Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,2,Mod(36,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("185.36");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 185.c (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: | \(1.47723243739\) |
| 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: |
\( x^{12} + 21x^{10} + 162x^{8} + 574x^{6} + 985x^{4} + 765x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{12} + 21x^{10} + 162x^{8} + 574x^{6} + 985x^{4} + 765x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{11} + 14\nu^{9} + 36\nu^{7} - 182\nu^{5} - 709\nu^{3} - 488\nu ) / 56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{10} + 20\nu^{8} + 142\nu^{6} + 432\nu^{4} + 553\nu^{2} + 220 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{11} + 77\nu^{9} + 515\nu^{7} + 1421\nu^{5} + 1581\nu^{3} + 526\nu ) / 28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{10} - 19\nu^{8} - 125\nu^{6} - 337\nu^{4} - 362\nu^{2} - 112 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -11\nu^{11} - 210\nu^{9} - 1376\nu^{7} - 3626\nu^{5} - 3709\nu^{3} - 1100\nu ) / 56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{11} - 210\nu^{9} - 1376\nu^{7} - 3598\nu^{5} - 3429\nu^{3} - 568\nu ) / 56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5\nu^{11} + 98\nu^{9} + 670\nu^{7} + 1890\nu^{5} + 2097\nu^{3} + 696\nu ) / 28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -3\nu^{10} - 58\nu^{8} - 388\nu^{6} - 1058\nu^{4} - 1121\nu^{2} - 316 ) / 8 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3\nu^{10} + 58\nu^{8} + 388\nu^{6} + 1058\nu^{4} + 1129\nu^{2} + 340 ) / 8 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -\nu^{10} - 19\nu^{8} - 124\nu^{6} - 327\nu^{4} - 341\nu^{2} - 104 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{9} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - 9\beta_{10} - 8\beta_{9} + \beta_{5} + \beta_{3} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{8} - 8\beta_{7} - 2\beta_{6} - 10\beta_{2} + 41\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{11} + 69\beta_{10} + 59\beta_{9} - 14\beta_{5} - 10\beta_{3} - 95 \)
|
| \(\nu^{7}\) | \(=\) |
\( 81\beta_{8} + 59\beta_{7} + 28\beta_{6} + 8\beta_{4} + 83\beta_{2} - 292\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -109\beta_{11} - 509\beta_{10} - 434\beta_{9} + 147\beta_{5} + 83\beta_{3} + 655 \)
|
| \(\nu^{9}\) | \(=\) |
\( -618\beta_{8} - 434\beta_{7} - 290\beta_{6} - 140\beta_{4} - 664\beta_{2} + 2107\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 908\beta_{11} + 3717\beta_{10} + 3205\beta_{9} - 1384\beta_{5} - 664\beta_{3} - 4651 \)
|
| \(\nu^{11}\) | \(=\) |
\( 4625\beta_{8} + 3205\beta_{7} + 2688\beta_{6} + 1672\beta_{4} + 5253\beta_{2} - 15290\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).
| \(n\) | \(76\) | \(112\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 36.1 |
|
− | 2.75497i | 2.91885 | −5.58988 | − | 1.00000i | − | 8.04135i | 1.45148 | 9.89002i | 5.51969 | −2.75497 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.2 | − | 2.59891i | −0.695617 | −4.75431 | 1.00000i | 1.80784i | −3.94647 | 7.15821i | −2.51612 | 2.59891 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.3 | − | 1.66705i | 0.792969 | −0.779058 | − | 1.00000i | − | 1.32192i | −0.457139 | − | 2.03537i | −2.37120 | −1.66705 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.4 | − | 1.36551i | 2.79310 | 0.135372 | 1.00000i | − | 3.81402i | −4.67865 | − | 2.91588i | 4.80143 | 1.36551 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.5 | − | 1.23687i | −2.43200 | 0.470165 | − | 1.00000i | 3.00806i | −3.53789 | − | 3.05526i | 2.91464 | −1.23687 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.6 | − | 0.694469i | −2.37730 | 1.51771 | 1.00000i | 1.65096i | 2.16868 | − | 2.44294i | 2.65157 | 0.694469 