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Properties

Label 168.4.t.a
Level $168$
Weight $4$
Character orbit 168.t
Analytic conductor $9.912$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,4,Mod(19,168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 5])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("168.19"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 2 q^{2} - 10 q^{4} - 16 q^{8} + 432 q^{9} + 18 q^{10} + 40 q^{11} + 274 q^{14} - 26 q^{16} - 18 q^{18} + 4 q^{22} + 270 q^{24} - 1200 q^{25} + 750 q^{26} + 922 q^{28} + 168 q^{30} - 108 q^{32} + 456 q^{35}+ \cdots + 720 q^{99}+O(q^{100}) \) 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 −2.82552 + 0.128292i −2.59808 + 1.50000i 7.96708 − 0.724984i −8.82092 + 15.2783i 7.14847 − 4.57159i −18.4632 − 1.45285i −22.4181 + 3.07057i 4.50000 − 7.79423i 22.9636 − 44.3007i
19.2 −2.82155 − 0.197133i −2.59808 + 1.50000i 7.92228 + 1.11244i −1.03601 + 1.79443i 7.62630 − 3.72016i 17.7288 + 5.35612i −22.1338 − 4.70055i 4.50000 − 7.79423i 3.27690 − 4.85883i
19.3 −2.80302 + 0.378234i 2.59808 − 1.50000i 7.71388 − 2.12040i −7.30307 + 12.6493i −6.71512 + 5.18721i −0.782311 + 18.5037i −20.8202 + 8.86116i 4.50000 − 7.79423i 15.6863 − 38.2185i
19.4 −2.77443 − 0.550031i 2.59808 − 1.50000i 7.39493 + 3.05204i 4.82508 − 8.35729i −8.03323 + 2.73262i 14.2799 − 11.7934i −18.8380 − 12.5351i 4.50000 − 7.79423i −17.9836 + 20.5328i
19.5 −2.61501 − 1.07784i −2.59808 + 1.50000i 5.67651 + 5.63713i 8.86587 − 15.3561i 8.41075 − 1.12219i −14.8898 + 11.0134i −8.76816 − 20.8595i 4.50000 − 7.79423i −39.7358 + 30.6004i
19.6 −2.58902 + 1.13884i −2.59808 + 1.50000i 5.40608 − 5.89697i 10.5886 − 18.3401i 5.01822 − 6.84233i 3.62295 − 18.1624i −7.28075 + 21.4241i 4.50000 − 7.79423i −6.52781 + 59.5416i
19.7 −2.52997 − 1.26461i 2.59808 − 1.50000i 4.80152 + 6.39886i 1.10134 − 1.90757i −8.46998 + 0.509406i −7.13695 + 17.0899i −4.05567 − 22.2610i 4.50000 − 7.79423i −5.19868 + 3.43334i
19.8 −2.45618 + 1.40256i 2.59808 − 1.50000i 4.06565 − 6.88988i −6.01586 + 10.4198i −4.27751 + 7.32823i 9.34266 − 15.9911i −0.322521 + 22.6251i 4.50000 − 7.79423i 0.161692 − 34.0305i
19.9 −2.31785 + 1.62097i −2.59808 + 1.50000i 2.74489 − 7.51436i 1.39001 − 2.40757i 3.59050 − 7.68819i −15.3983 + 10.2904i 5.81831 + 21.8666i 4.50000 − 7.79423i 0.680764 + 7.83358i
19.10 −2.28546 + 1.66633i 2.59808 − 1.50000i 2.44666 − 7.61668i 7.16825 − 12.4158i −3.43830 + 7.75746i 12.0691 + 14.0477i 7.10021 + 21.4846i 4.50000 − 7.79423i 4.30608 + 40.3204i
19.11 −2.25437 − 1.70816i −2.59808 + 1.50000i 2.16435 + 7.70166i −1.27195 + 2.20309i 8.41927 + 1.05639i −2.37801 − 18.3670i 8.27645 − 21.0594i 4.50000 − 7.79423i 6.63069 − 2.79387i
19.12 −2.23123 − 1.73828i 2.59808 − 1.50000i 1.95677 + 7.75700i −4.53700 + 7.85831i −8.40432 − 1.16934i −18.3201 − 2.71568i 9.11784 − 20.7091i 4.50000 − 7.79423i 23.7830 − 9.64712i
19.13 −1.82256 + 2.16293i −2.59808 + 1.50000i −1.35652 − 7.88415i −2.37822 + 4.11919i 1.49077 − 8.35330i 14.8436 + 11.0755i 19.5252 + 11.4353i 4.50000 − 7.79423i −4.57507 − 12.6514i
19.14 −1.74973 − 2.22226i −2.59808 + 1.50000i −1.87689 + 7.77671i −5.68130 + 9.84030i 7.87932 + 3.14901i 1.38512 + 18.4684i 20.5659 − 9.43622i 4.50000 − 7.79423i 31.8084 − 4.59254i
19.15 −1.72076 + 2.24477i 2.59808 − 1.50000i −2.07798 − 7.72541i 4.08960 − 7.08340i −1.10351 + 8.41322i −18.1007 − 3.91988i 20.9175 + 8.62897i 4.50000 − 7.79423i 8.86339 + 21.3690i
19.16 −1.40242 − 2.45626i 2.59808 − 1.50000i −4.06643 + 6.88942i −6.00810 + 10.4063i −7.32799 − 4.27792i 8.17074 − 16.6204i 22.6251 + 0.326351i 4.50000 − 7.79423i 33.9865 + 0.163401i
19.17 −1.21695 − 2.55324i 2.59808 − 1.50000i −5.03804 + 6.21435i 9.44320 − 16.3561i −6.99160 − 4.80807i −10.2846 − 15.4022i 21.9978 + 5.30073i 4.50000 − 7.79423i −53.2530 − 4.20609i
19.18 −1.08365 + 2.61261i 2.59808 − 1.50000i −5.65141 − 5.66229i −4.08960 + 7.08340i 1.10351 + 8.41322i 18.1007 + 3.91988i 20.9175 − 8.62897i 4.50000 − 7.79423i −14.0744 − 18.3604i
19.19 −0.961869 + 2.65985i −2.59808 + 1.50000i −6.14962 − 5.11686i 2.37822 − 4.11919i −1.49077 − 8.35330i −14.8436 − 11.0755i 19.5252 − 11.4353i 4.50000 − 7.79423i 8.66891 + 10.2878i
19.20 −0.475069 − 2.78824i 2.59808 − 1.50000i −7.54862 + 2.64922i −2.37912 + 4.12075i −5.41663 − 6.53147i 11.6311 + 14.4124i 10.9728 + 19.7888i 4.50000 − 7.79423i 12.6199 + 4.67592i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.t.a ✓ 96
4.b odd 2 1 672.4.bb.a 96
7.d odd 6 1 inner 168.4.t.a ✓ 96
8.b even 2 1 672.4.bb.a 96
8.d odd 2 1 inner 168.4.t.a ✓ 96
28.f even 6 1 672.4.bb.a 96
56.j odd 6 1 672.4.bb.a 96
56.m even 6 1 inner 168.4.t.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.t.a ✓ 96 1.a even 1 1 trivial
168.4.t.a ✓ 96 7.d odd 6 1 inner
168.4.t.a ✓ 96 8.d odd 2 1 inner
168.4.t.a ✓ 96 56.m even 6 1 inner
672.4.bb.a 96 4.b odd 2 1
672.4.bb.a 96 8.b even 2 1
672.4.bb.a 96 28.f even 6 1
672.4.bb.a 96 56.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(168, [\chi])\).