Properties

Label 21.33.h.b
Level $21$
Weight $33$
Character orbit 21.h
Analytic conductor $136.220$
Analytic rank $0$
Dimension $164$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(136.219975799\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164q + 43046720q^{3} + 176093659134q^{4} + 3915256954876q^{6} - 41868332499126q^{7} + 1232720974221986q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 164q + 43046720q^{3} + 176093659134q^{4} + 3915256954876q^{6} - 41868332499126q^{7} + 1232720974221986q^{9} - 653609167486978q^{10} - 80283770812252892q^{12} + 930128021838524592q^{13} + 295985739739384604q^{15} - \)\(35\!\cdots\!98\)\(q^{16} - \)\(28\!\cdots\!84\)\(q^{18} - \)\(55\!\cdots\!32\)\(q^{19} - \)\(63\!\cdots\!60\)\(q^{21} - \)\(13\!\cdots\!96\)\(q^{22} + \)\(10\!\cdots\!46\)\(q^{24} + \)\(37\!\cdots\!44\)\(q^{25} - \)\(46\!\cdots\!76\)\(q^{27} + \)\(47\!\cdots\!58\)\(q^{28} - \)\(33\!\cdots\!80\)\(q^{30} + \)\(52\!\cdots\!26\)\(q^{31} - \)\(73\!\cdots\!26\)\(q^{33} - \)\(22\!\cdots\!96\)\(q^{34} - \)\(58\!\cdots\!40\)\(q^{36} - \)\(47\!\cdots\!40\)\(q^{37} + \)\(80\!\cdots\!20\)\(q^{39} - \)\(45\!\cdots\!26\)\(q^{40} + \)\(10\!\cdots\!78\)\(q^{42} - \)\(82\!\cdots\!28\)\(q^{43} + \)\(88\!\cdots\!80\)\(q^{45} + \)\(30\!\cdots\!92\)\(q^{46} - \)\(10\!\cdots\!00\)\(q^{48} + \)\(42\!\cdots\!90\)\(q^{49} + \)\(89\!\cdots\!16\)\(q^{51} + \)\(39\!\cdots\!84\)\(q^{52} + \)\(16\!\cdots\!78\)\(q^{54} + \)\(55\!\cdots\!00\)\(q^{55} - \)\(88\!\cdots\!80\)\(q^{57} + \)\(63\!\cdots\!34\)\(q^{58} + \)\(69\!\cdots\!10\)\(q^{60} - \)\(49\!\cdots\!48\)\(q^{61} + \)\(38\!\cdots\!20\)\(q^{63} - \)\(17\!\cdots\!24\)\(q^{64} + \)\(20\!\cdots\!84\)\(q^{66} + \)\(41\!\cdots\!36\)\(q^{67} - \)\(54\!\cdots\!68\)\(q^{69} - \)\(18\!\cdots\!50\)\(q^{70} + \)\(88\!\cdots\!12\)\(q^{72} + \)\(14\!\cdots\!40\)\(q^{73} - \)\(13\!\cdots\!24\)\(q^{75} - \)\(13\!\cdots\!12\)\(q^{76} - \)\(15\!\cdots\!20\)\(q^{78} + \)\(11\!\cdots\!98\)\(q^{79} + \)\(41\!\cdots\!98\)\(q^{81} - \)\(17\!\cdots\!72\)\(q^{82} + \)\(23\!\cdots\!36\)\(q^{84} - \)\(11\!\cdots\!64\)\(q^{85} + \)\(25\!\cdots\!98\)\(q^{87} - \)\(10\!\cdots\!90\)\(q^{88} - \)\(13\!\cdots\!64\)\(q^{90} + \)\(33\!\cdots\!84\)\(q^{91} - \)\(88\!\cdots\!00\)\(q^{93} - \)\(96\!\cdots\!24\)\(q^{94} + \)\(14\!\cdots\!10\)\(q^{96} - \)\(26\!\cdots\!24\)\(q^{97} - \)\(24\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −111997. 64661.2i 4.13267e7 + 1.20467e7i 6.21466e9 + 1.07641e10i 1.71366e11 + 9.89381e10i −3.84949e12 4.02143e12i 2.06412e13 + 2.60455e13i 1.05196e15i 1.56277e15 + 9.95702e14i −1.27949e16 2.21615e16i
