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: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 16x^{14} + 145x^{12} - 976x^{10} + 5296x^{8} - 24400x^{6} + 90625x^{4} - 250000x^{2} + 390625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{16} - 16x^{14} + 145x^{12} - 976x^{10} + 5296x^{8} - 24400x^{6} + 90625x^{4} - 250000x^{2} + 390625 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 44 \nu^{14} - 28729 \nu^{12} + 107280 \nu^{10} - 180944 \nu^{8} + 885424 \nu^{6} + \cdots + 336734375 ) / 186750000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1567 \nu^{14} - 242897 \nu^{12} + 1523040 \nu^{10} - 9443392 \nu^{8} + 53796032 \nu^{6} + \cdots - 1179640625 ) / 933750000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 9011 \nu^{14} + 63901 \nu^{12} - 160320 \nu^{10} - 463264 \nu^{8} + 12176144 \nu^{6} + \cdots - 792171875 ) / 933750000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4349 \nu^{15} - 63109 \nu^{13} + 888880 \nu^{11} - 3642624 \nu^{9} + 19512704 \nu^{7} + \cdots - 876703125 \nu ) / 1556250000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16061 \nu^{15} + 161549 \nu^{13} - 2185680 \nu^{11} + 15272464 \nu^{9} - 89721344 \nu^{7} + \cdots + 3677328125 \nu ) / 4668750000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 577 \nu^{14} - 4232 \nu^{12} + 30540 \nu^{10} - 143152 \nu^{8} + 744542 \nu^{6} - 3328800 \nu^{4} + \cdots - 15875000 ) / 23343750 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6363 \nu^{15} + 90892 \nu^{13} - 727440 \nu^{11} + 4863712 \nu^{9} - 15526752 \nu^{7} + \cdots + 443937500 \nu ) / 1556250000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 69347 \nu^{14} + 790627 \nu^{12} - 6200640 \nu^{10} + 36236672 \nu^{8} - 177640912 \nu^{6} + \cdots + 4664171875 ) / 933750000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13367 \nu^{15} - 169072 \nu^{13} + 1194540 \nu^{11} - 6245192 \nu^{9} + 32279332 \nu^{7} + \cdots - 653312500 \nu ) / 1167187500 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 75019 \nu^{14} + 1022204 \nu^{12} - 8621280 \nu^{10} + 48696544 \nu^{8} - 247775024 \nu^{6} + \cdots + 6113187500 ) / 933750000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 16697 \nu^{14} + 151852 \nu^{12} - 1130640 \nu^{10} + 6810272 \nu^{8} - 33044512 \nu^{6} + \cdots + 587687500 ) / 186750000 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 75019 \nu^{15} - 1022204 \nu^{13} + 8621280 \nu^{11} - 48696544 \nu^{9} + 247775024 \nu^{7} + \cdots - 2378187500 \nu ) / 4668750000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 97372 \nu^{15} + 1140527 \nu^{13} - 10322640 \nu^{11} + 66769072 \nu^{9} + \cdots + 15696359375 \nu ) / 4668750000 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 24896 \nu^{15} + 254311 \nu^{13} - 1903020 \nu^{11} + 11552996 \nu^{9} + \cdots + 1293296875 \nu ) / 1167187500 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 123904 \nu^{15} + 1002989 \nu^{13} - 7284480 \nu^{11} + 41418304 \nu^{9} + \cdots + 4978578125 \nu ) / 4668750000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} + 2\beta_{12} - \beta_{9} + \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{11} + 2\beta_{10} - 5\beta_{8} + 3\beta_{3} - 2\beta_{2} + 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + 5\beta_{14} - 3\beta_{13} + 3\beta_{12} + \beta_{9} - \beta_{7} + 6\beta_{5} - 2\beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{11} + 2\beta_{10} - 5\beta_{8} - 10\beta_{6} + 13\beta_{3} - 12\beta_{2} + 14\beta _1 - 18 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 44\beta_{15} - 35\beta_{14} - 4\beta_{13} + 17\beta_{9} + 22\beta_{7} + 13\beta_{5} + 3\beta_{4} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -26\beta_{11} - 10\beta_{10} + 52\beta_{8} + 55\beta_{6} + 68\beta_{3} + 16\beta_{2} + 16\beta _1 - 26 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 