Properties

Label 64.8.a.i
Level $64$
Weight $8$
Character orbit 64.a
Self dual yes
Analytic conductor $19.993$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(19.9926416310\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 32)
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 \(\beta = 16\sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 70 q^{5} - 18 \beta q^{7} + 1653 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 70 q^{5} - 18 \beta q^{7} + 1653 q^{9} - 117 \beta q^{11} - 13758 q^{13} - 70 \beta q^{15} + 16994 q^{17} + 549 \beta q^{19} - 69120 q^{21} + 522 \beta q^{23} - 73225 q^{25} - 534 \beta q^{27} - 34190 q^{29} + 1944 \beta q^{31} - 449280 q^{33} + 1260 \beta q^{35} - 35206 q^{37} - 13758 \beta q^{39} - 484550 q^{41} - 10845 \beta q^{43} - 115710 q^{45} + 19476 \beta q^{47} + 420617 q^{49} + 16994 \beta q^{51} - 851702 q^{53} + 8190 \beta q^{55} + 2108160 q^{57} + 11223 \beta q^{59} - 71630 q^{61} - 29754 \beta q^{63} + 963060 q^{65} - 4959 \beta q^{67} + 2004480 q^{69} + 12222 \beta q^{71} + 3912042 q^{73} - 73225 \beta q^{75} + 8087040 q^{77} - 5076 \beta q^{79} - 5665671 q^{81} - 24795 \beta q^{83} - 1189580 q^{85} - 34190 \beta q^{87} - 2510630 q^{89} + 247644 \beta q^{91} + 7464960 q^{93} - 38430 \beta q^{95} - 50094 q^{97} - 193401 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 140 q^{5} + 3306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 140 q^{5} + 3306 q^{9} - 27516 q^{13} + 33988 q^{17} - 138240 q^{21} - 146450 q^{25} - 68380 q^{29} - 898560 q^{33} - 70412 q^{37} - 969100 q^{41} - 231420 q^{45} + 841234 q^{49} - 1703404 q^{53} + 4216320 q^{57} - 143260 q^{61} + 1926120 q^{65} + 4008960 q^{69} + 7824084 q^{73} + 16174080 q^{77} - 11331342 q^{81} - 2379160 q^{85} - 5021260 q^{89} + 14929920 q^{93} - 100188 q^{97}+O(q^{100}) \) 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
−3.87298
3.87298
0 −61.9677 0 −70.0000 0 1115.42 0 1653.00 0
1.2 0 61.9677 0 −70.0000 0 −1115.42 0 1653.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 64.8.a.i 2
3.b odd 2 1 576.8.a.bk 2
4.b odd 2 1 inner 64.8.a.i 2
8.b even 2 1 32.8.a.c 2
8.d odd 2 1 32.8.a.c 2
12.b even 2 1 576.8.a.bk 2
16.e even 4 2 256.8.b.i 4
16.f odd 4 2 256.8.b.i 4
24.f even 2 1 288.8.a.k 2
24.h odd 2 1 288.8.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.a.c 2 8.b even 2 1
32.8.a.c 2 8.d odd 2 1
64.8.a.i 2 1.a even 1 1 trivial
64.8.a.i 2 4.b odd 2 1 inner
256.8.b.i 4 16.e even 4 2
256.8.b.i 4 16.f odd 4 2
288.8.a.k 2 24.f even 2 1
288.8.a.k 2 24.h odd 2 1
576.8.a.bk 2 3.b odd 2 1
576.8.a.bk 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3840 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(64))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 3840 \) Copy content Toggle raw display
$5$ \( (T + 70)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1244160 \) Copy content Toggle raw display
$11$ \( T^{2} - 52565760 \) Copy content Toggle raw display
$13$ \( (T + 13758)^{2} \) Copy content Toggle raw display
$17$ \( (T - 16994)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 1157379840 \) Copy content Toggle raw display
$23$ \( T^{2} - 1046338560 \) Copy content Toggle raw display
$29$ \( (T + 34190)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 14511882240 \) Copy content Toggle raw display
$37$ \( (T + 35206)^{2} \) Copy content Toggle raw display
$41$ \( (T + 484550)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 451637856000 \) Copy content Toggle raw display
$47$ \( T^{2} - 1456567971840 \) Copy content Toggle raw display
$53$ \( (T + 851702)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 483669999360 \) Copy content Toggle raw display
$61$ \( (T + 71630)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 94432055040 \) Copy content Toggle raw display
$71$ \( T^{2} - 573608770560 \) Copy content Toggle raw display
$73$ \( (T - 3912042)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 98940579840 \) Copy content Toggle raw display
$83$ \( T^{2} - 2360801376000 \) Copy content Toggle raw display
$89$ \( (T + 2510630)^{2} \) Copy content Toggle raw display
$97$ \( (T + 50094)^{2} \) Copy content Toggle raw display
show more
show less