Newspace parameters
Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 25.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.1844652515\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{10}\cdot 3^{6}\cdot 5^{12} \) |
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 \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000 \) :
\(\beta_{1}\) | \(=\) | \( ( - 449334647 \nu^{11} + 12061467443 \nu^{10} - 45928215186 \nu^{9} + 33350532992 \nu^{8} + 485787845405 \nu^{7} + \cdots - 93665468020000 ) / 55\!\cdots\!00 \) |
\(\beta_{2}\) | \(=\) | \( ( - 381080397548 \nu^{11} + 6650619527307 \nu^{10} - 37023789923414 \nu^{9} + 65801651136208 \nu^{8} + \cdots + 89\!\cdots\!00 ) / 83\!\cdots\!00 \) |
\(\beta_{3}\) | \(=\) | \( ( - 435927172721 \nu^{11} + 7533818317764 \nu^{10} - 43325832090128 \nu^{9} + 78139416210016 \nu^{8} + \cdots + 92\!\cdots\!00 ) / 83\!\cdots\!00 \) |
\(\beta_{4}\) | \(=\) | \( ( - 207349351901 \nu^{11} + 3608923885464 \nu^{10} - 20095877149928 \nu^{9} + 35763821961616 \nu^{8} + \cdots + 47\!\cdots\!00 ) / 13\!\cdots\!00 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2727925523508 \nu^{11} + 14877294336467 \nu^{10} - 35907038981134 \nu^{9} - 337439888728952 \nu^{8} + \cdots - 11\!\cdots\!00 ) / 83\!\cdots\!00 \) |
\(\beta_{6}\) | \(=\) | \( ( 4175127571641 \nu^{11} - 21153383834384 \nu^{10} + 49390928549668 \nu^{9} + 514100031045104 \nu^{8} + \cdots + 12\!\cdots\!00 ) / 83\!\cdots\!00 \) |
\(\beta_{7}\) | \(=\) | \( ( - 1550015596341 \nu^{11} + 8356009540724 \nu^{10} - 20060042759548 \nu^{9} - 191658965219744 \nu^{8} + \cdots - 61\!\cdots\!00 ) / 13\!\cdots\!00 \) |
\(\beta_{8}\) | \(=\) | \( ( 572380746383 \nu^{11} - 2036265285124 \nu^{10} + 181744291018 \nu^{9} + 95492643662604 \nu^{8} + 909244071727231 \nu^{7} + \cdots + 38\!\cdots\!00 ) / 111630409428400 \) |
\(\beta_{9}\) | \(=\) | \( ( 39828320474 \nu^{11} - 141775272712 \nu^{10} + 13469724484 \nu^{9} + 6653307104727 \nu^{8} + 63244284483538 \nu^{7} + \cdots + 29\!\cdots\!75 ) / 6976900589275 \) |
\(\beta_{10}\) | \(=\) | \( ( 3672375956607 \nu^{11} - 24100275366453 \nu^{10} + 80120961677856 \nu^{9} + 333658706564268 \nu^{8} + \cdots + 27\!\cdots\!00 ) / 558152047142000 \) |
\(\beta_{11}\) | \(=\) | \( ( 8224243606507 \nu^{11} - 54768226676503 \nu^{10} + 182306064041706 \nu^{9} + 741572009869568 \nu^{8} + \cdots + 61\!\cdots\!00 ) / 11\!\cdots\!00 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{8} + 12\beta_{4} + 100\beta _1 + 100 ) / 300 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{11} - 6\beta_{10} + 15\beta_{6} - 45\beta_{5} - 15\beta_{3} - 45\beta_{2} + 2850\beta_1 ) / 150 \) |
\(\nu^{3}\) | \(=\) | \( ( 17\beta_{11} - 11\beta_{10} + 17\beta_{9} - 11\beta_{8} + 144\beta_{7} - 225\beta_{5} + 2525\beta _1 - 2525 ) / 75 \) |
\(\nu^{4}\) | \(=\) | \( ( 73 \beta_{9} - 144 \beta_{8} + 264 \beta_{7} + 390 \beta_{6} - 1470 \beta_{5} - 264 \beta_{4} + 390 \beta_{3} + 1470 \beta_{2} - 54825 ) / 75 \) |
