↑

Properties

Label 8005.2.a.e
Level $8005$
Weight $2$
Character orbit 8005.a
Self dual yes
Analytic conductor $63.920$
Analytic rank $1$
Dimension $126$
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] = [8005,2,Mod(1,8005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8005 = 5 \cdot 1601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8005.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: \(63.9202468180\)
Analytic rank: \(1\)
Dimension: \(126\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9} - 15 q^{10} - 35 q^{11} - 86 q^{12} - 36 q^{13} - 31 q^{14} - 46 q^{15} + 101 q^{16} - 62 q^{17} - 45 q^{18} - 29 q^{19} + 119 q^{20} - 16 q^{21} - 67 q^{22} - 107 q^{23} - 9 q^{24} + 126 q^{25} - 53 q^{26} - 181 q^{27} - 100 q^{28} - 39 q^{29} - 10 q^{30} - 29 q^{31} - 99 q^{32} - 72 q^{33} - 18 q^{34} - 60 q^{35} + 93 q^{36} - 72 q^{37} - 93 q^{38} - 8 q^{39} - 57 q^{40} - 28 q^{41} - 22 q^{42} - 103 q^{43} - 47 q^{44} + 112 q^{45} + q^{46} - 130 q^{47} - 134 q^{48} + 116 q^{49} - 15 q^{50} - 46 q^{51} - 117 q^{52} - 103 q^{53} + 11 q^{54} - 35 q^{55} - 84 q^{56} - 70 q^{57} - 77 q^{58} - 219 q^{59} - 86 q^{60} - 4 q^{61} - 77 q^{62} - 145 q^{63} + 51 q^{64} - 36 q^{65} + 16 q^{66} - 150 q^{67} - 130 q^{68} - 21 q^{69} - 31 q^{70} - 92 q^{71} - 115 q^{72} - 79 q^{73} - 57 q^{74} - 46 q^{75} - 38 q^{76} - 75 q^{77} - 23 q^{78} - 20 q^{79} + 101 q^{80} + 142 q^{81} - 63 q^{82} - 243 q^{83} - 2 q^{84} - 62 q^{85} - 14 q^{86} - 107 q^{87} - 121 q^{88} - 84 q^{89} - 45 q^{90} - 82 q^{91} - 228 q^{92} - 149 q^{93} - 29 q^{95} + 38 q^{96} - 85 q^{97} - 48 q^{98} - 33 q^{99}+O(q^{100}) \) 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 −2.77468 −2.03994 5.69885 1.00000 5.66018 −0.694570 −10.2631 1.16135 −2.77468
1.2 −2.76854 −3.01214 5.66482 1.00000 8.33923 −2.50261 −10.1462 6.07298 −2.76854
1.3 −2.76679 0.0783895 5.65511 1.00000 −0.216887 3.44587 −10.1129 −2.99386 −2.76679
1.4 −2.75799 0.297267 5.60652 1.00000 −0.819859 −0.723666 −9.94674 −2.91163 −2.75799
1.5 −2.69183 −0.785166 5.24594 1.00000 2.11353 −4.09093 −8.73752 −2.38351 −2.69183
1.6 −2.68284 2.48419 5.19763 1.00000 −6.66470 0.389624 −8.57874 3.17122 −2.68284
1.7 −2.65121 2.20817 5.02889 1.00000 −5.85432 1.13274 −8.03022 1.87603 −2.65121
1.8 −2.60227 −2.65544 4.77178 1.00000 6.91016 2.75161 −7.21291 4.05137 −2.60227
1.9 −2.59696 −3.31155 4.74420 1.00000 8.59996 3.53678 −7.12657 7.96636 −2.59696
1.10 −2.55329 −1.12143 4.51928 1.00000 2.86335 −4.52576 −6.43246 −1.74239 −2.55329
1.11 −2.45517 1.07973 4.02786 1.00000 −2.65092 3.17620 −4.97873 −1.83418 −2.45517
1.12 −2.44890 0.480075 3.99711 1.00000 −1.17566 −2.53410 −4.89071 −2.76953 −2.44890
1.13 −2.42783 −3.02578 3.89435 1.00000 7.34609 −3.63249 −4.59917 6.15537 −2.42783
1.14 −2.42700 2.14529 3.89034 1.00000 −5.20661 0.410687 −4.58786 1.60225 −2.42700
1.15 −2.36778 2.79221 3.60639 1.00000 −6.61135 −4.83880 −3.80359 4.79644 −2.36778
1.16 −2.32335 0.155880 3.39795 1.00000 −0.362164 0.705093 −3.24794 −2.97570 −2.32335
1.17 −2.30625 1.53084 3.31877 1.00000 −3.53050 −2.53384 −3.04142 −0.656523 −2.30625
1.18 −2.30618 −3.20957 3.31845 1.00000 7.40183 −1.98846 −3.04058 7.30133 −2.30618
1.19 −2.23772 −0.862309 3.00741 1.00000 1.92961 4.70956 −2.25429 −2.25642 −2.23772
1.20 −2.18184 −0.884770 2.76044 1.00000 1.93043 1.24169 −1.65916 −2.21718 −2.18184
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.126
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1601\) \(-1\)

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 8005.2.a.e ✓ 126
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8005.2.a.e ✓ 126 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):

\( T_{2}^{126} + 15 T_{2}^{125} - 73 T_{2}^{124} - 2181 T_{2}^{123} - 1425 T_{2}^{122} + 150549 T_{2}^{121} + \cdots + 61165793 \) Copy content Toggle raw display
\( T_{3}^{126} + 46 T_{3}^{125} + 813 T_{3}^{124} + 5105 T_{3}^{123} - 36484 T_{3}^{122} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display