↑

Properties

Label 120.4.b.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 3 q^{4} - 120 q^{5} + 11 q^{6} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 3 q^{4} - 120 q^{5} + 11 q^{6} - 21 q^{8} - 15 q^{10} - 33 q^{12} + 54 q^{14} + 153 q^{16} + 59 q^{18} + 12 q^{19} + 15 q^{20} + 4 q^{21} - 102 q^{22} - 228 q^{23} - 27 q^{24} + 600 q^{25} - 336 q^{26} + 132 q^{27} - 186 q^{28} - 55 q^{30} - 177 q^{32} + 116 q^{33} + 408 q^{34} + 641 q^{36} - 312 q^{38} + 656 q^{39} + 105 q^{40} - 1042 q^{42} + 450 q^{44} - 1104 q^{46} + 924 q^{47} - 717 q^{48} - 816 q^{49} + 75 q^{50} - 700 q^{51} - 1548 q^{52} - 528 q^{53} + 987 q^{54} + 390 q^{56} - 172 q^{57} + 1410 q^{58} + 165 q^{60} + 978 q^{62} - 476 q^{63} + 1137 q^{64} - 582 q^{66} + 1632 q^{67} + 1608 q^{68} - 980 q^{69} - 270 q^{70} - 216 q^{71} - 589 q^{72} - 216 q^{73} - 768 q^{74} - 1812 q^{76} - 324 q^{78} - 765 q^{80} + 152 q^{81} + 2244 q^{82} - 134 q^{84} + 2808 q^{86} - 252 q^{87} + 2622 q^{88} - 295 q^{90} - 1800 q^{91} + 1836 q^{92} - 1968 q^{94} - 60 q^{95} + 1445 q^{96} + 792 q^{97} - 4851 q^{98} - 1328 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.81366 − 0.288693i −3.93281 + 3.39603i 7.83331 + 1.62457i −5.00000 12.0460 − 8.41989i − 20.9207i −21.5712 − 6.83239i 3.93395 − 26.7119i 14.0683 + 1.44347i
11.2 −2.81366 + 0.288693i −3.93281 − 3.39603i 7.83331 − 1.62457i −5.00000 12.0460 + 8.41989i 20.9207i −21.5712 + 6.83239i 3.93395 + 26.7119i 14.0683 − 1.44347i
11.3 −2.71159 − 0.804541i 2.58903 − 4.50521i 6.70543 + 4.36317i −5.00000 −10.6450 + 10.1333i − 6.99225i −14.6720 − 17.2259i −13.5939 − 23.3282i 13.5579 + 4.02271i
11.4 −2.71159 + 0.804541i 2.58903 + 4.50521i 6.70543 − 4.36317i −5.00000 −10.6450 − 10.1333i 6.99225i −14.6720 + 17.2259i −13.5939 + 23.3282i 13.5579 − 4.02271i
11.5 −2.03621 − 1.96312i 5.02993 + 1.30378i 0.292289 + 7.99466i −5.00000 −7.68250 − 12.5291i 8.16056i 15.0993 − 16.8526i 23.6003 + 13.1158i 10.1810 + 9.81562i
11.6 −2.03621 + 1.96312i 5.02993 − 1.30378i 0.292289 − 7.99466i −5.00000 −7.68250 + 12.5291i − 8.16056i 15.0993 + 16.8526i 23.6003 − 13.1158i 10.1810 − 9.81562i
11.7 −1.49121 − 2.40339i −3.04181 + 4.21277i −3.55258 + 7.16793i −5.00000 14.6609 + 1.02854i 13.6956i 22.5250 − 2.15066i −8.49478 − 25.6289i 7.45606 + 12.0170i
11.8 −1.49121 + 2.40339i −3.04181 − 4.21277i −3.55258 − 7.16793i −5.00000 14.6609 − 1.02854i − 13.6956i 22.5250 + 2.15066i −8.49478 + 25.6289i 7.45606 − 12.0170i
