Properties

Label 64.9.c.b
Level $64$
Weight $9$
Character orbit 64.c
Analytic conductor $26.072$
Analytic rank $0$
Dimension $2$
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] = [64,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(26.0722310439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-39}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
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 \(\beta = 16\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 610 q^{5} + 14 \beta q^{7} - 3423 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 610 q^{5} + 14 \beta q^{7} - 3423 q^{9} + 185 \beta q^{11} + 5470 q^{13} + 610 \beta q^{15} + 73090 q^{17} + 195 \beta q^{19} + 139776 q^{21} - 2374 \beta q^{23} - 18525 q^{25} - 3138 \beta q^{27} + 128222 q^{29} - 680 \beta q^{31} + 1847040 q^{33} - 8540 \beta q^{35} + 3472030 q^{37} - 5470 \beta q^{39} + 2146882 q^{41} + 59329 \beta q^{43} + 2088030 q^{45} + 76324 \beta q^{47} + 3807937 q^{49} - 73090 \beta q^{51} - 824290 q^{53} - 112850 \beta q^{55} + 1946880 q^{57} + 37285 \beta q^{59} + 14746078 q^{61} - 47922 \beta q^{63} - 3336700 q^{65} - 152689 \beta q^{67} - 23702016 q^{69} - 11970 \beta q^{71} - 5725630 q^{73} + 18525 \beta q^{75} - 25858560 q^{77} + 359420 \beta q^{79} - 53788095 q^{81} + 520019 \beta q^{83} - 44584900 q^{85} - 128222 \beta q^{87} - 83324222 q^{89} + 76580 \beta q^{91} - 6789120 q^{93} - 118950 \beta q^{95} + 120619010 q^{97} - 633255 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1220 q^{5} - 6846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1220 q^{5} - 6846 q^{9} + 10940 q^{13} + 146180 q^{17} + 279552 q^{21} - 37050 q^{25} + 256444 q^{29} + 3694080 q^{33} + 6944060 q^{37} + 4293764 q^{41} + 4176060 q^{45} + 7615874 q^{49} - 1648580 q^{53} + 3893760 q^{57} + 29492156 q^{61} - 6673400 q^{65} - 47404032 q^{69} - 11451260 q^{73} - 51717120 q^{77} - 107576190 q^{81} - 89169800 q^{85} - 166648444 q^{89} - 13578240 q^{93} + 241238020 q^{97}+O(q^{100}) \) 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
0.500000 + 3.12250i
0.500000 3.12250i
0 99.9200i 0 −610.000 0 1398.88i 0 −3423.00 0
63.2 0 99.9200i 0 −610.000 0 1398.88i 0 −3423.00 0
\(n\): e.g. 2-40 or 990-1000
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 64.9.c.b 2
4.b odd 2 1 inner 64.9.c.b 2
8.b even 2 1 4.9.b.b 2
8.d odd 2 1 4.9.b.b 2
16.e even 4 2 256.9.d.e 4
16.f odd 4 2 256.9.d.e 4
24.f even 2 1 36.9.d.b 2
24.h odd 2 1 36.9.d.b 2
40.e odd 2 1 100.9.b.c 2
40.f even 2 1 100.9.b.c 2
40.i odd 4 2 100.9.d.b 4
40.k even 4 2 100.9.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 8.b even 2 1
4.9.b.b 2 8.d odd 2 1
36.9.d.b 2 24.f even 2 1
36.9.d.b 2 24.h odd 2 1
64.9.c.b 2 1.a even 1 1 trivial
64.9.c.b 2 4.b odd 2 1 inner
100.9.b.c 2 40.e odd 2 1
100.9.b.c 2 40.f even 2 1
100.9.d.b 4 40.i odd 4 2
100.9.d.b 4 40.k even 4 2
256.9.d.e 4 16.e even 4 2
256.9.d.e 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9984 \) acting on \(S_{9}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9984 \) Copy content Toggle raw display
$5$ \( (T + 610)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1956864 \) Copy content Toggle raw display
$11$ \( T^{2} + 341702400 \) Copy content Toggle raw display
$13$ \( (T - 5470)^{2} \) Copy content Toggle raw display
$17$ \( (T - 73090)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 379641600 \) Copy content Toggle raw display
$23$ \( T^{2} + 56268585984 \) Copy content Toggle raw display
$29$ \( (T - 128222)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4616601600 \) Copy content Toggle raw display
$37$ \( (T - 3472030)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2146882)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 35142983526144 \) Copy content Toggle raw display
$47$ \( T^{2} + 58160324112384 \) Copy content Toggle raw display
$53$ \( (T + 824290)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 13879469510400 \) Copy content Toggle raw display
$61$ \( (T - 14746078)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 232766284318464 \) Copy content Toggle raw display
$71$ \( T^{2} + 1430516505600 \) Copy content Toggle raw display
$73$ \( (T + 5725630)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 26\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 83324222)^{2} \) Copy content Toggle raw display
$97$ \( (T - 120619010)^{2} \) Copy content Toggle raw display
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