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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 54 q^{3} - 96 q^{4} + 30 q^{5} - 216 q^{6} - 212 q^{7} - 384 q^{8} - 486 q^{9} - 320 q^{10} + 578 q^{11} - 864 q^{12} + 964 q^{13} + 1312 q^{14} - 1350 q^{15} - 1536 q^{16} + 3318 q^{17} + 7776 q^{18} + 3240 q^{19} - 2400 q^{20} - 1998 q^{21} - 48 q^{22} - 2566 q^{23} + 13824 q^{24} + 16120 q^{25} - 2544 q^{26} - 4374 q^{27} - 3552 q^{28} + 2220 q^{29} + 4320 q^{30} + 12918 q^{31} + 24576 q^{32} - 108 q^{33} - 5968 q^{34} - 17190 q^{35} - 7776 q^{36} + 5738 q^{37} - 3600 q^{38} + 8676 q^{39} + 7680 q^{40} - 18732 q^{41} - 7992 q^{42} - 21176 q^{43} - 192 q^{44} + 6480 q^{45} + 20736 q^{46} + 37488 q^{47} - 13824 q^{48} + 62048 q^{49} + 7880 q^{50} - 32868 q^{51} + 15424 q^{52} + 49414 q^{53} - 17496 q^{54} - 83140 q^{55} - 14208 q^{56} - 42120 q^{57} + 8880 q^{58} - 37280 q^{59} + 11520 q^{60} + 65988 q^{61} - 20008 q^{62} + 26568 q^{63} - 24576 q^{64} - 62940 q^{65} + 20808 q^{66} + 38328 q^{67} - 58432 q^{68} + 46656 q^{69} + 33360 q^{70} + 199868 q^{71} - 31104 q^{72} - 50656 q^{73} + 2912 q^{74} - 82620 q^{75} - 74880 q^{76} - 83224 q^{77} - 23256 q^{78} + 132930 q^{79} + 20480 q^{80} - 39366 q^{81} + 15712 q^{82} - 304696 q^{83} + 47232 q^{84} + 12360 q^{85} + 9256 q^{86} + 93600 q^{87} + 36992 q^{88} - 343880 q^{89} + 9720 q^{90} + 453528 q^{91} + 82944 q^{92} - 142488 q^{93} + 149952 q^{94} + 321730 q^{95} - 55296 q^{96} + 153648 q^{97} + 125152 q^{98} - 91692 q^{99}+O(q^{100}) \) 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 −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 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.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 \( T_{7}^{12} + 106 T_{7}^{11} - 110736 T_{7}^{10} - 11273520 T_{7}^{9} + 4480317260 T_{7}^{8} + 428230842606 T_{7}^{7} - 79253486789869 T_{7}^{6} + \cdots - 97\!\cdots\!24 \) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display