Newspace parameters
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.o (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.7442857984.4 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( 2\nu^{7} + 3\nu^{6} + 52\nu^{5} + 60\nu^{4} + 374\nu^{3} + 219\nu^{2} + 612\nu - 162 ) / 432 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{7} + 3\nu^{6} - 8\nu^{5} + 24\nu^{4} + 155\nu^{3} - 249\nu^{2} + 774\nu - 810 ) / 432 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{7} - 3\nu^{6} + 52\nu^{5} - 60\nu^{4} + 374\nu^{3} - 219\nu^{2} + 612\nu + 162 ) / 432 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} - 3\nu^{6} - 8\nu^{5} - 24\nu^{4} + 155\nu^{3} + 249\nu^{2} + 1206\nu + 810 ) / 432 \) |
\(\beta_{6}\) | \(=\) | \( ( -5\nu^{7} - 112\nu^{5} - 665\nu^{3} - 954\nu ) / 432 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{6} + 20\nu^{4} + 97\nu^{2} + 114 ) / 24 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} + 3\beta_{4} - 3\beta_{2} - 7 \) |
\(\nu^{3}\) | \(=\) | \( -6\beta_{6} + 3\beta_{5} - 6\beta_{4} + 3\beta_{3} - 6\beta_{2} - 10\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( -13\beta_{7} + 6\beta_{5} - 45\beta_{4} - 6\beta_{3} + 45\beta_{2} - 6\beta _1 + 73 \) |
\(\nu^{5}\) | \(=\) | \( 114\beta_{6} - 45\beta_{5} + 120\beta_{4} - 45\beta_{3} + 120\beta_{2} + 118\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 187\beta_{7} - 120\beta_{5} + 609\beta_{4} + 120\beta_{3} - 609\beta_{2} + 120\beta _1 - 895 \) |
\(\nu^{7}\) | \(=\) | \( -1842\beta_{6} + 609\beta_{5} - 1890\beta_{4} + 609\beta_{3} - 1890\beta_{2} - 1504\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
\(n\) | \(157\) | \(365\) | \(379\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
265.1 |
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−0.707107 | + | 0.707107i | − | 1.00000i | − | 1.00000i | −0.0951965 | − | 0.0951965i | 0.707107 | + | 0.707107i | −0.0951965 | − | 2.64404i | 0.707107 | + | 0.707107i | −1.00000 | 0.134628 | ||||||||||||||||||||||||||||||
265.2 | −0.707107 | + | 0.707107i | − | 1.00000i | − | 1.00000i | 1.80230 | + | 1.80230i | 0.707107 | + | 0.707107i | 1.80230 | + | 1.93693i | 0.707107 | + | 0.707107i | −1.00000 | −2.54884 | |||||||||||||||||||||||||||||||
265.3 | 0.707107 | − | 0.707107i | − | 1.00000i | − | 1.00000i | −2.27220 | − | 2.27220i | −0.707107 | − | 0.707107i | −2.27220 | + | 1.35539i | −0.707107 | − | 0.707107i | −1.00000 | −3.21338 | |||||||||||||||||||||||||||||||
265.4 | 0.707107 | − | 0.707107i | − | 1.00000i | − | 1.00000i | 2.56510 | + | 2.56510i | −0.707107 | − | 0.707107i | 2.56510 | − | 0.648285i | −0.707107 | − | 0.707107i | −1.00000 | 3.62760 | |||||||||||||||||||||||||||||||
307.1 | −0.707107 | − | 0.707107i | 1.00000i | 1.00000i | −0.0951965 | + | 0.0951965i | 0.707107 | − | 0.707107i | −0.0951965 | + | 2.64404i | 0.707107 | − | 0.707107i | −1.00000 | 0.134628 | |||||||||||||||||||||||||||||||||
