↑

Properties

Label 165.4.s.a
Level $165$
Weight $4$
Character orbit 165.s
Analytic conductor $9.735$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,4,Mod(4,165)]
 
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([0, 5, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.s (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: \(144\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 144 q^{4} + 16 q^{5} - 24 q^{6} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 144 q^{4} + 16 q^{5} - 24 q^{6} + 324 q^{9} + 48 q^{10} + 84 q^{11} - 128 q^{14} + 102 q^{15} - 736 q^{16} + 392 q^{19} - 826 q^{20} + 672 q^{21} + 288 q^{24} - 728 q^{25} - 428 q^{26} + 564 q^{30} - 612 q^{31} - 1648 q^{34} - 632 q^{35} - 1296 q^{36} + 600 q^{39} - 1174 q^{40} - 1716 q^{41} + 7908 q^{44} - 144 q^{45} - 3104 q^{46} + 336 q^{49} - 2194 q^{50} - 1320 q^{51} - 864 q^{54} - 3776 q^{55} + 432 q^{56} - 816 q^{59} - 180 q^{60} - 4152 q^{61} + 7604 q^{64} + 4696 q^{65} + 3204 q^{66} + 1968 q^{69} - 3644 q^{70} + 3248 q^{71} + 8024 q^{74} - 1392 q^{75} + 17904 q^{76} + 1924 q^{79} - 762 q^{80} - 2916 q^{81} + 1944 q^{84} + 3436 q^{85} - 7744 q^{86} + 1400 q^{89} + 1188 q^{90} - 2660 q^{91} + 17068 q^{94} + 3440 q^{95} - 2688 q^{96} - 756 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −3.25446 + 4.47938i 2.85317 + 0.927051i −7.00120 − 21.5475i −10.1078 − 4.77836i −13.4381 + 9.76338i 21.6666 − 7.03992i 77.1779 + 25.0766i 7.28115 + 5.29007i 54.2995 − 29.7256i
4.2 −3.12616 + 4.30279i −2.85317 − 0.927051i −6.26898 − 19.2939i 9.23509 + 6.30184i 12.9084 − 9.37847i −2.59291 + 0.842489i 62.1497 + 20.1937i 7.28115 + 5.29007i −55.9858 + 20.0361i
4.3 −2.98174 + 4.10401i 2.85317 + 0.927051i −5.48001 − 16.8657i 10.3405 − 4.25135i −12.3120 + 8.94522i −28.8232 + 9.36524i 46.9606 + 15.2584i 7.28115 + 5.29007i −13.3851 + 55.1140i
4.4 −2.85444 + 3.92879i −2.85317 − 0.927051i −4.81548 − 14.8205i 0.378522 − 11.1739i 11.7864 − 8.56331i 1.07463 − 0.349169i 35.0237 + 11.3799i 7.28115 + 5.29007i 42.8196 + 33.3824i
4.5 −2.71567 + 3.73780i 2.85317 + 0.927051i −4.12415 − 12.6928i 6.51640 + 9.08496i −11.2134 + 8.14701i 18.0893 − 5.87758i 23.4907 + 7.63258i 7.28115 + 5.29007i −51.6542 − 0.314771i
4.6 −2.40488 + 3.31003i −2.85317 − 0.927051i −2.70072 − 8.31197i −10.7977 − 2.89977i 9.93009 − 7.21463i −31.4178 + 10.2083i 2.87838 + 0.935243i 7.28115 + 5.29007i 35.5656 − 28.7673i
4.7 −2.11250 + 2.90761i 2.85317 + 0.927051i −1.51940 − 4.67624i −8.78066 − 6.92098i −8.72284 + 6.33751i −7.70018 + 2.50194i −10.5384 − 3.42413i 7.28115 + 5.29007i 38.6727 − 10.9102i
4.8 −1.99961 + 2.75223i 2.85317 + 0.927051i −1.10418 − 3.39831i −7.91257 + 7.89881i −8.25668 + 5.99883i 3.18278 − 1.03415i −14.3226 − 4.65371i 7.28115 + 5.29007i −5.91726 − 37.5717i
