Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,3,Mod(11,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.j (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: | \(2.04360198270\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) 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.
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 | −2.26850 | − | 3.12232i | 0.549348 | − | 2.94927i | −3.36674 | + | 10.3618i | −3.89421 | + | 3.13610i | −10.4548 | + | 4.97519i | −5.08243 | 25.3081 | − | 8.22311i | −8.39643 | − | 3.24035i | 18.6259 | + | 5.04472i | ||
11.2 | −2.06309 | − | 2.83960i | 2.44039 | + | 1.74485i | −2.57091 | + | 7.91245i | 4.77936 | − | 1.46892i | −0.0800814 | − | 10.5295i | 6.80412 | 14.4196 | − | 4.68521i | 2.91102 | + | 8.51622i | −14.0314 | − | 10.5409i | ||
11.3 | −1.71799 | − | 2.36461i | −2.67836 | + | 1.35143i | −1.40382 | + | 4.32053i | 2.09404 | + | 4.54038i | 7.79701 | + | 4.01154i | 3.73689 | 1.50905 | − | 0.490320i | 5.34726 | − | 7.23926i | 7.13868 | − | 12.7519i | ||
11.4 | −1.49360 | − | 2.05577i | −0.154094 | + | 2.99604i | −0.759266 | + | 2.33678i | −4.43527 | − | 2.30832i | 6.38932 | − | 4.15811i | −8.13443 | −3.72888 | + | 1.21159i | −8.95251 | − | 0.923341i | 1.87916 | + | 12.5656i | ||
11.5 | −1.37909 | − | 1.89816i | −1.85450 | − | 2.35814i | −0.465043 | + | 1.43126i | 3.05671 | − | 3.95684i | −1.91859 | + | 6.77224i | −3.37306 | −5.56759 | + | 1.80902i | −2.12163 | + | 8.74635i | −11.7262 | − | 0.345267i | ||
11.6 | −1.06164 | − | 1.46122i | 2.88836 | − | 0.810770i | 0.227982 | − | 0.701657i | −4.39665 | − | 2.38107i | −4.25111 | − | 3.35979i | 6.57297 | −8.13837 | + | 2.64432i | 7.68530 | − | 4.68360i | 1.18838 | + | 8.95230i | ||
11.7 | −0.733747 | − | 1.00992i | 1.70510 | − | 2.46833i | 0.754522 | − | 2.32218i | 3.80815 | + | 3.24006i | −3.74392 | + | 0.0891202i | −0.783313 | −7.64775 | + | 2.48490i | −3.18527 | − | 8.41748i | 0.477973 | − | 6.22330i | ||
11.8 | −0.146095 | − | 0.201083i | −2.97010 | − | 0.422474i | 1.21698 | − | 3.74547i | −3.70242 | + | 3.36037i | 0.348966 | + | 0.658958i | −11.1634 | −1.87649 | + | 0.609710i | 8.64303 | + | 2.50958i | 1.21662 | + | 0.253561i | ||
11.9 | −0.0749426 | − | 0.103150i | 1.19040 | + | 2.75372i | 1.23104 | − | 3.78877i | −0.601131 | + | 4.96373i | 0.194834 | − | 0.329160i | 8.18661 | −0.968106 | + | 0.314557i | −6.16592 | + | 6.55603i | 0.557058 | − | 0.309989i | ||
11.10 | 0.0749426 | + | 0.103150i | −2.58164 | + | 1.52811i | 1.23104 | − | 3.78877i | 0.601131 | − | 4.96373i | −0.351099 | − | 0.151775i | 8.18661 | 0.968106 | − | 0.314557i | 4.32978 | − | 7.89006i | 0.557058 | − | 0.309989i | ||
11.11 | 0.146095 | + | 0.201083i | 2.65119 | + | 1.40399i | 1.21698 | − | 3.74547i | 3.70242 | − | 3.36037i | 0.105007 | + | 0.738225i | −11.1634 | 1.87649 | − | 0.609710i | 5.05760 | + | 7.44451i | 1.21662 | + | 0.253561i | ||
11.12 | 0.733747 | + | 1.00992i | 0.0713918 | − | 2.99915i | 0.754522 | − | 2.32218i | −3.80815 | − | 3.24006i | 3.08127 | − | 2.12852i | −0.783313 | 7.64775 | − | 2.48490i | −8.98981 | − | 0.428229i | 0.477973 | − | 6.22330i | ||
