Newspace parameters
Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1840.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(50.1363686423\) |
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: | \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
Coefficient ring index: | \( 2^{9} \) |
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 \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 249266 \nu^{15} - 19450521 \nu^{13} - 540300528 \nu^{11} - 7075107694 \nu^{9} - 46294879486 \nu^{7} - 143284317912 \nu^{5} + \cdots - 76734676377 \nu ) / 1473983980 \) |
\(\beta_{3}\) | \(=\) | \( ( 2028139777 \nu^{15} + 159386627629 \nu^{13} + 4483504013264 \nu^{11} + 59956611391678 \nu^{9} + 407018725514790 \nu^{7} + \cdots + 890813001228721 \nu ) / 3401955025840 \) |
\(\beta_{4}\) | \(=\) | \( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \) |
\(\beta_{5}\) | \(=\) | \( ( 651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + \cdots - 93749086185413 \nu ) / 485993575120 \) |
\(\beta_{6}\) | \(=\) | \( ( - 328071 \nu^{14} - 25254583 \nu^{12} - 684448696 \nu^{10} - 8586649314 \nu^{8} - 51832164194 \nu^{6} - 133890163508 \nu^{4} + \cdots - 885659459 ) / 252679840 \) |
\(\beta_{7}\) | \(=\) | \( ( 229620557 \nu^{15} + 17699955559 \nu^{13} + 480902133464 \nu^{11} + 6059544792898 \nu^{9} + 36877647447520 \nu^{7} + \cdots + 4166619657821 \nu ) / 121498393780 \) |
\(\beta_{8}\) | \(=\) | \( ( - 3312811321 \nu^{15} - 255261199013 \nu^{13} - 6931819756236 \nu^{11} - 87315212918534 \nu^{9} - 531693729398674 \nu^{7} + \cdots - 105891864922749 \nu ) / 1700977512920 \) |
\(\beta_{9}\) | \(=\) | \( ( - 14365389855 \nu^{14} - 1105876709199 \nu^{12} - 29972322851736 \nu^{10} - 375976473404450 \nu^{8} + \cdots + 44834433467221 ) / 6803910051680 \) |
\(\beta_{10}\) | \(=\) | \( ( - 11655955868 \nu^{15} + 10501228463 \nu^{14} - 902137906104 \nu^{13} + 808854205323 \nu^{12} - 24691247910968 \nu^{11} + \cdots + 41218966537631 ) / 6803910051680 \) |
\(\beta_{11}\) | \(=\) | \( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8504894308903 \nu ) / 57175714720 \) |
\(\beta_{12}\) | \(=\) | \( ( - 2913988967 \nu^{15} + 3973258621 \nu^{14} - 225534476526 \nu^{13} + 305594639244 \nu^{12} - 6172811977742 \nu^{11} + \cdots - 2167987268690 ) / 1700977512920 \) |
\(\beta_{13}\) | \(=\) | \( ( 33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + \cdots - 392465157363513 \nu ) / 6803910051680 \) |
\(\beta_{14}\) | \(=\) | \( ( 5827977934 \nu^{15} + 9278078059 \nu^{14} + 451068953052 \nu^{13} + 713586557253 \nu^{12} + 12345623955484 \nu^{11} + \cdots - 1041206773123 ) / 3401955025840 \) |