130\beta_{15} - 9\beta_{14} + 52\beta_{12} + 199\beta_{9} + 260\beta_{7} - 169\beta_{5} - 121\beta_{4} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 212\beta_{11} - 78\beta_{10} + 195\beta_{8} + 1300\beta_{6} + 533\beta_{3} + 78\beta_{2} + 212\beta _1 - 78 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 219 \beta_{15} + 890 \beta_{14} - 63 \beta_{13} + 63 \beta_{12} + 866 \beta_{9} + 219 \beta_{7} + \cdots + 78 \beta_{4} \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 1888 \beta_{11} - 1888 \beta_{10} + 565 \beta_{8} + 2190 \beta_{6} - 867 \beta_{3} - 2062 \beta_{2} + \cdots + 1432 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( -256\beta_{15} - 435\beta_{14} + 1606\beta_{13} - 1043\beta_{9} - 128\beta_{7} + 563\beta_{5} + 10003\beta_{4} ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( 304\beta_{11} - 4480\beta_{10} + 4312\beta_{8} - 320\beta_{6} - 472\beta_{3} - 4784\beta_{2} - 4784\beta _1 + 8799 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 8320 \beta_{15} - 22609 \beta_{14} + 16382 \beta_{12} - 61711 \beta_{9} + 16640 \beta_{7} + \cdots + 33969 \beta_{4} ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 11778 \beta_{11} + 27742 \beta_{10} - 20155 \beta_{8} + 83200 \beta_{6} + 34013 \beta_{3} + \cdots + 27742 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 26689 \beta_{15} + 27075 \beta_{14} - 32973 \beta_{13} + 32973 \beta_{12} - 68969 \beta_{9} + \cdots - 3142 \beta_{4} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(-1\) |
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 | −2.05289 | + | 0.886375i | 0 | 0.887499 | − | 0.887499i | 0.707107 | − | 0.707107i | 0 | 2.07837 | + | 0.824847i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.2 | −0.707107 | − | 0.707107i | 0 | 1.00000i | −1.26353 | − | 1.84485i | 0 | −1.24299 | + | 1.24299i | 0.707107 | − | 0.707107i | 0 | −0.411058 | + | 2.19796i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.3 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 1.79407 | − | 1.33466i | 0 | −2.25352 | + | 2.25352i | 0.707107 | − | 0.707107i | 0 | −2.21235 | − | 0.324847i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.4 | −0.707107 | − | 0.707107i | 0 | 1.00000i | 2.22946 | + | 0.171822i | 0 | 1.60902 | − | 1.60902i | 0.707107 | − | 0.707107i | 0 | −1.45497 | − | 1.69796i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.5 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −2.22946 | − | 0.171822i | 0 | 1.60902 | − | 1.60902i | −0.707107 | + | 0.707107i | 0 | −1.45497 | − | 1.69796i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.6 | 0.707107 | + | 0.707107i | 0 | 1.00000i | −1.79407 | + | 1.33466i | 0 | −2.25352 | + | 2.25352i | −0.707107 | + | 0.707107i | 0 | −2.21235 | − | 0.324847i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.7 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 1.26353 | + | 1.84485i | 0 | −1.24299 | + | 1.24299i | −0.707107 | + | 0.707107i | 0 | −0.411058 | + | 2.19796i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 323.8 | 0.707107 | + | 0.707107i | 0 | 1.00000i | 2.05289 | − | 0.886375i | 0 | 0.887499 | − | 0.887499i | −0.707107 | + | 0.707107i | 0 | 2.07837 | + | 0.824847i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.1 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −2.05289 | − | 0.886375i | 0 | 0.887499 | + | 0.887499i | 0.707107 | + | 0.707107i | 0 | 2.07837 | − | 0.824847i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.2 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | −1.26353 | + | 1.84485i | 0 | −1.24299 | − | 1.24299i | 0.707107 | + | 0.707107i | 0 | −0.411058 | − | 2.19796i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.3 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 1.79407 | + | 1.33466i | 0 | −2.25352 | − | 2.25352i | 0.707107 | + | 0.707107i | 