Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.598878015160\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{6})\) |
Defining polynomial: |
\( x^{4} + 9 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
\(n\) | \(26\) | \(52\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 |
|
0 | −1.22474 | + | 1.22474i | 2.00000i | 0 | 0 | 1.22474 | + | 1.22474i | 0 | − | 3.00000i | 0 | |||||||||||||||||||||||||
32.2 | 0 | 1.22474 | − | 1.22474i | 2.00000i | 0 | 0 | −1.22474 | − | 1.22474i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
68.1 | 0 | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 0 | 1.22474 | − | 1.22474i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||
68.2 | 0 | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | 0 | −1.22474 | + | 1.22474i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
15.d | odd | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.2.e.b | ✓ | 4 |
3.b | odd | 2 | 1 | CM | 75.2.e.b | ✓ | 4 |
4.b | odd | 2 | 1 | 1200.2.v.f | 4 | ||
5.b | even | 2 | 1 | inner | 75.2.e.b | ✓ | 4 |
5.c | odd | 4 | 2 | inner | 75.2.e.b | ✓ | 4 |
12.b | even | 2 | 1 | 1200.2.v.f | 4 | ||
15.d | odd | 2 | 1 | inner | 75.2.e.b | ✓ | 4 |
15.e | even | 4 | 2 | inner | 75.2.e.b | ✓ | 4 |
20.d | odd | 2 | 1 | 1200.2.v.f | 4 | ||
20.e | even | 4 | 2 | 1200.2.v.f | 4 | ||
60.h | even | 2 | 1 | 1200.2.v.f | 4 | ||
60.l | odd | 4 | 2 | 1200.2.v.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.2.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
75.2.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | CM |
75.2.e.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
75.2.e.b | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
75.2.e.b | ✓ | 4 | 15.d | odd | 2 | 1 | inner |
75.2.e.b | ✓ | 4 | 15.e | even | 4 | 2 | inner |
1200.2.v.f | 4 | 4.b | odd | 2 | 1 | ||
1200.2.v.f | 4 | 12.b | even | 2 | 1 | ||
1200.2.v.f | 4 | 20.d | odd | 2 | 1 | ||
1200.2.v.f | 4 | 20.e | even | 4 | 2 | ||
1200.2.v.f | 4 | 60.h | even | 2 | 1 | ||
1200.2.v.f | 4 | 60.l | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 9 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 9 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 729 \)
$17$
\( T^{4} \)
$19$
\( (T^{2} + 1)^{2} \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( (T - 7)^{4} \)
$37$
\( T^{4} + 2304 \)
$41$
\( T^{4} \)
$43$
\( T^{4} + 21609 \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} \)
$61$
\( (T + 13)^{4} \)
$67$
\( T^{4} + 59049 \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 36864 \)
$79$
\( (T^{2} + 16)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} + 131769 \)
show more
show less