Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [900,3,Mod(101,900)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 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("900.101");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 900.p (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: | \(24.5232237924\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 3^{6} \) |
| Twist minimal: | no (minimal twist has level 180) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 412 \nu^{11} - 17123 \nu^{10} + 22072 \nu^{9} + 48327 \nu^{8} - 317739 \nu^{7} + 1679139 \nu^{6} + \cdots + 368052417 ) / 95482233 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1264 \nu^{11} + 7514 \nu^{10} - 7570 \nu^{9} - 69045 \nu^{8} + 447600 \nu^{7} + \cdots - 255564072 ) / 95482233 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 262 \nu^{11} - 2342 \nu^{10} + 7018 \nu^{9} + 7395 \nu^{8} - 133365 \nu^{7} + 477666 \nu^{6} + \cdots + 188917434 ) / 10609137 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3622 \nu^{11} - 28592 \nu^{10} + 70732 \nu^{9} + 135600 \nu^{8} - 1647885 \nu^{7} + \cdots + 1669374279 ) / 95482233 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 554 \nu^{11} + 2386 \nu^{10} - 755 \nu^{9} - 32880 \nu^{8} + 119670 \nu^{7} - 115038 \nu^{6} + \cdots - 74637936 ) / 10609137 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5945 \nu^{11} + 23266 \nu^{10} - 2231 \nu^{9} - 288570 \nu^{8} + 1334805 \nu^{7} + \cdots - 967281669 ) / 95482233 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 956 \nu^{11} + 6430 \nu^{10} + 589 \nu^{9} - 82677 \nu^{8} + 330720 \nu^{7} - 477891 \nu^{6} + \cdots - 205903863 ) / 13640319 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1031 \nu^{11} + 5815 \nu^{10} - 26 \nu^{9} - 46335 \nu^{8} + 226131 \nu^{7} - 446544 \nu^{6} + \cdots - 222575364 ) / 10609137 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 599 \nu^{11} - 3664 \nu^{10} + 7577 \nu^{9} + 21270 \nu^{8} - 191652 \nu^{7} + 479619 \nu^{6} + \cdots + 145614834 ) / 4546773 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1417 \nu^{11} - 5975 \nu^{10} + 5086 \nu^{9} + 56283 \nu^{8} - 326775 \nu^{7} + 552132 \nu^{6} + \cdots + 189606339 ) / 10609137 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} - 5\beta_{3} + \beta_{2} - 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots + 44 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7 \beta_{11} - 11 \beta_{10} + 9 \beta_{9} + 7 \beta_{8} - 17 \beta_{7} + \beta_{6} - 14 \beta_{5} + \cdots - 43 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16 \beta_{11} + 16 \beta_{10} + 21 \beta_{9} - 38 \beta_{8} - 59 \beta_{7} + 79 \beta_{6} + 28 \beta_{5} + \cdots + 80 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 56 \beta_{11} - 74 \beta_{10} - 12 \beta_{9} + 106 \beta_{8} - 128 \beta_{7} - 284 \beta_{6} + \cdots - 1246 \)
|
| \(\nu^{8}\) | \(=\) |
\( 178 \beta_{11} - 74 \beta_{10} + 270 \beta_{9} - 362 \beta_{8} + 370 \beta_{7} - 692 \beta_{6} + \cdots - 88 \)
|
| \(\nu^{9}\) | \(=\) |
\( 34 \beta_{11} + 34 \beta_{10} + 1254 \beta_{9} + 898 \beta_{8} + 220 \beta_{7} - 3071 \beta_{6} + \cdots - 64 \)
|
| \(\nu^{10}\) | \(=\) |
\( 376 \beta_{11} + 1114 \beta_{10} + 3669 \beta_{9} - 2216 \beta_{8} - 6275 \beta_{7} + 2677 \beta_{6} + \cdots + 6305 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1501 \beta_{11} + 7837 \beta_{10} - 4473 \beta_{9} + 11491 \beta_{8} - 19466 \beta_{7} - 3806 \beta_{6} + \cdots - 12220 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(451\) | \(577\) |
| \(\chi(n)\) | \(1 + \beta_{3}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
0 | −2.89597 | − | 0.783177i | 0 | 0 | 0 | 2.35650 | − | 4.08158i | 0 | 7.77327 | + | 4.53611i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | 0 | −2.65605 | + | 1.39478i | 0 | 0 | 0 | −1.41583 | + | 2.45229i | 0 | 5.10917 | − | 7.40921i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | 0 | −0.841761 | − | 2.87949i | 0 | 0 | 0 | 5.98742 | − | 10.3705i | 0 | −7.58288 | + | 4.84768i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | 0 | −0.459278 | + | 2.96464i | 0 | 0 | 0 | −4.13490 | + | 7.16186i | 0 | −8.57813 | − | 2.72318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 101.5 | 0 | 2.85525 | + | 0.920635i | 0 | 0 | 0 | −0.594587 | + | 