Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [116,2,Mod(25,116)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(116, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("116.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 116 = 2^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 116.g (of order \(7\), degree \(6\), 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: | \(0.926264663447\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{7})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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
\( x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2666113593 \nu^{11} - 103378288493 \nu^{10} + 730364594638 \nu^{9} + \cdots - 87085829035390 ) / 87597662955238 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 76569689033 \nu^{11} - 329670261352 \nu^{10} + 1009877030172 \nu^{9} + \cdots + 235137749968678 ) / 175195325910476 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 80877989929 \nu^{11} + 560470291448 \nu^{10} - 1931233965956 \nu^{9} + \cdots + 92982388988572 ) / 175195325910476 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 154305296456 \nu^{11} - 821175970637 \nu^{10} + 2060526694663 \nu^{9} + \cdots + 132826473739629 ) / 175195325910476 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 182774232211 \nu^{11} + 793118206506 \nu^{10} - 1236048179004 \nu^{9} + \cdots - 277769000014142 ) / 175195325910476 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 103550331790 \nu^{11} - 520417772543 \nu^{10} + 1139225692987 \nu^{9} + \cdots + 108496393085623 ) / 87597662955238 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 230492168800 \nu^{11} + 1461546568214 \nu^{10} - 4597897267111 \nu^{9} + \cdots + 9060333114751 ) / 175195325910476 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 232650030836 \nu^{11} + 1290368348160 \nu^{10} - 3776871348421 \nu^{9} + \cdots - 303156139498967 ) / 175195325910476 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 309085724214 \nu^{11} - 1831991241511 \nu^{10} + 4952928996914 \nu^{9} + \cdots + 193843913960800 ) / 175195325910476 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 513040509027 \nu^{11} + 2862030349339 \nu^{10} - 7361090329451 \nu^{9} + \cdots - 456441142019925 ) / 175195325910476 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} - 4\beta_{5} \)
|
| \(\nu^{3}\) | \(=\) |
\( - 6 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} - 10 \beta_{5} - 4 \beta_{4} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 13 \beta_{11} - 11 \beta_{10} + 13 \beta_{9} + 8 \beta_{8} + 13 \beta_{7} - 26 \beta_{5} - 26 \beta_{4} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 29 \beta_{11} - 16 \beta_{10} + 50 \beta_{9} + 34 \beta_{8} + 63 \beta_{7} + 14 \beta_{6} - 63 \beta_{5} + \cdots + 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 49 \beta_{11} + 115 \beta_{9} + 153 \beta_{8} + 229 \beta_{7} + 76 \beta_{6} - 125 \beta_{5} + \cdots - 49 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 192 \beta_{10} + 192 \beta_{9} + 536 \beta_{8} + 536 \beta_{7} + 262 \beta_{6} - 262 \beta_{4} + \cdots - 147 \)
|
| \(\nu^{8}\) | \(=\) |
\( 606 \beta_{11} + 1214 \beta_{10} + 1498 \beta_{8} + 920 \beta_{7} + 920 \beta_{6} + 1396 \beta_{5} + \cdots - 790 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3829 \beta_{11} + 5015 \beta_{10} - 2134 \beta_{9} + 2729 \beta_{8} + 2729 \beta_{6} + 8366 \beta_{5} + \cdots - 3032 \)
|
| \(\nu^{10}\) | \(=\) |
\( 16110 \beta_{11} + 16110 \beta_{10} - 12910 \beta_{9} - 10231 \beta_{7} + 5253 \beta_{6} + 34746 \beta_{5} + \cdots - 10231 \)
|
| \(\nu^{11}\) | \(=\) |
\( 52909 \beta_{11} + 41607 \beta_{10} - 52909 \beta_{9} - 28208 \beta_{8} - 61701 \beta_{7} + 114610 \beta_{5} + \cdots - 28208 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).
