Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,4,Mod(13,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.13");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.f (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(6.01819482059\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 886x^{10} + 292945x^{8} + 42943904x^{6} + 2387634208x^{4} + 5944075264x^{2} + 2089586944 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{12} + 886x^{10} + 292945x^{8} + 42943904x^{6} + 2387634208x^{4} + 5944075264x^{2} + 2089586944 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4620737 \nu^{10} + 3075587556 \nu^{8} + 680488574645 \nu^{6} + 51539422181486 \nu^{4} + \cdots + 21\!\cdots\!48 ) / 58311087831360 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 21456529 \nu^{10} - 14028236490 \nu^{8} - 2994446731273 \nu^{6} - 202782251169076 \nu^{4} + \cdots + 88\!\cdots\!48 ) / 233244351325440 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 26731 \nu^{11} - 28609134 \nu^{9} - 11077887571 \nu^{7} - 1852898922076 \nu^{5} + \cdots - 180688762446016 \nu ) / 86293387514880 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17000529 \nu^{11} - 228959980 \nu^{10} + 15034390098 \nu^{9} - 141697829040 \nu^{8} + \cdots + 12\!\cdots\!40 ) / 38\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17000529 \nu^{11} + 228959980 \nu^{10} + 15034390098 \nu^{9} + 141697829040 \nu^{8} + \cdots - 12\!\cdots\!40 ) / 38\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 245960509 \nu^{11} + 257038576 \nu^{10} - 156732219138 \nu^{9} + 129049981464 \nu^{8} + \cdots - 85\!\cdots\!64 ) / 11\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 245960509 \nu^{11} - 257038576 \nu^{10} - 156732219138 \nu^{9} - 129049981464 \nu^{8} + \cdots + 85\!\cdots\!64 ) / 11\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 21173180457 \nu^{11} + 128982830536 \nu^{10} - 18320856009378 \nu^{9} + \cdots + 10\!\cdots\!52 ) / 26\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 21173180457 \nu^{11} + 128982830536 \nu^{10} + 18320856009378 \nu^{9} + \cdots + 10\!\cdots\!52 ) / 26\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 29931261965 \nu^{11} + 25174910445162 \nu^{9} + \cdots + 97\!\cdots\!24 \nu ) / 13\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 60688217789 \nu^{11} + 51233528430450 \nu^{9} + \cdots + 19\!\cdots\!12 \nu ) / 26\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} - \beta_{3} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{9} - 3\beta_{8} - 18\beta_{5} + 18\beta_{4} + 4\beta_{2} + 11\beta _1 - 443 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 227 \beta_{11} - 218 \beta_{10} + 12 \beta_{9} - 12 \beta_{8} + 51 \beta_{7} + 51 \beta_{6} + \cdots + 941 \beta_{3} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 303 \beta_{9} + 303 \beta_{8} - 14 \beta_{7} + 14 \beta_{6} + 2184 \beta_{5} - 2184 \beta_{4} + \cdots + 33175 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 53861 \beta_{11} + 47918 \beta_{10} - 2082 \beta_{9} + 2082 \beta_{8} - 22059 \beta_{7} + \cdots - 344447 \beta_{3} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 254049 \beta_{9} - 254049 \beta_{8} - 11316 \beta_{7} + 11316 \beta_{6} - 2037918 \beta_{5} + \cdots - 22852723 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12960715 \beta_{11} - 10727686 \beta_{10} + 196488 \beta_{9} - 196488 \beta_{8} + \cdots + 106860325 \beta_{3} ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( 22861251 \beta_{9} + 22861251 \beta_{8} + 4482294 \beta_{7} - 4482294 \beta_{6} + 196166036 \beta_{5} + \cdots + 1778373551 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 3155425309 \beta_{11} + 2441548486 \beta_{10} + 45030834 \beta_{9} - 45030834 \beta_{8} + \cdots - 30827734399 \beta_{3} ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 18147474453 \beta_{9} - 18147474453 \beta_{8} - 6815362680 \beta_{7} + 6815362680 \beta_{6} + \cdots - 1263469595483 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 775888742627 \beta_{11} - 563789280158 \beta_{10} - 36597869268 \beta_{9} + \cdots + 8559381040661 \beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
− | 2.00000i | −2.12132 | − | 2.12132i | −4.00000 | −6.87346 | − | 6.87346i | −4.24264 | + | 4.24264i | 13.1106 | − | 13.1106i | 8.00000i | 9.00000i | −13.7469 | + | 13.7469i | |||||||||||||||||||||||||||||||||||||||||||
