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Properties

Label 8464.2.a.bt
Level $8464$
Weight $2$
Character orbit 8464.a
Self dual yes
Analytic conductor $67.585$
Analytic rank $1$
Dimension $5$
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] = [8464,2,Mod(1,8464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8464, 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("8464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.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: \(67.5853802708\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + ( - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + ( - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{9} + ( - \beta_{4} + \beta_1 - 3) q^{11} + (2 \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 2) q^{13}+ \cdots + (3 \beta_{4} - 4 \beta_{3} - 5 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} + 7 q^{5} - 8 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} + 7 q^{5} - 8 q^{7} - q^{9} - 13 q^{11} + 4 q^{13} - 5 q^{15} + 16 q^{17} - 10 q^{19} + q^{21} - 2 q^{25} + 13 q^{27} + 7 q^{29} - 5 q^{31} + 3 q^{33} - 9 q^{35} - 7 q^{37} + 5 q^{39} + 9 q^{41} + 3 q^{45} + 9 q^{47} - 9 q^{49} - 13 q^{51} + 2 q^{53} - 16 q^{55} - 18 q^{57} + 16 q^{59} + 4 q^{61} - 5 q^{63} - q^{65} + 17 q^{67} + 7 q^{71} - 18 q^{73} - 8 q^{75} + 12 q^{77} - 13 q^{79} - 11 q^{81} - 13 q^{83} + 7 q^{85} + 17 q^{87} + 15 q^{89} - 2 q^{91} + 2 q^{93} + 8 q^{95} - 16 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) 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
1.30972
−0.830830
−1.68251
0.284630
1.91899
0 −2.20362 0 2.59435 0 −3.39788 0 1.85592 0
1.2 0 −1.59435 0 1.47889 0 1.22871 0 −0.458044 0
1.3 0 −0.478891 0 −1.51334 0 −2.54620 0 −2.77066 0
1.4 0 −0.236479 0 3.20362 0 −0.911844 0 −2.94408 0
1.5 0 2.51334 0 1.23648 0 −2.37279 0 3.31686 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(-1\)

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 8464.2.a.bt 5
4.b odd 2 1 529.2.a.j 5
12.b even 2 1 4761.2.a.bn 5
23.b odd 2 1 8464.2.a.bs 5
23.d odd 22 2 368.2.m.c 10
92.b even 2 1 529.2.a.i 5
92.g odd 22 2 529.2.c.a 10
92.g odd 22 2 529.2.c.c 10
92.g odd 22 2 529.2.c.e 10
92.g odd 22 2 529.2.c.f 10
92.g odd 22 2 529.2.c.h 10
92.h even 22 2 23.2.c.a ✓ 10
92.h even 22 2 529.2.c.b 10
92.h even 22 2 529.2.c.d 10
92.h even 22 2 529.2.c.g 10
92.h even 22 2 529.2.c.i 10
276.h odd 2 1 4761.2.a.bo 5
276.j odd 22 2 207.2.i.c 10
460.o even 22 2 575.2.k.b 10
460.v odd 44 4 575.2.p.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.c.a ✓ 10 92.h even 22 2
207.2.i.c 10 276.j odd 22 2
368.2.m.c 10 23.d odd 22 2
529.2.a.i 5 92.b even 2 1
529.2.a.j 5 4.b odd 2 1
529.2.c.a 10 92.g odd 22 2
529.2.c.b 10 92.h even 22 2
529.2.c.c 10 92.g odd 22 2
529.2.c.d 10 92.h even 22 2
529.2.c.e 10 92.g odd 22 2
529.2.c.f 10 92.g odd 22 2
529.2.c.g 10 92.h even 22 2
529.2.c.h 10 92.g odd 22 2
529.2.c.i 10 92.h even 22 2
575.2.k.b 10 460.o even 22 2
575.2.p.b 20 460.v odd 44 4
4761.2.a.bn 5 12.b even 2 1
4761.2.a.bo 5 276.h odd 2 1
8464.2.a.bs 5 23.b odd 2 1
8464.2.a.bt 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8464))\):

\( T_{3}^{5} + 2T_{3}^{4} - 5T_{3}^{3} - 13T_{3}^{2} - 7T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{5} - 7T_{5}^{4} + 13T_{5}^{3} + 6T_{5}^{2} - 35T_{5} + 23 \) Copy content Toggle raw display
\( T_{7}^{5} + 8T_{7}^{4} + 19T_{7}^{3} + 4T_{7}^{2} - 32T_{7} - 23 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 9T_{13}^{3} + 38T_{13}^{2} - 2T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 2 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{5} - 7 T^{4} + \cdots + 23 \) Copy content Toggle raw display
$7$ \( T^{5} + 8 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$11$ \( T^{5} + 13 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{5} - 16 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$19$ \( T^{5} + 10 T^{4} + \cdots + 736 \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 7 T^{4} + \cdots - 2221 \) Copy content Toggle raw display
$31$ \( T^{5} + 5 T^{4} + \cdots - 131 \) Copy content Toggle raw display
$37$ \( T^{5} + 7 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$41$ \( T^{5} - 9 T^{4} + \cdots + 43 \) Copy content Toggle raw display
$43$ \( T^{5} - 33 T^{3} + \cdots - 253 \) Copy content Toggle raw display
$47$ \( T^{5} - 9 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$53$ \( T^{5} - 2 T^{4} + \cdots - 22747 \) Copy content Toggle raw display
$59$ \( T^{5} - 16 T^{4} + \cdots - 67 \) Copy content Toggle raw display
$61$ \( T^{5} - 4 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$67$ \( T^{5} - 17 T^{4} + \cdots - 33373 \) Copy content Toggle raw display
$71$ \( T^{5} - 7 T^{4} + \cdots + 23 \) Copy content Toggle raw display
$73$ \( T^{5} + 18 T^{4} + \cdots - 991 \) Copy content Toggle raw display
$79$ \( T^{5} + 13 T^{4} + \cdots + 22747 \) Copy content Toggle raw display
$83$ \( T^{5} + 13 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$89$ \( T^{5} - 15 T^{4} + \cdots - 279841 \) Copy content Toggle raw display
$97$ \( T^{5} + 16 T^{4} + \cdots + 1541 \) Copy content Toggle raw display
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