Newspace parameters
Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 180.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.90464475849\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.85100625.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | no (minimal twist has level 60) |
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 \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{7} + 3\nu^{6} + 4\nu^{5} + 7\nu^{4} - 17\nu^{3} - 8\nu^{2} + 24\nu + 8 ) / 16 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} - 3\nu^{6} + 4\nu^{5} + \nu^{4} + \nu^{3} - 8\nu^{2} + 8\nu + 8 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 5\nu^{4} - 5\nu^{3} + 16\nu^{2} - 8\nu - 24 ) / 16 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + \nu^{4} + 3\nu^{3} + 2\nu^{2} - 8\nu - 8 ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -5\nu^{7} - \nu^{6} + 4\nu^{5} + 3\nu^{4} - 5\nu^{3} - 24\nu^{2} + 24\nu + 56 ) / 16 \) |
\(\beta_{6}\) | \(=\) | \( ( 7\nu^{7} + 3\nu^{6} - 12\nu^{5} - 9\nu^{4} + 15\nu^{3} + 32\nu^{2} - 8\nu - 88 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( ( -15\nu^{7} - 11\nu^{6} + 20\nu^{5} + 25\nu^{4} - 23\nu^{3} - 48\nu^{2} + 40\nu + 168 ) / 16 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + 2\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 8 \) |
\(\nu^{2}\) | \(=\) | \( ( 3\beta_{7} + 2\beta_{6} - 6\beta_{5} - 3\beta_{4} - \beta_{2} - \beta _1 + 7 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - 8\beta_{3} + \beta_{2} - 3\beta _1 + 5 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{7} + 2\beta_{6} - 6\beta_{5} + 5\beta_{4} - 8\beta_{3} - \beta_{2} + 7\beta _1 + 7 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( \beta_{7} - 2\beta_{6} - 18\beta_{5} - 9\beta_{4} - 8\beta_{3} + 5\beta_{2} - 3\beta _1 - 3 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( -5\beta_{7} - 2\beta_{6} - 6\beta_{5} - 3\beta_{4} - 16\beta_{3} - 5\beta_{2} + 7\beta _1 + 23 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( -7\beta_{7} - 2\beta_{6} - 2\beta_{5} + 7\beta_{4} + 13\beta_{2} + 13\beta _1 + 53 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(91\) | \(101\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 |
|
−1.87477 | − | 0.696577i | 0 | 3.02956 | + | 2.61185i | 2.23607 | 0 | − | 5.46770i | −3.86039 | − | 7.00695i | 0 | −4.19212 | − | 1.55759i | |||||||||||||||||||||||||||||||||
91.2 | −1.87477 | + | 0.696577i | 0 | 3.02956 | − | 2.61185i | 2.23607 | 0 | 5.46770i | −3.86039 | + | 7.00695i | 0 | −4.19212 | + | 1.55759i | |||||||||||||||||||||||||||||||||||
91.3 | −1.67986 | − | 1.08539i | 0 | 1.64388 | + | 3.64660i | −2.23607 | 0 | − | 0.596540i | 1.19648 | − | 7.91002i | 0 | 3.75629 | + | 2.42700i | ||||||||||||||||||||||||||||||||||
91.4 | −1.67986 | + | 1.08539i | 0 | 1.64388 | − | 3.64660i | −2.23607 | 0 | 0.596540i | 1.19648 | + | 7.91002i | 0 | 3.75629 | − | 2.42700i | |||||||||||||||||||||||||||||||||||
91.5 | −0.438172 | − | 1.95141i | 0 | −3.61601 | + | 1.71011i | −2.23607 | 0 | 6.33166i | 4.92155 | + | 6.30701i | 0 | 0.979781 | + | 4.36349i | |||||||||||||||||||||||||||||||||||
91.6 | −0.438172 | + | 1.95141i | 0 | −3.61601 | − | 1.71011i | −2.23607 | 0 | − | 6.33166i | 4.92155 | − | 6.30701i | 0 | 0.979781 | − | 4.36349i | ||||||||||||||||||||||||||||||||||
