Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,12,Mod(2,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.2");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 15.e (of order \(4\), 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: | \(11.5251477084\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −61.8751 | + | 61.8751i | 3.69942 | − | 420.872i | − | 5609.07i | −4347.55 | − | 5470.55i | 25812.6 | + | 26270.4i | −11712.7 | − | 11712.7i | 220342. | + | 220342.i | −177120. | − | 3113.96i | 607496. | + | 69485.9i | |
2.2 | −57.5586 | + | 57.5586i | 345.894 | + | 239.801i | − | 4577.97i | 1295.31 | + | 6866.61i | −33711.7 | + | 6106.58i | 53787.8 | + | 53787.8i | 145622. | + | 145622.i | 62138.3 | + | 165891.i | −469788. | − | 320675.i | |
2.3 | −48.4604 | + | 48.4604i | −342.528 | + | 244.585i | − | 2648.82i | −5084.66 | + | 4793.15i | 4746.35 | − | 28451.7i | −47877.8 | − | 47877.8i | 29116.1 | + | 29116.1i | 57503.4 | − | 167554.i | 14126.7 | − | 478683.i | |
2.4 | −43.6110 | + | 43.6110i | −418.328 | − | 46.3532i | − | 1755.85i | 6653.83 | − | 2134.16i | 20265.2 | − | 16222.2i | 35410.1 | + | 35410.1i | −12741.2 | − | 12741.2i | 172850. | + | 38781.7i | −197108. | + | 383253.i | |
2.5 | −37.4902 | + | 37.4902i | 369.789 | − | 201.005i | − | 763.031i | 6987.38 | − | 67.8135i | −6327.78 | + | 21399.2i | −46363.7 | − | 46363.7i | −48173.8 | − | 48173.8i | 96341.3 | − | 148659.i | −259416. | + | 264501.i | |
2.6 | −37.2586 | + | 37.2586i | 104.521 | + | 407.704i | − | 728.404i | −1506.89 | − | 6823.30i | −19084.8 | − | 11296.2i | −5795.67 | − | 5795.67i | −49166.3 | − | 49166.3i | −155298. | + | 85226.8i | 310371. | + | 198082.i | |
2.7 | −26.1390 | + | 26.1390i | −159.262 | − | 389.593i | 681.505i | −2800.46 | + | 6402.00i | 14346.5 | + | 6020.60i | 5513.09 | + | 5513.09i | −71346.5 | − | 71346.5i | −126418. | + | 124095.i | −94140.6 | − | 240543.i | ||
2.8 | −21.4344 | + | 21.4344i | 367.459 | − | 205.234i | 1129.13i | −6189.39 | − | 3243.38i | −3477.19 | + | 12275.3i | 45393.8 | + | 45393.8i | −68099.9 | − | 68099.9i | 92905.0 | − | 150830.i | 202186. | − | 63146.0i | ||
2.9 | −3.46044 | + | 3.46044i | −124.374 | + | 402.092i | 2024.05i | 6126.08 | + | 3361.44i | −961.029 | − | 1821.80i | 10635.2 | + | 10635.2i | −14091.1 | − | 14091.1i | −146209. | − | 100019.i | −32831.0 | + | 9566.86i | ||
2.10 | −1.50709 | + | 1.50709i | 357.838 | + | 221.583i | 2043.46i | −3168.43 | + | 6228.10i | −873.238 | + | 205.348i | −31114.0 | − | 31114.0i | −6166.19 | − | 6166.19i | 78948.9 | + | 158582.i | −4611.19 | − | 14161.4i | ||
2.11 | 1.50709 | − | 1.50709i | −221.583 | − | 357.838i | 2043.46i | 3168.43 | − | 6228.10i | −873.238 | − | 205.348i | −31114.0 | − | 31114.0i | 6166.19 | + | 6166.19i | −78948.9 | + | 158582.i | −4611.19 | − | 14161.4i | ||
