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Properties

Label 165.4.p.a
Level $165$
Weight $4$
Character orbit 165.p
Analytic conductor $9.735$
Analytic rank $0$
Dimension $192$
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] = [165,4,Mod(41,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.p (of order \(10\), 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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 4 q^{3} - 192 q^{4} - 100 q^{6} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 4 q^{3} - 192 q^{4} - 100 q^{6} + 108 q^{9} + 48 q^{12} - 60 q^{15} - 1248 q^{16} - 460 q^{18} + 900 q^{19} + 888 q^{22} + 1200 q^{24} + 1200 q^{25} + 568 q^{27} - 1680 q^{28} - 576 q^{31} - 1402 q^{33} + 264 q^{34} - 2364 q^{36} + 24 q^{37} + 1220 q^{39} + 4438 q^{42} - 360 q^{46} + 1946 q^{48} + 1848 q^{49} + 930 q^{51} + 840 q^{55} - 4170 q^{57} + 4320 q^{58} - 830 q^{60} - 1560 q^{61} - 7900 q^{63} - 11700 q^{64} - 4214 q^{66} + 168 q^{67} - 4992 q^{69} - 1620 q^{70} + 6720 q^{72} - 360 q^{73} + 150 q^{75} + 13972 q^{78} + 4740 q^{79} + 916 q^{81} + 6420 q^{82} + 6880 q^{84} + 2868 q^{88} + 8736 q^{91} - 5696 q^{93} - 840 q^{94} - 12470 q^{96} - 7356 q^{97} - 736 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −4.37775 + 3.18062i 5.19597 + 0.0430816i 6.57619 − 20.2394i 2.93893 − 4.04508i −22.8837 + 16.3378i −17.6902 − 5.74790i 22.2078 + 68.3486i 26.9963 + 0.447702i 27.0560i
41.2 −4.33342 + 3.14842i 1.85440 + 4.85399i 6.39390 − 19.6784i −2.93893 + 4.04508i −23.3183 − 15.1960i 16.5829 + 5.38812i 21.0066 + 64.6515i −20.1224 + 18.0025i − 26.7820i
41.3 −4.31401 + 3.13431i −4.16964 − 3.10066i 6.31465 − 19.4345i 2.93893 − 4.04508i 27.7063 + 0.307340i 6.06872 + 1.97185i 20.4899 + 63.0614i 7.77178 + 25.8573i 26.6621i
41.4 −4.12943 + 3.00021i 1.71341 − 4.90553i 5.57881 − 17.1698i −2.93893 + 4.04508i 7.64218 + 25.3976i −24.5008 − 7.96079i 15.8573 + 48.8036i −21.1284 − 16.8104i − 25.5213i
41.5 −3.95634 + 2.87445i −3.12055 + 4.15478i 4.91804 − 15.1362i 2.93893 − 4.04508i 0.403271 − 25.4076i 12.3288 + 4.00588i 11.9612 + 36.8128i −7.52432 − 25.9304i 24.4515i
41.6 −3.94368 + 2.86525i −4.41355 + 2.74237i 4.87081 − 14.9908i −2.93893 + 4.04508i 9.54802 − 23.4610i −18.8401 − 6.12152i 11.6927 + 35.9864i 11.9588 − 24.2072i − 24.3733i
41.7 −3.55039 + 2.57951i −3.58250 − 3.76374i 3.47925 − 10.7080i −2.93893 + 4.04508i 22.4278 + 4.12164i 8.73323 + 2.83760i 4.41974 + 13.6026i −1.33143 + 26.9672i − 21.9426i
41.8 −3.32358 + 2.41473i 4.15843 + 3.11568i 2.74318 − 8.44263i 2.93893 − 4.04508i −21.3444 − 0.313780i 11.3617 + 3.69163i 1.11348 + 3.42695i 7.58503 + 25.9127i 20.5409i
