Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9075,2,Mod(1,9075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, 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("9075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9075.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: | \(72.4642398343\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6 \nu^{11} + 46 \nu^{10} - 212 \nu^{9} - 784 \nu^{8} + 1980 \nu^{7} + 5252 \nu^{6} - 6883 \nu^{5} + \cdots - 5310 ) / 1033 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 29 \nu^{11} - 122 \nu^{10} - 336 \nu^{9} + 1720 \nu^{8} + 1306 \nu^{7} - 8360 \nu^{6} - 2450 \nu^{5} + \cdots + 160 ) / 1033 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32 \nu^{11} - 99 \nu^{10} - 442 \nu^{9} + 1328 \nu^{8} + 2296 \nu^{7} - 5734 \nu^{6} - 6408 \nu^{5} + \cdots + 604 ) / 1033 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 102 \nu^{11} + 251 \nu^{10} + 1538 \nu^{9} - 3200 \nu^{8} - 8868 \nu^{7} + 12983 \nu^{6} + \cdots + 1432 ) / 1033 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 241 \nu^{11} + 907 \nu^{10} + 3006 \nu^{9} - 12584 \nu^{8} - 12385 \nu^{7} + 59002 \nu^{6} + \cdots - 8810 ) / 1033 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 563 \nu^{11} + 2226 \nu^{10} + 5775 \nu^{9} - 29046 \nu^{8} - 12246 \nu^{7} + 123544 \nu^{6} + \cdots - 2750 ) / 1033 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 662 \nu^{11} + 2500 \nu^{10} + 7207 \nu^{9} - 32638 \nu^{8} - 20124 \nu^{7} + 139153 \nu^{6} + \cdots - 8105 ) / 1033 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 665 \nu^{11} - 2477 \nu^{10} - 7313 \nu^{9} + 32246 \nu^{8} + 21114 \nu^{7} - 136527 \nu^{6} + \cdots + 7516 ) / 1033 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 849 \nu^{11} - 2788 \nu^{10} - 10371 \nu^{9} + 36783 \nu^{8} + 39481 \nu^{7} - 159684 \nu^{6} + \cdots + 13055 ) / 1033 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{8} + \beta_{6} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{10} + 7\beta_{9} + 2\beta_{8} + 2\beta_{6} - 9\beta_{5} + 9\beta_{4} + \beta_{3} + 3\beta_{2} + 36\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13 \beta_{10} + 2 \beta_{9} + 11 \beta_{8} + 12 \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + \cdots + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( 69 \beta_{10} + 45 \beta_{9} + 24 \beta_{8} + \beta_{7} + 24 \beta_{6} - 62 \beta_{5} + 70 \beta_{4} + \cdots + 81 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{11} + 121 \beta_{10} + 29 \beta_{9} + 93 \beta_{8} + \beta_{7} + 108 \beta_{6} - 12 \beta_{5} + \cdots + 510 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{11} + 506 \beta_{10} + 293 \beta_{9} + 214 \beta_{8} + 18 \beta_{7} + 221 \beta_{6} + \cdots + 624 \)
|
| \(\nu^{10}\) | \(=\) |
\( 20 \beta_{11} + 1000 \beta_{10} + 298 \beta_{9} + 720 \beta_{8} + 28 \beta_{7} + 876 \beta_{6} + \cdots + 3180 \)
|
| \(\nu^{11}\) | \(=\) |
\( 48 \beta_{11} + 3652 \beta_{10} + 1960 \beta_{9} + 1720 \beta_{8} + 222 \beta_{7} + 1856 \beta_{6} + \cdots + 4702 \)
|
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 |
|
−2.72592 | 1.00000 | 5.43063 | 0 | −2.72592 | −0.486331 | −9.35163 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.59201 | 1.00000 | 4.71851 | 0 | −2.59201 | −3.41835 | −7.04639 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.10651 | 1.00000 | 2.43738 | 0 | −2.10651 | 3.31633 | −0.921348 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.90743 | 1.00000 | 1.63829 | 0 | −1.90743 | −4.45734 | 0.689943 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.803366 | 1.00000 | −1.35460 | 0 | −0.803366 | 0.508505 | 2.69497 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.488299 | 1.00000 | −1.76156 | 0 | −0.488299 | −5.10895 | 1.83677 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.298064 | 1.00000 | −1.91116 | 