Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6042,2,Mod(1,6042)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, 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("6042.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6042.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: | \(48.2456129013\) |
| 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: |
\( x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} + \cdots + 1534 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 2 x^{10} - 35 x^{9} + 77 x^{8} + 394 x^{7} - 994 x^{6} - 1477 x^{5} + 4683 x^{4} + 563 x^{3} + \cdots + 1534 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 61365 \nu^{10} - 19979 \nu^{9} + 2101724 \nu^{8} + 162971 \nu^{7} - 23813959 \nu^{6} + \cdots + 40606678 ) / 34480 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 84083 \nu^{10} - 26993 \nu^{9} + 2879648 \nu^{8} + 209481 \nu^{7} - 32622537 \nu^{6} + \cdots + 55455562 ) / 34480 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 77556 \nu^{10} - 25031 \nu^{9} + 2656416 \nu^{8} + 198432 \nu^{7} - 30099019 \nu^{6} + \cdots + 51286814 ) / 17240 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 50505 \nu^{10} - 16163 \nu^{9} + 1730043 \nu^{8} + 124862 \nu^{7} - 19605108 \nu^{6} + \cdots + 33382236 ) / 8620 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 210451 \nu^{10} + 67875 \nu^{9} - 7208340 \nu^{8} - 537653 \nu^{7} + 81674783 \nu^{6} + \cdots - 138742022 ) / 34480 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 192480 \nu^{10} + 61607 \nu^{9} - 6593832 \nu^{8} - 477308 \nu^{7} + 74728307 \nu^{6} + \cdots - 127192774 ) / 17240 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 107137 \nu^{10} + 34407 \nu^{9} - 3669944 \nu^{8} - 269099 \nu^{7} + 41587847 \nu^{6} + \cdots - 70791910 ) / 6896 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 647795 \nu^{10} + 207847 \nu^{9} - 22190632 \nu^{8} - 1620833 \nu^{7} + 251475487 \nu^{6} + \cdots - 428293574 ) / 34480 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 365939 \nu^{10} + 117420 \nu^{9} - 12535140 \nu^{8} - 915157 \nu^{7} + 142049012 \nu^{6} + \cdots - 241753888 ) / 17240 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{10} + \beta_{9} - \beta_{3} - \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} + 11\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 14 \beta_{10} + 16 \beta_{9} - 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \cdots + 80 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19 \beta_{10} - 17 \beta_{9} + 5 \beta_{8} + 17 \beta_{7} - 23 \beta_{6} + 48 \beta_{5} + 6 \beta_{4} + \cdots - 73 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 201 \beta_{10} + 245 \beta_{9} - 64 \beta_{8} + 20 \beta_{7} + 38 \beta_{6} + 39 \beta_{5} + \cdots + 1076 \)
|
| \(\nu^{7}\) | \(=\) |
\( 304 \beta_{10} - 218 \beta_{9} + 86 \beta_{8} + 257 \beta_{7} - 406 \beta_{6} + 896 \beta_{5} + \cdots - 997 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3020 \beta_{10} + 3714 \beta_{9} - 1050 \beta_{8} + 372 \beta_{7} + 593 \beta_{6} + 607 \beta_{5} + \cdots + 15505 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4602 \beta_{10} - 2352 \beta_{9} + 1003 \beta_{8} + 3834 \beta_{7} - 6611 \beta_{6} + 15577 \beta_{5} + \cdots - 11958 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 46515 \beta_{10} + 56273 \beta_{9} - 15844 \beta_{8} + 6913 \beta_{7} + 8667 \beta_{6} + \cdots + 230908 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −3.92287 | 1.00000 | −2.55808 | −1.00000 | 1.00000 | 3.92287 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −3.61806 | 1.00000 | 4.03393 | −1.00000 | 1.00000 | 3.61806 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | −2.56652 | 1.00000 | −0.973492 | −1.00000 | 1.00000 | 2.56652 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | −1.23681 | 1.00000 | 2.50061 | −1.00000 | 1.00000 | 1.23681 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | −0.425735 | 1.00000 | −2.24729 | −1.00000 | 1.00000 | 0.425735 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 0.864610 | 1.00000 | 2.90002 | −1.00000 | 1.00000 | −0.864610 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 1.67126 | 1.00000 | −4.35376 | −1.00000 | 1.00000 | −1.67126 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 2.32043 | 1.00000 | −0.928685 | −1.00000 | 1.00000 | −2.32043 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | −1.00000 | 1.00000 | 2.32145 | 1.00000 | −4.55788 | −1.00000 | 1.00000 | −2.32145 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | −1.00000 | 1.00000 | 2.52811 | 1.00000 | 0.198222 | −1.00000 | 1.00000 | −2.52811 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | −1.00000 | 1.00000 | 4.06414 | 1.00000 | 3.98638 | −1.00000 | 1.00000 | −4.06414 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(19\) | \(-1\) |
| \(53\) | \(-1\) |
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 | 6042.2.a.bd | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6042.2.a.bd | ✓ | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):
|
\( T_{5}^{11} - 2 T_{5}^{10} - 35 T_{5}^{9} + 77 T_{5}^{8} + 394 T_{5}^{7} - 994 T_{5}^{6} - 1477 T_{5}^{5} + \cdots + 1534 \)
|
|
\( T_{7}^{11} + 2 T_{7}^{10} - 48 T_{7}^{9} - 86 T_{7}^{8} + 804 T_{7}^{7} + 1346 T_{7}^{6} - 5376 T_{7}^{5} + \cdots - 2384 \)
|
|
\( T_{11}^{11} + 6 T_{11}^{10} - 61 T_{11}^{9} - 441 T_{11}^{8} + 785 T_{11}^{7} + 9498 T_{11}^{6} + \cdots + 270080 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{11} \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 2 T^{10} + \cdots + 1534 \)
|
| $7$ |
\( T^{11} + 2 T^{10} + \cdots - 2384 \)
|
| $11$ |
\( T^{11} + 6 T^{10} + \cdots + 270080 \)
|
| $13$ |
\( T^{11} + 7 T^{10} + \cdots + 287664 \)
|
| $17$ |
\( T^{11} + 16 T^{10} + \cdots + 151968 \)
|
| $19$ |
\( (T - 1)^{11} \)
|
| $23$ |
\( T^{11} + 11 T^{10} + \cdots + 101280 \)
|
| $29$ |
\( T^{11} + 7 T^{10} + \cdots + 3308640 \)
|
| $31$ |
\( T^{11} - 12 T^{10} + \cdots + 24909088 \)
|
| $37$ |
\( T^{11} + \cdots - 143826272 \)
|
| $41$ |
\( T^{11} - 11 T^{10} + \cdots - 80800000 \)
|
| $43$ |
\( T^{11} - 7 T^{10} + \cdots - 53440 \)
|
| $47$ |
\( T^{11} + 37 T^{10} + \cdots - 1085152 \)
|
| $53$ |
\( (T - 1)^{11} \)
|
| $59$ |
\( T^{11} + 16 T^{10} + \cdots - 2438768 \)
|
| $61$ |
\( T^{11} - 24 T^{10} + \cdots - 974624 \)
|
| $67$ |
\( T^{11} + 28 T^{10} + \cdots + 3554304 \)
|
| $71$ |
\( T^{11} + \cdots - 332669952 \)
|
| $73$ |
\( T^{11} + \cdots + 1284435968 \)
|
| $79$ |
\( T^{11} - 5 T^{10} + \cdots - 3987136 \)
|
| $83$ |
\( T^{11} + \cdots + 139594752 \)
|
| $89$ |
\( T^{11} + 10 T^{10} + \cdots - 5919040 \)
|
| $97$ |
\( T^{11} + \cdots + 526132224 \)
|
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