Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,2,Mod(323,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.323");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.f (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: | \(6.46788256372\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | 16.0.9349208943630483456.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 90) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 62 \nu^{14} + 434 \nu^{13} - 2541 \nu^{12} + 9604 \nu^{11} - 30137 \nu^{10} + 72992 \nu^{9} + \cdots - 5945 ) / 65 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 88 \nu^{14} - 616 \nu^{13} + 3568 \nu^{12} - 13400 \nu^{11} + 41486 \nu^{10} - 99278 \nu^{9} + \cdots + 5230 ) / 65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6912 \nu^{15} - 51840 \nu^{14} + 303752 \nu^{13} - 1188148 \nu^{12} + 3758744 \nu^{11} + \cdots - 268015 ) / 17095 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10 \nu^{14} + 70 \nu^{13} - 406 \nu^{12} + 1526 \nu^{11} - 4732 \nu^{10} + 11340 \nu^{9} - 22581 \nu^{8} + \cdots - 670 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5682 \nu^{15} + 71282 \nu^{14} - 457752 \nu^{13} + 2187961 \nu^{12} - 7734306 \nu^{11} + \cdots + 2380210 ) / 17095 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4521 \nu^{15} + 83746 \nu^{14} - 546491 \nu^{13} + 2790379 \nu^{12} - 10004954 \nu^{11} + \cdots + 2733755 ) / 17095 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1065 \nu^{15} - 72291 \nu^{14} + 495870 \nu^{13} - 2782795 \nu^{12} + 10328207 \nu^{11} + \cdots - 3797055 ) / 17095 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 14724 \nu^{15} - 139097 \nu^{14} + 852995 \nu^{13} - 3728513 \nu^{12} + 12573585 \nu^{11} + \cdots - 2536435 ) / 17095 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 17709 \nu^{15} - 163457 \nu^{14} + 991985 \nu^{13} - 4285895 \nu^{12} + 14291742 \nu^{11} + \cdots - 2291185 ) / 17095 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 21276 \nu^{15} + 186922 \nu^{14} - 1126878 \nu^{13} + 4767802 \nu^{12} - 15742346 \nu^{11} + \cdots + 1868580 ) / 17095 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 4446 \nu^{15} - 40446 \nu^{14} + 244733 \nu^{13} - 1050316 \nu^{12} + 3488215 \nu^{11} + \cdots - 506866 ) / 3419 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} + \cdots - 1312670 ) / 17095 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 10652 \nu^{15} + 79890 \nu^{14} - 470562 \nu^{13} + 1846988 \nu^{12} - 5882478 \nu^{11} + \cdots + 370592 ) / 3419 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + \cdots + 1951760 ) / 17095 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 62302 \nu^{15} - 467265 \nu^{14} + 2748053 \nu^{13} - 10775492 \nu^{12} + 34251669 \nu^{11} + \cdots - 2026280 ) / 17095 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} + \beta_{13} - \beta_{8} - \beta_{5} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} - 2 \beta_{7} - \beta_{6} + \cdots - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 7 \beta_{15} - 7 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - \beta_{9} + 3 \beta_{8} + \cdots - 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 12 \beta_{15} - 12 \beta_{14} - 12 \beta_{13} - 6 \beta_{12} + 4 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + \cdots + 15 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 49 \beta_{15} - 13 \beta_{14} + 46 \beta_{13} - 37 \beta_{12} + 9 \beta_{11} - 12 \beta_{10} + 11 \beta_{9} + \cdots + 31 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 140 \beta_{15} + 86 \beta_{14} + 131 \beta_{13} - \beta_{12} + 26 \beta_{11} - 46 \beta_{10} + 49 \beta_{9} + \cdots - 70 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 262 \beta_{15} + 188 \beta_{14} - 246 \beta_{13} + 288 \beta_{12} - 2 \beta_{11} + 38 \beta_{10} + \cdots - 223 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 1352 \beta_{15} - 488 \beta_{14} - 1246 \beta_{13} + 332 \beta_{12} - 236 \beta_{11} + 376 \beta_{10} + \cdots + 345 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 759 \beta_{15} - 1928 \beta_{14} + 757 \beta_{13} - 1846 \beta_{12} - 411 \beta_{11} + 102 \beta_{10} + \cdots + 1780 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 11309 \beta_{15} + 1911 \beta_{14} + 10441 \beta_{13} - 4468 \beta_{12} + 1405 \beta_{11} - 2652 \beta_{10} + \cdots - 1279 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 5113 \beta_{15} + 16763 \beta_{14} + 4179 \beta_{13} + 9571 \beta_{12} + 5359 \beta_{11} - 3307 \beta_{10} + \cdots - 14343 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 83210 \beta_{15} + 1490 \beta_{14} - 77319 \beta_{13} + 44050 \beta_{12} - 4900 \beta_{11} + \cdots - 2151 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 121950 \beta_{15} - 128542 \beta_{14} - 108471 \beta_{13} - 29395 \beta_{12} - 49208 \beta_{11} + \cdots + 108590 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 527958 \beta_{15} - 137578 \beta_{14} + 495522 \beta_{13} - 369682 \beta_{12} - 16030 \beta_{11} + \cdots + 115489 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1474449 \beta_{15} + 863619 \beta_{14} + 1335768 \beta_{13} - 142417 \beta_{12} + 373829 \beta_{11} + \cdots - 736075 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(-\beta_{13}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 323.1 |
|
−0.707107 | − | 0.707107i | 0 | 1.00000i | −2.18058 | + | 0.495044i | 0 | −2.74939 | + | 2.74939i | 0.707107 | − | 0.707107i | 0 | 1.89195 | + | 1.19185i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.2 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 0.930573 | − | 2.03323i | 0 | −1.42594 | + | 1.42594i | 0.707107 | − | 0.707107i | 0 | −2.09573 | + | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.3 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 1.86274 | + | 1.23701i | 0 | −1.70010 | + | 1.70010i | 0.707107 | − | 0.707107i | 0 | −0.442462 | − | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.4 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 2.21569 | + | 0.301182i | 0 | 1.87542 | − | 1.87542i | 0.707107 | − | 0.707107i | 0 | −1.35376 | − | 1.77970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.5 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −2.21569 | − | 0.301182i | 0 | 1.87542 | − | 1.87542i | −0.707107 | + | 0.707107i | 0 | −1.35376 | − | 1.77970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.6 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −1.86274 | − | 1.23701i | 0 | −1.70010 | + | 1.70010i | −0.707107 | + | 0.707107i | 0 | −0.442462 | − | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.7 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −0.930573 | + | 2.03323i | 0 | −1.42594 | + | 1.42594i | −0.707107 | + | 0.707107i | 0 | −2.09573 | + | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.8 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 2.18058 | − | 0.495044i | 0 | −2.74939 | + | 2.74939i | −0.707107 | + | 0.707107i | 0 | 1.89195 | + | 1.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −2.18058 | − | 0.495044i | 0 | −2.74939 | − | 2.74939i | 0.707107 | + | 0.707107i | 0 | 1.89195 | − | 1.19185i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.2 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 0.930573 | + | 2.03323i | 0 | −1.42594 | − | 1.42594i | 0.707107 | + | 0.707107i | 0 | −2.09573 | − | 0.779698i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.3 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 1.86274 | − | 1.23701i | 0 | −1.70010 | − | 1.70010i | 0.707107 | + | 0.707107i | 0 | −0.442462 | + | 2.19185i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.4 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 2.21569 | − | 0.301182i | 0 | 1.87542 | + | 1.87542i | 0.707107 | + | 0.707107i | 0 | −1.35376 | + | 1.77970i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.5 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −2.21569 | + | 0.301182i | 0 | 1.87542 | + | 1.87542i | −0.707107 | − | 0.707107i | 0 | −1.35376 | + | 1.77970i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.6 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −1.86274 | + | 1.23701i | 0 | −1.70010 | − | 1.70010i | −0.707107 | − | 0.707107i | 0 | −0.442462 | + | 2.19185i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.7 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −0.930573 | − | 2.03323i | 0 | −1.42594 | − | 1.42594i | −0.707107 | − | 0.707107i | 0 | −2.09573 | − | 0.779698i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.8 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 2.18058 | + | 0.495044i | 0 | −2.74939 | − | 2.74939i | −0.707107 | − | 0.707107i | 0 | 1.89195 | − | 1.19185i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 810.2.f.c | 16 | |
