Properties

Label 3330.2.d.k
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
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] = [3330,2,Mod(1999,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1999");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.d (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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{8}^{2} q^{2} - q^{4} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{2} q^{2} - q^{4} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{8} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{10} + ( - \zeta_{8}^{3} + \zeta_{8} + 5) q^{11} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + 3 \zeta_{8}) q^{13} + ( - \zeta_{8}^{3} + \zeta_{8} + 1) q^{14} + q^{16} + ( - \zeta_{8}^{3} + 5 \zeta_{8}^{2} - \zeta_{8}) q^{17} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 5) q^{19} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{20} + ( - \zeta_{8}^{3} - 5 \zeta_{8}^{2} - \zeta_{8}) q^{22} - 3 \zeta_{8}^{2} q^{23} + ( - 3 \zeta_{8}^{2} - 4) q^{25} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} + 1) q^{26} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8}) q^{28} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{29} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 2) q^{31} - \zeta_{8}^{2} q^{32} + (\zeta_{8}^{3} - \zeta_{8} + 5) q^{34} + (\zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8} - 3) q^{35} + \zeta_{8}^{2} q^{37} + (2 \zeta_{8}^{3} - 5 \zeta_{8}^{2} + 2 \zeta_{8}) q^{38} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{40} + (\zeta_{8}^{3} - \zeta_{8} + 8) q^{41} + (6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 6 \zeta_{8}) q^{43} + (\zeta_{8}^{3} - \zeta_{8} - 5) q^{44} - 3 q^{46} + ( - 7 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 7 \zeta_{8}) q^{47} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 4) q^{49} + (4 \zeta_{8}^{2} - 3) q^{50} + ( - 3 \zeta_{8}^{3} - \zeta_{8}^{2} - 3 \zeta_{8}) q^{52} + (2 \zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8}) q^{53} + (10 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 5 \zeta_{8} - 1) q^{55} + (\zeta_{8}^{3} - \zeta_{8} - 1) q^{56} + ( - 5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{58} + (5 \zeta_{8}^{3} - 5 \zeta_{8} - 6) q^{59} + (6 \zeta_{8}^{3} - 6 \zeta_{8} + 2) q^{61} + (4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{62} - q^{64} + (\zeta_{8}^{3} - 3 \zeta_{8}^{2} - 2 \zeta_{8} - 9) q^{65} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{67} + (\zeta_{8}^{3} - 5 \zeta_{8}^{2} + \zeta_{8}) q^{68} + (2 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{70} + ( - \zeta_{8}^{3} + \zeta_{8} + 8) q^{71} + ( - 2 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 2 \zeta_{8}) q^{73} + q^{74} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 5) q^{76} + (6 \zeta_{8}^{3} + 7 \zeta_{8}^{2} + 6 \zeta_{8}) q^{77} + (3 \zeta_{8}^{3} - 3 \zeta_{8} - 8) q^{79} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{80} + (\zeta_{8}^{3} - 8 \zeta_{8}^{2} + \zeta_{8}) q^{82} + ( - 7 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 7 \zeta_{8}) q^{83} + (5 \zeta_{8}^{3} + \zeta_{8}^{2} - 10 \zeta_{8} + 3) q^{85} + ( - 6 \zeta_{8}^{3} + 6 \zeta_{8} + 2) q^{86} + (\zeta_{8}^{3} + 5 \zeta_{8}^{2} + \zeta_{8}) q^{88} + ( - 7 \zeta_{8}^{3} + 7 \zeta_{8} - 7) q^{89} + (4 \zeta_{8}^{3} - 4 \zeta_{8} - 7) q^{91} + 3 \zeta_{8}^{2} q^{92} + (7 \zeta_{8}^{3} - 7 \zeta_{8} - 2) q^{94} + (10 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 5 \zeta_{8} + 2) q^{95} + (9 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 9 \zeta_{8}) q^{97} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 20 q^{11} + 4 q^{14} + 4 q^{16} + 20 q^{19} - 16 q^{25} + 4 q^{26} + 8 q^{31} + 20 q^{34} - 12 q^{35} + 32 q^{41} - 20 q^{44} - 12 q^{46} + 16 q^{49} - 12 q^{50} - 4 q^{55} - 4 q^{56} - 24 q^{59} + 8 q^{61} - 4 q^{64} - 36 q^{65} - 4 q^{70} + 32 q^{71} + 4 q^{74} - 20 q^{76} - 32 q^{79} + 12 q^{85} + 8 q^{86} - 28 q^{89} - 28 q^{91} - 8 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
1.00000i 0 −1.00000 −0.707107 + 2.12132i 0 2.41421i 1.00000i 0 2.12132 + 0.707107i
1999.2 1.00000i 0 −1.00000 0.707107 2.12132i 0 0.414214i 1.00000i 0 −2.12132 0.707107i
1999.3 1.00000i 0 −1.00000 −0.707107 2.12132i 0 2.41421i 1.00000i 0 2.12132 0.707107i
1999.4 1.00000i 0 −1.00000 0.707107 + 2.12132i 0 0.414214i 1.00000i 0 −2.12132 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.d.k 4
3.b odd 2 1 1110.2.d.g 4
5.b even 2 1 inner 3330.2.d.k 4
15.d odd 2 1 1110.2.d.g 4
15.e even 4 1 5550.2.a.bs 2
15.e even 4 1 5550.2.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.g 4 3.b odd 2 1
1110.2.d.g 4 15.d odd 2 1
3330.2.d.k 4 1.a even 1 1 trivial
3330.2.d.k 4 5.b even 2 1 inner
5550.2.a.bs 2 15.e even 4 1
5550.2.a.cb 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{4} + 6T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 10T_{11} + 23 \) Copy content Toggle raw display
\( T_{17}^{4} + 54T_{17}^{2} + 529 \) Copy content Toggle raw display
\( T_{29}^{2} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 10 T + 23)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$17$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 17)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$47$ \( T^{4} + 204T^{2} + 8836 \) Copy content Toggle raw display
$53$ \( T^{4} + 18T^{2} + 49 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T - 14)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T + 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 66T^{2} + 289 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16 T + 46)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 214T^{2} + 7921 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T - 49)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 396 T^{2} + 15876 \) Copy content Toggle raw display
show more
show less