Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1600,2,Mod(49,1600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.q (of order \(4\), degree \(2\), not 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: | \(12.7760643234\) |
| 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: |
\( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| Twist minimal: | no (minimal twist has level 80) |
| 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
\( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 65 \nu^{15} - 604 \nu^{14} + 534 \nu^{13} + 720 \nu^{12} + 1271 \nu^{11} - 2160 \nu^{10} + \cdots - 60352 ) / 1344 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 39 \nu^{15} + 74 \nu^{14} + 66 \nu^{13} + 16 \nu^{12} - 337 \nu^{11} - 454 \nu^{10} + 654 \nu^{9} + \cdots - 256 ) / 448 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 55 \nu^{15} - 38 \nu^{14} + 232 \nu^{13} + 304 \nu^{12} - 129 \nu^{11} - 1398 \nu^{10} + \cdots - 12672 ) / 384 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 20 \nu^{15} + 20 \nu^{14} + 41 \nu^{13} + 44 \nu^{12} - 108 \nu^{11} - 276 \nu^{10} + 95 \nu^{9} + \cdots - 576 ) / 96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 396 \nu^{15} + 1201 \nu^{14} - 256 \nu^{13} - 460 \nu^{12} - 3508 \nu^{11} - 489 \nu^{10} + \cdots + 83072 ) / 1344 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 556 \nu^{15} - 1613 \nu^{14} + 234 \nu^{13} + 492 \nu^{12} + 4780 \nu^{11} + 1317 \nu^{10} + \cdots - 109568 ) / 1344 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 526 \nu^{15} + 281 \nu^{14} + 1644 \nu^{13} + 1572 \nu^{12} - 3082 \nu^{11} - 9945 \nu^{10} + \cdots - 65920 ) / 1344 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + \cdots + 176384 ) / 2688 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 222 \nu^{15} - 593 \nu^{14} - 16 \nu^{13} + 148 \nu^{12} + 1914 \nu^{11} + 977 \nu^{10} - 5588 \nu^{9} + \cdots - 32384 ) / 448 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1367 \nu^{15} - 3586 \nu^{14} - 12 \nu^{13} + 720 \nu^{12} + 11729 \nu^{11} + 6366 \nu^{10} + \cdots - 196096 ) / 2688 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 355 \nu^{15} + 1041 \nu^{14} - 124 \nu^{13} - 400 \nu^{12} - 3209 \nu^{11} - 837 \nu^{10} + \cdots + 64864 ) / 672 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 106 \nu^{15} + 397 \nu^{14} - 200 \nu^{13} - 260 \nu^{12} - 1014 \nu^{11} + 483 \nu^{10} + \cdots + 33600 ) / 192 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 996 \nu^{15} + 1703 \nu^{14} + 1258 \nu^{13} + 988 \nu^{12} - 7004 \nu^{11} - 9999 \nu^{10} + \cdots + 36160 ) / 1344 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1167 \nu^{15} + 4247 \nu^{14} - 1916 \nu^{13} - 2612 \nu^{12} - 11105 \nu^{11} + 3909 \nu^{10} + \cdots + 342400 ) / 1344 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1175 \nu^{15} + 3065 \nu^{14} + 8 \nu^{13} - 676 \nu^{12} - 9945 \nu^{11} - 5133 \nu^{10} + \cdots + 168960 ) / 1344 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \cdots + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{6} + 4 \beta_{5} + \cdots + 3 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{15} - 3 \beta_{13} + \beta_{12} - \beta_{11} + 5 \beta_{10} + 5 \beta_{8} - \beta_{7} + \cdots - 1 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{15} + \beta_{14} - 3 \beta_{13} - 4 \beta_{12} - 3 \beta_{11} + \beta_{9} + 3 \beta_{8} + \cdots - 11 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4 \beta_{15} - 3 \beta_{13} - \beta_{12} + 4 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} + 4 \beta_{7} + \cdots - 1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 8 \beta_{15} - \beta_{14} - 3 \beta_{13} + 7 \beta_{11} + 10 \beta_{10} + 9 \beta_{9} + 