Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(368,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 9, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.368");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.ca (of order \(12\), degree \(4\), not 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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(176\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 315) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
368.1 | −2.55373 | + | 0.684269i | 0 | 4.32125 | − | 2.49487i | −2.01493 | − | 0.969559i | 0 | −2.63527 | − | 0.235267i | −5.58921 | + | 5.58921i | 0 | 5.80903 | + | 1.09723i | ||||||
368.2 | −2.42876 | + | 0.650783i | 0 | 3.74329 | − | 2.16119i | 1.67326 | + | 1.48331i | 0 | 2.32199 | − | 1.26821i | −4.12912 | + | 4.12912i | 0 | −5.02925 | − | 2.51367i | ||||||
368.3 | −2.41263 | + | 0.646461i | 0 | 3.67080 | − | 2.11934i | 2.07655 | − | 0.829434i | 0 | −1.09925 | − | 2.40658i | −3.95386 | + | 3.95386i | 0 | −4.47373 | + | 3.34352i | ||||||
368.4 | −2.40632 | + | 0.644772i | 0 | 3.64261 | − | 2.10306i | −0.935544 | + | 2.03095i | 0 | −1.30393 | + | 2.30212i | −3.88619 | + | 3.88619i | 0 | 0.941721 | − | 5.49033i | ||||||
368.5 | −2.31583 | + | 0.620525i | 0 | 3.24597 | − | 1.87406i | −0.708123 | − | 2.12098i | 0 | 2.37173 | − | 1.17256i | −2.96360 | + | 2.96360i | 0 | 2.95602 | + | 4.47242i | ||||||
368.6 | −2.14761 | + | 0.575451i | 0 | 2.54905 | − | 1.47169i | 2.10613 | + | 0.751132i | 0 | 1.83889 | + | 1.90223i | −1.48316 | + | 1.48316i | 0 | −4.95540 | − | 0.401163i | ||||||
368.7 | −1.93179 | + | 0.517620i | 0 | 1.73181 | − | 0.999863i | 1.43910 | − | 1.71143i | 0 | 0.522555 | + | 2.59363i | 0.000387571 | 0 | 0.000387571i | 0 | −1.89416 | + | 4.05102i | ||||||
368.8 | −1.81917 | + | 0.487445i | 0 | 1.33973 | − | 0.773492i | −0.601260 | + | 2.15371i | 0 | −2.54492 | + | 0.723460i | 0.603293 | − | 0.603293i | 0 | 0.0439768 | − | 4.21106i | ||||||
368.9 | −1.77167 | + | 0.474718i | 0 | 1.18141 | − | 0.682088i | −1.69841 | + | 1.45445i | 0 | 2.23175 | − | 1.42102i | 0.824634 | − | 0.824634i | 0 | 2.31857 | − | 3.38306i | ||||||
368.10 | −1.57017 | + | 0.420725i | 0 | 0.556367 | − | 0.321219i | −2.22760 | − | 0.194383i | 0 | −0.460381 | − | 2.60539i | 1.56044 | − | 1.56044i | 0 | 3.57949 | − | 0.631994i | ||||||
368.11 | −1.56147 | + | 0.418395i | 0 | 0.531086 | − | 0.306623i | 2.02767 | + | 0.942623i | 0 | −1.29986 | − | 2.30442i | 1.58517 | − | 1.58517i | 0 | −3.56054 | − | 0.623511i | ||||||
368.12 | −1.47958 | + | 0.396452i | 0 | 0.299929 | − | 0.173164i | 1.06067 | − | 1.96850i | 0 | −2.39893 | − | 1.11585i | 1.79114 | − | 1.79114i | 0 | −0.788924 | + | 3.33305i | ||||||
368.13 | −1.39960 | + | 0.375022i | 0 | 0.0861894 | − | 0.0497615i | −2.22170 | − | 0.253107i | 0 | 1.21622 | + | 2.34964i | 1.94719 | − | 1.94719i | 0 | 3.20441 | − | 0.478936i | ||||||