2.12 | 3.46044 | − | 3.46044i | −402.092 | + | 124.374i | 2024.05i | −6126.08 | − | 3361.44i | −961.029 | + | 1821.80i | 10635.2 | + | 10635.2i | 14091.1 | + | 14091.1i | 146209. | − | 100019.i | −32831.0 | + | 9566.86i | ||
2.13 | 21.4344 | − | 21.4344i | 205.234 | − | 367.459i | 1129.13i | 6189.39 | + | 3243.38i | −3477.19 | − | 12275.3i | 45393.8 | + | 45393.8i | 68099.9 | + | 68099.9i | −92905.0 | − | 150830.i | 202186. | − | 63146.0i | ||
2.14 | 26.1390 | − | 26.1390i | 389.593 | + | 159.262i | 681.505i | 2800.46 | − | 6402.00i | 14346.5 | − | 6020.60i | 5513.09 | + | 5513.09i | 71346.5 | + | 71346.5i | 126418. | + | 124095.i | −94140.6 | − | 240543.i | ||
2.15 | 37.2586 | − | 37.2586i | −407.704 | − | 104.521i | − | 728.404i | 1506.89 | + | 6823.30i | −19084.8 | + | 11296.2i | −5795.67 | − | 5795.67i | 49166.3 | + | 49166.3i | 155298. | + | 85226.8i | 310371. | + | 198082.i | |
2.16 | 37.4902 | − | 37.4902i | 201.005 | − | 369.789i | − | 763.031i | −6987.38 | + | 67.8135i | −6327.78 | − | 21399.2i | −46363.7 | − | 46363.7i | 48173.8 | + | 48173.8i | −96341.3 | − | 148659.i | −259416. | + | 264501.i | |
2.17 | 43.6110 | − | 43.6110i | 46.3532 | + | 418.328i | − | 1755.85i | −6653.83 | + | 2134.16i | 20265.2 | + | 16222.2i | 35410.1 | + | 35410.1i | 12741.2 | + | 12741.2i | −172850. | + | 38781.7i | −197108. | + | 383253.i | |
2.18 | 48.4604 | − | 48.4604i | −244.585 | + | 342.528i | − | 2648.82i | 5084.66 | − | 4793.15i | 4746.35 | + | 28451.7i | −47877.8 | − | 47877.8i | −29116.1 | − | 29116.1i | −57503.4 | − | 167554.i | 14126.7 | − | 478683.i | |
2.19 | 57.5586 | − | 57.5586i | −239.801 | − | 345.894i | − | 4577.97i | −1295.31 | − | 6866.61i | −33711.7 | − | 6106.58i | 53787.8 | + | 53787.8i | −145622. | − | 145622.i | −62138.3 | + | 165891.i | −469788. | − | 320675.i | |
2.20 | 61.8751 | − | 61.8751i | 420.872 | − | 3.69942i | − | 5609.07i | 4347.55 | + | 5470.55i | 25812.6 | − | 26270.4i | −11712.7 | − | 11712.7i | −220342. | − | 220342.i | 177120. | − | 3113.96i | 607496. | + | 69485.9i | |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 15.12.e.a | ✓ | 40 |
3.b | odd | 2 | 1 | inner | 15.12.e.a | ✓ | 40 |
5.b | even | 2 | 1 | 75.12.e.d | 40 | ||
5.c | odd | 4 | 1 | inner | 15.12.e.a | ✓ | 40 |
5.c | odd | 4 | 1 | 75.12.e.d | 40 | ||
15.d | odd | 2 | 1 | 75.12.e.d | 40 | ||
15.e | even | 4 | 1 | inner | 15.12.e.a | ✓ | 40 |
15.e | even | 4 | 1 | 75.12.e.d | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
15.12.e.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
15.12.e.a | ✓ | 40 | 3.b | odd | 2 | 1 | inner |
15.12.e.a | ✓ | 40 | 5.c | odd | 4 | 1 | inner |
15.12.e.a | ✓ | 40 | 15.e | even | 4 | 1 | inner |
75.12.e.d | 40 | 5.b | even | 2 | 1 | ||
75.12.e.d | 40 | 5.c | odd | 4 | 1 | ||
75.12.e.d | 40 | 15.d | odd | 2 | 1 | ||
75.12.e.d | 40 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(15, [\chi])\).