Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [115,6,Mod(1,115)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("115.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.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: | \(18.4441392785\) |
| 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: |
\( x^{12} - 4 x^{11} - 329 x^{10} + 1059 x^{9} + 41059 x^{8} - 99023 x^{7} - 2392947 x^{6} + 3889937 x^{5} + \cdots + 4039776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{12} - 4 x^{11} - 329 x^{10} + 1059 x^{9} + 41059 x^{8} - 99023 x^{7} - 2392947 x^{6} + 3889937 x^{5} + \cdots + 4039776 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4105114832381 \nu^{11} + 201360999557039 \nu^{10} + \cdots - 78\!\cdots\!52 ) / 49\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4485970522399 \nu^{11} - 13077055877179 \nu^{10} + \cdots + 42\!\cdots\!52 ) / 49\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1306807350427 \nu^{11} + 24571239904313 \nu^{10} + \cdots + 38\!\cdots\!36 ) / 12\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12427339108299 \nu^{11} - 7270386628201 \nu^{10} + \cdots - 14\!\cdots\!72 ) / 24\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 36432839572533 \nu^{11} - 115989961322647 \nu^{10} + \cdots - 43\!\cdots\!84 ) / 49\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 32723488673401 \nu^{11} + 19321474848899 \nu^{10} + \cdots + 50\!\cdots\!48 ) / 24\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 6009477378277 \nu^{11} - 21549568159577 \nu^{10} + \cdots + 11\!\cdots\!96 ) / 38\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 100656069415571 \nu^{11} - 543196049549311 \nu^{10} + \cdots + 59\!\cdots\!28 ) / 49\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 116538806587371 \nu^{11} + 502501164538551 \nu^{10} + \cdots - 87\!\cdots\!48 ) / 49\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 56 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} - \beta_{10} + \beta_{6} + 2\beta_{4} + \beta_{2} + 84\beta _1 + 42 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \cdots + 4683 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 147 \beta_{11} - 157 \beta_{10} + 2 \beta_{9} + 124 \beta_{6} - 3 \beta_{5} + 377 \beta_{4} + \cdots + 7309 \)
|
| \(\nu^{6}\) | \(=\) |
\( 164 \beta_{11} + 42 \beta_{10} - 262 \beta_{9} + 750 \beta_{8} + 688 \beta_{7} + 375 \beta_{6} + \cdots + 436251 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 17202 \beta_{11} - 19728 \beta_{10} + 850 \beta_{9} + 476 \beta_{8} + 928 \beta_{7} + 13386 \beta_{6} + \cdots + 996504 \)
|
| \(\nu^{8}\) | \(=\) |
\( 19130 \beta_{11} - 11362 \beta_{10} - 23402 \beta_{9} + 107278 \beta_{8} + 94540 \beta_{7} + \cdots + 42918402 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1882579 \beta_{11} - 2322275 \beta_{10} + 181432 \beta_{9} + 147218 \beta_{8} + 266624 \beta_{7} + \cdots + 125078532 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1861399 \beta_{11} - 3471513 \beta_{10} - 1533108 \beta_{9} + 14006586 \beta_{8} + 12065488 \beta_{7} + \cdots + 4356449237 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 201720793 \beta_{11} - 267429131 \beta_{10} + 30247242 \beta_{9} + 30042510 \beta_{8} + \cdots + 15167057835 \)
|
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 |
|
−9.80811 | −29.7028 | 64.1990 | 25.0000 | 291.328 | −68.7900 | −315.812 | 639.254 | −245.203 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.88408 | −9.77577 | 46.9269 | 25.0000 | 86.8487 | 148.975 | −132.612 | −147.434 | −222.102 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.44196 | 28.1750 | 39.2667 | 25.0000 | −237.853 | −83.6202 | −61.3452 | 550.833 | −211.049 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.41367 | 6.74608 | −2.69220 | 25.0000 | −36.5211 | 167.896 | 187.812 | −197.490 | −135.342 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.85960 | −12.6293 | −17.1035 | 25.0000 | 48.7440 | −113.441 | 189.520 | −83.5011 | −96.4901 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.713402 | 22.8326 | −31.4911 | 25.0000 | 16.2889 | 23.6618 | −45.2947 | 278.329 | 17.8351 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.02304 | −8.89333 | −30.9534 | 25.0000 | −9.09827 | −228.319 | −64.4041 | −163.909 | 25.5761 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.18883 | −22.3016 | −5.07606 | 25.0000 | −115.719 | 6.29344 | −192.381 | 254.359 | 129.721 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 7.78525 | 27.6988 | 28.6101 | 25.0000 | 215.642 | 167.772 | −26.3911 | 524.222 | 194.631 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 8.26134 | 12.5950 | 36.2497 | 25.0000 | 104.052 | 63.2337 | 35.1081 | −84.3653 | 206.533 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 10.3165 | −11.9831 | 74.4302 | 25.0000 | −123.623 | 148.504 | 437.732 | −99.4063 | 257.913 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 11.1191 | 19.2382 | 91.6335 | 25.0000 | 213.911 | −216.165 | 663.068 | 127.109 | 277.976 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 115.6.a.e | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1035.6.a.m | 12 | ||
| 5.b | even | 2 | 1 | 575.6.a.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.6.a.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 575.6.a.g | 12 | 5.b | even | 2 | 1 | ||
| 1035.6.a.m | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 8 T_{2}^{11} - 307 T_{2}^{10} + 2231 T_{2}^{9} + 35620 T_{2}^{8} - 227565 T_{2}^{7} + \cdots - 429434688 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(115))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 429434688 \)
|
| $3$ |
\( T^{12} + \cdots + 253857988051200 \)
|
| $5$ |
\( (T - 25)^{12} \)
|
| $7$ |
\( T^{12} + \cdots - 18\!\cdots\!04 \)
|
| $11$ |
\( T^{12} + \cdots + 25\!\cdots\!80 \)
|
| $13$ |
\( T^{12} + \cdots - 11\!\cdots\!24 \)
|
| $17$ |
\( T^{12} + \cdots - 19\!\cdots\!00 \)
|
| $19$ |
\( T^{12} + \cdots - 19\!\cdots\!00 \)
|
| $23$ |
\( (T + 529)^{12} \)
|
| $29$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 69\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 23\!\cdots\!48 \)
|
| $41$ |
\( T^{12} + \cdots - 44\!\cdots\!40 \)
|
| $43$ |
\( T^{12} + \cdots + 42\!\cdots\!00 \)
|
| $47$ |
\( T^{12} + \cdots - 34\!\cdots\!60 \)
|
| $53$ |
\( T^{12} + \cdots - 38\!\cdots\!28 \)
|
| $59$ |
\( T^{12} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots + 11\!\cdots\!04 \)
|
| $67$ |
\( T^{12} + \cdots + 35\!\cdots\!20 \)
|
| $71$ |
\( T^{12} + \cdots - 37\!\cdots\!00 \)
|
| $73$ |
\( T^{12} + \cdots - 11\!\cdots\!88 \)
|
| $79$ |
\( T^{12} + \cdots + 12\!\cdots\!20 \)
|
| $83$ |
\( T^{12} + \cdots + 47\!\cdots\!44 \)
|
| $89$ |
\( T^{12} + \cdots - 18\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 31\!\cdots\!96 \)
|
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