Properties

Label 1232.4.a.bc
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} + 2) q^{5} - 7 q^{7} + (\beta_{3} + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} + 2) q^{5} - 7 q^{7} + (\beta_{3} + 15) q^{9} - 11 q^{11} + (\beta_{5} - \beta_{2} + 4) q^{13} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 9) q^{15}+ \cdots + ( - 11 \beta_{3} - 165) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35} + 255 q^{37} + 60 q^{39} + 508 q^{41} - 656 q^{43} + 466 q^{45} + 18 q^{47} + 343 q^{49} - 850 q^{51} + 1806 q^{53} - 121 q^{55} + 1154 q^{57} - 665 q^{59} + 608 q^{61} - 756 q^{63} + 588 q^{65} - 669 q^{67} + 2067 q^{69} - 1169 q^{71} + 380 q^{73} - 1254 q^{75} + 539 q^{77} + 110 q^{79} + 2847 q^{81} + 496 q^{83} + 3446 q^{85} + 582 q^{87} + 1321 q^{89} - 182 q^{91} + 1493 q^{93} + 50 q^{95} + 3927 q^{97} - 1188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{6} - 13\nu^{5} - 180\nu^{4} + 2006\nu^{3} - 8748\nu^{2} - 59944\nu + 152160 ) / 12672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 13\nu^{5} + 180\nu^{4} + 2218\nu^{3} + 8748\nu^{2} - 248408\nu - 177504 ) / 12672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\nu^{6} + 31\nu^{5} - 6340\nu^{4} - 2834\nu^{3} + 207908\nu^{2} + 110776\nu - 1043232 ) / 12672 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -51\nu^{6} + 221\nu^{5} + 7284\nu^{4} - 25654\nu^{3} - 248340\nu^{2} + 453032\nu + 1126176 ) / 12672 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} + 3\beta_{2} + 73\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{6} - 6\beta_{4} + 94\beta_{3} + 45\beta_{2} - 12\beta _1 + 3057 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{6} + 54\beta_{5} + 348\beta_{4} - 60\beta_{3} + 267\beta_{2} + 6004\beta _1 - 405 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 375\beta_{6} + 234\beta_{5} - 858\beta_{4} + 8296\beta_{3} + 6075\beta_{2} - 3534\beta _1 + 249405 \) 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
9.28858
8.14310
3.19046
0.517294
−2.69688
−5.79047
−9.65209
0 −9.28858 0 −8.74645 0 −7.00000 0 59.2778 0
1.2 0 −8.14310 0 18.9749 0 −7.00000 0 39.3101 0
1.3 0 −3.19046 0 8.53193 0 −7.00000 0 −16.8209 0
1.4 0 −0.517294 0 −7.39668 0 −7.00000 0 −26.7324 0
1.5 0 2.69688 0 −14.1250 0 −7.00000 0 −19.7268 0
1.6 0 5.79047 0 16.8493 0 −7.00000 0 6.52956 0
1.7 0 9.65209 0 −3.08797 0 −7.00000 0 66.1628 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bc 7
4.b odd 2 1 616.4.a.k 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.k 7 4.b odd 2 1
1232.4.a.bc 7 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{7} + 3T_{3}^{6} - 144T_{3}^{5} - 354T_{3}^{4} + 5172T_{3}^{3} + 6504T_{3}^{2} - 34432T_{3} - 18816 \) Copy content Toggle raw display
\( T_{5}^{7} - 11T_{5}^{6} - 468T_{5}^{5} + 2820T_{5}^{4} + 72712T_{5}^{3} - 41856T_{5}^{2} - 3193632T_{5} - 7697280 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 3 T^{6} + \cdots - 18816 \) Copy content Toggle raw display
$5$ \( T^{7} - 11 T^{6} + \cdots - 7697280 \) Copy content Toggle raw display
$7$ \( (T + 7)^{7} \) Copy content Toggle raw display
$11$ \( (T + 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 175897990400 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 1907883940864 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots - 4643111424 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 604187873280 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 28\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 328802478283264 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 97\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 46\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 50\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 73\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 46\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots + 54\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 27\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
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