Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7595,2,Mod(1,7595)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7595, 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("7595.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7595 = 5 \cdot 7^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7595.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: | \(60.6463803352\) |
| 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: |
\( x^{11} - 14x^{9} + 67x^{7} - 130x^{5} - 2x^{4} + 90x^{3} + 4x^{2} - 14x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{11} - 14x^{9} + 67x^{7} - 130x^{5} - 2x^{4} + 90x^{3} + 4x^{2} - 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 11 \nu^{10} - 2 \nu^{9} + 153 \nu^{8} + 28 \nu^{7} - 724 \nu^{6} - 134 \nu^{5} + 1374 \nu^{4} + \cdots + 88 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11 \nu^{10} + 2 \nu^{9} - 153 \nu^{8} - 28 \nu^{7} + 724 \nu^{6} + 134 \nu^{5} - 1374 \nu^{4} + \cdots - 94 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13 \nu^{10} + 3 \nu^{9} - 181 \nu^{8} - 41 \nu^{7} + 858 \nu^{6} + 190 \nu^{5} - 1634 \nu^{4} + \cdots - 112 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13 \nu^{10} - 3 \nu^{9} + 181 \nu^{8} + 41 \nu^{7} - 858 \nu^{6} - 190 \nu^{5} + 1634 \nu^{4} + \cdots + 112 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 17 \nu^{10} + 3 \nu^{9} - 237 \nu^{8} - 43 \nu^{7} + 1124 \nu^{6} + 212 \nu^{5} - 2134 \nu^{4} + \cdots - 142 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17 \nu^{10} + 5 \nu^{9} - 237 \nu^{8} - 69 \nu^{7} + 1126 \nu^{6} + 322 \nu^{5} - 2154 \nu^{4} + \cdots - 156 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( 10 \nu^{10} + 2 \nu^{9} - 139 \nu^{8} - 28 \nu^{7} + 657 \nu^{6} + 134 \nu^{5} - 1244 \nu^{4} + \cdots - 81 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21 \nu^{10} - 4 \nu^{9} + 293 \nu^{8} + 56 \nu^{7} - 1392 \nu^{6} - 268 \nu^{5} + 2652 \nu^{4} + \cdots + 174 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( - 19 \nu^{10} - 4 \nu^{9} + 265 \nu^{8} + 56 \nu^{7} - 1259 \nu^{6} - 268 \nu^{5} + 2403 \nu^{4} + \cdots + 166 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 6\beta_{3} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 8\beta_{5} + 9\beta_{4} - 2\beta_{3} + \beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{10} - 9\beta_{9} + 9\beta_{8} - 11\beta_{6} + 10\beta_{5} - \beta_{4} + 37\beta_{3} + 44\beta_{2} - \beta _1 + 84 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 10 \beta_{10} + 10 \beta_{9} + 10 \beta_{8} - 11 \beta_{7} + 53 \beta_{5} + 66 \beta_{4} - 20 \beta_{3} + \cdots + 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 11 \beta_{10} - 63 \beta_{9} + 66 \beta_{8} - 87 \beta_{6} + 76 \beta_{5} - 11 \beta_{4} + 231 \beta_{3} + \cdots + 493 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 74 \beta_{10} + 74 \beta_{9} + 76 \beta_{8} - 87 \beta_{7} + 335 \beta_{5} + 450 \beta_{4} + \cdots + 30 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 87 \beta_{10} - 409 \beta_{9} + 450 \beta_{8} - 611 \beta_{6} + 526 \beta_{5} - 85 \beta_{4} + \cdots + 2967 \)
|
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 |
|
