Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,5,Mod(159,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 2]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.159");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.q (of order \(6\), degree \(2\), 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: | \(31.4244687775\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
159.1 | 0 | −14.5223 | + | 8.38447i | 0 | 14.2737 | + | 24.7229i | 0 | 68.5423i | 0 | 100.099 | − | 173.376i | 0 | ||||||||||||
159.2 | 0 | −11.6173 | + | 6.70724i | 0 | −20.2503 | − | 35.0746i | 0 | 70.8433i | 0 | 49.4742 | − | 85.6918i | 0 | ||||||||||||
159.3 | 0 | −10.9589 | + | 6.32715i | 0 | −14.7491 | − | 25.5461i | 0 | − | 60.9439i | 0 | 39.5656 | − | 68.5296i | 0 | |||||||||||
159.4 | 0 | −9.53297 | + | 5.50386i | 0 | 0.698568 | + | 1.20995i | 0 | − | 40.9545i | 0 | 20.0850 | − | 34.7882i | 0 | |||||||||||
159.5 | 0 | −7.34804 | + | 4.24239i | 0 | 6.30754 | + | 10.9250i | 0 | − | 23.4029i | 0 | −4.50418 | + | 7.80147i | 0 | |||||||||||
159.6 | 0 | −5.08005 | + | 2.93297i | 0 | 21.4753 | + | 37.1962i | 0 | − | 1.12615i | 0 | −23.2954 | + | 40.3488i | 0 | |||||||||||
159.7 | 0 | −1.32793 | + | 0.766679i | 0 | −3.49962 | − | 6.06153i | 0 | 1.93586i | 0 | −39.3244 | + | 68.1119i | 0 | ||||||||||||
159.8 | 0 | 1.59371 | − | 0.920131i | 0 | 2.29460 | + | 3.97436i | 0 | 89.4704i | 0 | −38.8067 | + | 67.2152i | 0 | ||||||||||||
159.9 | 0 | 2.94613 | − | 1.70095i | 0 | −15.4219 | − | 26.7115i | 0 | 25.4325i | 0 | −34.7135 | + | 60.1256i | 0 | ||||||||||||
159.10 | 0 | 6.36283 | − | 3.67358i | 0 | 11.8247 | + | 20.4810i | 0 | − | 89.1614i | 0 | −13.5096 | + | 23.3993i | 0 | |||||||||||
159.11 | 0 | 6.47147 | − | 3.73630i | 0 | −22.5885 | − | 39.1244i | 0 | − | 61.0442i | 0 | −12.5801 | + | 21.7893i | 0 | |||||||||||
159.12 | 0 | 9.74583 | − | 5.62676i | 0 | 2.35475 | + | 4.07855i | 0 | − | 19.2049i | 0 | 22.8209 | − | 39.5269i | 0 | |||||||||||
159.13 | 0 | 9.78032 | − | 5.64667i | 0 | 20.8784 | + | 36.1624i | 0 | 40.9622i | 0 | 23.2698 | − | 40.3044i | 0 | ||||||||||||
159.14 | 0 | 14.4872 | − | 8.36421i | 0 | −8.09816 | − | 14.0264i | 0 | 22.9000i | 0 | 99.4200 | − | 172.200i | 0 | ||||||||||||
239.1 | 0 | −14.5223 | − | 8.38447i | 0 | 14.2737 | − | 24.7229i | 0 | − | 68.5423i | 0 | 100.099 | + | 173.376i | 0 | |||||||||||
239.2 | 0 | −11.6173 | − | 6.70724i | 0 | −20.2503 | + | 35.0746i | 0 | − | 70.8433i | 0 | 49.4742 | + | 85.6918i | 0 | |||||||||||
239.3 | 0 | −10.9589 | − | 6.32715i | 0 | −14.7491 | + | 25.5461i | 0 | 60.9439i | 0 | 39.5656 | + | 68.5296i | 0 | ||||||||||||
239.4 | 0 | −9.53297 | − | 5.50386i | 0 | 0.698568 | − | 1.20995i | 0 | 40.9545i | 0 | 20.0850 | + | 34.7882i | 0 | ||||||||||||
239.5 | 0 | −7.34804 | − | 4.24239i | 0 | 6.30754 | − | 10.9250i | 0 | 23.4029i | 0 | −4.50418 | − | 7.80147i | 0 | ||||||||||||
239.6 | 0 | −5.08005 | − | 2.93297i | 0 | 21.4753 | − | 37.1962i | 0 | 1.12615i | 0 | −23.2954 | − | 40.3488i | 0 | ||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
76.g | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 304.5.q.b | ✓ | 28 |
4.b | odd | 2 | 1 | 304.5.q.c | yes | 28 | |
19.c | even | 3 | 1 | 304.5.q.c | yes | 28 | |
76.g | odd | 6 | 1 | inner | 304.5.q.b | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
304.5.q.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
304.5.q.b | ✓ | 28 | 76.g | odd | 6 | 1 | inner |
304.5.q.c | yes | 28 | 4.b | odd | 2 | 1 | |
304.5.q.c | yes | 28 | 19.c | even | 3 | 1 |