Properties

Label 5070.2.a.bp
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + ( -2 + \beta_{1} + \beta_{2} ) q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + ( -2 + \beta_{1} + \beta_{2} ) q^{7} - q^{8} + q^{9} - q^{10} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{11} + q^{12} + ( 2 - \beta_{1} - \beta_{2} ) q^{14} + q^{15} + q^{16} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{17} - q^{18} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{19} + q^{20} + ( -2 + \beta_{1} + \beta_{2} ) q^{21} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( \beta_{1} - 2 \beta_{2} ) q^{23} - q^{24} + q^{25} + q^{27} + ( -2 + \beta_{1} + \beta_{2} ) q^{28} + ( 4 - 7 \beta_{1} + 5 \beta_{2} ) q^{29} - q^{30} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{31} - q^{32} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{33} + ( 4 - 3 \beta_{1} + 2 \beta_{2} ) q^{34} + ( -2 + \beta_{1} + \beta_{2} ) q^{35} + q^{36} + ( -4 - 5 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{38} - q^{40} + ( -7 + 3 \beta_{1} - \beta_{2} ) q^{41} + ( 2 - \beta_{1} - \beta_{2} ) q^{42} + ( 4 - 4 \beta_{1} - \beta_{2} ) q^{43} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{44} + q^{45} + ( -\beta_{1} + 2 \beta_{2} ) q^{46} + ( -5 + 6 \beta_{1} - 2 \beta_{2} ) q^{47} + q^{48} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{49} - q^{50} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{51} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{53} - q^{54} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{55} + ( 2 - \beta_{1} - \beta_{2} ) q^{56} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{57} + ( -4 + 7 \beta_{1} - 5 \beta_{2} ) q^{58} + ( -4 + 4 \beta_{1} - \beta_{2} ) q^{59} + q^{60} + ( 2 + \beta_{2} ) q^{61} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{62} + ( -2 + \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{66} + ( -4 - 3 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{68} + ( \beta_{1} - 2 \beta_{2} ) q^{69} + ( 2 - \beta_{1} - \beta_{2} ) q^{70} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{71} - q^{72} + ( -6 + 7 \beta_{1} - 8 \beta_{2} ) q^{73} + ( 4 + 5 \beta_{1} - 3 \beta_{2} ) q^{74} + q^{75} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{76} + ( -6 + 6 \beta_{1} - 5 \beta_{2} ) q^{77} + ( 9 - 3 \beta_{1} + 7 \beta_{2} ) q^{79} + q^{80} + q^{81} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{82} + ( 3 - 8 \beta_{1} + 7 \beta_{2} ) q^{83} + ( -2 + \beta_{1} + \beta_{2} ) q^{84} + ( -4 + 3 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -4 + 4 \beta_{1} + \beta_{2} ) q^{86} + ( 4 - 7 \beta_{1} + 5 \beta_{2} ) q^{87} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{88} + ( -3 \beta_{1} - 6 \beta_{2} ) q^{89} - q^{90} + ( \beta_{1} - 2 \beta_{2} ) q^{92} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{93} + ( 5 - 6 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{95} - q^{96} + ( -6 - \beta_{1} + 5 \beta_{2} ) q^{97} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{98} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 3q^{12} + 6q^{14} + 3q^{15} + 3q^{16} - 7q^{17} - 3q^{18} + q^{19} + 3q^{20} - 6q^{21} + 3q^{23} - 3q^{24} + 3q^{25} + 3q^{27} - 6q^{28} - 3q^{30} - 16q^{31} - 3q^{32} + 7q^{34} - 6q^{35} + 3q^{36} - 20q^{37} - q^{38} - 3q^{40} - 17q^{41} + 6q^{42} + 9q^{43} + 3q^{45} - 3q^{46} - 7q^{47} + 3q^{48} + 5q^{49} - 3q^{50} - 7q^{51} + 3q^{53} - 3q^{54} + 6q^{56} + q^{57} - 7q^{59} + 3q^{60} + 5q^{61} + 16q^{62} - 6q^{63} + 3q^{64} - 17q^{67} - 7q^{68} + 3q^{69} + 6q^{70} + 7q^{71} - 3q^{72} - 3q^{73} + 20q^{74} + 3q^{75} + q^{76} - 7q^{77} + 17q^{79} + 3q^{80} + 3q^{81} + 17q^{82} - 6q^{83} - 6q^{84} - 7q^{85} - 9q^{86} + 3q^{89} - 3q^{90} + 3q^{92} - 16q^{93} + 7q^{94} + q^{95} - 3q^{96} - 24q^{97} - 5q^{98} + O(q^{100}) \)

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
−1.24698
0.445042
1.80194
−1.00000 1.00000 1.00000 1.00000 −1.00000 −3.69202 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −3.35690 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 1.04892 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 5070.2.a.bp 3
13.b even 2 1 5070.2.a.bw yes 3
13.d odd 4 2 5070.2.b.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bp 3 1.a even 1 1 trivial
5070.2.a.bw yes 3 13.b even 2 1
5070.2.b.z 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{3} + 6 T_{7}^{2} + 5 T_{7} - 13 \)
\( T_{11}^{3} - 7 T_{11} - 7 \)
\( T_{17}^{3} + 7 T_{17}^{2} - 7 \)
\( T_{31}^{3} + 16 T_{31}^{2} + 69 T_{31} + 83 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -13 + 5 T + 6 T^{2} + T^{3} \)
$11$ \( -7 - 7 T + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -7 + 7 T^{2} + T^{3} \)
$19$ \( -13 - 16 T - T^{2} + T^{3} \)
$23$ \( -1 - 4 T - 3 T^{2} + T^{3} \)
$29$ \( -203 - 91 T + T^{3} \)
$31$ \( 83 + 69 T + 16 T^{2} + T^{3} \)
$37$ \( -97 + 89 T + 20 T^{2} + T^{3} \)
$41$ \( 113 + 80 T + 17 T^{2} + T^{3} \)
$43$ \( 169 - 22 T - 9 T^{2} + T^{3} \)
$47$ \( 49 - 49 T + 7 T^{2} + T^{3} \)
$53$ \( 13 - 18 T - 3 T^{2} + T^{3} \)
$59$ \( -7 - 14 T + 7 T^{2} + T^{3} \)
$61$ \( -1 + 6 T - 5 T^{2} + T^{3} \)
$67$ \( 71 + 80 T + 17 T^{2} + T^{3} \)
$71$ \( 49 - 7 T^{2} + T^{3} \)
$73$ \( -433 - 130 T + 3 T^{2} + T^{3} \)
$79$ \( 587 + 10 T - 17 T^{2} + T^{3} \)
$83$ \( -349 - 121 T + 6 T^{2} + T^{3} \)
$89$ \( 783 - 144 T - 3 T^{2} + T^{3} \)
$97$ \( 169 + 143 T + 24 T^{2} + T^{3} \)
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