368.14 | −1.23263 | + | 0.330283i | 0 | −0.321754 | + | 0.185765i | 1.27451 | + | 1.83729i | 0 | −1.80378 | + | 1.93556i | 2.13995 | − | 2.13995i | 0 | −2.17783 | − | 1.84375i | ||||||
368.15 | −0.971193 | + | 0.260230i | 0 | −0.856554 | + | 0.494532i | 0.675134 | − | 2.13171i | 0 | 2.27467 | − | 1.35126i | 2.12511 | − | 2.12511i | 0 | −0.100949 | + | 2.24599i | ||||||
368.16 | −0.657323 | + | 0.176129i | 0 | −1.33100 | + | 0.768453i | −0.742479 | + | 2.10920i | 0 | 2.39381 | + | 1.12679i | 1.70194 | − | 1.70194i | 0 | 0.116557 | − | 1.51720i | ||||||
368.17 | −0.652018 | + | 0.174708i | 0 | −1.33745 | + | 0.772175i | −1.39591 | − | 1.74683i | 0 | −2.59171 | − | 0.532031i | 1.69176 | − | 1.69176i | 0 | 1.21535 | + | 0.895092i | ||||||
368.18 | −0.576351 | + | 0.154433i | 0 | −1.42372 | + | 0.821985i | 0.594062 | − | 2.15571i | 0 | −0.0615728 | + | 2.64503i | 1.53746 | − | 1.53746i | 0 | −0.00947561 | + | 1.33419i | ||||||
368.19 | −0.534767 | + | 0.143291i | 0 | −1.46661 | + | 0.846746i | 1.50743 | + | 1.65156i | 0 | 1.59225 | + | 2.11299i | 1.44592 | − | 1.44592i | 0 | −1.04278 | − | 0.667199i | ||||||
368.20 | −0.112117 | + | 0.0300417i | 0 | −1.72038 | + | 0.993264i | −1.26170 | + | 1.84611i | 0 | −0.423666 | − | 2.61161i | 0.327196 | − | 0.327196i | 0 | 0.0859974 | − | 0.244884i | ||||||
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.n | odd | 6 | 1 | inner |
315.bx | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.ca.a | 176 | |
3.b | odd | 2 | 1 | 315.2.bx.a | yes | 176 | |
5.c | odd | 4 | 1 | inner | 945.2.ca.a | 176 | |
7.c | even | 3 | 1 | 945.2.by.a | 176 | ||
9.c | even | 3 | 1 | 315.2.bv.a | ✓ | 176 | |
9.d | odd | 6 | 1 | 945.2.by.a | 176 | ||
15.e | even | 4 | 1 | 315.2.bx.a | yes | 176 | |
21.h | odd | 6 | 1 | 315.2.bv.a | ✓ | 176 | |
35.l | odd | 12 | 1 | 945.2.by.a | 176 | ||
45.k | odd | 12 | 1 | 315.2.bv.a | ✓ | 176 | |
45.l | even | 12 | 1 | 945.2.by.a | 176 | ||
63.g | even | 3 | 1 | 315.2.bx.a | yes | 176 | |
63.n | odd | 6 | 1 | inner | 945.2.ca.a | 176 | |
105.x | even | 12 | 1 | 315.2.bv.a | ✓ | 176 | |
315.bx | even | 12 | 1 | inner | 945.2.ca.a | 176 | |
315.ch | odd | 12 | 1 | 315.2.bx.a | yes | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bv.a | ✓ | 176 | 9.c | even | 3 | 1 | |
315.2.bv.a | ✓ | 176 | 21.h | odd | 6 | 1 | |
315.2.bv.a | ✓ | 176 | 45.k | odd | 12 | 1 | |
315.2.bv.a | ✓ | 176 | 105.x | even | 12 | 1 | |
315.2.bx.a | yes | 176 | 3.b | odd | 2 | 1 | |
315.2.bx.a | yes | 176 | 15.e | even | 4 | 1 | |
315.2.bx.a | yes | 176 | 63.g | even | 3 | 1 | |
315.2.bx.a | yes | 176 | 315.ch | odd | 12 | 1 | |
945.2.by.a | 176 | 7.c | even | 3 | 1 | ||
945.2.by.a | 176 | 9.d | odd | 6 | 1 | ||
945.2.by.a | 176 | 35.l | odd | 12 | 1 | ||
945.2.by.a | 176 | 45.l | even | 12 | 1 | ||
945.2.ca.a | 176 | 1.a | even | 1 | 1 | trivial | |
945.2.ca.a | 176 | 5.c | odd | 4 | 1 | inner | |
945.2.ca.a | 176 | 63.n | odd | 6 | 1 | inner | |
945.2.ca.a | 176 | 315.bx | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).