Newspace parameters
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.3052138789\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - x^{9} + 13x^{8} + 2x^{7} + 118x^{6} + 8x^{5} + 403x^{4} + 299x^{3} + 931x^{2} + 186x + 36 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{4}\cdot 3 \) |
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 \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} - x^{9} + 13x^{8} + 2x^{7} + 118x^{6} + 8x^{5} + 403x^{4} + 299x^{3} + 931x^{2} + 186x + 36 \) :
\(\beta_{1}\) | \(=\) | \( ( 1889 \nu^{9} + 8166 \nu^{8} - 23137 \nu^{7} + 138911 \nu^{6} - 137461 \nu^{5} + 817961 \nu^{4} - 4016082 \nu^{3} + 2483825 \nu^{2} + 498126 \nu - 2557980 ) / 13246392 \) |
\(\beta_{2}\) | \(=\) | \( ( - 1889 \nu^{9} - 8166 \nu^{8} + 23137 \nu^{7} - 138911 \nu^{6} + 137461 \nu^{5} - 817961 \nu^{4} + 4016082 \nu^{3} - 2483825 \nu^{2} + 25994658 \nu + 2557980 ) / 13246392 \) |
\(\beta_{3}\) | \(=\) | \( ( - 4083 \nu^{9} + 27930 \nu^{8} - 79135 \nu^{7} + 249637 \nu^{6} - 470155 \nu^{5} + 2797655 \nu^{4} - 2967548 \nu^{3} + 8495375 \nu^{2} + 1703730 \nu + 31870992 ) / 6623196 \) |
\(\beta_{4}\) | \(=\) | \( ( 11639 \nu^{9} + 4734 \nu^{8} - 13413 \nu^{7} + 306007 \nu^{6} - 79689 \nu^{5} + 474189 \nu^{4} - 6473584 \nu^{3} + 1439925 \nu^{2} + 288774 \nu - 22233324 ) / 6623196 \) |
\(\beta_{5}\) | \(=\) | \( ( - 12709 \nu^{9} - 48512 \nu^{8} - 46527 \nu^{7} - 1097617 \nu^{6} - 471225 \nu^{5} - 9458727 \nu^{4} + 1615680 \nu^{3} - 40328629 \nu^{2} + \cdots - 45901620 ) / 6623196 \) |
\(\beta_{6}\) | \(=\) | \( ( - 41651 \nu^{9} + 155956 \nu^{8} - 625853 \nu^{7} + 1259669 \nu^{6} - 3913103 \nu^{5} + 11022151 \nu^{4} - 10888160 \nu^{3} + 21863721 \nu^{2} + \cdots + 46713888 ) / 6623196 \) |
\(\beta_{7}\) | \(=\) | \( ( 71055 \nu^{9} - 69166 \nu^{8} + 931881 \nu^{7} + 118973 \nu^{6} + 8523401 \nu^{5} + 430979 \nu^{4} + 29453126 \nu^{3} + 17229363 \nu^{2} + 68636030 \nu + 7091160 ) / 6623196 \) |
\(\beta_{8}\) | \(=\) | \( ( - 80735 \nu^{9} + 116728 \nu^{8} - 1066640 \nu^{7} + 162904 \nu^{6} - 9324028 \nu^{5} + 2125416 \nu^{4} - 32315787 \nu^{3} - 22632176 \nu^{2} + \cdots - 7779900 ) / 3311598 \) |
\(\beta_{9}\) | \(=\) | \( ( 361247 \nu^{9} - 365594 \nu^{8} + 4715403 \nu^{7} + 484139 \nu^{6} + 42949699 \nu^{5} + 175201 \nu^{4} + 146217096 \nu^{3} + 93381657 \nu^{2} + \cdots + 34142808 ) / 6623196 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{9} - 5\beta_{7} + \beta_{3} + \beta_{2} - \beta _1 - 5 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{4} + \beta_{3} - 8\beta _1 - 3 \) |
\(\nu^{4}\) | \(=\) | \( ( - 9 \beta_{9} - 3 \beta_{8} + 39 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 11 \beta_{3} - 13 \beta_{2} - 13 \beta _1 - 38 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 16 \beta_{9} - 15 \beta_{8} + 56 \beta_{7} + 13 \beta_{6} + 11 \beta_{5} - 13 \beta_{4} - 18 \beta_{3} - 73 \beta_{2} + 73 \beta _1 + 43 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 13\beta_{6} - 13\beta_{5} - 20\beta_{4} - 114\beta_{3} + 152\beta _1 + 329 \) |
\(\nu^{7}\) | \(=\) | \( ( 198 \beta_{9} + 180 \beta_{8} - 672 \beta_{7} - 100 \beta_{6} - 140 \beta_{5} - 140 \beta_{4} - 238 \beta_{3} + 709 \beta_{2} + 709 \beta _1 + 532 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 889 \beta_{9} + 558 \beta_{8} - 3347 \beta_{7} - 278 \beta_{6} + 2 \beta_{5} + 278 \beta_{4} + 1169 \beta_{3} + 1717 \beta_{2} - 1717 \beta _1 - 3069 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( -278\beta_{6} + 278\beta_{5} + 1449\beta_{4} + 2831\beta_{3} - 7124\beta _1 - 6247 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(176\) | \(451\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
76.1 |
