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Properties

Label 2013.4.a.b
Level $2013$
Weight $4$
Character orbit 2013.a
Self dual yes
Analytic conductor $118.771$
Analytic rank $1$
Dimension $36$
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: \(1\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 108 q^{3} + 118 q^{4} - 5 q^{5} - 6 q^{6} - 63 q^{7} + 3 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 108 q^{3} + 118 q^{4} - 5 q^{5} - 6 q^{6} - 63 q^{7} + 3 q^{8} + 324 q^{9} - 45 q^{10} - 396 q^{11} - 354 q^{12} - 13 q^{13} + 82 q^{14} + 15 q^{15} + 262 q^{16} + 204 q^{17} + 18 q^{18} - 431 q^{19} + 354 q^{20} + 189 q^{21} - 22 q^{22} - 179 q^{23} - 9 q^{24} + 711 q^{25} + 331 q^{26} - 972 q^{27} - 296 q^{28} + 478 q^{29} + 135 q^{30} - 574 q^{31} - 149 q^{32} + 1188 q^{33} + 276 q^{34} - 194 q^{35} + 1062 q^{36} - 12 q^{37} + 325 q^{38} + 39 q^{39} - 185 q^{40} + 900 q^{41} - 246 q^{42} - 1053 q^{43} - 1298 q^{44} - 45 q^{45} - 407 q^{46} - 653 q^{47} - 786 q^{48} + 753 q^{49} - 1520 q^{50} - 612 q^{51} + 60 q^{52} + 735 q^{53} - 54 q^{54} + 55 q^{55} - 809 q^{56} + 1293 q^{57} - 1399 q^{58} - 1127 q^{59} - 1062 q^{60} + 2196 q^{61} - 1795 q^{62} - 567 q^{63} - 2133 q^{64} + 1886 q^{65} + 66 q^{66} - 989 q^{67} + 10 q^{68} + 537 q^{69} - 2130 q^{70} + 61 q^{71} + 27 q^{72} - 1471 q^{73} - 122 q^{74} - 2133 q^{75} - 4064 q^{76} + 693 q^{77} - 993 q^{78} - 1853 q^{79} + 2197 q^{80} + 2916 q^{81} - 2566 q^{82} - 3523 q^{83} + 888 q^{84} - 449 q^{85} - 771 q^{86} - 1434 q^{87} - 33 q^{88} + 2209 q^{89} - 405 q^{90} - 1668 q^{91} - 1999 q^{92} + 1722 q^{93} - 2844 q^{94} + 1220 q^{95} + 447 q^{96} - 3622 q^{97} + 3846 q^{98} - 3564 q^{99}+O(q^{100}) \) 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 −5.32772 −3.00000 20.3846 9.51582 15.9832 0.376884 −65.9820 9.00000 −50.6977
1.2 −4.97623 −3.00000 16.7629 −4.44765 14.9287 −23.2908 −43.6062 9.00000 22.1326
1.3 −4.89062 −3.00000 15.9181 15.5621 14.6718 16.9638 −38.7245 9.00000 −76.1085
1.4 −4.67500 −3.00000 13.8556 −8.51569 14.0250 7.22973 −27.3750 9.00000 39.8109
1.5 −4.58366 −3.00000 13.0099 −16.2221 13.7510 3.97810 −22.9636 9.00000 74.3567
1.6 −4.33380 −3.00000 10.7818 18.1421 13.0014 −4.33011 −12.0559 9.00000 −78.6243
1.7 −3.49679 −3.00000 4.22755 −13.5195 10.4904 −10.1391 13.1915 9.00000 47.2749
1.8 −3.42477 −3.00000 3.72902 −0.330135 10.2743 −34.8727 14.6271 9.00000 1.13064
1.9 −3.21541 −3.00000 2.33885 2.50702 9.64623 5.55272 18.2029 9.00000 −8.06108
1.10 −2.49518 −3.00000 −1.77407 21.0229 7.48554 −29.8474 24.3881 9.00000 −52.4560
1.11 −2.20148 −3.00000 −3.15347 −19.8720 6.60445 −18.3617 24.5542 9.00000 43.7479
1.12 −2.13143 −3.00000 −3.45699 −8.60507 6.39430 26.5144 24.4198 9.00000 18.3411
1.13 −2.00969 −3.00000 −3.96116 6.20179 6.02906 18.8550 24.0382 9.00000 −12.4637
1.14 −1.80221 −3.00000 −4.75204 13.7156 5.40663 7.13162 22.9818 9.00000 −24.7184
1.15 −1.71127 −3.00000 −5.07156 −5.95743 5.13381 −9.43303 22.3690 9.00000 10.1948
1.16 −0.691390 −3.00000 −7.52198 −5.77110 2.07417 9.57605 10.7317 9.00000 3.99008
1.17 −0.158900 −3.00000 −7.97475 7.59417 0.476701 −16.2139 2.53840 9.00000 −1.20672
1.18 0.0588759 −3.00000 −7.99653 −14.4051 −0.176628 36.0147 −0.941811 9.00000 −0.848116
1.19 0.402086 −3.00000 −7.83833 4.05035 −1.20626 4.43625 −6.36837 9.00000 1.62859
1.20 0.857145 −3.00000 −7.26530 −0.0808778 −2.57143 −28.7264 −13.0846 9.00000 −0.0693240
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
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.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.4.a.b ✓ 36 1.a even 1 1 trivial