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Properties

Label 8015.2.a.n
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $0$
Dimension $68$
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: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 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 −2.76919 2.35151 5.66840 −1.00000 −6.51177 1.00000 −10.1585 2.52960 2.76919
1.2 −2.73044 0.281096 5.45528 −1.00000 −0.767515 1.00000 −9.43443 −2.92099 2.73044
1.3 −2.66015 −2.46219 5.07642 −1.00000 6.54979 1.00000 −8.18374 3.06236 2.66015
1.4 −2.55606 −1.84104 4.53346 −1.00000 4.70581 1.00000 −6.47568 0.389420 2.55606
1.5 −2.40811 3.06409 3.79900 −1.00000 −7.37867 1.00000 −4.33220 6.38865 2.40811
1.6 −2.40730 0.0150324 3.79510 −1.00000 −0.0361876 1.00000 −4.32135 −2.99977 2.40730
1.7 −2.33888 −1.00822 3.47038 −1.00000 2.35811 1.00000 −3.43905 −1.98350 2.33888
1.8 −2.33656 2.32181 3.45950 −1.00000 −5.42504 1.00000 −3.41021 2.39080 2.33656
1.9 −2.27050 −2.16931 3.15517 −1.00000 4.92541 1.00000 −2.62281 1.70590 2.27050
1.10 −2.17774 0.311923 2.74253 −1.00000 −0.679286 1.00000 −1.61704 −2.90270 2.17774
1.11 −2.12688 0.933356 2.52363 −1.00000 −1.98514 1.00000 −1.11369 −2.12885 2.12688
1.12 −2.04947 −1.65528 2.20033 −1.00000 3.39245 1.00000 −0.410568 −0.260041 2.04947
1.13 −1.92788 −3.19402 1.71671 −1.00000 6.15768 1.00000 0.546142 7.20176 1.92788
1.14 −1.76060 2.09671 1.09972 −1.00000 −3.69147 1.00000 1.58503 1.39619 1.76060
1.15 −1.66934 1.36319 0.786696 −1.00000 −2.27563 1.00000 2.02542 −1.14171 1.66934
1.16 −1.64550 3.30855 0.707660 −1.00000 −5.44420 1.00000 2.12654 7.94648 1.64550
1.17 −1.48933 −2.36326 0.218100 −1.00000 3.51967 1.00000 2.65384 2.58500 1.48933
1.18 −1.39879 −1.98679 −0.0433739 −1.00000 2.77911 1.00000 2.85826 0.947327 1.39879
1.19 −1.37224 1.58108 −0.116968 −1.00000 −2.16961 1.00000 2.90498 −0.500189 1.37224
1.20 −1.36878 0.541505 −0.126449 −1.00000 −0.741199 1.00000 2.91064 −2.70677 1.36878
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.68
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.n ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.n ✓ 68 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{68} - 9 T_{2}^{67} - 69 T_{2}^{66} + 842 T_{2}^{65} + 1744 T_{2}^{64} - 37136 T_{2}^{63} + \cdots - 15565411 \) Copy content Toggle raw display
\( T_{3}^{68} - 145 T_{3}^{66} - 4 T_{3}^{65} + 9978 T_{3}^{64} + 547 T_{3}^{63} - 433678 T_{3}^{62} + \cdots - 504199168 \) Copy content Toggle raw display