Properties

Label 5070.2.b.a
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} - q^{4} + i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} - q^{3} - q^{4} + i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} + q^{9} - q^{10} -i q^{11} + q^{12} -3 q^{14} -i q^{15} + q^{16} + i q^{18} -5 i q^{19} -i q^{20} -3 i q^{21} + q^{22} + 4 q^{23} + i q^{24} - q^{25} - q^{27} -3 i q^{28} + q^{30} + 10 i q^{31} + i q^{32} + i q^{33} -3 q^{35} - q^{36} + i q^{37} + 5 q^{38} + q^{40} + 6 i q^{41} + 3 q^{42} + 2 q^{43} + i q^{44} + i q^{45} + 4 i q^{46} + 9 i q^{47} - q^{48} -2 q^{49} -i q^{50} -13 q^{53} -i q^{54} + q^{55} + 3 q^{56} + 5 i q^{57} -4 i q^{59} + i q^{60} -2 q^{61} -10 q^{62} + 3 i q^{63} - q^{64} - q^{66} -12 i q^{67} -4 q^{69} -3 i q^{70} -2 i q^{71} -i q^{72} + 16 i q^{73} - q^{74} + q^{75} + 5 i q^{76} + 3 q^{77} -10 q^{79} + i q^{80} + q^{81} -6 q^{82} + 12 i q^{83} + 3 i q^{84} + 2 i q^{86} - q^{88} -i q^{89} - q^{90} -4 q^{92} -10 i q^{93} -9 q^{94} + 5 q^{95} -i q^{96} + 12 i q^{97} -2 i q^{98} -i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} + 2q^{12} - 6q^{14} + 2q^{16} + 2q^{22} + 8q^{23} - 2q^{25} - 2q^{27} + 2q^{30} - 6q^{35} - 2q^{36} + 10q^{38} + 2q^{40} + 6q^{42} + 4q^{43} - 2q^{48} - 4q^{49} - 26q^{53} + 2q^{55} + 6q^{56} - 4q^{61} - 20q^{62} - 2q^{64} - 2q^{66} - 8q^{69} - 2q^{74} + 2q^{75} + 6q^{77} - 20q^{79} + 2q^{81} - 12q^{82} - 2q^{88} - 2q^{90} - 8q^{92} - 18q^{94} + 10q^{95} + O(q^{100}) \)

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

\( T_{7}^{2} + 9 \)
\( T_{11}^{2} + 1 \)
\( T_{17} \)
\( T_{31}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( 1 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( 25 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( 100 + T^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( 36 + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( ( 13 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 4 + T^{2} \)
$73$ \( 256 + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( 1 + T^{2} \)
$97$ \( 144 + T^{2} \)
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