Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,2,Mod(518,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.518");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.k (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: | \(6.58765816676\) |
| 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: |
\( x^{16} - 4 x^{15} + 8 x^{14} + 19 x^{12} - 80 x^{11} + 168 x^{10} + 28 x^{9} + 119 x^{8} - 432 x^{7} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{16} - 4 x^{15} + 8 x^{14} + 19 x^{12} - 80 x^{11} + 168 x^{10} + 28 x^{9} + 119 x^{8} - 432 x^{7} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4128071754399 \nu^{15} + 11898447714838 \nu^{14} - 15277599976658 \nu^{13} + \cdots - 80\!\cdots\!20 ) / 27\!\cdots\!44 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21143084361889 \nu^{15} - 76316193938758 \nu^{14} + 145347779465436 \nu^{13} + \cdots - 19\!\cdots\!44 ) / 54\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 38450704565045 \nu^{15} - 59618622420401 \nu^{14} - 46616013291644 \nu^{13} + \cdots + 14\!\cdots\!96 ) / 54\!\cdots\!88 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 106357174118413 \nu^{15} - 451134746013694 \nu^{14} + 911994273881980 \nu^{13} + \cdots - 12\!\cdots\!60 ) / 10\!\cdots\!76 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 157213841010915 \nu^{15} - 509728662885302 \nu^{14} + 841883624893756 \nu^{13} + \cdots + 16\!\cdots\!72 ) / 10\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 207551406788003 \nu^{15} - 560257236979352 \nu^{14} + 726910331340020 \nu^{13} + \cdots + 29\!\cdots\!80 ) / 10\!\cdots\!76 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 247784897113907 \nu^{15} - 933231595130810 \nu^{14} + \cdots - 40\!\cdots\!32 ) / 10\!\cdots\!76 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 249727967307343 \nu^{15} - 956625700505594 \nu^{14} + \cdots - 14\!\cdots\!96 ) / 10\!\cdots\!76 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 83688943960991 \nu^{15} - 319218427676056 \nu^{14} + 612574342939380 \nu^{13} + \cdots - 12\!\cdots\!88 ) / 27\!\cdots\!44 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 426823538486029 \nu^{15} + \cdots + 58\!\cdots\!96 ) / 10\!\cdots\!76 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 233776061987612 \nu^{15} - 998483250651247 \nu^{14} + \cdots - 28\!\cdots\!44 ) / 54\!\cdots\!88 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 67717762917308 \nu^{15} + 258235473611088 \nu^{14} - 497634782511666 \nu^{13} + \cdots + 41\!\cdots\!60 ) / 13\!\cdots\!22 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 626947109265783 \nu^{15} + \cdots + 25\!\cdots\!04 ) / 10\!\cdots\!76 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 627014941431923 \nu^{15} + \cdots - 41\!\cdots\!28 ) / 10\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{13} + 2\beta_{9} + \beta_{3} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{9} + \beta_{5} + 4\beta_{3} - \beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{14} - \beta_{8} + \beta_{5} + \beta_{3} - 6\beta_{2} - \beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} + \beta_{14} - 8 \beta_{13} + \beta_{12} - \beta_{11} - 10 \beta_{9} - 8 \beta_{8} + \cdots - 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 37 \beta_{13} + 4 \beta_{12} - 9 \beta_{11} - \beta_{10} - 52 \beta_{9} - 11 \beta_{8} + \cdots - 47 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 11 \beta_{15} - 9 \beta_{14} - 57 \beta_{13} + 15 \beta_{12} - 13 \beta_{11} - 77 \beta_{9} + \cdots + 134 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 53 \beta_{15} - 53 \beta_{14} - 11 \beta_{11} + 15 \beta_{10} + 92 \beta_{8} + 15 \beta_{7} + \cdots + 567 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 62 \beta_{15} - 92 \beta_{14} + 394 \beta_{13} - 154 \beta_{12} + 92 \beta_{11} + 60 \beta_{10} + \cdots + 948 \)
|
| \(\nu^{10}\) | \(=\) |
\( 60 \beta_{15} - 60 \beta_{14} + 1543 \beta_{13} - 612 \beta_{12} + 496 \beta_{11} + 154 \beta_{10} + \cdots + 2149 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 702 \beta_{15} + 394 \beta_{14} + 2697 \beta_{13} - 1356 \beta_{12} + 1006 \beta_{11} + 3893 \beta_{9} + \cdots - 6590 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2227 \beta_{15} + 2227 \beta_{14} + 744 \beta_{11} - 1356 \beta_{10} - 5149 \beta_{8} - 1356 \beta_{7} + \cdots - 25195 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2437 \beta_{15} + 5149 \beta_{14} - 18424 \beta_{13} + 11019 \beta_{12} - 5149 \beta_{11} - 5336 \beta_{10} + \cdots - 45538 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 5336 \beta_{15} + 5336 \beta_{14} - 69179 \beta_{13} + 42920 \beta_{12} - 25741 \beta_{11} + \cdots - 99995 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 37029 \beta_{15} - 14991 \beta_{14} - 126005 \beta_{13} + 85363 \beta_{12} - 57911 \beta_{11} + \cdots + 314250 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).
