Properties

Label 3330.2.d.b
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
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: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} - q^{4} + (i - 2) q^{5} + 2 i q^{7} + i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{4} + (i - 2) q^{5} + 2 i q^{7} + i q^{8} + (2 i + 1) q^{10} - 4 q^{11} + 6 i q^{13} + 2 q^{14} + q^{16} + 6 i q^{17} - 2 q^{19} + ( - i + 2) q^{20} + 4 i q^{22} - 4 i q^{23} + ( - 4 i + 3) q^{25} + 6 q^{26} - 2 i q^{28} - 8 q^{29} - i q^{32} + 6 q^{34} + ( - 4 i - 2) q^{35} - i q^{37} + 2 i q^{38} + ( - 2 i - 1) q^{40} + 2 q^{41} + 12 i q^{43} + 4 q^{44} - 4 q^{46} - 6 i q^{47} + 3 q^{49} + ( - 3 i - 4) q^{50} - 6 i q^{52} - 14 i q^{53} + ( - 4 i + 8) q^{55} - 2 q^{56} + 8 i q^{58} + 14 q^{59} - 12 q^{61} - q^{64} + ( - 12 i - 6) q^{65} - 4 i q^{67} - 6 i q^{68} + (2 i - 4) q^{70} - 16 i q^{73} - q^{74} + 2 q^{76} - 8 i q^{77} + 8 q^{79} + (i - 2) q^{80} - 2 i q^{82} + 16 i q^{83} + ( - 12 i - 6) q^{85} + 12 q^{86} - 4 i q^{88} - 10 q^{89} - 12 q^{91} + 4 i q^{92} - 6 q^{94} + ( - 2 i + 4) q^{95} - 6 i q^{97} - 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{5} + 2 q^{10} - 8 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{20} + 6 q^{25} + 12 q^{26} - 16 q^{29} + 12 q^{34} - 4 q^{35} - 2 q^{40} + 4 q^{41} + 8 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{50} + 16 q^{55} - 4 q^{56} + 28 q^{59} - 24 q^{61} - 2 q^{64} - 12 q^{65} - 8 q^{70} - 2 q^{74} + 4 q^{76} + 16 q^{79} - 4 q^{80} - 12 q^{85} + 24 q^{86} - 20 q^{89} - 24 q^{91} - 12 q^{94} + 8 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 2.00000i 1.00000i 0 1.00000 + 2.00000i
1999.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 2.00000i 1.00000i 0 1.00000 2.00000i
\(n\): 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.b 2
3.b odd 2 1 1110.2.d.d 2
5.b even 2 1 inner 3330.2.d.b 2
15.d odd 2 1 1110.2.d.d 2
15.e even 4 1 5550.2.a.p 1
15.e even 4 1 5550.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.d 2 3.b odd 2 1
1110.2.d.d 2 15.d odd 2 1
3330.2.d.b 2 1.a even 1 1 trivial
3330.2.d.b 2 5.b even 2 1 inner
5550.2.a.p 1 15.e even 4 1
5550.2.a.bc 1 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 \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) Copy content Toggle raw display
\( T_{29} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 196 \) Copy content Toggle raw display
$59$ \( (T - 14)^{2} \) Copy content Toggle raw display
$61$ \( (T + 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 36 \) Copy content Toggle raw display
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