| \(n\) | \(59\) | \(89\) |
| \(\chi(n)\) | \(1\) | \(\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
0 | −2.04546 | + | 0.985042i | 0 | −0.299629 | + | 1.31276i | 0 | −3.84740 | + | 1.85281i | 0 | 1.34313 | − | 1.68423i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | 0 | 2.22391 | − | 1.07098i | 0 | −0.768902 | + | 3.36878i | 0 | 0.421971 | − | 0.203210i | 0 | 1.92831 | − | 2.41802i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 45.1 | 0 | −1.92469 | + | 2.41349i | 0 | −3.16565 | − | 1.52449i | 0 | −0.677712 | + | 0.849824i | 0 | −1.45292 | − | 6.36565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 45.2 | 0 | −0.0997674 | + | 0.125104i | 0 | 1.58623 | + | 0.763888i | 0 | 1.14721 | − | 1.43856i | 0 | 0.661865 | + | 2.89982i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.1 | 0 | −1.92469 | − | 2.41349i | 0 | −3.16565 | + | 1.52449i | 0 | −0.677712 | − | 0.849824i | 0 | −1.45292 | + | 6.36565i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −0.0997674 | − | 0.125104i | 0 | 1.58623 | − | 0.763888i | 0 | 1.14721 | + | 1.43856i | 0 | 0.661865 | − | 2.89982i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 53.1 | 0 | −0.242969 | − | 1.06452i | 0 | 2.52737 | + | 3.16923i | 0 | −0.688011 | − | 3.01437i | 0 | 1.62874 | − | 0.784360i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 53.2 | 0 | 0.588980 | + | 2.58049i | 0 | −0.379425 | − | 0.475783i | 0 | 0.143938 | + | 0.630635i | 0 | −3.60913 | + | 1.73806i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.1 | 0 | −2.04546 | − | 0.985042i | 0 | −0.299629 | − | 1.31276i | 0 | −3.84740 | − | 1.85281i | 0 | 1.34313 | + | 1.68423i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | 2.22391 | + | 1.07098i | 0 | −0.768902 | − | 3.36878i | 0 | 0.421971 | + | 0.203210i | 0 | 1.92831 | + | 2.41802i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 81.1 | 0 | −0.242969 | + | 1.06452i | 0 | 2.52737 | − | 3.16923i | 0 | −0.688011 | + | 3.01437i | 0 | 1.62874 | + | 0.784360i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 81.2 | 0 | 0.588980 | − | 2.58049i | 0 | −0.379425 | + | 0.475783i | 0 | 0.143938 | − | 0.630635i | 0 | −3.60913 | − | 1.73806i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.d | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 116.2.g.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1044.2.u.c | 12 | ||
| 4.b | odd | 2 | 1 | 464.2.u.g | 12 | ||
| 29.d | even | 7 | 1 | inner | 116.2.g.b | ✓ | 12 |
| 29.d | even | 7 | 1 | 3364.2.a.m | 6 | ||
| 29.e | even | 14 | 1 | 3364.2.a.p | 6 | ||
| 29.f | odd | 28 | 2 | 3364.2.c.j | 12 | ||
| 87.j | odd | 14 | 1 | 1044.2.u.c | 12 | ||
| 116.j | odd | 14 | 1 | 464.2.u.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 116.2.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 116.2.g.b | ✓ | 12 | 29.d | even | 7 | 1 | inner |
| 464.2.u.g | 12 | 4.b | odd | 2 | 1 | ||
| 464.2.u.g | 12 | 116.j | odd | 14 | 1 | ||
| 1044.2.u.c | 12 | 3.b | odd | 2 | 1 | ||
| 1044.2.u.c | 12 | 87.j | odd | 14 | 1 | ||
| 3364.2.a.m | 6 | 29.d | even | 7 | 1 | ||
| 3364.2.a.p | 6 | 29.e | even | 14 | 1 | ||
| 3364.2.c.j | 12 | 29.f | odd | 28 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 3 T_{3}^{11} + 7 T_{3}^{10} + T_{3}^{9} + 17 T_{3}^{8} - 21 T_{3}^{7} - 55 T_{3}^{6} + \cdots + 64 \)
acting on \(S_{2}^{\mathrm{new}}(116, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 3 T^{11} + \cdots + 64 \)
|
| $5$ |
\( T^{12} + T^{11} + \cdots + 5041 \)
|
| $7$ |
\( T^{12} + 7 T^{11} + \cdots + 64 \)
|
| $11$ |
\( T^{12} - 3 T^{11} + \cdots + 46656 \)
|
| $13$ |
\( T^{12} + 21 T^{11} + \cdots + 1 \)
|
| $17$ |
\( (T^{6} + 5 T^{5} + \cdots + 344)^{2} \)
|
| $19$ |
\( T^{12} + \cdots + 285745216 \)
|
| $23$ |
\( T^{12} + 19 T^{11} + \cdots + 1032256 \)
|
| $29$ |
\( T^{12} + \cdots + 594823321 \)
|
| $31$ |
\( T^{12} + 27 T^{11} + \cdots + 53824 \)
|
| $37$ |
\( T^{12} + \cdots + 161620369 \)
|
| $41$ |
\( (T^{6} + T^{5} - 142 T^{4} + \cdots - 1624)^{2} \)
|
| $43$ |
\( T^{12} + 9 T^{11} + \cdots + 11235904 \)
|
| $47$ |
\( T^{12} + \cdots + 16533845056 \)
|
| $53$ |
\( (T^{6} - 11 T^{5} + \cdots + 841)^{2} \)
|
| $59$ |
\( (T^{6} - 18 T^{5} + \cdots + 19264)^{2} \)
|
| $61$ |
\( T^{12} + \cdots + 2812499089 \)
|
| $67$ |
\( T^{12} + \cdots + 5027377216 \)
|
| $71$ |
\( T^{12} + 35 T^{11} + \cdots + 96983104 \)
|
| $73$ |
\( T^{12} - 34 T^{11} + \cdots + 37442161 \)
|
| $79$ |
\( T^{12} + \cdots + 397313387584 \)
|
| $83$ |
\( T^{12} + T^{11} + \cdots + 13601344 \)
|
| $89$ |
\( T^{12} + \cdots + 876692881 \)
|
| $97$ |
\( T^{12} + \cdots + 1819798302001 \)
|
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