2.2 −110635. 63875.0i 2.07987e7 + 3.76886e7i 6.01253e9 + 1.04140e10i −2.09739e11 1.21093e11i 1.06302e11 5.49819e12i −3.25771e13 6.56963e12i 9.87520e14i −9.87847e14 + 1.56775e15i 1.54696e16 + 2.67941e16i
2.3 −107705. 62183.5i −3.45696e7 + 2.56508e7i 5.58609e9 + 9.67539e9i 4.70485e10 + 2.71635e10i 5.31837e12 6.13057e11i −5.64089e12 3.27507e13i 8.55298e14i 5.37095e14 1.77347e15i −3.37824e15 5.85128e15i
2.4 −103808. 59933.5i −9.97571e6 4.18749e7i 5.03657e9 + 8.72360e9i −1.39609e11 8.06034e10i −1.47415e12 + 4.94482e12i 2.67513e13 + 1.97179e13i 6.92614e14i −1.65399e15 + 8.35463e14i 9.66169e15 + 1.67345e16i
2.5 −102259. 59039.4i 1.92280e7 3.85137e7i 4.82383e9 + 8.35512e9i 7.05959e10 + 4.07586e10i −4.24007e12 + 2.80317e12i −3.32321e13 + 2.35838e11i 6.32040e14i −1.11359e15 1.48108e15i −4.81273e15 8.33589e15i
2.6 −100224. 57864.6i −4.15312e7 + 1.13217e7i 4.54913e9 + 7.87933e9i −9.99516e10 5.77071e10i 4.81756e12 + 1.26847e12i 1.10609e13 + 3.13382e13i 5.55881e14i 1.59666e15 9.40407e14i 6.67839e15 + 1.15673e16i
2.7 −100095. 57789.7i −3.45400e7 2.56906e7i 4.53182e9 + 7.84935e9i −4.47113e9 2.58141e9i 1.97262e12 + 4.56755e12i −8.18737e12 3.22086e13i 5.51161e14i 5.33007e14 + 1.77471e15i 2.98358e14 + 5.16770e14i
2.8 −95187.3 54956.4i −6.01559e6 + 4.26243e7i 3.89294e9 + 6.74277e9i 9.27499e10 + 5.35492e10i 2.91509e12 3.72670e12i 3.32287e13 5.30154e11i 3.83696e14i −1.78065e15 5.12821e14i −5.88575e15 1.01944e16i
2.9 −95143.0 54930.9i 4.10114e7 1.30799e7i 3.88731e9 + 6.73302e9i −1.11850e11 6.45768e10i −4.62044e12 1.00833e12i 2.02925e13 2.63181e13i 3.82281e14i 1.51085e15 1.07285e15i 7.09452e15 + 1.22881e16i
2.10 −92637.1 53484.1i −4.06397e7 1.41929e7i 3.57361e9 + 6.18967e9i 2.33257e11 + 1.34671e11i 3.00565e12 + 3.48836e12i −1.79704e13 + 2.79552e13i 3.05100e14i 1.45014e15 + 1.15359e15i −1.44055e16 2.49510e16i
2.11 −89987.0 51954.0i −1.34905e6 4.30256e7i 3.25096e9 + 5.63082e9i 2.01263e11 + 1.16199e11i −2.11396e12 + 3.94183e12i 2.96316e13 1.50465e13i 2.29320e14i −1.84938e15 + 1.16087e14i −1.20740e16 2.09128e16i
2.12 −85963.2 49630.9i 3.66132e7 2.26384e7i 2.77897e9 + 4.81331e9i −1.43172e11 8.26606e10i −4.27095e12 + 1.28924e11i −1.23935e13 + 3.08355e13i 1.25364e14i 8.28028e14 1.65773e15i 8.20504e15 + 1.42115e16i
