Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(30.5533957546\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{6})\) |
Defining polynomial: |
\( x^{4} + 9 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
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 \(1,\beta_1,\beta_2,\beta_3\) 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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−7.34847 | − | 7.34847i | −33.0681 | + | 33.0681i | − | 148.000i | 0 | 486.000 | −2872.03 | − | 2872.03i | −2968.78 | + | 2968.78i | − | 2187.00i | 0 | ||||||||||||||||||||
7.2 | 7.34847 | + | 7.34847i | 33.0681 | − | 33.0681i | − | 148.000i | 0 | 486.000 | 2872.03 | + | 2872.03i | 2968.78 | − | 2968.78i | − | 2187.00i | 0 | |||||||||||||||||||||
43.1 | −7.34847 | + | 7.34847i | −33.0681 | − | 33.0681i | 148.000i | 0 | 486.000 | −2872.03 | + | 2872.03i | −2968.78 | − | 2968.78i | 2187.00i | 0 | |||||||||||||||||||||||
43.2 | 7.34847 | − | 7.34847i | 33.0681 | + | 33.0681i | 148.000i | 0 | 486.000 | 2872.03 | − | 2872.03i | 2968.78 | + | 2968.78i | 2187.00i | 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 | 75.9.f.a | ✓ | 4 |
5.b | even | 2 | 1 | inner | 75.9.f.a | ✓ | 4 |
5.c | odd | 4 | 2 | inner | 75.9.f.a | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.9.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
75.9.f.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
75.9.f.a | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 11664 \)
acting on \(S_{9}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 11664 \)
$3$
\( T^{4} + 4782969 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + \cdots + 272153483555625 \)
$11$
\( (T + 234)^{4} \)
$13$
\( T^{4} + 81\!\cdots\!49 \)
$17$
\( T^{4} + 25\!\cdots\!04 \)
$19$
\( (T^{2} + 33012346249)^{2} \)
$23$
\( T^{4} + 21\!\cdots\!00 \)
$29$
\( (T^{2} + 57683550276)^{2} \)
$31$
\( (T - 836725)^{4} \)
$37$
\( T^{4} + 54\!\cdots\!00 \)
$41$
\( (T - 2822220)^{4} \)
$43$
\( T^{4} + 24\!\cdots\!49 \)
$47$
\( T^{4} + 33\!\cdots\!24 \)
$53$
\( T^{4} + 80\!\cdots\!00 \)
$59$
\( (T^{2} + 163480721977764)^{2} \)
$61$
\( (T - 517403)^{4} \)
$67$
\( T^{4} + 72\!\cdots\!69 \)
$71$
\( (T + 20828628)^{4} \)
$73$
\( T^{4} + 27\!\cdots\!00 \)
$79$
\( (T^{2} + 17\!\cdots\!00)^{2} \)
$83$
\( T^{4} + 79\!\cdots\!24 \)
$89$
\( (T^{2} + 90\!\cdots\!84)^{2} \)
$97$
\( T^{4} + 10\!\cdots\!69 \)
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