| \(n\) | \(376\) | \(551\) | \(727\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 518.1 |
|
−1.31757 | − | 1.31757i | −1.52275 | − | 0.825374i | 1.47196i | 0 | 0.918834 | + | 3.09381i | 2.09381 | − | 2.09381i | −0.695724 | + | 0.695724i | 1.63751 | + | 2.51367i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.2 | −1.14633 | − | 1.14633i | 0.582649 | − | 1.63111i | 0.628149i | 0 | −2.53770 | + | 1.20188i | 0.201884 | − | 0.201884i | −1.57260 | + | 1.57260i | −2.32104 | − | 1.90073i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.3 | −0.688858 | − | 0.688858i | 1.67171 | + | 0.453204i | − | 1.05095i | 0 | −0.839376 | − | 1.46376i | −2.46376 | + | 2.46376i | −2.10167 | + | 2.10167i | 2.58921 | + | 1.51525i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.4 | 0.178913 | + | 0.178913i | −0.720382 | − | 1.57513i | − | 1.93598i | 0 | 0.152926 | − | 0.410697i | −1.41070 | + | 1.41070i | 0.704196 | − | 0.704196i | −1.96210 | + | 2.26940i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.5 | 0.416745 | + | 0.416745i | −0.554354 | + | 1.64094i | − | 1.65265i | 0 | −0.914879 | + | 0.452830i | −0.547170 | + | 0.547170i | 1.52222 | − | 1.52222i | −2.38538 | − | 1.81933i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.6 | 1.05652 | + | 1.05652i | 1.71860 | + | 0.215468i | 0.232479i | 0 | 1.58809 | + | 2.04338i | 1.04338 | − | 1.04338i | 1.86743 | − | 1.86743i | 2.90715 | + | 0.740604i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.7 | 1.63522 | + | 1.63522i | 1.37033 | − | 1.05933i | 3.34791i | 0 | 3.97305 | + | 0.508557i | −0.491443 | + | 0.491443i | −2.20413 | + | 2.20413i | 0.755630 | − | 2.90328i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 518.8 | 1.86535 | + | 1.86535i | −1.54580 | + | 0.781338i | 4.95908i | 0 | −4.34094 | − | 1.42600i | −2.42600 | + | 2.42600i | −5.51973 | + | 5.51973i | 1.77902 | − | 2.41559i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.1 | −1.31757 | + | 1.31757i | −1.52275 | + | 0.825374i | − | 1.47196i | 0 | 0.918834 | − | 3.09381i | 2.09381 | + | 2.09381i | −0.695724 | − | 0.695724i | 1.63751 | − | 2.51367i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.2 | −1.14633 | + | 1.14633i | 0.582649 | + | 1.63111i | − | 0.628149i | 0 | −2.53770 | − | 1.20188i | 0.201884 | + | 0.201884i | −1.57260 | − | 1.57260i | −2.32104 | + | 1.90073i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.3 | −0.688858 | + | 0.688858i | 1.67171 | − | 0.453204i | 1.05095i | 0 | −0.839376 | + | 1.46376i | −2.46376 | − | 2.46376i | −2.10167 | − | 2.10167i | 2.58921 | − | 1.51525i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.4 | 0.178913 | − | 0.178913i | −0.720382 | + | 1.57513i | 1.93598i | 0 | 0.152926 | + | 0.410697i | −1.41070 | − | 1.41070i | 0.704196 | + | 0.704196i | −1.96210 | − | 2.26940i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.5 | 0.416745 | − | 0.416745i | −0.554354 | − | 1.64094i | 1.65265i | 0 | −0.914879 | − | 0.452830i | −0.547170 | − | 0.547170i | 1.52222 | + | 1.52222i | −2.38538 | + | 1.81933i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.6 | 1.05652 | − | 1.05652i | 1.71860 | − | 0.215468i | − | 0.232479i | 0 | 1.58809 | − | 2.04338i | 1.04338 | + | 1.04338i | 1.86743 | + | 1.86743i | 2.90715 | − | 0.740604i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.7 | 1.63522 | − | 1.63522i | 1.37033 | + | 1.05933i | − | 3.34791i | 0 | 3.97305 | − | 0.508557i | −0.491443 | − | 0.491443i | −2.20413 | − | 2.20413i | 0.755630 | + | 2.90328i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 782.8 | 1.86535 | − | 1.86535i | −1.54580 | − | 0.781338i | − | 4.95908i | 0 | −4.34094 | + | 1.42600i | −2.42600 | − | 2.42600i | −5.51973 | − | 5.51973i | 1.77902 | + | 2.41559i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 825.2.k.j | 16 | |
| 3.b | odd | 2 | 1 | 825.2.k.i | 16 | ||
| 5.b | even | 2 | 1 | 165.2.k.c | ✓ | 16 | |
| 5.c | odd | 4 | 1 | 165.2.k.d | yes | 16 | |
| 5.c | odd | 4 | 1 | 825.2.k.i | 16 | ||
| 15.d | odd | 2 | 1 | 165.2.k.d | yes | 16 | |
| 15.e | even | 4 | 1 | 165.2.k.c | ✓ | 16 | |
| 15.e | even | 4 | 1 | inner | 825.2.k.j | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.k.c | ✓ | 16 | 5.b | even | 2 | 1 | |
| 165.2.k.c | ✓ | 16 | 15.e | even | 4 | 1 | |
| 165.2.k.d | yes | 16 | 5.c | odd | 4 | 1 | |
| 165.2.k.d | yes | 16 | 15.d | odd | 2 | 1 | |
| 825.2.k.i | 16 | 3.b | odd | 2 | 1 | ||
| 825.2.k.i | 16 | 5.c | odd | 4 | 1 | ||
| 825.2.k.j | 16 | 1.a | even | 1 | 1 | trivial | |
| 825.2.k.j | 16 | 15.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):
|
\( T_{2}^{16} - 4 T_{2}^{15} + 8 T_{2}^{14} + 19 T_{2}^{12} - 80 T_{2}^{11} + 168 T_{2}^{10} + 28 T_{2}^{9} + \cdots + 16 \)
|
|
\( T_{7}^{16} + 8 T_{7}^{15} + 32 T_{7}^{14} + 56 T_{7}^{13} + 128 T_{7}^{12} + 648 T_{7}^{11} + 2656 T_{7}^{10} + \cdots + 256 \)
|
|
\( T_{13}^{16} + 40 T_{13}^{13} + 1304 T_{13}^{12} + 1112 T_{13}^{11} + 800 T_{13}^{10} + 21472 T_{13}^{9} + \cdots + 9216 \)
|
|
\( T_{29}^{8} + 8T_{29}^{7} - 70T_{29}^{6} - 504T_{29}^{5} + 956T_{29}^{4} + 8528T_{29}^{3} + 3784T_{29}^{2} - 20576T_{29} - 3392 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 4 T^{15} + \cdots + 16 \)
|
| $3$ |
\( T^{16} - 2 T^{15} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + 8 T^{15} + \cdots + 256 \)
|
| $11$ |
\( (T^{2} + 1)^{8} \)
|
| $13$ |
\( T^{16} + 40 T^{13} + \cdots + 9216 \)
|
| $17$ |
\( T^{16} + 8 T^{15} + \cdots + 256 \)
|
| $19$ |
\( T^{16} + 228 T^{14} + \cdots + 35426304 \)
|
| $23$ |
\( T^{16} + \cdots + 1615396864 \)
|
| $29$ |
\( (T^{8} + 8 T^{7} + \cdots - 3392)^{2} \)
|
| $31$ |
\( (T^{8} - 4 T^{7} + \cdots + 218512)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 214369000000 \)
|
| $41$ |
\( T^{16} + \cdots + 514444693504 \)
|
| $43$ |
\( T^{16} + \cdots + 1923348736 \)
|
| $47$ |
\( T^{16} + \cdots + 171409248256 \)
|
| $53$ |
\( T^{16} - 4 T^{15} + \cdots + 51609856 \)
|
| $59$ |
\( (T^{8} + 34 T^{7} + \cdots + 1225520)^{2} \)
|
| $61$ |
\( (T^{8} + 4 T^{7} + \cdots - 441344)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 3634642944 \)
|
| $71$ |
\( T^{16} + \cdots + 210027890944 \)
|
| $73$ |
\( T^{16} + \cdots + 15170339487744 \)
|
| $79$ |
\( T^{16} + \cdots + 14903526400 \)
|
| $83$ |
\( T^{16} + \cdots + 4917936384 \)
|
| $89$ |
\( (T^{8} - 16 T^{7} + \cdots + 112304)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 629744570362944 \)
|
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