Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2009,4,Mod(1,2009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2009.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2009 = 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2009.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: | \(118.534837202\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 4 x^{11} - 56 x^{10} + 209 x^{9} + 1107 x^{8} - 3746 x^{7} - 9090 x^{6} + 27681 x^{5} + \cdots - 16544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 287) |
| 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
\( x^{12} - 4 x^{11} - 56 x^{10} + 209 x^{9} + 1107 x^{8} - 3746 x^{7} - 9090 x^{6} + 27681 x^{5} + \cdots - 16544 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 53 \nu^{11} + 285 \nu^{10} + 2535 \nu^{9} - 13264 \nu^{8} - 39087 \nu^{7} + 190573 \nu^{6} + \cdots - 217824 ) / 34304 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 51 \nu^{11} - 163 \nu^{10} - 2811 \nu^{9} + 8076 \nu^{8} + 54271 \nu^{7} - 132545 \nu^{6} + \cdots + 710208 ) / 34304 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9 \nu^{11} + 281 \nu^{10} - 907 \nu^{9} - 12492 \nu^{8} + 55799 \nu^{7} + 161631 \nu^{6} + \cdots + 1391360 ) / 34304 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 39 \nu^{11} - 503 \nu^{10} - 447 \nu^{9} + 23580 \nu^{8} - 37133 \nu^{7} - 343693 \nu^{6} + \cdots - 53184 ) / 34304 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 47 \nu^{11} - 388 \nu^{10} - 1353 \nu^{9} + 17867 \nu^{8} - 7620 \nu^{7} - 251659 \nu^{6} + \cdots - 466176 ) / 17152 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 96 \nu^{11} + 563 \nu^{10} + 4038 \nu^{9} - 26249 \nu^{8} - 44415 \nu^{7} + 380178 \nu^{6} + \cdots + 251968 ) / 17152 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 143 \nu^{11} + 817 \nu^{10} + 5994 \nu^{9} - 37952 \nu^{8} - 65002 \nu^{7} + 546010 \nu^{6} + \cdots + 262544 ) / 17152 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 355 \nu^{11} + 2627 \nu^{10} + 11779 \nu^{9} - 121292 \nu^{8} - 16263 \nu^{7} + 1719481 \nu^{6} + \cdots + 3302976 ) / 34304 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 443 \nu^{11} + 2769 \nu^{10} + 17926 \nu^{9} - 128732 \nu^{8} - 172066 \nu^{7} + 1849390 \nu^{6} + \cdots + 1717264 ) / 17152 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{8} - \beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{2} + 19\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{10} - 2\beta_{9} + \beta_{8} + 3\beta_{7} + \beta_{6} - \beta_{5} + 23\beta_{2} + 28\beta _1 + 177 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 30 \beta_{10} + 2 \beta_{9} + 29 \beta_{8} - 33 \beta_{7} - \beta_{6} + 51 \beta_{5} - 22 \beta_{4} + \cdots + 155 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 8 \beta_{11} + 77 \beta_{10} - 70 \beta_{9} + 52 \beta_{8} + 104 \beta_{7} + 33 \beta_{6} + \cdots + 3567 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 724 \beta_{10} + 44 \beta_{9} + 714 \beta_{8} - 898 \beta_{7} - 12 \beta_{6} + 1070 \beta_{5} + \cdots + 4122 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 368 \beta_{11} + 2220 \beta_{10} - 1956 \beta_{9} + 1828 \beta_{8} + 2788 \beta_{7} + 920 \beta_{6} + \cdots + 75366 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 40 \beta_{11} - 16489 \beta_{10} + 464 \beta_{9} + 17169 \beta_{8} - 22721 \beta_{7} + 224 \beta_{6} + \cdots + 97964 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 11984 \beta_{11} + 58314 \beta_{10} - 51186 \beta_{9} + 54825 \beta_{8} + 68827 \beta_{7} + \cdots + 1627313 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 3024 \beta_{11} - 367566 \beta_{10} - 7086 \beta_{9} + 413605 \beta_{8} - 554025 \beta_{7} + \cdots + 2226019 \)
