Properties

Label 5070.2.b.l
Level $5070$
Weight $2$
Character orbit 5070.b
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.00000i 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.l 2
13.b even 2 1 inner 5070.2.b.l 2
13.d odd 4 1 5070.2.a.i 1
13.d odd 4 1 5070.2.a.x 1
13.f odd 12 2 390.2.i.d 2
39.k even 12 2 1170.2.i.g 2
65.o even 12 2 1950.2.z.j 4
65.s odd 12 2 1950.2.i.i 2
65.t even 12 2 1950.2.z.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 13.f odd 12 2
1170.2.i.g 2 39.k even 12 2
1950.2.i.i 2 65.s odd 12 2
1950.2.z.j 4 65.o even 12 2
1950.2.z.j 4 65.t even 12 2
5070.2.a.i 1 13.d odd 4 1
5070.2.a.x 1 13.d odd 4 1
5070.2.b.l 2 1.a even 1 1 trivial
5070.2.b.l 2 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 \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{2} + 25 \)
\( T_{11}^{2} + 9 \)
\( T_{17} - 8 \)
\( T_{31}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( 9 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -8 + T )^{2} \)
$19$ \( 25 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( 4 + T^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( 36 + T^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( ( -1 + T )^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( 4 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( 121 + T^{2} \)
$97$ \( T^{2} \)
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