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Properties

Label 1008.2.ca.d
Level $1008$
Weight $2$
Character orbit 1008.ca
Analytic conductor $8.049$
Analytic rank $0$
Dimension $16$
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] = [1008,2,Mod(257,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.ca (of order \(6\), degree \(2\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{3} + \beta_{9} q^{5} + ( - \beta_{15} + \beta_{2}) q^{7} + \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{3} + \beta_{9} q^{5} + ( - \beta_{15} + \beta_{2}) q^{7} + \beta_{11} q^{9} + ( - \beta_{14} + \beta_{12} + \cdots + \beta_1) q^{11}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{13} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + 3 q^{15} + 9 q^{17} + 6 q^{21} - 21 q^{23} - 8 q^{25} - 9 q^{27} + 6 q^{29} + 15 q^{35} + q^{37} + 3 q^{39} - 6 q^{41} + 2 q^{43} - 30 q^{45} + 36 q^{47} - 5 q^{49} + 33 q^{51} + 15 q^{57} + 30 q^{59} + 15 q^{63} - 14 q^{67} + 21 q^{69} + 57 q^{75} + 3 q^{77} - 2 q^{79} + 18 q^{81} + 6 q^{85} - 48 q^{87} + 21 q^{89} - 9 q^{91} + 21 q^{93} - 3 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} + \cdots + 19787976 ) / 621108 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + \cdots + 46300977 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6613 \nu^{15} - 4165 \nu^{14} - 58592 \nu^{13} + 79853 \nu^{12} - 42655 \nu^{11} + \cdots + 63188991 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1730 \nu^{15} + 4365 \nu^{14} - 8834 \nu^{13} + 21044 \nu^{12} - 39051 \nu^{11} + 29320 \nu^{10} + \cdots + 1850931 ) / 207036 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8480 \nu^{15} + 13039 \nu^{14} - 27196 \nu^{13} + 146632 \nu^{12} - 38759 \nu^{11} + \cdots + 50810571 ) / 621108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9197 \nu^{15} + 929 \nu^{14} - 62076 \nu^{13} + 49233 \nu^{12} - 112105 \nu^{11} + \cdots + 20080305 ) / 414072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} + \cdots - 61443765 ) / 621108 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10991 \nu^{15} + 435 \nu^{14} - 52928 \nu^{13} + 71471 \nu^{12} - 66663 \nu^{11} + \cdots + 23210631 ) / 414072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + \cdots + 23648031 ) / 621108 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17318 \nu^{15} + 11843 \nu^{14} + 72850 \nu^{13} - 74620 \nu^{12} + 24959 \nu^{11} + \cdots - 20594979 ) / 621108 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8906 \nu^{15} - 13891 \nu^{14} + 29326 \nu^{13} - 153874 \nu^{12} + 48131 \nu^{11} + \cdots - 52673895 ) / 310554 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20668 \nu^{15} + 11675 \nu^{14} - 91538 \nu^{13} + 200162 \nu^{12} - 134761 \nu^{11} + \cdots + 65179161 ) / 621108 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 7981 \nu^{15} + 3790 \nu^{14} - 34392 \nu^{13} + 96837 \nu^{12} - 29234 \nu^{11} + \cdots + 39766950 ) / 207036 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25616 \nu^{15} - 1811 \nu^{14} - 72430 \nu^{13} + 225742 \nu^{12} + 48469 \nu^{11} + \cdots + 67869171 ) / 621108 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19601 \nu^{15} + 19537 \nu^{14} - 71044 \nu^{13} + 245365 \nu^{12} - 112385 \nu^{11} + \cdots + 77064777 ) / 414072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{4} - \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} - 3 \beta_{13} + 7 \beta_{12} - 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} - 5 \beta_{9} + 9 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{15} - 3 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 12 \beta_{10} + 16 \beta_{9} + \cdots - 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7 \beta_{15} + 7 \beta_{14} - 11 \beta_{13} + 2 \beta_{12} - 11 \beta_{11} + 5 \beta_{10} + 11 \beta_{9} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4 \beta_{15} + 6 \beta_{14} + 11 \beta_{13} - 39 \beta_{12} - 15 \beta_{11} - 6 \beta_{10} + \cdots + 116 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 51 \beta_{15} + 43 \beta_{14} - 43 \beta_{13} - 37 \beta_{12} + 10 \beta_{11} + 2 \beta_{10} + \cdots - 116 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4 \beta_{15} - 48 \beta_{14} - 4 \beta_{13} + 35 \beta_{12} + 126 \beta_{11} - 33 \beta_{10} + \cdots - 177 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 41 \beta_{15} + 60 \beta_{14} - 71 \beta_{13} + 164 \beta_{12} - 57 \beta_{11} + 6 \beta_{10} + \cdots + 47 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 275 \beta_{15} + 134 \beta_{14} - 31 \beta_{13} + 20 \beta_{12} - 181 \beta_{11} + 58 \beta_{10} + \cdots + 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 438 \beta_{15} + 208 \beta_{14} + 101 \beta_{13} + 9 \beta_{12} + 133 \beta_{11} - 184 \beta_{10} + \cdots - 1389 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4 \beta_{15} - 144 \beta_{14} + 283 \beta_{13} - 602 \beta_{12} + 243 \beta_{11} + 237 \beta_{10} + \cdots - 1269 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 679 \beta_{15} + 815 \beta_{14} + 2796 \beta_{13} - 2187 \beta_{12} - 229 \beta_{11} + 331 \beta_{10} + \cdots + 1187 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1166 \beta_{15} + 2420 \beta_{14} - 2349 \beta_{13} + 662 \beta_{12} + 1109 \beta_{11} + 2848 \beta_{10} + \cdots - 3948 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.544978 − 1.64408i
−0.811340 − 1.53027i
1.68124 + 0.416458i
−0.213160 + 1.71888i
−0.268067 + 1.71118i
1.68042 − 0.419752i
−1.61108 − 0.635951i
1.08696 − 1.34852i
−0.544978 + 1.64408i
−0.811340 + 1.53027i
1.68124 − 0.416458i
−0.213160 − 1.71888i
−0.268067 − 1.71118i
1.68042 + 0.419752i
−1.61108 + 0.635951i
1.08696 + 1.34852i
0 −1.70992 − 0.276016i 0 −1.95741 − 3.39033i 0 −0.554241 + 2.58705i 0 2.84763 + 0.943929i 0
257.2 0 −1.68085 − 0.418028i 0 1.37166 + 2.37578i 0 2.60476 − 0.463945i 0 2.65051 + 1.40528i 0
257.3 0 −1.06740 + 1.36406i 0 0.349828 + 0.605920i 0 −2.48683 − 0.903137i 0 −0.721326 − 2.91199i 0
257.4 0 −0.106783 − 1.72876i 0 1.43402 + 2.48379i 0 −2.56899 + 0.632668i 0 −2.97719 + 0.369204i 0
257.5 0 0.134439 + 1.72683i 0 −0.842869 − 1.45989i 0 2.27938 + 1.34329i 0 −2.96385 + 0.464306i 0
257.6 0 1.36511 − 1.06606i 0 −1.48494 − 2.57199i 0 0.200279 − 2.63816i 0 0.727031 − 2.91057i 0
257.7 0 1.43204 + 0.974295i 0 1.09150 + 1.89054i 0 1.25859 − 2.32722i 0 1.10150 + 2.79047i 0
257.8 0 1.63336 − 0.576322i 0 0.0382122 + 0.0661855i 0 −0.232935 + 2.63548i 0 2.33571 − 1.88268i 0
353.1 0 −1.70992 + 0.276016i 0 −1.95741 + 3.39033i 0 −0.554241 − 2.58705i 0 2.84763 − 0.943929i 0
353.2 0 −1.68085 + 0.418028i 0 1.37166 − 2.37578i 0 2.60476 + 0.463945i 0 2.65051 − 1.40528i 0
353.3 0 −1.06740 − 1.36406i 0 0.349828 − 0.605920i 0 −2.48683 + 0.903137i 0 −0.721326 + 2.91199i 0
353.4 0 −0.106783 + 1.72876i 0 1.43402 − 2.48379i 0 −2.56899 − 0.632668i 0 −2.97719 − 0.369204i 0
353.5 0 0.134439 − 1.72683i 0 −0.842869 + 1.45989i 0 2.27938 − 1.34329i 0 −2.96385 − 0.464306i 0
353.6 0 1.36511 + 1.06606i 0 −1.48494 + 2.57199i 0 0.200279 + 2.63816i 0 0.727031 + 2.91057i 0
353.7 0 1.43204 − 0.974295i 0 1.09150 − 1.89054i 0 1.25859 + 2.32722i 0 1.10150 − 2.79047i 0
353.8 0 1.63336 + 0.576322i 0 0.0382122 − 0.0661855i 0 −0.232935 − 2.63548i 0 2.33571 + 1.88268i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.ca.d 16
3.b odd 2 1 3024.2.ca.d 16
4.b odd 2 1 252.2.w.a ✓ 16
7.d odd 6 1 1008.2.df.d 16
9.c even 3 1 3024.2.df.d 16
9.d odd 6 1 1008.2.df.d 16
12.b even 2 1 756.2.w.a 16
21.g even 6 1 3024.2.df.d 16
28.d even 2 1 1764.2.w.b 16
28.f even 6 1 252.2.bm.a yes 16