1.02985i | 0 | 7.30486 | + | 5.25728i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 101.6 | 0 | 2.99781 | + | 0.114662i | 0 | 0 | 0 | 0.801399 | − | 1.38806i | 0 | 8.97371 | + | 0.687471i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.1 | 0 | −2.89597 | + | 0.783177i | 0 | 0 | 0 | 2.35650 | + | 4.08158i | 0 | 7.77327 | − | 4.53611i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.2 | 0 | −2.65605 | − | 1.39478i | 0 | 0 | 0 | −1.41583 | − | 2.45229i | 0 | 5.10917 | + | 7.40921i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.3 | 0 | −0.841761 | + | 2.87949i | 0 | 0 | 0 | 5.98742 | + | 10.3705i | 0 | −7.58288 | − | 4.84768i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.4 | 0 | −0.459278 | − | 2.96464i | 0 | 0 | 0 | −4.13490 | − | 7.16186i | 0 | −8.57813 | + | 2.72318i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.5 | 0 | 2.85525 | − | 0.920635i | 0 | 0 | 0 | −0.594587 | − | 1.02985i | 0 | 7.30486 | − | 5.25728i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 401.6 | 0 | 2.99781 | − | 0.114662i | 0 | 0 | 0 | 0.801399 | + | 1.38806i | 0 | 8.97371 | − | 0.687471i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 900.3.p.c | 12 | |
| 3.b | odd | 2 | 1 | 2700.3.p.c | 12 | ||
| 5.b | even | 2 | 1 | 180.3.o.b | ✓ | 12 | |
| 5.c | odd | 4 | 2 | 900.3.u.c | 24 | ||
| 9.c | even | 3 | 1 | 2700.3.p.c | 12 | ||
| 9.d | odd | 6 | 1 | inner | 900.3.p.c | 12 | |
| 15.d | odd | 2 | 1 | 540.3.o.b | 12 | ||
| 15.e | even | 4 | 2 | 2700.3.u.c | 24 | ||
| 20.d | odd | 2 | 1 | 720.3.bs.b | 12 | ||
| 45.h | odd | 6 | 1 | 180.3.o.b | ✓ | 12 | |
| 45.h | odd | 6 | 1 | 1620.3.g.b | 12 | ||
| 45.j | even | 6 | 1 | 540.3.o.b | 12 | ||
| 45.j | even | 6 | 1 | 1620.3.g.b | 12 | ||
| 45.k | odd | 12 | 2 | 2700.3.u.c | 24 | ||
| 45.l | even | 12 | 2 | 900.3.u.c | 24 | ||
| 60.h | even | 2 | 1 | 2160.3.bs.b | 12 | ||
| 180.n | even | 6 | 1 | 720.3.bs.b | 12 | ||
| 180.p | odd | 6 | 1 | 2160.3.bs.b | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 180.3.o.b | ✓ | 12 | 5.b | even | 2 | 1 | |
| 180.3.o.b | ✓ | 12 | 45.h | odd | 6 | 1 | |
| 540.3.o.b | 12 | 15.d | odd | 2 | 1 | ||
| 540.3.o.b | 12 | 45.j | even | 6 | 1 | ||
| 720.3.bs.b | 12 | 20.d | odd | 2 | 1 | ||
| 720.3.bs.b | 12 | 180.n | even | 6 | 1 | ||
| 900.3.p.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 900.3.p.c | 12 | 9.d | odd | 6 | 1 | inner | |
| 900.3.u.c | 24 | 5.c | odd | 4 | 2 | ||
| 900.3.u.c | 24 | 45.l | even | 12 | 2 | ||
| 1620.3.g.b | 12 | 45.h | odd | 6 | 1 | ||
| 1620.3.g.b | 12 | 45.j | even | 6 | 1 | ||
| 2160.3.bs.b | 12 | 60.h | even | 2 | 1 | ||
| 2160.3.bs.b | 12 | 180.p | odd | 6 | 1 | ||
| 2700.3.p.c | 12 | 3.b | odd | 2 | 1 | ||
| 2700.3.p.c | 12 | 9.c | even | 3 | 1 | ||
| 2700.3.u.c | 24 | 15.e | even | 4 | 2 | ||
| 2700.3.u.c | 24 | 45.k | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{12} - 6 T_{7}^{11} + 141 T_{7}^{10} + 50 T_{7}^{9} + 11340 T_{7}^{8} - 14346 T_{7}^{7} + \cdots + 6345361 \)
acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 2 T^{11} + \cdots + 531441 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 6345361 \)
|
| $11$ |
\( T^{12} + \cdots + 416649744 \)
|
| $13$ |
\( T^{12} + \cdots + 15716943091600 \)
|
| $17$ |
\( T^{12} + \cdots + 191070131587344 \)
|
| $19$ |
\( (T^{6} - 36 T^{5} + \cdots - 19580204)^{2} \)
|
| $23$ |
\( T^{12} + \cdots + 13626529936569 \)
|
| $29$ |
\( T^{12} + \cdots + 3840796442025 \)
|
| $31$ |
\( T^{12} + \cdots + 177690926165776 \)
|
| $37$ |
\( (T^{6} - 6 T^{5} + \cdots - 2144769884)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 29\!\cdots\!25 \)
|
| $43$ |
\( T^{12} + \cdots + 67\!\cdots\!96 \)
|
| $47$ |
\( T^{12} + \cdots + 47\!\cdots\!49 \)
|
| $53$ |
\( T^{12} + \cdots + 14\!\cdots\!24 \)
|
| $59$ |
\( T^{12} + \cdots + 29\!\cdots\!84 \)
|
| $61$ |
\( T^{12} + \cdots + 11\!\cdots\!41 \)
|
| $67$ |
\( T^{12} + \cdots + 50\!\cdots\!41 \)
|
| $71$ |
\( T^{12} + \cdots + 28\!\cdots\!64 \)
|
| $73$ |
\( (T^{6} - 12 T^{5} + \cdots - 2761132736)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 86\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 15\!\cdots\!29 \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!29 \)
|
| $97$ |
\( T^{12} + \cdots + 60\!\cdots\!16 \)
|
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