Newspace parameters
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.350464.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
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\(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
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\(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
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\(\beta_{5}\) | \(=\) |
\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
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\(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} \)
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\(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \)
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\(\nu^{4}\) | \(=\) |
\( -\beta_{2} + 5\beta _1 - 7 \)
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\(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(106\) | \(157\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 |
|
− | 2.17009i | 1.00000 | −2.70928 | 0.630898i | − | 2.17009i | − | 1.00000i | 1.53919i | 1.00000 | 1.36910 | |||||||||||||||||||||||||||||||||
64.2 | − | 1.48119i | 1.00000 | −0.193937 | 4.15633i | − | 1.48119i | 1.00000i | − | 2.67513i | 1.00000 | 6.15633 | ||||||||||||||||||||||||||||||||||
64.3 | − | 0.311108i | 1.00000 | 1.90321 | 1.52543i | − | 0.311108i | − | 1.00000i | − | 1.21432i | 1.00000 | 0.474572 | |||||||||||||||||||||||||||||||||
64.4 | 0.311108i | 1.00000 | 1.90321 | − | 1.52543i | 0.311108i | 1.00000i | 1.21432i | 1.00000 | 0.474572 | ||||||||||||||||||||||||||||||||||||
64.5 | 1.48119i | 1.00000 | −0.193937 | − | 4.15633i | 1.48119i | − | 1.00000i | 2.67513i | 1.00000 | 6.15633 | |||||||||||||||||||||||||||||||||||
64.6 | 2.17009i | 1.00000 | −2.70928 | − | 0.630898i | 2.17009i | 1.00000i | − | 1.53919i | 1.00000 | 1.36910 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.c.b | ✓ | 6 |
3.b | odd | 2 | 1 | 819.2.c.c | 6 | ||
4.b | odd | 2 | 1 | 4368.2.h.o | 6 | ||
7.b | odd | 2 | 1 | 1911.2.c.h | 6 | ||
13.b | even | 2 | 1 | inner | 273.2.c.b | ✓ | 6 |
13.d | odd | 4 | 1 | 3549.2.a.k | 3 | ||
13.d | odd | 4 | 1 | 3549.2.a.q | 3 | ||
39.d | odd | 2 | 1 | 819.2.c.c | 6 | ||
52.b | odd | 2 | 1 | 4368.2.h.o | 6 | ||
91.b | odd | 2 | 1 | 1911.2.c.h | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
273.2.c.b | ✓ | 6 | 13.b | even | 2 | 1 | inner |
819.2.c.c | 6 | 3.b | odd | 2 | 1 | ||
819.2.c.c | 6 | 39.d | odd | 2 | 1 | ||
1911.2.c.h | 6 | 7.b | odd | 2 | 1 | ||
1911.2.c.h | 6 | 91.b | odd | 2 | 1 | ||
3549.2.a.k | 3 | 13.d | odd | 4 | 1 | ||
3549.2.a.q | 3 | 13.d | odd | 4 | 1 | ||
4368.2.h.o | 6 | 4.b | odd | 2 | 1 | ||
4368.2.h.o | 6 | 52.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 7 T^{4} + 11 T^{2} + 1 \)
$3$
\( (T - 1)^{6} \)
$5$
\( T^{6} + 20 T^{4} + 48 T^{2} + 16 \)
$7$
\( (T^{2} + 1)^{3} \)
$11$
\( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \)
$13$
\( T^{6} + 2 T^{5} + 27 T^{4} + \cdots + 2197 \)
$17$
\( (T^{3} - 16 T + 16)^{2} \)
$19$
\( T^{6} + 64 T^{4} + 512 T^{2} + \cdots + 1024 \)
$23$
\( (T^{3} - 2 T^{2} - 20 T + 8)^{2} \)
$29$
\( (T^{3} + 2 T^{2} - 52 T - 40)^{2} \)
$31$
\( T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400 \)
$37$
\( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \)
$41$
\( T^{6} + 132 T^{4} + 464 T^{2} + \cdots + 400 \)
$43$
\( (T^{3} + 16 T^{2} + 32 T - 128)^{2} \)
$47$
\( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704 \)
$53$
\( (T + 6)^{6} \)
$59$
\( T^{6} + 40 T^{4} + 80 T^{2} + 16 \)
$61$
\( (T^{3} - 14 T^{2} - 172 T + 2392)^{2} \)
$67$
\( T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 1024 \)
$71$
\( T^{6} + 332 T^{4} + 30784 T^{2} + \cdots + 547600 \)
$73$
\( T^{6} + 272 T^{4} + 6720 T^{2} + \cdots + 43264 \)
$79$
\( (T^{3} + 16 T^{2} + 72 T + 80)^{2} \)
$83$
\( T^{6} + 408 T^{4} + 49616 T^{2} + \cdots + 1567504 \)
$89$
\( T^{6} + 212 T^{4} + 6832 T^{2} + \cdots + 5776 \)
$97$
\( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \)
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