Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7581,2,Mod(1,7581)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, 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("7581.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7581.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: | \(60.5345897723\) |
| 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: |
\( x^{12} - x^{11} - 20 x^{10} + 17 x^{9} + 146 x^{8} - 93 x^{7} - 485 x^{6} + 165 x^{5} + 765 x^{4} + \cdots + 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 20 x^{10} + 17 x^{9} + 146 x^{8} - 93 x^{7} - 485 x^{6} + 165 x^{5} + 765 x^{4} + \cdots + 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{11} - 340 \nu^{10} + 518 \nu^{9} + 5555 \nu^{8} - 8200 \nu^{7} - 27963 \nu^{6} + 38559 \nu^{5} + \cdots - 12539 ) / 3019 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} - 340 \nu^{10} + 518 \nu^{9} + 5555 \nu^{8} - 8200 \nu^{7} - 27963 \nu^{6} + 38559 \nu^{5} + \cdots - 15558 ) / 3019 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 33 \nu^{11} + 856 \nu^{10} - 1020 \nu^{9} - 15939 \nu^{8} + 10167 \nu^{7} + 100662 \nu^{6} + \cdots - 15279 ) / 12076 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 215 \nu^{11} - 644 \nu^{10} - 3352 \nu^{9} + 10877 \nu^{8} + 15191 \nu^{7} - 61596 \nu^{6} + \cdots + 27253 ) / 6038 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 889 \nu^{11} - 360 \nu^{10} - 16500 \nu^{9} + 5349 \nu^{8} + 103731 \nu^{7} - 21794 \nu^{6} + \cdots - 1023 ) / 12076 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 855 \nu^{11} + 876 \nu^{10} + 15998 \nu^{9} - 15733 \nu^{8} - 104783 \nu^{7} + 97512 \nu^{6} + \cdots - 26795 ) / 6038 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 2579 \nu^{11} + 7388 \nu^{10} + 46780 \nu^{9} - 131007 \nu^{8} - 296157 \nu^{7} + 780626 \nu^{6} + \cdots - 122207 ) / 12076 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2603 \nu^{11} + 3472 \nu^{10} + 46424 \nu^{9} - 59035 \nu^{8} - 280497 \nu^{7} + 328670 \nu^{6} + \cdots - 62791 ) / 12076 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 2409 \nu^{11} - 3930 \nu^{10} - 44270 \nu^{9} + 68205 \nu^{8} + 283303 \nu^{7} - 392390 \nu^{6} + \cdots + 43929 ) / 6038 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + \beta_{8} - \beta_{7} - \beta_{3} + 7\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 2\beta_{5} + 9\beta_{4} - 10\beta_{3} + \beta_{2} + 31\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{11} - 10 \beta_{10} + \beta_{9} + 11 \beta_{8} - 10 \beta_{7} - \beta_{6} - 3 \beta_{5} + \cdots + 103 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{11} - \beta_{10} + 13 \beta_{9} + 12 \beta_{8} - 17 \beta_{7} + 11 \beta_{6} - 23 \beta_{5} + \cdots + 60 \)
|
| \(\nu^{8}\) | \(=\) |
\( 14 \beta_{11} - 82 \beta_{10} + 16 \beta_{9} + 92 \beta_{8} - 82 \beta_{7} - 17 \beta_{6} - 54 \beta_{5} + \cdots + 714 \)
|
| \(\nu^{9}\) | \(=\) |
\( 125 \beta_{11} - 19 \beta_{10} + 129 \beta_{9} + 105 \beta_{8} - 194 \beta_{7} + 94 \beta_{6} + \cdots + 467 \)
|
| \(\nu^{10}\) | \(=\) |
\( 140 \beta_{11} - 641 \beta_{10} + 179 \beta_{9} + 703 \beta_{8} - 640 \beta_{7} - 202 \beta_{6} + \cdots + 5101 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1084 \beta_{11} - 237 \beta_{10} + 1162 \beta_{9} + 823 \beta_{8} - 1888 \beta_{7} + 741 \beta_{6} + \cdots + 3731 \)
|
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 |
|
