Properties

Label 1500.2.o.c
Level $1500$
Weight $2$
Character orbit 1500.o
Analytic conductor $11.978$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.o (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 6q^{9} - 6q^{11} - 10q^{17} + 10q^{19} - 4q^{21} - 40q^{23} + 4q^{29} + 6q^{31} - 10q^{33} - 10q^{41} + 40q^{47} - 56q^{49} + 16q^{51} + 60q^{53} - 36q^{59} - 12q^{61} + 10q^{63} - 20q^{67} + 4q^{69} + 40q^{71} - 60q^{73} + 40q^{77} + 8q^{79} - 6q^{81} + 50q^{83} + 20q^{87} - 30q^{91} + 40q^{97} - 4q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −0.951057 + 0.309017i 0 0 0 3.78808i 0 0.809017 0.587785i 0
49.2 0 −0.951057 + 0.309017i 0 0 0 1.31873i 0 0.809017 0.587785i 0
49.3 0 −0.951057 + 0.309017i 0 0 0 3.54704i 0 0.809017 0.587785i 0
49.4 0 0.951057 0.309017i 0 0 0 1.04684i 0 0.809017 0.587785i 0
49.5 0 0.951057 0.309017i 0 0 0 0.595901i 0 0.809017 0.587785i 0
49.6 0 0.951057 0.309017i 0 0 0 4.62675i 0 0.809017 0.587785i 0
349.1 0 −0.587785 0.809017i 0 0 0 2.44380i 0 −0.309017 + 0.951057i 0
349.2 0 −0.587785 0.809017i 0 0 0 4.13266i 0 −0.309017 + 0.951057i 0
349.3 0 −0.587785 0.809017i 0 0 0 4.41540i 0 −0.309017 + 0.951057i 0
349.4 0 0.587785 + 0.809017i 0 0 0 3.80992i 0 −0.309017 + 0.951057i 0
349.5 0 0.587785 + 0.809017i 0 0 0 0.957526i 0 −0.309017 + 0.951057i 0
349.6 0 0.587785 + 0.809017i 0 0 0 1.57893i 0 −0.309017 + 0.951057i 0
649.1 0 −0.587785 + 0.809017i 0 0 0 2.44380i 0 −0.309017 0.951057i 0
649.2 0 −0.587785 + 0.809017i 0 0 0 4.13266i 0 −0.309017 0.951057i 0
649.3 0 −0.587785 + 0.809017i 0 0 0 4.41540i 0 −0.309017 0.951057i 0
649.4 0 0.587785 0.809017i 0 0 0 3.80992i 0 −0.309017 0.951057i 0
649.5 0 0.587785 0.809017i 0 0 0 0.957526i 0 −0.309017 0.951057i 0
649.6 0 0.587785 0.809017i 0 0 0 1.57893i 0 −0.309017 0.951057i 0
949.1 0 −0.951057 0.309017i 0 0 0 3.78808i 0 0.809017 + 0.587785i 0
949.2 0 −0.951057 0.309017i 0 0 0 1.31873i 0 0.809017 + 0.587785i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1500.2.o.c 24
5.b even 2 1 300.2.o.a 24
5.c odd 4 1 1500.2.m.c 24
5.c odd 4 1 1500.2.m.d 24
15.d odd 2 1 900.2.w.c 24
25.d even 5 1 300.2.o.a 24
25.d even 5 1 7500.2.d.g 24
25.e even 10 1 inner 1500.2.o.c 24
25.e even 10 1 7500.2.d.g 24
25.f odd 20 1 1500.2.m.c 24
25.f odd 20 1 1500.2.m.d 24
25.f odd 20 1 7500.2.a.m 12
25.f odd 20 1 7500.2.a.n 12
75.j odd 10 1 900.2.w.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.o.a 24 5.b even 2 1
300.2.o.a 24 25.d even 5 1
900.2.w.c 24 15.d odd 2 1
900.2.w.c 24 75.j odd 10 1
1500.2.m.c 24 5.c odd 4 1
1500.2.m.c 24 25.f odd 20 1
1500.2.m.d 24 5.c odd 4 1
1500.2.m.d 24 25.f odd 20 1
1500.2.o.c 24 1.a even 1 1 trivial
1500.2.o.c 24 25.e even 10 1 inner
7500.2.a.m 12 25.f odd 20 1
7500.2.a.n 12 25.f odd 20 1
7500.2.d.g 24 25.d even 5 1
7500.2.d.g 24 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\).