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Properties

Label 2015.2.a.l
Level $2015$
Weight $2$
Character orbit 2015.a
Self dual yes
Analytic conductor $16.090$
Analytic rank $0$
Dimension $22$
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] = [2015,2,Mod(1,2015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2015.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: \(16.0898560073\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 5 q^{2} + 4 q^{3} + 31 q^{4} + 22 q^{5} - 7 q^{6} + 5 q^{7} + 9 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 5 q^{2} + 4 q^{3} + 31 q^{4} + 22 q^{5} - 7 q^{6} + 5 q^{7} + 9 q^{8} + 42 q^{9} + 5 q^{10} + 12 q^{11} + 6 q^{12} - 22 q^{13} + 25 q^{14} + 4 q^{15} + 61 q^{16} + 2 q^{17} + 32 q^{18} + 8 q^{19} + 31 q^{20} + 6 q^{21} - q^{22} + 3 q^{23} + 22 q^{25} - 5 q^{26} + 7 q^{27} + 6 q^{28} + 28 q^{29} - 7 q^{30} - 22 q^{31} + 8 q^{32} - q^{33} + 5 q^{34} + 5 q^{35} + 46 q^{36} + 18 q^{37} + 13 q^{38} - 4 q^{39} + 9 q^{40} + 39 q^{41} + 21 q^{42} + 12 q^{43} + 38 q^{44} + 42 q^{45} - 38 q^{46} + 3 q^{47} - 59 q^{48} + 69 q^{49} + 5 q^{50} + 20 q^{51} - 31 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{55} + 93 q^{56} + 3 q^{57} + 8 q^{58} + 46 q^{59} + 6 q^{60} + 41 q^{61} - 5 q^{62} + 16 q^{63} + 133 q^{64} - 22 q^{65} - 61 q^{66} + 13 q^{67} - 55 q^{68} + 68 q^{69} + 25 q^{70} + 46 q^{71} + 10 q^{72} - 7 q^{73} - 58 q^{74} + 4 q^{75} - 29 q^{76} - 38 q^{77} + 7 q^{78} + 18 q^{79} + 61 q^{80} + 106 q^{81} + 22 q^{82} + 2 q^{83} - q^{84} + 2 q^{85} - 78 q^{86} + 16 q^{87} + 11 q^{88} + 33 q^{89} + 32 q^{90} - 5 q^{91} - 92 q^{92} - 4 q^{93} + 45 q^{94} + 8 q^{95} - 58 q^{96} + 55 q^{97} - 14 q^{98} - 12 q^{99}+O(q^{100}) \) 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 −2.80211 −2.07248 5.85184 1.00000 5.80733 −0.584705 −10.7933 1.29518 −2.80211
1.2 −2.76992 1.79224 5.67245 1.00000 −4.96436 −5.11218 −10.1724 0.212127 −2.76992
1.3 −2.12740 0.161318 2.52584 1.00000 −0.343187 −0.411943 −1.11868 −2.97398 −2.12740
1.4 −2.04109 3.04107 2.16603 1.00000 −6.20708 4.34888 −0.338882 6.24810 −2.04109
1.5 −1.77849 −0.614278 1.16304 1.00000 1.09249 2.73573 1.48853 −2.62266 −1.77849
1.6 −1.51285 2.86334 0.288701 1.00000 −4.33179 −4.20074 2.58893 5.19870 −1.51285
1.7 −1.07130 −2.90938 −0.852317 1.00000 3.11682 −0.609202 3.05569 5.46452 −1.07130
1.8 −0.802845 2.11595 −1.35544 1.00000 −1.69878 3.29759 2.69390 1.47723 −0.802845
1.9 −0.414001 −0.0596484 −1.82860 1.00000 0.0246945 1.37645 1.58505 −2.99644 −0.414001
1.10 0.0156266 −0.642037 −1.99976 1.00000 −0.0100329 −4.79628 −0.0625027 −2.58779 0.0156266
1.11 0.149185 −0.122146 −1.97774 1.00000 −0.0182224 −2.03839 −0.593420 −2.98508 0.149185
1.12 0.524389 2.55681 −1.72502 1.00000 1.34076 4.00960 −1.95336 3.53726 0.524389
1.13 0.763611 −3.39028 −1.41690 1.00000 −2.58885 3.22243 −2.60918 8.49398 0.763611
1.14 1.14097 3.32499 −0.698177 1.00000 3.79373 −2.66542 −3.07855 8.05557 1.14097
1.15 1.36079 −2.47539 −0.148263 1.00000 −3.36847 −4.84366 −2.92333 3.12754 1.36079
1.16 1.56300 −2.37379 0.442981 1.00000 −3.71025 4.14002 −2.43363 2.63489 1.56300
1.17 2.18281 1.66528 2.76464 1.00000 3.63498 1.10265 1.66907 −0.226842 2.18281
1.18 2.25541 3.24132 3.08686 1.00000 7.31049 2.29814 2.45132 7.50614 2.25541
1.19 2.26081 −0.979957 3.11127 1.00000 −2.21550 1.84324 2.51236 −2.03968 2.26081
1.20 2.53893 1.96468 4.44617 1.00000 4.98819 −1.49124 6.21065 0.859981 2.53893
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(1\)

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 2015.2.a.l ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.2.a.l ✓ 22 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\):

\( T_{2}^{22} - 5 T_{2}^{21} - 25 T_{2}^{20} + 157 T_{2}^{19} + 203 T_{2}^{18} - 2026 T_{2}^{17} + \cdots - 16 \) Copy content Toggle raw display
\( T_{3}^{22} - 4 T_{3}^{21} - 46 T_{3}^{20} + 195 T_{3}^{19} + 858 T_{3}^{18} - 3954 T_{3}^{17} + \cdots - 64 \) Copy content Toggle raw display