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Properties

Label 161.4.m.a
Level $161$
Weight $4$
Character orbit 161.m
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(2,161)]
 
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([22, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.2");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 161.m (of order \(33\), 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 \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 7 q^{2} - 9 q^{3} + 157 q^{4} + 3 q^{5} - 52 q^{6} - 22 q^{7} - 32 q^{8} + 397 q^{9} - q^{10} - 29 q^{11} - 101 q^{12} - 36 q^{13} + 134 q^{14} - 320 q^{15} + 1157 q^{16} - q^{17} - 81 q^{18}+ \cdots + 3640 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −4.85520 + 2.50303i −6.56184 + 5.16029i 12.6674 − 17.7888i 12.2615 − 11.6913i 18.9427 − 41.4787i −0.325027 − 18.5174i −10.7575 + 74.8203i 10.0637 − 41.4832i −30.2682 + 87.4543i
2.2 −4.80405 + 2.47666i 5.12364 − 4.02927i 12.3046 − 17.2794i 11.5320 − 10.9957i −14.6351 + 32.0463i 16.2574 + 8.87112i −10.1632 + 70.6863i 3.65114 − 15.0502i −28.1675 + 81.3846i
2.3 −4.65056 + 2.39753i 0.936935 − 0.736814i 11.2391 − 15.7831i −12.4464 + 11.8677i −2.59074 + 5.67293i 10.5941 − 15.1910i −8.47071 + 58.9151i −6.03054 + 24.8582i 29.4299 − 85.0320i
2.4 −4.47175 + 2.30535i −2.87636 + 2.26200i 10.0415 − 14.1013i −2.66271 + 2.53889i 7.64769 − 16.7461i −4.66951 + 17.9219i −6.66670 + 46.3679i −3.20866 + 13.2263i 6.05396 − 17.4918i
2.5 −4.34018 + 2.23752i 2.80876 − 2.20883i 9.19019 − 12.9058i 1.41622 − 1.35036i −7.24821 + 15.8714i −18.4754 − 1.28782i −5.45069 + 37.9104i −3.35530 + 13.8307i −3.12518 + 9.02963i
2.6 −4.06577 + 2.09605i 7.36412 − 5.79121i 7.49662 − 10.5275i −10.5890 + 10.0966i −17.8022 + 38.9813i −11.1382 + 14.7966i −3.20542 + 22.2942i 14.3267 − 59.0554i 21.8896 − 63.2458i
2.7 −4.00732 + 2.06592i −5.05989 + 3.97914i 7.15013 − 10.0409i −1.97159 + 1.87991i 12.0560 − 26.3990i 10.7443 + 15.0851i −2.77606 + 19.3079i 3.40342 − 14.0291i 4.01705 − 11.6065i
2.8 −3.63328 + 1.87308i −0.975873 + 0.767435i 5.05179 − 7.09425i 1.64575 − 1.56922i 2.10815 − 4.61620i −7.53074 − 16.9200i −0.412531 + 2.86921i −6.00212 + 24.7411i −3.04019 + 8.78405i
2.9 −3.18997 + 1.64454i 1.01302 − 0.796648i 2.83092 − 3.97548i 10.8076 − 10.3051i −1.92138 + 4.20724i 17.9683 − 4.48767i 1.59335 − 11.0820i −5.97393 + 24.6249i −17.5289 + 50.6464i
2.10 −3.13367 + 1.61552i 6.77618 − 5.32885i 2.56951 − 3.60837i 8.19577 − 7.81465i −12.6254 + 27.6459i −9.08513 − 16.1388i 1.79135 − 12.4591i 11.1545 − 45.9795i −13.0581 + 37.7289i
2.11 −3.07645 + 1.58602i −6.09210 + 4.79088i 2.30861 − 3.24199i −9.31482 + 8.88166i 11.1436 − 24.4011i 15.4357 − 10.2343i 1.98019 − 13.7725i 7.79564 − 32.1341i 14.5701 − 42.0974i
2.12 −3.06140 + 1.57826i 3.84012 − 3.01990i 2.24082 − 3.14679i −4.25064 + 4.05298i −6.98995 + 15.3058i 15.3839 + 10.3120i 2.02779 − 14.1036i −0.738799 + 3.04537i 6.61626 − 19.1164i
2.13 −2.97661 + 1.53455i −2.96858 + 2.33451i 1.86489 − 2.61887i 13.9057 − 13.2591i 5.25386 − 11.5043i −8.94313 + 16.2179i 2.28050 − 15.8613i −3.00301 + 12.3786i −21.0452 + 60.8060i
2.14 −2.70406 + 1.39404i −7.01556 + 5.51710i 0.728138 − 1.02253i −1.33033 + 1.26847i 11.2794 − 24.6985i −18.4474 − 1.64096i 2.92017 − 20.3102i 12.4142 − 51.1722i 1.82900 − 5.28454i
2.15 −2.29786 + 1.18463i −0.596175 + 0.468837i −0.763631 + 1.07237i −14.5434 + 13.8671i 0.814530 − 1.78357i −18.2947 + 2.88150i 3.42771 − 23.8403i −6.22987 + 25.6799i 16.9914 − 49.0933i
2.16 −1.91000 + 0.984675i 5.74622 − 4.51888i −1.96193 + 2.75514i −9.27838 + 8.84692i −6.52568 + 14.2892i 9.25410 − 16.0425i 3.48091 − 24.2103i 6.23330 − 25.6940i 9.01040 − 26.0338i
2.17 −1.74505 + 0.899636i 3.31948 − 2.61047i −2.40460 + 3.37679i 8.65609 − 8.25357i −3.44419 + 7.54172i −10.3067 + 15.3874i 3.39352 − 23.6025i −2.16109 + 8.90812i −7.68011 + 22.1902i
2.18 −1.07511 + 0.554255i −7.35523 + 5.78422i −3.79180 + 5.32484i 12.3247 − 11.7516i 4.70171 − 10.2953i 17.1659 + 6.95215i 2.50238 − 17.4044i 14.2767 − 58.8495i −6.73700 + 19.4653i
2.19 −1.02687 + 0.529389i −1.21057 + 0.952002i −3.86624 + 5.42938i 3.86419 − 3.68450i 0.739119 − 1.61845i 15.0072 − 10.8528i 2.41121 − 16.7704i −5.80632 + 23.9340i −2.01750 + 5.82917i
2.20 −1.00046 + 0.515774i −3.36024 + 2.64252i −3.90555 + 5.48458i 4.56107 − 4.34897i 1.99885 − 4.37686i −9.04432 − 16.1617i 2.36006 − 16.4145i −2.05721 + 8.47992i −2.32009 + 6.70346i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.46
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.4.m.a ✓ 920
7.c even 3 1 inner 161.4.m.a ✓ 920
23.c even 11 1 inner 161.4.m.a ✓ 920
161.m even 33 1 inner 161.4.m.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.m.a ✓ 920 1.a even 1 1 trivial
161.4.m.a ✓ 920 7.c even 3 1 inner
161.4.m.a ✓ 920 23.c even 11 1 inner
161.4.m.a ✓ 920 161.m even 33 1 inner

Hecke kernels

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