3 \beta_{8} + \cdots - 15 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2 \beta_{14} + 6 \beta_{13} - 8 \beta_{12} + \beta_{11} - 13 \beta_{10} + 20 \beta_{9} - \beta_{8} + \cdots - 10 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2 \beta_{15} - 20 \beta_{14} - \beta_{13} + 7 \beta_{12} + 28 \beta_{11} - 16 \beta_{10} + 20 \beta_{9} + \cdots - 33 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 34 \beta_{15} + 30 \beta_{14} + 30 \beta_{13} - 8 \beta_{12} + 13 \beta_{11} - 7 \beta_{10} + \cdots + 22 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 6 \beta_{15} - 37 \beta_{14} + 10 \beta_{13} - 15 \beta_{12} + 53 \beta_{11} - 40 \beta_{10} + 21 \beta_{9} + \cdots + 2 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 90 \beta_{15} - 2 \beta_{14} + 41 \beta_{13} + 7 \beta_{12} + 6 \beta_{11} - 78 \beta_{10} + \cdots - 49 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 80 \beta_{15} + 75 \beta_{14} - 2 \beta_{13} - 49 \beta_{12} + 71 \beta_{11} + 32 \beta_{10} + \cdots + 102 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 46 \beta_{15} + 14 \beta_{14} + 49 \beta_{13} - 63 \beta_{12} - 55 \beta_{11} + \beta_{10} + \cdots + 31 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 98 \beta_{15} + 8 \beta_{14} - 147 \beta_{13} - 67 \beta_{12} + 74 \beta_{11} - 26 \beta_{10} + \cdots - 167 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1151\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −2.32624 | − | 2.32624i | 0 | 0 | 0 | −0.982011 | 0 | 7.82281i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −1.66366 | − | 1.66366i | 0 | 0 | 0 | 2.89402 | 0 | 2.53555i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | −0.720673 | − | 0.720673i | 0 | 0 | 0 | 4.02840 | 0 | − | 1.96126i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | −0.120009 | − | 0.120009i | 0 | 0 | 0 | −2.66881 | 0 | − | 2.97120i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 0.209571 | + | 0.209571i | 0 | 0 | 0 | −1.73696 | 0 | − | 2.91216i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 1.37027 | + | 1.37027i | 0 | 0 | 0 | −2.73482 | 0 | 0.755274i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | 0 | 1.42313 | + | 1.42313i | 0 | 0 | 0 | 0.690576 | 0 | 1.05061i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | 0 | 1.82762 | + | 1.82762i | 0 | 0 | 0 | 4.50961 | 0 | 3.68037i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.1 | 0 | −2.32624 | + | 2.32624i | 0 | 0 | 0 | −0.982011 | 0 | − | 7.82281i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.2 | 0 | −1.66366 | + | 1.66366i | 0 | 0 | 0 | 2.89402 | 0 | − | 2.53555i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.3 | 0 | −0.720673 | + | 0.720673i | 0 | 0 | 0 | 4.02840 | 0 | 1.96126i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.4 | 0 | −0.120009 | + | 0.120009i | 0 | 0 | 0 | −2.66881 | 0 | 2.97120i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.5 | 0 | 0.209571 | − | 0.209571i | 0 | 0 | 0 | −1.73696 | 0 | 2.91216i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.6 | 0 | 1.37027 | − | 1.37027i | 0 | 0 | 0 | −2.73482 | 0 | − | 0.755274i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.7 | 0 | 1.42313 | − | 1.42313i | 0 | 0 | 0 | 0.690576 | 0 | − | 1.05061i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 849.8 | 0 | 1.82762 | − | 1.82762i | 0 | 0 | 0 | 4.50961 | 0 | − | 3.68037i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.q | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1600.2.q.h | 16 | |
| 4.b | odd | 2 | 1 | 400.2.q.g | 16 | ||
| 5.b | even | 2 | 1 | 1600.2.q.g | 16 | ||
| 5.c | odd | 4 | 1 | 320.2.l.a | 16 | ||
| 5.c | odd | 4 | 1 | 1600.2.l.i | 16 | ||
| 15.e | even | 4 | 1 | 2880.2.t.c | 16 | ||
| 16.e | even | 4 | 1 | 1600.2.q.g | 16 | ||
