Properties

Label 5070.2.a.bu
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
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: \(0\)
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.

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

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
0.445042
1.80194
−1.24698
1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.69202 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.35690 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.04892 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.bu yes 3
13.b even 2 1 5070.2.a.bj 3
13.d odd 4 2 5070.2.b.t 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bj 3 13.b even 2 1
5070.2.a.bu yes 3 1.a even 1 1 trivial
5070.2.b.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{3} - 7 T_{7} - 7 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 15 T_{11} + 13 \)
\( T_{17}^{3} - 3 T_{17}^{2} - 4 T_{17} + 13 \)
\( T_{31}^{3} - 4 T_{31}^{2} + 3 T_{31} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -7 - 7 T + T^{3} \)
$11$ \( 13 - 15 T + 2 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 4 T - 3 T^{2} + T^{3} \)
$19$ \( -1 + 6 T - 5 T^{2} + T^{3} \)
$23$ \( 27 - 18 T - 3 T^{2} + T^{3} \)
$29$ \( -41 + 17 T + 10 T^{2} + T^{3} \)
$31$ \( 1 + 3 T - 4 T^{2} + T^{3} \)
$37$ \( -1 - T + 2 T^{2} + T^{3} \)
$41$ \( 167 + 52 T - 17 T^{2} + T^{3} \)
$43$ \( -113 + 90 T - 19 T^{2} + T^{3} \)
$47$ \( 287 - 49 T - 7 T^{2} + T^{3} \)
$53$ \( 83 + 62 T - 19 T^{2} + T^{3} \)
$59$ \( -71 + 76 T - 19 T^{2} + T^{3} \)
$61$ \( -251 - 86 T - T^{2} + T^{3} \)
$67$ \( -169 - 72 T + T^{2} + T^{3} \)
$71$ \( -421 - 114 T + T^{2} + T^{3} \)
$73$ \( -2143 - 142 T + 15 T^{2} + T^{3} \)
$79$ \( -13 + 54 T - 15 T^{2} + T^{3} \)
$83$ \( 139 - 37 T - 6 T^{2} + T^{3} \)
$89$ \( -223 - 120 T - 9 T^{2} + T^{3} \)
$97$ \( -1 - 11 T + 4 T^{2} + T^{3} \)
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