41.9 −3.12619 + 2.27131i 5.16730 − 0.546811i 2.14208 − 6.59266i −2.93893 + 4.04508i −14.9120 + 13.4460i 16.5110 + 5.36474i −1.27538 − 3.92522i 26.4020 − 5.65107i − 19.3209i
41.10 −2.84801 + 2.06920i 4.14644 − 3.13162i 1.35744 − 4.17778i 2.93893 − 4.04508i −5.32916 + 17.4987i −7.01306 − 2.27868i −3.92409 − 12.0771i 7.38594 − 25.9701i 17.6017i
41.11 −2.81316 + 2.04388i −1.01719 − 5.09562i 1.26428 − 3.89106i 2.93893 − 4.04508i 13.2763 + 12.2558i −23.9271 − 7.77440i −4.20002 − 12.9263i −24.9307 + 10.3664i 17.3863i
41.12 −2.44441 + 1.77597i 5.06426 + 1.16331i 0.348949 − 1.07396i −2.93893 + 4.04508i −14.4451 + 6.15036i −25.0543 − 8.14063i −6.41512 − 19.7437i 24.2934 + 11.7826i − 15.1073i
41.13 −2.41966 + 1.75799i −5.15998 + 0.612021i 0.292106 − 0.899011i −2.93893 + 4.04508i 11.4095 − 10.5521i 29.5600 + 9.60462i −6.52018 − 20.0670i 26.2509 − 6.31604i − 14.9543i
41.14 −2.41426 + 1.75406i −1.09514 + 5.07944i 0.279774 − 0.861056i 2.93893 − 4.04508i −6.26568 − 14.1840i 9.59193 + 3.11661i −6.54242 − 20.1355i −24.6013 − 11.1254i 14.9209i
41.15 −2.06223 + 1.49830i −1.66099 + 4.92353i −0.464244 + 1.42880i −2.93893 + 4.04508i −3.95157 − 12.6421i −1.91226 − 0.621330i −7.48499 − 23.0364i −21.4822 − 16.3558i − 12.7453i
41.16 −1.85179 + 1.34540i −0.851496 − 5.12591i −0.853127 + 2.62565i 2.93893 − 4.04508i 8.47320 + 8.34649i 28.0892 + 9.12673i −7.61131 − 23.4252i −25.5499 + 8.72938i 11.4447i
41.17 −1.78814 + 1.29916i 2.75584 − 4.40515i −0.962500 + 2.96227i −2.93893 + 4.04508i 0.795157 + 11.4573i −4.06416 − 1.32053i −7.59147 − 23.3641i −11.8106 − 24.2798i − 11.0513i
41.18 −1.35979 + 0.987942i −5.19037 + 0.245075i −1.59915 + 4.92167i 2.93893 − 4.04508i 6.81567 − 5.46103i 18.3598 + 5.96546i −6.84296 − 21.0605i 26.8799 − 2.54406i 8.40394i
41.19 −1.01327 + 0.736186i 3.14372 + 4.13727i −1.98738 + 6.11654i −2.93893 + 4.04508i −6.23125 − 1.87782i 29.0946 + 9.45342i −5.58543 − 17.1902i −7.23403 + 26.0129i − 6.26237i
41.20 −0.923458 + 0.670932i 2.66996 + 4.45772i −2.06951 + 6.36930i 2.93893 − 4.04508i −5.45643 − 2.32516i −17.6104 − 5.72198i −5.18409 − 15.9550i −12.7426 + 23.8039i 5.70729i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.p.a ✓ 192
3.b odd 2 1 inner 165.4.p.a ✓ 192
11.d odd 10 1 inner 165.4.p.a ✓ 192
33.f even 10 1 inner 165.4.p.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.p.a ✓ 192 1.a even 1 1 trivial
165.4.p.a ✓ 192 3.b odd 2 1 inner
165.4.p.a ✓ 192 11.d odd 10 1 inner
165.4.p.a ✓ 192 33.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(165, [\chi])\).