Properties

Label 1008.6.a.bw
Level $1008$
Weight $6$
Character orbit 1008.a
Self dual yes
Analytic conductor $161.667$
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] = [1008,6,Mod(1,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, 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("1008.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.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: \(161.666890371\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
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 \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta + 39) q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta + 39) q^{5} + 49 q^{7} + (37 \beta - 185) q^{11} + ( - 10 \beta - 528) q^{13} + ( - 3 \beta + 513) q^{17} + ( - 48 \beta - 140) q^{19} + (163 \beta + 965) q^{23} + ( - 390 \beta + 3221) q^{25} + (48 \beta - 1278) q^{29} + (160 \beta - 4704) q^{31} + ( - 245 \beta + 1911) q^{35} + (922 \beta - 84) q^{37} + ( - 845 \beta + 4083) q^{41} + ( - 112 \beta - 13276) q^{43} + (202 \beta + 16662) q^{47} + 2401 q^{49} + (522 \beta - 17952) q^{53} + (2368 \beta - 42920) q^{55} + (2150 \beta - 9794) q^{59} + (128 \beta - 19514) q^{61} + (2250 \beta - 10942) q^{65} + (602 \beta - 28846) q^{67} + ( - 3299 \beta + 4779) q^{71} + (5350 \beta + 4356) q^{73} + (1813 \beta - 9065) q^{77} + ( - 1114 \beta - 16710) q^{79} + ( - 240 \beta + 8876) q^{83} + ( - 2682 \beta + 22902) q^{85} + (1395 \beta + 83859) q^{89} + ( - 490 \beta - 25872) q^{91} + ( - 1172 \beta + 40860) q^{95} + (3130 \beta - 23464) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 78 q^{5} + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 78 q^{5} + 98 q^{7} - 370 q^{11} - 1056 q^{13} + 1026 q^{17} - 280 q^{19} + 1930 q^{23} + 6442 q^{25} - 2556 q^{29} - 9408 q^{31} + 3822 q^{35} - 168 q^{37} + 8166 q^{41} - 26552 q^{43} + 33324 q^{47} + 4802 q^{49} - 35904 q^{53} - 85840 q^{55} - 19588 q^{59} - 39028 q^{61} - 21884 q^{65} - 57692 q^{67} + 9558 q^{71} + 8712 q^{73} - 18130 q^{77} - 33420 q^{79} + 17752 q^{83} + 45804 q^{85} + 167718 q^{89} - 51744 q^{91} + 81720 q^{95} - 46928 q^{97}+O(q^{100}) \) 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
7.44622
−6.44622
0 0 0 −30.4622 0 49.0000 0 0 0
1.2 0 0 0 108.462 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 1008.6.a.bw 2
3.b odd 2 1 336.6.a.v 2
4.b odd 2 1 504.6.a.v 2
12.b even 2 1 168.6.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.g 2 12.b even 2 1
336.6.a.v 2 3.b odd 2 1
504.6.a.v 2 4.b odd 2 1
1008.6.a.bw 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 78T_{5} - 3304 \) Copy content Toggle raw display
\( T_{11}^{2} + 370T_{11} - 229992 \) 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
$5$ \( T^{2} - 78T - 3304 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 370T - 229992 \) Copy content Toggle raw display
$13$ \( T^{2} + 1056 T + 259484 \) Copy content Toggle raw display
$17$ \( T^{2} - 1026 T + 261432 \) Copy content Toggle raw display
$19$ \( T^{2} + 280T - 425072 \) Copy content Toggle raw display
$23$ \( T^{2} - 1930 T - 4196592 \) Copy content Toggle raw display
$29$ \( T^{2} + 2556 T + 1188612 \) Copy content Toggle raw display
$31$ \( T^{2} + 9408 T + 17186816 \) Copy content Toggle raw display
$37$ \( T^{2} + 168 T - 164059156 \) Copy content Toggle raw display
$41$ \( T^{2} - 8166 T - 121135936 \) Copy content Toggle raw display
$43$ \( T^{2} + 26552 T + 173831184 \) Copy content Toggle raw display
$47$ \( T^{2} - 33324 T + 269747072 \) Copy content Toggle raw display
$53$ \( T^{2} + 35904 T + 269684892 \) Copy content Toggle raw display
$59$ \( T^{2} + 19588 T - 796220064 \) Copy content Toggle raw display
$61$ \( T^{2} + 39028 T + 377634084 \) Copy content Toggle raw display
$67$ \( T^{2} + 57692 T + 762147744 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 2077657552 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 5505167764 \) Copy content Toggle raw display
$79$ \( T^{2} + 33420 T + 39711872 \) Copy content Toggle raw display
$83$ \( T^{2} - 17752 T + 67666576 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 6656749056 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1340242404 \) Copy content Toggle raw display
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