Properties

Label 2013.4.a.e
Level 2013
Weight 4
Character orbit 2013.a
Self dual yes
Analytic conductor 118.771
Analytic rank 0
Dimension 38
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38q - 2q^{2} - 114q^{3} + 142q^{4} + 15q^{5} + 6q^{6} + 63q^{7} - 45q^{8} + 342q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 38q - 2q^{2} - 114q^{3} + 142q^{4} + 15q^{5} + 6q^{6} + 63q^{7} - 45q^{8} + 342q^{9} + 95q^{10} - 418q^{11} - 426q^{12} + 13q^{13} + 26q^{14} - 45q^{15} + 486q^{16} - 224q^{17} - 18q^{18} + 367q^{19} + 18q^{20} - 189q^{21} + 22q^{22} + 51q^{23} + 135q^{24} + 773q^{25} - 439q^{26} - 1026q^{27} + 22q^{28} - 462q^{29} - 285q^{30} + 234q^{31} - 597q^{32} + 1254q^{33} + 956q^{34} - 522q^{35} + 1278q^{36} + 954q^{37} + 705q^{38} - 39q^{39} + 1495q^{40} - 740q^{41} - 78q^{42} + 1441q^{43} - 1562q^{44} + 135q^{45} + 581q^{46} + 1003q^{47} - 1458q^{48} + 2707q^{49} + 388q^{50} + 672q^{51} + 788q^{52} + 735q^{53} + 54q^{54} - 165q^{55} + 1059q^{56} - 1101q^{57} + 177q^{58} + 261q^{59} - 54q^{60} - 2318q^{61} + 1251q^{62} + 567q^{63} + 5571q^{64} - 1354q^{65} - 66q^{66} + 3495q^{67} - 1856q^{68} - 153q^{69} + 542q^{70} - 873q^{71} - 405q^{72} + 989q^{73} - 3406q^{74} - 2319q^{75} + 1712q^{76} - 693q^{77} + 1317q^{78} + 2313q^{79} + 1593q^{80} + 3078q^{81} + 5170q^{82} + 569q^{83} - 66q^{84} - 1271q^{85} + 3065q^{86} + 1386q^{87} + 495q^{88} - 2917q^{89} + 855q^{90} + 2740q^{91} + 1083q^{92} - 702q^{93} + 3272q^{94} + 2696q^{95} + 1791q^{96} + 4250q^{97} + 5952q^{98} - 3762q^{99} + O(q^{100}) \)

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 −5.60876 −3.00000 23.4582 −9.03443 16.8263 14.8484 −86.7015 9.00000 50.6720
1.2 −5.27624 −3.00000 19.8387 −1.30000 15.8287 18.3929 −62.4637 9.00000 6.85913
1.3 −5.14871 −3.00000 18.5092 14.6569 15.4461 −22.4177 −54.1088 9.00000 −75.4639
1.4 −4.79323 −3.00000 14.9751 −18.8616 14.3797 −18.5258 −33.4332 9.00000 90.4081
1.5 −4.76869 −3.00000 14.7404 6.42496 14.3061 −27.3976 −32.1428 9.00000 −30.6386
1.6 −4.20688 −3.00000 9.69783 −5.61420 12.6206 32.2171 −7.14255 9.00000 23.6183
1.7 −3.95658 −3.00000 7.65452 8.88627 11.8697 −1.22023 1.36690 9.00000 −35.1592
1.8 −3.35127 −3.00000 3.23102 −13.0308 10.0538 23.8563 15.9822 9.00000 43.6697
1.9 −3.18056 −3.00000 2.11597 11.4382 9.54169 18.3064 18.7145 9.00000 −36.3799
1.10 −3.08546 −3.00000 1.52008 1.21167 9.25639 −4.33357 19.9936 9.00000 −3.73856
1.11 −3.03304 −3.00000 1.19935 −11.5262 9.09913 −17.2788 20.6267 9.00000 34.9594
1.12 −2.90911 −3.00000 0.462945 17.7603 8.72734 −6.04713 21.9262 9.00000 −51.6667
1.13 −2.30626 −3.00000 −2.68118 −4.34363 6.91877 −19.3387 24.6335 9.00000 10.0175
1.14 −2.21634 −3.00000 −3.08782 −20.1616 6.64903 13.2153 24.5744 9.00000 44.6850
1.15 −1.82980 −3.00000 −4.65183 19.9990 5.48940 32.4689 23.1503 9.00000 −36.5943
1.16 −0.789172 −3.00000 −7.37721 13.9316 2.36752 −21.1583 12.1353 9.00000 −10.9944
1.17 −0.747694 −3.00000 −7.44095 −10.4401 2.24308 −14.9879 11.5451 9.00000 7.80602
1.18 −0.565755 −3.00000 −7.67992 11.9466 1.69726 27.9962 8.87099 9.00000 −6.75887
1.19 −0.495609 −3.00000 −7.75437 −15.1035 1.48683 2.90433 7.80801 9.00000 7.48545
1.20 −0.449187 −3.00000 −7.79823 5.67323 1.34756 0.979067 7.09636 9.00000 −2.54834
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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.e 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.e 38 1.a even 1 1 trivial

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database