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Properties

Label 252.2.w.a
Level $252$
Weight $2$
Character orbit 252.w
Analytic conductor $2.012$
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] = [252,2,Mod(5,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("252.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.w (of order \(6\), degree \(2\), 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: \(2.01223013094\)
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: yes
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{8} + \beta_{3}) q^{7} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{8} + \beta_{3}) q^{7} - \beta_{4} q^{9} + (\beta_{15} - \beta_{14} + \cdots + \beta_1) q^{11}+ \cdots + (2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \cdots - 2) q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 1307 \nu^{15} + 5068 \nu^{14} + 824 \nu^{13} + 49267 \nu^{12} + 2716 \nu^{11} + 77018 \nu^{10} + \cdots + 19166868 ) / 621108 \) 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
\(\beta_{3}\)\(=\) \( ( - 6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + \cdots - 5872095 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 818 \nu^{15} - 2453 \nu^{14} - 9016 \nu^{13} - 10490 \nu^{12} - 12167 \nu^{11} + 10856 \nu^{10} + \cdots - 2858409 ) / 103518 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4659 \nu^{15} - 5459 \nu^{14} + 29980 \nu^{13} - 59671 \nu^{12} + 62083 \nu^{11} - 182738 \nu^{10} + \cdots - 25313067 ) / 414072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7279 \nu^{15} + 11960 \nu^{14} - 9158 \nu^{13} + 168785 \nu^{12} + 36326 \nu^{11} + \cdots + 70277058 ) / 621108 \) 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
\(\beta_{8}\)\(=\) \( ( 27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} + \cdots - 47062053 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7241 \nu^{15} + 4357 \nu^{14} - 32353 \nu^{13} + 38470 \nu^{12} - 78698 \nu^{11} + \cdots + 1180980 ) / 310554 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5228 \nu^{15} + 1525 \nu^{14} + 20418 \nu^{13} - 46974 \nu^{12} - 5255 \nu^{11} - 122904 \nu^{10} + \cdots - 19887849 ) / 207036 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16952 \nu^{15} - 9175 \nu^{14} + 73804 \nu^{13} - 123904 \nu^{12} + 128807 \nu^{11} + \cdots - 23648031 ) / 621108 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11445 \nu^{15} - 10925 \nu^{14} + 48460 \nu^{13} - 170497 \nu^{12} + 54037 \nu^{11} + \cdots - 69748533 ) / 414072 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19793 \nu^{15} + 1430 \nu^{14} + 108688 \nu^{13} - 147133 \nu^{12} + 121322 \nu^{11} + \cdots - 66410442 ) / 621108 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 59447 \nu^{15} + 8969 \nu^{14} + 284176 \nu^{13} - 415519 \nu^{12} + 220139 \nu^{11} + \cdots - 158830875 ) / 1242216 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24909 \nu^{15} - 6205 \nu^{14} + 135476 \nu^{13} - 180977 \nu^{12} + 218117 \nu^{11} + \cdots - 60703101 ) / 414072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - 2\beta_{14} + \beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{14} - \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 