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Properties

Label 663.2.i.f
Level $663$
Weight $2$
Character orbit 663.i
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(256,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.256");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.i (of order \(3\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 12 x^{14} + 100 x^{12} - 5 x^{11} + 418 x^{10} - 110 x^{9} + 1260 x^{8} - 340 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{4} - 1) q^{4} + ( - \beta_{15} + \beta_{5}) q^{5} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{7} - \beta_{3}) q^{7} + ( - \beta_{14} + \beta_{11} - \beta_{5}) q^{8} + (\beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{7} + \beta_{4} - 1) q^{4} + ( - \beta_{15} + \beta_{5}) q^{5} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{7} - \beta_{3}) q^{7} + ( - \beta_{14} + \beta_{11} - \beta_{5}) q^{8} + (\beta_{4} - 1) q^{9} + (\beta_{14} - \beta_{12} + \cdots + 2 \beta_{4}) q^{10}+ \cdots + (\beta_{13} - \beta_{11}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 8 q^{4} - 2 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} - 8 q^{4} - 2 q^{5} + 2 q^{7} - 8 q^{9} + 13 q^{10} - q^{11} - 16 q^{12} - 2 q^{13} - 2 q^{14} - q^{15} + 8 q^{17} - 2 q^{19} + 3 q^{20} + 4 q^{21} + q^{22} - 5 q^{23} + 22 q^{25} - 26 q^{26} - 16 q^{27} - 31 q^{28} + 17 q^{29} - 13 q^{30} - 32 q^{31} + 25 q^{32} + q^{33} - 7 q^{35} - 8 q^{36} + 4 q^{37} + 30 q^{38} - q^{39} - 32 q^{40} - q^{42} + q^{43} + 38 q^{44} + q^{45} - 19 q^{46} + 22 q^{47} - 2 q^{49} - 4 q^{50} + 16 q^{51} - 42 q^{52} + 22 q^{53} + 23 q^{55} - 13 q^{56} - 4 q^{57} + 52 q^{58} + 3 q^{59} + 6 q^{60} + 27 q^{61} + 24 q^{62} + 2 q^{63} + 20 q^{64} - 38 q^{65} + 2 q^{66} - 15 q^{67} + 8 q^{68} + 5 q^{69} - 60 q^{70} - 5 q^{71} + 38 q^{73} - 25 q^{74} + 11 q^{75} + 46 q^{76} - 4 q^{77} - 19 q^{78} - 26 q^{79} - 41 q^{80} - 8 q^{81} + 17 q^{82} + 58 q^{83} + 31 q^{84} - q^{85} + 38 q^{86} - 17 q^{87} + 21 q^{88} - 19 q^{89} - 26 q^{90} - 12 q^{91} - 44 q^{92} - 16 q^{93} + 41 q^{94} - 15 q^{95} + 50 q^{96} - 16 q^{97} - 44 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 12 x^{14} + 100 x^{12} - 5 x^{11} + 418 x^{10} - 110 x^{9} + 1260 x^{8} - 340 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 155245608835805 \nu^{15} - 277620304968378 \nu^{14} + \cdots + 10\!\cdots\!38 ) / 38\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37\!\cdots\!79 \nu^{15} + \cdots + 66\!\cdots\!04 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!25 \nu^{15} + \cdots + 70\!\cdots\!12 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{15} - 310491217671610 \nu^{14} + \cdots - 22\!\cdots\!00 ) / 77\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!71 \nu^{15} + \cdots + 21\!\cdots\!48 ) / 77\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!75 \nu^{15} + \cdots - 26\!\cdots\!12 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!27 \nu^{15} + \cdots + 26\!\cdots\!28 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!91 \nu^{15} + \cdots + 98\!\cdots\!56 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!37 \nu^{15} + \cdots - 15\!\cdots\!20 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 78\!\cdots\!55 \nu^{15} + \cdots + 16\!\cdots\!16 ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!53 \nu^{15} + \cdots + 14\!\cdots\!24 ) / 61\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!41 \nu^{15} + \cdots - 69\!\cdots\!84 ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!25 \nu^{15} + \cdots + 24\!\cdots\!16 ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!45 \nu^{15} + \cdots - 38\!