Properties

Label 13.8.b.a
Level $13$
Weight $8$
Character orbit 13.b
Analytic conductor $4.061$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,8,Mod(12,13)]
 
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([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.12");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 13.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.06100533129\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 449x^{4} + 37224x^{2} + 205776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 9) q^{3} + ( - \beta_{4} - 22) q^{4} + \beta_{5} q^{5} + ( - \beta_{3} - 10 \beta_1) q^{6} + (\beta_{5} - \beta_{3} - 11 \beta_1) q^{7} + ( - 2 \beta_{5} - 3 \beta_{3} - 26 \beta_1) q^{8} + ( - 6 \beta_{4} - 5 \beta_{2} - 192) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 9) q^{3} + ( - \beta_{4} - 22) q^{4} + \beta_{5} q^{5} + ( - \beta_{3} - 10 \beta_1) q^{6} + (\beta_{5} - \beta_{3} - 11 \beta_1) q^{7} + ( - 2 \beta_{5} - 3 \beta_{3} - 26 \beta_1) q^{8} + ( - 6 \beta_{4} - 5 \beta_{2} - 192) q^{9} + ( - 7 \beta_{4} + 6 \beta_{2} - 72) q^{10} + (12 \beta_{5} + 9 \beta_{3} + 133 \beta_1) q^{11} + (33 \beta_{4} - 44 \beta_{2} + 342) q^{12} + ( - 13 \beta_{5} - 26 \beta_{4} + \cdots - 832) q^{13}+ \cdots + (1428 \beta_{5} - 4139 \beta_{3} - 414983 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29} - 60114 q^{30} - 546720 q^{35} + 747772 q^{36} - 243492 q^{38} + 511160 q^{39} + 792058 q^{40} - 1138134 q^{42} - 1005576 q^{43} - 2807426 q^{48} + 3246846 q^{49} + 297984 q^{51} + 3252964 q^{52} + 1705524 q^{53} - 5320576 q^{55} - 2282730 q^{56} - 8237572 q^{61} + 9625800 q^{62} + 5070862 q^{64} + 7139028 q^{65} + 9289056 q^{66} - 16801086 q^{68} - 6017400 q^{69} - 28760826 q^{74} + 20426072 q^{75} + 4093560 q^{77} + 14088594 q^{78} + 15147200 q^{79} - 21318442 q^{81} - 9130240 q^{82} - 36474240 q^{87} + 38008000 q^{88} + 4769676 q^{90} + 21281832 q^{91} + 36462348 q^{92} - 34229770 q^{94} - 22075632 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 449x^{4} + 37224x^{2} + 205776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 365\nu^{2} + 12324 ) / 240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 365\nu^{3} + 12084\nu ) / 240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{2} - 150 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 445\nu^{3} - 34644\nu ) / 160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} - 150 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} - 3\beta_{3} - 282\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 365\beta_{4} + 240\beta_{2} + 42426 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 730\beta_{5} + 1335\beta_{3} + 90846\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
18.4900i
10.0583i
2.43912i
2.43912i
10.0583i
18.4900i
18.4900i −27.4160 −213.880 70.5606i 506.922i 454.852i 1587.93i −1435.36 −1304.67
12.2 10.0583i 50.8656 26.8297 8.88672i 511.623i 510.452i 1557.33i 400.306 −89.3858
12.3 2.43912i −51.4496 122.051 488.312i 125.492i 616.243i 609.904i 460.056 1191.05
12.4 2.43912i −51.4496 122.051 488.312i 125.492i 616.243i 609.904i 460.056 1191.05
12.5 10.0583i 50.8656 26.8297 8.88672i 511.623i 510.452i 1557.33i 400.306 −89.3858
12.6 18.4900i −27.4160 −213.880 70.5606i 506.922i 454.852i 1587.93i −1435.36 −1304.67
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.8.b.a 6
3.b odd 2 1 117.8.b.b 6
4.b odd 2 1 208.8.f.a 6
13.b even 2 1 inner 13.8.b.a 6
13.d odd 4 2 169.8.a.d 6
39.d odd 2 1 117.8.b.b 6
52.b odd 2 1 208.8.f.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.b.a 6 1.a even 1 1 trivial
13.8.b.a 6 13.b even 2 1 inner
117.8.b.b 6 3.b odd 2 1
117.8.b.b 6 39.d odd 2 1
169.8.a.d 6 13.d odd 4 2
208.8.f.a 6 4.b odd 2 1
208.8.f.a 6 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 449 T^{4} + \cdots + 205776 \) Copy content Toggle raw display
$3$ \( (T^{3} + 28 T^{2} + \cdots - 71748)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 93756690000 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( (T^{3} - 6576 T^{2} + \cdots - 976631438250)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{3} + \cdots - 38560806996192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots + 18\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
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