Properties

Label 64.9.c.c
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: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 16)
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 \(\beta = 8\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - 258 q^{5} + 238 \beta q^{7} + 6369 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - 258 q^{5} + 238 \beta q^{7} + 6369 q^{9} - 1671 \beta q^{11} - 19138 q^{13} + 258 \beta q^{15} - 58686 q^{17} + 11011 \beta q^{19} + 45696 q^{21} - 19494 \beta q^{23} - 324061 q^{25} - 12930 \beta q^{27} - 842178 q^{29} + 75864 \beta q^{31} - 320832 q^{33} - 61404 \beta q^{35} - 2548610 q^{37} + 19138 \beta q^{39} - 4324158 q^{41} + 147009 \beta q^{43} - 1643202 q^{45} - 522012 \beta q^{47} - 5110847 q^{49} + 58686 \beta q^{51} - 1192194 q^{53} + 431118 \beta q^{55} + 2114112 q^{57} - 24411 \beta q^{59} - 8414786 q^{61} + 1515822 \beta q^{63} + 4937604 q^{65} - 1257521 \beta q^{67} - 3742848 q^{69} - 2228322 \beta q^{71} + 12735874 q^{73} + 324061 \beta q^{75} + 76358016 q^{77} - 453700 \beta q^{79} + 39304449 q^{81} + 5995347 \beta q^{83} + 15140988 q^{85} + 842178 \beta q^{87} - 16802814 q^{89} - 4554844 \beta q^{91} + 14565888 q^{93} - 2840838 \beta q^{95} + 120994882 q^{97} - 10642599 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 516 q^{5} + 12738 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 516 q^{5} + 12738 q^{9} - 38276 q^{13} - 117372 q^{17} + 91392 q^{21} - 648122 q^{25} - 1684356 q^{29} - 641664 q^{33} - 5097220 q^{37} - 8648316 q^{41} - 3286404 q^{45} - 10221694 q^{49} - 2384388 q^{53} + 4228224 q^{57} - 16829572 q^{61} + 9875208 q^{65} - 7485696 q^{69} + 25471748 q^{73} + 152716032 q^{77} + 78608898 q^{81} + 30281976 q^{85} - 33605628 q^{89} + 29131776 q^{93} + 241989764 q^{97}+O(q^{100}) \) 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 + 0.866025i
0.500000 0.866025i
0 13.8564i 0 −258.000 0 3297.82i 0 6369.00 0
63.2 0 13.8564i 0 −258.000 0 3297.82i 0 6369.00 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.c 2
4.b odd 2 1 inner 64.9.c.c 2
8.b even 2 1 16.9.c.b 2
8.d odd 2 1 16.9.c.b 2
16.e even 4 2 256.9.d.d 4
16.f odd 4 2 256.9.d.d 4
24.f even 2 1 144.9.g.d 2
24.h odd 2 1 144.9.g.d 2
40.e odd 2 1 400.9.b.e 2
40.f even 2 1 400.9.b.e 2
40.i odd 4 2 400.9.h.a 4
40.k even 4 2 400.9.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.b 2 8.b even 2 1
16.9.c.b 2 8.d odd 2 1
64.9.c.c 2 1.a even 1 1 trivial
64.9.c.c 2 4.b odd 2 1 inner
144.9.g.d 2 24.f even 2 1
144.9.g.d 2 24.h odd 2 1
256.9.d.d 4 16.e even 4 2
256.9.d.d 4 16.f odd 4 2
400.9.b.e 2 40.e odd 2 1
400.9.b.e 2 40.f even 2 1
400.9.h.a 4 40.i odd 4 2
400.9.h.a 4 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 192 \) 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
$3$ \( T^{2} + 192 \) Copy content Toggle raw display
$5$ \( (T + 258)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10875648 \) Copy content Toggle raw display
$11$ \( T^{2} + 536110272 \) Copy content Toggle raw display
$13$ \( (T + 19138)^{2} \) Copy content Toggle raw display
$17$ \( (T + 58686)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 23278487232 \) Copy content Toggle raw display
$23$ \( T^{2} + 72963078912 \) Copy content Toggle raw display
$29$ \( (T + 842178)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1105026527232 \) Copy content Toggle raw display
$37$ \( (T + 2548610)^{2} \) Copy content Toggle raw display
$41$ \( (T + 4324158)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4149436047552 \) Copy content Toggle raw display
$47$ \( T^{2} + 52319333403648 \) Copy content Toggle raw display
$53$ \( (T + 1192194)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 114412208832 \) Copy content Toggle raw display
$61$ \( (T + 8414786)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 303620940564672 \) Copy content Toggle raw display
$71$ \( T^{2} + 953360435651328 \) Copy content Toggle raw display
$73$ \( (T - 12735874)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 39521988480000 \) Copy content Toggle raw display
$83$ \( T^{2} + 69\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( (T + 16802814)^{2} \) Copy content Toggle raw display
$97$ \( (T - 120994882)^{2} \) Copy content Toggle raw display
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