Properties

Label 2013.4.a.g
Level 2013
Weight 4
Character orbit 2013.a
Self dual yes
Analytic conductor 118.771
Analytic rank 0
Dimension 39
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: \(0\)
Dimension: \(39\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39q + 4q^{2} - 117q^{3} + 182q^{4} + 5q^{5} - 12q^{6} + 77q^{7} + 27q^{8} + 351q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 39q + 4q^{2} - 117q^{3} + 182q^{4} + 5q^{5} - 12q^{6} + 77q^{7} + 27q^{8} + 351q^{9} + 95q^{10} + 429q^{11} - 546q^{12} + 169q^{13} + 46q^{14} - 15q^{15} + 822q^{16} + 294q^{17} + 36q^{18} + 259q^{19} + 426q^{20} - 231q^{21} + 44q^{22} + 177q^{23} - 81q^{24} + 1388q^{25} + 695q^{26} - 1053q^{27} + 1104q^{28} - 18q^{29} - 285q^{30} + 422q^{31} + 55q^{32} - 1287q^{33} + 364q^{34} + 906q^{35} + 1638q^{36} + 424q^{37} + 9q^{38} - 507q^{39} + 1067q^{40} + 16q^{41} - 138q^{42} + 1013q^{43} + 2002q^{44} + 45q^{45} + 9q^{46} + 1615q^{47} - 2466q^{48} + 2024q^{49} - 1342q^{50} - 882q^{51} + 1298q^{52} - 541q^{53} - 108q^{54} + 55q^{55} - 161q^{56} - 777q^{57} + 1061q^{58} + 1019q^{59} - 1278q^{60} + 2379q^{61} + 879q^{62} + 693q^{63} + 1055q^{64} - 1134q^{65} - 132q^{66} + 1917q^{67} + 3526q^{68} - 531q^{69} + 758q^{70} - 479q^{71} + 243q^{72} + 3319q^{73} - 332q^{74} - 4164q^{75} + 692q^{76} + 847q^{77} - 2085q^{78} + 651q^{79} + 2973q^{80} + 3159q^{81} - 826q^{82} + 4001q^{83} - 3312q^{84} + 3595q^{85} - 6247q^{86} + 54q^{87} + 297q^{88} - 1625q^{89} + 855q^{90} + 2048q^{91} - 507q^{92} - 1266q^{93} - 2436q^{94} + 1400q^{95} - 165q^{96} + 2176q^{97} - 1396q^{98} + 3861q^{99} + O(q^{100}) \)

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 −5.55718 −3.00000 22.8823 18.5292 16.6716 4.29084 −82.7037 9.00000 −102.970
1.2 −5.31049 −3.00000 20.2013 −16.8641 15.9315 32.3654 −64.7946 9.00000 89.5564
1.3 −5.05192 −3.00000 17.5219 −10.1219 15.1558 −15.0658 −48.1037 9.00000 51.1353
1.4 −4.92702 −3.00000 16.2755 10.2503 14.7811 17.4569 −40.7737 9.00000 −50.5033
1.5 −4.64624 −3.00000 13.5875 −15.1595 13.9387 14.1108 −25.9610 9.00000 70.4347
1.6 −4.33185 −3.00000 10.7649 6.98313 12.9956 −22.4328 −11.9772 9.00000 −30.2499
1.7 −4.31511 −3.00000 10.6201 3.60789 12.9453 −12.6536 −11.3062 9.00000 −15.5684
1.8 −3.60515 −3.00000 4.99712 5.59172 10.8155 26.5819 10.8259 9.00000 −20.1590
1.9 −3.49324 −3.00000 4.20275 11.6764 10.4797 −11.9890 13.2647 9.00000 −40.7884
1.10 −3.44435 −3.00000 3.86353 −21.1953 10.3330 −30.8393 14.2475 9.00000 73.0040
1.11 −3.25584 −3.00000 2.60046 −8.42660 9.76751 −19.3408 17.5800 9.00000 27.4356
1.12 −2.91984 −3.00000 0.525437 16.8265 8.75951 26.0578 21.8245 9.00000 −49.1305
1.13 −2.78323 −3.00000 −0.253631 1.24118 8.34969 31.9108 22.9718 9.00000 −3.45448
1.14 −1.48743 −3.00000 −5.78756 8.78636 4.46228 −7.23392 20.5080 9.00000 −13.0691
1.15 −1.23835 −3.00000 −6.46649 −19.0439 3.71505 −16.0361 17.9146 9.00000 23.5830
1.16 −1.12400 −3.00000 −6.73663 −4.49934 3.37199 −12.1013 16.5639 9.00000 5.05724
1.17 −0.779300 −3.00000 −7.39269 18.3794 2.33790 12.7504 11.9955 9.00000 −14.3230
1.18 −0.405092 −3.00000 −7.83590 −18.0089 1.21528 23.3188 6.41500 9.00000 7.29527
1.19 −0.275199 −3.00000 −7.92427 15.4654 0.825597 −21.6439 4.38234 9.00000 −4.25605
1.20 0.296271 −3.00000 −7.91222 −4.98422 −0.888812 −5.17231 −4.71432 9.00000 −1.47668
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
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.g 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.g 39 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database