Properties

Label 1232.4.a.q
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(3\)
Coefficient field: 3.3.7636.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 16x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 154)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + ( - \beta_{2} + 9) q^{5} + 7 q^{7} + (2 \beta_{2} + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{3} + ( - \beta_{2} + 9) q^{5} + 7 q^{7} + (2 \beta_{2} + 19) q^{9} - 11 q^{11} + (6 \beta_{2} + 5 \beta_1 + 22) q^{13} + (6 \beta_{2} + 6 \beta_1 - 6) q^{15} + (5 \beta_{2} + 11 \beta_1 + 17) q^{17} + (3 \beta_{2} - 6 \beta_1 + 15) q^{19} + (7 \beta_1 - 14) q^{21} + ( - 2 \beta_{2} + 6 \beta_1 + 82) q^{23} + ( - 20 \beta_{2} - 12 \beta_1 + 67) q^{25} + ( - 12 \beta_{2} - 2 \beta_1 - 8) q^{27} + ( - 6 \beta_{2} + 22 \beta_1 + 68) q^{29} + (13 \beta_{2} - 25 \beta_1 - 129) q^{31} + ( - 11 \beta_1 + 22) q^{33} + ( - 7 \beta_{2} + 63) q^{35} + (2 \beta_{2} - 32 \beta_1 + 36) q^{37} + ( - 26 \beta_{2} + 50 \beta_1 + 94) q^{39} + ( - 27 \beta_{2} - 13 \beta_1 - 207) q^{41} + ( - 10 \beta_{2} - 22 \beta_1 + 170) q^{43} + (3 \beta_{2} + 24 \beta_1 - 51) q^{45} + ( - 37 \beta_{2} + 41 \beta_1 + 121) q^{47} + 49 q^{49} + ( - 8 \beta_{2} + 54 \beta_1 + 368) q^{51} + (8 \beta_{2} + 20 \beta_1 + 66) q^{53} + (11 \beta_{2} - 99) q^{55} + ( - 30 \beta_{2} + 12 \beta_1 - 318) q^{57} + (26 \beta_{2} + 61 \beta_1 + 80) q^{59} + ( - 34 \beta_{2} - 77 \beta_1 + 114) q^{61} + (14 \beta_{2} + 133) q^{63} + (64 \beta_{2} + 102 \beta_1 - 408) q^{65} + (20 \beta_{2} + 18 \beta_1 - 92) q^{67} + (24 \beta_{2} + 88 \beta_1 + 112) q^{69} + ( - 38 \beta_{2} - 58 \beta_1 + 174) q^{71} + (25 \beta_{2} + 117 \beta_1 - 351) q^{73} + (96 \beta_{2} - 17 \beta_1 - 398) q^{75} - 77 q^{77} + ( - 8 \beta_{2} - 74 \beta_1 + 360) q^{79} + (14 \beta_{2} - 48 \beta_1 - 437) q^{81} + ( - 33 \beta_{2} - 2 \beta_1 - 293) q^{83} + (82 \beta_{2} + 126 \beta_1 - 270) q^{85} + (80 \beta_{2} + 94 \beta_1 + 860) q^{87} + ( - 8 \beta_{2} - 4 \beta_1 + 1138) q^{89} + (42 \beta_{2} + 35 \beta_1 + 154) q^{91} + ( - 128 \beta_{2} - 140 \beta_1 - 948) q^{93} + ( - 6 \beta_{2} - 270) q^{95} + ( - 4 \beta_{2} - 34 \beta_1 + 682) q^{97} + ( - 22 \beta_{2} - 209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9} - 33 q^{11} + 72 q^{13} - 12 q^{15} + 56 q^{17} + 48 q^{19} - 42 q^{21} + 244 q^{23} + 181 q^{25} - 36 q^{27} + 198 q^{29} - 374 q^{31} + 66 q^{33} + 182 q^{35} + 110 q^{37} + 256 q^{39} - 648 q^{41} + 500 q^{43} - 150 q^{45} + 326 q^{47} + 147 q^{49} + 1096 q^{51} + 206 q^{53} - 286 q^{55} - 984 q^{57} + 266 q^{59} + 308 q^{61} + 413 q^{63} - 1160 q^{65} - 256 q^{67} + 360 q^{69} + 484 q^{71} - 1028 q^{73} - 1098 q^{75} - 231 q^{77} + 1072 q^{79} - 1297 q^{81} - 912 q^{83} - 728 q^{85} + 2660 q^{87} + 3406 q^{89} + 504 q^{91} - 2972 q^{93} - 816 q^{95} + 2042 q^{97} - 649 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 16x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 21 ) / 2 \) 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
