Newspace parameters
Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2205.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.10043835286\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.0.2048.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} + 4x^{2} + 2 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{8}\) |
Projective field: | Galois closure of 8.2.8338372875.1 |
$q$-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 \( x^{4} + 4x^{2} + 2 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 3\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 3\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).
\(n\) | \(442\) | \(1081\) | \(1226\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
244.1 |
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− | 1.84776i | 0 | −2.41421 | 1.00000 | 0 | 0 | 2.61313i | 0 | − | 1.84776i | ||||||||||||||||||||||||||||
244.2 | − | 0.765367i | 0 | 0.414214 | 1.00000 | 0 | 0 | − | 1.08239i | 0 | − | 0.765367i | ||||||||||||||||||||||||||||
244.3 | 0.765367i | 0 | 0.414214 | 1.00000 | 0 | 0 | 1.08239i | 0 | 0.765367i | |||||||||||||||||||||||||||||||
244.4 | 1.84776i | 0 | −2.41421 | 1.00000 | 0 | 0 | − | 2.61313i | 0 | 1.84776i | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
21.c | even | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2205.1.e.b | yes | 4 |
3.b | odd | 2 | 1 | 2205.1.e.a | ✓ | 4 | |
5.b | even | 2 | 1 | 2205.1.e.a | ✓ | 4 | |
7.b | odd | 2 | 1 | 2205.1.e.a | ✓ | 4 | |
7.c | even | 3 | 2 | 2205.1.bi.a | 8 | ||
7.d | odd | 6 | 2 | 2205.1.bi.b | 8 | ||
15.d | odd | 2 | 1 | CM | 2205.1.e.b | yes | 4 |
21.c | even | 2 | 1 | inner | 2205.1.e.b | yes | 4 |
21.g | even | 6 | 2 | 2205.1.bi.a | 8 | ||
21.h | odd | 6 | 2 | 2205.1.bi.b | 8 | ||
35.c | odd | 2 | 1 | inner | 2205.1.e.b | yes | 4 |
35.i | odd | 6 | 2 | 2205.1.bi.a | 8 | ||
35.j | even | 6 | 2 | 2205.1.bi.b | 8 | ||
105.g | even | 2 | 1 | 2205.1.e.a | ✓ | 4 | |
105.o | odd | 6 | 2 | 2205.1.bi.a | 8 | ||
105.p | even | 6 | 2 | 2205.1.bi.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2205.1.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
2205.1.e.a | ✓ | 4 | 5.b | even | 2 | 1 | |
2205.1.e.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
2205.1.e.a | ✓ | 4 | 105.g | even | 2 | 1 | |
2205.1.e.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
2205.1.e.b | yes | 4 | 15.d | odd | 2 | 1 | CM |
2205.1.e.b | yes | 4 | 21.c | even | 2 | 1 | inner |
2205.1.e.b | yes | 4 | 35.c | odd | 2 | 1 | inner |
2205.1.bi.a | 8 | 7.c | even | 3 | 2 | ||
2205.1.bi.a | 8 | 21.g | even | 6 | 2 | ||
2205.1.bi.a | 8 | 35.i | odd | 6 | 2 | ||
2205.1.bi.a | 8 | 105.o | odd | 6 | 2 | ||
2205.1.bi.b | 8 | 7.d | odd | 6 | 2 | ||
2205.1.bi.b | 8 | 21.h | odd | 6 | 2 | ||
2205.1.bi.b | 8 | 35.j | even | 6 | 2 | ||
2205.1.bi.b | 8 | 105.p | even | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{167} + 2 \)
acting on \(S_{1}^{\mathrm{new}}(2205, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 4T^{2} + 2 \)
$3$
\( T^{4} \)
$5$
\( (T - 1)^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( (T^{2} - 2)^{2} \)
$19$
\( T^{4} + 4T^{2} + 2 \)
$23$
\( T^{4} + 4T^{2} + 2 \)
$29$
\( T^{4} \)
$31$
\( T^{4} + 4T^{2} + 2 \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( (T^{2} - 2)^{2} \)
$53$
\( T^{4} + 4T^{2} + 2 \)
$59$
\( T^{4} \)
$61$
\( T^{4} + 4T^{2} + 2 \)
$67$
\( T^{4} \)
$71$
\( T^{4} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} - 2)^{2} \)
$83$
\( (T^{2} - 2)^{2} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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