Properties

Label 1666.2.a.y
Level $1666$
Weight $2$
Character orbit 1666.a
Self dual yes
Analytic conductor $13.303$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.4352.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \) 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
−1.74912
2.68554
−1.27133
0.334904
1.00000 −2.22274 1.00000 3.74912 −2.22274 0 1.00000 1.94059 3.74912
1.2 1.00000 0.887611 1.00000 −0.685544 0.887611 0 1.00000 −2.21215 −0.685544
1.3 1.00000 2.52660 1.00000 3.27133 2.52660 0 1.00000 3.38372 3.27133
1.4 1.00000 2.80853 1.00000 1.66510 2.80853 0 1.00000 4.88784 1.66510
\(n\): e.g. 2-40 or 990-1000
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.y yes 4
7.b odd 2 1 1666.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1666.2.a.x 4 7.b odd 2 1
1666.2.a.y yes 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{4} - 4T_{3}^{3} - 2T_{3}^{2} + 20T_{3} - 14 \) Copy content Toggle raw display
\( T_{5}^{4} - 8T_{5}^{3} + 18T_{5}^{2} - 4T_{5} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} - 2 T^{2} + 20 T - 14 \) Copy content Toggle raw display
$5$ \( T^{4} - 8 T^{3} + 18 T^{2} - 4 T - 14 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 10 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} - 20 T^{2} + 208 T - 92 \) Copy content Toggle raw display
$17$ \( (T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} - 20 T^{2} + 272 T - 508 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 20 T^{2} + 80 T - 56 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 74 T^{2} - 260 T - 14 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} - 140 T^{2} + \cdots + 4424 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 202 T^{2} + \cdots + 866 \) Copy content Toggle raw display
$41$ \( T^{4} - 20 T^{3} + 132 T^{2} + \cdots + 328 \) Copy content Toggle raw display
$43$ \( T^{4} - 56 T^{2} - 96 T + 32 \) Copy content Toggle raw display
$47$ \( T^{4} - 92 T^{2} + 256 T + 196 \) Copy content Toggle raw display
$53$ \( T^{4} - 120 T^{2} + 32 T + 3104 \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + 4 T^{2} + \cdots - 1148 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 102 T^{2} - 300 T + 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 128 T^{2} - 320 T + 1168 \) Copy content Toggle raw display
$71$ \( T^{4} - 152 T^{2} + 96 T + 1568 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 192 T^{2} + \cdots + 272 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} - 80 T^{2} + \cdots - 8128 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} - 36 T^{2} + 80 T + 196 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} - 244 T^{2} + \cdots + 8804 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} - 324 T^{2} + \cdots + 9352 \) Copy content Toggle raw display
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