Properties

Label 1339.2.a.e
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $21$
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: \(1\)
Dimension: \(21\)
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.

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

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 −2.50015 −2.63774 4.25075 −1.17684 6.59474 4.82219 −5.62720 3.95767 2.94227
1.2 −2.31499 1.57292 3.35919 −1.84365 −3.64130 3.43971 −3.14652 −0.525924 4.26804
1.3 −2.12469 0.347667 2.51433 −4.40399 −0.738685 −0.0796229 −1.09279 −2.87913 9.35713
1.4 −1.94515 1.90358 1.78363 1.71445 −3.70276 −2.09421 0.420881 0.623614 −3.33487
1.5 −1.91337 −1.36048 1.66097 1.75251 2.60311 −4.42851 0.648686 −1.14908 −3.35319
1.6 −1.47455 −1.76266 0.174289 0.466891 2.59913 −0.582262 2.69210 0.106981 −0.688453
1.7 −1.10076 0.263891 −0.788329 3.16171 −0.290481 1.73698 3.06928 −2.93036 −3.48028
1.8 −0.791299 0.744814 −1.37385 −0.897428 −0.589370 −1.66332 2.66972 −2.44525 0.710134
1.9 −0.784481 −1.61992 −1.38459 −4.15260 1.27079 4.61578 2.65515 −0.375870 3.25764
1.10 −0.615393 2.88796 −1.62129 −1.40822 −1.77723 −0.315755 2.22852 5.34034 0.866607
1.11 −0.0264721 −1.87769 −1.99930 2.52751 0.0497063 −1.16160 0.105870 0.525708 −0.0669085
1.12 0.106842 −3.24890 −1.98858 −2.21706 −0.347120 −0.711342 −0.426149 7.55538 −0.236876
1.13 0.267251 2.13493 −1.92858 −0.961056 0.570561 −2.53524 −1.04992 1.55791 −0.256843
1.14 0.499344 0.601343 −1.75066 −1.13095 0.300277 3.15035 −1.87287 −2.63839 −0.564734
1.15 1.33589 2.00110 −0.215389 −3.88298 2.67326 2.54624 −2.95952 1.00441 −5.18725
1.16 1.40414 1.06219 −0.0283926 0.529418 1.49147 −2.26187 −2.84815 −1.87175 0.743377
1.17 1.85697 −2.04069 1.44833 0.337566 −3.78949 −0.868388 −1.02443 1.16441 0.626849
1.18 2.08679 −1.93108 2.35469 1.87136 −4.02977 −0.548741 0.740165 0.729089 3.90513
1.19 2.21898 0.899770 2.92386 −2.41101 1.99657 −4.38690 2.05003 −2.19041 −5.34997
1.20 2.31727 −3.32603 3.36976 −2.87188 −7.70732 2.40638 3.17410 8.06248 −6.65492
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.e 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.e 21 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\(T_{2}^{21} + \cdots\)
\(T_{3}^{21} + \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database