Properties

Label 4730.2.a.bb
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 22 x^{9} + 21 x^{8} + 165 x^{7} - 130 x^{6} - 535 x^{5} + 323 x^{4} + 710 x^{3} - 343 x^{2} - 261 x + 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 179 \nu^{10} - 98 \nu^{9} - 3948 \nu^{8} + 1857 \nu^{7} + 29718 \nu^{6} - 7762 \nu^{5} - 95629 \nu^{4} + 4348 \nu^{3} + 121284 \nu^{2} + 10075 \nu - 37066 \)\()/106\)
\(\beta_{4}\)\(=\)\((\)\( 147 \nu^{10} - 79 \nu^{9} - 3228 \nu^{8} + 1504 \nu^{7} + 24177 \nu^{6} - 6282 \nu^{5} - 77497 \nu^{4} + 3571 \nu^{3} + 98197 \nu^{2} + 8106 \nu - 30135 \)\()/53\)
\(\beta_{5}\)\(=\)\((\)\( 337 \nu^{10} - 162 \nu^{9} - 7450 \nu^{8} + 3113 \nu^{7} + 56338 \nu^{6} - 12764 \nu^{5} - 182201 \nu^{4} + 5074 \nu^{3} + 232742 \nu^{2} + 20953 \nu - 72106 \)\()/106\)
\(\beta_{6}\)\(=\)\((\)\( 177 \nu^{10} - 67 \nu^{9} - 3956 \nu^{8} + 1295 \nu^{7} + 30382 \nu^{6} - 4834 \nu^{5} - 99703 \nu^{4} - 1090 \nu^{3} + 128967 \nu^{2} + 12963 \nu - 40313 \)\()/53\)
\(\beta_{7}\)\(=\)\((\)\( -421 \nu^{10} + 192 \nu^{9} + 9340 \nu^{8} - 3715 \nu^{7} - 70956 \nu^{6} + 15218 \nu^{5} + 230195 \nu^{4} - 5464 \nu^{3} - 294094 \nu^{2} - 26115 \nu + 90598 \)\()/106\)
\(\beta_{8}\)\(=\)\((\)\( -325 \nu^{10} + 135 \nu^{9} + 7233 \nu^{8} - 2603 \nu^{7} - 55234 \nu^{6} + 10089 \nu^{5} + 180357 \nu^{4} - 271 \nu^{3} - 232306 \nu^{2} - 23017 \nu + 72296 \)\()/53\)
\(\beta_{9}\)\(=\)\((\)\( 325 \nu^{10} - 135 \nu^{9} - 7233 \nu^{8} + 2603 \nu^{7} + 55234 \nu^{6} - 10089 \nu^{5} - 180357 \nu^{4} + 324 \nu^{3} + 232306 \nu^{2} + 22646 \nu - 72296 \)\()/53\)
\(\beta_{10}\)\(=\)\((\)\( -474 \nu^{10} + 192 \nu^{9} + 10559 \nu^{8} - 3715 \nu^{7} - 80708 \nu^{6} + 14370 \nu^{5} + 263479 \nu^{4} + 48 \nu^{3} - 338879 \nu^{2} - 33429 \nu + 105438 \)\()/53\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} + 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} + 10 \beta_{2} - 3 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(-\beta_{10} + 13 \beta_{9} + 13 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 4 \beta_{2} + 60 \beta_{1} - 4\)
\(\nu^{6}\)\(=\)\(2 \beta_{10} - 16 \beta_{9} - 32 \beta_{8} - 15 \beta_{7} - 13 \beta_{6} - 42 \beta_{5} + 14 \beta_{4} - \beta_{3} + 98 \beta_{2} - 59 \beta_{1} + 211\)
\(\nu^{7}\)\(=\)\(-14 \beta_{10} + 145 \beta_{9} + 149 \beta_{8} + 31 \beta_{7} - 26 \beta_{6} + 58 \beta_{5} - 35 \beta_{4} + 13 \beta_{3} - 84 \beta_{2} + 571 \beta_{1} - 113\)
\(\nu^{8}\)\(=\)\(35 \beta_{10} - 216 \beta_{9} - 405 \beta_{8} - 176 \beta_{7} - 135 \beta_{6} - 475 \beta_{5} + 164 \beta_{4} - 23 \beta_{3} + 979 \beta_{2} - 861 \beta_{1} + 1929\)
\(\nu^{9}\)\(=\)\(-164 \beta_{10} + 1554 \beta_{9} + 1658 \beta_{8} + 392 \beta_{7} - 253 \beta_{6} + 820 \beta_{5} - 452 \beta_{4} + 130 \beta_{3} - 1281 \beta_{2} + 5744 \beta_{1} - 1947\)
\(\nu^{10}\)\(=\)\(452 \beta_{10} - 2756 \beta_{9} - 4787 \beta_{8} - 1946 \beta_{7} - 1309 \beta_{6} - 5129 \beta_{5} + 1856 \beta_{4} - 361 \beta_{3} + 9984 \beta_{2} - 11201 \beta_{1} + 18835\)

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
2.87555
2.78036
2.17792
1.48284
0.580450
0.577714
−0.797340
−1.56687
−1.80931
−1.95364
−3.34767
−1.00000 −2.87555 1.00000 −1.00000 2.87555 −0.638121 −1.00000 5.26880 1.00000
1.2 −1.00000 −2.78036 1.00000 −1.00000 2.78036 −2.23633 −1.00000 4.73038 1.00000
1.3 −1.00000 −2.17792 1.00000 −1.00000 2.17792 4.63803 −1.00000 1.74332 1.00000
1.4 −1.00000 −1.48284 1.00000 −1.00000 1.48284 4.82678 −1.00000 −0.801199 1.00000
1.5 −1.00000 −0.580450 1.00000 −1.00000 0.580450 1.64698 −1.00000 −2.66308 1.00000
1.6 −1.00000 −0.577714 1.00000 −1.00000 0.577714 −2.69986 −1.00000 −2.66625 1.00000
1.7 −1.00000 0.797340 1.00000 −1.00000 −0.797340 −1.54185 −1.00000 −2.36425 1.00000
1.8 −1.00000 1.56687 1.00000 −1.00000 −1.56687 −4.55535 −1.00000 −0.544925 1.00000
1.9 −1.00000 1.80931 1.00000 −1.00000 −1.80931 4.53794 −1.00000 0.273608 1.00000
1.10 −1.00000 1.95364 1.00000 −1.00000 −1.95364 0.00701571 −1.00000 0.816706 1.00000
1.11 −1.00000 3.34767 1.00000 −1.00000 −3.34767 2.01478 −1.00000 8.20687 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.bb 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bb 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\(T_{3}^{11} + \cdots\)
\(T_{7}^{11} - \cdots\)
\(T_{13}^{11} - \cdots\)