Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7605,2,Mod(1,7605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7605, 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("7605.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7605.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: | \(60.7262307372\) |
| 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: |
\( x^{12} - x^{11} - 18 x^{10} + 16 x^{9} + 118 x^{8} - 90 x^{7} - 339 x^{6} + 212 x^{5} + 388 x^{4} + \cdots + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{12} - x^{11} - 18 x^{10} + 16 x^{9} + 118 x^{8} - 90 x^{7} - 339 x^{6} + 212 x^{5} + 388 x^{4} + \cdots + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20 \nu^{11} + 158 \nu^{10} + 77 \nu^{9} - 3172 \nu^{8} + 2172 \nu^{7} + 23339 \nu^{6} + \cdots - 9423 ) / 2018 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 169 \nu^{11} + 427 \nu^{10} + 2820 \nu^{9} - 7027 \nu^{8} - 16558 \nu^{7} + 40870 \nu^{6} + \cdots - 9852 ) / 2018 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 169 \nu^{11} + 427 \nu^{10} + 2820 \nu^{9} - 7027 \nu^{8} - 16558 \nu^{7} + 40870 \nu^{6} + \cdots - 15906 ) / 2018 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 258 \nu^{11} - 222 \nu^{10} - 4323 \nu^{9} + 3384 \nu^{8} + 25660 \nu^{7} - 17645 \nu^{6} + \cdots + 1183 ) / 2018 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 258 \nu^{11} - 222 \nu^{10} - 4323 \nu^{9} + 3384 \nu^{8} + 25660 \nu^{7} - 17645 \nu^{6} + \cdots + 1183 ) / 2018 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 279 \nu^{11} + 287 \nu^{10} + 4757 \nu^{9} - 4293 \nu^{8} - 28828 \nu^{7} + 22425 \nu^{6} + \cdots - 3661 ) / 2018 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 354 \nu^{11} + 375 \nu^{10} + 6307 \nu^{9} - 6098 \nu^{8} - 40863 \nu^{7} + 35028 \nu^{6} + \cdots - 6457 ) / 2018 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 396 \nu^{11} + 505 \nu^{10} + 7175 \nu^{9} - 7916 \nu^{8} - 47199 \nu^{7} + 42570 \nu^{6} + \cdots - 1323 ) / 2018 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 440 \nu^{11} + 449 \nu^{10} + 7748 \nu^{9} - 7226 \nu^{8} - 49080 \nu^{7} + 41246 \nu^{6} + \cdots - 8533 ) / 1009 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 696 \nu^{11} + 857 \nu^{10} + 12366 \nu^{9} - 14127 \nu^{8} - 79195 \nu^{7} + 82892 \nu^{6} + \cdots - 16543 ) / 1009 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 6\beta_{4} - 7\beta_{3} + \beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 9\beta_{6} + 8\beta_{5} - \beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{10} + 9\beta_{9} + 11\beta_{8} + 12\beta_{7} + 2\beta_{6} + 36\beta_{4} - 46\beta_{3} + 10\beta_{2} + 83 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{11} - 10 \beta_{10} - 10 \beta_{9} - 18 \beta_{8} - \beta_{7} - 67 \beta_{6} + 55 \beta_{5} + \cdots - 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{11} - 77 \beta_{10} + 66 \beta_{9} + 90 \beta_{8} + 107 \beta_{7} + 28 \beta_{6} + 220 \beta_{4} + \cdots + 485 \)
|
| \(\nu^{9}\) | \(=\) |
\( 107 \beta_{11} - 76 \beta_{10} - 77 \beta_{9} - 195 \beta_{8} - 15 \beta_{7} - 464 \beta_{6} + 363 \beta_{5} + \cdots - 49 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 15 \beta_{11} - 546 \beta_{10} + 457 \beta_{9} + 661 \beta_{8} + 848 \beta_{7} + 276 \beta_{6} + \cdots + 2934 \)
|
| \(\nu^{11}\) | \(=\) |
\( 848 \beta_{11} - 528 \beta_{10} - 544 \beta_{9} - 1728 \beta_{8} - 148 \beta_{7} - 3103 \beta_{6} + \cdots - 534 \)
|
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 |
|
