Properties

Label 588.4.a.i
Level $588$
Weight $4$
Character orbit 588.a
Self dual yes
Analytic conductor $34.693$
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] = [588,4,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, 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("588.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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 \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta + 6) q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta + 6) q^{5} + 9 q^{9} + (7 \beta - 6) q^{11} + ( - 5 \beta + 5) q^{13} + ( - 3 \beta + 18) q^{15} + (4 \beta + 48) q^{17} + ( - 3 \beta + 35) q^{19} + ( - 20 \beta + 48) q^{23} + ( - 11 \beta - 41) q^{25} + 27 q^{27} + (11 \beta + 132) q^{29} + ( - 20 \beta + 191) q^{31} + (21 \beta - 18) q^{33} + (45 \beta - 25) q^{37} + ( - 15 \beta + 15) q^{39} + (18 \beta - 90) q^{41} + (3 \beta + 359) q^{43} + ( - 9 \beta + 54) q^{45} + ( - 36 \beta - 90) q^{47} + (12 \beta + 144) q^{51} + ( - 9 \beta + 252) q^{53} + (41 \beta - 372) q^{55} + ( - 9 \beta + 105) q^{57} + (53 \beta + 60) q^{59} + (40 \beta - 286) q^{61} + ( - 30 \beta + 270) q^{65} + (77 \beta + 17) q^{67} + ( - 60 \beta + 144) q^{69} + ( - 44 \beta + 822) q^{71} + (53 \beta + 581) q^{73} + ( - 33 \beta - 123) q^{75} + ( - 62 \beta + 761) q^{79} + 81 q^{81} + ( - 101 \beta - 654) q^{83} + ( - 28 \beta + 96) q^{85} + (33 \beta + 396) q^{87} + (42 \beta - 1008) q^{89} + ( - 60 \beta + 573) q^{93} + ( - 50 \beta + 354) q^{95} + (29 \beta + 266) q^{97} + (63 \beta - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 11 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 11 q^{5} + 18 q^{9} - 5 q^{11} + 5 q^{13} + 33 q^{15} + 100 q^{17} + 67 q^{19} + 76 q^{23} - 93 q^{25} + 54 q^{27} + 275 q^{29} + 362 q^{31} - 15 q^{33} - 5 q^{37} + 15 q^{39} - 162 q^{41} + 721 q^{43} + 99 q^{45} - 216 q^{47} + 300 q^{51} + 495 q^{53} - 703 q^{55} + 201 q^{57} + 173 q^{59} - 532 q^{61} + 510 q^{65} + 111 q^{67} + 228 q^{69} + 1600 q^{71} + 1215 q^{73} - 279 q^{75} + 1460 q^{79} + 162 q^{81} - 1409 q^{83} + 164 q^{85} + 825 q^{87} - 1974 q^{89} + 1086 q^{93} + 658 q^{95} + 561 q^{97} - 45 q^{99}+O(q^{100}) \) 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
7.44622
−6.44622
0 3.00000 0 −1.44622 0 0 0 9.00000 0
1.2 0 3.00000 0 12.4462 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

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 588.4.a.i 2
3.b odd 2 1 1764.4.a.o 2
4.b odd 2 1 2352.4.a.bt 2
7.b odd 2 1 588.4.a.f 2
7.c even 3 2 84.4.i.a 4
7.d odd 6 2 588.4.i.j 4
21.c even 2 1 1764.4.a.y 2
21.g even 6 2 1764.4.k.q 4
21.h odd 6 2 252.4.k.f 4
28.d even 2 1 2352.4.a.bx 2
28.g odd 6 2 336.4.q.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 7.c even 3 2
252.4.k.f 4 21.h odd 6 2
336.4.q.i 4 28.g odd 6 2
588.4.a.f 2 7.b odd 2 1
588.4.a.i 2 1.a even 1 1 trivial
588.4.i.j 4 7.d odd 6 2
1764.4.a.o 2 3.b odd 2 1
1764.4.a.y 2 21.c even 2 1
1764.4.k.q 4 21.g even 6 2
2352.4.a.bt 2 4.b odd 2 1
2352.4.a.bx 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 11T_{5} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 11T - 18 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5T - 2358 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 1200 \) Copy content Toggle raw display
$17$ \( T^{2} - 100T + 1728 \) Copy content Toggle raw display
$19$ \( T^{2} - 67T + 688 \) Copy content Toggle raw display
$23$ \( T^{2} - 76T - 17856 \) Copy content Toggle raw display
$29$ \( T^{2} - 275T + 13068 \) Copy content Toggle raw display
$31$ \( T^{2} - 362T + 13461 \) Copy content Toggle raw display
$37$ \( T^{2} + 5T - 97700 \) Copy content Toggle raw display
$41$ \( T^{2} + 162T - 9072 \) Copy content Toggle raw display
$43$ \( T^{2} - 721T + 129526 \) Copy content Toggle raw display
$47$ \( T^{2} + 216T - 50868 \) Copy content Toggle raw display
$53$ \( T^{2} - 495T + 57348 \) Copy content Toggle raw display
$59$ \( T^{2} - 173T - 128052 \) Copy content Toggle raw display
$61$ \( T^{2} + 532T - 6444 \) Copy content Toggle raw display
$67$ \( T^{2} - 111T - 282994 \) Copy content Toggle raw display
$71$ \( T^{2} - 1600 T + 546588 \) Copy content Toggle raw display
$73$ \( T^{2} - 1215 T + 233522 \) Copy content Toggle raw display
$79$ \( T^{2} - 1460 T + 347427 \) Copy content Toggle raw display
$83$ \( T^{2} + 1409T + 4122 \) Copy content Toggle raw display
$89$ \( T^{2} + 1974 T + 889056 \) Copy content Toggle raw display
$97$ \( T^{2} - 561T + 38102 \) Copy content Toggle raw display
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