Properties

Label 120.4.b.a
Level $120$
Weight $4$
Character orbit 120.b
Analytic conductor $7.080$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,4,Mod(11,120)]
 
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([1, 1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 120.b (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: \(7.08022920069\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{4} + 120 q^{5} + 19 q^{6} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{4} + 120 q^{5} + 19 q^{6} + 21 q^{8} - 15 q^{10} + 65 q^{12} - 54 q^{14} + 153 q^{16} - 175 q^{18} + 12 q^{19} - 15 q^{20} - 4 q^{21} - 102 q^{22} + 228 q^{23} - 407 q^{24} + 600 q^{25} + 336 q^{26} + 132 q^{27} - 186 q^{28} + 95 q^{30} + 177 q^{32} + 116 q^{33} + 408 q^{34} + 673 q^{36} + 312 q^{38} - 656 q^{39} + 105 q^{40} - 990 q^{42} - 450 q^{44} - 1104 q^{46} - 924 q^{47} - 535 q^{48} - 816 q^{49} - 75 q^{50} - 700 q^{51} - 1548 q^{52} + 528 q^{53} + 1331 q^{54} - 390 q^{56} - 172 q^{57} + 1410 q^{58} + 325 q^{60} - 978 q^{62} + 476 q^{63} + 1137 q^{64} - 2794 q^{66} + 1632 q^{67} - 1608 q^{68} + 980 q^{69} - 270 q^{70} + 216 q^{71} - 3699 q^{72} - 216 q^{73} + 768 q^{74} - 1812 q^{76} + 4140 q^{78} + 765 q^{80} + 152 q^{81} + 2244 q^{82} + 5086 q^{84} - 2808 q^{86} + 252 q^{87} + 2622 q^{88} - 875 q^{90} - 1800 q^{91} - 1836 q^{92} - 1968 q^{94} + 60 q^{95} - 5455 q^{96} + 792 q^{97} + 4851 q^{98} - 1328 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.78614 0.487238i 4.99995 + 1.41438i 7.52520 + 2.71503i 5.00000 −13.2414 6.37683i 35.4366i −19.6434 11.2310i 22.9991 + 14.1437i −13.9307 2.43619i
11.2 −2.78614 + 0.487238i 4.99995 1.41438i 7.52520 2.71503i 5.00000 −13.2414 + 6.37683i 35.4366i −19.6434 + 11.2310i 22.9991 14.1437i −13.9307 + 2.43619i
11.3 −2.65744 0.968509i −4.88069 + 1.78294i 6.12398 + 5.14751i 5.00000 14.6969 0.0110603i 12.1872i −11.2887 19.6103i 20.6423 17.4039i −13.2872 4.84254i
11.4 −2.65744 + 0.968509i −4.88069 1.78294i 6.12398 5.14751i 5.00000 14.6969 + 0.0110603i 12.1872i −11.2887 + 19.6103i 20.6423 + 17.4039i −13.2872 + 4.84254i
11.5 −2.40201 1.49343i −1.49310 4.97701i 3.53933 + 7.17448i 5.00000 −3.84639 + 14.1847i 14.5956i 2.21306 22.5189i −22.5413 + 14.8623i −12.0101 7.46715i
11.6 −2.40201 + 1.49343i −1.49310 + 4.97701i 3.53933 7.17448i 5.00000 −3.84639 14.1847i 14.5956i 2.21306 + 22.5189i −22.5413 14.8623i −12.0101 + 7.46715i
11.7 −1.54070 2.37197i 3.91264 3.41925i −3.25251 + 7.30898i 5.00000 −14.1386 4.01264i 2.12151i 22.3478 3.54604i 3.61744 26.7566i −7.70348 11.8599i
11.8 −1.54070 + 2.37197i 3.91264 + 3.41925i −3.25251 7.30898i 5.00000 −14.1386 + 4.01264i 2.12151i 22.3478 + 3.54604i 3.61744 + 26.7566i −7.70348 + 11.8599i
