# Properties

 Label 690.2.j.b Level $690$ Weight $2$ Character orbit 690.j Analytic conductor $5.510$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 24 q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 24 q^{6} + 16 q^{13} - 24 q^{16} + 16 q^{23} - 16 q^{25} + 16 q^{31} + 24 q^{36} + 8 q^{46} + 40 q^{47} - 8 q^{50} + 16 q^{52} - 56 q^{55} - 16 q^{58} - 8 q^{62} + 32 q^{70} + 64 q^{71} - 16 q^{73} + 32 q^{75} + 16 q^{77} + 16 q^{78} - 24 q^{81} + 24 q^{82} - 48 q^{85} + 16 q^{87} + 16 q^{92} - 8 q^{93} + 24 q^{95} - 24 q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
367.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.98078 1.03755i 1.00000 0.124800 0.124800i 0.707107 0.707107i 1.00000i 0.666968 + 2.13428i
367.2 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −1.07661 + 1.95982i 1.00000 3.43458 3.43458i 0.707107 0.707107i 1.00000i 2.14708 0.624526i
367.3 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −0.597287 2.15482i 1.00000 1.47923 1.47923i 0.707107 0.707107i 1.00000i −1.10134 + 1.94603i
367.4 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0.597287 + 2.15482i 1.00000 −1.47923 + 1.47923i 0.707107 0.707107i 1.00000i 1.10134 1.94603i
367.5 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.07661 1.95982i 1.00000 −3.43458 + 3.43458i 0.707107 0.707107i 1.00000i −2.14708 + 0.624526i
367.6 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.98078 + 1.03755i 1.00000 −0.124800 + 0.124800i 0.707107 0.707107i 1.00000i −0.666968 2.13428i
367.7 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.19853 0.408021i 1.00000 −1.64584 + 1.64584i −0.707107 + 0.707107i 1.00000i −1.26608 1.84311i
367.8 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.52094 1.63913i 1.00000 2.68754 2.68754i −0.707107 + 0.707107i 1.00000i 0.0835670 2.23451i
367.9 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.643329 2.14152i 1.00000 −2.01701 + 2.01701i −0.707107 + 0.707107i 1.00000i 1.05938 1.96919i
367.10 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0.643329 + 2.14152i 1.00000 2.01701 2.01701i −0.707107 + 0.707107i 1.00000i −1.05938 + 1.96919i
367.11 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.52094 + 1.63913i 1.00000 −2.68754 + 2.68754i −0.707107 + 0.707107i 1.00000i −0.0835670 + 2.23451i
367.12 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 2.19853 + 0.408021i 1.00000 1.64584 1.64584i −0.707107 + 0.707107i 1.00000i 1.26608 + 1.84311i
643.1 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.98078 + 1.03755i 1.00000 0.124800 + 0.124800i 0.707107 + 0.707107i 1.00000i 0.666968 2.13428i
643.2 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.07661 1.95982i 1.00000 3.43458 + 3.43458i 0.707107 + 0.707107i 1.00000i 2.14708 + 0.624526i
643.3 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −0.597287 + 2.15482i 1.00000 1.47923 + 1.47923i 0.707107 + 0.707107i 1.00000i −1.10134 1.94603i
643.4 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.597287 2.15482i 1.00000 −1.47923 1.47923i 0.707107 + 0.707107i 1.00000i 1.10134 + 1.94603i
643.5 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.07661 + 1.95982i 1.00000 −3.43458 3.43458i 0.707107 + 0.707107i 1.00000i −2.14708 0.624526i
643.6 −0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.98078 1.03755i 1.00000 −0.124800 0.124800i 0.707107 + 0.707107i 1.00000i −0.666968 + 2.13428i
643.7 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.19853 + 0.408021i 1.00000 −1.64584 1.64584i −0.707107 0.707107i 1.00000i −1.26608 + 1.84311i
643.8 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.52094 + 1.63913i 1.00000 2.68754 + 2.68754i −0.707107 0.707107i 1.00000i 0.0835670 + 2.23451i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 643.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.j.b 24
5.c odd 4 1 inner 690.2.j.b 24
23.b odd 2 1 inner 690.2.j.b 24
115.e even 4 1 inner 690.2.j.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.j.b 24 1.a even 1 1 trivial
690.2.j.b 24 5.c odd 4 1 inner
690.2.j.b 24 23.b odd 2 1 inner
690.2.j.b 24 115.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{24} + 880 T_{7}^{20} + 207712 T_{7}^{16} + 16248576 T_{7}^{12} + 466759936 T_{7}^{8} + 4322951168 T_{7}^{4} + 4194304$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.