↑

Properties

Label 120.3.n.a
Level $120$
Weight $3$
Character orbit 120.n
Analytic conductor $3.270$
Analytic rank $0$
Dimension $32$
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] = [120,3,Mod(101,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 120.n (of order \(2\), degree \(1\), 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: \(3.26976317232\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{4} + 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{4} + 6 q^{6} + 10 q^{10} - 40 q^{12} - 30 q^{16} - 92 q^{18} - 92 q^{22} + 78 q^{24} + 160 q^{25} + 172 q^{28} + 40 q^{30} - 64 q^{31} - 16 q^{33} + 120 q^{34} - 142 q^{36} - 192 q^{39} - 70 q^{40} - 232 q^{42} - 264 q^{46} + 172 q^{48} + 192 q^{49} + 248 q^{52} + 126 q^{54} - 80 q^{57} + 84 q^{58} - 70 q^{60} + 320 q^{63} - 262 q^{64} - 240 q^{66} - 60 q^{70} + 400 q^{72} - 160 q^{73} + 64 q^{76} - 40 q^{78} - 640 q^{79} - 96 q^{81} + 520 q^{82} + 88 q^{84} - 384 q^{87} + 220 q^{88} - 90 q^{90} + 408 q^{94} + 14 q^{96} - 224 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.98480 − 0.246104i 0.244768 + 2.99000i 3.87887 + 0.976934i 2.23607 0.250035 − 5.99479i 6.71504 −7.45835 − 2.89362i −8.88018 + 1.46371i −4.43815 − 0.550305i
101.2 −1.98480 + 0.246104i 0.244768 − 2.99000i 3.87887 − 0.976934i 2.23607 0.250035 + 5.99479i 6.71504 −7.45835 + 2.89362i −8.88018 − 1.46371i −4.43815 + 0.550305i
101.3 −1.90771 − 0.600523i −1.97120 − 2.26149i 3.27874 + 2.29125i −2.23607 2.40241 + 5.49804i −5.76274 −4.87895 − 6.34002i −1.22872 + 8.91573i 4.26578 + 1.34281i
101.4 −1.90771 + 0.600523i −1.97120 + 2.26149i 3.27874 − 2.29125i −2.23607 2.40241 − 5.49804i −5.76274 −4.87895 + 6.34002i −1.22872 − 8.91573i 4.26578 − 1.34281i
101.5 −1.63912 − 1.14598i 2.49632 − 1.66385i 1.37344 + 3.75681i −2.23607 −5.99851 − 0.133485i 10.8275 2.05400 − 7.73182i 3.46321 − 8.30700i 3.66519 + 2.56250i
101.6 −1.63912 + 1.14598i 2.49632 + 1.66385i 1.37344 − 3.75681i −2.23607 −5.99851 + 0.133485i 10.8275 2.05400 + 7.73182i 3.46321 + 8.30700i 3.66519 − 2.56250i
101.7 −1.61470 − 1.18015i −2.96788 + 0.437795i 1.21448 + 3.81117i 2.23607 5.30889 + 2.79565i −2.01094 2.53675 − 7.58715i 8.61667 − 2.59865i −3.61057 − 2.63890i
101.8 −1.61470 + 1.18015i −2.96788 − 0.437795i 1.21448 − 3.81117i 2.23607 5.30889 − 2.79565i −2.01094 2.53675 + 7.58715i 8.61667 + 2.59865i −3.61057 + 2.63890i
101.9 −1.43092 − 1.39731i 0.503261 + 2.95749i 0.0950762 + 3.99887i −2.23607 3.41239 − 4.93514i −4.13108 5.45160 − 5.85492i −8.49346 + 2.97677i 3.19964 + 3.12447i
101.10 −1.43092 + 1.39731i 0.503261 − 2.95749i 0.0950762 − 3.99887i −2.23607 3.41239 + 4.93514i −4.13108 5.45160 + 5.85492i −8.49346 − 2.97677i 3.19964 − 3.12447i
101.11 −1.00245 − 1.73064i 2.76627 + 1.16093i −1.99021 + 3.46974i 2.23607 −0.763889 − 5.95117i −1.01226 7.99993 − 0.0338958i 6.30449 + 6.42288i −2.24154 − 3.86982i
101.12 −1.00245 + 1.73064i 2.76627 − 1.16093i −1.99021 − 3.46974i 2.23607 −0.763889 + 5.95117i −1.01226 7.99993 + 0.0338958i 6.30449 − 6.42288i −2.24154 + 3.86982i
101.13 −0.528201 − 1.92899i −2.56911 + 1.54910i −3.44201 + 2.03779i −2.23607 4.34519 + 4.13755i 7.97394 5.74894 + 5.56324i 4.20061 − 7.95958i 1.18109 + 4.31335i
101.14 −0.528201 + 1.92899i −2.56911 − 1.54910i −3.44201 − 2.03779i −2.23607 4.34519 − 4.13755i 7.97394 5.74894 − 5.56324i 4.20061 + 7.95958i 1.18109 − 4.31335i
101.15 −0.214016 − 1.98852i 1.58388 − 2.54781i −3.90839 + 0.851148i −2.23607 −5.40533 − 2.60431i −12.5995 2.52898 + 7.58975i −3.98263 − 8.07085i 0.478554 + 4.44646i
101.16 −0.214016 + 1.98852i 1.58388 + 2.54781i −3.90839 − 0.851148i −2.23607 −5.40533 + 2.60431i −12.5995 2.52898 − 7.58975i −3.98263 + 8.07085i 0.478554 − 4.44646i
101.17 0.214016 − 1.98852i −1.58388 + 2.54781i −3.90839 − 0.851148i 2.23607 4.72738 + 3.69485i −12.5995 −2.52898 + 7.58975i −3.98263 − 8.07085i 0.478554 − 4.44646i
101.18 0.214016 + 1.98852i −1.58388 − 2.54781i −3.90839 + 0.851148i 2.23607 4.72738 − 3.69485i −12.5995 −2.52898 − 7.58975i −3.98263 + 8.07085i 0.478554 + 4.44646i
101.19 0.528201 − 1.92899i 2.56911 − 1.54910i −3.44201 − 2.03779i 2.23607 −1.63119 − 5.77401i 7.97394 −5.74894 + 5.56324i 4.20061 − 7.95958i 1.18109 − 4.31335i
101.20 0.528201 + 1.92899i 2.56911 + 1.54910i −3.44201 + 2.03779i 2.23607 −1.63119 + 5.77401i 7.97394 −5.74894 − 5.56324i 4.20061 + 7.95958i 1.18109 + 4.31335i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.3.n.a ✓ 32
3.b odd 2 1 inner 120.3.n.a ✓ 32
4.b odd 2 1 480.3.n.a 32
8.b even 2 1 inner 120.3.n.a ✓ 32
8.d odd 2 1 480.3.n.a 32
12.b even 2 1 480.3.n.a 32
24.f even 2 1 480.3.n.a 32
24.h odd 2 1 inner 120.3.n.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.n.a ✓ 32 1.a even 1 1 trivial
120.3.n.a ✓ 32 3.b odd 2 1 inner
120.3.n.a ✓ 32 8.b even 2 1 inner
120.3.n.a ✓ 32 24.h odd 2 1 inner
480.3.n.a 32 4.b odd 2 1
480.3.n.a 32 8.d odd 2 1
480.3.n.a 32 12.b even 2 1
480.3.n.a 32 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(120, [\chi])\).