Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7007,2,Mod(1,7007)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7007, 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("7007.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7007 = 7^{2} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7007.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: | \(55.9511766963\) |
| 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: |
\( x^{11} - x^{10} - 18x^{9} + 15x^{8} + 117x^{7} - 78x^{6} - 326x^{5} + 167x^{4} + 348x^{3} - 143x^{2} - 74x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1001) |
| 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
\( x^{11} - x^{10} - 18x^{9} + 15x^{8} + 117x^{7} - 78x^{6} - 326x^{5} + 167x^{4} + 348x^{3} - 143x^{2} - 74x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 13\nu^{7} + 10\nu^{6} + 52\nu^{5} - 28\nu^{4} - 58\nu^{3} + 19\nu^{2} - 14\nu + 8 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{10} - 2 \nu^{9} - 16 \nu^{8} + 31 \nu^{7} + 86 \nu^{6} - 148 \nu^{5} - 178 \nu^{4} + 201 \nu^{3} + \cdots - 40 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{10} - 2 \nu^{9} - 8 \nu^{8} + 23 \nu^{7} - 18 \nu^{6} - 84 \nu^{5} + 254 \nu^{4} + 105 \nu^{3} + \cdots + 88 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{10} + 2 \nu^{9} + 16 \nu^{8} - 31 \nu^{7} - 86 \nu^{6} + 164 \nu^{5} + 162 \nu^{4} - 329 \nu^{3} + \cdots - 40 ) / 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{10} - 18\nu^{8} - 3\nu^{7} + 122\nu^{6} + 28\nu^{5} - 370\nu^{4} - 67\nu^{3} + 433\nu^{2} + 26\nu - 88 ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - \nu^{10} + \nu^{9} + 17 \nu^{8} - 14 \nu^{7} - 104 \nu^{6} + 68 \nu^{5} + 274 \nu^{4} - 131 \nu^{3} + \cdots + 48 ) / 8 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2 \nu^{10} + \nu^{9} + 35 \nu^{8} - 11 \nu^{7} - 218 \nu^{6} + 32 \nu^{5} + 564 \nu^{4} - 8 \nu^{3} + \cdots + 64 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{5} + 8\beta_{3} + 28\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} + 10\beta_{9} + 13\beta_{8} - 10\beta_{7} + \beta_{5} + \beta_{3} + 35\beta_{2} + \beta _1 + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2 \beta_{10} + 12 \beta_{9} + 16 \beta_{8} - 3 \beta_{7} + 13 \beta_{5} + 2 \beta_{4} + 55 \beta_{3} + \cdots + 26 \)
|
| \(\nu^{8}\) | \(=\) |
\( 15 \beta_{10} + 80 \beta_{9} + 123 \beta_{8} - 79 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + \cdots + 524 \)
|
| \(\nu^{9}\) | \(=\) |
\( 31 \beta_{10} + 112 \beta_{9} + 177 \beta_{8} - 46 \beta_{7} + 2 \beta_{6} + 123 \beta_{5} + 36 \beta_{4} + \cdots + 253 \)
|
| \(\nu^{10}\) | \(=\) |
\( 154 \beta_{10} + 598 \beta_{9} + 1034 \beta_{8} - 581 \beta_{7} + 36 \beta_{6} + 177 \beta_{5} + \cdots + 3301 \)
|
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 |
|
−2.69905 | 3.23405 | 5.28489 | 1.09400 | −8.72887 | 0 | −8.86610 | 7.45906 | −2.95277 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.42100 | −2.69707 | 3.86122 | −3.65517 | 6.52959 | 0 | −4.50601 | 4.27418 | 8.84916 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.14071 | −0.698070 | 2.58264 | 3.13099 | 1.49437 | 0 | −1.24727 | −2.51270 | −6.70256 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35489 | 0.152758 | −0.164270 | −3.54377 | −0.206971 | 0 | 2.93235 | −2.97666 | 4.80142 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.610570 | 1.93031 | −1.62720 | −1.75578 | −1.17859 | 0 | 2.21466 | 0.726080 | 1.07202 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.282248 | −3.11943 | −1.92034 | −0.477922 | 0.880455 | 0 | 1.10651 | 6.73086 | 0.134893 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.501958 | −0.372196 | −1.74804 | 1.99673 | −0.186827 | 0 | −1.88136 | −2.86147 | 1.00228 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.46230 | 2.15409 | 0.138330 | −0.136889 | 3.14993 | 0 | −2.72233 | 1.64009 | −0.200174 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.72345 | −1.94436 | 0.970287 | −2.87921 | −3.35102 | 0 | −1.77466 | 0.780553 | −4.96218 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.37413 | −2.26106 | 3.63647 | 2.08089 | −5.36804 | 0 | 3.88519 | 2.11240 | 4.94030 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.44663 | 1.62099 | 3.98600 | −2.85388 | 3.96597 | 0 | 4.85901 | −0.372378 | −6.98239 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 7007.2.a.w | 11 | |
| 7.b | odd | 2 | 1 | 1001.2.a.n | ✓ | 11 | |
| 21.c | even | 2 | 1 | 9009.2.a.bs | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1001.2.a.n | ✓ | 11 | 7.b | odd | 2 | 1 | |
| 7007.2.a.w | 11 | 1.a | even | 1 | 1 | trivial | |
| 9009.2.a.bs | 11 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7007))\):
|
\( T_{2}^{11} + T_{2}^{10} - 18 T_{2}^{9} - 15 T_{2}^{8} + 117 T_{2}^{7} + 78 T_{2}^{6} - 326 T_{2}^{5} + \cdots - 24 \)
|
|
\( T_{3}^{11} + 2 T_{3}^{10} - 22 T_{3}^{9} - 42 T_{3}^{8} + 165 T_{3}^{7} + 289 T_{3}^{6} - 514 T_{3}^{5} + \cdots - 32 \)
|
|
\( T_{5}^{11} + 7 T_{5}^{10} - 8 T_{5}^{9} - 138 T_{5}^{8} - 95 T_{5}^{7} + 896 T_{5}^{6} + 1011 T_{5}^{5} + \cdots + 174 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + T^{10} + \cdots - 24 \)
|
| $3$ |
\( T^{11} + 2 T^{10} + \cdots - 32 \)
|
| $5$ |
\( T^{11} + 7 T^{10} + \cdots + 174 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( (T - 1)^{11} \)
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| $13$ |
\( (T + 1)^{11} \)
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| $17$ |
\( T^{11} + 7 T^{10} + \cdots + 4272 \)
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| $19$ |
\( T^{11} + 22 T^{10} + \cdots - 93440 \)
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| $23$ |
\( T^{11} - 3 T^{10} + \cdots + 1092096 \)
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| $29$ |
\( T^{11} + 6 T^{10} + \cdots - 1261440 \)
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| $31$ |
\( T^{11} + 28 T^{10} + \cdots + 8046592 \)
|
| $37$ |
\( T^{11} - T^{10} + \cdots - 10120192 \)
|
| $41$ |
\( T^{11} - 4 T^{10} + \cdots - 897024 \)
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| $43$ |
\( T^{11} + 8 T^{10} + \cdots + 774352 \)
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| $47$ |
\( T^{11} + 22 T^{10} + \cdots - 12413952 \)
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| $53$ |
\( T^{11} - 9 T^{10} + \cdots + 9351948 \)
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| $59$ |
\( T^{11} + \cdots + 634513920 \)
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| $61$ |
\( T^{11} + \cdots + 104898944 \)
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| $67$ |
\( T^{11} - 23 T^{10} + \cdots + 20600344 \)
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| $71$ |
\( T^{11} - 3 T^{10} + \cdots + 3265536 \)
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| $73$ |
\( T^{11} + \cdots - 272272384 \)
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| $79$ |
\( T^{11} + \cdots - 789727040 \)
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| $83$ |
\( T^{11} + 9 T^{10} + \cdots + 17918904 \)
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| $89$ |
\( T^{11} + \cdots + 122805330 \)
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| $97$ |
\( T^{11} + \cdots + 6467409952 \)
|
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