Properties

Label 2013.4.a.f
Level 2013
Weight 4
Character orbit 2013.a
Self dual yes
Analytic conductor 118.771
Analytic rank 0
Dimension 38
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: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38q + 14q^{2} + 114q^{3} + 170q^{4} + 35q^{5} + 42q^{6} + 105q^{7} + 147q^{8} + 342q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 38q + 14q^{2} + 114q^{3} + 170q^{4} + 35q^{5} + 42q^{6} + 105q^{7} + 147q^{8} + 342q^{9} + 99q^{10} + 418q^{11} + 510q^{12} + 209q^{13} + 128q^{14} + 105q^{15} + 798q^{16} + 512q^{17} + 126q^{18} + 487q^{19} + 328q^{20} + 315q^{21} + 154q^{22} + 417q^{23} + 441q^{24} + 925q^{25} + 177q^{26} + 1026q^{27} + 902q^{28} + 626q^{29} + 297q^{30} + 300q^{31} + 1625q^{32} + 1254q^{33} - 180q^{34} + 1086q^{35} + 1530q^{36} + 554q^{37} + 845q^{38} + 627q^{39} + 329q^{40} + 1378q^{41} + 384q^{42} + 1979q^{43} + 1870q^{44} + 315q^{45} + 937q^{46} + 1345q^{47} + 2394q^{48} + 2635q^{49} + 800q^{50} + 1536q^{51} + 2006q^{52} + 1497q^{53} + 378q^{54} + 385q^{55} + 415q^{56} + 1461q^{57} + 1241q^{58} + 2827q^{59} + 984q^{60} - 2318q^{61} + 509q^{62} + 945q^{63} + 1003q^{64} + 2810q^{65} + 462q^{66} + 369q^{67} + 3936q^{68} + 1251q^{69} + 922q^{70} + 965q^{71} + 1323q^{72} + 3081q^{73} + 722q^{74} + 2775q^{75} + 2210q^{76} + 1155q^{77} + 531q^{78} + 3795q^{79} + 3793q^{80} + 3078q^{81} - 1678q^{82} + 3869q^{83} + 2706q^{84} + 3553q^{85} + 3305q^{86} + 1878q^{87} + 1617q^{88} + 2849q^{89} + 891q^{90} + 1252q^{91} + 4519q^{92} + 900q^{93} + 340q^{94} + 1504q^{95} + 4875q^{96} + 2562q^{97} + 6164q^{98} + 3762q^{99} + O(q^{100}) \)

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 −5.20141 3.00000 19.0547 −1.81138 −15.6042 3.78751 −57.5001 9.00000 9.42176
1.2 −5.11974 3.00000 18.2117 −8.33272 −15.3592 −0.123891 −52.2812 9.00000 42.6613
1.3 −4.92345 3.00000 16.2404 19.7094 −14.7704 4.38553 −40.5713 9.00000 −97.0385
1.4 −4.79323 3.00000 14.9751 12.3054 −14.3797 26.5090 −33.4332 9.00000 −58.9827
1.5 −4.44696 3.00000 11.7755 −6.52314 −13.3409 −19.5475 −16.7894 9.00000 29.0081
1.6 −4.08981 3.00000 8.72654 −14.1681 −12.2694 −18.6667 −2.97140 9.00000 57.9450
1.7 −4.00574 3.00000 8.04594 −13.8167 −12.0172 32.5179 −0.184008 9.00000 55.3460
1.8 −3.50711 3.00000 4.29983 10.2444 −10.5213 1.41773 12.9769 9.00000 −35.9282
1.9 −3.25945 3.00000 2.62403 10.3368 −9.77836 −26.6613 17.5227 9.00000 −33.6924
1.10 −2.86674 3.00000 0.218214 1.45116 −8.60023 27.7506 22.3084 9.00000 −4.16011
1.11 −2.38315 3.00000 −2.32059 14.5269 −7.14945 22.4384 24.5955 9.00000 −34.6198
1.12 −2.31501 3.00000 −2.64074 −15.6387 −6.94502 −20.9070 24.6334 9.00000 36.2036
1.13 −1.93536 3.00000 −4.25439 −18.7593 −5.80607 6.86594 23.7166 9.00000 36.3060
1.14 −1.15474 3.00000 −6.66657 −7.99519 −3.46423 12.3786 16.9361 9.00000 9.23239
1.15 −0.933922 3.00000 −7.12779 16.8720 −2.80177 −20.1785 14.1282 9.00000 −15.7571
1.16 −0.740360 3.00000 −7.45187 7.63307 −2.22108 1.52940 11.4399 9.00000 −5.65122
1.17 −0.477206 3.00000 −7.77227 5.50183 −1.43162 −27.2643 7.52663 9.00000 −2.62551
1.18 −0.0387605 3.00000 −7.99850 −3.27851 −0.116282 −8.05520 0.620110 9.00000 0.127077
1.19 0.801460 3.00000 −7.35766 19.8509 2.40438 26.4285 −12.3086 9.00000 15.9097
1.20 0.812552 3.00000 −7.33976 −6.24435 2.43766 23.9419 −12.4644 9.00000 −5.07386
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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.f 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.f 38 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database