Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [184,2,Mod(93,184)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(184, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("184.93");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 184 = 2^{3} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 184.b (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: | \(1.46924739719\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | 12.0.656213229924891664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2x^{10} - x^{9} + 8x^{8} - 2x^{7} - 14x^{6} - 4x^{5} + 32x^{4} - 8x^{3} - 32x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{12} - 2x^{10} - x^{9} + 8x^{8} - 2x^{7} - 14x^{6} - 4x^{5} + 32x^{4} - 8x^{3} - 32x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{11} - 26 \nu^{10} + 42 \nu^{9} + 31 \nu^{8} - 54 \nu^{7} - 126 \nu^{6} + 182 \nu^{5} + \cdots + 160 ) / 224 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} + 4\nu^{10} - \nu^{8} + 4\nu^{7} + 28\nu^{6} - 14\nu^{5} - 4\nu^{4} - 12\nu^{3} + 56\nu^{2} - 32\nu - 16 ) / 56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{11} + 3\nu^{10} + \nu^{8} - 11\nu^{7} - 14\nu^{5} + 18\nu^{4} + 12\nu^{3} - 28\nu^{2} - 24\nu + 72 ) / 56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{11} + 2 \nu^{10} - 2 \nu^{9} - 3 \nu^{8} - 2 \nu^{7} + 22 \nu^{6} - 22 \nu^{5} - 16 \nu^{4} + \cdots - 32 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 9 \nu^{11} + 6 \nu^{10} + 14 \nu^{9} - 19 \nu^{8} - 22 \nu^{7} + 70 \nu^{6} + 98 \nu^{5} + \cdots - 640 ) / 224 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13 \nu^{11} - 18 \nu^{10} - 14 \nu^{9} + 71 \nu^{8} + 66 \nu^{7} - 182 \nu^{6} - 98 \nu^{5} + \cdots + 800 ) / 224 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13 \nu^{11} + 18 \nu^{10} + 14 \nu^{9} - 15 \nu^{8} - 66 \nu^{7} + 70 \nu^{6} + 42 \nu^{5} + \cdots - 128 ) / 224 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -\nu^{11} + 2\nu^{9} + \nu^{8} - 8\nu^{7} + 2\nu^{6} + 14\nu^{5} + 4\nu^{4} - 32\nu^{3} + 8\nu^{2} + 32\nu ) / 16 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11 \nu^{11} - 12 \nu^{10} - 14 \nu^{9} + 17 \nu^{8} + 44 \nu^{7} - 70 \nu^{6} - 70 \nu^{5} + \cdots + 160 ) / 112 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{7} + \beta_{5} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{11} - \beta_{9} - \beta_{7} - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} - 2\beta_{9} + 2\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 3\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} - 2\beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} - 3\beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( -3\beta_{11} - 2\beta_{10} + \beta_{9} + 2\beta_{8} - \beta_{7} - 4\beta_{5} - 2\beta_{2} \)
|
| \(\nu^{8}\) | \(=\) |
\( 4 \beta_{11} + 3 \beta_{10} + 4 \beta_{8} + 10 \beta_{7} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \cdots + 7 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4 \beta_{11} - 7 \beta_{10} + 4 \beta_{9} - 5 \beta_{7} + \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + \cdots - 7 \)
|
| \(\nu^{10}\) | \(=\) |
\( - \beta_{11} - 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} - 8 \beta_{6} + 8 \beta_{5} + \cdots - 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( 16 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 9 \beta_{6} + 13 \beta_{5} + \cdots + 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(93\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 93.1 |
|
−1.38645 | − | 0.278834i | − | 0.830960i | 1.84450 | + | 0.773181i | 0.450506i | −0.231700 | + | 1.15209i | 1.45785 | −2.34173 | − | 1.58629i | 2.30951 | 0.125616 | − | 0.624605i | |||||||||||||||||||||||||||||||||||||||||||
| 93.2 | −1.38645 | + | 0.278834i | 0.830960i | 1.84450 | − | 0.773181i | − | 0.450506i | −0.231700 | − | 1.15209i | 1.45785 | −2.34173 | + | 1.58629i | 2.30951 | 0.125616 | + | 0.624605i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.3 | −1.03725 | − | 0.961312i | 0.398699i | 0.151759 | + | 1.99423i | − | 2.62719i | 0.383274 | − | 0.413549i | −4.58940 | 1.75967 | − | 2.21440i | 2.84104 | −2.52555 | + | 2.72504i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.4 | −1.03725 | + | 0.961312i | − | 0.398699i | 0.151759 | − | 1.99423i | 2.62719i | 0.383274 | + | 0.413549i | −4.58940 | 1.75967 | + | 2.21440i | 2.84104 | −2.52555 | − | 2.72504i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.5 | −0.747935 | − | 1.20025i | − | 2.91627i | −0.881186 | + | 1.79541i | − | 3.30948i | −3.50024 | + | 2.18118i | 3.13483 | 2.81401 | − | 0.285211i | −5.50461 | −3.97219 | + | 2.47527i | |||||||||||||||||||||||||||||||||||||||||||
