Newspace parameters
Level: | \( N \) | \(=\) | \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 2070.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(56.4034147226\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
Defining polynomial: |
\( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 2^{7} \) |
Twist minimal: | no (minimal twist has level 230) |
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
\( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \)
:
\(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + \cdots + 76734676377 \nu ) / 1473983980 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 3655930587 \nu^{15} - 279912880683 \nu^{13} - 7519929098840 \nu^{11} - 93142815057018 \nu^{9} - 550737169153146 \nu^{7} + \cdots + 192163143584185 \nu ) / 13607820103360 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 2028139777 \nu^{15} - 159386627629 \nu^{13} - 4483504013264 \nu^{11} - 59956611391678 \nu^{9} - 407018725514790 \nu^{7} + \cdots - 890813001228721 \nu ) / 3401955025840 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + \cdots + 885659459 ) / 505359680 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + \cdots - 93749086185413 \nu ) / 485993575120 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8504894308903 \nu ) / 114351429440 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} + \cdots - 4166619657821 \nu ) / 121498393780 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + \cdots - 44834433467221 ) / 6803910051680 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 45221309975 \nu^{15} - 21002456926 \nu^{14} + 3479923444575 \nu^{13} - 1617708410646 \nu^{12} + 94261526625960 \nu^{11} + \cdots - 82437933075262 ) / 13607820103360 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + \cdots - 392465157363513 \nu ) / 6803910051680 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 45221309975 \nu^{15} - 28278344419 \nu^{14} + 3479923444575 \nu^{13} - 2174316072571 \nu^{12} + 94261526625960 \nu^{11} + \cdots + 28012979344985 ) / 13607820103360 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 45221309975 \nu^{15} + 40620036785 \nu^{14} - 3479923444575 \nu^{13} + 3124787270393 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 6504254102973 ) / 13607820103360 \)
|
\(\beta_{15}\) | \(=\) |
\( ( - 45221309975 \nu^{15} + 68821388859 \nu^{14} - 3479923444575 \nu^{13} + 5299632441843 \nu^{12} - 94261526625960 \nu^{11} + \cdots + 78181554830047 ) / 13607820103360 \)
|
\(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{14} + \beta_{12} - \beta_{11} - 3\beta_{10} + \beta_{7} + 5\beta_{6} - 3\beta_{5} - \beta_{4} - \beta_{2} - 18 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( 7\beta_{12} + 5\beta_{9} - 19\beta_{8} + 2\beta_{7} - 7\beta_{4} + 5\beta_{3} + 10\beta_{2} - 25\beta_1 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 31 \beta_{12} + 43 \beta_{11} + 79 \beta_{10} - 31 \beta_{7} - 179 \beta_{6} + 117 \beta_{5} + 31 \beta_{4} + 31 \beta_{2} + 400 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( - 349 \beta_{12} - 193 \beta_{9} + 929 \beta_{8} - 28 \beta_{7} + 313 \beta_{4} - 223 \beta_{3} - 342 \beta_{2} + 834 \beta_1 ) / 2 \)
|
\(\nu^{6}\) | \(=\) |
\( - 359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 543 \beta_{12} - 872 \beta_{11} - 1235 \beta_{10} + 543 \beta_{7} + 3226 \beta_{6} - 2129 \beta_{5} - 543 \beta_{4} - 543 \beta_{2} - 5998 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 13910 \beta_{12} + 6966 \beta_{9} - 36654 \beta_{8} + 52 \beta_{7} - 11798 \beta_{4} + 8514 \beta_{3} + 12044 \beta_{2} - 29577 \beta_1 ) / 2 \)
|
\(\nu^{8}\) | \(=\) |
\( ( 28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 39353 \beta_{12} + 66439 \beta_{11} + 84671 \beta_{10} - 39353 \beta_{7} - 232271 \beta_{6} + 154427 \beta_{5} + 39353 \beta_{4} + \cdots + 406476 ) / 2 \)
|
\(\nu^{9}\) | \(=\) |
\( ( - 520505 \beta_{12} - 250339 \beta_{9} + 1366801 \beta_{8} + 12974 \beta_{7} + 430885 \beta_{4} - 313151 \beta_{3} - 434946 \beta_{2} + 1065837 \beta_1 ) / 2 \)
|
\(\nu^{10}\) | \(=\) |
\( ( - 1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 1431797 \beta_{12} - 2458153 \beta_{11} - 3010165 \beta_{10} + 1431797 \beta_{7} + 8384705 \beta_{6} + \cdots - 14401944 ) / 2 \)
|
\(\nu^{11}\) | \(=\) |
\( ( 19083917 \beta_{12} + 9029821 \beta_{9} - 50060453 \beta_{8} - 679532 \beta_{7} - 15643385 \beta_{4} + 11400387 \beta_{3} + 15800198 \beta_{2} - 38587470 \beta_1 ) / 2 \)
|
\(\nu^{12}\) | \(=\) |
\( 19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 26024800 \beta_{12} + 44933550 \beta_{11} + 54218408 \beta_{10} - 26024800 \beta_{7} - 151719058 \beta_{6} + \cdots + 259136963 \)
|
\(\nu^{13}\) | \(=\) |
\( ( - 694927048 \beta_{12} - 326698624 \beta_{9} + 1822403440 \beta_{8} + 27487912 \beta_{7} + 567460000 \beta_{4} - 413915040 \beta_{3} - 574139472 \beta_{2} + 1399069645 \beta_1 ) / 2 \)
|
\(\nu^{14}\) | \(=\) |
\( ( - 1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 1890378909 \beta_{12} - 3270129841 \beta_{11} - 3924573439 \beta_{10} + \cdots - 18751391662 ) / 2 \)
|
\(\nu^{15}\) | \(=\) |
\( ( 25246432839 \beta_{12} + 11839142477 \beta_{9} - 66202882547 \beta_{8} - 1035217254 \beta_{7} - 20585412727 \beta_{4} + 15018968005 \beta_{3} + \cdots - 50751416961 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).
