Newspace parameters
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.2806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.0.613376.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 58x^{2} + 599 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 46) |
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 \( x^{4} + 58x^{2} + 599 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 29\nu ) / 11 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} + 29 ) / 11 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 51\nu ) / 11 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( 11\beta_{2} - 29 \) |
\(\nu^{3}\) | \(=\) | \( ( -29\beta_{3} + 51\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).
\(n\) | \(47\) | \(235\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91.1 |
|
−1.41421 | 0 | 2.00000 | − | 5.18530i | 0 | − | 12.5184i | −2.82843 | 0 | 7.33312i | ||||||||||||||||||||||||||||
91.2 | −1.41421 | 0 | 2.00000 | 5.18530i | 0 | 12.5184i | −2.82843 | 0 | − | 7.33312i | ||||||||||||||||||||||||||||||
91.3 | 1.41421 | 0 | 2.00000 | − | 9.43995i | 0 | 3.91016i | 2.82843 | 0 | − | 13.3501i | |||||||||||||||||||||||||||||
91.4 | 1.41421 | 0 | 2.00000 | 9.43995i | 0 | − | 3.91016i | 2.82843 | 0 | 13.3501i | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.3.b.a | 4 | |
3.b | odd | 2 | 1 | 46.3.b.a | ✓ | 4 | |
12.b | even | 2 | 1 | 368.3.f.c | 4 | ||
15.d | odd | 2 | 1 | 1150.3.d.a | 4 | ||
15.e | even | 4 | 2 | 1150.3.c.a | 8 | ||
23.b | odd | 2 | 1 | inner | 414.3.b.a | 4 | |
24.f | even | 2 | 1 | 1472.3.f.c | 4 | ||
24.h | odd | 2 | 1 | 1472.3.f.f | 4 | ||
69.c | even | 2 | 1 | 46.3.b.a | ✓ | 4 | |
276.h | odd | 2 | 1 | 368.3.f.c | 4 | ||
345.h | even | 2 | 1 | 1150.3.d.a | 4 | ||
345.l | odd | 4 | 2 | 1150.3.c.a | 8 | ||
552.b | even | 2 | 1 | 1472.3.f.f | 4 | ||
552.h | odd | 2 | 1 | 1472.3.f.c | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
46.3.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
46.3.b.a | ✓ | 4 | 69.c | even | 2 | 1 | |
368.3.f.c | 4 | 12.b | even | 2 | 1 | ||
368.3.f.c | 4 | 276.h | odd | 2 | 1 | ||
414.3.b.a | 4 | 1.a | even | 1 | 1 | trivial | |
414.3.b.a | 4 | 23.b | odd | 2 | 1 | inner | |
1150.3.c.a | 8 | 15.e | even | 4 | 2 | ||
1150.3.c.a | 8 | 345.l | odd | 4 | 2 | ||
1150.3.d.a | 4 | 15.d | odd | 2 | 1 | ||
1150.3.d.a | 4 | 345.h | even | 2 | 1 | ||
1472.3.f.c | 4 | 24.f | even | 2 | 1 | ||
1472.3.f.c | 4 | 552.h | odd | 2 | 1 | ||
1472.3.f.f | 4 | 24.h | odd | 2 | 1 | ||
1472.3.f.f | 4 | 552.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 116T_{5}^{2} + 2396 \)
acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 116T^{2} + 2396 \)
$7$
\( T^{4} + 172T^{2} + 2396 \)
$11$
\( T^{4} + 172T^{2} + 2396 \)
$13$
\( (T^{2} + 2 T - 127)^{2} \)
$17$
\( T^{4} + 692 T^{2} + 117404 \)
$19$
\( T^{4} + 928 T^{2} + 153344 \)
$23$
\( T^{4} + 52 T^{3} + 1342 T^{2} + \cdots + 279841 \)
$29$
\( (T^{2} + 42 T + 241)^{2} \)
$31$
\( (T^{2} - 58 T + 823)^{2} \)
$37$
\( T^{4} + 4148 T^{2} + \cdots + 4027676 \)
$41$
\( (T^{2} + 10 T - 47)^{2} \)
$43$
\( T^{4} + 2992 T^{2} + \cdots + 1878464 \)
$47$
\( (T^{2} - 14 T - 529)^{2} \)
$53$
\( T^{4} + 6704 T^{2} + \cdots + 11079104 \)
$59$
\( (T^{2} - 92 T + 1724)^{2} \)
$61$
\( T^{4} + 2752 T^{2} + 613376 \)
$67$
\( T^{4} + 4716 T^{2} + 194076 \)
$71$
\( (T^{2} - 50 T - 6817)^{2} \)
$73$
\( (T^{2} + 74 T - 5831)^{2} \)
$79$
\( T^{4} + 14432 T^{2} + \cdots + 7513856 \)
$83$
\( T^{4} + 18700 T^{2} + \cdots + 73377500 \)
$89$
\( T^{4} + 17200 T^{2} + \cdots + 23960000 \)
$97$
\( T^{4} + 5684 T^{2} + \cdots + 5752796 \)
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