Properties

Label 2013.4.a.c
Level 2013
Weight 4
Character orbit 2013.a
Self dual yes
Analytic conductor 118.771
Analytic rank 1
Dimension 37
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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37q - 8q^{2} + 111q^{3} + 130q^{4} - 35q^{5} - 24q^{6} - 35q^{7} - 117q^{8} + 333q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 37q - 8q^{2} + 111q^{3} + 130q^{4} - 35q^{5} - 24q^{6} - 35q^{7} - 117q^{8} + 333q^{9} - 41q^{10} - 407q^{11} + 390q^{12} + 51q^{13} - 228q^{14} - 105q^{15} + 462q^{16} - 190q^{17} - 72q^{18} - 51q^{19} - 720q^{20} - 105q^{21} + 88q^{22} - 583q^{23} - 351q^{24} + 598q^{25} - 1019q^{26} + 999q^{27} - 498q^{28} - 566q^{29} - 123q^{30} - 696q^{31} - 859q^{32} - 1221q^{33} - 348q^{34} - 1102q^{35} + 1170q^{36} - 1022q^{37} - 455q^{38} + 153q^{39} - 503q^{40} - 790q^{41} - 684q^{42} - 87q^{43} - 1430q^{44} - 315q^{45} - 303q^{46} - 1603q^{47} + 1386q^{48} + 110q^{49} - 1926q^{50} - 570q^{51} + 736q^{52} - 2619q^{53} - 216q^{54} + 385q^{55} - 4937q^{56} - 153q^{57} - 1099q^{58} - 2471q^{59} - 2160q^{60} - 2257q^{61} - 2909q^{62} - 315q^{63} - 265q^{64} - 1970q^{65} + 264q^{66} - 3033q^{67} - 1956q^{68} - 1749q^{69} + 2410q^{70} - 3891q^{71} - 1053q^{72} + 391q^{73} - 532q^{74} + 1794q^{75} + 1554q^{76} + 385q^{77} - 3057q^{78} + 67q^{79} - 5111q^{80} + 2997q^{81} - 4818q^{82} - 5315q^{83} - 1494q^{84} - 2747q^{85} - 5195q^{86} - 1698q^{87} + 1287q^{88} - 8945q^{89} - 369q^{90} - 4432q^{91} - 4701q^{92} - 2088q^{93} - 372q^{94} - 3388q^{95} - 2577q^{96} - 3784q^{97} - 4502q^{98} - 3663q^{99} + O(q^{100}) \)

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 −5.41306 3.00000 21.3012 −13.4657 −16.2392 11.1590 −72.0002 9.00000 72.8907
1.2 −5.39202 3.00000 21.0739 2.00966 −16.1761 −20.9762 −70.4947 9.00000 −10.8361
1.3 −5.14443 3.00000 18.4652 9.21599 −15.4333 10.6441 −53.8373 9.00000 −47.4110
1.4 −5.05961 3.00000 17.5996 −15.9011 −15.1788 34.5776 −48.5703 9.00000 80.4535
1.5 −4.48960 3.00000 12.1565 5.33009 −13.4688 16.1084 −18.6612 9.00000 −23.9300
1.6 −4.28216 3.00000 10.3369 12.4880 −12.8465 −32.6569 −10.0068 9.00000 −53.4757
1.7 −4.20330 3.00000 9.66773 −8.42122 −12.6099 −13.3084 −7.00999 9.00000 35.3969
1.8 −3.67739 3.00000 5.52321 10.7113 −11.0322 −9.56351 9.10812 9.00000 −39.3898
1.9 −3.61193 3.00000 5.04607 −17.1653 −10.8358 13.1780 10.6694 9.00000 61.9998
1.10 −3.59523 3.00000 4.92567 14.8330 −10.7857 14.0315 11.0529 9.00000 −53.3279
1.11 −3.15750 3.00000 1.96979 −18.8998 −9.47249 −24.1141 19.0404 9.00000 59.6762
1.12 −2.45234 3.00000 −1.98602 −1.35025 −7.35703 −17.8806 24.4891 9.00000 3.31128
1.13 −2.43775 3.00000 −2.05739 −9.35530 −7.31324 19.3013 24.5174 9.00000 22.8059
1.14 −1.88493 3.00000 −4.44705 7.51572 −5.65478 1.38168 23.4618 9.00000 −14.1666
1.15 −1.41755 3.00000 −5.99056 8.60687 −4.25264 2.21724 19.8323 9.00000 −12.2006
1.16 −0.951961 3.00000 −7.09377 20.9675 −2.85588 2.74716 14.3687 9.00000 −19.9602
1.17 −0.624353 3.00000 −7.61018 −20.5187 −1.87306 −20.9935 9.74626 9.00000 12.8109
1.18 −0.476963 3.00000 −7.77251 17.0373 −1.43089 −27.8002 7.52291 9.00000 −8.12615
1.19 −0.169683 3.00000 −7.97121 0.479910 −0.509049 32.7035 2.71004 9.00000 −0.0814325
1.20 −0.108103 3.00000 −7.98831 −6.73266 −0.324309 −18.3209 1.72838 9.00000 0.727821
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
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.c 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.c 37 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database