Newspace parameters
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.44263734204\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{12}\cdot 3^{3} \) |
Twist minimal: | no (minimal twist has level 24) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) :
\(\beta_{1}\) | \(=\) | \( ( -280\nu^{7} + 1019\nu^{6} - 5633\nu^{5} + 20827\nu^{4} - 39823\nu^{3} + 101396\nu^{2} + 61086\nu - 151668 ) / 32244 \) |
\(\beta_{2}\) | \(=\) | \( ( 279\nu^{7} + 165\nu^{6} + 6371\nu^{5} + 2845\nu^{4} + 48212\nu^{3} + 31512\nu^{2} + 12660\nu + 24108 ) / 5374 \) |
\(\beta_{3}\) | \(=\) | \( ( - 3503 \nu^{7} + 1511 \nu^{6} - 72229 \nu^{5} + 63997 \nu^{4} - 468590 \nu^{3} + 172202 \nu^{2} + 106164 \nu - 714696 ) / 64488 \) |
\(\beta_{4}\) | \(=\) | \( ( - 2777 \nu^{7} - 920 \nu^{6} - 51808 \nu^{5} - 2998 \nu^{4} - 279485 \nu^{3} - 188894 \nu^{2} + 623634 \nu + 30708 ) / 32244 \) |
\(\beta_{5}\) | \(=\) | \( ( 1138 \nu^{7} - 1205 \nu^{6} + 22683 \nu^{5} - 33805 \nu^{4} + 145327 \nu^{3} - 164668 \nu^{2} - 126166 \nu + 86776 ) / 10748 \) |
\(\beta_{6}\) | \(=\) | \( ( 7859 \nu^{7} - 5349 \nu^{6} + 162511 \nu^{5} - 154275 \nu^{4} + 1119560 \nu^{3} - 619098 \nu^{2} + 96696 \nu + 543984 ) / 64488 \) |
\(\beta_{7}\) | \(=\) | \( ( 11979 \nu^{7} + 1537 \nu^{6} + 244245 \nu^{5} - 42709 \nu^{4} + 1439514 \nu^{3} + 205894 \nu^{2} - 1578300 \nu + 127056 ) / 64488 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 7\beta_{6} - 7\beta_{5} + \beta_{4} - 2\beta_{3} - 6\beta_{2} + \beta _1 + 4 ) / 72 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} - 10\beta_{3} - 14\beta _1 - 124 ) / 24 \) |
\(\nu^{3}\) | \(=\) | \( ( -37\beta_{7} - 25\beta_{6} + 61\beta_{5} - 25\beta_{4} + 20\beta_{3} + 60\beta_{2} + 83\beta _1 + 158 ) / 72 \) |
\(\nu^{4}\) | \(=\) | \( ( 19\beta_{7} + 43\beta_{6} - 13\beta_{5} + 6\beta_{4} + 118\beta_{3} - 12\beta_{2} - 18\beta _1 + 976 ) / 24 \) |
\(\nu^{5}\) | \(=\) | \( ( 541\beta_{7} - 119\beta_{6} - 529\beta_{5} + 589\beta_{4} - 452\beta_{3} - 312\beta_{2} - 1427\beta _1 - 6674 ) / 72 \) |
\(\nu^{6}\) | \(=\) | \( ( -105\beta_{7} - 269\beta_{6} + 311\beta_{5} + 6\beta_{4} - 382\beta_{3} + 132\beta_{2} + 890\beta _1 - 680 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( - 5689 \beta_{7} + 7175 \beta_{6} + 2293 \beta_{5} - 8713 \beta_{4} + 8768 \beta_{3} - 1920 \beta_{2} + 18755 \beta _1 + 134474 ) / 72 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(55\) | \(65\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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19.1 |
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−2.93333 | − | 2.71948i | 0 | 1.20889 | + | 15.9543i | 3.88698i | 0 | − | 23.9200i | 39.8412 | − | 50.0867i | 0 | 10.5706 | − | 11.4018i | |||||||||||||||||||||||||||||||||
19.2 | −2.93333 | + | 2.71948i | 0 | 1.20889 | − | 15.9543i | − | 3.88698i | 0 | 23.9200i | 39.8412 | + | 50.0867i | 0 | 10.5706 | + | 11.4018i | ||||||||||||||||||||||||||||||||||
19.3 | −1.03857 | − | 3.86282i | 0 | −13.8428 | + | 8.02360i | − | 34.2464i | 0 | − | 9.22331i | 45.3703 | + | 45.1390i | 0 | −132.288 | + | 35.5672i | |||||||||||||||||||||||||||||||||
19.4 | −1.03857 | + | 3.86282i | 0 | −13.8428 | − | 8.02360i | 34.2464i | 0 | 9.22331i | 45.3703 | − | 45.1390i | 0 | −132.288 | − | 35.5672i | |||||||||||||||||||||||||||||||||||
