Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,2,Mod(249,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 310.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: | \(2.47536246266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.619810816.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(251\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
− | 1.00000i | − | 3.44579i | −1.00000 | −1.82630 | + | 1.29021i | −3.44579 | 2.40146i | 1.00000i | −8.87345 | 1.29021 | + | 1.82630i | ||||||||||||||||||||||||||||||||||||
| 249.2 | − | 1.00000i | − | 1.79682i | −1.00000 | 1.60536 | + | 1.55654i | −1.79682 | − | 3.47817i | 1.00000i | −0.228545 | 1.55654 | − | 1.60536i | ||||||||||||||||||||||||||||||||||||
| 249.3 | − | 1.00000i | 0.370326i | −1.00000 | 1.45220 | − | 1.70032i | 0.370326 | 4.27844i | 1.00000i | 2.86286 | −1.70032 | − | 1.45220i | ||||||||||||||||||||||||||||||||||||||
| 249.4 | − | 1.00000i | 0.872276i | −1.00000 | −2.23127 | − | 0.146426i | 0.872276 | 2.79827i | 1.00000i | 2.23913 | −0.146426 | + | 2.23127i | ||||||||||||||||||||||||||||||||||||||
| 249.5 | 1.00000i | − | 0.872276i | −1.00000 | −2.23127 | + | 0.146426i | 0.872276 | − | 2.79827i | − | 1.00000i | 2.23913 | −0.146426 | − | 2.23127i | ||||||||||||||||||||||||||||||||||||
| 249.6 | 1.00000i | − | 0.370326i | −1.00000 | 1.45220 | + | 1.70032i | 0.370326 | − | 4.27844i | − | 1.00000i | 2.86286 | −1.70032 | + | 1.45220i | ||||||||||||||||||||||||||||||||||||
| 249.7 | 1.00000i | 1.79682i | −1.00000 | 1.60536 | − | 1.55654i | −1.79682 | 3.47817i | − | 1.00000i | −0.228545 | 1.55654 | + | 1.60536i | ||||||||||||||||||||||||||||||||||||||
| 249.8 | 1.00000i | 3.44579i | −1.00000 | −1.82630 | − | 1.29021i | −3.44579 | − | 2.40146i | − | 1.00000i | −8.87345 | 1.29021 | − | 1.82630i | |||||||||||||||||||||||||||||||||||||
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 | 310.2.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2790.2.d.m | 8 | ||
| 4.b | odd | 2 | 1 | 2480.2.d.d | 8 | ||
| 5.b | even | 2 | 1 | inner | 310.2.b.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 1550.2.a.o | 4 | ||
| 5.c | odd | 4 | 1 | 1550.2.a.p | 4 | ||
| 15.d | odd | 2 | 1 | 2790.2.d.m | 8 | ||
| 20.d | odd | 2 | 1 | 2480.2.d.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.2.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 310.2.b.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 1550.2.a.o | 4 | 5.c | odd | 4 | 1 | ||
| 1550.2.a.p | 4 | 5.c | odd | 4 | 1 | ||
| 2480.2.d.d | 8 | 4.b | odd | 2 | 1 | ||
| 2480.2.d.d | 8 | 20.d | odd | 2 | 1 | ||
| 2790.2.d.m | 8 | 3.b | odd | 2 | 1 | ||
| 2790.2.d.m | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 16T_{3}^{6} + 52T_{3}^{4} + 36T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(310, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} + 16 T^{6} + \cdots + 4 \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 625 \)
|
| $7$ |
\( T^{8} + 44 T^{6} + \cdots + 10000 \)
|
| $11$ |
\( (T^{4} + 8 T^{3} + 2 T^{2} + \cdots - 82)^{2} \)
|
| $13$ |
\( T^{8} + 84 T^{6} + \cdots + 36100 \)
|
| $17$ |
\( T^{8} + 128 T^{6} + \cdots + 2704 \)
|
| $19$ |
\( (T^{4} + 6 T^{3} + 4 T^{2} + \cdots - 8)^{2} \)
|
| $23$ |
\( T^{8} + 32 T^{6} + \cdots + 16 \)
|
| $29$ |
\( (T^{4} - 6 T^{3} - 38 T^{2} + \cdots - 2)^{2} \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} + 152 T^{6} + \cdots + 37636 \)
|
| $41$ |
\( (T^{4} - 104 T^{2} + \cdots + 1468)^{2} \)
|
| $43$ |
\( T^{8} + 172 T^{6} + \cdots + 6724 \)
|
| $47$ |
\( T^{8} + 172 T^{6} + \cdots + 795664 \)
|
| $53$ |
\( T^{8} + 148 T^{6} + \cdots + 3364 \)
|
| $59$ |
\( (T^{4} - 4 T^{3} - 64 T^{2} + \cdots - 64)^{2} \)
|
| $61$ |
\( (T^{4} - 2 T^{3} + \cdots + 574)^{2} \)
|
| $67$ |
\( T^{8} + 308 T^{6} + \cdots + 16321600 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 16)^{2} \)
|
| $73$ |
\( T^{8} + 400 T^{6} + \cdots + 2999824 \)
|
| $79$ |
\( (T^{4} - 16 T^{3} + \cdots + 2800)^{2} \)
|
| $83$ |
\( T^{8} + 168 T^{6} + \cdots + 36100 \)
|
| $89$ |
\( (T^{4} + 12 T^{3} + \cdots + 10048)^{2} \)
|
| $97$ |
\( T^{8} + 436 T^{6} + \cdots + 4080400 \)
|
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