\cdots\!28 ) / 15\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{11} - 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{11} - \beta_{10} - \beta_{9} - 7\beta_{7} - \beta_{6} - 14\beta_{4} - \beta_{3} + 7\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 7 \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + 29 \beta_{5} - 3 \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{15} + \beta_{14} + \beta_{13} - 12\beta_{11} + 9\beta_{6} + 24\beta_{5} - 44\beta_{2} + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{15} + 44 \beta_{14} - 44 \beta_{12} - 64 \beta_{11} + 12 \beta_{10} + 12 \beta_{9} + \cdots + 179 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{13} - 17 \beta_{12} + 76 \beta_{10} + 64 \beta_{9} - 12 \beta_{8} + 275 \beta_{7} + \cdots - 457 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105\beta_{15} - 275\beta_{14} - 76\beta_{13} + 432\beta_{11} - 105\beta_{6} - 1141\beta_{5} + 261\beta_{2} - 424 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 532 \beta_{15} - 185 \beta_{14} + 185 \beta_{12} + 822 \beta_{11} - 532 \beta_{10} - 432 \beta_{9} + \cdots - 1729 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 532 \beta_{13} + 1738 \beta_{12} - 827 \beta_{10} - 822 \beta_{9} + 532 \beta_{8} - 2204 \beta_{7} + \cdots + 3606 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3624 \beta_{15} + 1677 \beta_{14} + 827 \beta_{13} - 6118 \beta_{11} + 2897 \beta_{6} + 13181 \beta_{5} + \cdots + 18084 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6228 \beta_{15} + 11139 \beta_{14} - 11139 \beta_{12} - 19537 \beta_{11} + 6228 \beta_{10} + \cdots + 49184 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6228 \beta_{13} - 13838 \beta_{12} + 24505 \beta_{10} + 19537 \beta_{9} - 6228 \beta_{8} + \cdots - 117949 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 45831 \beta_{15} - 72372 \beta_{14} - 24505 \beta_{13} + 132772 \beta_{11} - 44351 \beta_{6} + \cdots - 214939 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
256.1
−1.13819 + 1.97140i
−1.09919 + 1.90385i
−0.685045 + 1.18653i
−0.255338 + 0.442258i
0.439022 − 0.760409i
0.606516 − 1.05052i
0.813100 − 1.40833i
1.31912 − 2.28479i
−1.13819 − 1.97140i
−1.09919 − 1.90385i
−0.685045 − 1.18653i
−0.255338 − 0.442258i
0.439022 + 0.760409i
0.606516 + 1.05052i
0.813100 + 1.40833i
1.31912 + 2.28479i
−1.13819 + 1.97140i 0.500000 − 0.866025i −1.59095 − 2.75560i 0.699710 1.13819 + 1.97140i −0.723097 − 1.25244i 2.69044 −0.500000 − 0.866025i −0.796403 + 1.37941i
256.2 −1.09919 + 1.90385i 0.500000 − 0.866025i −1.41644 − 2.45335i −3.54069 1.09919 + 1.90385i −1.70923 − 2.96048i 1.83099 −0.500000 − 0.866025i 3.89190 − 6.74096i
256.3 −0.685045 + 1.18653i 0.500000 − 0.866025i 0.0614264 + 0.106394i −2.44219 0.685045 + 1.18653i 2.31372 + 4.00748i −2.90850 −0.500000 − 0.866025i 1.67301 − 2.89774i
256.4 −0.255338 + 0.442258i 0.500000 − 0.866025i 0.869605 + 1.50620i 2.55838 0.255338 + 0.442258i 1.31329 + 2.27469i −1.90952 −0.500000 − 0.866025i −0.653252 + 1.13147i
256.5 0.439022 − 0.760409i 0.500000 − 0.866025i 0.614519 + 1.06438i −2.07739 −0.439022 − 0.760409i 1.65924 + 2.87388i 2.83524 −0.500000 − 0.866025i −0.912022 + 1.57967i
256.6 0.606516 − 1.05052i 0.500000 − 0.866025i 0.264276 + 0.457740i 3.73505 −0.606516 − 1.05052i −0.681382 − 1.18019i 3.06722 −0.500000 − 0.866025i 2.26537 − 3.92373i
256.7 0.813100 − 1.40833i 0.500000 − 0.866025i −0.322264 − 0.558178i −1.86326 −0.813100 − 1.40833i −0.395409 − 0.684868i 2.20427 −0.500000 − 0.866025i −1.51502 + 2.62408i
256.8 1.31912 − 2.28479i 0.500000 − 0.866025i −2.48017 − 4.29579i 1.93039 −1.31912 − 2.28479i −0.777126 − 1.34602i −7.81013 −0.500000 − 0.866025i 2.54642 − 4.41053i
