Properties

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

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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.

\(\operatorname{Tr}(f)(q) = \) \( 104q - 19q^{2} - 26q^{3} + 99q^{4} + 2q^{5} + 2q^{6} - 104q^{7} - 54q^{8} + 90q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 104q - 19q^{2} - 26q^{3} + 99q^{4} + 2q^{5} + 2q^{6} - 104q^{7} - 54q^{8} + 90q^{9} + 3q^{10} - 54q^{11} - 38q^{12} + 7q^{13} + 19q^{14} - 33q^{15} + 93q^{16} - 7q^{17} - 55q^{18} - 12q^{19} - 24q^{20} + 26q^{21} - 22q^{22} - 69q^{23} + 78q^{25} - 11q^{26} - 95q^{27} - 99q^{28} - 41q^{29} - 26q^{30} - 12q^{31} - 127q^{32} - 6q^{33} - 17q^{34} - 2q^{35} + 71q^{36} - 47q^{37} - 32q^{38} - 57q^{39} + 6q^{40} + 10q^{41} - 2q^{42} - 41q^{43} - 120q^{44} + 23q^{45} - 31q^{46} - 99q^{47} - 84q^{48} + 104q^{49} - 104q^{50} - 74q^{51} + 14q^{52} - 74q^{53} + 19q^{54} - 32q^{55} + 54q^{56} - 47q^{57} - 36q^{58} - 76q^{59} - 99q^{60} + 49q^{61} - 55q^{62} - 90q^{63} + 86q^{64} - 70q^{65} + 61q^{66} - 117q^{67} - 30q^{68} + 51q^{69} - 3q^{70} - 125q^{71} - 147q^{72} - 20q^{73} - 75q^{74} - 124q^{75} + 4q^{76} + 54q^{77} - 70q^{78} - 72q^{79} - 69q^{80} + 76q^{81} - 37q^{82} - 98q^{83} + 38q^{84} - 33q^{85} - 64q^{86} - 8q^{87} - 62q^{88} - 26q^{89} + 11q^{90} - 7q^{91} - 162q^{92} - 81q^{93} + 31q^{94} - 116q^{95} + 20q^{96} - 61q^{97} - 19q^{98} - 158q^{99} + O(q^{100}) \)

