Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.j (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} + 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 105) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \( x^{4} + 2x^{2} + 4 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} ) / 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 2\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 |
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−0.707107 | − | 1.22474i | 0 | 0 | −0.500000 | − | 0.866025i | 0 | −2.62132 | + | 0.358719i | −2.82843 | 0 | −0.707107 | + | 1.22474i | ||||||||||||||||||||||
46.2 | 0.707107 | + | 1.22474i | 0 | 0 | −0.500000 | − | 0.866025i | 0 | 1.62132 | − | 2.09077i | 2.82843 | 0 | 0.707107 | − | 1.22474i | |||||||||||||||||||||||
226.1 | −0.707107 | + | 1.22474i | 0 | 0 | −0.500000 | + | 0.866025i | 0 | −2.62132 | − | 0.358719i | −2.82843 | 0 | −0.707107 | − | 1.22474i | |||||||||||||||||||||||
226.2 | 0.707107 | − | 1.22474i | 0 | 0 | −0.500000 | + | 0.866025i | 0 | 1.62132 | + | 2.09077i | 2.82843 | 0 | 0.707107 | + | 1.22474i | |||||||||||||||||||||||
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 | 315.2.j.d | 4 | |
3.b | odd | 2 | 1 | 105.2.i.c | ✓ | 4 | |
7.c | even | 3 | 1 | inner | 315.2.j.d | 4 | |
7.c | even | 3 | 1 | 2205.2.a.u | 2 | ||
7.d | odd | 6 | 1 | 2205.2.a.s | 2 | ||
12.b | even | 2 | 1 | 1680.2.bg.p | 4 | ||
15.d | odd | 2 | 1 | 525.2.i.g | 4 | ||
15.e | even | 4 | 2 | 525.2.r.g | 8 | ||
21.c | even | 2 | 1 | 735.2.i.j | 4 | ||
21.g | even | 6 | 1 | 735.2.a.j | 2 | ||
21.g | even | 6 | 1 | 735.2.i.j | 4 | ||
21.h | odd | 6 | 1 | 105.2.i.c | ✓ | 4 | |
21.h | odd | 6 | 1 | 735.2.a.i | 2 | ||
84.n | even | 6 | 1 | 1680.2.bg.p | 4 | ||
105.o | odd | 6 | 1 | 525.2.i.g | 4 | ||
105.o | odd | 6 | 1 | 3675.2.a.z | 2 | ||
105.p | even | 6 | 1 | 3675.2.a.x | 2 | ||
105.x | even | 12 | 2 | 525.2.r.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.i.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
105.2.i.c | ✓ | 4 | 21.h | odd | 6 | 1 | |
315.2.j.d | 4 | 1.a | even | 1 | 1 | trivial | |
315.2.j.d | 4 | 7.c | even | 3 | 1 | inner | |
525.2.i.g | 4 | 15.d | odd | 2 | 1 | ||
525.2.i.g | 4 | 105.o | odd | 6 | 1 | ||
525.2.r.g | 8 | 15.e | even | 4 | 2 | ||
525.2.r.g | 8 | 105.x | even | 12 | 2 | ||
735.2.a.i | 2 | 21.h | odd | 6 | 1 | ||
735.2.a.j | 2 | 21.g | even | 6 | 1 | ||
735.2.i.j | 4 | 21.c | even | 2 | 1 | ||
735.2.i.j | 4 | 21.g | even | 6 | 1 | ||
1680.2.bg.p | 4 | 12.b | even | 2 | 1 | ||
1680.2.bg.p | 4 | 84.n | even | 6 | 1 | ||
2205.2.a.s | 2 | 7.d | odd | 6 | 1 | ||
2205.2.a.u | 2 | 7.c | even | 3 | 1 | ||
3675.2.a.x | 2 | 105.p | even | 6 | 1 | ||
3675.2.a.z | 2 | 105.o | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 2T^{2} + 4 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + T + 1)^{2} \)
$7$
\( T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49 \)
$11$
\( T^{4} - 4 T^{3} + 14 T^{2} - 8 T + 4 \)
$13$
\( (T^{2} + 6 T + 7)^{2} \)
$17$
\( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \)
$19$
\( T^{4} - 2 T^{3} + 35 T^{2} + 62 T + 961 \)
$23$
\( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \)
$29$
\( (T^{2} + 8 T - 2)^{2} \)
$31$
\( T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1 \)
$37$
\( T^{4} - 14 T^{3} + 149 T^{2} + \cdots + 2209 \)
$41$
\( (T^{2} - 4 T - 14)^{2} \)
$43$
\( (T^{2} + 18 T + 79)^{2} \)
$47$
\( T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376 \)
$53$
\( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \)
$59$
\( T^{4} + 2T^{2} + 4 \)
$61$
\( T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136 \)
$67$
\( T^{4} - 2 T^{3} + 165 T^{2} + \cdots + 25921 \)
$71$
\( (T^{2} - 4 T + 2)^{2} \)
$73$
\( T^{4} - 10 T^{3} + 125 T^{2} + \cdots + 625 \)
$79$
\( T^{4} + 2 T^{3} + 35 T^{2} - 62 T + 961 \)
$83$
\( (T^{2} + 8 T + 14)^{2} \)
$89$
\( T^{4} + 16 T^{3} + 210 T^{2} + \cdots + 2116 \)
$97$
\( (T^{2} + 16 T + 56)^{2} \)
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