Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,2,Mod(43,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.bd (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(2.17991597518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{14} + 20\nu^{12} + 147\nu^{10} + 480\nu^{8} + 646\nu^{6} + 183\nu^{4} - 105\nu^{2} + 26\nu - 6 ) / 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{14} - 20\nu^{12} - 147\nu^{10} - 480\nu^{8} - 646\nu^{6} - 183\nu^{4} + 105\nu^{2} + 26\nu + 6 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1107\nu^{2} - 23 ) / 104 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1003\nu^{2} + 289 ) / 104 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7 \nu^{15} + 5 \nu^{14} + 149 \nu^{13} + 107 \nu^{12} + 1210 \nu^{11} + 870 \nu^{10} + 4680 \nu^{9} + \cdots + 23 ) / 208 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5 \nu^{15} - 13 \nu^{14} + 115 \nu^{13} - 283 \nu^{12} + 1054 \nu^{11} - 2362 \nu^{10} + 4964 \nu^{9} + \cdots - 215 ) / 208 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 15 \nu^{14} + 323 \nu^{12} + 2656 \nu^{10} + 10400 \nu^{8} + 19631 \nu^{6} + 15823 \nu^{4} + \cdots + 203 ) / 104 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 9 \nu^{15} + 197 \nu^{13} + 2 \nu^{12} + 1662 \nu^{11} + 46 \nu^{10} + 6796 \nu^{9} + 406 \nu^{8} + \cdots + 160 ) / 104 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9 \nu^{15} + 197 \nu^{13} - 2 \nu^{12} + 1662 \nu^{11} - 46 \nu^{10} + 6796 \nu^{9} - 406 \nu^{8} + \cdots - 160 ) / 104 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 23 \nu^{15} + 7 \nu^{14} - 501 \nu^{13} + 149 \nu^{12} - 4194 \nu^{11} + 1210 \nu^{10} - 16932 \nu^{9} + \cdots - 167 ) / 208 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 23 \nu^{15} - 7 \nu^{14} - 501 \nu^{13} - 149 \nu^{12} - 4194 \nu^{11} - 1210 \nu^{10} - 16932 \nu^{9} + \cdots + 167 ) / 208 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 6\nu^{15} + 133\nu^{13} + 1142\nu^{11} + 4791\nu^{9} + 10194\nu^{7} + 10354\nu^{5} + 4323\nu^{3} + 471\nu + 26 ) / 52 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 41 \nu^{15} + 5 \nu^{14} - 895 \nu^{13} + 119 \nu^{12} - 7518 \nu^{11} + 1094 \nu^{10} - 30524 \nu^{9} + \cdots - 317 ) / 208 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 41 \nu^{15} - 5 \nu^{14} - 895 \nu^{13} - 119 \nu^{12} - 7518 \nu^{11} - 1094 \nu^{10} - 30524 \nu^{9} + \cdots + 317 ) / 208 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 59 \nu^{15} + 5 \nu^{14} + 1293 \nu^{13} + 107 \nu^{12} + 10934 \nu^{11} + 870 \nu^{10} + 44928 \nu^{9} + \cdots - 185 ) / 208 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 5\beta_{2} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{7} - 6\beta_{4} + 8\beta_{3} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 9 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} - \beta_{13} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{7} + 35 \beta_{4} - 57 \beta_{3} + \cdots - 84 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4 \beta_{15} + 66 \beta_{14} + 66 \beta_{13} + 24 \beta_{12} - 79 \beta_{11} - 79 \beta_{10} + \cdots - 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 13 \beta_{14} + 13 \beta_{13} + 57 \beta_{11} - 57 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 500 \)
|
| \(\nu^{9}\) | \(=\) |
\( 60 \beta_{15} - 453 \beta_{14} - 453 \beta_{13} - 218 \beta_{12} + 569 \beta_{11} + 569 \beta_{10} + \cdots + 79 \)
|
| \(\nu^{10}\) | \(=\) |
