Newspace parameters
Level: | \( N \) | \(=\) | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1386.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.0672657201\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{9}\cdot 3^{2} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 \) :
\(\beta_{1}\) | \(=\) | \( ( 1381965647 \nu^{15} - 16053899943 \nu^{13} + 91281494448 \nu^{11} + 1854613004958 \nu^{9} + 11898941379431 \nu^{7} + \cdots - 7903986405426 \nu ) / 8313550996212 \) |
\(\beta_{2}\) | \(=\) | \( ( 3221232346 \nu^{15} - 39189954813 \nu^{13} + 239020561026 \nu^{11} + 4127085846450 \nu^{9} + 25935569845462 \nu^{7} + \cdots - 2619108000162 \nu ) / 8313550996212 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3221232346 \nu^{15} + 39189954813 \nu^{13} - 239020561026 \nu^{11} - 4127085846450 \nu^{9} - 25935569845462 \nu^{7} + \cdots + 19246209992586 \nu ) / 8313550996212 \) |
\(\beta_{4}\) | \(=\) | \( ( 133007 \nu^{14} - 1743582 \nu^{12} + 11322414 \nu^{10} + 162109944 \nu^{8} + 903956675 \nu^{6} + 1249135962 \nu^{4} - 217432278 \nu^{2} + \cdots - 2021921136 ) / 481024764 \) |
\(\beta_{5}\) | \(=\) | \( ( - 12219569 \nu^{14} + 143946111 \nu^{12} - 844183530 \nu^{10} - 16075743516 \nu^{8} - 103907059937 \nu^{6} - 248202840507 \nu^{4} + \cdots + 22892859036 ) / 26476277058 \) |
\(\beta_{6}\) | \(=\) | \( ( 2097307895 \nu^{15} - 24781419255 \nu^{13} + 146288587464 \nu^{11} + 2746984641030 \nu^{9} + 17784897420335 \nu^{7} + \cdots - 2157576356394 \nu ) / 2771183665404 \) |
\(\beta_{7}\) | \(=\) | \( ( 19480188302 \nu^{15} + 13882041737 \nu^{14} - 234062963571 \nu^{13} - 161840906400 \nu^{12} + 1400972065902 \nu^{11} + \cdots - 26508114581148 ) / 16627101992424 \) |
\(\beta_{8}\) | \(=\) | \( ( 19480188302 \nu^{15} - 13882041737 \nu^{14} - 234062963571 \nu^{13} + 161840906400 \nu^{12} + 1400972065902 \nu^{11} + \cdots + 26508114581148 ) / 16627101992424 \) |
\(\beta_{9}\) | \(=\) | \( ( 39894432671 \nu^{15} + 384169521 \nu^{14} - 475048585479 \nu^{13} - 4764275946 \nu^{12} + 2827831403844 \nu^{11} + \cdots + 30719356487784 ) / 16627101992424 \) |
\(\beta_{10}\) | \(=\) | \( ( - 39894432671 \nu^{15} + 384169521 \nu^{14} + 475048585479 \nu^{13} - 4764275946 \nu^{12} - 2827831403844 \nu^{11} + \cdots + 30719356487784 ) / 16627101992424 \) |
\(\beta_{11}\) | \(=\) | \( ( 27614604307 \nu^{15} + 17650113884 \nu^{14} - 335527493352 \nu^{13} - 212774422656 \nu^{12} + 2043949302306 \nu^{11} + \cdots - 31653237281040 ) / 16627101992424 \) |
\(\beta_{12}\) | \(=\) | \( ( - 27614604307 \nu^{15} + 17650113884 \nu^{14} + 335527493352 \nu^{13} - 212774422656 \nu^{12} - 2043949302306 \nu^{11} + \cdots - 31653237281040 ) / 16627101992424 \) |
\(\beta_{13}\) | \(=\) | \( ( 66695 \nu^{14} - 802932 \nu^{12} + 4850514 \nu^{10} + 86012388 \nu^{8} + 548291507 \nu^{6} + 1249477428 \nu^{4} + 1468931886 \nu^{2} + \cdots - 120049236 ) / 32561172 \) |
\(\beta_{14}\) | \(=\) | \( ( - 54470935301 \nu^{15} - 1838672364 \nu^{14} + 645220229322 \nu^{13} + 23712216120 \nu^{12} - 3821988100914 \nu^{11} + \cdots + 19609438426344 ) / 16627101992424 \) |