| 13.2 | − | 2.00000i | −2.12132 | − | 2.12132i | −4.00000 | 2.87400 | + | 2.87400i | −4.24264 | + | 4.24264i | −23.5451 | + | 23.5451i | 8.00000i | 9.00000i | 5.74799 | − | 5.74799i | ||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | − | 2.00000i | −2.12132 | − | 2.12132i | −4.00000 | 12.9492 | + | 12.9492i | −4.24264 | + | 4.24264i | 17.3340 | − | 17.3340i | 8.00000i | 9.00000i | 25.8984 | − | 25.8984i | ||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | − | 2.00000i | 2.12132 | + | 2.12132i | −4.00000 | −11.8379 | − | 11.8379i | 4.24264 | − | 4.24264i | −13.2307 | + | 13.2307i | 8.00000i | 9.00000i | −23.6758 | + | 23.6758i | ||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | − | 2.00000i | 2.12132 | + | 2.12132i | −4.00000 | 0.627312 | + | 0.627312i | 4.24264 | − | 4.24264i | 13.3596 | − | 13.3596i | 8.00000i | 9.00000i | 1.25462 | − | 1.25462i | ||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | − | 2.00000i | 2.12132 | + | 2.12132i | −4.00000 | 10.2608 | + | 10.2608i | 4.24264 | − | 4.24264i | −13.0284 | + | 13.0284i | 8.00000i | 9.00000i | 20.5217 | − | 20.5217i | ||||||||||||||||||||||||||||||||||||||||||||
| 55.1 | 2.00000i | −2.12132 | + | 2.12132i | −4.00000 | −6.87346 | + | 6.87346i | −4.24264 | − | 4.24264i | 13.1106 | + | 13.1106i | − | 8.00000i | − | 9.00000i | −13.7469 | − | 13.7469i | |||||||||||||||||||||||||||||||||||||||||||
| 55.2 | 2.00000i | −2.12132 | + | 2.12132i | −4.00000 | 2.87400 | − | 2.87400i | −4.24264 | − | 4.24264i | −23.5451 | − | 23.5451i | − | 8.00000i | − | 9.00000i | 5.74799 | + | 5.74799i | |||||||||||||||||||||||||||||||||||||||||||
| 55.3 | 2.00000i | −2.12132 | + | 2.12132i | −4.00000 | 12.9492 | − | 12.9492i | −4.24264 | − | 4.24264i | 17.3340 | + | 17.3340i | − | 8.00000i | − | 9.00000i | 25.8984 | + | 25.8984i | |||||||||||||||||||||||||||||||||||||||||||
| 55.4 | 2.00000i | 2.12132 | − | 2.12132i | −4.00000 | −11.8379 | + | 11.8379i | 4.24264 | + | 4.24264i | −13.2307 | − | 13.2307i | − | 8.00000i | − | 9.00000i | −23.6758 | − | 23.6758i | |||||||||||||||||||||||||||||||||||||||||||
| 55.5 | 2.00000i | 2.12132 | − | 2.12132i | −4.00000 | 0.627312 | − | 0.627312i | 4.24264 | + | 4.24264i | 13.3596 | + | 13.3596i | − | 8.00000i | − | 9.00000i | 1.25462 | + | 1.25462i | |||||||||||||||||||||||||||||||||||||||||||
| 55.6 | 2.00000i | 2.12132 | − | 2.12132i | −4.00000 | 10.2608 | − | 10.2608i | 4.24264 | + | 4.24264i | −13.0284 | − | 13.0284i | − | 8.00000i | − | 9.00000i | 20.5217 | + | 20.5217i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.c | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 102.4.f.c | ✓ | 12 |
| 3.b | odd | 2 | 1 | 306.4.g.g | 12 | ||
| 17.c | even | 4 | 1 | inner | 102.4.f.c | ✓ | 12 |
| 17.d | even | 8 | 1 | 1734.4.a.bd | 6 | ||
| 17.d | even | 8 | 1 | 1734.4.a.be | 6 | ||
| 51.f | odd | 4 | 1 | 306.4.g.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.4.f.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 102.4.f.c | ✓ | 12 | 17.c | even | 4 | 1 | inner |
| 306.4.g.g | 12 | 3.b | odd | 2 | 1 | ||
| 306.4.g.g | 12 | 51.f | odd | 4 | 1 | ||
| 1734.4.a.bd | 6 | 17.d | even | 8 | 1 | ||
| 1734.4.a.be | 6 | 17.d | even | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{12} - 16 T_{5}^{11} + 128 T_{5}^{10} + 1040 T_{5}^{9} + 97449 T_{5}^{8} - 1376384 T_{5}^{7} + \cdots + 24314788624 \)
acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{6} \)
|
| $3$ |
\( (T^{4} + 81)^{3} \)
|
| $5$ |
\( T^{12} + \cdots + 24314788624 \)
|
| $7$ |
\( T^{12} + \cdots + 97\!\cdots\!24 \)
|
| $11$ |
\( T^{12} + \cdots + 24\!\cdots\!76 \)
|
| $13$ |
\( (T^{6} + 34 T^{5} + \cdots - 1867227264)^{2} \)
|
| $17$ |
\( T^{12} + \cdots + 14\!\cdots\!09 \)
|
| $19$ |
\( T^{12} + \cdots + 72\!\cdots\!04 \)
|
| $23$ |
\( T^{12} + \cdots + 20\!\cdots\!36 \)
|
| $29$ |
\( T^{12} + \cdots + 63\!\cdots\!56 \)
|
| $31$ |
\( T^{12} + \cdots + 57\!\cdots\!84 \)
|
| $37$ |
\( T^{12} + \cdots + 74\!\cdots\!16 \)
|
| $41$ |
\( T^{12} + \cdots + 15\!\cdots\!04 \)
|
| $43$ |
\( T^{12} + \cdots + 23\!\cdots\!44 \)
|
| $47$ |
\( (T^{6} + \cdots + 114689624134144)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 77\!\cdots\!44 \)
|
| $59$ |
\( T^{12} + \cdots + 15\!\cdots\!56 \)
|
| $61$ |
\( T^{12} + \cdots + 68\!\cdots\!96 \)
|
| $67$ |
\( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 16\!\cdots\!24 \)
|
| $73$ |
\( T^{12} + \cdots + 35\!\cdots\!76 \)
|
| $79$ |
\( T^{12} + \cdots + 25\!\cdots\!24 \)
|
| $83$ |
\( T^{12} + \cdots + 78\!\cdots\!00 \)
|
| $89$ |
\( (T^{6} + \cdots + 46\!\cdots\!28)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 95\!\cdots\!36 \)
|
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