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.
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 |
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 \(S_{4}^{\mathrm{new}}(120, [\chi])\).