Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,2,Mod(31,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([0, 34]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("430.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 430.q (of order \(21\), degree \(12\), 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: | \(3.43356728692\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −0.900969 | + | 0.433884i | −2.80831 | + | 1.91467i | 0.623490 | − | 0.781831i | 0.733052 | + | 0.680173i | 1.69945 | − | 2.94354i | −0.338333 | − | 0.586011i | −0.222521 | + | 0.974928i | 3.12461 | − | 7.96138i | −0.955573 | − | 0.294755i |
31.2 | −0.900969 | + | 0.433884i | −0.993471 | + | 0.677337i | 0.623490 | − | 0.781831i | 0.733052 | + | 0.680173i | 0.601201 | − | 1.04131i | −0.895490 | − | 1.55103i | −0.222521 | + | 0.974928i | −0.567823 | + | 1.44679i | −0.955573 | − | 0.294755i |
31.3 | −0.900969 | + | 0.433884i | 0.953461 | − | 0.650059i | 0.623490 | − | 0.781831i | 0.733052 | + | 0.680173i | −0.576989 | + | 0.999374i | 1.53817 | + | 2.66418i | −0.222521 | + | 0.974928i | −0.609511 | + | 1.55301i | −0.955573 | − | 0.294755i |
31.4 | −0.900969 | + | 0.433884i | 2.64426 | − | 1.80282i | 0.623490 | − | 0.781831i | 0.733052 | + | 0.680173i | −1.60018 | + | 2.77159i | −0.952135 | − | 1.64915i | −0.222521 | + | 0.974928i | 2.64590 | − | 6.74164i | −0.955573 | − | 0.294755i |
81.1 | −0.222521 | + | 0.974928i | −1.77929 | − | 0.548838i | −0.900969 | − | 0.433884i | −0.365341 | − | 0.930874i | 0.931007 | − | 1.61255i | 1.00410 | + | 1.73915i | 0.623490 | − | 0.781831i | 0.385931 | + | 0.263123i | 0.988831 | − | 0.149042i |
81.2 | −0.222521 | + | 0.974928i | 0.269177 | + | 0.0830302i | −0.900969 | − | 0.433884i | −0.365341 | − | 0.930874i | −0.140846 | + | 0.243953i | −0.703462 | − | 1.21843i | 0.623490 | − | 0.781831i | −2.41315 | − | 1.64526i | 0.988831 | − | 0.149042i |
81.3 | −0.222521 | + | 0.974928i | 1.03154 | + | 0.318188i | −0.900969 | − | 0.433884i | −0.365341 | − | 0.930874i | −0.539749 | + | 0.934873i | −1.69214 | − | 2.93087i | 0.623490 | − | 0.781831i | −1.51589 | − | 1.03351i | 0.988831 | − | 0.149042i |
81.4 | −0.222521 | + | 0.974928i | 3.15603 | + | 0.973506i | −0.900969 | − | 0.433884i | −0.365341 | − | 0.930874i | −1.65138 | + | 2.86027i | 0.781939 | + | 1.35436i | 0.623490 | − | 0.781831i | 6.53408 | + | 4.45486i | 0.988831 | − | 0.149042i |
101.1 | 0.623490 | − | 0.781831i | −1.07289 | + | 2.73367i | −0.222521 | − | 0.974928i | −0.0747301 | − | 0.997204i | 1.46834 | + | 2.54323i | −1.49192 | + | 2.58408i | −0.900969 | − | 0.433884i | −4.12272 | − | 3.82532i | −0.826239 | − | 0.563320i |
101.2 | 0.623490 | − | 0.781831i | −0.320555 | + | 0.816762i | −0.222521 | − | 0.974928i | −0.0747301 | − | 0.997204i | 0.438707 | + | 0.759863i | −0.337906 | + | 0.585270i | −0.900969 | − | 0.433884i | 1.63481 | + | 1.51688i | −0.826239 | − | 0.563320i |
