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Properties

Label 8015.2.a.i
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $1$
Dimension $44$
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] = [8015,2,Mod(1,8015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, 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("8015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8015.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: \(64.0000972201\)
Analytic rank: \(1\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} + 30 q^{4} + 44 q^{5} - 7 q^{6} - 44 q^{7} - 3 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{2} + 30 q^{4} + 44 q^{5} - 7 q^{6} - 44 q^{7} - 3 q^{8} + 16 q^{9} - 2 q^{10} - 15 q^{11} - 3 q^{12} - 17 q^{13} + 2 q^{14} + 10 q^{16} - 7 q^{17} - 16 q^{18} - 32 q^{19} + 30 q^{20} - 14 q^{22} + 8 q^{23} - 35 q^{24} + 44 q^{25} - 27 q^{26} + 6 q^{27} - 30 q^{28} - 42 q^{29} - 7 q^{30} - 43 q^{31} - 8 q^{32} - 33 q^{33} - 33 q^{34} - 44 q^{35} - 11 q^{36} - 44 q^{37} + 4 q^{38} + 3 q^{39} - 3 q^{40} - 62 q^{41} + 7 q^{42} - 7 q^{43} - 45 q^{44} + 16 q^{45} - 15 q^{46} + 2 q^{47} - 26 q^{48} + 44 q^{49} - 2 q^{50} - 25 q^{51} - 35 q^{52} - 25 q^{53} - 76 q^{54} - 15 q^{55} + 3 q^{56} - 7 q^{57} - 2 q^{58} - 35 q^{59} - 3 q^{60} - 86 q^{61} - 23 q^{62} - 16 q^{63} - 5 q^{64} - 17 q^{65} - 6 q^{66} + 2 q^{67} - q^{68} - 75 q^{69} + 2 q^{70} - 54 q^{71} - 3 q^{72} - 52 q^{73} - 22 q^{74} - 77 q^{76} + 15 q^{77} + 2 q^{78} + 46 q^{79} + 10 q^{80} - 72 q^{81} - 16 q^{82} + 26 q^{83} + 3 q^{84} - 7 q^{85} - 33 q^{86} - 8 q^{87} - 23 q^{88} - 105 q^{89} - 16 q^{90} + 17 q^{91} - 41 q^{92} - 11 q^{93} - 47 q^{94} - 32 q^{95} - 39 q^{96} - 80 q^{97} - 2 q^{98} - 53 q^{99}+O(q^{100}) \) 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 −2.79746 −0.603519 5.82580 1.00000 1.68832 −1.00000 −10.7025 −2.63576 −2.79746
1.2 −2.53948 2.62786 4.44894 1.00000 −6.67339 −1.00000 −6.21901 3.90566 −2.53948
1.3 −2.32161 −1.61048 3.38989 1.00000 3.73891 −1.00000 −3.22677 −0.406353 −2.32161
1.4 −2.27226 −1.26810 3.16315 1.00000 2.88144 −1.00000 −2.64298 −1.39193 −2.27226
1.5 −2.26027 2.68549 3.10882 1.00000 −6.06994 −1.00000 −2.50623 4.21188 −2.26027
1.6 −2.24100 −0.455413 3.02210 1.00000 1.02058 −1.00000 −2.29052 −2.79260 −2.24100
1.7 −2.23576 0.480979 2.99861 1.00000 −1.07535 −1.00000 −2.23266 −2.76866 −2.23576
1.8 −1.87978 3.08968 1.53356 1.00000 −5.80791 −1.00000 0.876807 6.54615 −1.87978
1.9 −1.84251 −2.60277 1.39485 1.00000 4.79563 −1.00000 1.11499 3.77440 −1.84251
1.10 −1.65863 1.21505 0.751045 1.00000 −2.01532 −1.00000 2.07155 −1.52365 −1.65863
1.11 −1.54745 −2.71325 0.394610 1.00000 4.19863 −1.00000 2.48426 4.36174 −1.54745
1.12 −1.41331 0.0185355 −0.00254977 1.00000 −0.0261965 −1.00000 2.83023 −2.99966 −1.41331
1.13 −1.25477 1.22958 −0.425541 1.00000 −1.54284 −1.00000 3.04351 −1.48814 −1.25477
1.14 −1.22585 2.00155 −0.497288 1.00000 −2.45360 −1.00000 3.06130 1.00619 −1.22585
1.15 −1.22552 −2.22296 −0.498095 1.00000 2.72428 −1.00000 3.06147 1.94153 −1.22552
1.16 −1.03648 −2.16444 −0.925705 1.00000 2.24341 −1.00000 3.03244 1.68482 −1.03648
1.17 −0.631186 1.05743 −1.60160 1.00000 −0.667437 −1.00000 2.27328 −1.88184 −0.631186
1.18 −0.612476 −1.57448 −1.62487 1.00000 0.964330 −1.00000 2.22015 −0.521024 −0.612476
1.19 −0.583301 −0.502435 −1.65976 1.00000 0.293071 −1.00000 2.13474 −2.74756 −0.583301
1.20 −0.508100 2.08594 −1.74183 1.00000 −1.05987 −1.00000 1.90123 1.35113 −0.508100
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(229\) \(-1\)

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 8015.2.a.i ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.i ✓ 44 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{44} + 2 T_{2}^{43} - 57 T_{2}^{42} - 113 T_{2}^{41} + 1495 T_{2}^{40} + 2937 T_{2}^{39} + \cdots - 583 \) Copy content Toggle raw display
\( T_{3}^{44} - 74 T_{3}^{42} - 2 T_{3}^{41} + 2522 T_{3}^{40} + 131 T_{3}^{39} - 52563 T_{3}^{38} + \cdots + 2560 \) Copy content Toggle raw display