Properties

Label 4730.2.a.bf
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 32 x^{11} - 5 x^{10} + 376 x^{9} + 100 x^{8} - 1985 x^{7} - 576 x^{6} + 4708 x^{5} + 889 x^{4} - 4774 x^{3} - 296 x^{2} + 1648 x - 128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(-1209723 \nu^{12} + 95374748 \nu^{11} - 33549200 \nu^{10} - 2743522569 \nu^{9} + 925347580 \nu^{8} + 27643360772 \nu^{7} - 4575966069 \nu^{6} - 113746877852 \nu^{5} - 29379276 \nu^{4} + 169384509677 \nu^{3} + 4304466878 \nu^{2} - 83293017552 \nu + 1796287264\)\()/ 2054583056 \)
\(\beta_{4}\)\(=\)\((\)\(-5481053 \nu^{12} + 43657412 \nu^{11} + 142029872 \nu^{10} - 1180471903 \nu^{9} - 1458988540 \nu^{8} + 10768673468 \nu^{7} + 8502787741 \nu^{6} - 36350024564 \nu^{5} - 29481549172 \nu^{4} + 27932078459 \nu^{3} + 39779836946 \nu^{2} + 9035066960 \nu - 10321684480\)\()/ 2054583056 \)
\(\beta_{5}\)\(=\)\((\)\(-1552491 \nu^{12} + 12856529 \nu^{11} + 28305288 \nu^{10} - 353658669 \nu^{9} - 66962761 \nu^{8} + 3432214148 \nu^{7} - 1065584245 \nu^{6} - 14002567981 \nu^{5} + 5222853248 \nu^{4} + 22264625561 \nu^{3} - 4442731421 \nu^{2} - 10640613778 \nu - 126650044\)\()/ 513645764 \)
\(\beta_{6}\)\(=\)\((\)\(-7498623 \nu^{12} + 119840596 \nu^{11} + 152682496 \nu^{10} - 3489939941 \nu^{9} - 1110548172 \nu^{8} + 36121754996 \nu^{7} + 5494069471 \nu^{6} - 157358169460 \nu^{5} - 21132901100 \nu^{4} + 263223860793 \nu^{3} + 12404294110 \nu^{2} - 133095724096 \nu + 9257057216\)\()/ 2054583056 \)
\(\beta_{7}\)\(=\)\((\)\(727017 \nu^{12} + 1618382 \nu^{11} - 22069016 \nu^{10} - 54545229 \nu^{9} + 235343334 \nu^{8} + 638419143 \nu^{7} - 1043909357 \nu^{6} - 3006216222 \nu^{5} + 1923637015 \nu^{4} + 4831785188 \nu^{3} - 2184380390 \nu^{2} - 1579243760 \nu + 617584666\)\()/ 128411441 \)
\(\beta_{8}\)\(=\)\((\)\(37859597 \nu^{12} + 18959624 \nu^{11} - 1105975856 \nu^{10} - 870485553 \nu^{9} + 11145550800 \nu^{8} + 12205431316 \nu^{7} - 43669040893 \nu^{6} - 63677149080 \nu^{5} + 50088512132 \nu^{4} + 107832991333 \nu^{3} - 7926495062 \nu^{2} - 53582557768 \nu + 1147504160\)\()/ 2054583056 \)
\(\beta_{9}\)\(=\)\((\)\(-21709793 \nu^{12} - 1131240 \nu^{11} + 660508136 \nu^{10} + 179115477 \nu^{9} - 7186292024 \nu^{8} - 3343610108 \nu^{7} + 33252100577 \nu^{6} + 19863052416 \nu^{5} - 60782498060 \nu^{4} - 36532740473 \nu^{3} + 36788542462 \nu^{2} + 18443062896 \nu - 2778200784\)\()/ 1027291528 \)
\(\beta_{10}\)\(=\)\((\)\(-57521037 \nu^{12} + 2964296 \nu^{11} + 1781628496 \nu^{10} + 152007089 \nu^{9} - 19772954064 \nu^{8} - 3198910436 \nu^{7} + 93374155469 \nu^{6} + 10580600648 \nu^{5} - 172412437988 \nu^{4} + 31477383355 \nu^{3} + 96187853574 \nu^{2} - 52892942840 \nu + 4872284112\)\()/ 2054583056 \)
\(\beta_{11}\)\(=\)\((\)\(-58907007 \nu^{12} - 134757996 \nu^{11} + 1894620016 \nu^{10} + 4238230907 \nu^{9} - 21372141628 \nu^{8} - 46129163084 \nu^{7} + 98233765135 \nu^{6} + 198773038460 \nu^{5} - 161245957452 \nu^{4} - 281557192999 \nu^{3} + 89669389822 \nu^{2} + 115425178880 \nu - 18968508576\)\()/ 2054583056 \)
\(\beta_{12}\)\(=\)\((\)\(-34294587 \nu^{12} - 28183806 \nu^{11} + 1057712496 \nu^{10} + 1016626975 \nu^{9} - 11529953354 \nu^{8} - 12220311388 \nu^{7} + 52055942691 \nu^{6} + 55935151726 \nu^{5} - 87424696364 \nu^{4} - 78411099859 \nu^{3} + 49091138148 \nu^{2} + 28811682716 \nu - 4427223552\)\()/ 1027291528 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{3} + 12 \beta_{2} + 2 \beta_{1} + 42\)
