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Properties

Label 161.4.k.b
Level $161$
Weight $4$
Character orbit 161.k
Analytic conductor $9.499$
Analytic rank $0$
Dimension $440$
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] = [161,4,Mod(20,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.20");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.k (of order \(22\), degree \(10\), 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: \(440\)
Relative dimension: \(44\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 440 q - 6 q^{2} - 150 q^{4} - 11 q^{7} + 86 q^{8} + 470 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q - 6 q^{2} - 150 q^{4} - 11 q^{7} + 86 q^{8} + 470 q^{9} - 22 q^{11} - 11 q^{14} - 22 q^{15} - 1750 q^{16} + 238 q^{18} - 671 q^{21} + 382 q^{23} - 1210 q^{25} + 341 q^{28} - 590 q^{29} - 1518 q^{30} + 1314 q^{32} - 297 q^{35} + 3800 q^{36} + 1166 q^{37} - 1454 q^{39} + 3619 q^{42} - 1738 q^{43} - 4598 q^{44} + 742 q^{46} + 1019 q^{49} + 1824 q^{50} - 682 q^{51} - 22 q^{53} - 9394 q^{56} + 3806 q^{57} + 4786 q^{58} - 22 q^{60} + 2079 q^{63} - 2674 q^{64} - 8646 q^{65} - 22 q^{67} + 8394 q^{70} + 1570 q^{71} - 4736 q^{72} + 10142 q^{74} - 517 q^{77} + 11314 q^{78} + 3674 q^{79} - 4958 q^{81} - 11363 q^{84} - 3794 q^{85} + 10450 q^{86} + 6006 q^{88} - 15236 q^{92} + 2724 q^{93} + 13358 q^{95} + 18154 q^{98} - 23122 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.782279 − 5.44087i −3.10677 − 1.41881i −21.3152 + 6.25869i −10.3864 + 11.9866i −5.28922 + 18.0134i 11.9827 − 14.1214i 32.4594 + 71.0763i −10.0423 − 11.5894i 73.3424 + 47.1343i
20.2 −0.782279 − 5.44087i 3.10677 + 1.41881i −21.3152 + 6.25869i 10.3864 − 11.9866i 5.28922 − 18.0134i −17.8231 + 5.03358i 32.4594 + 71.0763i −10.0423 − 11.5894i −73.3424 − 47.1343i
20.3 −0.705756 − 4.90864i −7.55804 − 3.45164i −15.9207 + 4.67475i 4.32777 − 4.99451i −11.6087 + 39.5358i 9.69711 + 15.7787i 17.7021 + 38.7622i 27.5290 + 31.7701i −27.5706 − 17.7186i
20.4 −0.705756 − 4.90864i 7.55804 + 3.45164i −15.9207 + 4.67475i −4.32777 + 4.99451i 11.6087 − 39.5358i 10.3245 + 15.3755i 17.7021 + 38.7622i 27.5290 + 31.7701i 27.5706 + 17.7186i
20.5 −0.655918 − 4.56201i −2.62030 − 1.19665i −12.7058 + 3.73075i −3.30555 + 3.81481i −3.74043 + 12.7387i −16.8096 + 7.77416i 10.0367 + 21.9774i −12.2473 − 14.1341i 19.5714 + 12.5777i
20.6 −0.655918 − 4.56201i 2.62030 + 1.19665i −12.7058 + 3.73075i 3.30555 − 3.81481i 3.74043 − 12.7387i 14.0546 − 12.0610i 10.0367 + 21.9774i −12.2473 − 14.1341i −19.5714 − 12.5777i
20.7 −0.584725 − 4.06685i −4.80823 − 2.19584i −8.52145 + 2.50212i 9.64911 − 11.1357i −6.11869 + 20.8383i −6.73700 − 17.2515i 1.50403 + 3.29337i 0.616097 + 0.711014i −50.9292 − 32.7302i
20.8 −0.584725 − 4.06685i 4.80823 + 2.19584i −8.52145 + 2.50212i −9.64911 + 11.1357i 6.11869 − 20.8383i −12.8938 − 13.2947i 1.50403 + 3.29337i 0.616097 + 0.711014i 50.9292 + 32.7302i