\( 124 \beta_{14} - 124 \beta_{13} - 403 \beta_{11} + 403 \beta_{10} + 30 \beta_{9} - 30 \beta_{8} + \cdots - 3091 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 620 \beta_{15} + 3026 \beta_{14} + 3026 \beta_{13} + 1778 \beta_{12} - 3927 \beta_{11} - 3927 \beta_{10} + \cdots - 579 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 1055 \beta_{14} + 1055 \beta_{13} + 2861 \beta_{11} - 2861 \beta_{10} - 310 \beta_{9} + 310 \beta_{8} + \cdots + 19574 \)
|
| \(\nu^{13}\) | \(=\) |
\( 5500 \beta_{15} - 19978 \beta_{14} - 19978 \beta_{13} - 13702 \beta_{12} + 26552 \beta_{11} + \cdots + 4101 \)
|
| \(\nu^{14}\) | \(=\) |
\( 8466 \beta_{14} - 8466 \beta_{13} - 20354 \beta_{11} + 20354 \beta_{10} + 2750 \beta_{9} - 2750 \beta_{8} + \cdots - 125893 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 45024 \beta_{15} + 131294 \beta_{14} + 131294 \beta_{13} + 102072 \beta_{12} - 177792 \beta_{11} + \cdots - 28524 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
| \(n\) | \(92\) | \(106\) | \(157\) |
| \(\chi(n)\) | \(1\) | \(\beta_{12}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−2.25861 | − | 1.30401i | −0.500000 | + | 0.866025i | 2.40088 | + | 4.15844i | − | 1.50528i | 2.25861 | − | 1.30401i | −0.866025 | + | 0.500000i | − | 7.30704i | −0.500000 | − | 0.866025i | −1.96290 | + | 3.39983i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −1.54092 | − | 0.889651i | −0.500000 | + | 0.866025i | 0.582956 | + | 1.00971i | − | 0.681820i | 1.54092 | − | 0.889651i | 0.866025 | − | 0.500000i | 1.48409i | −0.500000 | − | 0.866025i | −0.606581 | + | 1.05063i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −0.671904 | − | 0.387924i | −0.500000 | + | 0.866025i | −0.699030 | − | 1.21076i | 3.03444i | 0.671904 | − | 0.387924i | 0.866025 | − | 0.500000i | 2.63638i | −0.500000 | − | 0.866025i | 1.17713 | − | 2.03885i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −0.420879 | − | 0.242995i | −0.500000 | + | 0.866025i | −0.881907 | − | 1.52751i | 1.06536i | 0.420879 | − | 0.242995i | −0.866025 | + | 0.500000i | 1.82917i | −0.500000 | − | 0.866025i | 0.258876 | − | 0.448387i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | 0.0921099 | + | 0.0531797i | −0.500000 | + | 0.866025i | −0.994344 | − | 1.72225i | − | 1.41292i | −0.0921099 | + | 0.0531797i | −0.866025 | + | 0.500000i | − | 0.424234i | −0.500000 | − | 0.866025i | 0.0751388 | − | 0.130144i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 0.954423 | + | 0.551037i | −0.500000 | + | 0.866025i | −0.392717 | − | 0.680206i | − | 3.28432i | −0.954423 | + | 0.551037i | 0.866025 | − | 0.500000i | − | 3.06975i | −0.500000 | − | 0.866025i | 1.80978 | − | 3.13463i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 1.72135 | + | 0.993824i | −0.500000 | + | 0.866025i | 0.975372 | + | 1.68939i | 2.85284i | −1.72135 | + | 0.993824i | −0.866025 | + | 0.500000i | − | 0.0979034i | −0.500000 | − | 0.866025i | −2.83522 | + | 4.91075i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 2.12443 | + | 1.22654i | −0.500000 | + | 0.866025i | 2.00879 | + | 3.47933i | − | 0.0682999i | −2.12443 | + | 1.22654i | 0.866025 | − | 0.500000i | 4.94928i | −0.500000 | − | 0.866025i | 0.0837724 | − | 0.145098i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.1 | −2.25861 | + | 1.30401i | −0.500000 | − | 0.866025i | 2.40088 | − | 4.15844i | 1.50528i | 2.25861 | + | 1.30401i | −0.866025 | − | 0.500000i | 7.30704i | −0.500000 | + | 0.866025i | −1.96290 | − | 3.39983i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.2 | −1.54092 | + | 0.889651i | −0.500000 | − | 0.866025i | 0.582956 | − | 1.00971i | 0.681820i | 1.54092 | + | 0.889651i | 0.866025 | + | 0.500000i | − | 1.48409i | −0.500000 | + | 0.866025i | −0.606581 | − | 1.05063i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.3 | −0.671904 | + | 0.387924i | −0.500000 | − | 0.866025i | −0.699030 | + | 