11.9 −0.638417 − 2.75544i −0.899959 − 5.11762i −7.18485 + 3.51823i −5.00000 −13.5267 + 5.74695i 23.1184i 14.2812 + 17.5513i −25.3801 + 9.21130i 3.19208 + 13.7772i
11.10 −0.638417 + 2.75544i −0.899959 + 5.11762i −7.18485 − 3.51823i −5.00000 −13.5267 − 5.74695i − 23.1184i 14.2812 − 17.5513i −25.3801 − 9.21130i 3.19208 − 13.7772i
11.11 −0.165720 − 2.82357i 3.31357 + 4.00253i −7.94507 + 0.935845i −5.00000 10.7523 − 10.0194i − 30.8728i 3.95908 + 22.2784i −5.04045 + 26.5253i 0.828601 + 14.1178i
11.12 −0.165720 + 2.82357i 3.31357 − 4.00253i −7.94507 − 0.935845i −5.00000 10.7523 + 10.0194i 30.8728i 3.95908 − 22.2784i −5.04045 − 26.5253i 0.828601 − 14.1178i
11.13 0.620366 − 2.75956i −5.19284 + 0.185395i −7.23029 − 3.42387i −5.00000 −2.70986 + 14.4450i − 11.8996i −13.9338 + 17.8283i 26.9313 − 1.92545i −3.10183 + 13.7978i
11.14 0.620366 + 2.75956i −5.19284 − 0.185395i −7.23029 + 3.42387i −5.00000 −2.70986 − 14.4450i 11.8996i −13.9338 − 17.8283i 26.9313 + 1.92545i −3.10183 − 13.7978i
11.15 1.35014 − 2.48538i −0.403912 + 5.18043i −4.35424 − 6.71123i −5.00000 12.3300 + 7.99819i 23.0707i −22.5588 + 1.76083i −26.6737 − 4.18488i −6.75071 + 12.4269i
11.16 1.35014 + 2.48538i −0.403912 − 5.18043i −4.35424 + 6.71123i −5.00000 12.3300 − 7.99819i − 23.0707i −22.5588 − 1.76083i −26.6737 + 4.18488i −6.75071 − 12.4269i
11.17 1.54070 − 2.37197i 3.91264 − 3.41925i −3.25251 − 7.30898i −5.00000 −2.08219 − 14.5487i 2.12151i −22.3478 − 3.54604i 3.61744 − 26.7566i −7.70348 + 11.8599i
11.18 1.54070 + 2.37197i 3.91264 + 3.41925i −3.25251 + 7.30898i −5.00000 −2.08219 + 14.5487i − 2.12151i −22.3478 + 3.54604i 3.61744 + 26.7566i −7.70348 − 11.8599i
11.19 2.40201 − 1.49343i −1.49310 − 4.97701i 3.53933 − 7.17448i −5.00000 −11.0193 − 9.72502i − 14.5956i −2.21306 − 22.5189i −22.5413 + 14.8623i −12.0101 + 7.46715i
11.20 2.40201 + 1.49343i −1.49310 + 4.97701i 3.53933 + 7.17448i −5.00000 −11.0193 + 9.72502i 14.5956i −2.21306 + 22.5189i −22.5413 − 14.8623i −12.0101 − 7.46715i
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.b yes 24
3.b odd 2 1 120.4.b.a ✓ 24
4.b odd 2 1 480.4.b.a 24
8.b even 2 1 480.4.b.b 24
8.d odd 2 1 120.4.b.a ✓ 24
12.b even 2 1 480.4.b.b 24
24.f even 2 1 inner 120.4.b.b yes 24
24.h odd 2 1 480.4.b.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.b.a ✓ 24 3.b odd 2 1
120.4.b.a ✓ 24 8.d odd 2 1
120.4.b.b yes 24 1.a even 1 1 trivial
120.4.b.b yes 24 24.f even 2 1 inner
480.4.b.a 24 4.b odd 2 1
480.4.b.a 24 24.h odd 2 1
480.4.b.b 24 8.b even 2 1
480.4.b.b 24 12.b even 2 1

Hecke kernels

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