Properties

Label 3380.2.a.n
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
Defining polynomial: \(x^{3} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + q^{5} + \beta_{2} q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + q^{5} + \beta_{2} q^{7} + ( 1 + \beta_{2} ) q^{9} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + \beta_{1} q^{15} + ( 3 \beta_{1} - \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} ) q^{21} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{23} + q^{25} + 2 q^{27} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} ) q^{31} + ( -2 + 4 \beta_{1} - \beta_{2} ) q^{33} + \beta_{2} q^{35} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -2 - 3 \beta_{1} ) q^{43} + ( 1 + \beta_{2} ) q^{45} + ( 4 + 4 \beta_{1} - \beta_{2} ) q^{47} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 - \beta_{1} + \beta_{2} ) q^{55} + ( 10 - 2 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -8 + \beta_{1} - \beta_{2} ) q^{59} + ( -2 - 3 \beta_{2} ) q^{61} + ( 8 + 2 \beta_{1} - \beta_{2} ) q^{63} + ( -4 + 2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 6 \beta_{1} - \beta_{2} ) q^{69} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{73} + \beta_{1} q^{75} + 6 q^{77} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{81} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( -6 - 2 \beta_{2} ) q^{87} + ( 8 - 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -6 - \beta_{2} ) q^{93} + ( 3 \beta_{1} - \beta_{2} ) q^{95} + ( 6 - 2 \beta_{2} ) q^{97} + ( 8 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{5} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{5} + 3q^{9} + 6q^{11} + 6q^{21} + 6q^{23} + 3q^{25} + 6q^{27} - 6q^{29} + 6q^{31} - 6q^{33} + 12q^{37} - 6q^{41} - 6q^{43} + 3q^{45} + 12q^{47} + 3q^{49} - 6q^{53} + 6q^{55} + 30q^{57} - 24q^{59} - 6q^{61} + 24q^{63} - 12q^{67} + 24q^{73} + 18q^{77} - 12q^{79} - 9q^{81} + 12q^{83} - 18q^{87} + 24q^{89} - 18q^{93} + 18q^{97} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
−2.26180
−0.339877
2.60168
0 −2.26180 0 1.00000 0 1.11575 0 2.11575 0
1.2 0 −0.339877 0 1.00000 0 −3.88448 0 −2.88448 0
1.3 0 2.60168 0 1.00000 0 2.76873 0 3.76873 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.n 3
13.b even 2 1 3380.2.a.m 3
13.d odd 4 2 260.2.f.a 6
39.f even 4 2 2340.2.c.d 6
52.f even 4 2 1040.2.k.c 6
65.f even 4 2 1300.2.d.d 6
65.g odd 4 2 1300.2.f.e 6
65.k even 4 2 1300.2.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 13.d odd 4 2
1040.2.k.c 6 52.f even 4 2
1300.2.d.c 6 65.k even 4 2
1300.2.d.d 6 65.f even 4 2
1300.2.f.e 6 65.g odd 4 2
2340.2.c.d 6 39.f even 4 2
3380.2.a.m 3 13.b even 2 1
3380.2.a.n 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{3} - 6 T_{3} - 2 \)
\( T_{7}^{3} - 12 T_{7} + 12 \)
\( T_{19}^{3} - 48 T_{19} + 114 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -2 - 6 T + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 12 - 12 T + T^{3} \)
$11$ \( 18 - 6 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( T^{3} \)
$19$ \( 114 - 48 T + T^{3} \)
$23$ \( 174 - 30 T - 6 T^{2} + T^{3} \)
$29$ \( -84 - 12 T + 6 T^{2} + T^{3} \)
$31$ \( 66 - 12 T - 6 T^{2} + T^{3} \)
$37$ \( 252 - 12 T^{2} + T^{3} \)
$41$ \( -72 - 36 T + 6 T^{2} + T^{3} \)
$43$ \( -46 - 42 T + 6 T^{2} + T^{3} \)
$47$ \( 468 - 36 T - 12 T^{2} + T^{3} \)
$53$ \( 24 - 84 T + 6 T^{2} + T^{3} \)
$59$ \( 414 + 180 T + 24 T^{2} + T^{3} \)
$61$ \( -532 - 96 T + 6 T^{2} + T^{3} \)
$67$ \( -588 - 48 T + 12 T^{2} + T^{3} \)
$71$ \( -702 - 216 T + T^{3} \)
$73$ \( -252 + 144 T - 24 T^{2} + T^{3} \)
$79$ \( -1696 - 144 T + 12 T^{2} + T^{3} \)
$83$ \( 252 - 12 T^{2} + T^{3} \)
$89$ \( 2016 - 24 T^{2} + T^{3} \)
$97$ \( -24 + 60 T - 18 T^{2} + T^{3} \)
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