Newspace parameters
Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 175.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.39738203537\) |
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: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
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 \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
51.1 |
|
−0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 0 | −1.00000 | −0.500000 | + | 2.59808i | −3.00000 | 1.00000 | − | 1.73205i | 0 | ||||||||||||||
151.1 | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 0 | −1.00000 | −0.500000 | − | 2.59808i | −3.00000 | 1.00000 | + | 1.73205i | 0 | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 175.2.e.a | 2 | |
5.b | even | 2 | 1 | 175.2.e.b | 2 | ||
5.c | odd | 4 | 2 | 35.2.j.a | ✓ | 4 | |
7.c | even | 3 | 1 | inner | 175.2.e.a | 2 | |
7.c | even | 3 | 1 | 1225.2.a.f | 1 | ||
7.d | odd | 6 | 1 | 1225.2.a.g | 1 | ||
15.e | even | 4 | 2 | 315.2.bf.a | 4 | ||
20.e | even | 4 | 2 | 560.2.bw.b | 4 | ||
35.f | even | 4 | 2 | 245.2.j.c | 4 | ||
35.i | odd | 6 | 1 | 1225.2.a.b | 1 | ||
35.j | even | 6 | 1 | 175.2.e.b | 2 | ||
35.j | even | 6 | 1 | 1225.2.a.d | 1 | ||
35.k | even | 12 | 2 | 245.2.b.b | 2 | ||
35.k | even | 12 | 2 | 245.2.j.c | 4 | ||
35.l | odd | 12 | 2 | 35.2.j.a | ✓ | 4 | |
35.l | odd | 12 | 2 | 245.2.b.c | 2 | ||
105.w | odd | 12 | 2 | 2205.2.d.e | 2 | ||
105.x | even | 12 | 2 | 315.2.bf.a | 4 | ||
105.x | even | 12 | 2 | 2205.2.d.d | 2 | ||
140.w | even | 12 | 2 | 560.2.bw.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.j.a | ✓ | 4 | 5.c | odd | 4 | 2 | |
35.2.j.a | ✓ | 4 | 35.l | odd | 12 | 2 | |
175.2.e.a | 2 | 1.a | even | 1 | 1 | trivial | |
175.2.e.a | 2 | 7.c | even | 3 | 1 | inner | |
175.2.e.b | 2 | 5.b | even | 2 | 1 | ||
175.2.e.b | 2 | 35.j | even | 6 | 1 | ||
245.2.b.b | 2 | 35.k | even | 12 | 2 | ||
245.2.b.c | 2 | 35.l | odd | 12 | 2 | ||
245.2.j.c | 4 | 35.f | even | 4 | 2 | ||
245.2.j.c | 4 | 35.k | even | 12 | 2 | ||
315.2.bf.a | 4 | 15.e | even | 4 | 2 | ||
315.2.bf.a | 4 | 105.x | even | 12 | 2 | ||
560.2.bw.b | 4 | 20.e | even | 4 | 2 | ||
560.2.bw.b | 4 | 140.w | even | 12 | 2 | ||
1225.2.a.b | 1 | 35.i | odd | 6 | 1 | ||
1225.2.a.d | 1 | 35.j | even | 6 | 1 | ||
1225.2.a.f | 1 | 7.c | even | 3 | 1 | ||
1225.2.a.g | 1 | 7.d | odd | 6 | 1 | ||
2205.2.d.d | 2 | 105.x | even | 12 | 2 | ||
2205.2.d.e | 2 | 105.w | odd | 12 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T + 1 \)
$3$
\( T^{2} - T + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + T + 7 \)
$11$
\( T^{2} \)
$13$
\( (T - 2)^{2} \)
$17$
\( T^{2} + 2T + 4 \)
$19$
\( T^{2} + 6T + 36 \)
$23$
\( T^{2} - 3T + 9 \)
$29$
\( (T - 7)^{2} \)
$31$
\( T^{2} + 2T + 4 \)
$37$
\( T^{2} - 8T + 64 \)
$41$
\( (T - 5)^{2} \)
$43$
\( (T - 7)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} + 6T + 36 \)
$59$
\( T^{2} + 10T + 100 \)
$61$
\( T^{2} + 7T + 49 \)
$67$
\( T^{2} - 5T + 25 \)
$71$
\( (T + 2)^{2} \)
$73$
\( T^{2} - 6T + 36 \)
$79$
\( T^{2} - 2T + 4 \)
$83$
\( (T + 11)^{2} \)
$89$
\( T^{2} + 9T + 81 \)
$97$
\( (T + 16)^{2} \)
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