Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1175,4,Mod(1,1175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1175.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1175 = 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1175.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: | \(69.3272442567\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 50x^{6} + 124x^{5} + 844x^{4} - 1549x^{3} - 5393x^{2} + 5418x + 10316 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 47) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 50x^{6} + 124x^{5} + 844x^{4} - 1549x^{3} - 5393x^{2} + 5418x + 10316 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 457 \nu^{7} - 14084 \nu^{6} + 27766 \nu^{5} + 629470 \nu^{4} - 346298 \nu^{3} - 7251681 \nu^{2} + \cdots + 12792628 ) / 1179168 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 457 \nu^{7} - 14084 \nu^{6} + 27766 \nu^{5} + 629470 \nu^{4} - 346298 \nu^{3} - 8430849 \nu^{2} + \cdots + 28121812 ) / 1179168 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 281 \nu^{7} - 1876 \nu^{6} - 8902 \nu^{5} + 52882 \nu^{4} + 151866 \nu^{3} - 345919 \nu^{2} + \cdots + 89372 ) / 196528 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2575 \nu^{7} - 11596 \nu^{6} - 73882 \nu^{5} + 371294 \nu^{4} + 341174 \nu^{3} - 3349641 \nu^{2} + \cdots + 8061140 ) / 589584 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3305 \nu^{7} - 4580 \nu^{6} - 154358 \nu^{5} + 120514 \nu^{4} + 2188330 \nu^{3} - 282063 \nu^{2} + \cdots - 4690820 ) / 589584 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8845 \nu^{7} + 4244 \nu^{6} - 377422 \nu^{5} - 280438 \nu^{4} + 4762082 \nu^{3} + 4139301 \nu^{2} + \cdots - 15378628 ) / 1179168 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{2} + 19\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{7} - 4\beta_{6} - 2\beta_{5} - 2\beta_{4} - 21\beta_{3} + 30\beta_{2} + 6\beta _1 + 237 \)
|
| \(\nu^{5}\) | \(=\) |
\( 40\beta_{7} - 50\beta_{6} - 21\beta_{5} + 59\beta_{4} - 17\beta_{3} + 49\beta_{2} + 381\beta _1 + 258 \)
|
| \(\nu^{6}\) | \(=\) |
\( 234\beta_{7} - 200\beta_{6} - 94\beta_{5} - 68\beta_{4} - 434\beta_{3} + 760\beta_{2} + 295\beta _1 + 4882 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1348\beta_{7} - 1626\beta_{6} - 376\beta_{5} + 1410\beta_{4} - 715\beta_{3} + 1671\beta_{2} + 8074\beta _1 + 8067 \)
|
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 |
|
−5.13061 | −2.12464 | 18.3231 | 0 | 10.9007 | 25.1804 | −52.9639 | −22.4859 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.52397 | 5.04907 | 12.4663 | 0 | −22.8418 | 6.74296 | −20.2056 | −1.50693 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.53023 | −2.59326 | 4.46255 | 0 | 9.15480 | −31.0883 | 12.4880 | −20.2750 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.18922 | −8.24989 | −3.20731 | 0 | 18.0608 | 18.6902 | 24.5353 | 41.0606 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.20418 | −2.76301 | −6.54994 | 0 | −3.32718 | −21.2611 | −17.5208 | −19.3658 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.61832 | 8.85615 | −1.14438 | 0 | 23.1883 | −2.58900 | −23.9430 | 51.4314 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.06926 | −9.82442 | 8.55884 | 0 | −39.9781 | −4.03804 | 2.27405 | 69.5192 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.48227 | 4.64999 | 12.0908 | 0 | 20.8425 | −30.6370 | 18.3360 | −5.37761 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(47\) | \(-1\) |
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 | 1175.4.a.b | 8 | |
| 5.b | even | 2 | 1 | 47.4.a.b | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 423.4.a.g | 8 | ||
| 20.d | odd | 2 | 1 | 752.4.a.i | 8 | ||
| 35.c | odd | 2 | 1 | 2303.4.a.b | 8 | ||
| 235.b | odd | 2 | 1 | 2209.4.a.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.4.a.b | ✓ | 8 | 5.b | even | 2 | 1 | |
| 423.4.a.g | 8 | 15.d | odd | 2 | 1 | ||
| 752.4.a.i | 8 | 20.d | odd | 2 | 1 | ||
| 1175.4.a.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 2209.4.a.f | 8 | 235.b | odd | 2 | 1 | ||
| 2303.4.a.b | 8 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{7} - 50T_{2}^{6} - 124T_{2}^{5} + 844T_{2}^{4} + 1549T_{2}^{3} - 5393T_{2}^{2} - 5418T_{2} + 10316 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1175))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{7} + \cdots + 10316 \)
|
| $3$ |
\( T^{8} + 7 T^{7} + \cdots - 256552 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 39 T^{7} + \cdots - 671832924 \)
|
| $11$ |
\( T^{8} + \cdots - 13840895616 \)
|
| $13$ |
\( T^{8} + \cdots - 1112419583616 \)
|
| $17$ |
\( T^{8} + \cdots + 48362680514304 \)
|
| $19$ |
\( T^{8} + \cdots + 25\!\cdots\!72 \)
|
| $23$ |
\( T^{8} + \cdots - 320479281547776 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!44 \)
|
| $31$ |
\( T^{8} + \cdots - 31\!\cdots\!72 \)
|
| $37$ |
\( T^{8} + \cdots + 15\!\cdots\!32 \)
|
| $41$ |
\( T^{8} + \cdots + 91\!\cdots\!28 \)
|
| $43$ |
\( T^{8} + \cdots + 44\!\cdots\!16 \)
|
| $47$ |
\( (T - 47)^{8} \)
|
| $53$ |
\( T^{8} + \cdots - 12\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots - 18\!\cdots\!32 \)
|
| $61$ |
\( T^{8} + \cdots - 28\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots - 19\!\cdots\!88 \)
|
| $71$ |
\( T^{8} + \cdots + 30\!\cdots\!16 \)
|
| $73$ |
\( T^{8} + \cdots - 23\!\cdots\!72 \)
|
| $79$ |
\( T^{8} + \cdots + 58\!\cdots\!52 \)
|
| $83$ |
\( T^{8} + \cdots - 26\!\cdots\!48 \)
|
| $89$ |
\( T^{8} + \cdots - 93\!\cdots\!88 \)
|
| $97$ |
\( T^{8} + \cdots + 79\!\cdots\!12 \)
|
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