−2.76059 | −1.00000 | 5.62086 | −3.18723 | 2.76059 | 1.00000 | −9.99572 | 1.00000 | 8.79863 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.54211 | −1.00000 | 4.46230 | 2.00813 | 2.54211 | 1.00000 | −6.25944 | 1.00000 | −5.10488 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.91042 | −1.00000 | 1.64970 | −3.88718 | 1.91042 | 1.00000 | 0.669226 | 1.00000 | 7.42613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.79689 | −1.00000 | 1.22882 | 2.19840 | 1.79689 | 1.00000 | 1.38572 | 1.00000 | −3.95029 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.22530 | −1.00000 | −0.498629 | 3.29670 | 1.22530 | 1.00000 | 3.06158 | 1.00000 | −4.03946 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.161527 | −1.00000 | −1.97391 | −0.715976 | 0.161527 | 1.00000 | 0.641894 | 1.00000 | 0.115650 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.631493 | −1.00000 | −1.60122 | −0.722964 | −0.631493 | 1.00000 | −2.27414 | 1.00000 | −0.456546 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.850096 | −1.00000 | −1.27734 | 2.22154 | −0.850096 | 1.00000 | −2.78605 | 1.00000 | 1.88852 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.22966 | −1.00000 | −0.487931 | −1.26334 | −1.22966 | 1.00000 | −3.05931 | 1.00000 | −1.55349 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.42326 | −1.00000 | 0.0256637 | 4.02370 | −1.42326 | 1.00000 | −2.80999 | 1.00000 | 5.72676 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.57821 | −1.00000 | 4.64715 | −1.68016 | −2.57821 | 1.00000 | 6.82489 | 1.00000 | −4.33180 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.68412 | −1.00000 | 5.20452 | −4.29162 | −2.68412 | 1.00000 | 8.60134 | 1.00000 | −11.5192 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 7581.2.a.bk | ✓ | 12 |
| 19.b | odd | 2 | 1 | 7581.2.a.bn | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7581.2.a.bk | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 7581.2.a.bn | yes | 12 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):
|
\( T_{2}^{12} + T_{2}^{11} - 20 T_{2}^{10} - 17 T_{2}^{9} + 146 T_{2}^{8} + 93 T_{2}^{7} - 485 T_{2}^{6} + \cdots + 31 \)
|
|
\( T_{5}^{12} + 2 T_{5}^{11} - 43 T_{5}^{10} - 73 T_{5}^{9} + 673 T_{5}^{8} + 950 T_{5}^{7} - 4678 T_{5}^{6} + \cdots - 7600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 31 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots - 7600 \)
|
| $7$ |
\( (T - 1)^{12} \)
|
| $11$ |
\( T^{12} + 8 T^{11} + \cdots - 3856 \)
|
| $13$ |
\( T^{12} + 12 T^{11} + \cdots + 25920 \)
|
| $17$ |
\( T^{12} + 4 T^{11} + \cdots - 23104 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} + 8 T^{11} + \cdots - 116 \)
|
| $29$ |
\( T^{12} + 14 T^{11} + \cdots - 39640036 \)
|
| $31$ |
\( T^{12} + 21 T^{11} + \cdots + 31737280 \)
|
| $37$ |
\( T^{12} + \cdots + 188972624 \)
|
| $41$ |
\( T^{12} + 35 T^{11} + \cdots - 89733520 \)
|
| $43$ |
\( T^{12} - 14 T^{11} + \cdots + 33558004 \)
|
| $47$ |
\( T^{12} - 4 T^{11} + \cdots - 19456 \)
|
| $53$ |
\( T^{12} + 7 T^{11} + \cdots - 45664799 \)
|
| $59$ |
\( T^{12} + \cdots - 338637056 \)
|
| $61$ |
\( T^{12} + 17 T^{11} + \cdots + 4045264 \)
|
| $67$ |
\( T^{12} + \cdots + 152857520 \)
|
| $71$ |
\( T^{12} + \cdots - 98397082624 \)
|
| $73$ |
\( T^{12} + \cdots - 108550480880 \)
|
| $79$ |
\( T^{12} + \cdots + 420146224 \)
|
| $83$ |
\( T^{12} + \cdots + 108905865920 \)
|
| $89$ |
\( T^{12} + \cdots - 443024580784 \)
|
| $97$ |
\( T^{12} + \cdots - 7465913584 \)
|
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