Properties

Label 21.30.e.a
Level $21$
Weight $30$
Character orbit 21.e
Analytic conductor $111.884$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 38q + 25025q^{2} + 90876411q^{3} - 5448583733q^{4} - 2101591778q^{5} + 239387598450q^{6} - 1147384421099q^{7} - 53840580588510q^{8} - 434659056644259q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 38q + 25025q^{2} + 90876411q^{3} - 5448583733q^{4} - 2101591778q^{5} + 239387598450q^{6} - 1147384421099q^{7} - 53840580588510q^{8} - 434659056644259q^{9} - 226333527764279q^{10} + 968182184010316q^{11} + 26060407088843277q^{12} + 30248752986495458q^{13} + 188292059868669736q^{14} - 20103696649657764q^{15} - 1414287713687569841q^{16} + 102438111755912556q^{17} + 572491731185399025q^{18} - 6872193027481150651q^{19} + 16866263496288505610q^{20} - 5139459612060640272q^{21} + 84243517982597870158q^{22} + \)\(18\!\cdots\!16\)\(q^{23} - \)\(12\!\cdots\!95\)\(q^{24} - \)\(38\!\cdots\!11\)\(q^{25} + \)\(27\!\cdots\!88\)\(q^{26} - \)\(41\!\cdots\!42\)\(q^{27} + \)\(44\!\cdots\!39\)\(q^{28} + \)\(28\!\cdots\!44\)\(q^{29} + \)\(10\!\cdots\!51\)\(q^{30} - \)\(49\!\cdots\!83\)\(q^{31} + \)\(88\!\cdots\!35\)\(q^{32} - \)\(46\!\cdots\!04\)\(q^{33} + \)\(50\!\cdots\!84\)\(q^{34} - \)\(81\!\cdots\!66\)\(q^{35} + \)\(24\!\cdots\!26\)\(q^{36} + \)\(89\!\cdots\!69\)\(q^{37} - \)\(71\!\cdots\!56\)\(q^{38} + \)\(72\!\cdots\!01\)\(q^{39} + \)\(23\!\cdots\!35\)\(q^{40} - \)\(87\!\cdots\!24\)\(q^{41} + \)\(42\!\cdots\!17\)\(q^{42} - \)\(99\!\cdots\!58\)\(q^{43} - \)\(27\!\cdots\!17\)\(q^{44} - \)\(48\!\cdots\!58\)\(q^{45} + \)\(14\!\cdots\!52\)\(q^{46} + \)\(66\!\cdots\!02\)\(q^{47} - \)\(13\!\cdots\!58\)\(q^{48} + \)\(12\!\cdots\!37\)\(q^{49} - \)\(18\!\cdots\!48\)\(q^{50} - \)\(48\!\cdots\!64\)\(q^{51} + \)\(31\!\cdots\!64\)\(q^{52} - \)\(77\!\cdots\!12\)\(q^{53} - \)\(27\!\cdots\!25\)\(q^{54} + \)\(10\!\cdots\!56\)\(q^{55} + \)\(43\!\cdots\!53\)\(q^{56} - \)\(65\!\cdots\!38\)\(q^{57} + \)\(10\!\cdots\!03\)\(q^{58} + \)\(11\!\cdots\!10\)\(q^{59} + \)\(40\!\cdots\!45\)\(q^{60} - \)\(13\!\cdots\!86\)\(q^{61} + \)\(11\!\cdots\!42\)\(q^{62} + \)\(16\!\cdots\!71\)\(q^{63} - \)\(14\!\cdots\!22\)\(q^{64} + \)\(64\!\cdots\!54\)\(q^{65} + \)\(20\!\cdots\!51\)\(q^{66} - \)\(56\!\cdots\!17\)\(q^{67} - \)\(69\!\cdots\!52\)\(q^{68} + \)\(17\!\cdots\!08\)\(q^{69} + \)\(43\!\cdots\!57\)\(q^{70} - \)\(18\!\cdots\!64\)\(q^{71} + \)\(61\!\cdots\!55\)\(q^{72} + \)\(17\!