Properties

Label 588.6.a.k
Level $588$
Weight $6$
Character orbit 588.a
Self dual yes
Analytic conductor $94.306$
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] = [588,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5569}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1392 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta + 3) q^{5} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta + 3) q^{5} + 81 q^{9} + (7 \beta - 45) q^{11} + (6 \beta - 384) q^{13} + (9 \beta + 27) q^{15} + ( - \beta - 963) q^{17} + (24 \beta - 1124) q^{19} + (\beta + 3177) q^{23} + (6 \beta + 2453) q^{25} + 729 q^{27} + ( - 40 \beta + 5286) q^{29} + ( - 72 \beta + 1656) q^{31} + (63 \beta - 405) q^{33} + (150 \beta + 1052) q^{37} + (54 \beta - 3456) q^{39} + (33 \beta - 633) q^{41} + ( - 240 \beta - 2884) q^{43} + (81 \beta + 243) q^{45} + (162 \beta - 7806) q^{47} + ( - 9 \beta - 8667) q^{51} + (234 \beta + 8256) q^{53} + ( - 24 \beta + 38848) q^{55} + (216 \beta - 10116) q^{57} + ( - 50 \beta + 6570) q^{59} + ( - 264 \beta + 2898) q^{61} + ( - 366 \beta + 32262) q^{65} + ( - 462 \beta - 28058) q^{67} + (9 \beta + 28593) q^{69} + (895 \beta + 5511) q^{71} + (174 \beta + 42692) q^{73} + (54 \beta + 22077) q^{75} + (78 \beta - 9810) q^{79} + 6561 q^{81} + (320 \beta + 22212) q^{83} + ( - 966 \beta - 8458) q^{85} + ( - 360 \beta + 47574) q^{87} + ( - 399 \beta - 105609) q^{89} + ( - 648 \beta + 14904) q^{93} + ( - 1052 \beta + 130284) q^{95} + ( - 1614 \beta - 22432) q^{97} + (567 \beta - 3645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 6 q^{5} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 18 q^{3} + 6 q^{5} + 162 q^{9} - 90 q^{11} - 768 q^{13} + 54 q^{15} - 1926 q^{17} - 2248 q^{19} + 6354 q^{23} + 4906 q^{25} + 1458 q^{27} + 10572 q^{29} + 3312 q^{31} - 810 q^{33} + 2104 q^{37} - 6912 q^{39} - 1266 q^{41} - 5768 q^{43} + 486 q^{45} - 15612 q^{47} - 17334 q^{51} + 16512 q^{53} + 77696 q^{55} - 20232 q^{57} + 13140 q^{59} + 5796 q^{61} + 64524 q^{65} - 56116 q^{67} + 57186 q^{69} + 11022 q^{71} + 85384 q^{73} + 44154 q^{75} - 19620 q^{79} + 13122 q^{81} + 44424 q^{83} - 16916 q^{85} + 95148 q^{87} - 211218 q^{89} + 29808 q^{93} + 260568 q^{95} - 44864 q^{97} - 7290 q^{99}+O(q^{100}) \) 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
−36.8129
37.8129
0 9.00000 0 −71.6257 0 0 0 81.0000 0
1.2 0 9.00000 0 77.6257 0 0 0 81.0000 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 588.6.a.k 2
7.b odd 2 1 84.6.a.c 2
7.c even 3 2 588.6.i.i 4
7.d odd 6 2 588.6.i.l 4
21.c even 2 1 252.6.a.h 2
28.d even 2 1 336.6.a.x 2
84.h odd 2 1 1008.6.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.c 2 7.b odd 2 1
252.6.a.h 2 21.c even 2 1
336.6.a.x 2 28.d even 2 1
588.6.a.k 2 1.a even 1 1 trivial
588.6.i.i 4 7.c even 3 2
588.6.i.l 4 7.d odd 6 2
1008.6.a.bo 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6T_{5} - 5560 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 5560 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 90T - 270856 \) Copy content Toggle raw display
$13$ \( T^{2} + 768T - 53028 \) Copy content Toggle raw display
$17$ \( T^{2} + 1926 T + 921800 \) Copy content Toggle raw display
$19$ \( T^{2} + 2248 T - 1944368 \) Copy content Toggle raw display
$23$ \( T^{2} - 6354 T + 10087760 \) Copy content Toggle raw display
$29$ \( T^{2} - 10572 T + 19031396 \) Copy content Toggle raw display
$31$ \( T^{2} - 3312 T - 26127360 \) Copy content Toggle raw display
$37$ \( T^{2} - 2104 T - 124195796 \) Copy content Toggle raw display
$41$ \( T^{2} + 1266 T - 5663952 \) Copy content Toggle raw display
$43$ \( T^{2} + 5768 T - 312456944 \) Copy content Toggle raw display
$47$ \( T^{2} + 15612 T - 85219200 \) Copy content Toggle raw display
$53$ \( T^{2} - 16512 T - 236774628 \) Copy content Toggle raw display
$59$ \( T^{2} - 13140 T + 29242400 \) Copy content Toggle raw display
$61$ \( T^{2} - 5796 T - 379738620 \) Copy content Toggle raw display
$67$ \( T^{2} + 56116 T - 401418272 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 4430537104 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1653999820 \) Copy content Toggle raw display
$79$ \( T^{2} + 19620 T + 62354304 \) Copy content Toggle raw display
$83$ \( T^{2} - 44424 T - 76892656 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10266670512 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 14004028100 \) Copy content Toggle raw display
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