Properties

Label 168.4.v.a
Level $168$
Weight $4$
Character orbit 168.v
Analytic conductor $9.912$
Analytic rank $0$
Dimension $184$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,4,Mod(11,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.v (of order \(6\), degree \(2\), 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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(184\)
Relative dimension: \(92\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 2 q^{3} - 2 q^{4} - 20 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 184 q - 2 q^{3} - 2 q^{4} - 20 q^{6} - 2 q^{9} + 30 q^{10} - 2 q^{12} + 118 q^{16} - 52 q^{18} - 4 q^{19} - 340 q^{22} - 224 q^{24} - 1904 q^{25} - 8 q^{27} + 282 q^{28} + 210 q^{30} - 110 q^{33} - 256 q^{34} + 604 q^{36} - 498 q^{40} + 748 q^{42} - 16 q^{43} - 360 q^{46} - 416 q^{48} + 352 q^{49} + 590 q^{51} + 600 q^{52} - 1420 q^{54} + 100 q^{57} + 1050 q^{58} + 460 q^{60} + 1828 q^{64} + 1198 q^{66} + 1628 q^{67} + 3390 q^{70} - 914 q^{72} + 212 q^{73} - 144 q^{75} - 3496 q^{76} + 4084 q^{78} + 614 q^{81} - 3412 q^{82} - 2576 q^{84} + 2270 q^{88} - 3736 q^{90} + 936 q^{91} + 2604 q^{94} + 4274 q^{96} - 4672 q^{97} - 2924 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.82802 0.0477234i 0.799088 + 5.13434i 7.99544 + 0.269926i 0.139084 + 0.240900i −2.01481 14.5582i 4.18856 18.0404i −22.5984 1.14493i −25.7229 + 8.20558i −0.381836 0.687910i
11.2 −2.82363 0.164748i −0.360706 5.18362i 7.94572 + 0.930371i 10.4965 + 18.1805i 0.164511 + 14.6960i −18.5109 0.588189i −22.2824 3.93606i −26.7398 + 3.73953i −26.6430 53.0642i
11.3 −2.82183 0.193129i −5.18144 0.390781i 7.92540 + 1.08995i 7.95416 + 13.7770i 14.5456 + 2.10340i 18.4515 1.59414i −22.1536 4.60628i 26.6946 + 4.04961i −19.7845 40.4125i
11.4 −2.81518 0.273405i 4.60128 + 2.41418i 7.85050 + 1.53937i −7.18350 12.4422i −12.2934 8.05436i −15.0501 + 10.7932i −21.6797 6.47998i 15.3435 + 22.2166i 16.8211 + 36.9910i
11.5 −2.79847 0.410550i −5.18237 + 0.378250i 7.66290 + 2.29783i −9.90213 17.1510i 14.6580 + 1.06910i 1.86933 18.4257i −20.5010 9.57641i 26.7139 3.92046i 20.6695 + 52.0619i
11.6 −2.78006 + 0.520843i 1.08553 + 5.08150i 7.45745 2.89595i 6.68614 + 11.5807i −5.66451 13.5615i 4.08452 + 18.0642i −19.2238 + 11.9351i −24.6432 + 11.0323i −24.6196 28.7127i
11.7 −2.74932 + 0.664275i 4.62157 2.37509i 7.11748 3.65260i 3.04263 + 5.26998i −11.1284 + 9.59988i 17.0330 7.27154i −17.1419 + 14.7701i 15.7179 21.9533i −11.8659 12.4677i
11.8 −2.72868 + 0.744517i 1.45164 4.98926i 6.89139 4.06310i −6.36074 11.0171i −0.246461 + 14.6949i −10.4252 15.3073i −15.7793 + 16.2177i −22.7855 14.4852i 25.5589 + 25.3265i
11.9 −2.72824 0.746131i −3.42298 + 3.90938i 6.88658 + 4.07125i −0.646968 1.12058i 12.2556 8.11173i −16.4591 + 8.49107i −15.7505 16.2456i −3.56646 26.7634i 0.928982 + 3.53993i
