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Properties

Label 168.4.ba.c
Level $168$
Weight $4$
Character orbit 168.ba
Analytic conductor $9.912$
Analytic rank $0$
Dimension $176$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,4,Mod(5,168)] 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([0, 3, 3, 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("168.5"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [176,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q + 30 q^{4} - 76 q^{7} - 110 q^{9} + 330 q^{10} + 162 q^{12} - 332 q^{15} + 374 q^{16} + 18 q^{18} - 212 q^{22} - 84 q^{24} + 1112 q^{25} + 30 q^{28} - 104 q^{30} + 408 q^{31} + 2262 q^{33} + 1372 q^{36}+ \cdots - 1782 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.82592 − 0.118988i 1.50779 + 4.97258i 7.97168 + 0.672504i −8.85750 − 5.11388i −3.66921 − 14.2315i 14.2172 − 11.8690i −22.4473 − 2.84898i −22.4532 + 14.9952i 24.4221 + 15.5054i
5.2 −2.82548 + 0.129008i −5.18277 − 0.372721i 7.96671 − 0.729019i 3.84884 + 2.22213i 14.6919 + 0.384499i −14.6025 − 11.3916i −22.4158 + 3.08760i 26.7222 + 3.86345i −11.1615 − 5.78206i
5.3 −2.81133 + 0.310546i −4.34325 + 2.85240i 7.80712 − 1.74609i −7.95876 − 4.59499i 11.3245 − 9.36780i −6.65880 + 17.2818i −21.4061 + 7.33330i 10.7276 − 24.7774i 23.8016 + 10.4465i
5.4 −2.80104 − 0.392670i 1.32188 − 5.02520i 7.69162 + 2.19977i 4.36593 + 2.52067i −5.67587 + 13.5567i −8.52080 + 16.4437i −20.6807 − 9.18190i −23.5053 − 13.2854i −11.2393 − 8.77487i
5.5 −2.79792 + 0.414328i −1.10888 − 5.07645i 7.65666 − 2.31851i −16.5761 − 9.57020i 5.20586 + 13.7441i −3.98090 − 18.0874i −20.4621 + 9.65937i −24.5408 + 11.2583i 50.3437 + 19.9087i
5.6 −2.78765 + 0.478565i −2.84692 − 4.34684i 7.54195 − 2.66814i 14.4750 + 8.35717i 10.0165 + 10.7550i 18.3930 − 2.16763i −19.7474 + 11.0471i −10.7901 + 24.7502i −44.3508 − 16.3696i
5.7 −2.75382 + 0.645369i 5.19422 + 0.141752i 7.16700 − 3.55445i −2.58266 − 1.49110i −14.3954 + 2.96183i 13.1902 − 13.0008i −17.4427 + 14.4137i 26.9598 + 1.47258i 8.07449 + 2.43945i
5.8 −2.73440 − 0.723237i 5.18464 + 0.345678i 6.95386 + 3.95523i −16.5916 − 9.57915i −13.9269 − 4.69494i −14.3399 + 11.7204i −16.1540 − 15.8445i 26.7610 + 3.58443i 38.4399 + 38.1928i
5.9 −2.71145 + 0.805025i 4.75868 + 2.08685i 6.70387 − 4.36556i 16.1287 + 9.31191i −14.5829 − 1.82753i −17.3676 + 6.43172i −14.6628 + 17.2338i 18.2901 + 19.8613i −51.2284 − 12.2647i
5.10 −2.66484 − 0.947964i −3.53117 + 3.81194i 6.20273 + 5.05234i 11.9160 + 6.87970i 13.0236 − 6.81077i 18.5103 − 0.606943i −11.7398 − 19.3436i −2.06172 − 26.9212i −25.2325 − 29.6292i
5.11 −2.66319 + 0.952587i 0.850712 + 5.12604i 6.18516 − 5.07384i 4.25602 + 2.45721i −7.14860 − 12.8412i 7.65432 + 16.8645i −11.6390 + 19.4045i −25.5526 + 8.72156i −13.6753 − 2.48980i
