Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(331,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.331");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.l (of order \(3\), 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: | \(5.03057532734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6 \nu^{15} + 47 \nu^{14} - 161 \nu^{13} + 13 \nu^{12} - 287 \nu^{11} + 1486 \nu^{10} - 1109 \nu^{9} + \cdots - 188811 ) / 43740 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 193 \nu^{15} + 520 \nu^{14} - 2882 \nu^{13} + 5890 \nu^{12} - 14819 \nu^{11} + 17755 \nu^{10} + \cdots - 448335 ) / 787320 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 119 \nu^{15} + 1900 \nu^{14} + 586 \nu^{13} + 1870 \nu^{12} - 22943 \nu^{11} + 10315 \nu^{10} + \cdots + 3881925 ) / 787320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 53 \nu^{15} - 173 \nu^{14} + 37 \nu^{13} - 317 \nu^{12} + 1474 \nu^{11} - 899 \nu^{10} + \cdots - 39366 ) / 131220 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 92 \nu^{15} + 341 \nu^{14} - 208 \nu^{13} + 209 \nu^{12} - 736 \nu^{11} - 2242 \nu^{10} + \cdots + 428652 ) / 196830 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 349 \nu^{15} - 532 \nu^{14} + 1106 \nu^{13} - 718 \nu^{12} + 767 \nu^{11} - 12511 \nu^{10} + \cdots + 3833811 ) / 787320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} - 5 \nu^{11} - 2 \nu^{10} + 35 \nu^{9} - 81 \nu^{8} + \cdots + 2187 ) / 2187 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 247 \nu^{15} + 1105 \nu^{14} - 3143 \nu^{13} + 3145 \nu^{12} - 7466 \nu^{11} + 20005 \nu^{10} + \cdots - 2777490 ) / 393660 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 229 \nu^{15} - 385 \nu^{14} - 349 \nu^{13} - 1345 \nu^{12} + 32 \nu^{11} + 4205 \nu^{10} + \cdots - 174960 ) / 393660 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 727 \nu^{15} + 982 \nu^{14} - 4148 \nu^{13} + 9448 \nu^{12} - 3971 \nu^{11} - 899 \nu^{10} + \cdots + 234009 ) / 787320 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 619 \nu^{15} - 1520 \nu^{14} + 514 \nu^{13} + 6550 \nu^{12} + 4903 \nu^{11} - 935 \nu^{10} + \cdots - 2219805 ) / 787320 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 503 \nu^{15} + 905 \nu^{14} + 1193 \nu^{13} - 55 \nu^{12} + 4346 \nu^{11} - 24115 \nu^{10} + \cdots + 3171150 ) / 393660 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 499 \nu^{15} - 367 \nu^{14} + 1091 \nu^{13} - 3523 \nu^{12} + 5522 \nu^{11} - 7801 \nu^{10} + \cdots + 1570266 ) / 393660 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1291 \nu^{15} - 88 \nu^{14} - 286 \nu^{13} - 2902 \nu^{12} - 11947 \nu^{11} + 33671 \nu^{10} + \cdots - 640791 ) / 787320 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 376 \nu^{15} - 883 \nu^{14} + 269 \nu^{13} - 1717 \nu^{12} + 3413 \nu^{11} + 3176 \nu^{10} + \cdots - 968841 ) / 196830 \)
|
| \(\nu\) | \(=\) |
\( ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} - 4 \beta_{5} + \cdots - 4 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 6 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 7 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{15} - \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \cdots - 7 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 7 \beta_{15} - 2 \beta_{14} - \beta_{13} - 9 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} - 10 \beta_{9} + \cdots - 23 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{15} - 22 \beta_{14} + \beta_{13} - 12 \beta_{12} - 18 \beta_{11} - 4 \beta_{10} - 23 \beta_{9} + \cdots + 32 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 40 \beta_{15} + 37 \beta_{14} + 2 \beta_{13} - 21 \beta_{12} + 15 \beta_{11} - 5 \beta_{10} + \cdots - 26 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 23 \beta_{15} + 47 \beta_{14} + 25 \beta_{13} + 27 \beta_{12} + 36 \beta_{11} - 52 \beta_{10} + \cdots - 163 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 80 \beta_{15} - 59 \beta_{14} + 95 \beta_{13} - 60 \beta_{12} + 37 \beta_{10} - 55 \beta_{9} + 28 \beta_{8} + \cdots + 97 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 82 \beta_{15} + 197 \beta_{14} - 32 \beta_{13} + 147 \beta_{12} + 171 \beta_{11} - 91 \beta_{10} + \cdots + 101 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 106 \beta_{15} + 520 \beta_{14} + 8 \beta_{13} + 117 \beta_{12} + 228 \beta_{11} + 52 \beta_{10} + \cdots - 59 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 232 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 483 \beta_{12} + 18 \beta_{11} + 482 \beta_{10} + \cdots + 743 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 598 \beta_{15} + 478 \beta_{14} + 929 \beta_{13} - 435 \beta_{12} + 1032 \beta_{11} + 157 \beta_{10} + \cdots + 532 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 32 \beta_{15} + 713 \beta_{14} + 934 \beta_{13} + 1575 \beta_{12} + 1368 \beta_{11} - 403 \beta_{10} + \cdots - 2152 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 2627 \beta_{15} - 2318 \beta_{14} + 617 \beta_{13} + 174 \beta_{12} - 1548 \beta_{11} + 3871 \beta_{10} + \cdots - 1532 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 331.1 |
|
0.500000 | + | 0.866025i | −1.73057 | + | 0.0717234i | −0.500000 | + | 0.866025i | −1.00000 | −0.927397 | − | 1.46285i | −2.59109 | + | 0.535003i | −1.00000 | 2.98971 | − | 0.248244i | −0.500000 | − | 0.866025i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.2 | 0.500000 | + | 0.866025i | −1.71973 | − | 0.206189i | −0.500000 | + | 0.866025i | −1.00000 | −0.681302 | − | 1.59243i | 2.63512 | + | 0.236999i | −1.00000 | 2.91497 | + | 0.709180i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.3 | 0.500000 | + | 0.866025i | −0.978280 | + | 1.42932i | −0.500000 | + | 0.866025i | −1.00000 | −1.72697 | − | 0.132553i | 1.60537 | − | 2.10304i | −1.00000 | −1.08594 | − | 2.79656i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.4 | 0.500000 | + | 0.866025i | −0.852741 | − | 1.50759i | −0.500000 | + | 0.866025i | −1.00000 | 0.879242 | − | 1.49229i | −1.78630 | − | 1.95170i | −1.00000 | −1.54566 | + | 2.57117i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.5 | 0.500000 | + | 0.866025i | 0.461735 | + | 1.66937i | −0.500000 | + | 0.866025i | −1.00000 | −1.21485 | + | 1.23456i | −1.53750 | + | 2.15316i | −1.00000 | −2.57360 | + | 1.54161i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.6 | 0.500000 | + | 0.866025i | 1.12401 | − | 1.31780i | −0.500000 | + | 0.866025i | −1.00000 | 1.70326 | + | 0.314515i | −0.445734 | − | 2.60793i | −1.00000 | −0.473220 | − | 2.96244i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.7 | 0.500000 | + | 0.866025i | 1.47328 | + | 0.910737i | −0.500000 | + | 0.866025i | −1.00000 | −0.0520808 | + | 1.73127i | 2.16962 | − | 1.51418i | −1.00000 | 1.34112 | + | 2.68354i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 331.8 | 0.500000 | + | 0.866025i | 1.72230 | − | 0.183545i | −0.500000 | + | 0.866025i | −1.00000 | 1.02010 | + | 1.39978i | 1.95051 | + | 1.78760i | −1.00000 | 2.93262 | − | 0.632239i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.1 | 0.500000 | − | 0.866025i | −1.73057 | − | 0.0717234i | −0.500000 | − | 0.866025i | −1.00000 | −0.927397 | + | 1.46285i | −2.59109 | − | 0.535003i | −1.00000 | 2.98971 | + | 0.248244i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.2 | 0.500000 | − | 0.866025i | −1.71973 | + | 0.206189i | −0.500000 | − | 0.866025i | −1.00000 | −0.681302 | + | 1.59243i | 2.63512 | − | 0.236999i | −1.00000 | 2.91497 | − | 0.709180i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.3 | 0.500000 | − | 0.866025i | −0.978280 | − | 