Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1775,2,Mod(1,1775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1775.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1775 = 5^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1775.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: | \(14.1734463588\) |
| 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: |
\( x^{12} - x^{11} - 17 x^{10} + 15 x^{9} + 104 x^{8} - 72 x^{7} - 290 x^{6} + 129 x^{5} + 365 x^{4} + \cdots - 3 \)
|
| 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
\( x^{12} - x^{11} - 17 x^{10} + 15 x^{9} + 104 x^{8} - 72 x^{7} - 290 x^{6} + 129 x^{5} + 365 x^{4} + \cdots - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25 \nu^{11} - 10 \nu^{10} - 431 \nu^{9} + 153 \nu^{8} + 2643 \nu^{7} - 690 \nu^{6} - 7115 \nu^{5} + \cdots - 912 ) / 122 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37 \nu^{11} - 27 \nu^{10} - 572 \nu^{9} + 346 \nu^{8} + 3043 \nu^{7} - 1192 \nu^{6} - 7102 \nu^{5} + \cdots - 669 ) / 122 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 22 \nu^{11} + 21 \nu^{10} + 350 \nu^{9} - 303 \nu^{8} - 1933 \nu^{7} + 1388 \nu^{6} + 4663 \nu^{5} + \cdots + 195 ) / 61 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 53 \nu^{11} - 70 \nu^{10} - 882 \nu^{9} + 1071 \nu^{8} + 5264 \nu^{7} - 5379 \nu^{6} - 14486 \nu^{5} + \cdots - 1199 ) / 122 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 56 \nu^{11} - 59 \nu^{10} - 963 \nu^{9} + 921 \nu^{8} + 5974 \nu^{7} - 4742 \nu^{6} - 16877 \nu^{5} + \cdots - 1611 ) / 122 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 103 \nu^{11} + 90 \nu^{10} + 1744 \nu^{9} - 1377 \nu^{8} - 10550 \nu^{7} + 6881 \nu^{6} + \cdots + 1681 ) / 122 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 104 \nu^{11} + 127 \nu^{10} + 1771 \nu^{9} - 1937 \nu^{8} - 10868 \nu^{7} + 9678 \nu^{6} + \cdots + 2347 ) / 122 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 65 \nu^{11} + 87 \nu^{10} + 1084 \nu^{9} - 1325 \nu^{8} - 6457 \nu^{7} + 6613 \nu^{6} + 17462 \nu^{5} + \cdots + 1078 ) / 61 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 133 \nu^{11} - 163 \nu^{10} - 2249 \nu^{9} + 2500 \nu^{8} + 13624 \nu^{7} - 12589 \nu^{6} + \cdots - 2812 ) / 122 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + 6\beta_{2} - 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{11} - \beta_{10} + 9 \beta_{9} + 10 \beta_{7} - 10 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + \cdots - 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9 \beta_{11} - \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + \beta_{6} + 21 \beta_{5} + \cdots + 80 \)
|
| \(\nu^{7}\) | \(=\) |
\( 55 \beta_{11} - 12 \beta_{10} + 68 \beta_{9} + 82 \beta_{7} - 79 \beta_{6} + 69 \beta_{5} + 69 \beta_{4} + \cdots - 96 \)
|
| \(\nu^{8}\) | \(=\) |
\( 63 \beta_{11} - 14 \beta_{10} + 80 \beta_{9} - 82 \beta_{8} - 95 \beta_{7} + 19 \beta_{6} + 171 \beta_{5} + \cdots + 517 \)
|
| \(\nu^{9}\) | \(=\) |
\( 369 \beta_{11} - 108 \beta_{10} + 494 \beta_{9} + 2 \beta_{8} + 632 \beta_{7} - 583 \beta_{6} + \cdots - 780 \)
|
| \(\nu^{10}\) | \(=\) |
\( 405 \beta_{11} - 133 \beta_{10} + 588 \beta_{9} - 633 \beta_{8} - 764 \beta_{7} + 230 \beta_{6} + \cdots + 3570 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2486 \beta_{11} - 873 \beta_{10} + 3548 \beta_{9} + 41 \beta_{8} + 4750 \beta_{7} - 4213 \beta_{6} + \cdots - 6143 \)
