Properties

Label 5070.2.b.z
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: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

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
1.80194i
0.445042i
1.24698i
1.24698i
0.445042i
1.80194i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.04892i 1.00000i 1.00000 1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.35690i 1.00000i 1.00000 1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.69202i 1.00000i 1.00000 1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.69202i 1.00000i 1.00000 1.00000
1351.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.35690i 1.00000i 1.00000 1.00000
1351.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 1.04892i 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.z 6
13.b even 2 1 inner 5070.2.b.z 6
13.d odd 4 1 5070.2.a.bp 3
13.d odd 4 1 5070.2.a.bw yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bp 3 13.d odd 4 1
5070.2.a.bw yes 3 13.d odd 4 1
5070.2.b.z 6 1.a even 1 1 trivial
5070.2.b.z 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 \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{6} + 26T_{7}^{4} + 181T_{7}^{2} + 169 \) Copy content Toggle raw display
\( T_{11}^{6} + 14T_{11}^{4} + 49T_{11}^{2} + 49 \) Copy content Toggle raw display
\( T_{17}^{3} - 7T_{17}^{2} + 7 \) Copy content Toggle raw display
\( T_{31}^{6} + 118T_{31}^{4} + 2105T_{31}^{2} + 6889 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 26 T^{4} + 181 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{6} + 14 T^{4} + 49 T^{2} + 49 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 7 T^{2} + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 33 T^{4} + 230 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( (T^{3} + 3 T^{2} - 4 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 91 T - 203)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 118 T^{4} + 2105 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$37$ \( T^{6} + 222 T^{4} + 11801 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$41$ \( T^{6} + 129 T^{4} + 2558 T^{2} + \cdots + 12769 \) Copy content Toggle raw display
$43$ \( (T^{3} + 9 T^{2} - 22 T - 169)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 147 T^{4} + 1715 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T^{2} - 18 T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 77 T^{4} + 294 T^{2} + \cdots + 49 \) Copy content Toggle raw display
$61$ \( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 129 T^{4} + 3986 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$71$ \( T^{6} + 49 T^{4} + 686 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$73$ \( T^{6} + 269 T^{4} + 19498 T^{2} + \cdots + 187489 \) Copy content Toggle raw display
$79$ \( (T^{3} - 17 T^{2} + 10 T + 587)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 278 T^{4} + 18829 T^{2} + \cdots + 121801 \) Copy content Toggle raw display
$89$ \( T^{6} + 297 T^{4} + 25434 T^{2} + \cdots + 613089 \) Copy content Toggle raw display
$97$ \( T^{6} + 290 T^{4} + 12337 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
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