Properties

Label 1666.2.b.a
Level $1666$
Weight $2$
Character orbit 1666.b
Analytic conductor $13.303$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1666,2,Mod(883,1666)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1666.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1666.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: \(13.3030769767\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 238)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1666\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.00000i
1.00000i
−1.00000 3.00000i 1.00000 0 3.00000i 0 −1.00000 −6.00000 0
883.2 −1.00000 3.00000i 1.00000 0 3.00000i 0 −1.00000 −6.00000 0
\(n\): e.g. 2-40 or 990-1000
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 1666.2.b.a 2
7.b odd 2 1 1666.2.b.b 2
7.d odd 6 2 238.2.j.a 4
17.b even 2 1 inner 1666.2.b.a 2
119.d odd 2 1 1666.2.b.b 2
119.h odd 6 2 238.2.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.j.a 4 7.d odd 6 2
238.2.j.a 4 119.h odd 6 2
1666.2.b.a 2 1.a even 1 1 trivial
1666.2.b.a 2 17.b even 2 1 inner
1666.2.b.b 2 7.b odd 2 1
1666.2.b.b 2 119.d odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( T^{2} + 49 \) Copy content Toggle raw display
$83$ \( (T - 16)^{2} \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
show more
show less