# Properties

 Label 6012.2.h.a Level $6012$ Weight $2$ Character orbit 6012.h Analytic conductor $48.006$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.0060616952$$ Analytic rank: $$0$$ Dimension: $$56$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3005.1 0 0 0 −4.09297 0 2.29241 0 0 0
3005.2 0 0 0 −4.09297 0 2.29241 0 0 0
3005.3 0 0 0 −3.84494 0 −2.14597 0 0 0
3005.4 0 0 0 −3.84494 0 −2.14597 0 0 0
3005.5 0 0 0 −3.47368 0 3.93299 0 0 0
3005.6 0 0 0 −3.47368 0 3.93299 0 0 0
3005.7 0 0 0 −3.30258 0 −4.47594 0 0 0
3005.8 0 0 0 −3.30258 0 −4.47594 0 0 0
3005.9 0 0 0 −3.04368 0 0.233711 0 0 0
3005.10 0 0 0 −3.04368 0 0.233711 0 0 0
3005.11 0 0 0 −2.76473 0 0.618008 0 0 0
3005.12 0 0 0 −2.76473 0 0.618008 0 0 0
3005.13 0 0 0 −2.21423 0 −1.10661 0 0 0
3005.14 0 0 0 −2.21423 0 −1.10661 0 0 0
3005.15 0 0 0 −2.08738 0 3.97701 0 0 0
3005.16 0 0 0 −2.08738 0 3.97701 0 0 0
3005.17 0 0 0 −1.39469 0 −4.57475 0 0 0
3005.18 0 0 0 −1.39469 0 −4.57475 0 0 0
3005.19 0 0 0 −1.15018 0 −0.123775 0 0 0
3005.20 0 0 0 −1.15018 0 −0.123775 0 0 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3005.56 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
167.b odd 2 1 inner
501.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6012.2.h.a 56
3.b odd 2 1 inner 6012.2.h.a 56
167.b odd 2 1 inner 6012.2.h.a 56
501.c even 2 1 inner 6012.2.h.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.h.a 56 1.a even 1 1 trivial
6012.2.h.a 56 3.b odd 2 1 inner
6012.2.h.a 56 167.b odd 2 1 inner
6012.2.h.a 56 501.c even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(6012, [\chi])$$.