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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,6,Mod(31,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.31");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 31.6
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 \( T_{7}^{12} + 227 T_{7}^{11} - 81229 T_{7}^{10} - 19105670 T_{7}^{9} + 2328673420 T_{7}^{8} + 587745624282 T_{7}^{7} - 28690653762651 T_{7}^{6} + \cdots + 24\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display