Properties

Label 13.8.a.a
Level $13$
Weight $8$
Character orbit 13.a
Self dual yes
Analytic conductor $4.061$
Analytic rank $1$
Dimension $1$
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] = [13,8,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(4.06100533129\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 10 q^{2} - 73 q^{3} - 28 q^{4} - 295 q^{5} - 730 q^{6} + 1373 q^{7} - 1560 q^{8} + 3142 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 10 q^{2} - 73 q^{3} - 28 q^{4} - 295 q^{5} - 730 q^{6} + 1373 q^{7} - 1560 q^{8} + 3142 q^{9} - 2950 q^{10} - 7646 q^{11} + 2044 q^{12} + 2197 q^{13} + 13730 q^{14} + 21535 q^{15} - 12016 q^{16} - 4147 q^{17} + 31420 q^{18} - 3186 q^{19} + 8260 q^{20} - 100229 q^{21} - 76460 q^{22} - 17784 q^{23} + 113880 q^{24} + 8900 q^{25} + 21970 q^{26} - 69715 q^{27} - 38444 q^{28} - 93322 q^{29} + 215350 q^{30} - 124484 q^{31} + 79520 q^{32} + 558158 q^{33} - 41470 q^{34} - 405035 q^{35} - 87976 q^{36} + 273661 q^{37} - 31860 q^{38} - 160381 q^{39} + 460200 q^{40} + 585816 q^{41} - 1002290 q^{42} - 533559 q^{43} + 214088 q^{44} - 926890 q^{45} - 177840 q^{46} - 530055 q^{47} + 877168 q^{48} + 1061586 q^{49} + 89000 q^{50} + 302731 q^{51} - 61516 q^{52} - 615288 q^{53} - 697150 q^{54} + 2255570 q^{55} - 2141880 q^{56} + 232578 q^{57} - 933220 q^{58} - 392514 q^{59} - 602980 q^{60} + 1878064 q^{61} - 1244840 q^{62} + 4313966 q^{63} + 2333248 q^{64} - 648115 q^{65} + 5581580 q^{66} - 3971438 q^{67} + 116116 q^{68} + 1298232 q^{69} - 4050350 q^{70} - 3746601 q^{71} - 4901520 q^{72} + 2485802 q^{73} + 2736610 q^{74} - 649700 q^{75} + 89208 q^{76} - 10497958 q^{77} - 1603810 q^{78} - 1264456 q^{79} + 3544720 q^{80} - 1782359 q^{81} + 5858160 q^{82} + 434308 q^{83} + 2806412 q^{84} + 1223365 q^{85} - 5335590 q^{86} + 6812506 q^{87} + 11927760 q^{88} + 5830810 q^{89} - 9268900 q^{90} + 3016481 q^{91} + 497952 q^{92} + 9087332 q^{93} - 5300550 q^{94} + 939870 q^{95} - 5804960 q^{96} - 2045330 q^{97} + 10615860 q^{98} - 24023732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
10.0000 −73.0000 −28.0000 −295.000 −730.000 1373.00 −1560.00 3142.00 −2950.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 13.8.a.a 1
3.b odd 2 1 117.8.a.a 1
4.b odd 2 1 208.8.a.d 1
5.b even 2 1 325.8.a.a 1
13.b even 2 1 169.8.a.a 1
13.d odd 4 2 169.8.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.a 1 1.a even 1 1 trivial
117.8.a.a 1 3.b odd 2 1
169.8.a.a 1 13.b even 2 1
169.8.b.a 2 13.d odd 4 2
208.8.a.d 1 4.b odd 2 1
325.8.a.a 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 10 \) Copy content Toggle raw display
$3$ \( T + 73 \) Copy content Toggle raw display
$5$ \( T + 295 \) Copy content Toggle raw display
$7$ \( T - 1373 \) Copy content Toggle raw display
$11$ \( T + 7646 \) Copy content Toggle raw display
$13$ \( T - 2197 \) Copy content Toggle raw display
$17$ \( T + 4147 \) Copy content Toggle raw display
$19$ \( T + 3186 \) Copy content Toggle raw display
$23$ \( T + 17784 \) Copy content Toggle raw display
$29$ \( T + 93322 \) Copy content Toggle raw display
$31$ \( T + 124484 \) Copy content Toggle raw display
$37$ \( T - 273661 \) Copy content Toggle raw display
$41$ \( T - 585816 \) Copy content Toggle raw display
$43$ \( T + 533559 \) Copy content Toggle raw display
$47$ \( T + 530055 \) Copy content Toggle raw display
$53$ \( T + 615288 \) Copy content Toggle raw display
$59$ \( T + 392514 \) Copy content Toggle raw display
$61$ \( T - 1878064 \) Copy content Toggle raw display
$67$ \( T + 3971438 \) Copy content Toggle raw display
$71$ \( T + 3746601 \) Copy content Toggle raw display
$73$ \( T - 2485802 \) Copy content Toggle raw display
$79$ \( T + 1264456 \) Copy content Toggle raw display
$83$ \( T - 434308 \) Copy content Toggle raw display
$89$ \( T - 5830810 \) Copy content Toggle raw display
$97$ \( T + 2045330 \) Copy content Toggle raw display
show more
show less