Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,6,Mod(1,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 975.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: | \(156.374224318\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3\cdot 5^{3}\cdot 11 \) |
| Twist minimal: | yes |
| 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
\( x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 86\nu - 21 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1286527 \nu^{10} - 90509002 \nu^{9} - 346680258 \nu^{8} + 23125223382 \nu^{7} + \cdots - 10\!\cdots\!00 ) / 1971602843264 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20385315 \nu^{10} + 30245988 \nu^{9} + 4762321298 \nu^{8} - 1281153890 \nu^{7} + \cdots - 17\!\cdots\!32 ) / 3943205686528 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 12146105 \nu^{10} - 237424970 \nu^{9} - 1829394126 \nu^{8} + 63617203634 \nu^{7} + \cdots + 22\!\cdots\!84 ) / 1971602843264 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 27730735 \nu^{10} - 626840400 \nu^{9} - 7674408554 \nu^{8} + 164251378850 \nu^{7} + \cdots + 45\!\cdots\!68 ) / 3943205686528 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29180789 \nu^{10} + 5549056 \nu^{9} - 8355371630 \nu^{8} - 2320337258 \nu^{7} + \cdots - 11\!\cdots\!92 ) / 3943205686528 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 60411079 \nu^{10} + 315478660 \nu^{9} + 15624013090 \nu^{8} - 76119492426 \nu^{7} + \cdots - 784025003900992 ) / 1971602843264 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 105564843 \nu^{10} + 501938874 \nu^{9} + 27968787146 \nu^{8} - 113547750174 \nu^{7} + \cdots + 34\!\cdots\!28 ) / 1971602843264 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 86\beta _1 + 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 118\beta_{2} + 125\beta _1 + 4480 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} - \beta_{6} - 9 \beta_{5} + 11 \beta_{4} + 140 \beta_{3} + \cdots + 2799 \)
|
| \(\nu^{6}\) | \(=\) |
\( 176 \beta_{10} - 352 \beta_{9} - 160 \beta_{8} - 18 \beta_{7} - 172 \beta_{6} - 166 \beta_{5} + \cdots + 442426 \)
|
| \(\nu^{7}\) | \(=\) |
\( 732 \beta_{10} - 400 \beta_{9} + 2144 \beta_{8} + 24 \beta_{7} - 40 \beta_{6} - 1848 \beta_{5} + \cdots + 347191 \)
|
| \(\nu^{8}\) | \(=\) |
\( 24839 \beta_{10} - 49910 \beta_{9} - 22131 \beta_{8} - 5076 \beta_{7} - 22923 \beta_{6} + \cdots + 46147118 \)
|
| \(\nu^{9}\) | \(=\) |
\( 126407 \beta_{10} - 64186 \beta_{9} + 406779 \beta_{8} + 844 \beta_{7} + 10251 \beta_{6} + \cdots + 47264721 \)
|
| \(\nu^{10}\) | \(=\) |
\( 3254970 \beta_{10} - 6569668 \beta_{9} - 2776370 \beta_{8} - 940778 \beta_{7} - 2811870 \beta_{6} + \cdots + 4960185176 \)
|
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 |
|
−10.5321 | −9.00000 | 78.9248 | 0 | 94.7888 | 88.0004 | −494.216 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.62146 | −9.00000 | 60.5725 | 0 | 86.5931 | −139.353 | −274.909 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.81158 | −9.00000 | 1.77443 | 0 | 52.3042 | 186.151 | 175.658 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.77830 | −9.00000 | −9.16786 | 0 | 43.0047 | 26.6297 | 196.712 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.50720 | −9.00000 | −19.6996 | 0 | 31.5648 | −147.807 | 181.321 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.592612 | −9.00000 | −31.6488 | 0 | −5.33351 | 209.544 | −37.7190 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.29989 | −9.00000 | −26.7105 | 0 | −20.6990 | −228.917 | −135.028 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.89893 | −9.00000 | −8.00051 | 0 | −44.0903 | 56.8551 | −195.960 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.18506 | −9.00000 | 34.9953 | 0 | −73.6656 | −143.020 | 24.5164 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 9.28847 | −9.00000 | 54.2758 | 0 | −83.5963 | 85.6782 | 206.908 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 10.9857 | −9.00000 | 88.6845 | 0 | −98.8709 | −48.7613 | 622.716 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 975.6.a.u | yes | 11 |
| 5.b | even | 2 | 1 | 975.6.a.r | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 975.6.a.r | ✓ | 11 | 5.b | even | 2 | 1 | |
| 975.6.a.u | yes | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(975))\):
|
\( T_{2}^{11} - 2 T_{2}^{10} - 286 T_{2}^{9} + 442 T_{2}^{8} + 28715 T_{2}^{7} - 29138 T_{2}^{6} + \cdots + 55036800 \)
|
|
\( T_{7}^{11} + 55 T_{7}^{10} - 105532 T_{7}^{9} - 4590338 T_{7}^{8} + 3818034718 T_{7}^{7} + \cdots + 14\!\cdots\!24 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots + 55036800 \)
|
| $3$ |
\( (T + 9)^{11} \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} + \cdots + 14\!\cdots\!24 \)
|
| $11$ |
\( T^{11} + \cdots + 53\!\cdots\!80 \)
|
| $13$ |
\( (T - 169)^{11} \)
|
| $17$ |
\( T^{11} + \cdots + 48\!\cdots\!56 \)
|
| $19$ |
\( T^{11} + \cdots + 27\!\cdots\!40 \)
|
| $23$ |
\( T^{11} + \cdots + 94\!\cdots\!60 \)
|
| $29$ |
\( T^{11} + \cdots - 10\!\cdots\!95 \)
|
| $31$ |
\( T^{11} + \cdots - 11\!\cdots\!76 \)
|
| $37$ |
\( T^{11} + \cdots + 10\!\cdots\!76 \)
|
| $41$ |
\( T^{11} + \cdots + 13\!\cdots\!04 \)
|
| $43$ |
\( T^{11} + \cdots + 83\!\cdots\!88 \)
|
| $47$ |
\( T^{11} + \cdots - 19\!\cdots\!75 \)
|
| $53$ |
\( T^{11} + \cdots - 72\!\cdots\!05 \)
|
| $59$ |
\( T^{11} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{11} + \cdots - 70\!\cdots\!40 \)
|
| $67$ |
\( T^{11} + \cdots + 40\!\cdots\!89 \)
|
| $71$ |
\( T^{11} + \cdots - 43\!\cdots\!60 \)
|
| $73$ |
\( T^{11} + \cdots + 12\!\cdots\!68 \)
|
| $79$ |
\( T^{11} + \cdots + 10\!\cdots\!64 \)
|
| $83$ |
\( T^{11} + \cdots - 17\!\cdots\!52 \)
|
| $89$ |
\( T^{11} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{11} + \cdots + 13\!\cdots\!00 \)
|
| show more | |
| show less |