 19.1184 52.5308i 88.6724 18.3600i 20.5525i −148.112 + 104.072i 13.2465 −45.6540 + 312.915i
29.3 −5.32791 1.90088i 21.6833 24.7733 + 20.2555i 36.8367 42.0483i −115.527 41.2175i 236.448i −93.4865 155.011i 227.167 −276.192 + 154.007i
29.4 −5.32791 + 1.90088i 21.6833 24.7733 20.2555i 36.8367 + 42.0483i −115.527 + 41.2175i 236.448i −93.4865 + 155.011i 227.167 −276.192 154.007i
29.5 −5.08581 2.47679i 10.5561 19.7310 + 25.1930i −52.7686 + 18.4521i −53.6865 26.1453i 47.9937i −37.9506 176.996i −131.568 314.073 + 36.8527i
29.6 −5.08581 + 2.47679i 10.5561 19.7310 25.1930i −52.7686 18.4521i −53.6865 + 26.1453i 47.9937i −37.9506 + 176.996i −131.568 314.073 36.8527i
29.7 −3.53479 4.41648i −21.4357 −7.01055 + 31.2226i −48.3867 27.9951i 75.7708 + 94.6704i 39.9262i 162.675 79.4034i 216.491 47.3969 + 312.656i
29.8 −3.53479 + 4.41648i −21.4357 −7.01055 31.2226i −48.3867 + 27.9951i 75.7708 94.6704i 39.9262i 162.675 + 79.4034i 216.491 47.3969 312.656i
29.9 −3.25299 4.62796i 1.29818 −10.8361 + 30.1095i 51.3939 21.9924i −4.22299 6.00795i 170.399i 174.595 47.7971i −241.315 −268.964 166.308i
29.10 −3.25299 + 4.62796i 1.29818 −10.8361 30.1095i 51.3939 + 21.9924i −4.22299 + 6.00795i 170.399i 174.595 + 47.7971i −241.315 −268.964 + 166.308i
29.11 −1.42014 5.47569i 7.17847 −27.9664 + 15.5525i 1.28331 + 55.8870i −10.1944 39.3071i 146.905i 124.877 + 131.049i −191.470 304.197 86.3942i
29.12 −1.42014 + 5.47569i 7.17847 −27.9664 15.5525i 1.28331 55.8870i −10.1944 + 39.3071i 146.905i 124.877 131.049i −191.470 304.197 + 86.3942i
29.13 −0.685452 5.61517i 28.9041 −31.0603 + 7.69787i −40.5064 38.5257i −19.8124 162.302i 128.100i 64.5152 + 169.132i 592.448 −188.563 + 253.858i
29.14 −0.685452 + 5.61517i 28.9041 −31.0603 7.69787i −40.5064 + 38.5257i −19.8124 + 162.302i 128.100i 64.5152 169.132i 592.448 −188.563 253.858i
29.15 0.685452 5.61517i −28.9041 −31.0603 7.69787i 40.5064 + 38.5257i −19.8124 + 162.302i 128.100i −64.5152 + 169.132i 592.448 244.094 201.043i
29.16 0.685452 + 5.61517i −28.9041 −31.0603 + 7.69787i 40.5064 38.5257i −19.8124 162.302i 128.100i −64.5152 169.132i 592.448 244.094 + 201.043i
29.17 1.42014 5.47569i −7.17847 −27.9664 15.5525i −1.28331 55.8870i −10.1944 + 39.3071i 146.905i −124.877 + 131.049i −191.470 −307.842 72.3401i
29.18 1.42014 + 5.47569i −7.17847 −27.9664 + 15.5525i −1.28331 + 55.8870i −10.1944 39.3071i 146.905i −124.877 131.049i −191.470 −307.842 + 72.3401i
29.19 3.25299 4.62796i −1.29818 −10.8361 30.1095i −51.3939 + 21.9924i −4.22299 + 6.00795i 170.399i −174.595 47.7971i −241.315 −65.4039 + 309.390i
29.20 3.25299 + 4.62796i −1.29818 −10.8361 + 30.1095i −51.3939 21.9924i −4.22299 6.00795i 170.399i −174.595 + 47.7971i −241.315 −65.4039 309.390i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.6.f.a 28
4.b odd 2 1 160.6.f.a 28
5.b even 2 1 inner 40.6.f.a 28
5.c odd 4 2 200.6.d.e 28
8.b even 2 1 inner 40.6.f.a 28
8.d odd 2 1 160.6.f.a 28
20.d odd 2 1 160.6.f.a 28
20.e even 4 2 800.6.d.e 28
40.e odd 2 1 160.6.f.a 28
40.f even 2 1 inner 40.6.f.a 28
40.i odd 4 2 200.6.d.e 28
40.k even 4 2 800.6.d.e 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.f.a 28 1.a even 1 1 trivial
40.6.f.a 28 5.b even 2 1 inner
40.6.f.a 28 8.b even 2 1 inner
40.6.f.a 28 40.f even 2 1 inner
160.6.f.a 28 4.b odd 2 1
160.6.f.a 28 8.d odd 2 1
160.6.f.a 28 20.d odd 2 1
160.6.f.a 28 40.e odd 2 1
200.6.d.e 28 5.c odd 4 2
200.6.d.e 28 40.i odd 4 2
800.6.d.e 28 20.e even 4 2
800.6.d.e 28 40.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database