Newspace parameters
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.39738203537\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{10})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 25 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
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 \( x^{4} + 25 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 5\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 5\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
118.1 |
|
1.00000 | − | 1.00000i | −1.58114 | + | 1.58114i | 0 | 0 | 3.16228i | 0.581139 | + | 2.58114i | 2.00000 | + | 2.00000i | − | 2.00000i | 0 | |||||||||||||||||||||
118.2 | 1.00000 | − | 1.00000i | 1.58114 | − | 1.58114i | 0 | 0 | − | 3.16228i | −2.58114 | − | 0.581139i | 2.00000 | + | 2.00000i | − | 2.00000i | 0 | |||||||||||||||||||||
132.1 | 1.00000 | + | 1.00000i | −1.58114 | − | 1.58114i | 0 | 0 | − | 3.16228i | 0.581139 | − | 2.58114i | 2.00000 | − | 2.00000i | 2.00000i | 0 | ||||||||||||||||||||||
132.2 | 1.00000 | + | 1.00000i | 1.58114 | + | 1.58114i | 0 | 0 | 3.16228i | −2.58114 | + | 0.581139i | 2.00000 | − | 2.00000i | 2.00000i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.2.f.c | 4 | |
5.b | even | 2 | 1 | 35.2.f.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 35.2.f.a | ✓ | 4 | |
5.c | odd | 4 | 1 | inner | 175.2.f.c | 4 | |
7.b | odd | 2 | 1 | inner | 175.2.f.c | 4 | |
15.d | odd | 2 | 1 | 315.2.p.c | 4 | ||
15.e | even | 4 | 1 | 315.2.p.c | 4 | ||
20.d | odd | 2 | 1 | 560.2.bj.a | 4 | ||
20.e | even | 4 | 1 | 560.2.bj.a | 4 | ||
35.c | odd | 2 | 1 | 35.2.f.a | ✓ | 4 | |
35.f | even | 4 | 1 | 35.2.f.a | ✓ | 4 | |
35.f | even | 4 | 1 | inner | 175.2.f.c | 4 | |
35.i | odd | 6 | 2 | 245.2.l.c | 8 | ||
35.j | even | 6 | 2 | 245.2.l.c | 8 | ||
35.k | even | 12 | 2 | 245.2.l.c | 8 | ||
35.l | odd | 12 | 2 | 245.2.l.c | 8 | ||
105.g | even | 2 | 1 | 315.2.p.c | 4 | ||
105.k | odd | 4 | 1 | 315.2.p.c | 4 | ||
140.c | even | 2 | 1 | 560.2.bj.a | 4 | ||
140.j | odd | 4 | 1 | 560.2.bj.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.f.a | ✓ | 4 | 5.b | even | 2 | 1 | |
35.2.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
35.2.f.a | ✓ | 4 | 35.c | odd | 2 | 1 | |
35.2.f.a | ✓ | 4 | 35.f | even | 4 | 1 | |
175.2.f.c | 4 | 1.a | even | 1 | 1 | trivial | |
175.2.f.c | 4 | 5.c | odd | 4 | 1 | inner | |
175.2.f.c | 4 | 7.b | odd | 2 | 1 | inner | |
175.2.f.c | 4 | 35.f | even | 4 | 1 | inner | |
245.2.l.c | 8 | 35.i | odd | 6 | 2 | ||
245.2.l.c | 8 | 35.j | even | 6 | 2 | ||
245.2.l.c | 8 | 35.k | even | 12 | 2 | ||
245.2.l.c | 8 | 35.l | odd | 12 | 2 | ||
315.2.p.c | 4 | 15.d | odd | 2 | 1 | ||
315.2.p.c | 4 | 15.e | even | 4 | 1 | ||
315.2.p.c | 4 | 105.g | even | 2 | 1 | ||
315.2.p.c | 4 | 105.k | odd | 4 | 1 | ||
560.2.bj.a | 4 | 20.d | odd | 2 | 1 | ||
560.2.bj.a | 4 | 20.e | even | 4 | 1 | ||
560.2.bj.a | 4 | 140.c | even | 2 | 1 | ||
560.2.bj.a | 4 | 140.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 2)^{2} \)
$3$
\( T^{4} + 25 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 4 T^{3} + 8 T^{2} + 28 T + 49 \)
$11$
\( (T + 1)^{4} \)
$13$
\( T^{4} + 25 \)
$17$
\( T^{4} + 25 \)
$19$
\( (T^{2} - 10)^{2} \)
$23$
\( (T^{2} + 4 T + 8)^{2} \)
$29$
\( (T^{2} + 9)^{2} \)
$31$
\( (T^{2} + 10)^{2} \)
$37$
\( (T^{2} - 12 T + 72)^{2} \)
$41$
\( (T^{2} + 90)^{2} \)
$43$
\( (T^{2} - 6 T + 18)^{2} \)
$47$
\( T^{4} + 2025 \)
$53$
\( (T^{2} + 2 T + 2)^{2} \)
$59$
\( (T^{2} - 90)^{2} \)
$61$
\( (T^{2} + 40)^{2} \)
$67$
\( (T^{2} - 2 T + 2)^{2} \)
$71$
\( (T + 6)^{4} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} + 169)^{2} \)
$83$
\( T^{4} + 400 \)
$89$
\( (T^{2} - 40)^{2} \)
$97$
\( T^{4} + 25 \)
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