Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [66,7,Mod(43,66)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66.43");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 66.d (of order \(2\), degree \(1\), 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: | \(15.1835695189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 42116 x^{10} + 664942422 x^{8} + 4757582650496 x^{6} + \cdots + 52\!\cdots\!16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{10} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{12} + 42116 x^{10} + 664942422 x^{8} + 4757582650496 x^{6} + \cdots + 52\!\cdots\!16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1911583217765 \nu^{10} + \cdots - 10\!\cdots\!42 ) / 70\!\cdots\!96 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\!\cdots\!19 \nu^{10} + \cdots - 12\!\cdots\!88 ) / 57\!\cdots\!24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 38\!\cdots\!51 \nu^{10} + \cdots + 19\!\cdots\!98 ) / 57\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!99 \nu^{10} + \cdots + 75\!\cdots\!46 ) / 11\!\cdots\!48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 18\!\cdots\!45 \nu^{11} + \cdots - 85\!\cdots\!18 \nu ) / 10\!\cdots\!94 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 20\!\cdots\!27 \nu^{11} + \cdots - 10\!\cdots\!78 \nu ) / 36\!\cdots\!08 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 65\!\cdots\!81 \nu^{11} + \cdots - 32\!\cdots\!08 \nu ) / 39\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 65\!\cdots\!41 \nu^{11} + \cdots - 30\!\cdots\!08 \nu ) / 39\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 663665155046844 \nu^{11} + \cdots - 32\!\cdots\!26 \nu ) / 24\!\cdots\!97 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 53\!\cdots\!02 \nu^{11} + \cdots + 20\!\cdots\!32 ) / 13\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 16\!\cdots\!06 \nu^{11} + \cdots - 60\!\cdots\!96 ) / 39\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( 7\beta_{9} - 36\beta_{6} + 9\beta_{5} ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -45\beta_{11} + 45\beta_{10} - 45\beta_{5} - 84\beta_{4} + 324\beta_{2} + 526\beta _1 - 126270 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 16913\beta_{9} + 3216\beta_{8} - 3696\beta_{7} + 127692\beta_{6} - 651399\beta_{5} ) / 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 492705 \beta_{11} - 492705 \beta_{10} + 492705 \beta_{5} + 1216344 \beta_{4} - 12570 \beta_{3} + \cdots + 1330280970 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 29361600 \beta_{11} - 29361600 \beta_{10} - 1230012811 \beta_{9} - 161332296 \beta_{8} + \cdots + 30980430435 \beta_{5} ) / 72 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 1800832755 \beta_{11} + 1800832755 \beta_{10} - 1800832755 \beta_{5} - 5264087664 \beta_{4} + \cdots - 4955988473490 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 578459117760 \beta_{11} + 578459117760 \beta_{10} + 18720046770121 \beta_{9} + \cdots - 416965827642663 \beta_{5} ) / 72 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 60214425113115 \beta_{11} - 60214425113115 \beta_{10} + 60214425113115 \beta_{5} + \cdots + 17\!\cdots\!02 ) / 18 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 946242045314560 \beta_{11} - 946242045314560 \beta_{10} + \cdots + 59\!\cdots\!69 \beta_{5} ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 67\!\cdots\!55 \beta_{11} + \cdots - 19\!\cdots\!10 ) / 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 11\!\cdots\!60 \beta_{11} + \cdots - 66\!\cdots\!45 \beta_{5} ) / 72 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
