Newspace parameters
Level: | \( N \) | \(=\) | \( 927 = 3^{2} \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 927.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.40213226737\) |
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: | \( x^{16} - 34x^{14} + 564x^{12} - 5204x^{10} + 29740x^{8} - 99088x^{6} + 214644x^{4} - 222312x^{2} + 84100 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{7} \) |
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 \( x^{16} - 34x^{14} + 564x^{12} - 5204x^{10} + 29740x^{8} - 99088x^{6} + 214644x^{4} - 222312x^{2} + 84100 \) :
\(\beta_{1}\) | \(=\) | \( ( - 20114813547 \nu^{14} + 657518183386 \nu^{12} - 10224082165378 \nu^{10} + 82617743997017 \nu^{8} - 369286265262418 \nu^{6} + \cdots - 29\!\cdots\!16 ) / 19\!\cdots\!74 \) |
\(\beta_{2}\) | \(=\) | \( ( 31852621799 \nu^{14} - 927485431885 \nu^{12} + 13336321445016 \nu^{10} - 98606458894395 \nu^{8} + 433829284161384 \nu^{6} + \cdots + 44\!\cdots\!68 ) / 19\!\cdots\!74 \) |
\(\beta_{3}\) | \(=\) | \( ( 67192712276 \nu^{14} - 2088492923645 \nu^{12} + 31391992019638 \nu^{10} - 245522293476599 \nu^{8} + \cdots + 32\!\cdots\!40 ) / 19\!\cdots\!74 \) |
\(\beta_{4}\) | \(=\) | \( ( 161774372946 \nu^{14} - 5091711741364 \nu^{12} + 78990248483794 \nu^{10} - 659778233769221 \nu^{8} + \cdots - 15\!\cdots\!12 ) / 19\!\cdots\!74 \) |
\(\beta_{5}\) | \(=\) | \( ( - 73887296181 \nu^{14} + 2422756571858 \nu^{12} - 38744050645988 \nu^{10} + 338216353003786 \nu^{8} + \cdots + 63\!\cdots\!72 ) / 175709793597834 \) |
\(\beta_{6}\) | \(=\) | \( ( - 939520497824 \nu^{14} + 31142130627676 \nu^{12} - 503664966944388 \nu^{10} + \cdots + 82\!\cdots\!38 ) / 19\!\cdots\!74 \) |
\(\beta_{7}\) | \(=\) | \( ( - 1612023264355 \nu^{14} + 53278766175273 \nu^{12} - 858756502358368 \nu^{10} + \cdots + 14\!\cdots\!86 ) / 19\!\cdots\!74 \) |
\(\beta_{8}\) | \(=\) | \( ( 3564940540838 \nu^{15} - 67038268505007 \nu^{13} + 186895041080732 \nu^{11} + \cdots + 34\!\cdots\!04 \nu ) / 56\!\cdots\!60 \) |
\(\beta_{9}\) | \(=\) | \( ( - 6816403149681 \nu^{15} + 221102481148514 \nu^{13} + \cdots + 79\!\cdots\!12 \nu ) / 56\!\cdots\!60 \) |
\(\beta_{10}\) | \(=\) | \( ( - 6816403149681 \nu^{15} + 221102481148514 \nu^{13} + \cdots + 19\!\cdots\!32 \nu ) / 56\!\cdots\!60 \) |
\(\beta_{11}\) | \(=\) | \( ( 2096532504071 \nu^{15} - 50836045660766 \nu^{13} + 526979880771778 \nu^{11} - 732288998761202 \nu^{9} + \cdots + 77\!\cdots\!04 \nu ) / 11\!\cdots\!92 \) |
\(\beta_{12}\) | \(=\) | \( ( 4109972533687 \nu^{15} - 138855224047101 \nu^{13} + \cdots - 12\!\cdots\!88 \nu ) / 11\!\cdots\!92 \) |
\(\beta_{13}\) | \(=\) | \( ( - 21754241947832 \nu^{15} + 729304952533983 \nu^{13} + \cdots + 17\!\cdots\!64 \nu ) / 56\!\cdots\!60 \) |
