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Properties

Label 8015.2.a.h
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $1$
Dimension $38$
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: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 6 q^{2} - 9 q^{3} + 24 q^{4} + 38 q^{5} - 10 q^{6} + 38 q^{7} - 21 q^{8} + 13 q^{9} - 6 q^{10} - 18 q^{11} - 20 q^{12} - 25 q^{13} - 6 q^{14} - 9 q^{15} - 21 q^{17} - 7 q^{18} - 14 q^{19} + 24 q^{20} - 9 q^{21} - 11 q^{22} - 16 q^{23} - 10 q^{24} + 38 q^{25} - 21 q^{26} - 15 q^{27} + 24 q^{28} - 52 q^{29} - 10 q^{30} - 30 q^{31} - 8 q^{32} - 13 q^{33} - 9 q^{34} + 38 q^{35} - 16 q^{36} - 47 q^{37} - 10 q^{38} - 50 q^{39} - 21 q^{40} - 35 q^{41} - 10 q^{42} - 18 q^{43} - 22 q^{44} + 13 q^{45} - 17 q^{46} - 23 q^{47} - 13 q^{48} + 38 q^{49} - 6 q^{50} - 5 q^{51} - 41 q^{52} - 12 q^{53} + 35 q^{54} - 18 q^{55} - 21 q^{56} - 3 q^{57} - 16 q^{58} - 16 q^{59} - 20 q^{60} - 47 q^{61} + 14 q^{62} + 13 q^{63} - 55 q^{64} - 25 q^{65} - 6 q^{66} - 54 q^{67} + 31 q^{68} - 95 q^{69} - 6 q^{70} - 41 q^{71} - 59 q^{73} - q^{74} - 9 q^{75} - 23 q^{76} - 18 q^{77} + 19 q^{78} - 117 q^{79} - 38 q^{81} + 19 q^{82} - 21 q^{83} - 20 q^{84} - 21 q^{85} - 12 q^{86} - 14 q^{87} - 40 q^{88} - 96 q^{89} - 7 q^{90} - 25 q^{91} + 53 q^{92} - 53 q^{93} - 28 q^{94} - 14 q^{95} + 40 q^{96} - 70 q^{97} - 6 q^{98} - 52 q^{99}+O(q^{100}) \) 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 −2.60567 −1.96600 4.78952 1.00000 5.12274 1.00000 −7.26856 0.865151 −2.60567
1.2 −2.59504 0.990389 4.73422 1.00000 −2.57010 1.00000 −7.09539 −2.01913 −2.59504
1.3 −2.51011 −2.35176 4.30066 1.00000 5.90318 1.00000 −5.77493 2.53077 −2.51011
1.4 −2.46620 0.936734 4.08216 1.00000 −2.31018 1.00000 −5.13504 −2.12253 −2.46620
1.5 −2.38102 2.43622 3.66928 1.00000 −5.80069 1.00000 −3.97459 2.93515 −2.38102
1.6 −2.02615 0.680619 2.10529 1.00000 −1.37904 1.00000 −0.213342 −2.53676 −2.02615
1.7 −1.89678 −2.50257 1.59778 1.00000 4.74683 1.00000 0.762920 3.26285 −1.89678
1.8 −1.84626 −2.48565 1.40867 1.00000 4.58916 1.00000 1.09175 3.17847 −1.84626
1.9 −1.83849 1.40018 1.38005 1.00000 −2.57423 1.00000 1.13977 −1.03949 −1.83849
1.10 −1.74109 −0.00460406 1.03141 1.00000 0.00801611 1.00000 1.68640 −2.99998 −1.74109
1.11 −1.66719 2.43876 0.779527 1.00000 −4.06588 1.00000 2.03476 2.94756 −1.66719
1.12 −1.65687 −0.430057 0.745215 1.00000 0.712549 1.00000 2.07901 −2.81505 −1.65687
1.13 −1.23429 −1.22031 −0.476531 1.00000 1.50621 1.00000 3.05675 −1.51085 −1.23429
1.14 −1.08292 −0.496145 −0.827289 1.00000 0.537284 1.00000 3.06172 −2.75384 −1.08292
1.15 −0.697175 1.52901 −1.51395 1.00000 −1.06599 1.00000 2.44984 −0.662121 −0.697175
1.16 −0.656564 2.76581 −1.56892 1.00000 −1.81593 1.00000 2.34323 4.64973 −0.656564
1.17 −0.520713 −3.35310 −1.72886 1.00000 1.74600 1.00000 1.94166 8.24329 −0.520713
1.18 −0.217856 −2.35998 −1.95254 1.00000 0.514135 1.00000 0.861085 2.56949 −0.217856
1.19 −0.0912753 1.39308 −1.99167 1.00000 −0.127154 1.00000 0.364341 −1.05934 −0.0912753
1.20 −0.0830831 2.10839 −1.99310 1.00000 −0.175171 1.00000 0.331759 1.44530 −0.0830831
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
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.h ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.h ✓ 38 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{38} + 6 T_{2}^{37} - 32 T_{2}^{36} - 249 T_{2}^{35} + 382 T_{2}^{34} + 4672 T_{2}^{33} - 1265 T_{2}^{32} + \cdots + 3 \) Copy content Toggle raw display
\( T_{3}^{38} + 9 T_{3}^{37} - 23 T_{3}^{36} - 427 T_{3}^{35} - 266 T_{3}^{34} + 8797 T_{3}^{33} + \cdots - 80 \) Copy content Toggle raw display