Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,2,Mod(37,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 112.w (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: | \(0.894324502638\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.40462 | + | 0.164410i | −0.839165 | − | 0.224854i | 1.94594 | − | 0.461868i | −3.16320 | + | 0.847576i | 1.21568 | + | 0.177868i | 0.654939 | + | 2.56341i | −2.65738 | + | 0.968682i | −1.94444 | − | 1.12262i | 4.30375 | − | 1.71059i |
37.2 | −1.36184 | − | 0.381311i | −1.60845 | − | 0.430984i | 1.70920 | + | 1.03857i | 1.94900 | − | 0.522232i | 2.02612 | + | 1.20025i | −1.89075 | − | 1.85069i | −1.93164 | − | 2.06610i | −0.196696 | − | 0.113563i | −2.85335 | − | 0.0319777i |
37.3 | −1.25281 | + | 0.656100i | 1.95279 | + | 0.523249i | 1.13907 | − | 1.64394i | 0.959072 | − | 0.256983i | −2.78978 | + | 0.625694i | 0.292831 | − | 2.62950i | −0.348448 | + | 2.80688i | 0.941530 | + | 0.543593i | −1.03293 | + | 0.951197i |
37.4 | −0.892252 | + | 1.09722i | −2.20008 | − | 0.589510i | −0.407774 | − | 1.95799i | 2.32205 | − | 0.622192i | 2.60985 | − | 1.88798i | 2.41058 | + | 1.09047i | 2.51218 | + | 1.29960i | 1.89476 | + | 1.09394i | −1.38918 | + | 3.10295i |
37.5 | −0.805226 | − | 1.16259i | 1.16533 | + | 0.312249i | −0.703221 | + | 1.87229i | 0.980942 | − | 0.262843i | −0.575337 | − | 1.60623i | 2.60251 | + | 0.476386i | 2.74296 | − | 0.690063i | −1.33758 | − | 0.772254i | −1.09546 | − | 0.928784i |
37.6 | −0.246810 | + | 1.39251i | 2.51077 | + | 0.672759i | −1.87817 | − | 0.687371i | −2.91203 | + | 0.780276i | −1.55651 | + | 3.33023i | 1.41955 | + | 2.23268i | 1.42072 | − | 2.44572i | 3.25328 | + | 1.87828i | −0.367824 | − | 4.24761i |
37.7 | 0.236566 | + | 1.39429i | 0.827840 | + | 0.221819i | −1.88807 | + | 0.659683i | 4.10878 | − | 1.10095i | −0.113440 | + | 1.20672i | −2.50325 | + | 0.856573i | −1.36644 | − | 2.47646i | −1.96196 | − | 1.13274i | 2.50703 | + | 5.46838i |
37.8 | 0.262311 | − | 1.38967i | 2.91496 | + | 0.781062i | −1.86239 | − | 0.729053i | −0.745506 | + | 0.199758i | 1.85005 | − | 3.84597i | −2.51194 | − | 0.830765i | −1.50167 | + | 2.39687i | 5.28887 | + | 3.05353i | 0.0820438 | + | 1.08841i |
37.9 | 0.713936 | + | 1.22078i | −3.04092 | − | 0.814813i | −0.980592 | + | 1.74311i | −1.87270 | + | 0.501787i | −1.17632 | − | 4.29401i | −1.89831 | + | 1.84294i | −2.82803 | + | 0.0473857i | 5.98523 | + | 3.45557i | −1.94955 | − | 1.92790i |
