# Properties

 Label 1600.2.s.e Level $1600$ Weight $2$ Character orbit 1600.s Analytic conductor $12.776$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.s (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 400) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 40q^{9} + 20q^{11} - 12q^{19} - 8q^{29} - 20q^{51} + 8q^{59} - 48q^{61} + 64q^{69} + 16q^{71} - 104q^{79} + 48q^{81} - 96q^{89} - 64q^{91} + 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
207.1 0 −3.25766 0 0 0 2.54012 + 2.54012i 0 7.61238 0
207.2 0 −2.70780 0 0 0 −1.60911 1.60911i 0 4.33218 0
207.3 0 −2.35800 0 0 0 −2.66357 2.66357i 0 2.56018 0
207.4 0 −1.86755 0 0 0 0.719989 + 0.719989i 0 0.487737 0
207.5 0 −0.790153 0 0 0 −0.139907 0.139907i 0 −2.37566 0
207.6 0 −0.619018 0 0 0 1.82373 + 1.82373i 0 −2.61682 0
207.7 0 0.619018 0 0 0 −1.82373 1.82373i 0 −2.61682 0
207.8 0 0.790153 0 0 0 0.139907 + 0.139907i 0 −2.37566 0
207.9 0 1.86755 0 0 0 −0.719989 0.719989i 0 0.487737 0
207.10 0 2.35800 0 0 0 2.66357 + 2.66357i 0 2.56018 0
207.11 0 2.70780 0 0 0 1.60911 + 1.60911i 0 4.33218 0
207.12 0 3.25766 0 0 0 −2.54012 2.54012i 0 7.61238 0
943.1 0 −3.25766 0 0 0 2.54012 2.54012i 0 7.61238 0
943.2 0 −2.70780 0 0 0 −1.60911 + 1.60911i 0 4.33218 0
943.3 0 −2.35800 0 0 0 −2.66357 + 2.66357i 0 2.56018 0
943.4 0 −1.86755 0 0 0 0.719989 0.719989i 0 0.487737 0
943.5 0 −0.790153 0 0 0 −0.139907 + 0.139907i 0 −2.37566 0
943.6 0 −0.619018 0 0 0 1.82373 1.82373i 0 −2.61682 0
943.7 0 0.619018 0 0 0 −1.82373 + 1.82373i 0 −2.61682 0
943.8 0 0.790153 0 0 0 0.139907 0.139907i 0 −2.37566 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.12 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.b even 2 1 inner
80.j even 4 1 inner
80.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.s.e 24
4.b odd 2 1 400.2.s.e yes 24
5.b even 2 1 inner 1600.2.s.e 24
5.c odd 4 2 1600.2.j.e 24
16.e even 4 1 400.2.j.e 24
16.f odd 4 1 1600.2.j.e 24
20.d odd 2 1 400.2.s.e yes 24
20.e even 4 2 400.2.j.e 24
80.i odd 4 1 400.2.s.e yes 24
80.j even 4 1 inner 1600.2.s.e 24
80.k odd 4 1 1600.2.j.e 24
80.q even 4 1 400.2.j.e 24
80.s even 4 1 inner 1600.2.s.e 24
80.t odd 4 1 400.2.s.e yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.e 24 16.e even 4 1
400.2.j.e 24 20.e even 4 2
400.2.j.e 24 80.q even 4 1
400.2.s.e yes 24 4.b odd 2 1
400.2.s.e yes 24 20.d odd 2 1
400.2.s.e yes 24 80.i odd 4 1
400.2.s.e yes 24 80.t odd 4 1
1600.2.j.e 24 5.c odd 4 2
1600.2.j.e 24 16.f odd 4 1
1600.2.j.e 24 80.k odd 4 1
1600.2.s.e 24 1.a even 1 1 trivial
1600.2.s.e 24 5.b even 2 1 inner
1600.2.s.e 24 80.j even 4 1 inner
1600.2.s.e 24 80.s even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 28 T_{3}^{10} + 287 T_{3}^{8} - 1320 T_{3}^{6} + 2631 T_{3}^{4} - 1772 T_{3}^{2} + 361$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.