101.3 | 0.623490 | − | 0.781831i | 0.808920 | − | 2.06109i | −0.222521 | − | 0.974928i | −0.0747301 | − | 0.997204i | −1.10708 | − | 1.91751i | 1.63742 | − | 2.83609i | −0.900969 | − | 0.433884i | −1.39460 | − | 1.29400i | −0.826239 | − | 0.563320i |
101.4 | 0.623490 | − | 0.781831i | 1.11246 | − | 2.83449i | −0.222521 | − | 0.974928i | −0.0747301 | − | 0.997204i | −1.52249 | − | 2.63703i | −2.19740 | + | 3.80600i | −0.900969 | − | 0.433884i | −4.59762 | − | 4.26597i | −0.826239 | − | 0.563320i |
111.1 | −0.900969 | − | 0.433884i | −2.80831 | − | 1.91467i | 0.623490 | + | 0.781831i | 0.733052 | − | 0.680173i | 1.69945 | + | 2.94354i | −0.338333 | + | 0.586011i | −0.222521 | − | 0.974928i | 3.12461 | + | 7.96138i | −0.955573 | + | 0.294755i |
111.2 | −0.900969 | − | 0.433884i | −0.993471 | − | 0.677337i | 0.623490 | + | 0.781831i | 0.733052 | − | 0.680173i | 0.601201 | + | 1.04131i | −0.895490 | + | 1.55103i | −0.222521 | − | 0.974928i | −0.567823 | − | 1.44679i | −0.955573 | + | 0.294755i |
111.3 | −0.900969 | − | 0.433884i | 0.953461 | + | 0.650059i | 0.623490 | + | 0.781831i | 0.733052 | − | 0.680173i | −0.576989 | − | 0.999374i | 1.53817 | − | 2.66418i | −0.222521 | − | 0.974928i | −0.609511 | − | 1.55301i | −0.955573 | + | 0.294755i |
111.4 | −0.900969 | − | 0.433884i | 2.64426 | + | 1.80282i | 0.623490 | + | 0.781831i | 0.733052 | − | 0.680173i | −1.60018 | − | 2.77159i | −0.952135 | + | 1.64915i | −0.222521 | − | 0.974928i | 2.64590 | + | 6.74164i | −0.955573 | + | 0.294755i |
181.1 | 0.623490 | + | 0.781831i | −2.67211 | − | 0.402755i | −0.222521 | + | 0.974928i | −0.826239 | − | 0.563320i | −1.35114 | − | 2.34025i | 0.0929644 | − | 0.161019i | −0.900969 | + | 0.433884i | 4.11122 | + | 1.26814i | −0.0747301 | − | 0.997204i |
181.2 | 0.623490 | + | 0.781831i | −0.933730 | − | 0.140737i | −0.222521 | + | 0.974928i | −0.826239 | − | 0.563320i | −0.472138 | − | 0.817768i | −0.0334660 | + | 0.0579648i | −0.900969 | + | 0.433884i | −2.01467 | − | 0.621444i | −0.0747301 | − | 0.997204i |
181.3 | 0.623490 | + | 0.781831i | −0.399412 | − | 0.0602017i | −0.222521 | + | 0.974928i | −0.826239 | − | 0.563320i | −0.201962 | − | 0.349808i | −2.60651 | + | 4.51462i | −0.900969 | + | 0.433884i | −2.71081 | − | 0.836175i | −0.0747301 | − | 0.997204i |
181.4 | 0.623490 | + | 0.781831i | 2.57635 | + | 0.388322i | −0.222521 | + | 0.974928i | −0.826239 | − | 0.563320i | 1.30272 | + | 2.25638i | 1.51139 | − | 2.61780i | −0.900969 | + | 0.433884i | 3.62005 | + | 1.11664i | −0.0747301 | − | 0.997204i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.g | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 430.2.q.b | ✓ | 48 |
43.g | even | 21 | 1 | inner | 430.2.q.b | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
430.2.q.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
430.2.q.b | ✓ | 48 | 43.g | even | 21 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + T_{3}^{47} - 5 T_{3}^{46} - 24 T_{3}^{45} + 35 T_{3}^{44} - 63 T_{3}^{43} + \cdots + 2832474841 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\).