Properties

Label 13.5.d.a
Level $13$
Weight $5$
Character orbit 13.d
Analytic conductor $1.344$
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] = [13,5,Mod(5,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 13.d (of order \(4\), degree \(2\), 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: \(1.34380952009\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.53039932416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{5}+ \cdots + (\beta_{4} + 9 \beta_{3} + 9 \beta_1 - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{5}+ \cdots + ( - 8 \beta_{5} - 8 \beta_{4} + \cdots + 3314) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 4 q^{3} - 14 q^{5} + 32 q^{6} + 48 q^{7} - 96 q^{8} - 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 4 q^{3} - 14 q^{5} + 32 q^{6} + 48 q^{7} - 96 q^{8} - 58 q^{9} - 32 q^{11} - 244 q^{14} + 404 q^{15} + 1044 q^{16} - 802 q^{18} + 732 q^{19} + 428 q^{20} - 2128 q^{21} - 1632 q^{22} - 24 q^{24} + 910 q^{26} + 236 q^{27} + 1884 q^{28} + 4184 q^{29} - 3468 q^{31} + 2092 q^{32} + 2324 q^{33} - 5304 q^{34} - 4204 q^{35} - 1758 q^{37} + 1196 q^{39} - 708 q^{40} + 4750 q^{41} + 9532 q^{42} - 3956 q^{44} + 830 q^{45} + 516 q^{46} - 6872 q^{47} - 9436 q^{48} - 322 q^{50} + 3900 q^{52} + 2108 q^{53} - 184 q^{54} + 6408 q^{55} - 5800 q^{57} + 6516 q^{58} + 4372 q^{59} + 1324 q^{60} + 5988 q^{61} - 652 q^{63} - 5018 q^{65} - 4592 q^{66} + 72 q^{67} - 10572 q^{68} + 7368 q^{70} - 14672 q^{71} - 7980 q^{72} + 5874 q^{73} + 1544 q^{74} + 3576 q^{76} + 5720 q^{78} + 2616 q^{79} - 12080 q^{80} - 19450 q^{81} + 19264 q^{83} + 6296 q^{84} + 4164 q^{85} + 29376 q^{86} + 35584 q^{87} - 986 q^{89} - 30888 q^{91} + 5304 q^{92} - 9520 q^{93} - 36156 q^{94} + 20720 q^{96} - 23154 q^{97} - 41426 q^{98} + 17492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} - 12x^{3} + 529x^{2} - 1334x + 1682 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -581\nu^{5} + 437\nu^{4} + 114\nu^{3} - 11153\nu^{2} - 302999\nu + 392573 ) / 424531 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{5} + 44\nu^{4} - 625\nu^{3} + 150\nu^{2} - 13189\nu + 33698 ) / 14639 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31\nu^{5} + 531\nu^{4} + 775\nu^{3} - 186\nu^{2} - 1798\nu + 173115 ) / 14639 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9152\nu^{5} + 6153\nu^{4} + 20063\nu^{3} + 230580\nu^{2} - 4343971\nu + 5696499 ) / 424531 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 17\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 21\beta_{3} + 12\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} + 31\beta_{3} + 31\beta _1 - 367 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{5} + 19\beta_{4} - 402\beta_{2} - 479\beta _1 + 402 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
3.18200 3.18200i
1.30633 1.30633i
−3.48832 + 3.48832i
3.18200 + 3.18200i
1.30633 + 1.30633i
−3.48832 3.48832i
−3.18200 + 3.18200i −10.6142 4.25023i −9.43223 + 9.43223i 33.7744 33.7744i 54.8891 + 54.8891i −37.3878 37.3878i 31.6617 60.0267i
5.2 −1.30633 + 1.30633i 9.97438 12.5870i 9.28070 9.28070i −13.0298 + 13.0298i −46.1782 46.1782i −37.3440 37.3440i 18.4882 24.2472i
5.3 3.48832 3.48832i −1.36015 8.33680i −6.84848 + 6.84848i −4.74466 + 4.74466i 15.2891 + 15.2891i 26.7317 + 26.7317i −79.1500 47.7794i
8.1 −3.18200 3.18200i −10.6142 4.25023i −9.43223 9.43223i 33.7744 + 33.7744i 54.8891 54.8891i −37.3878 + 37.3878i 31.6617 60.0267i
8.2 −1.30633 1.30633i 9.97438 12.5870i 9.28070 + 9.28070i −13.0298 13.0298i −46.1782 + 46.1782i −37.3440 + 37.3440i 18.4882 24.2472i
8.3 3.48832 + 3.48832i −1.36015 8.33680i −6.84848 6.84848i −4.74466 4.74466i 15.2891 15.2891i 26.7317 26.7317i −79.1500 47.7794i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.5.d.a 6
3.b odd 2 1 117.5.j.a 6
4.b odd 2 1 208.5.t.c 6
13.b even 2 1 169.5.d.a 6
13.d odd 4 1 inner 13.5.d.a 6
13.d odd 4 1 169.5.d.a 6
39.f even 4 1 117.5.j.a 6
52.f even 4 1 208.5.t.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.d.a 6 1.a even 1 1 trivial
13.5.d.a 6 13.d odd 4 1 inner
117.5.j.a 6 3.b odd 2 1
117.5.j.a 6 39.f even 4 1
169.5.d.a 6 13.b even 2 1
169.5.d.a 6 13.d odd 4 1
208.5.t.c 6 4.b odd 2 1
208.5.t.c 6 52.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 1682 \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} + \cdots - 144)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 14 T^{5} + \cdots + 2875202 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 12014360072 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 88136171552 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 26482633392384 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2092 T^{2} + \cdots + 695715376)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 11\!\cdots\!18 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T^{3} - 1054 T^{2} + \cdots - 7634592356)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{3} - 2994 T^{2} + \cdots + 1197889048)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 55\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( (T^{3} - 1308 T^{2} + \cdots - 7469664296)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 75\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
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