# Properties

 Label 21.30.e.b Level $21$ Weight $30$ Character orbit 21.e Analytic conductor $111.884$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## 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: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 7743q^{2} - 95659380q^{3} - 5985454645q^{4} + 14215352760q^{5} + 74069057934q^{6} - 368001036334q^{7} + 49989480496086q^{8} - 457535849099220q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 7743q^{2} - 95659380q^{3} - 5985454645q^{4} + 14215352760q^{5} + 74069057934q^{6} - 368001036334q^{7} + 49989480496086q^{8} - 457535849099220q^{9} - 167879133295903q^{10} - 3154300361224314q^{11} - 28628244017941005q^{12} + 27686480502978220q^{13} - 214257133773063450q^{14} - 135983183150288880q^{15} - 1764048935658629353q^{16} + 41802108452009436q^{17} - 177135003978763023q^{18} - 7235175939726809174q^{19} - 2944336803535185762q^{20} + 9048396870023397228q^{21} + 89086825354637664038q^{22} + 44902952138264019852q^{23} -$$$$11\!\cdots\!67$$$$q^{24} -$$$$85\!\cdots\!74$$$$q^{25} +$$$$74\!\cdots\!54$$$$q^{26} +$$$$43\!\cdots\!60$$$$q^{27} +$$$$83\!\cdots\!43$$$$q^{28} +$$$$30\!\cdots\!64$$$$q^{29} -$$$$80\!\cdots\!07$$$$q^{30} -$$$$71\!\cdots\!18$$$$q^{31} -$$$$32\!\cdots\!99$$$$q^{32} -$$$$15\!\cdots\!66$$$$q^{33} -$$$$40\!\cdots\!64$$$$q^{34} +$$$$22\!\cdots\!92$$$$q^{35} +$$$$27\!\cdots\!90$$$$q^{36} -$$$$32\!\cdots\!42$$$$q^{37} -$$$$21\!\cdots\!74$$$$q^{38} -$$$$66\!\cdots\!90$$$$q^{39} -$$$$56\!\cdots\!13$$$$q^{40} -$$$$40\!\cdots\!60$$$$q^{41} +$$$$26\!\cdots\!59$$$$q^{42} -$$$$74\!\cdots\!76$$$$q^{43} -$$$$86\!\cdots\!51$$$$q^{44} +$$$$32\!\cdots\!60$$$$q^{45} -$$$$36\!\cdots\!80$$$$q^{46} +$$$$50\!\cdots\!56$$$$q^{47} +$$$$16\!\cdots\!14$$$$q^{48} -$$$$11\!\cdots\!44$$$$q^{49} +$$$$20\!\cdots\!96$$$$q^{50} +$$$$19\!\cdots\!84$$$$q^{51} -$$$$15\!\cdots\!36$$$$q^{52} +$$$$45\!\cdots\!24$$$$q^{53} -$$$$84\!\cdots\!87$$$$q^{54} -$$$$26\!\cdots\!92$$$$q^{55} +$$$$23\!\cdots\!07$$$$q^{56} +$$$$69\!\cdots\!12$$$$q^{57} -$$$$19\!\cdots\!41$$$$q^{58} -$$$$10\!\cdots\!06$$$$q^{59} +$$$$70\!\cdots\!89$$$$q^{60} -$$$$55\!\cdots\!44$$$$q^{61} +$$$$59\!\cdots\!38$$$$q^{62} -$$$$34\!\cdots\!58$$$$q^{63} +$$$$25\!\cdots\!02$$$$q^{64} +$$$$11\!\cdots\!16$$$$q^{65} -$$$$21\!\cdots\!11$$$$q^{66} -$$$$39\!\cdots\!14$$$$q^{67} -$$$$34\!\cdots\!16$$$$q^{68} -$$$$42\!\cdots\!76$$$$q^{69} +$$$$31\!\cdots\!73$$$$q^{70} -$$$$35\!\cdots\!92$$$$q^{71} -$$$$57\!