\(\beta_{14}\) | \(=\) | \( ( 11911187 \nu^{15} - 394390138 \nu^{13} + 6361271963 \nu^{11} - 56162895403 \nu^{9} + 301920423430 \nu^{7} - 893755136446 \nu^{5} + \cdots - 993728799594 \nu ) / 88294348010 \) |
\(\beta_{15}\) | \(=\) | \( ( - 144495421804111 \nu^{15} + \cdots + 12\!\cdots\!92 \nu ) / 56\!\cdots\!60 \) |
\(\nu\) | \(=\) | \( ( \beta_{10} - \beta_{9} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 3\beta _1 + 10 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 8\beta_{15} + 11\beta_{14} - 8\beta_{13} + \beta_{12} + \beta_{11} + 4\beta_{10} - 22\beta_{9} - \beta_{8} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( -10\beta_{7} + 10\beta_{6} + 9\beta_{5} + 3\beta_{4} + 4\beta_{3} - 2\beta_{2} + 5\beta _1 + 6 \) |
\(\nu^{5}\) | \(=\) | \( ( 133\beta_{15} + 205\beta_{14} - 93\beta_{13} - 7\beta_{12} - 27\beta_{10} - 241\beta_{9} + 3\beta_{8} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -103\beta_{7} + 114\beta_{6} + 74\beta_{5} + 4\beta_{4} - 115\beta_{3} + 146\beta_{2} - 181\beta _1 - 417 \) |
\(\nu^{7}\) | \(=\) | \( 526 \beta_{15} + 880 \beta_{14} - 199 \beta_{13} - 110 \beta_{12} - 130 \beta_{11} - 431 \beta_{10} - 757 \beta_{9} + 185 \beta_{8} \) |
\(\nu^{8}\) | \(=\) | \( -418\beta_{7} + 544\beta_{6} + 180\beta_{5} - 92\beta_{4} - 2464\beta_{3} + 2454\beta_{2} - 4268\beta _1 - 7962 \) |
\(\nu^{9}\) | \(=\) | \( - 442 \beta_{15} - 315 \beta_{14} + 1244 \beta_{13} - 1317 \beta_{12} - 2457 \beta_{11} - 5687 \beta_{10} + 1671 \beta_{9} + 3495 \beta_{8} \) |
\(\nu^{10}\) | \(=\) | \( 6952 \beta_{7} - 7640 \beta_{6} - 5099 \beta_{5} - 406 \beta_{4} - 27249 \beta_{3} + 23475 \beta_{2} - 51378 \beta _1 - 85946 \) |
\(\nu^{11}\) | \(=\) | \( - 84667 \beta_{15} - 140704 \beta_{14} + 33297 \beta_{13} - 9188 \beta_{12} - 23959 \beta_{11} - 44459 \beta_{10} + 125659 \beta_{9} + 34550 \beta_{8} \) |
\(\nu^{12}\) | \(=\) | \( 183964 \beta_{7} - 222330 \beta_{6} - 105600 \beta_{5} + 13846 \beta_{4} - 147172 \beta_{3} + 106182 \beta_{2} - 303754 \beta _1 - 456678 \) |
\(\nu^{13}\) | \(=\) | \( - 1470719 \beta_{15} - 2522011 \beta_{14} + 373873 \beta_{13} - 291 \beta_{12} - 54284 \beta_{11} - 39329 \beta_{10} + 2003829 \beta_{9} + 81133 \beta_{8} \) |
\(\nu^{14}\) | \(=\) | \( 2337014 \beta_{7} - 2925858 \beta_{6} - 1195382 \beta_{5} + 293856 \beta_{4} + 1136828 \beta_{3} - 1090018 \beta_{2} + 2009156 \beta _1 + 3636584 \) |
\(\nu^{15}\) | \(=\) | \( - 14469820 \beta_{15} - 25228700 \beta_{14} + 2580414 \beta_{13} + 1142700 \beta_{12} + 2637840 \beta_{11} + 5298572 \beta_{10} + 18771308 \beta_{9} - 3792314 \beta_{8} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).
