Newspace parameters
Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 840.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.70743376979\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.3058043990573056.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{6} \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + \nu^{5} - 8 ) / 4 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{10} + \nu^{8} - 8\nu^{3} - 16 ) / 16 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{10} + \nu^{8} - 8\nu^{5} + 16\nu + 16 ) / 16 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{11} + \nu^{9} - 8\nu^{4} - 16\nu ) / 32 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{11} - \nu^{9} + 8\nu^{6} + 16\nu^{4} - 16\nu ) / 32 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} - \nu^{5} + 8\nu^{2} + 8 ) / 4 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{10} - \nu^{8} + 8\nu^{5} + 16\nu^{3} - 16 ) / 16 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 4\nu ) / 4 \) |
\(\beta_{9}\) | \(=\) | \( ( \nu^{11} + 5\nu^{9} + 4\nu^{7} - 8\nu^{6} - 16\nu^{4} - 32\nu^{2} + 16\nu ) / 32 \) |
\(\beta_{10}\) | \(=\) | \( ( \nu^{8} + \nu^{6} - 4\nu^{3} - 12\nu ) / 4 \) |
\(\beta_{11}\) | \(=\) | \( ( \nu^{11} - 3\nu^{9} - 12\nu^{7} - 8\nu^{6} - 8\nu^{5} + 16\nu^{4} + 64\nu^{2} + 16\nu + 64 ) / 32 \) |
\(\nu\) | \(=\) | \( ( -\beta_{10} + \beta_{8} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{10} - \beta_{8} + 2\beta_{7} + 2\beta_{3} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 2\beta_{11} + 2\beta_{9} - \beta_{6} + 4\beta_{5} + \beta_1 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{10} + \beta_{8} + 2\beta_{7} - 2\beta_{3} + 4\beta_{2} + 8 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( -2\beta_{11} - 4\beta_{10} - 2\beta_{9} + 4\beta_{8} + \beta_{6} + 4\beta_{5} + 8\beta_{4} - \beta_1 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( \beta_{10} - \beta_{8} - 2\beta_{7} + 2\beta_{3} - 4\beta_{2} + 8\beta _1 + 8 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 8 \beta_{7} - \beta_{6} - 4 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + \beta_1 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( -\beta_{10} + 16\beta_{9} + \beta_{8} + 2\beta_{7} + 8\beta_{6} + 16\beta_{5} - 2\beta_{3} + 4\beta_{2} - 8 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 8 \beta_{7} + \beta_{6} + 4 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} + 32 \beta_{2} - \beta _1 + 32 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 16 \beta_{11} - 15 \beta_{10} + 15 \beta_{8} - 2 \beta_{7} - 16 \beta_{6} + 16 \beta_{5} + 64 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 8 \beta _1 + 8 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
\(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
421.1 |
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−1.23149 | − | 0.695292i | − | 1.00000i | 1.03314 | + | 1.71249i | 1.00000i | −0.695292 | + | 1.23149i | 1.00000 | −0.0816198 | − | 2.82725i | −1.00000 | 0.695292 | − | 1.23149i | |||||||||||||||||||||||||||||||||||||||||||
421.2 | −1.23149 | + | 0.695292i | 1.00000i | 1.03314 | − | 1.71249i | − | 1.00000i | −0.695292 | − | 1.23149i | 1.00000 | −0.0816198 | + | 2.82725i | −1.00000 | 0.695292 | + | 1.23149i | ||||||||||||||||||||||||||||||||||||||||||||
421.3 | −1.11568 | − | 0.869059i | 1.00000i | 0.489471 | + | 1.93918i | − | 1.00000i | 0.869059 | − | 1.11568i | 1.00000 | 1.13917 | − | 2.58888i | −1.00000 | −0.869059 | + | 1.11568i | ||||||||||||||||||||||||||||||||||||||||||||
421.4 | −1.11568 | + | 0.869059i | − | 1.00000i | 0.489471 | − | 1.93918i | 1.00000i | 0.869059 | + | 1.11568i | 1.00000 | 1.13917 | + | 2.58888i | −1.00000 | −0.869059 | − | 1.11568i | ||||||||||||||||||||||||||||||||||||||||||||
