Properties

Label 432.8.a.w
Level $432$
Weight $8$
Character orbit 432.a
Self dual yes
Analytic conductor $134.950$
Analytic rank $1$
Dimension $4$
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] = [432,8,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(134.950331009\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 271x^{2} - 1425x - 1692 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 216)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 47) q^{5} + (\beta_1 + 45) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 47) q^{5} + (\beta_1 + 45) q^{7} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 365) q^{11} + (\beta_{3} + 6 \beta_{2} - 5 \beta_1 - 110) q^{13} + ( - 2 \beta_{3} - 59 \beta_{2} - 3 \beta_1 - 2482) q^{17} + (6 \beta_{3} + 37 \beta_{2} + 19 \beta_1 + 508) q^{19} + (7 \beta_{3} + 174 \beta_{2} - 7 \beta_1 + 6056) q^{23} + (7 \beta_{3} + 60 \beta_{2} + 7 \beta_1 + 6074) q^{25} + (8 \beta_{3} - 167 \beta_{2} - 75 \beta_1 - 14520) q^{29} + ( - 20 \beta_{3} + 25 \beta_{2} - 90 \beta_1 - 15983) q^{31} + ( - 28 \beta_{3} - 92 \beta_{2} - 288 \beta_1 - 16065) q^{35} + ( - 37 \beta_{3} + 458 \beta_{2} + 137 \beta_1 + 87900) q^{37} + (14 \beta_{3} + 699 \beta_{2} - 391 \beta_1 + 2460) q^{41} + (28 \beta_{3} + 2168 \beta_{2} - 106 \beta_1 - 35950) q^{43} + (84 \beta_{3} - 1352 \beta_{2} + 36 \beta_1 - 94014) q^{47} + (13 \beta_{3} + 3656 \beta_{2} - 263 \beta_1 + 426548) q^{49} + ( - 124 \beta_{3} + 1932 \beta_{2} + 829 \beta_1 + 124729) q^{53} + ( - 160 \beta_{3} + 1904 \beta_{2} + 785 \beta_1 - 355627) q^{55} + ( - 98 \beta_{3} - 1808 \beta_{2} + 1406 \beta_1 - 109840) q^{59} + (6 \beta_{3} - 2528 \beta_{2} - 1106 \beta_1 + 738484) q^{61} + (258 \beta_{3} - 2007 \beta_{2} + 873 \beta_1 - 392558) q^{65} + (408 \beta_{3} - 6512 \beta_{2} + 282 \beta_1 - 1867550) q^{67} + ( - 187 \beta_{3} + 8318 \beta_{2} - 185 \beta_1 - 28610) q^{71} + (177 \beta_{3} - 5608 \beta_{2} + 2677 \beta_1 + 2080675) q^{73} + ( - 252 \beta_{3} - 12339 \beta_{2} + 1382 \beta_1 - 928467) q^{77} + ( - 184 \beta_{3} + 3617 \beta_{2} - 4161 \beta_1 - 3238390) q^{79} + (429 \beta_{3} + 5928 \beta_{2} + 2321 \beta_1 - 101111) q^{83} + (177 \beta_{3} + 7938 \beta_{2} + 2327 \beta_1 + 4946990) q^{85} + (44 \beta_{3} + 9834 \beta_{2} - 3888 \beta_1 - 442590) q^{89} + (454 \beta_{3} - 8677 \beta_{2} + 3131 \beta_1 - 6408954) q^{91} + (169 \beta_{3} - 15526 \beta_{2} - 8881 \beta_1 - 3175784) q^{95} + ( - 540 \beta_{3} - 27240 \beta_{2} - 7148 \beta_1 + 6359015) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 188 q^{5} + 180 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 188 q^{5} + 180 q^{7} + 1460 q^{11} - 440 q^{13} - 9928 q^{17} + 2032 q^{19} + 24224 q^{23} + 24296 q^{25} - 58080 q^{29} - 63932 q^{31} - 64260 q^{35} + 351600 q^{37} + 9840 q^{41} - 143800 q^{43} - 376056 q^{47} + 1706192 q^{49} + 498916 q^{53} - 1422508 q^{55} - 439360 q^{59} + 2953936 q^{61} - 1570232 q^{65} - 7470200 q^{67} - 114440 q^{71} + 8322700 q^{73} - 3713868 q^{77} - 12953560 q^{79} - 404444 q^{83} + 19787960 q^{85} - 1770360 q^{89} - 25635816 q^{91} - 12703136 q^{95} + 25436060 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 271x^{2} - 1425x - 1692 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\nu^{3} - 148\nu^{2} - 4111\nu - 3054 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu^{2} - 277\nu - 660 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 165\nu^{3} - 516\nu^{2} - 41961\nu - 129384 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 29\beta_{2} + 2\beta _1 + 216 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} - 59\beta_{2} - 13\beta _1 + 29322 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 341\beta_{3} - 8113\beta_{2} + 346\beta _1 + 1099224 ) / 864 \) Copy content Toggle raw display

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
−3.98165
−12.4503
−1.77687
19.2088
0 0 0 −363.380 0 1440.56 0 0 0
1.2 0 0 0 −285.771 0 −1595.12 0 0 0
1.3 0 0 0 139.046 0 570.265 0 0 0
1.4 0 0 0 322.106 0 −235.705 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

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 432.8.a.w 4
3.b odd 2 1 432.8.a.z 4
4.b odd 2 1 216.8.a.e 4
12.b even 2 1 216.8.a.h yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.8.a.e 4 4.b odd 2 1
216.8.a.h yes 4 12.b even 2 1
432.8.a.w 4 1.a even 1 1 trivial
432.8.a.z 4 3.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 \(S_{8}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5}^{4} + 188T_{5}^{3} - 150726T_{5}^{2} - 18813820T_{5} + 4650878725 \) Copy content Toggle raw display
\( T_{7}^{4} - 180T_{7}^{3} - 2483982T_{7}^{2} + 747996660T_{7} + 308864705229 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 188 T^{3} + \cdots + 4650878725 \) Copy content Toggle raw display
$7$ \( T^{4} - 180 T^{3} + \cdots + 308864705229 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 479126207656201 \) Copy content Toggle raw display
$13$ \( T^{4} + 440 T^{3} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} + 9928 T^{3} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{4} - 2032 T^{3} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{4} - 24224 T^{3} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{4} + 58080 T^{3} + \cdots + 97\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{4} + 63932 T^{3} + \cdots + 34\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{4} - 351600 T^{3} + \cdots - 42\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{4} - 9840 T^{3} + \cdots - 59\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{4} + 143800 T^{3} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + 376056 T^{3} + \cdots - 61\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} - 498916 T^{3} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{4} + 439360 T^{3} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{4} - 2953936 T^{3} + \cdots - 36\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + 7470200 T^{3} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{4} + 114440 T^{3} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{4} - 8322700 T^{3} + \cdots - 23\!\cdots\!59 \) Copy content Toggle raw display
$79$ \( T^{4} + 12953560 T^{3} + \cdots - 80\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{4} + 404444 T^{3} + \cdots - 16\!\cdots\!71 \) Copy content Toggle raw display
$89$ \( T^{4} + 1770360 T^{3} + \cdots - 27\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{4} - 25436060 T^{3} + \cdots - 29\!\cdots\!11 \) Copy content Toggle raw display
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