| 3.b | odd | 2 | 1 | inner | 810.2.f.c | 16 | |
| 5.c | odd | 4 | 1 | inner | 810.2.f.c | 16 | |
| 9.c | even | 3 | 1 | 90.2.l.b | ✓ | 16 | |
| 9.c | even | 3 | 1 | 270.2.m.b | 16 | ||
| 9.d | odd | 6 | 1 | 90.2.l.b | ✓ | 16 | |
| 9.d | odd | 6 | 1 | 270.2.m.b | 16 | ||
| 15.e | even | 4 | 1 | inner | 810.2.f.c | 16 | |
| 36.f | odd | 6 | 1 | 720.2.cu.b | 16 | ||
| 36.h | even | 6 | 1 | 720.2.cu.b | 16 | ||
| 45.h | odd | 6 | 1 | 450.2.p.h | 16 | ||
| 45.h | odd | 6 | 1 | 1350.2.q.h | 16 | ||
| 45.j | even | 6 | 1 | 450.2.p.h | 16 | ||
| 45.j | even | 6 | 1 | 1350.2.q.h | 16 | ||
| 45.k | odd | 12 | 1 | 90.2.l.b | ✓ | 16 | |
| 45.k | odd | 12 | 1 | 270.2.m.b | 16 | ||
| 45.k | odd | 12 | 1 | 450.2.p.h | 16 | ||
| 45.k | odd | 12 | 1 | 1350.2.q.h | 16 | ||
| 45.l | even | 12 | 1 | 90.2.l.b | ✓ | 16 | |
| 45.l | even | 12 | 1 | 270.2.m.b | 16 | ||
| 45.l | even | 12 | 1 | 450.2.p.h | 16 | ||
| 45.l | even | 12 | 1 | 1350.2.q.h | 16 | ||
| 180.v | odd | 12 | 1 | 720.2.cu.b | 16 | ||
| 180.x | even | 12 | 1 | 720.2.cu.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.2.l.b | ✓ | 16 | 9.c | even | 3 | 1 | |
| 90.2.l.b | ✓ | 16 | 9.d | odd | 6 | 1 | |
| 90.2.l.b | ✓ | 16 | 45.k | odd | 12 | 1 | |
| 90.2.l.b | ✓ | 16 | 45.l | even | 12 | 1 | |
| 270.2.m.b | 16 | 9.c | even | 3 | 1 | ||
| 270.2.m.b | 16 | 9.d | odd | 6 | 1 | ||
| 270.2.m.b | 16 | 45.k | odd | 12 | 1 | ||
| 270.2.m.b | 16 | 45.l | even | 12 | 1 | ||
| 450.2.p.h | 16 | 45.h | odd | 6 | 1 | ||
| 450.2.p.h | 16 | 45.j | even | 6 | 1 | ||
| 450.2.p.h | 16 | 45.k | odd | 12 | 1 | ||
| 450.2.p.h | 16 | 45.l | even | 12 | 1 | ||
| 720.2.cu.b | 16 | 36.f | odd | 6 | 1 | ||
| 720.2.cu.b | 16 | 36.h | even | 6 | 1 | ||
| 720.2.cu.b | 16 | 180.v | odd | 12 | 1 | ||
| 720.2.cu.b | 16 | 180.x | even | 12 | 1 | ||
| 810.2.f.c | 16 | 1.a | even | 1 | 1 | trivial | |
| 810.2.f.c | 16 | 3.b | odd | 2 | 1 | inner | |
| 810.2.f.c | 16 | 5.c | odd | 4 | 1 | inner | |
| 810.2.f.c | 16 | 15.e | even | 4 | 1 | inner | |
| 1350.2.q.h | 16 | 45.h | odd | 6 | 1 | ||
| 1350.2.q.h | 16 | 45.j | even | 6 | 1 | ||
| 1350.2.q.h | 16 | 45.k | odd | 12 | 1 | ||
| 1350.2.q.h | 16 | 45.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 8T_{7}^{7} + 32T_{7}^{6} + 56T_{7}^{5} + 100T_{7}^{4} + 400T_{7}^{3} + 1568T_{7}^{2} + 2800T_{7} + 2500 \)
acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 16 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( (T^{8} + 8 T^{7} + \cdots + 2500)^{2} \)
|
| $11$ |
\( (T^{8} + 44 T^{6} + \cdots + 1849)^{2} \)
|
| $13$ |
\( (T^{8} + 24 T^{5} + \cdots + 36)^{2} \)
|
| $17$ |
\( T^{16} + 4132 T^{12} + \cdots + 390625 \)
|
| $19$ |
\( (T^{8} + 52 T^{6} + \cdots + 10201)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 82538991616 \)
|
| $29$ |
\( (T^{8} - 80 T^{6} + \cdots + 6400)^{2} \)
|
| $31$ |
\( (T^{4} - 4 T^{3} + \cdots + 670)^{4} \)
|
| $37$ |
\( (T^{8} - 576 T^{5} + \cdots + 82944)^{2} \)
|
| $41$ |
\( (T^{8} + 164 T^{6} + \cdots + 555025)^{2} \)
|
| $43$ |
\( (T^{8} + 48 T^{5} + \cdots + 189225)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 2981133747216 \)
|
| $53$ |
\( T^{16} + \cdots + 3906250000 \)
|
| $59$ |
\( (T^{8} - 44 T^{6} + \cdots + 25)^{2} \)
|
| $61$ |
\( (T^{4} - 12 T^{3} + \cdots - 6774)^{4} \)
|
| $67$ |
\( (T^{8} - 16 T^{7} + \cdots + 1849)^{2} \)
|
| $71$ |
\( (T^{8} + 272 T^{6} + \cdots + 14032516)^{2} \)
|
| $73$ |
\( (T^{8} - 8 T^{7} + \cdots + 966289)^{2} \)
|
| $79$ |
\( (T^{8} + 432 T^{6} + \cdots + 41602500)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 21743271936 \)
|
| $89$ |
\( (T^{4} - 28 T^{2} + 100)^{4} \)
|
| $97$ |
\( (T^{8} - 48 T^{7} + \cdots + 4422609)^{2} \)
|
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