Newspace parameters
Level: | \( N \) | \(=\) | \( 170 = 2 \cdot 5 \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 170.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.35745683436\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.5161984.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
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 \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \) |
\(\beta_{3}\) | \(=\) | \( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \) |
\(\beta_{4}\) | \(=\) | \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) |
\(\beta_{5}\) | \(=\) | \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{4} - 5\beta_{2} + 2 \) |
\(\nu^{4}\) | \(=\) | \( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \) |
\(\nu^{5}\) | \(=\) | \( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).
\(n\) | \(71\) | \(137\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
69.1 |
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− | 1.00000i | − | 2.62620i | −1.00000 | −2.19388 | + | 0.432320i | −2.62620 | − | 0.864641i | 1.00000i | −3.89692 | 0.432320 | + | 2.19388i | |||||||||||||||||||||||||||||
69.2 | − | 1.00000i | 0.484862i | −1.00000 | 1.80487 | + | 1.32001i | 0.484862 | − | 2.64002i | 1.00000i | 2.76491 | 1.32001 | − | 1.80487i | |||||||||||||||||||||||||||||||
69.3 | − | 1.00000i | 3.14134i | −1.00000 | 1.38900 | − | 1.75233i | 3.14134 | 3.50466i | 1.00000i | −6.86799 | −1.75233 | − | 1.38900i | ||||||||||||||||||||||||||||||||
69.4 | 1.00000i | − | 3.14134i | −1.00000 | 1.38900 | + | 1.75233i | 3.14134 | − | 3.50466i | − | 1.00000i | −6.86799 | −1.75233 | + | 1.38900i | ||||||||||||||||||||||||||||||
69.5 | 1.00000i | − | 0.484862i | −1.00000 | 1.80487 | − | 1.32001i | 0.484862 | 2.64002i | − | 1.00000i | 2.76491 | 1.32001 | + | 1.80487i | |||||||||||||||||||||||||||||||
69.6 | 1.00000i | 2.62620i | −1.00000 | −2.19388 | − | 0.432320i | −2.62620 | 0.864641i | − | 1.00000i | −3.89692 | 0.432320 | − | 2.19388i | ||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 170.2.c.b | ✓ | 6 |
3.b | odd | 2 | 1 | 1530.2.d.g | 6 | ||
4.b | odd | 2 | 1 | 1360.2.e.c | 6 | ||
5.b | even | 2 | 1 | inner | 170.2.c.b | ✓ | 6 |
5.c | odd | 4 | 1 | 850.2.a.p | 3 | ||
5.c | odd | 4 | 1 | 850.2.a.q | 3 | ||
15.d | odd | 2 | 1 | 1530.2.d.g | 6 | ||
15.e | even | 4 | 1 | 7650.2.a.dj | 3 | ||
15.e | even | 4 | 1 | 7650.2.a.do | 3 | ||
20.d | odd | 2 | 1 | 1360.2.e.c | 6 | ||
20.e | even | 4 | 1 | 6800.2.a.bk | 3 | ||
20.e | even | 4 | 1 | 6800.2.a.bp | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
170.2.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
170.2.c.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
850.2.a.p | 3 | 5.c | odd | 4 | 1 | ||
850.2.a.q | 3 | 5.c | odd | 4 | 1 | ||
1360.2.e.c | 6 | 4.b | odd | 2 | 1 | ||
1360.2.e.c | 6 | 20.d | odd | 2 | 1 | ||
1530.2.d.g | 6 | 3.b | odd | 2 | 1 | ||
1530.2.d.g | 6 | 15.d | odd | 2 | 1 | ||
6800.2.a.bk | 3 | 20.e | even | 4 | 1 | ||
6800.2.a.bp | 3 | 20.e | even | 4 | 1 | ||
7650.2.a.dj | 3 | 15.e | even | 4 | 1 | ||
7650.2.a.do | 3 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 17T_{3}^{4} + 72T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{3} \)
$3$
\( T^{6} + 17 T^{4} + 72 T^{2} + 16 \)
$5$
\( T^{6} - 2 T^{5} - 3 T^{4} + 24 T^{3} + \cdots + 125 \)
$7$
\( T^{6} + 20 T^{4} + 100 T^{2} + \cdots + 64 \)
$11$
\( (T - 2)^{6} \)
$13$
\( T^{6} + 17 T^{4} + 72 T^{2} + 16 \)
$17$
\( (T^{2} + 1)^{3} \)
$19$
\( (T^{3} + T^{2} - 24 T + 20)^{2} \)
$23$
\( T^{6} + 68 T^{4} + 676 T^{2} + \cdots + 1024 \)
$29$
\( (T^{3} + 17 T^{2} + 66 T - 50)^{2} \)
$31$
\( (T^{3} - 3 T^{2} - 46 T - 74)^{2} \)
$37$
\( T^{6} + 308 T^{4} + 31140 T^{2} + \cdots + 1032256 \)
$41$
\( (T^{3} - 16 T^{2} + 60 T + 16)^{2} \)
$43$
\( T^{6} + 72 T^{4} + 1040 T^{2} + \cdots + 1024 \)
$47$
\( T^{6} + 257 T^{4} + 20176 T^{2} + \cdots + 440896 \)
$53$
\( T^{6} + 265 T^{4} + 22040 T^{2} + \cdots + 583696 \)
$59$
\( (T^{3} + 9 T^{2} - 16 T - 160)^{2} \)
$61$
\( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \)
$67$
\( T^{6} + 132 T^{4} + 2816 T^{2} + \cdots + 4096 \)
$71$
\( (T^{3} + T^{2} - 118 T + 334)^{2} \)
$73$
\( T^{6} + 17 T^{4} + 72 T^{2} + 16 \)
$79$
\( (T^{3} + 4 T^{2} - 74 T - 160)^{2} \)
$83$
\( T^{6} + 328 T^{4} + 6416 T^{2} + \cdots + 16384 \)
$89$
\( (T^{3} - 9 T^{2} - 16 T + 160)^{2} \)
$97$
\( T^{6} + 89 T^{4} + 2200 T^{2} + \cdots + 10000 \)
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