# Properties

 Label 756.2.ba.a Level $756$ Weight $2$ Character orbit 756.ba Analytic conductor $6.037$ Analytic rank $0$ Dimension $72$ 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.ba (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 252) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72 q + 42 q^{20} + 36 q^{25} - 30 q^{32} - 12 q^{34} - 12 q^{40} + 60 q^{41} - 24 q^{46} + 36 q^{49} + 78 q^{50} - 18 q^{52} - 18 q^{58} - 60 q^{64} - 24 q^{65} - 78 q^{68} - 24 q^{73} + 12 q^{76} - 36 q^{82} - 30 q^{86} + 24 q^{88} + 114 q^{92} + 42 q^{94} - 12 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1 −1.41075 + 0.0989244i 0 1.98043 0.279115i −0.795659 0.459374i 0 0.866025 0.500000i −2.76628 + 0.589674i 0 1.16792 + 0.569351i
71.2 −1.40971 + 0.112741i 0 1.97458 0.317866i 2.73233 + 1.57751i 0 −0.866025 + 0.500000i −2.74775 + 0.670717i 0 −4.02965 1.91579i
71.3 −1.37785 0.318634i 0 1.79694 + 0.878061i −2.51813 1.45385i 0 −0.866025 + 0.500000i −2.19614 1.78241i 0 3.00637 + 2.80555i
71.4 −1.34671 + 0.431716i 0 1.62724 1.16279i −1.40424 0.810740i 0 −0.866025 + 0.500000i −1.68942 + 2.26845i 0 2.24112 + 0.485596i
71.5 −1.31979 + 0.508077i 0 1.48372 1.34111i 2.42137 + 1.39798i 0 0.866025 0.500000i −1.27681 + 2.52384i 0 −3.90599 0.614803i
71.6 −1.24611 0.668739i 0 1.10558 + 1.66664i 0.245511 + 0.141746i 0 0.866025 0.500000i −0.263117 2.81616i 0 −0.211142 0.340813i
71.7 −1.24162 0.677033i 0 1.08325 + 1.68124i 0.948457 + 0.547592i 0 0.866025 0.500000i −0.206735 2.82086i 0 −0.806888 1.32204i
71.8 −1.09990 + 0.888937i 0 0.419581 1.95549i 2.42137 + 1.39798i 0 −0.866025 + 0.500000i 1.27681 + 2.52384i 0 −3.90599 + 0.614803i
71.9 −1.04723 + 0.950425i 0 0.193385 1.99063i −1.40424 0.810740i 0 0.866025 0.500000i 1.68942 + 2.26845i 0 2.24112 0.485596i
71.10 −0.908765 1.08358i 0 −0.348292 + 1.96944i −2.24261 1.29477i 0 −0.866025 + 0.500000i 2.45056 1.41236i 0 0.635018 + 3.60669i
71.11 −0.802493 + 1.16448i 0 −0.712009 1.86897i 2.73233 + 1.57751i 0 0.866025 0.500000i 2.74775 + 0.670717i 0 −4.02965 + 1.91579i
71.12 −0.791046 + 1.17228i 0 −0.748493 1.85466i −0.795659 0.459374i 0 −0.866025 + 0.500000i 2.76628 + 0.589674i 0 1.16792 0.569351i
71.13 −0.726513 1.21333i 0 −0.944358 + 1.76301i −0.256612 0.148155i 0 −0.866025 + 0.500000i 2.82520 0.135024i 0 0.00667042 + 0.418992i
71.14 −0.498620 1.32340i 0 −1.50276 + 1.31974i −2.06874 1.19439i 0 0.866025 0.500000i 2.49585 + 1.33069i 0 −0.549133 + 3.33330i
71.15 −0.412980 + 1.35257i 0 −1.65890 1.11717i −2.51813 1.45385i 0 0.866025 0.500000i 2.19614 1.78241i 0 3.00637 2.80555i
71.16 −0.246614 1.39255i 0 −1.87836 + 0.686842i 3.28588 + 1.89710i 0 −0.866025 + 0.500000i 1.41969 + 2.44632i 0 1.83146 5.04359i
71.17 −0.0878075 1.41148i 0 −1.98458 + 0.247878i −2.57586 1.48718i 0 0.866025 0.500000i 0.524137 + 2.77944i 0 −1.87295 + 3.76638i
71.18 −0.0439092 + 1.41353i 0 −1.99614 0.124134i 0.245511 + 0.141746i 0 −0.866025 + 0.500000i 0.263117 2.81616i 0 −0.211142 + 0.340813i
71.19 −0.0344832 + 1.41379i 0 −1.99762 0.0975041i 0.948457 + 0.547592i 0 −0.866025 + 0.500000i 0.206735 2.82086i 0 −0.806888 + 1.32204i
71.20 0.0405943 1.41363i 0 −1.99670 0.114771i 2.16781 + 1.25159i 0 0.866025 0.500000i −0.243298 + 2.81794i 0 1.85728 3.01367i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 575.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h 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.ba.a 72
3.b odd 2 1 252.2.ba.a 72
4.b odd 2 1 inner 756.2.ba.a 72
9.c even 3 1 252.2.ba.a 72
9.d odd 6 1 inner 756.2.ba.a 72
12.b even 2 1 252.2.ba.a 72
36.f odd 6 1 252.2.ba.a 72
36.h even 6 1 inner 756.2.ba.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.ba.a 72 3.b odd 2 1
252.2.ba.a 72 9.c even 3 1
252.2.ba.a 72 12.b even 2 1
252.2.ba.a 72 36.f odd 6 1
756.2.ba.a 72 1.a even 1 1 trivial
756.2.ba.a 72 4.b odd 2 1 inner
756.2.ba.a 72 9.d odd 6 1 inner
756.2.ba.a 72 36.h even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.