Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6975,2,Mod(1,6975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6975.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6975.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: | \(55.6956554098\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 465) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 3x^{10} - 14x^{9} + 44x^{8} + 61x^{7} - 211x^{6} - 83x^{5} + 369x^{4} + 10x^{3} - 168x^{2} - 31x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + 3\nu^{8} + 15\nu^{7} - 42\nu^{6} - 76\nu^{5} + 183\nu^{4} + 149\nu^{3} - 252\nu^{2} - 79\nu + 30 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{10} + 19\nu^{8} + 3\nu^{7} - 127\nu^{6} - 40\nu^{5} + 343\nu^{4} + 160\nu^{3} - 295\nu^{2} - 177\nu - 5 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 2 \nu^{10} + 4 \nu^{9} + 31 \nu^{8} - 54 \nu^{7} - 161 \nu^{6} + 219 \nu^{5} + 309 \nu^{4} + \cdots - 35 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{10} + 4 \nu^{9} + 31 \nu^{8} - 54 \nu^{7} - 161 \nu^{6} + 219 \nu^{5} + 314 \nu^{4} - 246 \nu^{3} + \cdots - 5 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 5 \nu^{10} + 8 \nu^{9} + 81 \nu^{8} - 105 \nu^{7} - 454 \nu^{6} + 403 \nu^{5} + 1021 \nu^{4} + \cdots + 15 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 11 \nu^{10} + 20 \nu^{9} + 174 \nu^{8} - 272 \nu^{7} - 937 \nu^{6} + 1120 \nu^{5} + 1963 \nu^{4} + \cdots - 5 ) / 10 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{10} - 18 \nu^{9} - 180 \nu^{8} + 242 \nu^{7} + 1021 \nu^{6} - 978 \nu^{5} - 2329 \nu^{4} + \cdots - 85 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{10} - \beta_{9} - \beta_{4} + 9\beta_{3} + \beta_{2} + 38\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{10} - \beta_{9} - \beta_{8} + 12 \beta_{7} - 10 \beta_{6} + \beta_{5} - \beta_{4} + 10 \beta_{3} + \cdots + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( -12\beta_{10} - 14\beta_{9} + \beta_{8} + 3\beta_{7} - 12\beta_{4} + 69\beta_{3} + 13\beta_{2} + 251\beta _1 + 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 16 \beta_{10} - 16 \beta_{9} - 15 \beta_{8} + 109 \beta_{7} - 78 \beta_{6} + 13 \beta_{5} + \cdots + 545 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 110 \beta_{10} - 140 \beta_{9} + 12 \beta_{8} + 51 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 103 \beta_{4} + \cdots + 30 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 173 \beta_{10} - 179 \beta_{9} - 155 \beta_{8} + 899 \beta_{7} - 555 \beta_{6} + 115 \beta_{5} + \cdots + 3567 \)
|
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 |
|
−2.61420 | 0 | 4.83402 | 0 | 0 | −4.28641 | −7.40869 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.11614 | 0 | 2.47806 | 0 | 0 | −1.31151 | −1.01163 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.65822 | 0 | 0.749697 | 0 | 0 | −0.225788 | 2.07328 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.523109 | 0 | −1.72636 | 0 | 0 | −3.11438 | 1.94929 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.364910 | 0 | −1.86684 | 0 | 0 | −0.715024 | 1.41105 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.104188 | 0 | −1.98914 | 