Properties

Label 704.4.a.d
Level $704$
Weight $4$
Character orbit 704.a
Self dual yes
Analytic conductor $41.537$
Analytic rank $1$
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] = [704,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(41.5373446440\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{3} - 14 q^{5} - 8 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{3} - 14 q^{5} - 8 q^{7} - 11 q^{9} + 11 q^{11} + 50 q^{13} + 56 q^{15} + 130 q^{17} + 108 q^{19} + 32 q^{21} - 96 q^{23} + 71 q^{25} + 152 q^{27} - 142 q^{29} + 40 q^{31} - 44 q^{33} + 112 q^{35} - 382 q^{37} - 200 q^{39} - 118 q^{41} - 220 q^{43} + 154 q^{45} + 520 q^{47} - 279 q^{49} - 520 q^{51} - 238 q^{53} - 154 q^{55} - 432 q^{57} + 852 q^{59} - 190 q^{61} + 88 q^{63} - 700 q^{65} + 12 q^{67} + 384 q^{69} - 112 q^{71} - 6 q^{73} - 284 q^{75} - 88 q^{77} + 304 q^{79} - 311 q^{81} - 820 q^{83} - 1820 q^{85} + 568 q^{87} + 202 q^{89} - 400 q^{91} - 160 q^{93} - 1512 q^{95} - 1406 q^{97} - 121 q^{99}+O(q^{100}) \) 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 −4.00000 0 −14.0000 0 −8.00000 0 −11.0000 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 704.4.a.d 1
4.b odd 2 1 704.4.a.i 1
8.b even 2 1 22.4.a.b 1
8.d odd 2 1 176.4.a.b 1
24.f even 2 1 1584.4.a.b 1
24.h odd 2 1 198.4.a.d 1
40.f even 2 1 550.4.a.k 1
40.i odd 4 2 550.4.b.b 2
56.h odd 2 1 1078.4.a.a 1
88.b odd 2 1 242.4.a.f 1
88.g even 2 1 1936.4.a.g 1
88.o even 10 4 242.4.c.h 4
88.p odd 10 4 242.4.c.b 4
264.m even 2 1 2178.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.b 1 8.b even 2 1
176.4.a.b 1 8.d odd 2 1
198.4.a.d 1 24.h odd 2 1
242.4.a.f 1 88.b odd 2 1
242.4.c.b 4 88.p odd 10 4
242.4.c.h 4 88.o even 10 4
550.4.a.k 1 40.f even 2 1
550.4.b.b 2 40.i odd 4 2
704.4.a.d 1 1.a even 1 1 trivial
704.4.a.i 1 4.b odd 2 1
1078.4.a.a 1 56.h odd 2 1
1584.4.a.b 1 24.f even 2 1
1936.4.a.g 1 88.g even 2 1
2178.4.a.a 1 264.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(704))\):

\( T_{3} + 4 \) Copy content Toggle raw display
\( T_{5} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 4 \) Copy content Toggle raw display
$5$ \( T + 14 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 50 \) Copy content Toggle raw display
$17$ \( T - 130 \) Copy content Toggle raw display
$19$ \( T - 108 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T + 142 \) Copy content Toggle raw display
$31$ \( T - 40 \) Copy content Toggle raw display
$37$ \( T + 382 \) Copy content Toggle raw display
$41$ \( T + 118 \) Copy content Toggle raw display
$43$ \( T + 220 \) Copy content Toggle raw display
$47$ \( T - 520 \) Copy content Toggle raw display
$53$ \( T + 238 \) Copy content Toggle raw display
$59$ \( T - 852 \) Copy content Toggle raw display
$61$ \( T + 190 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 112 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T - 304 \) Copy content Toggle raw display
$83$ \( T + 820 \) Copy content Toggle raw display
$89$ \( T - 202 \) Copy content Toggle raw display
$97$ \( T + 1406 \) Copy content Toggle raw display
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