2.13 −85030.6 49092.5i −1.46438e6 + 4.30218e7i 2.67265e9 + 4.62917e9i 9.94676e10 + 5.74276e10i 2.23656e12 3.58628e12i −2.61520e13 + 2.05061e13i 1.03128e14i −1.84873e15 1.26000e14i −5.63853e15 9.76621e15i
2.14 −81147.0 46850.3i 2.92478e7 + 3.15846e7i 2.24241e9 + 3.88397e9i 1.46995e10 + 8.48674e9i −8.93631e11 3.93326e12i 5.53322e12 3.27691e13i 1.77891e13i −1.42148e14 + 1.84756e15i −7.95212e14 1.37735e15i
2.15 −80445.4 46445.2i 4.22184e7 + 8.40380e6i 2.16683e9 + 3.75305e9i 1.68281e11 + 9.71571e10i −3.00596e12 2.63689e12i −2.65601e13 1.99748e13i 3.59344e12i 1.71177e15 + 7.09591e14i −9.02495e15 1.56317e16i
2.16 −77640.0 44825.5i −2.98329e7 3.10325e7i 1.87116e9 + 3.24095e9i −1.87248e11 1.08108e11i 9.25177e11 + 3.74664e12i −3.25948e13 6.48148e12i 4.95447e13i −7.30174e13 + 1.85158e15i 9.69196e15 + 1.67870e16i
2.17 −76577.7 44212.2i −1.26540e7 + 4.11448e7i 1.76195e9 + 3.05178e9i −2.57749e11 1.48811e11i 2.78812e12 2.59132e12i 3.15580e13 1.04173e13i 6.81818e13i −1.53277e15 1.04129e15i 1.31586e16 + 2.27913e16i
2.18 −74754.1 43159.3i 3.41923e7 + 2.61516e7i 1.57796e9 + 2.73311e9i −9.12997e10 5.27119e10i −1.42733e12 3.43066e12i 3.71380e12 + 3.30248e13i 9.83202e13i 4.85204e14 + 1.78837e15i 4.55002e15 + 7.88086e15i
2.19 −68807.1 39725.8i −3.56249e7 + 2.41638e7i 1.00879e9 + 1.74728e9i −1.14635e11 6.61847e10i 3.41117e12 2.47415e11i −3.31594e13 + 2.20914e12i 1.80942e14i 6.85242e14 1.72166e15i 5.25848e15 + 9.10795e15i
2.20 −65272.0 37684.8i −3.95790e7 1.69270e7i 6.92806e8 + 1.19998e9i 3.97688e10 + 2.29605e10i 1.94551e12 + 2.59639e12i 3.31510e13 + 2.33200e12i 2.19277e14i 1.27997e15 + 1.33991e15i −1.73053e15 2.99736e15i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.33.h.b 164
3.b odd 2 1 inner 21.33.h.b 164
7.c even 3 1 inner 21.33.h.b 164
21.h odd 6 1 inner 21.33.h.b 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.h.b 164 1.a even 1 1 trivial
21.33.h.b 164 3.b odd 2 1 inner
21.33.h.b 164 7.c even 3 1 inner
21.33.h.b 164 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(36\!\cdots\!83\)\( T_{2}^{160} - \)\(34\!\cdots\!54\)\( T_{2}^{158} + \)\(24\!\cdots\!56\)\( T_{2}^{156} - \)\(14\!\cdots\!08\)\( T_{2}^{154} + \)\(71\!\cdots\!24\)\( T_{2}^{152} - \)\(30\!\cdots\!48\)\( T_{2}^{150} + \)\(11\!\cdots\!20\)\( T_{2}^{148} - \)\(38\!\cdots\!08\)\( T_{2}^{146} + \)\(11\!\cdots\!68\)\( T_{2}^{144} - \)\(32\!\cdots\!36\)\( T_{2}^{142} + \)\(81\!\cdots\!92\)\( T_{2}^{140} - \)\(18\!\cdots\!80\)\( T_{2}^{138} + \)\(40\!\cdots\!12\)\( T_{2}^{136} - \)\(80\!\cdots\!76\)\( T_{2}^{134} + \)\(14\!\cdots\!96\)\( T_{2}^{132} - \)\(25\!