Properties

Label 2013.4.a.b
Level 2013
Weight 4
Character orbit 2013.a
Self dual yes
Analytic conductor 118.771
Analytic rank 1
Dimension 36
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(118.770844842\)
Analytic rank: \(1\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36q + 2q^{2} - 108q^{3} + 118q^{4} - 5q^{5} - 6q^{6} - 63q^{7} + 3q^{8} + 324q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 2q^{2} - 108q^{3} + 118q^{4} - 5q^{5} - 6q^{6} - 63q^{7} + 3q^{8} + 324q^{9} - 45q^{10} - 396q^{11} - 354q^{12} - 13q^{13} + 82q^{14} + 15q^{15} + 262q^{16} + 204q^{17} + 18q^{18} - 431q^{19} + 354q^{20} + 189q^{21} - 22q^{22} - 179q^{23} - 9q^{24} + 711q^{25} + 331q^{26} - 972q^{27} - 296q^{28} + 478q^{29} + 135q^{30} - 574q^{31} - 149q^{32} + 1188q^{33} + 276q^{34} - 194q^{35} + 1062q^{36} - 12q^{37} + 325q^{38} + 39q^{39} - 185q^{40} + 900q^{41} - 246q^{42} - 1053q^{43} - 1298q^{44} - 45q^{45} - 407q^{46} - 653q^{47} - 786q^{48} + 753q^{49} - 1520q^{50} - 612q^{51} + 60q^{52} + 735q^{53} - 54q^{54} + 55q^{55} - 809q^{56} + 1293q^{57} - 1399q^{58} - 1127q^{59} - 1062q^{60} + 2196q^{61} - 1795q^{62} - 567q^{63} - 2133q^{64} + 1886q^{65} + 66q^{66} - 989q^{67} + 10q^{68} + 537q^{69} - 2130q^{70} + 61q^{71} + 27q^{72} - 1471q^{73} - 122q^{74} - 2133q^{75} - 4064q^{76} + 693q^{77} - 993q^{78} - 1853q^{79} + 2197q^{80} + 2916q^{81} - 2566q^{82} - 3523q^{83} + 888q^{84} - 449q^{85} - 771q^{86} - 1434q^{87} - 33q^{88} + 2209q^{89} - 405q^{90} - 1668q^{91} - 1999q^{92} + 1722q^{93} - 2844q^{94} + 1220q^{95} + 447q^{96} - 3622q^{97} + 3846q^{98} - 3564q^{99} + O(q^{100}) \)

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 −5.32772 −3.00000 20.3846 9.51582 15.9832 0.376884 −65.9820 9.00000 −50.6977
1.2 −4.97623 −3.00000 16.7629 −4.44765 14.9287 −23.2908 −43.6062 9.00000 22.1326
1.3 −4.89062 −3.00000 15.9181 15.5621 14.6718 16.9638 −38.7245 9.00000 −76.1085
1.4 −4.67500 −3.00000 13.8556 −8.51569 14.0250 7.22973 −27.3750 9.00000 39.8109
1.5 −4.58366 −3.00000 13.0099 −16.2221 13.7510 3.97810 −22.9636 9.00000 74.3567
1.6 −4.33380 −3.00000 10.7818 18.1421 13.0014 −4.33011 −12.0559 9.00000 −78.6243
1.7 −3.49679 −3.00000 4.22755 −13.5195 10.4904 −10.1391 13.1915 9.00000 47.2749
1.8 −3.42477 −3.00000 3.72902 −0.330135 10.2743 −34.8727 14.6271 9.00000 1.13064
1.9 −3.21541 −3.00000 2.33885 2.50702 9.64623 5.55272 18.2029 9.00000 −8.06108
1.10 −2.49518 −3.00000 −1.77407 21.0229 7.48554 −29.8474 24.3881 9.00000 −52.4560
1.11 −2.20148 −3.00000 −3.15347 −19.8720 6.60445 −18.3617 24.5542 9.00000 43.7479
1.12 −2.13143 −3.00000 −3.45699 −8.60507 6.39430 26.5144 24.4198 9.00000 18.3411
1.13 −2.00969 −3.00000 −3.96116 6.20179 6.02906 18.8550 24.0382 9.00000 −12.4637
1.14 −1.80221 −3.00000 −4.75204 13.7156 5.40663 7.13162 22.9818 9.00000 −24.7184
1.15 −1.71127 −3.00000 −5.07156 −5.95743 5.13381 −9.43303 22.3690 9.00000 10.1948
1.16 −0.691390 −3.00000 −7.52198 −5.77110 2.07417 9.57605 10.7317 9.00000 3.99008
1.17 −0.158900 −3.00000 −7.97475 7.59417 0.476701 −16.2139 2.53840 9.00000 −1.20672
1.18 0.0588759 −3.00000 −7.99653 −14.4051 −0.176628 36.0147 −0.941811 9.00000 −0.848116
1.19 0.402086 −3.00000 −7.83833 4.05035 −1.20626 4.43625 −6.36837 9.00000 1.62859
1.20 0.857145 −3.00000 −7.26530 −0.0808778 −2.57143 −28.7264 −13.0846 9.00000 −0.0693240
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(-1\)

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 2013.4.a.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.b 36 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database