# Properties

 Label 756.2.ca.a Level 756 Weight 2 Character orbit 756.ca Analytic conductor 6.037 Analytic rank 0 Dimension 144 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.ca (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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q - 12q^{9} + 12q^{11} + 12q^{15} - 3q^{21} - 15q^{23} - 6q^{29} - 42q^{39} + 18q^{45} - 54q^{47} - 36q^{49} + 18q^{51} + 45q^{53} + 3q^{57} + 54q^{61} + 39q^{63} - 3q^{65} + 36q^{69} + 36q^{71} + 93q^{77} - 18q^{79} - 36q^{81} + 36q^{85} - 18q^{91} + 60q^{93} + 6q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
173.1 0 −1.73005 0.0832183i 0 0.849177 0.712544i 0 −0.774464 + 2.52986i 0 2.98615 + 0.287944i 0
173.2 0 −1.69481 0.357227i 0 1.90051 1.59472i 0 1.42097 2.23178i 0 2.74478 + 1.21086i 0
173.3 0 −1.38241 1.04352i 0 −2.01034 + 1.68687i 0 −2.51700 0.815311i 0 0.822127 + 2.88515i 0
173.4 0 −1.34169 1.09539i 0 −2.77975 + 2.33249i 0 2.63835 0.197791i 0 0.600250 + 2.93934i 0
173.5 0 −1.31760 + 1.12424i 0 −2.86428 + 2.40342i 0 −1.48545 + 2.18939i 0 0.472151 2.96261i 0
173.6 0 −1.31609 + 1.12602i 0 −0.481690 + 0.404186i 0 2.64564 + 0.0243961i 0 0.464163 2.96387i 0
173.7 0 −1.16896 + 1.27810i 0 0.898527 0.753954i 0 −1.31798 2.29411i 0 −0.267080 2.98809i 0
173.8 0 −1.04679 1.37994i 0 0.700470 0.587764i 0 −0.673988 + 2.55846i 0 −0.808445 + 2.88902i 0
173.9 0 −0.586799 1.62962i 0 3.01033 2.52597i 0 −2.34045 1.23381i 0 −2.31133 + 1.91252i 0
173.10 0 −0.568097 + 1.63624i 0 3.32029 2.78605i 0 1.31877 + 2.29366i 0 −2.35453 1.85908i 0
173.11 0 −0.385495 1.68861i 0 −1.16198 + 0.975020i 0 0.818663 2.51591i 0 −2.70279 + 1.30190i 0
173.12 0 −0.267657 + 1.71125i 0 0.977821 0.820489i 0 −2.63440 0.244842i 0 −2.85672 0.916052i 0
173.13 0 0.0527582 + 1.73125i 0 −3.03928 + 2.55026i 0 0.308602 2.62769i 0 −2.99443 + 0.182675i 0
173.14 0 0.280100 1.70925i 0 −0.516309 + 0.433235i 0 1.59408 + 2.11161i 0 −2.84309 0.957524i 0
173.15 0 0.294282 + 1.70687i 0 −0.796960 + 0.668729i 0 0.995586 + 2.45129i 0 −2.82680 + 1.00460i 0
173.16 0 0.785443 1.54372i 0 1.98604 1.66648i 0 2.56407 0.652322i 0 −1.76616 2.42501i 0
173.17 0 0.839673 1.51491i 0 −1.81938 + 1.52664i 0 −2.37423 + 1.16749i 0 −1.58990 2.54406i 0
173.18 0 1.12890 + 1.31362i 0 0.424429 0.356139i 0 −2.49967 + 0.866968i 0 −0.451185 + 2.96588i 0
173.19 0 1.37133 + 1.05805i 0 −1.39267 + 1.16859i 0 0.270511 2.63189i 0 0.761074 + 2.90186i 0
173.20 0 1.47752 + 0.903842i 0 1.37147 1.15080i 0 2.42417 1.05990i 0 1.36614 + 2.67089i 0
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 689.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.bd 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.ca.a 144
7.d odd 6 1 756.2.ck.a yes 144
27.f odd 18 1 756.2.ck.a yes 144
189.bd even 18 1 inner 756.2.ca.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.ca.a 144 1.a even 1 1 trivial
756.2.ca.a 144 189.bd even 18 1 inner
756.2.ck.a yes 144 7.d odd 6 1
756.2.ck.a yes 144 27.f odd 18 1

## 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