Newspace parameters
Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1150.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(31.3352304014\) |
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: | \( 2^{9} \) |
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
\( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + \cdots + 76734676377 \nu ) / 736991990 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 1452833849 \nu^{15} + 114494825161 \nu^{13} + 3236490394640 \nu^{11} + 43627262833926 \nu^{9} + 300170143661102 \nu^{7} + \cdots + 713709368150605 \nu ) / 3401955025840 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 5957154299 \nu^{15} - 459480090555 \nu^{13} - 12507983573336 \nu^{11} - 158460209288026 \nu^{9} + \cdots - 543467028934999 \nu ) / 13607820103360 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + \cdots + 885659459 ) / 505359680 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8733597167783 \nu ) / 114351429440 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} + \cdots - 4166619657821 \nu ) / 121498393780 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 3600464285 \nu^{15} - 277707100247 \nu^{13} - 7555326565548 \nu^{11} - 95479887197410 \nu^{9} - 585118020325518 \nu^{7} + \cdots - 191041726435967 \nu ) / 1700977512920 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + \cdots - 44834433467221 ) / 6803910051680 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 5871432519 \nu^{15} + 451130209015 \nu^{13} + 12184818665176 \nu^{11} + 151934673265506 \nu^{9} + 905732893319778 \nu^{7} + \cdots - 230413141490301 \nu ) / 1943974300480 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 43531256921 \nu^{15} - 3343956125401 \nu^{13} - 90283361017128 \nu^{11} + \cdots + 17\!\cdots\!83 \nu ) / 13607820103360 \)
|
\(\beta_{12}\) | \(=\) |
\( ( - 29269066035 \nu^{15} + 21002456926 \nu^{14} - 2263755902763 \nu^{13} + 1617708410646 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 82437933075262 ) / 13607820103360 \)
|
\(\beta_{13}\) | \(=\) |
\( ( - 29269066035 \nu^{15} - 28278344419 \nu^{14} - 2263755902763 \nu^{13} - 2174316072571 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 28012979344985 ) / 13607820103360 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 29269066035 \nu^{15} + 40620036785 \nu^{14} - 2263755902763 \nu^{13} + 3124787270393 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 6504254102973 ) / 13607820103360 \)
|
\(\beta_{15}\) | \(=\) |
\( ( - 29269066035 \nu^{15} + 68821388859 \nu^{14} - 2263755902763 \nu^{13} + 5299632441843 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 78181554830047 ) / 13607820103360 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{8} + \beta_{6} - \beta_{3} ) / 4 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 18 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -7\beta_{11} + 2\beta_{10} - 3\beta_{8} + 5\beta_{7} - 17\beta_{6} + 8\beta_{3} + 7\beta_{2} + 11\beta_1 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 43 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 79 \beta_{9} + 45 \beta_{8} - 45 \beta_{6} - 179 \beta_{5} + 117 \beta_{4} - 45 \beta_{3} - 45 \beta_{2} - 45 \beta _1 + 400 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 698 \beta_{11} - 56 \beta_{10} + 3 \beta_{8} - 386 \beta_{7} + 1219 \beta_{6} - 449 \beta_{3} - 626 \beta_{2} - 878 \beta_1 ) / 4 \)
|
\(\nu^{6}\) | \(=\) |
\( - 359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 872 \beta_{12} - 868 \beta_{11} - 868 \beta_{10} - 1235 \beta_{9} - 868 \beta_{8} + 868 \beta_{6} + 3226 \beta_{5} - 2129 \beta_{4} + 868 \beta_{3} + 868 \beta_{2} + \cdots - 5998 \)
|
\(\nu^{7}\) | \(=\) |
\( ( - 27820 \beta_{11} + 104 \beta_{10} + 1733 \beta_{8} + 13932 \beta_{7} - 43859 \beta_{6} + 15295 \beta_{3} + 23596 \beta_{2} + 32356 \beta_1 ) / 4 \)
|
\(\nu^{8}\) | \(=\) |
\( ( 28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 66439 \beta_{12} + 64379 \beta_{11} + 64379 \beta_{10} + 84671 \beta_{9} + 64379 \beta_{8} - 64379 \beta_{6} - 232271 \beta_{5} + 154427 \beta_{4} + \cdots + 406476 ) / 2 \)
|
\(\nu^{9}\) | \(=\) |
\( ( 520505 \beta_{11} + 12974 \beta_{10} - 40318 \beta_{8} - 250339 \beta_{7} + 793004 \beta_{6} - 272833 \beta_{3} - 430885 \beta_{2} - 589491 \beta_1 ) / 2 \)
|
\(\nu^{10}\) | \(=\) |
\( ( - 1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 2458153 \beta_{12} - 2355559 \beta_{11} - 2355559 \beta_{10} - 3010165 \beta_{9} - 2355559 \beta_{8} + \cdots - 14401944 ) / 2 \)
|
\(\nu^{11}\) | \(=\) |
\( ( - 38167834 \beta_{11} - 1359064 \beta_{10} + 3109921 \beta_{8} + 18059642 \beta_{7} - 57484087 \beta_{6} + 19690853 \beta_{3} + 31286770 \beta_{2} + 42843970 \beta_1 ) / 4 \)
|
\(\nu^{12}\) | \(=\) |
\( 19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 44933550 \beta_{12} + 42873518 \beta_{11} + 42873518 \beta_{10} + 54218408 \beta_{9} + 42873518 \beta_{8} + \cdots + 259136963 \)
|
\(\nu^{13}\) | \(=\) |
\( ( 1389854096 \beta_{11} + 54975824 \beta_{10} - 114833875 \beta_{8} - 653397248 \beta_{7} + 2085143085 \beta_{6} - 712996205 \beta_{3} - 1134920000 \beta_{2} - 1555514512 \beta_1 ) / 4 \)
|
\(\nu^{14}\) | \(=\) |
