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Properties

Label 6422.2.a.bk
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

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: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{7} - 1) q^{3} + q^{4} - \beta_{4} q^{5} + (\beta_{7} - 1) q^{6} + (\beta_{3} + \beta_1 - 2) q^{7} + q^{8} + (\beta_{8} + \beta_{5} + \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{7} - 1) q^{3} + q^{4} - \beta_{4} q^{5} + (\beta_{7} - 1) q^{6} + (\beta_{3} + \beta_1 - 2) q^{7} + q^{8} + (\beta_{8} + \beta_{5} + \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - 4 \beta_{6} + \beta_{5} + 9 \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{8} - 3\nu^{7} - 9\nu^{6} + 23\nu^{5} + 31\nu^{4} - 42\nu^{3} - 41\nu^{2} - 7\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{8} - 6\nu^{7} - 18\nu^{6} + 46\nu^{5} + 63\nu^{4} - 87\nu^{3} - 87\nu^{2} - 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{8} - 5\nu^{7} - 22\nu^{6} + 42\nu^{5} + 86\nu^{4} - 81\nu^{3} - 118\nu^{2} - 16\nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 3\nu^{8} - 8\nu^{7} - 31\nu^{6} + 64\nu^{5} + 120\nu^{4} - 119\nu^{3} - 171\nu^{2} - 26\nu + 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{8} + 11\nu^{7} + 40\nu^{6} - 87\nu^{5} - 151\nu^{4} + 162\nu^{3} + 212\nu^{2} + 27\nu - 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\nu^{8} + 17\nu^{7} + 58\nu^{6} - 133\nu^{5} - 214\nu^{4} + 249\nu^{3} + 298\nu^{2} + 31\nu - 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 10\nu^{8} - 28\nu^{7} - 97\nu^{6} + 215\nu^{5} + 366\nu^{4} - 388\nu^{3} - 527\nu^{2} - 78\nu + 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} + 8\beta_{6} + 3\beta_{5} - 4\beta_{3} + \beta_{2} + 16\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} + 16\beta_{6} + 12\beta_{5} + \beta_{4} + 8\beta_{2} + 45\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} - 26\beta_{7} + 67\beta_{6} + 34\beta_{5} + 5\beta_{4} - 13\beta_{3} + 16\beta_{2} + 125\beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{8} - 32\beta_{7} + 173\beta_{6} + 109\beta_{5} + 25\beta_{4} + 8\beta_{3} + 67\beta_{2} + 352\beta _1 + 143 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21 \beta_{8} - 147 \beta_{7} + 589 \beta_{6} + 306 \beta_{5} + 97 \beta_{4} - 10 \beta_{3} + 173 \beta_{2} + \cdots + 492 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.816316
−1.85435
−1.89727
1.88062
2.69813
0.102176
−0.785289
2.96112
−0.288819
1.00000 −3.11887 1.00000 0.453021 −3.11887 −1.01438 1.00000 6.72735 0.453021
1.2 1.00000 −3.07508 1.00000 −1.48707 −3.07508 −5.10133 1.00000 6.45613 −1.48707
1.3 1.00000 −2.42044 1.00000 4.26312 −2.42044 −3.45222 1.00000 2.85851 4.26312
1.4 1.00000 −1.60929 1.00000 −4.22571 −1.60929 0.325660 1.00000 −0.410201 −4.22571
1.5 1.00000 0.162557 1.00000 2.16373 0.162557 −0.548851 1.00000 −2.97358 2.16373
1.6 1.00000 0.519384 1.00000 −0.0567031 0.519384 −0.0958867 1.00000 −2.73024 −0.0567031
1.7 1.00000 0.782742 1.00000 1.76453 0.782742 −2.34025 1.00000 −2.38731 1.76453
1.8 1.00000 1.04453 1.00000 −1.64330 1.04453 2.76306 1.00000 −1.90896 −1.64330
1.9 1.00000 2.71446 1.00000 −0.231615 2.71446 −3.53580 1.00000 4.36830 −0.231615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)
\(19\) \(-1\)

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 6422.2.a.bk yes 9
13.b even 2 1 6422.2.a.bi ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.bi ✓ 9 13.b even 2 1
6422.2.a.bk yes 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{9} + 5T_{3}^{8} - 6T_{3}^{7} - 53T_{3}^{6} - 11T_{3}^{5} + 136T_{3}^{4} - T_{3}^{3} - 126T_{3}^{2} + 63T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{9} - T_{5}^{8} - 24T_{5}^{7} + 21T_{5}^{6} + 118T_{5}^{5} - 64T_{5}^{4} - 175T_{5}^{3} + 32T_{5}^{2} + 20T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{9} + 13T_{7}^{8} + 52T_{7}^{7} + 14T_{7}^{6} - 380T_{7}^{5} - 863T_{7}^{4} - 587T_{7}^{3} - 7T_{7}^{2} + 77T_{7} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 5 T^{8} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} - 24 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{9} + 13 T^{8} + \cdots + 7 \) Copy content Toggle raw display
$11$ \( T^{9} + 13 T^{8} + \cdots - 4459 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 12 T^{8} + \cdots + 47783 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 22 T^{8} + \cdots + 796109 \) Copy content Toggle raw display
$29$ \( T^{9} - 12 T^{8} + \cdots + 628069 \) Copy content Toggle raw display
$31$ \( T^{9} - T^{8} + \cdots - 34307 \) Copy content Toggle raw display
$37$ \( T^{9} + 25 T^{8} + \cdots + 2899 \) Copy content Toggle raw display
$41$ \( T^{9} - 11 T^{8} + \cdots + 3295669 \) Copy content Toggle raw display
$43$ \( T^{9} + 10 T^{8} + \cdots + 1362073 \) Copy content Toggle raw display
$47$ \( T^{9} + 12 T^{8} + \cdots - 419 \) Copy content Toggle raw display
$53$ \( T^{9} - 9 T^{8} + \cdots - 451543 \) Copy content Toggle raw display
$59$ \( T^{9} - 10 T^{8} + \cdots + 1221571 \) Copy content Toggle raw display
$61$ \( T^{9} - 32 T^{8} + \cdots + 2927897 \) Copy content Toggle raw display
$67$ \( T^{9} + 73 T^{8} + \cdots + 35377399 \) Copy content Toggle raw display
$71$ \( T^{9} + 51 T^{8} + \cdots - 48682591 \) Copy content Toggle raw display
$73$ \( T^{9} + 14 T^{8} + \cdots + 219983 \) Copy content Toggle raw display
$79$ \( T^{9} + 28 T^{8} + \cdots - 11213 \) Copy content Toggle raw display
$83$ \( T^{9} + 22 T^{8} + \cdots - 510299 \) Copy content Toggle raw display
$89$ \( T^{9} - 3 T^{8} + \cdots + 28903 \) Copy content Toggle raw display
$97$ \( T^{9} - 196 T^{7} + \cdots + 1628677 \) Copy content Toggle raw display
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