Properties

Label 663.2.b.e
Level $663$
Weight $2$
Character orbit 663.b
Analytic conductor $5.294$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [663,2,Mod(103,663)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.b (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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 28x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1) q^{5} + \beta_1 q^{6} - \beta_{7} q^{7} + \beta_{3} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{4} - \beta_1) q^{5} + \beta_1 q^{6} - \beta_{7} q^{7} + \beta_{3} q^{8} + q^{9} + (\beta_{5} - \beta_{2} + 3) q^{10} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{11} + (\beta_{2} - 1) q^{12} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{13} + (\beta_{5} + 1) q^{14} + (\beta_{4} - \beta_1) q^{15} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{16} + q^{17} + \beta_1 q^{18} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{19} + (\beta_{7} + \beta_{4} - \beta_{3} + 2 \beta_1) q^{20} - \beta_{7} q^{21} + ( - \beta_{6} + \beta_{5} + 3 \beta_{2} - 3) q^{22} + (\beta_{6} + \beta_{5} - \beta_{2} + 3) q^{23} + \beta_{3} q^{24} + ( - 2 \beta_{5} - 1) q^{25} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{26} + q^{27} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{28} + ( - \beta_{6} + 3 \beta_{5} + \beta_{2} - 1) q^{29} + (\beta_{5} - \beta_{2} + 3) q^{30} + (2 \beta_{7} - \beta_{3} - \beta_1) q^{31} + \beta_{4} q^{32} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{33} + \beta_1 q^{34} + (\beta_{6} - 2 \beta_{5} - 2) q^{35} + (\beta_{2} - 1) q^{36} + ( - 4 \beta_{7} - \beta_{3} + \beta_1) q^{37} + ( - \beta_{6} + 2 \beta_{5} + 4 \beta_{2} - 4) q^{38} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{39} + ( - \beta_{6} + 3 \beta_{5} + 3 \beta_{2} - 3) q^{40} + ( - 2 \beta_{7} - 3 \beta_{4} + 2 \beta_{3} + \beta_1) q^{41} + (\beta_{5} + 1) q^{42} + ( - 2 \beta_{6} + \beta_{5} - \beta_{2} + 3) q^{43} + (2 \beta_{7} + \beta_{4} + 2 \beta_{3} - 5 \beta_1) q^{44} + (\beta_{4} - \beta_1) q^{45} + (2 \beta_{7} - \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{46} + (2 \beta_{7} + \beta_{4} + \beta_{3}) q^{47} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{48} + ( - \beta_{5} - \beta_{2} + 4) q^{49} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_1) q^{50} + q^{51} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{52} + (\beta_{6} - 2 \beta_{5} - 4) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{5} - 4 \beta_{2} + 2) q^{55} + (2 \beta_{5} + \beta_{2}) q^{56} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{57} + (2 \beta_{7} - 3 \beta_{4} + 2 \beta_{3} - \beta_1) q^{58} + (2 \beta_{7} - 3 \beta_{4} + \beta_{3}) q^{59} + (\beta_{7} + \beta_{4} - \beta_{3} + 2 \beta_1) q^{60} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 4) q^{61} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} - 1) q^{62} - \beta_{7} q^{63} + (2 \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{64} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{65} + ( - \beta_{6} + \beta_{5} + 3 \beta_{2} - 3) q^{66} + (\beta_{7} - 2 \beta_{4} + \beta_{3} - 3 \beta_1) q^{67} + (\beta_{2} - 1) q^{68} + (\beta_{6} + \beta_{5} - \beta_{2} + 3) q^{69} + ( - \beta_{7} + 2 \beta_{4} - \beta_{3} - 3 \beta_1) q^{70} + ( - \beta_{7} - 3 \beta_{4} - \beta_{3} + 4 \beta_1) q^{71} + \beta_{3} q^{72} + (2 \beta_{7} + \beta_{3} - \beta_1) q^{73} + ( - \beta_{6} + 5 \beta_{5} + 4 \beta_{2} - 1) q^{74} + ( - 2 \beta_{5} - 1) q^{75} + ( - \beta_{7} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_1) q^{76} + (\beta_{6} - 2 \beta_{5} + 2) q^{77} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{78} - 4 \beta_{2} q^{79} + (4 \beta_{7} - \beta_{4} + 2 \beta_{3} - \beta_1) q^{80} + q^{81} + (2 \beta_{6} - 3 \beta_{5} - 5 \beta_{2} + 3) q^{82} + ( - 3 \beta_{7} + \beta_{4} - 5 \beta_1) q^{83} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{84} + (\beta_{4} - \beta_1) q^{85} + ( - \beta_{7} - \beta_{4} + \beta_{3} + 6 \beta_1) q^{86} + ( - \beta_{6} + 3 \beta_{5} + \beta_{2} - 1) q^{87} + ( - \beta_{5} - 5 \beta_{2} + 11) q^{88} + (5 \beta_{7} - \beta_{4} - 5 \beta_1) q^{89} + (\beta_{5} - \beta_{2} + 3) q^{90} + (2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{91} + (\beta_{5} + 7 \beta_{2} - 9) q^{92} + (2 \beta_{7} - \beta_{3} - \beta_1) q^{93} + (\beta_{6} - 2 \beta_{5} - 3 \beta_{2}) q^{94} + (2 \beta_{6} - 4 \beta_{5} - 4 \beta_{2} + 4) q^{95} + \beta_{4} q^{96} + ( - 3 \beta_{7} - 4 \beta_{4} + 2 \beta_1) q^{97} + ( - \beta_{7} + \beta_{4} - \beta_{3} + 5 \beta_1) q^{98} + (\beta_{7} + \beta_{4} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 4 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 4 q^{4} + 8 q^{9} + 20 q^{10} - 4 q^{12} - 4 q^{13} + 8 q^{14} + 8 q^{17} - 16 q^{22} + 24 q^{23} - 8 q^{25} - 20 q^{26} + 8 q^{27} - 8 q^{29} + 20 q^{30} - 12 q^{35} - 4 q^{36} - 20 q^{38} - 4 q^{39} - 16 q^{40} + 8 q^{42} + 12 q^{43} + 28 q^{49} + 8 q^{51} - 28 q^{52} - 28 q^{53} + 4 q^{56} - 20 q^{61} - 4 q^{62} - 8 q^{65} - 16 q^{66} - 4 q^{68} + 24 q^{69} + 4 q^{74} - 8 q^{75} + 20 q^{77} - 20 q^{78} - 16 q^{79} + 8 q^{81} + 12 q^{82} - 8 q^{87} + 68 q^{88} + 20 q^{90} + 4 q^{91} - 44 q^{92} - 8 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 28x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} + 8\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 8\nu^{4} + 12\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 9\nu^{4} + 19\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 9\nu^{5} + 20\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} - 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{4} - 8\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{6} + 9\beta_{5} + 44\beta_{2} - 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 9\beta_{4} + 52\beta_{3} - 112\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1
2.44735i
1.53025i
1.27474i
0.209470i
0.209470i
1.27474i
1.53025i
2.44735i
2.44735i 1.00000 −3.98951 2.54977i 2.44735i 0.511026i 4.86902i 1.00000 6.24017
103.2 1.53025i 1.00000 −0.341661 3.44293i 1.53025i 2.56617i 2.53767i 1.00000 5.26854
103.3 1.27474i 1.00000 0.375049 0.816926i 1.27474i 1.30719i 3.02756i 1.00000 −1.04137
103.4 0.209470i 1.00000 1.95612 2.23105i 0.209470i 2.33343i 0.828690i 1.00000 −0.467338
103.5 0.209470i 1.00000 1.95612 2.23105i 0.209470i 2.33343i 0.828690i 1.00000 −0.467338
103.6 1.27474i 1.00000 0.375049 0.816926i 1.27474i 1.30719i 3.02756i 1.00000 −1.04137
103.7 1.53025i 1.00000 −0.341661 3.44293i 1.53025i 2.56617i 2.53767i 1.00000 5.26854
103.8 2.44735i 1.00000 −3.98951 2.54977i 2.44735i 0.511026i 4.86902i 1.00000 6.24017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.b.e 8
3.b odd 2 1 1989.2.b.h 8
13.b even 2 1 inner 663.2.b.e 8
13.d odd 4 2 8619.2.a.bg 8
39.d odd 2 1 1989.2.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.b.e 8 1.a even 1 1 trivial
663.2.b.e 8 13.b even 2 1 inner
1989.2.b.h 8 3.b odd 2 1
1989.2.b.h 8 39.d odd 2 1
8619.2.a.bg 8 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10T_{2}^{6} + 28T_{2}^{4} + 24T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{6} + 28 T^{4} + 24 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{6} + 184 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + 14 T^{6} + 60 T^{4} + 76 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + 824 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + 4 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T - 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 68 T^{6} + 1120 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{4} - 12 T^{3} + 4 T^{2} + 356 T - 992)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 68 T^{2} - 468 T - 776)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 74 T^{6} + 1436 T^{4} + \cdots + 8464 \) Copy content Toggle raw display
$37$ \( T^{8} + 266 T^{6} + 21940 T^{4} + \cdots + 4227136 \) Copy content Toggle raw display
$41$ \( T^{8} + 232 T^{6} + 18168 T^{4} + \cdots + 4596736 \) Copy content Toggle raw display
$43$ \( (T^{4} - 6 T^{3} - 128 T^{2} + 1244 T - 2768)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 118 T^{6} + 4288 T^{4} + \cdots + 204304 \) Copy content Toggle raw display
$53$ \( (T^{4} + 14 T^{3} + 24 T^{2} - 144 T - 352)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 230 T^{6} + 9616 T^{4} + \cdots + 258064 \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} - 80 T^{2} - 832 T - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 196 T^{6} + 9504 T^{4} + \cdots + 541696 \) Copy content Toggle raw display
$71$ \( T^{8} + 356 T^{6} + 36136 T^{4} + \cdots + 7441984 \) Copy content Toggle raw display
$73$ \( T^{8} + 106 T^{6} + 3732 T^{4} + \cdots + 30976 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 128 T^{2} - 1152 T - 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 522 T^{6} + 72832 T^{4} + \cdots + 234256 \) Copy content Toggle raw display
$89$ \( T^{8} + 434 T^{6} + \cdots + 69488896 \) Copy content Toggle raw display
$97$ \( T^{8} + 294 T^{6} + 7396 T^{4} + \cdots + 123904 \) Copy content Toggle raw display
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