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Properties

Label 152.4.t.a
Level $152$
Weight $4$
Character orbit 152.t
Analytic conductor $8.968$
Analytic rank $0$
Dimension $348$
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] = [152,4,Mod(5,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.t (of order \(18\), degree \(6\), 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: \(8.96829032087\)
Analytic rank: \(0\)
Dimension: \(348\)
Relative dimension: \(58\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 348 q - 6 q^{2} + 12 q^{4} - 24 q^{6} - 6 q^{7} - 3 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 348 q - 6 q^{2} + 12 q^{4} - 24 q^{6} - 6 q^{7} - 3 q^{8} - 12 q^{9} + 75 q^{10} - 3 q^{12} - 81 q^{14} - 12 q^{15} + 192 q^{16} - 12 q^{17} - 12 q^{18} + 558 q^{20} - 30 q^{22} - 12 q^{23} - 108 q^{24} - 12 q^{25} - 411 q^{26} + 1140 q^{28} - 282 q^{30} - 1122 q^{31} + 39 q^{32} - 174 q^{33} + 1536 q^{34} - 2835 q^{36} + 966 q^{38} - 24 q^{39} - 1068 q^{40} + 48 q^{41} + 3015 q^{42} - 1203 q^{44} - 528 q^{46} - 624 q^{47} + 4755 q^{48} - 6768 q^{49} - 1074 q^{50} - 1089 q^{52} + 615 q^{54} - 762 q^{55} + 2046 q^{56} - 12 q^{57} + 4092 q^{58} + 4050 q^{60} + 3120 q^{62} + 2046 q^{63} + 2355 q^{64} - 6 q^{65} + 1716 q^{66} - 1176 q^{68} - 2973 q^{70} - 12 q^{71} - 7122 q^{72} + 636 q^{73} - 9363 q^{74} - 2454 q^{76} - 6897 q^{78} - 12 q^{79} - 6885 q^{80} + 1206 q^{81} - 7311 q^{82} + 1947 q^{84} + 7452 q^{86} - 6 q^{87} - 1131 q^{88} - 12 q^{89} + 4014 q^{90} + 7656 q^{92} + 11898 q^{94} + 1848 q^{95} - 18966 q^{96} - 12 q^{97} + 8055 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.82713 + 0.0857083i −6.31470 − 1.11345i 7.98531 − 0.484617i 13.6906 − 16.3158i 17.9479 + 2.60665i 2.50109 + 4.33202i −22.5340 + 2.05448i 13.2639 + 4.82767i −37.3066 + 47.3002i
5.2 −2.81612 − 0.263534i −0.304087 − 0.0536187i 7.86110 + 1.48429i −12.5870 + 15.0006i 0.842215 + 0.231134i −3.26800 − 5.66034i −21.7467 − 6.25161i −25.2821 − 9.20193i 39.3997 − 38.9264i
5.3 −2.81245 + 0.300165i 4.63211 + 0.816766i 7.81980 − 1.68840i 0.297222 − 0.354215i −13.2728 − 0.906719i −12.9666 − 22.4588i −21.4860 + 7.09579i −4.58236 − 1.66784i −0.729600 + 1.08543i
5.4 −2.80737 − 0.344520i −6.82434 − 1.20332i 7.76261 + 1.93439i −1.31007 + 1.56128i 18.7439 + 5.72927i −8.09624 − 14.0231i −21.1261 − 8.10492i 19.7520 + 7.18914i 4.21573 − 3.93174i
5.5 −2.79657 + 0.423345i 2.55460 + 0.450445i 7.64156 − 2.36782i −0.869689 + 1.03646i −7.33480 − 0.178223i 9.58545 + 16.6025i −20.3677 + 9.85678i −19.0486 − 6.93313i 1.99337 − 3.26669i
5.6 −2.73220 + 0.731475i 9.54931 + 1.68380i 6.92989 − 3.99708i 8.11803 − 9.67470i −27.3223 + 2.38460i 10.6991 + 18.5314i −16.0101 + 15.9899i 62.9825 + 22.9237i −15.1033 + 32.3714i
5.7 −2.56317 + 1.19588i −9.54931 − 1.68380i 5.13972 − 6.13052i −8.11803 + 9.67470i 26.4902 − 7.10400i 10.6991 + 18.5314i −5.84259 + 21.8601i 62.9825 + 22.9237i 9.23811 − 34.5062i
