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Properties

Label 1421.2.a.r
Level $1421$
Weight $2$
Character orbit 1421.a
Self dual yes
Analytic conductor $11.347$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [1421,2,Mod(1,1421)] 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("1421.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1421, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1421.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,-5,6,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.3467421272\)
Analytic rank: \(1\)
Dimension: \(8\)
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: \( x^{8} - 2x^{7} - 9x^{6} + 16x^{5} + 20x^{4} - 25x^{3} - 14x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 203)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} + 1) q^{4} + \beta_{7} q^{5} + (\beta_{7} - \beta_{5}) q^{6} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{7} + 4 \beta_{5} + 4 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 5 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{8} + 9 q^{9} - 7 q^{10} - 7 q^{11} + 8 q^{12} - 9 q^{13} - 5 q^{15} + 14 q^{16} - 4 q^{17} - 7 q^{18} - 16 q^{19} - 8 q^{20} - 21 q^{22} + 6 q^{23}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 16x^{5} + 20x^{4} - 25x^{3} - 14x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 16\nu^{4} + 27\nu^{3} - 24\nu^{2} - 21\nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + 4\nu^{6} + 17\nu^{5} - 32\nu^{4} - 30\nu^{3} + 51\nu^{2} + 6\nu - 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} + 24\nu^{5} - 55\nu^{4} - 37\nu^{3} + 82\nu^{2} + 11\nu - 11 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 16\nu^{4} + 20\nu^{3} - 25\nu^{2} - 14\nu + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} + 24\nu^{5} - 65\nu^{4} - 38\nu^{3} + 104\nu^{2} + 16\nu - 13 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - 2\beta_{3} + 6\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{6} + 7\beta_{4} - 10\beta_{3} + \beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 11\beta_{6} + 17\beta_{5} - 9\beta_{4} - 21\beta_{3} + 38\beta_{2} + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + 59\beta_{6} + 2\beta_{5} + 41\beta_{4} - 80\beta_{3} + 14\beta_{2} + 166\beta _1 + 17 \) 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
2.64104
2.14340
1.46161
0.316147
−0.213157
−0.538529
−1.35791
−2.45258
−2.64104 0.117685 4.97507 0.721511 −0.310811 0 −7.85726 −2.98615 −1.90554
1.2 −2.14340 2.20146 2.59414 −3.01825 −4.71859 0 −1.27349 1.84641 6.46930
1.3 −1.46161 −3.03231 0.136290 4.20019 4.43203 0 2.72401 6.19488 −6.13903
1.4 −0.316147 −1.73876 −1.90005 0.228604 0.549706 0 1.23299 0.0233033 −0.0722724
1.5 0.213157 −3.24810 −1.95456 −3.82037 −0.692354 0 −0.842942 7.55014 −0.814336
1.6 0.538529 0.635773 −1.70999 2.72009 0.342382 0 −1.99794 −2.59579 1.46485
1.7 1.35791 1.60774 −0.156073 −1.30883 2.18318 0 −2.92776 −0.415157 −1.77727
1.8 2.45258 −1.54349 4.01517 −1.72296 −3.78554 0 4.94238 −0.617636 −4.22571
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(29\) \( -1 \)

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 1421.2.a.r 8
7.b odd 2 1 1421.2.a.s 8
7.d odd 6 2 203.2.e.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
203.2.e.d ✓ 16 7.d odd 6 2
1421.2.a.r 8 1.a even 1 1 trivial
1421.2.a.s 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1421))\):

\( T_{2}^{8} + 2T_{2}^{7} - 9T_{2}^{6} - 16T_{2}^{5} + 20T_{2}^{4} + 25T_{2}^{3} - 14T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 5T_{3}^{7} - 4T_{3}^{6} - 43T_{3}^{5} - 11T_{3}^{4} + 103T_{3}^{3} + 35T_{3}^{2} - 65T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 25T_{5}^{6} - 54T_{5}^{5} + 140T_{5}^{4} + 315T_{5}^{3} - 67T_{5}^{2} - 217T_{5} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots + 7 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + \cdots - 2009 \) Copy content Toggle raw display
$13$ \( T^{8} + 9 T^{7} + \cdots + 196 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots - 3479 \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 57187 \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 35 T^{7} + \cdots - 198401 \) Copy content Toggle raw display
$37$ \( T^{8} - 14 T^{7} + \cdots + 3231 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 28 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + \cdots - 386352 \) Copy content Toggle raw display
$47$ \( T^{8} - 3 T^{7} + \cdots + 336511 \) Copy content Toggle raw display
$53$ \( T^{8} + 22 T^{7} + \cdots - 42437 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + \cdots + 5015647 \) Copy content Toggle raw display
$61$ \( T^{8} + 43 T^{7} + \cdots - 1830717 \) Copy content Toggle raw display
$67$ \( T^{8} + 7 T^{7} + \cdots - 478169 \) Copy content Toggle raw display
$71$ \( T^{8} - 13 T^{7} + \cdots + 3136 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + \cdots - 6209 \) Copy content Toggle raw display
$79$ \( T^{8} - T^{7} + \cdots + 4911667 \) Copy content Toggle raw display
$83$ \( T^{8} - 17 T^{7} + \cdots - 318416 \) Copy content Toggle raw display
$89$ \( T^{8} + 31 T^{7} + \cdots - 80717 \) Copy content Toggle raw display
$97$ \( T^{8} + 22 T^{7} + \cdots - 39452 \) Copy content Toggle raw display
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