Properties

Label 3380.2.f.e
Level $3380$
Weight $2$
Character orbit 3380.f
Analytic conductor $26.989$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,2,Mod(3041,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3380.3041");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.f (of order \(2\), degree \(1\), not 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: \(26.9894358832\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
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.

\(f(q)\) \(=\) \( q + q^{3} - i q^{5} - i q^{7} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - i q^{5} - i q^{7} - 2 q^{9} + 3 i q^{11} - i q^{15} + 3 q^{17} + 7 i q^{19} - i q^{21} + 3 q^{23} - q^{25} - 5 q^{27} + 3 q^{29} + 4 i q^{31} + 3 i q^{33} - q^{35} - 7 i q^{37} + 9 i q^{41} - 11 q^{43} + 2 i q^{45} + 6 q^{49} + 3 q^{51} - 6 q^{53} + 3 q^{55} + 7 i q^{57} - 3 i q^{59} + 11 q^{61} + 2 i q^{63} + 7 i q^{67} + 3 q^{69} + 3 i q^{71} + 2 i q^{73} - q^{75} + 3 q^{77} + 8 q^{79} + q^{81} + 12 i q^{83} - 3 i q^{85} + 3 q^{87} + 15 i q^{89} + 4 i q^{93} + 7 q^{95} + 7 i q^{97} - 6 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{9} + 6 q^{17} + 6 q^{23} - 2 q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{35} - 22 q^{43} + 12 q^{49} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 22 q^{61} + 6 q^{69} - 2 q^{75} + 6 q^{77} + 16 q^{79} + 2 q^{81} + 6 q^{87} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
1.00000i
1.00000i
0 1.00000 0 1.00000i 0 1.00000i 0 −2.00000 0
3041.2 0 1.00000 0 1.00000i 0 1.00000i 0 −2.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.e 2
13.b even 2 1 inner 3380.2.f.e 2
13.d odd 4 1 3380.2.a.g 1
13.d odd 4 1 3380.2.a.h 1
13.f odd 12 2 260.2.i.b 2
39.k even 12 2 2340.2.q.b 2
52.l even 12 2 1040.2.q.j 2
65.o even 12 2 1300.2.bb.a 4
65.s odd 12 2 1300.2.i.e 2
65.t even 12 2 1300.2.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.b 2 13.f odd 12 2
1040.2.q.j 2 52.l even 12 2
1300.2.i.e 2 65.s odd 12 2
1300.2.bb.a 4 65.o even 12 2
1300.2.bb.a 4 65.t even 12 2
2340.2.q.b 2 39.k even 12 2
3380.2.a.g 1 13.d odd 4 1
3380.2.a.h 1 13.d odd 4 1
3380.2.f.e 2 1.a even 1 1 trivial
3380.2.f.e 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{19}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 49 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 81 \) Copy content Toggle raw display
$43$ \( (T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T - 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 49 \) Copy content Toggle raw display
$71$ \( T^{2} + 9 \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 225 \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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