4.9 −1.85880 + 2.55841i −2.85317 − 0.927051i −0.618217 − 1.90268i 11.1377 − 0.975393i 7.67524 − 5.57639i −3.61245 + 1.17376i −18.0438 − 5.86278i 7.28115 + 5.29007i −18.2073 + 30.3079i
4.10 −1.85387 + 2.55163i 2.85317 + 0.927051i −0.601856 − 1.85232i 6.28941 − 9.24356i −7.65489 + 5.56161i −5.73245 + 1.86259i −18.1548 − 5.89884i 7.28115 + 5.29007i 11.9264 + 33.1846i
4.11 −1.78981 + 2.46346i −2.85317 − 0.927051i −0.393074 − 1.20976i −10.7362 − 3.12005i 7.39037 − 5.36942i 32.3795 − 10.5207i −19.4840 − 6.33074i 7.28115 + 5.29007i 26.9017 − 20.8638i
4.12 −1.51439 + 2.08437i −2.85317 − 0.927051i 0.420886 + 1.29535i 0.00429668 + 11.1803i 6.25312 − 4.54316i −9.07467 + 2.94854i −22.9400 − 7.45366i 7.28115 + 5.29007i −23.3105 − 16.9224i
4.13 −1.13933 + 1.56815i 2.85317 + 0.927051i 1.31111 + 4.03517i 1.25995 + 11.1091i −4.70446 + 3.41799i −24.0173 + 7.80371i −22.5693 − 7.33322i 7.28115 + 5.29007i −18.8563 − 10.6811i
4.14 −0.959091 + 1.32008i −2.85317 − 0.927051i 1.64939 + 5.07631i 4.58171 − 10.1984i 3.96023 − 2.87727i −0.793875 + 0.257946i −20.6978 − 6.72511i 7.28115 + 5.29007i 9.06842 + 15.8294i
4.15 −0.871639 + 1.19971i 2.85317 + 0.927051i 1.79259 + 5.51703i 2.07879 − 10.9854i −3.59912 + 2.61492i 31.1180 − 10.1109i −19.4640 − 6.32425i 7.28115 + 5.29007i 11.3673 + 12.0692i
4.16 −0.373069 + 0.513485i 2.85317 + 0.927051i 2.34765 + 7.22532i 8.72345 + 6.99295i −1.54045 + 1.11921i 7.15479 − 2.32473i −9.41503 − 3.05913i 7.28115 + 5.29007i −6.84522 + 1.87051i
4.17 −0.223145 + 0.307132i −2.85317 − 0.927051i 2.42760 + 7.47138i 3.58385 + 10.5904i 0.921397 − 0.669434i 20.3192 − 6.60211i −5.72486 − 1.86012i 7.28115 + 5.29007i −4.05236 − 1.26247i
4.18 −0.220056 + 0.302881i −2.85317 − 0.927051i 2.42882 + 7.47515i −9.66636 + 5.61796i 0.908642 − 0.660167i −17.9728 + 5.83970i −5.64702 − 1.83483i 7.28115 + 5.29007i 0.425567 − 4.16402i
4.19 0.220056 − 0.302881i 2.85317 + 0.927051i 2.42882 + 7.47515i −8.33007 + 7.45721i 0.908642 − 0.660167i 17.9728 − 5.83970i 5.64702 + 1.83483i 7.28115 + 5.29007i 0.425567 + 4.16402i
4.20 0.223145 − 0.307132i 2.85317 + 0.927051i 2.42760 + 7.47138i −8.96457 − 6.68105i 0.921397 − 0.669434i −20.3192 + 6.60211i 5.72486 + 1.86012i 7.28115 + 5.29007i −4.05236 + 1.26247i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j 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.s.a ✓ 144
5.b even 2 1 inner 165.4.s.a ✓ 144
11.c even 5 1 inner 165.4.s.a ✓ 144
55.j even 10 1 inner 165.4.s.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.s.a ✓ 144 1.a even 1 1 trivial
165.4.s.a ✓ 144 5.b even 2 1 inner
165.4.s.a ✓ 144 11.c even 5 1 inner
165.4.s.a ✓ 144 55.j even 10 1 inner

Hecke kernels

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