Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,3,Mod(11,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.i (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: | \(1.22616118962\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{14} + 43 \nu^{12} + 715 \nu^{10} + 5777 \nu^{8} + 23051 \nu^{6} + 39821 \nu^{4} + 25605 \nu^{2} + \cdots + 28755 ) / 4752 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{14} - 43 \nu^{12} - 715 \nu^{10} - 5777 \nu^{8} - 23051 \nu^{6} - 39821 \nu^{4} - 20853 \nu^{2} + \cdots - 243 ) / 4752 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7 \nu^{15} - 68 \nu^{14} + 369 \nu^{13} - 3056 \nu^{12} + 7299 \nu^{11} - 53504 \nu^{10} + \cdots - 532116 ) / 175824 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53 \nu^{15} + 155 \nu^{14} + 2741 \nu^{13} + 6659 \nu^{12} + 54989 \nu^{11} + 108827 \nu^{10} + \cdots + 1364877 ) / 351648 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 215 \nu^{15} + 195 \nu^{14} + 10139 \nu^{13} + 9423 \nu^{12} + 187691 \nu^{11} + 177387 \nu^{10} + \cdots + 1868049 ) / 351648 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 3 \nu^{15} - 143 \nu^{13} - 2693 \nu^{11} - 25397 \nu^{9} - 125029 \nu^{7} - 304045 \nu^{5} + \cdots - 2376 ) / 4752 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 261 \nu^{15} + 145 \nu^{14} - 12289 \nu^{13} + 6301 \nu^{12} - 227389 \nu^{11} + 106117 \nu^{10} + \cdots + 1084239 ) / 351648 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 247 \nu^{15} - 77 \nu^{14} + 11847 \nu^{13} - 3245 \nu^{12} + 225075 \nu^{11} - 52613 \nu^{10} + \cdots + 678645 ) / 351648 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 247 \nu^{15} - 77 \nu^{14} - 11847 \nu^{13} - 3245 \nu^{12} - 225075 \nu^{11} - 52613 \nu^{10} + \cdots + 678645 ) / 351648 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 67 \nu^{15} - 5 \nu^{14} + 3257 \nu^{13} - 290 \nu^{12} + 62816 \nu^{11} - 6572 \nu^{10} + \cdots - 202257 ) / 87912 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 245 \nu^{15} + 19 \nu^{14} - 11805 \nu^{13} + 769 \nu^{12} - 225273 \nu^{11} + 12253 \nu^{10} + \cdots + 336609 ) / 175824 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 161 \nu^{15} - 16 \nu^{14} + 7747 \nu^{13} - 632 \nu^{12} + 147749 \nu^{11} - 9072 \nu^{10} + \cdots + 92304 ) / 58608 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 17 \nu^{15} - 815 \nu^{13} - 15443 \nu^{11} - 146605 \nu^{9} - 727123 \nu^{7} - 1784449 \nu^{5} + \cdots - 14256 ) / 4752 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1361 \nu^{15} - 13 \nu^{14} - 65433 \nu^{13} - 421 \nu^{12} - 1245657 \nu^{11} - 2413 \nu^{10} + \cdots + 1470933 ) / 351648 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - \beta_{14} + 2\beta_{11} - \beta_{8} + 3\beta_{7} - 2\beta_{4} - \beta_{2} - 10\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{12} + \beta_{11} + 3\beta_{10} + 3\beta_{9} - \beta_{7} - 18\beta_{3} - 12\beta_{2} + 3\beta _1 + 60 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 18 \beta_{15} + 26 \beta_{14} + 6 \beta_{13} + 4 \beta_{12} - 34 \beta_{11} - 6 \beta_{10} + \cdots - 42 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 23 \beta_{12} - 27 \beta_{11} - 57 \beta_{10} - 59 \beta_{9} + \cdots - 681 \)
