Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.3052138789\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
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: | \( x^{16} + 36x^{14} + 424x^{12} + 2232x^{10} + 5580x^{8} + 6288x^{6} + 2704x^{4} + 384x^{2} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{14} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{16} + 36x^{14} + 424x^{12} + 2232x^{10} + 5580x^{8} + 6288x^{6} + 2704x^{4} + 384x^{2} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( - 121 \nu^{15} + 560 \nu^{14} - 4114 \nu^{13} + 20200 \nu^{12} - 43076 \nu^{11} + 238710 \nu^{10} - 182710 \nu^{9} + 1261180 \nu^{8} - 273460 \nu^{7} + 3155940 \nu^{6} + \cdots + 106880 ) / 19360 \) |
\(\beta_{2}\) | \(=\) | \( ( 121 \nu^{15} + 560 \nu^{14} + 4114 \nu^{13} + 20200 \nu^{12} + 43076 \nu^{11} + 238710 \nu^{10} + 182710 \nu^{9} + 1261180 \nu^{8} + 273460 \nu^{7} + 3155940 \nu^{6} + \cdots + 106880 ) / 19360 \) |
\(\beta_{3}\) | \(=\) | \( ( 221 \nu^{14} + 7894 \nu^{12} + 91796 \nu^{10} + 477150 \nu^{8} + 1185760 \nu^{6} + 1361988 \nu^{4} + 630808 \nu^{2} + 65688 ) / 4840 \) |
\(\beta_{4}\) | \(=\) | \( ( 65 \nu^{15} + 2336 \nu^{13} + 27554 \nu^{11} + 147584 \nu^{9} + 388868 \nu^{7} + 497584 \nu^{5} + 285008 \nu^{3} + 64816 \nu ) / 1936 \) |
\(\beta_{5}\) | \(=\) | \( ( - 401 \nu^{14} - 14214 \nu^{12} - 163036 \nu^{10} - 831450 \nu^{8} - 1997840 \nu^{6} - 2108628 \nu^{4} - 763608 \nu^{2} - 49568 ) / 4840 \) |
\(\beta_{6}\) | \(=\) | \( ( 1873 \nu^{15} - 186 \nu^{14} + 67242 \nu^{13} - 6934 \nu^{12} + 787218 \nu^{11} - 85876 \nu^{10} + 4094660 \nu^{9} - 468960 \nu^{8} + 9982380 \nu^{7} - 1179240 \nu^{6} + \cdots - 10608 ) / 19360 \) |
\(\beta_{7}\) | \(=\) | \( ( - 1873 \nu^{15} - 67242 \nu^{13} - 787218 \nu^{11} - 4094660 \nu^{9} - 9982380 \nu^{7} - 10598184 \nu^{5} - 3804224 \nu^{3} - 298704 \nu ) / 19360 \) |
\(\beta_{8}\) | \(=\) | \( ( - 1873 \nu^{15} - 186 \nu^{14} - 67242 \nu^{13} - 6934 \nu^{12} - 787218 \nu^{11} - 85876 \nu^{10} - 4094660 \nu^{9} - 468960 \nu^{8} - 9982380 \nu^{7} - 1179240 \nu^{6} + \cdots - 10608 ) / 19360 \) |
\(\beta_{9}\) | \(=\) | \( ( 1108 \nu^{15} - 512 \nu^{14} + 39777 \nu^{13} - 18313 \nu^{12} + 465693 \nu^{11} - 212977 \nu^{10} + 2423115 \nu^{9} - 1097730 \nu^{8} + 5916360 \nu^{7} - 2639870 \nu^{6} + \cdots - 81816 ) / 9680 \) |
\(\beta_{10}\) | \(=\) | \( ( 494 \nu^{14} + 17681 \nu^{12} + 205974 \nu^{10} + 1065930 \nu^{8} + 2587460 \nu^{6} + 2738812 \nu^{4} + 978712 \nu^{2} + 74232 ) / 2420 \) |
\(\beta_{11}\) | \(=\) | \( ( - 503 \nu^{15} - 18118 \nu^{13} - 213650 \nu^{11} - 1127568 \nu^{9} - 2835700 \nu^{7} - 3240784 \nu^{5} - 1419984 \nu^{3} - 150624 \nu ) / 3872 \) |
\(\beta_{12}\) | \(=\) | \( ( - 1108 \nu^{15} - 512 \nu^{14} - 39777 \nu^{13} - 18313 \nu^{12} - 465693 \nu^{11} - 212977 \nu^{10} - 2423115 \nu^{9} - 1097730 \nu^{8} - 5916360 \nu^{7} - 2639870 \nu^{6} + \cdots - 81816 ) / 9680 \) |
