Properties

Label 4730.2.a.y
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{9} - q^{10} - q^{11} + \beta_{1} q^{12} + ( 1 + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} ) q^{14} + \beta_{1} q^{15} + q^{16} + ( \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{18} + ( 1 + \beta_{2} + \beta_{5} - \beta_{7} ) q^{19} + q^{20} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{21} + q^{22} + ( 1 - \beta_{3} - \beta_{5} - \beta_{7} ) q^{23} -\beta_{1} q^{24} + q^{25} + ( -1 - \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{26} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{27} + ( 1 + \beta_{1} - \beta_{2} ) q^{28} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{29} -\beta_{1} q^{30} + ( 1 + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} - q^{32} -\beta_{1} q^{33} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{34} + ( 1 + \beta_{1} - \beta_{2} ) q^{35} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{36} + ( 1 + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{37} + ( -1 - \beta_{2} - \beta_{5} + \beta_{7} ) q^{38} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{39} - q^{40} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{41} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{42} - q^{43} - q^{44} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{45} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{46} + ( 1 - 3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{47} + \beta_{1} q^{48} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{49} - q^{50} + ( 3 + \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{51} + ( 1 + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{52} + ( -1 - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{54} - q^{55} + ( -1 - \beta_{1} + \beta_{2} ) q^{56} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{57} + ( \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{58} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{7} ) q^{59} + \beta_{1} q^{60} + ( 1 - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{61} + ( -1 - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{62} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{63} + q^{64} + ( 1 + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{65} + \beta_{1} q^{66} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} ) q^{68} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{69} + ( -1 - \beta_{1} + \beta_{2} ) q^{70} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{71} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{72} + ( 5 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} + ( -1 - \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{74} + \beta_{1} q^{75} + ( 1 + \beta_{2} + \beta_{5} - \beta_{7} ) q^{76} + ( -1 - \beta_{1} + \beta_{2} ) q^{77} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{78} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{79} + q^{80} + ( -3 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{81} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{82} + ( 4 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{83} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{84} + ( \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} ) q^{85} + q^{86} + ( 5 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{87} + q^{88} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{89} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{90} + ( 1 + 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - 5 \beta_{6} ) q^{91} + ( 1 - \beta_{3} - \beta_{5} - \beta_{7} ) q^{92} + ( 5 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} + ( -1 + 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{94} + ( 1 + \beta_{2} + \beta_{5} - \beta_{7} ) q^{95} -\beta_{1} q^{96} + ( 3 + \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{97} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{98} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 3q^{3} + 8q^{4} + 8q^{5} - 3q^{6} + 11q^{7} - 8q^{8} + 3q^{9} + O(q^{10}) \) \( 8q - 8q^{2} + 3q^{3} + 8q^{4} + 8q^{5} - 3q^{6} + 11q^{7} - 8q^{8} + 3q^{9} - 8q^{10} - 8q^{11} + 3q^{12} + 6q^{13} - 11q^{14} + 3q^{15} + 8q^{16} + 9q^{17} - 3q^{18} + 9q^{19} + 8q^{20} + 21q^{21} + 8q^{22} - 2q^{23} - 3q^{24} + 8q^{25} - 6q^{26} + 24q^{27} + 11q^{28} - 10q^{29} - 3q^{30} + 6q^{31} - 8q^{32} - 3q^{33} - 9q^{34} + 11q^{35} + 3q^{36} + 16q^{37} - 9q^{38} + 6q^{39} - 8q^{40} - 4q^{41} - 21q^{42} - 8q^{43} - 8q^{44} + 3q^{45} + 2q^{46} - 11q^{47} + 3q^{48} + 11q^{49} - 8q^{50} + 38q^{51} + 6q^{52} - 15q^{53} - 24q^{54} - 8q^{55} - 11q^{56} - 2q^{57} + 10q^{58} - 7q^{59} + 3q^{60} + 2q^{61} - 6q^{62} + 33q^{63} + 8q^{64} + 6q^{65} + 3q^{66} + 4q^{67} + 9q^{68} - 4q^{69} - 11q^{70} - 3q^{71} - 3q^{72} + 32q^{73} - 16q^{74} + 3q^{75} + 9q^{76} - 11q^{77} - 6q^{78} + 45q^{79} + 8q^{80} + 20q^{81} + 4q^{82} + 39q^{83} + 21q^{84} + 9q^{85} + 8q^{86} + 2q^{87} + 8q^{88} - 22q^{89} - 3q^{90} + 18q^{91} - 2q^{92} + 18q^{93} + 11q^{94} + 9q^{95} - 3q^{96} + 30q^{97} - 11q^{98} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 13 x^{6} - 6 x^{5} + 46 x^{4} + 26 x^{3} - 52 x^{2} - 20 x + 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{7} + 2 \nu^{6} + 23 \nu^{5} - 10 \nu^{4} - 71 \nu^{3} + 16 \nu^{2} + 56 \nu - 18 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} - 3 \nu^{6} - 22 \nu^{5} + 21 \nu^{4} + 66 \nu^{3} - 44 \nu^{2} - 50 \nu + 30 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} + 23 \nu^{5} - 23 \nu^{4} - 75 \nu^{3} + 56 \nu^{2} + 68 \nu - 42 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} + 23 \nu^{5} - 23 \nu^{4} - 75 \nu^{3} + 58 \nu^{2} + 68 \nu - 50 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 3 \nu^{6} + 35 \nu^{5} - 17 \nu^{4} - 108 \nu^{3} + 34 \nu^{2} + 82 \nu - 34 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 4 \nu^{6} - 34 \nu^{5} + 28 \nu^{4} + 105 \nu^{3} - 66 \nu^{2} - 86 \nu + 54 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 4 \nu^{6} - 34 \nu^{5} + 28 \nu^{4} + 105 \nu^{3} - 68 \nu^{2} - 82 \nu + 62 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 8\)\()/2\)
\(\nu^{4}\)\(=\)\(2 \beta_{7} + \beta_{6} + \beta_{5} + 12 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 27\)
\(\nu^{5}\)\(=\)\((\)\(35 \beta_{7} - 5 \beta_{6} + 22 \beta_{5} + 87 \beta_{4} - 79 \beta_{3} - 22 \beta_{2} - 18 \beta_{1} + 108\)\()/2\)
\(\nu^{6}\)\(=\)\(30 \beta_{7} + 13 \beta_{6} + 17 \beta_{5} + 128 \beta_{4} - 113 \beta_{3} - 30 \beta_{2} - 6 \beta_{1} + 231\)
\(\nu^{7}\)\(=\)\((\)\(293 \beta_{7} + 37 \beta_{6} + 206 \beta_{5} + 861 \beta_{4} - 749 \beta_{3} - 222 \beta_{2} - 150 \beta_{1} + 1196\)\()/2\)

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
−1.19586
−1.81167
0.939990
−1.27816
1.82453
3.17194
−2.24275
0.591975
−1.00000 −2.26092 1.00000 1.00000 2.26092 1.24841 −1.00000 2.11175 −1.00000
1.2 −1.00000 −1.26740 1.00000 1.00000 1.26740 −2.82729 −1.00000 −1.39369 −1.00000
1.3 −1.00000 −0.519165 1.00000 1.00000 0.519165 1.27228 −1.00000 −2.73047 −1.00000
1.4 −1.00000 −0.232604 1.00000 1.00000 0.232604 5.22723 −1.00000 −2.94590 −1.00000
1.5 −1.00000 −0.206773 1.00000 1.00000 0.206773 −1.71176 −1.00000 −2.95724 −1.00000
1.6 −1.00000 1.68714 1.00000 1.00000 −1.68714 1.43903 −1.00000 −0.153574 −1.00000
1.7 −1.00000 2.54582 1.00000 1.00000 −2.54582 1.88573 −1.00000 3.48118 −1.00000
1.8 −1.00000 3.25391 1.00000 1.00000 −3.25391 4.46635 −1.00000 7.58795 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 4730.2.a.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.y 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{13}^{8} - \cdots\)