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Properties

Label 150.4.h.a
Level $150$
Weight $4$
Character orbit 150.h
Analytic conductor $8.850$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [150,4,Mod(19,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.19"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 9])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} - 36 q^{6} + 54 q^{9} + 128 q^{11} - 140 q^{13} - 28 q^{14} + 120 q^{15} - 96 q^{16} - 460 q^{17} - 150 q^{19} + 120 q^{20} + 42 q^{21} - 140 q^{22} + 370 q^{23} - 576 q^{24} + 220 q^{25}+ \cdots + 2088 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.90211 + 0.618034i 1.76336 + 2.42705i 3.23607 − 2.35114i −7.32204 − 8.44913i −4.85410 − 3.52671i − 0.297306i −4.70228 + 6.47214i −2.78115 + 8.55951i 19.1492 + 11.5459i
19.2 −1.90211 + 0.618034i 1.76336 + 2.42705i 3.23607 − 2.35114i 8.00740 − 7.80267i −4.85410 − 3.52671i − 15.6288i −4.70228 + 6.47214i −2.78115 + 8.55951i −10.4087 + 19.7904i
19.3 −1.90211 + 0.618034i 1.76336 + 2.42705i 3.23607 − 2.35114i 8.04003 + 7.76904i −4.85410 − 3.52671i 16.8776i −4.70228 + 6.47214i −2.78115 + 8.55951i −20.0946 − 9.80857i
19.4 1.90211 − 0.618034i −1.76336 − 2.42705i 3.23607 − 2.35114i −11.0137 − 1.92299i −4.85410 − 3.52671i − 2.38187i 4.70228 − 6.47214i −2.78115 + 8.55951i −22.1378 + 3.14911i
19.5 1.90211 − 0.618034i −1.76336 − 2.42705i 3.23607 − 2.35114i 1.73370 + 11.0451i −4.85410 − 3.52671i 24.2738i 4.70228 − 6.47214i −2.78115 + 8.55951i 10.1239 + 19.9376i
19.6 1.90211 − 0.618034i −1.76336 − 2.42705i 3.23607 − 2.35114i 9.49891 − 5.89667i −4.85410 − 3.52671i − 6.94049i 4.70228 − 6.47214i −2.78115 + 8.55951i 14.4237 − 17.0868i
79.1 −1.90211 − 0.618034i 1.76336 − 2.42705i 3.23607 + 2.35114i −7.32204 + 8.44913i −4.85410 + 3.52671i 0.297306i −4.70228 − 6.47214i −2.78115 − 8.55951i 19.1492 − 11.5459i
79.2 −1.90211 − 0.618034i 1.76336 − 2.42705i 3.23607 + 2.35114i 8.00740 + 7.80267i −4.85410 + 3.52671i 15.6288i −4.70228 − 6.47214i −2.78115 − 8.55951i −10.4087 − 19.7904i
79.3 −1.90211 − 0.618034i 1.76336 − 2.42705i 3.23607 + 2.35114i 8.04003 − 7.76904i −4.85410 + 3.52671i − 16.8776i −4.70228 − 6.47214i −2.78115 − 8.55951i −20.0946 + 9.80857i
79.4 1.90211 + 0.618034i −1.76336 + 2.42705i 3.23607 + 2.35114i −11.0137 + 1.92299i −4.85410 + 3.52671i 2.38187i 4.70228 + 6.47214i −2.78115 − 8.55951i −22.1378 − 3.14911i
79.5 1.90211 + 0.618034i −1.76336 + 2.42705i 3.23607 + 2.35114i 1.73370 − 11.0451i −4.85410 + 3.52671i − 24.2738i 4.70228 + 6.47214i −2.78115 − 8.55951i 10.1239 − 19.9376i
79.6 1.90211 + 0.618034i −1.76336 + 2.42705i 3.23607 + 2.35114i 9.49891 + 5.89667i −4.85410 + 3.52671i 6.94049i 4.70228 + 6.47214i −2.78115 − 8.55951i 14.4237 + 17.0868i
109.1 −1.17557 + 1.61803i −2.85317 + 0.927051i −1.23607 − 3.80423i −9.78158 + 5.41486i 1.85410 − 5.70634i 31.3951i 7.60845 + 2.47214i 7.28115 − 5.29007i 2.73751 − 22.1925i
109.2 −1.17557 + 1.61803i −2.85317 + 0.927051i −1.23607 − 3.80423i −5.30270 − 9.84283i 1.85410 − 5.70634i 5.42104i 7.60845 + 2.47214i 7.28115 − 5.29007i 22.1597 + 2.99099i
109.3 −1.17557 + 1.61803i −2.85317 + 0.927051i −1.23607 − 3.80423i 7.98349 + 7.82712i 1.85410 − 5.70634i − 8.44386i 7.60845 + 2.47214i 7.28115 − 5.29007i −22.0497 + 3.71623i
109.4 1.17557 − 1.61803i 2.85317 − 0.927051i −1.23607 − 3.80423i −11.1611 − 0.656454i 1.85410 − 5.70634i − 11.3168i −7.60845 − 2.47214i 7.28115 − 5.29007i −14.1828 + 17.2873i
109.5 1.17557 − 1.61803i 2.85317 − 0.927051i −1.23607 − 3.80423i −0.468516 + 11.1705i 1.85410 − 5.70634i 33.2818i −7.60845 − 2.47214i 7.28115 − 5.29007i 17.5235 + 13.8898i
109.6 1.17557 − 1.61803i 2.85317 − 0.927051i −1.23607 − 3.80423i 9.78609 − 5.40671i 1.85410 − 5.70634i − 7.59274i −7.60845 − 2.47214i 7.28115 − 5.29007i 2.75600 − 22.1902i
139.1 −1.17557 − 1.61803i −2.85317 − 0.927051i −1.23607 + 3.80423i −9.78158 − 5.41486i 1.85410 + 5.70634i − 31.3951i 7.60845 − 2.47214i 7.28115 + 5.29007i 2.73751 + 22.1925i
139.2 −1.17557 − 1.61803i −2.85317 − 0.927051i −1.23607 + 3.80423i −5.30270 + 9.84283i 1.85410 + 5.70634i − 5.42104i 7.60845 − 2.47214i 7.28115 + 5.29007i 22.1597 − 2.99099i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.4.h.a ✓ 24
25.e even 10 1 inner 150.4.h.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.h.a ✓ 24 1.a even 1 1 trivial
150.4.h.a ✓ 24 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 3552 T_{7}^{22} + 4951116 T_{7}^{20} + 3502082070 T_{7}^{18} + 1379650581030 T_{7}^{16} + \cdots + 16\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display