91.7 | 1.99281 | − | 0.169449i | 0 | 3.94257 | − | 0.675358i | 2.23607 | 0 | 12.3959i | 7.74236 | − | 2.01392i | 0 | 4.45606 | − | 0.378899i | |||||||||||||||||||||||||||||||||||
91.8 | 1.99281 | + | 0.169449i | 0 | 3.94257 | + | 0.675358i | 2.23607 | 0 | − | 12.3959i | 7.74236 | + | 2.01392i | 0 | 4.45606 | + | 0.378899i | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 180.3.c.b | 8 | |
3.b | odd | 2 | 1 | 60.3.c.a | ✓ | 8 | |
4.b | odd | 2 | 1 | inner | 180.3.c.b | 8 | |
5.b | even | 2 | 1 | 900.3.c.u | 8 | ||
5.c | odd | 4 | 2 | 900.3.f.f | 16 | ||
8.b | even | 2 | 1 | 2880.3.e.j | 8 | ||
8.d | odd | 2 | 1 | 2880.3.e.j | 8 | ||
12.b | even | 2 | 1 | 60.3.c.a | ✓ | 8 | |
15.d | odd | 2 | 1 | 300.3.c.d | 8 | ||
15.e | even | 4 | 2 | 300.3.f.b | 16 | ||
20.d | odd | 2 | 1 | 900.3.c.u | 8 | ||
20.e | even | 4 | 2 | 900.3.f.f | 16 | ||
24.f | even | 2 | 1 | 960.3.e.c | 8 | ||
24.h | odd | 2 | 1 | 960.3.e.c | 8 | ||
60.h | even | 2 | 1 | 300.3.c.d | 8 | ||
60.l | odd | 4 | 2 | 300.3.f.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.3.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
60.3.c.a | ✓ | 8 | 12.b | even | 2 | 1 | |
180.3.c.b | 8 | 1.a | even | 1 | 1 | trivial | |
180.3.c.b | 8 | 4.b | odd | 2 | 1 | inner | |
300.3.c.d | 8 | 15.d | odd | 2 | 1 | ||
300.3.c.d | 8 | 60.h | even | 2 | 1 | ||
300.3.f.b | 16 | 15.e | even | 4 | 2 | ||
300.3.f.b | 16 | 60.l | odd | 4 | 2 | ||
900.3.c.u | 8 | 5.b | even | 2 | 1 | ||
900.3.c.u | 8 | 20.d | odd | 2 | 1 | ||
900.3.f.f | 16 | 5.c | odd | 4 | 2 | ||
900.3.f.f | 16 | 20.e | even | 4 | 2 | ||
960.3.e.c | 8 | 24.f | even | 2 | 1 | ||
960.3.e.c | 8 | 24.h | odd | 2 | 1 | ||
2880.3.e.j | 8 | 8.b | even | 2 | 1 | ||
2880.3.e.j | 8 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 224T_{7}^{6} + 12032T_{7}^{4} + 188416T_{7}^{2} + 65536 \)
acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 4 T^{7} + 3 T^{6} - 16 T^{5} + \cdots + 256 \)
$3$
\( T^{8} \)
$5$
\( (T^{2} - 5)^{4} \)
$7$
\( T^{8} + 224 T^{6} + 12032 T^{4} + \cdots + 65536 \)
$11$
\( (T^{4} + 208 T^{2} + 10496)^{2} \)
$13$
\( (T^{4} - 8 T^{3} - 472 T^{2} + 5792 T - 12464)^{2} \)
$17$
\( (T^{4} - 424 T^{2} - 3840 T - 8816)^{2} \)
$19$
\( T^{8} + 1696 T^{6} + \cdots + 6544162816 \)
$23$
\( T^{8} + 3616 T^{6} + \cdots + 101419319296 \)
$29$
\( (T^{4} + 32 T^{3} - 2152 T^{2} + \cdots + 1334416)^{2} \)
$31$
\( T^{8} + 5408 T^{6} + \cdots + 59895709696 \)
$37$
\( (T^{4} + 56 T^{3} - 1528 T^{2} + \cdots - 244784)^{2} \)
$41$
\( (T^{4} - 8 T^{3} - 1800 T^{2} + \cdots + 87184)^{2} \)
$43$
\( T^{8} + 10816 T^{6} + \cdots + 33624411406336 \)
$47$
\( T^{8} + 8032 T^{6} + \cdots + 1056981385216 \)
$53$
\( (T^{4} + 176 T^{3} + 9752 T^{2} + \cdots - 478064)^{2} \)
$59$
\( T^{8} + 4896 T^{6} + \cdots + 173909016576 \)
$61$
\( (T^{4} + 88 T^{3} - 2536 T^{2} + \cdots - 2142704)^{2} \)
$67$
\( T^{8} + 16064 T^{6} + \cdots + 281086590976 \)
$71$
\( T^{8} + 13952 T^{6} + \cdots + 16079971680256 \)
$73$
\( (T^{4} + 120 T^{3} - 1576 T^{2} + \cdots + 4962064)^{2} \)
$79$
\( T^{8} + 41888 T^{6} + \cdots + 31\!\cdots\!36 \)
$83$
\( T^{8} + 36928 T^{6} + \cdots + 42\!\cdots\!36 \)
$89$
\( (T^{4} + 40 T^{3} - 20584 T^{2} + \cdots + 70652944)^{2} \)
$97$
\( (T^{4} - 216 T^{3} + 5880 T^{2} + \cdots - 59281776)^{2} \)
show more
show less