\(n\) | \(461\) | \(1657\) | \(1891\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 |
|
−1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | − | 7.10180i | −2.82843 | 0 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.2 | −1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | − | 1.47532i | −2.82843 | 0 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.3 | −1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | 1.16919i | −2.82843 | 0 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.4 | −1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | 10.1866i | −2.82843 | 0 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.5 | −1.41421 | 0 | 2.00000 | 2.23607i | 0 | − | 10.1866i | −2.82843 | 0 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.6 | −1.41421 | 0 | 2.00000 | 2.23607i | 0 | − | 1.16919i | −2.82843 | 0 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.7 | −1.41421 | 0 | 2.00000 | 2.23607i | 0 | 1.47532i | −2.82843 | 0 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.8 | −1.41421 | 0 | 2.00000 | 2.23607i | 0 | 7.10180i | −2.82843 | 0 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.9 | 1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | − | 8.51262i | 2.82843 | 0 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.10 | 1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | − | 8.24199i | 2.82843 | 0 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.11 | 1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | − | 7.05858i | 2.82843 | 0 | − | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.12 | 1.41421 | 0 | 2.00000 | − | 2.23607i | 0 | 7.61815i | 2.82843 | 0 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.13 | 1.41421 | 0 | 2.00000 | 2.23607i | 0 | − | 7.61815i | 2.82843 | 0 | 3.16228i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.14 | 1.41421 | 0 | 2.00000 | 2.23607i | 0 | 7.05858i | 2.82843 | 0 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.15 | 1.41421 | 0 | 2.00000 | 2.23607i | 0 | 8.24199i | 2.82843 | 0 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
91.16 | 1.41421 | 0 | 2.00000 | 2.23607i | 0 | 8.51262i | 2.82843 | 0 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2070.3.c.a | 16 | |
3.b | odd | 2 | 1 | 230.3.d.a | ✓ | 16 | |
12.b | even | 2 | 1 | 1840.3.k.d | 16 | ||
15.d | odd | 2 | 1 | 1150.3.d.b | 16 | ||
15.e | even | 4 | 2 | 1150.3.c.c | 32 | ||
23.b | odd | 2 | 1 | inner | 2070.3.c.a | 16 | |
69.c | even | 2 | 1 | 230.3.d.a | ✓ | 16 | |
276.h | odd | 2 | 1 | 1840.3.k.d | 16 | ||
345.h | even | 2 | 1 | 1150.3.d.b | 16 | ||
345.l | odd | 4 | 2 | 1150.3.c.c | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.3.d.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
230.3.d.a | ✓ | 16 | 69.c | even | 2 | 1 | |
1150.3.c.c | 32 | 15.e | even | 4 | 2 | ||
1150.3.c.c | 32 | 345.l | odd | 4 | 2 | ||
1150.3.d.b | 16 | 15.d | odd | 2 | 1 | ||
1150.3.d.b | 16 | 345.h | even | 2 | 1 | ||
1840.3.k.d | 16 | 12.b | even | 2 | 1 | ||
1840.3.k.d | 16 | 276.h | odd | 2 | 1 | ||
2070.3.c.a | 16 | 1.a | even | 1 | 1 | trivial | |
2070.3.c.a | 16 | 23.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 406 T_{7}^{14} + 67901 T_{7}^{12} + 6012916 T_{7}^{10} + 299518572 T_{7}^{8} + 8103516608 T_{7}^{6} + 100475819056 T_{7}^{4} + 285091956800 T_{7}^{2} + \cdots + 221645107264 \)
acting on \(S_{3}^{\mathrm{new}}(2070, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{8} \)
$3$
\( T^{16} \)
$5$
\( (T^{2} + 5)^{8} \)
$7$
\( T^{16} + 406 T^{14} + \cdots + 221645107264 \)
$11$
\( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \)
$13$
\( (T^{8} - 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \)
$17$
\( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \)
$19$
\( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \)
$23$
\( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \)
$29$
\( (T^{8} - 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \)
$31$
\( (T^{8} + 58 T^{7} + \cdots + 229759835104)^{2} \)
$37$
\( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \)
$41$
\( (T^{8} - 78 T^{7} + \cdots - 212194449184)^{2} \)
$43$
\( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \)
$47$
\( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \)
$53$
\( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \)
$59$
\( (T^{8} + 102 T^{7} + \cdots - 42922529206784)^{2} \)
$61$
\( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \)
$67$
\( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \)
$71$
\( (T^{8} + 118 T^{7} + \cdots - 24390990617024)^{2} \)
$73$
\( (T^{8} + 56 T^{7} + \cdots + 1317400530416)^{2} \)
$79$
\( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \)
$83$
\( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \)
$89$
\( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \)
$97$
\( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \)
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