Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,2,Mod(451,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.451");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 700.p (of order \(6\), 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: | \(5.58952814149\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 140) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
451.1 | −1.40479 | + | 0.163027i | 0.940044 | − | 1.62820i | 1.94684 | − | 0.458035i | 0 | −1.05512 | + | 2.44053i | 2.57589 | − | 0.603960i | −2.66023 | + | 0.960828i | −0.267367 | − | 0.463092i | 0 | ||||
451.2 | −1.34238 | − | 0.444985i | −1.29809 | + | 2.24836i | 1.60398 | + | 1.19468i | 0 | 2.74302 | − | 2.44053i | 0.603960 | + | 2.57589i | −1.62154 | − | 2.31746i | −1.87009 | − | 3.23909i | 0 | ||||
451.3 | −1.24320 | − | 0.674125i | −0.366453 | + | 0.634715i | 1.09111 | + | 1.67615i | 0 | 0.883452 | − | 0.542044i | 0.664037 | − | 2.56107i | −0.226536 | − | 2.81934i | 1.23142 | + | 2.13289i | 0 | ||||
451.4 | −1.00031 | − | 0.999687i | 0.739583 | − | 1.28100i | 0.00125109 | + | 2.00000i | 0 | −2.02041 | + | 0.542044i | −2.56107 | − | 0.664037i | 1.99812 | − | 2.00188i | 0.406034 | + | 0.703271i | 0 | ||||
451.5 | −0.843578 | + | 1.13507i | 0.940044 | − | 1.62820i | −0.576753 | − | 1.91503i | 0 | 1.05512 | + | 2.44053i | 2.57589 | − | 0.603960i | 2.66023 | + | 0.960828i | −0.267367 | − | 0.463092i | 0 | ||||
451.6 | −0.365598 | − | 1.36614i | −0.739583 | + | 1.28100i | −1.73268 | + | 0.998916i | 0 | 2.02041 | + | 0.542044i | 2.56107 | + | 0.664037i | 1.99812 | + | 2.00188i | 0.406034 | + | 0.703271i | 0 | ||||
451.7 | −0.285823 | + | 1.38503i | −1.29809 | + | 2.24836i | −1.83661 | − | 0.791746i | 0 | −2.74302 | − | 2.44053i | 0.603960 | + | 2.57589i | 1.62154 | − | 2.31746i | −1.87009 | − | 3.23909i | 0 | ||||
451.8 | −0.0377920 | + | 1.41371i | −0.366453 | + | 0.634715i | −1.99714 | − | 0.106854i | 0 | −0.883452 | − | 0.542044i | 0.664037 | − | 2.56107i | 0.226536 | − | 2.81934i | 1.23142 | + | 2.13289i | 0 | ||||
451.9 | 0.0377920 | − | 1.41371i | 0.366453 | − | 0.634715i | −1.99714 | − | 0.106854i | 0 | −0.883452 | − | 0.542044i | −0.664037 | + | 2.56107i | −0.226536 | + | 2.81934i | 1.23142 | + | 2.13289i | 0 | ||||
451.10 | 0.285823 | − | 1.38503i | 1.29809 | − | 2.24836i | −1.83661 | − | 0.791746i | 0 | −2.74302 | − | 2.44053i | −0.603960 | − | 2.57589i | −1.62154 | + | 2.31746i | −1.87009 | − | 3.23909i | 0 | ||||
451.11 | 0.365598 | + | 1.36614i | 0.739583 | − | 1.28100i | −1.73268 | + | 0.998916i | 0 | 2.02041 | + | 0.542044i | −2.56107 | − | 0.664037i | −1.99812 | − | 2.00188i | 0.406034 | + | 0.703271i | 0 | ||||
451.12 | 0.843578 | − | 1.13507i | −0.940044 | + | 1.62820i | −0.576753 | − | 1.91503i | 0 | 1.05512 | + | 2.44053i | −2.57589 | + | 0.603960i | −2.66023 | − | 0.960828i | −0.267367 | − | 0.463092i | 0 | ||||
451.13 | 1.00031 | + | 0.999687i | −0.739583 | + | 1.28100i | 0.00125109 | + | 2.00000i | 0 | −2.02041 | + | 0.542044i | 2.56107 | + | 0.664037i | −1.99812 | + | 2.00188i | 0.406034 | + | 0.703271i | 0 | ||||
451.14 | 1.24320 | + | 0.674125i | 0.366453 | − | 0.634715i | 1.09111 | + | 1.67615i | 0 | 0.883452 | − | 0.542044i | −0.664037 | + | 2.56107i | 0.226536 | + | 2.81934i | 1.23142 | + | 2.13289i | 0 | ||||
451.15 | 1.34238 | + | 0.444985i | 1.29809 | − | 2.24836i | 1.60398 | + | 1.19468i | 0 | 2.74302 | − | 2.44053i | −0.603960 | − | 2.57589i | 1.62154 | + | 2.31746i | −1.87009 | − | 3.23909i | 0 | ||||