|
| \(\nu^{7}\) | \(=\) |
\( 267 \beta_{15} - 471 \beta_{14} - 144 \beta_{13} - 96 \beta_{12} + 504 \beta_{11} + 186 \beta_{10} + \cdots + 684 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 108 \beta_{15} + 108 \beta_{14} - 48 \beta_{13} + 432 \beta_{12} + 528 \beta_{11} + 900 \beta_{10} + \cdots + 8187 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3806 \beta_{15} + 7550 \beta_{14} + 2568 \beta_{13} + 1746 \beta_{12} - 7258 \beta_{11} + \cdots - 10198 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2088 \beta_{15} - 2088 \beta_{14} + 924 \beta_{13} - 7298 \beta_{12} - 9146 \beta_{11} - 13506 \beta_{10} + \cdots - 102042 \)
|
| \(\nu^{11}\) | \(=\) |
\( 53703 \beta_{15} - 114319 \beta_{14} - 41040 \beta_{13} - 28598 \beta_{12} + 103604 \beta_{11} + \cdots + 146268 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 35528 \beta_{15} + 35528 \beta_{14} - 16156 \beta_{13} + 115609 \beta_{12} + 147921 \beta_{11} + \cdots + 1306266 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 755544 \beta_{15} + 1678848 \beta_{14} + 622506 \beta_{13} + 443874 \beta_{12} - 1472856 \beta_{11} + \cdots - 2059032 \)
|
| \(\nu^{14}\) | \(=\) |
\( 566496 \beta_{15} - 566496 \beta_{14} + 265242 \beta_{13} - 1758429 \beta_{12} - 2288913 \beta_{11} + \cdots - 17072361 \)
|
| \(\nu^{15}\) | \(=\) |
\( 10619317 \beta_{15} - 24221449 \beta_{14} - 9179076 \beta_{13} - 6672042 \beta_{12} + 20879918 \beta_{11} + \cdots + 28720238 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1 + \beta_{7}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−3.23594 | − | 1.86827i | 2.62117 | + | 1.45927i | 4.98088 | + | 8.62715i | 1.93649 | − | 1.11803i | −5.75562 | − | 9.61919i | 3.63061 | − | 6.28840i | − | 22.2764i | 4.74103 | + | 7.65001i | −8.35517 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −2.83245 | − | 1.63532i | −2.26486 | − | 1.96733i | 3.34853 | + | 5.79983i | −1.93649 | + | 1.11803i | 3.19790 | + | 9.27615i | −3.16931 | + | 5.48940i | − | 8.82112i | 1.25919 | + | 8.91148i | 7.31337 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −1.20554 | − | 0.696021i | 2.33148 | − | 1.88791i | −1.03111 | − | 1.78593i | −1.93649 | + | 1.11803i | −4.12473 | + | 0.653207i | 4.41004 | − | 7.63842i | 8.43886i | 1.87156 | − | 8.80325i | 3.11270 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −0.0412321 | − | 0.0238054i | −1.76580 | − | 2.42528i | −1.99887 | − | 3.46214i | 1.93649 | − | 1.11803i | 0.0150729 | + | 0.142035i | 1.90756 | − | 3.30399i | 0.380778i | −2.76393 | + | 8.56508i | −0.106461 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 0.600071 | + | 0.346451i | 2.96601 | + | 0.450312i | −1.75994 | − | 3.04831i | 1.93649 | − | 1.11803i | 1.62381 | + | 1.29780i | −6.14112 | + | 10.6367i | − | 5.21055i | 8.59444 | + | 2.67126i | 1.54938 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 1.58822 | + | 0.916957i | 0.822398 | + | 2.88508i | −0.318381 | − | 0.551452i | −1.93649 | + | 1.11803i | −1.33934 | + | 5.33623i | 3.23398 | − | 5.60142i | − | 8.50342i | −7.64732 | + | 4.74536i | −4.10076 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 2.44978 | + | 1.41438i | 0.110987 | − | 2.99795i | 2.00096 | + | 3.46576i | −1.93649 | + | 1.11803i | 4.51214 | − | 7.18734i | −3.97472 | + | 6.88441i | 0.00541780i | −8.97536 | − | 0.665467i | −6.32531 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 2.67710 | + | 1.54563i | −2.82138 | + | 1.01971i | 2.77793 | + | 4.81151i | 1.93649 | − | 1.11803i | −9.12922 | − | 1.63094i | 1.10296 | − | 1.91037i | 4.80953i | 6.92040 | − | 5.75396i | 6.91225 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | −3.23594 | + | 1.86827i | 2.62117 | − | 1.45927i | 4.98088 | − | 8.62715i | 1.93649 | + | 1.11803i | −5.75562 | + | 9.61919i | 3.63061 | + | 6.28840i | 22.2764i | 4.74103 | − | 7.65001i | −8.35517 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | −2.83245 | + | 1.63532i | −2.26486 | + | 1.96733i | 3.34853 | − | 5.79983i | −1.93649 | − | 1.11803i | 3.19790 | − | 9.27615i | −3.16931 | − | 5.48940i | 8.82112i | 1.25919 | − | 8.91148i | 7.31337 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | −1.20554 | + | 0.696021i | 2.33148 | + | 1.88791i | −1.03111 | + | 1.78593i | −1.93649 | − | 1.11803i | −4.12473 | − | 0.653207i | 4.41004 | + | 7.63842i | − | 8.43886i | 1.87156 | + | 8.80325i | 3.11270 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | −0.0412321 | + | 0.0238054i | −1.76580 | + | 2.42528i | −1.99887 | + | 3.46214i | 1.93649 | + | 1.11803i | 0.0150729 | − | 0.142035i | 1.90756 | + | 3.30399i | − | 0.380778i | −2.76393 | − | 8.56508i | −0.106461 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.5 | 0.600071 | − | 0.346451i | 2.96601 | − | 0.450312i | −1.75994 | + | 3.04831i | 1.93649 | + | 1.11803i | 1.62381 | − | 1.29780i | −6.14112 | − | 10.6367i | 5.21055i | 8.59444 | − | 2.67126i | 1.54938 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.6 | 1.58822 | − | 0.916957i | 0.822398 | − | 2.88508i | −0.318381 | + | 0.551452i | −1.93649 | − | 1.11803i | −1.33934 | − | 5.33623i | 3.23398 | + | 5.60142i | 8.50342i | −7.64732 | − | 4.74536i | −4.10076 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.7 | 2.44978 | − | 1.41438i | 0.110987 | + | 2.99795i | 2.00096 | − | 3.46576i | −1.93649 | − | 1.11803i | 4.51214 | + | 7.18734i | −3.97472 | − | 6.88441i | − | 0.00541780i | −8.97536 | + | 0.665467i | −6.32531 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.8 | 2.67710 | − | 1.54563i | −2.82138 | − | 1.01971i | 2.77793 | − | 4.81151i | 1.93649 | + | 1.11803i | −9.12922 | + | 1.63094i | 1.10296 | + | 1.91037i | − | 4.80953i | 6.92040 | + | 5.75396i | 6.91225 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.3.i.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 135.3.i.a | 16 | ||
