Properties

Label 756.2.be.d
Level 756
Weight 2
Character orbit 756.be
Analytic conductor 6.037
Analytic rank 0
Dimension 28
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.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 4q^{4} + 2q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 4q^{4} + 2q^{7} + 4q^{10} + 8q^{13} + 12q^{16} - 42q^{19} + 4q^{22} + 6q^{25} + 24q^{28} + 30q^{31} + 24q^{34} + 12q^{37} + 24q^{46} - 14q^{49} - 24q^{52} - 44q^{58} + 6q^{61} + 8q^{64} + 24q^{67} - 32q^{70} - 22q^{73} + 48q^{79} + 36q^{82} - 24q^{85} - 4q^{88} + 16q^{91} + 60q^{94} - 4q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.41404 + 0.0220675i 0 1.99903 0.0624087i 0.303357 + 0.175143i 0 2.63538 + 0.234036i −2.82533 + 0.132362i 0 −0.432824 0.240965i
107.2 −1.33158 + 0.476336i 0 1.54621 1.26856i 2.47695 + 1.43007i 0 −2.52686 0.784194i −1.45464 + 2.42570i 0 −3.97945 0.724388i
107.3 −1.16654 + 0.799486i 0 0.721643 1.86527i −3.44992 1.99182i 0 −0.212727 + 2.63719i 0.649430 + 2.75286i 0 5.61691 0.434631i
107.4 −0.943439 1.05353i 0 −0.219845 + 1.98788i 2.75822 + 1.59246i 0 0.944233 2.47152i 2.30170 1.64383i 0 −0.924512 4.40826i
107.5 −0.498187 1.32356i 0 −1.50362 + 1.31876i −0.936239 0.540538i 0 0.749282 + 2.53743i 2.49454 + 1.33314i 0 −0.249012 + 1.50846i
107.6 −0.373943 + 1.36388i 0 −1.72033 1.02003i −1.28692 0.743002i 0 1.36953 2.26371i 2.03450 1.96490i 0 1.49460 1.47736i
107.7 −0.297424 + 1.38258i 0 −1.82308 0.822428i 0.479813 + 0.277020i 0 −2.45883 + 0.976797i 1.67930 2.27595i 0 −0.525712 + 0.580990i
107.8 0.297424 1.38258i 0 −1.82308 0.822428i −0.479813 0.277020i 0 −2.45883 + 0.976797i −1.67930 + 2.27595i 0 −0.525712 + 0.580990i
107.9 0.373943 1.36388i 0 −1.72033 1.02003i 1.28692 + 0.743002i 0 1.36953 2.26371i −2.03450 + 1.96490i 0 1.49460 1.47736i
107.10 0.498187 + 1.32356i 0 −1.50362 + 1.31876i 0.936239 + 0.540538i 0 0.749282 + 2.53743i −2.49454 1.33314i 0 −0.249012 + 1.50846i
107.11 0.943439 + 1.05353i 0 −0.219845 + 1.98788i −2.75822 1.59246i 0 0.944233 2.47152i −2.30170 + 1.64383i 0 −0.924512 4.40826i
107.12 1.16654 0.799486i 0 0.721643 1.86527i 3.44992 + 1.99182i 0 −0.212727 + 2.63719i −0.649430 2.75286i 0 5.61691 0.434631i
107.13 1.33158 0.476336i 0 1.54621 1.26856i −2.47695 1.43007i 0 −2.52686 0.784194i 1.45464 2.42570i 0 −3.97945 0.724388i
107.14 1.41404 0.0220675i 0 1.99903 0.0624087i −0.303357 0.175143i 0 2.63538 + 0.234036i 2.82533 0.132362i 0 −0.432824 0.240965i
431.1 −1.41404 0.0220675i 0 1.99903 + 0.0624087i 0.303357 0.175143i 0 2.63538 0.234036i −2.82533 0.132362i 0 −0.432824 + 0.240965i
431.2 −1.33158 0.476336i 0 1.54621 + 1.26856i 2.47695 1.43007i 0 −2.52686 + 0.784194i −1.45464 2.42570i 0 −3.97945 + 0.724388i
431.3 −1.16654 0.799486i 0 0.721643 + 1.86527i −3.44992 + 1.99182i 0 −0.212727 2.63719i 0.649430 2.75286i 0 5.61691 + 0.434631i
431.4 −0.943439 + 1.05353i 0 −0.219845 1.98788i 2.75822 1.59246i 0 0.944233 + 2.47152i 2.30170 + 1.64383i 0 −0.924512 + 4.40826i
431.5 −0.498187 + 1.32356i 0 −1.50362 1.31876i −0.936239 + 0.540538i 0 0.749282 2.53743i 2.49454 1.33314i 0 −0.249012 1.50846i
431.6 −0.373943 1.36388i 0 −1.72033 + 1.02003i −1.28692 + 0.743002i 0 1.36953 + 2.26371i 2.03450 + 1.96490i 0 1.49460 + 1.47736i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.be.d yes 28
3.b odd 2 1 inner 756.2.be.d yes 28
4.b odd 2 1 756.2.be.c 28
7.c even 3 1 756.2.be.c 28
12.b even 2 1 756.2.be.c 28
21.h odd 6 1 756.2.be.c 28
28.g odd 6 1 inner 756.2.be.d yes 28
84.n even 6 1 inner 756.2.be.d yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.c 28 4.b odd 2 1
756.2.be.c 28 7.c even 3 1
756.2.be.c 28 12.b even 2 1
756.2.be.c 28 21.h odd 6 1
756.2.be.d yes 28 1.a even 1 1 trivial
756.2.be.d yes 28 3.b odd 2 1 inner
756.2.be.d yes 28 28.g odd 6 1 inner
756.2.be.d yes 28 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\(T_{5}^{28} - \cdots\)
\(T_{19}^{14} + \cdots\)

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database