\cdots\!20\)\( T_{2}^{130} + \)\(41\!\cdots\!76\)\( T_{2}^{128} - \)\(62\!\cdots\!12\)\( T_{2}^{126} + \)\(88\!\cdots\!24\)\( T_{2}^{124} - \)\(11\!\cdots\!44\)\( T_{2}^{122} + \)\(14\!\cdots\!96\)\( T_{2}^{120} - \)\(17\!\cdots\!56\)\( T_{2}^{118} + \)\(19\!\cdots\!88\)\( T_{2}^{116} - \)\(20\!\cdots\!28\)\( T_{2}^{114} + \)\(20\!\cdots\!80\)\( T_{2}^{112} - \)\(19\!\cdots\!44\)\( T_{2}^{110} + \)\(17\!\cdots\!88\)\( T_{2}^{108} - \)\(14\!\cdots\!00\)\( T_{2}^{106} + \)\(11\!\cdots\!72\)\( T_{2}^{104} - \)\(88\!\cdots\!88\)\( T_{2}^{102} + \)\(63\!\cdots\!92\)\( T_{2}^{100} - \)\(42\!\cdots\!92\)\( T_{2}^{98} + \)\(27\!\cdots\!56\)\( T_{2}^{96} - \)\(16\!\cdots\!04\)\( T_{2}^{94} + \)\(95\!\cdots\!44\)\( T_{2}^{92} - \)\(51\!\cdots\!92\)\( T_{2}^{90} + \)\(26\!\cdots\!28\)\( T_{2}^{88} - \)\(12\!\cdots\!88\)\( T_{2}^{86} + \)\(58\!\cdots\!16\)\( T_{2}^{84} - \)\(25\!\cdots\!52\)\( T_{2}^{82} + \)\(10\!\cdots\!92\)\( T_{2}^{80} - \)\(39\!\cdots\!16\)\( T_{2}^{78} + \)\(14\!\cdots\!04\)\( T_{2}^{76} - \)\(49\!\cdots\!96\)\( T_{2}^{74} + \)\(15\!\cdots\!08\)\( T_{2}^{72} - \)\(47\!\cdots\!64\)\( T_{2}^{70} + \)\(13\!\cdots\!16\)\( T_{2}^{68} - \)\(34\!\cdots\!36\)\( T_{2}^{66} + \)\(86\!\cdots\!80\)\( T_{2}^{64} - \)\(19\!\cdots\!08\)\( T_{2}^{62} + \)\(42\!\cdots\!68\)\( T_{2}^{60} - \)\(85\!\cdots\!64\)\( T_{2}^{58} + \)\(15\!\cdots\!96\)\( T_{2}^{56} - \)\(27\!\cdots\!04\)\( T_{2}^{54} + \)\(43\!\cdots\!92\)\( T_{2}^{52} - \)\(64\!\cdots\!36\)\( T_{2}^{50} + \)\(88\!\cdots\!24\)\( T_{2}^{48} - \)\(11\!\cdots\!64\)\( T_{2}^{46} + \)\(12\!\cdots\!76\)\( T_{2}^{44} - \)\(13\!\cdots\!40\)\( T_{2}^{42} + \)\(12\!\cdots\!40\)\( T_{2}^{40} - \)\(11\!\cdots\!00\)\( T_{2}^{38} + \)\(87\!\cdots\!00\)\( T_{2}^{36} - \)\(62\!\cdots\!00\)\( T_{2}^{34} + \)\(39\!\cdots\!00\)\( T_{2}^{32} - \)\(22\!\cdots\!00\)\( T_{2}^{30} + \)\(11\!\cdots\!00\)\( T_{2}^{28} - \)\(51\!\cdots\!00\)\( T_{2}^{26} + \)\(19\!\cdots\!00\)\( T_{2}^{24} - \)\(67\!\cdots\!00\)\( T_{2}^{22} + \)\(19\!\cdots\!00\)\( T_{2}^{20} - \)\(47\!\cdots\!00\)\( T_{2}^{18} + \)\(96\!\cdots\!00\)\( T_{2}^{16} - \)\(15\!\cdots\!00\)\( T_{2}^{14} + \)\(19\!\cdots\!00\)\( T_{2}^{12} - \)\(16\!\cdots\!00\)\( T_{2}^{10} + \)\(86\!\cdots\!00\)\( T_{2}^{8} - \)\(12\!\cdots\!00\)\( T_{2}^{6} + \)\(13\!\cdots\!00\)\( T_{2}^{4} - \)\(48\!\cdots\!00\)\( T_{2}^{2} + \)\(13\!\cdots\!00\)\( \)">\(T_{2}^{164} - \cdots\) acting on \(S_{33}^{\mathrm{new}}(21, [\chi])\).