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Properties

Label 161.4.o.a
Level $161$
Weight $4$
Character orbit 161.o
Analytic conductor $9.499$
Analytic rank $0$
Dimension $920$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,Mod(5,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([55, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.o (of order \(66\), degree \(20\), 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: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(920\)
Relative dimension: \(46\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 11 q^{2} - 27 q^{3} + 157 q^{4} - 33 q^{5} - 22 q^{7} - 104 q^{8} - 359 q^{9} - 33 q^{10} - 11 q^{11} - 75 q^{12} - 22 q^{14} - 44 q^{15} + 1157 q^{16} - 33 q^{17} + 179 q^{18} - 33 q^{19} - 1342 q^{21}+ \cdots + 23056 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −4.01594 + 3.82919i 0.670881 − 3.48086i 1.08442 − 22.7647i −3.02350 − 2.37771i 10.6347 + 16.5479i −4.17776 − 18.0429i 53.7453 + 62.0253i 13.3996 + 5.36440i 21.2469 − 2.02883i
5.2 −3.96636 + 3.78192i −1.39802 + 7.25362i 1.04846 − 22.0100i 14.2516 + 11.2076i −21.8876 − 34.0577i 18.0932 + 3.95415i 50.3701 + 58.1302i −25.5947 − 10.2466i −98.9135 + 9.44509i
5.3 −3.63378 + 3.46481i 1.69687 − 8.80421i 0.818849 − 17.1898i 5.19287 + 4.08372i 24.3388 + 37.8719i −2.37837 + 18.3669i 30.2798 + 34.9448i −49.5687 − 19.8443i −33.0191 + 3.15294i
5.4 −3.49780 + 3.33514i −1.40095 + 7.26882i 0.730752 − 15.3404i −6.73839 − 5.29912i −19.3423 − 30.0972i −15.8934 + 9.50796i 23.2868 + 26.8744i −25.8072 − 10.3316i 41.2428 − 3.93821i
5.5 −3.31841 + 3.16409i 0.136938 − 0.710501i 0.619675 − 13.0086i −1.58854 − 1.24924i 1.79368 + 2.79102i −2.21383 + 18.3875i 15.0831 + 17.4068i 24.5799 + 9.84030i 9.22416 − 0.880801i
5.6 −3.25729 + 3.10582i −0.129179 + 0.670244i 0.583169 − 12.2422i 14.7588 + 11.6065i −1.66088 − 2.58439i −16.8947 − 7.58743i 12.5441 + 14.4766i 24.6334 + 9.86172i −84.1216 + 8.03264i
5.7 −3.24194 + 3.09119i −0.982886 + 5.09970i 0.574105 − 12.0520i −12.2883 − 9.66364i −12.5776 − 19.5712i 9.38388 − 15.9669i 11.9262 + 13.7636i 0.0251004 + 0.0100487i 69.7101 − 6.65651i
5.8 −2.96783 + 2.82982i 1.06599 − 5.53086i 0.419477 − 8.80592i 8.09793 + 6.36828i 12.4877 + 19.4312i 15.1371 − 10.6709i 2.19102 + 2.52857i −4.38811 − 1.75673i −42.0544 + 4.01571i
5.9 −2.85879 + 2.72585i −0.426732 + 2.21410i 0.361762 − 7.59433i 1.05636 + 0.830727i −4.81536 − 7.49285i 17.9199 + 4.67723i −1.02709 − 1.18532i 20.3458 + 8.14523i −5.28434 + 0.504593i
5.10 −2.83997 + 2.70790i 1.53801 − 7.97996i 0.352019 − 7.38979i −16.5667 − 13.0282i 17.2411 + 26.8276i 18.4164 + 1.95822i −1.54654 − 1.78480i −36.2484 − 14.5117i 82.3279 − 7.86136i
5.11 −2.69164 + 2.56647i 0.850115 − 4.41082i 0.277483 − 5.82509i −12.7324 − 10.0129i 9.03204 + 14.0541i −18.2990 − 2.85403i −5.28088 − 6.09446i 6.33332 + 2.53548i 59.9689 − 5.72633i
5.12 −2.18667 + 2.08499i −0.825120 + 4.28113i 0.0537055 − 1.12742i 7.78943 + 6.12567i −7.12183 − 11.0818i −13.2313 − 12.9589i −13.5954 − 15.6899i 7.41869 + 2.97000i −29.8048 + 2.84602i
5.13 −2.00341 + 1.91025i −1.84003 + 9.54697i −0.0160482 + 0.336894i 2.35504 + 1.85203i −14.5508 − 22.6414i 11.5607 − 14.4689i −15.1135 − 17.4419i −62.6929 − 25.0985i −8.25596 + 0.788348i
5.14 −1.79024 + 1.70699i −1.61276 + 8.36780i −0.0895119 + 1.87909i 7.16920 + 5.63792i −11.3965 − 17.7334i −11.1422 + 14.7936i −16.0063 − 18.4723i −42.3532 − 16.9557i −22.4585 + 2.14452i
5.15 −1.67621 + 1.59826i 1.64234 − 8.52127i −0.125421 + 2.63292i 3.99249 + 3.13973i 10.8663 + 16.9083i −9.33068 − 15.9981i −16.1314 − 18.6166i −44.8488 − 17.9547i −11.7104 + 1.11820i
5.16 −1.62148 + 1.54608i 0.945638 − 4.90644i −0.141816 + 2.97708i 16.3400 + 12.8499i 6.05239 + 9.41771i −2.90032 + 18.2918i −16.1102 − 18.5922i 1.88704 + 0.755457i −46.3618 + 4.42701i
5.17 −1.44983 + 1.38241i 0.623641 − 3.23576i −0.189705 + 3.98240i −4.51835 − 3.55327i 3.56898 + 5.55344i −14.6712 + 11.3030i −15.7252 − 18.1478i 14.9847 + 5.99898i 11.4629 − 1.09458i
5.18 −1.34276 + 1.28032i −0.215084 + 1.11596i −0.216868 + 4.55262i −1.16338 − 0.914890i −1.13998 − 1.77384i 13.3347 + 12.8524i −15.2574 − 17.6080i 23.8668 + 9.55484i 2.73348 − 0.261016i
5.19 −1.29091 + 1.23088i −0.218672 + 1.13458i −0.229273 + 4.81304i −6.70162 − 5.27021i −1.11424 − 1.73380i −3.99105 − 18.0851i −14.9728 − 17.2795i 23.8265 + 9.53868i 15.1382 − 1.44552i
5.20 −1.23458 + 1.17717i −1.25565 + 6.51495i −0.242196 + 5.08432i −16.1878 − 12.7302i −6.11899 − 9.52134i 10.1554 + 15.4877i −14.6228 − 16.8756i −15.8019 − 6.32615i 34.9708 − 3.33930i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.o.a ✓ 920
7.d odd 6 1 inner 161.4.o.a ✓ 920
23.d odd 22 1 inner 161.4.o.a ✓ 920
161.o even 66 1 inner 161.4.o.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.o.a ✓ 920 1.a even 1 1 trivial
161.4.o.a ✓ 920 7.d odd 6 1 inner
161.4.o.a ✓ 920 23.d odd 22 1 inner
161.4.o.a ✓ 920 161.o even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(161, [\chi])\).