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Properties

Label 1548.2.f.b
Level $1548$
Weight $2$
Character orbit 1548.f
Analytic conductor $12.361$
Analytic rank $0$
Dimension $8$
CM discriminant -43
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1548,2,Mod(773,1548)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1548.773"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1548, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1548 = 2^{2} \cdot 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1548.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.3608422329\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3}, \sqrt{43})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 38x^{6} - 12x^{5} + 655x^{4} - 288x^{3} - 5034x^{2} + 4428x + 21222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{11} - \beta_1 q^{13} + \beta_{6} q^{17} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{23} - 5 q^{25} + ( - 2 \beta_{2} + \beta_1) q^{31} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{41} + ( - \beta_{2} - \beta_1) q^{43}+ \cdots - \beta_{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{25} + 56 q^{49} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 38x^{6} - 12x^{5} + 655x^{4} - 288x^{3} - 5034x^{2} + 4428x + 21222 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25694 \nu^{7} + 350843 \nu^{6} + 6748582 \nu^{5} - 17272363 \nu^{4} - 205085342 \nu^{3} + \cdots - 2603302455 ) / 1040385795 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 350776 \nu^{7} - 420163 \nu^{6} + 8557808 \nu^{5} + 24691298 \nu^{4} - 90801148 \nu^{3} + \cdots + 1384470780 ) / 1040385795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 837302 \nu^{7} - 6341962 \nu^{6} - 33313170 \nu^{5} + 169588323 \nu^{4} + 551090005 \nu^{3} + \cdots + 16130142141 ) / 1040385795 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 768 \nu^{7} + 1061 \nu^{6} + 20949 \nu^{5} - 44986 \nu^{4} - 292629 \nu^{3} + 1126211 \nu^{2} + \cdots - 5686860 ) / 633995 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} + 102\nu^{6} - 596\nu^{5} - 2715\nu^{4} + 6256\nu^{3} + 14850\nu^{2} - 64380\nu + 82947 ) / 12765 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1473597 \nu^{7} - 2928831 \nu^{6} - 41889272 \nu^{5} + 96289255 \nu^{4} + 672062852 \nu^{3} + \cdots + 13614735549 ) / 1040385795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2078485 \nu^{7} + 4175266 \nu^{6} + 73529603 \nu^{5} - 82900628 \nu^{4} - 1132606618 \nu^{3} + \cdots - 11516265642 ) / 1040385795 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 4\beta_{2} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 4\beta_{6} - \beta_{5} - 4\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 28 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{7} + 51\beta_{6} - 6\beta_{5} + 43\beta_{4} - 33\beta_{3} + 48\beta_{2} - 6\beta _1 + 24 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{7} - 92\beta_{6} - 11\beta_{5} - 136\beta_{4} - 28\beta_{3} + 92\beta_{2} - 38\beta _1 + 95 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -41\beta_{7} + 1001\beta_{6} - 220\beta_{5} + 619\beta_{4} - 611\beta_{3} - 44\beta_{2} - 112\beta _1 + 2680 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -54\beta_{7} - 678\beta_{6} + 156\beta_{5} - 1894\beta_{4} - 762\beta_{3} + 3075\beta_{2} - 510\beta _1 - 3282 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 715 \beta_{7} + 6217 \beta_{6} - 4928 \beta_{5} - 4313 \beta_{4} - 9955 \beta_{3} - 14152 \beta_{2} + \cdots + 81452 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1548\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(433\) \(775\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
773.1
4.14474 − 0.517638i
−2.41269 + 0.517638i
2.41269 − 1.93185i
−4.14474 + 1.93185i
−4.14474 − 1.93185i
2.41269 + 1.93185i
−2.41269 − 0.517638i
4.14474 + 0.517638i
0 0 0 0 0 0 0 0 0
773.2 0 0 0 0 0 0 0 0 0
773.3 0 0 0 0 0 0 0 0 0
773.4 0 0 0 0 0 0 0 0 0
773.5 0 0 0 0 0 0 0 0 0
773.6 0 0 0 0 0 0 0 0 0
773.7 0 0 0 0 0 0 0 0 0
773.8 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 773.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 CM by \(\Q(\sqrt{-43}) \)
3.b odd 2 1 inner
129.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1548.2.f.b ✓ 8
3.b odd 2 1 inner 1548.2.f.b ✓ 8
4.b odd 2 1 6192.2.l.i 8
12.b even 2 1 6192.2.l.i 8
43.b odd 2 1 CM 1548.2.f.b ✓ 8
129.d even 2 1 inner 1548.2.f.b ✓ 8
172.d even 2 1 6192.2.l.i 8
516.h odd 2 1 6192.2.l.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1548.2.f.b ✓ 8 1.a even 1 1 trivial
1548.2.f.b ✓ 8 3.b odd 2 1 inner
1548.2.f.b ✓ 8 43.b odd 2 1 CM
1548.2.f.b ✓ 8 129.d even 2 1 inner
6192.2.l.i 8 4.b odd 2 1
6192.2.l.i 8 12.b even 2 1
6192.2.l.i 8 172.d even 2 1
6192.2.l.i 8 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(1548, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 88 T^{6} + \cdots + 6084 \) Copy content Toggle raw display
$13$ \( (T^{4} - 35 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 136 T^{6} + \cdots + 617796 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 184 T^{6} + \cdots + 2490084 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 143 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 328 T^{6} + \cdots + 12404484 \) Copy content Toggle raw display
$43$ \( (T^{2} - 43)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 188 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 424 T^{6} + \cdots + 19873764 \) Copy content Toggle raw display
$59$ \( (T^{4} + 236 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 359 T^{2} + 24964)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 172)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 664 T^{6} + \cdots + 30669444 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - T - 290)^{4} \) Copy content Toggle raw display
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