2 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{10} - \beta_{9} - 2 \beta_{7} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} - \beta_{13} + 2 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \cdots + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{15} + 5 \beta_{14} - 7 \beta_{13} + 9 \beta_{12} - 3 \beta_{11} + 10 \beta_{10} - 16 \beta_{9} + \cdots - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} + 7 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} - 12 \beta_{11} - 25 \beta_{10} - 3 \beta_{9} + \cdots - 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35 \beta_{15} + 4 \beta_{14} - 29 \beta_{13} - 8 \beta_{12} - 27 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} + \cdots + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 73 \beta_{15} - 51 \beta_{14} - 30 \beta_{13} + 4 \beta_{12} - 63 \beta_{11} + 52 \beta_{10} - 24 \beta_{9} + \cdots - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 132 \beta_{15} + 4 \beta_{14} - 100 \beta_{13} + 15 \beta_{12} + 48 \beta_{11} + 61 \beta_{10} + \cdots - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 79 \beta_{15} + 41 \beta_{14} - 17 \beta_{13} - 71 \beta_{12} + 15 \beta_{11} - 174 \beta_{10} + \cdots - 463 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5 \beta_{15} + 275 \beta_{14} - 241 \beta_{13} + 185 \beta_{12} - 225 \beta_{11} - 63 \beta_{10} + \cdots - 80 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 995 \beta_{15} + 438 \beta_{14} - 26 \beta_{13} + 764 \beta_{12} + 51 \beta_{11} + 139 \beta_{10} + \cdots - 198 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 376 \beta_{15} - 4 \beta_{14} + 276 \beta_{13} - 699 \beta_{12} + 408 \beta_{11} - 924 \beta_{10} + \cdots - 1935 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 453 \beta_{15} + 679 \beta_{14} - 1199 \beta_{13} - 40 \beta_{12} - 1635 \beta_{11} + 2518 \beta_{10} + \cdots - 1881 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1349 \beta_{15} - 1166 \beta_{14} + 1447 \beta_{13} + 472 \beta_{12} - 1887 \beta_{11} + 861 \beta_{10} + \cdots + 611 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1 + \beta_{1}\) \(1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.08696 + 1.34852i
−1.61108 + 0.635951i
1.68042 + 0.419752i
−0.268067 − 1.71118i
−0.213160 − 1.71888i
1.68124 − 0.416458i
−0.811340 + 1.53027i
−0.544978 + 1.64408i
1.08696 − 1.34852i
−1.61108 − 0.635951i
1.68042 − 0.419752i
−0.268067 + 1.71118i
−0.213160 + 1.71888i
1.68124 + 0.416458i
−0.811340 − 1.53027i
−0.544978 − 1.64408i
0 −1.63336 + 0.576322i 0 0.0382122 + 0.0661855i 0 0.232935 − 2.63548i 0 2.33571 − 1.88268i 0
5.2 0 −1.43204 − 0.974295i 0 1.09150 + 1.89054i 0 −1.25859 + 2.32722i 0 1.10150 + 2.79047i 0
5.3 0 −1.36511 + 1.06606i 0 −1.48494 − 2.57199i 0 −0.200279 + 2.63816i 0 0.727031 − 2.91057i 0
5.4 0 −0.134439 − 1.72683i 0 −0.842869 − 1.45989i 0 −2.27938 − 1.34329i 0 −2.96385 + 0.464306i 0
5.5 0 0.106783 + 1.72876i 0 1.43402 + 2.48379i 0 2.56899 − 0.632668i 0 −2.97719 + 0.369204i 0
5.6 0 1.06740 − 1.36406i 0 0.349828 + 0.605920i 0 2.48683 + 0.903137i 0 −0.721326 − 2.91199i 0
5.7 0 1.68085 + 0.418028i 0 1.37166 + 2.37578i 0 −2.60476 + 0.463945i 0 2.65051 + 1.40528i 0
