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Properties

Label 663.2.f.b
Level $663$
Weight $2$
Character orbit 663.f
Analytic conductor $5.294$
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] = [663,2,Mod(118,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.f (of order \(2\), degree \(1\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 20 x^{13} + 12 x^{12} - 34 x^{11} + 188 x^{10} - 162 x^{9} + 42 x^{8} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{9} q^{3} + ( - \beta_{5} + 1) q^{4} + ( - \beta_{9} + \beta_{8}) q^{5} - \beta_{11} q^{6} + ( - \beta_{13} + \beta_{9}) q^{7} + ( - \beta_{7} + \beta_{4} - 1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{9} q^{3} + ( - \beta_{5} + 1) q^{4} + ( - \beta_{9} + \beta_{8}) q^{5} - \beta_{11} q^{6} + ( - \beta_{13} + \beta_{9}) q^{7} + ( - \beta_{7} + \beta_{4} - 1) q^{8} - q^{9} + ( - \beta_{14} + \beta_{11} + \cdots + \beta_{8}) q^{10}+ \cdots + (\beta_{14} - \beta_{13} + \cdots - \beta_{11}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 12 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 12 q^{8} - 16 q^{9} + 16 q^{13} + 12 q^{15} - 12 q^{17} + 8 q^{19} - 8 q^{21} - 8 q^{25} - 16 q^{30} + 16 q^{32} - 12 q^{33} - 12 q^{34} + 28 q^{35} - 16 q^{36} + 52 q^{38} + 12 q^{42} + 48 q^{43} - 4 q^{47} + 8 q^{49} - 64 q^{50} + 8 q^{51} + 16 q^{52} + 20 q^{53} + 16 q^{55} - 12 q^{59} - 12 q^{60} - 40 q^{64} + 20 q^{66} - 64 q^{67} - 44 q^{68} - 12 q^{69} - 24 q^{70} + 12 q^{72} - 16 q^{76} + 4 q^{77} + 16 q^{81} - 24 q^{83} - 28 q^{84} + 16 q^{85} + 68 q^{86} + 20 q^{87} + 32 q^{89} - 4 q^{93} - 56 q^{94} + 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 20 x^{13} + 12 x^{12} - 34 x^{11} + 188 x^{10} - 162 x^{9} + 42 x^{8} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 205117091616394 \nu^{15} + \cdots - 18\!\cdots\!54 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 223014738891678 \nu^{15} + \cdots - 60\!\cdots\!14 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 230558646410955 \nu^{15} + \cdots - 17\!\cdots\!75 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 310218627506165 \nu^{15} + \cdots - 27\!\cdots\!23 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 680788515166546 \nu^{15} + \cdots - 14\!\cdots\!11 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 685218668592406 \nu^{15} + \cdots - 10\!\cdots\!65 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 914043689715494 \nu^{15} + \cdots - 20\!\cdots\!32 ) / 78\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!27 \nu^{15} + \cdots + 16\!\cdots\!50 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 56\!\cdots\!61 \nu^{15} + \cdots + 21\!\cdots\!99 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 89\!\cdots\!69 \nu^{15} + \cdots + 41\!\cdots\!17 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!56 \nu^{15} + \cdots + 48\!\cdots\!47 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!35 \nu^{15} + \cdots - 56\!\cdots\!46 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!63 \nu^{15} + \cdots + 58\!\cdots\!50 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!96 \nu^{15} + \cdots + 77\!\cdots\!77 ) / 23\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!54 \nu^{15} + \cdots - 10\!\cdots\!01 ) / 23\!\cdots\!91 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 3\beta_{9} + \beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} + 3\beta_{11} - 7\beta_{9} + 5\beta_{8} + \beta_{4} - 3\beta_{3} - 5\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{4} - 8\beta_{3} - 8\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{14} - \beta_{13} + \beta_{12} - 29 \beta_{11} + 4 \beta_{10} + 48 \beta_{9} - 32 \beta_{8} + \cdots - 47 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{14} - 2\beta_{13} + 3\beta_{12} - 61\beta_{11} + 12\beta_{10} + 93\beta_{9} - 58\beta_{8} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{15} + 95 \beta_{14} - 12 \beta_{13} + 18 \beta_{12} - 232 \beta_{11} + 50 \beta_{10} + \cdots + 313 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 24\beta_{7} + 44\beta_{6} - 114\beta_{5} - 188\beta_{4} + 456\beta_{3} + 419\beta_{2} - 3\beta _1 + 582 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20 \beta_{15} - 758 \beta_{14} + 108 \beta_{13} - 208 \beta_{12} + 1764 \beta_{11} - 470 \beta_{10} + \cdots + 2105 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 50 \beta_{15} - 1486 \beta_{14} + 215 \beta_{13} - 463 \beta_{12} + 3425 \beta_{11} + \cdots + 3043 \beta_{8} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 248 \beta_{15} - 5847 \beta_{14} + 886 \beta_{13} - 2012 \beta_{12} + 13181 \beta_{11} - 3998 \beta_{10} + \cdots - 14341 