Properties

Label 1339.2.a.g
Level $1339$
Weight $2$
Character orbit 1339.a
Self dual yes
Analytic conductor $10.692$
Analytic rank $0$
Dimension $30$
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: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30q + 5q^{3} + 34q^{4} + 17q^{5} + 3q^{6} + 2q^{7} + 3q^{8} + 49q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q + 5q^{3} + 34q^{4} + 17q^{5} + 3q^{6} + 2q^{7} + 3q^{8} + 49q^{9} + 5q^{10} + q^{11} + 15q^{12} - 30q^{13} + 24q^{14} + 6q^{15} + 38q^{16} + 17q^{17} + 8q^{18} + 9q^{19} + 31q^{20} + 27q^{21} + 2q^{22} + 14q^{23} + 26q^{24} + 47q^{25} + 14q^{27} - 6q^{28} + 53q^{29} + 25q^{30} + 19q^{31} - 4q^{32} + q^{33} - 22q^{34} + 9q^{35} + 61q^{36} + 46q^{38} - 5q^{39} - 35q^{40} + 28q^{41} - 7q^{42} + 6q^{43} - 12q^{44} + 68q^{45} + 7q^{46} + 12q^{47} + 13q^{48} + 54q^{49} - 18q^{50} + 10q^{51} - 34q^{52} + 37q^{53} + 22q^{54} + 11q^{55} + 67q^{56} - 57q^{57} - 5q^{58} + 61q^{59} - 102q^{60} + 16q^{61} - 2q^{62} - 7q^{63} + 29q^{64} - 17q^{65} - 83q^{66} - 2q^{67} + 57q^{68} + 98q^{69} - 10q^{70} + 50q^{71} - 8q^{72} - 10q^{73} - 13q^{74} + 5q^{75} - 19q^{76} + 54q^{77} - 3q^{78} + 3q^{79} + 76q^{80} + 118q^{81} + 7q^{82} + 6q^{83} + 44q^{84} + 33q^{85} - 29q^{86} - 10q^{87} - 13q^{88} + 77q^{89} + 38q^{90} - 2q^{91} - 3q^{92} + 34q^{93} - 25q^{94} + 24q^{95} - 28q^{96} + 12q^{97} - 14q^{98} - 58q^{99} + O(q^{100}) \)

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 −2.71030 −2.41604 5.34571 3.72865 6.54819 −2.88700 −9.06786 2.83725 −10.1057
1.2 −2.70794 2.09535 5.33295 −1.11757 −5.67408 −3.78835 −9.02545 1.39048 3.02631
1.3 −2.44933 −0.0495252 3.99920 1.48835 0.121303 2.04936 −4.89669 −2.99755 −3.64544
1.4 −2.25031 3.33769 3.06389 0.684997 −7.51084 1.16005 −2.39409 8.14018 −1.54146
1.5 −2.18260 −0.879021 2.76374 3.84044 1.91855 1.42534 −1.66693 −2.22732 −8.38214
1.6 −2.15890 −0.0701034 2.66085 −2.25465 0.151346 −1.38084 −1.42671 −2.99509 4.86756
1.7 −1.71336 −2.73021 0.935608 −2.57910 4.67784 0.316675 1.82369 4.45405 4.41893
1.8 −1.52011 −3.31256 0.310729 0.643321 5.03545 −3.98605 2.56787 7.97304 −0.977918
1.9 −1.43550 2.33343 0.0606568 2.88716 −3.34964 3.63915 2.78392 2.44491 −4.14452
1.10 −1.25596 2.77119 −0.422559 −3.18202 −3.48051 3.63505 3.04264 4.67947 3.99650
1.11 −1.13041 1.24412 −0.722174 −3.15829 −1.40637 −4.57377 3.07717 −1.45216 3.57016
1.12 −0.736639 2.43572 −1.45736 4.11528 −1.79424 −5.11924 2.54683 2.93271 −3.03147
1.13 −0.660355 0.482681 −1.56393 −1.20794 −0.318741 3.96472 2.35346 −2.76702 0.797669
1.14 −0.496151 −2.36213 −1.75383 0.249222 1.17197 2.50523 1.86247 2.57965 −0.123652
1.15 −0.319351 0.543955 −1.89802 1.90715 −0.173713 0.747001 1.24483 −2.70411 −0.609049
1.16 0.397347 −1.55160 −1.84212 0.426444 −0.616523 −5.04601 −1.52665 −0.592540 0.169446
1.17 0.430342 −0.376968 −1.81481 −3.33541 −0.162225 −1.83693 −1.64167 −2.85790 −1.43537
1.18 0.508360 −0.408426 −1.74157 4.07667 −0.207627 0.983456 −1.90207 −2.83319 2.07242
1.19 0.903670 2.25573 −1.18338 1.13079 2.03843 2.22819 −2.87673 2.08831 1.02186
1.20 0.947942 3.31104 −1.10141 4.30053 3.13867 1.61201 −2.93995 7.96298 4.07665
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
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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.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.g 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):

\(T_{2}^{30} - \cdots\)
\(T_{3}^{30} - \cdots\)

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database