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Properties

Label 64.25.c.d
Level $64$
Weight $25$
Character orbit 64.c
Analytic conductor $233.579$
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] = [64,25,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, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 25 \)
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: \(233.578977445\)
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: \( x^{10} + 7350292833 x^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{8}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 5675810) q^{5} + (\beta_{6} - 757 \beta_1) q^{7} + (\beta_{4} + \beta_{3} + \cdots - 93905456568) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 5675810) q^{5} + (\beta_{6} - 757 \beta_1) q^{7} + (\beta_{4} + \beta_{3} + \cdots - 93905456568) q^{9}+ \cdots + ( - 5143871667 \beta_{9} + \cdots - 15\!\cdots\!20 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 56758100 q^{5} - 939054565686 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 56758100 q^{5} - 939054565686 q^{9} + 24798065342764 q^{13} - 16\!\cdots\!64 q^{17}+ \cdots + 22\!\cdots\!36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7350292833 x^{8} + \cdots + 68\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 16\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 98\!\cdots\!44 \nu^{8} + \cdots - 59\!\cdots\!60 ) / 69\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!52 \nu^{8} + \cdots + 57\!\cdots\!10 ) / 11\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 72\!\cdots\!04 \nu^{8} + \cdots + 16\!\cdots\!25 ) / 38\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 57\!\cdots\!88 \nu^{8} + \cdots - 43\!\cdots\!18 ) / 69\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!72 \nu^{9} + \cdots - 16\!\cdots\!08 \nu ) / 59\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!96 \nu^{9} + \cdots - 48\!\cdots\!80 \nu ) / 69\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!56 \nu^{9} + \cdots - 52\!\cdots\!60 \nu ) / 41\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!16 \nu^{9} + \cdots - 61\!\cdots\!20 \nu ) / 23\!\cdots\!65 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} + 114\beta_{2} - 376334993049 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -759\beta_{9} + 942\beta_{8} + 37302\beta_{7} - 1553397\beta_{6} - 629705987850\beta_1 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 39579409161 \beta_{5} - 568281007893 \beta_{4} - 321124963098 \beta_{3} - 133112576394396 \beta_{2} + 11\!\cdots\!65 ) / 32768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 479259988925031 \beta_{9} - 533291936214150 \beta_{8} + \cdots + 24\!\cdots\!39 \beta_1 ) / 524288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 75\!\cdots\!63 \beta_{5} + \cdots - 11\!\cdots\!20 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 61\!\cdots\!43 \beta_{9} + \cdots - 26\!\cdots\!68 \beta_1 ) / 16777216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11\!\cdots\!39 \beta_{5} + \cdots + 12\!\cdots\!85 ) / 33554432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 74\!\cdots\!09 \beta_{9} + \cdots + 29\!\cdots\!67 \beta_1 ) / 536870912 \) Copy content Toggle raw display

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

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
− 60174.2i
− 41641.2i
− 35566.9i
− 24156.2i
− 12118.0i
12118.0i
24156.2i
35566.9i
41641.2i
60174.2i
0 − 962787.i 0 −1.24317e8 0 3.87327e9i 0 −6.44529e11 0
63.2 0 − 666259.i 0 −9.84947e7 0 − 1.39617e10i 0 −1.61471e11 0
63.3 0 − 569070.i 0 2.26723e8 0 − 6.45252e9i 0 −4.14111e10 0
63.4 0 − 386500.i 0 3.41994e8 0 1.51829e10i 0 1.33047e11 0
63.5 0 − 193889.i 0 −3.74285e8 0 2.47654e10i 0 2.44837e11 0
63.6 0 193889.i 0 −3.74285e8 0 − 2.47654e10i 0 2.44837e11 0
63.7 0 386500.i 0 3.41994e8 0 − 1.51829e10i 0 1.33047e11 0
63.8 0 569070.i 0 2.26723e8 0 6.45252e9i 0 −4.14111e10 0
63.9 0 666259.i 0 −9.84947e7 0 1.39617e10i 0 −1.61471e11 0
63.10 0 962787.i 0 −1.24317e8 0 − 3.87327e9i 0 −6.44529e11 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 64.25.c.d 10
4.b odd 2 1 inner 64.25.c.d 10
8.b even 2 1 4.25.b.b ✓ 10
8.d odd 2 1 4.25.b.b ✓ 10
24.f even 2 1 36.25.d.b 10
24.h odd 2 1 36.25.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.25.b.b ✓ 10 8.b even 2 1
4.25.b.b ✓ 10 8.d odd 2 1
36.25.d.b 10 24.f even 2 1
36.25.d.b 10 24.h odd 2 1
64.25.c.d 10 1.a even 1 1 trivial
64.25.c.d 10 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 1881674965248 T_{3}^{8} + \cdots + 74\!\cdots\!00 \) acting on \(S_{25}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{5} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 91\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
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