Properties

Label 5070.2.b.g
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.

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

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 2.00000i 1.00000i 1.00000 1.00000
1351.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 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.g 2
13.b even 2 1 inner 5070.2.b.g 2
13.d odd 4 1 5070.2.a.f 1
13.d odd 4 1 5070.2.a.o 1
13.f odd 12 2 390.2.i.a 2
39.k even 12 2 1170.2.i.k 2
65.o even 12 2 1950.2.z.h 4
65.s odd 12 2 1950.2.i.s 2
65.t even 12 2 1950.2.z.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 13.f odd 12 2
1170.2.i.k 2 39.k even 12 2
1950.2.i.s 2 65.s odd 12 2
1950.2.z.h 4 65.o even 12 2
1950.2.z.h 4 65.t even 12 2
5070.2.a.f 1 13.d odd 4 1
5070.2.a.o 1 13.d odd 4 1
5070.2.b.g 2 1.a even 1 1 trivial
5070.2.b.g 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 \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 1 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -3 + T )^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( 9 + T^{2} \)
$37$ \( 25 + T^{2} \)
$41$ \( 100 + T^{2} \)
$43$ \( ( 5 + T )^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( ( -14 + T )^{2} \)
$59$ \( 25 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( 16 + T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 100 + T^{2} \)
$97$ \( 100 + T^{2} \)
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