Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4012,2,Mod(237,4012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4012.237");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4012 = 2^{2} \cdot 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4012.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: | \(32.0359812909\) |
Analytic rank: | \(0\) |
Dimension: | \(46\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
237.1 | 0 | − | 3.24132i | 0 | 0.276658i | 0 | − | 5.07344i | 0 | −7.50616 | 0 | ||||||||||||||||
237.2 | 0 | − | 3.20235i | 0 | − | 1.49915i | 0 | 2.89750i | 0 | −7.25507 | 0 | ||||||||||||||||
237.3 | 0 | − | 3.18403i | 0 | − | 1.45906i | 0 | − | 1.23253i | 0 | −7.13805 | 0 | |||||||||||||||
237.4 | 0 | − | 3.13247i | 0 | − | 3.91603i | 0 | 2.98385i | 0 | −6.81236 | 0 | ||||||||||||||||
237.5 | 0 | − | 2.81557i | 0 | 1.77357i | 0 | − | 1.01302i | 0 | −4.92746 | 0 | ||||||||||||||||
237.6 | 0 | − | 2.74502i | 0 | 1.20235i | 0 | − | 3.93843i | 0 | −4.53516 | 0 | ||||||||||||||||
237.7 | 0 | − | 2.57223i | 0 | 2.31913i | 0 | 3.71324i | 0 | −3.61638 | 0 | |||||||||||||||||
237.8 | 0 | − | 2.34087i | 0 | − | 2.91532i | 0 | − | 3.43927i | 0 | −2.47969 | 0 | |||||||||||||||
237.9 | 0 | − | 2.27623i | 0 | 2.55864i | 0 | − | 1.00518i | 0 | −2.18122 | 0 | ||||||||||||||||
237.10 | 0 | − | 1.82354i | 0 | − | 0.284793i | 0 | − | 0.645277i | 0 | −0.325298 | 0 | |||||||||||||||
237.11 | 0 | − | 1.79340i | 0 | 1.19915i | 0 | − | 1.20458i | 0 | −0.216279 | 0 | ||||||||||||||||
237.12 | 0 | − | 1.77972i | 0 | 1.91545i | 0 | 2.45686i | 0 | −0.167415 | 0 | |||||||||||||||||
237.13 | 0 | − | 1.58213i | 0 | − | 3.58424i | 0 | − | 1.89303i | 0 | 0.496852 | 0 | |||||||||||||||
237.14 | 0 | − | 1.50352i | 0 | 4.31120i | 0 | − | 4.93216i | 0 | 0.739426 | 0 | ||||||||||||||||
237.15 | 0 | − | 1.45665i | 0 | − | 3.69601i | 0 | 3.33013i | 0 | 0.878183 | 0 | ||||||||||||||||
237.16 | 0 | − | 1.43806i | 0 | 2.66437i | 0 | 3.93282i | 0 | 0.931993 | 0 | |||||||||||||||||
237.17 | 0 | − | 1.38906i | 0 | − | 2.35139i | 0 | − | 0.244801i | 0 | 1.07052 | 0 | |||||||||||||||
237.18 | 0 | − | 1.13278i | 0 | − | 2.09684i | 0 | 2.71039i | 0 | 1.71682 | 0 | ||||||||||||||||
237.19 | 0 | − | 0.593296i | 0 | 3.87533i | 0 | 2.92807i | 0 | 2.64800 | 0 | |||||||||||||||||
237.20 | 0 | − | 0.405081i | 0 | 1.20499i | 0 | − | 0.122460i | 0 | 2.83591 | 0 | ||||||||||||||||
See all 46 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4012.2.b.b | ✓ | 46 |
17.b | even | 2 | 1 | inner | 4012.2.b.b | ✓ | 46 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4012.2.b.b | ✓ | 46 | 1.a | even | 1 | 1 | trivial |
4012.2.b.b | ✓ | 46 | 17.b | even | 2 | 1 | inner |