Properties

Label 64.22.a.m
Level $64$
Weight $22$
Character orbit 64.a
Self dual yes
Analytic conductor $178.866$
Analytic rank $0$
Dimension $3$
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] = [64,22,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(178.865500344\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 32255) q^{3} + ( - \beta_{2} - 9 \beta_1 + 8037261) q^{5} + ( - 4 \beta_{2} + 4014 \beta_1 - 98664098) q^{7} + ( - 306 \beta_{2} - 120578 \beta_1 + 6281605939) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 32255) q^{3} + ( - \beta_{2} - 9 \beta_1 + 8037261) q^{5} + ( - 4 \beta_{2} + 4014 \beta_1 - 98664098) q^{7} + ( - 306 \beta_{2} - 120578 \beta_1 + 6281605939) q^{9} + (2856 \beta_{2} - 241515 \beta_1 - 13444955723) q^{11} + (8551 \beta_{2} - 1219041 \beta_1 - 44577735635) q^{13} + (21060 \beta_{2} - 24214790 \beta_1 + 407720224330) q^{15} + (206750 \beta_{2} - 78703794 \beta_1 + 2599270326072) q^{17} + ( - 319144 \beta_{2} + 10179315 \beta_1 + 11929396534227) q^{19} + (1323540 \beta_{2} + 391662132 \beta_1 - 66179872171740) q^{21} + ( - 5214236 \beta_{2} - 1679798214 \beta_1 - 64589693893622) q^{23} + (292556 \beta_{2} + 2369077164 \beta_1 + 375854023120779) q^{25} + ( - 29609784 \beta_{2} - 11178145762 \beta_1 + 17\!\cdots\!38) q^{27}+ \cdots + ( - 15464975405472 \beta_{2} + \cdots - 31\!\cdots\!83) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 96764 q^{3} + 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 96764 q^{3} + 24111774 q^{5} - 295988280 q^{7} + 18844697239 q^{9} - 40335108684 q^{11} - 133734425946 q^{13} + 1223136458200 q^{15} + 7797732274422 q^{17} + 35788199781996 q^{19} - 198539224853088 q^{21} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 94\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4963x + 96223 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 64\nu^{2} + 640\nu - 211989 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 8896\nu^{2} + 662400\nu - 29657664 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 139\beta _1 + 191193 ) / 573440 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 1035\beta _1 + 189750951 ) / 57344 \) 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
−78.2002
57.9766
21.2235
0 −97085.2 0 3.39292e7 0 5.28731e8 0 −1.03483e9 0
1.2 0 −7983.67 0 −3.09730e7 0 −9.17385e7 0 −1.03966e10 0
1.3 0 201833. 0 2.11555e7 0 −7.32981e8 0 3.02761e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 64.22.a.m 3
4.b odd 2 1 64.22.a.l 3
8.b even 2 1 16.22.a.f 3
8.d odd 2 1 8.22.a.b 3
24.f even 2 1 72.22.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.b 3 8.d odd 2 1
16.22.a.f 3 8.b even 2 1
64.22.a.l 3 4.b odd 2 1
64.22.a.m 3 1.a even 1 1 trivial
72.22.a.f 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 96764T_{3}^{2} - 20431242576T_{3} - 156439878638400 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(64))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + \cdots - 156439878638400 \) Copy content Toggle raw display
$5$ \( T^{3} - 24111774 T^{2} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + 295988280 T^{2} + \cdots - 35\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{3} + 40335108684 T^{2} + \cdots - 59\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{3} + 133734425946 T^{2} + \cdots + 64\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{3} - 7797732274422 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{3} - 35788199781996 T^{2} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{3} + 193770761479080 T^{2} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 88\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 79\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 42\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 28\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 44\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
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