Properties

Label 3360.2.t.f
Level $3360$
Weight $2$
Character orbit 3360.t
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.t (of order \(2\), degree \(1\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} + 6 q^{11} + 6 i q^{13} + ( 2 - i ) q^{15} -2 q^{19} - q^{21} -8 i q^{23} + ( -3 + 4 i ) q^{25} + i q^{27} + 10 q^{29} + 2 q^{31} -6 i q^{33} + ( 2 - i ) q^{35} -8 i q^{37} + 6 q^{39} -6 q^{41} + 12 i q^{43} + ( -1 - 2 i ) q^{45} + 8 i q^{47} - q^{49} + 10 i q^{53} + ( 6 + 12 i ) q^{55} + 2 i q^{57} -4 q^{59} -2 q^{61} + i q^{63} + ( -12 + 6 i ) q^{65} -8 i q^{67} -8 q^{69} + 10 q^{71} + 6 i q^{73} + ( 4 + 3 i ) q^{75} -6 i q^{77} + 8 q^{79} + q^{81} -10 i q^{87} -6 q^{89} + 6 q^{91} -2 i q^{93} + ( -2 - 4 i ) q^{95} -10 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{9} + 12 q^{11} + 4 q^{15} - 4 q^{19} - 2 q^{21} - 6 q^{25} + 20 q^{29} + 4 q^{31} + 4 q^{35} + 12 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{49} + 12 q^{55} - 8 q^{59} - 4 q^{61} - 24 q^{65} - 16 q^{69} + 20 q^{71} + 8 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{89} + 12 q^{91} - 4 q^{95} - 12 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 1.00000i 0 −1.00000 0
2689.2 0 1.00000i 0 1.00000 2.00000i 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
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 3360.2.t.f yes 2
4.b odd 2 1 3360.2.t.c 2
5.b even 2 1 inner 3360.2.t.f yes 2
20.d odd 2 1 3360.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.t.c 2 4.b odd 2 1
3360.2.t.c 2 20.d odd 2 1
3360.2.t.f yes 2 1.a even 1 1 trivial
3360.2.t.f yes 2 5.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 \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11} - 6 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( -10 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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