↑

Properties

Label 8015.2.a.j
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $1$
Dimension $45$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(45\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9} + 6 q^{10} - q^{11} - 3 q^{12} - 21 q^{13} - 6 q^{14} + 8 q^{16} - 7 q^{17} - 36 q^{18} - 20 q^{19} - 34 q^{20} - 34 q^{22} - 22 q^{23} - 11 q^{24} + 45 q^{25} - q^{26} + 12 q^{27} + 34 q^{28} + 10 q^{29} - q^{30} - 27 q^{31} - 26 q^{32} - 39 q^{33} - 13 q^{34} - 45 q^{35} - 3 q^{36} - 72 q^{37} + 2 q^{38} - 37 q^{39} + 15 q^{40} - 4 q^{41} + q^{42} - 49 q^{43} + 5 q^{44} - 29 q^{45} - 67 q^{46} + 2 q^{47} + 8 q^{48} + 45 q^{49} - 6 q^{50} - 49 q^{51} - 47 q^{52} - 35 q^{53} - 12 q^{54} + q^{55} - 15 q^{56} - 77 q^{57} - 50 q^{58} + 4 q^{59} + 3 q^{60} - 36 q^{61} + 17 q^{62} + 29 q^{63} + 5 q^{64} + 21 q^{65} - 8 q^{66} - 80 q^{67} + 27 q^{68} + 9 q^{69} + 6 q^{70} - 12 q^{71} - 97 q^{72} - 55 q^{73} + 32 q^{74} - 37 q^{76} - q^{77} + 20 q^{78} - 94 q^{79} - 8 q^{80} - 19 q^{81} - 36 q^{82} + 24 q^{83} - 3 q^{84} + 7 q^{85} - 3 q^{86} - 4 q^{87} - 95 q^{88} + q^{89} + 36 q^{90} - 21 q^{91} - 65 q^{92} - 71 q^{93} - 53 q^{94} + 20 q^{95} - 13 q^{96} - 110 q^{97} - 6 q^{98} - 27 q^{99}+O(q^{100}) \) 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 −2.78224 −1.94372 5.74083 −1.00000 5.40788 1.00000 −10.4079 0.778035 2.78224
1.2 −2.58820 1.46821 4.69879 −1.00000 −3.80002 1.00000 −6.98503 −0.844367 2.58820
1.3 −2.51360 3.19000 4.31820 −1.00000 −8.01840 1.00000 −5.82702 7.17611 2.51360
1.4 −2.46499 0.580032 4.07617 −1.00000 −1.42977 1.00000 −5.11774 −2.66356 2.46499
1.5 −2.28900 0.276088 3.23954 −1.00000 −0.631967 1.00000 −2.83731 −2.92378 2.28900
1.6 −2.26339 2.16114 3.12292 −1.00000 −4.89149 1.00000 −2.54160 1.67051 2.26339
1.7 −2.24619 −2.43741 3.04539 −1.00000 5.47490 1.00000 −2.34814 2.94098 2.24619
1.8 −2.21589 −3.14559 2.91017 −1.00000 6.97028 1.00000 −2.01683 6.89472 2.21589
1.9 −1.81008 −0.874461 1.27639 −1.00000 1.58284 1.00000 1.30979 −2.23532 1.81008
1.10 −1.69745 0.851741 0.881321 −1.00000 −1.44578 1.00000 1.89890 −2.27454 1.69745
1.11 −1.67660 −1.48567 0.810994 −1.00000 2.49087 1.00000 1.99349 −0.792790 1.67660
1.12 −1.65958 −1.94100 0.754194 −1.00000 3.22125 1.00000 2.06751 0.767498 1.65958
1.13 −1.60365 2.74735 0.571694 −1.00000 −4.40579 1.00000 2.29050 4.54792 1.60365
1.14 −1.48089 −0.908664 0.193023 −1.00000 1.34563 1.00000 2.67593 −2.17433 1.48089
1.15 −1.27565 −2.40768 −0.372716 −1.00000 3.07136 1.00000 3.02676 2.79693 1.27565
1.16 −1.13794 1.06234 −0.705083 −1.00000 −1.20889 1.00000 3.07823 −1.87143 1.13794
1.17 −0.903684 3.04554 −1.18335 −1.00000 −2.75220 1.00000 2.87675 6.27529 0.903684
1.18 −0.883459 2.68305 −1.21950 −1.00000 −2.37036 1.00000 2.84430 4.19874 0.883459
1.19 −0.647127 −0.415940 −1.58123 −1.00000 0.269166 1.00000 2.31751 −2.82699 0.647127
1.20 −0.556743 −0.855264 −1.69004 −1.00000 0.476163 1.00000 2.05440 −2.26852 0.556743
See all 45 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.45
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.j ✓ 45
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.j ✓ 45 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{45} + 6 T_{2}^{44} - 44 T_{2}^{43} - 323 T_{2}^{42} + 795 T_{2}^{41} + 7988 T_{2}^{40} + \cdots + 104 \) Copy content Toggle raw display
\( T_{3}^{45} - 82 T_{3}^{43} - 4 T_{3}^{42} + 3099 T_{3}^{41} + 295 T_{3}^{40} - 71657 T_{3}^{39} + \cdots + 2816 \) Copy content Toggle raw display