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1751 \beta_{7} - 4259 \beta_{6} + 8107 \beta_{5} + 11389 \beta_{4} - 24865 \beta_{3} - 22192 \beta_{2} + \cdots - 26632 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2508 \beta_{15} + 44361 \beta_{14} - 6981 \beta_{13} + 17751 \beta_{12} - 97817 \beta_{11} + \cdots - 98883 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5385 \beta_{15} + 86046 \beta_{14} - 13703 \beta_{13} + 36470 \beta_{12} - 188234 \beta_{11} + \cdots - 162266 \beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 22767 \beta_{15} + 333515 \beta_{14} - 53850 \beta_{13} + 148330 \beta_{12} - 723666 \beta_{11} + \cdots + 688939 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.647720 + 0.647720i
0.647720 − 0.647720i
0.113543 + 0.113543i
0.113543 − 0.113543i
1.23240 + 1.23240i
1.23240 − 1.23240i
−0.628261 − 0.628261i
−0.628261 + 0.628261i
1.71975 + 1.71975i
1.71975 − 1.71975i
1.91819 + 1.91819i
1.91819 − 1.91819i
−0.929738 − 0.929738i
−0.929738 + 0.929738i
−1.07361 − 1.07361i
−1.07361 + 1.07361i
−2.45636 − 1.00000i 4.03369 1.29544i 2.45636i − 5.01662i −4.99548 −1.00000 − 3.18206i
118.2 −2.45636 1.00000i 4.03369 − 1.29544i − 2.45636i 5.01662i −4.99548 −1.00000 3.18206i
118.3 −2.20130 − 1.00000i 2.84573 0.227085i 2.20130i 1.61263i −1.86170 −1.00000 − 0.499883i
118.4 −2.20130 1.00000i 2.84573 − 0.227085i − 2.20130i − 1.61263i −1.86170 −1.00000 0.499883i
118.5 −1.42717 − 1.00000i 0.0368194 2.46481i 1.42717i 0.246460i 2.80180 −1.00000 − 3.51770i
118.6 −1.42717 1.00000i 0.0368194 − 2.46481i − 1.42717i − 0.246460i 2.80180 −1.00000 3.51770i
118.7 0.0459450 − 1.00000i −1.99789 − 1.25652i − 0.0459450i 2.96916i −0.183683 −1.00000 − 0.0577309i
118.8 0.0459450 1.00000i −1.99789 1.25652i 0.0459450i − 2.96916i −0.183683 −1.00000 0.0577309i
118.9 0.475560 − 1.00000i −1.77384 3.43949i − 0.475560i − 2.07406i −1.79469 −1.00000 1.63568i
118.10 0.475560 1.00000i −1.77384 − 3.43949i 0.475560i 2.07406i −1.79469 −1.00000 − 1.63568i
118.11 1.52255 − 1.00000i 0.318156 3.83639i − 1.52255i − 0.0785372i −2.56069 −1.00000 5.84109i
118.12 1.52255 1.00000i 0.318156 − 3.83639i 1.52255i 0.0785372i −2.56069 −1.00000 − 5.84109i
118.13 1.58830 − 1.00000i 0.522703 − 1.85948i − 1.58830i − 3.02863i −2.34639 −1.00000 − 2.95341i
118.14 1.58830 1.00000i 0.522703 1.85948i 1.58830i 3.02863i −2.34639 −1.00000 2.95341i
118.15 2.45247 − 1.00000i 4.01463 − 2.14721i − 2.45247i 1.36959i 4.94084 −1.00000 − 5.26599i
118.16 2.45247 1.00000i 4.01463 2.14721i 2.45247i − 1.36959i 4.94084 −1.00000 5.26599i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.f.b ✓ 16
3.b odd 2 1 1989.2.f.i 16
17.b even 2 1 inner 663.2.f.b ✓ 16
51.c odd 2 1 1989.2.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.f.b ✓ 16 1.a even 1 1 trivial
663.2.f.b ✓ 16 17.b even 2 1 inner
1989.2.f.i 16 3.b odd 2 1
1989.2.f.i 16 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 12T_{2}^{6} + 2T_{2}^{5} + 44T_{2}^{4} - 16T_{2}^{3} - 48T_{2}^{2} + 24T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 12 T^{6} + 2 T^{5} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 44 T^{14} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{16} + 52 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{16} + 92 T^{14} + \cdots + 2310400 \) Copy content Toggle raw display
$13$ \( (T - 1)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + \cdots - 1600)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 81229560064 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 1320304896 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 351187600 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1287087376 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 8656441600 \) Copy content Toggle raw display
$43$ \( (T^{8} - 24 T^{7} + \cdots - 20624)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 2 T^{7} + \cdots - 663244)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 10 T^{7} + \cdots + 26201664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 6 T^{7} + \cdots - 3660)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1130734489600 \) Copy content Toggle raw display
$67$ \( (T^{8} + 32 T^{7} + \cdots + 34164288)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1530374400 \) Copy content Toggle raw display
$73$ \( T^{16} + 464 T^{14} + \cdots + 2108304 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 244874543104 \) Copy content Toggle raw display
$83$ \( (T^{8} + 12 T^{7} + \cdots - 863484)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 16 T^{7} + \cdots - 4200020)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2360930587024 \) Copy content Toggle raw display
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