Properties

 Label 756.2.bx.a Level 756 Weight 2 Character orbit 756.bx Analytic conductor 6.037 Analytic rank 0 Dimension 144 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.bx (of order $$18$$, degree $$6$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$24$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{9} - 6q^{11} + 12q^{15} - 30q^{21} + 48q^{23} + 12q^{29} - 27q^{35} - 42q^{39} + 18q^{49} + 18q^{51} + 30q^{57} - 15q^{63} + 42q^{65} + 36q^{71} - 51q^{77} + 36q^{79} + 18q^{81} + 36q^{85} + 9q^{91} + 96q^{93} - 48q^{95} + 36q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −1.71962 + 0.207109i 0 −1.79713 1.50797i 0 1.89497 + 1.84637i 0 2.91421 0.712300i 0
41.2 0 −1.70123 0.325320i 0 2.96273 + 2.48603i 0 1.79607 + 1.94271i 0 2.78833 + 1.10688i 0
41.3 0 −1.65267 0.518332i 0 −1.29455 1.08625i 0 −2.25956 + 1.37637i 0 2.46266 + 1.71327i 0
41.4 0 −1.60018 + 0.662888i 0 1.50391 + 1.26193i 0 −2.43356 1.03816i 0 2.12116 2.12148i 0
41.5 0 −1.48751 + 0.887305i 0 0.296683 + 0.248947i 0 1.95697 1.78053i 0 1.42538 2.63975i 0
41.6 0 −1.36115 1.07111i 0 −2.64973 2.22338i 0 0.252743 2.63365i 0 0.705464 + 2.91587i 0
41.7 0 −0.885631 + 1.48851i 0 0.580812 + 0.487359i 0 −1.90191 + 1.83923i 0 −1.43132 2.63654i 0
41.8 0 −0.803760 + 1.53427i 0 −3.10123 2.60224i 0 −0.828412 2.51271i 0 −1.70794 2.46636i 0
41.9 0 −0.801144 1.53563i 0 −0.0128291 0.0107649i 0 2.63948 0.182048i 0 −1.71634 + 2.46053i 0
41.10 0 −0.686580 1.59016i 0 1.16122 + 0.974380i 0 −2.63730 0.211289i 0 −2.05722 + 2.18354i 0
41.11 0 −0.481075 + 1.66390i 0 −1.62248 1.36143i 0 −0.358949 + 2.62129i 0 −2.53713 1.60092i 0
41.12 0 −0.304497 + 1.70508i 0 2.29372 + 1.92466i 0 2.03395 1.69205i 0 −2.81456 1.03838i 0
41.13 0 0.304497 1.70508i 0 −2.29372 1.92466i 0 0.470465 + 2.60359i 0 −2.81456 1.03838i 0
41.14 0 0.481075 1.66390i 0 1.62248 + 1.36143i 0 1.40996 2.23875i 0 −2.53713 1.60092i 0
41.15 0 0.686580 + 1.59016i 0 −1.16122 0.974380i 0 −2.15610 1.53337i 0 −2.05722 + 2.18354i 0
41.16 0 0.801144 + 1.53563i 0 0.0128291 + 0.0107649i 0 1.90494 + 1.83608i 0 −1.71634 + 2.46053i 0
41.17 0 0.803760 1.53427i 0 3.10123 + 2.60224i 0 −2.24974 + 1.39236i 0 −1.70794 2.46636i 0
41.18 0 0.885631 1.48851i 0 −0.580812 0.487359i 0 −0.274713 2.63145i 0 −1.43132 2.63654i 0
41.19 0 1.36115 + 1.07111i 0 2.64973 + 2.22338i 0 −1.49927 + 2.17995i 0 0.705464 + 2.91587i 0
41.20 0 1.48751 0.887305i 0 −0.296683 0.248947i 0 0.354626 + 2.62188i 0 1.42538 2.63975i 0
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 713.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
27.f odd 18 1 inner
189.be even 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bx.a 144
7.b odd 2 1 inner 756.2.bx.a 144
27.f odd 18 1 inner 756.2.bx.a 144
189.be even 18 1 inner 756.2.bx.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bx.a 144 1.a even 1 1 trivial
756.2.bx.a 144 7.b odd 2 1 inner
756.2.bx.a 144 27.f odd 18 1 inner
756.2.bx.a 144 189.be even 18 1 inner

Hecke kernels

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

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database