Properties

Label 1008.6.a.bt
Level $1008$
Weight $6$
Character orbit 1008.a
Self dual yes
Analytic conductor $161.667$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) 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 \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta + 31) q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta + 31) q^{5} - 49 q^{7} + (6 \beta + 486) q^{11} + (71 \beta + 39) q^{13} + (6 \beta - 280) q^{17} + (37 \beta - 1321) q^{19} + (248 \beta + 1136) q^{23} + ( - 310 \beta + 2261) q^{25} + ( - 14 \beta + 3904) q^{29} + (130 \beta - 2722) q^{31} + (245 \beta - 1519) q^{35} + ( - 650 \beta + 288) q^{37} + (378 \beta + 8444) q^{41} + ( - 338 \beta + 4198) q^{43} + ( - 1238 \beta - 2266) q^{47} + 2401 q^{49} + ( - 1152 \beta - 710) q^{53} + ( - 2244 \beta + 9756) q^{55} + ( - 325 \beta + 17073) q^{59} + ( - 3347 \beta + 9553) q^{61} + (2006 \beta - 61626) q^{65} + (1240 \beta - 28476) q^{67} + ( - 700 \beta - 3612) q^{71} + (1260 \beta + 64414) q^{73} + ( - 294 \beta - 23814) q^{77} + ( - 1548 \beta - 26404) q^{79} + ( - 5209 \beta - 42243) q^{83} + (1586 \beta - 13990) q^{85} + ( - 2312 \beta + 65486) q^{89} + ( - 3479 \beta - 1911) q^{91} + (7752 \beta - 73696) q^{95} + ( - 350 \beta + 97312) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 62 q^{5} - 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 62 q^{5} - 98 q^{7} + 972 q^{11} + 78 q^{13} - 560 q^{17} - 2642 q^{19} + 2272 q^{23} + 4522 q^{25} + 7808 q^{29} - 5444 q^{31} - 3038 q^{35} + 576 q^{37} + 16888 q^{41} + 8396 q^{43} - 4532 q^{47} + 4802 q^{49} - 1420 q^{53} + 19512 q^{55} + 34146 q^{59} + 19106 q^{61} - 123252 q^{65} - 56952 q^{67} - 7224 q^{71} + 128828 q^{73} - 47628 q^{77} - 52808 q^{79} - 84486 q^{83} - 27980 q^{85} + 130972 q^{89} - 3822 q^{91} - 147392 q^{95} + 194624 q^{97}+O(q^{100}) \) 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
7.15207
−6.15207
0 0 0 −35.5207 0 −49.0000 0 0 0
1.2 0 0 0 97.5207 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.bt 2
3.b odd 2 1 112.6.a.k 2
4.b odd 2 1 504.6.a.s 2
12.b even 2 1 56.6.a.c 2
21.c even 2 1 784.6.a.p 2
24.f even 2 1 448.6.a.z 2
24.h odd 2 1 448.6.a.q 2
84.h odd 2 1 392.6.a.f 2
84.j odd 6 2 392.6.i.g 4
84.n even 6 2 392.6.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.c 2 12.b even 2 1
112.6.a.k 2 3.b odd 2 1
392.6.a.f 2 84.h odd 2 1
392.6.i.g 4 84.j odd 6 2
392.6.i.l 4 84.n even 6 2
448.6.a.q 2 24.h odd 2 1
448.6.a.z 2 24.f even 2 1
504.6.a.s 2 4.b odd 2 1
784.6.a.p 2 21.c even 2 1
1008.6.a.bt 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 62T_{5} - 3464 \) Copy content Toggle raw display
\( T_{11}^{2} - 972T_{11} + 229824 \) 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
$5$ \( T^{2} - 62T - 3464 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 972T + 229824 \) Copy content Toggle raw display
$13$ \( T^{2} - 78T - 890736 \) Copy content Toggle raw display
$17$ \( T^{2} + 560T + 72028 \) Copy content Toggle raw display
$19$ \( T^{2} + 2642 T + 1502728 \) Copy content Toggle raw display
$23$ \( T^{2} - 2272 T - 9595712 \) Copy content Toggle raw display
$29$ \( T^{2} - 7808 T + 15206524 \) Copy content Toggle raw display
$31$ \( T^{2} + 5444 T + 4417984 \) Copy content Toggle raw display
$37$ \( T^{2} - 576 T - 74699556 \) Copy content Toggle raw display
$41$ \( T^{2} - 16888 T + 46010668 \) Copy content Toggle raw display
$43$ \( T^{2} - 8396 T - 2597984 \) Copy content Toggle raw display
$47$ \( T^{2} + 4532 T - 266143232 \) Copy content Toggle raw display
$53$ \( T^{2} + 1420 T - 234393308 \) Copy content Toggle raw display
$59$ \( T^{2} - 34146 T + 272791704 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 1891566584 \) Copy content Toggle raw display
$67$ \( T^{2} + 56952 T + 538727376 \) Copy content Toggle raw display
$71$ \( T^{2} + 7224 T - 73683456 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 3868158196 \) Copy content Toggle raw display
$79$ \( T^{2} + 52808 T + 273025408 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 3018190488 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3342290308 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 9447942844 \) Copy content Toggle raw display
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