# Properties

 Label 1512.2.q.c Level $1512$ Weight $2$ Character orbit 1512.q Analytic conductor $12.073$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1512.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 3q^{5} - 5q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 3q^{5} - 5q^{7} + 3q^{11} - 3q^{13} - 7q^{17} - q^{19} - 2q^{23} - 10q^{25} - 9q^{29} + 8q^{31} - 14q^{35} + 2q^{37} - 16q^{41} + 10q^{47} + 15q^{49} - 11q^{53} + 22q^{55} - 38q^{59} + 26q^{61} + 26q^{65} - 52q^{67} + 48q^{71} - 35q^{73} - 17q^{77} - 20q^{79} + 28q^{83} - 20q^{85} - 6q^{89} - 37q^{91} + 24q^{95} - 29q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
793.1 0 0 0 −1.76479 3.05671i 0 −2.63986 0.176417i 0 0 0
793.2 0 0 0 −1.71796 2.97559i 0 −0.727932 2.54364i 0 0 0
793.3 0 0 0 −1.26145 2.18490i 0 2.63136 + 0.275550i 0 0 0
793.4 0 0 0 −0.918286 1.59052i 0 −0.361656 + 2.62092i 0 0 0
793.5 0 0 0 −0.790938 1.36994i 0 2.57645 0.601597i 0 0 0
793.6 0 0 0 −0.240694 0.416893i 0 −1.92765 + 1.81223i 0 0 0
793.7 0 0 0 0.170100 + 0.294622i 0 −2.63360 0.253251i 0 0 0
793.8 0 0 0 0.841578 + 1.45766i 0 1.65502 + 2.06419i 0 0 0
793.9 0 0 0 0.927957 + 1.60727i 0 0.900017 2.48796i 0 0 0
793.10 0 0 0 1.33401 + 2.31057i 0 0.581213 + 2.58112i 0 0 0
793.11 0 0 0 1.92048 + 3.32636i 0 −2.55336 0.693065i 0 0 0
1369.1 0 0 0 −1.76479 + 3.05671i 0 −2.63986 + 0.176417i 0 0 0
1369.2 0 0 0 −1.71796 + 2.97559i 0 −0.727932 + 2.54364i 0 0 0
1369.3 0 0 0 −1.26145 + 2.18490i 0 2.63136 0.275550i 0 0 0
1369.4 0 0 0 −0.918286 + 1.59052i 0 −0.361656 2.62092i 0 0 0
1369.5 0 0 0 −0.790938 + 1.36994i 0 2.57645 + 0.601597i 0 0 0
1369.6 0 0 0 −0.240694 + 0.416893i 0 −1.92765 1.81223i 0 0 0
1369.7 0 0 0 0.170100 0.294622i 0 −2.63360 + 0.253251i 0 0 0
1369.8 0 0 0 0.841578 1.45766i 0 1.65502 2.06419i 0 0 0
1369.9 0 0 0 0.927957 1.60727i 0 0.900017 + 2.48796i 0 0 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1369.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.q.c 22
3.b odd 2 1 504.2.q.d 22
4.b odd 2 1 3024.2.q.k 22
7.c even 3 1 1512.2.t.d 22
9.c even 3 1 1512.2.t.d 22
9.d odd 6 1 504.2.t.d yes 22
12.b even 2 1 1008.2.q.k 22
21.h odd 6 1 504.2.t.d yes 22
28.g odd 6 1 3024.2.t.l 22
36.f odd 6 1 3024.2.t.l 22
36.h even 6 1 1008.2.t.k 22
63.h even 3 1 inner 1512.2.q.c 22
63.j odd 6 1 504.2.q.d 22
84.n even 6 1 1008.2.t.k 22
252.u odd 6 1 3024.2.q.k 22
252.bb even 6 1 1008.2.q.k 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d 22 3.b odd 2 1
504.2.q.d 22 63.j odd 6 1
504.2.t.d yes 22 9.d odd 6 1
504.2.t.d yes 22 21.h odd 6 1
1008.2.q.k 22 12.b even 2 1
1008.2.q.k 22 252.bb even 6 1
1008.2.t.k 22 36.h even 6 1
1008.2.t.k 22 84.n even 6 1
1512.2.q.c 22 1.a even 1 1 trivial
1512.2.q.c 22 63.h even 3 1 inner
1512.2.t.d 22 7.c even 3 1
1512.2.t.d 22 9.c even 3 1
3024.2.q.k 22 4.b odd 2 1
3024.2.q.k 22 252.u odd 6 1
3024.2.t.l 22 28.g odd 6 1
3024.2.t.l 22 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{22} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.