Properties

Label 578.2.a.d
Level $578$
Weight $2$
Character orbit 578.a
Self dual yes
Analytic conductor $4.615$
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] = [578,2,Mod(1,578)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(578, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("578.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 578.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: \(4.61535323683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 34)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} - \beta q^{5} + \beta q^{6} + q^{8} + 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta q^{3} + q^{4} - \beta q^{5} + \beta q^{6} + q^{8} + 5 q^{9} - \beta q^{10} + \beta q^{11} + \beta q^{12} + 2 q^{13} - 8 q^{15} + q^{16} + 5 q^{18} + 4 q^{19} - \beta q^{20} + \beta q^{22} - 2 \beta q^{23} + \beta q^{24} + 3 q^{25} + 2 q^{26} + 2 \beta q^{27} - \beta q^{29} - 8 q^{30} + q^{32} + 8 q^{33} + 5 q^{36} - 3 \beta q^{37} + 4 q^{38} + 2 \beta q^{39} - \beta q^{40} - 2 \beta q^{41} + 4 q^{43} + \beta q^{44} - 5 \beta q^{45} - 2 \beta q^{46} + \beta q^{48} - 7 q^{49} + 3 q^{50} + 2 q^{52} - 6 q^{53} + 2 \beta q^{54} - 8 q^{55} + 4 \beta q^{57} - \beta q^{58} - 12 q^{59} - 8 q^{60} - 3 \beta q^{61} + q^{64} - 2 \beta q^{65} + 8 q^{66} - 4 q^{67} - 16 q^{69} + 2 \beta q^{71} + 5 q^{72} - 3 \beta q^{74} + 3 \beta q^{75} + 4 q^{76} + 2 \beta q^{78} + 6 \beta q^{79} - \beta q^{80} + q^{81} - 2 \beta q^{82} + 12 q^{83} + 4 q^{86} - 8 q^{87} + \beta q^{88} + 6 q^{89} - 5 \beta q^{90} - 2 \beta q^{92} - 4 \beta q^{95} + \beta q^{96} + 6 \beta q^{97} - 7 q^{98} + 5 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} + 4 q^{13} - 16 q^{15} + 2 q^{16} + 10 q^{18} + 8 q^{19} + 6 q^{25} + 4 q^{26} - 16 q^{30} + 2 q^{32} + 16 q^{33} + 10 q^{36} + 8 q^{38} + 8 q^{43} - 14 q^{49} + 6 q^{50} + 4 q^{52} - 12 q^{53} - 16 q^{55} - 24 q^{59} - 16 q^{60} + 2 q^{64} + 16 q^{66} - 8 q^{67} - 32 q^{69} + 10 q^{72} + 8 q^{76} + 2 q^{81} + 24 q^{83} + 8 q^{86} - 16 q^{87} + 12 q^{89} - 14 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
1.00000 −2.82843 1.00000 2.82843 −2.82843 0 1.00000 5.00000 2.82843
1.2 1.00000 2.82843 1.00000 −2.82843 2.82843 0 1.00000 5.00000 −2.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 578.2.a.d 2
3.b odd 2 1 5202.2.a.u 2
4.b odd 2 1 4624.2.a.s 2
17.b even 2 1 inner 578.2.a.d 2
17.c even 4 2 34.2.b.a 2
17.d even 8 2 578.2.c.a 2
17.d even 8 2 578.2.c.d 2
17.e odd 16 8 578.2.d.g 8
51.c odd 2 1 5202.2.a.u 2
51.f odd 4 2 306.2.b.d 2
68.d odd 2 1 4624.2.a.s 2
68.f odd 4 2 272.2.b.a 2
85.f odd 4 2 850.2.d.i 4
85.i odd 4 2 850.2.d.i 4
85.j even 4 2 850.2.b.f 2
119.f odd 4 2 1666.2.b.c 2
136.i even 4 2 1088.2.b.b 2
136.j odd 4 2 1088.2.b.a 2
204.l even 4 2 2448.2.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 17.c even 4 2
272.2.b.a 2 68.f odd 4 2
306.2.b.d 2 51.f odd 4 2
578.2.a.d 2 1.a even 1 1 trivial
578.2.a.d 2 17.b even 2 1 inner
578.2.c.a 2 17.d even 8 2
578.2.c.d 2 17.d even 8 2
578.2.d.g 8 17.e odd 16 8
850.2.b.f 2 85.j even 4 2
850.2.d.i 4 85.f odd 4 2
850.2.d.i 4 85.i odd 4 2
1088.2.b.a 2 136.j odd 4 2
1088.2.b.b 2 136.i even 4 2
1666.2.b.c 2 119.f odd 4 2
2448.2.c.n 2 204.l even 4 2
4624.2.a.s 2 4.b odd 2 1
4624.2.a.s 2 68.d odd 2 1
5202.2.a.u 2 3.b odd 2 1
5202.2.a.u 2 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(578))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 32 \) Copy content Toggle raw display
$29$ \( T^{2} - 8 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} - 32 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 72 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 288 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 288 \) Copy content Toggle raw display
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