11.9 −1.35014 2.48538i −0.403912 + 5.18043i −4.35424 + 6.71123i 5.00000 13.4207 5.99044i 23.0707i 22.5588 + 1.76083i −26.6737 4.18488i −6.75071 12.4269i
11.10 −1.35014 + 2.48538i −0.403912 5.18043i −4.35424 6.71123i 5.00000 13.4207 + 5.99044i 23.0707i 22.5588 1.76083i −26.6737 + 4.18488i −6.75071 + 12.4269i
11.11 −0.620366 2.75956i −5.19284 + 0.185395i −7.23029 + 3.42387i 5.00000 3.73307 + 14.2149i 11.8996i 13.9338 + 17.8283i 26.9313 1.92545i −3.10183 13.7978i
11.12 −0.620366 + 2.75956i −5.19284 0.185395i −7.23029 3.42387i 5.00000 3.73307 14.2149i 11.8996i 13.9338 17.8283i 26.9313 + 1.92545i −3.10183 + 13.7978i
11.13 0.165720 2.82357i 3.31357 + 4.00253i −7.94507 0.935845i 5.00000 11.8505 8.69280i 30.8728i −3.95908 + 22.2784i −5.04045 + 26.5253i 0.828601 14.1178i
11.14 0.165720 + 2.82357i 3.31357 4.00253i −7.94507 + 0.935845i 5.00000 11.8505 + 8.69280i 30.8728i −3.95908 22.2784i −5.04045 26.5253i 0.828601 + 14.1178i
11.15 0.638417 2.75544i −0.899959 5.11762i −7.18485 3.51823i 5.00000 −14.6758 0.787399i 23.1184i −14.2812 + 17.5513i −25.3801 + 9.21130i 3.19208 13.7772i
11.16 0.638417 + 2.75544i −0.899959 + 5.11762i −7.18485 + 3.51823i 5.00000 −14.6758 + 0.787399i 23.1184i −14.2812 17.5513i −25.3801 9.21130i 3.19208 + 13.7772i
11.17 1.49121 2.40339i −3.04181 + 4.21277i −3.55258 7.16793i 5.00000 5.58894 + 13.5928i 13.6956i −22.5250 2.15066i −8.49478 25.6289i 7.45606 12.0170i
11.18 1.49121 + 2.40339i −3.04181 4.21277i −3.55258 + 7.16793i 5.00000 5.58894 13.5928i 13.6956i −22.5250 + 2.15066i −8.49478 + 25.6289i 7.45606 + 12.0170i
11.19 2.03621 1.96312i 5.02993 + 1.30378i 0.292289 7.99466i 5.00000 12.8015 7.21960i 8.16056i −15.0993 16.8526i 23.6003 + 13.1158i 10.1810 9.81562i
11.20 2.03621 + 1.96312i 5.02993 1.30378i 0.292289 + 7.99466i 5.00000 12.8015 + 7.21960i 8.16056i −15.0993 + 16.8526i 23.6003 13.1158i 10.1810 + 9.81562i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.4.b.a 24
3.b odd 2 1 120.4.b.b yes 24
4.b odd 2 1 480.4.b.b 24
8.b even 2 1 480.4.b.a 24
8.d odd 2 1 120.4.b.b yes 24
12.b even 2 1 480.4.b.a 24
24.f even 2 1 inner 120.4.b.a 24
24.h odd 2 1 480.4.b.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.b.a 24 1.a even 1 1 trivial
120.4.b.a 24 24.f even 2 1 inner
120.4.b.b yes 24 3.b odd 2 1
120.4.b.b yes 24 8.d odd 2 1
480.4.b.a 24 8.b even 2 1
480.4.b.a 24 12.b even 2 1
480.4.b.b 24 4.b odd 2 1
480.4.b.b 24 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{12} - 114 T_{23}^{11} - 74952 T_{23}^{10} + 7325588 T_{23}^{9} + 2230771920 T_{23}^{8} - 173720020368 T_{23}^{7} - 33369206298896 T_{23}^{6} + \cdots + 12\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display