\(\beta_{15}\) | \(=\) | \( ( - 5827977934 \nu^{15} + 19413839169 \nu^{14} - 451068953052 \nu^{13} + 1494915649571 \nu^{12} - 12345623955484 \nu^{11} + \cdots + 25507426770635 ) / 3401955025840 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta _1 - 18 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 7\beta_{13} - 7\beta_{11} + 5\beta_{8} - 5\beta_{7} + 2\beta_{5} + 7\beta_{3} - 10\beta_{2} - 20\beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 14 \beta_{15} + 14 \beta_{14} - 45 \beta_{13} - 47 \beta_{12} - 45 \beta_{11} - 43 \beta_{10} - 79 \beta_{9} - 90 \beta_{8} + 127 \beta_{6} + 45 \beta_{5} + 117 \beta_{4} - 45 \beta_{3} - 45 \beta_{2} - 90 \beta _1 + 400 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 349 \beta_{13} + 353 \beta_{11} - 223 \beta_{8} + 193 \beta_{7} - 28 \beta_{5} - 313 \beta_{3} + 342 \beta_{2} + 611 \beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( - 359 \beta_{15} - 325 \beta_{14} + 868 \beta_{13} + 898 \beta_{12} + 868 \beta_{11} + 872 \beta_{10} + 1235 \beta_{9} + 1736 \beta_{8} - 2404 \beta_{6} - 868 \beta_{5} - 2129 \beta_{4} + 868 \beta_{3} + 868 \beta_{2} + \cdots - 5998 \) |
\(\nu^{7}\) | \(=\) | \( ( 13910 \beta_{13} - 14070 \beta_{11} + 8514 \beta_{8} - 6966 \beta_{7} + 52 \beta_{5} + 11798 \beta_{3} - 12044 \beta_{2} - 21063 \beta_1 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 28822 \beta_{15} + 25026 \beta_{14} - 64379 \beta_{13} - 66115 \beta_{12} - 64379 \beta_{11} - 66439 \beta_{10} - 84671 \beta_{9} - 128758 \beta_{8} + 176117 \beta_{6} + 64379 \beta_{5} + \cdots + 406476 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 520505 \beta_{13} + 526825 \beta_{11} - 313151 \beta_{8} + 250339 \beta_{7} + 12974 \beta_{5} - 430885 \beta_{3} + 434946 \beta_{2} + 752686 \beta_1 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 1081278 \beta_{15} - 923762 \beta_{14} + 2355559 \beta_{13} + 2410481 \beta_{12} + 2355559 \beta_{11} + 2458153 \beta_{10} + 3010165 \beta_{9} + 4711118 \beta_{8} + \cdots - 14401944 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 19083917 \beta_{13} - 19330033 \beta_{11} + 11400387 \beta_{8} - 9029821 \beta_{7} - 679532 \beta_{5} + 15643385 \beta_{3} - 15800198 \beta_{2} - 27187083 \beta_1 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( 19844966 \beta_{15} + 16848718 \beta_{14} - 42873518 \beta_{13} - 43809734 \beta_{12} - 42873518 \beta_{11} - 44933550 \beta_{10} - 54218408 \beta_{9} - 85747036 \beta_{8} + \cdots + 259136963 \) |
\(\nu^{13}\) | \(=\) | \( ( - 694927048 \beta_{13} + 704244200 \beta_{11} - 413915040 \beta_{8} + 326698624 \beta_{7} + 27487912 \beta_{5} - 567460000 \beta_{3} + 574139472 \beta_{2} + 985154605 \beta_1 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( - 1446079476 \beta_{15} - 1224795972 \beta_{14} + 3115174881 \beta_{13} + 3181503425 \beta_{12} + 3115174881 \beta_{11} + 3270129841 \beta_{10} + \cdots - 18751391662 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( 25246432839 \beta_{13} - 25591957271 \beta_{11} + 15018968005 \beta_{8} - 11839142477 \beta_{7} - 1035217254 \beta_{5} + 20585412727 \beta_{3} + \cdots - 35732448956 \beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).
