Properties

Label 5070.2.a.y
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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.

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

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 −2.00000 −1.00000 1.00000 1.00000
1.2 −1.00000 −1.00000 1.00000 −1.00000 1.00000 −2.00000 −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.y 2
13.b even 2 1 5070.2.a.bg 2
13.d odd 4 2 5070.2.b.o 4
13.f odd 12 2 390.2.bb.b 4
39.k even 12 2 1170.2.bs.e 4
65.o even 12 2 1950.2.y.f 4
65.s odd 12 2 1950.2.bc.b 4
65.t even 12 2 1950.2.y.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.b 4 13.f odd 12 2
1170.2.bs.e 4 39.k even 12 2
1950.2.y.c 4 65.t even 12 2
1950.2.y.f 4 65.o even 12 2
1950.2.bc.b 4 65.s odd 12 2
5070.2.a.y 2 1.a even 1 1 trivial
5070.2.a.bg 2 13.b even 2 1
5070.2.b.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7} + 2 \)
\( T_{11}^{2} + 6 T_{11} - 3 \)
\( T_{17} - 4 \)
\( T_{31}^{2} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( -3 + 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( 4 + 8 T + T^{2} \)
$23$ \( 1 + 4 T + T^{2} \)
$29$ \( 1 - 4 T + T^{2} \)
$31$ \( -3 + T^{2} \)
$37$ \( -11 + 8 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -23 + 10 T + T^{2} \)
$47$ \( 37 + 14 T + T^{2} \)
$53$ \( -12 + 12 T + T^{2} \)
$59$ \( 13 - 10 T + T^{2} \)
$61$ \( -108 + T^{2} \)
$67$ \( 52 + 16 T + T^{2} \)
$71$ \( -104 - 4 T + T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( 1 + 14 T + T^{2} \)
$83$ \( -44 + 4 T + T^{2} \)
$89$ \( 4 - 8 T + T^{2} \)
$97$ \( 4 + 8 T + T^{2} \)
show more
show less