0 | −2.21235 | + | 0.324847i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.4 | −0.707107 | + | 0.707107i | 0 | − | 1.00000i | 2.22946 | − | 0.171822i | 0 | 1.60902 | + | 1.60902i | 0.707107 | + | 0.707107i | 0 | −1.45497 | + | 1.69796i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.5 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −2.22946 | + | 0.171822i | 0 | 1.60902 | + | 1.60902i | −0.707107 | − | 0.707107i | 0 | −1.45497 | + | 1.69796i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.6 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | −1.79407 | − | 1.33466i | 0 | −2.25352 | − | 2.25352i | −0.707107 | − | 0.707107i | 0 | −2.21235 | + | 0.324847i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.7 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 1.26353 | − | 1.84485i | 0 | −1.24299 | − | 1.24299i | −0.707107 | − | 0.707107i | 0 | −0.411058 | − | 2.19796i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 647.8 | 0.707107 | − | 0.707107i | 0 | − | 1.00000i | 2.05289 | + | 0.886375i | 0 | 0.887499 | + | 0.887499i | −0.707107 | − | 0.707107i | 0 | 2.07837 | − | 0.824847i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 810.2.f.d | ✓ | 16 |
| 5.c | odd | 4 | 1 | inner | 810.2.f.d | ✓ | 16 |
| 9.c | even | 3 | 1 | 810.2.m.i | 16 | ||
| 9.c | even | 3 | 1 | 810.2.m.j | 16 | ||
| 9.d | odd | 6 | 1 | 810.2.m.i | 16 | ||
| 9.d | odd | 6 | 1 | 810.2.m.j | 16 | ||
| 15.e | even | 4 | 1 | inner | 810.2.f.d | ✓ | 16 |
| 45.k | odd | 12 | 1 | 810.2.m.i | 16 | ||
| 45.k | odd | 12 | 1 | 810.2.m.j | 16 | ||
| 45.l | even | 12 | 1 | 810.2.m.i | 16 | ||
| 45.l | even | 12 | 1 | 810.2.m.j | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 810.2.f.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 810.2.f.d | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 810.2.f.d | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 810.2.f.d | ✓ | 16 | 15.e | even | 4 | 1 | inner |
| 810.2.m.i | 16 | 9.c | even | 3 | 1 | ||
| 810.2.m.i | 16 | 9.d | odd | 6 | 1 | ||
| 810.2.m.i | 16 | 45.k | odd | 12 | 1 | ||
| 810.2.m.i | 16 | 45.l | even | 12 | 1 | ||
| 810.2.m.j | 16 | 9.c | even | 3 | 1 | ||
| 810.2.m.j | 16 | 9.d | odd | 6 | 1 | ||
| 810.2.m.j | 16 | 45.k | odd | 12 | 1 | ||
| 810.2.m.j | 16 | 45.l | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 10T_{7}^{5} + 49T_{7}^{4} + 40T_{7}^{3} + 32T_{7}^{2} - 128T_{7} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{4} \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 16 T^{14} + \cdots + 390625 \)
|
| $7$ |
\( (T^{8} + 2 T^{7} + \cdots + 256)^{2} \)
|
| $11$ |
\( (T^{8} + 68 T^{6} + \cdots + 4096)^{2} \)
|
| $13$ |
\( (T^{8} - 24 T^{5} + \cdots + 19881)^{2} \)
|
| $17$ |
\( T^{16} + 874 T^{12} + \cdots + 6250000 \)
|
| $19$ |
\( (T^{8} + 46 T^{6} + \cdots + 256)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 409600000000 \)
|
| $29$ |
\( (T^{8} - 164 T^{6} + \cdots + 65536)^{2} \)
|
| $31$ |
\( (T^{4} + 2 T^{3} + \cdots + 736)^{4} \)
|
| $37$ |
\( (T^{8} + 6 T^{7} + \cdots + 324)^{2} \)
|
| $41$ |
\( (T^{8} + 176 T^{6} + \cdots + 487204)^{2} \)
|
| $43$ |
\( (T^{8} - 24 T^{7} + \cdots + 589824)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 429981696 \)
|
| $53$ |
\( T^{16} + \cdots + 260120641601536 \)
|
| $59$ |
\( (T^{8} - 236 T^{6} + \cdots + 495616)^{2} \)
|
| $61$ |
\( (T^{4} + 18 T^{3} + \cdots - 3378)^{4} \)
|
| $67$ |
\( (T^{8} - 4 T^{7} + \cdots + 4096)^{2} \)
|
| $71$ |
\( (T^{8} + 140 T^{6} + \cdots + 1024)^{2} \)
|
| $73$ |
\( (T^{8} + 34 T^{7} + \cdots + 952576)^{2} \)
|
| $79$ |
\( (T^{8} + 600 T^{6} + \cdots + 147456)^{2} \)
|
| $83$ |
\( T^{16} + 7032 T^{12} + \cdots + 84934656 \)
|
| $89$ |
\( (T^{8} - 176 T^{6} + \cdots + 1865956)^{2} \)
|
| $97$ |
\( (T^{8} - 12 T^{7} + \cdots + 2304)^{2} \)
|
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