421.5 | −0.210663 | − | 1.39844i | − | 1.00000i | −1.91124 | + | 0.589197i | 1.00000i | −1.39844 | + | 0.210663i | 1.00000 | 1.22658 | + | 2.54863i | −1.00000 | 1.39844 | − | 0.210663i | ||||||||||||||||||||||||||||||||||||||||||||
421.6 | −0.210663 | + | 1.39844i | 1.00000i | −1.91124 | − | 0.589197i | − | 1.00000i | −1.39844 | − | 0.210663i | 1.00000 | 1.22658 | − | 2.54863i | −1.00000 | 1.39844 | + | 0.210663i | ||||||||||||||||||||||||||||||||||||||||||||
421.7 | 0.422872 | − | 1.34951i | 1.00000i | −1.64236 | − | 1.14134i | − | 1.00000i | 1.34951 | + | 0.422872i | 1.00000 | −2.23476 | + | 1.73374i | −1.00000 | −1.34951 | − | 0.422872i | ||||||||||||||||||||||||||||||||||||||||||||
421.8 | 0.422872 | + | 1.34951i | − | 1.00000i | −1.64236 | + | 1.14134i | 1.00000i | 1.34951 | − | 0.422872i | 1.00000 | −2.23476 | − | 1.73374i | −1.00000 | −1.34951 | + | 0.422872i | ||||||||||||||||||||||||||||||||||||||||||||
421.9 | 0.723626 | − | 1.21506i | − | 1.00000i | −0.952732 | − | 1.75849i | 1.00000i | −1.21506 | − | 0.723626i | 1.00000 | −2.82609 | − | 0.114868i | −1.00000 | 1.21506 | + | 0.723626i | ||||||||||||||||||||||||||||||||||||||||||||
421.10 | 0.723626 | + | 1.21506i | 1.00000i | −0.952732 | + | 1.75849i | − | 1.00000i | −1.21506 | + | 0.723626i | 1.00000 | −2.82609 | + | 0.114868i | −1.00000 | 1.21506 | − | 0.723626i | ||||||||||||||||||||||||||||||||||||||||||||
421.11 | 1.41133 | − | 0.0902148i | 1.00000i | 1.98372 | − | 0.254646i | − | 1.00000i | 0.0902148 | + | 1.41133i | 1.00000 | 2.77672 | − | 0.538352i | −1.00000 | −0.0902148 | − | 1.41133i | ||||||||||||||||||||||||||||||||||||||||||||
421.12 | 1.41133 | + | 0.0902148i | − | 1.00000i | 1.98372 | + | 0.254646i | 1.00000i | 0.0902148 | − | 1.41133i | 1.00000 | 2.77672 | + | 0.538352i | −1.00000 | −0.0902148 | + | 1.41133i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 840.2.g.c | ✓ | 12 |
4.b | odd | 2 | 1 | 3360.2.g.b | 12 | ||
8.b | even | 2 | 1 | inner | 840.2.g.c | ✓ | 12 |
8.d | odd | 2 | 1 | 3360.2.g.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.g.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
840.2.g.c | ✓ | 12 | 8.b | even | 2 | 1 | inner |
3360.2.g.b | 12 | 4.b | odd | 2 | 1 | ||
3360.2.g.b | 12 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{12} + 72T_{11}^{10} + 1664T_{11}^{8} + 14336T_{11}^{6} + 55360T_{11}^{4} + 98304T_{11}^{2} + 65536 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + T^{10} - 8 T^{7} - 16 T^{5} + \cdots + 64 \)
$3$
\( (T^{2} + 1)^{6} \)
$5$
\( (T^{2} + 1)^{6} \)
$7$
\( (T - 1)^{12} \)
$11$
\( T^{12} + 72 T^{10} + 1664 T^{8} + \cdots + 65536 \)
$13$
\( T^{12} + 64 T^{10} + 1232 T^{8} + \cdots + 9216 \)
$17$
\( (T^{6} - 48 T^{4} + 64 T^{3} + 544 T^{2} + \cdots + 768)^{2} \)
$19$
\( T^{12} + 64 T^{10} + 1232 T^{8} + \cdots + 1024 \)
$23$
\( (T^{6} - 4 T^{5} - 52 T^{4} + 16 T^{3} + \cdots - 128)^{2} \)
$29$
\( T^{12} + 168 T^{10} + \cdots + 40144896 \)
$31$
\( (T^{6} + 20 T^{5} + 100 T^{4} - 136 T^{3} + \cdots - 1024)^{2} \)
$37$
\( T^{12} + 264 T^{10} + \cdots + 373571584 \)
$41$
\( (T^{6} - 156 T^{4} - 320 T^{3} + \cdots - 49408)^{2} \)
$43$
\( T^{12} + 352 T^{10} + \cdots + 967458816 \)
$47$
\( (T^{6} + 8 T^{5} - 116 T^{4} - 864 T^{3} + \cdots - 26176)^{2} \)
$53$
\( T^{12} + 464 T^{10} + \cdots + 94745764864 \)
$59$
\( T^{12} + 656 T^{10} + \cdots + 510504534016 \)
$61$
\( T^{12} + 160 T^{10} + 5472 T^{8} + \cdots + 16384 \)
$67$
\( T^{12} + 544 T^{10} + \cdots + 1882865664 \)
$71$
\( (T^{6} - 28 T^{5} + 252 T^{4} - 744 T^{3} + \cdots - 8768)^{2} \)
$73$
\( (T^{6} - 272 T^{4} - 1016 T^{3} + \cdots + 152416)^{2} \)
$79$
\( (T^{6} - 8 T^{5} - 56 T^{4} + 384 T^{3} + \cdots + 512)^{2} \)
$83$
\( T^{12} + 672 T^{10} + \cdots + 84934656 \)
$89$
\( (T^{6} - 44 T^{4} + 448 T^{2} + 256 T - 512)^{2} \)
$97$
\( (T^{6} - 400 T^{4} + 1320 T^{3} + \cdots - 652576)^{2} \)
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