Newspace parameters
Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 495.k (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.95259490005\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(i, \sqrt{11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - 5x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 55) |
Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 \( x^{4} - 5x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} - 2\nu ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{2} - 3 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 3 \) |
\(\nu^{3}\) | \(=\) | \( 3\beta_{2} + 2\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) | \(397\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
208.1 |
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0 | 0 | − | 2.00000i | −1.65831 | − | 1.50000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||
208.2 | 0 | 0 | − | 2.00000i | 1.65831 | − | 1.50000i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
307.1 | 0 | 0 | 2.00000i | −1.65831 | + | 1.50000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
307.2 | 0 | 0 | 2.00000i | 1.65831 | + | 1.50000i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
5.c | odd | 4 | 1 | inner |
55.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 495.2.k.a | 4 | |
3.b | odd | 2 | 1 | 55.2.e.b | ✓ | 4 | |
5.c | odd | 4 | 1 | inner | 495.2.k.a | 4 | |
11.b | odd | 2 | 1 | CM | 495.2.k.a | 4 | |
12.b | even | 2 | 1 | 880.2.bd.d | 4 | ||
15.d | odd | 2 | 1 | 275.2.e.a | 4 | ||
15.e | even | 4 | 1 | 55.2.e.b | ✓ | 4 | |
15.e | even | 4 | 1 | 275.2.e.a | 4 | ||
33.d | even | 2 | 1 | 55.2.e.b | ✓ | 4 | |
33.f | even | 10 | 4 | 605.2.m.a | 16 | ||
33.h | odd | 10 | 4 | 605.2.m.a | 16 | ||
55.e | even | 4 | 1 | inner | 495.2.k.a | 4 | |
60.l | odd | 4 | 1 | 880.2.bd.d | 4 | ||
132.d | odd | 2 | 1 | 880.2.bd.d | 4 | ||
165.d | even | 2 | 1 | 275.2.e.a | 4 | ||
165.l | odd | 4 | 1 | 55.2.e.b | ✓ | 4 | |
165.l | odd | 4 | 1 | 275.2.e.a | 4 | ||
165.u | odd | 20 | 4 | 605.2.m.a | 16 | ||
165.v | even | 20 | 4 | 605.2.m.a | 16 | ||
660.q | even | 4 | 1 | 880.2.bd.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
55.2.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
55.2.e.b | ✓ | 4 | 15.e | even | 4 | 1 | |
55.2.e.b | ✓ | 4 | 33.d | even | 2 | 1 | |
55.2.e.b | ✓ | 4 | 165.l | odd | 4 | 1 | |
275.2.e.a | 4 | 15.d | odd | 2 | 1 | ||
275.2.e.a | 4 | 15.e | even | 4 | 1 | ||
275.2.e.a | 4 | 165.d | even | 2 | 1 | ||
275.2.e.a | 4 | 165.l | odd | 4 | 1 | ||
495.2.k.a | 4 | 1.a | even | 1 | 1 | trivial | |
495.2.k.a | 4 | 5.c | odd | 4 | 1 | inner | |
495.2.k.a | 4 | 11.b | odd | 2 | 1 | CM | |
495.2.k.a | 4 | 55.e | even | 4 | 1 | inner | |
605.2.m.a | 16 | 33.f | even | 10 | 4 | ||
605.2.m.a | 16 | 33.h | odd | 10 | 4 | ||
605.2.m.a | 16 | 165.u | odd | 20 | 4 | ||
605.2.m.a | 16 | 165.v | even | 20 | 4 | ||
880.2.bd.d | 4 | 12.b | even | 2 | 1 | ||
880.2.bd.d | 4 | 60.l | odd | 4 | 1 | ||
880.2.bd.d | 4 | 132.d | odd | 2 | 1 | ||
880.2.bd.d | 4 | 660.q | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - T^{2} + 25 \)
$7$
\( T^{4} \)
$11$
\( (T^{2} - 11)^{2} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1225 \)
$29$
\( T^{4} \)
$31$
\( (T^{2} - 99)^{2} \)
$37$
\( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 625 \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 2500 \)
$53$
\( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 4900 \)
$59$
\( (T^{2} + 11)^{2} \)
$61$
\( T^{4} \)
$67$
\( T^{4} - 26 T^{3} + 338 T^{2} + \cdots + 1225 \)
$71$
\( (T - 3)^{4} \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( (T^{2} + 81)^{2} \)
$97$
\( T^{4} + 34 T^{3} + 578 T^{2} + \cdots + 9025 \)
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