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Properties

Label 1008.3.f.j
Level $1008$
Weight $3$
Character orbit 1008.f
Analytic conductor $27.466$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(433,1008)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.433");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.551252791296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 17x^{6} + 26x^{5} + 270x^{4} - 302x^{3} - 1007x^{2} - 3502x + 10609 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + (\beta_{4} - \beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + (\beta_{4} - \beta_{3} + 1) q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 3) q^{11}+ \cdots + ( - 6 \beta_{7} + 4 \beta_{6} + \cdots - 6 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 16 q^{11} + 96 q^{23} + 64 q^{29} - 120 q^{35} - 40 q^{37} - 136 q^{43} + 112 q^{49} - 112 q^{53} + 208 q^{65} + 8 q^{67} + 16 q^{71} - 288 q^{77} - 104 q^{79} - 424 q^{85} + 192 q^{91} - 240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 17x^{6} + 26x^{5} + 270x^{4} - 302x^{3} - 1007x^{2} - 3502x + 10609 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 114\nu^{7} - 370\nu^{6} - 1748\nu^{5} - 1292\nu^{4} + 18236\nu^{3} + 29944\nu^{2} - 19570\nu - 1063218 ) / 155365 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16713 \nu^{7} + 21473 \nu^{6} - 147028 \nu^{5} - 234962 \nu^{4} + 2989140 \nu^{3} + \cdots - 62320871 ) / 6401038 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2626 \nu^{7} - 1855 \nu^{6} + 41552 \nu^{5} + 40698 \nu^{4} - 628474 \nu^{3} - 929726 \nu^{2} + \cdots + 11807817 ) / 695765 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 222581 \nu^{7} + 121235 \nu^{6} - 3351792 \nu^{5} - 2698858 \nu^{4} + 47714004 \nu^{3} + \cdots - 823434427 ) / 32005190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10504 \nu^{7} + 7420 \nu^{6} - 166208 \nu^{5} - 162792 \nu^{4} + 2513896 \nu^{3} + 3718904 \nu^{2} + \cdots - 44448208 ) / 695765 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 99915 \nu^{7} - 37173 \nu^{6} + 1532416 \nu^{5} + 996498 \nu^{4} - 23098276 \nu^{3} + \cdots + 392734777 ) / 6401038 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -366\nu^{7} - 298\nu^{6} + 5604\nu^{5} + 6964\nu^{4} - 85636\nu^{3} - 126368\nu^{2} + 83870\nu + 1532022 ) / 16583 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} - 4\beta_{3} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} + 4\beta_{6} + 9\beta_{5} + 4\beta_{2} + 6\beta _1 + 40 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} - 32\beta_{6} - 18\beta_{5} - 72\beta_{4} - 19\beta _1 + 32 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 90\beta_{7} + 115\beta_{5} + 44\beta_{4} + 32\beta_{3} + 44\beta_{2} - 68\beta _1 - 416 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -205\beta_{7} - 372\beta_{6} - 219\beta_{5} - 744\beta_{4} + 744\beta_{3} + 372\beta_{2} - 19\beta _1 - 504 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 119\beta_{7} + 252\beta_{6} + 175\beta_{5} + 700\beta_{4} - 1295\beta _1 - 6128 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -322\beta_{7} - 575\beta_{5} + 5152\beta_{4} + 11084\beta_{3} + 5152\beta_{2} + 2898\beta _1 + 7452 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
3.27281 + 2.12870i
−1.39023 − 2.19650i
2.59734 − 0.105729i
−3.47991 + 1.76998i
−3.47991 − 1.76998i
2.59734 + 0.105729i
−1.39023 + 2.19650i
3.27281 − 2.12870i
0 0 0 − 8.47037i 0 6.54561 − 2.48091i 0 0 0
433.2 0 0 0 − 3.76315i 0 −2.78047 − 6.42409i 0 0 0
433.3 0 0 0 − 2.74854i 0 5.19468 − 4.69204i 0 0 0
433.4 0 0 0 − 2.55678i 0 −6.95983 − 0.748863i 0 0 0
433.5 0 0 0 2.55678i 0 −6.95983 + 0.748863i 0 0 0
433.6 0 0 0 2.74854i 0 5.19468 + 4.69204i 0 0 0
433.7 0 0 0 3.76315i 0 −2.78047 + 6.42409i 0 0 0
433.8 0 0 0 8.47037i 0 6.54561 + 2.48091i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.f.j 8
3.b odd 2 1 336.3.f.d 8
4.b odd 2 1 504.3.f.c 8
7.b odd 2 1 inner 1008.3.f.j 8
12.b even 2 1 168.3.f.a ✓ 8
21.c even 2 1 336.3.f.d 8
24.f even 2 1 1344.3.f.g 8
24.h odd 2 1 1344.3.f.h 8
28.d even 2 1 504.3.f.c 8
84.h odd 2 1 168.3.f.a ✓ 8
84.j odd 6 1 1176.3.z.a 8
84.j odd 6 1 1176.3.z.f 8
84.n even 6 1 1176.3.z.a 8
84.n even 6 1 1176.3.z.f 8
168.e odd 2 1 1344.3.f.g 8
168.i even 2 1 1344.3.f.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.f.a ✓ 8 12.b even 2 1
168.3.f.a ✓ 8 84.h odd 2 1
336.3.f.d 8 3.b odd 2 1
336.3.f.d 8 21.c even 2 1
504.3.f.c 8 4.b odd 2 1
504.3.f.c 8 28.d even 2 1
1008.3.f.j 8 1.a even 1 1 trivial
1008.3.f.j 8 7.b odd 2 1 inner
1176.3.z.a 8 84.j odd 6 1
1176.3.z.a 8 84.n even 6 1
1176.3.z.f 8 84.j odd 6 1
1176.3.z.f 8 84.n even 6 1
1344.3.f.g 8 24.f even 2 1
1344.3.f.g 8 168.e odd 2 1
1344.3.f.h 8 24.h odd 2 1
1344.3.f.h 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{8} + 100T_{5}^{6} + 2276T_{5}^{4} + 18560T_{5}^{2} + 50176 \) Copy content Toggle raw display
\( T_{11}^{4} + 8T_{11}^{3} - 298T_{11}^{2} - 3896T_{11} - 12152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 100 T^{6} + \cdots + 50176 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + \cdots - 12152)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 592 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{8} + 1124 T^{6} + \cdots + 125440000 \) Copy content Toggle raw display
$19$ \( T^{8} + 760 T^{6} + \cdots + 1183744 \) Copy content Toggle raw display
$23$ \( (T^{4} - 48 T^{3} + \cdots - 257912)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} + \cdots + 96400)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 604860841984 \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + \cdots + 2984896)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1137577984 \) Copy content Toggle raw display
$43$ \( (T^{4} + 68 T^{3} + \cdots - 312608)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 98660515840000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 56 T^{3} + \cdots + 1097104)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 11808 T^{2} + 12278016)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots - 4707392)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 5867272)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21012469252096 \) Copy content Toggle raw display
$79$ \( (T^{4} + 52 T^{3} + \cdots + 269824)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 106468057808896 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 6532972417024 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 3160089296896 \) Copy content Toggle raw display
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