Properties

Label 3380.2.a.i
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 1.00000 0 1.00000 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.a.i 1
13.b even 2 1 3380.2.a.f 1
13.d odd 4 2 3380.2.f.d 2
13.e even 6 2 260.2.i.a 2
39.h odd 6 2 2340.2.q.f 2
52.i odd 6 2 1040.2.q.i 2
65.l even 6 2 1300.2.i.d 2
65.r odd 12 4 1300.2.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 13.e even 6 2
1040.2.q.i 2 52.i odd 6 2
1300.2.i.d 2 65.l even 6 2
1300.2.bb.b 4 65.r odd 12 4
2340.2.q.f 2 39.h odd 6 2
3380.2.a.f 1 13.b even 2 1
3380.2.a.i 1 1.a even 1 1 trivial
3380.2.f.d 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{19} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T - 9 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T + 3 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 9 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T + 9 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 3 \) Copy content Toggle raw display
$97$ \( T + 17 \) Copy content Toggle raw display
show more
show less