Properties

Label 2013.4.a.a
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

Related objects

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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.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 14q^{2} + 108q^{3} + 146q^{4} - 65q^{5} - 42q^{6} - 105q^{7} - 189q^{8} + 324q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 14q^{2} + 108q^{3} + 146q^{4} - 65q^{5} - 42q^{6} - 105q^{7} - 189q^{8} + 324q^{9} - 161q^{10} + 396q^{11} + 438q^{12} - 233q^{13} - 264q^{14} - 195q^{15} + 574q^{16} - 556q^{17} - 126q^{18} - 615q^{19} - 136q^{20} - 315q^{21} - 154q^{22} - 457q^{23} - 567q^{24} + 863q^{25} + 115q^{26} + 972q^{27} - 424q^{28} - 754q^{29} - 483q^{30} - 508q^{31} - 1511q^{32} + 1188q^{33} - 860q^{34} - 826q^{35} + 1314q^{36} - 412q^{37} - 599q^{38} - 699q^{39} - 2791q^{40} - 2066q^{41} - 792q^{42} - 2063q^{43} + 1606q^{44} - 585q^{45} - 787q^{46} - 1815q^{47} + 1722q^{48} + 2825q^{49} + 808q^{50} - 1668q^{51} - 2882q^{52} - 759q^{53} - 378q^{54} - 715q^{55} - 1749q^{56} - 1845q^{57} - 335q^{58} - 2337q^{59} - 408q^{60} + 2196q^{61} - 1689q^{62} - 945q^{63} + 4723q^{64} - 3550q^{65} - 462q^{66} - 1331q^{67} - 6166q^{68} - 1371q^{69} - 1750q^{70} - 361q^{71} - 1701q^{72} - 4627q^{73} - 3394q^{74} + 2589q^{75} - 7214q^{76} - 1155q^{77} + 345q^{78} - 2583q^{79} - 2643q^{80} + 2916q^{81} + 1090q^{82} - 6123q^{83} - 1272q^{84} + 295q^{85} + 613q^{86} - 2262q^{87} - 2079q^{88} - 2485q^{89} - 1449q^{90} - 3156q^{91} - 6291q^{92} - 1524q^{93} - 1744q^{94} - 5572q^{95} - 4533q^{96} - 2558q^{97} - 2314q^{98} + 3564q^{99} + O(q^{100}) \)

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 −5.61297 3.00000 23.5055 13.7147 −16.8389 34.2924 −87.0318 9.00000 −76.9805
1.2 −5.53692 3.00000 22.6575 4.34863 −16.6108 −33.2757 −81.1573 9.00000 −24.0780
1.3 −5.51047 3.00000 22.3653 −14.2389 −16.5314 3.43977 −79.1594 9.00000 78.4630
1.4 −4.75663 3.00000 14.6255 7.60303 −14.2699 −25.1377 −31.5151 9.00000 −36.1648
1.5 −4.51714 3.00000 12.4046 19.2722 −13.5514 −8.39412 −19.8961 9.00000 −87.0554
1.6 −4.38409 3.00000 11.2202 −10.2931 −13.1523 22.1567 −14.1179 9.00000 45.1257
1.7 −4.37392 3.00000 11.1312 9.50672 −13.1218 −4.68752 −13.6956 9.00000 −41.5816
1.8 −4.25496 3.00000 10.1047 −22.0823 −12.7649 8.60879 −8.95532 9.00000 93.9591
1.9 −3.72734 3.00000 5.89305 10.0247 −11.1820 19.7213 7.85332 9.00000 −37.3656
1.10 −3.33494 3.00000 3.12184 0.253130 −10.0048 −0.691781 16.2684 9.00000 −0.844174
1.11 −3.06792 3.00000 1.41213 −10.3390 −9.20376 −32.3876 20.2111 9.00000 31.7193
1.12 −3.06030 3.00000 1.36541 −15.4248 −9.18089 12.8287 20.3038 9.00000 47.2044
1.13 −3.05325 3.00000 1.32235 −6.59737 −9.15976 −7.71882 20.3886 9.00000 20.1434
1.14 −1.93393 3.00000 −4.25991 7.94786 −5.80179 −25.7974 23.7098 9.00000 −15.3706
1.15 −1.70047 3.00000 −5.10841 −3.95974 −5.10140 33.3215 22.2904 9.00000 6.73341
1.16 −1.31102 3.00000 −6.28122 5.66783 −3.93306 −0.126112 18.7230 9.00000 −7.43064
1.17 −1.11376 3.00000 −6.75954 −13.8886 −3.34127 −3.42861 16.4385 9.00000 15.4685
1.18 −0.942036 3.00000 −7.11257 1.29514 −2.82611 21.4949 14.2366 9.00000 −1.22007
1.19 −0.121017 3.00000 −7.98535 12.2818 −0.363050 1.74252 1.93450 9.00000 −1.48630
1.20 0.439845 3.00000 −7.80654 −18.8775 1.31954 −35.5540 −6.95243 9.00000 −8.30317
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.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.a 36 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database