Properties

Label 6013.2.a.d
Level $6013$
Weight $2$
Character orbit 6013.a
Self dual yes
Analytic conductor $48.014$
Analytic rank $1$
Dimension $104$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6013 = 7 \cdot 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6013.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0140467354\)
Analytic rank: \(1\)
Dimension: \(104\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 104q - 17q^{2} - 34q^{3} + 99q^{4} - 46q^{5} - 18q^{6} + 104q^{7} - 48q^{8} + 98q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 104q - 17q^{2} - 34q^{3} + 99q^{4} - 46q^{5} - 18q^{6} + 104q^{7} - 48q^{8} + 98q^{9} - 17q^{10} - 46q^{11} - 62q^{12} - 37q^{13} - 17q^{14} - 17q^{15} + 93q^{16} - 75q^{17} - 41q^{18} - 40q^{19} - 106q^{20} - 34q^{21} - 14q^{22} - 57q^{23} - 38q^{24} + 102q^{25} - 45q^{26} - 121q^{27} + 99q^{28} - 29q^{29} + 26q^{30} - 42q^{31} - 113q^{32} - 42q^{33} - 7q^{34} - 46q^{35} + 99q^{36} - 31q^{37} - 62q^{38} - 9q^{39} - 36q^{40} - 106q^{41} - 18q^{42} - 29q^{43} - 60q^{44} - 121q^{45} + 21q^{46} - 141q^{47} - 88q^{48} + 104q^{49} - 70q^{50} - 2q^{51} - 58q^{52} - 70q^{53} - 49q^{54} - 14q^{55} - 48q^{56} - 11q^{57} - 28q^{58} - 202q^{59} + 17q^{60} - 79q^{61} - 41q^{62} + 98q^{63} + 110q^{64} - 34q^{65} - 21q^{66} - 57q^{67} - 180q^{68} - 99q^{69} - 17q^{70} - 109q^{71} - 73q^{72} - 46q^{73} - 7q^{74} - 146q^{75} - 54q^{76} - 46q^{77} - 42q^{78} - 12q^{79} - 187q^{80} + 100q^{81} - 45q^{82} - 156q^{83} - 62q^{84} - 13q^{85} - 8q^{86} - 76q^{87} - 6q^{88} - 140q^{89} - 5q^{90} - 37q^{91} - 98q^{92} - q^{93} - 17q^{94} - 48q^{95} - 12q^{96} - 63q^{97} - 17q^{98} - 78q^{99} + O(q^{100}) \)

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 −2.81212 2.27146 5.90799 −1.69764 −6.38761 1.00000 −10.9897 2.15954 4.77395
1.2 −2.80884 0.402205 5.88959 2.13491 −1.12973 1.00000 −10.9252 −2.83823 −5.99663
1.3 −2.74380 −2.93976 5.52843 −4.29158 8.06610 1.00000 −9.68130 5.64218 11.7752
1.4 −2.73548 −2.83530 5.48288 2.55962 7.75591 1.00000 −9.52736 5.03891 −7.00181
1.5 −2.73004 −0.655778 5.45311 −0.775046 1.79030 1.00000 −9.42711 −2.56996 2.11590
1.6 −2.67480 1.18584 5.15457 −2.80512 −3.17188 1.00000 −8.43786 −1.59379 7.50315
1.7 −2.63784 3.06244 4.95819 −3.52332 −8.07822 1.00000 −7.80322 6.37853 9.29393
1.8 −2.57571 −1.85963 4.63428 3.27062 4.78986 1.00000 −6.78514 0.458214 −8.42416
1.9 −2.50413 −2.14527 4.27066 −3.15595 5.37203 1.00000 −5.68604 1.60217 7.90290
1.10 −2.49952 2.02668 4.24760 1.29364 −5.06572 1.00000 −5.61792 1.10742 −3.23348
1.11 −2.42761 −1.54319 3.89329 −3.03492 3.74626 1.00000 −4.59617 −0.618564 7.36759
1.12 −2.40887 −1.48249 3.80264 1.73279 3.57112 1.00000 −4.34233 −0.802232 −4.17407
1.13 −2.35495 −3.13271 3.54579 −0.919184 7.37736 1.00000 −3.64026 6.81384 2.16463
1.14 −2.31949 1.94621 3.38001 1.57325 −4.51421 1.00000 −3.20092 0.787746 −3.64913
1.15 −2.30883 −0.594888 3.33072 −3.14300 1.37350 1.00000 −3.07241 −2.64611 7.25667
1.16 −2.23888 0.907516 3.01259 −2.38757 −2.03182 1.00000 −2.26707 −2.17641 5.34549
1.17 −2.13035 −2.00060 2.53838 1.16499 4.26197 1.00000 −1.14694 1.00240 −2.48184
1.18 −2.12186 −3.29762 2.50229 4.04257 6.99709 1.00000 −1.06579 7.87430 −8.57776
1.19 −2.10979 2.86366 2.45121 −1.33833 −6.04172 1.00000 −0.951963 5.20055 2.82359
1.20 −2.08457 2.65051 2.34543 3.08275 −5.52518 1.00000 −0.720080 4.02522 −6.42620
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.104
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(859\) \(-1\)

