Properties

Label 64.23.c.e
Level $64$
Weight $23$
Character orbit 64.c
Analytic conductor $196.293$
Analytic rank $0$
Dimension $10$
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] = [64,23,Mod(63,64)]
 
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([1, 0]))
 
N = Newforms(chi, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 23 \)
Character orbit: \([\chi]\) \(=\) 64.c (of order \(2\), degree \(1\), not 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: \(196.292758299\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{180}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 4)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} + 1709110) q^{5} + ( - \beta_{3} + 749 \beta_1) q^{7} + (\beta_{4} + 5 \beta_{2} - 10430961370) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{2} + 1709110) q^{5} + ( - \beta_{3} + 749 \beta_1) q^{7} + (\beta_{4} + 5 \beta_{2} - 10430961370) q^{9} + ( - \beta_{5} + 20 \beta_{3} + 174367 \beta_1) q^{11} + (\beta_{6} + 5 \beta_{4} - 3886 \beta_{2} + 53123035655) q^{13} + (\beta_{9} - 16 \beta_{5} + 762 \beta_{3} - 1305198 \beta_1) q^{15} + ( - \beta_{7} - 18 \beta_{6} - 279 \beta_{4} + \cdots + 1405817811498) q^{17}+ \cdots + ( - 481885866 \beta_{9} + \cdots - 91\!\cdots\!08 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 17091100 q^{5} - 104309613702 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 17091100 q^{5} - 104309613702 q^{9} + 531230356540 q^{13} + 14058178115540 q^{17} + 313135665760512 q^{21} + 77\!\cdots\!70 q^{25}+ \cdots - 20\!\cdots\!40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3059527 \nu^{9} - 245264621 \nu^{8} - 1074014129748 \nu^{7} - 789332238025400 \nu^{6} + \cdots - 13\!\cdots\!20 ) / 69\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7421 \nu^{9} - 3965313 \nu^{8} - 1074075428 \nu^{7} + 247102967144 \nu^{6} + 93623286784016 \nu^{5} + \cdots + 10\!\cdots\!60 ) / 23\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29522529089 \nu^{9} - 2403948356587 \nu^{8} + \cdots - 13\!\cdots\!60 ) / 69\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8071135 \nu^{9} - 4302794123 \nu^{8} - 1156022538892 \nu^{7} + 260365540875128 \nu^{6} + \cdots + 11\!\cdots\!84 ) / 69\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1531236400253 \nu^{9} + 119955483142399 \nu^{8} + \cdots + 69\!\cdots\!20 ) / 34\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48080309 \nu^{9} - 25824561817 \nu^{8} - 6649123206788 \nu^{7} + \cdots + 68\!\cdots\!72 ) / 34\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67890143 \nu^{9} + 35085255051 \nu^{8} + 9417410667148 \nu^{7} + \cdots - 96\!\cdots\!88 ) / 23\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31140071174407 \nu^{9} + \cdots + 15\!\cdots\!80 ) / 69\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2580417204287 \nu^{9} - 128014907406997 \nu^{8} + \cdots - 11\!\cdots\!20 ) / 38\!\cdots\!08 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - 8 \beta_{7} - 2 \beta_{5} + 48 \beta_{4} - 2354 \beta_{3} - 88328 \beta_{2} + 32539539 \beta _1 + 34359738376 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 1024 \beta_{9} - 59 \beta_{8} + 536 \beta_{7} + 8192 \beta_{6} + 10358 \beta_{5} - 11408 \beta_{4} + 262790 \beta_{3} - 25645800 \beta_{2} - 7804446433 \beta _1 + 870737617772008 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 113664 \beta_{9} + 5335 \beta_{8} - 22264 \beta_{7} - 1449984 \beta_{6} - 19845550 \beta_{5} - 99079728 \beta_{4} - 2825766142 \beta_{3} + \cdots + 94\!\cdots\!72 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 116798464 \beta_{9} + 8656297 \beta_{8} + 65538552 \beta_{7} - 1175609344 \beta_{6} + 8075467950 \beta_{5} + 67522143280 \beta_{4} + \cdots - 89\!\cdots\!16 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 47893785600 \beta_{9} + 46781584543 \beta_{8} + 369506982344 \beta_{7} + 4451033088 \beta_{6} - 3928865435966 \beta_{5} + \cdots - 33\!\cdots\!00 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9436276083712 \beta_{9} - 2969748351543 \beta_{8} + 5777265680632 \beta_{7} - 157365071929344 \beta_{6} + \cdots - 59\!\cdots\!20 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18\!\cdots\!60 \beta_{9} + \cdots - 90\!\cdots\!56 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 33\!\cdots\!52 \beta_{9} + \cdots - 40\!\cdots\!52 ) / 68719476736 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 51\!\cdots\!84 \beta_{9} + \cdots + 90\!\cdots\!00 ) / 68719476736 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
−313.209 + 431.746i
407.912 + 251.607i
−501.982 216.483i
408.476 250.605i
1.30158 510.489i
1.30158 + 510.489i
408.476 + 250.605i
−501.982 + 216.483i
407.912 251.607i
−313.209 431.746i
0 308212.i 0 6.05072e7 0 1.97415e9i 0 −6.36133e10 0
63.2 0 267899.i 0 −8.57934e7 0 2.80689e8i 0 −4.03887e10 0
63.3 0 142172.i 0 −1.40852e7 0 7.81741e8i 0 1.11682e10 0
63.4 0 127855.i 0 6.52379e7 0 3.27904e9i 0 1.50341e10 0
63.5 0 75737.5i 0 −1.73209e7 0 2.02930e9i 0 2.56449e10 0
63.6 0 75737.5i 0 −1.73209e7 0 2.02930e9i 0 2.56449e10 0
63.7 0 127855.i 0 6.52379e7 0 3.27904e9i 0 1.50341e10 0
63.8 0 142172.i 0 −1.40852e7 0 7.81741e8i 0 1.11682e10 0
63.9 0 267899.i 0 −8.57934e7 0 2.80689e8i 0 −4.03887e10 0
63.10 0 308212.i 0 6.05072e7 0 1.97415e9i 0 −6.36133e10 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.23.c.e 10
4.b odd 2 1 inner 64.23.c.e 10
8.b even 2 1 4.23.b.a 10
8.d odd 2 1 4.23.b.a 10
24.f even 2 1 36.23.d.c 10
24.h odd 2 1 36.23.d.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.23.b.a 10 8.b even 2 1
4.23.b.a 10 8.d odd 2 1
36.23.d.c 10 24.f even 2 1
36.23.d.c 10 24.h odd 2 1
64.23.c.e 10 1.a even 1 1 trivial
64.23.c.e 10 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 209060104896 T_{3}^{8} + \cdots + 12\!\cdots\!00 \) acting on \(S_{23}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 209060104896 T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{5} - 8545550 T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} - 265615178270 T^{4} + \cdots - 41\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} - 7029089057770 T^{4} + \cdots + 27\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 79\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 22\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 80\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 46\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
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