Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [20,12,Mod(3,20)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20.3");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 20.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: | \(15.3668636112\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(30\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −45.1846 | + | 2.52028i | −18.0064 | + | 18.0064i | 2035.30 | − | 227.755i | 6954.24 | − | 683.153i | 768.231 | − | 858.993i | −18984.5 | − | 18984.5i | −91390.1 | + | 15420.5i | 176499.i | −312503. | + | 48394.6i | ||
3.2 | −44.9401 | − | 5.32801i | 359.350 | − | 359.350i | 1991.22 | + | 478.883i | −6959.92 | − | 622.606i | −18063.8 | + | 14234.6i | −2235.42 | − | 2235.42i | −86934.3 | − | 32130.3i | − | 81117.8i | 309462. | + | 65062.5i | |
3.3 | −43.4480 | − | 12.6598i | −513.533 | + | 513.533i | 1727.46 | + | 1100.09i | −3227.70 | − | 6197.58i | 28813.2 | − | 15810.8i | −25537.2 | − | 25537.2i | −61127.8 | − | 69665.8i | − | 350285.i | 61777.1 | + | 310135.i | |
3.4 | −42.2496 | + | 16.2165i | −165.660 | + | 165.660i | 1522.05 | − | 1370.28i | −4210.13 | + | 5577.00i | 4312.64 | − | 9685.50i | 9765.23 | + | 9765.23i | −42084.8 | + | 82576.1i | 122260.i | 87436.6 | − | 303899.i | ||
3.5 | −37.1216 | + | 25.8841i | 544.922 | − | 544.922i | 708.031 | − | 1921.72i | 5461.94 | + | 4358.36i | −6123.60 | + | 34333.2i | 41303.5 | + | 41303.5i | 23458.6 | + | 89664.0i | − | 416734.i | −315568. | − | 20412.0i | |
3.6 | −36.3945 | − | 26.8968i | −266.570 | + | 266.570i | 601.119 | + | 1957.79i | 205.209 | + | 6984.70i | 16871.6 | − | 2531.79i | 18557.1 | + | 18557.1i | 30781.1 | − | 87421.2i | 35028.2i | 180398. | − | 259724.i | ||
3.7 | −35.4618 | − | 28.1151i | 228.106 | − | 228.106i | 467.078 | + | 1994.03i | 2265.99 | − | 6610.10i | −14502.3 | + | 1675.81i | 43448.3 | + | 43448.3i | 39498.9 | − | 83843.7i | 73082.7i | −266200. | + | 170697.i | ||
3.8 | −34.0099 | + | 29.8551i | −257.870 | + | 257.870i | 265.350 | − | 2030.74i | −1405.66 | − | 6844.87i | 1071.41 | − | 16468.9i | 55896.5 | + | 55896.5i | 51603.3 | + | 76987.2i | 44152.8i | 252161. | + | 190827.i | ||
3.9 | −29.8551 | + | 34.0099i | 257.870 | − | 257.870i | −265.350 | − | 2030.74i | −1405.66 | − | 6844.87i | 1071.41 | + | 16468.9i | −55896.5 | − | 55896.5i | 76987.2 | + | 51603.3i | 44152.8i | 274760. | + | 156548.i | ||
3.10 | −29.1822 | − | 34.5890i | 418.357 | − | 418.357i | −344.796 | + | 2018.77i | 4148.35 | + | 5623.11i | −26679.1 | − | 2261.96i | −45215.7 | − | 45215.7i | 79889.0 | − | 46986.0i | − | 172898.i | 73439.5 | − | 307582.i | |
3.11 | −25.8841 | + | 37.1216i | −544.922 | + | 544.922i | −708.031 | − | 1921.72i | 5461.94 | + | 4358.36i | −6123.60 | − | 34333.2i | −41303.5 | − | 41303.5i | 89664.0 | + | 23458.6i | − | 416734.i | −303167. | + | 89944.3i | |
