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Properties

Label 1008.3.cg.n
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,3,Mod(145,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.145"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-6,0,4,0,0,0,14,0,0,0,0,0,-12,0,-78] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.126303473664.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{5} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots + 5) q^{11} + (3 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} + \cdots + 4) q^{13}+ \cdots + ( - 13 \beta_{7} + 2 \beta_{6} + \cdots + 37) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5} + 4 q^{7} + 14 q^{11} - 12 q^{17} - 78 q^{19} - 6 q^{25} + 4 q^{29} + 24 q^{31} - 156 q^{35} + 50 q^{37} + 20 q^{43} + 12 q^{47} + 220 q^{49} + 50 q^{53} - 186 q^{59} - 240 q^{61} + 148 q^{65}+ \cdots + 144 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\nu^{7} + 81\nu^{6} + 32\nu^{5} + 486\nu^{4} + 418\nu^{3} + 1120\nu^{2} + 5733\nu - 10633 ) / 10976 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -73\nu^{7} + 55\nu^{6} - 288\nu^{5} + 526\nu^{4} - 5330\nu^{3} - 672\nu^{2} + 21217\nu + 57281 ) / 21952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 99\nu^{7} + 75\nu^{6} - 800\nu^{5} - 1818\nu^{4} + 4950\nu^{3} + 16800\nu^{2} - 10731\nu - 111475 ) / 21952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{7} + 15\nu^{6} + 96\nu^{5} + 54\nu^{4} - 418\nu^{3} - 1504\nu^{2} - 819\nu + 9065 ) / 3136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -195\nu^{7} - 163\nu^{6} + 1440\nu^{5} + 2970\nu^{4} - 1798\nu^{3} - 23520\nu^{2} + 15435\nu + 125195 ) / 21952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -135\nu^{7} - 129\nu^{6} + 1376\nu^{5} + 2194\nu^{4} - 8514\nu^{3} - 17920\nu^{2} + 23471\nu + 125881 ) / 10976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -49\nu^{7} - 39\nu^{6} + 288\nu^{5} + 862\nu^{4} - 1870\nu^{3} - 6304\nu^{2} + 1673\nu + 42287 ) / 1568 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} - \beta_{4} - 3\beta_{3} + 2\beta_{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 12\beta_{3} + \beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 4\beta_{6} + 12\beta_{5} - \beta_{4} - 3\beta_{3} + \beta _1 + 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - \beta_{5} - 4\beta_{4} + 16\beta_{3} - \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} + 54\beta_{6} + 30\beta_{4} + 112\beta_{3} - 66\beta_{2} + 27\beta _1 + 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14\beta_{7} - 28\beta_{6} - 44\beta_{5} + 113\beta_{4} - 141\beta_{3} + 127\beta _1 + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -57\beta_{7} - 390\beta_{5} - 281\beta_{4} - 1067\beta_{3} - 390\beta_{2} + 224\beta _1 + 390 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−1.43536 − 2.22255i
2.18070 + 1.49818i
2.64247 + 0.131782i
−2.38781 − 1.13946i
−1.43536 + 2.22255i
2.18070 − 1.49818i
2.64247 − 0.131782i
−2.38781 + 1.13946i
0 0 0 −7.79369 − 4.49969i 0 6.97403 + 0.602461i 0 0 0
145.2 0 0 0 0.425150 + 0.245461i 0 6.25375 + 3.14494i 0 0 0
145.3 0 0 0 2.05105 + 1.18418i 0 −4.55981 + 5.31113i 0 0 0
145.4 0 0 0 2.31749 + 1.33800i 0 −6.66796 − 2.13033i 0 0 0
577.1 0 0 0 −7.79369 + 4.49969i 0 6.97403 − 0.602461i 0 0 0
577.2 0 0 0 0.425150 − 0.245461i 0 6.25375 − 3.14494i 0 0 0
577.3 0 0 0 2.05105 − 1.18418i 0 −4.55981 − 5.31113i 0 0 0
577.4 0 0 0 2.31749 − 1.33800i 0 −6.66796 + 2.13033i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.n 8
3.b odd 2 1 336.3.bh.h 8
4.b odd 2 1 504.3.by.a 8
7.d odd 6 1 inner 1008.3.cg.n 8
12.b even 2 1 168.3.z.a ✓ 8
21.g even 6 1 336.3.bh.h 8
21.g even 6 1 2352.3.f.k 8
21.h odd 6 1 2352.3.f.k 8
28.f even 6 1 504.3.by.a 8
28.f even 6 1 3528.3.f.f 8
28.g odd 6 1 3528.3.f.f 8
84.h odd 2 1 1176.3.z.d 8
84.j odd 6 1 168.3.z.a ✓ 8
84.j odd 6 1 1176.3.f.a 8
84.n even 6 1 1176.3.f.a 8
84.n even 6 1 1176.3.z.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.a ✓ 8 12.b even 2 1
168.3.z.a ✓ 8 84.j odd 6 1
336.3.bh.h 8 3.b odd 2 1
336.3.bh.h 8 21.g even 6 1
504.3.by.a 8 4.b odd 2 1
504.3.by.a 8 28.f even 6 1
1008.3.cg.n 8 1.a even 1 1 trivial
1008.3.cg.n 8 7.d odd 6 1 inner
1176.3.f.a 8 84.j odd 6 1
1176.3.f.a 8 84.n even 6 1
1176.3.z.d 8 84.h odd 2 1
1176.3.z.d 8 84.n even 6 1
2352.3.f.k 8 21.g even 6 1
2352.3.f.k 8 21.h odd 6 1
3528.3.f.f 8 28.f even 6 1
3528.3.f.f 8 28.g odd 6 1

Hecke kernels

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

\( T_{5}^{8} + 6T_{5}^{7} - 29T_{5}^{6} - 246T_{5}^{5} + 1973T_{5}^{4} - 5412T_{5}^{3} + 6956T_{5}^{2} - 3696T_{5} + 784 \) Copy content Toggle raw display
\( T_{11}^{8} - 14 T_{11}^{7} + 263 T_{11}^{6} - 2030 T_{11}^{5} + 29773 T_{11}^{4} - 225652 T_{11}^{3} + \cdots + 20322064 \) Copy content Toggle raw display
\( T_{13}^{8} + 1318T_{13}^{6} + 584489T_{13}^{4} + 98460224T_{13}^{2} + 4781999104 \) 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
$5$ \( T^{8} + 6 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 14 T^{7} + \cdots + 20322064 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4781999104 \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 123878657296 \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots + 278272)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 4336090801 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 6736469776 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 22503866142976 \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + \cdots + 994948)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 107206475776 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 618123019264 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 562746188841984 \) Copy content Toggle raw display
$61$ \( (T^{2} + 60 T + 1200)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 583524876544 \) Copy content Toggle raw display
$71$ \( (T^{4} - 212 T^{3} + \cdots - 11002304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 97154396416 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 450927645793249 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 53944676847616 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 656906741956864 \) Copy content Toggle raw display
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