↑

Properties

Label 26.5.f.b
Level $26$
Weight $5$
Character orbit 26.f
Analytic conductor $2.688$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [26,5,Mod(7,26)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(26, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("26.7"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 26.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68761904018\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{6} + 2 \beta_{3}) q^{2} + ( - \beta_{6} + 3 \beta_{5} + \beta_1) q^{3} - 8 \beta_{5} q^{4} + ( - \beta_{7} - 5 \beta_{6} - 9 \beta_{5} + \cdots + 5) q^{5} + (2 \beta_{7} - 6 \beta_{6} + 4 \beta_{5} + \cdots - 4) q^{6}+ \cdots + (664 \beta_{7} - 6782 \beta_{6} + \cdots - 7096) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 18 q^{5} - 36 q^{6} + 4 q^{7} + 128 q^{8} - 198 q^{9} + 348 q^{10} + 144 q^{11} - 318 q^{13} - 40 q^{14} - 924 q^{15} + 256 q^{16} - 390 q^{17} - 792 q^{18} + 2278 q^{19} + 624 q^{20} - 678 q^{21}+ \cdots - 55296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 494x^{6} + 90747x^{4} - 7343258x^{2} + 221087161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 247\nu^{2} + 26\nu + 14869 ) / 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 247\nu^{2} + 26\nu - 14869 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 494\nu^{5} - 75878\nu^{3} + 3670615\nu + 386594 ) / 773188 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 370\nu^{4} + 45498\nu^{2} - 1858625 ) / 1262 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1625 \nu^{7} - 14869 \nu^{6} - 609453 \nu^{5} + 5501530 \nu^{4} + 75943985 \nu^{3} + \cdots + 25309075053 ) / 487881628 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1625 \nu^{7} - 14869 \nu^{6} + 609453 \nu^{5} + 5501530 \nu^{4} - 75943985 \nu^{3} + \cdots + 25309075053 ) / 487881628 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 13\nu^{6} - 370\nu^{5} + 4810\nu^{4} + 44236\nu^{3} - 591474\nu^{2} - 1702137\nu + 24162125 ) / 32812 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{6} + 13\beta_{5} + \beta_{4} + 124 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -26\beta_{7} - 124\beta_{6} + 124\beta_{5} - 13\beta_{4} + 26\beta_{3} + 123\beta_{2} + 123\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3211\beta_{6} + 3211\beta_{5} + 247\beta_{4} - 26\beta_{2} + 26\beta _1 + 15759 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6370 \beta_{7} - 31642 \beta_{6} + 31642 \beta_{5} - 3185 \beta_{4} + 12870 \beta_{3} + 15512 \beta_{2} + \cdots - 6435 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 596596\beta_{6} + 596596\beta_{5} + 47154\beta_{4} - 9620\beta_{2} + 9620\beta _1 + 2047703 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1173952 \beta_{7} - 6222276 \beta_{6} + 6222276 \beta_{5} - 586976 \beta_{4} + 3611764 \beta_{3} + \cdots - 1805882 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(-\beta_{5}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−10.0615 + 0.500000i
10.0615 + 0.500000i
−12.0941 + 0.500000i
12.0941 + 0.500000i
−10.0615 − 0.500000i
10.0615 − 0.500000i
−12.0941 − 0.500000i
12.0941 − 0.500000i
2.73205 − 0.732051i −6.32978 + 10.9635i 6.92820 − 4.00000i 24.0516 + 24.0516i −9.26744 + 34.5865i −4.96375 − 1.33003i 16.0000 − 16.0000i −39.6322 − 68.6449i 83.3172 + 48.1032i
7.2 2.73205 − 0.732051i 3.73170 − 6.46350i 6.92820 − 4.00000i −3.43687 − 3.43687i 5.46359 − 20.3904i 5.09773 + 1.36593i 16.0000 − 16.0000i 12.6488 + 21.9084i −11.9057 − 6.87373i
11.1 −0.732051 + 2.73205i −4.74800 − 8.22378i −6.92820 − 4.00000i −19.2341 − 19.2341i 25.9436 − 6.95155i −5.11403 − 19.0858i 16.0000 − 16.0000i −4.58700 + 7.94492i 66.6289 − 38.4682i
11.2 −0.732051 + 2.73205i 7.34608 + 12.7238i −6.92820 − 4.00000i −10.3806 − 10.3806i −40.1397 + 10.7554i 6.98005 + 26.0499i 16.0000 − 16.0000i −67.4297 + 116.792i 35.9596 − 20.7613i
15.1 2.73205 + 0.732051i −6.32978 − 10.9635i 6.92820 + 4.00000i 24.0516 − 24.0516i −9.26744 − 34.5865i −4.96375 + 1.33003i 16.0000 + 16.0000i −39.6322 + 68.6449i 83.3172 − 48.1032i
15.2 2.73205 + 0.732051i 3.73170 + 6.46350i 6.92820 + 4.00000i −3.43687 + 3.43687i 5.46359 + 20.3904i 5.09773 − 1.36593i 16.0000 + 16.0000i 12.6488 − 21.9084i −11.9057 + 6.87373i
19.1 −0.732051 − 2.73205i −4.74800 + 8.22378i −6.92820 + 4.00000i −19.2341 + 19.2341i 25.9436 + 6.95155i −5.11403 + 19.0858i 16.0000 + 16.0000i −4.58700 − 7.94492i 66.6289 + 38.4682i
19.2 −0.732051 − 2.73205i 7.34608 − 12.7238i −6.92820 + 4.00000i −10.3806 + 10.3806i −40.1397 − 10.7554i 6.98005 − 26.0499i 16.0000 + 16.0000i −67.4297 − 116.792i 35.9596 + 20.7613i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.5.f.b ✓ 8
3.b odd 2 1 234.5.bb.a 8
13.c even 3 1 338.5.d.f 8
13.e even 6 1 338.5.d.g 8
13.f odd 12 1 inner 26.5.f.b ✓ 8
13.f odd 12 1 338.5.d.f 8
13.f odd 12 1 338.5.d.g 8
39.k even 12 1 234.5.bb.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.5.f.b ✓ 8 1.a even 1 1 trivial
26.5.f.b ✓ 8 13.f odd 12 1 inner
234.5.bb.a 8 3.b odd 2 1
234.5.bb.a 8 39.k even 12 1
338.5.d.f 8 13.c even 3 1
338.5.d.f 8 13.f odd 12 1
338.5.d.g 8 13.e even 6 1
338.5.d.g 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 261T_{3}^{6} + 468T_{3}^{5} + 54939T_{3}^{4} + 61074T_{3}^{3} + 3495258T_{3}^{2} - 3084588T_{3} + 173765124 \) acting on \(S_{5}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 261 T^{6} + \cdots + 173765124 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 4358376324 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 208860304 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 46\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 99\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 43\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 43469639270772)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 393593299652832)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
show more
show less