Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(157,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([4, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.cg (of order \(12\), degree \(4\), 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: | \(160\) |
Relative dimension: | \(40\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 | −2.59469 | + | 0.695245i | 0.350395 | − | 1.69624i | 4.51700 | − | 2.60789i | −0.619852 | + | 2.14844i | 0.270133 | + | 4.64482i | 1.50779 | + | 2.17407i | −6.10819 | + | 6.10819i | −2.75445 | − | 1.18871i | 0.114633 | − | 6.00548i |
157.2 | −2.57821 | + | 0.690830i | 1.61836 | + | 0.617171i | 4.43789 | − | 2.56221i | −2.21752 | − | 0.287408i | −4.59885 | − | 0.473186i | −1.61180 | − | 2.09812i | −5.89699 | + | 5.89699i | 2.23820 | + | 1.99762i | 5.91579 | − | 0.790931i |
157.3 | −2.57260 | + | 0.689326i | −1.64466 | + | 0.543229i | 4.41104 | − | 2.54672i | 0.495489 | − | 2.18048i | 3.85658 | − | 2.53122i | 2.23839 | − | 1.41053i | −5.82578 | + | 5.82578i | 2.40980 | − | 1.78685i | 0.228366 | + | 5.95105i |
157.4 | −2.33005 | + | 0.624336i | 1.68393 | + | 0.405458i | 3.30730 | − | 1.90947i | 2.23418 | − | 0.0919089i | −4.17678 | + | 0.106596i | 2.03168 | + | 1.69478i | −3.10261 | + | 3.10261i | 2.67121 | + | 1.36552i | −5.14837 | + | 1.60903i |
157.5 | −2.27594 | + | 0.609836i | −1.68624 | − | 0.395739i | 3.07594 | − | 1.77589i | 0.427382 | + | 2.19484i | 4.07910 | − | 0.127650i | −2.23213 | − | 1.42042i | −2.58544 | + | 2.58544i | 2.68678 | + | 1.33462i | −2.31119 | − | 4.73470i |
157.6 | −2.17690 | + | 0.583298i | −0.853583 | + | 1.50712i | 2.66660 | − | 1.53956i | −2.19471 | + | 0.428081i | 0.979065 | − | 3.77873i | −1.03156 | + | 2.43637i | −1.71970 | + | 1.71970i | −1.54279 | − | 2.57289i | 4.52796 | − | 2.21206i |
157.7 | −1.97684 | + | 0.529692i | −1.08684 | − | 1.34862i | 1.89527 | − | 1.09423i | 1.45689 | − | 1.69631i | 2.86285 | + | 2.09032i | −0.793015 | + | 2.52411i | −0.272736 | + | 0.272736i | −0.637576 | + | 2.93147i | −1.98151 | + | 4.12504i |
157.8 | −1.89474 | + | 0.507694i | −0.298455 | − | 1.70614i | 1.60024 | − | 0.923896i | −1.92207 | − | 1.14265i | 1.43169 | + | 3.08117i | 0.974477 | − | 2.45976i | 0.211119 | − | 0.211119i | −2.82185 | + | 1.01841i | 4.22194 | + | 1.18920i |
157.9 | −1.84711 | + | 0.494931i | 1.20304 | − | 1.24607i | 1.43480 | − | 0.828381i | 2.21218 | + | 0.325952i | −1.60543 | + | 2.89705i | −1.00496 | − | 2.44746i | 0.464119 | − | 0.464119i | −0.105377 | − | 2.99815i | −4.24746 | + | 0.492809i |
157.10 | −1.58439 | + | 0.424536i | 0.483103 | + | 1.66331i | 0.598007 | − | 0.345260i | −0.829835 | − | 2.07638i | −1.47156 | − | 2.43024i | 2.60741 | + | 0.448788i | 1.51881 | − | 1.51881i | −2.53322 | + | 1.60710i | 2.19628 | + | 2.93751i |
157.11 | −1.35792 | + | 0.363854i | −1.38606 | + | 1.03867i | −0.0204859 | + | 0.0118276i | 1.77376 | + | 1.36153i | 1.50424 | − | 1.91476i | 2.14673 | + | 1.54646i | 2.01165 | − | 2.01165i | 0.842327 | − | 2.87932i | −2.90403 | − | 1.20346i |
157.12 | −1.33121 | + | 0.356696i | 1.23400 | + | 1.21542i | −0.0871697 | + | 0.0503274i | 0.497684 | + | 2.17998i | −2.07625 | − | 1.17781i | −2.13219 | + | 1.56645i | 2.04711 | − | 2.04711i | 0.0455283 | + | 2.99965i | −1.44011 | − | 2.72448i |
157.13 | −1.30553 | + | 0.349816i | 1.70101 | − | 0.326450i | −0.150007 | + | 0.0866067i | −1.14755 | + | 1.91915i | −2.10652 | + | 1.02123i | 2.22736 | − | 1.42789i | 2.07698 | − | 2.07698i | 2.78686 | − | 1.11059i | 0.826810 | − | 2.90694i |