37.10 | 0.945375 | − | 1.05179i | −0.0543752 | − | 0.0145698i | −0.212530 | − | 1.98868i | 1.25781 | − | 0.337028i | −0.0667293 | + | 0.0434174i | 0.230738 | + | 2.63567i | −2.29259 | − | 1.65651i | −2.59533 | − | 1.49842i | 0.834616 | − | 1.64157i |
37.11 | 1.39181 | − | 0.250740i | −2.46036 | − | 0.659252i | 1.87426 | − | 0.697963i | 1.87532 | − | 0.502490i | −3.58965 | − | 0.300642i | 0.364450 | − | 2.62053i | 2.43360 | − | 1.44138i | 3.02070 | + | 1.74400i | 2.48409 | − | 1.16959i |
37.12 | 1.41356 | + | 0.0428532i | 0.831674 | + | 0.222846i | 1.99633 | + | 0.121151i | −2.02749 | + | 0.543265i | 1.16607 | + | 0.350647i | −2.63544 | + | 0.233350i | 2.81674 | + | 0.256804i | −1.95606 | − | 1.12933i | −2.88927 | + | 0.681055i |
53.1 | −1.41419 | + | 0.00789795i | 0.814813 | + | 3.04092i | 1.99988 | − | 0.0223384i | 0.501787 | − | 1.87270i | −1.17632 | − | 4.29401i | 1.89831 | + | 1.84294i | −2.82803 | + | 0.0473857i | −5.98523 | + | 3.45557i | −0.694833 | + | 2.65231i |
53.2 | −1.32577 | − | 0.492271i | −0.221819 | − | 0.827840i | 1.51534 | + | 1.30528i | −1.10095 | + | 4.10878i | −0.113440 | + | 1.20672i | 2.50325 | + | 0.856573i | −1.36644 | − | 2.47646i | 1.96196 | − | 1.13274i | 3.48224 | − | 4.90534i |
53.3 | −1.08254 | − | 0.909999i | −0.672759 | − | 2.51077i | 0.343804 | + | 1.97023i | 0.780276 | − | 2.91203i | −1.55651 | + | 3.33023i | −1.41955 | + | 2.23268i | 1.42072 | − | 2.44572i | −3.25328 | + | 1.87828i | −3.49463 | + | 2.44235i |
53.4 | −0.743894 | + | 1.20276i | −0.222846 | − | 0.831674i | −0.893243 | − | 1.78945i | 0.543265 | − | 2.02749i | 1.16607 | + | 0.350647i | 2.63544 | + | 0.233350i | 2.81674 | + | 0.256804i | 1.95606 | − | 1.12933i | 2.03445 | + | 2.16165i |
53.5 | −0.504093 | − | 1.32132i | 0.589510 | + | 2.20008i | −1.49178 | + | 1.33214i | −0.622192 | + | 2.32205i | 2.60985 | − | 1.88798i | −2.41058 | + | 1.09047i | 2.51218 | + | 1.29960i | −1.89476 | + | 1.09394i | 3.38182 | − | 0.348414i |
53.6 | −0.478757 | + | 1.33071i | 0.659252 | + | 2.46036i | −1.54158 | − | 1.27417i | −0.502490 | + | 1.87532i | −3.58965 | − | 0.300642i | −0.364450 | − | 2.62053i | 2.43360 | − | 1.44138i | −3.02070 | + | 1.74400i | −2.25494 | − | 1.56649i |
53.7 | 0.0582062 | − | 1.41302i | −0.523249 | − | 1.95279i | −1.99322 | − | 0.164492i | −0.256983 | + | 0.959072i | −2.78978 | + | 0.625694i | −0.292831 | − | 2.62950i | −0.348448 | + | 2.80688i | −0.941530 | + | 0.543593i | 1.34023 | + | 0.418944i |
53.8 | 0.438190 | + | 1.34461i | 0.0145698 | + | 0.0543752i | −1.61598 | + | 1.17839i | −0.337028 | + | 1.25781i | −0.0667293 | + | 0.0434174i | −0.230738 | + | 2.63567i | −2.29259 | − | 1.65651i | 2.59533 | − | 1.49842i | −1.83895 | + | 0.0979855i |
