Properties

Label 1232.4.a.j
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 3) q^{3} + ( - \beta + 13) q^{5} - 7 q^{7} + (6 \beta + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 3) q^{3} + ( - \beta + 13) q^{5} - 7 q^{7} + (6 \beta + 19) q^{9} - 11 q^{11} + (\beta + 1) q^{13} + ( - 10 \beta - 2) q^{15} + ( - 4 \beta - 44) q^{17} + (19 \beta - 21) q^{19} + (7 \beta + 21) q^{21} + ( - 4 \beta - 80) q^{23} + ( - 26 \beta + 81) q^{25} + ( - 10 \beta - 198) q^{27} + (6 \beta + 12) q^{29} + (14 \beta - 124) q^{31} + (11 \beta + 33) q^{33} + (7 \beta - 91) q^{35} + (28 \beta - 26) q^{37} + ( - 4 \beta - 40) q^{39} + (24 \beta + 264) q^{41} + ( - 14 \beta + 102) q^{43} + (59 \beta + 25) q^{45} + (90 \beta - 12) q^{47} + 49 q^{49} + (56 \beta + 280) q^{51} + ( - 46 \beta + 124) q^{53} + (11 \beta - 143) q^{55} + ( - 36 \beta - 640) q^{57} + ( - 27 \beta - 285) q^{59} + (23 \beta - 669) q^{61} + ( - 42 \beta - 133) q^{63} + (12 \beta - 24) q^{65} + (34 \beta + 90) q^{67} + (92 \beta + 388) q^{69} + (150 \beta + 30) q^{71} + ( - 14 \beta - 2) q^{73} + ( - 3 \beta + 719) q^{75} + 77 q^{77} + (60 \beta + 964) q^{79} + (66 \beta + 451) q^{81} + ( - 145 \beta + 271) q^{83} + ( - 8 \beta - 424) q^{85} + ( - 30 \beta - 258) q^{87} + (208 \beta - 70) q^{89} + ( - 7 \beta - 7) q^{91} + (82 \beta - 146) q^{93} + (268 \beta - 976) q^{95} + ( - 82 \beta - 1356) q^{97} + ( - 66 \beta - 209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9} - 22 q^{11} + 2 q^{13} - 4 q^{15} - 88 q^{17} - 42 q^{19} + 42 q^{21} - 160 q^{23} + 162 q^{25} - 396 q^{27} + 24 q^{29} - 248 q^{31} + 66 q^{33} - 182 q^{35} - 52 q^{37} - 80 q^{39} + 528 q^{41} + 204 q^{43} + 50 q^{45} - 24 q^{47} + 98 q^{49} + 560 q^{51} + 248 q^{53} - 286 q^{55} - 1280 q^{57} - 570 q^{59} - 1338 q^{61} - 266 q^{63} - 48 q^{65} + 180 q^{67} + 776 q^{69} + 60 q^{71} - 4 q^{73} + 1438 q^{75} + 154 q^{77} + 1928 q^{79} + 902 q^{81} + 542 q^{83} - 848 q^{85} - 516 q^{87} - 140 q^{89} - 14 q^{91} - 292 q^{93} - 1952 q^{95} - 2712 q^{97} - 418 q^{99}+O(q^{100}) \) 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
3.54138
−2.54138
0 −9.08276 0 6.91724 0 −7.00000 0 55.4966 0
1.2 0 3.08276 0 19.0828 0 −7.00000 0 −17.4966 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.j 2
4.b odd 2 1 154.4.a.g 2
12.b even 2 1 1386.4.a.u 2
28.d even 2 1 1078.4.a.i 2
44.c even 2 1 1694.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.g 2 4.b odd 2 1
1078.4.a.i 2 28.d even 2 1
1232.4.a.j 2 1.a even 1 1 trivial
1386.4.a.u 2 12.b even 2 1
1694.4.a.p 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{2} + 6T_{3} - 28 \) Copy content Toggle raw display
\( T_{5}^{2} - 26T_{5} + 132 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T - 28 \) Copy content Toggle raw display
$5$ \( T^{2} - 26T + 132 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 88T + 1344 \) Copy content Toggle raw display
$19$ \( T^{2} + 42T - 12916 \) Copy content Toggle raw display
$23$ \( T^{2} + 160T + 5808 \) Copy content Toggle raw display
$29$ \( T^{2} - 24T - 1188 \) Copy content Toggle raw display
$31$ \( T^{2} + 248T + 8124 \) Copy content Toggle raw display
$37$ \( T^{2} + 52T - 28332 \) Copy content Toggle raw display
$41$ \( T^{2} - 528T + 48384 \) Copy content Toggle raw display
$43$ \( T^{2} - 204T + 3152 \) Copy content Toggle raw display
$47$ \( T^{2} + 24T - 299556 \) Copy content Toggle raw display
$53$ \( T^{2} - 248T - 62916 \) Copy content Toggle raw display
$59$ \( T^{2} + 570T + 54252 \) Copy content Toggle raw display
$61$ \( T^{2} + 1338 T + 427988 \) Copy content Toggle raw display
$67$ \( T^{2} - 180T - 34672 \) Copy content Toggle raw display
$71$ \( T^{2} - 60T - 831600 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 7248 \) Copy content Toggle raw display
$79$ \( T^{2} - 1928 T + 796096 \) Copy content Toggle raw display
$83$ \( T^{2} - 542T - 704484 \) Copy content Toggle raw display
$89$ \( T^{2} + 140 T - 1595868 \) Copy content Toggle raw display
$97$ \( T^{2} + 2712 T + 1589948 \) Copy content Toggle raw display
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