562.1 −1.13819 − 1.97140i 0.500000 + 0.866025i −1.59095 + 2.75560i 0.699710 1.13819 − 1.97140i −0.723097 + 1.25244i 2.69044 −0.500000 + 0.866025i −0.796403 − 1.37941i
562.2 −1.09919 − 1.90385i 0.500000 + 0.866025i −1.41644 + 2.45335i −3.54069 1.09919 − 1.90385i −1.70923 + 2.96048i 1.83099 −0.500000 + 0.866025i 3.89190 + 6.74096i
562.3 −0.685045 − 1.18653i 0.500000 + 0.866025i 0.0614264 − 0.106394i −2.44219 0.685045 − 1.18653i 2.31372 − 4.00748i −2.90850 −0.500000 + 0.866025i 1.67301 + 2.89774i
562.4 −0.255338 − 0.442258i 0.500000 + 0.866025i 0.869605 − 1.50620i 2.55838 0.255338 − 0.442258i 1.31329 − 2.27469i −1.90952 −0.500000 + 0.866025i −0.653252 − 1.13147i
562.5 0.439022 + 0.760409i 0.500000 + 0.866025i 0.614519 − 1.06438i −2.07739 −0.439022 + 0.760409i 1.65924 − 2.87388i 2.83524 −0.500000 + 0.866025i −0.912022 − 1.57967i
562.6 0.606516 + 1.05052i 0.500000 + 0.866025i 0.264276 − 0.457740i 3.73505 −0.606516 + 1.05052i −0.681382 + 1.18019i 3.06722 −0.500000 + 0.866025i 2.26537 + 3.92373i
562.7 0.813100 + 1.40833i 0.500000 + 0.866025i −0.322264 + 0.558178i −1.86326 −0.813100 + 1.40833i −0.395409 + 0.684868i 2.20427 −0.500000 + 0.866025i −1.51502 − 2.62408i
562.8 1.31912 + 2.28479i 0.500000 + 0.866025i −2.48017 + 4.29579i 1.93039 −1.31912 + 2.28479i −0.777126 + 1.34602i −7.81013 −0.500000 + 0.866025i 2.54642 + 4.41053i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 256.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.i.f ✓ 16
13.c even 3 1 inner 663.2.i.f ✓ 16
13.c even 3 1 8619.2.a.bd 8
13.e even 6 1 8619.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.i.f ✓ 16 1.a even 1 1 trivial
663.2.i.f ✓ 16 13.c even 3 1 inner
8619.2.a.bd 8 13.c even 3 1
8619.2.a.be 8 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):

\( T_{2}^{16} + 12 T_{2}^{14} + 100 T_{2}^{12} - 5 T_{2}^{11} + 418 T_{2}^{10} - 110 T_{2}^{9} + 1260 T_{2}^{8} + \cdots + 256 \) Copy content Toggle raw display
\( T_{5}^{8} + T_{5}^{7} - 25T_{5}^{6} - 26T_{5}^{5} + 193T_{5}^{4} + 187T_{5}^{3} - 540T_{5}^{2} - 387T_{5} + 432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 12 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + T^{7} - 25 T^{6} + \cdots + 432)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 111556 \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots + 3564544 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 273042576 \) Copy content Toggle raw display
$23$ \( T^{16} + 5 T^{15} + \cdots + 47524 \) Copy content Toggle raw display
$29$ \( T^{16} - 17 T^{15} + \cdots + 19307236 \) Copy content Toggle raw display
$31$ \( (T^{8} + 16 T^{7} + \cdots - 52)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 471410944 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 139052918404 \) Copy content Toggle raw display
$43$ \( T^{16} - T^{15} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( (T^{8} - 11 T^{7} + \cdots - 21046712)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 11 T^{7} + \cdots - 530764)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 147581978896 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 3699558976 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 525278507121 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 194658304 \) Copy content Toggle raw display
$73$ \( (T^{8} - 19 T^{7} + \cdots - 973192)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 13 T^{7} + \cdots + 7099804)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 29 T^{7} + \cdots + 993112)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1903781810176 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 7545701956 \) Copy content Toggle raw display
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