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 −2.80449 −2.76312 5.86515 0.655993 7.74914 −1.00000 −10.8398 4.63484 −1.83973
1.2 −2.79954 −0.206703 5.83741 2.43879 0.578672 −1.00000 −10.7430 −2.95727 −6.82749
1.3 −2.79135 −2.87879 5.79164 −2.47936 8.03571 −1.00000 −10.5838 5.28744 6.92075
1.4 −2.76981 0.446127 5.67184 −3.40223 −1.23569 −1.00000 −10.1703 −2.80097 9.42353
1.5 −2.68972 2.91852 5.23459 0.347371 −7.85000 −1.00000 −8.70012 5.51776 −0.934329
1.6 −2.66314 −0.352926 5.09230 −3.56407 0.939890 −1.00000 −8.23520 −2.87544 9.49161
1.7 −2.57881 1.45096 4.65028 4.21886 −3.74175 −1.00000 −6.83459 −0.894725 −10.8797
1.8 −2.50501 −2.73412 4.27508 −0.209181 6.84900 −1.00000 −5.69911 4.47541 0.524002
1.9 −2.49724 2.33621 4.23620 −3.93107 −5.83407 −1.00000 −5.58433 2.45787 9.81682
1.10 −2.47703 −3.45031 4.13569 3.27478 8.54653 −1.00000 −5.29018 8.90465 −8.11174
1.11 −2.46187 0.832905 4.06078 1.64347 −2.05050 −1.00000 −5.07337 −2.30627 −4.04600
1.12 −2.45278 −0.285776 4.01611 1.53542 0.700944 −1.00000 −4.94507 −2.91833 −3.76604
1.13 −2.42052 −1.81832 3.85894 −2.20687 4.40128 −1.00000 −4.49961 0.306272 5.34179
1.14 −2.41334 3.15271 3.82421 −0.650667 −7.60855 −1.00000 −4.40244 6.93956 1.57028
1.15 −2.37810 0.188335 3.65538 0.902654 −0.447879 −1.00000 −3.93667 −2.96453 −2.14661
1.16 −2.33681 0.433467 3.46066 −2.70266 −1.01293 −1.00000 −3.41328 −2.81211 6.31560
1.17 −2.32315 1.73935 3.39700 3.06159 −4.04076 −1.00000 −3.24544 0.0253336 −7.11252
1.18 −2.13208 −1.43746 2.54578 0.297217 3.06479 −1.00000 −1.16366 −0.933703 −0.633692
1.19 −2.03131 −2.52319 2.12623 0.0590478 5.12539 −1.00000 −0.256407 3.36649 −0.119944
1.20 −2.02779 2.00075 2.11193 −3.78073 −4.05710 −1.00000 −0.226966 1.00300 7.66652
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.c 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6013.2.a.c 104 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(10\!\cdots\!98\)\( T_{2}^{88} - \)\(13\!\cdots\!62\)\( T_{2}^{87} + 525940739058 T_{2}^{86} + \)\(15\!\cdots\!63\)\( T_{2}^{85} + \)\(12\!\cdots\!34\)\( T_{2}^{84} - \)\(15\!\cdots\!97\)\( T_{2}^{83} - \)\(23\!\cdots\!99\)\( T_{2}^{82} + \)\(12\!\cdots\!84\)\( T_{2}^{81} + \)\(28\!\cdots\!78\)\( T_{2}^{80} - \)\(79\!\cdots\!15\)\( T_{2}^{79} - \)\(25\!\cdots\!35\)\( T_{2}^{78} + \)\(40\!\cdots\!21\)\( T_{2}^{77} + \)\(19\!\cdots\!83\)\( T_{2}^{76} - \)\(15\!\cdots\!56\)\( T_{2}^{75} - \)\(12\!\cdots\!39\)\( T_{2}^{74} + \)\(26\!\cdots\!42\)\( T_{2}^{73} + \)\(66\!\cdots\!61\)\( T_{2}^{72} + \)\(16\!\cdots\!27\)\( T_{2}^{71} - \)\(30\!\cdots\!47\)\( T_{2}^{70} - \)\(20\!\cdots\!67\)\( T_{2}^{69} + \)\(12\!\cdots\!48\)\( T_{2}^{68} + \)\(12\!\cdots\!02\)\( T_{2}^{67} - \)\(43\!\cdots\!43\)\( T_{2}^{66} - \)\(62\!\cdots\!68\)\( T_{2}^{65} + \)\(12\!\cdots\!24\)\( T_{2}^{64} + \)\(24\!\cdots\!77\)\( T_{2}^{63} - \)\(32\!\cdots\!19\)\( T_{2}^{62} - \)\(83\!\cdots\!44\)\( T_{2}^{61} + \)\(66\!\cdots\!18\)\( T_{2}^{60} + \)\(24\!\cdots\!77\)\( T_{2}^{59} - \)\(10\!\cdots\!06\)\( T_{2}^{58} - \)\(61\!\cdots\!36\)\( T_{2}^{57} + \)\(76\!\cdots\!68\)\( T_{2}^{56} + \)\(13\!\cdots\!55\)\( T_{2}^{55} + \)\(16\!\cdots\!28\)\( T_{2}^{54} - \)\(26\!\cdots\!45\)\( T_{2}^{53} - \)\(87\!\cdots\!36\)\( T_{2}^{52} + \)\(44\!\cdots\!21\)\( T_{2}^{51} + \)\(23\!\cdots\!50\)\( T_{2}^{50} - \)\(65\!\cdots\!68\)\( T_{2}^{49} - \)\(47\!\cdots\!34\)\( T_{2}^{48} + \)\(83\!\cdots\!15\)\( T_{2}^{47} + \)\(78\!\cdots\!71\)\( T_{2}^{46} - \)\(92\!\cdots\!29\)\( T_{2}^{45} - \)\(10\!\cdots\!36\)\( T_{2}^{44} + \)\(88\!\cdots\!50\)\( T_{2}^{43} + \)\(13\!\cdots\!92\)\( T_{2}^{42} - \)\(70\!\cdots\!45\)\( T_{2}^{41} - \)\(13\!\cdots\!04\)\( T_{2}^{40} + \)\(45\!\cdots\!99\)\( T_{2}^{39} + \)\(11\!\cdots\!72\)\( T_{2}^{38} - \)\(22\!\cdots\!79\)\( T_{2}^{37} - \)\(89\!\cdots\!54\)\( T_{2}^{36} + \)\(59\!\cdots\!99\)\( T_{2}^{35} + \)\(58\!\cdots\!17\)\( T_{2}^{34} + \)\(18\!\cdots\!22\)\( T_{2}^{33} - \)\(32\!\cdots\!43\)\( T_{2}^{32} - \)\(36\!\cdots\!48\)\( T_{2}^{31} + \)\(15\!\cdots\!32\)\( T_{2}^{30} + \)\(27\!\cdots\!77\)\( T_{2}^{29} - \)\(64\!\cdots\!49\)\( T_{2}^{28} - \)\(14\!\cdots\!68\)\( T_{2}^{27} + \)\(22\!\cdots\!98\)\( T_{2}^{26} + \)\(58\!\cdots\!93\)\( T_{2}^{25} - \)\(64\!\cdots\!47\)\( T_{2}^{24} - \)\(18\!\cdots\!34\)\( T_{2}^{23} + \)\(15\!\cdots\!31\)\( T_{2}^{22} + \)\(45\!\cdots\!00\)\( T_{2}^{21} - \)\(31\!\cdots\!79\)\( T_{2}^{20} - \)\(87\!\cdots\!71\)\( T_{2}^{19} + \)\(50\!\cdots\!61\)\( T_{2}^{18} + \)\(12\!\cdots\!37\)\( T_{2}^{17} - \)\(66\!\cdots\!33\)\( T_{2}^{16} - \)\(14\!\cdots\!71\)\( T_{2}^{15} + \)\(67\!\cdots\!26\)\( T_{2}^{14} + \)\(10\!\cdots\!33\)\( T_{2}^{13} - \)\(50\!\cdots\!87\)\( T_{2}^{12} - \)\(53\!\cdots\!14\)\( T_{2}^{11} + \)\(26\!\cdots\!26\)\( T_{2}^{10} + \)\(13\!\cdots\!95\)\( T_{2}^{9} - \)\(85\!\cdots\!79\)\( T_{2}^{8} - \)\(35\!\cdots\!13\)\( T_{2}^{7} + \)\(13\!\cdots\!13\)\( T_{2}^{6} - 281638005238 T_{2}^{5} - 92293383864 T_{2}^{4} + 3431331926 T_{2}^{3} + 165071800 T_{2}^{2} - 8802344 T_{2} + 86336 \)">\(T_{2}^{104} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6013))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database