Newspace parameters
Level: | \( N \) | \(=\) | \( 440 = 2^{3} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 440.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.51341768894\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.192526503153664.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 2x^{11} - x^{10} + 2x^{9} + 6x^{8} - 20x^{6} + 24x^{4} + 16x^{3} - 16x^{2} - 64x + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
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 \( x^{12} - 2x^{11} - x^{10} + 2x^{9} + 6x^{8} - 20x^{6} + 24x^{4} + 16x^{3} - 16x^{2} - 64x + 64 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{11} - 2\nu^{10} - \nu^{9} + 2\nu^{8} + 6\nu^{7} - 20\nu^{5} + 24\nu^{3} + 16\nu^{2} - 16\nu - 64 ) / 32 \) |
\(\beta_{4}\) | \(=\) | \( ( - 3 \nu^{11} + 7 \nu^{9} + 8 \nu^{8} - 14 \nu^{7} - 28 \nu^{6} + 20 \nu^{5} + 56 \nu^{4} + 24 \nu^{3} - 64 \nu^{2} - 80 \nu + 96 ) / 32 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{11} + \nu^{10} + 3\nu^{9} - \nu^{8} - 8\nu^{7} - 6\nu^{6} + 20\nu^{5} + 20\nu^{4} - 24\nu^{3} - 40\nu^{2} + 80 ) / 16 \) |
\(\beta_{6}\) | \(=\) | \( ( - \nu^{11} - 4 \nu^{10} + \nu^{9} + 12 \nu^{8} + 18 \nu^{7} - 12 \nu^{6} - 60 \nu^{5} - 8 \nu^{4} + 88 \nu^{3} + 96 \nu^{2} - 48 \nu - 192 ) / 32 \) |
\(\beta_{7}\) | \(=\) | \( ( 3 \nu^{10} + 2 \nu^{9} - 7 \nu^{8} - 14 \nu^{7} + 6 \nu^{6} + 36 \nu^{5} + 12 \nu^{4} - 48 \nu^{3} - 56 \nu^{2} + 16 \nu + 96 ) / 16 \) |
\(\beta_{8}\) | \(=\) | \( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} + 4 \nu^{8} - 30 \nu^{7} - 44 \nu^{6} + 52 \nu^{5} + 88 \nu^{4} - 8 \nu^{3} - 128 \nu^{2} - 112 \nu + 192 ) / 32 \) |
\(\beta_{9}\) | \(=\) | \( ( 3 \nu^{11} - \nu^{10} - 7 \nu^{9} - 7 \nu^{8} + 14 \nu^{7} + 30 \nu^{6} - 16 \nu^{5} - 52 \nu^{4} - 16 \nu^{3} + 56 \nu^{2} + 80 \nu - 80 ) / 16 \) |
\(\beta_{10}\) | \(=\) | \( ( - 3 \nu^{11} + 3 \nu^{10} + 9 \nu^{9} + 5 \nu^{8} - 20 \nu^{7} - 34 \nu^{6} + 32 \nu^{5} + 68 \nu^{4} + 8 \nu^{3} - 88 \nu^{2} - 96 \nu + 112 ) / 16 \) |
\(\beta_{11}\) | \(=\) | \( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 22 \nu^{7} - 24 \nu^{6} + 44 \nu^{5} + 56 \nu^{4} - 24 \nu^{3} - 96 \nu^{2} - 64 \nu + 160 ) / 16 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta _1 + 1 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 1 \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{11} + \beta_{10} + 2\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 - 3 \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{11} + \beta_{9} - 4\beta_{8} + 2\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{2} - \beta _1 + 1 \) |
\(\nu^{7}\) | \(=\) | \( \beta_{11} + 3 \beta_{10} + 6 \beta_{9} - \beta_{8} - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta _1 - 1 \) |
\(\nu^{8}\) | \(=\) | \( - 2 \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - 8 \beta_{3} - 3 \beta_{2} - 3 \beta _1 - 1 \) |
\(\nu^{9}\) | \(=\) | \( - 9 \beta_{11} + 5 \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 5 \beta _1 + 9 \) |
\(\nu^{10}\) | \(=\) | \( 6 \beta_{11} + 8 \beta_{10} + \beta_{9} - 16 \beta_{8} + 2 \beta_{7} + 7 \beta_{6} + \beta_{5} - 5 \beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 3 \beta _1 + 9 \) |
\(\nu^{11}\) | \(=\) | \( - 19 \beta_{11} + 15 \beta_{10} - 6 \beta_{9} - \beta_{8} + \beta_{7} - 11 \beta_{6} - 10 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta _1 + 15 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).
