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: \( x^{3} - x^{2} - 2x + 1 \) 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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( - \beta_1 - 3) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( - \beta_1 - 3) q^{7} + q^{8} + q^{9} + q^{10} + (\beta_{2} - \beta_1 - 2) q^{11} + q^{12} + ( - \beta_1 - 3) q^{14} + q^{15} + q^{16} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{17} + q^{18} + (2 \beta_{2} + 3 \beta_1 - 4) q^{19} + q^{20} + ( - \beta_1 - 3) q^{21} + (\beta_{2} - \beta_1 - 2) q^{22} + ( - \beta_{2} + 2 \beta_1 - 4) q^{23} + q^{24} + q^{25} + q^{27} + ( - \beta_1 - 3) q^{28} + (6 \beta_{2} - 3 \beta_1 + 3) q^{29} + q^{30} + ( - \beta_{2} + \beta_1 - 6) q^{31} + q^{32} + (\beta_{2} - \beta_1 - 2) q^{33} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{34} + ( - \beta_1 - 3) q^{35} + q^{36} + (2 \beta_{2} - 5 \beta_1 - 1) q^{37} + (2 \beta_{2} + 3 \beta_1 - 4) q^{38} + q^{40} + (2 \beta_{2} + 3 \beta_1 - 2) q^{41} + ( - \beta_1 - 3) q^{42} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{43} + (\beta_{2} - \beta_1 - 2) q^{44} + q^{45} + ( - \beta_{2} + 2 \beta_1 - 4) q^{46} + (4 \beta_{2} - 8 \beta_1 + 3) q^{47} + q^{48} + (\beta_{2} + 6 \beta_1 + 4) q^{49} + q^{50} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{51} + (2 \beta_{2} - 3 \beta_1 - 4) q^{53} + q^{54} + (\beta_{2} - \beta_1 - 2) q^{55} + ( - \beta_1 - 3) q^{56} + (2 \beta_{2} + 3 \beta_1 - 4) q^{57} + (6 \beta_{2} - 3 \beta_1 + 3) q^{58} + ( - 9 \beta_{2} + 7 \beta_1 - 9) q^{59} + q^{60} + (\beta_{2} + 3 \beta_1 + 3) q^{61} + ( - \beta_{2} + \beta_1 - 6) q^{62} + ( - \beta_1 - 3) q^{63} + q^{64} + (\beta_{2} - \beta_1 - 2) q^{66} + ( - 3 \beta_{2} + 6 \beta_1 - 8) q^{67} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{68} + ( - \beta_{2} + 2 \beta_1 - 4) q^{69} + ( - \beta_1 - 3) q^{70} + ( - 3 \beta_{2} - 4 \beta_1 - 4) q^{71} + q^{72} + ( - 11 \beta_{2} + 2 \beta_1 - 4) q^{73} + (2 \beta_{2} - 5 \beta_1 - 1) q^{74} + q^{75} + (2 \beta_{2} + 3 \beta_1 - 4) q^{76} + ( - 3 \beta_{2} + 5 \beta_1 + 7) q^{77} - 3 \beta_1 q^{79} + q^{80} + q^{81} + (2 \beta_{2} + 3 \beta_1 - 2) q^{82} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{83} + ( - \beta_1 - 3) q^{84} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{85} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{86} + (6 \beta_{2} - 3 \beta_1 + 3) q^{87} + (\beta_{2} - \beta_1 - 2) q^{88} + (9 \beta_{2} + 2 \beta_1 + 2) q^{89} + q^{90} + ( - \beta_{2} + 2 \beta_1 - 4) q^{92} + ( - \beta_{2} + \beta_1 - 6) q^{93} + (4 \beta_{2} - 8 \beta_1 + 3) q^{94} + (2 \beta_{2} + 3 \beta_1 - 4) q^{95} + q^{96} + ( - 8 \beta_{2} - 5 \beta_1 + 1) q^{97} + (\beta_{2} + 6 \beta_1 + 4) q^{98} + (\beta_{2} - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 10 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 10 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 10 q^{14} + 3 q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{18} - 11 q^{19} + 3 q^{20} - 10 q^{21} - 8 q^{22} - 9 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} - 10 q^{28} + 3 q^{30} - 16 q^{31} + 3 q^{32} - 8 q^{33} - 3 q^{34} - 10 q^{35} + 3 q^{36} - 10 q^{37} - 11 q^{38} + 3 q^{40} - 5 q^{41} - 10 q^{42} - 3 q^{43} - 8 q^{44} + 3 q^{45} - 9 q^{46} - 3 q^{47} + 3 q^{48} + 17 q^{49} + 3 q^{50} - 3 q^{51} - 17 q^{53} + 3 q^{54} - 8 q^{55} - 10 q^{56} - 11 q^{57} - 11 q^{59} + 3 q^{60} + 11 q^{61} - 16 q^{62} - 10 q^{63} + 3 q^{64} - 8 q^{66} - 15 q^{67} - 3 q^{68} - 9 q^{69} - 10 q^{70} - 13 q^{71} + 3 q^{72} + q^{73} - 10 q^{74} + 3 q^{75} - 11 q^{76} + 29 q^{77} - 3 q^{79} + 3 q^{80} + 3 q^{81} - 5 q^{82} - 12 q^{83} - 10 q^{84} - 3 q^{85} - 3 q^{86} - 8 q^{88} - q^{89} + 3 q^{90} - 9 q^{92} - 16 q^{93} - 3 q^{94} - 11 q^{95} + 3 q^{96} + 6 q^{97} + 17 q^{98} - 8 q^{99}+O(q^{100}) \) 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
1.80194
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

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