\(\nu^{5}\) | \(=\) | \( ( - 747 \beta_{11} + 586 \beta_{10} + 747 \beta_{9} - 586 \beta_{8} - 7212 \beta_{4} + 1125 \beta_{3} + 15000 \beta_{2} - 163975 \beta _1 - 163975 ) / 75 \) |
\(\nu^{6}\) | \(=\) | \( ( - 1583 \beta_{11} + 2424 \beta_{10} - 7320 \beta_{7} - 6210 \beta_{6} + 28380 \beta_{5} - 7320 \beta_{4} + 6210 \beta_{3} + 28380 \beta_{2} - 865775 \beta_1 ) / 25 \) |
\(\nu^{7}\) | \(=\) | \( ( - 36613 \beta_{11} + 33104 \beta_{10} - 36613 \beta_{9} + 33104 \beta_{8} - 367476 \beta_{7} - 99225 \beta_{6} + 875700 \beta_{5} - 9996725 \beta _1 + 9996725 ) / 75 \) |
\(\nu^{8}\) | \(=\) | \( ( - 290297 \beta_{9} + 386016 \beta_{8} - 1424736 \beta_{7} - 920760 \beta_{6} + 4776480 \beta_{5} + 1424736 \beta_{4} - 920760 \beta_{3} - 4776480 \beta_{2} + 133078425 ) / 75 \) |
\(\nu^{9}\) | \(=\) | \( ( 1898923 \beta_{11} - 1876624 \beta_{10} - 1898923 \beta_{9} + 1876624 \beta_{8} + 19240308 \beta_{4} - 6627825 \beta_{3} - 49553100 \beta_{2} + 588239275 \beta _1 + 588239275 ) / 75 \) |
\(\nu^{10}\) | \(=\) | \( ( 17056301 \beta_{11} - 20989728 \beta_{10} + 85237080 \beta_{7} + 47378670 \beta_{6} - 265898760 \beta_{5} + 85237080 \beta_{4} - 47378670 \beta_{3} - 265898760 \beta_{2} + 7082438925 \beta_1 ) / 75 \) |
\(\nu^{11}\) | \(=\) | \( ( 101596857 \beta_{11} - 105815456 \beta_{10} + 101596857 \beta_{9} - 105815456 \beta_{8} + 1030566804 \beta_{7} + 402640425 \beta_{6} - 2776867500 \beta_{5} + \cdots - 33828064025 ) / 75 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
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−21.8383 | − | 21.8383i | −42.5089 | + | 42.5089i | 697.824i | 0 | 1856.64 | −432.570 | − | 432.570i | 9648.70 | − | 9648.70i | 2946.99i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
7.2 | −18.1270 | − | 18.1270i | 95.3921 | − | 95.3921i | 401.173i | 0 | −3458.34 | −1478.49 | − | 1478.49i | 2631.55 | − | 2631.55i | − | 11638.3i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
7.3 | −0.0371264 | − | 0.0371264i | −36.2471 | + | 36.2471i | − | 255.997i | 0 | 2.69145 | 1325.17 | + | 1325.17i | −19.0086 | + | 19.0086i | 3933.30i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
7.4 | 0.0371264 | + | 0.0371264i | 36.2471 | − | 36.2471i | − | 255.997i | 0 | 2.69145 | −1325.17 | − | 1325.17i | 19.0086 | − | 19.0086i | 3933.30i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
7.5 | 18.1270 | + | 18.1270i | −95.3921 | + | 95.3921i | 401.173i | 0 | −3458.34 | 1478.49 | + | 1478.49i | −2631.55 | + | 2631.55i | − | 11638.3i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
7.6 | 21.8383 | + | 21.8383i | 42.5089 | − | 42.5089i | 697.824i | 0 | 1856.64 | 432.570 | + | 432.570i | −9648.70 | + | 9648.70i | 2946.99i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
18.1 | −21.8383 | + | 21.8383i | −42.5089 | − | 42.5089i | − | 697.824i | 0 | 1856.64 | −432.570 | + | 432.570i | 9648.70 | + | 9648.70i | − | 2946.99i | 0 | |||||||||||||||||||||||||||||||||||||||||||||
