Newspace parameters
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.k (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(69.0162250860\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.469950251728896.64 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 98x^{6} - 12x^{5} + 3571x^{4} + 576x^{3} + 53874x^{2} + 20916x + 343098 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{12}\cdot 3^{2}\cdot 5^{4} \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 98x^{6} - 12x^{5} + 3571x^{4} + 576x^{3} + 53874x^{2} + 20916x + 343098 \) :
\(\beta_{1}\) | \(=\) | \( ( 1992230 \nu^{7} + 6919103 \nu^{6} + 273872758 \nu^{5} + 823212151 \nu^{4} + 13841503414 \nu^{3} + 28892850095 \nu^{2} + \cdots + 341852125167 ) / 195543175135 \) |
\(\beta_{2}\) | \(=\) | \( ( 1490 \nu^{7} - 4586 \nu^{6} + 109894 \nu^{5} - 575137 \nu^{4} + 2716082 \nu^{3} - 17767115 \nu^{2} + 15960150 \nu - 174345129 ) / 61407885 \) |
\(\beta_{3}\) | \(=\) | \( ( - 40406470 \nu^{7} + 4736544 \nu^{6} - 2908689491 \nu^{5} + 1002126303 \nu^{4} - 66384131413 \nu^{3} - 19303151040 \nu^{2} + \cdots - 97724147409 ) / 586629525405 \) |
\(\beta_{4}\) | \(=\) | \( ( 488 \nu^{7} + 290524 \nu^{6} + 29712 \nu^{5} + 21402716 \nu^{4} - 4687192 \nu^{3} + 513139420 \nu^{2} + 124879372 \nu + 3235482684 ) / 4093859 \) |
\(\beta_{5}\) | \(=\) | \( ( 14964200 \nu^{7} - 1311680 \nu^{6} + 1538887960 \nu^{5} - 317149060 \nu^{4} + 62283286760 \nu^{3} + 2073057700 \nu^{2} + \cdots + 220339276380 ) / 5586947861 \) |
\(\beta_{6}\) | \(=\) | \( ( - 36224 \nu^{7} - 89472 \nu^{6} - 3547756 \nu^{5} - 3517188 \nu^{4} - 107094404 \nu^{3} + 201588480 \nu^{2} - 1013561676 \nu + 5942434428 ) / 12281577 \) |
\(\beta_{7}\) | \(=\) | \( ( - 1152 \nu^{7} - 4656 \nu^{6} - 112608 \nu^{5} - 326664 \nu^{4} - 3341952 \nu^{3} - 8668080 \nu^{2} - 30894048 \nu - 72152556 ) / 66871 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} - 60\beta_{2} - 120\beta_1 ) / 120 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} + 6\beta_{6} + 120\beta_{3} + 360\beta_{2} - 2940 ) / 120 \) |
\(\nu^{3}\) | \(=\) | \( ( 9\beta_{7} - 25\beta_{5} + 9\beta_{4} - 360\beta_{3} + 4380\beta_{2} + 8820\beta _1 + 540 ) / 120 \) |
\(\nu^{4}\) | \(=\) | \( ( 49\beta_{7} - 300\beta_{6} + 18\beta_{5} - 36\beta_{4} - 17520\beta_{3} - 52920\beta_{2} - 720\beta _1 + 73860 ) / 120 \) |
\(\nu^{5}\) | \(=\) | \( ( - 750 \beta_{7} + 90 \beta_{6} + 604 \beta_{5} - 735 \beta_{4} + 88200 \beta_{3} - 177360 \beta_{2} - 373620 \beta _1 - 131400 ) / 120 \) |
\(\nu^{6}\) | \(=\) | \( ( - 338 \beta_{7} + 2304 \beta_{6} - 441 \beta_{5} + 900 \beta_{4} + 215424 \beta_{3} + 673812 \beta_{2} + 52560 \beta _1 - 316344 ) / 24 \) |
\(\nu^{7}\) | \(=\) | \( ( 40698 \beta_{7} - 15435 \beta_{6} - 9526 \beta_{5} + 37758 \beta_{4} - 7865460 \beta_{3} + 4920360 \beta_{2} + 13294680 \beta _1 + 11332440 ) / 120 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(151\) | \(277\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
