Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,3,Mod(107,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.107");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.l (of order \(4\), 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: | \(8.17440793081\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(32\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −1.99335 | + | 0.162980i | −2.88075 | + | 0.837414i | 3.94688 | − | 0.649751i | 0 | 5.60586 | − | 2.13876i | −1.76134 | + | 1.76134i | −7.76160 | + | 1.93844i | 7.59748 | − | 4.82477i | 0 | ||||
107.2 | −1.99335 | + | 0.162980i | −0.837414 | + | 2.88075i | 3.94688 | − | 0.649751i | 0 | 1.19975 | − | 5.87883i | 1.76134 | − | 1.76134i | −7.76160 | + | 1.93844i | −7.59748 | − | 4.82477i | 0 | ||||
107.3 | −1.95438 | − | 0.424734i | 1.76787 | − | 2.42377i | 3.63920 | + | 1.66018i | 0 | −4.48454 | + | 3.98609i | 8.59996 | − | 8.59996i | −6.40725 | − | 4.79032i | −2.74930 | − | 8.56979i | 0 | ||||
107.4 | −1.95438 | − | 0.424734i | 2.42377 | − | 1.76787i | 3.63920 | + | 1.66018i | 0 | −5.48784 | + | 2.42562i | −8.59996 | + | 8.59996i | −6.40725 | − | 4.79032i | 2.74930 | − | 8.56979i | 0 | ||||
107.5 | −1.72739 | − | 1.00803i | −2.43345 | − | 1.75452i | 1.96775 | + | 3.48252i | 0 | 2.43491 | + | 5.48372i | 3.08744 | − | 3.08744i | 0.111402 | − | 7.99922i | 2.84335 | + | 8.53905i | 0 | ||||
107.6 | −1.72739 | − | 1.00803i | 1.75452 | + | 2.43345i | 1.96775 | + | 3.48252i | 0 | −0.577744 | − | 5.97212i | −3.08744 | + | 3.08744i | 0.111402 | − | 7.99922i | −2.84335 | + | 8.53905i | 0 | ||||
107.7 | −1.45557 | + | 1.37162i | −0.658656 | − | 2.92680i | 0.237343 | − | 3.99295i | 0 | 4.97316 | + | 3.35673i | −4.73351 | + | 4.73351i | 5.13133 | + | 6.13755i | −8.13235 | + | 3.85551i | 0 | ||||
107.8 | −1.45557 | + | 1.37162i | 2.92680 | + | 0.658656i | 0.237343 | − | 3.99295i | 0 | −5.16358 | + | 3.05573i | 4.73351 | − | 4.73351i | 5.13133 | + | 6.13755i | 8.13235 | + | 3.85551i | 0 | ||||
107.9 | −1.37162 | + | 1.45557i | −2.92680 | − | 0.658656i | −0.237343 | − | 3.99295i | 0 | 4.97316 | − | 3.35673i | −4.73351 | + | 4.73351i | 6.13755 | + | 5.13133i | 8.13235 | + | 3.85551i | 0 | ||||
107.10 | −1.37162 | + | 1.45557i | 0.658656 | + | 2.92680i | −0.237343 | − | 3.99295i | 0 | −5.16358 | − | 3.05573i | 4.73351 | − | 4.73351i | 6.13755 | + | 5.13133i | −8.13235 | + | 3.85551i | 0 | ||||
107.11 | −1.00803 | − | 1.72739i | −2.43345 | − | 1.75452i | −1.96775 | + | 3.48252i | 0 | −0.577744 | + | 5.97212i | 3.08744 | − | 3.08744i | 7.99922 | − | 0.111402i | 2.84335 | + | 8.53905i | 0 | ||||
107.12 | −1.00803 | − | 1.72739i | 1.75452 | + | 2.43345i | −1.96775 | + | 3.48252i | 0 | 2.43491 | − | 5.48372i | −3.08744 | + | 3.08744i | 7.99922 | − | 0.111402i | −2.84335 | + | 8.53905i | 0 | ||||
107.13 | −0.424734 | − | 1.95438i | 1.76787 | − | 2.42377i | −3.63920 | + | 1.66018i | 0 | −5.48784 | − | 2.42562i | 8.59996 | − | 8.59996i | 4.79032 | + | 6.40725i | −2.74930 | − | 8.56979i | 0 | ||||
107.14 | −0.424734 | − | 1.95438i | 2.42377 | − | 1.76787i | −3.63920 | + | 1.66018i | 0 | −4.48454 | − | 3.98609i | −8.59996 | + | 8.59996i | 4.79032 | + | 6.40725i | 2.74930 | − | 8.56979i | 0 | ||||
