Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,3,Mod(31,222)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(222, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 222.f (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: | \(6.04906186880\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 24 x^{10} - 76 x^{9} - 544 x^{8} - 304 x^{7} + 20552 x^{6} + 155648 x^{5} + 830496 x^{4} + \cdots + 2085136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 24 x^{10} - 76 x^{9} - 544 x^{8} - 304 x^{7} + 20552 x^{6} + 155648 x^{5} + 830496 x^{4} + \cdots + 2085136 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 15484701974314 \nu^{11} - 130214495109911 \nu^{10} + 86623973605528 \nu^{9} + \cdots - 33\!\cdots\!44 ) / 40\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 207083384 \nu^{11} + 928143046 \nu^{10} + 2586340237 \nu^{9} - 479611224 \nu^{8} + \cdots + 99718494746744 ) / 170137986927480 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 82714904240 \nu^{11} - 152070972562 \nu^{10} - 1590178151863 \nu^{9} + \cdots + 10\!\cdots\!68 ) / 58\!\cdots\!08 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1712699232512 \nu^{11} - 2106699968338 \nu^{10} - 40729587898891 \nu^{9} + \cdots + 13\!\cdots\!48 ) / 11\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 48849634 \nu^{11} + 125455841 \nu^{10} + 853576232 \nu^{9} + 1469867436 \nu^{8} + \cdots - 22726158893696 ) / 17909261781840 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 48849634 \nu^{11} - 125455841 \nu^{10} - 853576232 \nu^{9} - 1469867436 \nu^{8} + \cdots + 22726158893696 ) / 17909261781840 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 897925614930856 \nu^{11} + \cdots - 22\!\cdots\!76 ) / 25\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 195253651055026 \nu^{11} + 417144007457879 \nu^{10} + \cdots - 24\!\cdots\!44 ) / 40\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 582827654405123 \nu^{11} + \cdots - 10\!\cdots\!68 ) / 76\!\cdots\!32 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 99333500638003 \nu^{11} - 20196899088986 \nu^{10} + \cdots - 33\!\cdots\!00 ) / 80\!\cdots\!56 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 162612730755641 \nu^{11} + 258583507432594 \nu^{10} + \cdots - 24\!\cdots\!84 ) / 12\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} - \beta_{10} + 2\beta_{6} + 2\beta_{5} + 19\beta_{4} - 19\beta_{3} + 3\beta_{2} + 4\beta _1 + 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{6} + 19\beta_{4} - 19\beta_{3} + 19\beta_{2} + 28\beta _1 + 19 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 6 \beta_{11} - 2 \beta_{10} + 14 \beta_{9} - 4 \beta_{8} + 14 \beta_{7} + 55 \beta_{6} - 47 \beta_{5} + \cdots + 282 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 38 \beta_{11} + 19 \beta_{9} - 150 \beta_{8} + 19 \beta_{7} + 492 \beta_{6} - 492 \beta_{5} + \cdots + 893 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 832 \beta_{11} + 224 \beta_{9} - 2272 \beta_{8} + 2272 \beta_{6} - 3024 \beta_{5} - 20488 \beta_{3} + \cdots + 3024 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2584 \beta_{11} + 152 \beta_{10} + 1368 \beta_{9} - 19604 \beta_{8} - 1064 \beta_{7} + 8258 \beta_{6} + \cdots - 37240 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 16566 \beta_{11} + 5522 \beta_{10} + 14436 \beta_{9} - 123920 \beta_{8} - 14436 \beta_{7} + \cdots - 416652 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 85728 \beta_{11} + 60192 \beta_{10} + 60192 \beta_{9} - 820528 \beta_{8} - 85728 \beta_{7} + \cdots - 3288444 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 568664 \beta_{11} + 568664 \beta_{10} + 264344 \beta_{9} - 5738432 \beta_{8} - 793032 \beta_{7} + \cdots - 27934360 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 2029656 \beta_{11} + 3543728 \beta_{10} + 757036 \beta_{9} - 36815336 \beta_{8} - 6330420 \beta_{7} + \cdots - 221981484 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/222\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(187\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
1.00000 | + | 1.00000i | − | 1.73205i | 2.00000i | −3.93737 | + | 3.93737i | 1.73205 | − | 1.73205i | 0.503427 | −2.00000 | + | 2.00000i | −3.00000 | −7.87474 | |||||||||||||||||||||||||||||||||||||||||||||
