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Properties

Label 4624.2.a.bp
Level $4624$
Weight $2$
Character orbit 4624.a
Self dual yes
Analytic conductor $36.923$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4624,2,Mod(1,4624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4624 = 2^{4} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4624.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: \(36.9228258946\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{3} - \beta_1 q^{5} + ( - \beta_{3} + \beta_1) q^{7} + (2 \beta_{2} + 1) q^{9} + (\beta_{3} - \beta_1) q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{2} - 2) q^{15} + ( - 2 \beta_{2} - 2) q^{19} + 2 \beta_{2} q^{21} + ( - 3 \beta_{3} - \beta_1) q^{23} + (\beta_{2} - 3) q^{25} + ( - 2 \beta_{3} + 2 \beta_1) q^{27} + ( - \beta_{3} - 2 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_1) q^{31} - 2 \beta_{2} q^{33} - 2 q^{35} - 5 \beta_{3} q^{37} - 2 \beta_1 q^{39} + (\beta_{3} + 4 \beta_1) q^{41} + (2 \beta_{2} + 2) q^{43} + ( - 2 \beta_{3} - 3 \beta_1) q^{45} + ( - 2 \beta_{2} - 8) q^{47} + ( - 2 \beta_{2} - 3) q^{49} - \beta_{2} q^{53} + 2 q^{55} + ( - 2 \beta_{3} - 6 \beta_1) q^{57} - 6 q^{59} + 5 \beta_1 q^{61} + (3 \beta_{3} + \beta_1) q^{63} + (\beta_{3} + \beta_1) q^{65} + (2 \beta_{2} + 4) q^{67} + ( - 2 \beta_{2} - 8) q^{69} + ( - 5 \beta_{3} - 5 \beta_1) q^{71} + 7 \beta_{3} q^{73} + ( - 3 \beta_{3} - \beta_1) q^{75} + (2 \beta_{2} - 4) q^{77} + ( - 3 \beta_{3} - \beta_1) q^{79} + ( - 2 \beta_{2} - 3) q^{81} + (4 \beta_{2} - 6) q^{83} + ( - 4 \beta_{2} - 6) q^{87} + (\beta_{2} + 8) q^{89} - 2 \beta_{3} q^{91} - 6 \beta_{2} q^{93} + (2 \beta_{3} + 4 \beta_1) q^{95} + (6 \beta_{3} + \beta_1) q^{97} + ( - 3 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} - 8 q^{15} - 8 q^{19} - 12 q^{25} - 8 q^{35} + 8 q^{43} - 32 q^{47} - 12 q^{49} + 8 q^{55} - 24 q^{59} + 16 q^{67} - 32 q^{69} - 16 q^{77} - 12 q^{81} - 24 q^{83} - 24 q^{87} + 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.84776
0.765367
−0.765367
1.84776
0 −2.61313 0 1.84776 0 −1.08239 0 3.82843 0
1.2 0 −1.08239 0 −0.765367 0 2.61313 0 −1.82843 0
1.3 0 1.08239 0 0.765367 0 −2.61313 0 −1.82843 0
1.4 0 2.61313 0 −1.84776 0 1.08239 0 3.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4624.2.a.bp 4
4.b odd 2 1 289.2.a.f 4
12.b even 2 1 2601.2.a.bb 4
17.b even 2 1 inner 4624.2.a.bp 4
17.e odd 16 2 272.2.v.d 4
20.d odd 2 1 7225.2.a.u 4
68.d odd 2 1 289.2.a.f 4
68.f odd 4 2 289.2.b.b 4
68.g odd 8 4 289.2.c.c 8
68.i even 16 2 17.2.d.a ✓ 4
68.i even 16 2 289.2.d.a 4
68.i even 16 2 289.2.d.b 4
68.i even 16 2 289.2.d.c 4
204.h even 2 1 2601.2.a.bb 4
204.t odd 16 2 153.2.l.c 4
340.d odd 2 1 7225.2.a.u 4
340.bc odd 16 2 425.2.n.b 4
340.bg even 16 2 425.2.m.a 4
340.bj odd 16 2 425.2.n.a 4
476.bf odd 16 2 833.2.l.a 4
476.bk odd 48 4 833.2.v.a 8
476.bm even 48 4 833.2.v.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a ✓ 4 68.i even 16 2
153.2.l.c 4 204.t odd 16 2
272.2.v.d 4 17.e odd 16 2
289.2.a.f 4 4.b odd 2 1
289.2.a.f 4 68.d odd 2 1
289.2.b.b 4 68.f odd 4 2
289.2.c.c 8 68.g odd 8 4
289.2.d.a 4 68.i even 16 2
289.2.d.b 4 68.i even 16 2
289.2.d.c 4 68.i even 16 2
425.2.m.a 4 340.bg even 16 2
425.2.n.a 4 340.bj odd 16 2
425.2.n.b 4 340.bc odd 16 2
833.2.l.a 4 476.bf odd 16 2
833.2.v.a 8 476.bk odd 48 4
833.2.v.b 8 476.bm even 48 4
2601.2.a.bb 4 12.b even 2 1
2601.2.a.bb 4 204.h even 2 1
4624.2.a.bp 4 1.a even 1 1 trivial
4624.2.a.bp 4 17.b even 2 1 inner
7225.2.a.u 4 20.d odd 2 1
7225.2.a.u 4 340.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4624))\):

\( T_{3}^{4} - 8T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{4} - 8T^{2} + 8 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 40T^{2} + 392 \) Copy content Toggle raw display
$29$ \( T^{4} - 20T^{2} + 2 \) Copy content Toggle raw display
$31$ \( T^{4} - 72T^{2} + 648 \) Copy content Toggle raw display
$37$ \( T^{4} - 100T^{2} + 1250 \) Copy content Toggle raw display
$41$ \( T^{4} - 68T^{2} + 98 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 100T^{2} + 1250 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 200T^{2} + 5000 \) Copy content Toggle raw display
$73$ \( T^{4} - 196T^{2} + 4802 \) Copy content Toggle raw display
$79$ \( T^{4} - 40T^{2} + 392 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 148T^{2} + 4418 \) Copy content Toggle raw display
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