Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,16,Mod(1,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.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: | \(32.8195061730\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 4 x^{11} - 68942 x^{10} - 977032 x^{9} + 1644150380 x^{8} + 50352376602 x^{7} + \cdots + 34\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{22}\cdot 3^{6}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 68942 x^{10} - 977032 x^{9} + 1644150380 x^{8} + 50352376602 x^{7} + \cdots + 34\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10\!\cdots\!41 \nu^{11} + \cdots + 36\!\cdots\!20 ) / 22\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\!\cdots\!41 \nu^{11} + \cdots - 47\!\cdots\!40 ) / 11\!\cdots\!60 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!07 \nu^{11} + \cdots - 90\!\cdots\!16 ) / 88\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 54\!\cdots\!55 \nu^{11} + \cdots + 14\!\cdots\!36 ) / 29\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 52\!\cdots\!35 \nu^{11} + \cdots + 48\!\cdots\!56 ) / 88\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 36\!\cdots\!23 \nu^{11} + \cdots - 27\!\cdots\!20 ) / 44\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 17\!\cdots\!29 \nu^{11} + \cdots - 24\!\cdots\!36 ) / 14\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 39\!\cdots\!59 \nu^{11} + \cdots - 13\!\cdots\!12 ) / 29\!\cdots\!60 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 65\!\cdots\!25 \nu^{11} + \cdots - 11\!\cdots\!96 ) / 44\!\cdots\!40 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 31\!\cdots\!45 \nu^{11} + \cdots - 52\!\cdots\!56 ) / 14\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 53\beta _1 + 45984 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} + \cdots + 2533939 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 83 \beta_{11} - 241 \beta_{10} + 244 \beta_{9} - 25 \beta_{8} + 168 \beta_{7} + 403 \beta_{6} + \cdots + 1967634047 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17043 \beta_{11} - 93077 \beta_{10} - 11312 \beta_{9} + 78053 \beta_{8} - 71950 \beta_{7} + \cdots + 98912518029 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 761645 \beta_{11} - 8352331 \beta_{10} + 10510980 \beta_{9} + 603293 \beta_{8} + 3078664 \beta_{7} + \cdots + 49619449579813 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 64786283 \beta_{11} - 2332805471 \beta_{10} - 383670104 \beta_{9} + 1685964919 \beta_{8} + \cdots + 34\!\cdots\!75 ) / 8 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 64803741831 \beta_{11} - 224240419747 \beta_{10} + 354381826612 \beta_{9} + 38523685577 \beta_{8} + \cdots + 13\!\cdots\!41 ) / 8 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 19433414604075 \beta_{11} - 53468004726027 \beta_{10} - 8195782569304 \beta_{9} + \cdots + 11\!\cdots\!87 ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 47\!\cdots\!31 \beta_{11} + \cdots + 37\!\cdots\!53 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 10\!\cdots\!55 \beta_{11} + \cdots + 38\!\cdots\!11 ) / 8 \)
|
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 |
|
−333.711 | −2416.34 | 78595.0 | 165276. | 806359. | −2.92451e6 | −1.52930e7 | −8.51021e6 | −5.51544e7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −306.914 | 1666.99 | 61428.0 | 64027.3 | −511621. | 1.64942e6 | −8.79615e6 | −1.15701e7 | −1.96509e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −196.556 | −4944.68 | 5866.22 | −184879. | 971906. | −2.97859e6 | 5.28770e6 | 1.01009e7 | 3.63390e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −150.613 | 3985.78 | −10083.6 | 347827. | −600312. | −3.26400e6 | 6.45403e6 | 1.53751e6 | −5.23874e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −145.471 | 254.987 | −11606.2 | −233086. | −37093.1 | 2.20456e6 | 6.45516e6 | −1.42839e7 | 3.39071e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −91.4073 | 6050.26 | −24412.7 | −98557.7 | −553038. | 1.04995e6 | 5.22673e6 | 2.22568e7 | 9.00889e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −15.5252 | −2381.89 | −32527.0 | 172933. | 36979.3 | −570604. | 1.01372e6 | −8.67549e6 | −2.68481e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 8.28020 | −6881.62 | −32699.4 | −91711.3 | −56981.2 | 1.74802e6 | −542083. | 3.30077e7 | −759388. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 166.320 | 5379.02 | −5105.54 | −246835. | 894641. | 345495. | −6.29914e6 | 1.45850e7 | −4.10537e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 174.185 | 1818.41 | −2427.63 | 106224. | 316739. | −648096. | −6.13055e6 | −1.10423e7 | 1.85026e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 302.300 | 1384.15 | 58617.0 | −149481. | 418427. | −3.04303e6 | 7.81414e6 | −1.24330e7 | −4.51879e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 333.112 | −5660.06 | 78195.9 | −56214.4 | −1.88544e6 | 714042. | 1.51326e7 | 1.76874e7 | −1.87257e7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(-1\) |
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 | 23.16.a.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.16.a.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 256 T_{2}^{11} - 245760 T_{2}^{10} - 66355616 T_{2}^{9} + 18854461712 T_{2}^{8} + \cdots - 15\!\cdots\!56 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 15\!\cdots\!56 \)
|
| $3$ |
\( T^{12} + \cdots - 15\!\cdots\!00 \)
|
| $5$ |
\( T^{12} + \cdots - 54\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots + 52\!\cdots\!72 \)
|
| $11$ |
\( T^{12} + \cdots - 58\!\cdots\!60 \)
|
| $13$ |
\( T^{12} + \cdots - 45\!\cdots\!64 \)
|
| $17$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $19$ |
\( T^{12} + \cdots + 20\!\cdots\!00 \)
|
| $23$ |
\( (T - 3404825447)^{12} \)
|
| $29$ |
\( T^{12} + \cdots - 31\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 19\!\cdots\!60 \)
|
| $37$ |
\( T^{12} + \cdots + 26\!\cdots\!44 \)
|
| $41$ |
\( T^{12} + \cdots + 41\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots + 50\!\cdots\!28 \)
|
| $47$ |
\( T^{12} + \cdots - 29\!\cdots\!60 \)
|
| $53$ |
\( T^{12} + \cdots - 96\!\cdots\!84 \)
|
| $59$ |
\( T^{12} + \cdots - 59\!\cdots\!80 \)
|
| $61$ |
\( T^{12} + \cdots + 39\!\cdots\!36 \)
|
| $67$ |
\( T^{12} + \cdots + 24\!\cdots\!32 \)
|
| $71$ |
\( T^{12} + \cdots - 62\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots + 17\!\cdots\!80 \)
|
| $79$ |
\( T^{12} + \cdots - 29\!\cdots\!08 \)
|
| $83$ |
\( T^{12} + \cdots - 37\!\cdots\!08 \)
|
| $89$ |
\( T^{12} + \cdots - 17\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots - 49\!\cdots\!00 \)
|
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