Newspace parameters
Level: | \( N \) | \(=\) | \( 153 = 3^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 153.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.02729223088\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.0.4669632.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 74x^{2} + 1072 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 17) |
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 \( x^{4} + 74x^{2} + 1072 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{2} + 40 ) / 6 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} + 46\nu ) / 6 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( 6\beta_{2} - 40 \) |
\(\nu^{3}\) | \(=\) | \( 6\beta_{3} - 46\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).
\(n\) | \(37\) | \(137\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
118.1 |
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−2.37228 | 0 | −2.37228 | − | 19.4389i | 0 | − | 14.9929i | 24.6060 | 0 | 46.1145i | ||||||||||||||||||||||||||||
118.2 | −2.37228 | 0 | −2.37228 | 19.4389i | 0 | 14.9929i | 24.6060 | 0 | − | 46.1145i | ||||||||||||||||||||||||||||||
118.3 | 3.37228 | 0 | 3.37228 | − | 10.1060i | 0 | − | 17.4703i | −15.6060 | 0 | − | 34.0802i | ||||||||||||||||||||||||||||
118.4 | 3.37228 | 0 | 3.37228 | 10.1060i | 0 | 17.4703i | −15.6060 | 0 | 34.0802i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 153.4.d.b | 4 | |
3.b | odd | 2 | 1 | 17.4.b.a | ✓ | 4 | |
12.b | even | 2 | 1 | 272.4.b.d | 4 | ||
15.d | odd | 2 | 1 | 425.4.d.c | 4 | ||
15.e | even | 4 | 2 | 425.4.c.c | 8 | ||
17.b | even | 2 | 1 | inner | 153.4.d.b | 4 | |
51.c | odd | 2 | 1 | 17.4.b.a | ✓ | 4 | |
51.f | odd | 4 | 2 | 289.4.a.e | 4 | ||
204.h | even | 2 | 1 | 272.4.b.d | 4 | ||
255.h | odd | 2 | 1 | 425.4.d.c | 4 | ||
255.o | even | 4 | 2 | 425.4.c.c | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
17.4.b.a | ✓ | 4 | 51.c | odd | 2 | 1 | |
153.4.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
153.4.d.b | 4 | 17.b | even | 2 | 1 | inner | |
272.4.b.d | 4 | 12.b | even | 2 | 1 | ||
272.4.b.d | 4 | 204.h | even | 2 | 1 | ||
289.4.a.e | 4 | 51.f | odd | 4 | 2 | ||
425.4.c.c | 8 | 15.e | even | 4 | 2 | ||
425.4.c.c | 8 | 255.o | even | 4 | 2 | ||
425.4.d.c | 4 | 15.d | odd | 2 | 1 | ||
425.4.d.c | 4 | 255.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 8 \)
acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T - 8)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 480 T^{2} + 38592 \)
$7$
\( T^{4} + 530 T^{2} + 68608 \)
$11$
\( T^{4} + 3626 T^{2} + \cdots + 2573872 \)
$13$
\( (T^{2} - 70 T - 392)^{2} \)
$17$
\( T^{4} + 112 T^{3} + \cdots + 24137569 \)
$19$
\( (T + 28)^{4} \)
$23$
\( T^{4} + 28322 T^{2} + \cdots + 10295488 \)
$29$
\( T^{4} + 23520 T^{2} + \cdots + 92659392 \)
$31$
\( T^{4} + 4274 T^{2} + \cdots + 4390912 \)
$37$
\( T^{4} + 86240 T^{2} + \cdots + 1245754048 \)
$41$
\( T^{4} + 116960 T^{2} + \cdots + 2799480832 \)
$43$
\( (T^{2} + 260 T - 30752)^{2} \)
$47$
\( (T^{2} + 476 T + 50176)^{2} \)
$53$
\( (T^{2} - 288 T - 49092)^{2} \)
$59$
\( (T^{2} - 588 T - 75264)^{2} \)
$61$
\( T^{4} + 482528 T^{2} + \cdots + 7146728128 \)
$67$
\( (T^{2} + 120 T - 30192)^{2} \)
$71$
\( T^{4} + 52626 T^{2} + \cdots + 92659392 \)
$73$
\( T^{4} + 1611264 T^{2} + \cdots + 647466319872 \)
$79$
\( T^{4} + 689234 T^{2} + \cdots + 70925616832 \)
$83$
\( (T^{2} - 1428 T + 451584)^{2} \)
$89$
\( (T^{2} + 994 T + 232456)^{2} \)
$97$
\( T^{4} + 275552 T^{2} + \cdots + 16878665728 \)
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