0 | −0.298064 | 2.32107 | 1.16578 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.288813 | 1.00000 | −1.91659 | 0 | 0.288813 | 3.66740 | −1.13116 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.784131 | 1.00000 | −1.38514 | 0 | 0.784131 | −1.51801 | −2.65439 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.27093 | 1.00000 | −0.384732 | 0 | 1.27093 | 0.483291 | −3.03083 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.21395 | 1.00000 | 2.90157 | 0 | 2.21395 | −1.59339 | 1.99602 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.36377 | 1.00000 | 3.58741 | 0 | 2.36377 | −1.71423 | 3.75227 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
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 | 9075.2.a.dy | 12 | |
| 5.b | even | 2 | 1 | 9075.2.a.dz | 12 | ||
| 5.c | odd | 4 | 2 | 1815.2.c.j | 24 | ||
| 11.b | odd | 2 | 1 | 9075.2.a.ea | 12 | ||
| 11.c | even | 5 | 2 | 825.2.n.p | 24 | ||
| 55.d | odd | 2 | 1 | 9075.2.a.dx | 12 | ||
| 55.e | even | 4 | 2 | 1815.2.c.k | 24 | ||
| 55.j | even | 10 | 2 | 825.2.n.o | 24 | ||
| 55.k | odd | 20 | 4 | 165.2.s.a | ✓ | 48 | |
| 165.v | even | 20 | 4 | 495.2.ba.c | 48 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.s.a | ✓ | 48 | 55.k | odd | 20 | 4 | |
| 495.2.ba.c | 48 | 165.v | even | 20 | 4 | ||
| 825.2.n.o | 24 | 55.j | even | 10 | 2 | ||
| 825.2.n.p | 24 | 11.c | even | 5 | 2 | ||
| 1815.2.c.j | 24 | 5.c | odd | 4 | 2 | ||
| 1815.2.c.k | 24 | 55.e | even | 4 | 2 | ||
| 9075.2.a.dx | 12 | 55.d | odd | 2 | 1 | ||
| 9075.2.a.dy | 12 | 1.a | even | 1 | 1 | trivial | |
| 9075.2.a.dz | 12 | 5.b | even | 2 | 1 | ||
| 9075.2.a.ea | 12 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):
|
\( T_{2}^{12} + 4 T_{2}^{11} - 10 T_{2}^{10} - 52 T_{2}^{9} + 18 T_{2}^{8} + 220 T_{2}^{7} + 61 T_{2}^{6} + \cdots - 5 \)
|
|
\( T_{7}^{12} + 8 T_{7}^{11} - 16 T_{7}^{10} - 240 T_{7}^{9} - 135 T_{7}^{8} + 2248 T_{7}^{7} + 3233 T_{7}^{6} + \cdots - 1089 \)
|
|
\( T_{13}^{12} + 18 T_{13}^{11} + 93 T_{13}^{10} - 186 T_{13}^{9} - 3561 T_{13}^{8} - 11538 T_{13}^{7} + \cdots - 21101 \)
|
|
\( T_{17}^{12} + 18 T_{17}^{11} + 79 T_{17}^{10} - 380 T_{17}^{9} - 4133 T_{17}^{8} - 7728 T_{17}^{7} + \cdots - 66481 \)
|
|
\( T_{19}^{12} + 16 T_{19}^{11} + 16 T_{19}^{10} - 942 T_{19}^{9} - 4470 T_{19}^{8} + 10358 T_{19}^{7} + \cdots + 734525 \)
|
|
\( T_{23}^{12} - 161 T_{23}^{10} - 60 T_{23}^{9} + 9560 T_{23}^{8} + 5890 T_{23}^{7} - 268150 T_{23}^{6} + \cdots + 41299351 \)
|
|
\( T_{37}^{12} - 231 T_{37}^{10} + 190 T_{37}^{9} + 15718 T_{37}^{8} - 34920 T_{37}^{7} - 372590 T_{37}^{6} + \cdots + 1618831 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{11} + \cdots - 5 \)
|
| $3$ |
\( (T - 1)^{12} \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots - 1089 \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + 18 T^{11} + \cdots - 21101 \)
|
| $17$ |
\( T^{12} + 18 T^{11} + \cdots - 66481 \)
|
| $19$ |
\( T^{12} + 16 T^{11} + \cdots + 734525 \)
|
| $23$ |
\( T^{12} - 161 T^{10} + \cdots + 41299351 \)
|
| $29$ |
\( T^{12} - 155 T^{10} + \cdots + 1084880 \)
|
| $31$ |
\( T^{12} - 226 T^{10} + \cdots + 11486981 \)
|
| $37$ |
\( T^{12} - 231 T^{10} + \cdots + 1618831 \)
|
| $41$ |
\( T^{12} - 132 T^{10} + \cdots - 87119 \)
|
| $43$ |
\( T^{12} + 32 T^{11} + \cdots - 972401 \)
|
| $47$ |
\( T^{12} + \cdots + 1113195775 \)
|
| $53$ |
\( T^{12} + 12 T^{11} + \cdots - 47459081 \)
|
| $59$ |
\( T^{12} + \cdots - 965103975 \)
|
| $61$ |
\( T^{12} + 20 T^{11} + \cdots - 48108995 \)
|
| $67$ |
\( T^{12} + \cdots + 474528995 \)
|
| $71$ |
\( T^{12} + \cdots + 517162301 \)
|
| $73$ |
\( T^{12} + 26 T^{11} + \cdots - 306704 \)
|
| $79$ |
\( T^{12} + \cdots - 1150699275 \)
|
| $83$ |
\( T^{12} + \cdots + 119614891 \)
|
| $89$ |
\( T^{12} + \cdots - 619209755 \)
|
| $97$ |
\( T^{12} + \cdots + 8411891531 \)
|
| show more | |
| show less |