451.16 | 1.40479 | − | 0.163027i | −0.940044 | + | 1.62820i | 1.94684 | − | 0.458035i | 0 | −1.05512 | + | 2.44053i | −2.57589 | + | 0.603960i | 2.66023 | − | 0.960828i | −0.267367 | − | 0.463092i | 0 | ||||
551.1 | −1.40479 | − | 0.163027i | 0.940044 | + | 1.62820i | 1.94684 | + | 0.458035i | 0 | −1.05512 | − | 2.44053i | 2.57589 | + | 0.603960i | −2.66023 | − | 0.960828i | −0.267367 | + | 0.463092i | 0 | ||||
551.2 | −1.34238 | + | 0.444985i | −1.29809 | − | 2.24836i | 1.60398 | − | 1.19468i | 0 | 2.74302 | + | 2.44053i | 0.603960 | − | 2.57589i | −1.62154 | + | 2.31746i | −1.87009 | + | 3.23909i | 0 | ||||
551.3 | −1.24320 | + | 0.674125i | −0.366453 | − | 0.634715i | 1.09111 | − | 1.67615i | 0 | 0.883452 | + | 0.542044i | 0.664037 | + | 2.56107i | −0.226536 | + | 2.81934i | 1.23142 | − | 2.13289i | 0 | ||||
551.4 | −1.00031 | + | 0.999687i | 0.739583 | + | 1.28100i | 0.00125109 | − | 2.00000i | 0 | −2.02041 | − | 0.542044i | −2.56107 | + | 0.664037i | 1.99812 | + | 2.00188i | 0.406034 | − | 0.703271i | 0 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.f | even | 6 | 1 | inner |
35.i | odd | 6 | 1 | inner |
140.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 700.2.p.e | 32 | |
4.b | odd | 2 | 1 | inner | 700.2.p.e | 32 | |
5.b | even | 2 | 1 | inner | 700.2.p.e | 32 | |
5.c | odd | 4 | 2 | 140.2.s.b | ✓ | 32 | |
7.d | odd | 6 | 1 | inner | 700.2.p.e | 32 | |
20.d | odd | 2 | 1 | inner | 700.2.p.e | 32 | |
20.e | even | 4 | 2 | 140.2.s.b | ✓ | 32 | |
28.f | even | 6 | 1 | inner | 700.2.p.e | 32 | |
35.f | even | 4 | 2 | 980.2.s.e | 32 | ||
35.i | odd | 6 | 1 | inner | 700.2.p.e | 32 | |
35.k | even | 12 | 2 | 140.2.s.b | ✓ | 32 | |
35.k | even | 12 | 2 | 980.2.c.d | 32 | ||
35.l | odd | 12 | 2 | 980.2.c.d | 32 | ||
35.l | odd | 12 | 2 | 980.2.s.e | 32 | ||
140.j | odd | 4 | 2 | 980.2.s.e | 32 | ||
140.s | even | 6 | 1 | inner | 700.2.p.e | 32 | |
140.w | even | 12 | 2 | 980.2.c.d | 32 | ||
140.w | even | 12 | 2 | 980.2.s.e | 32 | ||
140.x | odd | 12 | 2 | 140.2.s.b | ✓ | 32 | |
140.x | odd | 12 | 2 | 980.2.c.d | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
140.2.s.b | ✓ | 32 | 5.c | odd | 4 | 2 | |
140.2.s.b | ✓ | 32 | 20.e | even | 4 | 2 | |
140.2.s.b | ✓ | 32 | 35.k | even | 12 | 2 | |
140.2.s.b | ✓ | 32 | 140.x | odd | 12 | 2 | |
700.2.p.e | 32 | 1.a | even | 1 | 1 | trivial | |
700.2.p.e | 32 | 4.b | odd | 2 | 1 | inner | |
700.2.p.e | 32 | 5.b | even | 2 | 1 | inner | |
700.2.p.e | 32 | 7.d | odd | 6 | 1 | inner | |
700.2.p.e | 32 | 20.d | odd | 2 | 1 | inner | |
700.2.p.e | 32 | 28.f | even | 6 | 1 | inner | |
700.2.p.e | 32 | 35.i | odd | 6 | 1 | inner | |
700.2.p.e | 32 | 140.s | even | 6 | 1 | inner | |
980.2.c.d | 32 | 35.k | even | 12 | 2 | ||
980.2.c.d | 32 | 35.l | odd | 12 | 2 | ||
980.2.c.d | 32 | 140.w | even | 12 | 2 | ||
980.2.c.d | 32 | 140.x | odd | 12 | 2 | ||
980.2.s.e | 32 | 35.f | even | 4 | 2 | ||
980.2.s.e | 32 | 35.l | odd | 12 | 2 | ||
980.2.s.e | 32 | 140.j | odd | 4 | 2 | ||
980.2.s.e | 32 | 140.w | even | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):
\( T_{3}^{16} + 13T_{3}^{14} + 116T_{3}^{12} + 535T_{3}^{10} + 1780T_{3}^{8} + 3353T_{3}^{6} + 4445T_{3}^{4} + 2156T_{3}^{2} + 784 \) |
\( T_{17}^{16} - 85 T_{17}^{14} + 5369 T_{17}^{12} - 135448 T_{17}^{10} + 2480860 T_{17}^{8} - 18050816 T_{17}^{6} + 95473040 T_{17}^{4} - 174212096 T_{17}^{2} + 243859456 \) |