|
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 |
|
−2.73638 | 1.93147 | 5.48780 | 0 | −5.28525 | 4.77963 | −9.54396 | 0.730584 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.84707 | −1.24468 | 1.41167 | 0 | 2.29902 | 2.95160 | 1.08668 | −1.45076 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.66291 | 1.30941 | 0.765264 | 0 | −2.17742 | 0.173513 | 2.05325 | −1.28545 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.36761 | −0.270307 | −0.129629 | 0 | 0.369677 | −3.32166 | 2.91251 | −2.92693 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.503289 | 2.97086 | −1.74670 | 0 | −1.49520 | 2.38749 | 1.88567 | 5.82599 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.201042 | −3.37681 | −1.95958 | 0 | 0.678879 | 3.09353 | 0.796041 | 8.40283 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.130351 | 1.95353 | −1.98301 | 0 | −0.254644 | −2.73938 | 0.519189 | 0.816282 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.13303 | −1.45111 | −0.716254 | 0 | −1.64414 | 2.02730 | −3.07758 | −0.894292 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.61466 | −1.86733 | 0.607138 | 0 | −3.01511 | 2.29682 | −2.24900 | 0.486925 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.85792 | 3.11866 | 1.45188 | 0 | 5.79423 | −1.80633 | −1.01837 | 6.72605 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.21676 | 1.71610 | 2.91404 | 0 | 3.80419 | 4.45729 | 2.02622 | −0.0550008 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.62629 | −0.789794 | 4.89738 | 0 | −2.07422 | −0.299804 | 7.60935 | −2.37623 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(71\) | \(1\) |
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 | 1775.2.a.p | yes | 12 |
| 5.b | even | 2 | 1 | 1775.2.a.o | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1775.2.a.o | ✓ | 12 | 5.b | even | 2 | 1 | |
| 1775.2.a.p | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - T_{2}^{11} - 17 T_{2}^{10} + 15 T_{2}^{9} + 104 T_{2}^{8} - 72 T_{2}^{7} - 290 T_{2}^{6} + \cdots - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1775))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - T^{11} + \cdots - 3 \)
|
| $3$ |
\( T^{12} - 4 T^{11} + \cdots + 191 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} - 14 T^{11} + \cdots + 1849 \)
|
| $11$ |
\( T^{12} - 69 T^{10} + \cdots + 3 \)
|
| $13$ |
\( T^{12} - 22 T^{11} + \cdots + 36481 \)
|
| $17$ |
\( T^{12} - 19 T^{11} + \cdots + 10029 \)
|
| $19$ |
\( T^{12} - T^{11} + \cdots + 21401 \)
|
| $23$ |
\( T^{12} - 12 T^{11} + \cdots - 324327 \)
|
| $29$ |
\( T^{12} - 7 T^{11} + \cdots - 33279 \)
|
| $31$ |
\( T^{12} + 5 T^{11} + \cdots + 36055697 \)
|
| $37$ |
\( T^{12} - 15 T^{11} + \cdots + 21501643 \)
|
| $41$ |
\( T^{12} + \cdots - 238424913 \)
|
| $43$ |
\( T^{12} - 11 T^{11} + \cdots + 2871269 \)
|
| $47$ |
\( T^{12} - 12 T^{11} + \cdots + 54808128 \)
|
| $53$ |
\( T^{12} - 11 T^{11} + \cdots - 161709 \)
|
| $59$ |
\( T^{12} - 12 T^{11} + \cdots - 1332483 \)
|
| $61$ |
\( T^{12} + \cdots + 144499869 \)
|
| $67$ |
\( T^{12} + \cdots + 480519209 \)
|
| $71$ |
\( (T + 1)^{12} \)
|
| $73$ |
\( T^{12} + \cdots - 746258993 \)
|
| $79$ |
\( T^{12} + \cdots - 45037399031 \)
|
| $83$ |
\( T^{12} + 8 T^{11} + \cdots - 7215099 \)
|
| $89$ |
\( T^{12} + 7 T^{11} + \cdots + 70613184 \)
|
| $97$ |
\( T^{12} + \cdots - 4887612007 \)
|
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