3.12 | −16.2165 | + | 42.2496i | 165.660 | − | 165.660i | −1522.05 | − | 1370.28i | −4210.13 | + | 5577.00i | 4312.64 | + | 9685.50i | −9765.23 | − | 9765.23i | 82576.1 | − | 42084.8i | 122260.i | −167352. | − | 268316.i | ||
3.13 | −16.1888 | − | 42.2602i | −59.6583 | + | 59.6583i | −1523.84 | + | 1368.29i | −6580.51 | − | 2350.54i | 3486.97 | + | 1555.37i | −33756.3 | − | 33756.3i | 82493.3 | + | 42246.9i | 170029.i | 7196.45 | + | 316146.i | ||
3.14 | −9.80346 | − | 44.1802i | −358.943 | + | 358.943i | −1855.78 | + | 866.239i | 6343.79 | − | 2929.93i | 19377.1 | + | 12339.3i | 14614.2 | + | 14614.2i | 56463.7 | + | 73496.8i | − | 80532.8i | −191636. | − | 251547.i | |
3.15 | −2.52028 | + | 45.1846i | 18.0064 | − | 18.0064i | −2035.30 | − | 227.755i | 6954.24 | − | 683.153i | 768.231 | + | 858.993i | 18984.5 | + | 18984.5i | 15420.5 | − | 91390.1i | 176499.i | 13341.4 | + | 315946.i | ||
3.16 | 1.55298 | − | 45.2282i | 281.443 | − | 281.443i | −2043.18 | − | 140.477i | −4194.88 | + | 5588.48i | −12292.1 | − | 13166.2i | 51844.9 | + | 51844.9i | −9526.55 | + | 92191.0i | 18726.5i | 246242. | + | 198406.i | ||
3.17 | 5.32801 | + | 44.9401i | −359.350 | + | 359.350i | −1991.22 | + | 478.883i | −6959.92 | − | 622.606i | −18063.8 | − | 14234.6i | 2235.42 | + | 2235.42i | −32130.3 | − | 86934.3i | − | 81117.8i | −9102.56 | − | 316097.i | |
3.18 | 9.12236 | − | 44.3259i | 433.162 | − | 433.162i | −1881.57 | − | 808.713i | 4261.88 | − | 5537.55i | −15248.8 | − | 23151.7i | −8187.22 | − | 8187.22i | −53011.2 | + | 76024.6i | − | 198112.i | −206578. | − | 239427.i | |
3.19 | 12.6598 | + | 43.4480i | 513.533 | − | 513.533i | −1727.46 | + | 1100.09i | −3227.70 | − | 6197.58i | 28813.2 | + | 15810.8i | 25537.2 | + | 25537.2i | −69665.8 | − | 61127.8i | − | 350285.i | 228411. | − | 218697.i | |
3.20 | 19.1088 | − | 41.0226i | −116.548 | + | 116.548i | −1317.71 | − | 1567.78i | 4797.95 | + | 5080.14i | 2554.02 | + | 7008.21i | −39201.4 | − | 39201.4i | −89494.4 | + | 24097.3i | 149980.i | 300084. | − | 99748.9i | ||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 20.12.e.b | ✓ | 60 |
4.b | odd | 2 | 1 | inner | 20.12.e.b | ✓ | 60 |
5.b | even | 2 | 1 | 100.12.e.e | 60 | ||
5.c | odd | 4 | 1 | inner | 20.12.e.b | ✓ | 60 |
5.c | odd | 4 | 1 | 100.12.e.e | 60 | ||
20.d | odd | 2 | 1 | 100.12.e.e | 60 | ||
20.e | even | 4 | 1 | inner | 20.12.e.b | ✓ | 60 |
20.e | even | 4 | 1 | 100.12.e.e | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.12.e.b | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
20.12.e.b | ✓ | 60 | 4.b | odd | 2 | 1 | inner |
20.12.e.b | ✓ | 60 | 5.c | odd | 4 | 1 | inner |
20.12.e.b | ✓ | 60 | 20.e | even | 4 | 1 | inner |
100.12.e.e | 60 | 5.b | even | 2 | 1 | ||
100.12.e.e | 60 | 5.c | odd | 4 | 1 | ||
100.12.e.e | 60 | 20.d | odd | 2 | 1 | ||
100.12.e.e | 60 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{60} + 1410707871720 T_{3}^{56} + \cdots + 10\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(20, [\chi])\).