\( ( - 1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 3270129841 \beta_{12} - 3115174881 \beta_{11} - 3115174881 \beta_{10} + \cdots - 18751391662 ) / 2 \)
|
\(\nu^{15}\) | \(=\) |
\( ( - 25246432839 \beta_{11} - 1035217254 \beta_{10} + 2094391749 \beta_{8} + 11839142477 \beta_{7} - 37826840705 \beta_{6} + 12924576256 \beta_{3} + \cdots + 28227614047 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(277\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
551.1 |
|
−1.41421 | −4.30716 | 2.00000 | 0 | 6.09125 | − | 1.47532i | −2.82843 | 9.55167 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.2 | −1.41421 | −4.30716 | 2.00000 | 0 | 6.09125 | 1.47532i | −2.82843 | 9.55167 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.3 | −1.41421 | −1.43837 | 2.00000 | 0 | 2.03417 | − | 10.1866i | −2.82843 | −6.93108 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.4 | −1.41421 | −1.43837 | 2.00000 | 0 | 2.03417 | 10.1866i | −2.82843 | −6.93108 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.5 | −1.41421 | 3.36596 | 2.00000 | 0 | −4.76019 | − | 1.16919i | −2.82843 | 2.32968 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.6 | −1.41421 | 3.36596 | 2.00000 | 0 | −4.76019 | 1.16919i | −2.82843 | 2.32968 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.7 | −1.41421 | 3.79379 | 2.00000 | 0 | −5.36524 | − | 7.10180i | −2.82843 | 5.39287 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.8 | −1.41421 | 3.79379 | 2.00000 | 0 | −5.36524 | 7.10180i | −2.82843 | 5.39287 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.9 | 1.41421 | −4.76369 | 2.00000 | 0 | −6.73687 | − | 7.05858i | 2.82843 | 13.6927 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.10 | 1.41421 | −4.76369 | 2.00000 | 0 | −6.73687 | 7.05858i | 2.82843 | 13.6927 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.11 | 1.41421 | −2.34854 | 2.00000 | 0 | −3.32134 | − | 7.61815i | 2.82843 | −3.48436 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.12 | 1.41421 | −2.34854 | 2.00000 | 0 | −3.32134 | 7.61815i | 2.82843 | −3.48436 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.13 | 1.41421 | 0.278523 | 2.00000 | 0 | 0.393890 | − | 8.51262i | 2.82843 | −8.92243 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.14 | 1.41421 | 0.278523 | 2.00000 | 0 | 0.393890 | 8.51262i | 2.82843 | −8.92243 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.15 | 1.41421 | 5.41949 | 2.00000 | 0 | 7.66432 | − | 8.24199i | 2.82843 | 20.3709 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
551.16 | 1.41421 | 5.41949 | 2.00000 | 0 | 7.66432 | 8.24199i | 2.82843 | 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 | 1150.3.d.b | 16 | |
5.b | even | 2 | 1 | 230.3.d.a | ✓ | 16 | |
5.c | odd | 4 | 2 | 1150.3.c.c | 32 | ||
15.d | odd | 2 | 1 | 2070.3.c.a | 16 | ||
20.d | odd | 2 | 1 | 1840.3.k.d | 16 | ||
23.b | odd | 2 | 1 | inner | 1150.3.d.b | 16 | |
115.c | odd | 2 | 1 | 230.3.d.a | ✓ | 16 | |
115.e | even | 4 | 2 | 1150.3.c.c | 32 | ||
345.h | even | 2 | 1 | 2070.3.c.a | 16 | ||
460.g | even | 2 | 1 | 1840.3.k.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.3.d.a | ✓ | 16 | 5.b | even | 2 | 1 | |
230.3.d.a | ✓ | 16 | 115.c | odd | 2 | 1 | |
1150.3.c.c | 32 | 5.c | odd | 4 | 2 | ||
1150.3.c.c | 32 | 115.e | even | 4 | 2 | ||
1150.3.d.b | 16 | 1.a | even | 1 | 1 | trivial | |
1150.3.d.b | 16 | 23.b | odd | 2 | 1 | inner | |
1840.3.k.d | 16 | 20.d | odd | 2 | 1 | ||
1840.3.k.d | 16 | 460.g | even | 2 | 1 | ||
2070.3.c.a | 16 | 15.d | odd | 2 | 1 | ||
2070.3.c.a | 16 | 345.h | even | 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 \(S_{3}^{\mathrm{new}}(1150, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{8} \)
$3$
\( (T^{8} - 52 T^{6} - 16 T^{5} + 829 T^{4} + \cdots + 1336)^{2} \)
$5$
\( T^{16} \)
$7$
\( T^{16} + 406 T^{14} + \cdots + 221645107264 \)
$11$
\( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \)
$13$
\( (T^{8} + 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \)
$17$
\( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \)
$19$
\( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \)
$23$
\( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \)
$29$
\( (T^{8} + 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \)
$31$
\( (T^{8} + 58 T^{7} + \cdots + 229759835104)^{2} \)
$37$
\( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \)
$41$
\( (T^{8} + 78 T^{7} + \cdots - 212194449184)^{2} \)
$43$
\( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \)
$47$
\( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \)
$53$
\( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \)
$59$
\( (T^{8} - 102 T^{7} + \cdots - 42922529206784)^{2} \)
$61$
\( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \)
$67$
\( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \)
$71$
\( (T^{8} - 118 T^{7} + \cdots - 24390990617024)^{2} \)
$73$
\( (T^{8} - 56 T^{7} + \cdots + 1317400530416)^{2} \)
$79$
\( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \)
$83$
\( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \)
$89$
\( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \)
$97$
\( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \)
show more
show less