Properties

Label 5070.2.a.be
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: \( x^{2} - 3 \) Copy content Toggle raw display
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.

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

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.be 2
13.b even 2 1 5070.2.a.ba 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.d 4
65.s odd 12 2 1950.2.bc.a 4
65.t even 12 2 1950.2.y.e 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.o even 12 2
1950.2.y.e 4 65.t even 12 2
1950.2.bc.a 4 65.s odd 12 2
5070.2.a.ba 2 13.b even 2 1
5070.2.a.be 2 1.a even 1 1 trivial
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 1 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{31}^{2} - 4T_{31} - 44 \) Copy content Toggle raw display

Hecke characteristic polynomials

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