\(n\) | \(737\) | \(1151\) | \(1201\) | \(1381\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
321.1 |
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0 | −4.76369 | 0 | − | 2.23607i | 0 | − | 7.05858i | 0 | 13.6927 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.2 | 0 | −4.76369 | 0 | 2.23607i | 0 | 7.05858i | 0 | 13.6927 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.3 | 0 | −4.30716 | 0 | − | 2.23607i | 0 | − | 1.47532i | 0 | 9.55167 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.4 | 0 | −4.30716 | 0 | 2.23607i | 0 | 1.47532i | 0 | 9.55167 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.5 | 0 | −2.34854 | 0 | − | 2.23607i | 0 | 7.61815i | 0 | −3.48436 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.6 | 0 | −2.34854 | 0 | 2.23607i | 0 | − | 7.61815i | 0 | −3.48436 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.7 | 0 | −1.43837 | 0 | − | 2.23607i | 0 | 10.1866i | 0 | −6.93108 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.8 | 0 | −1.43837 | 0 | 2.23607i | 0 | − | 10.1866i | 0 | −6.93108 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.9 | 0 | 0.278523 | 0 | − | 2.23607i | 0 | − | 8.51262i | 0 | −8.92243 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.10 | 0 | 0.278523 | 0 | 2.23607i | 0 | 8.51262i | 0 | −8.92243 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.11 | 0 | 3.36596 | 0 | − | 2.23607i | 0 | 1.16919i | 0 | 2.32968 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.12 | 0 | 3.36596 | 0 | 2.23607i | 0 | − | 1.16919i | 0 | 2.32968 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.13 | 0 | 3.79379 | 0 | − | 2.23607i | 0 | − | 7.10180i | 0 | 5.39287 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.14 | 0 | 3.79379 | 0 | 2.23607i | 0 | 7.10180i | 0 | 5.39287 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.15 | 0 | 5.41949 | 0 | − | 2.23607i | 0 | − | 8.24199i | 0 | 20.3709 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
321.16 | 0 | 5.41949 | 0 | 2.23607i | 0 | 8.24199i | 0 | 20.3709 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 1840.3.k.d | 16 | |
4.b | odd | 2 | 1 | 230.3.d.a | ✓ | 16 | |
12.b | even | 2 | 1 | 2070.3.c.a | 16 | ||
20.d | odd | 2 | 1 | 1150.3.d.b | 16 | ||
20.e | even | 4 | 2 | 1150.3.c.c | 32 | ||
23.b | odd | 2 | 1 | inner | 1840.3.k.d | 16 | |
92.b | even | 2 | 1 | 230.3.d.a | ✓ | 16 | |
276.h | odd | 2 | 1 | 2070.3.c.a | 16 | ||
460.g | even | 2 | 1 | 1150.3.d.b | 16 | ||
460.k | 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 | 4.b | odd | 2 | 1 | |
230.3.d.a | ✓ | 16 | 92.b | even | 2 | 1 | |
1150.3.c.c | 32 | 20.e | even | 4 | 2 | ||
1150.3.c.c | 32 | 460.k | odd | 4 | 2 | ||
1150.3.d.b | 16 | 20.d | odd | 2 | 1 | ||
1150.3.d.b | 16 | 460.g | even | 2 | 1 | ||
1840.3.k.d | 16 | 1.a | even | 1 | 1 | trivial | |
1840.3.k.d | 16 | 23.b | odd | 2 | 1 | inner | |
2070.3.c.a | 16 | 12.b | even | 2 | 1 | ||
2070.3.c.a | 16 | 276.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 52T_{3}^{6} - 16T_{3}^{5} + 829T_{3}^{4} + 456T_{3}^{3} - 4114T_{3}^{2} - 3704T_{3} + 1336 \)
acting on \(S_{3}^{\mathrm{new}}(1840, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( (T^{8} - 52 T^{6} - 16 T^{5} + 829 T^{4} + \cdots + 1336)^{2} \)
$5$
\( (T^{2} + 5)^{8} \)
$7$
\( T^{16} + 406 T^{14} + \cdots + 221645107264 \)
$11$
\( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \)
$13$
\( (T^{8} - 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \)
$17$
\( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \)
$19$
\( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \)
$23$
\( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \)
$29$
\( (T^{8} + 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \)
$31$
\( (T^{8} - 58 T^{7} + \cdots + 229759835104)^{2} \)
$37$
\( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \)
$41$
\( (T^{8} + 78 T^{7} + \cdots - 212194449184)^{2} \)
$43$
\( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \)
$47$
\( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \)
$53$
\( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \)
$59$
\( (T^{8} + 102 T^{7} + \cdots - 42922529206784)^{2} \)
$61$
\( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \)
$67$
\( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \)
$71$
\( (T^{8} + 118 T^{7} + \cdots - 24390990617024)^{2} \)
$73$
\( (T^{8} + 56 T^{7} + \cdots + 1317400530416)^{2} \)
$79$
\( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \)
$83$
\( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \)
$89$
\( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \)
$97$
\( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \)
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