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Properties

Label 1120.2.bz.f
Level $1120$
Weight $2$
Character orbit 1120.bz
Analytic conductor $8.943$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,2,Mod(271,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bz (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 280)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 12 q^{5} + 10 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 12 q^{5} + 10 q^{7} + 12 q^{9} - 8 q^{11} - 20 q^{13} + 6 q^{17} - 18 q^{19} + 26 q^{21} + 18 q^{23} - 12 q^{25} - 6 q^{31} + 12 q^{33} + 8 q^{35} - 18 q^{39} - 32 q^{43} - 12 q^{45} + 8 q^{49} + 22 q^{51} - 30 q^{53} - 16 q^{55} - 44 q^{57} + 18 q^{59} - 22 q^{61} - 12 q^{63} - 10 q^{65} + 8 q^{67} + 12 q^{69} + 30 q^{73} + 12 q^{75} + 32 q^{77} + 6 q^{79} - 4 q^{81} + 14 q^{87} - 60 q^{89} - 18 q^{91} + 18 q^{93} - 18 q^{95} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1 0 −2.75363 + 1.58981i 0 0.500000 − 0.866025i 0 −1.04250 − 2.43170i 0 3.55500 − 6.15745i 0
271.2 0 −2.26702 + 1.30886i 0 0.500000 − 0.866025i 0 1.72615 + 2.00510i 0 1.92625 − 3.33637i 0
271.3 0 −1.94732 + 1.12428i 0 0.500000 − 0.866025i 0 2.47009 − 0.947962i 0 1.02803 − 1.78060i 0
271.4 0 −1.90624 + 1.10057i 0 0.500000 − 0.866025i 0 −0.584379 − 2.58041i 0 0.922503 − 1.59782i 0
271.5 0 −1.75472 + 1.01309i 0 0.500000 − 0.866025i 0 −1.63843 + 2.07739i 0 0.552704 − 0.957311i 0
271.6 0 −0.784482 + 0.452921i 0 0.500000 − 0.866025i 0 1.23347 + 2.34063i 0 −1.08973 + 1.88746i 0
271.7 0 −0.725648 + 0.418953i 0 0.500000 − 0.866025i 0 −2.36913 + 1.17781i 0 −1.14896 + 1.99005i 0
271.8 0 0.219454 − 0.126702i 0 0.500000 − 0.866025i 0 0.978876 − 2.45801i 0 −1.46789 + 2.54247i 0
271.9 0 0.502680 − 0.290223i 0 0.500000 − 0.866025i 0 −2.63362 + 0.253028i 0 −1.33154 + 2.30630i 0
271.10 0 0.908317 − 0.524417i 0 0.500000 − 0.866025i 0 2.14799 + 1.54472i 0 −0.949974 + 1.64540i 0
271.11 0 1.84104 − 1.06293i 0 0.500000 − 0.866025i 0 2.17552 + 1.50569i 0 0.759621 − 1.31570i 0
271.12 0 2.66758 − 1.54013i 0 0.500000 − 0.866025i 0 2.53597 − 0.754231i 0 3.24397 − 5.61873i 0
591.1 0 −2.75363 − 1.58981i 0 0.500000 + 0.866025i 0 −1.04250 + 2.43170i 0 3.55500 + 6.15745i 0
591.2 0 −2.26702 − 1.30886i 0 0.500000 + 0.866025i 0 1.72615 − 2.00510i 0 1.92625 + 3.33637i 0
591.3 0 −1.94732 − 1.12428i 0 0.500000 + 0.866025i 0 2.47009 + 0.947962i 0 1.02803 + 1.78060i 0
591.4 0 −1.90624 − 1.10057i 0 0.500000 + 0.866025i 0 −0.584379 + 2.58041i 0 0.922503 + 1.59782i 0
591.5 0 −1.75472 − 1.01309i 0 0.500000 + 0.866025i 0 −1.63843 − 2.07739i 0 0.552704 + 0.957311i 0
591.6 0 −0.784482 − 0.452921i 0 0.500000 + 0.866025i 0 1.23347 − 2.34063i 0 −1.08973 − 1.88746i 0
591.7 0 −0.725648 − 0.418953i 0 0.500000 + 0.866025i 0 −2.36913 − 1.17781i 0 −1.14896 − 1.99005i 0
591.8 0 0.219454 + 0.126702i 0 0.500000 + 0.866025i 0 0.978876 + 2.45801i 0 −1.46789 − 2.54247i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.bz.f 24
4.b odd 2 1 280.2.bj.e ✓ 24
7.d odd 6 1 1120.2.bz.e 24
8.b even 2 1 280.2.bj.f yes 24
8.d odd 2 1 1120.2.bz.e 24
28.f even 6 1 280.2.bj.f yes 24
56.j odd 6 1 280.2.bj.e ✓ 24
56.m even 6 1 inner 1120.2.bz.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bj.e ✓ 24 4.b odd 2 1
280.2.bj.e ✓ 24 56.j odd 6 1
280.2.bj.f yes 24 8.b even 2 1
280.2.bj.f yes 24 28.f even 6 1
1120.2.bz.e 24 7.d odd 6 1
1120.2.bz.e 24 8.d odd 2 1
1120.2.bz.f 24 1.a even 1 1 trivial
1120.2.bz.f 24 56.m even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\):

\( T_{3}^{24} + 12 T_{3}^{23} + 48 T_{3}^{22} - 494 T_{3}^{20} - 492 T_{3}^{19} + 5528 T_{3}^{18} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{13}^{12} + 10 T_{13}^{11} - 34 T_{13}^{10} - 512 T_{13}^{9} + 221 T_{13}^{8} + 9034 T_{13}^{7} + \cdots + 81088 \) Copy content Toggle raw display