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Properties

Label 1472.4.a.bl
Level $1472$
Weight $4$
Character orbit 1472.a
Self dual yes
Analytic conductor $86.851$
Analytic rank $0$
Dimension $9$
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] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1472.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,14,0,-30,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(86.8508115285\)
Analytic rank: \(0\)
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: \( x^{9} - 4x^{8} - 155x^{7} + 342x^{6} + 7139x^{5} - 8520x^{4} - 113229x^{3} + 12582x^{2} + 530388x + 301320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 736)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + (\beta_{4} - 3) q^{5} + ( - \beta_{6} + 3) q^{7} + (\beta_{2} - 2 \beta_1 + 12) q^{9} + ( - \beta_{8} + \beta_{6} - \beta_1 - 4) q^{11} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots - 8) q^{13}+ \cdots + (5 \beta_{8} + 27 \beta_{7} + \cdots + 227) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 14 q^{3} - 30 q^{5} + 28 q^{7} + 103 q^{9} - 40 q^{11} - 76 q^{13} + 72 q^{15} + 166 q^{17} + 148 q^{19} - 96 q^{21} - 207 q^{23} + 339 q^{25} + 326 q^{27} - 252 q^{29} - 258 q^{31} + 460 q^{33} + 276 q^{35}+ \cdots + 2024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 155x^{7} + 342x^{6} + 7139x^{5} - 8520x^{4} - 113229x^{3} + 12582x^{2} + 530388x + 301320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 621 \nu^{8} - 5822 \nu^{7} - 79879 \nu^{6} + 681976 \nu^{5} + 2838855 \nu^{4} - 22394254 \nu^{3} + \cdots + 10485468 ) / 1150056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2733 \nu^{8} + 17906 \nu^{7} + 365743 \nu^{6} - 1850680 \nu^{5} - 13064103 \nu^{4} + \cdots - 172591236 ) / 3450168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2343 \nu^{8} - 14404 \nu^{7} - 319127 \nu^{6} + 1472234 \nu^{5} + 11703027 \nu^{4} + \cdots + 236206800 ) / 862542 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8173 \nu^{8} - 57744 \nu^{7} - 1096251 \nu^{6} + 6167860 \nu^{5} + 40424135 \nu^{4} + \cdots + 844490448 ) / 1725084 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14803 \nu^{8} + 101305 \nu^{7} + 2000534 \nu^{6} - 10748574 \nu^{5} - 74296283 \nu^{4} + \cdots - 1626979122 ) / 2587626 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 93093 \nu^{8} - 647446 \nu^{7} - 12542447 \nu^{6} + 68940824 \nu^{5} + 464534391 \nu^{4} + \cdots + 9559052244 ) / 3450168 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - 6\beta_{7} - \beta_{6} - 3\beta_{5} - 7\beta_{4} + 3\beta_{2} + 71\beta _1 + 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{8} - 36\beta_{7} + 12\beta_{6} - 12\beta_{5} - 18\beta_{4} - 3\beta_{3} + 104\beta_{2} + 372\beta _1 + 2479 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 158 \beta_{8} - 804 \beta_{7} - 17 \beta_{6} - 318 \beta_{5} - 737 \beta_{4} + 45 \beta_{3} + \cdots + 11751 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1568 \beta_{8} - 6366 \beta_{7} + 1792 \beta_{6} - 2142 \beta_{5} - 4190 \beta_{4} - 186 \beta_{3} + \cdots + 234781 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20437 \beta_{8} - 96882 \beta_{7} + 5219 \beta_{6} - 34443 \beta_{5} - 79321 \beta_{4} + \cdots + 1749290 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 214697 \beta_{8} - 895998 \beta_{7} + 207184 \beta_{6} - 302538 \beta_{5} - 643388 \beta_{4} + \cdots + 25713041 \) 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
11.1809
5.85314
5.45105
2.86977
−0.626091
−2.90761
−2.92721
−6.97389
−7.92010
0 −9.18094 0 −11.3755 0 −1.86775 0 57.2896 0
1.2 0 −3.85314 0 4.56877 0 17.3356 0 −12.1533 0
1.3 0 −3.45105 0 −17.8073 0 29.6935 0 −15.0902 0
1.4 0 −0.869773 0 5.39127 0 −27.1854 0 −26.2435 0
1.5 0 2.62609 0 17.8295 0 15.0724 0 −20.1036 0
1.6 0 4.90761 0 −14.9865 0 5.67238 0 −2.91537 0
1.7 0 4.92721 0 −19.3859 0 −20.7764 0 −2.72262 0
1.8 0 8.97389 0 6.92259 0 34.1625 0 53.5307 0
1.9 0 9.92010 0 −1.15689 0 −24.1069 0 71.4084 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(23\) \( +1 \)

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 1472.4.a.bl 9
4.b odd 2 1 1472.4.a.bi 9
8.b even 2 1 736.4.a.g ✓ 9
8.d odd 2 1 736.4.a.j yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
736.4.a.g ✓ 9 8.b even 2 1
736.4.a.j yes 9 8.d odd 2 1
1472.4.a.bi 9 4.b odd 2 1
1472.4.a.bl 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 14 T_{3}^{8} - 75 T_{3}^{7} + 1604 T_{3}^{6} - 1553 T_{3}^{5} - 39542 T_{3}^{4} + \cdots - 600256 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 14 T^{8} + \cdots - 600256 \) Copy content Toggle raw display
$5$ \( T^{9} + 30 T^{8} + \cdots + 206988160 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 38235540992 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 30010477590720 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 320529677593800 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 848815162683840 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 13\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( (T + 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 97\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 46\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 48\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 39\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 47\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 48\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 87\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 18\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 35\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 46\!\cdots\!40 \) Copy content Toggle raw display
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