Properties

Label 704.2.a.p
Level $704$
Weight $2$
Character orbit 704.a
Self dual yes
Analytic conductor $5.621$
Analytic rank $0$
Dimension $2$
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] = [704,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + (\beta - 2) q^{5} + 2 \beta q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + (\beta - 2) q^{5} + 2 \beta q^{7} + (\beta + 1) q^{9} - q^{11} + ( - 2 \beta + 2) q^{13} + ( - \beta + 4) q^{15} + 2 q^{17} - 4 q^{19} + (2 \beta + 8) q^{21} + ( - \beta - 4) q^{23} + ( - 3 \beta + 3) q^{25} + ( - \beta + 4) q^{27} + ( - 2 \beta + 2) q^{29} + ( - \beta + 4) q^{31} - \beta q^{33} + ( - 2 \beta + 8) q^{35} + ( - \beta + 6) q^{37} - 8 q^{39} + (2 \beta + 2) q^{41} + (2 \beta - 4) q^{43} + 2 q^{45} - 8 q^{47} + (4 \beta + 9) q^{49} + 2 \beta q^{51} + (4 \beta - 6) q^{53} + ( - \beta + 2) q^{55} - 4 \beta q^{57} - 5 \beta q^{59} + (2 \beta + 2) q^{61} + (4 \beta + 8) q^{63} + (4 \beta - 12) q^{65} + ( - \beta + 8) q^{67} + ( - 5 \beta - 4) q^{69} + ( - 3 \beta + 4) q^{71} + ( - 2 \beta + 2) q^{73} - 12 q^{75} - 2 \beta q^{77} + ( - 2 \beta + 8) q^{79} - 7 q^{81} + (2 \beta + 4) q^{83} + (2 \beta - 4) q^{85} - 8 q^{87} + ( - 3 \beta - 2) q^{89} - 16 q^{91} + (3 \beta - 4) q^{93} + ( - 4 \beta + 8) q^{95} + ( - \beta + 14) q^{97} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 18 q^{21} - 9 q^{23} + 3 q^{25} + 7 q^{27} + 2 q^{29} + 7 q^{31} - q^{33} + 14 q^{35} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 4 q^{45} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 3 q^{55} - 4 q^{57} - 5 q^{59} + 6 q^{61} + 20 q^{63} - 20 q^{65} + 15 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} - 24 q^{75} - 2 q^{77} + 14 q^{79} - 14 q^{81} + 10 q^{83} - 6 q^{85} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} + 12 q^{95} + 27 q^{97} - 3 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.56155 0 −3.56155 0 −3.12311 0 −0.561553 0
1.2 0 2.56155 0 0.561553 0 5.12311 0 3.56155 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 704.2.a.p 2
3.b odd 2 1 6336.2.a.cx 2
4.b odd 2 1 704.2.a.m 2
8.b even 2 1 176.2.a.d 2
8.d odd 2 1 88.2.a.b 2
11.b odd 2 1 7744.2.a.cl 2
12.b even 2 1 6336.2.a.cu 2
16.e even 4 2 2816.2.c.p 4
16.f odd 4 2 2816.2.c.w 4
24.f even 2 1 792.2.a.h 2
24.h odd 2 1 1584.2.a.t 2
40.e odd 2 1 2200.2.a.o 2
40.f even 2 1 4400.2.a.bp 2
40.i odd 4 2 4400.2.b.v 4
40.k even 4 2 2200.2.b.g 4
44.c even 2 1 7744.2.a.by 2
56.e even 2 1 4312.2.a.n 2
56.h odd 2 1 8624.2.a.cb 2
88.b odd 2 1 1936.2.a.r 2
88.g even 2 1 968.2.a.j 2
88.k even 10 4 968.2.i.q 8
88.l odd 10 4 968.2.i.r 8
264.p odd 2 1 8712.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b 2 8.d odd 2 1
176.2.a.d 2 8.b even 2 1
704.2.a.m 2 4.b odd 2 1
704.2.a.p 2 1.a even 1 1 trivial
792.2.a.h 2 24.f even 2 1
968.2.a.j 2 88.g even 2 1
968.2.i.q 8 88.k even 10 4
968.2.i.r 8 88.l odd 10 4
1584.2.a.t 2 24.h odd 2 1
1936.2.a.r 2 88.b odd 2 1
2200.2.a.o 2 40.e odd 2 1
2200.2.b.g 4 40.k even 4 2
2816.2.c.p 4 16.e even 4 2
2816.2.c.w 4 16.f odd 4 2
4312.2.a.n 2 56.e even 2 1
4400.2.a.bp 2 40.f even 2 1
4400.2.b.v 4 40.i odd 4 2
6336.2.a.cu 2 12.b even 2 1
6336.2.a.cx 2 3.b odd 2 1
7744.2.a.by 2 44.c even 2 1
7744.2.a.cl 2 11.b odd 2 1
8624.2.a.cb 2 56.h odd 2 1
8712.2.a.bb 2 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(704))\):

\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$97$ \( T^{2} - 27T + 178 \) Copy content Toggle raw display
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