11.10 −2.70695 + 0.820003i −3.91011 3.42214i 6.65519 4.43942i −2.88032 4.98886i 13.3907 + 6.05726i −5.96675 + 17.5328i −14.3749 + 17.4746i 3.57796 + 26.7619i 11.8878 + 11.1427i
11.11 −2.68148 0.899817i 3.33150 3.98762i 6.38066 + 4.82568i −3.71534 6.43516i −12.5215 + 7.69498i 3.81353 + 18.1234i −12.7674 18.6814i −4.80222 26.5695i 4.17215 + 20.5989i
11.12 −2.62607 1.05059i 5.17720 0.443396i 5.79251 + 5.51787i 4.08846 + 7.08141i −14.0615 4.27474i −7.44249 16.9591i −9.41451 20.5759i 26.6068 4.59110i −3.29690 22.8916i
11.13 −2.61525 1.07724i −1.39745 5.00471i 5.67911 + 5.63451i −1.03843 1.79862i −1.73659 + 14.5940i 18.4984 + 0.899114i −8.78260 20.8534i −23.0943 + 13.9877i 0.778223 + 5.82248i
11.14 −2.50772 + 1.30819i 4.70191 + 2.21179i 4.57730 6.56112i −6.88079 11.9179i −14.6845 + 0.604435i 18.3466 + 2.52991i −2.89542 + 22.4414i 17.2160 + 20.7993i 32.8459 + 20.8853i
11.15 −2.49348 + 1.33513i −4.07521 + 3.22376i 4.43487 6.65822i −4.58233 7.93683i 5.85732 13.4793i 15.9950 + 9.33591i −2.16868 + 22.5233i 6.21469 26.2750i 22.0226 + 13.6723i
11.16 −2.40299 + 1.49185i −4.07521 + 3.22376i 3.54875 7.16982i 4.58233 + 7.93683i 4.98333 13.8263i −15.9950 9.33591i 2.16868 + 22.5233i 6.21469 26.2750i −22.8519 12.2360i
11.17 −2.38678 + 1.51765i 4.70191 + 2.21179i 3.39345 7.24462i 6.88079 + 11.9179i −14.5792 + 1.85683i −18.3466 2.52991i 2.89542 + 22.4414i 17.2160 + 20.7993i −34.5102 18.0027i
11.18 −2.28518 1.66672i 4.42614 + 2.72200i 2.44409 + 7.61751i 7.35113 + 12.7325i −5.57772 13.5974i 12.0391 + 14.0734i 7.11106 21.4810i 12.1815 + 24.0959i 4.42291 41.3484i
11.19 −2.21188 1.76283i −0.903574 + 5.11699i 1.78486 + 7.79835i −7.29019 12.6270i 11.0190 9.72533i 15.6852 + 9.84762i 9.79926 20.3955i −25.3671 9.24716i −6.13417 + 40.7808i
11.20 −2.21019 1.76495i −5.16921 0.528471i 1.76987 + 7.80177i 2.23983 + 3.87950i 10.4922 + 10.2914i −11.5640 + 14.4663i 9.85801 20.3671i 26.4414 + 5.46355i 1.89669 12.5276i
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.d odd 2 1 inner
21.h odd 6 1 inner
24.f even 2 1 inner
56.k odd 6 1 inner
168.v 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.v.a 184
3.b odd 2 1 inner 168.4.v.a 184
7.c even 3 1 inner 168.4.v.a 184
8.d odd 2 1 inner 168.4.v.a 184
21.h odd 6 1 inner 168.4.v.a 184
24.f even 2 1 inner 168.4.v.a 184
56.k odd 6 1 inner 168.4.v.a 184
168.v even 6 1 inner 168.4.v.a 184
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.v.a 184 1.a even 1 1 trivial
168.4.v.a 184 3.b odd 2 1 inner
168.4.v.a 184 7.c even 3 1 inner
168.4.v.a 184 8.d odd 2 1 inner
168.4.v.a 184 21.h odd 6 1 inner
168.4.v.a 184 24.f even 2 1 inner
168.4.v.a 184 56.k odd 6 1 inner
168.4.v.a 184 168.v even 6 1 inner

Hecke kernels

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