Properties

Label 1232.4.a.p
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{137})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{3} + ( - \beta - 3) q^{5} - 7 q^{7} + ( - 5 \beta + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{3} + ( - \beta - 3) q^{5} - 7 q^{7} + ( - 5 \beta + 16) q^{9} + 11 q^{11} + ( - 10 \beta + 28) q^{13} + (\beta + 25) q^{15} + ( - 4 \beta - 42) q^{17} + ( - 14 \beta + 30) q^{19} + (7 \beta - 21) q^{21} + ( - 15 \beta + 65) q^{23} + (7 \beta - 82) q^{25} + (\beta + 137) q^{27} + ( - 16 \beta - 178) q^{29} + (3 \beta + 67) q^{31} + ( - 11 \beta + 33) q^{33} + (7 \beta + 21) q^{35} + ( - 5 \beta - 7) q^{37} + ( - 48 \beta + 424) q^{39} + ( - 20 \beta - 210) q^{41} + (8 \beta - 100) q^{43} + (4 \beta + 122) q^{45} + (24 \beta - 376) q^{47} + 49 q^{49} + (34 \beta + 10) q^{51} + (20 \beta - 414) q^{53} + ( - 11 \beta - 33) q^{55} + ( - 58 \beta + 566) q^{57} + ( - 71 \beta + 53) q^{59} + (100 \beta + 106) q^{61} + (35 \beta - 112) q^{63} + (12 \beta + 256) q^{65} + ( - 153 \beta + 227) q^{67} + ( - 95 \beta + 705) q^{69} + ( - 125 \beta + 131) q^{71} + ( - 80 \beta + 434) q^{73} + (96 \beta - 484) q^{75} - 77 q^{77} + ( - 6 \beta + 186) q^{79} - 55 q^{81} + (20 \beta + 1152) q^{83} + (58 \beta + 262) q^{85} + (146 \beta + 10) q^{87} + (21 \beta + 607) q^{89} + (70 \beta - 196) q^{91} + ( - 61 \beta + 99) q^{93} + (26 \beta + 386) q^{95} + ( - 181 \beta + 445) q^{97} + ( - 55 \beta + 176) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9} + 22 q^{11} + 46 q^{13} + 51 q^{15} - 88 q^{17} + 46 q^{19} - 35 q^{21} + 115 q^{23} - 157 q^{25} + 275 q^{27} - 372 q^{29} + 137 q^{31} + 55 q^{33} + 49 q^{35} - 19 q^{37} + 800 q^{39} - 440 q^{41} - 192 q^{43} + 248 q^{45} - 728 q^{47} + 98 q^{49} + 54 q^{51} - 808 q^{53} - 77 q^{55} + 1074 q^{57} + 35 q^{59} + 312 q^{61} - 189 q^{63} + 524 q^{65} + 301 q^{67} + 1315 q^{69} + 137 q^{71} + 788 q^{73} - 872 q^{75} - 154 q^{77} + 366 q^{79} - 110 q^{81} + 2324 q^{83} + 582 q^{85} + 166 q^{87} + 1235 q^{89} - 322 q^{91} + 137 q^{93} + 798 q^{95} + 709 q^{97} + 297 q^{99}+O(q^{100}) \) 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
6.35235
−5.35235
0 −3.35235 0 −9.35235 0 −7.00000 0 −15.7617 0
1.2 0 8.35235 0 2.35235 0 −7.00000 0 42.7617 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.p 2
4.b odd 2 1 154.4.a.f 2
12.b even 2 1 1386.4.a.ba 2
28.d even 2 1 1078.4.a.j 2
44.c even 2 1 1694.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.f 2 4.b odd 2 1
1078.4.a.j 2 28.d even 2 1
1232.4.a.p 2 1.a even 1 1 trivial
1386.4.a.ba 2 12.b even 2 1
1694.4.a.l 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{2} - 5T_{3} - 28 \) Copy content Toggle raw display
\( T_{5}^{2} + 7T_{5} - 22 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 28 \) Copy content Toggle raw display
$5$ \( T^{2} + 7T - 22 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 46T - 2896 \) Copy content Toggle raw display
$17$ \( T^{2} + 88T + 1388 \) Copy content Toggle raw display
$19$ \( T^{2} - 46T - 6184 \) Copy content Toggle raw display
$23$ \( T^{2} - 115T - 4400 \) Copy content Toggle raw display
$29$ \( T^{2} + 372T + 25828 \) Copy content Toggle raw display
$31$ \( T^{2} - 137T + 4384 \) Copy content Toggle raw display
$37$ \( T^{2} + 19T - 766 \) Copy content Toggle raw display
$41$ \( T^{2} + 440T + 34700 \) Copy content Toggle raw display
$43$ \( T^{2} + 192T + 7024 \) Copy content Toggle raw display
$47$ \( T^{2} + 728T + 112768 \) Copy content Toggle raw display
$53$ \( T^{2} + 808T + 149516 \) Copy content Toggle raw display
$59$ \( T^{2} - 35T - 172348 \) Copy content Toggle raw display
$61$ \( T^{2} - 312T - 318164 \) Copy content Toggle raw display
$67$ \( T^{2} - 301T - 779108 \) Copy content Toggle raw display
$71$ \( T^{2} - 137T - 530464 \) Copy content Toggle raw display
$73$ \( T^{2} - 788T - 63964 \) Copy content Toggle raw display
$79$ \( T^{2} - 366T + 32256 \) Copy content Toggle raw display
$83$ \( T^{2} - 2324 T + 1336544 \) Copy content Toggle raw display
$89$ \( T^{2} - 1235 T + 366202 \) Copy content Toggle raw display
$97$ \( T^{2} - 709T - 996394 \) Copy content Toggle raw display
show more
show less