Properties

Label 5070.2.b.p
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

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
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 1.00000i 1.00000 1.00000
1351.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.00000i 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.p 4
13.b even 2 1 inner 5070.2.b.p 4
13.c even 3 1 390.2.bb.a 4
13.d odd 4 1 5070.2.a.ba 2
13.d odd 4 1 5070.2.a.be 2
13.e even 6 1 390.2.bb.a 4
39.h odd 6 1 1170.2.bs.d 4
39.i odd 6 1 1170.2.bs.d 4
65.l even 6 1 1950.2.bc.a 4
65.n even 6 1 1950.2.bc.a 4
65.q odd 12 1 1950.2.y.d 4
65.q odd 12 1 1950.2.y.e 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 13.c even 3 1
390.2.bb.a 4 13.e even 6 1
1170.2.bs.d 4 39.h odd 6 1
1170.2.bs.d 4 39.i odd 6 1
1950.2.y.d 4 65.q odd 12 1
1950.2.y.d 4 65.r odd 12 1
1950.2.y.e 4 65.q odd 12 1
1950.2.y.e 4 65.r odd 12 1
1950.2.bc.a 4 65.l even 6 1
1950.2.bc.a 4 65.n even 6 1
5070.2.a.ba 2 13.d odd 4 1
5070.2.a.be 2 13.d odd 4 1
5070.2.b.p 4 1.a even 1 1 trivial
5070.2.b.p 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 \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{2} + 9 \)
\( T_{11}^{4} + 14 T_{11}^{2} + 1 \)
\( T_{17} - 4 \)
\( T_{31}^{4} + 104 T_{31}^{2} + 1936 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 9 + T^{2} )^{2} \)
$11$ \( 1 + 14 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -4 + T )^{4} \)
$19$ \( 169 + 38 T^{2} + T^{4} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( ( -8 + 4 T + T^{2} )^{2} \)
$31$ \( 1936 + 104 T^{2} + T^{4} \)
$37$ \( 2209 + 98 T^{2} + T^{4} \)
$41$ \( ( 16 + T^{2} )^{2} \)
$43$ \( ( -6 + T )^{4} \)
$47$ \( 9 + 42 T^{2} + T^{4} \)
$53$ \( ( 1 + 4 T + T^{2} )^{2} \)
$59$ \( 2704 + 152 T^{2} + T^{4} \)
$61$ \( ( 4 + 8 T + T^{2} )^{2} \)
$67$ \( 64 + 32 T^{2} + T^{4} \)
$71$ \( 144 + 168 T^{2} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 52 - 20 T + T^{2} )^{2} \)
$83$ \( 576 + 96 T^{2} + T^{4} \)
$89$ \( 20449 + 302 T^{2} + T^{4} \)
$97$ \( 10816 + 224 T^{2} + T^{4} \)
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