|
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 |
|
−4.83064 | 3.24764 | 15.3351 | −9.39894 | −15.6882 | 0 | −35.4330 | −16.4528 | 45.4028 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.65637 | −2.44489 | 13.6818 | 12.9284 | 11.3843 | 0 | −26.4563 | −21.0225 | −60.1995 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.52428 | −2.40971 | 12.4691 | −19.2604 | 10.9022 | 0 | −20.2194 | −21.1933 | 87.1392 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.19862 | 5.32629 | −3.16605 | 4.39258 | −11.7105 | 0 | 24.5500 | 1.36932 | −9.65764 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.40588 | −3.55493 | −6.02351 | 2.36466 | 4.99779 | 0 | 19.7153 | −14.3625 | −3.32442 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.16795 | 8.79259 | −6.63589 | 13.4158 | −10.2693 | 0 | 17.0940 | 50.3097 | −15.6690 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.341329 | −6.31189 | −7.88349 | 0.157781 | 2.15443 | 0 | 5.42150 | 12.8399 | −0.0538551 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.939569 | 4.94120 | −7.11721 | −10.8628 | 4.64260 | 0 | −14.2037 | −2.58450 | −10.2063 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.72218 | 0.0898893 | −0.589730 | −2.04207 | 0.244695 | 0 | −23.3828 | −26.9919 | −5.55887 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.72743 | −4.44609 | −0.561142 | 13.3319 | −12.1264 | 0 | −23.3499 | −7.23233 | 36.3617 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 3.95973 | 10.1649 | 7.67949 | −6.01337 | 40.2502 | 0 | −1.26912 | 76.3247 | −23.8113 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 4.77615 | 3.60502 | 14.8116 | 14.9864 | 17.2181 | 0 | 32.5334 | −14.0038 | 71.5772 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 2009.4.a.e | 12 | |
| 7.b | odd | 2 | 1 | 287.4.a.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.4.a.c | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 2009.4.a.e | 12 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2009))\):
|
\( T_{2}^{12} + 4 T_{2}^{11} - 56 T_{2}^{10} - 209 T_{2}^{9} + 1107 T_{2}^{8} + 3746 T_{2}^{7} + \cdots - 16544 \)
|
|
\( T_{3}^{12} - 17 T_{3}^{11} - 26 T_{3}^{10} + 1523 T_{3}^{9} - 2294 T_{3}^{8} - 48902 T_{3}^{7} + \cdots - 1454968 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{11} + \cdots - 16544 \)
|
| $3$ |
\( T^{12} - 17 T^{11} + \cdots - 1454968 \)
|
| $5$ |
\( T^{12} + \cdots - 1371392992 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} + \cdots + 36\!\cdots\!24 \)
|
| $13$ |
\( T^{12} + \cdots + 24\!\cdots\!64 \)
|
| $17$ |
\( T^{12} + \cdots + 18\!\cdots\!12 \)
|
| $19$ |
\( T^{12} + \cdots - 54\!\cdots\!28 \)
|
| $23$ |
\( T^{12} + \cdots + 19\!\cdots\!32 \)
|
| $29$ |
\( T^{12} + \cdots + 13\!\cdots\!36 \)
|
| $31$ |
\( T^{12} + \cdots + 61\!\cdots\!80 \)
|
| $37$ |
\( T^{12} + \cdots - 99\!\cdots\!08 \)
|
| $41$ |
\( (T - 41)^{12} \)
|
| $43$ |
\( T^{12} + \cdots - 10\!\cdots\!20 \)
|
| $47$ |
\( T^{12} + \cdots + 23\!\cdots\!36 \)
|
| $53$ |
\( T^{12} + \cdots + 10\!\cdots\!48 \)
|
| $59$ |
\( T^{12} + \cdots + 62\!\cdots\!56 \)
|
| $61$ |
\( T^{12} + \cdots - 43\!\cdots\!28 \)
|
| $67$ |
\( T^{12} + \cdots + 94\!\cdots\!96 \)
|
| $71$ |
\( T^{12} + \cdots + 49\!\cdots\!44 \)
|
| $73$ |
\( T^{12} + \cdots + 52\!\cdots\!12 \)
|
| $79$ |
\( T^{12} + \cdots - 23\!\cdots\!68 \)
|
| $83$ |
\( T^{12} + \cdots + 18\!\cdots\!00 \)
|
| $89$ |
\( T^{12} + \cdots - 71\!\cdots\!36 \)
|
| $97$ |
\( T^{12} + \cdots + 37\!\cdots\!96 \)
|
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