\cdots\!23$$$$q^{72} +$$$$15\!\cdots\!06$$$$q^{73} -$$$$25\!\cdots\!74$$$$q^{74} -$$$$40\!\cdots\!06$$$$q^{75} +$$$$24\!\cdots\!00$$$$q^{76} -$$$$78\!\cdots\!80$$$$q^{77} -$$$$70\!\cdots\!52$$$$q^{78} +$$$$56\!\cdots\!66$$$$q^{79} -$$$$39\!\cdots\!99$$$$q^{80} -$$$$10\!\cdots\!20$$$$q^{81} -$$$$14\!\cdots\!64$$$$q^{82} +$$$$21\!\cdots\!48$$$$q^{83} -$$$$13\!\cdots\!24$$$$q^{84} +$$$$89\!\cdots\!08$$$$q^{85} -$$$$15\!\cdots\!00$$$$q^{86} -$$$$72\!\cdots\!08$$$$q^{87} -$$$$63\!\cdots\!25$$$$q^{88} -$$$$11\!\cdots\!12$$$$q^{89} +$$$$76\!\cdots\!66$$$$q^{90} -$$$$15\!\cdots\!90$$$$q^{91} +$$$$15\!\cdots\!48$$$$q^{92} -$$$$34\!\cdots\!42$$$$q^{93} -$$$$13\!\cdots\!06$$$$q^{94} -$$$$15\!\cdots\!84$$$$q^{95} -$$$$15\!\cdots\!31$$$$q^{96} -$$$$39\!\cdots\!32$$$$q^{97} +$$$$23\!\cdots\!19$$$$q^{98} +$$$$14\!\cdots\!08$$$$q^{99} + O(q^{100})$$

## 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 −22377.3 38758.7i −2.39148e6 + 4.14217e6i −7.33056e8 + 1.26969e9i 9.58959e9 + 1.66096e10i 2.14060e11 1.32627e12 1.20868e12i 4.15879e13 −1.14384e13 1.98119e13i 4.29179e14 7.43360e14i
4.2 −21754.4 37679.7i −2.39148e6 + 4.14217e6i −6.78069e8 + 1.17445e9i −1.24919e10 2.16367e10i 2.08101e11 −9.19987e11 1.54063e12i 3.56453e13 −1.14384e13 1.98119e13i −5.43509e14 + 9.41385e14i
4.3 −17878.7 30966.9i −2.39148e6 + 4.14217e6i −3.70862e8 + 6.42353e8i 2.47312e9 + 4.28357e9i 1.71027e11 −1.35165e12 + 1.18023e12i 7.32506e12 −1.14384e13 1.98119e13i 8.84324e13 1.53169e14i
4.4 −17562.2 30418.7i −2.39148e6 + 4.14217e6i −3.48428e8 + 6.03495e8i −6.19331e9 1.07271e10i 1.67999e11 8.11811e11 + 1.60027e12i 5.61938e12 −1.14384e13 1.98119e13i −2.17537e14 + 3.76784e14i
4.5 −15572.5 26972.3i −2.39148e6 + 4.14217e6i −2.16568e8 + 3.75107e8i 3.77327e9 + 6.53549e9i 1.48965e11 1.66300e12 6.74056e11i −3.23080e12 −1.14384e13 1.98119e13i 1.17518e14 2.03547e14i
4.6 −12378.0 21439.3i −2.39148e6 + 4.14217e6i −3.79927e7 + 6.58054e7i 6.22725e9 + 1.07859e10i 1.18407e11 −1.28632e12 1.25111e12i −1.14097e13 −1.14384e13 1.98119e13i 1.54161e14 2.67016e14i
4.7 −9025.62 15632.8i −2.39148e6 + 4.14217e6i 1.05512e8 1.82752e8i −6.88316e9 1.19220e10i 8.63385e10 −1.50044e12 9.84161e11i −1.35004e13 −1.14384e13 1.98119e13i −1.24249e14 + 2.15206e14i
4.8 −7408.63 12832.1i −2.39148e6 + 4.14217e6i 1.58660e8 2.74807e8i 1.23671e10 + 2.14205e10i 7.08705e10 1.23011e12 + 1.30642e12i −1.26568e13 −1.14384e13 1.98119e13i 1.83247e14 3.17393e14i
