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 $4$

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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:

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 \(S_{3}^{\mathrm{new}}(1050, [\chi])\).