Properties

Label 900.3.u.c
Level $900$
Weight $3$
Character orbit 900.u
Analytic conductor $24.523$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 180)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 52q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 52q^{9} + 96q^{11} - 144q^{19} - 256q^{21} - 300q^{29} - 24q^{31} - 80q^{39} + 180q^{41} - 96q^{49} - 288q^{51} - 96q^{59} - 156q^{61} + 300q^{69} - 240q^{79} + 868q^{81} + 240q^{91} + 240q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 0 −2.96464 + 0.459278i 0 0 0 7.16186 4.13490i 0 8.57813 2.72318i 0
149.2 0 −2.87949 0.841761i 0 0 0 10.3705 5.98742i 0 7.58288 + 4.84768i 0
149.3 0 −1.39478 + 2.65605i 0 0 0 2.45229 1.41583i 0 −5.10917 7.40921i 0
149.4 0 −0.920635 2.85525i 0 0 0 1.02985 0.594587i 0 −7.30486 + 5.25728i 0
149.5 0 −0.783177 2.89597i 0 0 0 4.08158 2.35650i 0 −7.77327 + 4.53611i 0
149.6 0 −0.114662 2.99781i 0 0 0 −1.38806 + 0.801399i 0 −8.97371 + 0.687471i 0
149.7 0 0.114662 + 2.99781i 0 0 0 1.38806 0.801399i 0 −8.97371 + 0.687471i 0
149.8 0 0.783177 + 2.89597i 0 0 0 −4.08158 + 2.35650i 0 −7.77327 + 4.53611i 0
149.9 0 0.920635 + 2.85525i 0 0 0 −1.02985 + 0.594587i 0 −7.30486 + 5.25728i 0
149.10 0 1.39478 2.65605i 0 0 0 −2.45229 + 1.41583i 0 −5.10917 7.40921i 0
149.11 0 2.87949 + 0.841761i 0 0 0 −10.3705 + 5.98742i 0 7.58288 + 4.84768i 0
149.12 0 2.96464 0.459278i 0 0 0 −7.16186 + 4.13490i 0 8.57813 2.72318i 0
749.1 0 −2.96464 0.459278i 0 0 0 7.16186 + 4.13490i 0 8.57813 + 2.72318i 0
749.2 0 −2.87949 + 0.841761i 0 0 0 10.3705 + 5.98742i 0 7.58288 4.84768i 0
749.3 0 −1.39478 2.65605i 0 0 0 2.45229 + 1.41583i 0 −5.10917 + 7.40921i 0
749.4 0 −0.920635 + 2.85525i 0 0 0 1.02985 + 0.594587i 0 −7.30486 5.25728i 0
749.5 0 −0.783177 + 2.89597i 0 0 0 4.08158 + 2.35650i 0 −7.77327 4.53611i 0
749.6 0 −0.114662 + 2.99781i 0 0 0 −1.38806 0.801399i 0 −8.97371 0.687471i 0
749.7 0 0.114662 2.99781i 0 0 0 1.38806 + 0.801399i 0 −8.97371 0.687471i 0
749.8 0 0.783177 2.89597i 0 0 0 −4.08158 2.35650i 0 −7.77327 4.53611i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.c 24
3.b odd 2 1 2700.3.u.c 24
5.b even 2 1 inner 900.3.u.c 24
5.c odd 4 1 180.3.o.b 12
5.c odd 4 1 900.3.p.c 12
9.c even 3 1 2700.3.u.c 24
9.d odd 6 1 inner 900.3.u.c 24
15.d odd 2 1 2700.3.u.c 24
15.e even 4 1 540.3.o.b 12
15.e even 4 1 2700.3.p.c 12
20.e even 4 1 720.3.bs.b 12
45.h odd 6 1 inner 900.3.u.c 24
45.j even 6 1 2700.3.u.c 24
45.k odd 12 1 540.3.o.b 12
45.k odd 12 1 1620.3.g.b 12
45.k odd 12 1 2700.3.p.c 12
45.l even 12 1 180.3.o.b 12
45.l even 12 1 900.3.p.c 12
45.l even 12 1 1620.3.g.b 12
60.l odd 4 1 2160.3.bs.b 12
180.v odd 12 1 720.3.bs.b 12
180.x even 12 1 2160.3.bs.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b 12 5.c odd 4 1
180.3.o.b 12 45.l even 12 1
540.3.o.b 12 15.e even 4 1
540.3.o.b 12 45.k odd 12 1
720.3.bs.b 12 20.e even 4 1
720.3.bs.b 12 180.v odd 12 1
900.3.p.c 12 5.c odd 4 1
900.3.p.c 12 45.l even 12 1
900.3.u.c 24 1.a even 1 1 trivial
900.3.u.c 24 5.b even 2 1 inner
900.3.u.c 24 9.d odd 6 1 inner
900.3.u.c 24 45.h odd 6 1 inner
1620.3.g.b 12 45.k odd 12 1
1620.3.g.b 12 45.l even 12 1
2160.3.bs.b 12 60.l odd 4 1
2160.3.bs.b 12 180.x even 12 1
2700.3.p.c 12 15.e even 4 1
2700.3.p.c 12 45.k odd 12 1
2700.3.u.c 24 3.b odd 2 1
2700.3.u.c 24 9.c even 3 1
2700.3.u.c 24 15.d odd 2 1
2700.3.u.c 24 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(64\!\cdots\!30\)\( T_{7}^{8} - \)\(20\!\cdots\!90\)\( T_{7}^{6} + \)\(46\!\cdots\!61\)\( T_{7}^{4} - \)\(51\!\cdots\!26\)\( T_{7}^{2} + \)\(40\!\cdots\!21\)\( \)">\(T_{7}^{24} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).