Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1073,2,Mod(1,1073)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1073, 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("1073.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1073 = 29 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1073.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: | \(8.56794813688\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 28 x^{15} + 322 x^{13} - 8 x^{12} - 1970 x^{11} + 146 x^{10} + 6947 x^{9} - 1000 x^{8} + \cdots - 936 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 28 x^{15} + 322 x^{13} - 8 x^{12} - 1970 x^{11} + 146 x^{10} + 6947 x^{9} - 1000 x^{8} + \cdots - 936 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2 \nu^{16} - 3 \nu^{15} + 42 \nu^{14} + 97 \nu^{13} - 313 \nu^{12} - 1222 \nu^{11} + 897 \nu^{10} + \cdots - 6087 ) / 43 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 49 \nu^{16} - 78 \nu^{15} - 1105 \nu^{14} + 1683 \nu^{13} + 9505 \nu^{12} - 13910 \nu^{11} + \cdots + 16143 ) / 129 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 21 \nu^{16} - 31 \nu^{15} - 509 \nu^{14} + 725 \nu^{13} + 4897 \nu^{12} - 6807 \nu^{11} - 23845 \nu^{10} + \cdots - 4630 ) / 86 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 56 \nu^{16} - 117 \nu^{15} - 1289 \nu^{14} + 2670 \nu^{13} + 11510 \nu^{12} - 23962 \nu^{11} + \cdots - 2070 ) / 129 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 41 \nu^{16} - 183 \nu^{15} - 857 \nu^{14} + 4377 \nu^{13} + 6521 \nu^{12} - 41581 \nu^{11} + \cdots - 47838 ) / 258 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 19 \nu^{16} + 183 \nu^{15} + 229 \nu^{14} - 4305 \nu^{13} + 767 \nu^{12} + 39809 \nu^{11} + \cdots + 30246 ) / 258 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{16} + 6 \nu^{15} - 311 \nu^{14} - 131 \nu^{13} + 3569 \nu^{12} + 972 \nu^{11} - 21383 \nu^{10} + \cdots - 3003 ) / 43 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5 \nu^{16} - 35 \nu^{15} - 108 \nu^{14} + 867 \nu^{13} + 892 \nu^{12} - 8596 \nu^{11} - 3378 \nu^{10} + \cdots - 14257 ) / 43 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 61 \nu^{16} - 101 \nu^{15} - 1435 \nu^{14} + 2295 \nu^{13} + 13219 \nu^{12} - 20599 \nu^{11} + \cdots - 5928 ) / 86 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 83 \nu^{16} + 237 \nu^{15} + 2021 \nu^{14} - 5883 \nu^{13} - 19619 \nu^{12} + 59035 \nu^{11} + \cdots + 125802 ) / 258 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 63 \nu^{16} + 81 \nu^{15} + 1545 \nu^{14} - 1881 \nu^{13} - 15065 \nu^{12} + 17569 \nu^{11} + \cdots + 20494 ) / 86 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 41 \nu^{16} + 45 \nu^{15} + 1033 \nu^{14} - 1059 \nu^{13} - 10439 \nu^{12} + 10118 \nu^{11} + \cdots + 13425 ) / 43 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 145 \nu^{16} - 219 \nu^{15} - 3550 \nu^{14} + 5136 \nu^{13} + 34594 \nu^{12} - 48362 \nu^{11} + \cdots - 52245 ) / 129 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{12} + \beta_{10} - \beta_{9} + \beta_{5} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{16} - 2 \beta_{12} + 2 \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 31 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{15} + \beta_{14} - 12 \beta_{12} - 2 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 2 \beta_{8} + \cdots + 78 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9 \beta_{16} - 3 \beta_{15} + \beta_{14} - \beta_{13} - 24 \beta_{12} + 22 \beta_{11} + \beta_{10} + \cdots + 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{16} - 14 \beta_{15} + 11 \beta_{14} + 2 \beta_{13} - 111 \beta_{12} - 26 \beta_{11} + 105 \beta_{10} + \cdots + 474 \)
|
| \(\nu^{9}\) | \(=\) |
\( 60 \beta_{16} - 48 \beta_{15} + 10 \beta_{14} - 9 \beta_{13} - 219 \beta_{12} + 182 \beta_{11} + \cdots + 89 \)
|
| \(\nu^{10}\) | \(=\) |
\( 14 \beta_{16} - 142 \beta_{15} + 79 \beta_{14} + 38 \beta_{13} - 935 \beta_{12} - 238 \beta_{11} + \cdots + 3039 \)
|
| \(\nu^{11}\) | \(=\) |
\( 351 \beta_{16} - 532 \beta_{15} + 55 \beta_{14} - 31 \beta_{13} - 1828 \beta_{12} + 1356 \beta_{11} + \cdots + 1068 \)
|
| \(\nu^{12}\) | \(=\) |
\( 121 \beta_{16} - 1289 \beta_{15} + 437 \beta_{14} + 476 \beta_{13} - 7530 \beta_{12} - 1903 \beta_{11} + \cdots + 20213 \)
|
| \(\nu^{13}\) | \(=\) |
\( 1840 \beta_{16} - 5090 \beta_{15} + 82 \beta_{14} + 260 \beta_{13} - 14712 \beta_{12} + 9591 \beta_{11} + \cdots + 10843 \)
|
| \(\nu^{14}\) | \(=\) |
\( 774 \beta_{16} - 11147 \beta_{15} + 1662 \beta_{14} + 4996 \beta_{13} - 59175 \beta_{12} - 14250 \beta_{11} + \cdots + 138014 \)
|
| \(\nu^{15}\) | \(=\) |
