Properties

Label 1008.2.v.e
Level 1008
Weight 2
Character orbit 1008.v
Analytic conductor 8.049
Analytic rank 0
Dimension 40
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1008.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 40q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 40q^{7} + 48q^{10} - 24q^{13} + 12q^{16} - 32q^{19} - 8q^{22} - 56q^{34} - 8q^{37} + 32q^{43} - 52q^{46} + 40q^{49} - 8q^{52} + 48q^{55} + 56q^{58} - 24q^{61} + 48q^{64} + 48q^{70} - 24q^{76} - 64q^{82} + 64q^{85} - 120q^{88} - 24q^{91} - 128q^{94} + 64q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.40729 0.139782i 0 1.96092 + 0.393427i −1.17902 1.17902i 0 1.00000 −2.70459 0.827766i 0 1.49442 + 1.82403i
323.2 −1.39680 + 0.221233i 0 1.90211 0.618037i −2.51504 2.51504i 0 1.00000 −2.52014 + 1.28408i 0 4.06942 + 2.95660i
323.3 −1.37088 0.347400i 0 1.75863 + 0.952488i 0.111394 + 0.111394i 0 1.00000 −2.07997 1.91669i 0 −0.114009 0.191406i
323.4 −1.13420 0.844744i 0 0.572816 + 1.91622i 2.12043 + 2.12043i 0 1.00000 0.969023 2.65725i 0 −0.613769 4.19620i
323.5 −1.11731 + 0.866958i 0 0.496766 1.93732i 0.925496 + 0.925496i 0 1.00000 1.12454 + 2.59527i 0 −1.83643 0.231700i
323.6 −0.890883 1.09833i 0 −0.412656 + 1.95697i −1.65702 1.65702i 0 1.00000 2.51702 1.29019i 0 −0.343745 + 3.29617i
323.7 −0.614527 + 1.27372i 0 −1.24471 1.56547i −2.62814 2.62814i 0 1.00000 2.75887 0.623393i 0 4.96257 1.73245i
323.8 −0.579268 + 1.29014i 0 −1.32890 1.49467i −0.667815 0.667815i 0 1.00000 2.69811 0.848644i 0 1.24842 0.474728i
323.9 −0.351970 1.36971i 0 −1.75223 + 0.964197i 2.96859 + 2.96859i 0 1.00000 1.93741 + 2.06069i 0 3.02127 5.11098i
323.10 −0.153718 1.40583i 0 −1.95274 + 0.432203i 0.0893433 + 0.0893433i 0 1.00000 0.907777 + 2.67879i 0 0.111868 0.139335i
323.11 0.153718 + 1.40583i 0 −1.95274 + 0.432203i −0.0893433 0.0893433i 0 1.00000 −0.907777 2.67879i 0 0.111868 0.139335i
323.12 0.351970 + 1.36971i 0 −1.75223 + 0.964197i −2.96859 2.96859i 0 1.00000 −1.93741 2.06069i 0 3.02127 5.11098i
323.13 0.579268 1.29014i 0 −1.32890 1.49467i 0.667815 + 0.667815i 0 1.00000 −2.69811 + 0.848644i 0 1.24842 0.474728i
323.14 0.614527 1.27372i 0 −1.24471 1.56547i 2.62814 + 2.62814i 0 1.00000 −2.75887 + 0.623393i 0 4.96257 1.73245i
323.15 0.890883 + 1.09833i 0 −0.412656 + 1.95697i 1.65702 + 1.65702i 0 1.00000 −2.51702 + 1.29019i 0 −0.343745 + 3.29617i
323.16 1.11731 0.866958i 0 0.496766 1.93732i −0.925496 0.925496i 0 1.00000 −1.12454 2.59527i 0 −1.83643 0.231700i
323.17 1.13420 + 0.844744i 0 0.572816 + 1.91622i −2.12043 2.12043i 0 1.00000 −0.969023 + 2.65725i 0 −0.613769 4.19620i
323.18 1.37088 + 0.347400i 0 1.75863 + 0.952488i −0.111394 0.111394i 0 1.00000 2.07997 + 1.91669i 0 −0.114009 0.191406i
323.19 1.39680 0.221233i 0 1.90211 0.618037i 2.51504 + 2.51504i 0 1.00000 2.52014 1.28408i 0 4.06942 + 2.95660i
323.20 1.40729 + 0.139782i 0 1.96092 + 0.393427i 1.17902 + 1.17902i 0 1.00000 2.70459 + 0.827766i 0 1.49442 + 1.82403i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 827.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.v.e 40
3.b odd 2 1 inner 1008.2.v.e 40
4.b odd 2 1 4032.2.v.e 40
12.b even 2 1 4032.2.v.e 40
16.e even 4 1 4032.2.v.e 40
16.f odd 4 1 inner 1008.2.v.e 40
48.i odd 4 1 4032.2.v.e 40
48.k even 4 1 inner 1008.2.v.e 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.e 40 1.a even 1 1 trivial
1008.2.v.e 40 3.b odd 2 1 inner
1008.2.v.e 40 16.f odd 4 1 inner
1008.2.v.e 40 48.k even 4 1 inner
4032.2.v.e 40 4.b odd 2 1
4032.2.v.e 40 12.b even 2 1
4032.2.v.e 40 16.e even 4 1
4032.2.v.e 40 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\(T_{5}^{40} + \cdots\)
\(T_{11}^{40} + \cdots\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database