Properties

Label 1148.2.r.a
Level $1148$
Weight $2$
Character orbit 1148.r
Analytic conductor $9.167$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 56q + 4q^{5} + 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q + 4q^{5} + 32q^{9} - 6q^{21} + 2q^{23} - 24q^{25} - 4q^{31} + 10q^{33} + 10q^{37} + 10q^{39} + 20q^{41} + 8q^{43} - 22q^{45} - 4q^{49} + 18q^{51} + 28q^{57} - 16q^{59} + 16q^{61} + 2q^{73} + 34q^{77} - 28q^{81} - 48q^{83} - 2q^{87} - 42q^{91} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −2.89464 1.67122i 0 1.42784 + 2.47308i 0 −1.36322 + 2.26752i 0 4.08596 + 7.07710i 0
81.2 0 −2.68013 1.54737i 0 0.0658562 + 0.114066i 0 −0.113033 2.64334i 0 3.28873 + 5.69624i 0
81.3 0 −2.67190 1.54262i 0 −1.10096 1.90691i 0 2.60761 + 0.447649i 0 3.25937 + 5.64540i 0
81.4 0 −1.90271 1.09853i 0 −1.32827 2.30063i 0 −2.55338 0.693005i 0 0.913540 + 1.58230i 0
81.5 0 −1.85927 1.07345i 0 −0.972000 1.68355i 0 −1.16501 + 2.37545i 0 0.804601 + 1.39361i 0
81.6 0 −1.79608 1.03697i 0 −0.904853 1.56725i 0 2.10654 1.60077i 0 0.650613 + 1.12689i 0
81.7 0 −1.64009 0.946908i 0 1.83233 + 3.17368i 0 0.288265 2.63000i 0 0.293269 + 0.507958i 0
81.8 0 −1.59496 0.920850i 0 1.70544 + 2.95391i 0 2.57642 0.601730i 0 0.195929 + 0.339358i 0
81.9 0 −1.48683 0.858424i 0 1.14679 + 1.98629i 0 −2.58388 + 0.568830i 0 −0.0262152 0.0454060i 0
81.10 0 −1.32068 0.762496i 0 0.729632 + 1.26376i 0 0.676296 + 2.55786i 0 −0.337200 0.584047i 0
81.11 0 −0.900937 0.520156i 0 −0.292521 0.506661i 0 −2.31637 1.27845i 0 −0.958875 1.66082i 0
81.12 0 −0.507835 0.293199i 0 1.03735 + 1.79674i 0 1.44101 + 2.21889i 0 −1.32807 2.30028i 0
81.13 0 −0.388672 0.224400i 0 −0.254464 0.440744i 0 2.49203 0.888703i 0 −1.39929 2.42364i 0
81.14 0 −0.294024 0.169755i 0 −2.09216 3.62373i 0 0.0873039 2.64431i 0 −1.44237 2.49825i 0
81.15 0 0.294024 + 0.169755i 0 −2.09216 3.62373i 0 −0.0873039 + 2.64431i 0 −1.44237 2.49825i 0
81.16 0 0.388672 + 0.224400i 0 −0.254464 0.440744i 0 −2.49203 + 0.888703i 0 −1.39929 2.42364i 0
81.17 0 0.507835 + 0.293199i 0 1.03735 + 1.79674i 0 −1.44101 2.21889i 0 −1.32807 2.30028i 0
81.18 0 0.900937 + 0.520156i 0 −0.292521 0.506661i 0 2.31637 + 1.27845i 0 −0.958875 1.66082i 0
81.19 0 1.32068 + 0.762496i 0 0.729632 + 1.26376i 0 −0.676296 2.55786i 0 −0.337200 0.584047i 0
81.20 0 1.48683 + 0.858424i 0 1.14679 + 1.98629i 0 2.58388 0.568830i 0 −0.0262152 0.0454060i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.b even 2 1 inner
287.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.r.a 56
7.c even 3 1 inner 1148.2.r.a 56
41.b even 2 1 inner 1148.2.r.a 56
287.j even 6 1 inner 1148.2.r.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.r.a 56 1.a even 1 1 trivial
1148.2.r.a 56 7.c even 3 1 inner
1148.2.r.a 56 41.b even 2 1 inner
1148.2.r.a 56 287.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1148, [\chi])\).