Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,2,Mod(97,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("637.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 637.bd (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: | \(5.08647060876\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 91) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
97.1 | −0.573722 | − | 2.14116i | −0.959879 | − | 0.554186i | −2.52336 | + | 1.45686i | −2.14446 | − | 2.14446i | −0.635898 | + | 2.37321i | 0 | 1.43221 | + | 1.43221i | −0.885755 | − | 1.53417i | −3.36131 | + | 5.82195i | ||
97.2 | −0.360315 | − | 1.34471i | 1.25446 | + | 0.724265i | 0.0536242 | − | 0.0309600i | −0.470766 | − | 0.470766i | 0.521927 | − | 1.94786i | 0 | −2.02975 | − | 2.02975i | −0.450880 | − | 0.780947i | −0.463421 | + | 0.802669i | ||
97.3 | −0.0990633 | − | 0.369709i | −0.792292 | − | 0.457430i | 1.60518 | − | 0.926751i | 2.62172 | + | 2.62172i | −0.0906291 | + | 0.338232i | 0 | −1.04293 | − | 1.04293i | −1.08152 | − | 1.87324i | 0.709559 | − | 1.22899i | ||
97.4 | −0.0707322 | − | 0.263976i | −2.17048 | − | 1.25313i | 1.66737 | − | 0.962657i | −1.04692 | − | 1.04692i | −0.177273 | + | 0.661591i | 0 | −0.758543 | − | 0.758543i | 1.64065 | + | 2.84169i | −0.202311 | + | 0.350412i | ||
97.5 | 0.179714 | + | 0.670702i | 2.71085 | + | 1.56511i | 1.31451 | − | 0.758931i | −0.0263009 | − | 0.0263009i | −0.562545 | + | 2.09944i | 0 | 1.72723 | + | 1.72723i | 3.39913 | + | 5.88747i | 0.0129135 | − | 0.0223668i | ||
97.6 | 0.419805 | + | 1.56673i | 0.445073 | + | 0.256963i | −0.546366 | + | 0.315445i | 1.07946 | + | 1.07946i | −0.215749 | + | 0.805186i | 0 | 1.57027 | + | 1.57027i | −1.36794 | − | 2.36934i | −1.23806 | + | 2.14439i | ||
97.7 | 0.638288 | + | 2.38212i | 0.146239 | + | 0.0844309i | −3.53505 | + | 2.04096i | −1.74479 | − | 1.74479i | −0.107782 | + | 0.402250i | 0 | −3.63054 | − | 3.63054i | −1.48574 | − | 2.57338i | 3.04263 | − | 5.26998i | ||
293.1 | −2.06932 | + | 0.554474i | −0.0170886 | − | 0.00986613i | 2.24261 | − | 1.29477i | 0.984647 | − | 0.984647i | 0.0408324 | + | 0.0109410i | 0 | −0.893066 | + | 0.893066i | −1.49981 | − | 2.59774i | −1.49159 | + | 2.58351i | ||
293.2 | −1.75471 | + | 0.470172i | 2.65867 | + | 1.53499i | 1.12588 | − | 0.650030i | −0.787258 | + | 0.787258i | −5.38690 | − | 1.44342i | 0 | 0.899098 | − | 0.899098i | 3.21237 | + | 5.56398i | 1.01126 | − | 1.75155i | ||
293.3 | −0.278342 | + | 0.0745816i | −0.923129 | − | 0.532969i | −1.66014 | + | 0.958482i | −0.365574 | + | 0.365574i | 0.296695 | + | 0.0794993i | 0 | 0.798123 | − | 0.798123i | −0.931889 | − | 1.61408i | 0.0744896 | − | 0.129020i | ||
293.4 | 0.474142 | − | 0.127046i | 2.11812 | + | 1.22290i | −1.52338 | + | 0.879524i | 2.54420 | − | 2.54420i | 1.15965 | + | 0.310728i | 0 | −1.30475 | + | 1.30475i | 1.49096 | + | 2.58241i | 0.883082 | − | 1.52954i | ||
293.5 | 1.19734 | − | 0.320827i | −1.92717 | − | 1.11265i | −0.401352 | + | 0.231720i | 1.84228 | − | 1.84228i | −2.66446 | − | 0.713939i | 0 | −2.15924 | + | 2.15924i | 0.975997 | + | 1.69048i | 1.61479 | − | 2.79689i | ||
293.6 | 1.94658 | − | 0.521585i | −1.25027 | − | 0.721843i | 1.78508 | − | 1.03062i | −2.32001 | + | 2.32001i | −2.81025 | − | 0.753004i | 0 | 0.0872533 | − | 0.0872533i | −0.457887 | − | 0.793083i | −3.30601 | + | 5.72618i | ||
293.7 | 2.35033 | − | 0.629770i | 1.70689 | + | 0.985473i | 3.39540 | − | 1.96034i | −0.166234 | + | 0.166234i | 4.63238 | + | 1.24124i | 0 | 3.30464 | − | 3.30464i | 0.442313 | + | 0.766108i | −0.286016 | + | 0.495394i | ||
