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Properties

Label 1380.4.f.a
Level $1380$
Weight $4$
Character orbit 1380.f
Analytic conductor $81.423$
Analytic rank $0$
Dimension $30$
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] = [1380,4,Mod(829,1380)] 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("1380.829"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.4226358079\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 12 q^{5} - 270 q^{9} + 42 q^{15} - 72 q^{19} + 144 q^{21} - 162 q^{25} + 912 q^{29} - 248 q^{31} - 68 q^{35} + 168 q^{39} - 416 q^{41} - 108 q^{45} - 130 q^{49} + 36 q^{51} - 132 q^{55} + 532 q^{59}+ \cdots - 556 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1 0 − 3.00000i 0 −11.1229 − 1.13228i 0 9.07273i 0 −9.00000 0
829.2 0 − 3.00000i 0 −11.1204 − 1.15628i 0 32.1488i 0 −9.00000 0
829.3 0 − 3.00000i 0 −10.0006 + 4.99881i 0 9.12680i 0 −9.00000 0
829.4 0 − 3.00000i 0 −6.10192 + 9.36838i 0 − 26.6185i 0 −9.00000 0
829.5 0 − 3.00000i 0 −4.13755 − 10.3866i 0 − 9.23139i 0 −9.00000 0
829.6 0 − 3.00000i 0 −4.03185 + 10.4280i 0 − 9.41961i 0 −9.00000 0
829.7 0 − 3.00000i 0 −0.777786 + 11.1533i 0 23.0546i 0 −9.00000 0
829.8 0 − 3.00000i 0 −0.614312 − 11.1635i 0 4.08781i 0 −9.00000 0
829.9 0 − 3.00000i 0 1.53272 − 11.0748i 0 − 33.2009i 0 −9.00000 0
829.10 0 − 3.00000i 0 5.70477 + 9.61538i 0 12.5577i 0 −9.00000 0
829.11 0 − 3.00000i 0 8.02323 − 7.78638i 0 30.1456i 0 −9.00000 0
829.12 0 − 3.00000i 0 8.50013 + 7.26277i 0 − 11.2260i 0 −9.00000 0
829.13 0 − 3.00000i 0 8.57672 − 7.17216i 0 4.35948i 0 −9.00000 0
829.14 0 − 3.00000i 0 10.3897 + 4.12975i 0 − 15.6119i 0 −9.00000 0
829.15 0 − 3.00000i 0 11.1800 − 0.0844906i 0 4.75481i 0 −9.00000 0
829.16 0 3.00000i 0 −11.1229 + 1.13228i 0 − 9.07273i 0 −9.00000 0
829.17 0 3.00000i 0 −11.1204 + 1.15628i 0 − 32.1488i 0 −9.00000 0
829.18 0 3.00000i 0 −10.0006 − 4.99881i 0 − 9.12680i 0 −9.00000 0
829.19 0 3.00000i 0 −6.10192 − 9.36838i 0 26.6185i 0 −9.00000 0
829.20 0 3.00000i 0 −4.13755 + 10.3866i 0 9.23139i 0 −9.00000 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.4.f.a ✓ 30
5.b even 2 1 inner 1380.4.f.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.f.a ✓ 30 1.a even 1 1 trivial
1380.4.f.a ✓ 30 5.b even 2 1 inner