Properties

Label 3136.2.b.h
Level $3136$
Weight $2$
Character orbit 3136.b
Analytic conductor $25.041$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,2,Mod(1569,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 448)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_1 q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_1 q^{5} + (\beta_{2} - 1) q^{9} + 2 \beta_1 q^{11} + (\beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{2} + 4) q^{15} - 2 q^{17} - \beta_{3} q^{19} + \beta_{2} q^{23} + (\beta_{2} + 1) q^{25} + ( - \beta_{3} + \beta_1) q^{27} + ( - \beta_{3} - \beta_1) q^{29} - 4 q^{31} + ( - 2 \beta_{2} + 8) q^{33} + (\beta_{3} + \beta_1) q^{37} + (3 \beta_{2} - 8) q^{39} + 2 q^{41} + 2 \beta_1 q^{43} + (\beta_{3} - 4 \beta_1) q^{45} + (2 \beta_{2} + 4) q^{47} + 2 \beta_1 q^{51} + (3 \beta_{3} - 3 \beta_1) q^{53} + (2 \beta_{2} - 8) q^{55} - \beta_{2} q^{57} + (\beta_{3} + 4 \beta_1) q^{59} + (2 \beta_{3} + 5 \beta_1) q^{61} + ( - 3 \beta_{2} + 8) q^{65} + ( - 2 \beta_{3} + 2 \beta_1) q^{67} + ( - \beta_{3} + 3 \beta_1) q^{69} + ( - 2 \beta_{2} - 4) q^{71} + (2 \beta_{2} - 6) q^{73} + ( - \beta_{3} + 2 \beta_1) q^{75} + (2 \beta_{2} - 4) q^{79} + (\beta_{2} + 1) q^{81} + (2 \beta_{3} + 3 \beta_1) q^{83} - 2 \beta_1 q^{85} - 4 q^{87} + (4 \beta_{2} + 2) q^{89} + 4 \beta_1 q^{93} + \beta_{2} q^{95} + (2 \beta_{2} - 2) q^{97} + (2 \beta_{3} - 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 16 q^{15} - 8 q^{17} + 4 q^{25} - 16 q^{31} + 32 q^{33} - 32 q^{39} + 8 q^{41} + 16 q^{47} - 32 q^{55} + 32 q^{65} - 16 q^{71} - 24 q^{73} - 16 q^{79} + 4 q^{81} - 16 q^{87} + 8 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 3\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 2.73205i 0 2.73205i 0 0 0 −4.46410 0
1569.2 0 0.732051i 0 0.732051i 0 0 0 2.46410 0
1569.3 0 0.732051i 0 0.732051i 0 0 0 2.46410 0
1569.4 0 2.73205i 0 2.73205i 0 0 0 −4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.b.h 4
4.b odd 2 1 3136.2.b.g 4
7.b odd 2 1 448.2.b.d yes 4
8.b even 2 1 inner 3136.2.b.h 4
8.d odd 2 1 3136.2.b.g 4
21.c even 2 1 4032.2.c.n 4
28.d even 2 1 448.2.b.c 4
56.e even 2 1 448.2.b.c 4
56.h odd 2 1 448.2.b.d yes 4
84.h odd 2 1 4032.2.c.k 4
112.j even 4 1 1792.2.a.k 2
112.j even 4 1 1792.2.a.q 2
112.l odd 4 1 1792.2.a.i 2
112.l odd 4 1 1792.2.a.s 2
168.e odd 2 1 4032.2.c.k 4
168.i even 2 1 4032.2.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.c 4 28.d even 2 1
448.2.b.c 4 56.e even 2 1
448.2.b.d yes 4 7.b odd 2 1
448.2.b.d yes 4 56.h odd 2 1
1792.2.a.i 2 112.l odd 4 1
1792.2.a.k 2 112.j even 4 1
1792.2.a.q 2 112.j even 4 1
1792.2.a.s 2 112.l odd 4 1
3136.2.b.g 4 4.b odd 2 1
3136.2.b.g 4 8.d odd 2 1
3136.2.b.h 4 1.a even 1 1 trivial
3136.2.b.h 4 8.b even 2 1 inner
4032.2.c.k 4 84.h odd 2 1
4032.2.c.k 4 168.e odd 2 1
4032.2.c.n 4 21.c even 2 1
4032.2.c.n 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$61$ \( T^{4} + 296 T^{2} + 21316 \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T - 188)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
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