\(\beta_{13}\) | \(=\) | \( ( - 1277 \nu^{15} - 45778 \nu^{13} - 534502 \nu^{11} - 2769275 \nu^{9} - 6705890 \nu^{7} - 7011566 \nu^{5} - 2391776 \nu^{3} - 145196 \nu ) / 4840 \) |
\(\beta_{14}\) | \(=\) | \( ( - 3430 \nu^{15} + 221 \nu^{14} - 123120 \nu^{13} + 7894 \nu^{12} - 1441680 \nu^{11} + 91191 \nu^{10} - 7512675 \nu^{9} + 459000 \nu^{8} - 18412650 \nu^{7} + 1039350 \nu^{6} + \cdots - 31112 ) / 4840 \) |
\(\beta_{15}\) | \(=\) | \( ( 3430 \nu^{15} + 221 \nu^{14} + 123120 \nu^{13} + 7894 \nu^{12} + 1441680 \nu^{11} + 91191 \nu^{10} + 7512675 \nu^{9} + 459000 \nu^{8} + 18412650 \nu^{7} + 1039350 \nu^{6} + \cdots - 31112 ) / 4840 \) |
\(\nu\) | \(=\) | \( ( \beta_{8} - 2\beta_{7} - \beta_{6} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{10} + 2\beta_{8} + 2\beta_{6} + 2\beta_{5} - 8 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( - \beta_{15} + \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{9} - 13 \beta_{8} + 26 \beta_{7} + 13 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3 \beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{12} - 8 \beta_{10} - 2 \beta_{9} - 22 \beta_{8} - 22 \beta_{6} - 19 \beta_{5} - \beta_{3} - 6 \beta_{2} - 6 \beta _1 + 47 \) |
\(\nu^{5}\) | \(=\) | \( ( 23 \beta_{15} - 23 \beta_{14} + 150 \beta_{13} - 20 \beta_{12} + 10 \beta_{11} + 20 \beta_{9} + 216 \beta_{8} - 482 \beta_{7} - 216 \beta_{6} - 60 \beta_{4} + 70 \beta_{2} - 70 \beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 49 \beta_{15} + 49 \beta_{14} + 40 \beta_{12} + 138 \beta_{10} + 40 \beta_{9} + 436 \beta_{8} + 436 \beta_{6} + 362 \beta_{5} + 28 \beta_{3} + 150 \beta_{2} + 150 \beta _1 - 778 \) |
\(\nu^{7}\) | \(=\) | \( - 230 \beta_{15} + 230 \beta_{14} - 1540 \beta_{13} + 178 \beta_{12} - 126 \beta_{11} - 178 \beta_{9} - 1978 \beta_{8} + 4642 \beta_{7} + 1978 \beta_{6} + 574 \beta_{4} - 708 \beta_{2} + 708 \beta_1 \) |
\(\nu^{8}\) | \(=\) | \( - 1000 \beta_{15} - 1000 \beta_{14} - 752 \beta_{12} - 2552 \beta_{10} - 752 \beta_{9} - 8464 \beta_{8} - 8464 \beta_{6} - 6946 \beta_{5} - 586 \beta_{3} - 3088 \beta_{2} - 3088 \beta _1 + 14350 \) |
\(\nu^{9}\) | \(=\) | \( 4473 \beta_{15} - 4473 \beta_{14} + 30210 \beta_{13} - 3304 \beta_{12} + 2586 \beta_{11} + 3304 \beta_{9} + 37522 \beta_{8} - 89590 \beta_{7} - 37522 \beta_{6} - 11016 \beta_{4} + 13842 \beta_{2} - 13842 \beta_1 \) |
\(\nu^{10}\) | \(=\) | \( 19578 \beta_{15} + 19578 \beta_{14} + 14320 \beta_{12} + 48538 \beta_{10} + 14320 \beta_{9} + 163476 \beta_{8} + 163476 \beta_{6} + 133684 \beta_{5} + 11532 \beta_{3} + 60660 \beta_{2} + \cdots - 273136 \) |
\(\nu^{11}\) | \(=\) | \( - 86420 \beta_{15} + 86420 \beta_{14} - 585288 \beta_{13} + 62858 \beta_{12} - 50688 \beta_{11} - 62858 \beta_{9} - 719798 \beta_{8} + 1727980 \beta_{7} + 719798 \beta_{6} + 212014 \beta_{4} + \cdots + 267958 \beta_1 \) |
\(\nu^{12}\) | \(=\) | \( - 379064 \beta_{15} - 379064 \beta_{14} - 274872 \beta_{12} - 931812 \beta_{10} - 274872 \beta_{9} - 3153048 \beta_{8} - 3153048 \beta_{6} - 2575664 \beta_{5} - 223528 \beta_{3} + \cdots + 5245872 \) |
