Properties

Label 5070.2.a.bt
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.

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

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

\( T_{7}^{3} - 7 T_{7} - 7 \)
\( T_{11}^{3} + 14 T_{11}^{2} + 63 T_{11} + 91 \)
\( T_{17}^{3} - 3 T_{17}^{2} - 4 T_{17} + 13 \)
\( T_{31}^{3} + 14 T_{31}^{2} + 35 T_{31} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -7 - 7 T + T^{3} \)
$11$ \( 91 + 63 T + 14 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 13 - 4 T - 3 T^{2} + T^{3} \)
$19$ \( -203 - 28 T + 7 T^{2} + T^{3} \)
$23$ \( -419 - 78 T + 5 T^{2} + T^{3} \)
$29$ \( 377 - 15 T - 12 T^{2} + T^{3} \)
$31$ \( -7 + 35 T + 14 T^{2} + T^{3} \)
$37$ \( 71 - 25 T - 4 T^{2} + T^{3} \)
$41$ \( -139 - 2 T + 13 T^{2} + T^{3} \)
$43$ \( -41 - 8 T + 5 T^{2} + T^{3} \)
$47$ \( 203 + 119 T + 21 T^{2} + T^{3} \)
$53$ \( 13 - 18 T - 3 T^{2} + T^{3} \)
$59$ \( 91 - 70 T + 7 T^{2} + T^{3} \)
$61$ \( -601 - 4 T + 17 T^{2} + T^{3} \)
$67$ \( 1 + 6 T + 5 T^{2} + T^{3} \)
$71$ \( -13 - 30 T + 13 T^{2} + T^{3} \)
$73$ \( 503 - 58 T - 13 T^{2} + T^{3} \)
$79$ \( -91 - 56 T + 7 T^{2} + T^{3} \)
$83$ \( -587 - 43 T + 12 T^{2} + T^{3} \)
$89$ \( -293 - 88 T + 3 T^{2} + T^{3} \)
$97$ \( -7 - 21 T + T^{3} \)
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