307.2 | −0.707107 | − | 0.707107i | 1.00000i | 1.00000i | 1.80230 | − | 1.80230i | 0.707107 | − | 0.707107i | 1.80230 | − | 1.93693i | 0.707107 | − | 0.707107i | −1.00000 | −2.54884 | |||||||||||||||||||||||||||||||||
307.3 | 0.707107 | + | 0.707107i | 1.00000i | 1.00000i | −2.27220 | + | 2.27220i | −0.707107 | + | 0.707107i | −2.27220 | − | 1.35539i | −0.707107 | + | 0.707107i | −1.00000 | −3.21338 | |||||||||||||||||||||||||||||||||
307.4 | 0.707107 | + | 0.707107i | 1.00000i | 1.00000i | 2.56510 | − | 2.56510i | −0.707107 | + | 0.707107i | 2.56510 | + | 0.648285i | −0.707107 | + | 0.707107i | −1.00000 | 3.62760 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.i | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.o.d | yes | 8 |
3.b | odd | 2 | 1 | 1638.2.x.b | 8 | ||
7.b | odd | 2 | 1 | 546.2.o.a | ✓ | 8 | |
13.d | odd | 4 | 1 | 546.2.o.a | ✓ | 8 | |
21.c | even | 2 | 1 | 1638.2.x.d | 8 | ||
39.f | even | 4 | 1 | 1638.2.x.d | 8 | ||
91.i | even | 4 | 1 | inner | 546.2.o.d | yes | 8 |
273.o | odd | 4 | 1 | 1638.2.x.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.o.a | ✓ | 8 | 7.b | odd | 2 | 1 | |
546.2.o.a | ✓ | 8 | 13.d | odd | 4 | 1 | |
546.2.o.d | yes | 8 | 1.a | even | 1 | 1 | trivial |
546.2.o.d | yes | 8 | 91.i | even | 4 | 1 | inner |
1638.2.x.b | 8 | 3.b | odd | 2 | 1 | ||
1638.2.x.b | 8 | 273.o | odd | 4 | 1 | ||
1638.2.x.d | 8 | 21.c | even | 2 | 1 | ||
1638.2.x.d | 8 | 39.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{6} + 4T_{5}^{5} + 113T_{5}^{4} - 424T_{5}^{3} + 800T_{5}^{2} + 160T_{5} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 1)^{2} \)
$3$
\( (T^{2} + 1)^{4} \)
$5$
\( T^{8} - 4 T^{7} + 8 T^{6} + 4 T^{5} + \cdots + 16 \)
$7$
\( T^{8} - 4 T^{7} + 6 T^{6} - 4 T^{5} + \cdots + 2401 \)
$11$
\( T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 64 \)
$13$
\( (T^{2} + 4 T + 13)^{4} \)
$17$
\( (T^{4} - 6 T^{3} - 35 T^{2} + 204 T - 92)^{2} \)
$19$
\( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 777924 \)
$23$
\( T^{8} + 102 T^{6} + 3265 T^{4} + \cdots + 135424 \)
$29$
\( (T^{4} + 6 T^{3} - 67 T^{2} - 260 T + 356)^{2} \)
$31$
\( T^{8} + 20 T^{7} + 200 T^{6} + \cdots + 5184 \)
$37$
\( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 777924 \)
$41$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 3936256 \)
$43$
\( T^{8} + 254 T^{6} + \cdots + 14961424 \)
$47$
\( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 65536 \)
$53$
\( (T^{4} + 12 T^{3} - 4 T^{2} - 128 T + 128)^{2} \)
$59$
\( T^{8} + 28 T^{7} + 392 T^{6} + \cdots + 73984 \)
$61$
\( T^{8} + 106 T^{6} + 3489 T^{4} + \cdots + 8464 \)
$67$
\( T^{8} - 8 T^{7} + 32 T^{6} + 800 T^{5} + \cdots + 16 \)
$71$
\( T^{8} - 52 T^{7} + 1352 T^{6} + \cdots + 1327104 \)
$73$
\( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 777924 \)
$79$
\( (T^{4} + 24 T^{3} - 2 T^{2} - 3480 T - 20744)^{2} \)
$83$
\( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 541696 \)
$89$
\( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 45050944 \)
$97$
\( T^{8} - 36 T^{7} + \cdots + 485409024 \)
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