440.1 | −0.573722 | + | 2.14116i | −0.959879 | + | 0.554186i | −2.52336 | − | 1.45686i | −2.14446 | + | 2.14446i | −0.635898 | − | 2.37321i | 0 | 1.43221 | − | 1.43221i | −0.885755 | + | 1.53417i | −3.36131 | − | 5.82195i | ||
440.2 | −0.360315 | + | 1.34471i | 1.25446 | − | 0.724265i | 0.0536242 | + | 0.0309600i | −0.470766 | + | 0.470766i | 0.521927 | + | 1.94786i | 0 | −2.02975 | + | 2.02975i | −0.450880 | + | 0.780947i | −0.463421 | − | 0.802669i | ||
440.3 | −0.0990633 | + | 0.369709i | −0.792292 | + | 0.457430i | 1.60518 | + | 0.926751i | 2.62172 | − | 2.62172i | −0.0906291 | − | 0.338232i | 0 | −1.04293 | + | 1.04293i | −1.08152 | + | 1.87324i | 0.709559 | + | 1.22899i | ||
440.4 | −0.0707322 | + | 0.263976i | −2.17048 | + | 1.25313i | 1.66737 | + | 0.962657i | −1.04692 | + | 1.04692i | −0.177273 | − | 0.661591i | 0 | −0.758543 | + | 0.758543i | 1.64065 | − | 2.84169i | −0.202311 | − | 0.350412i | ||
440.5 | 0.179714 | − | 0.670702i | 2.71085 | − | 1.56511i | 1.31451 | + | 0.758931i | −0.0263009 | + | 0.0263009i | −0.562545 | − | 2.09944i | 0 | 1.72723 | − | 1.72723i | 3.39913 | − | 5.88747i | 0.0129135 | + | 0.0223668i | ||
440.6 | 0.419805 | − | 1.56673i | 0.445073 | − | 0.256963i | −0.546366 | − | 0.315445i | 1.07946 | − | 1.07946i | −0.215749 | − | 0.805186i | 0 | 1.57027 | − | 1.57027i | −1.36794 | + | 2.36934i | −1.23806 | − | 2.14439i | ||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 637.2.bd.b | 28 | |
7.b | odd | 2 | 1 | 637.2.bd.a | 28 | ||
7.c | even | 3 | 1 | 91.2.ba.a | yes | 28 | |
7.c | even | 3 | 1 | 637.2.x.a | 28 | ||
7.d | odd | 6 | 1 | 91.2.w.a | ✓ | 28 | |
7.d | odd | 6 | 1 | 637.2.bb.a | 28 | ||
13.f | odd | 12 | 1 | 637.2.bd.a | 28 | ||
21.g | even | 6 | 1 | 819.2.gh.b | 28 | ||
21.h | odd | 6 | 1 | 819.2.et.b | 28 | ||
91.w | even | 12 | 1 | 91.2.ba.a | yes | 28 | |
91.x | odd | 12 | 1 | 91.2.w.a | ✓ | 28 | |
91.ba | even | 12 | 1 | 637.2.x.a | 28 | ||
91.bc | even | 12 | 1 | inner | 637.2.bd.b | 28 | |
91.bd | odd | 12 | 1 | 637.2.bb.a | 28 | ||
273.bv | even | 12 | 1 | 819.2.gh.b | 28 | ||
273.ch | odd | 12 | 1 | 819.2.et.b | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.w.a | ✓ | 28 | 7.d | odd | 6 | 1 | |
91.2.w.a | ✓ | 28 | 91.x | odd | 12 | 1 | |
91.2.ba.a | yes | 28 | 7.c | even | 3 | 1 | |
91.2.ba.a | yes | 28 | 91.w | even | 12 | 1 | |
637.2.x.a | 28 | 7.c | even | 3 | 1 | ||
637.2.x.a | 28 | 91.ba | even | 12 | 1 | ||
637.2.bb.a | 28 | 7.d | odd | 6 | 1 | ||
637.2.bb.a | 28 | 91.bd | odd | 12 | 1 | ||
637.2.bd.a | 28 | 7.b | odd | 2 | 1 | ||
637.2.bd.a | 28 | 13.f | odd | 12 | 1 | ||
637.2.bd.b | 28 | 1.a | even | 1 | 1 | trivial | |
637.2.bd.b | 28 | 91.bc | even | 12 | 1 | inner | |
819.2.et.b | 28 | 21.h | odd | 6 | 1 | ||
819.2.et.b | 28 | 273.ch | odd | 12 | 1 | ||
819.2.gh.b | 28 | 21.g | even | 6 | 1 | ||
819.2.gh.b | 28 | 273.bv | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):
\( T_{2}^{28} - 4 T_{2}^{27} + 5 T_{2}^{26} - 43 T_{2}^{24} + 134 T_{2}^{23} - 102 T_{2}^{22} + 972 T_{2}^{20} - 3530 T_{2}^{19} + 2728 T_{2}^{18} + 2302 T_{2}^{17} - 11391 T_{2}^{16} + 26088 T_{2}^{15} - 12471 T_{2}^{14} - 14534 T_{2}^{13} + \cdots + 9 \) |
\( T_{3}^{28} - 6 T_{3}^{27} - 6 T_{3}^{26} + 108 T_{3}^{25} - 12 T_{3}^{24} - 1212 T_{3}^{23} + 886 T_{3}^{22} + 8688 T_{3}^{21} - 7718 T_{3}^{20} - 45780 T_{3}^{19} + 41579 T_{3}^{18} + 176952 T_{3}^{17} - 125505 T_{3}^{16} - 511386 T_{3}^{15} + \cdots + 1 \) |