5.8 −2.54896 − 1.22590i −1.62482 − 0.286499i 4.99435 + 6.24952i 2.82361 − 3.36505i 3.79036 + 2.72213i 13.2108 + 22.8819i −5.06910 − 22.0523i −22.8138 − 8.30353i −11.3225 + 5.11590i
5.9 −2.50272 − 1.31772i 8.78143 + 1.54840i 4.52722 + 6.59578i −6.43807 + 7.67259i −19.9371 − 15.4467i −1.25984 − 2.18211i −2.63897 − 22.4730i 49.3442 + 17.9598i 26.2230 − 10.7188i
5.10 −2.49413 − 1.33392i 4.10931 + 0.724583i 4.44133 + 6.65392i 9.57274 − 11.4084i −9.28261 − 7.28868i −5.68735 − 9.85077i −2.20147 − 22.5201i −9.01028 − 3.27947i −39.0934 + 15.6846i
5.11 −2.41441 + 1.47330i −2.55460 − 0.450445i 3.65879 − 7.11430i 0.869689 − 1.03646i 6.83150 − 2.67612i 9.58545 + 16.6025i 1.64763 + 22.5674i −19.0486 − 6.93313i −0.572784 + 3.78374i
5.12 −2.34741 + 1.57787i −4.63211 − 0.816766i 3.02065 − 7.40781i −0.297222 + 0.354215i 12.1622 − 5.39159i −12.9666 − 22.4588i 4.59789 + 22.1553i −4.58236 − 1.66784i 0.138795 − 1.30047i
5.13 −2.22080 + 1.75159i 6.31470 + 1.11345i 1.86389 − 7.77984i −13.6906 + 16.3158i −15.9740 + 8.58799i 2.50109 + 4.33202i 9.48775 + 20.5422i 13.2639 + 4.82767i 1.82548 − 60.2142i
5.14 −2.19911 − 1.77874i −5.54183 − 0.977174i 1.67215 + 7.82329i −10.2575 + 12.2244i 10.4489 + 12.0064i 7.83774 + 13.5754i 10.2384 − 20.1786i 4.38529 + 1.59612i 44.3015 − 8.63736i
5.15 −2.00897 − 1.99099i −4.45779 − 0.786029i 0.0718986 + 7.99968i 2.48844 − 2.96560i 7.39058 + 10.4545i −17.1535 − 29.7107i 15.7829 − 16.2142i −6.11761 − 2.22663i −10.9037 + 1.00334i
5.16 −1.98788 + 2.01205i 0.304087 + 0.0536187i −0.0966730 − 7.99942i 12.5870 − 15.0006i −0.712371 + 0.505249i −3.26800 − 5.66034i 16.2874 + 15.7074i −25.2821 − 9.20193i 5.16049 + 55.1450i
5.17 −1.92911 + 2.06846i 6.82434 + 1.20332i −0.557038 − 7.98058i 1.31007 − 1.56128i −15.6539 + 11.7945i −8.09624 − 14.0231i 17.5821 + 14.2432i 19.7520 + 7.18914i 0.702169 + 5.72171i
5.18 −1.64798 − 2.29873i −9.17013 − 1.61694i −2.56833 + 7.57652i 6.10344 − 7.27380i 11.3953 + 23.7443i 6.67659 + 11.5642i 21.6489 − 6.58205i 56.1051 + 20.4206i −26.7789 − 2.04310i
5.19 −1.57035 − 2.35245i 4.57945 + 0.807480i −3.06800 + 7.38833i −6.51289 + 7.76175i −5.29178 − 12.0409i 1.91022 + 3.30860i 22.1985 − 4.38495i −5.05239 − 1.83892i 28.4866 + 3.13254i
5.20 −1.26778 − 2.52839i 0.365686 + 0.0644802i −4.78548 + 6.41087i −4.71526 + 5.61942i −0.300577 − 1.00634i −3.17618 − 5.50130i 22.2761 + 3.97197i −25.2421 − 9.18738i 20.1860 + 4.79781i
See next 80 embeddings (of 348 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
19.e even 9 1 inner
152.t even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.4.t.a ✓ 348
8.b even 2 1 inner 152.4.t.a ✓ 348
19.e even 9 1 inner 152.4.t.a ✓ 348
152.t even 18 1 inner 152.4.t.a ✓ 348
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.t.a ✓ 348 1.a even 1 1 trivial
152.4.t.a ✓ 348 8.b even 2 1 inner
152.4.t.a ✓ 348 19.e even 9 1 inner
152.4.t.a ✓ 348 152.t even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(152, [\chi])\).