\(n\) | \(722\) | \(829\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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926.1 |
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− | 2.10850i | 0 | −2.44579 | −3.56582 | 0 | 0.463913 | 0.939944i | 0 | 7.51854i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.2 | − | 2.10850i | 0 | −2.44579 | 3.56582 | 0 | 0.463913 | 0.939944i | 0 | − | 7.51854i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.3 | − | 1.67237i | 0 | −0.796815 | −3.92802 | 0 | 4.16190 | − | 2.01217i | 0 | 6.56910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.4 | − | 1.67237i | 0 | −0.796815 | 3.92802 | 0 | 4.16190 | − | 2.01217i | 0 | − | 6.56910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.5 | − | 0.793520i | 0 | 1.37033 | −2.13753 | 0 | 0.751881 | − | 2.67442i | 0 | 1.69617i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.6 | − | 0.793520i | 0 | 1.37033 | 2.13753 | 0 | 0.751881 | − | 2.67442i | 0 | − | 1.69617i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.7 | − | 0.357385i | 0 | 1.87228 | −3.35955 | 0 | −1.37769 | − | 1.38389i | 0 | 1.20065i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.8 | − | 0.357385i | 0 | 1.87228 | 3.35955 | 0 | −1.37769 | − | 1.38389i | 0 | − | 1.20065i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.9 | 0.357385i | 0 | 1.87228 | −3.35955 | 0 | −1.37769 | 1.38389i | 0 | − | 1.20065i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.10 | 0.357385i | 0 | 1.87228 | 3.35955 | 0 | −1.37769 | 1.38389i | 0 | 1.20065i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.11 | 0.793520i | 0 | 1.37033 | −2.13753 | 0 | 0.751881 | 2.67442i | 0 | − | 1.69617i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.12 | 0.793520i | 0 | 1.37033 | 2.13753 | 0 | 0.751881 | 2.67442i | 0 | 1.69617i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.13 | 1.67237i | 0 | −0.796815 | −3.92802 | 0 | 4.16190 | 2.01217i | 0 | − | 6.56910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.14 | 1.67237i | 0 | −0.796815 | 3.92802 | 0 | 4.16190 | 2.01217i | 0 | 6.56910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.15 | 2.10850i | 0 | −2.44579 | −3.56582 | 0 | 0.463913 | − | 0.939944i | 0 | − | 7.51854i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
926.16 | 2.10850i | 0 | −2.44579 | 3.56582 | 0 | 0.463913 | − | 0.939944i | 0 | 7.51854i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
103.b | odd | 2 | 1 | inner |
309.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 927.2.c.a | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 927.2.c.a | ✓ | 16 |
103.b | odd | 2 | 1 | inner | 927.2.c.a | ✓ | 16 |
309.c | even | 2 | 1 | inner | 927.2.c.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
927.2.c.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
927.2.c.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
927.2.c.a | ✓ | 16 | 103.b | odd | 2 | 1 | inner |
927.2.c.a | ✓ | 16 | 309.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 8T_{2}^{6} + 18T_{2}^{4} + 10T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} + 8 T^{6} + 18 T^{4} + 10 T^{2} + \cdots + 1)^{2} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 44 T^{6} + 694 T^{4} + \cdots + 10117)^{2} \)
$7$
\( (T^{4} - 4 T^{3} - 2 T^{2} + 6 T - 2)^{4} \)
$11$
\( (T^{8} - 82 T^{6} + 2072 T^{4} + \cdots + 40468)^{2} \)
$13$
\( (T^{4} + 2 T^{3} - 8 T^{2} - 18 T - 5)^{4} \)
$17$
\( (T^{8} + 24 T^{6} + 184 T^{4} + 512 T^{2} + \cdots + 400)^{2} \)
$19$
\( (T^{4} - 2 T^{3} - 24 T^{2} + 24 T - 4)^{4} \)
$23$
\( (T^{8} + 32 T^{6} + 334 T^{4} + 1226 T^{2} + \cdots + 841)^{2} \)
$29$
\( (T^{8} + 130 T^{6} + 4540 T^{4} + \cdots + 676)^{2} \)
$31$
\( (T^{8} + 128 T^{6} + 5410 T^{4} + \cdots + 495733)^{2} \)
$37$
\( (T^{8} + 120 T^{6} + 4204 T^{4} + \cdots + 40468)^{2} \)
$41$
\( (T^{8} + 166 T^{6} + 8464 T^{4} + \cdots + 448900)^{2} \)
$43$
\( (T^{8} + 252 T^{6} + 18962 T^{4} + \cdots + 1709773)^{2} \)
$47$
\( (T^{8} - 82 T^{6} + 2072 T^{4} + \cdots + 40468)^{2} \)
$53$
\( (T^{8} - 166 T^{6} + 8820 T^{4} + \cdots + 1011700)^{2} \)
$59$
\( (T^{8} + 248 T^{6} + 18622 T^{4} + \cdots + 5004169)^{2} \)
$61$
\( (T^{4} + 8 T^{3} - 48 T^{2} - 94 T - 29)^{4} \)
$67$
\( (T^{8} + 208 T^{6} + 13418 T^{4} + \cdots + 1709773)^{2} \)
$71$
\( (T^{8} - 208 T^{6} + 5832 T^{4} + \cdots + 161872)^{2} \)
$73$
\( (T^{8} + 388 T^{6} + 42236 T^{4} + \cdots + 1982932)^{2} \)
$79$
\( (T^{4} + 10 T^{3} - 48 T^{2} + 42 T + 2)^{4} \)
$83$
\( (T^{8} + 298 T^{6} + 29134 T^{4} + \cdots + 5004169)^{2} \)
$89$
\( (T^{8} - 470 T^{6} + 70652 T^{4} + \cdots + 89393812)^{2} \)
$97$
\( (T^{4} - 4 T^{3} - 242 T^{2} + 1548 T + 2617)^{4} \)
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