1.42932i | −0.500000 | − | 0.866025i | −1.00000 | −1.72697 | + | 0.132553i | 1.60537 | + | 2.10304i | −1.00000 | −1.08594 | + | 2.79656i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.4 | 0.500000 | − | 0.866025i | −0.852741 | + | 1.50759i | −0.500000 | − | 0.866025i | −1.00000 | 0.879242 | + | 1.49229i | −1.78630 | + | 1.95170i | −1.00000 | −1.54566 | − | 2.57117i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.5 | 0.500000 | − | 0.866025i | 0.461735 | − | 1.66937i | −0.500000 | − | 0.866025i | −1.00000 | −1.21485 | − | 1.23456i | −1.53750 | − | 2.15316i | −1.00000 | −2.57360 | − | 1.54161i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.6 | 0.500000 | − | 0.866025i | 1.12401 | + | 1.31780i | −0.500000 | − | 0.866025i | −1.00000 | 1.70326 | − | 0.314515i | −0.445734 | + | 2.60793i | −1.00000 | −0.473220 | + | 2.96244i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.7 | 0.500000 | − | 0.866025i | 1.47328 | − | 0.910737i | −0.500000 | − | 0.866025i | −1.00000 | −0.0520808 | − | 1.73127i | 2.16962 | + | 1.51418i | −1.00000 | 1.34112 | − | 2.68354i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 571.8 | 0.500000 | − | 0.866025i | 1.72230 | + | 0.183545i | −0.500000 | − | 0.866025i | −1.00000 | 1.02010 | − | 1.39978i | 1.95051 | − | 1.78760i | −1.00000 | 2.93262 | + | 0.632239i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.g | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.2.l.i | yes | 16 |
| 3.b | odd | 2 | 1 | 1890.2.l.i | 16 | ||
| 7.c | even | 3 | 1 | 630.2.i.i | ✓ | 16 | |
| 9.c | even | 3 | 1 | 630.2.i.i | ✓ | 16 | |
| 9.d | odd | 6 | 1 | 1890.2.i.i | 16 | ||
| 21.h | odd | 6 | 1 | 1890.2.i.i | 16 | ||
| 63.g | even | 3 | 1 | inner | 630.2.l.i | yes | 16 |
| 63.n | odd | 6 | 1 | 1890.2.l.i | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 630.2.i.i | ✓ | 16 | 7.c | even | 3 | 1 | |
| 630.2.i.i | ✓ | 16 | 9.c | even | 3 | 1 | |
| 630.2.l.i | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 630.2.l.i | yes | 16 | 63.g | even | 3 | 1 | inner |
| 1890.2.i.i | 16 | 9.d | odd | 6 | 1 | ||
| 1890.2.i.i | 16 | 21.h | odd | 6 | 1 | ||
| 1890.2.l.i | 16 | 3.b | odd | 2 | 1 | ||
| 1890.2.l.i | 16 | 63.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{8} + T_{11}^{7} - 52T_{11}^{6} + 32T_{11}^{5} + 654T_{11}^{4} - 1716T_{11}^{3} + 1557T_{11}^{2} - 522T_{11} + 54 \)
|
|
\( T_{13}^{16} - 2 T_{13}^{15} + 54 T_{13}^{14} - 8 T_{13}^{13} + 1834 T_{13}^{12} + 477 T_{13}^{11} + \cdots + 2298256 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{8} \)
|
| $3$ |
\( T^{16} + T^{15} + \cdots + 6561 \)
|
| $5$ |
\( (T + 1)^{16} \)
|
| $7$ |
\( T^{16} - 4 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} + T^{7} - 52 T^{6} + \cdots + 54)^{2} \)
|
| $13$ |
\( T^{16} - 2 T^{15} + \cdots + 2298256 \)
|
| $17$ |
\( T^{16} + \cdots + 8707129344 \)
|
| $19$ |
\( T^{16} + \cdots + 1174158756 \)
|
| $23$ |
\( (T^{8} + 11 T^{7} + \cdots + 30456)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 17639292969 \)
|
| $31$ |
\( T^{16} + \cdots + 744962436 \)
|
| $37$ |
\( T^{16} + \cdots + 147622500 \)
|
| $41$ |
\( T^{16} + \cdots + 86812992387801 \)
|
| $43$ |
\( T^{16} + \cdots + 285914281 \)
|
| $47$ |
\( T^{16} + \cdots + 178982301969 \)
|
| $53$ |
\( T^{16} + \cdots + 1238515248996 \)
|
| $59$ |
\( T^{16} - T^{15} + \cdots + 2125764 \)
|
| $61$ |
\( T^{16} + \cdots + 252373607424 \)
|
| $67$ |
\( T^{16} + \cdots + 16\!\cdots\!64 \)
|
| $71$ |
\( (T^{8} + 19 T^{7} + \cdots - 7422678)^{2} \)
|
| $73$ |
\( T^{16} + 8 T^{15} + \cdots + 2143296 \)
|
| $79$ |
\( T^{16} + \cdots + 165269893156 \)
|
| $83$ |
\( T^{16} + \cdots + 2857962683601 \)
|
| $89$ |
\( T^{16} + \cdots + 99588211776 \)
|
| $97$ |
\( T^{16} + \cdots + 1545178274704 \)
|
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