4.9 −5215.37 9033.29i −2.39148e6 + 4.14217e6i 2.14035e8 3.70720e8i −6.54948e9 1.13440e10i 4.98899e10 1.56106e12 8.84875e11i −1.00651e13 −1.14384e13 1.98119e13i −6.83160e13 + 1.18327e14i
4.10 −2969.08 5142.60i −2.39148e6 + 4.14217e6i 2.50805e8 4.34406e8i 3.53467e9 + 6.12222e9i 2.84020e10 −1.40680e12 + 1.11392e12i −6.16666e12 −1.14384e13 1.98119e13i 2.09894e13 3.63547e13i
4.11 3938.62 + 6821.90i −2.39148e6 + 4.14217e6i 2.37410e8 4.11206e8i 1.52926e8 + 2.64876e8i −3.76766e10 1.72284e12 + 5.01708e11i 7.96934e12 −1.14384e13 1.98119e13i −1.20464e12 + 2.08649e12i
4.12 6919.62 + 11985.1i −2.39148e6 + 4.14217e6i 1.72673e8 2.99079e8i 4.10288e9 + 7.10639e9i −6.61927e10 4.76232e10 1.79378e12i 1.22092e13 −1.14384e13 1.98119e13i −5.67807e13 + 9.83471e13i
4.13 7416.41 + 12845.6i −2.39148e6 + 4.14217e6i 1.58429e8 2.74407e8i −5.40182e9 9.35622e9i −7.09449e10 −8.77985e11 1.56494e12i 1.26632e13 −1.14384e13 1.98119e13i 8.01242e13 1.38779e14i
4.14 7798.16 + 13506.8i −2.39148e6 + 4.14217e6i 1.46813e8 2.54287e8i 5.50788e9 + 9.53994e9i −7.45967e10 −9.07383e11 + 1.54808e12i 1.29527e13 −1.14384e13 1.98119e13i −8.59027e13 + 1.48788e14i
4.15 10114.7 + 17519.3i −2.39148e6 + 4.14217e6i 6.38192e7 1.10538e8i −1.35079e10 2.33965e10i −9.67571e10 −1.40897e12 + 1.11118e12i 1.34427e13 −1.14384e13 1.98119e13i 2.73259e14 4.73298e14i
4.16 16123.2 + 27926.2i −2.39148e6 + 4.14217e6i −2.51480e8 + 4.35576e8i −4.30356e9 7.45399e9i −1.54234e11 1.08632e12 + 1.42822e12i 1.09353e12 −1.14384e13 1.98119e13i 1.38774e14 2.40364e14i
4.17 16582.6 + 28721.9i −2.39148e6 + 4.14217e6i −2.81530e8 + 4.87624e8i 1.20484e10 + 2.08684e10i −1.58628e11 1.59164e12 8.28603e11i −8.68562e11 −1.14384e13 1.98119e13i −3.99587e14 + 6.92105e14i
4.18 16888.6 + 29251.9i −2.39148e6 + 4.14217e6i −3.02013e8 + 5.23103e8i 8.22263e9 + 1.42420e10i −1.61555e11 −1.78996e12 1.26263e11i −2.26834e12 −1.14384e13 1.98119e13i −2.77737e14 + 4.81055e14i
4.19 19476.8 + 33734.8i −2.39148e6 + 4.14217e6i −4.90254e8 + 8.49146e8i −7.17536e9 1.24281e10i −1.86314e11 7.19771e11 1.64373e12i −1.72813e13 −1.14384e13 1.98119e13i 2.79506e14 4.84119e14i
4.20 23011.6 + 39857.2i −2.39148e6 + 4.14217e6i −7.90629e8 + 1.36941e9i 1.61456e9 + 2.79651e9i −2.20127e11 −4.94948e11 + 1.72480e12i −4.80660e13 −1.14384e13 1.98119e13i −7.43073e13 + 1.28704e14i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 40
7.c even 3 1 inner 21.30.e.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.e.b 40 1.a even 1 1 trivial
21.30.e.b 40 7.c even 3 1 inner