Newspace parameters
Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 725.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.78915414654\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 29) |
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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} + 4\nu \) |
\(\beta_{3}\) | \(=\) | \( 2\nu^{2} + 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} - 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{2} + 2\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).
\(n\) | \(176\) | \(552\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
724.1 |
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−2.23607 | −2.23607 | 3.00000 | 0 | 5.00000 | − | 2.00000i | −2.23607 | 2.00000 | 0 | |||||||||||||||||||||||||||||
724.2 | −2.23607 | −2.23607 | 3.00000 | 0 | 5.00000 | 2.00000i | −2.23607 | 2.00000 | 0 | |||||||||||||||||||||||||||||||
724.3 | 2.23607 | 2.23607 | 3.00000 | 0 | 5.00000 | − | 2.00000i | 2.23607 | 2.00000 | 0 | ||||||||||||||||||||||||||||||
724.4 | 2.23607 | 2.23607 | 3.00000 | 0 | 5.00000 | 2.00000i | 2.23607 | 2.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
29.b | even | 2 | 1 | inner |
145.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 725.2.d.a | 4 | |
5.b | even | 2 | 1 | inner | 725.2.d.a | 4 | |
5.c | odd | 4 | 1 | 29.2.b.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 725.2.c.c | 2 | ||
15.e | even | 4 | 1 | 261.2.c.a | 2 | ||
20.e | even | 4 | 1 | 464.2.e.a | 2 | ||
29.b | even | 2 | 1 | inner | 725.2.d.a | 4 | |
35.f | even | 4 | 1 | 1421.2.b.b | 2 | ||
40.i | odd | 4 | 1 | 1856.2.e.g | 2 | ||
40.k | even | 4 | 1 | 1856.2.e.f | 2 | ||
60.l | odd | 4 | 1 | 4176.2.o.k | 2 | ||
145.d | even | 2 | 1 | inner | 725.2.d.a | 4 | |
145.e | even | 4 | 1 | 841.2.a.b | 2 | ||
145.h | odd | 4 | 1 | 29.2.b.a | ✓ | 2 | |
145.h | odd | 4 | 1 | 725.2.c.c | 2 | ||
145.j | even | 4 | 1 | 841.2.a.b | 2 | ||
145.o | even | 28 | 6 | 841.2.d.h | 12 | ||
145.p | odd | 28 | 6 | 841.2.e.g | 12 | ||
145.q | odd | 28 | 6 | 841.2.e.g | 12 | ||
145.t | even | 28 | 6 | 841.2.d.h | 12 | ||
435.i | odd | 4 | 1 | 7569.2.a.i | 2 | ||
435.p | even | 4 | 1 | 261.2.c.a | 2 | ||
435.t | odd | 4 | 1 | 7569.2.a.i | 2 | ||
580.o | even | 4 | 1 | 464.2.e.a | 2 | ||
1015.l | even | 4 | 1 | 1421.2.b.b | 2 | ||
1160.bb | even | 4 | 1 | 1856.2.e.f | 2 | ||
1160.be | odd | 4 | 1 | 1856.2.e.g | 2 | ||
1740.v | odd | 4 | 1 | 4176.2.o.k | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.2.b.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
29.2.b.a | ✓ | 2 | 145.h | odd | 4 | 1 | |
261.2.c.a | 2 | 15.e | even | 4 | 1 | ||
261.2.c.a | 2 | 435.p | even | 4 | 1 | ||
464.2.e.a | 2 | 20.e | even | 4 | 1 | ||
464.2.e.a | 2 | 580.o | even | 4 | 1 | ||
725.2.c.c | 2 | 5.c | odd | 4 | 1 | ||
725.2.c.c | 2 | 145.h | odd | 4 | 1 | ||
725.2.d.a | 4 | 1.a | even | 1 | 1 | trivial | |
725.2.d.a | 4 | 5.b | even | 2 | 1 | inner | |
725.2.d.a | 4 | 29.b | even | 2 | 1 | inner | |
725.2.d.a | 4 | 145.d | even | 2 | 1 | inner | |
841.2.a.b | 2 | 145.e | even | 4 | 1 | ||
841.2.a.b | 2 | 145.j | even | 4 | 1 | ||
841.2.d.h | 12 | 145.o | even | 28 | 6 | ||
841.2.d.h | 12 | 145.t | even | 28 | 6 | ||
841.2.e.g | 12 | 145.p | odd | 28 | 6 | ||
841.2.e.g | 12 | 145.q | odd | 28 | 6 | ||
1421.2.b.b | 2 | 35.f | even | 4 | 1 | ||
1421.2.b.b | 2 | 1015.l | even | 4 | 1 | ||
1856.2.e.f | 2 | 40.k | even | 4 | 1 | ||
1856.2.e.f | 2 | 1160.bb | even | 4 | 1 | ||
1856.2.e.g | 2 | 40.i | odd | 4 | 1 | ||
1856.2.e.g | 2 | 1160.be | odd | 4 | 1 | ||
4176.2.o.k | 2 | 60.l | odd | 4 | 1 | ||
4176.2.o.k | 2 | 1740.v | odd | 4 | 1 | ||
7569.2.a.i | 2 | 435.i | odd | 4 | 1 | ||
7569.2.a.i | 2 | 435.t | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 5)^{2} \)
$3$
\( (T^{2} - 5)^{2} \)
$5$
\( T^{4} \)
$7$
\( (T^{2} + 4)^{2} \)
$11$
\( (T^{2} + 5)^{2} \)
$13$
\( (T^{2} + 1)^{2} \)
$17$
\( (T^{2} - 20)^{2} \)
$19$
\( T^{4} \)
$23$
\( (T^{2} + 36)^{2} \)
$29$
\( (T^{2} - 6 T + 29)^{2} \)
$31$
\( (T^{2} + 45)^{2} \)
$37$
\( T^{4} \)
$41$
\( (T^{2} + 20)^{2} \)
$43$
\( (T^{2} - 45)^{2} \)
$47$
\( (T^{2} - 5)^{2} \)
$53$
\( (T^{2} + 81)^{2} \)
$59$
\( (T + 6)^{4} \)
$61$
\( (T^{2} + 180)^{2} \)
$67$
\( (T^{2} + 64)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} + 45)^{2} \)
$83$
\( (T^{2} + 36)^{2} \)
$89$
\( (T^{2} + 20)^{2} \)
$97$
\( (T^{2} - 180)^{2} \)
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