Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,12,Mod(55,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.55");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 84.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: | \(64.5408271670\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 | −45.2221 | − | 1.72065i | −243.000 | 2042.08 | + | 155.623i | 479.067i | 10989.0 | + | 418.118i | 18444.0 | − | 40461.6i | −92079.3 | − | 10551.3i | 59049.0 | 824.307 | − | 21664.4i | ||||||
55.2 | −45.2221 | + | 1.72065i | −243.000 | 2042.08 | − | 155.623i | − | 479.067i | 10989.0 | − | 418.118i | 18444.0 | + | 40461.6i | −92079.3 | + | 10551.3i | 59049.0 | 824.307 | + | 21664.4i | |||||
55.3 | −44.4594 | − | 8.44758i | −243.000 | 1905.28 | + | 751.149i | 11354.1i | 10803.6 | + | 2052.76i | −41246.7 | − | 16614.4i | −78362.1 | − | 49490.6i | 59049.0 | 95914.4 | − | 504795.i | ||||||
55.4 | −44.4594 | + | 8.44758i | −243.000 | 1905.28 | − | 751.149i | − | 11354.1i | 10803.6 | − | 2052.76i | −41246.7 | + | 16614.4i | −78362.1 | + | 49490.6i | 59049.0 | 95914.4 | + | 504795.i | |||||
55.5 | −40.8700 | − | 19.4330i | −243.000 | 1292.72 | + | 1588.45i | − | 1211.18i | 9931.41 | + | 4722.22i | −25032.4 | + | 36752.0i | −21964.9 | − | 90041.5i | 59049.0 | −23536.9 | + | 49501.0i | |||||
55.6 | −40.8700 | + | 19.4330i | −243.000 | 1292.72 | − | 1588.45i | 1211.18i | 9931.41 | − | 4722.22i | −25032.4 | − | 36752.0i | −21964.9 | + | 90041.5i | 59049.0 | −23536.9 | − | 49501.0i | ||||||
55.7 | −40.1866 | − | 20.8095i | −243.000 | 1181.93 | + | 1672.53i | − | 11675.3i | 9765.35 | + | 5056.70i | −983.024 | − | 44456.3i | −12693.5 | − | 91808.6i | 59049.0 | −242957. | + | 469190.i | |||||
55.8 | −40.1866 | + | 20.8095i | −243.000 | 1181.93 | − | 1672.53i | 11675.3i | 9765.35 | − | 5056.70i | −983.024 | + | 44456.3i | −12693.5 | + | 91808.6i | 59049.0 | −242957. | − | 469190.i | ||||||
55.9 | −39.5815 | − | 21.9387i | −243.000 | 1085.38 | + | 1736.73i | 10289.8i | 9618.30 | + | 5331.11i | 43786.1 | + | 7752.48i | −4859.36 | − | 92554.4i | 59049.0 | 225745. | − | 407286.i | ||||||
55.10 | −39.5815 | + | 21.9387i | −243.000 | 1085.38 | − | 1736.73i | − | 10289.8i | 9618.30 | − | 5331.11i | 43786.1 | − | 7752.48i | −4859.36 | + | 92554.4i | 59049.0 | 225745. | + | 407286.i | |||||
55.11 | −32.5648 | − | 31.4250i | −243.000 | 72.9330 | + | 2046.70i | − | 4483.20i | 7913.25 | + | 7636.29i | 44244.4 | + | 4445.20i | 61942.6 | − | 68942.3i | 59049.0 | −140885. | + | 145994.i | |||||
55.12 | −32.5648 | + | 31.4250i | −243.000 | 72.9330 | − | 2046.70i | 4483.20i | 7913.25 | − | 7636.29i | 44244.4 | − | 4445.20i | 61942.6 | + | 68942.3i | 59049.0 | −140885. | − | 145994.i | ||||||
55.13 | −25.9452 | − | 37.0790i | −243.000 | −701.697 | + | 1924.04i | 5436.30i | 6304.67 | + | 9010.19i | −42645.9 | − | 12595.7i | 89547.0 | − | 23901.3i | 59049.0 | 201572. | − | 141046.i | ||||||
55.14 | −25.9452 | + | 37.0790i | −243.000 | −701.697 | − | 1924.04i | − | 5436.30i | 6304.67 | − | 9010.19i | −42645.9 | + | 12595.7i | 89547.0 | + | 23901.3i | 59049.0 | 201572. | + | 141046.i | |||||
55.15 | −23.3606 | − | 38.7593i | −243.000 | −956.568 | + | 1810.88i | − | 7022.22i | 5676.62 | + | 9418.51i | −17991.9 | + | 40664.7i | 92534.4 | − | 5227.27i | 59049.0 | −272177. | + | 164043.i | |||||
55.16 | −23.3606 | + | 38.7593i | −243.000 | −956.568 | − | 1810.88i | 7022.22i | 5676.62 | − | 9418.51i | −17991.9 | − | 40664.7i | 92534.4 | + | 5227.27i | 59049.0 | −272177. | − | 164043.i | ||||||
55.17 | −20.9262 | − | 40.1260i | −243.000 | −1172.19 | + | 1679.37i | 6779.01i | 5085.06 | + | 9750.61i | 9419.93 | − | 43457.9i | 91915.7 | + | 11892.5i | 59049.0 | 272015. | − | 141859.i | ||||||
55.18 | −20.9262 | + | 40.1260i | −243.000 | −1172.19 | − | 1679.37i | − | 6779.01i | 5085.06 | − | 9750.61i | 9419.93 | + | 43457.9i | 91915.7 | − | 11892.5i | 59049.0 | 272015. | + | 141859.i | |||||
55.19 | −8.52362 | − | 44.4449i | −243.000 | −1902.70 | + | 757.663i | − | 11942.9i | 2071.24 | + | 10800.1i | −30635.0 | − | 32230.8i | 49892.1 | + | 78107.1i | 59049.0 | −530802. | + | 101797.i | |||||
55.20 | −8.52362 | + | 44.4449i | −243.000 | −1902.70 | − | 757.663i | 11942.9i | 2071.24 | − | 10800.1i | −30635.0 | + | 32230.8i | 49892.1 | − | 78107.1i | 59049.0 | −530802. | − | 101797.i | ||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
28.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 84.12.b.a | ✓ | 44 |
4.b | odd | 2 | 1 | 84.12.b.b | yes | 44 | |
7.b | odd | 2 | 1 | 84.12.b.b | yes | 44 | |
28.d | even | 2 | 1 | inner | 84.12.b.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
84.12.b.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
84.12.b.a | ✓ | 44 | 28.d | even | 2 | 1 | inner |
84.12.b.b | yes | 44 | 4.b | odd | 2 | 1 | |
84.12.b.b | yes | 44 | 7.b | odd | 2 | 1 |