\(\beta_{15}\) | \(=\) | \( ( - 54470935301 \nu^{15} + 1838672364 \nu^{14} + 645220229322 \nu^{13} - 23712216120 \nu^{12} - 3821988100914 \nu^{11} + \cdots - 36236540418768 ) / 16627101992424 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 4 \beta_{5} - \beta_{4} + 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{15} + \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 13 \beta_{6} + 4 \beta_{3} + 4 \beta_{2} - 5 \beta _1 + 1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 8\beta_{13} - 17\beta_{12} - 17\beta_{11} + 28\beta_{8} - 28\beta_{7} - 144\beta_{5} ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 14 \beta_{15} + 14 \beta_{14} + 3 \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 17 \beta_{9} - 11 \beta_{8} - 11 \beta_{7} + 191 \beta_{6} - 74 \beta_{3} + 74 \beta_{2} + 43 \beta _1 + 14 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( - 202 \beta_{15} + 202 \beta_{14} + 117 \beta_{13} - 202 \beta_{12} - 202 \beta_{11} + 271 \beta_{10} + 271 \beta_{9} + 271 \beta_{8} - 271 \beta_{7} - 1390 \beta_{5} + 117 \beta_{4} - 1188 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 69 \beta_{15} + 69 \beta_{14} + 202 \beta_{12} - 202 \beta_{11} + 133 \beta_{10} - 133 \beta_{9} - 271 \beta_{8} - 271 \beta_{7} + 1168 \beta_{6} - 2869 \beta_{3} + 533 \beta_{2} + 1168 \beta _1 + 69 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( -4175\beta_{15} + 4175\beta_{14} + 6004\beta_{10} + 6004\beta_{9} + 2336\beta_{4} - 26389 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 1173 \beta_{15} - 1173 \beta_{14} + 3002 \beta_{12} - 3002 \beta_{11} - 1829 \beta_{10} + 1829 \beta_{9} - 4175 \beta_{8} - 4175 \beta_{7} - 17858 \beta_{6} - 43319 \beta_{3} - 7603 \beta_{2} + 17858 \beta _1 - 1173 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 45148 \beta_{15} + 45148 \beta_{14} - 25461 \beta_{13} + 45148 \beta_{12} + 45148 \beta_{11} + 63523 \beta_{10} + 63523 \beta_{9} - 63523 \beta_{8} + 63523 \beta_{7} + 324490 \beta_{5} + \cdots - 279342 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 45148 \beta_{15} - 45148 \beta_{14} + 18375 \beta_{12} - 18375 \beta_{11} - 63523 \beta_{10} + 63523 \beta_{9} - 26773 \beta_{8} - 26773 \beta_{7} - 654937 \beta_{6} - 270916 \beta_{3} + \cdots - 45148 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( - 541832 \beta_{13} + 962603 \beta_{12} + 962603 \beta_{11} - 1363420 \beta_{8} + 1363420 \beta_{7} + 6956172 \beta_{5} ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 681710 \beta_{15} - 681710 \beta_{14} - 280893 \beta_{12} + 280893 \beta_{11} - 962603 \beta_{10} + 962603 \beta_{9} + 400817 \beta_{8} + 400817 \beta_{7} - 9905483 \beta_{6} + \cdots - 681710 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( 10306300 \beta_{15} - 10306300 \beta_{14} - 5804109 \beta_{13} + 10306300 \beta_{12} + 10306300 \beta_{11} - 14568643 \beta_{10} - 14568643 \beta_{9} - 14568643 \beta_{8} + \cdots + 64051830 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( - 4262343 \beta_{15} - 4262343 \beta_{14} - 10306300 \beta_{12} + 10306300 \beta_{11} - 6043957 \beta_{10} + 6043957 \beta_{9} + 14568643 \beta_{8} + 14568643 \beta_{7} + \cdots - 4262343 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).