− | 5.65685i | −15.5885 | −32.0000 | −190.969 | 88.1816i | 301.363i | 181.019i | 243.000 | 1080.29i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | − | 5.65685i | −15.5885 | −32.0000 | −40.5118 | 88.1816i | − | 666.948i | 181.019i | 243.000 | 229.169i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | − | 5.65685i | −15.5885 | −32.0000 | 109.089 | 88.1816i | − | 42.6817i | 181.019i | 243.000 | − | 617.100i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | − | 5.65685i | 15.5885 | −32.0000 | −191.829 | − | 88.1816i | − | 226.333i | 181.019i | 243.000 | 1085.15i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | − | 5.65685i | 15.5885 | −32.0000 | 15.2366 | − | 88.1816i | − | 168.075i | 181.019i | 243.000 | − | 86.1913i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | − | 5.65685i | 15.5885 | −32.0000 | 74.9845 | − | 88.1816i | 412.353i | 181.019i | 243.000 | − | 424.176i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 5.65685i | −15.5885 | −32.0000 | −190.969 | − | 88.1816i | − | 301.363i | − | 181.019i | 243.000 | − | 1080.29i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 5.65685i | −15.5885 | −32.0000 | −40.5118 | − | 88.1816i | 666.948i | − | 181.019i | 243.000 | − | 229.169i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.9 | 5.65685i | −15.5885 | −32.0000 | 109.089 | − | 88.1816i | 42.6817i | − | 181.019i | 243.000 | 617.100i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.10 | 5.65685i | 15.5885 | −32.0000 | −191.829 | 88.1816i | 226.333i | − | 181.019i | 243.000 | − | 1085.15i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.11 | 5.65685i | 15.5885 | −32.0000 | 15.2366 | 88.1816i | 168.075i | − | 181.019i | 243.000 | 86.1913i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.12 | 5.65685i | 15.5885 | −32.0000 | 74.9845 | 88.1816i | − | 412.353i | − | 181.019i | 243.000 | 424.176i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 66.7.d.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 198.7.d.c | 12 | ||
| 4.b | odd | 2 | 1 | 528.7.j.a | 12 | ||
| 11.b | odd | 2 | 1 | inner | 66.7.d.a | ✓ | 12 |
| 33.d | even | 2 | 1 | 198.7.d.c | 12 | ||
| 44.c | even | 2 | 1 | 528.7.j.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.7.d.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 66.7.d.a | ✓ | 12 | 11.b | odd | 2 | 1 | inner |
| 198.7.d.c | 12 | 3.b | odd | 2 | 1 | ||
| 198.7.d.c | 12 | 33.d | even | 2 | 1 | ||
| 528.7.j.a | 12 | 4.b | odd | 2 | 1 | ||
| 528.7.j.a | 12 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(66, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32)^{6} \)
|
| $3$ |
\( (T^{2} - 243)^{6} \)
|
| $5$ |
\( (T^{6} + 224 T^{5} + \cdots - 184969070000)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 18\!\cdots\!24 \)
|
| $11$ |
\( T^{12} + \cdots + 30\!\cdots\!61 \)
|
| $13$ |
\( T^{12} + \cdots + 20\!\cdots\!24 \)
|
| $17$ |
\( T^{12} + \cdots + 16\!\cdots\!84 \)
|
| $19$ |
\( T^{12} + \cdots + 18\!\cdots\!04 \)
|
| $23$ |
\( (T^{6} + \cdots - 21\!\cdots\!64)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 72\!\cdots\!04 \)
|
| $31$ |
\( (T^{6} + \cdots - 11\!\cdots\!76)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots + 13\!\cdots\!64 \)
|
| $47$ |
\( (T^{6} + \cdots - 19\!\cdots\!32)^{2} \)
|
| $53$ |
\( (T^{6} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots - 57\!\cdots\!04)^{2} \)
|
| $61$ |
\( T^{12} + \cdots + 11\!\cdots\!24 \)
|
| $67$ |
\( (T^{6} + \cdots - 23\!\cdots\!56)^{2} \)
|
| $71$ |
\( (T^{6} + \cdots - 64\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 86\!\cdots\!64 \)
|
| $79$ |
\( T^{12} + \cdots + 23\!\cdots\!16 \)
|
| $83$ |
\( T^{12} + \cdots + 82\!\cdots\!04 \)
|
| $89$ |
\( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots - 84\!\cdots\!76)^{2} \)
|
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