Properties

Label 4730.2.a.ba
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 21 x^{8} + 22 x^{7} + 138 x^{6} - 154 x^{5} - 291 x^{4} + 327 x^{3} + 97 x^{2} - 124 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -21 \nu^{9} - 5 \nu^{8} + 445 \nu^{7} + 38 \nu^{6} - 3014 \nu^{5} + 338 \nu^{4} + 7253 \nu^{3} - 1627 \nu^{2} - 4673 \nu + 864 \)\()/214\)
\(\beta_{4}\)\(=\)\((\)\( -349 \nu^{9} - 185 \nu^{8} + 6621 \nu^{7} + 2048 \nu^{6} - 37474 \nu^{5} + 2876 \nu^{4} + 66987 \nu^{3} - 33663 \nu^{2} - 23529 \nu + 13136 \)\()/2354\)
\(\beta_{5}\)\(=\)\((\)\( 328 \nu^{9} + 501 \nu^{8} - 5855 \nu^{7} - 8002 \nu^{6} + 29324 \nu^{5} + 29990 \nu^{4} - 41116 \nu^{3} - 20073 \nu^{2} + 6765 \nu - 181 \)\()/1177\)
\(\beta_{6}\)\(=\)\((\)\( 1005 \nu^{9} + 1187 \nu^{8} - 18331 \nu^{7} - 18052 \nu^{6} + 96122 \nu^{5} + 57104 \nu^{4} - 151573 \nu^{3} - 6483 \nu^{2} + 53537 \nu - 15852 \)\()/2354\)
\(\beta_{7}\)\(=\)\((\)\( -53 \nu^{9} - 33 \nu^{8} + 1011 \nu^{7} + 422 \nu^{6} - 5747 \nu^{5} - 380 \nu^{4} + 10270 \nu^{3} - 3077 \nu^{2} - 3728 \nu + 1294 \)\()/107\)
\(\beta_{8}\)\(=\)\((\)\( 679 \nu^{9} + 768 \nu^{8} - 12605 \nu^{7} - 11893 \nu^{6} + 68527 \nu^{5} + 40574 \nu^{4} - 116564 \nu^{3} - 19797 \nu^{2} + 45342 \nu - 1828 \)\()/1177\)
\(\beta_{9}\)\(=\)\((\)\( -1497 \nu^{9} - 2527 \nu^{8} + 26525 \nu^{7} + 40648 \nu^{6} - 130692 \nu^{5} - 157408 \nu^{4} + 180291 \nu^{3} + 126633 \nu^{2} - 50149 \nu - 9182 \)\()/2354\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 10 \beta_{2} - \beta_{1} + 31\)
\(\nu^{5}\)\(=\)\(2 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} + 9 \beta_{5} - 17 \beta_{4} + 56 \beta_{1} - 11\)
\(\nu^{6}\)\(=\)\(-17 \beta_{9} + 11 \beta_{8} - 31 \beta_{6} - 29 \beta_{5} - 36 \beta_{4} + 12 \beta_{3} + 96 \beta_{2} - 12 \beta_{1} + 274\)
\(\nu^{7}\)\(=\)\(\beta_{9} + 36 \beta_{8} + 49 \beta_{7} - 109 \beta_{6} + 74 \beta_{5} - 208 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 491 \beta_{1} - 91\)
\(\nu^{8}\)\(=\)\(-217 \beta_{9} + 100 \beta_{8} - \beta_{7} - 385 \beta_{6} - 323 \beta_{5} - 468 \beta_{4} + 114 \beta_{3} + 932 \beta_{2} - 86 \beta_{1} + 2558\)
\(\nu^{9}\)\(=\)\(26 \beta_{9} + 488 \beta_{8} + 608 \beta_{7} - 1073 \beta_{6} + 614 \beta_{5} - 2299 \beta_{4} + 170 \beta_{3} + 226 \beta_{2} + 4545 \beta_{1} - 578\)

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
3.24563
2.19566
2.18792
1.10594
0.530117
0.156987
−0.721995
−1.74331
−2.86415
−3.09280
1.00000 −3.24563 1.00000 −1.00000 −3.24563 −2.81488 1.00000 7.53411 −1.00000
1.2 1.00000 −2.19566 1.00000 −1.00000 −2.19566 4.26177 1.00000 1.82093 −1.00000
1.3 1.00000 −2.18792 1.00000 −1.00000 −2.18792 −0.664484 1.00000 1.78700 −1.00000
1.4 1.00000 −1.10594 1.00000 −1.00000 −1.10594 2.85930 1.00000 −1.77690 −1.00000
1.5 1.00000 −0.530117 1.00000 −1.00000 −0.530117 −4.20387 1.00000 −2.71898 −1.00000
1.6 1.00000 −0.156987 1.00000 −1.00000 −0.156987 −3.72319 1.00000 −2.97536 −1.00000
1.7 1.00000 0.721995 1.00000 −1.00000 0.721995 3.23576 1.00000 −2.47872 −1.00000
1.8 1.00000 1.74331 1.00000 −1.00000 1.74331 3.90137 1.00000 0.0391392 −1.00000
1.9 1.00000 2.86415 1.00000 −1.00000 2.86415 −2.27749 1.00000 5.20338 −1.00000
1.10 1.00000 3.09280 1.00000 −1.00000 3.09280 2.42572 1.00000 6.56539 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.ba 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.ba 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{13}^{10} - \cdots\)