# 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 $4$

# Related objects

## 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.

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

## 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: 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
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 $$S_{2}^{\mathrm{new}}(756, [\chi])$$.