\cdots\!67\)\(q^{73} - \)\(42\!\cdots\!48\)\(q^{74} + \)\(18\!\cdots\!59\)\(q^{75} + \)\(15\!\cdots\!12\)\(q^{76} + \)\(68\!\cdots\!56\)\(q^{77} + \)\(26\!\cdots\!44\)\(q^{78} + \)\(31\!\cdots\!39\)\(q^{79} + \)\(46\!\cdots\!15\)\(q^{80} - \)\(99\!\cdots\!99\)\(q^{81} - \)\(15\!\cdots\!24\)\(q^{82} + \)\(12\!\cdots\!92\)\(q^{83} - \)\(10\!\cdots\!96\)\(q^{84} + \)\(28\!\cdots\!00\)\(q^{85} + \)\(14\!\cdots\!74\)\(q^{86} + \)\(67\!\cdots\!68\)\(q^{87} - \)\(20\!\cdots\!17\)\(q^{88} - \)\(26\!\cdots\!12\)\(q^{89} + \)\(10\!\cdots\!38\)\(q^{90} - \)\(22\!\cdots\!79\)\(q^{91} - \)\(37\!\cdots\!76\)\(q^{92} + \)\(23\!\cdots\!27\)\(q^{93} + \)\(73\!\cdots\!98\)\(q^{94} - \)\(56\!\cdots\!94\)\(q^{95} - \)\(42\!\cdots\!15\)\(q^{96} + \)\(23\!\cdots\!28\)\(q^{97} - \)\(29\!\cdots\!15\)\(q^{98} - \)\(44\!\cdots\!52\)\(q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −21682.7 37555.5i 2.39148e6 4.14217e6i −6.71844e8 + 1.16367e9i −3.79032e9 6.56503e9i −2.07415e11 −1.42775e12 + 1.08693e12i 3.49879e13 −1.14384e13 1.98119e13i −1.64369e14 + 2.84695e14i
4.2 −19318.8 33461.1i 2.39148e6 4.14217e6i −4.77996e8 + 8.27914e8i −7.79808e8 1.35067e9i −1.84802e11 −1.66873e11 1.78663e12i 1.61939e13 −1.14384e13 1.98119e13i −3.01299e13 + 5.21865e13i
4.3 −19095.2 33073.8i 2.39148e6 4.14217e6i −4.60816e8 + 7.98156e8i 9.98170e9 + 1.72888e10i −1.82663e11 1.32001e12 + 1.21552e12i 1.46941e13 −1.14384e13 1.98119e13i 3.81205e14 6.60266e14i
4.4 −13945.7 24154.6i 2.39148e6 4.14217e6i −1.20528e8 + 2.08761e8i −1.12432e10 1.94737e10i −1.33403e11 1.74806e12 + 4.05197e11i −8.25066e12 −1.14384e13 1.98119e13i −3.13587e14 + 5.43149e14i
4.5 −13747.8 23811.9i 2.39148e6 4.14217e6i −1.09569e8 + 1.89779e8i −5.45988e9 9.45678e9i −1.31511e11 −1.44405e12 + 1.06518e12i −8.73628e12 −1.14384e13 1.98119e13i −1.50123e14 + 2.60020e14i
4.6 −10085.3 17468.3i 2.39148e6 4.14217e6i 6.50070e7 1.12595e8i 4.83495e9 + 8.37438e9i −9.64758e10 −7.77849e11 + 1.61705e12i −1.34515e13 −1.14384e13 1.98119e13i 9.75244e13 1.68917e14i
4.7 −9506.59 16465.9i 2.39148e6 4.14217e6i 8.76848e7 1.51874e8i 4.01268e9 + 6.95016e9i −9.09395e10 1.51909e12 9.55123e11i −1.35420e13 −1.14384e13 1.98119e13i 7.62938e13 1.32145e14i
4.8 −5312.73 9201.91i 2.39148e6 4.14217e6i 2.11985e8 3.67169e8i 6.21129e8 + 1.07583e9i −5.08212e10 −1.19318e12 1.34023e12i −1.02094e13 −1.14384e13 1.98119e13i 6.59978e12 1.14312e13i
