Properties

Label 1008.6.a.bq
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: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
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 \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 \beta + 9) q^{5} - 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta + 9) q^{5} - 49 q^{7} + (62 \beta + 198) q^{11} + ( - 63 \beta - 175) q^{13} + ( - 38 \beta - 900) q^{17} + ( - 9 \beta + 1633) q^{19} + ( - 284 \beta + 1044) q^{23} + (90 \beta - 1619) q^{25} + (126 \beta - 3348) q^{29} + (270 \beta + 10) q^{31} + ( - 245 \beta - 441) q^{35} + ( - 270 \beta + 3116) q^{37} + ( - 546 \beta + 3024) q^{41} + ( - 2394 \beta + 1510) q^{43} + ( - 1874 \beta + 5850) q^{47} + 2401 q^{49} + ( - 104 \beta - 4734) q^{53} + (1548 \beta + 19452) q^{55} + (1025 \beta - 21969) q^{59} + (2403 \beta - 32377) q^{61} + ( - 1442 \beta - 19530) q^{65} + ( - 972 \beta - 12392) q^{67} + (2100 \beta + 48708) q^{71} + (2628 \beta + 8726) q^{73} + ( - 3038 \beta - 9702) q^{77} + (7452 \beta - 25628) q^{79} + ( - 7875 \beta + 58779) q^{83} + ( - 4842 \beta - 18930) q^{85} + ( - 11104 \beta - 42138) q^{89} + (3087 \beta + 8575) q^{91} + (8084 \beta + 12132) q^{95} + ( - 4410 \beta + 10388) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{5} - 98 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 18 q^{5} - 98 q^{7} + 396 q^{11} - 350 q^{13} - 1800 q^{17} + 3266 q^{19} + 2088 q^{23} - 3238 q^{25} - 6696 q^{29} + 20 q^{31} - 882 q^{35} + 6232 q^{37} + 6048 q^{41} + 3020 q^{43} + 11700 q^{47} + 4802 q^{49} - 9468 q^{53} + 38904 q^{55} - 43938 q^{59} - 64754 q^{61} - 39060 q^{65} - 24784 q^{67} + 97416 q^{71} + 17452 q^{73} - 19404 q^{77} - 51256 q^{79} + 117558 q^{83} - 37860 q^{85} - 84276 q^{89} + 17150 q^{91} + 24264 q^{95} + 20776 q^{97}+O(q^{100}) \) 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
−3.27492
4.27492
0 0 0 −28.7492 0 −49.0000 0 0 0
1.2 0 0 0 46.7492 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.bq 2
3.b odd 2 1 112.6.a.h 2
4.b odd 2 1 63.6.a.f 2
12.b even 2 1 7.6.a.b 2
21.c even 2 1 784.6.a.v 2
24.f even 2 1 448.6.a.w 2
24.h odd 2 1 448.6.a.u 2
28.d even 2 1 441.6.a.l 2
60.h even 2 1 175.6.a.c 2
60.l odd 4 2 175.6.b.c 4
84.h odd 2 1 49.6.a.f 2
84.j odd 6 2 49.6.c.d 4
84.n even 6 2 49.6.c.e 4
132.d odd 2 1 847.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 12.b even 2 1
49.6.a.f 2 84.h odd 2 1
49.6.c.d 4 84.j odd 6 2
49.6.c.e 4 84.n even 6 2
63.6.a.f 2 4.b odd 2 1
112.6.a.h 2 3.b odd 2 1
175.6.a.c 2 60.h even 2 1
175.6.b.c 4 60.l odd 4 2
441.6.a.l 2 28.d even 2 1
448.6.a.u 2 24.h odd 2 1
448.6.a.w 2 24.f even 2 1
784.6.a.v 2 21.c even 2 1
847.6.a.c 2 132.d odd 2 1
1008.6.a.bq 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 18T_{5} - 1344 \) Copy content Toggle raw display
\( T_{11}^{2} - 396T_{11} - 179904 \) 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
$5$ \( T^{2} - 18T - 1344 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 396T - 179904 \) Copy content Toggle raw display
$13$ \( T^{2} + 350T - 195608 \) Copy content Toggle raw display
$17$ \( T^{2} + 1800 T + 727692 \) Copy content Toggle raw display
$19$ \( T^{2} - 3266 T + 2662072 \) Copy content Toggle raw display
$23$ \( T^{2} - 2088 T - 3507456 \) Copy content Toggle raw display
$29$ \( T^{2} + 6696 T + 10304172 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 4155200 \) Copy content Toggle raw display
$37$ \( T^{2} - 6232 T + 5554156 \) Copy content Toggle raw display
$41$ \( T^{2} - 6048 T - 7848036 \) Copy content Toggle raw display
$43$ \( T^{2} - 3020 T - 324400352 \) Copy content Toggle raw display
$47$ \( T^{2} - 11700 T - 165954432 \) Copy content Toggle raw display
$53$ \( T^{2} + 9468 T + 21794244 \) Copy content Toggle raw display
$59$ \( T^{2} + 43938 T + 422751336 \) Copy content Toggle raw display
$61$ \( T^{2} + 64754 T + 719128816 \) Copy content Toggle raw display
$67$ \( T^{2} + 24784 T + 99708976 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2121099264 \) Copy content Toggle raw display
$73$ \( T^{2} - 17452 T - 317520812 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2508546944 \) Copy content Toggle raw display
$83$ \( T^{2} - 117558 T - 79919784 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 5252421468 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 1000631156 \) Copy content Toggle raw display
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