Properties

Label 5070.2.b.s
Level $5070$
Weight $2$
Character orbit 5070.b
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - q^{3} - q^{4} + \beta_{5} q^{5} -\beta_{5} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} -\beta_{5} q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} - q^{3} - q^{4} + \beta_{5} q^{5} -\beta_{5} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} -\beta_{5} q^{8} + q^{9} - q^{10} + ( -\beta_{1} + 5 \beta_{5} ) q^{11} + q^{12} + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{14} -\beta_{5} q^{15} + q^{16} + ( -1 - \beta_{2} + \beta_{4} ) q^{17} + \beta_{5} q^{18} + ( 5 \beta_{1} - 3 \beta_{3} - 5 \beta_{5} ) q^{19} -\beta_{5} q^{20} + ( -\beta_{1} - \beta_{3} ) q^{21} + ( -5 + \beta_{4} ) q^{22} + ( 5 - 7 \beta_{2} - 3 \beta_{4} ) q^{23} + \beta_{5} q^{24} - q^{25} - q^{27} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( 1 + 3 \beta_{2} + 6 \beta_{4} ) q^{29} + q^{30} + ( 3 \beta_{1} - 4 \beta_{3} - 7 \beta_{5} ) q^{31} + \beta_{5} q^{32} + ( \beta_{1} - 5 \beta_{5} ) q^{33} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{34} + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{35} - q^{36} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{37} + ( 2 + 3 \beta_{2} - 2 \beta_{4} ) q^{38} + q^{40} + ( -5 \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{41} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{42} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{43} + ( \beta_{1} - 5 \beta_{5} ) q^{44} + \beta_{5} q^{45} + ( 4 \beta_{1} - 7 \beta_{3} - 2 \beta_{5} ) q^{46} + ( 4 \beta_{1} - 2 \beta_{3} + 5 \beta_{5} ) q^{47} - q^{48} + ( 1 + \beta_{2} + 3 \beta_{4} ) q^{49} -\beta_{5} q^{50} + ( 1 + \beta_{2} - \beta_{4} ) q^{51} + ( 2 - 3 \beta_{2} ) q^{53} -\beta_{5} q^{54} + ( -5 + \beta_{4} ) q^{55} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{56} + ( -5 \beta_{1} + 3 \beta_{3} + 5 \beta_{5} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{3} + 4 \beta_{5} ) q^{58} + ( -3 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{59} + \beta_{5} q^{60} + ( -4 - 6 \beta_{2} + \beta_{4} ) q^{61} + ( 3 + 4 \beta_{2} + \beta_{4} ) q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} - q^{64} + ( 5 - \beta_{4} ) q^{66} + ( -\beta_{3} - 2 \beta_{5} ) q^{67} + ( 1 + \beta_{2} - \beta_{4} ) q^{68} + ( -5 + 7 \beta_{2} + 3 \beta_{4} ) q^{69} + ( -\beta_{1} - \beta_{3} ) q^{70} + ( 4 \beta_{1} - 7 \beta_{3} - 8 \beta_{5} ) q^{71} -\beta_{5} q^{72} + ( 7 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -1 + 3 \beta_{2} + 4 \beta_{4} ) q^{74} + q^{75} + ( -5 \beta_{1} + 3 \beta_{3} + 5 \beta_{5} ) q^{76} + ( 8 - 6 \beta_{2} - 11 \beta_{4} ) q^{77} + ( -6 + 5 \beta_{2} + 6 \beta_{4} ) q^{79} + \beta_{5} q^{80} + q^{81} + ( 6 - 5 \beta_{2} ) q^{82} + ( -5 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{83} + ( \beta_{1} + \beta_{3} ) q^{84} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{86} + ( -1 - 3 \beta_{2} - 6 \beta_{4} ) q^{87} + ( 5 - \beta_{4} ) q^{88} + ( 2 \beta_{1} + 5 \beta_{3} + 2 \beta_{5} ) q^{89} - q^{90} + ( -5 + 7 \beta_{2} + 3 \beta_{4} ) q^{92} + ( -3 \beta_{1} + 4 \beta_{3} + 7 \beta_{5} ) q^{93} + ( -7 + 2 \beta_{2} - 2 \beta_{4} ) q^{94} + ( 2 + 3 \beta_{2} - 2 \beta_{4} ) q^{95} -\beta_{5} q^{96} + ( 3 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{97} + ( 2 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{98} + ( -\beta_{1} + 5 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 28q^{22} + 10q^{23} - 6q^{25} - 6q^{27} + 24q^{29} + 6q^{30} - 6q^{36} + 14q^{38} + 6q^{40} + 10q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 6q^{53} - 28q^{55} - 34q^{61} + 28q^{62} - 6q^{64} + 28q^{66} + 6q^{68} - 10q^{69} + 8q^{74} + 6q^{75} + 14q^{77} - 14q^{79} + 6q^{81} + 26q^{82} - 24q^{87} + 28q^{88} - 6q^{90} - 10q^{92} - 42q^{94} + 14q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
0.445042i
1.80194i
1.24698i
1.24698i
1.80194i
0.445042i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.69202i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.35690i 1.00000i 1.00000 −1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.04892i 1.00000i 1.00000 −1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.04892i 1.00000i 1.00000 −1.00000
1351.5 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.35690i 1.00000i 1.00000 −1.00000
1351.6 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.69202i 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.s 6
13.b even 2 1 inner 5070.2.b.s 6
13.d odd 4 1 5070.2.a.bk 3
13.d odd 4 1 5070.2.a.bt yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bk 3 13.d odd 4 1
5070.2.a.bt yes 3 13.d odd 4 1
5070.2.b.s 6 1.a even 1 1 trivial
5070.2.b.s 6 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{6} + 14 T_{7}^{4} + 49 T_{7}^{2} + 49 \)
\( T_{11}^{6} + 70 T_{11}^{4} + 1421 T_{11}^{2} + 8281 \)
\( T_{17}^{3} + 3 T_{17}^{2} - 4 T_{17} - 13 \)
\( T_{31}^{6} + 126 T_{31}^{4} + 1421 T_{31}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( ( 1 + T^{2} )^{3} \)
$7$ \( 49 + 49 T^{2} + 14 T^{4} + T^{6} \)
$11$ \( 8281 + 1421 T^{2} + 70 T^{4} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( -13 - 4 T + 3 T^{2} + T^{3} )^{2} \)
$19$ \( 41209 + 3626 T^{2} + 105 T^{4} + T^{6} \)
$23$ \( ( 419 - 78 T - 5 T^{2} + T^{3} )^{2} \)
$29$ \( ( 377 - 15 T - 12 T^{2} + T^{3} )^{2} \)
$31$ \( 49 + 1421 T^{2} + 126 T^{4} + T^{6} \)
$37$ \( 5041 + 1193 T^{2} + 66 T^{4} + T^{6} \)
$41$ \( 19321 + 3618 T^{2} + 173 T^{4} + T^{6} \)
$43$ \( ( 41 - 8 T - 5 T^{2} + T^{3} )^{2} \)
$47$ \( 41209 + 5635 T^{2} + 203 T^{4} + T^{6} \)
$53$ \( ( 13 - 18 T - 3 T^{2} + T^{3} )^{2} \)
$59$ \( 8281 + 3626 T^{2} + 189 T^{4} + T^{6} \)
$61$ \( ( -601 - 4 T + 17 T^{2} + T^{3} )^{2} \)
$67$ \( 1 + 26 T^{2} + 13 T^{4} + T^{6} \)
$71$ \( 169 + 1238 T^{2} + 229 T^{4} + T^{6} \)
$73$ \( 253009 + 16442 T^{2} + 285 T^{4} + T^{6} \)
$79$ \( ( -91 - 56 T + 7 T^{2} + T^{3} )^{2} \)
$83$ \( 344569 + 15937 T^{2} + 230 T^{4} + T^{6} \)
$89$ \( 85849 + 9502 T^{2} + 185 T^{4} + T^{6} \)
$97$ \( 49 + 441 T^{2} + 42 T^{4} + T^{6} \)
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