Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4375,2,Mod(1,4375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4375, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4375.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4375 = 5^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4375.a (trivial) |
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: | yes |
Analytic conductor: | \(34.9345508843\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Twist minimal: | no (minimal twist has level 175) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.39162 | 2.75183 | 3.71983 | 0 | −6.58132 | −1.00000 | −4.11316 | 4.57256 | 0 | ||||||||||||||||||
1.2 | −2.25420 | −0.437412 | 3.08142 | 0 | 0.986013 | −1.00000 | −2.43773 | −2.80867 | 0 | ||||||||||||||||||
1.3 | −2.22283 | 0.876253 | 2.94096 | 0 | −1.94776 | −1.00000 | −2.09160 | −2.23218 | 0 | ||||||||||||||||||
1.4 | −2.15723 | −0.859279 | 2.65366 | 0 | 1.85367 | −1.00000 | −1.41009 | −2.26164 | 0 | ||||||||||||||||||
1.5 | −1.59572 | 2.59518 | 0.546330 | 0 | −4.14119 | −1.00000 | 2.31965 | 3.73495 | 0 | ||||||||||||||||||
1.6 | −1.50576 | −0.541589 | 0.267313 | 0 | 0.815502 | −1.00000 | 2.60901 | −2.70668 | 0 | ||||||||||||||||||
1.7 | −1.14814 | 2.58276 | −0.681763 | 0 | −2.96538 | −1.00000 | 3.07905 | 3.67063 | 0 | ||||||||||||||||||
1.8 | −1.08354 | −2.69351 | −0.825936 | 0 | 2.91854 | −1.00000 | 3.06202 | 4.25502 | 0 | ||||||||||||||||||
1.9 | −0.953107 | −1.09278 | −1.09159 | 0 | 1.04154 | −1.00000 | 2.94661 | −1.80582 | 0 | ||||||||||||||||||
1.10 | −0.775732 | 1.73672 | −1.39824 | 0 | −1.34723 | −1.00000 | 2.63612 | 0.0162108 | 0 | ||||||||||||||||||
1.11 | −0.407806 | 0.904287 | −1.83369 | 0 | −0.368774 | −1.00000 | 1.56340 | −2.18226 | 0 | ||||||||||||||||||
1.12 | −0.392266 | −2.78480 | −1.84613 | 0 | 1.09238 | −1.00000 | 1.50871 | 4.75511 | 0 | ||||||||||||||||||
1.13 | 0.198876 | 1.54520 | −1.96045 | 0 | 0.307303 | −1.00000 | −0.787638 | −0.612360 | 0 | ||||||||||||||||||
1.14 | 0.237222 | −2.06682 | −1.94373 | 0 | −0.490294 | −1.00000 | −0.935537 | 1.27174 | 0 | ||||||||||||||||||
1.15 | 0.429002 | 0.963349 | −1.81596 | 0 | 0.413279 | −1.00000 | −1.63705 | −2.07196 | 0 | ||||||||||||||||||
1.16 | 0.840072 | −0.242722 | −1.29428 | 0 | −0.203904 | −1.00000 | −2.76743 | −2.94109 | 0 | ||||||||||||||||||
1.17 | 1.02774 | 2.86976 | −0.943758 | 0 | 2.94936 | −1.00000 | −3.02541 | 5.23555 | 0 | ||||||||||||||||||
1.18 | 1.08158 | 3.19566 | −0.830195 | 0 | 3.45635 | −1.00000 | −3.06107 | 7.21224 | 0 | ||||||||||||||||||
1.19 | 1.18550 | −1.73464 | −0.594585 | 0 | −2.05642 | −1.00000 | −3.07589 | 0.00897058 | 0 | ||||||||||||||||||
1.20 | 1.25752 | −0.637250 | −0.418653 | 0 | −0.801353 | −1.00000 | −3.04150 | −2.59391 | 0 | ||||||||||||||||||
See all 28 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4375.2.a.p | 28 | |
5.b | even | 2 | 1 | 4375.2.a.o | 28 | ||
25.d | even | 5 | 2 | 875.2.h.d | 56 | ||
25.e | even | 10 | 2 | 875.2.h.e | 56 | ||
25.f | odd | 20 | 2 | 175.2.n.a | ✓ | 56 | |
25.f | odd | 20 | 2 | 875.2.n.c | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
175.2.n.a | ✓ | 56 | 25.f | odd | 20 | 2 | |
875.2.h.d | 56 | 25.d | even | 5 | 2 | ||
875.2.h.e | 56 | 25.e | even | 10 | 2 | ||
875.2.n.c | 56 | 25.f | odd | 20 | 2 | ||
4375.2.a.o | 28 | 5.b | even | 2 | 1 | ||
4375.2.a.p | 28 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4375))\):
\( T_{2}^{28} - 8 T_{2}^{27} - 8 T_{2}^{26} + 216 T_{2}^{25} - 258 T_{2}^{24} - 2408 T_{2}^{23} + 5326 T_{2}^{22} + 14012 T_{2}^{21} - 45273 T_{2}^{20} - 42344 T_{2}^{19} + 220154 T_{2}^{18} + 38276 T_{2}^{17} - 670685 T_{2}^{16} + \cdots + 205 \) |
\( T_{3}^{28} - 8 T_{3}^{27} - 21 T_{3}^{26} + 312 T_{3}^{25} - 84 T_{3}^{24} - 5166 T_{3}^{23} + 6774 T_{3}^{22} + 47508 T_{3}^{21} - 91708 T_{3}^{20} - 266884 T_{3}^{19} + 650500 T_{3}^{18} + 950622 T_{3}^{17} - 2828120 T_{3}^{16} + \cdots + 11321 \) |