Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [181,2,Mod(1,181)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(181, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("181.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 181 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 181.a (trivial) |
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: | yes |
| Analytic conductor: | \(1.44529227659\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 8\nu^{4} + 25\nu^{3} - 18\nu^{2} - 10\nu + 15 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 16\nu^{5} - 21\nu^{4} + 31\nu^{3} + 12\nu^{2} - 15\nu + 1 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 15\nu^{6} - 2\nu^{5} - 69\nu^{4} + 17\nu^{3} + 100\nu^{2} - 25\nu - 31 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 5\nu^{7} - 2\nu^{6} + 40\nu^{5} - 31\nu^{4} - 74\nu^{3} + 66\nu^{2} + 35\nu - 20 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 7\nu^{6} + 26\nu^{5} - 56\nu^{4} - 59\nu^{3} + 106\nu^{2} + 42\nu - 45 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} + 5\nu^{6} - 32\nu^{5} + 7\nu^{4} + 59\nu^{3} - 24\nu^{2} - 25\nu + 7 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} + 2\beta_{6} - \beta_{3} + \beta_{2} + 3\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} + 3\beta_{6} - \beta_{5} + \beta_{4} + 8\beta_{2} - 2\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{8} - 8\beta_{7} + 20\beta_{6} - 2\beta_{5} + \beta_{4} - 8\beta_{3} + 10\beta_{2} + 9\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta_{8} - 11\beta_{7} + 36\beta_{6} - 10\beta_{5} + 9\beta_{4} - \beta_{3} + 57\beta_{2} - 22\beta _1 + 100 \)
|
| \(\nu^{7}\) | \(=\) |
\( 81\beta_{8} - 58\beta_{7} + 162\beta_{6} - 22\beta_{5} + 11\beta_{4} - 52\beta_{3} + 86\beta_{2} + 19\beta _1 + 151 \)
|
| \(\nu^{8}\) | \(=\) |
\( 138 \beta_{8} - 97 \beta_{7} + 327 \beta_{6} - 81 \beta_{5} + 64 \beta_{4} - 16 \beta_{3} + 400 \beta_{2} + \cdots + 661 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.35228 | 0.154676 | 3.53320 | 2.68725 | −0.363841 | −1.31879 | −3.60652 | −2.97608 | −6.32114 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.35249 | −2.73889 | −0.170779 | −2.15664 | 3.70431 | −1.90242 | 2.93595 | 4.50151 | 2.91682 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.24377 | 3.09454 | −0.453038 | −1.97661 | −3.84889 | 1.79580 | 3.05101 | 6.57616 | 2.45845 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.730132 | 0.820975 | −1.46691 | 1.76532 | −0.599420 | 1.92280 | 2.53130 | −2.32600 | −1.28892 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.682920 | 2.48419 | −1.53362 | 1.99024 | 1.69650 | −0.932755 | −2.41318 | 3.17118 | 1.35918 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.22961 | −2.38474 | −0.488068 | 3.23480 | −2.93229 | 4.67891 | −3.05934 | 2.68698 | 3.97754 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.92391 | 0.621031 | 1.70143 | 0.00911680 | 1.19481 | 1.84046 | −0.574421 | −2.61432 | 0.0175399 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.14345 | 2.53942 | 2.59437 | −3.90648 | 5.44311 | −4.19031 | 1.27399 | 3.44866 | −8.37332 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.69878 | −1.59120 | 5.28341 | −0.647009 | −4.29429 | 0.106317 | 8.86121 | −0.468091 | −1.74614 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(181\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 181.2.a.b | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1629.2.a.g | 9 | ||
| 4.b | odd | 2 | 1 | 2896.2.a.m | 9 | ||
| 5.b | even | 2 | 1 | 4525.2.a.j | 9 | ||
| 7.b | odd | 2 | 1 | 8869.2.a.o | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 181.2.a.b | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1629.2.a.g | 9 | 3.b | odd | 2 | 1 | ||
| 2896.2.a.m | 9 | 4.b | odd | 2 | 1 | ||
| 4525.2.a.j | 9 | 5.b | even | 2 | 1 | ||
| 8869.2.a.o | 9 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 3T_{2}^{8} - 9T_{2}^{7} + 29T_{2}^{6} + 23T_{2}^{5} - 84T_{2}^{4} - 23T_{2}^{3} + 89T_{2}^{2} + 8T_{2} - 27 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(181))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 3 T^{8} + \cdots - 27 \)
|
| $3$ |
\( T^{9} - 3 T^{8} + \cdots + 16 \)
|
| $5$ |
\( T^{9} - T^{8} - 24 T^{7} + \cdots - 3 \)
|
| $7$ |
\( T^{9} - 2 T^{8} + \cdots - 31 \)
|
| $11$ |
\( T^{9} - 24 T^{8} + \cdots - 19056 \)
|
| $13$ |
\( T^{9} + 8 T^{8} + \cdots + 2993 \)
|
| $17$ |
\( T^{9} + T^{8} + \cdots - 503952 \)
|
| $19$ |
\( T^{9} - 4 T^{8} + \cdots + 5575 \)
|
| $23$ |
\( T^{9} - 14 T^{8} + \cdots + 32553 \)
|
| $29$ |
\( T^{9} - 13 T^{8} + \cdots - 1245 \)
|
| $31$ |
\( T^{9} + 7 T^{8} + \cdots - 1174577 \)
|
| $37$ |
\( T^{9} + 20 T^{8} + \cdots - 118307 \)
|
| $41$ |
\( T^{9} - 10 T^{8} + \cdots - 43344 \)
|
| $43$ |
\( T^{9} + 5 T^{8} + \cdots + 516752 \)
|
| $47$ |
\( T^{9} + 2 T^{8} + \cdots - 2500083 \)
|
| $53$ |
\( T^{9} + T^{8} + \cdots - 2156208 \)
|
| $59$ |
\( T^{9} - 24 T^{8} + \cdots + 3673680 \)
|
| $61$ |
\( T^{9} + 16 T^{8} + \cdots + 103472 \)
|
| $67$ |
\( T^{9} - 9 T^{8} + \cdots - 11584 \)
|
| $71$ |
\( T^{9} - 21 T^{8} + \cdots - 1005903 \)
|
| $73$ |
\( T^{9} + 5 T^{8} + \cdots + 488789 \)
|
| $79$ |
\( T^{9} - 10 T^{8} + \cdots - 148160 \)
|
| $83$ |
\( T^{9} - 23 T^{8} + \cdots - 52184619 \)
|
| $89$ |
\( T^{9} + T^{8} + \cdots + 36027600 \)
|
| $97$ |
\( T^{9} + 3 T^{8} + \cdots + 478352 \)
|
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