Newspace parameters
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.39738203537\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{17})\) |
Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9x^{2} + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
\(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
99.1 |
|
− | 2.56155i | − | 1.56155i | −4.56155 | 0 | −4.00000 | − | 1.00000i | 6.56155i | 0.561553 | 0 | |||||||||||||||||||||||||||
99.2 | − | 1.56155i | − | 2.56155i | −0.438447 | 0 | −4.00000 | 1.00000i | − | 2.43845i | −3.56155 | 0 | ||||||||||||||||||||||||||||
99.3 | 1.56155i | 2.56155i | −0.438447 | 0 | −4.00000 | − | 1.00000i | 2.43845i | −3.56155 | 0 | ||||||||||||||||||||||||||||||
99.4 | 2.56155i | 1.56155i | −4.56155 | 0 | −4.00000 | 1.00000i | − | 6.56155i | 0.561553 | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.2.b.b | 4 | |
3.b | odd | 2 | 1 | 1575.2.d.e | 4 | ||
4.b | odd | 2 | 1 | 2800.2.g.t | 4 | ||
5.b | even | 2 | 1 | inner | 175.2.b.b | 4 | |
5.c | odd | 4 | 1 | 35.2.a.b | ✓ | 2 | |
5.c | odd | 4 | 1 | 175.2.a.f | 2 | ||
7.b | odd | 2 | 1 | 1225.2.b.f | 4 | ||
15.d | odd | 2 | 1 | 1575.2.d.e | 4 | ||
15.e | even | 4 | 1 | 315.2.a.e | 2 | ||
15.e | even | 4 | 1 | 1575.2.a.p | 2 | ||
20.d | odd | 2 | 1 | 2800.2.g.t | 4 | ||
20.e | even | 4 | 1 | 560.2.a.i | 2 | ||
20.e | even | 4 | 1 | 2800.2.a.bi | 2 | ||
35.c | odd | 2 | 1 | 1225.2.b.f | 4 | ||
35.f | even | 4 | 1 | 245.2.a.d | 2 | ||
35.f | even | 4 | 1 | 1225.2.a.s | 2 | ||
35.k | even | 12 | 2 | 245.2.e.h | 4 | ||
35.l | odd | 12 | 2 | 245.2.e.i | 4 | ||
40.i | odd | 4 | 1 | 2240.2.a.bh | 2 | ||
40.k | even | 4 | 1 | 2240.2.a.bd | 2 | ||
55.e | even | 4 | 1 | 4235.2.a.m | 2 | ||
60.l | odd | 4 | 1 | 5040.2.a.bt | 2 | ||
65.h | odd | 4 | 1 | 5915.2.a.l | 2 | ||
105.k | odd | 4 | 1 | 2205.2.a.x | 2 | ||
140.j | odd | 4 | 1 | 3920.2.a.bs | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.a.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
175.2.a.f | 2 | 5.c | odd | 4 | 1 | ||
175.2.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
175.2.b.b | 4 | 5.b | even | 2 | 1 | inner | |
245.2.a.d | 2 | 35.f | even | 4 | 1 | ||
245.2.e.h | 4 | 35.k | even | 12 | 2 | ||
245.2.e.i | 4 | 35.l | odd | 12 | 2 | ||
315.2.a.e | 2 | 15.e | even | 4 | 1 | ||
560.2.a.i | 2 | 20.e | even | 4 | 1 | ||
1225.2.a.s | 2 | 35.f | even | 4 | 1 | ||
1225.2.b.f | 4 | 7.b | odd | 2 | 1 | ||
1225.2.b.f | 4 | 35.c | odd | 2 | 1 | ||
1575.2.a.p | 2 | 15.e | even | 4 | 1 | ||
1575.2.d.e | 4 | 3.b | odd | 2 | 1 | ||
1575.2.d.e | 4 | 15.d | odd | 2 | 1 | ||
2205.2.a.x | 2 | 105.k | odd | 4 | 1 | ||
2240.2.a.bd | 2 | 40.k | even | 4 | 1 | ||
2240.2.a.bh | 2 | 40.i | odd | 4 | 1 | ||
2800.2.a.bi | 2 | 20.e | even | 4 | 1 | ||
2800.2.g.t | 4 | 4.b | odd | 2 | 1 | ||
2800.2.g.t | 4 | 20.d | odd | 2 | 1 | ||
3920.2.a.bs | 2 | 140.j | odd | 4 | 1 | ||
4235.2.a.m | 2 | 55.e | even | 4 | 1 | ||
5040.2.a.bt | 2 | 60.l | odd | 4 | 1 | ||
5915.2.a.l | 2 | 65.h | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 9T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 9T^{2} + 16 \)
$3$
\( T^{4} + 9T^{2} + 16 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 1)^{2} \)
$11$
\( (T^{2} - T - 4)^{2} \)
$13$
\( T^{4} + 21T^{2} + 4 \)
$17$
\( T^{4} + 21T^{2} + 4 \)
$19$
\( (T^{2} - 6 T - 8)^{2} \)
$23$
\( T^{4} + 36T^{2} + 256 \)
$29$
\( (T^{2} + T - 38)^{2} \)
$31$
\( T^{4} \)
$37$
\( (T^{2} + 36)^{2} \)
$41$
\( (T^{2} - 2 T - 16)^{2} \)
$43$
\( T^{4} + 84T^{2} + 64 \)
$47$
\( T^{4} + 89T^{2} + 1024 \)
$53$
\( T^{4} + 36T^{2} + 256 \)
$59$
\( (T - 4)^{4} \)
$61$
\( (T^{2} - 6 T - 144)^{2} \)
$67$
\( T^{4} + 144T^{2} + 4096 \)
$71$
\( (T - 8)^{4} \)
$73$
\( T^{4} + 168T^{2} + 2704 \)
$79$
\( (T^{2} - 9 T + 16)^{2} \)
$83$
\( (T^{2} + 16)^{2} \)
$89$
\( (T^{2} + 6 T - 8)^{2} \)
$97$
\( T^{4} + 253T^{2} + 7396 \)
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