\(\nu^{13}\) | \(=\) | \( 1666896 \beta_{15} - 1666896 \beta_{14} + 11298976 \beta_{13} - 1206684 \beta_{12} + 981656 \beta_{11} + 1206684 \beta_{9} + 13855240 \beta_{8} - 33316440 \beta_{7} + \cdots - 5171820 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( 7316384 \beta_{15} + 7316384 \beta_{14} + 5291544 \beta_{12} + 17940100 \beta_{10} + 5291544 \beta_{9} + 60790304 \beta_{8} + 60790304 \beta_{6} + 49642176 \beta_{5} + \cdots - 101014744 \) |
\(\nu^{15}\) | \(=\) | \( - 32137472 \beta_{15} + 32137472 \beta_{14} - 217900032 \beta_{13} + 23231644 \beta_{12} - 18948216 \beta_{11} - 23231644 \beta_{9} - 266967868 \beta_{8} + \cdots + 99732060 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(176\) | \(451\) |
\(\chi(n)\) | \(-\beta_{7}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 |
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−2.59867 | + | 2.59867i | 1.22474 | + | 1.22474i | − | 9.50617i | 0 | −6.36542 | 1.87083 | − | 1.87083i | 14.3087 | + | 14.3087i | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.2 | −2.16577 | + | 2.16577i | −1.22474 | − | 1.22474i | − | 5.38111i | 0 | 5.30503 | 1.87083 | − | 1.87083i | 2.99117 | + | 2.99117i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.3 | −1.51969 | + | 1.51969i | 1.22474 | + | 1.22474i | − | 0.618887i | 0 | −3.72245 | −1.87083 | + | 1.87083i | −5.13823 | − | 5.13823i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.4 | −0.496903 | + | 0.496903i | 1.22474 | + | 1.22474i | 3.50617i | 0 | −1.21716 | 1.87083 | − | 1.87083i | −3.72984 | − | 3.72984i | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.5 | 0.496903 | − | 0.496903i | −1.22474 | − | 1.22474i | 3.50617i | 0 | −1.21716 | −1.87083 | + | 1.87083i | 3.72984 | + | 3.72984i | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.6 | 1.51969 | − | 1.51969i | −1.22474 | − | 1.22474i | − | 0.618887i | 0 | −3.72245 | 1.87083 | − | 1.87083i | 5.13823 | + | 5.13823i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.7 | 2.16577 | − | 2.16577i | 1.22474 | + | 1.22474i | − | 5.38111i | 0 | 5.30503 | −1.87083 | + | 1.87083i | −2.99117 | − | 2.99117i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
43.8 | 2.59867 | − | 2.59867i | −1.22474 | − | 1.22474i | − | 9.50617i | 0 | −6.36542 | −1.87083 | + | 1.87083i | −14.3087 | − | 14.3087i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.1 | −2.59867 | − | 2.59867i | 1.22474 | − | 1.22474i | 9.50617i | 0 | −6.36542 | 1.87083 | + | 1.87083i | 14.3087 | − | 14.3087i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.2 | −2.16577 | − | 2.16577i | −1.22474 | + | 1.22474i | 5.38111i | 0 | 5.30503 | 1.87083 | + | 1.87083i | 2.99117 | − | 2.99117i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.3 | −1.51969 | − | 1.51969i | 1.22474 | − | 1.22474i | 0.618887i | 0 | −3.72245 | −1.87083 | − | 1.87083i | −5.13823 | + | 5.13823i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.4 | −0.496903 | − | 0.496903i | 1.22474 | − | 1.22474i | − | 3.50617i | 0 | −1.21716 | 1.87083 | + | 1.87083i | −3.72984 | + | 3.72984i | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.5 | 0.496903 | + | 0.496903i | −1.22474 | + | 1.22474i | − | 3.50617i | 0 | −1.21716 | −1.87083 | − | 1.87083i | 3.72984 | − | 3.72984i | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.6 | 1.51969 | + | 1.51969i | −1.22474 | + | 1.22474i | 0.618887i | 0 | −3.72245 | 1.87083 | + | 1.87083i | 5.13823 | − | 5.13823i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.7 | 2.16577 | + | 2.16577i | 1.22474 | − | 1.22474i | 5.38111i | 0 | 5.30503 | −1.87083 | − | 1.87083i | −2.99117 | + | 2.99117i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
232.8 | 2.59867 | + | 2.59867i | −1.22474 | + | 1.22474i | 9.50617i | 0 | −6.36542 | −1.87083 | − | 1.87083i | −14.3087 | + | 14.3087i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.3.l.d | ✓ | 16 |
5.b | even | 2 | 1 | inner | 525.3.l.d | ✓ | 16 |
5.c | odd | 4 | 2 | inner | 525.3.l.d | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.3.l.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
525.3.l.d | ✓ | 16 | 5.b | even | 2 | 1 | inner |
525.3.l.d | ✓ | 16 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 292T_{2}^{12} + 21894T_{2}^{8} + 347812T_{2}^{4} + 83521 \)
acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 292 T^{12} + 21894 T^{8} + \cdots + 83521 \)
$3$
\( (T^{4} + 9)^{4} \)
$5$
\( T^{16} \)
$7$
\( (T^{4} + 49)^{4} \)
$11$
\( (T^{4} - 12 T^{3} - 168 T^{2} + 2064 T - 1776)^{4} \)
$13$
\( T^{16} + \cdots + 358719963529216 \)
$17$
\( T^{16} + 518272 T^{12} + \cdots + 35\!\cdots\!56 \)
$19$
\( (T^{8} + 1504 T^{6} + \cdots + 6194319616)^{2} \)
$23$
\( T^{16} + 2738496 T^{12} + \cdots + 16\!\cdots\!00 \)
$29$
\( (T^{8} + 4864 T^{6} + \cdots + 339133851904)^{2} \)
$31$
\( (T^{4} + 40 T^{3} - 1512 T^{2} + \cdots + 908560)^{4} \)
$37$
\( T^{16} + 1332352 T^{12} + \cdots + 94\!\cdots\!76 \)
$41$
\( (T^{4} + 16 T^{3} - 5376 T^{2} + \cdots + 275200)^{4} \)
$43$
\( T^{16} + 33104512 T^{12} + \cdots + 13\!\cdots\!36 \)
$47$
\( T^{16} + 48501376 T^{12} + \cdots + 50\!\cdots\!00 \)
$53$
\( T^{16} + 25340224 T^{12} + \cdots + 46\!\cdots\!16 \)
$59$
\( (T^{8} + 11968 T^{6} + \cdots + 2548544045056)^{2} \)
$61$
\( (T^{4} - 28 T^{3} - 3960 T^{2} + \cdots + 1423696)^{4} \)
$67$
\( T^{16} + 184288384 T^{12} + \cdots + 31\!\cdots\!16 \)
$71$
\( (T^{4} - 104 T^{3} + 504 T^{2} + \cdots + 161680)^{4} \)
$73$
\( T^{16} + 84926464 T^{12} + \cdots + 10\!\cdots\!96 \)
$79$
\( (T^{8} + 53392 T^{6} + \cdots + 18\!\cdots\!76)^{2} \)
$83$
\( T^{16} + 339533824 T^{12} + \cdots + 80\!\cdots\!76 \)
$89$
\( (T^{8} + 20416 T^{6} + \cdots + 340380655550464)^{2} \)
$97$
\( T^{16} + 549517312 T^{12} + \cdots + 10\!\cdots\!96 \)
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