Newspace parameters
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.3253342510\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.41421 | 4.65685 | 21.3137 | 0 | −25.2132 | 7.00000 | −72.0833 | −5.31371 | 0 | ||||||||||||||||||||||||
| 1.2 | −2.58579 | −6.65685 | −1.31371 | 0 | 17.2132 | 7.00000 | 24.0833 | 17.3137 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 175.4.a.c | 2 | |
| 3.b | odd | 2 | 1 | 1575.4.a.z | 2 | ||
| 5.b | even | 2 | 1 | 35.4.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 175.4.b.c | 4 | ||
| 7.b | odd | 2 | 1 | 1225.4.a.m | 2 | ||
| 15.d | odd | 2 | 1 | 315.4.a.f | 2 | ||
| 20.d | odd | 2 | 1 | 560.4.a.r | 2 | ||
| 35.c | odd | 2 | 1 | 245.4.a.k | 2 | ||
| 35.i | odd | 6 | 2 | 245.4.e.i | 4 | ||
| 35.j | even | 6 | 2 | 245.4.e.h | 4 | ||
| 40.e | odd | 2 | 1 | 2240.4.a.bo | 2 | ||
| 40.f | even | 2 | 1 | 2240.4.a.bn | 2 | ||
| 105.g | even | 2 | 1 | 2205.4.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 175.4.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 175.4.b.c | 4 | 5.c | odd | 4 | 2 | ||
| 245.4.a.k | 2 | 35.c | odd | 2 | 1 | ||
| 245.4.e.h | 4 | 35.j | even | 6 | 2 | ||
| 245.4.e.i | 4 | 35.i | odd | 6 | 2 | ||
| 315.4.a.f | 2 | 15.d | odd | 2 | 1 | ||
| 560.4.a.r | 2 | 20.d | odd | 2 | 1 | ||
| 1225.4.a.m | 2 | 7.b | odd | 2 | 1 | ||
| 1575.4.a.z | 2 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.u | 2 | 105.g | even | 2 | 1 | ||
| 2240.4.a.bn | 2 | 40.f | even | 2 | 1 | ||
| 2240.4.a.bo | 2 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8T_{2} + 14 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 8T + 14 \)
$3$
\( T^{2} + 2T - 31 \)
$5$
\( T^{2} \)
$7$
\( (T - 7)^{2} \)
$11$
\( T^{2} + 14T - 1999 \)
$13$
\( T^{2} + 50T + 593 \)
$17$
\( T^{2} - 50T - 3247 \)
$19$
\( T^{2} - 36T - 3548 \)
$23$
\( T^{2} + 244T + 5636 \)
$29$
\( T^{2} + 26T - 983 \)
$31$
\( T^{2} + 120T - 61200 \)
$37$
\( T^{2} + 564T + 72324 \)
$41$
\( T^{2} + 328T - 3856 \)
$43$
\( T^{2} - 260T + 7652 \)
$47$
\( T^{2} - 350T - 4223 \)
$53$
\( T^{2} - 56T - 31984 \)
$59$
\( (T + 616)^{2} \)
$61$
\( T^{2} - 336T + 4896 \)
$67$
\( T^{2} - 152T - 2416 \)
$71$
\( (T + 952)^{2} \)
$73$
\( T^{2} + 676T - 122428 \)
$79$
\( T^{2} - 1014 T + 134041 \)
$83$
\( T^{2} - 376T - 684656 \)
$89$
\( T^{2} + 216T + 7792 \)
$97$
\( T^{2} + 2742 T + 1782841 \)
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