Properties

Label 3380.2.a.p
Level $3380$
Weight $2$
Character orbit 3380.a
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} - q^{5} + ( -2 - \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} - q^{5} + ( -2 - \beta_{2} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} ) q^{9} -\beta_{3} q^{11} + ( -1 - \beta_{2} ) q^{15} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{19} + ( -4 - 2 \beta_{2} + \beta_{3} ) q^{21} + ( -2 - \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{27} + ( -\beta_{1} + 2 \beta_{3} ) q^{29} + ( -2 - 4 \beta_{2} + 2 \beta_{3} ) q^{31} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{33} + ( 2 + \beta_{2} - \beta_{3} ) q^{35} + ( -6 - 3 \beta_{2} + \beta_{3} ) q^{37} + ( -6 + \beta_{3} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( -1 - \beta_{1} ) q^{45} -2 \beta_{3} q^{47} + ( 1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{49} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{51} + ( 1 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{53} + \beta_{3} q^{55} + ( 4 - \beta_{2} - 3 \beta_{3} ) q^{57} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -3 - \beta_{1} - 2 \beta_{3} ) q^{63} + ( -4 + \beta_{2} - 5 \beta_{3} ) q^{67} + ( -7 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 3 \beta_{3} ) q^{71} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{73} + ( 1 + \beta_{2} ) q^{75} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{77} + ( -1 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -2 - \beta_{1} + 4 \beta_{3} ) q^{81} + ( 8 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{85} + ( \beta_{1} - 3 \beta_{2} ) q^{87} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -12 - 2 \beta_{1} + 2 \beta_{3} ) q^{93} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{95} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{97} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} - 6q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} - 6q^{7} + 4q^{9} - 2q^{15} + 6q^{17} - 12q^{21} - 6q^{23} + 4q^{25} + 2q^{27} - 6q^{33} + 6q^{35} - 18q^{37} - 24q^{41} + 10q^{43} - 4q^{45} - 4q^{49} + 12q^{53} + 18q^{57} - 12q^{59} + 4q^{61} - 12q^{63} - 18q^{67} - 24q^{69} - 12q^{71} - 24q^{73} + 2q^{75} - 6q^{77} - 8q^{79} - 8q^{81} + 36q^{83} - 6q^{85} + 6q^{87} - 12q^{89} - 48q^{93} - 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 2 \beta_{1} + 4\)

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
1.21969
−1.49551
−0.219687
2.49551
0 −2.33225 0 −1.00000 0 −0.399804 0 2.43937 0
1.2 0 −0.0947876 0 −1.00000 0 0.826838 0 −2.99102 0
1.3 0 1.60020 0 −1.00000 0 −4.33225 0 −0.439374 0
1.4 0 2.82684 0 −1.00000 0 −2.09479 0 4.99102 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.p 4
13.b even 2 1 3380.2.a.q 4
13.d odd 4 2 3380.2.f.i 8
13.f odd 12 2 260.2.x.a 8
39.k even 12 2 2340.2.dj.d 8
52.l even 12 2 1040.2.da.c 8
65.o even 12 2 1300.2.ba.c 8
65.s odd 12 2 1300.2.y.b 8
65.t even 12 2 1300.2.ba.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a 8 13.f odd 12 2
1040.2.da.c 8 52.l even 12 2
1300.2.y.b 8 65.s odd 12 2
1300.2.ba.b 8 65.t even 12 2
1300.2.ba.c 8 65.o even 12 2
2340.2.dj.d 8 39.k even 12 2
3380.2.a.p 4 1.a even 1 1 trivial
3380.2.a.q 4 13.b even 2 1
3380.2.f.i 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\):

\( T_{3}^{4} - 2 T_{3}^{3} - 6 T_{3}^{2} + 10 T_{3} + 1 \)
\( T_{7}^{4} + 6 T_{7}^{3} + 6 T_{7}^{2} - 6 T_{7} - 3 \)
\( T_{19}^{4} - 30 T_{19}^{2} + 33 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 10 T - 6 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( -3 - 6 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( ( -3 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( -27 + 54 T - 18 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 33 - 30 T^{2} + T^{4} \)
$23$ \( -3 - 30 T - 18 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( -39 - 96 T - 42 T^{2} + T^{4} \)
$31$ \( 2112 - 96 T^{2} + T^{4} \)
$37$ \( -207 - 162 T + 66 T^{2} + 18 T^{3} + T^{4} \)
$41$ \( ( 33 + 12 T + T^{2} )^{2} \)
$43$ \( -2243 + 914 T - 66 T^{2} - 10 T^{3} + T^{4} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( -624 + 1920 T - 156 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( -747 - 612 T - 42 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( -971 + 644 T - 102 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 1041 - 810 T - 6 T^{2} + 18 T^{3} + T^{4} \)
$71$ \( -6723 - 2484 T - 162 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( -1584 + 156 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( -368 - 1504 T - 180 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 576 - 1440 T + 408 T^{2} - 36 T^{3} + T^{4} \)
$89$ \( 117 - 324 T - 42 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( -3603 - 1830 T - 186 T^{2} + 6 T^{3} + T^{4} \)
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