\(n\) | \(111\) | \(177\) | \(221\) | \(321\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
221.1 |
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−1.12509 | − | 0.856842i | − | 2.25017i | 0.531643 | + | 1.92804i | 1.00000i | −1.92804 | + | 2.53164i | −4.09451 | 1.05388 | − | 2.62475i | −2.06329 | 0.856842 | − | 1.12509i | |||||||||||||||||||||||||||||||||||||||||||
221.2 | −1.12509 | + | 0.856842i | 2.25017i | 0.531643 | − | 1.92804i | − | 1.00000i | −1.92804 | − | 2.53164i | −4.09451 | 1.05388 | + | 2.62475i | −2.06329 | 0.856842 | + | 1.12509i | ||||||||||||||||||||||||||||||||||||||||||||
221.3 | −0.850428 | − | 1.12994i | 1.70086i | −0.553545 | + | 1.92187i | − | 1.00000i | 1.92187 | − | 1.44646i | 1.51507 | 2.64236 | − | 1.00894i | 0.107089 | −1.12994 | + | 0.850428i | ||||||||||||||||||||||||||||||||||||||||||||
221.4 | −0.850428 | + | 1.12994i | − | 1.70086i | −0.553545 | − | 1.92187i | 1.00000i | 1.92187 | + | 1.44646i | 1.51507 | 2.64236 | + | 1.00894i | 0.107089 | −1.12994 | − | 0.850428i | ||||||||||||||||||||||||||||||||||||||||||||
221.5 | −0.324093 | − | 1.37658i | − | 0.648186i | −1.78993 | + | 0.892278i | 1.00000i | −0.892278 | + | 0.210073i | 1.46292 | 1.80839 | + | 2.17479i | 2.57985 | 1.37658 | − | 0.324093i | ||||||||||||||||||||||||||||||||||||||||||||
221.6 | −0.324093 | + | 1.37658i | 0.648186i | −1.78993 | − | 0.892278i | − | 1.00000i | −0.892278 | − | 0.210073i | 1.46292 | 1.80839 | − | 2.17479i | 2.57985 | 1.37658 | + | 0.324093i | ||||||||||||||||||||||||||||||||||||||||||||
221.7 | 0.195848 | − | 1.40059i | − | 0.391695i | −1.92329 | − | 0.548603i | − | 1.00000i | −0.548603 | − | 0.0767126i | 5.05421 | −1.14504 | + | 2.58629i | 2.84657 | −1.40059 | − | 0.195848i | |||||||||||||||||||||||||||||||||||||||||||
221.8 | 0.195848 | + | 1.40059i | 0.391695i | −1.92329 | + | 0.548603i | 1.00000i | −0.548603 | + | 0.0767126i | 5.05421 | −1.14504 | − | 2.58629i | 2.84657 | −1.40059 | + | 0.195848i | |||||||||||||||||||||||||||||||||||||||||||||
221.9 | 0.773803 | − | 1.18373i | − | 1.54761i | −0.802457 | − | 1.83196i | − | 1.00000i | −1.83196 | − | 1.19754i | −3.13415 | −2.78949 | − | 0.467677i | 0.604914 | −1.18373 | − | 0.773803i | |||||||||||||||||||||||||||||||||||||||||||
221.10 | 0.773803 | + | 1.18373i | 1.54761i | −0.802457 | + | 1.83196i | 1.00000i | −1.83196 | + | 1.19754i | −3.13415 | −2.78949 | + | 0.467677i | 0.604914 | −1.18373 | + | 0.773803i | |||||||||||||||||||||||||||||||||||||||||||||
