Newspace parameters
Level: | \( N \) | \(=\) | \( 215 = 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 215.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.107298977716\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{14})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{3} - x^{2} - 2x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{7}\) |
Projective field: | Galois closure of 7.1.9938375.1 |
Artin image: | $D_7$ |
Artin field: | Galois closure of 7.1.9938375.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/215\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(87\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
214.1 |
|
−1.80194 | 1.24698 | 2.24698 | 1.00000 | −2.24698 | −0.445042 | −2.24698 | 0.554958 | −1.80194 | |||||||||||||||||||||||||||
214.2 | −0.445042 | −1.80194 | −0.801938 | 1.00000 | 0.801938 | 1.24698 | 0.801938 | 2.24698 | −0.445042 | ||||||||||||||||||||||||||||
214.3 | 1.24698 | −0.445042 | 0.554958 | 1.00000 | −0.554958 | −1.80194 | −0.554958 | −0.801938 | 1.24698 | ||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
215.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-215}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 215.1.d.a | ✓ | 3 |
3.b | odd | 2 | 1 | 1935.1.h.b | 3 | ||
4.b | odd | 2 | 1 | 3440.1.p.b | 3 | ||
5.b | even | 2 | 1 | 215.1.d.b | yes | 3 | |
5.c | odd | 4 | 2 | 1075.1.c.c | 6 | ||
15.d | odd | 2 | 1 | 1935.1.h.a | 3 | ||
20.d | odd | 2 | 1 | 3440.1.p.a | 3 | ||
43.b | odd | 2 | 1 | 215.1.d.b | yes | 3 | |
129.d | even | 2 | 1 | 1935.1.h.a | 3 | ||
172.d | even | 2 | 1 | 3440.1.p.a | 3 | ||
215.d | odd | 2 | 1 | CM | 215.1.d.a | ✓ | 3 |
215.g | even | 4 | 2 | 1075.1.c.c | 6 | ||
645.d | even | 2 | 1 | 1935.1.h.b | 3 | ||
860.b | even | 2 | 1 | 3440.1.p.b | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
215.1.d.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
215.1.d.a | ✓ | 3 | 215.d | odd | 2 | 1 | CM |
215.1.d.b | yes | 3 | 5.b | even | 2 | 1 | |
215.1.d.b | yes | 3 | 43.b | odd | 2 | 1 | |
1075.1.c.c | 6 | 5.c | odd | 4 | 2 | ||
1075.1.c.c | 6 | 215.g | even | 4 | 2 | ||
1935.1.h.a | 3 | 15.d | odd | 2 | 1 | ||
1935.1.h.a | 3 | 129.d | even | 2 | 1 | ||
1935.1.h.b | 3 | 3.b | odd | 2 | 1 | ||
1935.1.h.b | 3 | 645.d | even | 2 | 1 | ||
3440.1.p.a | 3 | 20.d | odd | 2 | 1 | ||
3440.1.p.a | 3 | 172.d | even | 2 | 1 | ||
3440.1.p.b | 3 | 4.b | odd | 2 | 1 | ||
3440.1.p.b | 3 | 860.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(215, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + T^{2} - 2T - 1 \)
$3$
\( T^{3} + T^{2} - 2T - 1 \)
$5$
\( (T - 1)^{3} \)
$7$
\( T^{3} + T^{2} - 2T - 1 \)
$11$
\( T^{3} + T^{2} - 2T - 1 \)
$13$
\( T^{3} \)
$17$
\( T^{3} \)
$19$
\( T^{3} \)
$23$
\( T^{3} \)
$29$
\( T^{3} \)
$31$
\( T^{3} + T^{2} - 2T - 1 \)
$37$
\( T^{3} + T^{2} - 2T - 1 \)
$41$
\( T^{3} + T^{2} - 2T - 1 \)
$43$
\( (T - 1)^{3} \)
$47$
\( T^{3} \)
$53$
\( T^{3} \)
$59$
\( T^{3} + T^{2} - 2T - 1 \)
$61$
\( T^{3} \)
$67$
\( T^{3} \)
$71$
\( T^{3} \)
$73$
\( T^{3} + T^{2} - 2T - 1 \)
$79$
\( T^{3} + T^{2} - 2T - 1 \)
$83$
\( T^{3} \)
$89$
\( T^{3} \)
$97$
\( T^{3} \)
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