Properties

Label 1050.3.q.e
Level $1050$
Weight $3$
Character orbit 1050.q
Analytic conductor $28.610$
Analytic rank $0$
Dimension $32$
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.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 210)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} - 48 q^{9} - 8 q^{11} - 16 q^{14} - 64 q^{16} + 144 q^{19} - 48 q^{21} - 144 q^{29} + 240 q^{31} - 192 q^{36} - 72 q^{39} + 16 q^{44} + 16 q^{46} + 80 q^{49} - 24 q^{51} + 32 q^{56} - 264 q^{59} + 192 q^{61} - 256 q^{64} + 144 q^{66} - 272 q^{71} + 224 q^{74} - 560 q^{79} - 144 q^{81} + 48 q^{84} - 176 q^{86} + 600 q^{89} - 544 q^{91} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.86123 6.38854i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 4.01037 5.73733i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.50174 + 2.59373i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.72455 1.94434i 2.82843i −1.50000 2.59808i 0
199.5 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 5.56601 + 4.24494i 2.82843i −1.50000 2.59808i 0
199.6 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.46925 + 2.67372i 2.82843i −1.50000 2.59808i 0
199.7 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 4.61524 5.26304i 2.82843i −1.50000 2.59808i 0
199.8 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 0.325616 + 6.99242i 2.82843i −1.50000 2.59808i 0
199.9 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −0.325616 6.99242i 2.82843i −1.50000 2.59808i 0
199.10 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 2.86123 + 6.38854i 2.82843i −1.50000 2.59808i 0
199.11 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −5.56601 4.24494i 2.82843i −1.50000 2.59808i 0
199.12 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.72455 + 1.94434i 2.82843i −1.50000 2.59808i 0
199.13 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.46925 2.67372i 2.82843i −1.50000 2.59808i 0
199.14 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.50174 2.59373i 2.82843i −1.50000 2.59808i 0
199.15 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −4.01037 + 5.73733i 2.82843i −1.50000 2.59808i 0
199.16 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −4.61524 + 5.26304i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.86123 + 6.38854i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.01037 + 5.73733i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.50174 2.59373i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.72455 + 1.94434i 2.82843i −1.50000 + 2.59808i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.16
Significant digits:
Format:

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.e 32
5.b even 2 1 inner 1050.3.q.e 32
5.c odd 4 1 210.3.o.b 16
5.c odd 4 1 1050.3.p.i 16
7.d odd 6 1 inner 1050.3.q.e 32
15.e even 4 1 630.3.v.c 16
35.i odd 6 1 inner 1050.3.q.e 32
35.k even 12 1 210.3.o.b 16
35.k even 12 1 1050.3.p.i 16
35.k even 12 1 1470.3.f.d 16
35.l odd 12 1 1470.3.f.d 16
105.w odd 12 1 630.3.v.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 5.c odd 4 1
210.3.o.b 16 35.k even 12 1
630.3.v.c 16 15.e even 4 1
630.3.v.c 16 105.w odd 12 1
1050.3.p.i 16 5.c odd 4 1
1050.3.p.i 16 35.k even 12 1
1050.3.q.e 32 1.a even 1 1 trivial
1050.3.q.e 32 5.b even 2 1 inner
1050.3.q.e 32 7.d odd 6 1 inner
1050.3.q.e 32 35.i odd 6 1 inner
1470.3.f.d 16 35.k even 12 1
1470.3.f.d 16 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} + 4 T_{11}^{15} + 604 T_{11}^{14} + 352 T_{11}^{13} + 252064 T_{11}^{12} + 129616 T_{11}^{11} + 49322848 T_{11}^{10} - 7989152 T_{11}^{9} + 6945139504 T_{11}^{8} + 676772544 T_{11}^{7} + \cdots + 9682651996416 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display