107.15 | −0.162980 | + | 1.99335i | 0.837414 | − | 2.88075i | −3.94688 | − | 0.649751i | 0 | 5.60586 | + | 2.13876i | −1.76134 | + | 1.76134i | 1.93844 | − | 7.76160i | −7.59748 | − | 4.82477i | 0 | ||||
107.16 | −0.162980 | + | 1.99335i | 2.88075 | − | 0.837414i | −3.94688 | − | 0.649751i | 0 | 1.19975 | + | 5.87883i | 1.76134 | − | 1.76134i | 1.93844 | − | 7.76160i | 7.59748 | − | 4.82477i | 0 | ||||
107.17 | 0.162980 | − | 1.99335i | −2.88075 | + | 0.837414i | −3.94688 | − | 0.649751i | 0 | 1.19975 | + | 5.87883i | −1.76134 | + | 1.76134i | −1.93844 | + | 7.76160i | 7.59748 | − | 4.82477i | 0 | ||||
107.18 | 0.162980 | − | 1.99335i | −0.837414 | + | 2.88075i | −3.94688 | − | 0.649751i | 0 | 5.60586 | + | 2.13876i | 1.76134 | − | 1.76134i | −1.93844 | + | 7.76160i | −7.59748 | − | 4.82477i | 0 | ||||
107.19 | 0.424734 | + | 1.95438i | −2.42377 | + | 1.76787i | −3.63920 | + | 1.66018i | 0 | −4.48454 | − | 3.98609i | 8.59996 | − | 8.59996i | −4.79032 | − | 6.40725i | 2.74930 | − | 8.56979i | 0 | ||||
107.20 | 0.424734 | + | 1.95438i | −1.76787 | + | 2.42377i | −3.63920 | + | 1.66018i | 0 | −5.48784 | − | 2.42562i | −8.59996 | + | 8.59996i | −4.79032 | − | 6.40725i | −2.74930 | − | 8.56979i | 0 | ||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
12.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
20.d | odd | 2 | 1 | inner |
20.e | even | 4 | 2 | inner |
60.h | even | 2 | 1 | inner |
60.l | odd | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.3.l.h | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
4.b | odd | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
5.b | even | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
5.c | odd | 4 | 2 | inner | 300.3.l.h | ✓ | 64 |
12.b | even | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
15.d | odd | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
15.e | even | 4 | 2 | inner | 300.3.l.h | ✓ | 64 |
20.d | odd | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
20.e | even | 4 | 2 | inner | 300.3.l.h | ✓ | 64 |
60.h | even | 2 | 1 | inner | 300.3.l.h | ✓ | 64 |
60.l | odd | 4 | 2 | inner | 300.3.l.h | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.3.l.h | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
300.3.l.h | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 4.b | odd | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 5.b | even | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 5.c | odd | 4 | 2 | inner |
300.3.l.h | ✓ | 64 | 12.b | even | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 15.d | odd | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 15.e | even | 4 | 2 | inner |
300.3.l.h | ✓ | 64 | 20.d | odd | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 20.e | even | 4 | 2 | inner |
300.3.l.h | ✓ | 64 | 60.h | even | 2 | 1 | inner |
300.3.l.h | ✓ | 64 | 60.l | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):
\( T_{7}^{16} + 24290T_{7}^{12} + 53553793T_{7}^{8} + 17995252896T_{7}^{4} + 614781446400 \) |
\( T_{17}^{16} + 597042T_{17}^{12} + 109730300913T_{17}^{8} + 5774819750046144T_{17}^{4} + 313717924683878400 \) |
\( T_{19}^{8} - 2024T_{19}^{6} + 1426642T_{19}^{4} - 416756448T_{19}^{2} + 43164127845 \) |