\(n\) | \(155\) | \(199\) | \(1135\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
881.1 |
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− | 1.00000i | 0 | −1.00000 | −3.90295 | 0 | 1.49520 | − | 2.18274i | 1.00000i | 0 | 3.90295i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.2 | − | 1.00000i | 0 | −1.00000 | −2.97672 | 0 | 2.55176 | − | 0.698947i | 1.00000i | 0 | 2.97672i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.3 | − | 1.00000i | 0 | −1.00000 | −2.79965 | 0 | −2.20231 | + | 1.46623i | 1.00000i | 0 | 2.79965i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.4 | − | 1.00000i | 0 | −1.00000 | −0.260874 | 0 | −1.84465 | + | 1.89664i | 1.00000i | 0 | 0.260874i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.5 | − | 1.00000i | 0 | −1.00000 | 0.260874 | 0 | −1.84465 | − | 1.89664i | 1.00000i | 0 | − | 0.260874i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.6 | − | 1.00000i | 0 | −1.00000 | 2.79965 | 0 | −2.20231 | − | 1.46623i | 1.00000i | 0 | − | 2.79965i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.7 | − | 1.00000i | 0 | −1.00000 | 2.97672 | 0 | 2.55176 | + | 0.698947i | 1.00000i | 0 | − | 2.97672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.8 | − | 1.00000i | 0 | −1.00000 | 3.90295 | 0 | 1.49520 | + | 2.18274i | 1.00000i | 0 | − | 3.90295i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.9 | 1.00000i | 0 | −1.00000 | −3.90295 | 0 | 1.49520 | + | 2.18274i | − | 1.00000i | 0 | − | 3.90295i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.10 | 1.00000i | 0 | −1.00000 | −2.97672 | 0 | 2.55176 | + | 0.698947i | − | 1.00000i | 0 | − | 2.97672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.11 | 1.00000i | 0 | −1.00000 | −2.79965 | 0 | −2.20231 | − | 1.46623i | − | 1.00000i | 0 | − | 2.79965i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.12 | 1.00000i | 0 | −1.00000 | −0.260874 | 0 | −1.84465 | − | 1.89664i | − | 1.00000i | 0 | − | 0.260874i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.13 | 1.00000i | 0 | −1.00000 | 0.260874 | 0 | −1.84465 | + | 1.89664i | − | 1.00000i | 0 | 0.260874i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.14 | 1.00000i | 0 | −1.00000 | 2.79965 | 0 | −2.20231 | + | 1.46623i | − | 1.00000i | 0 | 2.79965i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.15 | 1.00000i | 0 | −1.00000 | 2.97672 | 0 | 2.55176 | − | 0.698947i | − | 1.00000i | 0 | 2.97672i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
881.16 | 1.00000i | 0 | −1.00000 | 3.90295 | 0 | 1.49520 | − | 2.18274i | − | 1.00000i | 0 | 3.90295i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1386.2.g.b | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 1386.2.g.b | ✓ | 16 |
7.b | odd | 2 | 1 | inner | 1386.2.g.b | ✓ | 16 |
21.c | even | 2 | 1 | inner | 1386.2.g.b | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1386.2.g.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
1386.2.g.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
1386.2.g.b | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
1386.2.g.b | ✓ | 16 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 32T_{5}^{6} + 326T_{5}^{4} - 1080T_{5}^{2} + 72 \)
acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{8} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 32 T^{6} + 326 T^{4} - 1080 T^{2} + \cdots + 72)^{2} \)
$7$
\( (T^{8} - 6 T^{6} - 8 T^{5} + 66 T^{4} + \cdots + 2401)^{2} \)
$11$
\( (T^{2} + 1)^{8} \)
$13$
\( (T^{8} + 88 T^{6} + 2576 T^{4} + \cdots + 73728)^{2} \)
$17$
\( (T^{8} - 52 T^{6} + 674 T^{4} - 1488 T^{2} + \cdots + 288)^{2} \)
$19$
\( (T^{8} + 56 T^{6} + 230 T^{4} + 264 T^{2} + \cdots + 72)^{2} \)
$23$
\( T^{16} \)
$29$
\( (T^{8} + 188 T^{6} + 10420 T^{4} + \cdots + 254016)^{2} \)
$31$
\( (T^{8} + 92 T^{6} + 2834 T^{4} + \cdots + 73728)^{2} \)
$37$
\( (T^{4} + 4 T^{3} - 76 T^{2} - 224 T + 448)^{4} \)
$41$
\( (T^{8} - 116 T^{6} + 2018 T^{4} + \cdots + 288)^{2} \)
$43$
\( (T^{4} + 20 T^{3} + 10 T^{2} - 1680 T - 7792)^{4} \)
$47$
\( (T^{8} - 140 T^{6} + 4946 T^{4} + \cdots + 225792)^{2} \)
$53$
\( (T^{8} + 284 T^{6} + 26356 T^{4} + \cdots + 11451456)^{2} \)
$59$
\( (T^{8} - 416 T^{6} + 57440 T^{4} + \cdots + 40716288)^{2} \)
$61$
\( (T^{8} + 200 T^{6} + 5648 T^{4} + \cdots + 73728)^{2} \)
$67$
\( (T^{4} - 4 T^{3} - 86 T^{2} - 144 T + 144)^{4} \)
$71$
\( (T^{2} + 72)^{8} \)
$73$
\( (T^{8} + 484 T^{6} + 87266 T^{4} + \cdots + 206613792)^{2} \)
$79$
\( (T^{4} - 24 T^{3} + 68 T^{2} + 1040 T - 1564)^{4} \)
$83$
\( (T^{8} - 476 T^{6} + 55250 T^{4} + \cdots + 4608)^{2} \)
$89$
\( (T^{8} - 500 T^{6} + 64868 T^{4} + \cdots + 12221568)^{2} \)
$97$
\( (T^{8} + 256 T^{6} + 20864 T^{4} + \cdots + 294912)^{2} \)
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