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 6013.2.a.d 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.d 104 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(24\!\cdots\!24\)\( T_{2}^{88} - \)\(10\!\cdots\!12\)\( T_{2}^{87} + \)\(21\!\cdots\!18\)\( T_{2}^{86} + \)\(14\!\cdots\!13\)\( T_{2}^{85} - \)\(13\!\cdots\!98\)\( T_{2}^{84} - \)\(15\!\cdots\!29\)\( T_{2}^{83} + \)\(27\!\cdots\!95\)\( T_{2}^{82} + \)\(14\!\cdots\!20\)\( T_{2}^{81} + \)\(62\!\cdots\!50\)\( T_{2}^{80} - \)\(10\!\cdots\!71\)\( T_{2}^{79} - \)\(10\!\cdots\!47\)\( T_{2}^{78} + \)\(72\!\cdots\!49\)\( T_{2}^{77} + \)\(10\!\cdots\!69\)\( T_{2}^{76} - \)\(40\!\cdots\!30\)\( T_{2}^{75} - \)\(81\!\cdots\!37\)\( T_{2}^{74} + \)\(19\!\cdots\!66\)\( T_{2}^{73} + \)\(51\!\cdots\!77\)\( T_{2}^{72} - \)\(75\!\cdots\!93\)\( T_{2}^{71} - \)\(27\!\cdots\!07\)\( T_{2}^{70} + \)\(23\!\cdots\!67\)\( T_{2}^{69} + \)\(12\!\cdots\!10\)\( T_{2}^{68} - \)\(52\!\cdots\!06\)\( T_{2}^{67} - \)\(50\!\cdots\!59\)\( T_{2}^{66} + \)\(27\!\cdots\!40\)\( T_{2}^{65} + \)\(17\!\cdots\!60\)\( T_{2}^{64} + \)\(43\!\cdots\!09\)\( T_{2}^{63} - \)\(54\!\cdots\!13\)\( T_{2}^{62} - \)\(28\!\cdots\!66\)\( T_{2}^{61} + \)\(14\!\cdots\!34\)\( T_{2}^{60} + \)\(11\!\cdots\!75\)\( T_{2}^{59} - \)\(34\!\cdots\!40\)\( T_{2}^{58} - \)\(35\!\cdots\!28\)\( T_{2}^{57} + \)\(69\!\cdots\!08\)\( T_{2}^{56} + \)\(94\!\cdots\!71\)\( T_{2}^{55} - \)\(12\!\cdots\!52\)\( T_{2}^{54} - \)\(21\!\cdots\!73\)\( T_{2}^{53} + \)\(18\!\cdots\!54\)\( T_{2}^{52} + \)\(41\!\cdots\!71\)\( T_{2}^{51} - \)\(22\!\cdots\!66\)\( T_{2}^{50} - \)\(70\!\cdots\!22\)\( T_{2}^{49} + \)\(21\!\cdots\!70\)\( T_{2}^{48} + \)\(10\!\cdots\!67\)\( T_{2}^{47} - \)\(10\!\cdots\!15\)\( T_{2}^{46} - \)\(13\!\cdots\!77\)\( T_{2}^{45} - \)\(96\!\cdots\!90\)\( T_{2}^{44} + \)\(15\!\cdots\!54\)\( T_{2}^{43} + \)\(33\!\cdots\!26\)\( T_{2}^{42} - \)\(14\!\cdots\!95\)\( T_{2}^{41} - \)\(54\!\cdots\!52\)\( T_{2}^{40} + \)\(12\!\cdots\!23\)\( T_{2}^{39} + \)\(63\!\cdots\!48\)\( T_{2}^{38} - \)\(90\!\cdots\!15\)\( T_{2}^{37} - \)\(59\!\cdots\!94\)\( T_{2}^{36} + \)\(55\!\cdots\!27\)\( T_{2}^{35} + \)\(45\!\cdots\!77\)\( T_{2}^{34} - \)\(28\!\cdots\!40\)\( T_{2}^{33} - \)\(29\!\cdots\!21\)\( T_{2}^{32} + \)\(11\!\cdots\!32\)\( T_{2}^{31} + \)\(16\!\cdots\!44\)\( T_{2}^{30} - \)\(37\!\cdots\!31\)\( T_{2}^{29} - \)\(72\!\cdots\!43\)\( T_{2}^{28} + \)\(75\!\cdots\!84\)\( T_{2}^{27} + \)\(27\!\cdots\!34\)\( T_{2}^{26} + \)\(13\!\cdots\!69\)\( T_{2}^{25} - \)\(85\!\cdots\!19\)\( T_{2}^{24} - \)\(83\!\cdots\!52\)\( T_{2}^{23} + \)\(21\!\cdots\!99\)\( T_{2}^{22} + \)\(39\!\cdots\!22\)\( T_{2}^{21} - \)\(44\!\cdots\!09\)\( T_{2}^{20} - \)\(11\!\cdots\!61\)\( T_{2}^{19} + \)\(69\!\cdots\!37\)\( T_{2}^{18} + \)\(23\!\cdots\!37\)\( T_{2}^{17} - \)\(80\!\cdots\!45\)\( T_{2}^{16} - \)\(36\!\cdots\!97\)\( T_{2}^{15} + \)\(63\!\cdots\!24\)\( T_{2}^{14} + \)\(41\!\cdots\!87\)\( T_{2}^{13} - \)\(25\!\cdots\!43\)\( T_{2}^{12} - \)\(32\!\cdots\!98\)\( T_{2}^{11} - \)\(67\!\cdots\!82\)\( T_{2}^{10} + \)\(16\!\cdots\!61\)\( T_{2}^{9} + \)\(14\!\cdots\!15\)\( T_{2}^{8} - \)\(50\!\cdots\!59\)\( T_{2}^{7} - \)\(71\!\cdots\!95\)\( T_{2}^{6} + \)\(63\!\cdots\!06\)\( T_{2}^{5} + \)\(13\!\cdots\!86\)\( T_{2}^{4} + 6413236614 T_{2}^{3} - 6626596512 T_{2}^{2} - 133527780 T_{2} + 2948952 \)">\(T_{2}^{104} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database