Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,12,Mod(1,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.a (trivial) |
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: | yes |
Analytic conductor: | \(135.996742959\) |
Analytic rank: | \(1\) |
Dimension: | \(27\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −88.8229 | 243.000 | 5841.51 | 11395.4 | −21584.0 | 23008.9 | −336951. | 59049.0 | −1.01218e6 | ||||||||||||||||||
1.2 | −86.7335 | 243.000 | 5474.69 | −11727.0 | −21076.2 | 24236.6 | −297209. | 59049.0 | 1.01712e6 | ||||||||||||||||||
1.3 | −78.8162 | 243.000 | 4163.99 | 6894.11 | −19152.3 | 41129.0 | −166774. | 59049.0 | −543367. | ||||||||||||||||||
1.4 | −71.6191 | 243.000 | 3081.30 | −9710.01 | −17403.4 | 37645.0 | −74003.9 | 59049.0 | 695422. | ||||||||||||||||||
1.5 | −68.7593 | 243.000 | 2679.84 | 7324.59 | −16708.5 | −63928.8 | −43445.2 | 59049.0 | −503634. | ||||||||||||||||||
1.6 | −63.4822 | 243.000 | 1982.00 | −1374.17 | −15426.2 | −16112.8 | 4190.11 | 59049.0 | 87235.4 | ||||||||||||||||||
1.7 | −54.1540 | 243.000 | 884.652 | −9497.19 | −13159.4 | −21091.6 | 62999.9 | 59049.0 | 514310. | ||||||||||||||||||
1.8 | −47.2991 | 243.000 | 189.204 | 4204.59 | −11493.7 | 8235.98 | 87919.4 | 59049.0 | −198873. | ||||||||||||||||||
1.9 | −45.9608 | 243.000 | 64.3958 | −11937.5 | −11168.5 | −79874.8 | 91168.1 | 59049.0 | 548659. | ||||||||||||||||||
1.10 | −45.6492 | 243.000 | 35.8459 | −2292.40 | −11092.7 | 55379.8 | 91853.1 | 59049.0 | 104646. | ||||||||||||||||||
1.11 | −19.9033 | 243.000 | −1651.86 | 11169.4 | −4836.50 | −27660.5 | 73639.4 | 59049.0 | −222307. | ||||||||||||||||||
1.12 | −18.7578 | 243.000 | −1696.14 | −6326.10 | −4558.15 | −59668.2 | 70231.9 | 59049.0 | 118664. | ||||||||||||||||||
1.13 | −17.2441 | 243.000 | −1750.64 | 7994.91 | −4190.32 | 1325.62 | 65504.2 | 59049.0 | −137865. | ||||||||||||||||||
1.14 | −4.91496 | 243.000 | −2023.84 | −1117.68 | −1194.33 | −21416.7 | 20012.9 | 59049.0 | 5493.36 | ||||||||||||||||||
1.15 | −2.16187 | 243.000 | −2043.33 | 654.400 | −525.335 | 79533.7 | 8844.92 | 59049.0 | −1414.73 | ||||||||||||||||||
1.16 | 10.6976 | 243.000 | −1933.56 | 12608.0 | 2599.53 | −75685.5 | −42593.3 | 59049.0 | 134876. | ||||||||||||||||||
1.17 | 13.5218 | 243.000 | −1865.16 | −12002.6 | 3285.79 | −19689.8 | −52912.9 | 59049.0 | −162297. | ||||||||||||||||||
1.18 | 32.2053 | 243.000 | −1010.82 | −5509.44 | 7825.90 | −36737.1 | −98510.2 | 59049.0 | −177433. | ||||||||||||||||||
1.19 | 33.0193 | 243.000 | −957.729 | −9842.24 | 8023.68 | 66020.7 | −99246.9 | 59049.0 | −324984. | ||||||||||||||||||
1.20 | 35.0313 | 243.000 | −820.805 | 1509.53 | 8512.61 | 260.533 | −100498. | 59049.0 | 52880.9 | ||||||||||||||||||
See all 27 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(59\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.12.a.b | ✓ | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.12.a.b | ✓ | 27 | 1.a | even | 1 | 1 | trivial |