\(\nu^{5}\)\(=\)\(\beta_{11} - 2 \beta_{10} + 16 \beta_{9} + 15 \beta_{8} + \beta_{7} - 12 \beta_{6} + 15 \beta_{5} - 14 \beta_{4} + 12 \beta_{3} + 90 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(-14 \beta_{12} - 35 \beta_{11} + 34 \beta_{10} + 18 \beta_{9} - 11 \beta_{8} - \beta_{7} - 20 \beta_{6} + 7 \beta_{5} + \beta_{4} - 35 \beta_{3} + 136 \beta_{2} + 38 \beta_{1} + 403\)
\(\nu^{7}\)\(=\)\(-11 \beta_{12} + 18 \beta_{11} - 30 \beta_{10} + 214 \beta_{9} + 188 \beta_{8} + 15 \beta_{7} - 134 \beta_{6} + 195 \beta_{5} - 171 \beta_{4} + 125 \beta_{3} + 6 \beta_{2} + 949 \beta_{1} + 305\)
\(\nu^{8}\)\(=\)\(-173 \beta_{12} - 479 \beta_{11} + 467 \beta_{10} + 268 \beta_{9} - 88 \beta_{8} - 4 \beta_{7} - 320 \beta_{6} + 178 \beta_{5} + 12 \beta_{4} - 468 \beta_{3} + 1524 \beta_{2} + 561 \beta_{1} + 4093\)
\(\nu^{9}\)\(=\)\(-319 \beta_{12} + 234 \beta_{11} - 302 \beta_{10} + 2714 \beta_{9} + 2231 \beta_{8} + 177 \beta_{7} - 1542 \beta_{6} + 2442 \beta_{5} - 2003 \beta_{4} + 1262 \beta_{3} + 165 \beta_{2} + 10329 \beta_{1} + 3953\)
\(\nu^{10}\)\(=\)\(-2171 \beta_{12} - 5994 \beta_{11} + 5991 \beta_{10} + 3754 \beta_{9} - 466 \beta_{8} + 142 \beta_{7} - 4631 \beta_{6} + 3166 \beta_{5} + 69 \beta_{4} - 5715 \beta_{3} + 17032 \beta_{2} + 7657 \beta_{1} + 42977\)
\(\nu^{11}\)\(=\)\(-6101 \beta_{12} + 2654 \beta_{11} - 2223 \beta_{10} + 33580 \beta_{9} + 25975 \beta_{8} + 2029 \beta_{7} - 18340 \beta_{6} + 30181 \beta_{5} - 23026 \beta_{4} + 12628 \beta_{3} + 3111 \beta_{2} + 114645 \beta_{1} + 48941\)
\(\nu^{12}\)\(=\)\(-28049 \beta_{12} - 71881 \beta_{11} + 74580 \beta_{10} + 50863 \beta_{9} + 1043 \beta_{8} + 4072 \beta_{7} - 63229 \beta_{6} + 48708 \beta_{5} - 457 \beta_{4} - 67170 \beta_{3} + 190399 \beta_{2} + 100998 \beta_{1} + 461337\)

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
3.45880
3.26258
2.61361
1.30046
1.16663
0.743787
0.0802958
−0.921464
−1.13397
−2.15443
−2.24942
−2.86226
−3.30461
1.00000 −3.45880 1.00000 −1.00000 −3.45880 −3.54413 1.00000 8.96327 −1.00000
1.2 1.00000 −3.26258 1.00000 −1.00000 −3.26258 2.73152 1.00000 7.64446 −1.00000
1.3 1.00000 −2.61361 1.00000 −1.00000 −2.61361 0.881450 1.00000 3.83093 −1.00000
1.4 1.00000 −1.30046 1.00000 −1.00000 −1.30046 3.42310 1.00000 −1.30880 −1.00000
1.5 1.00000 −1.16663 1.00000 −1.00000 −1.16663 −3.85859 1.00000 −1.63898 −1.00000
1.6 1.00000 −0.743787 1.00000 −1.00000 −0.743787 2.37023 1.00000 −2.44678 −1.00000
1.7 1.00000 −0.0802958 1.00000 −1.00000 −0.0802958 2.53237 1.00000 −2.99355 −1.00000
1.8 1.00000 0.921464 1.00000 −1.00000 0.921464 −3.84192 1.00000 −2.15090 −1.00000
1.9 1.00000 1.13397 1.00000 −1.00000 1.13397 −0.848165 1.00000 −1.71412 −1.00000
1.10 1.00000 2.15443 1.00000 −1.00000 2.15443 5.20049 1.00000 1.64159 −1.00000
1.11 1.00000 2.24942 1.00000 −1.00000 2.24942 −1.43432 1.00000 2.05989 −1.00000
1.12 1.00000 2.86226 1.00000 −1.00000 2.86226 4.06491 1.00000 5.19253 −1.00000
1.13 1.00000 3.30461 1.00000 −1.00000 3.30461 −0.676945 1.00000 7.92047 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.bf 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.bf 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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