−3.22881
−1.24586
4.47467
0 −8.45761 0 −3.76558 0 7.00000 0 44.5312 0
1.2 0 −4.49172 0 21.9122 0 7.00000 0 −6.82442 0
1.3 0 6.94933 0 7.85338 0 7.00000 0 21.2932 0
\(n\): e.g. 2-40 or 990-1000
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.q 3
4.b odd 2 1 154.4.a.i 3
12.b even 2 1 1386.4.a.bc 3
28.d even 2 1 1078.4.a.p 3
44.c even 2 1 1694.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.i 3 4.b odd 2 1
1078.4.a.p 3 28.d even 2 1
1232.4.a.q 3 1.a even 1 1 trivial
1386.4.a.bc 3 12.b even 2 1
1694.4.a.s 3 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{3} + 6T_{3}^{2} - 52T_{3} - 264 \) Copy content Toggle raw display
\( T_{5}^{3} - 26T_{5}^{2} + 60T_{5} + 648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 6 T^{2} + \cdots - 264 \) Copy content Toggle raw display
$5$ \( T^{3} - 26 T^{2} + \cdots + 648 \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( (T + 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 72 T^{2} + \cdots + 331632 \) Copy content Toggle raw display
$17$ \( T^{3} - 56 T^{2} + \cdots + 88976 \) Copy content Toggle raw display
$19$ \( T^{3} - 48 T^{2} + \cdots - 28512 \) Copy content Toggle raw display
$23$ \( T^{3} - 244 T^{2} + \cdots - 219584 \) Copy content Toggle raw display
$29$ \( T^{3} - 198 T^{2} + \cdots + 3523896 \) Copy content Toggle raw display
$31$ \( T^{3} + 374 T^{2} + \cdots - 15721176 \) Copy content Toggle raw display
$37$ \( T^{3} - 110 T^{2} + \cdots + 5981784 \) Copy content Toggle raw display
$41$ \( T^{3} + 648 T^{2} + \cdots - 28837872 \) Copy content Toggle raw display
$43$ \( T^{3} - 500 T^{2} + \cdots + 2503232 \) Copy content Toggle raw display
$47$ \( T^{3} - 326 T^{2} + \cdots + 136297368 \) Copy content Toggle raw display
$53$ \( T^{3} - 206 T^{2} + \cdots + 863064 \) Copy content Toggle raw display
$59$ \( T^{3} - 266 T^{2} + \cdots + 4809816 \) Copy content Toggle raw display
$61$ \( T^{3} - 308 T^{2} + \cdots + 81057168 \) Copy content Toggle raw display
$67$ \( T^{3} + 256 T^{2} + \cdots + 1711488 \) Copy content Toggle raw display
$71$ \( T^{3} - 484 T^{2} + \cdots + 19953216 \) Copy content Toggle raw display
$73$ \( T^{3} + 1028 T^{2} + \cdots - 550878192 \) Copy content Toggle raw display
$79$ \( T^{3} - 1072 T^{2} + \cdots + 148401792 \) Copy content Toggle raw display
$83$ \( T^{3} + 912 T^{2} + \cdots - 33774048 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 1452062376 \) Copy content Toggle raw display
$97$ \( T^{3} - 2042 T^{2} + \cdots - 259709688 \) Copy content Toggle raw display
show more
show less