Properties

Label 1232.4.a.k
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: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta + 1) q^{3} + (2 \beta - 4) q^{5} + 7 q^{7} + (6 \beta + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta + 1) q^{3} + (2 \beta - 4) q^{5} + 7 q^{7} + (6 \beta + 19) q^{9} - 11 q^{11} + ( - 17 \beta - 9) q^{13} + ( - 10 \beta + 26) q^{15} + ( - 25 \beta - 37) q^{17} + ( - 4 \beta - 80) q^{19} + (21 \beta + 7) q^{21} + (30 \beta + 10) q^{23} + ( - 16 \beta - 89) q^{25} + ( - 18 \beta + 82) q^{27} + ( - 62 \beta + 36) q^{29} + ( - 9 \beta - 119) q^{31} + ( - 33 \beta - 11) q^{33} + (14 \beta - 28) q^{35} + ( - 170 \beta - 28) q^{37} + ( - 44 \beta - 264) q^{39} + ( - 73 \beta + 131) q^{41} + (92 \beta + 188) q^{43} + (14 \beta - 16) q^{45} + ( - 87 \beta + 39) q^{47} + 49 q^{49} + ( - 136 \beta - 412) q^{51} + (10 \beta + 248) q^{53} + ( - 22 \beta + 44) q^{55} + ( - 244 \beta - 140) q^{57} + ( - 89 \beta - 499) q^{59} + (9 \beta + 781) q^{61} + (42 \beta + 133) q^{63} + (50 \beta - 134) q^{65} + (34 \beta + 250) q^{67} + (60 \beta + 460) q^{69} + ( - 14 \beta - 1050) q^{71} + (405 \beta + 293) q^{73} + ( - 283 \beta - 329) q^{75} - 77 q^{77} + ( - 12 \beta + 8) q^{79} + (66 \beta - 701) q^{81} + (284 \beta + 824) q^{83} + (26 \beta - 102) q^{85} + (46 \beta - 894) q^{87} + (280 \beta - 470) q^{89} + ( - 119 \beta - 63) q^{91} + ( - 366 \beta - 254) q^{93} + ( - 144 \beta + 280) q^{95} + (130 \beta + 576) q^{97} + ( - 66 \beta - 209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 8 q^{5} + 14 q^{7} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 8 q^{5} + 14 q^{7} + 38 q^{9} - 22 q^{11} - 18 q^{13} + 52 q^{15} - 74 q^{17} - 160 q^{19} + 14 q^{21} + 20 q^{23} - 178 q^{25} + 164 q^{27} + 72 q^{29} - 238 q^{31} - 22 q^{33} - 56 q^{35} - 56 q^{37} - 528 q^{39} + 262 q^{41} + 376 q^{43} - 32 q^{45} + 78 q^{47} + 98 q^{49} - 824 q^{51} + 496 q^{53} + 88 q^{55} - 280 q^{57} - 998 q^{59} + 1562 q^{61} + 266 q^{63} - 268 q^{65} + 500 q^{67} + 920 q^{69} - 2100 q^{71} + 586 q^{73} - 658 q^{75} - 154 q^{77} + 16 q^{79} - 1402 q^{81} + 1648 q^{83} - 204 q^{85} - 1788 q^{87} - 940 q^{89} - 126 q^{91} - 508 q^{93} + 560 q^{95} + 1152 q^{97} - 418 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
0 −5.70820 0 −8.47214 0 7.00000 0 5.58359 0
1.2 0 7.70820 0 0.472136 0 7.00000 0 32.4164 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.k 2
4.b odd 2 1 616.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.c 2 4.b odd 2 1
1232.4.a.k 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{2} - 2T_{3} - 44 \) Copy content Toggle raw display
\( T_{5}^{2} + 8T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$5$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 18T - 1364 \) Copy content Toggle raw display
$17$ \( T^{2} + 74T - 1756 \) Copy content Toggle raw display
$19$ \( T^{2} + 160T + 6320 \) Copy content Toggle raw display
$23$ \( T^{2} - 20T - 4400 \) Copy content Toggle raw display
$29$ \( T^{2} - 72T - 17924 \) Copy content Toggle raw display
$31$ \( T^{2} + 238T + 13756 \) Copy content Toggle raw display
$37$ \( T^{2} + 56T - 143716 \) Copy content Toggle raw display
$41$ \( T^{2} - 262T - 9484 \) Copy content Toggle raw display
$43$ \( T^{2} - 376T - 6976 \) Copy content Toggle raw display
$47$ \( T^{2} - 78T - 36324 \) Copy content Toggle raw display
$53$ \( T^{2} - 496T + 61004 \) Copy content Toggle raw display
$59$ \( T^{2} + 998T + 209396 \) Copy content Toggle raw display
$61$ \( T^{2} - 1562 T + 609556 \) Copy content Toggle raw display
$67$ \( T^{2} - 500T + 56720 \) Copy content Toggle raw display
$71$ \( T^{2} + 2100 T + 1101520 \) Copy content Toggle raw display
$73$ \( T^{2} - 586T - 734276 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T - 656 \) Copy content Toggle raw display
$83$ \( T^{2} - 1648 T + 275696 \) Copy content Toggle raw display
$89$ \( T^{2} + 940T - 171100 \) Copy content Toggle raw display
$97$ \( T^{2} - 1152 T + 247276 \) Copy content Toggle raw display
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