Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,10,Mod(4,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.4");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.b (of order \(2\), degree \(1\), 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: | \(7.72553754246\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{12}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 201\nu^{6} + 179562\nu^{4} + 32600241\nu^{2} + 600254096 ) / 2360192 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1023\nu^{6} + 834630\nu^{4} + 129381687\nu^{2} + 3318414960 ) / 1180096 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 20285\nu^{7} + 18191922\nu^{5} + 3715480325\nu^{3} + 205760824848\nu ) / 9877403520 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 597\nu^{7} + 559746\nu^{5} + 132904173\nu^{3} + 10255365504\nu ) / 182914880 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 31883 \nu^{7} + 652860 \nu^{6} + 40544322 \nu^{5} + 486481140 \nu^{4} + 15087510287 \nu^{3} + \cdots - 2075307082560 ) / 2469350880 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31883 \nu^{7} - 652860 \nu^{6} + 40544322 \nu^{5} - 486481140 \nu^{4} + 15087510287 \nu^{3} + \cdots + 2075307082560 ) / 2469350880 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -7429\nu^{7} - 6671874\nu^{5} - 1369843021\nu^{3} - 67422946896\nu ) / 318625920 \)
|
| \(\nu\) | \(=\) |
\( ( 9\beta_{7} + 2\beta_{4} + 99\beta_{3} ) / 270 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{6} - 6\beta_{5} + 7\beta_{2} - 34\beta _1 - 21122 ) / 90 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -421\beta_{7} - 16\beta_{6} - 16\beta_{5} + 1070\beta_{4} - 6279\beta_{3} ) / 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2766\beta_{6} + 2766\beta_{5} - 4567\beta_{2} + 29314\beta _1 + 10036322 ) / 90 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 704463\beta_{7} + 46440\beta_{6} + 46440\beta_{5} - 2544506\beta_{4} + 11457813\beta_{3} ) / 90 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1497846\beta_{6} - 1497846\beta_{5} + 2944567\beta_{2} - 19616194\beta _1 - 5808868802 ) / 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -143622821\beta_{7} - 10952096\beta_{6} - 10952096\beta_{5} + 562412110\beta_{4} - 2372069319\beta_{3} ) / 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
− | 33.3847i | − | 81.0000i | −602.537 | −1324.50 | + | 445.892i | −2704.16 | 176.118i | 3022.54i | −6561.00 | 14886.0 | + | 44218.1i | ||||||||||||||||||||||||||||||||||||
| 4.2 | − | 29.7516i | 81.0000i | −373.156 | 343.445 | + | 1354.68i | 2409.88 | 10707.3i | − | 4130.82i | −6561.00 | 40304.0 | − | 10218.0i | |||||||||||||||||||||||||||||||||||||
| 4.3 | − | 20.9703i | 81.0000i | 72.2477 | −743.946 | − | 1183.08i | 1698.59 | − | 3575.78i | − | 12251.8i | −6561.00 | −24809.4 | + | 15600.8i | ||||||||||||||||||||||||||||||||||||
| 4.4 | − | 14.3372i | − | 81.0000i | 306.446 | 1380.00 | + | 220.717i | −1161.31 | − | 2878.61i | − | 11734.2i | −6561.00 | 3164.45 | − | 19785.3i | |||||||||||||||||||||||||||||||||||
| 4.5 | 14.3372i | 81.0000i | 306.446 | 1380.00 | − | 220.717i | −1161.31 | 2878.61i | 11734.2i | −6561.00 | 3164.45 | + | 19785.3i | |||||||||||||||||||||||||||||||||||||||
| 4.6 | 20.9703i | − | 81.0000i | 72.2477 | −743.946 | + | 1183.08i | 1698.59 | 3575.78i | 12251.8i | −6561.00 | −24809.4 | − | 15600.8i | ||||||||||||||||||||||||||||||||||||||
| 4.7 | 29.7516i | − | 81.0000i | −373.156 | 343.445 | − | 1354.68i | 2409.88 | − | 10707.3i | 4130.82i | −6561.00 | 40304.0 | + | 10218.0i | |||||||||||||||||||||||||||||||||||||
| 4.8 | 33.3847i | 81.0000i | −602.537 | −1324.50 | − | 445.892i | −2704.16 | − | 176.118i | − | 3022.54i | −6561.00 | 14886.0 | − | 44218.1i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 15.10.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 45.10.b.c | 8 | ||
| 4.b | odd | 2 | 1 | 240.10.f.c | 8 | ||
| 5.b | even | 2 | 1 | inner | 15.10.b.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 75.10.a.i | 4 | ||
| 5.c | odd | 4 | 1 | 75.10.a.l | 4 | ||
| 15.d | odd | 2 | 1 | 45.10.b.c | 8 | ||
| 15.e | even | 4 | 1 | 225.10.a.q | 4 | ||
| 15.e | even | 4 | 1 | 225.10.a.u | 4 | ||
| 20.d | odd | 2 | 1 | 240.10.f.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.10.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 15.10.b.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 45.10.b.c | 8 | 3.b | odd | 2 | 1 | ||
| 45.10.b.c | 8 | 15.d | odd | 2 | 1 | ||
| 75.10.a.i | 4 | 5.c | odd | 4 | 1 | ||
| 75.10.a.l | 4 | 5.c | odd | 4 | 1 | ||
| 225.10.a.q | 4 | 15.e | even | 4 | 1 | ||
| 225.10.a.u | 4 | 15.e | even | 4 | 1 | ||
| 240.10.f.c | 8 | 4.b | odd | 2 | 1 | ||
| 240.10.f.c | 8 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(15, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 89176293376 \)
|
| $3$ |
\( (T^{2} + 6561)^{4} \)
|
| $5$ |
\( T^{8} + \cdots + 14\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $11$ |
\( (T^{4} + \cdots + 69\!\cdots\!44)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 47\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $19$ |
\( (T^{4} + \cdots - 31\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 23\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 79\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 70\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} + \cdots - 62\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 48\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 63\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 31\!\cdots\!00 \)
|
| $59$ |
\( (T^{4} + \cdots - 71\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 30\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 86\!\cdots\!96 \)
|
| $71$ |
\( (T^{4} + \cdots - 70\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 48\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + \cdots - 29\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 43\!\cdots\!96 \)
|
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