−2.57467 | 0 | 4.62894 | −1.00000 | 0 | −5.28221 | −6.76866 | 0 | 2.57467 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16970 | 0 | 2.70760 | −1.00000 | 0 | −2.85355 | −1.53529 | 0 | 2.16970 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.83811 | 0 | 1.37867 | −1.00000 | 0 | −0.254073 | 1.14208 | 0 | 1.83811 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.52530 | 0 | 0.326546 | −1.00000 | 0 | 1.72579 | 2.55252 | 0 | 1.52530 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.320510 | 0 | −1.89727 | −1.00000 | 0 | −3.22920 | 1.24911 | 0 | 0.320510 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.262945 | 0 | −1.93086 | −1.00000 | 0 | −1.90149 | 1.03360 | 0 | 0.262945 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.393647 | 0 | −1.84504 | −1.00000 | 0 | −2.38899 | −1.51359 | 0 | −0.393647 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.656398 | 0 | −1.56914 | −1.00000 | 0 | 3.08138 | −2.34278 | 0 | −0.656398 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.67318 | 0 | 0.799537 | −1.00000 | 0 | 2.20111 | −2.00859 | 0 | −1.67318 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.94727 | 0 | 1.79185 | −1.00000 | 0 | −3.59812 | −0.405317 | 0 | −1.94727 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.45935 | 0 | 4.04839 | −1.00000 | 0 | 1.94541 | 5.03770 | 0 | −2.45935 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.56140 | 0 | 4.56078 | −1.00000 | 0 | −1.44606 | 6.55920 | 0 | −2.56140 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 7605.2.a.cx | yes | 12 |
| 3.b | odd | 2 | 1 | 7605.2.a.cv | ✓ | 12 | |
| 13.b | even | 2 | 1 | 7605.2.a.cw | yes | 12 | |
| 39.d | odd | 2 | 1 | 7605.2.a.cy | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7605.2.a.cv | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 7605.2.a.cw | yes | 12 | 13.b | even | 2 | 1 | |
| 7605.2.a.cx | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 7605.2.a.cy | yes | 12 | 39.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):
|
\( T_{2}^{12} - T_{2}^{11} - 18 T_{2}^{10} + 16 T_{2}^{9} + 118 T_{2}^{8} - 90 T_{2}^{7} - 339 T_{2}^{6} + \cdots + 7 \)
|
|
\( T_{7}^{12} + 12 T_{7}^{11} + 26 T_{7}^{10} - 190 T_{7}^{9} - 829 T_{7}^{8} + 520 T_{7}^{7} + 6561 T_{7}^{6} + \cdots + 6656 \)
|
|
\( T_{11}^{12} + 6 T_{11}^{11} - 61 T_{11}^{10} - 444 T_{11}^{9} + 671 T_{11}^{8} + 9082 T_{11}^{7} + \cdots + 832 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots + 7 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( T^{12} + 12 T^{11} + \cdots + 6656 \)
|
| $11$ |
\( T^{12} + 6 T^{11} + \cdots + 832 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} - 14 T^{11} + \cdots - 22464 \)
|
| $19$ |
\( T^{12} - 139 T^{10} + \cdots + 1469299 \)
|
| $23$ |
\( T^{12} - 28 T^{11} + \cdots + 665119 \)
|
| $29$ |
\( T^{12} - 24 T^{11} + \cdots - 34112 \)
|
| $31$ |
\( T^{12} + 2 T^{11} + \cdots - 30941833 \)
|
| $37$ |
\( T^{12} + 8 T^{11} + \cdots + 1590784 \)
|
| $41$ |
\( T^{12} + 4 T^{11} + \cdots + 71522368 \)
|
| $43$ |
\( T^{12} - 4 T^{11} + \cdots - 18337472 \)
|
| $47$ |
\( T^{12} - 22 T^{11} + \cdots + 84629063 \)
|
| $53$ |
\( T^{12} + \cdots - 620732099 \)
|
| $59$ |
\( T^{12} + 28 T^{11} + \cdots + 92845376 \)
|
| $61$ |
\( T^{12} + \cdots - 697311511 \)
|
| $67$ |
\( T^{12} + \cdots + 6823125504 \)
|
| $71$ |
\( T^{12} + \cdots + 7971825344 \)
|
| $73$ |
\( T^{12} + \cdots - 12660304384 \)
|
| $79$ |
\( T^{12} - 20 T^{11} + \cdots - 27272609 \)
|
| $83$ |
\( T^{12} + 26 T^{11} + \cdots - 54759313 \)
|
| $89$ |
\( T^{12} + \cdots + 3431900864 \)
|
| $97$ |
\( T^{12} + \cdots - 20011549376 \)
|
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