Properties

Label 1008.6.a.bi
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 56)
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.

\(f(q)\) \(=\) \( q + ( - 5 \beta - 21) q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta - 21) q^{5} + 49 q^{7} + ( - 14 \beta - 358) q^{11} + ( - 17 \beta - 357) q^{13} + (54 \beta + 672) q^{17} + ( - 93 \beta + 973) q^{19} + (112 \beta - 896) q^{23} + (210 \beta + 2141) q^{25} + (546 \beta + 600) q^{29} + (446 \beta + 3402) q^{31} + ( - 245 \beta - 1029) q^{35} + ( - 154 \beta + 7320) q^{37} + (442 \beta - 3948) q^{41} + ( - 518 \beta - 262) q^{43} + ( - 746 \beta - 9198) q^{47} + 2401 q^{49} + ( - 1008 \beta - 22566) q^{53} + (2084 \beta + 21028) q^{55} + (365 \beta + 11291) q^{59} + (1117 \beta - 26411) q^{61} + (2142 \beta + 23902) q^{65} + ( - 224 \beta - 4924) q^{67} + (4060 \beta - 420) q^{71} + ( - 1204 \beta - 61026) q^{73} + ( - 686 \beta - 17542) q^{77} + (1372 \beta - 15852) q^{79} + (4185 \beta + 18487) q^{83} + ( - 4494 \beta - 66222) q^{85} + ( - 1512 \beta + 105294) q^{89} + ( - 833 \beta - 17493) q^{91} + ( - 2912 \beta + 69312) q^{95} + (8882 \beta - 22120) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 42 q^{5} + 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 42 q^{5} + 98 q^{7} - 716 q^{11} - 714 q^{13} + 1344 q^{17} + 1946 q^{19} - 1792 q^{23} + 4282 q^{25} + 1200 q^{29} + 6804 q^{31} - 2058 q^{35} + 14640 q^{37} - 7896 q^{41} - 524 q^{43} - 18396 q^{47} + 4802 q^{49} - 45132 q^{53} + 42056 q^{55} + 22582 q^{59} - 52822 q^{61} + 47804 q^{65} - 9848 q^{67} - 840 q^{71} - 122052 q^{73} - 35084 q^{77} - 31704 q^{79} + 36974 q^{83} - 132444 q^{85} + 210588 q^{89} - 34986 q^{91} + 138624 q^{95} - 44240 q^{97}+O(q^{100}) \) 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 −90.4622 0 49.0000 0 0 0
1.2 0 0 0 48.4622 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.bi 2
3.b odd 2 1 112.6.a.j 2
4.b odd 2 1 504.6.a.m 2
12.b even 2 1 56.6.a.d 2
21.c even 2 1 784.6.a.q 2
24.f even 2 1 448.6.a.x 2
24.h odd 2 1 448.6.a.r 2
84.h odd 2 1 392.6.a.e 2
84.j odd 6 2 392.6.i.h 4
84.n even 6 2 392.6.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 12.b even 2 1
112.6.a.j 2 3.b odd 2 1
392.6.a.e 2 84.h odd 2 1
392.6.i.h 4 84.j odd 6 2
392.6.i.k 4 84.n even 6 2
448.6.a.r 2 24.h odd 2 1
448.6.a.x 2 24.f even 2 1
504.6.a.m 2 4.b odd 2 1
784.6.a.q 2 21.c even 2 1
1008.6.a.bi 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} + 42T_{5} - 4384 \) Copy content Toggle raw display
\( T_{11}^{2} + 716T_{11} + 90336 \) 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
$5$ \( T^{2} + 42T - 4384 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 716T + 90336 \) Copy content Toggle raw display
$13$ \( T^{2} + 714T + 71672 \) Copy content Toggle raw display
$17$ \( T^{2} - 1344 T - 111204 \) Copy content Toggle raw display
$19$ \( T^{2} - 1946 T - 722528 \) Copy content Toggle raw display
$23$ \( T^{2} + 1792 T - 1618176 \) Copy content Toggle raw display
$29$ \( T^{2} - 1200 T - 57176388 \) Copy content Toggle raw display
$31$ \( T^{2} - 6804 T - 26817184 \) Copy content Toggle raw display
$37$ \( T^{2} - 14640 T + 49005212 \) Copy content Toggle raw display
$41$ \( T^{2} + 7896 T - 22118548 \) Copy content Toggle raw display
$43$ \( T^{2} + 524 T - 51717888 \) Copy content Toggle raw display
$47$ \( T^{2} + 18396 T - 22804384 \) Copy content Toggle raw display
$53$ \( T^{2} + 45132 T + 313124004 \) Copy content Toggle raw display
$59$ \( T^{2} - 22582 T + 101774256 \) Copy content Toggle raw display
$61$ \( T^{2} + 52822 T + 456736944 \) Copy content Toggle raw display
$67$ \( T^{2} + 9848 T + 14561808 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 3181158400 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3444396788 \) Copy content Toggle raw display
$79$ \( T^{2} + 31704 T - 112014208 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3038476256 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10645600644 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 14736460932 \) Copy content Toggle raw display
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