# Properties

 Label 1503.2.c.a Level $1503$ Weight $2$ Character orbit 1503.c Analytic conductor $12.002$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1503 = 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1503.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0015154238$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 48q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 48q^{4} + 32q^{16} - 8q^{19} + 16q^{22} + 64q^{25} - 32q^{28} + 40q^{31} + 56q^{49} - 32q^{58} + 24q^{61} - 56q^{76} + 72q^{88} - 56q^{94} - 48q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1502.1 1.18160i 0 0.603817 −4.40626 0 −3.48779 3.07668i 0 5.20644i
1502.2 1.18160i 0 0.603817 −4.40626 0 −3.48779 3.07668i 0 5.20644i
1502.3 2.66906i 0 −5.12388 −3.16492 0 −0.263490 8.33782i 0 8.44736i
1502.4 2.66906i 0 −5.12388 −3.16492 0 −0.263490 8.33782i 0 8.44736i
1502.5 1.78744i 0 −1.19494 3.10541 0 3.91211 1.43899i 0 5.55073i
1502.6 1.78744i 0 −1.19494 3.10541 0 3.91211 1.43899i 0 5.55073i
1502.7 2.52227i 0 −4.36184 2.54201 0 2.74873 5.95720i 0 6.41163i
1502.8 2.52227i 0 −4.36184 2.54201 0 2.74873 5.95720i 0 6.41163i
1502.9 1.78653i 0 −1.19168 3.08210 0 4.41569 1.44408i 0 5.50625i
1502.10 1.78653i 0 −1.19168 3.08210 0 4.41569 1.44408i 0 5.50625i
1502.11 0.118553i 0 1.98595 3.37331 0 2.05489 0.472544i 0 0.399915i
1502.12 0.118553i 0 1.98595 3.37331 0 2.05489 0.472544i 0 0.399915i
1502.13 2.46644i 0 −4.08332 1.88017 0 −1.78260 5.13837i 0 4.63731i
1502.14 2.46644i 0 −4.08332 1.88017 0 −1.78260 5.13837i 0 4.63731i
1502.15 2.16730i 0 −2.69717 −1.88079 0 −3.73517 1.51098i 0 4.07624i
1502.16 2.16730i 0 −2.69717 −1.88079 0 −3.73517 1.51098i 0 4.07624i
1502.17 0.799957i 0 1.36007 −2.53587 0 −2.19364 2.68791i 0 2.02859i
1502.18 0.799957i 0 1.36007 −2.53587 0 −2.19364 2.68791i 0 2.02859i
1502.19 1.84274i 0 −1.39570 −1.15735 0 0.906437 1.11357i 0 2.13270i
1502.20 1.84274i 0 −1.39570 −1.15735 0 0.906437 1.11357i 0 2.13270i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1502.56 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 1503.2.c.a 56
3.b odd 2 1 inner 1503.2.c.a 56
167.b odd 2 1 inner 1503.2.c.a 56
501.c even 2 1 inner 1503.2.c.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1503.2.c.a 56 1.a even 1 1 trivial
1503.2.c.a 56 3.b odd 2 1 inner
1503.2.c.a 56 167.b odd 2 1 inner
1503.2.c.a 56 501.c even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(1503, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database