↑

Properties

Label 1320.2.z.d
Level $1320$
Weight $2$
Character orbit 1320.z
Analytic conductor $10.540$
Analytic rank $0$
Dimension $48$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1320,2,Mod(571,1320)] 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("1320.571"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.z (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0] 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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{3} - 4 q^{4} + 48 q^{9} + 8 q^{11} - 4 q^{12} - 12 q^{14} + 16 q^{16} + 4 q^{20} - 8 q^{22} - 48 q^{25} + 36 q^{26} + 48 q^{27} + 8 q^{33} - 44 q^{34} - 4 q^{36} + 48 q^{38} - 12 q^{42} + 24 q^{44}+ \cdots + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1 −1.40041 − 0.197127i 1.00000 1.92228 + 0.552116i − 1.00000i −1.40041 − 0.197127i −2.51962 −2.58314 − 1.15212i 1.00000 −0.197127 + 1.40041i
571.2 −1.40041 + 0.197127i 1.00000 1.92228 − 0.552116i 1.00000i −1.40041 + 0.197127i −2.51962 −2.58314 + 1.15212i 1.00000 −0.197127 − 1.40041i
571.3 −1.39378 − 0.239542i 1.00000 1.88524 + 0.667737i − 1.00000i −1.39378 − 0.239542i 3.76969 −2.46766 − 1.38227i 1.00000 −0.239542 + 1.39378i
571.4 −1.39378 + 0.239542i 1.00000 1.88524 − 0.667737i 1.00000i −1.39378 + 0.239542i 3.76969 −2.46766 + 1.38227i 1.00000 −0.239542 − 1.39378i
571.5 −1.38008 − 0.308819i 1.00000 1.80926 + 0.852392i 1.00000i −1.38008 − 0.308819i −1.18702 −2.23370 − 1.73511i 1.00000 0.308819 − 1.38008i
571.6 −1.38008 + 0.308819i 1.00000 1.80926 − 0.852392i − 1.00000i −1.38008 + 0.308819i −1.18702 −2.23370 + 1.73511i 1.00000 0.308819 + 1.38008i
571.7 −1.19110 − 0.762416i 1.00000 0.837444 + 1.81623i 1.00000i −1.19110 − 0.762416i −2.25729 0.387242 − 2.80179i 1.00000 0.762416 − 1.19110i
571.8 −1.19110 + 0.762416i 1.00000 0.837444 − 1.81623i − 1.00000i −1.19110 + 0.762416i −2.25729 0.387242 + 2.80179i 1.00000 0.762416 + 1.19110i
571.9 −1.10408 − 0.883740i 1.00000 0.438007 + 1.95145i − 1.00000i −1.10408 − 0.883740i −1.36573 1.24098 − 2.54165i 1.00000 −0.883740 + 1.10408i
571.10 −1.10408 + 0.883740i 1.00000 0.438007 − 1.95145i 1.00000i −1.10408 + 0.883740i −1.36573 1.24098 + 2.54165i 1.00000 −0.883740 − 1.10408i
571.11 −1.08042 − 0.912515i 1.00000 0.334632 + 1.97181i − 1.00000i −1.08042 − 0.912515i 3.94184 1.43776 − 2.43574i 1.00000 −0.912515 + 1.08042i
571.12 −1.08042 + 0.912515i 1.00000 0.334632 − 1.97181i 1.00000i −1.08042 + 0.912515i 3.94184 1.43776 + 2.43574i 1.00000 −0.912515 − 1.08042i
571.13 −1.04693 − 0.950753i 1.00000 0.192136 + 1.99075i 1.00000i −1.04693 − 0.950753i 2.47224 1.69156 − 2.26685i 1.00000 0.950753 − 1.04693i
571.14 −1.04693 + 0.950753i 1.00000 0.192136 − 1.99075i − 1.00000i −1.04693 + 0.950753i 2.47224 1.69156 + 2.26685i 1.00000 0.950753 + 1.04693i
571.15 −0.546258 − 1.30445i 1.00000 −1.40320 + 1.42514i − 1.00000i −0.546258 − 1.30445i 1.94035 2.62554 + 1.05192i 1.00000 −1.30445 + 0.546258i
571.16 −0.546258 + 1.30445i 1.00000 −1.40320 − 1.42514i 1.00000i −0.546258 + 1.30445i 1.94035 2.62554 − 1.05192i 1.00000 −1.30445 − 0.546258i
571.17 −0.492518 − 1.32568i 1.00000 −1.51485 + 1.30584i 1.00000i −0.492518 − 1.32568i −4.38734 2.47722 + 1.36506i 1.00000 1.32568 − 0.492518i
571.18 −0.492518 + 1.32568i 1.00000 −1.51485 − 1.30584i − 1.00000i −0.492518 + 1.32568i −4.38734 2.47722 − 1.36506i 1.00000 1.32568 + 0.492518i
571.19 −0.447186 − 1.34165i 1.00000 −1.60005 + 1.19993i − 1.00000i −0.447186 − 1.34165i 1.05594 2.32541 + 1.61011i 1.00000 −1.34165 + 0.447186i
571.20 −0.447186 + 1.34165i 1.00000 −1.60005 − 1.19993i 1.00000i −0.447186 + 1.34165i 1.05594 2.32541 − 1.61011i 1.00000 −1.34165 − 0.447186i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.b odd 2 1 inner
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.z.d ✓ 48
4.b odd 2 1 5280.2.z.d 48
8.b even 2 1 5280.2.z.d 48
8.d odd 2 1 inner 1320.2.z.d ✓ 48
11.b odd 2 1 inner 1320.2.z.d ✓ 48
44.c even 2 1 5280.2.z.d 48
88.b odd 2 1 5280.2.z.d 48
88.g even 2 1 inner 1320.2.z.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.z.d ✓ 48 1.a even 1 1 trivial
1320.2.z.d ✓ 48 8.d odd 2 1 inner
1320.2.z.d ✓ 48 11.b odd 2 1 inner
1320.2.z.d ✓ 48 88.g even 2 1 inner
5280.2.z.d 48 4.b odd 2 1
5280.2.z.d 48 8.b even 2 1
5280.2.z.d 48 44.c even 2 1
5280.2.z.d 48 88.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} - 96 T_{7}^{22} + 3924 T_{7}^{20} - 89568 T_{7}^{18} + 1261360 T_{7}^{16} - 11462656 T_{7}^{14} + \cdots + 110166016 \) acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\). Copy content Toggle raw display