19.5 | 3.40459 | − | 2.09970i | 0 | 7.18250 | − | 14.2973i | 13.7025i | 0 | − | 88.3075i | −5.56649 | − | 63.7575i | 0 | 28.7711 | + | 46.6513i | ||||||||||||||||||||||||||||||||||
19.6 | 3.40459 | + | 2.09970i | 0 | 7.18250 | + | 14.2973i | − | 13.7025i | 0 | 88.3075i | −5.56649 | + | 63.7575i | 0 | 28.7711 | − | 46.6513i | ||||||||||||||||||||||||||||||||||
19.7 | 3.56731 | − | 1.80951i | 0 | 9.45137 | − | 12.9101i | − | 38.1617i | 0 | 42.0542i | 10.3550 | − | 63.1567i | 0 | −69.0539 | − | 136.135i | ||||||||||||||||||||||||||||||||||
19.8 | 3.56731 | + | 1.80951i | 0 | 9.45137 | + | 12.9101i | 38.1617i | 0 | − | 42.0542i | 10.3550 | + | 63.1567i | 0 | −69.0539 | + | 136.135i | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 72.5.b.d | 8 | |
3.b | odd | 2 | 1 | 24.5.b.a | ✓ | 8 | |
4.b | odd | 2 | 1 | 288.5.b.d | 8 | ||
8.b | even | 2 | 1 | 288.5.b.d | 8 | ||
8.d | odd | 2 | 1 | inner | 72.5.b.d | 8 | |
12.b | even | 2 | 1 | 96.5.b.a | 8 | ||
24.f | even | 2 | 1 | 24.5.b.a | ✓ | 8 | |
24.h | odd | 2 | 1 | 96.5.b.a | 8 | ||
48.i | odd | 4 | 2 | 768.5.g.k | 16 | ||
48.k | even | 4 | 2 | 768.5.g.k | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.5.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
24.5.b.a | ✓ | 8 | 24.f | even | 2 | 1 | |
72.5.b.d | 8 | 1.a | even | 1 | 1 | trivial | |
72.5.b.d | 8 | 8.d | odd | 2 | 1 | inner | |
96.5.b.a | 8 | 12.b | even | 2 | 1 | ||
96.5.b.a | 8 | 24.h | odd | 2 | 1 | ||
288.5.b.d | 8 | 4.b | odd | 2 | 1 | ||
288.5.b.d | 8 | 8.b | even | 2 | 1 | ||
768.5.g.k | 16 | 48.i | odd | 4 | 2 | ||
768.5.g.k | 16 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 2832T_{5}^{6} + 2244192T_{5}^{4} + 353952000T_{5}^{2} + 4845166848 \)
acting on \(S_{5}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 6 T^{7} + 14 T^{6} + \cdots + 65536 \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 2832 T^{6} + \cdots + 4845166848 \)
$7$
\( T^{8} + 10224 T^{6} + \cdots + 671288262912 \)
$11$
\( (T^{4} + 96 T^{3} - 40064 T^{2} + \cdots + 193957888)^{2} \)
$13$
\( T^{8} + 155136 T^{6} + \cdots + 56\!\cdots\!12 \)
$17$
\( (T^{4} + 120 T^{3} - 96872 T^{2} + \cdots - 1023349232)^{2} \)
$19$
\( (T^{4} + 352 T^{3} - 220128 T^{2} + \cdots - 5712629504)^{2} \)
$23$
\( T^{8} + 1644480 T^{6} + \cdots + 64\!\cdots\!92 \)
$29$
\( T^{8} + 4596240 T^{6} + \cdots + 22\!\cdots\!88 \)
$31$
\( T^{8} + 2201712 T^{6} + \cdots + 15\!\cdots\!12 \)
$37$
\( T^{8} + 6796224 T^{6} + \cdots + 78\!\cdots\!68 \)
$41$
\( (T^{4} + 360 T^{3} + \cdots + 226033840912)^{2} \)
$43$
\( (T^{4} - 5024 T^{3} + \cdots + 2240667904)^{2} \)
$47$
\( T^{8} + 14015424 T^{6} + \cdots + 13\!\cdots\!08 \)
$53$
\( T^{8} + 27560592 T^{6} + \cdots + 86\!\cdots\!48 \)
$59$
\( (T^{4} + 6528 T^{3} + \cdots - 21853379868416)^{2} \)
$61$
\( T^{8} + 79948992 T^{6} + \cdots + 34\!\cdots\!12 \)
$67$
\( (T^{4} + 3328 T^{3} + \cdots - 30681172039424)^{2} \)
$71$
\( T^{8} + 82031040 T^{6} + \cdots + 64\!\cdots\!92 \)
$73$
\( (T^{4} + 8440 T^{3} + \cdots - 35664142829552)^{2} \)
$79$
\( T^{8} + 162047472 T^{6} + \cdots + 43\!\cdots\!52 \)
$83$
\( (T^{4} - 12000 T^{3} + \cdots + 15292937408512)^{2} \)
$89$
\( (T^{4} + 7800 T^{3} + \cdots + 219207569912848)^{2} \)
$97$
\( (T^{4} + 6088 T^{3} + \cdots - 29\!\cdots\!68)^{2} \)
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