Properties

Label 3360.2.j.a
Level $3360$
Weight $2$
Character orbit 3360.j
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -2 + i ) q^{5} -i q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -2 + i ) q^{5} -i q^{7} + q^{9} -6 q^{13} + ( 2 - i ) q^{15} -2 i q^{17} -4 i q^{19} + i q^{21} + 4 i q^{23} + ( 3 - 4 i ) q^{25} - q^{27} -6 i q^{29} + 8 q^{31} + ( 1 + 2 i ) q^{35} -2 q^{37} + 6 q^{39} -8 q^{41} + 6 q^{43} + ( -2 + i ) q^{45} + 2 i q^{47} - q^{49} + 2 i q^{51} -6 q^{53} + 4 i q^{57} + 6 i q^{59} + 10 i q^{61} -i q^{63} + ( 12 - 6 i ) q^{65} + 2 q^{67} -4 i q^{69} + 8 q^{71} + 6 i q^{73} + ( -3 + 4 i ) q^{75} -10 q^{79} + q^{81} -4 q^{83} + ( 2 + 4 i ) q^{85} + 6 i q^{87} + 6 i q^{91} -8 q^{93} + ( 4 + 8 i ) q^{95} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 12 q^{13} + 4 q^{15} + 6 q^{25} - 2 q^{27} + 16 q^{31} + 2 q^{35} - 4 q^{37} + 12 q^{39} - 16 q^{41} + 12 q^{43} - 4 q^{45} - 2 q^{49} - 12 q^{53} + 24 q^{65} + 4 q^{67} + 16 q^{71} - 6 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} - 16 q^{93} + 8 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 −1.00000 0 −2.00000 1.00000i 0 1.00000i 0 1.00000 0
1009.2 0 −1.00000 0 −2.00000 + 1.00000i 0 1.00000i 0 1.00000 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.j.a 2
4.b odd 2 1 840.2.j.b 2
5.b even 2 1 3360.2.j.d 2
8.b even 2 1 3360.2.j.d 2
8.d odd 2 1 840.2.j.c yes 2
20.d odd 2 1 840.2.j.c yes 2
40.e odd 2 1 840.2.j.b 2
40.f even 2 1 inner 3360.2.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.j.b 2 4.b odd 2 1
840.2.j.b 2 40.e odd 2 1
840.2.j.c yes 2 8.d odd 2 1
840.2.j.c yes 2 20.d odd 2 1
3360.2.j.a 2 1.a even 1 1 trivial
3360.2.j.a 2 40.f even 2 1 inner
3360.2.j.d 2 5.b even 2 1
3360.2.j.d 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11} \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( 100 + T^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( 4 + T^{2} \)
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