5.12 −2.66254 − 0.954414i 1.86846 + 4.84860i 6.17819 + 5.08232i 0.715919 + 0.413336i −0.347261 − 14.6928i −16.5388 − 8.33474i −11.5990 − 19.4284i −20.0177 + 18.1188i −1.51167 − 1.78380i
5.13 −2.65194 − 0.983470i 4.15215 − 3.12404i 6.06558 + 5.21621i 11.2278 + 6.48238i −14.0837 + 4.20125i −0.198886 − 18.5192i −10.9556 − 19.7984i 7.48076 − 25.9430i −23.4003 − 28.2331i
5.14 −2.57507 − 1.17004i −4.47107 − 2.64755i 5.26201 + 6.02588i −8.84675 − 5.10767i 8.41561 + 12.0490i 12.7989 + 13.3861i −6.49955 − 21.6739i 12.9810 + 23.6748i 16.8049 + 23.5037i
5.15 −2.49165 + 1.33854i 2.74818 − 4.40993i 4.41662 − 6.67034i −3.38400 − 1.95375i −0.944612 + 14.6666i 14.8660 + 11.0455i −2.07613 + 22.5320i −11.8951 − 24.2386i 11.0469 + 0.338444i
5.16 −2.38000 + 1.52826i −2.59084 + 4.50417i 3.32883 − 7.27454i 12.7681 + 7.37169i −0.717338 − 14.6794i −2.36716 − 18.3684i 3.19475 + 22.4007i −13.5751 − 23.3392i −41.6541 + 1.96841i
5.17 −2.30082 − 1.64506i 4.47107 + 2.64755i 2.58756 + 7.56998i 8.84675 + 5.10767i −5.93177 − 13.4467i 12.7989 + 13.3861i 6.49955 − 21.6739i 12.9810 + 23.6748i −11.9524 − 26.3053i
5.18 −2.26030 + 1.70030i 4.19123 − 3.07141i 2.21794 − 7.68640i −3.83548 − 2.21442i −4.25113 + 14.0687i −18.2781 − 2.98509i 8.05600 + 21.1448i 8.13288 − 25.7460i 12.4345 − 1.51623i
5.19 −2.17768 − 1.80491i −4.15215 + 3.12404i 1.48458 + 7.86104i −11.2278 − 6.48238i 14.6807 + 0.691119i −0.198886 − 18.5192i 10.9556 − 19.7984i 7.48076 − 25.9430i 12.7505 + 34.3818i
5.20 −2.16766 + 1.81693i −5.17339 − 0.485852i 1.39753 − 7.87699i −7.76020 − 4.48035i 12.0969 − 8.34652i 18.1053 − 3.89868i 11.2825 + 19.6139i 26.5279 + 5.02700i 24.9620 − 4.38784i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.88
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba 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.ba.c ✓ 176
3.b odd 2 1 inner 168.4.ba.c ✓ 176
7.d odd 6 1 inner 168.4.ba.c ✓ 176
8.b even 2 1 inner 168.4.ba.c ✓ 176
21.g even 6 1 inner 168.4.ba.c ✓ 176
24.h odd 2 1 inner 168.4.ba.c ✓ 176
56.j odd 6 1 inner 168.4.ba.c ✓ 176
168.ba even 6 1 inner 168.4.ba.c ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.ba.c ✓ 176 1.a even 1 1 trivial
168.4.ba.c ✓ 176 3.b odd 2 1 inner
168.4.ba.c ✓ 176 7.d odd 6 1 inner
168.4.ba.c ✓ 176 8.b even 2 1 inner
168.4.ba.c ✓ 176 21.g even 6 1 inner
168.4.ba.c ✓ 176 24.h odd 2 1 inner
168.4.ba.c ✓ 176 56.j odd 6 1 inner
168.4.ba.c ✓ 176 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{88} - 3028 T_{5}^{86} + 4967424 T_{5}^{84} - 5635337432 T_{5}^{82} + 4890926593820 T_{5}^{80} + \cdots + 70\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display