221.11 | 1.32996 | − | 0.480846i | 2.65991i | 1.53757 | − | 1.27901i | 1.00000i | 1.27901 | + | 3.53757i | 1.19647 | 1.42990 | − | 2.44037i | −4.07515 | 0.480846 | + | 1.32996i | |||||||||||||||||||||||||||||||||||||||||||||
221.12 | 1.32996 | + | 0.480846i | − | 2.65991i | 1.53757 | + | 1.27901i | − | 1.00000i | 1.27901 | − | 3.53757i | 1.19647 | 1.42990 | + | 2.44037i | −4.07515 | 0.480846 | − | 1.32996i | |||||||||||||||||||||||||||||||||||||||||||
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 | 440.2.g.b | ✓ | 12 |
4.b | odd | 2 | 1 | 1760.2.g.b | 12 | ||
8.b | even | 2 | 1 | inner | 440.2.g.b | ✓ | 12 |
8.d | odd | 2 | 1 | 1760.2.g.b | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
440.2.g.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
440.2.g.b | ✓ | 12 | 8.b | even | 2 | 1 | inner |
1760.2.g.b | 12 | 4.b | odd | 2 | 1 | ||
1760.2.g.b | 12 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 18T_{3}^{10} + 117T_{3}^{8} + 336T_{3}^{6} + 412T_{3}^{4} + 160T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 3 T^{10} - 2 T^{9} + 6 T^{8} + \cdots + 64 \)
$3$
\( T^{12} + 18 T^{10} + 117 T^{8} + \cdots + 16 \)
$5$
\( (T^{2} + 1)^{6} \)
$7$
\( (T^{6} - 2 T^{5} - 27 T^{4} + 44 T^{3} + \cdots + 172)^{2} \)
$11$
\( (T^{2} + 1)^{6} \)
$13$
\( T^{12} + 80 T^{10} + 2460 T^{8} + \cdots + 565504 \)
$17$
\( (T^{6} + 8 T^{5} - 5 T^{4} - 110 T^{3} + \cdots + 124)^{2} \)
$19$
\( T^{12} + 90 T^{10} + 1753 T^{8} + \cdots + 102400 \)
$23$
\( (T^{6} - 6 T^{5} - 58 T^{4} + 228 T^{3} + \cdots - 64)^{2} \)
$29$
\( T^{12} + 122 T^{10} + 4281 T^{8} + \cdots + 4096 \)
$31$
\( (T^{6} - 6 T^{5} - 95 T^{4} + 416 T^{3} + \cdots - 752)^{2} \)
$37$
\( T^{12} + 138 T^{10} + \cdots + 24127744 \)
$41$
\( (T^{6} - 32 T^{5} + 332 T^{4} - 1008 T^{3} + \cdots - 6016)^{2} \)
$43$
\( T^{12} + 240 T^{10} + \cdots + 266864896 \)
$47$
\( (T^{6} - 2 T^{5} - 194 T^{4} + 740 T^{3} + \cdots + 67552)^{2} \)
$53$
\( T^{12} + 410 T^{10} + \cdots + 9231366400 \)
$59$
\( T^{12} + 600 T^{10} + \cdots + 365628227584 \)
$61$
\( T^{12} + 418 T^{10} + \cdots + 491065600 \)
$67$
\( T^{12} + 568 T^{10} + \cdots + 255437246464 \)
$71$
\( (T^{6} + 26 T^{5} + 77 T^{4} + \cdots + 181904)^{2} \)
$73$
\( (T^{6} + 14 T^{5} - 94 T^{4} - 1268 T^{3} + \cdots - 18064)^{2} \)
$79$
\( (T^{6} + 4 T^{5} - 120 T^{4} - 240 T^{3} + \cdots - 5888)^{2} \)
$83$
\( T^{12} + 416 T^{10} + \cdots + 10653542656 \)
$89$
\( (T^{6} - 22 T^{5} - 75 T^{4} + 3712 T^{3} + \cdots - 14576)^{2} \)
$97$
\( (T^{6} + 12 T^{5} - 132 T^{4} + \cdots - 33728)^{2} \)
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