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Properties

Label 1148.2.i.e
Level $1148$
Weight $2$
Character orbit 1148.i
Analytic conductor $9.167$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30q + q^{3} - 3q^{5} + 3q^{7} - 30q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q + q^{3} - 3q^{5} + 3q^{7} - 30q^{9} - 9q^{11} + 14q^{13} + 4q^{15} - 3q^{17} - 7q^{19} - 3q^{21} + q^{23} - 32q^{25} + 22q^{27} + 36q^{29} - 30q^{31} + 16q^{33} - 47q^{35} - 23q^{37} - 5q^{39} - 30q^{41} + 24q^{43} + 13q^{45} + 16q^{47} - 31q^{49} - 29q^{51} - 33q^{53} + 74q^{55} + 32q^{57} + 10q^{59} - q^{61} - 75q^{63} - 16q^{65} - 20q^{67} + 42q^{69} + 10q^{71} + 3q^{73} + 51q^{75} - 15q^{77} - 25q^{79} - 43q^{81} + 36q^{83} + 72q^{85} + 53q^{87} + 11q^{89} - 41q^{91} - 65q^{93} + 30q^{95} + 32q^{97} - 36q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 0 −1.62107 + 2.80777i 0 −0.748394 − 1.29626i 0 2.53169 + 0.768468i 0 −3.75571 − 6.50509i 0
165.2 0 −1.45317 + 2.51697i 0 −0.253051 − 0.438298i 0 −2.53278 − 0.764863i 0 −2.72341 − 4.71709i 0
165.3 0 −1.36871 + 2.37067i 0 0.431527 + 0.747427i 0 −2.57022 + 0.627666i 0 −2.24672 − 3.89144i 0
165.4 0 −1.23867 + 2.14545i 0 1.74922 + 3.02973i 0 1.19623 − 2.35988i 0 −1.56862 − 2.71694i 0
165.5 0 −1.05884 + 1.83397i 0 −1.96828 − 3.40916i 0 1.39753 − 2.24653i 0 −0.742303 − 1.28571i 0
165.6 0 −0.370691 + 0.642056i 0 1.24331 + 2.15348i 0 0.670716 + 2.55932i 0 1.22518 + 2.12207i 0
165.7 0 −0.224795 + 0.389356i 0 0.393901 + 0.682257i 0 1.75718 + 1.97796i 0 1.39893 + 2.42303i 0
165.8 0 0.257541 − 0.446074i 0 −1.09982 − 1.90494i 0 −2.48493 + 0.908374i 0 1.36735 + 2.36831i 0
165.9 0 0.637220 − 1.10370i 0 −2.18714 − 3.78824i 0 1.89439 + 1.84696i 0 0.687901 + 1.19148i 0
165.10 0 0.702089 − 1.21605i 0 −1.80784 − 3.13127i 0 0.606375 − 2.57533i 0 0.514141 + 0.890519i 0
165.11 0 0.864978 − 1.49819i 0 0.0691753 + 0.119815i 0 0.329472 − 2.62516i 0 0.00362599 + 0.00628040i 0
165.12 0 0.966125 − 1.67338i 0 0.237733 + 0.411765i 0 −1.29492 + 2.30720i 0 −0.366793 − 0.635305i 0
165.13 0 1.14590 − 1.98477i 0 1.96151 + 3.39744i 0 −0.412096 + 2.61346i 0 −1.12620 − 1.95063i 0
165.14 0 1.55023 − 2.68508i 0 1.65684 + 2.86974i 0 −1.55882 − 2.13778i 0 −3.30645 − 5.72694i 0
165.15 0 1.71186 − 2.96502i 0 −1.17870 − 2.04157i 0 1.97017 − 1.76591i 0 −4.36091 − 7.55332i 0
821.1 0 −1.62107 − 2.80777i 0 −0.748394 + 1.29626i 0 2.53169 − 0.768468i 0 −3.75571 + 6.50509i 0
821.2 0 −1.45317 − 2.51697i 0 −0.253051 + 0.438298i 0 −2.53278 + 0.764863i 0 −2.72341 + 4.71709i 0
821.3 0 −1.36871 − 2.37067i 0 0.431527 − 0.747427i 0 −2.57022 − 0.627666i 0 −2.24672 + 3.89144i 0
821.4 0 −1.23867 − 2.14545i 0 1.74922 − 3.02973i 0 1.19623 + 2.35988i 0 −1.56862 + 2.71694i 0
821.5 0 −1.05884 − 1.83397i 0 −1.96828 + 3.40916i 0 1.39753 + 2.24653i 0 −0.742303 + 1.28571i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 821.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.e ✓ 30
7.c even 3 1 inner 1148.2.i.e ✓ 30
7.c even 3 1 8036.2.a.q 15
7.d odd 6 1 8036.2.a.r 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.e ✓ 30 1.a even 1 1 trivial
1148.2.i.e ✓ 30 7.c even 3 1 inner
8036.2.a.q 15 7.c even 3 1
8036.2.a.r 15 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\):

\(T_{3}^{30} - \cdots\)
\(16\!\cdots\!66\)\( T_{11}^{14} + \)\(35\!\cdots\!24\)\( T_{11}^{13} + \)\(93\!\cdots\!08\)\( T_{11}^{12} + \)\(14\!\cdots\!70\)\( T_{11}^{11} + \)\(34\!\cdots\!42\)\( T_{11}^{10} + \)\(28\!\cdots\!75\)\( T_{11}^{9} + \)\(59\!\cdots\!84\)\( T_{11}^{8} - \)\(70\!\cdots\!83\)\( T_{11}^{7} + \)\(85\!\cdots\!27\)\( T_{11}^{6} - \)\(15\!\cdots\!25\)\( T_{11}^{5} + \)\(36\!\cdots\!63\)\( T_{11}^{4} - \)\(36\!\cdots\!28\)\( T_{11}^{3} + \)\(11\!\cdots\!07\)\( T_{11}^{2} - \)\(15\!\cdots\!02\)\( T_{11} + 220991189409 \)">\(T_{11}^{30} + \cdots\)