Newspace parameters
Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 180.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.6203438010\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(91\) | \(101\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
61.1 |
|
0 | 4.50000 | − | 2.59808i | 0 | 2.50000 | + | 4.33013i | 0 | 3.50000 | − | 6.06218i | 0 | 13.5000 | − | 23.3827i | 0 | ||||||||||||||||
121.1 | 0 | 4.50000 | + | 2.59808i | 0 | 2.50000 | − | 4.33013i | 0 | 3.50000 | + | 6.06218i | 0 | 13.5000 | + | 23.3827i | 0 | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 180.4.i.a | ✓ | 2 |
3.b | odd | 2 | 1 | 540.4.i.a | 2 | ||
9.c | even | 3 | 1 | inner | 180.4.i.a | ✓ | 2 |
9.c | even | 3 | 1 | 1620.4.a.a | 1 | ||
9.d | odd | 6 | 1 | 540.4.i.a | 2 | ||
9.d | odd | 6 | 1 | 1620.4.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
180.4.i.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
180.4.i.a | ✓ | 2 | 9.c | even | 3 | 1 | inner |
540.4.i.a | 2 | 3.b | odd | 2 | 1 | ||
540.4.i.a | 2 | 9.d | odd | 6 | 1 | ||
1620.4.a.a | 1 | 9.c | even | 3 | 1 | ||
1620.4.a.b | 1 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 7T_{7} + 49 \)
acting on \(S_{4}^{\mathrm{new}}(180, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 9T + 27 \)
$5$
\( T^{2} - 5T + 25 \)
$7$
\( T^{2} - 7T + 49 \)
$11$
\( T^{2} - 30T + 900 \)
$13$
\( T^{2} - 22T + 484 \)
$17$
\( (T - 48)^{2} \)
$19$
\( (T - 68)^{2} \)
$23$
\( T^{2} + 111T + 12321 \)
$29$
\( T^{2} + 87T + 7569 \)
$31$
\( T^{2} + 20T + 400 \)
$37$
\( (T - 200)^{2} \)
$41$
\( T^{2} + 69T + 4761 \)
$43$
\( T^{2} - 232T + 53824 \)
$47$
\( T^{2} + 243T + 59049 \)
$53$
\( (T + 498)^{2} \)
$59$
\( T^{2} + 66T + 4356 \)
$61$
\( T^{2} + 359T + 128881 \)
$67$
\( T^{2} - 1063 T + 1129969 \)
$71$
\( (T + 618)^{2} \)
$73$
\( (T + 532)^{2} \)
$79$
\( T^{2} + 410T + 168100 \)
$83$
\( T^{2} + 693T + 480249 \)
$89$
\( (T + 1599)^{2} \)
$97$
\( T^{2} + 50T + 2500 \)
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