Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(16,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.k (of order \(3\), degree \(2\), 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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −1.17004 | − | 2.02657i | 1.32145 | + | 1.11972i | −1.73798 | + | 3.01027i | −1.00000 | 0.723033 | − | 3.98812i | −1.00344 | + | 2.44808i | 3.45385 | 0.492465 | + | 2.95930i | 1.17004 | + | 2.02657i | ||||
16.2 | −1.16277 | − | 2.01398i | 0.434379 | − | 1.67670i | −1.70408 | + | 2.95156i | −1.00000 | −3.88192 | + | 1.07479i | 0.459529 | − | 2.60554i | 3.27475 | −2.62263 | − | 1.45664i | 1.16277 | + | 2.01398i | ||||
16.3 | −1.08176 | − | 1.87367i | −1.55999 | + | 0.752618i | −1.34041 | + | 2.32167i | −1.00000 | 3.09769 | + | 2.10874i | 2.14928 | + | 1.54292i | 1.47299 | 1.86713 | − | 2.34815i | 1.08176 | + | 1.87367i | ||||
16.4 | −0.599034 | − | 1.03756i | 1.72885 | + | 0.105192i | 0.282317 | − | 0.488988i | −1.00000 | −0.926499 | − | 1.85680i | 2.04342 | − | 1.68061i | −3.07261 | 2.97787 | + | 0.363723i | 0.599034 | + | 1.03756i | ||||
16.5 | −0.259245 | − | 0.449026i | −1.70948 | + | 0.278682i | 0.865584 | − | 1.49923i | −1.00000 | 0.568312 | + | 0.695356i | −1.91816 | + | 1.82226i | −1.93458 | 2.84467 | − | 0.952806i | 0.259245 | + | 0.449026i | ||||
16.6 | −0.148731 | − | 0.257610i | 0.310785 | − | 1.70394i | 0.955758 | − | 1.65542i | −1.00000 | −0.485175 | + | 0.173367i | −2.64436 | + | 0.0857253i | −1.16353 | −2.80683 | − | 1.05912i | 0.148731 | + | 0.257610i | ||||
16.7 | 0.154039 | + | 0.266804i | 1.05264 | − | 1.37548i | 0.952544 | − | 1.64985i | −1.00000 | 0.529131 | + | 0.0689719i | 2.20433 | + | 1.46319i | 1.20307 | −0.783880 | − | 2.89578i | −0.154039 | − | 0.266804i | ||||
16.8 | 0.304907 | + | 0.528114i | −1.22862 | + | 1.22086i | 0.814064 | − | 1.41000i | −1.00000 | −1.01937 | − | 0.276604i | 1.83653 | − | 1.90451i | 2.21248 | 0.0190166 | − | 2.99994i | −0.304907 | − | 0.528114i | ||||
16.9 | 0.517769 | + | 0.896802i | −1.26722 | − | 1.18074i | 0.463831 | − | 0.803378i | −1.00000 | 0.402766 | − | 1.74780i | −1.07729 | − | 2.41649i | 3.03170 | 0.211690 | + | 2.99252i | −0.517769 | − | 0.896802i | ||||
16.10 | 0.805191 | + | 1.39463i | 1.04104 | + | 1.38428i | −0.296664 | + | 0.513837i | −1.00000 | −1.09233 | + | 2.56648i | 2.48389 | + | 0.911196i | 2.26528 | −0.832482 | + | 2.88218i | −0.805191 | − | 1.39463i | ||||
16.11 | 0.859635 | + | 1.48893i | −1.31121 | − | 1.13170i | −0.477944 | + | 0.827824i | −1.00000 | 0.557856 | − | 2.92514i | 0.594106 | + | 2.57818i | 1.79511 | 0.438533 | + | 2.96778i | −0.859635 | − | 1.48893i | ||||
16.12 | 1.28004 | + | 2.21710i | 1.68737 | − | 0.390872i | −2.27701 | + | 3.94390i | −1.00000 | 3.02651 | + | 3.24073i | −1.62783 | + | 2.08571i | −6.53853 | 2.69444 | − | 1.31909i | −1.28004 | − | 2.21710i | ||||
