Properties

Label 2013.4.a.h
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

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: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39q + 8q^{2} + 117q^{3} + 154q^{4} + 65q^{5} + 24q^{6} + 35q^{7} + 75q^{8} + 351q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 39q + 8q^{2} + 117q^{3} + 154q^{4} + 65q^{5} + 24q^{6} + 35q^{7} + 75q^{8} + 351q^{9} - 21q^{10} - 429q^{11} + 462q^{12} - 27q^{13} + 164q^{14} + 195q^{15} + 686q^{16} + 170q^{17} + 72q^{18} + 139q^{19} + 1056q^{20} + 105q^{21} - 88q^{22} + 291q^{23} + 225q^{24} + 1236q^{25} + 583q^{26} + 1053q^{27} + 976q^{28} + 374q^{29} - 63q^{30} + 232q^{31} + 933q^{32} - 1287q^{33} + 332q^{34} + 626q^{35} + 1386q^{36} + 232q^{37} + 989q^{38} - 81q^{39} - 263q^{40} + 1014q^{41} + 492q^{42} + 515q^{43} - 1694q^{44} + 585q^{45} - 371q^{46} + 2005q^{47} + 2058q^{48} + 2064q^{49} + 4582q^{50} + 510q^{51} + 216q^{52} + 1485q^{53} + 216q^{54} - 715q^{55} + 2307q^{56} + 417q^{57} + 573q^{58} + 2749q^{59} + 3168q^{60} + 2379q^{61} + 1837q^{62} + 315q^{63} + 7295q^{64} + 3630q^{65} - 264q^{66} + 3575q^{67} + 2630q^{68} + 873q^{69} + 4218q^{70} + 4723q^{71} + 675q^{72} + 859q^{73} + 4232q^{74} + 3708q^{75} + 2466q^{76} - 385q^{77} + 1749q^{78} - 1887q^{79} + 8933q^{80} + 3159q^{81} + 6806q^{82} + 5609q^{83} + 2928q^{84} - 565q^{85} + 5185q^{86} + 1122q^{87} - 825q^{88} + 6725q^{89} - 189q^{90} + 2808q^{91} + 3257q^{92} + 696q^{93} + 3184q^{94} + 3216q^{95} + 2799q^{96} + 3512q^{97} + 4464q^{98} - 3861q^{99} + O(q^{100}) \)

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 −5.58742 3.00000 23.2193 7.47722 −16.7623 8.26209 −85.0366 9.00000 −41.7784
1.2 −5.19520 3.00000 18.9901 20.2319 −15.5856 −14.1923 −57.0959 9.00000 −105.109
1.3 −4.97376 3.00000 16.7383 −4.79568 −14.9213 −12.7682 −43.4620 9.00000 23.8525
1.4 −4.73713 3.00000 14.4404 3.01069 −14.2114 21.8287 −30.5090 9.00000 −14.2620
1.5 −4.26095 3.00000 10.1557 −7.59416 −12.7828 −6.39265 −9.18529 9.00000 32.3583
1.6 −3.99434 3.00000 7.95479 10.8725 −11.9830 34.9110 0.180577 9.00000 −43.4284
1.7 −3.90025 3.00000 7.21198 −4.45780 −11.7008 14.5889 3.07349 9.00000 17.3866
1.8 −3.67115 3.00000 5.47734 14.7390 −11.0134 −28.2167 9.26107 9.00000 −54.1090
1.9 −3.32411 3.00000 3.04969 −5.05305 −9.97232 −19.7829 16.4554 9.00000 16.7969
1.10 −2.98423 3.00000 0.905618 18.2801 −8.95268 −1.48333 21.1713 9.00000 −54.5520
1.11 −2.88889 3.00000 0.345688 −8.48381 −8.66667 20.9827 22.1125 9.00000 24.5088
1.12 −2.16141 3.00000 −3.32830 7.38660 −6.48423 −25.4500 24.4851 9.00000 −15.9655
1.13 −2.09003 3.00000 −3.63175 −21.1504 −6.27010 13.8715 24.3108 9.00000 44.2051
1.14 −1.70549 3.00000 −5.09131 −1.03329 −5.11647 1.91271 22.3271 9.00000 1.76227
1.15 −1.69454 3.00000 −5.12852 −5.42092 −5.08363 −29.9078 22.2469 9.00000 9.18599
1.16 −1.01355 3.00000 −6.97272 12.1717 −3.04064 28.0678 15.1756 9.00000 −12.3366
1.17 −0.878549 3.00000 −7.22815 −10.2662 −2.63565 2.69154 13.3787 9.00000 9.01940
1.18 −0.382791 3.00000 −7.85347 10.1790 −1.14837 17.4416 6.06857 9.00000 −3.89641
1.19 0.0220344 3.00000 −7.99951 −17.2328 0.0661031 −5.99920 −0.352539 9.00000 −0.379714
1.20 0.170804 3.00000 −7.97083 −3.72511 0.512411 −19.4372 −2.72788 9.00000 −0.636262
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.h 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.h 39 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database