# Properties

 Label 150.6.g.b Level $150$ Weight $6$ Character orbit 150.g Analytic conductor $24.058$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$6$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} - 54q^{3} - 96q^{4} + 30q^{5} - 216q^{6} - 212q^{7} - 384q^{8} - 486q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{2} - 54q^{3} - 96q^{4} + 30q^{5} - 216q^{6} - 212q^{7} - 384q^{8} - 486q^{9} - 320q^{10} + 578q^{11} - 864q^{12} + 964q^{13} + 1312q^{14} - 1350q^{15} - 1536q^{16} + 3318q^{17} + 7776q^{18} + 3240q^{19} - 2400q^{20} - 1998q^{21} - 48q^{22} - 2566q^{23} + 13824q^{24} + 16120q^{25} - 2544q^{26} - 4374q^{27} - 3552q^{28} + 2220q^{29} + 4320q^{30} + 12918q^{31} + 24576q^{32} - 108q^{33} - 5968q^{34} - 17190q^{35} - 7776q^{36} + 5738q^{37} - 3600q^{38} + 8676q^{39} + 7680q^{40} - 18732q^{41} - 7992q^{42} - 21176q^{43} - 192q^{44} + 6480q^{45} + 20736q^{46} + 37488q^{47} - 13824q^{48} + 62048q^{49} + 7880q^{50} - 32868q^{51} + 15424q^{52} + 49414q^{53} - 17496q^{54} - 83140q^{55} - 14208q^{56} - 42120q^{57} + 8880q^{58} - 37280q^{59} + 11520q^{60} + 65988q^{61} - 20008q^{62} + 26568q^{63} - 24576q^{64} - 62940q^{65} + 20808q^{66} + 38328q^{67} - 58432q^{68} + 46656q^{69} + 33360q^{70} + 199868q^{71} - 31104q^{72} - 50656q^{73} + 2912q^{74} - 82620q^{75} - 74880q^{76} - 83224q^{77} - 23256q^{78} + 132930q^{79} + 20480q^{80} - 39366q^{81} + 15712q^{82} - 304696q^{83} + 47232q^{84} + 12360q^{85} + 9256q^{86} + 93600q^{87} + 36992q^{88} - 343880q^{89} + 9720q^{90} + 453528q^{91} + 82944q^{92} - 142488q^{93} + 149952q^{94} + 321730q^{95} - 55296q^{96} + 153648q^{97} + 125152q^{98} - 91692q^{99} + O(q^{100})$$

## 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 −55.5966 5.83204i −29.1246 21.1603i −184.364 −51.7771 37.6183i 25.0304 77.0356i −46.5348 218.711i
31.2 1.23607 + 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i −55.5791 + 5.99730i −29.1246 21.1603i 161.028 −51.7771 37.6183i 25.0304 77.0356i −91.5146 204.022i
31.3 1.23607 + 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i −10.1537 + 54.9718i −29.1246 21.1603i 197.545 −51.7771 37.6183i 25.0304 77.0356i −221.676 + 29.3220i
31.4 1.23607 + 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 37.7606 + 41.2206i −29.1246 21.1603i −165.301 −51.7771 37.6183i 25.0304 77.0356i −110.138 + 194.601i
31.5 1.23607 + 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 38.8459 40.1995i −29.1246 21.1603i −37.1005 −51.7771 37.6183i 25.0304 77.0356i 200.944 + 98.0892i
31.6 1.23607 + 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 55.5770 + 6.01600i −29.1246 21.1603i 98.1755 −51.7771 37.6183i 25.0304 77.0356i 45.8108 + 218.864i
61.1 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i −55.1199 + 9.31618i 11.1246 + 34.2380i 58.3864 19.7771 + 60.8676i −65.5304 47.6106i 156.468 159.743i
61.2 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i −44.9762 33.1985i 11.1246 + 34.2380i −54.5329 19.7771 + 60.8676i −65.5304 47.6106i 223.600 + 1.68726i
61.3 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i −8.34027 + 55.2760i 11.1246 + 34.2380i −188.522 19.7771 + 60.8676i −65.5304 47.6106i −102.972 198.486i
61.4 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i 10.3231 54.9403i 11.1246 + 34.2380i 175.692 19.7771 + 60.8676i −65.5304 47.6106i 95.7661 + 202.061i
61.5 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i 49.1876 26.5627i 11.1246 + 34.2380i −149.411 19.7771 + 60.8676i −65.5304 47.6106i −96.7217 + 201.606i
61.6 −3.23607 + 2.35114i 2.78115 8.55951i 4.94427 15.2169i 53.0716 + 17.5614i 11.1246 + 34.2380i −17.5965 19.7771 + 60.8676i −65.5304 47.6106i −213.033 + 67.9490i
91.1 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −55.1199 9.31618i 11.1246 34.2380i 58.3864 19.7771 60.8676i −65.5304 + 47.6106i 156.468 + 159.743i
91.2 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −44.9762 + 33.1985i 11.1246 34.2380i −54.5329 19.7771 60.8676i −65.5304 + 47.6106i 223.600 1.68726i
91.3 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −8.34027 55.2760i 11.1246 34.2380i −188.522 19.7771 60.8676i −65.5304 + 47.6106i −102.972 + 198.486i
91.4 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 10.3231 + 54.9403i 11.1246 34.2380i 175.692 19.7771 60.8676i −65.5304 + 47.6106i 95.7661 202.061i
91.5 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 49.1876 + 26.5627i 11.1246 34.2380i −149.411 19.7771 60.8676i −65.5304 + 47.6106i −96.7217 201.606i
91.6 −3.23607 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 53.0716 17.5614i 11.1246 34.2380i −17.5965 19.7771 60.8676i −65.5304 + 47.6106i −213.033 67.9490i
121.1 1.23607 3.80423i −7.28115 5.29007i −12.9443 9.40456i −55.5966 + 5.83204i −29.1246 + 21.1603i −184.364 −51.7771 + 37.6183i 25.0304 + 77.0356i −46.5348 + 218.711i
121.2 1.23607 3.80423i −7.28115 5.29007i −12.9443 9.40456i −55.5791 5.99730i −29.1246 + 21.1603i 161.028 −51.7771 + 37.6183i 25.0304 + 77.0356i −91.5146 + 204.022i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.6 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.b 24
25.d even 5 1 inner 150.6.g.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.g.b 24 1.a even 1 1 trivial
150.6.g.b 24 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$79\!\cdots\!69$$$$T_{7}^{6} -$$$$68\!\cdots\!16$$$$T_{7}^{5} +$$$$55\!\cdots\!65$$$$T_{7}^{4} +$$$$41\!\cdots\!80$$$$T_{7}^{3} -$$$$82\!\cdots\!16$$$$T_{7}^{2} -$$$$79\!\cdots\!36$$$$T_{7} -$$$$97\!\cdots\!24$$">$$T_{7}^{12} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(150, [\chi])$$.