\( 8205 \beta_{16} - 45131 \beta_{15} - 2560 \beta_{14} + 6401 \beta_{13} - 116454 \beta_{12} + \cdots + 100517 \)
|
| \(\nu^{16}\) | \(=\) |
\( 3276 \beta_{16} - 94008 \beta_{15} - 756 \beta_{14} + 47699 \beta_{13} - 458410 \beta_{12} + \cdots + 960649 \)
|
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 |
|
−2.75562 | −1.77902 | 5.59344 | −0.769283 | 4.90230 | 2.12904 | −9.90216 | 0.164903 | 2.11985 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.59235 | 1.23979 | 4.72029 | 3.97032 | −3.21397 | 4.17553 | −7.05194 | −1.46292 | −10.2925 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.06381 | 3.00005 | 2.25930 | −1.16896 | −6.19153 | 3.30543 | −0.535152 | 6.00030 | 2.41250 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00193 | −0.0619586 | 2.00771 | 4.00658 | 0.124037 | −2.75796 | −0.0154298 | −2.99616 | −8.02087 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.31600 | 0.651444 | −0.268143 | −3.15263 | −0.857300 | 4.78161 | 2.98488 | −2.57562 | 4.14886 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.21045 | −2.92837 | −0.534810 | 1.94178 | 3.54465 | 2.84198 | 3.06826 | 5.57537 | −2.35043 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.05002 | 2.52855 | −0.897453 | 0.590881 | −2.65504 | −1.36593 | 3.04239 | 3.39357 | −0.620438 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.713746 | −1.84163 | −1.49057 | 2.39639 | 1.31445 | 4.28927 | 2.49138 | 0.391594 | −1.71042 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.498011 | −0.252082 | −1.75199 | 1.54958 | 0.125539 | −2.11889 | 1.86853 | −2.93645 | −0.771709 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.854381 | −2.11534 | −1.27003 | 0.0250012 | −1.80731 | −2.33469 | −2.79385 | 1.47468 | 0.0213606 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.862616 | 1.92172 | −1.25589 | 3.23815 | 1.65771 | 1.84693 | −2.80859 | 0.693009 | 2.79328 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.41978 | −2.55413 | 0.0157674 | −3.91350 | −3.62629 | 3.57554 | −2.81717 | 3.52356 | −5.55629 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.78259 | 3.25966 | 1.17763 | 2.77931 | 5.81063 | −1.59281 | −1.46595 | 7.62536 | 4.95437 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.13343 | 1.58095 | 2.55151 | −2.05702 | 3.37284 | 3.60983 | 1.17660 | −0.500590 | −4.38849 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.13557 | −2.46000 | 2.56067 | −2.81485 | −5.25351 | −2.58412 | 1.19734 | 3.05160 | −6.01131 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.42128 | −1.42895 | 3.86260 | 2.83713 | −3.45988 | 4.63237 | 4.50987 | −0.958109 | 6.86949 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.59229 | 1.23931 | 4.71998 | 0.541113 | 3.21266 | 1.56687 | 7.05099 | −1.46410 | 1.40272 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(-1\) |
| \(37\) | \(1\) |
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 | 1073.2.a.f | ✓ | 17 |
| 3.b | odd | 2 | 1 | 9657.2.a.k | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1073.2.a.f | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 9657.2.a.k | 17 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 28 T_{2}^{15} + 322 T_{2}^{13} + 8 T_{2}^{12} - 1970 T_{2}^{11} - 146 T_{2}^{10} + \cdots + 936 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1073))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 28 T^{15} + \cdots + 936 \)
|
| $3$ |
\( T^{17} - 35 T^{15} + \cdots + 214 \)
|
| $5$ |
\( T^{17} - 10 T^{16} + \cdots - 1504 \)
|
| $7$ |
\( T^{17} - 24 T^{16} + \cdots - 22732544 \)
|
| $11$ |
\( T^{17} + 2 T^{16} + \cdots - 106562 \)
|
| $13$ |
\( T^{17} - 12 T^{16} + \cdots - 5225504 \)
|
| $17$ |
\( T^{17} - 10 T^{16} + \cdots + 38021632 \)
|
| $19$ |
\( T^{17} - 18 T^{16} + \cdots - 82427392 \)
|
| $23$ |
\( T^{17} + \cdots + 3900182528 \)
|
| $29$ |
\( (T - 1)^{17} \)
|
| $31$ |
\( T^{17} - 56 T^{16} + \cdots - 826400 \)
|
| $37$ |
\( (T + 1)^{17} \)
|
| $41$ |
\( T^{17} + \cdots - 5272772096 \)
|
| $43$ |
\( T^{17} - 5 T^{16} + \cdots + 2522816 \)
|
| $47$ |
\( T^{17} + \cdots + 29962544704 \)
|
| $53$ |
\( T^{17} - 41 T^{16} + \cdots - 5020508 \)
|
| $59$ |
\( T^{17} + \cdots + 40176209994668 \)
|
| $61$ |
\( T^{17} + \cdots + 239046680944 \)
|
| $67$ |
\( T^{17} + \cdots + 5805651328 \)
|
| $71$ |
\( T^{17} + \cdots + 460367615744 \)
|
| $73$ |
\( T^{17} + \cdots - 7695980148728 \)
|
| $79$ |
\( T^{17} + \cdots - 77\!\cdots\!12 \)
|
| $83$ |
\( T^{17} + \cdots + 344095332352 \)
|
| $89$ |
\( T^{17} + \cdots - 73107067176 \)
|
| $97$ |
\( T^{17} + \cdots - 2614086229248 \)
|
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