↑

Properties

Label 8047.2.a.b
Level $8047$
Weight $2$
Character orbit 8047.a
Self dual yes
Analytic conductor $64.256$
Analytic rank $1$
Dimension $142$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, 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("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(1\)
Dimension: \(142\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 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 −2.78532 −0.218444 5.75799 3.30077 0.608437 −0.196953 −10.4672 −2.95228 −9.19368
1.2 −2.78511 −1.73890 5.75683 −2.18885 4.84303 2.36535 −10.4632 0.0237834 6.09618
1.3 −2.74745 −2.94528 5.54847 1.95253 8.09201 3.66595 −9.74923 5.67469 −5.36448
1.4 −2.73051 1.50429 5.45570 −3.60561 −4.10747 2.19221 −9.43585 −0.737125 9.84515
1.5 −2.66381 1.33027 5.09589 −0.440221 −3.54360 2.49453 −8.24687 −1.23037 1.17267
1.6 −2.65986 −2.58294 5.07483 0.599360 6.87024 −3.90786 −8.17861 3.67156 −1.59421
1.7 −2.65232 2.65488 5.03481 −0.675897 −7.04159 −2.02772 −8.04928 4.04839 1.79269
1.8 −2.56842 0.368799 4.59680 −0.479572 −0.947232 4.01594 −6.66967 −2.86399 1.23174
1.9 −2.55403 1.62979 4.52309 1.02400 −4.16254 −2.90866 −6.44404 −0.343777 −2.61533
1.10 −2.52587 −3.25300 4.38004 −2.46689 8.21667 −3.38874 −6.01169 7.58202 6.23106
1.11 −2.52091 −0.134089 4.35498 2.66858 0.338025 −2.31114 −5.93668 −2.98202 −6.72724
1.12 −2.51095 −2.93259 4.30485 −3.06715 7.36357 3.82085 −5.78735 5.60007 7.70146
1.13 −2.50524 −0.199696 4.27622 −1.81098 0.500286 −1.67033 −5.70247 −2.96012 4.53694
1.14 −2.45771 2.98686 4.04034 −1.52748 −7.34083 5.18217 −5.01456 5.92132 3.75410
1.15 −2.43267 −0.839960 3.91786 −3.64522 2.04334 −2.93716 −4.66552 −2.29447 8.86761
1.16 −2.41564 2.60884 3.83533 1.90553 −6.30203 −0.0638029 −4.43349 3.80607 −4.60307
1.17 −2.27884 −1.20174 3.19313 −1.58295 2.73859 1.25281 −2.71895 −1.55581 3.60730
1.18 −2.23848 1.32587 3.01078 3.90699 −2.96792 −1.03210 −2.26261 −1.24208 −8.74571
1.19 −2.23735 −0.938044 3.00574 3.28954 2.09873 1.35040 −2.25019 −2.12007 −7.35985
1.20 −2.15511 1.98414 2.64449 −3.20612 −4.27605 1.44494 −1.38895 0.936828 6.90953
See next 80 embeddings (of 142 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.142
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(619\) \(-1\)

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 8047.2.a.b ✓ 142
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8047.2.a.b ✓ 142 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{142} + 13 T_{2}^{141} - 122 T_{2}^{140} - 2288 T_{2}^{139} + 4951 T_{2}^{138} + \cdots + 33720492213 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\). Copy content Toggle raw display