# Properties

 Label 756.2.be.e Level 756 Weight 2 Character orbit 756.be Analytic conductor 6.037 Analytic rank 0 Dimension 64 CM no Inner twists 8

# 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: $$64$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 16q^{13} + 8q^{16} - 28q^{22} + 36q^{25} + 26q^{28} - 56q^{34} - 8q^{37} + 22q^{40} - 18q^{46} + 28q^{49} - 26q^{52} - 36q^{58} + 16q^{61} - 12q^{64} - 18q^{70} + 32q^{73} - 144q^{76} + 34q^{82} + 32q^{85} - 20q^{88} - 78q^{94} + 56q^{97} + O(q^{100})$$

## 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 −1.41403 0.0228601i 0 1.99895 + 0.0646498i 0.158956 + 0.0917731i 0 1.36747 + 2.26495i −2.82510 0.137113i 0 −0.222670 0.133404i
107.2 −1.38037 0.307526i 0 1.81086 + 0.849002i 3.20305 + 1.84928i 0 −2.45439 0.987919i −2.23856 1.72882i 0 −3.85269 3.53771i
107.3 −1.37308 + 0.338582i 0 1.77072 0.929803i −3.53194 2.03916i 0 1.92589 1.81410i −2.11654 + 1.87623i 0 5.54007 + 1.60410i
107.4 −1.32269 + 0.500480i 0 1.49904 1.32396i 2.34508 + 1.35394i 0 1.75134 + 1.98313i −1.32015 + 2.50144i 0 −3.77945 0.617175i
107.5 −1.28676 0.586718i 0 1.31152 + 1.50994i −1.46375 0.845096i 0 −1.83823 + 1.90287i −0.801716 2.71243i 0 1.38767 + 1.94625i
107.6 −1.27826 + 0.605023i 0 1.26789 1.54675i −1.71449 0.989859i 0 −2.59406 0.520431i −0.684877 + 2.74426i 0 2.79044 + 0.227993i
107.7 −1.15149 0.821011i 0 0.651880 + 1.89078i −1.46375 0.845096i 0 1.83823 1.90287i 0.801716 2.71243i 0 0.991666 + 2.17488i
107.8 −1.04167 + 0.956516i 0 0.170154 1.99275i 0.835158 + 0.482179i 0 2.03771 1.68753i 1.72885 + 2.23854i 0 −1.33117 + 0.296571i
107.9 −0.956512 1.04167i 0 −0.170171 + 1.99275i 3.20305 + 1.84928i 0 2.45439 + 0.987919i 2.23856 1.72882i 0 −1.13740 5.10539i
107.10 −0.743813 + 1.20281i 0 −0.893484 1.78933i 1.64673 + 0.950741i 0 −0.905127 2.48611i 2.81680 + 0.256236i 0 −2.36842 + 1.27353i
107.11 −0.726812 1.21315i 0 −0.943489 + 1.76347i 0.158956 + 0.0917731i 0 −1.36747 2.26495i 2.82510 0.137113i 0 −0.00419589 0.259540i
107.12 −0.669754 + 1.24556i 0 −1.10286 1.66844i −1.64673 0.950741i 0 0.905127 + 2.48611i 2.81680 0.256236i 0 2.28712 1.41435i
107.13 −0.393322 1.35842i 0 −1.69060 + 1.06859i −3.53194 2.03916i 0 −1.92589 + 1.81410i 2.11654 + 1.87623i 0 −1.38085 + 5.59989i
107.14 −0.307532 + 1.38037i 0 −1.81085 0.849017i −0.835158 0.482179i 0 −2.03771 + 1.68753i 1.72885 2.23854i 0 0.922423 1.00454i
107.15 −0.227919 1.39573i 0 −1.89611 + 0.636225i 2.34508 + 1.35394i 0 −1.75134 1.98313i 1.32015 + 2.50144i 0 1.35523 3.58168i
107.16 −0.115165 1.40952i 0 −1.97347 + 0.324653i −1.71449 0.989859i 0 2.59406 + 0.520431i 0.684877 + 2.74426i 0 −1.19777 + 2.53059i
107.17 0.115165 + 1.40952i 0 −1.97347 + 0.324653i 1.71449 + 0.989859i 0 2.59406 + 0.520431i −0.684877 2.74426i 0 −1.19777 + 2.53059i
107.18 0.227919 + 1.39573i 0 −1.89611 + 0.636225i −2.34508 1.35394i 0 −1.75134 1.98313i −1.32015 2.50144i 0 1.35523 3.58168i
107.19 0.307532 1.38037i 0 −1.81085 0.849017i 0.835158 + 0.482179i 0 −2.03771 + 1.68753i −1.72885 + 2.23854i 0 0.922423 1.00454i
107.20 0.393322 + 1.35842i 0 −1.69060 + 1.06859i 3.53194 + 2.03916i 0 −1.92589 + 1.81410i −2.11654 1.87623i 0 −1.38085 + 5.59989i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.32 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
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 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.e 64
3.b odd 2 1 inner 756.2.be.e 64
4.b odd 2 1 inner 756.2.be.e 64
7.c even 3 1 inner 756.2.be.e 64
12.b even 2 1 inner 756.2.be.e 64
21.h odd 6 1 inner 756.2.be.e 64
28.g odd 6 1 inner 756.2.be.e 64
84.n even 6 1 inner 756.2.be.e 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.e 64 1.a even 1 1 trivial
756.2.be.e 64 3.b odd 2 1 inner
756.2.be.e 64 4.b odd 2 1 inner
756.2.be.e 64 7.c even 3 1 inner
756.2.be.e 64 12.b even 2 1 inner
756.2.be.e 64 21.h odd 6 1 inner
756.2.be.e 64 28.g odd 6 1 inner
756.2.be.e 64 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 $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{32} - \cdots$$ $$T_{19}^{32} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database