Properties

Label 1339.2.a.f
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6919688306\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28q + 6q^{2} + 5q^{3} + 32q^{4} + 27q^{5} + 9q^{6} + 6q^{7} + 21q^{8} + 39q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 6q^{2} + 5q^{3} + 32q^{4} + 27q^{5} + 9q^{6} + 6q^{7} + 21q^{8} + 39q^{9} + 5q^{10} + 9q^{11} + 15q^{12} + 28q^{13} + 20q^{14} + 4q^{15} + 32q^{16} - 7q^{17} + 10q^{18} - 3q^{19} + 53q^{20} + 25q^{21} - 14q^{22} - 6q^{23} + 10q^{24} + 45q^{25} + 6q^{26} + 8q^{27} - 12q^{28} + 25q^{29} - 43q^{30} + q^{31} + 28q^{32} + 17q^{33} + 10q^{34} + 11q^{35} + 27q^{36} + 18q^{37} + 2q^{38} + 5q^{39} + 37q^{40} + 56q^{41} - 29q^{42} - 30q^{43} + 30q^{44} + 38q^{45} - 27q^{46} + 40q^{47} + 19q^{48} + 36q^{49} + 20q^{50} - 18q^{51} + 32q^{52} + 51q^{53} - 36q^{54} - 5q^{55} - 13q^{56} + 31q^{57} - 29q^{58} + 29q^{59} + 48q^{60} + 16q^{61} - 38q^{62} - 23q^{63} - 5q^{64} + 27q^{65} + 25q^{66} - 2q^{67} - 43q^{68} + 38q^{69} + 10q^{70} + 34q^{71} - 28q^{72} + 14q^{73} + 21q^{74} + 19q^{75} + 5q^{76} + 34q^{77} + 9q^{78} + 3q^{79} + 58q^{80} + 4q^{81} - 9q^{82} + 38q^{83} - 78q^{84} - 27q^{85} + 49q^{86} + 8q^{87} - 25q^{88} + 125q^{89} - 74q^{90} + 6q^{91} - 35q^{92} + 36q^{93} - q^{94} - 12q^{95} + 26q^{96} - 2q^{97} + 18q^{98} + 56q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.50812 −2.94446 4.29069 −0.0167965 7.38506 −1.24099 −5.74532 5.66982 0.0421278
1.2 −2.50096 2.99924 4.25481 4.07294 −7.50098 −1.09357 −5.63921 5.99543 −10.1863
1.3 −2.35134 1.07411 3.52878 3.29713 −2.52560 3.84969 −3.59467 −1.84628 −7.75266
1.4 −2.21826 −0.565867 2.92070 −1.45314 1.25524 −4.80056 −2.04235 −2.67979 3.22344
1.5 −1.86443 −1.06486 1.47608 −1.98836 1.98536 3.02740 0.976806 −1.86606 3.70715
1.6 −1.73240 1.73806 1.00120 −1.26585 −3.01101 −1.31498 1.73032 0.0208578 2.19296
1.7 −1.61074 −0.965673 0.594496 4.40810 1.55545 −3.07617 2.26391 −2.06748 −7.10033
1.8 −1.08251 −0.806331 −0.828177 1.00289 0.872859 −2.77298 3.06152 −2.34983 −1.08564
1.9 −1.04956 2.85751 −0.898418 1.97843 −2.99913 2.56542 3.04207 5.16534 −2.07648
1.10 −0.779581 −1.24905 −1.39225 1.96235 0.973735 3.90578 2.64454 −1.43988 −1.52981
1.11 −0.656140 −3.17911 −1.56948 2.97924 2.08594 −2.69809 2.34208 7.10672 −1.95480
1.12 −0.413705 3.05093 −1.82885 0.683022 −1.26219 −1.51920 1.58402 6.30816 −0.282570
1.13 −0.175714 −2.57328 −1.96912 −0.872787 0.452162 1.26960 0.697432 3.62178 0.153361
1.14 0.203882 1.83284 −1.95843 3.17177 0.373682 2.89343 −0.807052 0.359299 0.646665
1.15 0.221198 1.18074 −1.95107 −3.72276 0.261177 −1.11234 −0.873968 −1.60585 −0.823467
1.16 0.612731 −0.480212 −1.62456 −0.793154 −0.294241 3.03298 −2.22088 −2.76940 −0.485990
1.17 0.835105 −2.05058 −1.30260 3.71011 −1.71245 −2.16987 −2.75802 1.20488 3.09833
1.18 0.980046 3.26733 −1.03951 −1.81818 3.20213 3.84583 −2.97886 7.67545 −1.78190
1.19 1.56395 −2.92532 0.445953 −2.35359 −4.57507 −3.99061 −2.43046 5.55751 −3.68091
1.20 1.80754 0.896394 1.26722 3.66177 1.62027 2.54556 −1.32454 −2.19648 6.61881
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(103\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1339.2.a.f 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.f 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\(T_{2}^{28} - \cdots\)
\(T_{3}^{28} - \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database