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Properties

Label 185.3.s.a
Level $185$
Weight $3$
Character orbit 185.s
Analytic conductor $5.041$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,3,Mod(47,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 185.s (of order \(12\), 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: \(5.04088489067\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} + 2 q^{5} - 16 q^{6} - 2 q^{7} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} + 2 q^{5} - 16 q^{6} - 2 q^{7} - 72 q^{8} + 4 q^{10} - 30 q^{12} + 2 q^{13} + 30 q^{15} + 236 q^{16} + 4 q^{17} - 82 q^{18} + 58 q^{20} + 20 q^{21} - 42 q^{22} + 28 q^{23} - 256 q^{26} - 312 q^{27} - 74 q^{28} - 196 q^{30} + 80 q^{31} + 158 q^{32} + 2 q^{33} - 80 q^{35} + 704 q^{36} - 198 q^{37} + 348 q^{38} + 70 q^{40} + 204 q^{41} + 16 q^{42} - 104 q^{43} + 544 q^{45} - 324 q^{46} - 316 q^{47} - 776 q^{48} + 140 q^{50} + 48 q^{51} + 138 q^{52} + 280 q^{53} + 222 q^{55} + 288 q^{56} - 94 q^{57} - 288 q^{58} - 944 q^{60} - 388 q^{61} + 70 q^{62} - 168 q^{63} + 220 q^{65} - 432 q^{66} - 416 q^{68} - 60 q^{70} + 92 q^{71} - 358 q^{72} + 780 q^{73} - 520 q^{75} - 268 q^{76} + 54 q^{77} - 244 q^{78} + 588 q^{80} + 136 q^{81} + 304 q^{82} - 168 q^{83} + 1272 q^{85} + 580 q^{86} - 56 q^{87} - 1116 q^{88} - 852 q^{90} + 660 q^{91} - 212 q^{92} - 226 q^{93} + 116 q^{95} - 56 q^{96} - 176 q^{97} + 646 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −3.81078 + 1.02109i −4.68176 − 1.25447i 10.0153 − 5.78232i 3.19520 − 3.84586i 19.1221 1.33219 − 4.97182i −21.1029 + 21.1029i 12.5509 + 7.24627i −8.24921 + 17.9183i
47.2 −3.68646 + 0.987785i 2.87296 + 0.769808i 9.15018 − 5.28286i 4.49595 + 2.18780i −11.3515 −2.49356 + 9.30609i −17.7188 + 17.7188i −0.132914 − 0.0767379i −18.7352 − 3.62420i
47.3 −3.48478 + 0.933743i 2.45222 + 0.657070i 7.80768 − 4.50777i −3.50600 + 3.56483i −9.15897 2.39647 − 8.94374i −12.7948 + 12.7948i −2.21259 − 1.27744i 8.88898 − 15.6963i
47.4 −3.27969 + 0.878790i −2.09480 − 0.561300i 6.51998 − 3.76431i −4.96975 − 0.549175i 7.36356 −2.89050 + 10.7875i −8.47188 + 8.47188i −3.72110 − 2.14838i 16.7818 − 2.56624i
47.5 −3.00828 + 0.806066i 1.84259 + 0.493722i 4.93590 − 2.84974i 2.92082 − 4.05818i −5.94101 1.48426 − 5.53933i −3.74264 + 3.74264i −4.64284 − 2.68054i −5.51549 + 14.5625i
47.6 −2.93452 + 0.786301i −3.34433 − 0.896112i 4.52902 − 2.61483i −0.588751 + 4.96522i 10.5186 1.32399 − 4.94120i −2.64157 + 2.64157i 2.58733 + 1.49380i −2.17646 − 15.0334i
47.7 −2.87616 + 0.770665i 5.49783 + 1.47314i 4.21428 − 2.43312i −2.38784 − 4.39298i −16.9480 −0.949396 + 3.54320i −1.82386 + 1.82386i 20.2618 + 11.6982i 10.2533 + 10.7947i