256.1 | −1.17004 | + | 2.02657i | 1.32145 | − | 1.11972i | −1.73798 | − | 3.01027i | −1.00000 | 0.723033 | + | 3.98812i | −1.00344 | − | 2.44808i | 3.45385 | 0.492465 | − | 2.95930i | 1.17004 | − | 2.02657i | ||||
256.2 | −1.16277 | + | 2.01398i | 0.434379 | + | 1.67670i | −1.70408 | − | 2.95156i | −1.00000 | −3.88192 | − | 1.07479i | 0.459529 | + | 2.60554i | 3.27475 | −2.62263 | + | 1.45664i | 1.16277 | − | 2.01398i | ||||
256.3 | −1.08176 | + | 1.87367i | −1.55999 | − | 0.752618i | −1.34041 | − | 2.32167i | −1.00000 | 3.09769 | − | 2.10874i | 2.14928 | − | 1.54292i | 1.47299 | 1.86713 | + | 2.34815i | 1.08176 | − | 1.87367i | ||||
256.4 | −0.599034 | + | 1.03756i | 1.72885 | − | 0.105192i | 0.282317 | + | 0.488988i | −1.00000 | −0.926499 | + | 1.85680i | 2.04342 | + | 1.68061i | −3.07261 | 2.97787 | − | 0.363723i | 0.599034 | − | 1.03756i | ||||
256.5 | −0.259245 | + | 0.449026i | −1.70948 | − | 0.278682i | 0.865584 | + | 1.49923i | −1.00000 | 0.568312 | − | 0.695356i | −1.91816 | − | 1.82226i | −1.93458 | 2.84467 | + | 0.952806i | 0.259245 | − | 0.449026i | ||||
256.6 | −0.148731 | + | 0.257610i | 0.310785 | + | 1.70394i | 0.955758 | + | 1.65542i | −1.00000 | −0.485175 | − | 0.173367i | −2.64436 | − | 0.0857253i | −1.16353 | −2.80683 | + | 1.05912i | 0.148731 | − | 0.257610i | ||||
256.7 | 0.154039 | − | 0.266804i | 1.05264 | + | 1.37548i | 0.952544 | + | 1.64985i | −1.00000 | 0.529131 | − | 0.0689719i | 2.20433 | − | 1.46319i | 1.20307 | −0.783880 | + | 2.89578i | −0.154039 | + | 0.266804i | ||||
256.8 | 0.304907 | − | 0.528114i | −1.22862 | − | 1.22086i | 0.814064 | + | 1.41000i | −1.00000 | −1.01937 | + | 0.276604i | 1.83653 | + | 1.90451i | 2.21248 | 0.0190166 | + | 2.99994i | −0.304907 | + | 0.528114i | ||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.k.b | ✓ | 24 |
3.b | odd | 2 | 1 | 945.2.k.b | 24 | ||
7.c | even | 3 | 1 | 315.2.l.b | yes | 24 | |
9.c | even | 3 | 1 | 315.2.l.b | yes | 24 | |
9.d | odd | 6 | 1 | 945.2.l.b | 24 | ||
21.h | odd | 6 | 1 | 945.2.l.b | 24 | ||
63.g | even | 3 | 1 | inner | 315.2.k.b | ✓ | 24 |
63.n | odd | 6 | 1 | 945.2.k.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.k.b | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
315.2.k.b | ✓ | 24 | 63.g | even | 3 | 1 | inner |
315.2.l.b | yes | 24 | 7.c | even | 3 | 1 | |
315.2.l.b | yes | 24 | 9.c | even | 3 | 1 | |
945.2.k.b | 24 | 3.b | odd | 2 | 1 | ||
945.2.k.b | 24 | 63.n | odd | 6 | 1 | ||
945.2.l.b | 24 | 9.d | odd | 6 | 1 | ||
945.2.l.b | 24 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + T_{2}^{23} + 16 T_{2}^{22} + 9 T_{2}^{21} + 157 T_{2}^{20} + 62 T_{2}^{19} + 930 T_{2}^{18} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).