Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [928,4,Mod(1,928)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(928, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("928.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 928 = 2^{5} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 928.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: | \(54.7537724853\) |
| 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: |
\( x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| 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
\( x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + \cdots + 14751072 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 8262965802853 \nu^{11} + 902081932283552 \nu^{10} + \cdots - 13\!\cdots\!68 ) / 18\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 9923410960919 \nu^{11} - 37676872998341 \nu^{10} + \cdots - 24\!\cdots\!64 ) / 18\!\cdots\!84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 31658966007779 \nu^{11} + 107796402154664 \nu^{10} + \cdots - 11\!\cdots\!72 ) / 37\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 79714148233592 \nu^{11} + 244066653715127 \nu^{10} + \cdots - 37\!\cdots\!08 ) / 94\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17168595976539 \nu^{11} - 61050049383782 \nu^{10} + \cdots + 69\!\cdots\!20 ) / 12\!\cdots\!56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29213196068751 \nu^{11} + 128904048786731 \nu^{10} + \cdots - 13\!\cdots\!64 ) / 10\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 558861445914547 \nu^{11} + \cdots + 38\!\cdots\!28 ) / 18\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 717031054281463 \nu^{11} + \cdots + 26\!\cdots\!32 ) / 18\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 520026572887418 \nu^{11} + \cdots - 12\!\cdots\!72 ) / 94\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 550701841585907 \nu^{11} + \cdots - 17\!\cdots\!48 ) / 62\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} - \beta_{4} + \beta_{3} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3 \beta_{11} - 3 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + 6 \beta_{4} + \cdots - 34 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 12 \beta_{11} + 4 \beta_{10} - 11 \beta_{9} - 24 \beta_{8} + \beta_{7} - 97 \beta_{6} + 7 \beta_{5} + \cdots + 2524 \)
|
| \(\nu^{5}\) | \(=\) |
\( 403 \beta_{11} - 335 \beta_{10} + 208 \beta_{9} + 215 \beta_{8} - 218 \beta_{7} + 447 \beta_{6} + \cdots - 6939 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2178 \beta_{11} + 908 \beta_{10} - 2191 \beta_{9} - 3382 \beta_{8} + 199 \beta_{7} - 9563 \beta_{6} + \cdots + 216826 \)
|
| \(\nu^{7}\) | \(=\) |
\( 45353 \beta_{11} - 34219 \beta_{10} + 27832 \beta_{9} + 30467 \beta_{8} - 20172 \beta_{7} + \cdots - 1017961 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 305376 \beta_{11} + 148482 \beta_{10} - 300869 \beta_{9} - 401240 \beta_{8} + 38209 \beta_{7} + \cdots + 20994312 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4965951 \beta_{11} - 3534029 \beta_{10} + 3361826 \beta_{9} + 3819385 \beta_{8} - 1875142 \beta_{7} + \cdots - 134215891 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 38971046 \beta_{11} + 20764400 \beta_{10} - 36942863 \beta_{9} - 45988898 \beta_{8} + \cdots + 2172946166 \)
|
| \(\nu^{11}\) | \(=\) |
\( 546331693 \beta_{11} - 373781107 \beta_{10} + 393952724 \beta_{9} + 457049227 \beta_{8} + \cdots - 16780997049 \)
|
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 |
|
0 | −9.74783 | 0 | −20.5968 | 0 | −29.4259 | 0 | 68.0201 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.22268 | 0 | 9.91795 | 0 | 18.1458 | 0 | 58.0578 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −6.28045 | 0 | 18.9942 | 0 | −32.3083 | 0 | 12.4440 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −6.24549 | 0 | 5.77776 | 0 | −10.3273 | 0 | 12.0061 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −5.35906 | 0 | −20.7045 | 0 | 26.2041 | 0 | 1.71955 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.61334 | 0 | −10.5322 | 0 | −13.7418 | 0 | −24.3972 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.323988 | 0 | −5.78413 | 0 | 34.7780 | 0 | −26.8950 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.71183 | 0 | 8.28780 | 0 | 9.40926 | 0 | −24.0696 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.34435 | 0 | 11.5573 | 0 | −8.54075 | 0 | −21.5040 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 5.04158 | 0 | 10.9552 | 0 | 1.11655 | 0 | −1.58244 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 5.97229 | 0 | −11.6005 | 0 | −8.29658 | 0 | 8.66827 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 9.72277 | 0 | −6.27199 | 0 | −31.0130 | 0 | 67.5323 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(29\) | \(-1\) |
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 | 928.4.a.h | ✓ | 12 |
| 4.b | odd | 2 | 1 | 928.4.a.j | yes | 12 | |
| 8.b | even | 2 | 1 | 1856.4.a.bl | 12 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.bj | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 928.4.a.h | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 928.4.a.j | yes | 12 | 4.b | odd | 2 | 1 | |
| 1856.4.a.bj | 12 | 8.d | odd | 2 | 1 | ||
| 1856.4.a.bl | 12 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 14 T_{3}^{11} - 129 T_{3}^{10} - 2360 T_{3}^{9} + 2402 T_{3}^{8} + 122236 T_{3}^{7} + \cdots - 11605016 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(928))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 14 T^{11} + \cdots - 11605016 \)
|
| $5$ |
\( T^{12} + \cdots + 2158837575300 \)
|
| $7$ |
\( T^{12} + \cdots - 51510514450432 \)
|
| $11$ |
\( T^{12} + \cdots - 51\!\cdots\!24 \)
|
| $13$ |
\( T^{12} + \cdots + 49\!\cdots\!32 \)
|
| $17$ |
\( T^{12} + \cdots + 95\!\cdots\!48 \)
|
| $19$ |
\( T^{12} + \cdots - 91\!\cdots\!96 \)
|
| $23$ |
\( T^{12} + \cdots - 11\!\cdots\!16 \)
|
| $29$ |
\( (T - 29)^{12} \)
|
| $31$ |
\( T^{12} + \cdots - 25\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 54\!\cdots\!16 \)
|
| $41$ |
\( T^{12} + \cdots + 67\!\cdots\!16 \)
|
| $43$ |
\( T^{12} + \cdots - 12\!\cdots\!64 \)
|
| $47$ |
\( T^{12} + \cdots + 46\!\cdots\!32 \)
|
| $53$ |
\( T^{12} + \cdots + 10\!\cdots\!36 \)
|
| $59$ |
\( T^{12} + \cdots + 40\!\cdots\!72 \)
|
| $61$ |
\( T^{12} + \cdots + 33\!\cdots\!64 \)
|
| $67$ |
\( T^{12} + \cdots + 31\!\cdots\!04 \)
|
| $71$ |
\( T^{12} + \cdots - 50\!\cdots\!52 \)
|
| $73$ |
\( T^{12} + \cdots - 76\!\cdots\!00 \)
|
| $79$ |
\( T^{12} + \cdots + 43\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots - 49\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots + 74\!\cdots\!04 \)
|
| $97$ |
\( T^{12} + \cdots - 53\!\cdots\!72 \)
|
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