Properties

Label 2013.4.a.d
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

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: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37q - 4q^{2} - 111q^{3} + 158q^{4} - 15q^{5} + 12q^{6} - 77q^{7} - 69q^{8} + 333q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 37q - 4q^{2} - 111q^{3} + 158q^{4} - 15q^{5} + 12q^{6} - 77q^{7} - 69q^{8} + 333q^{9} - 45q^{10} + 407q^{11} - 474q^{12} - 169q^{13} + 102q^{14} + 45q^{15} + 598q^{16} - 338q^{17} - 36q^{18} - 235q^{19} - 550q^{20} + 231q^{21} - 44q^{22} - 53q^{23} + 207q^{24} + 750q^{25} - 75q^{26} - 999q^{27} - 1378q^{28} - 30q^{29} + 135q^{30} - 506q^{31} - 841q^{32} - 1221q^{33} - 316q^{34} - 822q^{35} + 1422q^{36} - 830q^{37} - 371q^{38} + 507q^{39} - 613q^{40} + 16q^{41} - 306q^{42} - 1137q^{43} + 1738q^{44} - 135q^{45} - 659q^{46} - 489q^{47} - 1794q^{48} + 2214q^{49} + 1066q^{50} + 1014q^{51} - 2342q^{52} + 731q^{53} + 108q^{54} - 165q^{55} + 3051q^{56} + 705q^{57} - 611q^{58} - 425q^{59} + 1650q^{60} - 2257q^{61} + 453q^{62} - 693q^{63} + 4919q^{64} + 1346q^{65} + 132q^{66} - 1907q^{67} - 3236q^{68} + 159q^{69} - 1050q^{70} - 561q^{71} - 621q^{72} - 2397q^{73} - 1840q^{74} - 2250q^{75} - 3868q^{76} - 847q^{77} + 225q^{78} + 393q^{79} - 4031q^{80} + 2997q^{81} - 1946q^{82} - 4191q^{83} + 4134q^{84} - 2667q^{85} + 2405q^{86} + 90q^{87} - 759q^{88} + 1437q^{89} - 405q^{90} - 5192q^{91} - 737q^{92} + 1518q^{93} - 1960q^{94} + 1356q^{95} + 2523q^{96} - 2368q^{97} - 3014q^{98} + 3663q^{99} + O(q^{100}) \)

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 −5.57224 −3.00000 23.0498 −16.5315 16.7167 −30.4189 −83.8611 9.00000 92.1176
1.2 −5.34535 −3.00000 20.5728 5.60041 16.0360 11.9095 −67.2058 9.00000 −29.9362
1.3 −5.20750 −3.00000 19.1180 13.8219 15.6225 −32.6173 −57.8972 9.00000 −71.9777
1.4 −4.94279 −3.00000 16.4312 0.994625 14.8284 23.9386 −41.6735 9.00000 −4.91622
1.5 −4.85102 −3.00000 15.5324 −6.76910 14.5530 −14.4227 −36.5396 9.00000 32.8370
1.6 −4.27346 −3.00000 10.2624 −10.6639 12.8204 −5.91849 −9.66837 9.00000 45.5715
1.7 −4.00724 −3.00000 8.05801 20.5241 12.0217 −10.4521 −0.232470 9.00000 −82.2450
1.8 −3.99170 −3.00000 7.93363 2.24192 11.9751 2.67076 0.264912 9.00000 −8.94905
1.9 −3.73288 −3.00000 5.93441 −18.1575 11.1986 21.4866 7.71060 9.00000 67.7799
1.10 −2.84685 −3.00000 0.104547 −9.28768 8.54055 6.39681 22.4772 9.00000 26.4406
1.11 −2.75819 −3.00000 −0.392374 6.83862 8.27458 −1.91645 23.1478 9.00000 −18.8622
1.12 −2.70850 −3.00000 −0.664006 8.28724 8.12551 −34.8516 23.4665 9.00000 −22.4460
1.13 −2.29940 −3.00000 −2.71276 −16.8558 6.89820 −2.53212 24.6329 9.00000 38.7581
1.14 −1.94841 −3.00000 −4.20370 15.7553 5.84523 3.79858 23.7778 9.00000 −30.6978
1.15 −1.80549 −3.00000 −4.74021 −6.98611 5.41646 21.1508 23.0023 9.00000 12.6133
1.16 −1.54125 −3.00000 −5.62456 −4.20975 4.62374 −32.1268 20.9988 9.00000 6.48826
1.17 −1.30641 −3.00000 −6.29329 −3.39016 3.91923 25.4059 18.6729 9.00000 4.42894
1.18 −0.441122 −3.00000 −7.80541 14.6691 1.32336 −18.4288 6.97211 9.00000 −6.47085
1.19 0.197712 −3.00000 −7.96091 13.9720 −0.593137 33.5020 −3.15567 9.00000 2.76244
1.20 0.250375 −3.00000 −7.93731 9.04215 −0.751125 10.1810 −3.99031 9.00000 2.26393
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.d 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.d 37 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database