| 31.2 | 1.00000 | + | 1.00000i | − | 1.73205i | 2.00000i | −2.84688 | + | 2.84688i | 1.73205 | − | 1.73205i | −13.7085 | −2.00000 | + | 2.00000i | −3.00000 | −5.69377 | ||||||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 1.00000 | + | 1.00000i | − | 1.73205i | 2.00000i | 3.78425 | − | 3.78425i | 1.73205 | − | 1.73205i | 2.27682 | −2.00000 | + | 2.00000i | −3.00000 | 7.56851 | ||||||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 1.00000 | + | 1.00000i | 1.73205i | 2.00000i | −4.53592 | + | 4.53592i | −1.73205 | + | 1.73205i | −0.315153 | −2.00000 | + | 2.00000i | −3.00000 | −9.07183 | |||||||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 1.00000 | + | 1.00000i | 1.73205i | 2.00000i | −0.308548 | + | 0.308548i | −1.73205 | + | 1.73205i | −9.72032 | −2.00000 | + | 2.00000i | −3.00000 | −0.617096 | |||||||||||||||||||||||||||||||||||||||||||||||
| 31.6 | 1.00000 | + | 1.00000i | 1.73205i | 2.00000i | 1.84446 | − | 1.84446i | −1.73205 | + | 1.73205i | 12.9637 | −2.00000 | + | 2.00000i | −3.00000 | 3.68893 | |||||||||||||||||||||||||||||||||||||||||||||||
| 43.1 | 1.00000 | − | 1.00000i | − | 1.73205i | − | 2.00000i | −4.53592 | − | 4.53592i | −1.73205 | − | 1.73205i | −0.315153 | −2.00000 | − | 2.00000i | −3.00000 | −9.07183 | |||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | 1.00000 | − | 1.00000i | − | 1.73205i | − | 2.00000i | −0.308548 | − | 0.308548i | −1.73205 | − | 1.73205i | −9.72032 | −2.00000 | − | 2.00000i | −3.00000 | −0.617096 | |||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | 1.00000 | − | 1.00000i | − | 1.73205i | − | 2.00000i | 1.84446 | + | 1.84446i | −1.73205 | − | 1.73205i | 12.9637 | −2.00000 | − | 2.00000i | −3.00000 | 3.68893 | |||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | 1.00000 | − | 1.00000i | 1.73205i | − | 2.00000i | −3.93737 | − | 3.93737i | 1.73205 | + | 1.73205i | 0.503427 | −2.00000 | − | 2.00000i | −3.00000 | −7.87474 | ||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | 1.00000 | − | 1.00000i | 1.73205i | − | 2.00000i | −2.84688 | − | 2.84688i | 1.73205 | + | 1.73205i | −13.7085 | −2.00000 | − | 2.00000i | −3.00000 | −5.69377 | ||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 1.00000 | − | 1.00000i | 1.73205i | − | 2.00000i | 3.78425 | + | 3.78425i | 1.73205 | + | 1.73205i | 2.27682 | −2.00000 | − | 2.00000i | −3.00000 | 7.56851 | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 222.3.f.c | ✓ | 12 |
| 3.b | odd | 2 | 1 | 666.3.i.f | 12 | ||
| 37.d | odd | 4 | 1 | inner | 222.3.f.c | ✓ | 12 |
| 111.g | even | 4 | 1 | 666.3.i.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 222.3.f.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 222.3.f.c | ✓ | 12 | 37.d | odd | 4 | 1 | inner |
| 666.3.i.f | 12 | 3.b | odd | 2 | 1 | ||
| 666.3.i.f | 12 | 111.g | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} + 12 T_{5}^{11} + 72 T_{5}^{10} + 132 T_{5}^{9} + 884 T_{5}^{8} + 9936 T_{5}^{7} + \cdots + 767376 \)
acting on \(S_{3}^{\mathrm{new}}(222, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{6} \)
|
| $3$ |
\( (T^{2} + 3)^{6} \)
|
| $5$ |
\( T^{12} + 12 T^{11} + \cdots + 767376 \)
|
| $7$ |
\( (T^{6} + 8 T^{5} + \cdots - 624)^{2} \)
|
| $11$ |
\( T^{12} + \cdots + 17831863296 \)
|
| $13$ |
\( T^{12} + \cdots + 308915776 \)
|
| $17$ |
\( T^{12} + \cdots + 1748963930256 \)
|
| $19$ |
\( T^{12} + \cdots + 451540993024 \)
|
| $23$ |
\( T^{12} + \cdots + 16103610000 \)
|
| $29$ |
\( T^{12} + \cdots + 89528762737296 \)
|
| $31$ |
\( T^{12} + \cdots + 61798716998656 \)
|
| $37$ |
\( T^{12} + \cdots + 65\!\cdots\!81 \)
|
| $41$ |
\( T^{12} + \cdots + 512398636978176 \)
|
| $43$ |
\( T^{12} + \cdots + 72\!\cdots\!56 \)
|
| $47$ |
\( (T^{6} + 36 T^{5} + \cdots + 11305339776)^{2} \)
|
| $53$ |
\( (T^{6} - 10504 T^{4} + \cdots - 20692397856)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 20\!\cdots\!24 \)
|
| $61$ |
\( T^{12} + \cdots + 40\!\cdots\!84 \)
|
| $67$ |
\( T^{12} + \cdots + 96\!\cdots\!84 \)
|
| $71$ |
\( (T^{6} - 4 T^{5} + \cdots - 4520636544)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 22\!\cdots\!44 \)
|
| $79$ |
\( T^{12} + \cdots + 21\!\cdots\!04 \)
|
| $83$ |
\( (T^{6} - 13312 T^{4} + \cdots + 24502151904)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!24 \)
|
| $97$ |
\( T^{12} + \cdots + 60\!\cdots\!04 \)
|
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