18.2 | −18.1270 | + | 18.1270i | 95.3921 | + | 95.3921i | − | 401.173i | 0 | −3458.34 | −1478.49 | + | 1478.49i | 2631.55 | + | 2631.55i | 11638.3i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
18.3 | −0.0371264 | + | 0.0371264i | −36.2471 | − | 36.2471i | 255.997i | 0 | 2.69145 | 1325.17 | − | 1325.17i | −19.0086 | − | 19.0086i | − | 3933.30i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
18.4 | 0.0371264 | − | 0.0371264i | 36.2471 | + | 36.2471i | 255.997i | 0 | 2.69145 | −1325.17 | + | 1325.17i | 19.0086 | + | 19.0086i | − | 3933.30i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
18.5 | 18.1270 | − | 18.1270i | −95.3921 | − | 95.3921i | − | 401.173i | 0 | −3458.34 | 1478.49 | − | 1478.49i | −2631.55 | − | 2631.55i | 11638.3i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
18.6 | 21.8383 | − | 21.8383i | 42.5089 | + | 42.5089i | − | 697.824i | 0 | 1856.64 | 432.570 | − | 432.570i | −9648.70 | − | 9648.70i | − | 2946.99i | 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 | 25.9.c.c | ✓ | 12 |
5.b | even | 2 | 1 | inner | 25.9.c.c | ✓ | 12 |
5.c | odd | 4 | 2 | inner | 25.9.c.c | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
25.9.c.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
25.9.c.c | ✓ | 12 | 5.b | even | 2 | 1 | inner |
25.9.c.c | ✓ | 12 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 1341657T_{2}^{8} + 392912788608T_{2}^{4} + 2985984 \)
acting on \(S_{9}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 1341657 T^{8} + \cdots + 2985984 \)
$3$
\( T^{12} + 351180027 T^{8} + \cdots + 29\!\cdots\!29 \)
$5$
\( T^{12} \)
$7$
\( T^{12} + 31588550560512 T^{8} + \cdots + 33\!\cdots\!64 \)
$11$
\( (T^{3} + 29529 T^{2} + \cdots - 1580036164893)^{4} \)
$13$
\( T^{12} + \cdots + 10\!\cdots\!44 \)
$17$
\( T^{12} + \cdots + 63\!\cdots\!49 \)
$19$
\( (T^{6} + 59573487675 T^{4} + \cdots + 11\!\cdots\!25)^{2} \)
$23$
\( T^{12} + \cdots + 93\!\cdots\!84 \)
$29$
\( (T^{6} + 999419388300 T^{4} + \cdots + 97\!\cdots\!00)^{2} \)
$31$
\( (T^{3} - 805926 T^{2} + \cdots + 48\!\cdots\!12)^{4} \)
$37$
\( T^{12} + \cdots + 17\!\cdots\!44 \)
$41$
\( (T^{3} - 2735091 T^{2} + \cdots + 77\!\cdots\!27)^{4} \)
$43$
\( T^{12} + \cdots + 23\!\cdots\!64 \)
$47$
\( T^{12} + \cdots + 20\!\cdots\!04 \)
$53$
\( T^{12} + \cdots + 42\!\cdots\!04 \)
$59$
\( (T^{6} + 328367873378700 T^{4} + \cdots + 33\!\cdots\!00)^{2} \)
$61$
\( (T^{3} + 5899404 T^{2} + \cdots - 25\!\cdots\!68)^{4} \)
$67$
\( T^{12} + \cdots + 67\!\cdots\!49 \)
$71$
\( (T^{3} + 11967714 T^{2} + \cdots - 19\!\cdots\!28)^{4} \)
$73$
\( T^{12} + \cdots + 62\!\cdots\!09 \)
$79$
\( (T^{6} + \cdots + 67\!\cdots\!00)^{2} \)
$83$
\( T^{12} + \cdots + 13\!\cdots\!49 \)
$89$
\( (T^{6} + \cdots + 43\!\cdots\!25)^{2} \)
$97$
\( T^{12} + \cdots + 10\!\cdots\!04 \)
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