157.14 | −1.21428 | + | 0.325365i | −0.985125 | + | 1.42462i | −0.363441 | + | 0.209833i | 2.19872 | − | 0.406980i | 0.732697 | − | 2.05040i | −1.07100 | − | 2.41929i | 2.15087 | − | 2.15087i | −1.05906 | − | 2.80685i | −2.53744 | + | 1.20957i |
157.15 | −1.02124 | + | 0.273642i | −1.58003 | − | 0.709585i | −0.763991 | + | 0.441090i | −1.46795 | + | 1.68675i | 1.80777 | + | 0.292298i | 2.16930 | + | 1.51465i | 2.15473 | − | 2.15473i | 1.99298 | + | 2.24233i | 1.03757 | − | 2.12428i |
157.16 | −1.01365 | + | 0.271606i | −1.72630 | − | 0.141011i | −0.778338 | + | 0.449374i | −1.53386 | − | 1.62705i | 1.78816 | − | 0.325938i | −2.60738 | − | 0.448985i | 2.15099 | − | 2.15099i | 2.96023 | + | 0.486856i | 1.99671 | + | 1.23265i |
157.17 | −0.649326 | + | 0.173986i | −0.195991 | − | 1.72093i | −1.34070 | + | 0.774052i | 1.84508 | + | 1.26321i | 0.426680 | + | 1.08334i | −2.09066 | + | 1.62146i | 1.68656 | − | 1.68656i | −2.92317 | + | 0.674573i | −1.41784 | − | 0.499216i |
157.18 | −0.409914 | + | 0.109836i | 1.73004 | + | 0.0834602i | −1.57608 | + | 0.909953i | 0.861800 | − | 2.06332i | −0.718335 | + | 0.155809i | −2.64547 | − | 0.0385118i | 1.14627 | − | 1.14627i | 2.98607 | + | 0.288779i | −0.126636 | + | 0.940443i |
157.19 | −0.0182967 | + | 0.00490258i | −0.782329 | − | 1.54530i | −1.73174 | + | 0.999821i | 0.587591 | + | 2.15748i | 0.0218900 | + | 0.0244385i | 0.705053 | − | 2.55008i | 0.0535716 | − | 0.0535716i | −1.77592 | + | 2.41787i | −0.0213282 | − | 0.0365941i |
157.20 | 0.112795 | − | 0.0302234i | −0.803249 | + | 1.53453i | −1.72024 | + | 0.993182i | 0.634953 | − | 2.14402i | −0.0442239 | + | 0.197365i | −0.960898 | + | 2.46509i | −0.329161 | + | 0.329161i | −1.70958 | − | 2.46523i | 0.00682007 | − | 0.261026i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
63.k | odd | 6 | 1 | inner |
315.cg | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.cg.e | yes | 160 |
3.b | odd | 2 | 1 | 945.2.cj.e | 160 | ||
5.c | odd | 4 | 1 | inner | 315.2.cg.e | yes | 160 |
7.d | odd | 6 | 1 | 315.2.bs.e | ✓ | 160 | |
9.c | even | 3 | 1 | 315.2.bs.e | ✓ | 160 | |
9.d | odd | 6 | 1 | 945.2.bv.e | 160 | ||
15.e | even | 4 | 1 | 945.2.cj.e | 160 | ||
21.g | even | 6 | 1 | 945.2.bv.e | 160 | ||
35.k | even | 12 | 1 | 315.2.bs.e | ✓ | 160 | |
45.k | odd | 12 | 1 | 315.2.bs.e | ✓ | 160 | |
45.l | even | 12 | 1 | 945.2.bv.e | 160 | ||
63.k | odd | 6 | 1 | inner | 315.2.cg.e | yes | 160 |
63.s | even | 6 | 1 | 945.2.cj.e | 160 | ||
105.w | odd | 12 | 1 | 945.2.bv.e | 160 | ||
315.bw | odd | 12 | 1 | 945.2.cj.e | 160 | ||
315.cg | even | 12 | 1 | inner | 315.2.cg.e | yes | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bs.e | ✓ | 160 | 7.d | odd | 6 | 1 | |
315.2.bs.e | ✓ | 160 | 9.c | even | 3 | 1 | |
315.2.bs.e | ✓ | 160 | 35.k | even | 12 | 1 | |
315.2.bs.e | ✓ | 160 | 45.k | odd | 12 | 1 | |
315.2.cg.e | yes | 160 | 1.a | even | 1 | 1 | trivial |
315.2.cg.e | yes | 160 | 5.c | odd | 4 | 1 | inner |
315.2.cg.e | yes | 160 | 63.k | odd | 6 | 1 | inner |
315.2.cg.e | yes | 160 | 315.cg | even | 12 | 1 | inner |
945.2.bv.e | 160 | 9.d | odd | 6 | 1 | ||
945.2.bv.e | 160 | 21.g | even | 6 | 1 | ||
945.2.bv.e | 160 | 45.l | even | 12 | 1 | ||
945.2.bv.e | 160 | 105.w | odd | 12 | 1 | ||
945.2.cj.e | 160 | 3.b | odd | 2 | 1 | ||
945.2.cj.e | 160 | 15.e | even | 4 | 1 | ||
945.2.cj.e | 160 | 63.s | even | 6 | 1 | ||
945.2.cj.e | 160 | 315.bw | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):
\( T_{2}^{160} + 2 T_{2}^{159} + 2 T_{2}^{158} + 12 T_{2}^{157} - 277 T_{2}^{156} - 596 T_{2}^{155} + \cdots + 294499921 \) |
\( T_{11}^{40} - 8 T_{11}^{39} - 194 T_{11}^{38} + 1652 T_{11}^{37} + 16475 T_{11}^{36} + \cdots + 4204122386796 \) |