Properties

Label 1232.4.a.n
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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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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} - \beta q^{5} + 7 q^{7} - 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} - \beta q^{5} + 7 q^{7} - 23 q^{9} - 11 q^{11} + (2 \beta + 8) q^{13} - 2 \beta q^{15} + ( - 3 \beta - 6) q^{17} + (7 \beta + 42) q^{19} + 14 q^{21} + (14 \beta + 32) q^{23} + 23 q^{25} - 100 q^{27} + (6 \beta + 54) q^{29} + (\beta - 8) q^{31} - 22 q^{33} - 7 \beta q^{35} + ( - 8 \beta - 14) q^{37} + (4 \beta + 16) q^{39} + (21 \beta - 22) q^{41} + ( - 4 \beta - 428) q^{43} + 23 \beta q^{45} + ( - 17 \beta - 344) q^{47} + 49 q^{49} + ( - 6 \beta - 12) q^{51} + ( - 6 \beta - 66) q^{53} + 11 \beta q^{55} + (14 \beta + 84) q^{57} + ( - 18 \beta - 58) q^{59} + ( - 46 \beta - 60) q^{61} - 161 q^{63} + ( - 8 \beta - 296) q^{65} + (48 \beta - 456) q^{67} + (28 \beta + 64) q^{69} + (6 \beta + 392) q^{71} + (33 \beta - 298) q^{73} + 46 q^{75} - 77 q^{77} + ( - 42 \beta - 692) q^{79} + 421 q^{81} + (7 \beta - 1334) q^{83} + (6 \beta + 444) q^{85} + (12 \beta + 108) q^{87} + (12 \beta + 786) q^{89} + (14 \beta + 56) q^{91} + (2 \beta - 16) q^{93} + ( - 42 \beta - 1036) q^{95} + ( - 68 \beta - 210) q^{97} + 253 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 14 q^{7} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 14 q^{7} - 46 q^{9} - 22 q^{11} + 16 q^{13} - 12 q^{17} + 84 q^{19} + 28 q^{21} + 64 q^{23} + 46 q^{25} - 200 q^{27} + 108 q^{29} - 16 q^{31} - 44 q^{33} - 28 q^{37} + 32 q^{39} - 44 q^{41} - 856 q^{43} - 688 q^{47} + 98 q^{49} - 24 q^{51} - 132 q^{53} + 168 q^{57} - 116 q^{59} - 120 q^{61} - 322 q^{63} - 592 q^{65} - 912 q^{67} + 128 q^{69} + 784 q^{71} - 596 q^{73} + 92 q^{75} - 154 q^{77} - 1384 q^{79} + 842 q^{81} - 2668 q^{83} + 888 q^{85} + 216 q^{87} + 1572 q^{89} + 112 q^{91} - 32 q^{93} - 2072 q^{95} - 420 q^{97} + 506 q^{99}+O(q^{100}) \) 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 2.00000 0 −12.1655 0 7.00000 0 −23.0000 0
1.2 0 2.00000 0 12.1655 0 7.00000 0 −23.0000 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.n 2
4.b odd 2 1 616.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.4.a.b 2 4.b odd 2 1
1232.4.a.n 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 148 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 16T - 528 \) Copy content Toggle raw display
$17$ \( T^{2} + 12T - 1296 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 5488 \) Copy content Toggle raw display
$23$ \( T^{2} - 64T - 27984 \) Copy content Toggle raw display
$29$ \( T^{2} - 108T - 2412 \) Copy content Toggle raw display
$31$ \( T^{2} + 16T - 84 \) Copy content Toggle raw display
$37$ \( T^{2} + 28T - 9276 \) Copy content Toggle raw display
$41$ \( T^{2} + 44T - 64784 \) Copy content Toggle raw display
$43$ \( T^{2} + 856T + 180816 \) Copy content Toggle raw display
$47$ \( T^{2} + 688T + 75564 \) Copy content Toggle raw display
$53$ \( T^{2} + 132T - 972 \) Copy content Toggle raw display
$59$ \( T^{2} + 116T - 44588 \) Copy content Toggle raw display
$61$ \( T^{2} + 120T - 309568 \) Copy content Toggle raw display
$67$ \( T^{2} + 912T - 133056 \) Copy content Toggle raw display
$71$ \( T^{2} - 784T + 148336 \) Copy content Toggle raw display
$73$ \( T^{2} + 596T - 72368 \) Copy content Toggle raw display
$79$ \( T^{2} + 1384 T + 217792 \) Copy content Toggle raw display
$83$ \( T^{2} + 2668 T + 1772304 \) Copy content Toggle raw display
$89$ \( T^{2} - 1572 T + 596484 \) Copy content Toggle raw display
$97$ \( T^{2} + 420T - 640252 \) Copy content Toggle raw display
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