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Properties

Label 4015.2.a.h
Level $4015$
Weight $2$
Character orbit 4015.a
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $37$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(0\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9} + 5 q^{10} + 37 q^{11} + 6 q^{12} + 11 q^{13} + 11 q^{14} + 3 q^{15} + 43 q^{16} + 38 q^{17} + 11 q^{18} + 34 q^{19} + 43 q^{20} + 39 q^{21} + 5 q^{22} + 4 q^{23} + 25 q^{24} + 37 q^{25} - 9 q^{26} + 3 q^{27} + 14 q^{28} + 58 q^{29} + 9 q^{30} + 8 q^{31} + 14 q^{32} + 3 q^{33} + 8 q^{34} + 6 q^{35} + 20 q^{36} + 2 q^{37} + 15 q^{38} + 14 q^{39} + 12 q^{40} + 62 q^{41} - 13 q^{42} + 30 q^{43} + 43 q^{44} + 50 q^{45} + 31 q^{46} + 5 q^{47} - 25 q^{48} + 59 q^{49} + 5 q^{50} + 23 q^{51} - q^{52} + 18 q^{53} + 13 q^{54} + 37 q^{55} + 22 q^{56} + 5 q^{57} - 40 q^{58} + 15 q^{59} + 6 q^{60} + 57 q^{61} + 20 q^{62} - 29 q^{63} + 10 q^{64} + 11 q^{65} + 9 q^{66} - 14 q^{67} + 53 q^{68} + 24 q^{69} + 11 q^{70} + 8 q^{71} + 15 q^{72} + 37 q^{73} + 7 q^{74} + 3 q^{75} + 59 q^{76} + 6 q^{77} + q^{78} + 42 q^{79} + 43 q^{80} + 61 q^{81} - 22 q^{82} + 44 q^{83} + 66 q^{84} + 38 q^{85} - q^{86} - 26 q^{87} + 12 q^{88} + 69 q^{89} + 11 q^{90} - 10 q^{91} - 21 q^{92} - 26 q^{93} + 29 q^{94} + 34 q^{95} - 9 q^{96} + 37 q^{97} - 15 q^{98} + 50 q^{99}+O(q^{100}) \) 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 −2.70375 −1.27224 5.31028 1.00000 3.43983 −0.968765 −8.95019 −1.38140 −2.70375
1.2 −2.58511 −0.179292 4.68277 1.00000 0.463488 2.09214 −6.93525 −2.96785 −2.58511
1.3 −2.53170 2.11924 4.40953 1.00000 −5.36530 2.19751 −6.10021 1.49120 −2.53170
1.4 −2.41139 −3.18109 3.81483 1.00000 7.67086 −3.73754 −4.37626 7.11932 −2.41139
1.5 −2.14348 2.01146 2.59449 1.00000 −4.31152 −3.47503 −1.27427 1.04598 −2.14348
1.6 −1.99859 2.62046 1.99434 1.00000 −5.23721 2.62346 0.0113035 3.86680 −1.99859
1.7 −1.93175 −1.55022 1.73167 1.00000 2.99464 5.03826 0.518349 −0.596820 −1.93175
1.8 −1.86942 −2.63227 1.49473 1.00000 4.92082 −2.31866 0.944554 3.92884 −1.86942
1.9 −1.82541 0.572536 1.33213 1.00000 −1.04512 −3.97729 1.21914 −2.67220 −1.82541
1.10 −1.58931 −0.358295 0.525917 1.00000 0.569443 2.77249 2.34278 −2.87162 −1.58931
1.11 −1.16251 −1.28072 −0.648575 1.00000 1.48884 −0.641997 3.07899 −1.35976 −1.16251
1.12 −1.11064 3.24263 −0.766474 1.00000 −3.60140 2.31774 3.07256 7.51466 −1.11064
1.13 −0.745458 3.05905 −1.44429 1.00000 −2.28039 −3.66055 2.56757 6.35779 −0.745458
1.14 −0.742421 0.782455 −1.44881 1.00000 −0.580911 0.262449 2.56047 −2.38776 −0.742421
1.15 −0.446590 1.08482 −1.80056 1.00000 −0.484472 4.20380 1.69729 −1.82316 −0.446590
1.16 −0.421939 −2.05594 −1.82197 1.00000 0.867482 −0.0580639 1.61264 1.22690 −0.421939
1.17 −0.327392 −2.36457 −1.89281 1.00000 0.774141 2.46288 1.27448 2.59119 −0.327392
1.18 −0.199490 −3.23393 −1.96020 1.00000 0.645135 −5.08849 0.790020 7.45829 −0.199490
1.19 0.132711 1.97030 −1.98239 1.00000 0.261480 2.54454 −0.528506 0.882080 0.132711
1.20 0.503541 0.0692183 −1.74645 1.00000 0.0348542 −2.92313 −1.88649 −2.99521 0.503541
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(73\) \(-1\)

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 4015.2.a.h ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4015.2.a.h ✓ 37 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{37} - 5 T_{2}^{36} - 46 T_{2}^{35} + 261 T_{2}^{34} + 910 T_{2}^{33} - 6168 T_{2}^{32} + \cdots - 7872 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\). Copy content Toggle raw display