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: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + (\beta_{2} + \beta_1 - 2) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + (\beta_{2} + \beta_1 - 2) q^{7} - q^{8} + q^{9} - q^{10} + (\beta_{2} - 2 \beta_1 + 1) q^{11} + q^{12} + ( - \beta_{2} - \beta_1 + 2) q^{14} + q^{15} + q^{16} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{17} - q^{18} + ( - 3 \beta_{2} + \beta_1 - 1) q^{19} + q^{20} + (\beta_{2} + \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + \beta_1) q^{23} - q^{24} + q^{25} + q^{27} + (\beta_{2} + \beta_1 - 2) q^{28} + (5 \beta_{2} - 7 \beta_1 + 4) q^{29} - q^{30} + ( - \beta_{2} - 2 \beta_1 - 5) q^{31} - q^{32} + (\beta_{2} - 2 \beta_1 + 1) q^{33} + (2 \beta_{2} - 3 \beta_1 + 4) q^{34} + (\beta_{2} + \beta_1 - 2) q^{35} + q^{36} + (3 \beta_{2} - 5 \beta_1 - 4) q^{37} + (3 \beta_{2} - \beta_1 + 1) q^{38} - q^{40} + ( - \beta_{2} + 3 \beta_1 - 7) q^{41} + ( - \beta_{2} - \beta_1 + 2) q^{42} + ( - \beta_{2} - 4 \beta_1 + 4) q^{43} + (\beta_{2} - 2 \beta_1 + 1) q^{44} + q^{45} + (2 \beta_{2} - \beta_1) q^{46} + ( - 2 \beta_{2} + 6 \beta_1 - 5) q^{47} + q^{48} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{49} - q^{50} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{51} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{53} - q^{54} + (\beta_{2} - 2 \beta_1 + 1) q^{55} + ( - \beta_{2} - \beta_1 + 2) q^{56} + ( - 3 \beta_{2} + \beta_1 - 1) q^{57} + ( - 5 \beta_{2} + 7 \beta_1 - 4) q^{58} + ( - \beta_{2} + 4 \beta_1 - 4) q^{59} + q^{60} + (\beta_{2} + 2) q^{61} + (\beta_{2} + 2 \beta_1 + 5) q^{62} + (\beta_{2} + \beta_1 - 2) q^{63} + q^{64} + ( - \beta_{2} + 2 \beta_1 - 1) q^{66} + (2 \beta_{2} - 3 \beta_1 - 4) q^{67} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{68} + ( - 2 \beta_{2} + \beta_1) q^{69} + ( - \beta_{2} - \beta_1 + 2) q^{70} + ( - 2 \beta_{2} - \beta_1 + 2) q^{71} - q^{72} + ( - 8 \beta_{2} + 7 \beta_1 - 6) q^{73} + ( - 3 \beta_{2} + 5 \beta_1 + 4) q^{74} + q^{75} + ( - 3 \beta_{2} + \beta_1 - 1) q^{76} + ( - 5 \beta_{2} + 6 \beta_1 - 6) q^{77} + (7 \beta_{2} - 3 \beta_1 + 9) q^{79} + q^{80} + q^{81} + (\beta_{2} - 3 \beta_1 + 7) q^{82} + (7 \beta_{2} - 8 \beta_1 + 3) q^{83} + (\beta_{2} + \beta_1 - 2) q^{84} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{85} + (\beta_{2} + 4 \beta_1 - 4) q^{86} + (5 \beta_{2} - 7 \beta_1 + 4) q^{87} + ( - \beta_{2} + 2 \beta_1 - 1) q^{88} + ( - 6 \beta_{2} - 3 \beta_1) q^{89} - q^{90} + ( - 2 \beta_{2} + \beta_1) q^{92} + ( - \beta_{2} - 2 \beta_1 - 5) q^{93} + (2 \beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 3 \beta_{2} + \beta_1 - 1) q^{95} - q^{96} + (5 \beta_{2} - \beta_1 - 6) q^{97} + (2 \beta_{2} + 3 \beta_1 - 2) q^{98} + (\beta_{2} - 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} - 6 q^{28} - 3 q^{30} - 16 q^{31} - 3 q^{32} + 7 q^{34} - 6 q^{35} + 3 q^{36} - 20 q^{37} - q^{38} - 3 q^{40} - 17 q^{41} + 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} - 7 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} + 6 q^{56} + q^{57} - 7 q^{59} + 3 q^{60} + 5 q^{61} + 16 q^{62} - 6 q^{63} + 3 q^{64} - 17 q^{67} - 7 q^{68} + 3 q^{69} + 6 q^{70} + 7 q^{71} - 3 q^{72} - 3 q^{73} + 20 q^{74} + 3 q^{75} + q^{76} - 7 q^{77} + 17 q^{79} + 3 q^{80} + 3 q^{81} + 17 q^{82} - 6 q^{83} - 6 q^{84} - 7 q^{85} - 9 q^{86} + 3 q^{89} - 3 q^{90} + 3 q^{92} - 16 q^{93} + 7 q^{94} + q^{95} - 3 q^{96} - 24 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

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

Hecke characteristic polynomials

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