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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [611,2,Mod(1,611)] 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("611.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(611, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 611 = 13 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 611.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.87885956350\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 6x^{7} + 21x^{6} + 8x^{5} - 42x^{4} + 2x^{3} + 25x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_{8} - 1) q^{5} + \beta_{5} q^{6} + (\beta_{4} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + ( - \beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{9}+ \cdots + ( - 2 \beta_{8} - 2 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 7 q^{5} - 3 q^{6} - 7 q^{7} - 6 q^{8} + 5 q^{9} - 6 q^{10} - 2 q^{11} + 4 q^{12} - 9 q^{13} + 6 q^{14} - 4 q^{15} - 5 q^{16} - 6 q^{17} - 3 q^{18} - 18 q^{19} - 8 q^{20}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 6x^{7} + 21x^{6} + 8x^{5} - 42x^{4} + 2x^{3} + 25x^{2} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - \nu^{5} - 6\nu^{4} + 4\nu^{3} + 7\nu^{2} - 2\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 12\nu^{2} - 5\nu - 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 6\nu^{6} - 20\nu^{5} - 9\nu^{4} + 36\nu^{3} + 2\nu^{2} - 17\nu + 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - 2\nu^{7} - 8\nu^{6} + 14\nu^{5} + 20\nu^{4} - 27\nu^{3} - 16\nu^{2} + 14\nu + 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 6\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 8\beta_{5} + \beta_{4} + 9\beta_{3} + 33\beta_{2} + 34\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + \beta_{7} - 9\beta_{6} + 11\beta_{5} + 8\beta_{4} + 40\beta_{3} + 53\beta_{2} + 97\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{8} + 2\beta_{7} - 40\beta_{6} + 52\beta_{5} + 10\beta_{4} + 61\beta_{3} + 181\beta_{2} + 190\beta _1 + 192 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.38806
2.25744
1.53483
0.882091
0.430543
−0.114111
−1.04469
−1.33597
−1.99819
−2.38806 −0.958410 3.70283 0.364069 2.28874 −0.0415900 −4.06646 −2.08145 −0.869418
1.2 −2.25744 2.18258 3.09603 −2.22590 −4.92704 −3.18258 −2.47422 1.76365 5.02484
1.3 −1.53483 0.559039 0.355700 2.48985 −0.858029 −1.55904 2.52372 −2.68748 −3.82149
1.4 −0.882091 −3.09754 −1.22192 −3.27957 2.73232 2.09754 2.84202 6.59477 2.89288
1.5 −0.430543 −1.25685 −1.81463 1.74597 0.541129 0.256852 1.64236 −1.42032 −0.751715
1.6 0.114111 2.73797 −1.98698 −1.76267 0.312432 −3.73797 −0.454958 4.49646 −0.201141
1.7 1.04469 0.541880 −0.908628 −1.16701 0.566095 −1.54188 −3.03861 −2.70637 −1.21916
1.8 1.33597 −2.65285 −0.215183 1.10398 −3.54413 1.65285 −2.95942 4.03762 1.47488
1.9 1.99819 −0.0558073 1.99278 −4.26870 −0.111514 −0.944193 −0.0144362 −2.99689 −8.52968
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)
\(47\) \( +1 \)

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 611.2.a.c ✓ 9
3.b odd 2 1 5499.2.a.r 9
4.b odd 2 1 9776.2.a.w 9
13.b even 2 1 7943.2.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
611.2.a.c ✓ 9 1.a even 1 1 trivial
5499.2.a.r 9 3.b odd 2 1
7943.2.a.j 9 13.b even 2 1
9776.2.a.w 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 3T_{2}^{8} - 6T_{2}^{7} - 21T_{2}^{6} + 8T_{2}^{5} + 42T_{2}^{4} + 2T_{2}^{3} - 25T_{2}^{2} - 6T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(611))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 3 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} + 7 T^{8} + \cdots + 112 \) Copy content Toggle raw display
$7$ \( T^{9} + 7 T^{8} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{9} + 2 T^{8} + \cdots - 176 \) Copy content Toggle raw display
$13$ \( (T + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 6 T^{8} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{9} + 18 T^{8} + \cdots + 2384 \) Copy content Toggle raw display
$23$ \( T^{9} - 3 T^{8} + \cdots - 95792 \) Copy content Toggle raw display
$29$ \( T^{9} + 5 T^{8} + \cdots + 2709232 \) Copy content Toggle raw display
$31$ \( T^{9} + 20 T^{8} + \cdots - 1904 \) Copy content Toggle raw display
$37$ \( T^{9} + 25 T^{8} + \cdots - 3768521 \) Copy content Toggle raw display
$41$ \( T^{9} + 12 T^{8} + \cdots - 3344 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} + \cdots - 66032 \) Copy content Toggle raw display
$47$ \( (T + 1)^{9} \) Copy content Toggle raw display
$53$ \( T^{9} - 10 T^{8} + \cdots - 62791519 \) Copy content Toggle raw display
$59$ \( T^{9} + 17 T^{8} + \cdots + 939793 \) Copy content Toggle raw display
$61$ \( T^{9} + 44 T^{8} + \cdots - 6246869 \) Copy content Toggle raw display
$67$ \( T^{9} + 4 T^{8} + \cdots + 20056144 \) Copy content Toggle raw display
$71$ \( T^{9} - 9 T^{8} + \cdots + 210427 \) Copy content Toggle raw display
$73$ \( T^{9} + 22 T^{8} + \cdots + 61116976 \) Copy content Toggle raw display
$79$ \( T^{9} + 16 T^{8} + \cdots - 14167417 \) Copy content Toggle raw display
$83$ \( T^{9} - 19 T^{8} + \cdots + 1861888 \) Copy content Toggle raw display
$89$ \( T^{9} + 8 T^{8} + \cdots - 30482821 \) Copy content Toggle raw display
$97$ \( T^{9} + 47 T^{8} + \cdots - 64452223 \) Copy content Toggle raw display
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