Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [154,2,Mod(9,154)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(154, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([10, 18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("154.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 154 = 2 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 154.m (of order \(15\), degree \(8\), 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: | \(1.22969619113\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | −0.669131 | − | 0.743145i | −3.06161 | − | 1.36311i | −0.104528 | + | 0.994522i | 0.248798 | + | 0.0528836i | 1.03562 | + | 3.18732i | −2.64537 | − | 0.0451581i | 0.809017 | − | 0.587785i | 5.50796 | + | 6.11721i | −0.127178 | − | 0.220279i |
9.2 | −0.669131 | − | 0.743145i | −0.735907 | − | 0.327647i | −0.104528 | + | 0.994522i | 2.12235 | + | 0.451119i | 0.248929 | + | 0.766124i | 2.01873 | + | 1.71018i | 0.809017 | − | 0.587785i | −1.57319 | − | 1.74720i | −1.08488 | − | 1.87907i |
9.3 | −0.669131 | − | 0.743145i | 2.31936 | + | 1.03265i | −0.104528 | + | 0.994522i | 1.77235 | + | 0.376725i | −0.784551 | − | 2.41460i | −2.18977 | + | 1.48489i | 0.809017 | − | 0.587785i | 2.30570 | + | 2.56074i | −0.905975 | − | 1.56919i |
25.1 | −0.913545 | − | 0.406737i | −1.43491 | − | 0.304999i | 0.669131 | + | 0.743145i | −0.00806762 | + | 0.0767583i | 1.18680 | + | 0.862261i | −1.78535 | − | 1.95256i | −0.309017 | − | 0.951057i | −0.774697 | − | 0.344917i | 0.0385906 | − | 0.0668408i |
25.2 | −0.913545 | − | 0.406737i | 0.137294 | + | 0.0291827i | 0.669131 | + | 0.743145i | 0.368040 | − | 3.50166i | −0.113554 | − | 0.0825021i | 1.20864 | + | 2.35355i | −0.309017 | − | 0.951057i | −2.72264 | − | 1.21220i | −1.76048 | + | 3.04923i |
25.3 | −0.913545 | − | 0.406737i | 0.693087 | + | 0.147320i | 0.669131 | + | 0.743145i | −0.384648 | + | 3.65968i | −0.573246 | − | 0.416487i | −1.85490 | + | 1.88662i | −0.309017 | − | 0.951057i | −2.28197 | − | 1.01600i | 1.83992 | − | 3.18683i |
37.1 | −0.913545 | + | 0.406737i | −1.43491 | + | 0.304999i | 0.669131 | − | 0.743145i | −0.00806762 | − | 0.0767583i | 1.18680 | − | 0.862261i | −1.78535 | + | 1.95256i | −0.309017 | + | 0.951057i | −0.774697 | + | 0.344917i | 0.0385906 | + | 0.0668408i |
37.2 | −0.913545 | + | 0.406737i | 0.137294 | − | 0.0291827i | 0.669131 | − | 0.743145i | 0.368040 | + | 3.50166i | −0.113554 | + | 0.0825021i | 1.20864 | − | 2.35355i | −0.309017 | + | 0.951057i | −2.72264 | + | 1.21220i | −1.76048 | − | 3.04923i |
37.3 | −0.913545 | + | 0.406737i | 0.693087 | − | 0.147320i | 0.669131 | − | 0.743145i | −0.384648 | − | 3.65968i | −0.573246 | + | 0.416487i | −1.85490 | − | 1.88662i | −0.309017 | + | 0.951057i | −2.28197 | + | 1.01600i | 1.83992 | + | 3.18683i |
53.1 | 0.978148 | − | 0.207912i | −0.265383 | − | 2.52495i | 0.913545 | − | 0.406737i | −1.21243 | + | 1.34654i | −0.784551 | − | 2.41460i | 0.735534 | − | 2.54145i | 0.809017 | − | 0.587785i | −3.37052 | + | 0.716425i | −0.905975 | + | 1.56919i |
