# Properties

 Label 150.6.g.d Level $150$ Weight $6$ Character orbit 150.g Analytic conductor $24.058$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 150.g (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0575729719$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{2} + 63q^{3} - 112q^{4} + 11q^{5} + 252q^{6} - 62q^{7} - 448q^{8} - 567q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 28q^{2} + 63q^{3} - 112q^{4} + 11q^{5} + 252q^{6} - 62q^{7} - 448q^{8} - 567q^{9} + 44q^{10} - 135q^{11} + 1008q^{12} + 616q^{13} - 48q^{14} + 531q^{15} - 1792q^{16} - 3372q^{17} + 9072q^{18} + 488q^{19} - 944q^{20} - 387q^{21} - 1920q^{22} - 620q^{23} - 16128q^{24} + 6659q^{25} - 1616q^{26} + 5103q^{27} + 688q^{28} + 5682q^{29} + 2844q^{30} + 3339q^{31} + 28672q^{32} + 4320q^{33} + 14032q^{34} - 23984q^{35} - 9072q^{36} + 4506q^{37} - 1528q^{38} - 5544q^{39} - 5056q^{40} + 44962q^{41} - 1548q^{42} + 38028q^{43} - 7680q^{44} + 9396q^{45} + 7480q^{46} - 58300q^{47} + 16128q^{48} + 114402q^{49} + 7476q^{50} + 2448q^{51} + 9856q^{52} - 138q^{53} + 20412q^{54} + 96850q^{55} + 2752q^{56} + 1908q^{57} + 22728q^{58} + 40245q^{59} - 16704q^{60} + 79380q^{61} - 54504q^{62} - 972q^{63} - 28672q^{64} + 3532q^{65} + 4860q^{66} - 34352q^{67} - 4352q^{68} - 16830q^{69} - 69476q^{70} + 115860q^{71} - 36288q^{72} - 121214q^{73} - 37856q^{74} + 21969q^{75} - 3392q^{76} + 195005q^{77} + 14904q^{78} + 36224q^{79} + 29696q^{80} - 45927q^{81} + 34608q^{82} + 85051q^{83} + 1728q^{84} + 137496q^{85} - 18568q^{86} + 127062q^{87} - 8640q^{88} - 53826q^{89} + 3564q^{90} - 408104q^{91} + 29920q^{92} - 185166q^{93} - 233200q^{94} - 482004q^{95} + 64512q^{96} - 244169q^{97} - 395952q^{98} + 99630q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −54.4623 + 12.6040i 29.1246 + 21.1603i 92.2244 −51.7771 37.6183i 25.0304 77.0356i −115.268 191.607i
31.2 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −32.5575 + 45.4424i 29.1246 + 21.1603i −77.9335 −51.7771 37.6183i 25.0304 77.0356i −213.116 67.6864i
31.3 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −30.6277 46.7648i 29.1246 + 21.1603i −78.0347 −51.7771 37.6183i 25.0304 77.0356i 140.046 174.319i
31.4 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 25.5464 + 49.7230i 29.1246 + 21.1603i 186.658 −51.7771 37.6183i 25.0304 77.0356i −157.581 + 158.645i
31.5 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 31.6852 46.0549i 29.1246 + 21.1603i 113.911 −51.7771 37.6183i 25.0304 77.0356i 214.368 + 63.6106i
31.6 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 45.5005 + 32.4762i 29.1246 + 21.1603i −57.2345 −51.7771 37.6183i 25.0304 77.0356i −67.3052 + 213.237i
31.7 1.23607 + 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 47.2934 29.8050i 29.1246 + 21.1603i −207.389 −51.7771 37.6183i 25.0304 77.0356i 171.843 + 143.074i
61.1 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −55.6970 + 4.77930i −11.1246 34.2380i 235.720 19.7771 + 60.8676i −65.5304 47.6106i 169.003 146.418i
61.2 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −54.7742 + 11.1708i −11.1246 34.2380i −236.588 19.7771 + 60.8676i −65.5304 47.6106i 150.989 164.931i
61.3 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −24.3873 + 50.3017i −11.1246 34.2380i 94.8525 19.7771 + 60.8676i −65.5304 47.6106i −39.3474 220.118i
61.4 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −17.5819 53.0648i −11.1246 34.2380i 47.3035 19.7771 + 60.8676i −65.5304 47.6106i 181.659 + 130.384i
61.5 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 29.0425 + 47.7654i −11.1246 34.2380i −12.1985 19.7771 + 60.8676i −65.5304 47.6106i −206.287 86.2889i
61.6 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 41.0391 37.9577i −11.1246 34.2380i −217.896 19.7771 + 60.8676i −65.5304 47.6106i −43.5614 + 219.323i
61.7 −3.23607 + 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 55.4808 6.84686i −11.1246 34.2380i 85.6043 19.7771 + 60.8676i −65.5304 47.6106i −163.442 + 152.600i
91.1 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −55.6970 4.77930i −11.1246 + 34.2380i 235.720 19.7771 60.8676i −65.5304 + 47.6106i 169.003 + 146.418i
91.2 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −54.7742 11.1708i −11.1246 + 34.2380i −236.588 19.7771 60.8676i −65.5304 + 47.6106i 150.989 + 164.931i
91.3 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −24.3873 50.3017i −11.1246 + 34.2380i 94.8525 19.7771 60.8676i −65.5304 + 47.6106i −39.3474 + 220.118i
91.4 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −17.5819 + 53.0648i −11.1246 + 34.2380i 47.3035 19.7771 60.8676i −65.5304 + 47.6106i 181.659 130.384i
91.5 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 29.0425 47.7654i −11.1246 + 34.2380i −12.1985 19.7771 60.8676i −65.5304 + 47.6106i −206.287 + 86.2889i
91.6 −3.23607 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 41.0391 + 37.9577i −11.1246 + 34.2380i −217.896 19.7771 60.8676i −65.5304 + 47.6106i −43.5614 219.323i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.6.g.d 28
25.d even 5 1 inner 150.6.g.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.g.d 28 1.a even 1 1 trivial
150.6.g.d 28 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!87$$$$T_{7}^{8} +$$$$68\!\cdots\!13$$$$T_{7}^{7} +$$$$18\!\cdots\!48$$$$T_{7}^{6} -$$$$83\!\cdots\!07$$$$T_{7}^{5} -$$$$83\!\cdots\!99$$$$T_{7}^{4} +$$$$36\!\cdots\!56$$$$T_{7}^{3} +$$$$16\!\cdots\!56$$$$T_{7}^{2} -$$$$53\!\cdots\!24$$$$T_{7} -$$$$80\!\cdots\!84$$">$$T_{7}^{14} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(150, [\chi])$$.