Properties

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

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} + q^{10} + ( 1 - 2 \beta ) q^{11} + q^{12} + ( 1 + \beta ) q^{14} + q^{15} + q^{16} + 2 \beta q^{17} + q^{18} + ( -1 - \beta ) q^{19} + q^{20} + ( 1 + \beta ) q^{21} + ( 1 - 2 \beta ) q^{22} -3 \beta q^{23} + q^{24} + q^{25} + q^{27} + ( 1 + \beta ) q^{28} + ( 4 + \beta ) q^{29} + q^{30} + ( -2 + 3 \beta ) q^{31} + q^{32} + ( 1 - 2 \beta ) q^{33} + 2 \beta q^{34} + ( 1 + \beta ) q^{35} + q^{36} + ( -1 + 2 \beta ) q^{37} + ( -1 - \beta ) q^{38} + q^{40} + ( 6 - 4 \beta ) q^{41} + ( 1 + \beta ) q^{42} + ( 2 + \beta ) q^{43} + ( 1 - 2 \beta ) q^{44} + q^{45} -3 \beta q^{46} + 7 q^{47} + q^{48} + ( -2 + 3 \beta ) q^{49} + q^{50} + 2 \beta q^{51} + ( 7 - \beta ) q^{53} + q^{54} + ( 1 - 2 \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( -1 - \beta ) q^{57} + ( 4 + \beta ) q^{58} + ( 8 + \beta ) q^{59} + q^{60} + 6 q^{61} + ( -2 + 3 \beta ) q^{62} + ( 1 + \beta ) q^{63} + q^{64} + ( 1 - 2 \beta ) q^{66} + ( 4 + 4 \beta ) q^{67} + 2 \beta q^{68} -3 \beta q^{69} + ( 1 + \beta ) q^{70} + ( -10 + 2 \beta ) q^{71} + q^{72} -6 \beta q^{73} + ( -1 + 2 \beta ) q^{74} + q^{75} + ( -1 - \beta ) q^{76} + ( -7 - 3 \beta ) q^{77} + ( 10 - \beta ) q^{79} + q^{80} + q^{81} + ( 6 - 4 \beta ) q^{82} + ( -4 + 2 \beta ) q^{83} + ( 1 + \beta ) q^{84} + 2 \beta q^{85} + ( 2 + \beta ) q^{86} + ( 4 + \beta ) q^{87} + ( 1 - 2 \beta ) q^{88} + ( -11 + 5 \beta ) q^{89} + q^{90} -3 \beta q^{92} + ( -2 + 3 \beta ) q^{93} + 7 q^{94} + ( -1 - \beta ) q^{95} + q^{96} + ( -4 + 2 \beta ) q^{97} + ( -2 + 3 \beta ) q^{98} + ( 1 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} + 2q^{15} + 2q^{16} + 2q^{17} + 2q^{18} - 3q^{19} + 2q^{20} + 3q^{21} - 3q^{23} + 2q^{24} + 2q^{25} + 2q^{27} + 3q^{28} + 9q^{29} + 2q^{30} - q^{31} + 2q^{32} + 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} + 8q^{41} + 3q^{42} + 5q^{43} + 2q^{45} - 3q^{46} + 14q^{47} + 2q^{48} - q^{49} + 2q^{50} + 2q^{51} + 13q^{53} + 2q^{54} + 3q^{56} - 3q^{57} + 9q^{58} + 17q^{59} + 2q^{60} + 12q^{61} - q^{62} + 3q^{63} + 2q^{64} + 12q^{67} + 2q^{68} - 3q^{69} + 3q^{70} - 18q^{71} + 2q^{72} - 6q^{73} + 2q^{75} - 3q^{76} - 17q^{77} + 19q^{79} + 2q^{80} + 2q^{81} + 8q^{82} - 6q^{83} + 3q^{84} + 2q^{85} + 5q^{86} + 9q^{87} - 17q^{89} + 2q^{90} - 3q^{92} - q^{93} + 14q^{94} - 3q^{95} + 2q^{96} - 6q^{97} - q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
1.00000 1.00000 1.00000 1.00000 1.00000 −0.561553 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 3.56155 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.bi 2
13.b even 2 1 5070.2.a.bb 2
13.c even 3 2 390.2.i.g 4
13.d odd 4 2 5070.2.b.r 4
39.i odd 6 2 1170.2.i.o 4
65.n even 6 2 1950.2.i.bi 4
65.q odd 12 4 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 13.c even 3 2
1170.2.i.o 4 39.i odd 6 2
1950.2.i.bi 4 65.n even 6 2
1950.2.z.n 8 65.q odd 12 4
5070.2.a.bb 2 13.b even 2 1
5070.2.a.bi 2 1.a even 1 1 trivial
5070.2.b.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{2} - 3 T_{7} - 2 \)
\( T_{11}^{2} - 17 \)
\( T_{17}^{2} - 2 T_{17} - 16 \)
\( T_{31}^{2} + T_{31} - 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -2 - 3 T + T^{2} \)
$11$ \( -17 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -16 - 2 T + T^{2} \)
$19$ \( -2 + 3 T + T^{2} \)
$23$ \( -36 + 3 T + T^{2} \)
$29$ \( 16 - 9 T + T^{2} \)
$31$ \( -38 + T + T^{2} \)
$37$ \( -17 + T^{2} \)
$41$ \( -52 - 8 T + T^{2} \)
$43$ \( 2 - 5 T + T^{2} \)
$47$ \( ( -7 + T )^{2} \)
$53$ \( 38 - 13 T + T^{2} \)
$59$ \( 68 - 17 T + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -32 - 12 T + T^{2} \)
$71$ \( 64 + 18 T + T^{2} \)
$73$ \( -144 + 6 T + T^{2} \)
$79$ \( 86 - 19 T + T^{2} \)
$83$ \( -8 + 6 T + T^{2} \)
$89$ \( -34 + 17 T + T^{2} \)
$97$ \( -8 + 6 T + T^{2} \)
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