| 93.6 | −0.747935 | + | 1.20025i | 2.91627i | −0.881186 | − | 1.79541i | 3.30948i | −3.50024 | − | 2.18118i | 3.13483 | 2.81401 | + | 0.285211i | −5.50461 | −3.97219 | − | 2.47527i | |||||||||||||||||||||||||||||||||||||||||||||
| 93.7 | 0.832473 | − | 1.14324i | 1.30052i | −0.613977 | − | 1.90343i | 2.11819i | 1.48680 | + | 1.08265i | 3.62873 | −2.68718 | − | 0.882631i | 1.30865 | 2.42159 | + | 1.76333i | |||||||||||||||||||||||||||||||||||||||||||||
| 93.8 | 0.832473 | + | 1.14324i | − | 1.30052i | −0.613977 | + | 1.90343i | − | 2.11819i | 1.48680 | − | 1.08265i | 3.62873 | −2.68718 | + | 0.882631i | 1.30865 | 2.42159 | − | 1.76333i | |||||||||||||||||||||||||||||||||||||||||||
| 93.9 | 1.08708 | − | 0.904581i | 1.54336i | 0.363467 | − | 1.96670i | − | 4.24622i | 1.39609 | + | 1.67775i | −0.751283 | −1.38392 | − | 2.46673i | 0.618038 | −3.84105 | − | 4.61596i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.10 | 1.08708 | + | 0.904581i | − | 1.54336i | 0.363467 | + | 1.96670i | 4.24622i | 1.39609 | − | 1.67775i | −0.751283 | −1.38392 | + | 2.46673i | 0.618038 | −3.84105 | + | 4.61596i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.11 | 1.25209 | − | 0.657482i | − | 3.09397i | 1.13543 | − | 1.64645i | 2.72492i | −2.03423 | − | 3.87391i | 1.11928 | 0.339152 | − | 2.80802i | −6.57262 | 1.79158 | + | 3.41183i | ||||||||||||||||||||||||||||||||||||||||||||
| 93.12 | 1.25209 | + | 0.657482i | 3.09397i | 1.13543 | + | 1.64645i | − | 2.72492i | −2.03423 | + | 3.87391i | 1.11928 | 0.339152 | + | 2.80802i | −6.57262 | 1.79158 | − | 3.41183i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 184.2.b.c | ✓ | 12 |
| 4.b | odd | 2 | 1 | 736.2.b.c | 12 | ||
| 8.b | even | 2 | 1 | inner | 184.2.b.c | ✓ | 12 |
| 8.d | odd | 2 | 1 | 736.2.b.c | 12 | ||
| 16.e | even | 4 | 2 | 5888.2.a.bd | 12 | ||
| 16.f | odd | 4 | 2 | 5888.2.a.be | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.2.b.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 184.2.b.c | ✓ | 12 | 8.b | even | 2 | 1 | inner |
| 736.2.b.c | 12 | 4.b | odd | 2 | 1 | ||
| 736.2.b.c | 12 | 8.d | odd | 2 | 1 | ||
| 5888.2.a.bd | 12 | 16.e | even | 4 | 2 | ||
| 5888.2.a.be | 12 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 23T_{3}^{10} + 178T_{3}^{8} + 542T_{3}^{6} + 689T_{3}^{4} + 323T_{3}^{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(184, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 2 T^{10} + \cdots + 64 \)
|
| $3$ |
\( T^{12} + 23 T^{10} + \cdots + 36 \)
|
| $5$ |
\( T^{12} + 48 T^{10} + \cdots + 9216 \)
|
| $7$ |
\( (T^{6} - 4 T^{5} - 16 T^{4} + \cdots + 64)^{2} \)
|
| $11$ |
\( T^{12} + 104 T^{10} + \cdots + 614656 \)
|
| $13$ |
\( T^{12} + 107 T^{10} + \cdots + 781456 \)
|
| $17$ |
\( (T^{6} - 62 T^{4} + \cdots - 3088)^{2} \)
|
| $19$ |
\( T^{12} + 164 T^{10} + \cdots + 12845056 \)
|
| $23$ |
\( (T + 1)^{12} \)
|
| $29$ |
\( T^{12} + 203 T^{10} + \cdots + 633616 \)
|
| $31$ |
\( (T^{6} + 5 T^{5} + \cdots - 112)^{2} \)
|
| $37$ |
\( T^{12} + 212 T^{10} + \cdots + 25240576 \)
|
| $41$ |
\( (T^{6} - 3 T^{5} + \cdots + 554)^{2} \)
|
| $43$ |
\( T^{12} + \cdots + 5435817984 \)
|
| $47$ |
\( (T^{6} - 5 T^{5} + \cdots + 105392)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 3712221184 \)
|
| $59$ |
\( T^{12} + 312 T^{10} + \cdots + 4096 \)
|
| $61$ |
\( T^{12} + 92 T^{10} + \cdots + 147456 \)
|
| $67$ |
\( T^{12} + 228 T^{10} + \cdots + 1763584 \)
|
| $71$ |
\( (T^{6} - 3 T^{5} + \cdots + 7688)^{2} \)
|
| $73$ |
\( (T^{6} + 35 T^{5} + \cdots + 78022)^{2} \)
|
| $79$ |
\( (T^{6} + 8 T^{5} + \cdots + 2048)^{2} \)
|
| $83$ |
\( T^{12} + 336 T^{10} + \cdots + 4129024 \)
|
| $89$ |
\( (T^{6} + 20 T^{5} + \cdots + 768)^{2} \)
|
| $97$ |
\( (T^{6} - 26 T^{5} + \cdots + 166832)^{2} \)
|
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