Newspace parameters
| Level: | \( N \) | \(=\) | \( 169 = 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 169.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.34947179416\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.64827.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−1.12349 | + | 1.94594i | 0.277479 | − | 0.480608i | −1.52446 | − | 2.64044i | 1.44504 | 0.623490 | + | 1.07992i | 1.02446 | + | 1.77441i | 2.35690 | 1.34601 | + | 2.33136i | −1.62349 | + | 2.81197i | ||||||||||||||||||||||
| 22.2 | −0.277479 | + | 0.480608i | −0.400969 | + | 0.694498i | 0.846011 | + | 1.46533i | 2.80194 | −0.222521 | − | 0.385418i | −1.34601 | − | 2.33136i | −2.04892 | 1.17845 | + | 2.04113i | −0.777479 | + | 1.34663i | |||||||||||||||||||||||
| 22.3 | 0.400969 | − | 0.694498i | 1.12349 | − | 1.94594i | 0.678448 | + | 1.17511i | −0.246980 | −0.900969 | − | 1.56052i | −1.17845 | − | 2.04113i | 2.69202 | −1.02446 | − | 1.77441i | −0.0990311 | + | 0.171527i | |||||||||||||||||||||||
| 146.1 | −1.12349 | − | 1.94594i | 0.277479 | + | 0.480608i | −1.52446 | + | 2.64044i | 1.44504 | 0.623490 | − | 1.07992i | 1.02446 | − | 1.77441i | 2.35690 | 1.34601 | − | 2.33136i | −1.62349 | − | 2.81197i | |||||||||||||||||||||||
| 146.2 | −0.277479 | − | 0.480608i | −0.400969 | − | 0.694498i | 0.846011 | − | 1.46533i | 2.80194 | −0.222521 | + | 0.385418i | −1.34601 | + | 2.33136i | −2.04892 | 1.17845 | − | 2.04113i | −0.777479 | − | 1.34663i | |||||||||||||||||||||||
| 146.3 | 0.400969 | + | 0.694498i | 1.12349 | + | 1.94594i | 0.678448 | − | 1.17511i | −0.246980 | −0.900969 | + | 1.56052i | −1.17845 | + | 2.04113i | 2.69202 | −1.02446 | + | 1.77441i | −0.0990311 | − | 0.171527i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 169.2.c.b | 6 | |
| 13.b | even | 2 | 1 | 169.2.c.c | 6 | ||
| 13.c | even | 3 | 1 | 169.2.a.c | yes | 3 | |
| 13.c | even | 3 | 1 | inner | 169.2.c.b | 6 | |
| 13.d | odd | 4 | 2 | 169.2.e.b | 12 | ||
| 13.e | even | 6 | 1 | 169.2.a.b | ✓ | 3 | |
| 13.e | even | 6 | 1 | 169.2.c.c | 6 | ||
| 13.f | odd | 12 | 2 | 169.2.b.b | 6 | ||
| 13.f | odd | 12 | 2 | 169.2.e.b | 12 | ||
| 39.h | odd | 6 | 1 | 1521.2.a.r | 3 | ||
| 39.i | odd | 6 | 1 | 1521.2.a.o | 3 | ||
| 39.k | even | 12 | 2 | 1521.2.b.l | 6 | ||
| 52.i | odd | 6 | 1 | 2704.2.a.z | 3 | ||
| 52.j | odd | 6 | 1 | 2704.2.a.ba | 3 | ||
| 52.l | even | 12 | 2 | 2704.2.f.o | 6 | ||
| 65.l | even | 6 | 1 | 4225.2.a.bg | 3 | ||
| 65.n | even | 6 | 1 | 4225.2.a.bb | 3 | ||
