Properties

Label 5070.2.a.bx
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( -3 - \beta_{1} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( -3 - \beta_{1} ) q^{7} + q^{8} + q^{9} + q^{10} + ( -2 - \beta_{1} + \beta_{2} ) q^{11} + q^{12} + ( -3 - \beta_{1} ) q^{14} + q^{15} + q^{16} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{17} + q^{18} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{19} + q^{20} + ( -3 - \beta_{1} ) q^{21} + ( -2 - \beta_{1} + \beta_{2} ) q^{22} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{23} + q^{24} + q^{25} + q^{27} + ( -3 - \beta_{1} ) q^{28} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{29} + q^{30} + ( -6 + \beta_{1} - \beta_{2} ) q^{31} + q^{32} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{34} + ( -3 - \beta_{1} ) q^{35} + q^{36} + ( -1 - 5 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{38} + q^{40} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -3 - \beta_{1} ) q^{42} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -2 - \beta_{1} + \beta_{2} ) q^{44} + q^{45} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{46} + ( 3 - 8 \beta_{1} + 4 \beta_{2} ) q^{47} + q^{48} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{49} + q^{50} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{51} + ( -4 - 3 \beta_{1} + 2 \beta_{2} ) q^{53} + q^{54} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} + ( -3 - \beta_{1} ) q^{56} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{58} + ( -9 + 7 \beta_{1} - 9 \beta_{2} ) q^{59} + q^{60} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{61} + ( -6 + \beta_{1} - \beta_{2} ) q^{62} + ( -3 - \beta_{1} ) q^{63} + q^{64} + ( -2 - \beta_{1} + \beta_{2} ) q^{66} + ( -8 + 6 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{68} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{69} + ( -3 - \beta_{1} ) q^{70} + ( -4 - 4 \beta_{1} - 3 \beta_{2} ) q^{71} + q^{72} + ( -4 + 2 \beta_{1} - 11 \beta_{2} ) q^{73} + ( -1 - 5 \beta_{1} + 2 \beta_{2} ) q^{74} + q^{75} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 7 + 5 \beta_{1} - 3 \beta_{2} ) q^{77} -3 \beta_{1} q^{79} + q^{80} + q^{81} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -6 + 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -3 - \beta_{1} ) q^{84} + ( -4 + 4 \beta_{1} - 5 \beta_{2} ) q^{85} + ( -1 - 3 \beta_{1} - 3 \beta_{2} ) q^{86} + ( 3 - 3 \beta_{1} + 6 \beta_{2} ) q^{87} + ( -2 - \beta_{1} + \beta_{2} ) q^{88} + ( 2 + 2 \beta_{1} + 9 \beta_{2} ) q^{89} + q^{90} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{92} + ( -6 + \beta_{1} - \beta_{2} ) q^{93} + ( 3 - 8 \beta_{1} + 4 \beta_{2} ) q^{94} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{95} + q^{96} + ( 1 - 5 \beta_{1} - 8 \beta_{2} ) q^{97} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{98} + ( -2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} - 10q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} - 10q^{7} + 3q^{8} + 3q^{9} + 3q^{10} - 8q^{11} + 3q^{12} - 10q^{14} + 3q^{15} + 3q^{16} - 3q^{17} + 3q^{18} - 11q^{19} + 3q^{20} - 10q^{21} - 8q^{22} - 9q^{23} + 3q^{24} + 3q^{25} + 3q^{27} - 10q^{28} + 3q^{30} - 16q^{31} + 3q^{32} - 8q^{33} - 3q^{34} - 10q^{35} + 3q^{36} - 10q^{37} - 11q^{38} + 3q^{40} - 5q^{41} - 10q^{42} - 3q^{43} - 8q^{44} + 3q^{45} - 9q^{46} - 3q^{47} + 3q^{48} + 17q^{49} + 3q^{50} - 3q^{51} - 17q^{53} + 3q^{54} - 8q^{55} - 10q^{56} - 11q^{57} - 11q^{59} + 3q^{60} + 11q^{61} - 16q^{62} - 10q^{63} + 3q^{64} - 8q^{66} - 15q^{67} - 3q^{68} - 9q^{69} - 10q^{70} - 13q^{71} + 3q^{72} + q^{73} - 10q^{74} + 3q^{75} - 11q^{76} + 29q^{77} - 3q^{79} + 3q^{80} + 3q^{81} - 5q^{82} - 12q^{83} - 10q^{84} - 3q^{85} - 3q^{86} - 8q^{88} - q^{89} + 3q^{90} - 9q^{92} - 16q^{93} - 3q^{94} - 11q^{95} + 3q^{96} + 6q^{97} + 17q^{98} - 8q^{99} + O(q^{100}) \)

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
1.80194
0.445042
−1.24698
1.00000 1.00000 1.00000 1.00000 1.00000 −4.80194 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 −3.44504 1.00000 1.00000 1.00000
1.3 1.00000 1.00000 1.00000 1.00000 1.00000 −1.75302 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.bx yes 3
13.b even 2 1 5070.2.a.bo 3
13.d odd 4 2 5070.2.b.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bo 3 13.b even 2 1
5070.2.a.bx yes 3 1.a even 1 1 trivial
5070.2.b.w 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{3} + 10 T_{7}^{2} + 31 T_{7} + 29 \)
\( T_{11}^{3} + 8 T_{11}^{2} + 19 T_{11} + 13 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 46 T_{17} - 139 \)
\( T_{31}^{3} + 16 T_{31}^{2} + 83 T_{31} + 139 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( 29 + 31 T + 10 T^{2} + T^{3} \)
$11$ \( 13 + 19 T + 8 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -139 - 46 T + 3 T^{2} + T^{3} \)
$19$ \( -211 - 4 T + 11 T^{2} + T^{3} \)
$23$ \( 13 + 20 T + 9 T^{2} + T^{3} \)
$29$ \( 189 - 63 T + T^{3} \)
$31$ \( 139 + 83 T + 16 T^{2} + T^{3} \)
$37$ \( -223 - 11 T + 10 T^{2} + T^{3} \)
$41$ \( -167 - 36 T + 5 T^{2} + T^{3} \)
$43$ \( 127 - 60 T + 3 T^{2} + T^{3} \)
$47$ \( -559 - 109 T + 3 T^{2} + T^{3} \)
$53$ \( 71 + 80 T + 17 T^{2} + T^{3} \)
$59$ \( -1079 - 116 T + 11 T^{2} + T^{3} \)
$61$ \( 29 + 10 T - 11 T^{2} + T^{3} \)
$67$ \( -1 + 12 T + 15 T^{2} + T^{3} \)
$71$ \( -13 - 30 T + 13 T^{2} + T^{3} \)
$73$ \( -377 - 240 T - T^{2} + T^{3} \)
$79$ \( -27 - 18 T + 3 T^{2} + T^{3} \)
$83$ \( -27 + 27 T + 12 T^{2} + T^{3} \)
$89$ \( -1009 - 240 T + T^{2} + T^{3} \)
$97$ \( 2561 - 289 T - 6 T^{2} + T^{3} \)
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