−2.51271 | 2.36887 | 4.31369 | 1.00000 | −5.95227 | 0 | −5.81363 | 2.61154 | −2.51271 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88741 | −0.286056 | 1.56230 | 1.00000 | 0.539905 | 0 | 0.826115 | −2.91817 | −1.88741 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.68134 | 1.38018 | 0.826900 | 1.00000 | −2.32056 | 0 | 1.97238 | −1.09509 | −1.68134 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.08711 | −0.211543 | −0.818196 | 1.00000 | 0.229970 | 0 | 3.06368 | −2.95525 | −1.08711 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.231150 | −1.11685 | −1.94657 | 1.00000 | 0.258159 | 0 | 0.912248 | −1.75265 | −0.231150 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.223386 | −3.14923 | −1.95010 | 1.00000 | 0.703494 | 0 | 0.882397 | 6.91765 | −0.223386 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.634578 | 2.14316 | −1.59731 | 1.00000 | 1.36000 | 0 | −2.28277 | 1.59315 | 0.634578 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.864774 | −1.32988 | −1.25217 | 1.00000 | −1.15005 | 0 | −2.81239 | −1.23142 | 0.864774 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.63411 | 0.857552 | 0.670329 | 1.00000 | 1.40134 | 0 | −2.17283 | −2.26460 | 1.63411 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.00739 | −2.19254 | 2.02963 | 1.00000 | −4.40128 | 0 | 0.0594803 | 1.80721 | 2.00739 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.48223 | 0.536325 | 4.16149 | 1.00000 | 1.33128 | 0 | 5.36532 | −2.71236 | 2.48223 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(31\) | \(1\) |
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 | 7595.2.a.w | ✓ | 11 |
| 7.b | odd | 2 | 1 | 7595.2.a.x | yes | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7595.2.a.w | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 7595.2.a.x | yes | 11 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7595))\):
|
\( T_{2}^{11} - 14T_{2}^{9} + 67T_{2}^{7} - 130T_{2}^{5} + 2T_{2}^{4} + 90T_{2}^{3} - 4T_{2}^{2} - 14T_{2} - 2 \)
|
|
\( T_{3}^{11} + T_{3}^{10} - 15 T_{3}^{9} - 9 T_{3}^{8} + 73 T_{3}^{7} + 25 T_{3}^{6} - 133 T_{3}^{5} + \cdots - 2 \)
|
|
\( T_{11}^{11} + 3 T_{11}^{10} - 49 T_{11}^{9} - 161 T_{11}^{8} + 627 T_{11}^{7} + 2575 T_{11}^{6} + \cdots + 2458 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 14 T^{9} + \cdots - 2 \)
|
| $3$ |
\( T^{11} + T^{10} + \cdots - 2 \)
|
| $5$ |
\( (T - 1)^{11} \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} + 3 T^{10} + \cdots + 2458 \)
|
| $13$ |
\( T^{11} + 5 T^{10} + \cdots - 296 \)
|
| $17$ |
\( T^{11} + 15 T^{10} + \cdots - 184 \)
|
| $19$ |
\( T^{11} - 14 T^{10} + \cdots - 43904 \)
|
| $23$ |
\( T^{11} + 6 T^{10} + \cdots - 10768 \)
|
| $29$ |
\( T^{11} + 15 T^{10} + \cdots + 911728 \)
|
| $31$ |
\( (T + 1)^{11} \)
|
| $37$ |
\( T^{11} + 24 T^{10} + \cdots + 939952 \)
|
| $41$ |
\( T^{11} - 212 T^{9} + \cdots - 1995776 \)
|
| $43$ |
\( T^{11} + 30 T^{10} + \cdots - 37294144 \)
|
| $47$ |
\( T^{11} + T^{10} + \cdots - 93512048 \)
|
| $53$ |
\( T^{11} + 4 T^{10} + \cdots - 9428 \)
|
| $59$ |
\( T^{11} + 2 T^{10} + \cdots + 42872 \)
|
| $61$ |
\( T^{11} - 10 T^{10} + \cdots - 1999244 \)
|
| $67$ |
\( T^{11} - 8 T^{10} + \cdots - 401696 \)
|
| $71$ |
\( T^{11} + \cdots - 990081728 \)
|
| $73$ |
\( T^{11} + 14 T^{10} + \cdots + 9894752 \)
|
| $79$ |
\( T^{11} + \cdots + 238235054 \)
|
| $83$ |
\( T^{11} + \cdots - 683228216 \)
|
| $89$ |
\( T^{11} + 8 T^{10} + \cdots + 661136 \)
|
| $97$ |
\( T^{11} + \cdots + 6980189636 \)
|
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