Newspace parameters
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.18840615665\) |
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: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 31) |
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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 1 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 2\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).
\(n\) | \(251\) | \(652\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
249.1 |
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− | 1.61803i | − | 3.23607i | −0.618034 | 0 | −5.23607 | − | 0.236068i | − | 2.23607i | −7.47214 | 0 | ||||||||||||||||||||||||||
249.2 | − | 0.618034i | − | 1.23607i | 1.61803 | 0 | −0.763932 | − | 4.23607i | − | 2.23607i | 1.47214 | 0 | |||||||||||||||||||||||||||
249.3 | 0.618034i | 1.23607i | 1.61803 | 0 | −0.763932 | 4.23607i | 2.23607i | 1.47214 | 0 | |||||||||||||||||||||||||||||||
249.4 | 1.61803i | 3.23607i | −0.618034 | 0 | −5.23607 | 0.236068i | 2.23607i | −7.47214 | 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 | 775.2.b.d | 4 | |
5.b | even | 2 | 1 | inner | 775.2.b.d | 4 | |
5.c | odd | 4 | 1 | 31.2.a.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 775.2.a.d | 2 | ||
15.e | even | 4 | 1 | 279.2.a.a | 2 | ||
15.e | even | 4 | 1 | 6975.2.a.y | 2 | ||
20.e | even | 4 | 1 | 496.2.a.i | 2 | ||
35.f | even | 4 | 1 | 1519.2.a.a | 2 | ||
40.i | odd | 4 | 1 | 1984.2.a.r | 2 | ||
40.k | even | 4 | 1 | 1984.2.a.n | 2 | ||
55.e | even | 4 | 1 | 3751.2.a.b | 2 | ||
60.l | odd | 4 | 1 | 4464.2.a.bf | 2 | ||
65.h | odd | 4 | 1 | 5239.2.a.f | 2 | ||
85.g | odd | 4 | 1 | 8959.2.a.b | 2 | ||
155.f | even | 4 | 1 | 961.2.a.f | 2 | ||
155.o | odd | 12 | 2 | 961.2.c.e | 4 | ||
155.p | even | 12 | 2 | 961.2.c.c | 4 | ||
155.r | even | 20 | 2 | 961.2.d.a | 4 | ||
155.r | even | 20 | 2 | 961.2.d.g | 4 | ||
155.s | odd | 20 | 2 | 961.2.d.c | 4 | ||
155.s | odd | 20 | 2 | 961.2.d.d | 4 | ||
155.w | odd | 60 | 4 | 961.2.g.a | 8 | ||
155.w | odd | 60 | 4 | 961.2.g.h | 8 | ||
155.x | even | 60 | 4 | 961.2.g.d | 8 | ||
155.x | even | 60 | 4 | 961.2.g.e | 8 | ||
465.m | odd | 4 | 1 | 8649.2.a.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
31.2.a.a | ✓ | 2 | 5.c | odd | 4 | 1 | |
279.2.a.a | 2 | 15.e | even | 4 | 1 | ||
496.2.a.i | 2 | 20.e | even | 4 | 1 | ||
775.2.a.d | 2 | 5.c | odd | 4 | 1 | ||
775.2.b.d | 4 | 1.a | even | 1 | 1 | trivial | |
775.2.b.d | 4 | 5.b | even | 2 | 1 | inner | |
961.2.a.f | 2 | 155.f | even | 4 | 1 | ||
961.2.c.c | 4 | 155.p | even | 12 | 2 | ||
961.2.c.e | 4 | 155.o | odd | 12 | 2 | ||
961.2.d.a | 4 | 155.r | even | 20 | 2 | ||
961.2.d.c | 4 | 155.s | odd | 20 | 2 | ||
961.2.d.d | 4 | 155.s | odd | 20 | 2 | ||
961.2.d.g | 4 | 155.r | even | 20 | 2 | ||
961.2.g.a | 8 | 155.w | odd | 60 | 4 | ||
961.2.g.d | 8 | 155.x | even | 60 | 4 | ||
961.2.g.e | 8 | 155.x | even | 60 | 4 | ||
961.2.g.h | 8 | 155.w | odd | 60 | 4 | ||
1519.2.a.a | 2 | 35.f | even | 4 | 1 | ||
1984.2.a.n | 2 | 40.k | even | 4 | 1 | ||
1984.2.a.r | 2 | 40.i | odd | 4 | 1 | ||
3751.2.a.b | 2 | 55.e | even | 4 | 1 | ||
4464.2.a.bf | 2 | 60.l | odd | 4 | 1 | ||
5239.2.a.f | 2 | 65.h | odd | 4 | 1 | ||
6975.2.a.y | 2 | 15.e | even | 4 | 1 | ||
8649.2.a.c | 2 | 465.m | odd | 4 | 1 | ||
8959.2.a.b | 2 | 85.g | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 3T^{2} + 1 \)
$3$
\( T^{4} + 12T^{2} + 16 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 18T^{2} + 1 \)
$11$
\( (T - 2)^{4} \)
$13$
\( T^{4} + 12T^{2} + 16 \)
$17$
\( T^{4} + 28T^{2} + 16 \)
$19$
\( (T^{2} - 5)^{2} \)
$23$
\( T^{4} + 92T^{2} + 1936 \)
$29$
\( (T^{2} + 10 T + 20)^{2} \)
$31$
\( (T - 1)^{4} \)
$37$
\( (T^{2} + 4)^{2} \)
$41$
\( (T - 7)^{4} \)
$43$
\( T^{4} + 12T^{2} + 16 \)
$47$
\( T^{4} + 48T^{2} + 256 \)
$53$
\( T^{4} + 112T^{2} + 256 \)
$59$
\( (T^{2} - 5)^{2} \)
$61$
\( (T^{2} + 6 T - 116)^{2} \)
$67$
\( (T^{2} + 64)^{2} \)
$71$
\( (T^{2} - 4 T - 121)^{2} \)
$73$
\( T^{4} + 72T^{2} + 16 \)
$79$
\( (T^{2} - 10 T - 20)^{2} \)
$83$
\( T^{4} + 232T^{2} + 1936 \)
$89$
\( (T^{2} + 10 T - 20)^{2} \)
$97$
\( T^{4} + 258T^{2} + 961 \)
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