Properties

Label 756.2.e.a
Level 756
Weight 2
Character orbit 756.e
Analytic conductor 6.037
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 8q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 8q^{4} - 16q^{10} + 8q^{16} + 16q^{22} - 24q^{25} - 8q^{28} - 8q^{34} + 16q^{37} - 8q^{40} - 24q^{49} - 8q^{52} + 32q^{58} - 80q^{61} + 40q^{64} - 24q^{70} - 32q^{82} + 56q^{85} + 56q^{88} + 72q^{94} + O(q^{100}) \)

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 −1.40635 0.148963i 0 1.95562 + 0.418986i 1.05610i 0 1.00000i −2.68787 0.880554i 0 0.157319 1.48524i
323.2 −1.40635 + 0.148963i 0 1.95562 0.418986i 1.05610i 0 1.00000i −2.68787 + 0.880554i 0 0.157319 + 1.48524i
323.3 −1.25904 0.644062i 0 1.17037 + 1.62180i 3.70845i 0 1.00000i −0.429004 2.79570i 0 −2.38847 + 4.66909i
323.4 −1.25904 + 0.644062i 0 1.17037 1.62180i 3.70845i 0 1.00000i −0.429004 + 2.79570i 0 −2.38847 4.66909i
323.5 −0.904694 1.08698i 0 −0.363059 + 1.96677i 2.20652i 0 1.00000i 2.46630 1.38469i 0 2.39844 1.99622i
323.6 −0.904694 + 1.08698i 0 −0.363059 1.96677i 2.20652i 0 1.00000i 2.46630 + 1.38469i 0 2.39844 + 1.99622i
323.7 −0.575837 1.29167i 0 −1.33682 + 1.48758i 0.311265i 0 1.00000i 2.69126 + 0.870128i 0 −0.402052 + 0.179238i
323.8 −0.575837 + 1.29167i 0 −1.33682 1.48758i 0.311265i 0 1.00000i 2.69126 0.870128i 0 −0.402052 0.179238i
323.9 −0.436579 1.34514i 0 −1.61880 + 1.17452i 3.88017i 0 1.00000i 2.28662 + 1.66474i 0 −5.21937 + 1.69400i
323.10 −0.436579 + 1.34514i 0 −1.61880 1.17452i 3.88017i 0 1.00000i 2.28662 1.66474i 0 −5.21937 1.69400i
323.11 −0.310394 1.37973i 0 −1.80731 + 0.856520i 1.05392i 0 1.00000i 1.74275 + 2.22774i 0 1.45413 0.327131i
323.12 −0.310394 + 1.37973i 0 −1.80731 0.856520i 1.05392i 0 1.00000i 1.74275 2.22774i 0 1.45413 + 0.327131i
323.13 0.310394 1.37973i 0 −1.80731 0.856520i 1.05392i 0 1.00000i −1.74275 + 2.22774i 0 1.45413 + 0.327131i
323.14 0.310394 + 1.37973i 0 −1.80731 + 0.856520i 1.05392i 0 1.00000i −1.74275 2.22774i 0 1.45413 0.327131i
323.15 0.436579 1.34514i 0 −1.61880 1.17452i 3.88017i 0 1.00000i −2.28662 + 1.66474i 0 −5.21937 1.69400i
323.16 0.436579 + 1.34514i 0 −1.61880 + 1.17452i 3.88017i 0 1.00000i −2.28662 1.66474i 0 −5.21937 + 1.69400i
323.17 0.575837 1.29167i 0 −1.33682 1.48758i 0.311265i 0 1.00000i −2.69126 + 0.870128i 0 −0.402052 0.179238i
323.18 0.575837 + 1.29167i 0 −1.33682 + 1.48758i 0.311265i 0 1.00000i −2.69126 0.870128i 0 −0.402052 + 0.179238i
323.19 0.904694 1.08698i 0 −0.363059 1.96677i 2.20652i 0 1.00000i −2.46630 1.38469i 0 2.39844 + 1.99622i
323.20 0.904694 + 1.08698i 0 −0.363059 + 1.96677i 2.20652i 0 1.00000i −2.46630 + 1.38469i 0 2.39844 1.99622i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.e.a 24
3.b odd 2 1 inner 756.2.e.a 24
4.b odd 2 1 inner 756.2.e.a 24
12.b even 2 1 inner 756.2.e.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.e.a 24 1.a even 1 1 trivial
756.2.e.a 24 3.b odd 2 1 inner
756.2.e.a 24 4.b odd 2 1 inner
756.2.e.a 24 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 36 T_{5}^{10} + 427 T_{5}^{8} + 1864 T_{5}^{6} + 2851 T_{5}^{4} + 1508 T_{5}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database