# Properties

 Label 1050.3.h.c Level $1050$ Weight $3$ Character orbit 1050.h Analytic conductor $28.610$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 48 q^{4} + 72 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q - 48 q^{4} + 72 q^{9} - 32 q^{11} - 16 q^{14} + 96 q^{16} - 12 q^{21} - 96 q^{29} - 144 q^{36} + 24 q^{39} + 64 q^{44} + 160 q^{46} - 236 q^{49} + 144 q^{51} + 32 q^{56} - 192 q^{64} + 496 q^{71} + 128 q^{74} + 416 q^{79} + 216 q^{81} + 24 q^{84} + 256 q^{86} - 316 q^{91} - 96 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1 1.41421i −1.73205 −2.00000 0 2.44949i −4.78154 + 5.11242i 2.82843i 3.00000 0
349.2 1.41421i −1.73205 −2.00000 0 2.44949i −4.78154 5.11242i 2.82843i 3.00000 0
349.3 1.41421i 1.73205 −2.00000 0 2.44949i −5.85173 3.84151i 2.82843i 3.00000 0
349.4 1.41421i 1.73205 −2.00000 0 2.44949i −5.85173 + 3.84151i 2.82843i 3.00000 0
349.5 1.41421i −1.73205 −2.00000 0 2.44949i −3.49930 6.06258i 2.82843i 3.00000 0
349.6 1.41421i −1.73205 −2.00000 0 2.44949i −3.49930 + 6.06258i 2.82843i 3.00000 0
349.7 1.41421i −1.73205 −2.00000 0 2.44949i 6.69738 + 2.03595i 2.82843i 3.00000 0
349.8 1.41421i −1.73205 −2.00000 0 2.44949i 6.69738 2.03595i 2.82843i 3.00000 0
349.9 1.41421i −1.73205 −2.00000 0 2.44949i −1.46536 + 6.84490i 2.82843i 3.00000 0
349.10 1.41421i −1.73205 −2.00000 0 2.44949i −1.46536 6.84490i 2.82843i 3.00000 0
349.11 1.41421i −1.73205 −2.00000 0 2.44949i −1.07086 6.91761i 2.82843i 3.00000 0
349.12 1.41421i −1.73205 −2.00000 0 2.44949i −1.07086 + 6.91761i 2.82843i 3.00000 0
349.13 1.41421i 1.73205 −2.00000 0 2.44949i 1.07086 6.91761i 2.82843i 3.00000 0
349.14 1.41421i 1.73205 −2.00000 0 2.44949i 1.07086 + 6.91761i 2.82843i 3.00000 0
349.15 1.41421i −1.73205 −2.00000 0 2.44949i 5.85173 3.84151i 2.82843i 3.00000 0
349.16 1.41421i −1.73205 −2.00000 0 2.44949i 5.85173 + 3.84151i 2.82843i 3.00000 0
349.17 1.41421i 1.73205 −2.00000 0 2.44949i 1.46536 + 6.84490i 2.82843i 3.00000 0
349.18 1.41421i 1.73205 −2.00000 0 2.44949i 1.46536 6.84490i 2.82843i 3.00000 0
349.19 1.41421i 1.73205 −2.00000 0 2.44949i −6.69738 + 2.03595i 2.82843i 3.00000 0
349.20 1.41421i 1.73205 −2.00000 0 2.44949i −6.69738 2.03595i 2.82843i 3.00000 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.24 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.h.c 24
5.b even 2 1 inner 1050.3.h.c 24
5.c odd 4 1 1050.3.f.c 12
5.c odd 4 1 1050.3.f.d yes 12
7.b odd 2 1 inner 1050.3.h.c 24
35.c odd 2 1 inner 1050.3.h.c 24
35.f even 4 1 1050.3.f.c 12
35.f even 4 1 1050.3.f.d yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.f.c 12 5.c odd 4 1
1050.3.f.c 12 35.f even 4 1
1050.3.f.d yes 12 5.c odd 4 1
1050.3.f.d yes 12 35.f even 4 1
1050.3.h.c 24 1.a even 1 1 trivial
1050.3.h.c 24 5.b even 2 1 inner
1050.3.h.c 24 7.b odd 2 1 inner
1050.3.h.c 24 35.c odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{6} + 8 T_{11}^{5} - 462 T_{11}^{4} - 4780 T_{11}^{3} + 27217 T_{11}^{2} + 422940 T_{11} + 1141308$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.