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Properties

Label 360.4.b.a
Level $360$
Weight $4$
Character orbit 360.b
Analytic conductor $21.241$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [360,4,Mod(251,360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("360.251"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} + 6 q^{4} - 120 q^{5} + 6 q^{8} + 30 q^{10} - 108 q^{14} - 126 q^{16} - 24 q^{19} - 30 q^{20} + 240 q^{22} + 456 q^{23} + 600 q^{25} + 168 q^{26} + 120 q^{28} + 354 q^{32} - 420 q^{34} - 420 q^{38}+ \cdots + 1890 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −2.74252 − 0.691792i 0 7.04285 + 3.79451i −5.00000 0 25.7913i −16.6901 − 15.2787i 0 13.7126 + 3.45896i
251.2 −2.74252 + 0.691792i 0 7.04285 − 3.79451i −5.00000 0 − 25.7913i −16.6901 + 15.2787i 0 13.7126 − 3.45896i
251.3 −2.61333 − 1.08191i 0 5.65895 + 5.65476i −5.00000 0 − 24.2359i −8.67075 − 20.9002i 0 13.0666 + 5.40954i
251.4 −2.61333 + 1.08191i 0 5.65895 − 5.65476i −5.00000 0 24.2359i −8.67075 + 20.9002i 0 13.0666 − 5.40954i
251.5 −2.38691 − 1.51745i 0 3.39471 + 7.24403i −5.00000 0 27.9651i 2.88957 − 22.4422i 0 11.9346 + 7.58724i
251.6 −2.38691 + 1.51745i 0 3.39471 − 7.24403i −5.00000 0 − 27.9651i 2.88957 + 22.4422i 0 11.9346 − 7.58724i
251.7 −2.18228 − 1.79934i 0 1.52472 + 7.85336i −5.00000 0 − 12.2479i 10.8035 − 19.8817i 0 10.9114 + 8.99672i
251.8 −2.18228 + 1.79934i 0 1.52472 − 7.85336i −5.00000 0 12.2479i 10.8035 + 19.8817i 0 10.9114 − 8.99672i
251.9 −1.70898 − 2.25375i 0 −2.15877 + 7.70323i −5.00000 0 − 15.4755i 21.0504 − 8.29935i 0 8.54491 + 11.2687i
251.10 −1.70898 + 2.25375i 0 −2.15877 − 7.70323i −5.00000 0 15.4755i 21.0504 + 8.29935i 0 8.54491 − 11.2687i
251.11 −0.671247 − 2.74762i 0 −7.09886 + 3.68866i −5.00000 0 − 2.99301i 14.9001 + 17.0290i 0 3.35623 + 13.7381i
251.12 −0.671247 + 2.74762i 0 −7.09886 − 3.68866i −5.00000 0 2.99301i 14.9001 − 17.0290i 0 3.35623 − 13.7381i
251.13 −0.281738 − 2.81436i 0 −7.84125 + 1.58583i −5.00000 0 − 11.3970i 6.67226 + 21.6213i 0 1.40869 + 14.0718i
251.14 −0.281738 + 2.81436i 0 −7.84125 − 1.58583i −5.00000 0 11.3970i 6.67226 − 21.6213i 0 1.40869 − 14.0718i
251.15 0.666534 − 2.74877i 0 −7.11147 − 3.66429i −5.00000 0 22.7533i −14.8123 + 17.1054i 0 −3.33267 + 13.7438i
251.16 0.666534 + 2.74877i 0 −7.11147 + 3.66429i −5.00000 0 − 22.7533i −14.8123 − 17.1054i 0 −3.33267 − 13.7438i
251.17 1.51037 − 2.39140i 0 −3.43758 − 7.22378i −5.00000 0 − 35.1660i −22.4670 − 2.68996i 0 −7.55184 + 11.9570i
251.18 1.51037 + 2.39140i 0 −3.43758 + 7.22378i −5.00000 0 35.1660i −22.4670 + 2.68996i 0 −7.55184 − 11.9570i
251.19 2.19727 − 1.78101i 0 1.65600 − 7.82673i −5.00000 0 − 4.37103i −10.3008 − 20.1468i 0 −10.9864 + 8.90506i
251.20 2.19727 + 1.78101i 0 1.65600 + 7.82673i −5.00000 0 4.37103i −10.3008 + 20.1468i 0 −10.9864 − 8.90506i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 251.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.b.a ✓ 24
3.b odd 2 1 360.4.b.b yes 24
4.b odd 2 1 1440.4.b.a 24
8.b even 2 1 1440.4.b.b 24
8.d odd 2 1 360.4.b.b yes 24
12.b even 2 1 1440.4.b.b 24
24.f even 2 1 inner 360.4.b.a ✓ 24
24.h odd 2 1 1440.4.b.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.b.a ✓ 24 1.a even 1 1 trivial
360.4.b.a ✓ 24 24.f even 2 1 inner
360.4.b.b yes 24 3.b odd 2 1
360.4.b.b yes 24 8.d odd 2 1
1440.4.b.a 24 4.b odd 2 1
1440.4.b.a 24 24.h odd 2 1
1440.4.b.b 24 8.b even 2 1
1440.4.b.b 24 12.b even 2 1