Properties

Label 2268.4.f.a
Level $2268$
Weight $4$
Character orbit 2268.f
Analytic conductor $133.816$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2268.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(133.816331893\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 252)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 12q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 12q^{7} + 1200q^{25} + 336q^{37} - 168q^{43} - 636q^{49} + 1176q^{67} - 408q^{79} + 720q^{85} - 1080q^{91} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1133.1 0 0 0 −21.1141 0 −13.5800 + 12.5930i 0 0 0
1133.2 0 0 0 −21.1141 0 −13.5800 12.5930i 0 0 0
1133.3 0 0 0 −18.2402 0 18.5185 0.256813i 0 0 0
1133.4 0 0 0 −18.2402 0 18.5185 + 0.256813i 0 0 0
1133.5 0 0 0 −16.5975 0 −11.9446 14.1537i 0 0 0
1133.6 0 0 0 −16.5975 0 −11.9446 + 14.1537i 0 0 0
1133.7 0 0 0 −15.6490 0 −3.87814 18.1097i 0 0 0
1133.8 0 0 0 −15.6490 0 −3.87814 + 18.1097i 0 0 0
1133.9 0 0 0 −12.0714 0 14.8834 + 11.0221i 0 0 0
1133.10 0 0 0 −12.0714 0 14.8834 11.0221i 0 0 0
1133.11 0 0 0 −10.9938 0 11.4708 14.5403i 0 0 0
1133.12 0 0 0 −10.9938 0 11.4708 + 14.5403i 0 0 0
1133.13 0 0 0 −10.3297 0 2.73026 + 18.3179i 0 0 0
1133.14 0 0 0 −10.3297 0 2.73026 18.3179i 0 0 0
1133.15 0 0 0 −7.06893 0 −18.4258 1.86777i 0 0 0
1133.16 0 0 0 −7.06893 0 −18.4258 + 1.86777i 0 0 0
1133.17 0 0 0 −5.99994 0 −15.8480 9.58340i 0 0 0
1133.18 0 0 0 −5.99994 0 −15.8480 + 9.58340i 0 0 0
1133.19 0 0 0 −4.68539 0 −7.47522 16.9446i 0 0 0
1133.20 0 0 0 −4.68539 0 −7.47522 + 16.9446i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1133.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.4.f.a 48
3.b odd 2 1 inner 2268.4.f.a 48
7.b odd 2 1 inner 2268.4.f.a 48
9.c even 3 1 252.4.x.a 48
9.c even 3 1 756.4.x.a 48
9.d odd 6 1 252.4.x.a 48
9.d odd 6 1 756.4.x.a 48
21.c even 2 1 inner 2268.4.f.a 48
63.l odd 6 1 252.4.x.a 48
63.l odd 6 1 756.4.x.a 48
63.o even 6 1 252.4.x.a 48
63.o even 6 1 756.4.x.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.x.a 48 9.c even 3 1
252.4.x.a 48 9.d odd 6 1
252.4.x.a 48 63.l odd 6 1
252.4.x.a 48 63.o even 6 1
756.4.x.a 48 9.c even 3 1
756.4.x.a 48 9.d odd 6 1
756.4.x.a 48 63.l odd 6 1
756.4.x.a 48 63.o even 6 1
2268.4.f.a 48 1.a even 1 1 trivial
2268.4.f.a 48 3.b odd 2 1 inner
2268.4.f.a 48 7.b odd 2 1 inner
2268.4.f.a 48 21.c even 2 1 inner