↑

Properties

Label 64.19.c.e
Level $64$
Weight $19$
Character orbit 64.c
Analytic conductor $131.447$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 19 \)
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: \(131.447128134\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} + \cdots + 71\!\cdots\!30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{112}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} - 107610) q^{5} + (\beta_{3} + 202 \beta_1) q^{7} + (\beta_{4} - 15 \beta_{2} - 91192695) q^{9} + (\beta_{6} + 12 \beta_{3} - 5621 \beta_1) q^{11} + ( - \beta_{5} - 12 \beta_{4} + \cdots + 832347286) q^{13}+ \cdots + ( - 22498458 \beta_{7} + \cdots + 8858850972369 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 860880 q^{5} - 729541560 q^{9} + 6658778288 q^{13} + 213854181648 q^{17} + 771822339072 q^{21} + 528925732440 q^{25} - 11238056568912 q^{29} - 21541938424320 q^{33} + 158886968816432 q^{37} + 451509984725136 q^{41}+ \cdots - 12\!\cdots\!12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3 x^{7} - 10969 x^{6} - 835887 x^{5} + 20786973 x^{4} - 17082875145 x^{3} + \cdots + 71\!\cdots\!30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 349 \nu^{7} + 53248 \nu^{6} + 525867 \nu^{5} + 212100462 \nu^{4} - 9178232109 \nu^{3} + \cdots - 18\!\cdots\!30 ) / 5772436045824 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 203 \nu^{7} - 24576 \nu^{6} + 1893869 \nu^{5} + 118153602 \nu^{4} - 88763350907 \nu^{3} + \cdots + 32\!\cdots\!10 ) / 120259084288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 164099 \nu^{7} + 21180416 \nu^{6} + 1883223669 \nu^{5} - 147902791470 \nu^{4} + \cdots - 10\!\cdots\!90 ) / 1443109011456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1281 \nu^{7} + 6627328 \nu^{6} + 155568089 \nu^{5} - 70792780870 \nu^{4} + \cdots - 98\!\cdots\!78 ) / 120259084288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 103453 \nu^{7} - 1875968 \nu^{6} - 1446588053 \nu^{5} - 71155841554 \nu^{4} + \cdots + 13\!\cdots\!38 ) / 30064771072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10150111 \nu^{7} - 135335936 \nu^{6} + 74463550713 \nu^{5} + 9702533103354 \nu^{4} + \cdots - 16\!\cdots\!10 ) / 2886218022912 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81903265 \nu^{7} + 625807360 \nu^{6} - 985551982713 \nu^{5} - 91240663441914 \nu^{4} + \cdots + 67\!\cdots\!30 ) / 2886218022912 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 8\beta_{5} - 16\beta_{4} - 15\beta_{3} - 1872\beta_{2} + 5444\beta _1 + 100663296 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 7 \beta_{7} - 1031 \beta_{6} - 8 \beta_{5} - 8208 \beta_{4} + 22633 \beta_{3} + \cdots + 736419119104 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1171 \beta_{7} + 78995 \beta_{6} + 4712 \beta_{5} - 678704 \beta_{4} + 2464611 \beta_{3} + \cdots + 87456573554688 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 938323 \beta_{7} + 6058669 \beta_{6} - 7867688 \beta_{5} - 98957904 \beta_{4} + \cdots + 56\!\cdots\!48 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 161202677 \beta_{7} + 765179403 \beta_{6} - 1308895448 \beta_{5} - 4418474416 \beta_{4} + \cdots + 44\!\cdots\!56 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10493167501 \beta_{7} - 104457993331 \beta_{6} - 179426196648 \beta_{5} + 439762740912 \beta_{4} + \cdots + 68\!\cdots\!04 ) / 268435456 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4466073973275 \beta_{7} + 1854603989531 \beta_{6} - 8565850414808 \beta_{5} + \cdots + 41\!\cdots\!88 ) / 268435456 \) 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
130.763 + 7.78570i
−116.282 − 46.4306i
−67.8767 + 106.586i
54.8957 − 117.008i
54.8957 + 117.008i
−67.8767 − 106.586i
−116.282 + 46.4306i
130.763 − 7.78570i
0 − 30905.2i 0 −991387. 0 6.49702e7i 0 −5.67711e8 0
63.2 0 − 27088.6i 0 2.11965e6 0 − 3.35455e7i 0 −3.46370e8 0
63.3 0 − 14439.2i 0 −2.88086e6 0 − 4.40005e7i 0 1.78931e8 0
63.4 0 − 4128.11i 0 1.32216e6 0 − 1.88890e7i 0 3.70379e8 0
63.5 0 4128.11i 0 1.32216e6 0 1.88890e7i 0 3.70379e8 0
63.6 0 14439.2i 0 −2.88086e6 0 4.40005e7i 0 1.78931e8 0
63.7 0 27088.6i 0 2.11965e6 0 3.35455e7i 0 −3.46370e8 0
63.8 0 30905.2i 0 −991387. 0 − 6.49702e7i 0 −5.67711e8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.8
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 64.19.c.e 8
4.b odd 2 1 inner 64.19.c.e 8
8.b even 2 1 4.19.b.a ✓ 8
8.d odd 2 1 4.19.b.a ✓ 8
24.f even 2 1 36.19.d.c 8
24.h odd 2 1 36.19.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.19.b.a ✓ 8 8.b even 2 1
4.19.b.a ✓ 8 8.d odd 2 1
36.19.d.c 8 24.f even 2 1
36.19.d.c 8 24.h odd 2 1
64.19.c.e 8 1.a even 1 1 trivial
64.19.c.e 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1914452736 T_{3}^{6} + \cdots + 24\!\cdots\!00 \) acting on \(S_{19}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{4} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 57\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots - 57\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 21\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 18\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 45\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
show more
show less