Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,3,Mod(11,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, 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("90.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.h (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: | \(2.45232237924\) |
| 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: |
\( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| 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
\( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 41 \nu^{14} - 225 \nu^{12} + 1333 \nu^{10} + 31 \nu^{8} - 114072 \nu^{6} + 798716 \nu^{4} + \cdots + 15616332 ) / 4341600 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 289974 \nu^{15} - 348681 \nu^{14} + 3618395 \nu^{13} - 9875745 \nu^{12} - 23841698 \nu^{11} + \cdots - 355528882692 ) / 93092587200 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 399973 \nu^{15} - 1285313 \nu^{14} + 10390855 \nu^{13} + 16249060 \nu^{12} + \cdots + 398535722064 ) / 93092587200 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 334523 \nu^{15} + 505875 \nu^{14} - 4891765 \nu^{13} - 10514390 \nu^{12} + 54769019 \nu^{11} + \cdots - 217467572760 ) / 31030862400 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 72336 \nu^{15} - 137953 \nu^{14} - 727168 \nu^{13} + 1003727 \nu^{12} + 8534536 \nu^{11} + \cdots + 238211532 ) / 3723703488 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 389063 \nu^{15} - 2906172 \nu^{14} + 3910476 \nu^{13} + 32198997 \nu^{12} + \cdots + 254978122644 ) / 18618517440 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1043836 \nu^{15} + 64681 \nu^{14} - 7299935 \nu^{13} + 2483580 \nu^{12} + 85117948 \nu^{11} + \cdots - 1432070568 ) / 46546293600 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2122022 \nu^{15} - 3591180 \nu^{14} + 26082035 \nu^{13} + 40328710 \nu^{12} + \cdots + 527986862280 ) / 93092587200 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 215 \nu^{15} - 2296 \nu^{13} + 25907 \nu^{11} - 183272 \nu^{9} + 859816 \nu^{7} - 4023896 \nu^{5} + \cdots + 4110048 ) / 8220096 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 881988 \nu^{15} + 2003904 \nu^{14} - 8761925 \nu^{13} - 15591600 \nu^{12} + 105064844 \nu^{11} + \cdots + 1201238208 ) / 31030862400 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2939307 \nu^{15} + 1272107 \nu^{14} + 15171655 \nu^{13} - 28863275 \nu^{12} + \cdots - 476058785436 ) / 93092587200 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 9042 \nu^{15} + 90896 \nu^{13} - 1066817 \nu^{11} + 7184007 \nu^{9} - 35214371 \nu^{7} + \cdots + 642245346 \nu ) / 232731468 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 13164 \nu^{15} + 130775 \nu^{13} - 1568132 \nu^{11} + 10603551 \nu^{9} + \cdots + 1021382172 \nu ) / 231573600 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 6407642 \nu^{15} - 1841030 \nu^{14} + 51299115 \nu^{13} + 26308340 \nu^{12} + \cdots + 264953937600 ) / 93092587200 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 48776 \nu^{15} + 2911 \nu^{14} - 509200 \nu^{13} - 15975 \nu^{12} + 5779688 \nu^{11} + \cdots + 1108759572 ) / 616507200 \)
|
| \(\nu\) | \(=\) |
\( ( - 2 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 4 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2 \beta_{15} + 6 \beta_{14} - 3 \beta_{12} - 6 \beta_{11} - \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 12 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 5 \beta_{15} - 4 \beta_{14} - 18 \beta_{13} + 45 \beta_{12} - 4 \beta_{11} - 11 \beta_{10} + \cdots - 38 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 4 \beta_{15} + 6 \beta_{14} - 18 \beta_{13} - 33 \beta_{12} - 6 \beta_{11} - 37 \beta_{10} + \cdots - 90 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19 \beta_{15} - 52 \beta_{14} + 144 \beta_{13} - 27 \beta_{12} - 52 \beta_{11} - 47 \beta_{10} + \cdots - 176 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 118 \beta_{15} - 246 \beta_{14} - 90 \beta_{13} + 99 \beta_{12} + 246 \beta_{11} - 109 \beta_{10} + \cdots - 102 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 391 \beta_{15} + 290 \beta_{14} + 396 \beta_{13} - 2367 \beta_{12} + 290 \beta_{11} + 319 \beta_{10} + \cdots + 2230 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 