| 91.n | odd | 6 | 1 | 8281.2.a.bj | 3 | ||
| 91.t | odd | 6 | 1 | 8281.2.a.bf | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 169.2.a.b | ✓ | 3 | 13.e | even | 6 | 1 | |
| 169.2.a.c | yes | 3 | 13.c | even | 3 | 1 | |
| 169.2.b.b | 6 | 13.f | odd | 12 | 2 | ||
| 169.2.c.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 169.2.c.b | 6 | 13.c | even | 3 | 1 | inner | |
| 169.2.c.c | 6 | 13.b | even | 2 | 1 | ||
| 169.2.c.c | 6 | 13.e | even | 6 | 1 | ||
| 169.2.e.b | 12 | 13.d | odd | 4 | 2 | ||
| 169.2.e.b | 12 | 13.f | odd | 12 | 2 | ||
| 1521.2.a.o | 3 | 39.i | odd | 6 | 1 | ||
| 1521.2.a.r | 3 | 39.h | odd | 6 | 1 | ||
| 1521.2.b.l | 6 | 39.k | even | 12 | 2 | ||
| 2704.2.a.z | 3 | 52.i | odd | 6 | 1 | ||
| 2704.2.a.ba | 3 | 52.j | odd | 6 | 1 | ||
| 2704.2.f.o | 6 | 52.l | even | 12 | 2 | ||
| 4225.2.a.bb | 3 | 65.n | even | 6 | 1 | ||
| 4225.2.a.bg | 3 | 65.l | even | 6 | 1 | ||
| 8281.2.a.bf | 3 | 91.t | odd | 6 | 1 | ||
| 8281.2.a.bj | 3 | 91.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} + 5T_{2}^{4} + 3T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 2 T^{5} + 5 T^{4} + 3 T^{2} + \cdots + 1 \)
$3$
\( T^{6} - 2 T^{5} + 5 T^{4} + 3 T^{2} + \cdots + 1 \)
$5$
\( (T^{3} - 4 T^{2} + 3 T + 1)^{2} \)
$7$
\( T^{6} + 3 T^{5} + 13 T^{4} + 14 T^{3} + \cdots + 169 \)
$11$
\( T^{6} + 8 T^{5} + 45 T^{4} + 126 T^{3} + \cdots + 169 \)
$13$
\( T^{6} \)
$17$
\( T^{6} - 2 T^{5} + 19 T^{4} + 56 T^{3} + \cdots + 169 \)
$19$
\( T^{6} + 4 T^{5} + 27 T^{4} - 42 T^{3} + \cdots + 1 \)
$23$
\( T^{6} - 5 T^{5} + 26 T^{4} - 21 T^{3} + \cdots + 169 \)
$29$
\( T^{6} - T^{5} + 45 T^{4} + 210 T^{3} + \cdots + 6889 \)
$31$
\( (T^{3} - 5 T^{2} - 36 T + 167)^{2} \)
$37$
\( T^{6} - 12 T^{5} + 103 T^{4} + \cdots + 841 \)
$41$
\( T^{6} + 7 T^{5} + 98 T^{4} + \cdots + 2401 \)
$43$
\( T^{6} + 13 T^{5} + 129 T^{4} + \cdots + 169 \)
$47$
\( (T^{3} - 18 T^{2} + 101 T - 167)^{2} \)
$53$
\( (T^{3} - T^{2} - 86 T + 337)^{2} \)
$59$
\( T^{6} + 19 T^{5} + 278 T^{4} + 1575 T^{3} + \cdots + 1 \)
$61$
\( T^{6} + 4 T^{5} + 83 T^{4} + \cdots + 57121 \)
$67$
\( T^{6} + T^{5} + 73 T^{4} - 154 T^{3} + \cdots + 1681 \)
$71$
\( T^{6} + 27 T^{5} + 507 T^{4} + \cdots + 299209 \)
$73$
\( (T^{3} + 9 T^{2} - 120 T - 911)^{2} \)
$79$
\( (T^{3} + 5 T^{2} - 162 T + 127)^{2} \)
$83$
\( (T^{3} - 7 T^{2} - 140 T - 203)^{2} \)
$89$
\( T^{6} + 11 T^{5} + 195 T^{4} + \cdots + 78961 \)
$97$
\( T^{6} - 7 T^{5} + 133 T^{4} + \cdots + 90601 \)
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