Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,10,Mod(111,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.111");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.f (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: | \(57.6840136504\) |
| 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: |
\( x^{12} + 803377 x^{10} + 231385144768 x^{8} + \cdots + 49\!\cdots\!01 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{38}\cdot 3^{4}\cdot 7^{9} \) |
| 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
\( x^{12} + 803377 x^{10} + 231385144768 x^{8} + \cdots + 49\!\cdots\!01 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 21\!\cdots\!32 \nu^{10} + \cdots - 11\!\cdots\!35 ) / 46\!\cdots\!43 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 39\!\cdots\!00 \nu^{10} + \cdots + 95\!\cdots\!16 ) / 42\!\cdots\!87 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\!\cdots\!42 \nu^{11} + \cdots + 28\!\cdots\!66 \nu ) / 32\!\cdots\!07 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!42 \nu^{11} + \cdots + 15\!\cdots\!38 \nu ) / 32\!\cdots\!07 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15\!\cdots\!40 \nu^{11} + \cdots - 33\!\cdots\!27 ) / 68\!\cdots\!55 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\!\cdots\!40 \nu^{11} + \cdots - 49\!\cdots\!47 ) / 68\!\cdots\!55 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 15\!\cdots\!40 \nu^{11} + \cdots + 43\!\cdots\!29 ) / 68\!\cdots\!55 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 34\!\cdots\!26 \nu^{11} + \cdots + 52\!\cdots\!42 \nu ) / 39\!\cdots\!05 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 78\!\cdots\!00 \nu^{11} + \cdots - 29\!\cdots\!43 ) / 68\!\cdots\!55 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 42\!\cdots\!28 \nu^{11} + \cdots + 14\!\cdots\!84 \nu ) / 30\!\cdots\!45 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 39\!\cdots\!10 \nu^{11} + \cdots + 66\!\cdots\!54 ) / 68\!\cdots\!55 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + \beta_{2} - 7\beta _1 - 1071167 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 257 \beta_{11} + 2373 \beta_{10} + 7185 \beta_{9} + 6681 \beta_{8} - 65179 \beta_{5} + \cdots - 1838073 \beta_{3} ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 279387 \beta_{9} + 1365123 \beta_{7} + 1082957 \beta_{6} + 1679101 \beta_{5} - 631557 \beta_{2} + \cdots + 487048424712 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 63466757 \beta_{11} - 373901935 \beta_{10} - 1909387668 \beta_{9} - 1307146660 \beta_{8} + \cdots + 242399122964 \beta_{3} ) / 16 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 113831077383 \beta_{9} - 651951764667 \beta_{7} - 469369415661 \beta_{6} - 751737735921 \beta_{5} + \cdots - 12\!\cdots\!76 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 25204196989174 \beta_{11} + 103208989463542 \beta_{10} + 780110879882397 \beta_{9} + \cdots - 64\!\cdots\!13 \beta_{3} ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 20\!\cdots\!51 \beta_{9} + \cdots + 16\!\cdots\!44 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 18\!\cdots\!35 \beta_{11} + \cdots + 33\!\cdots\!37 \beta_{3} ) / 32 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 14\!\cdots\!51 \beta_{9} + \cdots - 82\!\cdots\!50 ) / 16 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 31\!\cdots\!81 \beta_{11} + \cdots - 40\!\cdots\!57 \beta_{3} ) / 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 111.1 |
|
0 | −207.658 | 0 | − | 631.636i | 0 | −5152.96 | + | 3714.92i | 0 | 23438.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.2 | 0 | −207.658 | 0 | 631.636i | 0 | −5152.96 | − | 3714.92i | 0 | 23438.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.3 | 0 | −188.894 | 0 | − | 2177.16i | 0 | 6318.34 | − | 657.438i | 0 | 15997.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.4 | 0 | −188.894 | 0 | 2177.16i | 0 | 6318.34 | + | 657.438i | 0 | 15997.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.5 | 0 | −18.7979 | 0 | − | 1169.26i | 0 | −2731.15 | − | 5735.37i | 0 | −19329.6 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.6 | 0 | −18.7979 | 0 | 1169.26i | 0 | −2731.15 | + | 5735.37i | 0 | −19329.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.7 | 0 | 18.7979 | 0 | − | 1169.26i | 0 | 2731.15 | + | 5735.37i | 0 | −19329.6 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.8 | 0 | 18.7979 | 0 | 1169.26i | 0 | 2731.15 | − | 5735.37i | 0 | −19329.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.9 | 0 | 188.894 | 0 | − | 2177.16i | 0 | −6318.34 | + | 657.438i | 0 | 15997.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.10 | 0 | 188.894 | 0 | 2177.16i | 0 | −6318.34 | − | 657.438i | 0 | 15997.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.11 | 0 | 207.658 | 0 | − | 631.636i | 0 | 5152.96 | − | 3714.92i | 0 | 23438.9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 111.12 | 0 | 207.658 | 0 | 631.636i | 0 | 5152.96 | + | 3714.92i | 0 | 23438.9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 28.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.10.f.a | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 112.10.f.a | ✓ | 12 |
| 7.b | odd | 2 | 1 | inner | 112.10.f.a | ✓ | 12 |
| 28.d | even | 2 | 1 | inner | 112.10.f.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 112.10.f.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 112.10.f.a | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 112.10.f.a | ✓ | 12 | 7.b | odd | 2 | 1 | inner |
| 112.10.f.a | ✓ | 12 | 28.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 79156T_{3}^{4} + 1566467280T_{3}^{2} - 543687971904 \)
acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{6} - 79156 T^{4} + \cdots - 543687971904)^{2} \)
|
| $5$ |
\( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{12} + \cdots + 43\!\cdots\!49 \)
|
| $11$ |
\( (T^{6} + \cdots + 39\!\cdots\!40)^{2} \)
|
| $13$ |
\( (T^{6} + \cdots + 98\!\cdots\!40)^{2} \)
|
| $17$ |
\( (T^{6} + \cdots + 72\!\cdots\!60)^{2} \)
|
| $19$ |
\( (T^{6} + \cdots - 22\!\cdots\!00)^{2} \)
|
| $23$ |
\( (T^{6} + \cdots + 22\!\cdots\!40)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots + 13\!\cdots\!92)^{4} \)
|
| $31$ |
\( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 16\!\cdots\!20)^{4} \)
|
| $41$ |
\( (T^{6} + \cdots + 10\!\cdots\!40)^{2} \)
|
| $43$ |
\( (T^{6} + \cdots + 11\!\cdots\!60)^{2} \)
|
| $47$ |
\( (T^{6} + \cdots - 14\!\cdots\!16)^{2} \)
|
| $53$ |
\( (T^{3} + \cdots - 14\!\cdots\!00)^{4} \)
|
| $59$ |
\( (T^{6} + \cdots - 60\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 47\!\cdots\!40)^{2} \)
|
| $67$ |
\( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{6} + \cdots + 70\!\cdots\!60)^{2} \)
|
| $73$ |
\( (T^{6} + \cdots + 65\!\cdots\!40)^{2} \)
|
| $79$ |
\( (T^{6} + \cdots + 11\!\cdots\!40)^{2} \)
|
| $83$ |
\( (T^{6} + \cdots - 10\!\cdots\!36)^{2} \)
|
| $89$ |
\( (T^{6} + \cdots + 12\!\cdots\!40)^{2} \)
|
| $97$ |
\( (T^{6} + \cdots + 26\!\cdots\!40)^{2} \)
|
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