# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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