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−3.26275 | − | 1.73205i | 6.64554 | 0 | 5.65125i | −5.10886 | − | 4.78534i | −8.63174 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
76.2 | −3.26275 | 1.73205i | 6.64554 | 0 | − | 5.65125i | −5.10886 | + | 4.78534i | −8.63174 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
76.3 | −2.17295 | − | 1.73205i | 0.721702 | 0 | 3.76366i | 6.11016 | + | 3.41555i | 7.12357 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
76.4 | −2.17295 | 1.73205i | 0.721702 | 0 | − | 3.76366i | 6.11016 | − | 3.41555i | 7.12357 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
76.5 | 0.203414 | − | 1.73205i | −3.95862 | 0 | − | 0.352323i | −3.03444 | + | 6.30811i | −1.61889 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
76.6 | 0.203414 | 1.73205i | −3.95862 | 0 | 0.352323i | −3.03444 | − | 6.30811i | −1.61889 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
76.7 | 1.55255 | − | 1.73205i | −1.58958 | 0 | − | 2.68910i | 3.69430 | − | 5.94577i | −8.67812 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
76.8 | 1.55255 | 1.73205i | −1.58958 | 0 | 2.68910i | 3.69430 | + | 5.94577i | −8.67812 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
76.9 | 2.67973 | − | 1.73205i | 3.18096 | 0 | − | 4.64143i | −6.16116 | − | 3.32267i | −2.19482 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
76.10 | 2.67973 | 1.73205i | 3.18096 | 0 | 4.64143i | −6.16116 | + | 3.32267i | −2.19482 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.3.h.b | ✓ | 10 |
5.b | even | 2 | 1 | 525.3.h.c | yes | 10 | |
5.c | odd | 4 | 2 | 525.3.e.b | 20 | ||
7.b | odd | 2 | 1 | inner | 525.3.h.b | ✓ | 10 |
35.c | odd | 2 | 1 | 525.3.h.c | yes | 10 | |
35.f | even | 4 | 2 | 525.3.e.b | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.3.e.b | 20 | 5.c | odd | 4 | 2 | ||
525.3.e.b | 20 | 35.f | even | 4 | 2 | ||
525.3.h.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
525.3.h.b | ✓ | 10 | 7.b | odd | 2 | 1 | inner |
525.3.h.c | yes | 10 | 5.b | even | 2 | 1 | |
525.3.h.c | yes | 10 | 35.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + T_{2}^{4} - 12T_{2}^{3} - 5T_{2}^{2} + 31T_{2} - 6 \)
acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{5} + T^{4} - 12 T^{3} - 5 T^{2} + 31 T - 6)^{2} \)
$3$
\( (T^{2} + 3)^{5} \)
$5$
\( T^{10} \)
$7$
\( T^{10} + 9 T^{9} + 37 T^{8} + \cdots + 282475249 \)
$11$
\( (T^{5} - 16 T^{4} - 207 T^{3} + 4226 T^{2} + \cdots + 8400)^{2} \)
$13$
\( T^{10} + 1269 T^{8} + \cdots + 11663315712 \)
$17$
\( T^{10} + 2298 T^{8} + \cdots + 874800000000 \)
$19$
\( T^{10} + 2163 T^{8} + \cdots + 128517193728 \)
$23$
\( (T^{5} + 16 T^{4} - 1542 T^{3} + \cdots + 3662082)^{2} \)
$29$
\( (T^{5} + 20 T^{4} - 1698 T^{3} + \cdots + 3075678)^{2} \)
$31$
\( T^{10} + 3333 T^{8} + \cdots + 338397758208 \)
$37$
\( (T^{5} - 108 T^{4} + 3761 T^{3} + \cdots + 46312)^{2} \)
$41$
\( T^{10} + 6654 T^{8} + \cdots + 1097349120000 \)
$43$
\( (T^{5} - 3 T^{4} - 4810 T^{3} + \cdots - 46365899)^{2} \)
$47$
\( T^{10} + 12324 T^{8} + \cdots + 17\!\cdots\!72 \)
$53$
\( (T^{5} + 16 T^{4} - 5463 T^{3} + \cdots + 22722000)^{2} \)
$59$
\( T^{10} + 24738 T^{8} + \cdots + 13\!\cdots\!72 \)
$61$
\( T^{10} + 17013 T^{8} + \cdots + 18172516320000 \)
$67$
\( (T^{5} + 13 T^{4} - 11477 T^{3} + \cdots - 304035028)^{2} \)
$71$
\( (T^{5} - 94 T^{4} - 7779 T^{3} + \cdots - 260099904)^{2} \)
$73$
\( T^{10} + 34008 T^{8} + \cdots + 16\!\cdots\!72 \)
$79$
\( (T^{5} - 88 T^{4} - 12131 T^{3} + \cdots + 272274352)^{2} \)
$83$
\( T^{10} + 25938 T^{8} + \cdots + 21\!\cdots\!68 \)
$89$
\( T^{10} + 80832 T^{8} + \cdots + 52\!\cdots\!72 \)
$97$
\( T^{10} + 48771 T^{8} + \cdots + 13\!\cdots\!88 \)
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