256 \beta_{15} - 714 \beta_{14} + 1170 \beta_{13} + 3039 \beta_{12} + 714 \beta_{11} + 2747 \beta_{10} + \cdots + 5430 ) / 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 2417 \beta_{15} + 4028 \beta_{14} - 9216 \beta_{13} - 6435 \beta_{12} + 4028 \beta_{11} + \cdots + 13552 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 3614 \beta_{15} + 11814 \beta_{14} + 6534 \beta_{13} - 1815 \beta_{12} - 11814 \beta_{11} + 10925 \beta_{10} + \cdots + 24930 ) / 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 46367 \beta_{15} - 3310 \beta_{14} - 52128 \beta_{13} + 85239 \beta_{12} - 3310 \beta_{11} + \cdots - 80726 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 22112 \beta_{15} + 81330 \beta_{14} - 73674 \beta_{13} - 220923 \beta_{12} - 81330 \beta_{11} + \cdots - 212358 ) / 6 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 50989 \beta_{15} - 258820 \beta_{14} + 353016 \beta_{13} + 572463 \beta_{12} - 258820 \beta_{11} + \cdots - 856784 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 150718 \beta_{15} - 451686 \beta_{14} - 517374 \beta_{13} - 380793 \beta_{12} + 451686 \beta_{11} + \cdots - 2222106 ) / 6 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 3029575 \beta_{15} - 660730 \beta_{14} + 3995640 \beta_{13} - 3532959 \beta_{12} - 660730 \beta_{11} + \cdots + 2510854 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1 - \beta_{9}\) | \(1\) |
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 |
|
−1.22474 | − | 0.707107i | −2.99847 | − | 0.0956863i | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 3.60470 | + | 2.23743i | −4.35057 | + | 7.53542i | − | 2.82843i | 8.98169 | + | 0.573826i | −3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.2 | −1.22474 | − | 0.707107i | −2.68332 | + | 1.34156i | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 4.23501 | + | 0.254321i | 6.95799 | − | 12.0516i | − | 2.82843i | 5.40042 | − | 7.19969i | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.3 | −1.22474 | − | 0.707107i | −0.129213 | − | 2.99722i | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | −1.96110 | + | 3.76219i | 3.31598 | − | 5.74345i | − | 2.82843i | −8.96661 | + | 0.774559i | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.4 | −1.22474 | − | 0.707107i | 1.13677 | + | 2.77628i | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 0.570872 | − | 4.20406i | −3.24916 | + | 5.62771i | − | 2.82843i | −6.41550 | + | 6.31200i | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.5 | 1.22474 | + | 0.707107i | −2.43483 | + | 1.75260i | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | −4.22132 | + | 0.424801i | −5.39499 | + | 9.34440i | 2.82843i | 2.85680 | − | 8.53456i | −3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.6 | 1.22474 | + | 0.707107i | 0.605881 | + | 2.93818i | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | −1.33556 | + | 4.02694i | 0.531482 | − | 0.920554i | 2.82843i | −8.26582 | + | 3.56038i | 3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.7 | 1.22474 | + | 0.707107i | 1.52181 | − | 2.58536i | 1.00000 | + | 1.73205i | 1.93649 | − | 1.11803i | 3.69195 | − | 2.09033i | −0.496891 | + | 0.860641i | 2.82843i | −4.36821 | − | 7.86884i | 3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11.8 | 1.22474 | + | 0.707107i | 2.98138 | + | 0.333742i | 1.00000 | + | 1.73205i | −1.93649 | + | 1.11803i | 3.41544 | + | 2.51690i | 0.686165 | − | 1.18847i | 2.82843i | 8.77723 | + | 1.99002i | −3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | −1.22474 | + | 0.707107i | −2.99847 | + | 0.0956863i | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 3.60470 | − | 2.23743i | −4.35057 | − | 7.53542i | 2.82843i | 8.98169 | − | 0.573826i | −3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | −1.22474 | + | 0.707107i | −2.68332 | − | 1.34156i | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 4.23501 | − | 0.254321i | 6.95799 | + | 12.0516i | 2.82843i | 5.40042 | + | 7.19969i | 3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | −1.22474 | + | 0.707107i | −0.129213 | + | 2.99722i | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | −1.96110 | − | 3.76219i | 3.31598 | + | 5.74345i | 2.82843i | −8.96661 | − | 0.774559i | −3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | −1.22474 | + | 0.707107i | 1.13677 | − | 2.77628i | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 0.570872 | + | 4.20406i | −3.24916 | − | 5.62771i | 2.82843i | −6.41550 | − | 6.31200i | 3.16228 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.5 | 1.22474 | − | 0.707107i | −2.43483 | − | 1.75260i | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | −4.22132 | − | 0.424801i | −5.39499 | − | 9.34440i | − | 2.82843i | 2.85680 | + | 8.53456i | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.6 | 1.22474 | − | 0.707107i | 0.605881 | − | 2.93818i | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | −1.33556 | − | 4.02694i | 0.531482 | + | 0.920554i | − | 2.82843i | −8.26582 | − | 3.56038i | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.7 | 1.22474 | − | 0.707107i | 1.52181 | + | 2.58536i | 1.00000 | − | 1.73205i | 1.93649 | + | 1.11803i | 3.69195 | + | 2.09033i | −0.496891 | − | 0.860641i | − | 2.82843i | −4.36821 | + | 7.86884i | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.8 | 1.22474 | − | 0.707107i | 2.98138 | − | 0.333742i | 1.00000 | − | 1.73205i | −1.93649 | − | 1.11803i | 3.41544 | − | 2.51690i | 0.686165 | + | 1.18847i | − | 2.82843i | 8.77723 | − | 1.99002i | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 90.3.h.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 270.3.h.a | 16 | ||
| 4.b | odd | 2 | 1 | 720.3.bs.d | 16 | ||
| 5.b | even | 2 | 1 | 450.3.i.g | 16 | ||
| 5.c | odd | 4 | 2 | 450.3.k.c | 32 | ||
| 9.c | even | 3 | 1 | 270.3.h.a | 16 | ||
| 9.c | even | 3 | 1 | 810.3.d.c | 16 | ||
| 9.d | odd | 6 | 1 | inner | 90.3.h.a | ✓ | 16 |
| 9.d | odd | 6 | 1 | 810.3.d.c | 16 | ||
| 12.b | even | 2 | 1 | 2160.3.bs.d | 16 | ||
| 15.d | odd | 2 | 1 | 1350.3.i.g | 16 | ||
| 15.e | even | 4 | 2 | 1350.3.k.b | 32 | ||
| 36.f | odd | 6 | 1 | 2160.3.bs.d | 16 | ||
| 36.h | even | 6 | 1 | 720.3.bs.d | 16 | ||
| 45.h | odd | 6 | 1 | 450.3.i.g | 16 | ||
| 45.j | even | 6 | 1 | 1350.3.i.g | 16 | ||
| 45.k | odd | 12 | 2 | 1350.3.k.b | 32 | ||
| 45.l | even | 12 | 2 | 450.3.k.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.3.h.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 90.3.h.a | ✓ | 16 | 9.d | odd | 6 | 1 | inner |
| 270.3.h.a | 16 | 3.b | odd | 2 | 1 | ||
| 270.3.h.a | 16 | 9.c | even | 3 | 1 | ||
| 450.3.i.g | 16 | 5.b | even | 2 | 1 | ||
| 450.3.i.g | 16 | 45.h | odd | 6 | 1 | ||
| 450.3.k.c | 32 | 5.c | odd | 4 | 2 | ||
| 450.3.k.c | 32 | 45.l | even | 12 | 2 | ||
| 720.3.bs.d | 16 | 4.b | odd | 2 | 1 | ||
| 720.3.bs.d | 16 | 36.h | even | 6 | 1 | ||
| 810.3.d.c | 16 | 9.c | even | 3 | 1 | ||
| 810.3.d.c | 16 | 9.d | odd | 6 | 1 | ||
| 1350.3.i.g | 16 | 15.d | odd | 2 | 1 | ||
| 1350.3.i.g | 16 | 45.j | even | 6 | 1 | ||
| 1350.3.k.b | 32 | 15.e | even | 4 | 2 | ||
| 1350.3.k.b | 32 | 45.k | odd | 12 | 2 | ||
| 2160.3.bs.d | 16 | 12.b | even | 2 | 1 | ||
| 2160.3.bs.d | 16 | 36.f | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{16} + 4 T^{15} + \cdots + 43046721 \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{4} \)
|
| $7$ |
\( T^{16} + \cdots + 6662640625 \)
|
| $11$ |
\( T^{16} + \cdots + 7901855928576 \)
|
| $13$ |
\( T^{16} + \cdots + 991374241321216 \)
|
| $17$ |
\( T^{16} + \cdots + 29\!\cdots\!00 \)
|
| $19$ |
\( (T^{8} - 40 T^{7} + \cdots + 4403341840)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 15\!\cdots\!61 \)
|
| $29$ |
\( T^{16} + \cdots + 35\!\cdots\!61 \)
|
| $31$ |
\( T^{16} + \cdots + 12\!\cdots\!56 \)
|
| $37$ |
\( (T^{8} + 44 T^{7} + \cdots - 221671664)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 23\!\cdots\!81 \)
|
| $43$ |
\( T^{16} + \cdots + 82\!\cdots\!00 \)
|
| $47$ |
\( T^{16} + \cdots + 64\!\cdots\!41 \)
|
| $53$ |
\( T^{16} + \cdots + 19\!\cdots\!76 \)
|
| $59$ |
\( T^{16} + \cdots + 51\!\cdots\!00 \)
|
| $61$ |
\( T^{16} + \cdots + 20\!\cdots\!61 \)
|
| $67$ |
\( T^{16} + \cdots + 52\!\cdots\!25 \)
|
| $71$ |
\( T^{16} + \cdots + 19\!\cdots\!76 \)
|
| $73$ |
\( (T^{8} + \cdots - 638114737319936)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 22\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 11\!\cdots\!21 \)
|
| $89$ |
\( T^{16} + \cdots + 95\!\cdots\!25 \)
|
| $97$ |
\( T^{16} + \cdots + 10\!\cdots\!56 \)
|
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