4.9 −803.934 1392.45i 2.39148e6 4.14217e6i 2.67143e8 4.62705e8i −1.04266e10 1.80594e10i −7.69038e9 −3.52235e11 1.75950e12i −1.72228e12 −1.14384e13 1.98119e13i −1.67646e13 + 2.90371e13i
4.10 1314.29 + 2276.42i 2.39148e6 4.14217e6i 2.64981e8 4.58960e8i 1.11285e10 + 1.92751e10i 1.25724e10 −1.79440e12 + 6.33421e9i 2.80425e12 −1.14384e13 1.98119e13i −2.92521e13 + 5.06661e13i
4.11 3171.13 + 5492.56i 2.39148e6 4.14217e6i 2.48323e8 4.30109e8i −5.04465e9 8.73759e9i 3.03349e10 7.64524e11 + 1.62339e12i 6.55484e12 −1.14384e13 1.98119e13i 3.19945e13 5.54161e13i
4.12 4961.84 + 8594.16i 2.39148e6 4.14217e6i 2.19196e8 3.79658e8i 8.80219e9 + 1.52458e10i 4.74646e10 1.78765e12 + 1.55597e11i 9.67819e12 −1.14384e13 1.98119e13i −8.73501e13 + 1.51295e14i
4.13 10661.0 + 18465.4i 2.39148e6 4.14217e6i 4.11213e7 7.12241e7i −5.16991e9 8.95455e9i 1.01983e11 −1.66601e12 + 6.66564e11i 1.32007e13 −1.14384e13 1.98119e13i 1.10233e14 1.90929e14i
4.14 11213.9 + 19423.1i 2.39148e6 4.14217e6i 1.69302e7 2.93240e7i 4.19413e9 + 7.26445e9i 1.07272e11 4.56481e11 1.73538e12i 1.28003e13 −1.14384e13 1.98119e13i −9.40656e13 + 1.62926e14i
4.15 14667.6 + 25405.0i 2.39148e6 4.14217e6i −1.61841e8 + 2.80316e8i −8.78577e9 1.52174e10i 1.40309e11 1.75086e12 3.92932e11i 6.25395e12 −1.14384e13 1.98119e13i 2.57732e14 4.46405e14i
4.16 17773.2 + 30784.0i 2.39148e6 4.14217e6i −3.63334e8 + 6.29314e8i 6.21359e9 + 1.07623e10i 1.70017e11 −3.33772e10 + 1.79410e12i −6.74661e12 −1.14384e13 1.98119e13i −2.20870e14 + 3.82558e14i
4.17 20203.6 + 34993.6i 2.39148e6 4.14217e6i −5.47933e8 + 9.49048e8i 5.22782e7 + 9.05485e7i 1.93266e11 1.78686e12 + 1.64388e11i −2.25874e13 −1.14384e13 1.98119e13i −2.11241e12 + 3.65880e12i
4.18 20330.6 + 35213.6i 2.39148e6 4.14217e6i −5.58229e8 + 9.66881e8i 9.23100e9 + 1.59886e10i 1.94481e11 −1.57660e12 8.56870e11i −2.35667e13 −1.14384e13 1.98119e13i −3.75343e14 + 6.50113e14i
4.19 21714.1 + 37610.0i 2.39148e6 4.14217e6i −6.74573e8 + 1.16839e9i −9.42284e9 1.63208e10i 2.07716e11 −1.27489e12 1.26276e12i −3.52757e13 −1.14384e13 1.98119e13i 4.09218e14 7.08786e14i
16.1 −21682.7 + 37555.5i 2.39148e6 + 4.14217e6i −6.71844e8 1.16367e9i −3.79032e9 + 6.56503e9i −2.07415e11 −1.42775e12 1.08693e12i 3.49879e13 −1.14384e13 + 1.98119e13i −1.64369e14 2.84695e14i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.30.e.a 38
7.c even 3 1 inner 21.30.e.a 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.e.a 38 1.a even 1 1 trivial
21.30.e.a 38 7.c even 3 1 inner