Properties

Label 1666.2.a.z
Level $1666$
Weight $2$
Character orbit 1666.a
Self dual yes
Analytic conductor $13.303$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1666,2,Mod(1,1666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1666, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1666.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1666.a (trivial)

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: yes
Analytic conductor: \(13.3030769767\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23949216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 13x^{3} - 2x^{2} + 24x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 238)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 13x^{3} - 2x^{2} + 24x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 9\nu^{2} - 20\nu - 4 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 15\nu^{2} - 20\nu + 26 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{4} + \nu^{3} + 12\nu^{2} - 7\nu - 14 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{4} - 11\beta_{3} + 13\beta_{2} + 2\beta _1 + 47 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.40920
1.26770
0.171872
−1.72247
−3.12631
1.00000 −3.40920 1.00000 −0.364940 −3.40920 0 1.00000 8.62267 −0.364940
1.2 1.00000 −1.26770 1.00000 −2.80049 −1.26770 0 1.00000 −1.39293 −2.80049
1.3 1.00000 −0.171872 1.00000 3.68841 −0.171872 0 1.00000 −2.97046 3.68841
1.4 1.00000 1.72247 1.00000 2.42127 1.72247 0 1.00000 −0.0331027 2.42127
1.5 1.00000 3.12631 1.00000 −3.94425 3.12631 0 1.00000 6.77382 −3.94425
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1666.2.a.z 5
7.b odd 2 1 1666.2.a.ba 5
7.c even 3 2 238.2.e.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.e.f 10 7.c even 3 2
1666.2.a.z 5 1.a even 1 1 trivial
1666.2.a.ba 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1666))\):

\( T_{3}^{5} - 13T_{3}^{3} + 2T_{3}^{2} + 24T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{5} + T_{5}^{4} - 21T_{5}^{3} - 15T_{5}^{2} + 96T_{5} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 13 T^{3} + 2 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$5$ \( T^{5} + T^{4} - 21 T^{3} - 15 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 10 T^{4} + 27 T^{3} - 60 T + 36 \) Copy content Toggle raw display
$13$ \( T^{5} - 2 T^{4} - 35 T^{3} + 16 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$17$ \( (T + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{5} - 3 T^{4} - 33 T^{3} + 91 T^{2} + \cdots - 108 \) Copy content Toggle raw display
$23$ \( T^{5} - 3 T^{4} - 103 T^{3} + \cdots - 1284 \) Copy content Toggle raw display
$29$ \( T^{5} - 12 T^{4} - 16 T^{3} + 324 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} - 48 T^{3} - 92 T^{2} + \cdots + 432 \) Copy content Toggle raw display
$37$ \( T^{5} - 9 T^{4} + 11 T^{3} + 55 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{5} - 14 T^{4} + 16 T^{3} + \cdots + 1008 \) Copy content Toggle raw display
$43$ \( T^{5} - T^{4} - 93 T^{3} - 323 T^{2} + \cdots + 432 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} - 84 T^{3} - 228 T^{2} + \cdots + 432 \) Copy content Toggle raw display
$53$ \( T^{5} - 20 T^{4} + 9 T^{3} + \cdots - 6264 \) Copy content Toggle raw display
$59$ \( T^{5} - 3 T^{4} - 253 T^{3} + \cdots - 53376 \) Copy content Toggle raw display
$61$ \( T^{5} - 16 T^{4} - 120 T^{3} + \cdots + 31104 \) Copy content Toggle raw display
$67$ \( T^{5} - 15 T^{4} - 37 T^{3} + \cdots - 6988 \) Copy content Toggle raw display
$71$ \( T^{5} - 7 T^{4} - 96 T^{3} + \cdots - 11259 \) Copy content Toggle raw display
$73$ \( T^{5} + 34 T^{4} + 400 T^{3} + \cdots - 7856 \) Copy content Toggle raw display
$79$ \( T^{5} + 12 T^{4} - 33 T^{3} + \cdots + 8748 \) Copy content Toggle raw display
$83$ \( T^{5} - 16 T^{4} - 120 T^{3} + \cdots + 48384 \) Copy content Toggle raw display
$89$ \( T^{5} - T^{4} - 294 T^{3} + \cdots - 29097 \) Copy content Toggle raw display
$97$ \( T^{5} + 18 T^{4} + 12 T^{3} + \cdots + 8064 \) Copy content Toggle raw display
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