157.1 |
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0 | −11.0227 | + | 11.0227i | 0 | 0 | 0 | −249.820 | − | 249.820i | 0 | − | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||
157.2 | 0 | −11.0227 | + | 11.0227i | 0 | 0 | 0 | 232.674 | + | 232.674i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
157.3 | 0 | 11.0227 | − | 11.0227i | 0 | 0 | 0 | −232.674 | − | 232.674i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
157.4 | 0 | 11.0227 | − | 11.0227i | 0 | 0 | 0 | 249.820 | + | 249.820i | 0 | − | 243.000i | 0 | ||||||||||||||||||||||||||||||||||||||
193.1 | 0 | −11.0227 | − | 11.0227i | 0 | 0 | 0 | −249.820 | + | 249.820i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
193.2 | 0 | −11.0227 | − | 11.0227i | 0 | 0 | 0 | 232.674 | − | 232.674i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
193.3 | 0 | 11.0227 | + | 11.0227i | 0 | 0 | 0 | −232.674 | + | 232.674i | 0 | 243.000i | 0 | |||||||||||||||||||||||||||||||||||||||
193.4 | 0 | 11.0227 | + | 11.0227i | 0 | 0 | 0 | 249.820 | − | 249.820i | 0 | 243.000i | 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 | 300.7.k.b | ✓ | 8 |
5.b | even | 2 | 1 | inner | 300.7.k.b | ✓ | 8 |
5.c | odd | 4 | 2 | inner | 300.7.k.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.7.k.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
300.7.k.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
300.7.k.b | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 27303292818T_{7}^{4} + 182648738100865680081 \)
acting on \(S_{7}^{\mathrm{new}}(300, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 59049)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 27303292818 T^{4} + \cdots + 18\!\cdots\!81 \)
$11$
\( (T^{2} - 1212 T - 1029564)^{4} \)
$13$
\( T^{8} + 94292537951250 T^{4} + \cdots + 77\!\cdots\!25 \)
$17$
\( T^{8} + \cdots + 11\!\cdots\!96 \)
$19$
\( (T^{4} + 72003650 T^{2} + \cdots + 75616502850625)^{2} \)
$23$
\( T^{8} + \cdots + 49\!\cdots\!56 \)
$29$
\( (T^{4} + 1139458248 T^{2} + \cdots + 29\!\cdots\!76)^{2} \)
$31$
\( (T^{2} + 4102 T - 904062599)^{4} \)
$37$
\( T^{8} + \cdots + 48\!\cdots\!16 \)
$41$
\( (T^{2} - 64920 T - 1294369200)^{4} \)
$43$
\( T^{8} + \cdots + 16\!\cdots\!01 \)
$47$
\( T^{8} + \cdots + 41\!\cdots\!36 \)
$53$
\( T^{8} + \cdots + 21\!\cdots\!96 \)
$59$
\( (T^{4} + 5469627528 T^{2} + \cdots + 74\!\cdots\!96)^{2} \)
$61$
\( (T^{2} - 211294 T + 9593729809)^{4} \)
$67$
\( T^{8} + \cdots + 81\!\cdots\!01 \)
$71$
\( (T^{2} - 148440 T - 78334311600)^{4} \)
$73$
\( T^{8} + \cdots + 53\!\cdots\!36 \)
$79$
\( (T^{4} + 1200234542792 T^{2} + \cdots + 30\!\cdots\!16)^{2} \)
$83$
\( T^{8} + \cdots + 25\!\cdots\!56 \)
$89$
\( (T^{4} + 2279902413600 T^{2} + \cdots + 12\!\cdots\!00)^{2} \)
$97$
\( T^{8} + \cdots + 27\!\cdots\!41 \)
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