1.21076i | − | 3.03444i | 0.671904 | + | 0.387924i | 0.866025 | + | 0.500000i | − | 2.63638i | −0.500000 | + | 0.866025i | 1.17713 | + | 2.03885i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.4 | −0.420879 | + | 0.242995i | −0.500000 | − | 0.866025i | −0.881907 | + | 1.52751i | − | 1.06536i | 0.420879 | + | 0.242995i | −0.866025 | − | 0.500000i | − | 1.82917i | −0.500000 | + | 0.866025i | 0.258876 | + | 0.448387i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.5 | 0.0921099 | − | 0.0531797i | −0.500000 | − | 0.866025i | −0.994344 | + | 1.72225i | 1.41292i | −0.0921099 | − | 0.0531797i | −0.866025 | − | 0.500000i | 0.424234i | −0.500000 | + | 0.866025i | 0.0751388 | + | 0.130144i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.6 | 0.954423 | − | 0.551037i | −0.500000 | − | 0.866025i | −0.392717 | + | 0.680206i | 3.28432i | −0.954423 | − | 0.551037i | 0.866025 | + | 0.500000i | 3.06975i | −0.500000 | + | 0.866025i | 1.80978 | + | 3.13463i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.7 | 1.72135 | − | 0.993824i | −0.500000 | − | 0.866025i | 0.975372 | − | 1.68939i | − | 2.85284i | −1.72135 | − | 0.993824i | −0.866025 | − | 0.500000i | 0.0979034i | −0.500000 | + | 0.866025i | −2.83522 | − | 4.91075i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 127.8 | 2.12443 | − | 1.22654i | −0.500000 | − | 0.866025i | 2.00879 | − | 3.47933i | 0.0682999i | −2.12443 | − | 1.22654i | 0.866025 | + | 0.500000i | − | 4.94928i | −0.500000 | + | 0.866025i | 0.0837724 | + | 0.145098i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 273.2.bd.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 819.2.ct.b | 16 | ||
| 13.e | even | 6 | 1 | inner | 273.2.bd.a | ✓ | 16 |
| 13.f | odd | 12 | 1 | 3549.2.a.bb | 8 | ||
| 13.f | odd | 12 | 1 | 3549.2.a.bd | 8 | ||
| 39.h | odd | 6 | 1 | 819.2.ct.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.bd.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 273.2.bd.a | ✓ | 16 | 13.e | even | 6 | 1 | inner |
| 819.2.ct.b | 16 | 3.b | odd | 2 | 1 | ||
| 819.2.ct.b | 16 | 39.h | odd | 6 | 1 | ||
| 3549.2.a.bb | 8 | 13.f | odd | 12 | 1 | ||
| 3549.2.a.bd | 8 | 13.f | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 11 T_{2}^{14} + 88 T_{2}^{12} - 6 T_{2}^{11} - 315 T_{2}^{10} + 12 T_{2}^{9} + 824 T_{2}^{8} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 11 T^{14} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} + T + 1)^{8} \)
|
| $5$ |
\( T^{16} + 34 T^{14} + \cdots + 9 \)
|
| $7$ |
\( (T^{4} - T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{16} + 12 T^{15} + \cdots + 144 \)
|
| $13$ |
\( T^{16} + \cdots + 815730721 \)
|
| $17$ |
\( T^{16} + \cdots + 2883582601 \)
|
| $19$ |
\( T^{16} + \cdots + 25335725584 \)
|
| $23$ |
\( T^{16} + \cdots + 1446433024 \)
|
| $29$ |
\( T^{16} + \cdots + 2538849769 \)
|
| $31$ |
\( T^{16} + 222 T^{14} + \cdots + 7929856 \)
|
| $37$ |
\( T^{16} + \cdots + 16292735449 \)
|
| $41$ |
\( T^{16} - 18 T^{15} + \cdots + 692224 \)
|
| $43$ |
\( T^{16} + \cdots + 15764309136 \)
|
| $47$ |
\( T^{16} + 222 T^{14} + \cdots + 1336336 \)
|
| $53$ |
\( (T^{8} - 6 T^{7} + \cdots - 76707)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 3887273104 \)
|
| $61$ |
\( T^{16} + \cdots + 103866332089 \)
|
| $67$ |
\( T^{16} + \cdots + 5866334464 \)
|
| $71$ |
\( T^{16} + \cdots + 84\!\cdots\!84 \)
|
| $73$ |
\( T^{16} + \cdots + 17309352523401 \)
|
| $79$ |
\( (T^{8} + 2 T^{7} + \cdots + 75240852)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 2140872301584 \)
|
| $89$ |
\( T^{16} + \cdots + 72\!\cdots\!16 \)
|
| $97$ |
\( T^{16} + \cdots + 941955655936 \)
|
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