# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 24 q^{4} - 36 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 24 q^{4} - 36 q^{9} - 8 q^{11} - 16 q^{14} - 48 q^{16} - 24 q^{19} + 36 q^{21} - 48 q^{26} + 48 q^{29} - 396 q^{31} - 144 q^{36} + 72 q^{39} + 16 q^{44} + 64 q^{46} - 56 q^{49} - 48 q^{51} + 80 q^{56} + 96 q^{59} + 372 q^{61} - 192 q^{64} + 72 q^{66} - 272 q^{71} + 128 q^{74} + 140 q^{79} - 108 q^{81} + 24 q^{84} - 416 q^{86} - 336 q^{89} + 584 q^{91} + 408 q^{94} + 48 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 2.33475 6.59916i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −1.64177 6.80475i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.74171 1.88399i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 0.264136 + 6.99501i 2.82843i −1.50000 2.59808i 0
199.5 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.78392 1.72580i 2.82843i −1.50000 2.59808i 0
199.6 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.34914 + 2.94762i 2.82843i −1.50000 2.59808i 0
199.7 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −2.33475 + 6.59916i 2.82843i −1.50000 2.59808i 0
199.8 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 1.64177 + 6.80475i 2.82843i −1.50000 2.59808i 0
199.9 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.34914 2.94762i 2.82843i −1.50000 2.59808i 0
199.10 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −0.264136 6.99501i 2.82843i −1.50000 2.59808i 0
199.11 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.74171 + 1.88399i 2.82843i −1.50000 2.59808i 0
199.12 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.78392 + 1.72580i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.33475 + 6.59916i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −1.64177 + 6.80475i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.74171 + 1.88399i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 0.264136 6.99501i 2.82843i −1.50000 + 2.59808i 0
649.5 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.78392 + 1.72580i 2.82843i −1.50000 + 2.59808i 0
649.6 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.34914 2.94762i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.33475 6.59916i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i 1.64177 6.80475i 2.82843i −1.50000 + 2.59808i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.12 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
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.d 24
5.b even 2 1 inner 1050.3.q.d 24
5.c odd 4 1 1050.3.p.e 12
5.c odd 4 1 1050.3.p.f yes 12
7.d odd 6 1 inner 1050.3.q.d 24
35.i odd 6 1 inner 1050.3.q.d 24
35.k even 12 1 1050.3.p.e 12
35.k even 12 1 1050.3.p.f yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.p.e 12 5.c odd 4 1
1050.3.p.e 12 35.k even 12 1
1050.3.p.f yes 12 5.c odd 4 1
1050.3.p.f yes 12 35.k even 12 1
1050.3.q.d 24 1.a even 1 1 trivial
1050.3.q.d 24 5.b even 2 1 inner
1050.3.q.d 24 7.d odd 6 1 inner
1050.3.q.d 24 35.i odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{12} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.