47.8 −2.49505 + 0.668548i −0.829958 − 0.222387i 2.31424 − 1.33613i 4.56540 + 2.03890i 2.21947 0.455557 − 1.70016i 2.42513 − 2.42513i −7.15485 − 4.13086i −12.7540 − 2.03498i
47.9 −2.20837 + 0.591732i −1.03840 − 0.278238i 1.06267 − 0.613532i 1.70919 − 4.69879i 2.45782 −1.89244 + 7.06268i 4.48285 − 4.48285i −6.79337 − 3.92216i −0.994105 + 11.3881i
47.10 −2.04573 + 0.548151i −3.83611 − 1.02788i 0.420430 − 0.242735i −3.20055 − 3.84142i 8.41107 2.32010 − 8.65875i 5.26328 − 5.26328i 5.86497 + 3.38614i 8.65312 + 6.10412i
47.11 −2.03208 + 0.544494i 3.22501 + 0.864138i 0.368777 − 0.212913i −2.08989 + 4.54229i −7.02399 −1.84108 + 6.87099i 5.31689 − 5.31689i 1.85971 + 1.07370i 1.77357 − 10.3682i
47.12 −1.80870 + 0.484641i −5.70356 − 1.52826i −0.427569 + 0.246857i 4.84815 + 1.22290i 11.0567 −3.12776 + 11.6730i 5.94996 − 5.94996i 22.4008 + 12.9331i −9.36153 + 0.137751i
47.13 −1.64454 + 0.440653i 1.65406 + 0.443204i −0.953770 + 0.550659i −4.99741 + 0.160845i −2.91546 1.17773 − 4.39536i 6.14140 − 6.14140i −5.25475 − 3.03383i 8.14756 − 2.46664i
47.14 −1.60919 + 0.431181i 4.78097 + 1.28106i −1.06053 + 0.612299i 4.07601 + 2.89588i −8.24585 2.72341 − 10.1639i 6.15461 − 6.15461i 13.4223 + 7.74939i −7.80772 − 2.90252i
47.15 −0.742943 + 0.199071i −3.48390 − 0.933507i −2.95177 + 1.70420i −3.43661 + 3.63177i 2.77417 −0.0901376 + 0.336398i 4.02923 − 4.02923i 3.47186 + 2.00448i 1.83022 − 3.38233i
47.16 −0.737280 + 0.197554i 2.34079 + 0.627212i −2.95955 + 1.70870i −3.38464 − 3.68024i −1.84972 −1.11063 + 4.14492i 4.00336 − 4.00336i −2.70835 − 1.56366i 3.22248 + 2.04472i
47.17 −0.283851 + 0.0760576i −0.902612 − 0.241854i −3.38932 + 1.95682i 4.28415 − 2.57800i 0.274602 −0.0444148 + 0.165758i 1.64440 − 1.64440i −7.03801 − 4.06340i −1.01998 + 1.05761i
47.18 −0.248085 + 0.0664743i −0.375544 − 0.100627i −3.40697 + 1.96702i 2.67979 + 4.22122i 0.0998561 0.177316 − 0.661751i 1.44091 − 1.44091i −7.66332 − 4.42442i −0.945420 − 0.869086i
47.19 −0.102826 + 0.0275522i 4.18323 + 1.12089i −3.45429 + 1.99433i 4.75788 − 1.53707i −0.461029 −2.89820 + 10.8162i 0.601340 − 0.601340i 8.44877 + 4.87790i −0.446886 + 0.289142i
47.20 −0.0186851 + 0.00500667i 2.39574 + 0.641937i −3.46378 + 1.99981i 0.361590 − 4.98691i −0.0479787 3.46004 − 12.9131i 0.109423 − 0.109423i −2.46674 − 1.42417i 0.0182114 + 0.0949915i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
37.c even 3 1 inner
185.s odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.3.s.a ✓ 144
5.c odd 4 1 inner 185.3.s.a ✓ 144
37.c even 3 1 inner 185.3.s.a ✓ 144
185.s odd 12 1 inner 185.3.s.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.3.s.a ✓ 144 1.a even 1 1 trivial
185.3.s.a ✓ 144 5.c odd 4 1 inner
185.3.s.a ✓ 144 37.c even 3 1 inner
185.3.s.a ✓ 144 185.s odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(185, [\chi])\).