11.13 | 1.06164 | + | 1.46122i | −1.86018 | − | 2.35367i | 0.227982 | − | 0.701657i | 4.39665 | + | 2.38107i | 1.46439 | − | 5.21687i | 6.57297 | 8.13837 | − | 2.64432i | −2.07948 | + | 8.75647i | 1.18838 | + | 8.95230i | ||
11.14 | 1.37909 | + | 1.89816i | 2.88640 | − | 0.817724i | −0.465043 | + | 1.43126i | −3.05671 | + | 3.95684i | 5.53279 | + | 4.35114i | −3.37306 | 5.56759 | − | 1.80902i | 7.66265 | − | 4.72056i | −11.7262 | − | 0.345267i | ||
11.15 | 1.49360 | + | 2.05577i | −1.63636 | + | 2.51442i | −0.759266 | + | 2.33678i | 4.43527 | + | 2.30832i | −7.61315 | + | 0.391563i | −8.13443 | 3.72888 | − | 1.21159i | −3.64463 | − | 8.22902i | 1.87916 | + | 12.5656i | ||
11.16 | 1.71799 | + | 2.36461i | 1.37249 | + | 2.66763i | −1.40382 | + | 4.32053i | −2.09404 | − | 4.54038i | −3.94999 | + | 7.82837i | 3.73689 | −1.50905 | + | 0.490320i | −5.23255 | + | 7.32260i | 7.13868 | − | 12.7519i | ||
11.17 | 2.06309 | + | 2.83960i | −2.99991 | − | 0.0228156i | −2.57091 | + | 7.91245i | −4.77936 | + | 1.46892i | −6.12430 | − | 8.56561i | 6.80412 | −14.4196 | + | 4.68521i | 8.99896 | + | 0.136890i | −14.0314 | − | 10.5409i | ||
11.18 | 2.26850 | + | 3.12232i | 1.28911 | − | 2.70891i | −3.36674 | + | 10.3618i | 3.89421 | − | 3.13610i | 11.3824 | − | 2.12015i | −5.08243 | −25.3081 | + | 8.22311i | −5.67640 | − | 6.98416i | 18.6259 | + | 5.04472i | ||
41.1 | −2.26850 | + | 3.12232i | 0.549348 | + | 2.94927i | −3.36674 | − | 10.3618i | −3.89421 | − | 3.13610i | −10.4548 | − | 4.97519i | −5.08243 | 25.3081 | + | 8.22311i | −8.39643 | + | 3.24035i | 18.6259 | − | 5.04472i | ||
41.2 | −2.06309 | + | 2.83960i | 2.44039 | − | 1.74485i | −2.57091 | − | 7.91245i | 4.77936 | + | 1.46892i | −0.0800814 | + | 10.5295i | 6.80412 | 14.4196 | + | 4.68521i | 2.91102 | − | 8.51622i | −14.0314 | + | 10.5409i | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.3.j.a | ✓ | 72 |
3.b | odd | 2 | 1 | inner | 75.3.j.a | ✓ | 72 |
5.b | even | 2 | 1 | 375.3.j.a | 72 | ||
5.c | odd | 4 | 2 | 375.3.h.b | 144 | ||
15.d | odd | 2 | 1 | 375.3.j.a | 72 | ||
15.e | even | 4 | 2 | 375.3.h.b | 144 | ||
25.d | even | 5 | 1 | inner | 75.3.j.a | ✓ | 72 |
25.e | even | 10 | 1 | 375.3.j.a | 72 | ||
25.f | odd | 20 | 2 | 375.3.h.b | 144 | ||
75.h | odd | 10 | 1 | 375.3.j.a | 72 | ||
75.j | odd | 10 | 1 | inner | 75.3.j.a | ✓ | 72 |
75.l | even | 20 | 2 | 375.3.h.b | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.3.j.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
75.3.j.a | ✓ | 72 | 3.b | odd | 2 | 1 | inner |
75.3.j.a | ✓ | 72 | 25.d | even | 5 | 1 | inner |
75.3.j.a | ✓ | 72 | 75.j | odd | 10 | 1 | inner |
375.3.h.b | 144 | 5.c | odd | 4 | 2 | ||
375.3.h.b | 144 | 15.e | even | 4 | 2 | ||
375.3.h.b | 144 | 25.f | odd | 20 | 2 | ||
375.3.h.b | 144 | 75.l | even | 20 | 2 | ||
375.3.j.a | 72 | 5.b | even | 2 | 1 | ||
375.3.j.a | 72 | 15.d | odd | 2 | 1 | ||
375.3.j.a | 72 | 25.e | even | 10 | 1 | ||
375.3.j.a | 72 | 75.h | odd | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(75, [\chi])\).