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Properties

Label 6422.2.a.bh
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} - \beta_1 q^{6} + \beta_{3} q^{7} + q^{8} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} - \beta_1 q^{6} + \beta_{3} q^{7} + q^{8} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{9}+ \cdots + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} - 2 q^{7} + 8 q^{8} - 2 q^{10} - 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} - 8 q^{19} - 2 q^{20} + 12 q^{21} - 10 q^{22} - 8 q^{23} - 4 q^{24} - 14 q^{25} - 22 q^{27} - 2 q^{28} + 8 q^{29} - 12 q^{31} + 8 q^{32} + 4 q^{33} + 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 12 q^{42} - 16 q^{43} - 10 q^{44} - 12 q^{45} - 8 q^{46} + 12 q^{47} - 4 q^{48} - 14 q^{49} - 14 q^{50} - 22 q^{51} - 24 q^{53} - 22 q^{54} - 10 q^{55} - 2 q^{56} + 4 q^{57} + 8 q^{58} - 4 q^{59} - 26 q^{61} - 12 q^{62} - 12 q^{63} + 8 q^{64} + 4 q^{66} + 10 q^{67} + 2 q^{68} + 6 q^{69} - 10 q^{70} - 36 q^{71} + 20 q^{73} + 6 q^{75} - 8 q^{76} - 12 q^{77} - 22 q^{79} - 2 q^{80} + 36 q^{81} + 2 q^{82} + 18 q^{83} + 12 q^{84} + 26 q^{85} - 16 q^{86} - 38 q^{87} - 10 q^{88} - 18 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} + 12 q^{94} + 2 q^{95} - 4 q^{96} - 8 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 11\nu^{5} + 12\nu^{4} + 36\nu^{3} - 7\nu^{2} - 19\nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} - 10\nu^{5} + 33\nu^{4} + 18\nu^{3} - 20\nu^{2} + 4\nu - 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 14\nu^{6} + 23\nu^{5} - 78\nu^{4} - 45\nu^{3} + 91\nu^{2} + \nu - 19 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 26\nu^{6} + 35\nu^{5} - 141\nu^{4} - 66\nu^{3} + 169\nu^{2} + 19\nu - 34 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} + 46\nu^{5} - 153\nu^{4} - 99\nu^{3} + 170\nu^{2} + 26\nu - 29 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} - 92\nu^{5} + 306\nu^{4} + 201\nu^{3} - 349\nu^{2} - 58\nu + 70 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{2} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} - 6\beta_{6} - 13\beta_{5} - 2\beta_{4} + 13\beta_{2} + 24\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{7} - 10\beta_{6} - 44\beta_{5} - 7\beta_{4} + 2\beta_{3} + 44\beta_{2} + 83\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -108\beta_{7} - 38\beta_{6} - 159\beta_{5} - 34\beta_{4} + 7\beta_{3} + 161\beta_{2} + 270\beta _1 + 91 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -350\beta_{7} - 85\beta_{6} - 545\beta_{5} - 121\beta_{4} + 36\beta_{3} + 552\beta_{2} + 910\beta _1 + 266 \) 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
3.36639
2.47489
0.898616
0.533995
0.394661
−0.516521
−1.40354
−1.74849
1.00000 −3.36639 1.00000 0.659053 −3.36639 −1.55483 1.00000 8.33259 0.659053
1.2 1.00000 −2.47489 1.00000 −2.86363 −2.47489 −1.77068 1.00000 3.12506 −2.86363
1.3 1.00000 −0.898616 1.00000 −0.684128 −0.898616 3.14167 1.00000 −2.19249 −0.684128
1.4 1.00000 −0.533995 1.00000 2.79724 −0.533995 −0.904890 1.00000 −2.71485 2.79724
1.5 1.00000 −0.394661 1.00000 −0.507765 −0.394661 −1.24955 1.00000 −2.84424 −0.507765
1.6 1.00000 0.516521 1.00000 1.53223 0.516521 −3.77281 1.00000 −2.73321 1.53223
1.7 1.00000 1.40354 1.00000 −0.425845 1.40354 2.92512 1.00000 −1.03007 −0.425845
1.8 1.00000 1.74849 1.00000 −2.50715 1.74849 1.18596 1.00000 0.0572044 −2.50715
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bh 8
13.b even 2 1 6422.2.a.bg 8
13.f odd 12 2 494.2.m.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.a ✓ 16 13.f odd 12 2
6422.2.a.bg 8 13.b even 2 1
6422.2.a.bh 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{8} + 4T_{3}^{7} - 4T_{3}^{6} - 22T_{3}^{5} + 3T_{3}^{4} + 28T_{3}^{3} + 7T_{3}^{2} - 6T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 11T_{5}^{6} - 20T_{5}^{5} + 25T_{5}^{4} + 36T_{5}^{3} - 2T_{5}^{2} - 12T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{8} + 2T_{7}^{7} - 19T_{7}^{6} - 38T_{7}^{5} + 86T_{7}^{4} + 220T_{7}^{3} + 5T_{7}^{2} - 240T_{7} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots - 3 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots - 128 \) Copy content Toggle raw display
$11$ \( T^{8} + 10 T^{7} + \cdots + 214 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots - 512 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 976 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} + \cdots - 10067 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots - 46016 \) Copy content Toggle raw display
$37$ \( T^{8} - 155 T^{6} + \cdots + 191808 \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 2597437 \) Copy content Toggle raw display
$43$ \( T^{8} + 16 T^{7} + \cdots - 12200 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots - 498 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + \cdots - 36195267 \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots + 989862 \) Copy content Toggle raw display
$61$ \( T^{8} + 26 T^{7} + \cdots - 15483 \) Copy content Toggle raw display
$67$ \( T^{8} - 10 T^{7} + \cdots - 195584 \) Copy content Toggle raw display
$71$ \( T^{8} + 36 T^{7} + \cdots + 745566 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots - 228467 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 63006 \) Copy content Toggle raw display
$83$ \( T^{8} - 18 T^{7} + \cdots - 2176946 \) Copy content Toggle raw display
$89$ \( T^{8} + 18 T^{7} + \cdots - 17288 \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots - 117248 \) Copy content Toggle raw display
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