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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,4,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.20141 3.00000 19.0547 −1.81138 −15.6042 3.78751 −57.5001 9.00000 9.42176
1.2 −5.11974 3.00000 18.2117 −8.33272 −15.3592 −0.123891 −52.2812 9.00000 42.6613
1.3 −4.92345 3.00000 16.2404 19.7094 −14.7704 4.38553 −40.5713 9.00000 −97.0385
1.4 −4.79323 3.00000 14.9751 12.3054 −14.3797 26.5090 −33.4332 9.00000 −58.9827
1.5 −4.44696 3.00000 11.7755 −6.52314 −13.3409 −19.5475 −16.7894 9.00000 29.0081
1.6 −4.08981 3.00000 8.72654 −14.1681 −12.2694 −18.6667 −2.97140 9.00000 57.9450
1.7 −4.00574 3.00000 8.04594 −13.8167 −12.0172 32.5179 −0.184008 9.00000 55.3460
1.8 −3.50711 3.00000 4.29983 10.2444 −10.5213 1.41773 12.9769 9.00000 −35.9282
1.9 −3.25945 3.00000 2.62403 10.3368 −9.77836 −26.6613 17.5227 9.00000 −33.6924
1.10 −2.86674 3.00000 0.218214 1.45116 −8.60023 27.7506 22.3084 9.00000 −4.16011
1.11 −2.38315 3.00000 −2.32059 14.5269 −7.14945 22.4384 24.5955 9.00000 −34.6198
1.12 −2.31501 3.00000 −2.64074 −15.6387 −6.94502 −20.9070 24.6334 9.00000 36.2036
1.13 −1.93536 3.00000 −4.25439 −18.7593 −5.80607 6.86594 23.7166 9.00000 36.3060
1.14 −1.15474 3.00000 −6.66657 −7.99519 −3.46423 12.3786 16.9361 9.00000 9.23239
1.15 −0.933922 3.00000 −7.12779 16.8720 −2.80177 −20.1785 14.1282 9.00000 −15.7571
1.16 −0.740360 3.00000 −7.45187 7.63307 −2.22108 1.52940 11.4399 9.00000 −5.65122
1.17 −0.477206 3.00000 −7.77227 5.50183 −1.43162 −27.2643 7.52663 9.00000 −2.62551
1.18 −0.0387605 3.00000 −7.99850 −3.27851 −0.116282 −8.05520 0.620110 9.00000 0.127077
1.19 0.801460 3.00000 −7.35766 19.8509 2.40438 26.4285 −12.3086 9.00000 15.9097
1.20 0.812552 3.00000 −7.33976 −6.24435 2.43766 23.9419 −12.4644 9.00000 −5.07386
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.f ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.f ✓ 38 1.a even 1 1 trivial