| 4.b | odd | 2 | 1 | 720.3.bs.c | 16 | ||
| 5.b | even | 2 | 1 | 225.3.j.b | 16 | ||
| 5.c | odd | 4 | 2 | 225.3.i.b | 32 | ||
| 9.c | even | 3 | 1 | 135.3.i.a | 16 | ||
| 9.c | even | 3 | 1 | 405.3.c.a | 16 | ||
| 9.d | odd | 6 | 1 | inner | 45.3.i.a | ✓ | 16 |
| 9.d | odd | 6 | 1 | 405.3.c.a | 16 | ||
| 12.b | even | 2 | 1 | 2160.3.bs.c | 16 | ||
| 15.d | odd | 2 | 1 | 675.3.j.b | 16 | ||
| 15.e | even | 4 | 2 | 675.3.i.c | 32 | ||
| 36.f | odd | 6 | 1 | 2160.3.bs.c | 16 | ||
| 36.h | even | 6 | 1 | 720.3.bs.c | 16 | ||
| 45.h | odd | 6 | 1 | 225.3.j.b | 16 | ||
| 45.j | even | 6 | 1 | 675.3.j.b | 16 | ||
| 45.k | odd | 12 | 2 | 675.3.i.c | 32 | ||
| 45.l | even | 12 | 2 | 225.3.i.b | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.3.i.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 45.3.i.a | ✓ | 16 | 9.d | odd | 6 | 1 | inner |
| 135.3.i.a | 16 | 3.b | odd | 2 | 1 | ||
| 135.3.i.a | 16 | 9.c | even | 3 | 1 | ||
| 225.3.i.b | 32 | 5.c | odd | 4 | 2 | ||
| 225.3.i.b | 32 | 45.l | even | 12 | 2 | ||
| 225.3.j.b | 16 | 5.b | even | 2 | 1 | ||
| 225.3.j.b | 16 | 45.h | odd | 6 | 1 | ||
| 405.3.c.a | 16 | 9.c | even | 3 | 1 | ||
| 405.3.c.a | 16 | 9.d | odd | 6 | 1 | ||
| 675.3.i.c | 32 | 15.e | even | 4 | 2 | ||
| 675.3.i.c | 32 | 45.k | odd | 12 | 2 | ||
| 675.3.j.b | 16 | 15.d | odd | 2 | 1 | ||
| 675.3.j.b | 16 | 45.j | even | 6 | 1 | ||
| 720.3.bs.c | 16 | 4.b | odd | 2 | 1 | ||
| 720.3.bs.c | 16 | 36.h | even | 6 | 1 | ||
| 2160.3.bs.c | 16 | 12.b | even | 2 | 1 | ||
| 2160.3.bs.c | 16 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 24 T^{14} + \cdots + 81 \)
|
| $3$ |
\( T^{16} - 4 T^{15} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{4} \)
|
| $7$ |
\( T^{16} + \cdots + 4654875290256 \)
|
| $11$ |
\( T^{16} + \cdots + 112356358416 \)
|
| $13$ |
\( T^{16} + \cdots + 69380636886016 \)
|
| $17$ |
\( T^{16} + \cdots + 17\!\cdots\!00 \)
|
| $19$ |
\( (T^{8} + 26 T^{7} + \cdots - 3133657244)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 64\!\cdots\!16 \)
|
| $29$ |
\( T^{16} + \cdots + 21\!\cdots\!56 \)
|
| $31$ |
\( T^{16} + \cdots + 396718580736 \)
|
| $37$ |
\( (T^{8} - 22 T^{7} + \cdots - 143779124336)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 44\!\cdots\!61 \)
|
| $43$ |
\( T^{16} + \cdots + 36\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 28\!\cdots\!76 \)
|
| $53$ |
\( T^{16} + \cdots + 32\!\cdots\!00 \)
|
| $59$ |
\( T^{16} + \cdots + 25\!\cdots\!76 \)
|
| $61$ |
\( T^{16} + \cdots + 18\!\cdots\!56 \)
|
| $67$ |
\( T^{16} + \cdots + 39\!\cdots\!61 \)
|
| $71$ |
\( T^{16} + \cdots + 34\!\cdots\!00 \)
|
| $73$ |
\( (T^{8} + \cdots - 3873480104384)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 15\!\cdots\!16 \)
|
| $83$ |
\( T^{16} + \cdots + 23\!\cdots\!96 \)
|
| $89$ |
\( T^{16} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 17\!\cdots\!76 \)
|
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