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Properties

Label 4015.2.a.d
Level $4015$
Weight $2$
Character orbit 4015.a
Self dual yes
Analytic conductor $32.060$
Analytic rank $0$
Dimension $27$
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: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 2 q^{2} + 3 q^{3} + 22 q^{4} - 27 q^{5} + 5 q^{6} + 3 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q + 2 q^{2} + 3 q^{3} + 22 q^{4} - 27 q^{5} + 5 q^{6} + 3 q^{8} + 24 q^{9} - 2 q^{10} - 27 q^{11} + 6 q^{12} + 3 q^{13} + 15 q^{14} - 3 q^{15} + 12 q^{16} + 32 q^{17} + 4 q^{18} - 20 q^{19} - 22 q^{20} + 2 q^{21} - 2 q^{22} + 6 q^{23} + 3 q^{24} + 27 q^{25} + 21 q^{26} + 15 q^{27} + 14 q^{28} + 6 q^{29} - 5 q^{30} + 10 q^{31} + 23 q^{32} - 3 q^{33} - 8 q^{34} + 13 q^{36} + 8 q^{37} + 27 q^{38} + 6 q^{39} - 3 q^{40} + 44 q^{41} + 67 q^{42} - 11 q^{43} - 22 q^{44} - 24 q^{45} - q^{46} + 31 q^{47} - 9 q^{48} + 9 q^{49} + 2 q^{50} + 23 q^{51} + 15 q^{52} + 46 q^{53} + 37 q^{54} + 27 q^{55} + 50 q^{56} + 15 q^{57} + 32 q^{58} + 15 q^{59} - 6 q^{60} - 11 q^{61} + 12 q^{62} + 29 q^{63} - 5 q^{64} - 3 q^{65} - 5 q^{66} + 33 q^{68} - 8 q^{69} - 15 q^{70} + 22 q^{71} + 72 q^{72} + 27 q^{73} + 33 q^{74} + 3 q^{75} - 11 q^{76} + 41 q^{78} - 6 q^{79} - 12 q^{80} + 7 q^{81} + 10 q^{82} + 42 q^{83} + 50 q^{84} - 32 q^{85} + 25 q^{86} - 21 q^{87} - 3 q^{88} + 89 q^{89} - 4 q^{90} - 22 q^{91} + 27 q^{92} + 22 q^{93} + 3 q^{94} + 20 q^{95} - 23 q^{96} + 57 q^{97} + 44 q^{98} - 24 q^{99}+O(q^{100}) \) 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 −2.51893 0.286970 4.34502 −1.00000 −0.722857 −1.12829 −5.90694 −2.91765 2.51893
1.2 −2.48708 −1.54524 4.18555 −1.00000 3.84314 −0.0646689 −5.43564 −0.612227 2.48708
1.3 −2.34234 2.45447 3.48657 −1.00000 −5.74921 −4.19016 −3.48205 3.02442 2.34234
1.4 −2.16109 1.09070 2.67032 −1.00000 −2.35711 3.24111 −1.44862 −1.81037 2.16109
1.5 −1.97880 −2.04684 1.91564 −1.00000 4.05029 −1.79662 0.166923 1.18956 1.97880
1.6 −1.83742 −2.64130 1.37612 −1.00000 4.85318 3.42628 1.14634 3.97646 1.83742
1.7 −1.32977 3.37143 −0.231701 −1.00000 −4.48324 1.05432 2.96766 8.36656 1.32977
1.8 −1.19427 −0.177671 −0.573715 −1.00000 0.212188 −3.85321 3.07372 −2.96843 1.19427
1.9 −1.14525 1.84204 −0.688402 −1.00000 −2.10960 1.31161 3.07889 0.393113 1.14525
1.10 −0.825600 −2.10128 −1.31839 −1.00000 1.73482 0.894122 2.73966 1.41539 0.825600
1.11 −0.694873 1.33419 −1.51715 −1.00000 −0.927094 −1.39137 2.44397 −1.21993 0.694873
1.12 −0.424275 −3.22417 −1.81999 −1.00000 1.36794 −0.404433 1.62073 7.39528 0.424275
1.13 −0.0970376 2.11047 −1.99058 −1.00000 −0.204795 −4.33197 0.387237 1.45410 0.0970376
1.14 0.242803 0.0860286 −1.94105 −1.00000 0.0208880 5.09755 −0.956898 −2.99260 −0.242803
1.15 0.295380 −0.929145 −1.91275 −1.00000 −0.274451 1.99781 −1.15575 −2.13669 −0.295380
1.16 0.775213 1.34744 −1.39904 −1.00000 1.04455 −2.13043 −2.63498 −1.18440 −0.775213
1.17 0.893971 −2.95026 −1.20082 −1.00000 −2.63745 −0.0849500 −2.86144 5.70403 −0.893971
1.18 1.00837 2.30979 −0.983185 −1.00000 2.32913 0.123981 −3.00816 2.33512 −1.00837
1.19 1.20214 −0.184405 −0.554868 −1.00000 −0.221680 −0.735046 −3.07130 −2.96599 −1.20214
1.20 1.43653 −0.576854 0.0636182 −1.00000 −0.828668 −4.80506 −2.78167 −2.66724 −1.43653
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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.d ✓ 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4015.2.a.d ✓ 27 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} - 2 T_{2}^{26} - 36 T_{2}^{25} + 71 T_{2}^{24} + 564 T_{2}^{23} - 1099 T_{2}^{22} - 5050 T_{2}^{21} + \cdots + 85 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\). Copy content Toggle raw display