Properties

Label 13.8.a.c
Level $13$
Weight $8$
Character orbit 13.a
Self dual yes
Analytic conductor $4.061$
Analytic rank $0$
Dimension $4$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 354x^{2} - 640x + 20912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 4) q^{2} + (\beta_{2} - 2 \beta_1 + 21) q^{3} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 63) q^{4} + ( - 6 \beta_{3} - \beta_{2} + \cdots + 63) q^{5}+ \cdots + (18 \beta_{3} + 31 \beta_{2} + \cdots + 882) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 4) q^{2} + (\beta_{2} - 2 \beta_1 + 21) q^{3} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 63) q^{4} + ( - 6 \beta_{3} - \beta_{2} + \cdots + 63) q^{5}+ \cdots + (136548 \beta_{3} - 77786 \beta_{2} + \cdots + 1572948) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16} + 11814 q^{17} - 2738 q^{18} + 27660 q^{19} - 30369 q^{20} + 69930 q^{21} + 59930 q^{22} + 172920 q^{23} + 26553 q^{24} + 219414 q^{25} - 32955 q^{26} + 174800 q^{27} - 170903 q^{28} + 133344 q^{29} - 624695 q^{30} - 231748 q^{31} - 459291 q^{32} - 33868 q^{33} - 480249 q^{34} + 322296 q^{35} - 157930 q^{36} + 248026 q^{37} - 85326 q^{38} - 175760 q^{39} + 379895 q^{40} + 588108 q^{41} + 132275 q^{42} + 309304 q^{43} + 1270806 q^{44} - 1773144 q^{45} + 574948 q^{46} + 557916 q^{47} + 837429 q^{48} - 767378 q^{49} + 5737920 q^{50} - 1816496 q^{51} - 555841 q^{52} + 2022348 q^{53} - 479 q^{54} - 3770208 q^{55} + 1391625 q^{56} - 5609844 q^{57} - 3027194 q^{58} + 1162668 q^{59} - 8100077 q^{60} - 1340572 q^{61} + 2715564 q^{62} + 4213476 q^{63} - 6734935 q^{64} - 566826 q^{65} + 8124226 q^{66} - 598484 q^{67} + 4515993 q^{68} - 108848 q^{69} - 4329415 q^{70} + 697860 q^{71} + 1607430 q^{72} - 13725816 q^{73} + 14884053 q^{74} + 16604592 q^{75} + 12055070 q^{76} - 5326932 q^{77} - 3469063 q^{78} + 20079576 q^{79} + 3710667 q^{80} - 8368396 q^{81} + 857160 q^{82} - 2024724 q^{83} - 13603483 q^{84} - 16146514 q^{85} - 22922499 q^{86} + 22170952 q^{87} - 2100618 q^{88} + 17646240 q^{89} - 42372890 q^{90} - 3717324 q^{91} + 14504892 q^{92} - 17910680 q^{93} - 3729785 q^{94} + 12508152 q^{95} - 37637751 q^{96} - 6329096 q^{97} - 22166262 q^{98} + 6200852 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 354x^{2} - 640x + 20912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 15\nu^{2} - 180\nu + 1956 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 13\nu^{2} - 188\nu + 1606 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta_{2} + 4\beta _1 + 175 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 15\beta_{3} - 26\beta_{2} + 240\beta _1 + 669 \) 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
18.6191
7.26058
−12.1994
−12.6802
−14.6191 −51.4405 85.7173 123.659 752.012 559.667 618.134 459.121 −1807.78
1.2 −3.26058 66.7551 −117.369 259.973 −217.660 1453.99 800.043 2269.24 −847.662
1.3 16.1994 71.3782 134.421 −532.467 1156.29 −6.33667 104.021 2907.85 −8625.66
1.4 16.6802 −6.69280 150.230 406.835 −111.637 −315.324 370.802 −2142.21 6786.11
\(n\): 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.c 4
3.b odd 2 1 117.8.a.e 4
4.b odd 2 1 208.8.a.k 4
5.b even 2 1 325.8.a.c 4
13.b even 2 1 169.8.a.c 4
13.d odd 4 2 169.8.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.c 4 1.a even 1 1 trivial
117.8.a.e 4 3.b odd 2 1
169.8.a.c 4 13.b even 2 1
169.8.b.c 8 13.d odd 4 2
208.8.a.k 4 4.b odd 2 1
325.8.a.c 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 15T_{2}^{3} - 270T_{2}^{2} + 3264T_{2} + 12880 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 15 T^{3} + \cdots + 12880 \) Copy content Toggle raw display
$3$ \( T^{4} - 80 T^{3} + \cdots + 1640448 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots - 6964113500 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 1625961532 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 19773464676784 \) Copy content Toggle raw display
$13$ \( (T + 2197)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 43\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 32\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 83\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 28\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 44\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 57\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 25\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
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