Properties

Label 704.6.a.c
Level $704$
Weight $6$
Character orbit 704.a
Self dual yes
Analytic conductor $112.910$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,6,Mod(1,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.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: \(112.910209148\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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)
 
\(f(q)\) \(=\) \( q - 15 q^{3} + 19 q^{5} - 10 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 15 q^{3} + 19 q^{5} - 10 q^{7} - 18 q^{9} - 121 q^{11} + 1148 q^{13} - 285 q^{15} + 686 q^{17} - 384 q^{19} + 150 q^{21} - 3709 q^{23} - 2764 q^{25} + 3915 q^{27} + 5424 q^{29} + 6443 q^{31} + 1815 q^{33} - 190 q^{35} - 12063 q^{37} - 17220 q^{39} - 1528 q^{41} - 4026 q^{43} - 342 q^{45} - 7168 q^{47} - 16707 q^{49} - 10290 q^{51} + 29862 q^{53} - 2299 q^{55} + 5760 q^{57} - 6461 q^{59} + 16980 q^{61} + 180 q^{63} + 21812 q^{65} + 29999 q^{67} + 55635 q^{69} - 31023 q^{71} + 1924 q^{73} + 41460 q^{75} + 1210 q^{77} - 65138 q^{79} - 54351 q^{81} - 102714 q^{83} + 13034 q^{85} - 81360 q^{87} + 17415 q^{89} - 11480 q^{91} - 96645 q^{93} - 7296 q^{95} + 66905 q^{97} + 2178 q^{99}+O(q^{100}) \) 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
0
0 −15.0000 0 19.0000 0 −10.0000 0 −18.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.6.a.c 1
4.b odd 2 1 704.6.a.h 1
8.b even 2 1 176.6.a.c 1
8.d odd 2 1 11.6.a.a 1
24.f even 2 1 99.6.a.c 1
40.e odd 2 1 275.6.a.a 1
40.k even 4 2 275.6.b.a 2
56.e even 2 1 539.6.a.c 1
88.g even 2 1 121.6.a.b 1
264.p odd 2 1 1089.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.a 1 8.d odd 2 1
99.6.a.c 1 24.f even 2 1
121.6.a.b 1 88.g even 2 1
176.6.a.c 1 8.b even 2 1
275.6.a.a 1 40.e odd 2 1
275.6.b.a 2 40.k even 4 2
539.6.a.c 1 56.e even 2 1
704.6.a.c 1 1.a even 1 1 trivial
704.6.a.h 1 4.b odd 2 1
1089.6.a.c 1 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 15 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 15 \) Copy content Toggle raw display
$5$ \( T - 19 \) Copy content Toggle raw display
$7$ \( T + 10 \) Copy content Toggle raw display
$11$ \( T + 121 \) Copy content Toggle raw display
$13$ \( T - 1148 \) Copy content Toggle raw display
$17$ \( T - 686 \) Copy content Toggle raw display
$19$ \( T + 384 \) Copy content Toggle raw display
$23$ \( T + 3709 \) Copy content Toggle raw display
$29$ \( T - 5424 \) Copy content Toggle raw display
$31$ \( T - 6443 \) Copy content Toggle raw display
$37$ \( T + 12063 \) Copy content Toggle raw display
$41$ \( T + 1528 \) Copy content Toggle raw display
$43$ \( T + 4026 \) Copy content Toggle raw display
$47$ \( T + 7168 \) Copy content Toggle raw display
$53$ \( T - 29862 \) Copy content Toggle raw display
$59$ \( T + 6461 \) Copy content Toggle raw display
$61$ \( T - 16980 \) Copy content Toggle raw display
$67$ \( T - 29999 \) Copy content Toggle raw display
$71$ \( T + 31023 \) Copy content Toggle raw display
$73$ \( T - 1924 \) Copy content Toggle raw display
$79$ \( T + 65138 \) Copy content Toggle raw display
$83$ \( T + 102714 \) Copy content Toggle raw display
$89$ \( T - 17415 \) Copy content Toggle raw display
$97$ \( T - 66905 \) Copy content Toggle raw display
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