Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,7,Mod(22,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.22");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.d (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: | \(15.8737317698\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
22.1 | −15.4071 | 15.5885 | 173.378 | − | 161.383i | −240.172 | − | 585.691i | −1685.19 | 243.000 | 2486.44i | ||||||||||||||||
22.2 | −15.4071 | 15.5885 | 173.378 | 161.383i | −240.172 | 585.691i | −1685.19 | 243.000 | − | 2486.44i | |||||||||||||||||
22.3 | −13.6749 | −15.5885 | 123.003 | − | 37.2861i | 213.170 | − | 322.993i | −806.853 | 243.000 | 509.883i | ||||||||||||||||
22.4 | −13.6749 | −15.5885 | 123.003 | 37.2861i | 213.170 | 322.993i | −806.853 | 243.000 | − | 509.883i | |||||||||||||||||
22.5 | −10.6305 | 15.5885 | 49.0083 | − | 66.6032i | −165.714 | 306.844i | 159.370 | 243.000 | 708.028i | |||||||||||||||||
22.6 | −10.6305 | 15.5885 | 49.0083 | 66.6032i | −165.714 | − | 306.844i | 159.370 | 243.000 | − | 708.028i | ||||||||||||||||
22.7 | −8.53823 | −15.5885 | 8.90142 | − | 233.104i | 133.098 | − | 155.980i | 470.445 | 243.000 | 1990.29i | ||||||||||||||||
22.8 | −8.53823 | −15.5885 | 8.90142 | 233.104i | 133.098 | 155.980i | 470.445 | 243.000 | − | 1990.29i | |||||||||||||||||
22.9 | −4.73591 | −15.5885 | −41.5712 | − | 38.4146i | 73.8255 | 655.803i | 499.975 | 243.000 | 181.928i | |||||||||||||||||
22.10 | −4.73591 | −15.5885 | −41.5712 | 38.4146i | 73.8255 | − | 655.803i | 499.975 | 243.000 | − | 181.928i | ||||||||||||||||
22.11 | −3.69366 | 15.5885 | −50.3569 | − | 218.943i | −57.5785 | − | 261.444i | 422.396 | 243.000 | 808.702i | ||||||||||||||||
22.12 | −3.69366 | 15.5885 | −50.3569 | 218.943i | −57.5785 | 261.444i | 422.396 | 243.000 | − | 808.702i | |||||||||||||||||
22.13 | −0.368531 | 15.5885 | −63.8642 | − | 113.928i | −5.74483 | 569.901i | 47.1219 | 243.000 | 41.9859i | |||||||||||||||||
22.14 | −0.368531 | 15.5885 | −63.8642 | 113.928i | −5.74483 | − | 569.901i | 47.1219 | 243.000 | − | 41.9859i | ||||||||||||||||
22.15 | 2.88321 | −15.5885 | −55.6871 | − | 133.948i | −44.9448 | 59.3150i | −345.083 | 243.000 | − | 386.201i | ||||||||||||||||
22.16 | 2.88321 | −15.5885 | −55.6871 | 133.948i | −44.9448 | − | 59.3150i | −345.083 | 243.000 | 386.201i | |||||||||||||||||
22.17 | 6.36312 | 15.5885 | −23.5107 | − | 60.2242i | 99.1913 | − | 233.613i | −556.841 | 243.000 | − | 383.214i | |||||||||||||||
22.18 | 6.36312 | 15.5885 | −23.5107 | 60.2242i | 99.1913 | 233.613i | −556.841 | 243.000 | 383.214i | ||||||||||||||||||
22.19 | 10.8996 | −15.5885 | 54.8005 | − | 162.476i | −169.907 | 219.782i | −100.271 | 243.000 | − | 1770.91i | ||||||||||||||||
22.20 | 10.8996 | −15.5885 | 54.8005 | 162.476i | −169.907 | − | 219.782i | −100.271 | 243.000 | 1770.91i | |||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.7.d.a | ✓ | 24 |
3.b | odd | 2 | 1 | 207.7.d.e | 24 | ||
23.b | odd | 2 | 1 | inner | 69.7.d.a | ✓ | 24 |
69.c | even | 2 | 1 | 207.7.d.e | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.7.d.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
69.7.d.a | ✓ | 24 | 23.b | odd | 2 | 1 | inner |
207.7.d.e | 24 | 3.b | odd | 2 | 1 | ||
207.7.d.e | 24 | 69.c | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).