Properties

Label 504.6.a.l
Level $504$
Weight $6$
Character orbit 504.a
Self dual yes
Analytic conductor $80.833$
Analytic rank $1$
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] = [504,6,Mod(1,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("504.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.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: \(80.8334451857\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(\beta = 2\sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta - 24) q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta - 24) q^{5} - 49 q^{7} + ( - 37 \beta - 184) q^{11} + (80 \beta - 78) q^{13} + (33 \beta + 1656) q^{17} + (48 \beta + 1868) q^{19} + (17 \beta - 776) q^{23} + (240 \beta + 1151) q^{25} + (192 \beta - 864) q^{29} + ( - 16 \beta - 1812) q^{31} + (245 \beta + 1176) q^{35} + (112 \beta + 3498) q^{37} + ( - 341 \beta - 13080) q^{41} + ( - 32 \beta - 15092) q^{43} + ( - 58 \beta + 5712) q^{47} + 2401 q^{49} + ( - 2322 \beta + 8688) q^{53} + (1808 \beta + 31796) q^{55} + (1810 \beta - 7504) q^{59} + ( - 1168 \beta + 17782) q^{61} + ( - 1530 \beta - 57328) q^{65} + ( - 512 \beta - 35252) q^{67} + (2435 \beta - 20376) q^{71} + ( - 1184 \beta - 26946) q^{73} + (1813 \beta + 9016) q^{77} + ( - 2720 \beta - 4872) q^{79} + (7800 \beta - 15680) q^{83} + ( - 9072 \beta - 64164) q^{85} + ( - 6129 \beta - 17976) q^{89} + ( - 3920 \beta + 3822) q^{91} + ( - 10492 \beta - 80352) q^{95} + (11680 \beta + 33326) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 48 q^{5} - 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 48 q^{5} - 98 q^{7} - 368 q^{11} - 156 q^{13} + 3312 q^{17} + 3736 q^{19} - 1552 q^{23} + 2302 q^{25} - 1728 q^{29} - 3624 q^{31} + 2352 q^{35} + 6996 q^{37} - 26160 q^{41} - 30184 q^{43} + 11424 q^{47} + 4802 q^{49} + 17376 q^{53} + 63592 q^{55} - 15008 q^{59} + 35564 q^{61} - 114656 q^{65} - 70504 q^{67} - 40752 q^{71} - 53892 q^{73} + 18032 q^{77} - 9744 q^{79} - 31360 q^{83} - 128328 q^{85} - 35952 q^{89} + 7644 q^{91} - 160704 q^{95} + 66652 q^{97}+O(q^{100}) \) 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
3.54138
−2.54138
0 0 0 −84.8276 0 −49.0000 0 0 0
1.2 0 0 0 36.8276 0 −49.0000 0 0 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 504.6.a.l 2
3.b odd 2 1 504.6.a.r yes 2
4.b odd 2 1 1008.6.a.bh 2
12.b even 2 1 1008.6.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.6.a.l 2 1.a even 1 1 trivial
504.6.a.r yes 2 3.b odd 2 1
1008.6.a.bh 2 4.b odd 2 1
1008.6.a.bs 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5}^{2} + 48T_{5} - 3124 \) Copy content Toggle raw display
\( T_{11}^{2} + 368T_{11} - 168756 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 48T - 3124 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 368T - 168756 \) Copy content Toggle raw display
$13$ \( T^{2} + 156T - 941116 \) Copy content Toggle raw display
$17$ \( T^{2} - 3312 T + 2581164 \) Copy content Toggle raw display
$19$ \( T^{2} - 3736 T + 3148432 \) Copy content Toggle raw display
$23$ \( T^{2} + 1552 T + 559404 \) Copy content Toggle raw display
$29$ \( T^{2} + 1728 T - 4709376 \) Copy content Toggle raw display
$31$ \( T^{2} + 3624 T + 3245456 \) Copy content Toggle raw display
$37$ \( T^{2} - 6996 T + 10379492 \) Copy content Toggle raw display
$41$ \( T^{2} + 26160 T + 153876812 \) Copy content Toggle raw display
$43$ \( T^{2} + 30184 T + 227616912 \) Copy content Toggle raw display
$47$ \( T^{2} - 11424 T + 32129072 \) Copy content Toggle raw display
$53$ \( T^{2} - 17376 T - 722487888 \) Copy content Toggle raw display
$59$ \( T^{2} + 15008 T - 428552784 \) Copy content Toggle raw display
$61$ \( T^{2} - 35564 T + 114294372 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1203906192 \) Copy content Toggle raw display
$71$ \( T^{2} + 40752 T - 462343924 \) Copy content Toggle raw display
$73$ \( T^{2} + 53892 T + 518612228 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 1071226816 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 8758457600 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5236430292 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 19079892924 \) Copy content Toggle raw display
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