Properties

Label 5070.2.a.ba
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.73205
−1.73205
−1.00000 −1.00000 1.00000 1.00000 1.00000 −3.00000 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 −3.00000 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.a.ba 2
13.b even 2 1 5070.2.a.be 2
13.d odd 4 2 5070.2.b.p 4
13.f odd 12 2 390.2.bb.a 4
39.k even 12 2 1170.2.bs.d 4
65.o even 12 2 1950.2.y.e 4
65.s odd 12 2 1950.2.bc.a 4
65.t even 12 2 1950.2.y.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 13.f odd 12 2
1170.2.bs.d 4 39.k even 12 2
1950.2.y.d 4 65.t even 12 2
1950.2.y.e 4 65.o even 12 2
1950.2.bc.a 4 65.s odd 12 2
5070.2.a.ba 2 1.a even 1 1 trivial
5070.2.a.be 2 13.b even 2 1
5070.2.b.p 4 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7} + 3 \)
\( T_{11}^{2} - 4 T_{11} + 1 \)
\( T_{17} + 4 \)
\( T_{31}^{2} + 4 T_{31} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( 1 - 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( 13 - 8 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( -8 + 4 T + T^{2} \)
$31$ \( -44 + 4 T + T^{2} \)
$37$ \( -47 - 2 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( -3 - 6 T + T^{2} \)
$53$ \( 1 + 4 T + T^{2} \)
$59$ \( 52 - 16 T + T^{2} \)
$61$ \( 4 + 8 T + T^{2} \)
$67$ \( -8 - 4 T + T^{2} \)
$71$ \( -12 + 12 T + T^{2} \)
$73$ \( -48 + T^{2} \)
$79$ \( 52 - 20 T + T^{2} \)
$83$ \( 24 - 12 T + T^{2} \)
$89$ \( -143 + 4 T + T^{2} \)
$97$ \( -104 + 4 T + T^{2} \)
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