Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [507,4,Mod(337,507)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(2))
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("507.337");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 507 = 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 507.b (of order \(2\), degree \(1\), not 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: | \(29.9139683729\) |
| 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: |
\( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 13^{2} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
\( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} - 40\nu^{4} - 329\nu^{2} + 178 ) / 78 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{7} - 518\nu^{5} - 6739\nu^{3} - 31348\nu ) / 11856 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23\nu^{7} + 1622\nu^{5} + 41575\nu^{3} + 349012\nu ) / 11856 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 118\nu^{4} + 2357\nu^{2} + 5048 ) / 156 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{7} + 397\nu^{5} + 6242\nu^{3} + 19814\nu ) / 1482 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{6} - 239\nu^{4} - 3166\nu^{2} - 8704 ) / 78 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} + 518\nu^{5} + 6739\nu^{3} + 19492\nu ) / 912 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - 13\beta_{2} ) / 13 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{4} + 9\beta _1 - 179 ) / 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24\beta_{7} - 13\beta_{5} + 13\beta_{3} + 273\beta_{2} ) / 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{6} + 6\beta_{4} - 17\beta _1 + 291 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -698\beta_{7} + 663\beta_{5} - 377\beta_{3} - 6487\beta_{2} ) / 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1422\beta_{6} - 2462\beta_{4} + 4865\beta _1 - 90115 ) / 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 21016\beta_{7} - 23257\beta_{5} + 9789\beta_{3} + 161265\beta_{2} ) / 13 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).
| \(n\) | \(170\) | \(340\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 5.33039i | −3.00000 | −20.4131 | 16.4131i | 15.9912i | 9.67968i | 66.1667i | 9.00000 | 87.4882 | |||||||||||||||||||||||||||||||||||||||||
| 337.2 | − | 4.22605i | −3.00000 | −9.85953 | − | 5.85953i | 12.6782i | 24.1254i | 7.85849i | 9.00000 | −24.7627 | |||||||||||||||||||||||||||||||||||||||||
| 337.3 | − | 2.36176i | −3.00000 | 2.42208 | − | 6.42208i | 7.08529i | − | 29.4938i | − | 24.6145i | 9.00000 | −15.1674 | |||||||||||||||||||||||||||||||||||||||
| 337.4 | − | 1.46610i | −3.00000 | 5.85055 | 9.85055i | 4.39830i | − | 29.9396i | − | 20.3063i | 9.00000 | 14.4419 | ||||||||||||||||||||||||||||||||||||||||
| 337.5 | 1.46610i | −3.00000 | 5.85055 | − | 9.85055i | − | 4.39830i | 29.9396i | 20.3063i | 9.00000 | 14.4419 | |||||||||||||||||||||||||||||||||||||||||
| 337.6 | 2.36176i | −3.00000 | 2.42208 | 6.42208i | − | 7.08529i | 29.4938i | 24.6145i | 9.00000 | −15.1674 | ||||||||||||||||||||||||||||||||||||||||||
| 337.7 | 4.22605i | −3.00000 | −9.85953 | 5.85953i | − | 12.6782i | − | 24.1254i | − | 7.85849i | 9.00000 | −24.7627 | ||||||||||||||||||||||||||||||||||||||||
| 337.8 | 5.33039i | −3.00000 | −20.4131 | − | 16.4131i | − | 15.9912i | − | 9.67968i | − | 66.1667i | 9.00000 | 87.4882 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 507.4.b.h | 8 | |
| 13.b | even | 2 | 1 | inner | 507.4.b.h | 8 | |
| 13.d | odd | 4 | 1 | 507.4.a.i | 4 | ||
| 13.d | odd | 4 | 1 | 507.4.a.m | 4 | ||
| 13.f | odd | 12 | 2 | 39.4.e.c | ✓ | 8 | |
| 39.f | even | 4 | 1 | 1521.4.a.v | 4 | ||
| 39.f | even | 4 | 1 | 1521.4.a.bb | 4 | ||
| 39.k | even | 12 | 2 | 117.4.g.e | 8 | ||
| 52.l | even | 12 | 2 | 624.4.q.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.e.c | ✓ | 8 | 13.f | odd | 12 | 2 | |
| 117.4.g.e | 8 | 39.k | even | 12 | 2 | ||
| 507.4.a.i | 4 | 13.d | odd | 4 | 1 | ||
| 507.4.a.m | 4 | 13.d | odd | 4 | 1 | ||
| 507.4.b.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 507.4.b.h | 8 | 13.b | even | 2 | 1 | inner | |
| 624.4.q.i | 8 | 52.l | even | 12 | 2 | ||
| 1521.4.a.v | 4 | 39.f | even | 4 | 1 | ||
| 1521.4.a.bb | 4 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):
|
\( T_{2}^{8} + 54T_{2}^{6} + 877T_{2}^{4} + 4476T_{2}^{2} + 6084 \)
|
|
\( T_{5}^{8} + 442T_{5}^{6} + 55249T_{5}^{4} + 2494440T_{5}^{2} + 37015056 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 54 T^{6} + \cdots + 6084 \)
|
| $3$ |
\( (T + 3)^{8} \)
|
| $5$ |
\( T^{8} + 442 T^{6} + \cdots + 37015056 \)
|
| $7$ |
\( T^{8} + \cdots + 42523388944 \)
|
| $11$ |
\( T^{8} + 4112 T^{6} + \cdots + 751198464 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( (T^{4} + 98 T^{3} + \cdots + 22571952)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 2959721107456 \)
|
| $23$ |
\( (T^{4} + 104 T^{3} + \cdots - 2571504)^{2} \)
|
| $29$ |
\( (T^{4} - 194 T^{3} + \cdots - 274591068)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $37$ |
\( T^{8} + \cdots + 738573457717264 \)
|
| $41$ |
\( T^{8} + \cdots + 10\!\cdots\!04 \)
|
| $43$ |
\( (T^{4} + 450 T^{3} + \cdots - 2362804828)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!04 \)
|
| $53$ |
\( (T^{4} - 262 T^{3} + \cdots + 744728256)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 44\!\cdots\!56 \)
|
| $61$ |
\( (T^{4} + 928 T^{3} + \cdots - 5230543711)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 10\!\cdots\!44 \)
|
| $71$ |
\( T^{8} + \cdots + 98\!\cdots\!16 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!09 \)
|
| $79$ |
\( (T^{4} + 746 T^{3} + \cdots + 680937616)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 34\!\cdots\!36 \)
|
| $89$ |
\( T^{8} + \cdots + 72\!\cdots\!96 \)
|
| $97$ |
\( T^{8} + \cdots + 19\!\cdots\!96 \)
|
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