Properties

Label 5070.2.a.ca
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 14 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(4 \beta_{2} + \beta_{1} + 10\)
\(\nu^{3}\)\(=\)\(4 \beta_{3} + 4 \beta_{2} + 15 \beta_{1} + 10\)

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
−3.64466
−1.32258
2.32258
4.64466
1.00000 1.00000 1.00000 −1.00000 1.00000 −3.64466 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 −1.32258 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 2.32258 1.00000 1.00000 −1.00000
1.4 1.00000 1.00000 1.00000 −1.00000 1.00000 4.64466 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.ca 4
13.b even 2 1 5070.2.a.bz 4
13.d odd 4 2 5070.2.b.ba 8
13.f odd 12 2 390.2.bb.c 8
39.k even 12 2 1170.2.bs.f 8
65.o even 12 2 1950.2.y.j 8
65.s odd 12 2 1950.2.bc.g 8
65.t even 12 2 1950.2.y.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 13.f odd 12 2
1170.2.bs.f 8 39.k even 12 2
1950.2.y.j 8 65.o even 12 2
1950.2.y.k 8 65.t even 12 2
1950.2.bc.g 8 65.s odd 12 2
5070.2.a.bz 4 13.b even 2 1
5070.2.a.ca 4 1.a even 1 1 trivial
5070.2.b.ba 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{4} - 2 T_{7}^{3} - 19 T_{7}^{2} + 20 T_{7} + 52 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 34 T_{11}^{2} - 62 T_{11} + 181 \)
\( T_{17} - 4 \)
\( T_{31}^{4} - 4 T_{31}^{3} - 79 T_{31}^{2} + 556 T_{31} - 956 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 52 + 20 T - 19 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( 181 - 62 T - 34 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -4 + T )^{4} \)
$19$ \( 468 - 72 T - 63 T^{2} + T^{4} \)
$23$ \( 52 + 16 T - 43 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 376 + 148 T - 39 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( -956 + 556 T - 79 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( -803 - 430 T - 34 T^{2} + 10 T^{3} + T^{4} \)
$41$ \( 832 + 160 T - 76 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( -572 + 836 T - 43 T^{2} - 14 T^{3} + T^{4} \)
$47$ \( -1103 + 776 T - 82 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( 52 + 4 T - 75 T^{2} - 8 T^{3} + T^{4} \)
$59$ \( -284 - 216 T - 31 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( ( 4 - 8 T + T^{2} )^{2} \)
$67$ \( 352 + 144 T - 52 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( -5024 + 1936 T - 100 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( -4544 + 1728 T - 124 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( 3508 - 860 T - 147 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( -5024 + 1936 T - 100 T^{2} - 16 T^{3} + T^{4} \)
$89$ \( -1004 - 784 T - 115 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( -5408 + 624 T + 380 T^{2} + 36 T^{3} + T^{4} \)
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