| 16.f | odd | 4 | 1 | 400.2.q.h | 16 | ||
| 20.d | odd | 2 | 1 | 400.2.q.h | 16 | ||
| 20.e | even | 4 | 1 | 80.2.l.a | ✓ | 16 | |
| 20.e | even | 4 | 1 | 400.2.l.h | 16 | ||
| 40.i | odd | 4 | 1 | 640.2.l.a | 16 | ||
| 40.k | even | 4 | 1 | 640.2.l.b | 16 | ||
| 60.l | odd | 4 | 1 | 720.2.t.c | 16 | ||
| 80.i | odd | 4 | 1 | 320.2.l.a | 16 | ||
| 80.j | even | 4 | 1 | 400.2.l.h | 16 | ||
| 80.j | even | 4 | 1 | 640.2.l.b | 16 | ||
| 80.k | odd | 4 | 1 | 400.2.q.g | 16 | ||
| 80.q | even | 4 | 1 | inner | 1600.2.q.h | 16 | |
| 80.s | even | 4 | 1 | 80.2.l.a | ✓ | 16 | |
| 80.t | odd | 4 | 1 | 640.2.l.a | 16 | ||
| 80.t | odd | 4 | 1 | 1600.2.l.i | 16 | ||
| 160.v | odd | 8 | 1 | 5120.2.a.t | 8 | ||
| 160.v | odd | 8 | 1 | 5120.2.a.u | 8 | ||
| 160.ba | even | 8 | 1 | 5120.2.a.s | 8 | ||
| 160.ba | even | 8 | 1 | 5120.2.a.v | 8 | ||
| 240.z | odd | 4 | 1 | 720.2.t.c | 16 | ||
| 240.bb | even | 4 | 1 | 2880.2.t.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.2.l.a | ✓ | 16 | 20.e | even | 4 | 1 | |
| 80.2.l.a | ✓ | 16 | 80.s | even | 4 | 1 | |
| 320.2.l.a | 16 | 5.c | odd | 4 | 1 | ||
| 320.2.l.a | 16 | 80.i | odd | 4 | 1 | ||
| 400.2.l.h | 16 | 20.e | even | 4 | 1 | ||
| 400.2.l.h | 16 | 80.j | even | 4 | 1 | ||
| 400.2.q.g | 16 | 4.b | odd | 2 | 1 | ||
| 400.2.q.g | 16 | 80.k | odd | 4 | 1 | ||
| 400.2.q.h | 16 | 16.f | odd | 4 | 1 | ||
| 400.2.q.h | 16 | 20.d | odd | 2 | 1 | ||
| 640.2.l.a | 16 | 40.i | odd | 4 | 1 | ||
| 640.2.l.a | 16 | 80.t | odd | 4 | 1 | ||
| 640.2.l.b | 16 | 40.k | even | 4 | 1 | ||
| 640.2.l.b | 16 | 80.j | even | 4 | 1 | ||
| 720.2.t.c | 16 | 60.l | odd | 4 | 1 | ||
| 720.2.t.c | 16 | 240.z | odd | 4 | 1 | ||
| 1600.2.l.i | 16 | 5.c | odd | 4 | 1 | ||
| 1600.2.l.i | 16 | 80.t | odd | 4 | 1 | ||
| 1600.2.q.g | 16 | 5.b | even | 2 | 1 | ||
| 1600.2.q.g | 16 | 16.e | even | 4 | 1 | ||
| 1600.2.q.h | 16 | 1.a | even | 1 | 1 | trivial | |
| 1600.2.q.h | 16 | 80.q | even | 4 | 1 | inner | |
| 2880.2.t.c | 16 | 15.e | even | 4 | 1 | ||
| 2880.2.t.c | 16 | 240.bb | even | 4 | 1 | ||
| 5120.2.a.s | 8 | 160.ba | even | 8 | 1 | ||
| 5120.2.a.t | 8 | 160.v | odd | 8 | 1 | ||
| 5120.2.a.u | 8 | 160.v | odd | 8 | 1 | ||
| 5120.2.a.v | 8 | 160.ba | even | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 8 T_{3}^{13} + 112 T_{3}^{12} - 80 T_{3}^{11} + 32 T_{3}^{10} - 176 T_{3}^{9} + 2632 T_{3}^{8} + \cdots + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 8 T^{13} + \cdots + 16 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( (T^{8} - 4 T^{7} + \cdots + 452)^{2} \)
|
| $11$ |
\( T^{16} - 8 T^{15} + \cdots + 1290496 \)
|
| $13$ |
\( T^{16} - 128 T^{13} + \cdots + 20647936 \)
|
| $17$ |
\( T^{16} + \cdots + 192876544 \)
|
| $19$ |
\( T^{16} + 8 T^{15} + \cdots + 614656 \)
|
| $23$ |
\( (T^{8} - 12 T^{7} + \cdots + 1316)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 3017085184 \)
|
| $31$ |
\( (T^{8} - 96 T^{6} + \cdots - 20224)^{2} \)
|
| $37$ |
\( T^{16} - 16 T^{15} + \cdots + 18939904 \)
|
| $41$ |
\( T^{16} + \cdots + 110660014336 \)
|
| $43$ |
\( T^{16} + 8 T^{15} + \cdots + 53640976 \)
|
| $47$ |
\( T^{16} + \cdots + 330675601936 \)
|
| $53$ |
\( T^{16} + \cdots + 383725735936 \)
|
| $59$ |
\( T^{16} + \cdots + 12227051776 \)
|
| $61$ |
\( T^{16} + \cdots + 1393986371584 \)
|
| $67$ |
\( T^{16} + \cdots + 46120451769616 \)
|
| $71$ |
\( T^{16} + \cdots + 3333516427264 \)
|
| $73$ |
\( (T^{8} - 280 T^{6} + \cdots - 125888)^{2} \)
|
| $79$ |
\( (T^{8} - 8 T^{7} + \cdots + 4352)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 2050640656 \)
|
| $89$ |
\( T^{16} + \cdots + 684153962496 \)
|
| $97$ |
\( T^{16} + \cdots + 73090735673344 \)
|
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