Properties

Label 4730.2.a.z
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 17 x^{8} + 21 x^{7} + 107 x^{6} - 45 x^{5} - 262 x^{4} - 47 x^{3} + 120 x^{2} - 2 x - 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -17 \nu^{9} + 65 \nu^{8} + 183 \nu^{7} - 745 \nu^{6} - 615 \nu^{5} + 2538 \nu^{4} + 657 \nu^{3} - 2504 \nu^{2} + 275 \nu + 159 \)\()/71\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{9} + 15 \nu^{8} - 209 \nu^{7} - 232 \nu^{6} + 1715 \nu^{5} + 1536 \nu^{4} - 4518 \nu^{3} - 3964 \nu^{2} + 1691 \nu + 348 \)\()/71\)
\(\beta_{5}\)\(=\)\((\)\( -26 \nu^{9} + 66 \nu^{8} + 401 \nu^{7} - 751 \nu^{6} - 2252 \nu^{5} + 2186 \nu^{4} + 4772 \nu^{3} - 501 \nu^{2} - 1037 \nu + 97 \)\()/71\)
\(\beta_{6}\)\(=\)\((\)\( 24 \nu^{9} - 50 \nu^{8} - 392 \nu^{7} + 513 \nu^{6} + 2330 \nu^{5} - 1002 \nu^{4} - 5104 \nu^{3} - 1531 \nu^{2} + 990 \nu + 189 \)\()/71\)
\(\beta_{7}\)\(=\)\((\)\( 46 \nu^{9} - 155 \nu^{8} - 562 \nu^{7} + 1711 \nu^{6} + 2466 \nu^{5} - 5151 \nu^{4} - 4363 \nu^{3} + 2858 \nu^{2} + 87 \nu + 167 \)\()/71\)
\(\beta_{8}\)\(=\)\((\)\( -42 \nu^{9} + 194 \nu^{8} + 331 \nu^{7} - 2158 \nu^{6} - 350 \nu^{5} + 7185 \nu^{4} - 1292 \nu^{3} - 7243 \nu^{2} + 2421 \nu + 326 \)\()/71\)
\(\beta_{9}\)\(=\)\((\)\( 95 \nu^{9} - 334 \nu^{8} - 1102 \nu^{7} + 3637 \nu^{6} + 4602 \nu^{5} - 10942 \nu^{4} - 8299 \nu^{3} + 7060 \nu^{2} + 1061 \nu - 450 \)\()/71\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 7 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(\beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 13 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(\beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 12 \beta_{6} + 2 \beta_{5} - 15 \beta_{4} + 14 \beta_{3} + 15 \beta_{2} + 59 \beta_{1} + 53\)
\(\nu^{6}\)\(=\)\(2 \beta_{9} + 18 \beta_{8} + 28 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} - 40 \beta_{4} + 34 \beta_{3} + 85 \beta_{2} + 141 \beta_{1} + 241\)
\(\nu^{7}\)\(=\)\(16 \beta_{9} + 58 \beta_{8} + 62 \beta_{7} + 137 \beta_{6} + 47 \beta_{5} - 188 \beta_{4} + 158 \beta_{3} + 192 \beta_{2} + 541 \beta_{1} + 596\)
\(\nu^{8}\)\(=\)\(42 \beta_{9} + 251 \beta_{8} + 331 \beta_{7} + 355 \beta_{6} + 225 \beta_{5} - 569 \beta_{4} + 433 \beta_{3} + 851 \beta_{2} + 1470 \beta_{1} + 2306\)
\(\nu^{9}\)\(=\)\(209 \beta_{9} + 836 \beta_{8} + 896 \beta_{7} + 1578 \beta_{6} + 742 \beta_{5} - 2241 \beta_{4} + 1693 \beta_{3} + 2288 \beta_{2} + 5211 \beta_{1} + 6509\)

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
3.31796
2.88983
2.27147
0.523103
0.274153
−0.216680
−1.41316
−1.53249
−1.68568
−2.42851
−1.00000 −2.31796 1.00000 1.00000 2.31796 4.25200 −1.00000 2.37293 −1.00000
1.2 −1.00000 −1.88983 1.00000 1.00000 1.88983 −3.45622 −1.00000 0.571445 −1.00000
1.3 −1.00000 −1.27147 1.00000 1.00000 1.27147 −0.316678 −1.00000 −1.38335 −1.00000
1.4 −1.00000 0.476897 1.00000 1.00000 −0.476897 2.32668 −1.00000 −2.77257 −1.00000
1.5 −1.00000 0.725847 1.00000 1.00000 −0.725847 −1.66243 −1.00000 −2.47315 −1.00000
1.6 −1.00000 1.21668 1.00000 1.00000 −1.21668 3.59770 −1.00000 −1.51969 −1.00000
1.7 −1.00000 2.41316 1.00000 1.00000 −2.41316 −5.00890 −1.00000 2.82333 −1.00000
1.8 −1.00000 2.53249 1.00000 1.00000 −2.53249 −0.583820 −1.00000 3.41349 −1.00000
1.9 −1.00000 2.68568 1.00000 1.00000 −2.68568 4.03613 −1.00000 4.21287 −1.00000
1.10 −1.00000 3.42851 1.00000 1.00000 −3.42851 −0.184466 −1.00000 8.75468 −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.z 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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