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
16.e | even | 4 | 1 | inner |
112.w | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 112.2.w.c | ✓ | 48 |
4.b | odd | 2 | 1 | 448.2.ba.c | 48 | ||
7.b | odd | 2 | 1 | 784.2.x.o | 48 | ||
7.c | even | 3 | 1 | inner | 112.2.w.c | ✓ | 48 |
7.c | even | 3 | 1 | 784.2.m.j | 24 | ||
7.d | odd | 6 | 1 | 784.2.m.k | 24 | ||
7.d | odd | 6 | 1 | 784.2.x.o | 48 | ||
8.b | even | 2 | 1 | 896.2.ba.f | 48 | ||
8.d | odd | 2 | 1 | 896.2.ba.e | 48 | ||
16.e | even | 4 | 1 | inner | 112.2.w.c | ✓ | 48 |
16.e | even | 4 | 1 | 896.2.ba.f | 48 | ||
16.f | odd | 4 | 1 | 448.2.ba.c | 48 | ||
16.f | odd | 4 | 1 | 896.2.ba.e | 48 | ||
28.g | odd | 6 | 1 | 448.2.ba.c | 48 | ||
56.k | odd | 6 | 1 | 896.2.ba.e | 48 | ||
56.p | even | 6 | 1 | 896.2.ba.f | 48 | ||
112.l | odd | 4 | 1 | 784.2.x.o | 48 | ||
112.u | odd | 12 | 1 | 448.2.ba.c | 48 | ||
112.u | odd | 12 | 1 | 896.2.ba.e | 48 | ||
112.w | even | 12 | 1 | inner | 112.2.w.c | ✓ | 48 |
112.w | even | 12 | 1 | 784.2.m.j | 24 | ||
112.w | even | 12 | 1 | 896.2.ba.f | 48 | ||
112.x | odd | 12 | 1 | 784.2.m.k | 24 | ||
112.x | odd | 12 | 1 | 784.2.x.o | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
112.2.w.c | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
112.2.w.c | ✓ | 48 | 7.c | even | 3 | 1 | inner |
112.2.w.c | ✓ | 48 | 16.e | even | 4 | 1 | inner |
112.2.w.c | ✓ | 48 | 112.w | even | 12 | 1 | inner |
448.2.ba.c | 48 | 4.b | odd | 2 | 1 | ||
448.2.ba.c | 48 | 16.f | odd | 4 | 1 | ||
448.2.ba.c | 48 | 28.g | odd | 6 | 1 | ||
448.2.ba.c | 48 | 112.u | odd | 12 | 1 | ||
784.2.m.j | 24 | 7.c | even | 3 | 1 | ||
784.2.m.j | 24 | 112.w | even | 12 | 1 | ||
784.2.m.k | 24 | 7.d | odd | 6 | 1 | ||
784.2.m.k | 24 | 112.x | odd | 12 | 1 | ||
784.2.x.o | 48 | 7.b | odd | 2 | 1 | ||
784.2.x.o | 48 | 7.d | odd | 6 | 1 | ||
784.2.x.o | 48 | 112.l | odd | 4 | 1 | ||
784.2.x.o | 48 | 112.x | odd | 12 | 1 | ||
896.2.ba.e | 48 | 8.d | odd | 2 | 1 | ||
896.2.ba.e | 48 | 16.f | odd | 4 | 1 | ||
896.2.ba.e | 48 | 56.k | odd | 6 | 1 | ||
896.2.ba.e | 48 | 112.u | odd | 12 | 1 | ||
896.2.ba.f | 48 | 8.b | even | 2 | 1 | ||
896.2.ba.f | 48 | 16.e | even | 4 | 1 | ||
896.2.ba.f | 48 | 56.p | even | 6 | 1 | ||
896.2.ba.f | 48 | 112.w | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 8 T_{3}^{45} - 162 T_{3}^{44} - 24 T_{3}^{43} + 32 T_{3}^{42} - 1116 T_{3}^{41} + \cdots + 194481 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\).