53.2 | 0.978148 | − | 0.207912i | 0.0842029 | + | 0.801137i | 0.913545 | − | 0.406737i | −1.45185 | + | 1.61245i | 0.248929 | + | 0.766124i | 2.25030 | + | 1.39145i | 0.809017 | − | 0.587785i | 2.29971 | − | 0.488819i | −1.08488 | + | 1.87907i |
53.3 | 0.978148 | − | 0.207912i | 0.350311 | + | 3.33299i | 0.913545 | − | 0.406737i | −0.170197 | + | 0.189023i | 1.03562 | + | 3.18732i | −0.860411 | − | 2.50194i | 0.809017 | − | 0.587785i | −8.05163 | + | 1.71143i | −0.127178 | + | 0.220279i |
81.1 | 0.104528 | − | 0.994522i | −0.474126 | − | 0.526571i | −0.978148 | − | 0.207912i | 3.36170 | + | 1.49673i | −0.573246 | + | 0.416487i | 0.391723 | − | 2.61659i | −0.309017 | + | 0.951057i | 0.261105 | − | 2.48424i | 1.83992 | − | 3.18683i |
81.2 | 0.104528 | − | 0.994522i | −0.0939198 | − | 0.104309i | −0.978148 | − | 0.207912i | −3.21655 | − | 1.43210i | −0.113554 | + | 0.0825021i | −2.36119 | − | 1.19364i | −0.309017 | + | 0.951057i | 0.311526 | − | 2.96397i | −1.76048 | + | 3.04923i |
81.3 | 0.104528 | − | 0.994522i | 0.981592 | + | 1.09017i | −0.978148 | − | 0.207912i | 0.0705085 | + | 0.0313924i | 1.18680 | − | 0.862261i | 2.59207 | + | 0.530253i | −0.309017 | + | 0.951057i | 0.0886414 | − | 0.843366i | 0.0385906 | − | 0.0668408i |
93.1 | 0.978148 | + | 0.207912i | −0.265383 | + | 2.52495i | 0.913545 | + | 0.406737i | −1.21243 | − | 1.34654i | −0.784551 | + | 2.41460i | 0.735534 | + | 2.54145i | 0.809017 | + | 0.587785i | −3.37052 | − | 0.716425i | −0.905975 | − | 1.56919i |
93.2 | 0.978148 | + | 0.207912i | 0.0842029 | − | 0.801137i | 0.913545 | + | 0.406737i | −1.45185 | − | 1.61245i | 0.248929 | − | 0.766124i | 2.25030 | − | 1.39145i | 0.809017 | + | 0.587785i | 2.29971 | + | 0.488819i | −1.08488 | − | 1.87907i |
93.3 | 0.978148 | + | 0.207912i | 0.350311 | − | 3.33299i | 0.913545 | + | 0.406737i | −0.170197 | − | 0.189023i | 1.03562 | − | 3.18732i | −0.860411 | + | 2.50194i | 0.809017 | + | 0.587785i | −8.05163 | − | 1.71143i | −0.127178 | − | 0.220279i |
135.1 | 0.104528 | + | 0.994522i | −0.474126 | + | 0.526571i | −0.978148 | + | 0.207912i | 3.36170 | − | 1.49673i | −0.573246 | − | 0.416487i | 0.391723 | + | 2.61659i | −0.309017 | − | 0.951057i | 0.261105 | + | 2.48424i | 1.83992 | + | 3.18683i |
135.2 | 0.104528 | + | 0.994522i | −0.0939198 | + | 0.104309i | −0.978148 | + | 0.207912i | −3.21655 | + | 1.43210i | −0.113554 | − | 0.0825021i | −2.36119 | + | 1.19364i | −0.309017 | − | 0.951057i | 0.311526 | + | 2.96397i | −1.76048 | − | 3.04923i |
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
77.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 154.2.m.b | ✓ | 24 |
7.c | even | 3 | 1 | inner | 154.2.m.b | ✓ | 24 |
11.c | even | 5 | 1 | inner | 154.2.m.b | ✓ | 24 |
77.m | even | 15 | 1 | inner | 154.2.m.b | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
154.2.m.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
154.2.m.b | ✓ | 24 | 7.c | even | 3 | 1 | inner |
154.2.m.b | ✓ | 24 | 11.c | even | 5 | 1 | inner |
154.2.m.b | ✓ | 24 | 77.m | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 3 T_{3}^{23} + 8 T_{3}^{22} + 27 T_{3}^{21} - 6 T_{3}^{20} - 194 T_{3}^{19} + 295 T_{3}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(154, [\chi])\).