28.f even 6 1 1764.2.x.b 16
28.g odd 6 1 1764.2.x.a 16
28.g odd 6 1 1764.2.bm.a 16
36.f odd 6 1 756.2.bm.a 16
36.f odd 6 1 2268.2.t.b 16
36.h even 6 1 252.2.bm.a yes 16
36.h even 6 1 2268.2.t.a 16
63.i even 6 1 inner 1008.2.ca.d 16
63.t odd 6 1 3024.2.ca.d 16
84.h odd 2 1 5292.2.w.b 16
84.j odd 6 1 756.2.bm.a 16
84.j odd 6 1 5292.2.x.b 16
84.n even 6 1 5292.2.x.a 16
84.n even 6 1 5292.2.bm.a 16
252.n even 6 1 2268.2.t.a 16
252.n even 6 1 5292.2.x.a 16
252.o even 6 1 1764.2.x.b 16
252.r odd 6 1 252.2.w.a ✓ 16
252.s odd 6 1 1764.2.bm.a 16
252.u odd 6 1 5292.2.w.b 16
252.bb even 6 1 1764.2.w.b 16
252.bi even 6 1 5292.2.bm.a 16
252.bj even 6 1 756.2.w.a 16
252.bl odd 6 1 5292.2.x.b 16
252.bn odd 6 1 1764.2.x.a 16
252.bn odd 6 1 2268.2.t.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a ✓ 16 4.b odd 2 1
252.2.w.a ✓ 16 252.r odd 6 1
252.2.bm.a yes 16 28.f even 6 1
252.2.bm.a yes 16 36.h even 6 1
756.2.w.a 16 12.b even 2 1
756.2.w.a 16 252.bj even 6 1
756.2.bm.a 16 36.f odd 6 1
756.2.bm.a 16 84.j odd 6 1
1008.2.ca.d 16 1.a even 1 1 trivial
1008.2.ca.d 16 63.i even 6 1 inner
1008.2.df.d 16 7.d odd 6 1
1008.2.df.d 16 9.d odd 6 1
1764.2.w.b 16 28.d even 2 1
1764.2.w.b 16 252.bb even 6 1
1764.2.x.a 16 28.g odd 6 1
1764.2.x.a 16 252.bn odd 6 1
1764.2.x.b 16 28.f even 6 1
1764.2.x.b 16 252.o even 6 1
1764.2.bm.a 16 28.g odd 6 1
1764.2.bm.a 16 252.s odd 6 1
2268.2.t.a 16 36.h even 6 1
2268.2.t.a 16 252.n even 6 1
2268.2.t.b 16 36.f odd 6 1
2268.2.t.b 16 252.bn odd 6 1
3024.2.ca.d 16 3.b odd 2 1
3024.2.ca.d 16 63.t odd 6 1
3024.2.df.d 16 9.c even 3 1
3024.2.df.d 16 21.g even 6 1
5292.2.w.b 16 84.h odd 2 1
5292.2.w.b 16 252.u odd 6 1
5292.2.x.a 16 84.n even 6 1
5292.2.x.a 16 252.n even 6 1
5292.2.x.b 16 84.j odd 6 1
5292.2.x.b 16 252.bl odd 6 1
5292.2.bm.a 16 84.n even 6 1
5292.2.bm.a 16 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 24 T_{5}^{14} - 24 T_{5}^{13} + 405 T_{5}^{12} - 423 T_{5}^{11} + 3600 T_{5}^{10} + \cdots + 324 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 3 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 24 T^{14} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{16} - T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 26244 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} + \cdots + 3337929 \) Copy content Toggle raw display
$17$ \( T^{16} - 9 T^{15} + \cdots + 13549761 \) Copy content Toggle raw display
$19$ \( T^{16} - 93 T^{14} + \cdots + 2099601 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15198451524 \) Copy content Toggle raw display
$29$ \( T^{16} - 6 T^{15} + \cdots + 15752961 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 3910251024 \) Copy content Toggle raw display
$37$ \( T^{16} - T^{15} + \cdots + 52765696 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 91647269289 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 28009034881 \) Copy content Toggle raw display
$47$ \( (T^{8} - 18 T^{7} + \cdots + 1404144)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} - 153 T^{14} + \cdots + 531441 \) Copy content Toggle raw display
$59$ \( (T^{8} - 15 T^{7} + \cdots - 406908)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1475481744 \) Copy content Toggle raw display
$67$ \( (T^{8} + 7 T^{7} + \cdots - 1454288)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 780959242139904 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 7523023152969 \) Copy content Toggle raw display
$79$ \( (T^{8} + T^{7} - 143 T^{6} + \cdots - 3248)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 669184533369 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 7161826993281 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 22864161681 \) Copy content Toggle raw display
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