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.7 | 0.694469i | −2.37730 | 1.51771 | − | 1.00000i | − | 1.65096i | 2.16868 | 2.44294i | 2.65157 | 0.694469 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.8 | 1.23687i | −2.43200 | 0.470165 | 1.00000i | − | 3.00806i | −3.53789 | 3.05526i | 2.91464 | −1.23687 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.9 | 1.36551i | 2.79310 | 0.135372 | − | 1.00000i | 3.81402i | −4.67865 | 2.91588i | 4.80143 | 1.36551 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.10 | 1.66705i | 0.792969 | −0.779058 | 1.00000i | 1.32192i | −0.457139 | 2.03537i | −2.37120 | −1.66705 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.11 | 2.59891i | −0.695617 | −4.75431 | − | 1.00000i | − | 1.80784i | −3.94647 | − | 7.15821i | −2.51612 | 2.59891 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36.12 | 2.75497i | 2.91885 | −5.58988 | 1.00000i | 8.04135i | 1.45148 | − | 9.89002i | 5.51969 | −2.75497 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 185.2.c.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1665.2.e.e | 12 | ||
| 4.b | odd | 2 | 1 | 2960.2.p.h | 12 | ||
| 5.b | even | 2 | 1 | 925.2.c.c | 12 | ||
| 5.c | odd | 4 | 1 | 925.2.d.e | 12 | ||
| 5.c | odd | 4 | 1 | 925.2.d.f | 12 | ||
| 37.b | even | 2 | 1 | inner | 185.2.c.b | ✓ | 12 |
| 37.d | odd | 4 | 1 | 6845.2.a.h | 6 | ||
| 37.d | odd | 4 | 1 | 6845.2.a.i | 6 | ||
| 111.d | odd | 2 | 1 | 1665.2.e.e | 12 | ||
| 148.b | odd | 2 | 1 | 2960.2.p.h | 12 | ||
| 185.d | even | 2 | 1 | 925.2.c.c | 12 | ||
| 185.h | odd | 4 | 1 | 925.2.d.e | 12 | ||
| 185.h | odd | 4 | 1 | 925.2.d.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.c.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 185.2.c.b | ✓ | 12 | 37.b | even | 2 | 1 | inner |
| 925.2.c.c | 12 | 5.b | even | 2 | 1 | ||
| 925.2.c.c | 12 | 185.d | even | 2 | 1 | ||
| 925.2.d.e | 12 | 5.c | odd | 4 | 1 | ||
| 925.2.d.e | 12 | 185.h | odd | 4 | 1 | ||
| 925.2.d.f | 12 | 5.c | odd | 4 | 1 | ||
| 925.2.d.f | 12 | 185.h | odd | 4 | 1 | ||
| 1665.2.e.e | 12 | 3.b | odd | 2 | 1 | ||
| 1665.2.e.e | 12 | 111.d | odd | 2 | 1 | ||
| 2960.2.p.h | 12 | 4.b | odd | 2 | 1 | ||
| 2960.2.p.h | 12 | 148.b | odd | 2 | 1 | ||
| 6845.2.a.h | 6 | 37.d | odd | 4 | 1 | ||
| 6845.2.a.i | 6 | 37.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 21T_{2}^{10} + 162T_{2}^{8} + 574T_{2}^{6} + 985T_{2}^{4} + 765T_{2}^{2} + 196 \)
acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 21 T^{10} + \cdots + 196 \)
|
| $3$ |
\( (T^{6} - T^{5} - 14 T^{4} + \cdots - 26)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{6} \)
|
| $7$ |
\( (T^{6} + 9 T^{5} + 12 T^{4} + \cdots + 94)^{2} \)
|
| $11$ |
\( (T^{6} - T^{5} - 44 T^{4} + \cdots - 64)^{2} \)
|
| $13$ |
\( T^{12} + 100 T^{10} + \cdots + 3136 \)
|
| $17$ |
\( T^{12} + 68 T^{10} + \cdots + 64 \)
|
| $19$ |
\( T^{12} + 120 T^{10} + \cdots + 274576 \)
|
| $23$ |
\( T^{12} + 96 T^{10} + \cdots + 4096 \)
|
| $29$ |
\( T^{12} + \cdots + 451477504 \)
|
| $31$ |
\( T^{12} + 144 T^{10} + \cdots + 1024 \)
|
| $37$ |
\( T^{12} + \cdots + 2565726409 \)
|
| $41$ |
\( (T^{6} + 5 T^{5} + \cdots - 1552)^{2} \)
|
| $43$ |
\( T^{12} + 132 T^{10} + \cdots + 50176 \)
|
| $47$ |
\( (T^{6} - T^{5} + \cdots - 41098)^{2} \)
|
| $53$ |
\( (T^{6} + 3 T^{5} + \cdots - 109184)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 618616384 \)
|
| $61$ |
\( T^{12} + \cdots + 20699152384 \)
|
| $67$ |
\( (T^{6} - 8 T^{5} + \cdots + 664)^{2} \)
|
| $71$ |
\( (T^{6} - 21 T^{5} + \cdots - 10624)^{2} \)
|
| $73$ |
\( (T^{6} + 23 T^{5} + \cdots - 13888)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 384316816 \)
|
| $83$ |
\( (T^{6} + 31 T^{5} + \cdots - 158806)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 629407744 \)
|
| $97$ |
\( T^{12} + \cdots + 18183983104 \)
|
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