Properties

Label 5070.2.b.r
Level $5070$
Weight $2$
Character orbit 5070.b
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

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

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
1.56155i
2.56155i
2.56155i
1.56155i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.561553i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.56155i 1.00000i 1.00000 −1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.56155i 1.00000i 1.00000 −1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.561553i 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
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.r 4
13.b even 2 1 inner 5070.2.b.r 4
13.d odd 4 1 5070.2.a.bb 2
13.d odd 4 1 5070.2.a.bi 2
13.f odd 12 2 390.2.i.g 4
39.k even 12 2 1170.2.i.o 4
65.o even 12 2 1950.2.z.n 8
65.s odd 12 2 1950.2.i.bi 4
65.t even 12 2 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 13.f odd 12 2
1170.2.i.o 4 39.k even 12 2
1950.2.i.bi 4 65.s odd 12 2
1950.2.z.n 8 65.o even 12 2
1950.2.z.n 8 65.t even 12 2
5070.2.a.bb 2 13.d odd 4 1
5070.2.a.bi 2 13.d odd 4 1
5070.2.b.r 4 1.a even 1 1 trivial
5070.2.b.r 4 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 \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{4} + 13 T_{7}^{2} + 4 \)
\( T_{11}^{2} + 17 \)
\( T_{17}^{2} + 2 T_{17} - 16 \)
\( T_{31}^{4} + 77 T_{31}^{2} + 1444 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 4 + 13 T^{2} + T^{4} \)
$11$ \( ( 17 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -16 + 2 T + T^{2} )^{2} \)
$19$ \( 4 + 13 T^{2} + T^{4} \)
$23$ \( ( -36 - 3 T + T^{2} )^{2} \)
$29$ \( ( 16 - 9 T + T^{2} )^{2} \)
$31$ \( 1444 + 77 T^{2} + T^{4} \)
$37$ \( ( 17 + T^{2} )^{2} \)
$41$ \( 2704 + 168 T^{2} + T^{4} \)
$43$ \( ( 2 + 5 T + T^{2} )^{2} \)
$47$ \( ( 49 + T^{2} )^{2} \)
$53$ \( ( 38 - 13 T + T^{2} )^{2} \)
$59$ \( 4624 + 153 T^{2} + T^{4} \)
$61$ \( ( -6 + T )^{4} \)
$67$ \( 1024 + 208 T^{2} + T^{4} \)
$71$ \( 4096 + 196 T^{2} + T^{4} \)
$73$ \( 20736 + 324 T^{2} + T^{4} \)
$79$ \( ( 86 - 19 T + T^{2} )^{2} \)
$83$ \( 64 + 52 T^{2} + T^{4} \)
$89$ \( 1156 + 357 T^{2} + T^{4} \)
$97$ \( 64 + 52 T^{2} + T^{4} \)
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