Properties

Label 1232.4.a.z
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $5$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 616)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 2) q^{5} + 7 q^{7} + (\beta_{4} + 2 \beta_{2} + 2 \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 2) q^{5} + 7 q^{7} + (\beta_{4} + 2 \beta_{2} + 2 \beta_1 + 14) q^{9} + 11 q^{11} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 39) q^{13}+ \cdots + (11 \beta_{4} + 22 \beta_{2} + \cdots + 154) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} - 14 q^{5} + 35 q^{7} + 79 q^{9} + 55 q^{11} + 194 q^{13} + 118 q^{15} - 248 q^{17} + 190 q^{19} - 14 q^{21} + 34 q^{23} + 529 q^{25} - 254 q^{27} - 334 q^{29} + 172 q^{31} - 22 q^{33} - 98 q^{35} + 360 q^{37} + 360 q^{39} - 852 q^{41} + 1016 q^{43} - 2164 q^{45} + 194 q^{47} + 245 q^{49} + 836 q^{51} - 934 q^{53} - 154 q^{55} - 980 q^{57} + 542 q^{59} + 838 q^{61} + 553 q^{63} - 1780 q^{65} + 1830 q^{67} + 50 q^{69} - 422 q^{71} - 68 q^{73} - 320 q^{75} + 385 q^{77} + 8 q^{79} + 1889 q^{81} + 366 q^{83} + 2056 q^{85} + 4384 q^{87} - 524 q^{89} + 1358 q^{91} - 2242 q^{93} + 2884 q^{95} - 1336 q^{97} + 869 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 105x^{3} + 92x^{2} + 1858x - 2576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{4} - 13\nu^{3} - 363\nu^{2} + 61\nu + 978 ) / 257 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{4} + 41\nu^{3} - 496\nu^{2} - 3395\nu + 7156 ) / 257 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{4} + 26\nu^{3} + 983\nu^{2} - 636\nu - 12493 ) / 257 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{2} + 2\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{4} + 5\beta_{3} + 4\beta_{2} + 75\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 83\beta_{4} + 13\beta_{3} + 207\beta_{2} + 328\beta _1 + 2885 \) 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
9.98555
4.06945
1.45755
−5.28837
−8.22418
0 −9.98555 0 −20.3895 0 7.00000 0 72.7112 0
1.2 0 −4.06945 0 10.6234 0 7.00000 0 −10.4396 0
1.3 0 −1.45755 0 −4.53944 0 7.00000 0 −24.8755 0
1.4 0 5.28837 0 17.5419 0 7.00000 0 0.966854 0
1.5 0 8.22418 0 −17.2362 0 7.00000 0 40.6371 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 1232.4.a.z 5
4.b odd 2 1 616.4.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.g 5 4.b odd 2 1
1232.4.a.z 5 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{5} + 2T_{3}^{4} - 105T_{3}^{3} - 92T_{3}^{2} + 1858T_{3} + 2576 \) Copy content Toggle raw display
\( T_{5}^{5} + 14T_{5}^{4} - 479T_{5}^{3} - 5256T_{5}^{2} + 52388T_{5} + 297296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 2 T^{4} + \cdots + 2576 \) Copy content Toggle raw display
$5$ \( T^{5} + 14 T^{4} + \cdots + 297296 \) Copy content Toggle raw display
$7$ \( (T - 7)^{5} \) Copy content Toggle raw display
$11$ \( (T - 11)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 194 T^{4} + \cdots + 165725120 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 1652853792 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 1871657024 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 1356850592 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 558300459168 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 7177964236 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 575625945376 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 260317168544 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 37111868928 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 8257603328 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 118934413056 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 50642124297976 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 865511238448 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots + 74164434252000 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 254347095519296 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 6750753157408 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 4640907655424 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 90576457428736 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 29444789628360 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
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