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Properties

Label 3013.2.a.d
Level $3013$
Weight $2$
Character orbit 3013.a
Self dual yes
Analytic conductor $24.059$
Analytic rank $0$
Dimension $68$
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] = [3013,2,Mod(1,3013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3013 = 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3013.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: \(24.0589261290\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 14 q^{2} + 22 q^{3} + 82 q^{4} + 9 q^{5} + 8 q^{6} + 7 q^{7} + 42 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 14 q^{2} + 22 q^{3} + 82 q^{4} + 9 q^{5} + 8 q^{6} + 7 q^{7} + 42 q^{8} + 90 q^{9} - 9 q^{10} + 35 q^{11} + 42 q^{12} + 14 q^{13} + 19 q^{14} + 18 q^{15} + 102 q^{16} + 11 q^{17} + 42 q^{18} + 8 q^{19} + 10 q^{20} + 5 q^{21} + 6 q^{22} + 68 q^{23} - 2 q^{24} + 101 q^{25} + 22 q^{26} + 79 q^{27} + 15 q^{28} + 31 q^{29} - 7 q^{30} + 18 q^{31} + 98 q^{32} + 20 q^{33} + 6 q^{34} + 82 q^{35} + 154 q^{36} + 2 q^{37} + 48 q^{38} + 26 q^{39} - 15 q^{40} + 16 q^{41} - 27 q^{42} + 52 q^{43} + 9 q^{44} - 5 q^{45} + 14 q^{46} + 59 q^{47} + 75 q^{48} + 91 q^{49} + 71 q^{50} + 39 q^{51} + 26 q^{52} + 90 q^{53} + 45 q^{54} + 7 q^{55} + 48 q^{56} - 6 q^{57} + 34 q^{58} + 84 q^{59} - 57 q^{60} - 20 q^{61} + 60 q^{62} + 7 q^{63} + 138 q^{64} - 12 q^{66} + 73 q^{67} + 70 q^{68} + 22 q^{69} - 43 q^{70} + 49 q^{71} + 100 q^{72} - 22 q^{73} + 68 q^{74} + 115 q^{75} - 26 q^{76} + 63 q^{77} + 19 q^{78} + q^{79} - 22 q^{80} + 168 q^{81} + 44 q^{82} + 91 q^{83} - 91 q^{84} + 42 q^{85} + 81 q^{86} + 38 q^{87} - 81 q^{88} + 40 q^{89} - 87 q^{90} - 27 q^{91} + 82 q^{92} + 14 q^{93} + 9 q^{94} + 29 q^{95} + 17 q^{96} - 28 q^{97} + 124 q^{98} + 64 q^{99}+O(q^{100}) \) 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 −2.72007 0.657432 5.39881 −1.35294 −1.78826 −1.82789 −9.24501 −2.56778 3.68010
1.2 −2.55652 2.78590 4.53581 2.81442 −7.12221 3.38667 −6.48286 4.76122 −7.19514
1.3 −2.53919 0.499158 4.44748 4.32288 −1.26746 −2.72925 −6.21462 −2.75084 −10.9766
1.4 −2.52043 3.37597 4.35255 −3.12983 −8.50889 −3.90858 −5.92943 8.39719 7.88851
1.5 −2.45283 −2.45767 4.01639 1.74201 6.02826 2.37492 −4.94587 3.04015 −4.27287
1.6 −2.45205 −1.30594 4.01256 −2.33911 3.20222 −2.18528 −4.93489 −1.29453 5.73561
1.7 −2.40220 −2.75836 3.77058 1.04232 6.62614 1.19932 −4.25329 4.60855 −2.50386
1.8 −2.28355 0.748497 3.21462 −0.339132 −1.70923 −0.629811 −2.77365 −2.43975 0.774426
1.9 −2.23002 2.12407 2.97298 0.816697 −4.73672 5.09342 −2.16977 1.51168 −1.82125
1.10 −2.21412 3.28095 2.90234 −0.447259 −7.26444 0.00870051 −1.99789 7.76466 0.990287
1.11 −2.00049 0.619846 2.00194 −3.42979 −1.23999 −2.88727 −0.00388177 −2.61579 6.86124
1.12 −1.95064 0.0733562 1.80501 3.17599 −0.143092 2.39340 0.380347 −2.99462 −6.19523
1.13 −1.77205 −1.82573 1.14015 −1.28540 3.23527 −1.80476 1.52370 0.333280 2.27779
1.14 −1.65160 −2.39219 0.727773 3.72695 3.95093 −1.09371 2.10121 2.72256 −6.15541
1.15 −1.57733 −1.41930 0.487982 −0.104689 2.23871 3.55696 2.38496 −0.985592 0.165129
1.16 −1.42842 2.52183 0.0403815 2.85973 −3.60223 −0.106869 2.79916 3.35963 −4.08490
1.17 −1.39890 1.36722 −0.0430760 2.54750 −1.91261 −1.34545 2.85806 −1.13071 −3.56370
1.18 −1.37804 −1.53490 −0.101012 −2.79428 2.11515 3.26996 2.89527 −0.644074 3.85062
1.19 −1.37027 0.988423 −0.122356 −3.18464 −1.35441 0.912098 2.90820 −2.02302 4.36383
1.20 −1.19272 −0.0949112 −0.577416 1.13513 0.113203 −5.09006 3.07414 −2.99099 −1.35390
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.68
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(-1\)
\(131\) \(1\)

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 3013.2.a.d ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3013.2.a.d ✓ 68 1.a even 1 1 trivial