5.8 0 1.70992 + 0.276016i 0 −1.95741 − 3.39033i 0 0.554241 − 2.58705i 0 2.84763 + 0.943929i 0
101.1 0 −1.63336 − 0.576322i 0 0.0382122 − 0.0661855i 0 0.232935 + 2.63548i 0 2.33571 + 1.88268i 0
101.2 0 −1.43204 + 0.974295i 0 1.09150 − 1.89054i 0 −1.25859 − 2.32722i 0 1.10150 − 2.79047i 0
101.3 0 −1.36511 − 1.06606i 0 −1.48494 + 2.57199i 0 −0.200279 − 2.63816i 0 0.727031 + 2.91057i 0
101.4 0 −0.134439 + 1.72683i 0 −0.842869 + 1.45989i 0 −2.27938 + 1.34329i 0 −2.96385 − 0.464306i 0
101.5 0 0.106783 − 1.72876i 0 1.43402 − 2.48379i 0 2.56899 + 0.632668i 0 −2.97719 − 0.369204i 0
101.6 0 1.06740 + 1.36406i 0 0.349828 − 0.605920i 0 2.48683 − 0.903137i 0 −0.721326 + 2.91199i 0
101.7 0 1.68085 − 0.418028i 0 1.37166 − 2.37578i 0 −2.60476 − 0.463945i 0 2.65051 − 1.40528i 0
101.8 0 1.70992 − 0.276016i 0 −1.95741 + 3.39033i 0 0.554241 + 2.58705i 0 2.84763 − 0.943929i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
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 252.2.w.a ✓ 16
3.b odd 2 1 756.2.w.a 16
4.b odd 2 1 1008.2.ca.d 16
7.b odd 2 1 1764.2.w.b 16
7.c even 3 1 1764.2.x.a 16
7.c even 3 1 1764.2.bm.a 16
7.d odd 6 1 252.2.bm.a yes 16
7.d odd 6 1 1764.2.x.b 16
9.c even 3 1 756.2.bm.a 16
9.c even 3 1 2268.2.t.b 16
9.d odd 6 1 252.2.bm.a yes 16
9.d odd 6 1 2268.2.t.a 16
12.b even 2 1 3024.2.ca.d 16
21.c even 2 1 5292.2.w.b 16
21.g even 6 1 756.2.bm.a 16
21.g even 6 1 5292.2.x.b 16
21.h odd 6 1 5292.2.x.a 16
21.h odd 6 1 5292.2.bm.a 16
28.f even 6 1 1008.2.df.d 16
36.f odd 6 1 3024.2.df.d 16
36.h even 6 1 1008.2.df.d 16
63.g even 3 1 5292.2.x.b 16
63.h even 3 1 5292.2.w.b 16
63.i even 6 1 inner 252.2.w.a ✓ 16
63.j odd 6 1 1764.2.w.b 16
63.k odd 6 1 2268.2.t.a 16
63.k odd 6 1 5292.2.x.a 16
63.l odd 6 1 5292.2.bm.a 16
63.n odd 6 1 1764.2.x.b 16
63.o even 6 1 1764.2.bm.a 16
63.s even 6 1 1764.2.x.a 16
63.s even 6 1 2268.2.t.b 16
63.t odd 6 1 756.2.w.a 16
84.j odd 6 1 3024.2.df.d 16
252.r odd 6 1 1008.2.ca.d 16
252.bj even 6 1 3024.2.ca.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a ✓ 16 1.a even 1 1 trivial
252.2.w.a ✓ 16 63.i even 6 1 inner
252.2.bm.a yes 16 7.d odd 6 1
252.2.bm.a yes 16 9.d odd 6 1
756.2.w.a 16 3.b odd 2 1
756.2.w.a 16 63.t odd 6 1
756.2.bm.a 16 9.c even 3 1
756.2.bm.a 16 21.g even 6 1
1008.2.ca.d 16 4.b odd 2 1
1008.2.ca.d 16 252.r odd 6 1
1008.2.df.d 16 28.f even 6 1
1008.2.df.d 16 36.h even 6 1
1764.2.w.b 16 7.b odd 2 1
1764.2.w.b 16 63.j odd 6 1
1764.2.x.a 16 7.c even 3 1
1764.2.x.a 16 63.s even 6 1
1764.2.x.b 16 7.d odd 6 1
1764.2.x.b 16 63.n odd 6 1
1764.2.bm.a 16 7.c even 3 1
1764.2.bm.a 16 63.o even 6 1
2268.2.t.a 16 9.d odd 6 1
2268.2.t.a 16 63.k odd 6 1
2268.2.t.b 16 9.c even 3 1
2268.2.t.b 16 63.s even 6 1
3024.2.ca.d 16 12.b even 2 1
3024.2.ca.d 16 252.bj even 6 1
3024.2.df.d 16 36.f odd 6 1
3024.2.df.d 16 84.j odd 6 1
5292.2.w.b 16 21.c even 2 1
5292.2.w.b 16 63.h even 3 1
5292.2.x.a 16 21.h odd 6 1
5292.2.x.a 16 63.k odd 6 1
5292.2.x.b 16 21.g even 6 1
5292.2.x.b 16 63.g even 3 1
5292.2.bm.a 16 21.h odd 6 1
5292.2.bm.a 16 63.l odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).

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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