Properties

Label 150.6.g.c
Level $150$
Weight $6$
Character orbit 150.g
Analytic conductor $24.058$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{2} - 54q^{3} - 96q^{4} + 80q^{5} + 216q^{6} - 454q^{7} + 384q^{8} - 486q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{2} - 54q^{3} - 96q^{4} + 80q^{5} + 216q^{6} - 454q^{7} + 384q^{8} - 486q^{9} + 120q^{10} + 125q^{11} - 864q^{12} - 578q^{13} - 344q^{14} + 855q^{15} - 1536q^{16} - 2664q^{17} - 7776q^{18} + 1424q^{19} + 1520q^{20} + 1269q^{21} + 1280q^{22} - 4810q^{23} - 13824q^{24} - 10880q^{25} + 6392q^{26} - 4374q^{27} + 2256q^{28} - 1554q^{29} + 4320q^{30} - 24513q^{31} - 24576q^{32} - 2880q^{33} - 4004q^{34} + 15310q^{35} - 7776q^{36} - 28123q^{37} + 8384q^{38} - 5202q^{39} + 7680q^{40} - 17914q^{41} - 5076q^{42} + 1256q^{43} - 5120q^{44} - 2025q^{45} - 13040q^{46} + 930q^{47} - 13824q^{48} + 24606q^{49} - 7880q^{50} + 29934q^{51} - 9248q^{52} - 49701q^{53} + 17496q^{54} - 99060q^{55} - 9024q^{56} + 12096q^{57} + 6216q^{58} - 14465q^{59} - 3600q^{60} - 52750q^{61} - 72528q^{62} + 6966q^{63} - 24576q^{64} - 34885q^{65} - 4500q^{66} + 199846q^{67} + 53216q^{68} + 29340q^{69} + 116820q^{70} - 42880q^{71} + 31104q^{72} + 35852q^{73} + 21832q^{74} - 28620q^{75} + 21504q^{76} - 114745q^{77} - 49572q^{78} - 236128q^{79} - 6400q^{80} - 39366q^{81} + 46216q^{82} - 97273q^{83} + 12384q^{84} + 372975q^{85} - 232304q^{86} + 21609q^{87} - 8000q^{88} + 278147q^{89} - 25920q^{90} - 103052q^{91} + 52160q^{92} + 114858q^{93} - 3720q^{94} + 79590q^{95} + 55296q^{96} - 249803q^{97} - 28164q^{98} + 31590q^{99} + O(q^{100}) \)

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 −49.3905 + 26.1836i 29.1246 + 21.1603i −91.4091 51.7771 + 37.6183i 25.0304 77.0356i 160.658 + 155.528i
31.2 −1.23607 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i −44.8368 33.3866i 29.1246 + 21.1603i 131.563 51.7771 + 37.6183i 25.0304 77.0356i −71.5889 + 211.837i
31.3 −1.23607 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i −5.54795 55.6257i 29.1246 + 21.1603i −223.096 51.7771 + 37.6183i 25.0304 77.0356i −204.755 + 89.8628i
31.4 −1.23607 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 22.4321 + 51.2035i 29.1246 + 21.1603i 146.638 51.7771 + 37.6183i 25.0304 77.0356i 167.062 148.628i
31.5 −1.23607 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 43.0312 35.6835i 29.1246 + 21.1603i 32.5493 51.7771 + 37.6183i 25.0304 77.0356i −188.938 119.593i
31.6 −1.23607 3.80423i −7.28115 + 5.29007i −12.9443 + 9.40456i 50.9577 + 22.9850i 29.1246 + 21.1603i −122.044 51.7771 + 37.6183i 25.0304 77.0356i 24.4527 222.266i
61.1 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i −41.7488 + 37.1757i −11.1246 34.2380i −164.912 −19.7771 60.8676i −65.5304 47.6106i −47.6968 + 218.461i
61.2 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i −21.1242 51.7568i −11.1246 34.2380i 92.5726 −19.7771 60.8676i −65.5304 47.6106i −190.047 117.822i
61.3 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i −10.7648 + 54.8554i −11.1246 34.2380i 36.7606 −19.7771 60.8676i −65.5304 47.6106i 94.1374 + 202.825i
61.4 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i 6.45828 55.5274i −11.1246 34.2380i −133.388 −19.7771 60.8676i −65.5304 47.6106i −109.653 194.875i
61.5 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i 36.5224 + 42.3216i −11.1246 34.2380i −114.252 −19.7771 60.8676i −65.5304 47.6106i 217.693 + 51.0861i
61.6 3.23607 2.35114i 2.78115 8.55951i 4.94427 15.2169i 54.0113 + 14.4147i −11.1246 34.2380i 182.018 −19.7771 60.8676i −65.5304 47.6106i 208.675 80.3412i
91.1 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −41.7488 37.1757i −11.1246 + 34.2380i −164.912 −19.7771 + 60.8676i −65.5304 + 47.6106i −47.6968 218.461i
91.2 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −21.1242 + 51.7568i −11.1246 + 34.2380i 92.5726 −19.7771 + 60.8676i −65.5304 + 47.6106i −190.047 + 117.822i
91.3 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i −10.7648 54.8554i −11.1246 + 34.2380i 36.7606 −19.7771 + 60.8676i −65.5304 + 47.6106i 94.1374 202.825i
91.4 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 6.45828 + 55.5274i −11.1246 + 34.2380i −133.388 −19.7771 + 60.8676i −65.5304 + 47.6106i −109.653 + 194.875i
91.5 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 36.5224 42.3216i −11.1246 + 34.2380i −114.252 −19.7771 + 60.8676i −65.5304 + 47.6106i 217.693 51.0861i
91.6 3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 + 15.2169i 54.0113 14.4147i −11.1246 + 34.2380i 182.018 −19.7771 + 60.8676i −65.5304 + 47.6106i 208.675 + 80.3412i
121.1 −1.23607 + 3.80423i −7.28115 5.29007i −12.9443 9.40456i −49.3905 26.1836i 29.1246 21.1603i −91.4091 51.7771 37.6183i 25.0304 + 77.0356i 160.658 155.528i
121.2 −1.23607 + 3.80423i −7.28115 5.29007i −12.9443 9.40456i −44.8368 + 33.3866i 29.1246 21.1603i 131.563 51.7771 37.6183i 25.0304 + 77.0356i −71.5889 211.837i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(28\!\cdots\!51\)\( T_{7}^{6} - \)\(81\!\cdots\!03\)\( T_{7}^{5} + \)\(15\!\cdots\!80\)\( T_{7}^{4} + \)\(49\!\cdots\!45\)\( T_{7}^{3} - \)\(59\!\cdots\!19\)\( T_{7}^{2} - \)\(10\!\cdots\!28\)\( T_{7} + \)\(24\!\cdots\!76\)\( \)">\(T_{7}^{12} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\).