0 | 0 | 3.88714 | −0.415622 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.31655 | 0 | −0.266704 | 0 | 0 | −4.78991 | −2.98422 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.34956 | 0 | −0.178679 | 0 | 0 | 3.66966 | −2.94027 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.15913 | 0 | 2.66183 | 0 | 0 | −2.73322 | 1.42896 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.60996 | 0 | 4.81187 | 0 | 0 | −1.35174 | 7.33885 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.73720 | 0 | 5.49226 | 0 | 0 | 2.97118 | 9.55900 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 6975.2.a.cj | 11 | |
| 3.b | odd | 2 | 1 | 2325.2.a.bc | 11 | ||
| 5.b | even | 2 | 1 | 6975.2.a.ci | 11 | ||
| 5.c | odd | 4 | 2 | 1395.2.c.h | 22 | ||
| 15.d | odd | 2 | 1 | 2325.2.a.bd | 11 | ||
| 15.e | even | 4 | 2 | 465.2.c.b | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.c.b | ✓ | 22 | 15.e | even | 4 | 2 | |
| 1395.2.c.h | 22 | 5.c | odd | 4 | 2 | ||
| 2325.2.a.bc | 11 | 3.b | odd | 2 | 1 | ||
| 2325.2.a.bd | 11 | 15.d | odd | 2 | 1 | ||
| 6975.2.a.ci | 11 | 5.b | even | 2 | 1 | ||
| 6975.2.a.cj | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{11} - 3 T_{2}^{10} - 14 T_{2}^{9} + 44 T_{2}^{8} + 61 T_{2}^{7} - 211 T_{2}^{6} - 83 T_{2}^{5} + \cdots + 5 \)
|
|
\( T_{7}^{11} + 8 T_{7}^{10} - 18 T_{7}^{9} - 278 T_{7}^{8} - 263 T_{7}^{7} + 2858 T_{7}^{6} + 6772 T_{7}^{5} + \cdots - 2120 \)
|
|
\( T_{11}^{11} - 72 T_{11}^{9} - 20 T_{11}^{8} + 1808 T_{11}^{7} + 984 T_{11}^{6} - 19296 T_{11}^{5} + \cdots - 81920 \)
|
|
\( T_{13}^{11} + 14 T_{13}^{10} + 4 T_{13}^{9} - 756 T_{13}^{8} - 3024 T_{13}^{7} + 7346 T_{13}^{6} + \cdots + 115840 \)
|
|
\( T_{17}^{11} - 12 T_{17}^{10} - 66 T_{17}^{9} + 1144 T_{17}^{8} - 52 T_{17}^{7} - 31324 T_{17}^{6} + \cdots + 81920 \)
|
|
\( T_{29}^{11} - 8 T_{29}^{10} - 170 T_{29}^{9} + 1748 T_{29}^{8} + 5438 T_{29}^{7} - 103710 T_{29}^{6} + \cdots - 8631680 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 3 T^{10} + \cdots + 5 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} + 8 T^{10} + \cdots - 2120 \)
|
| $11$ |
\( T^{11} - 72 T^{9} + \cdots - 81920 \)
|
| $13$ |
\( T^{11} + 14 T^{10} + \cdots + 115840 \)
|
| $17$ |
\( T^{11} - 12 T^{10} + \cdots + 81920 \)
|
| $19$ |
\( T^{11} - 12 T^{10} + \cdots + 2720000 \)
|
| $23$ |
\( T^{11} - 12 T^{10} + \cdots + 1408000 \)
|
| $29$ |
\( T^{11} - 8 T^{10} + \cdots - 8631680 \)
|
| $31$ |
\( (T - 1)^{11} \)
|
| $37$ |
\( T^{11} + \cdots + 190672000 \)
|
| $41$ |
\( T^{11} - 8 T^{10} + \cdots + 39773792 \)
|
| $43$ |
\( T^{11} + 28 T^{10} + \cdots + 5519360 \)
|
| $47$ |
\( T^{11} + \cdots + 385637120 \)
|
| $53$ |
\( T^{11} - 26 T^{10} + \cdots - 65536 \)
|
| $59$ |
\( T^{11} + \cdots + 118868000 \)
|
| $61$ |
\( T^{11} - 14 T^{10} + \cdots - 22200320 \)
|
| $67$ |
\( T^{11} + \cdots + 386024320 \)
|
| $71$ |
\( T^{11} + \cdots - 156292160 \)
|
| $73$ |
\( T^{11} + \cdots + 864434560 \)
|
| $79$ |
\( T^{11} + \cdots + 298624000 \)
|
| $83$ |
\( T^{11} - 48 T^{10} + \cdots + 25600 \)
|
| $89$ |
\( T^{11} + \cdots + 9081262720 \)
|
| $97$ |
\( T^{11} + 16 T^{10} + \cdots + 87159424 \)
|
| show more | |
| show less |