Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(24.6631136589\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 16849 x^{9} + 81012 x^{8} + 99148992 x^{7} - 680754100 x^{6} - 245202429236 x^{5} + \cdots - 72\!\cdots\!70 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 16849 x^{9} + 81012 x^{8} + 99148992 x^{7} - 680754100 x^{6} - 245202429236 x^{5} + \cdots - 72\!\cdots\!70 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19\!\cdots\!97 \nu^{10} + \cdots + 25\!\cdots\!90 ) / 22\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\!\cdots\!97 \nu^{10} + \cdots + 11\!\cdots\!10 ) / 11\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!23 \nu^{10} + \cdots + 58\!\cdots\!10 ) / 94\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11\!\cdots\!67 \nu^{10} + \cdots - 53\!\cdots\!30 ) / 87\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29\!\cdots\!42 \nu^{10} + \cdots + 78\!\cdots\!90 ) / 21\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 15\!\cdots\!27 \nu^{10} + \cdots - 73\!\cdots\!70 ) / 75\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!13 \nu^{10} + \cdots - 40\!\cdots\!30 ) / 56\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 32\!\cdots\!67 \nu^{10} + \cdots + 42\!\cdots\!10 ) / 11\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 17\!\cdots\!67 \nu^{10} + \cdots + 37\!\cdots\!90 ) / 22\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} - 11\beta _1 + 12256 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{10} - 10\beta_{9} - \beta_{8} + 5\beta_{7} + 5\beta_{4} - 15\beta_{3} + 308\beta_{2} + 20160\beta _1 - 143653 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 25 \beta_{10} + 42 \beta_{9} + 201 \beta_{8} + 27 \beta_{7} - 32 \beta_{6} - 152 \beta_{5} + \cdots + 124359005 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13381 \beta_{10} - 100174 \beta_{9} - 8645 \beta_{8} + 37241 \beta_{7} - 5380 \beta_{6} + \cdots - 1838363061 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 270565 \beta_{10} + 712282 \beta_{9} + 980461 \beta_{8} + 146075 \beta_{7} - 290560 \beta_{6} + \cdots + 372476995665 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 64910497 \beta_{10} - 421679956 \beta_{9} - 29397417 \beta_{8} + 116909617 \beta_{7} + \cdots - 8246444443991 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 3155420259 \beta_{10} + 10077975456 \beta_{9} + 7655315931 \beta_{8} + 1349437117 \beta_{7} + \cdots + 23\!\cdots\!29 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 1121115619027 \beta_{10} - 6719425968378 \beta_{9} - 403437347683 \beta_{8} + 1403958713239 \beta_{7} + \cdots - 13\!\cdots\!87 ) / 8 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 60943826818479 \beta_{10} + 217093886002298 \beta_{9} + 112071315696015 \beta_{8} + \cdots + 31\!\cdots\!43 ) / 8 \)
|
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 |
|
−178.406 | 1587.47 | 23636.6 | 41568.2 | −283214. | −112887. | −2.75540e6 | 925746. | −7.41600e6 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −159.538 | −495.925 | 17260.4 | −33288.2 | 79118.8 | −166559. | −1.44675e6 | −1.34838e6 | 5.31073e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −107.209 | 915.216 | 3301.81 | 23889.6 | −98119.6 | −262916. | 524273. | −756702. | −2.56118e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −64.3846 | −1685.52 | −4046.62 | −37754.0 | 108522. | 326916. | 787979. | 1.24666e6 | 2.43078e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −48.8684 | −1975.52 | −5803.88 | 27867.4 | 96540.6 | −438834. | 683956. | 2.30836e6 | −1.36184e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −23.6395 | 1012.45 | −7633.17 | −3436.13 | −23933.8 | 301626. | 374099. | −569269. | 81228.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 70.6751 | −1733.36 | −3197.03 | 38887.4 | −122505. | 410231. | −804921. | 1.41021e6 | 2.74837e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 77.1227 | 1906.74 | −2244.09 | −40583.6 | 147053. | −97829.0 | −804859. | 2.04134e6 | −3.12991e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 82.8983 | 503.467 | −1319.88 | 34875.3 | 41736.5 | −494362. | −788518. | −1.34084e6 | 2.89110e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 133.648 | −341.999 | 9669.68 | −2470.99 | −45707.3 | −224862. | 197489. | −1.47736e6 | −330243. | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 153.702 | −553.020 | 15432.2 | −59873.0 | −85000.2 | 328532. | 1.11284e6 | −1.28849e6 | −9.20259e6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 23.14.a.a | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.14.a.a | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 64 T_{2}^{10} - 65536 T_{2}^{9} - 2958888 T_{2}^{8} + 1530523104 T_{2}^{7} + \cdots - 21\!\cdots\!68 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots - 21\!\cdots\!68 \)
|
| $3$ |
\( T^{11} + \cdots - 76\!\cdots\!00 \)
|
| $5$ |
\( T^{11} + \cdots - 97\!\cdots\!00 \)
|
| $7$ |
\( T^{11} + \cdots + 31\!\cdots\!28 \)
|
| $11$ |
\( T^{11} + \cdots - 10\!\cdots\!80 \)
|
| $13$ |
\( T^{11} + \cdots + 33\!\cdots\!44 \)
|
| $17$ |
\( T^{11} + \cdots + 11\!\cdots\!00 \)
|
| $19$ |
\( T^{11} + \cdots + 15\!\cdots\!00 \)
|
| $23$ |
\( (T + 148035889)^{11} \)
|
| $29$ |
\( T^{11} + \cdots + 32\!\cdots\!00 \)
|
| $31$ |
\( T^{11} + \cdots - 41\!\cdots\!00 \)
|
| $37$ |
\( T^{11} + \cdots + 13\!\cdots\!24 \)
|
| $41$ |
\( T^{11} + \cdots - 42\!\cdots\!60 \)
|
| $43$ |
\( T^{11} + \cdots - 12\!\cdots\!00 \)
|
| $47$ |
\( T^{11} + \cdots - 46\!\cdots\!00 \)
|
| $53$ |
\( T^{11} + \cdots + 18\!\cdots\!92 \)
|
| $59$ |
\( T^{11} + \cdots + 24\!\cdots\!20 \)
|
| $61$ |
\( T^{11} + \cdots - 35\!\cdots\!44 \)
|
| $67$ |
\( T^{11} + \cdots - 27\!\cdots\!00 \)
|
| $71$ |
\( T^{11} + \cdots - 28\!\cdots\!00 \)
|
| $73$ |
\( T^{11} + \cdots - 33\!\cdots\!88 \)
|
| $79$ |
\( T^{11} + \cdots + 11\!\cdots\!80 \)
|
| $83$ |
\( T^{11} + \cdots + 49\!\cdots\!16 \)
|
| $89$ |
\( T^{11} + \cdots - 69\!\cdots\!00 \)
|
| $97$ |
\( T^{11} + \cdots - 27\!\cdots\!12 \)
|
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