20.9 −0.524717 − 3.64949i −9.28060 − 4.23831i −5.36748 + 1.57604i −8.92512 + 10.3001i −10.5980 + 36.0934i −16.3861 − 8.63117i −3.68500 − 8.06903i 50.4851 + 58.2629i 42.2734 + 27.1675i
20.10 −0.524717 − 3.64949i 9.28060 + 4.23831i −5.36748 + 1.57604i 8.92512 − 10.3001i 10.5980 − 36.0934i −1.04418 − 18.4908i −3.68500 − 8.06903i 50.4851 + 58.2629i −42.2734 − 27.1675i
20.11 −0.505901 − 3.51862i −2.43932 − 1.11400i −4.44880 + 1.30629i 6.05057 − 6.98273i −2.68569 + 9.14662i 15.8017 + 9.65945i −4.96677 − 10.8757i −12.9719 − 14.9704i −27.6306 − 17.7571i
20.12 −0.505901 − 3.51862i 2.43932 + 1.11400i −4.44880 + 1.30629i −6.05057 + 6.98273i 2.68569 − 9.14662i 2.22226 + 18.3865i −4.96677 − 10.8757i −12.9719 − 14.9704i 27.6306 + 17.7571i
20.13 −0.373888 − 2.60045i −3.35146 − 1.53056i 1.05341 − 0.309310i −13.3809 + 15.4424i −2.72707 + 9.28755i 13.3320 + 12.8552i −9.92919 − 21.7419i −8.79158 − 10.1460i 45.1600 + 29.0226i
20.14 −0.373888 − 2.60045i 3.35146 + 1.53056i 1.05341 − 0.309310i 13.3809 − 15.4424i 2.72707 − 9.28755i 6.15520 + 17.4675i −9.92919 − 21.7419i −8.79158 − 10.1460i −45.1600 − 29.0226i
20.15 −0.366731 − 2.55067i −4.04142 − 1.84566i 1.30453 − 0.383044i −4.87294 + 5.62367i −3.22554 + 10.9852i 10.0302 − 15.5691i −10.0193 − 21.9392i −4.75460 − 5.48710i 16.1312 + 10.3669i
20.16 −0.366731 − 2.55067i 4.04142 + 1.84566i 1.30453 − 0.383044i 4.87294 − 5.62367i 3.22554 − 10.9852i −18.3288 + 2.65613i −10.0193 − 21.9392i −4.75460 − 5.48710i −16.1312 − 10.3669i
20.17 −0.229702 − 1.59761i −4.62193 − 2.11076i 5.17635 − 1.51991i 1.73458 − 2.00182i −2.31051 + 7.86888i −15.7781 + 9.69796i −8.98121 − 19.6661i −0.774353 − 0.893651i −3.59656 − 2.31137i
20.18 −0.229702 − 1.59761i 4.62193 + 2.11076i 5.17635 − 1.51991i −1.73458 + 2.00182i 2.31051 − 7.86888i 15.3761 − 10.3236i −8.98121 − 19.6661i −0.774353 − 0.893651i 3.59656 + 2.31137i
20.19 −0.172734 − 1.20139i −7.89006 − 3.60327i 6.26244 − 1.83882i 8.67247 − 10.0086i −2.96606 + 10.1015i 15.1657 − 10.6302i −7.32455 − 16.0385i 31.5883 + 36.4548i −13.5222 − 8.69022i
20.20 −0.172734 − 1.20139i 7.89006 + 3.60327i 6.26244 − 1.83882i −8.67247 + 10.0086i 2.96606 − 10.1015i −15.9696 + 9.37929i −7.32455 − 16.0385i 31.5883 + 36.4548i 13.5222 + 8.69022i
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.d odd 22 1 inner
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.k.b ✓ 440
7.b odd 2 1 inner 161.4.k.b ✓ 440
23.d odd 22 1 inner 161.4.k.b ✓ 440
161.k even 22 1 inner 161.4.k.b ✓ 440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.k.b ✓ 440 1.a even 1 1 trivial
161.4.k.b ✓ 440 7.b odd 2 1 inner
161.4.k.b ✓ 440 23.d odd 22 1 inner
161.4.k.b ✓ 440 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{220} + 3 T_{2}^{219} + 130 T_{2}^{218} + 324 T_{2}^{217} + 10041 T_{2}^{216} + \cdots + 31\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display