Newspace parameters
Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 624.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(36.8171918436\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 156) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
\(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
337.1 |
|
0 | −3.00000 | 0 | − | 6.92820i | 0 | 10.3923i | 0 | 9.00000 | 0 | |||||||||||||||||||||||
337.2 | 0 | −3.00000 | 0 | 6.92820i | 0 | − | 10.3923i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 624.4.c.a | 2 | |
4.b | odd | 2 | 1 | 156.4.b.a | ✓ | 2 | |
12.b | even | 2 | 1 | 468.4.b.b | 2 | ||
13.b | even | 2 | 1 | inner | 624.4.c.a | 2 | |
52.b | odd | 2 | 1 | 156.4.b.a | ✓ | 2 | |
52.f | even | 4 | 2 | 2028.4.a.g | 2 | ||
156.h | even | 2 | 1 | 468.4.b.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
156.4.b.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
156.4.b.a | ✓ | 2 | 52.b | odd | 2 | 1 | |
468.4.b.b | 2 | 12.b | even | 2 | 1 | ||
468.4.b.b | 2 | 156.h | even | 2 | 1 | ||
624.4.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
624.4.c.a | 2 | 13.b | even | 2 | 1 | inner | |
2028.4.a.g | 2 | 52.f | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 48 \)
acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T + 3)^{2} \)
$5$
\( T^{2} + 48 \)
$7$
\( T^{2} + 108 \)
$11$
\( T^{2} + 300 \)
$13$
\( T^{2} + 26T + 2197 \)
$17$
\( (T - 18)^{2} \)
$19$
\( T^{2} + 2028 \)
$23$
\( (T + 24)^{2} \)
$29$
\( (T - 6)^{2} \)
$31$
\( T^{2} + 11532 \)
$37$
\( T^{2} + 34992 \)
$41$
\( T^{2} + 12288 \)
$43$
\( (T + 20)^{2} \)
$47$
\( T^{2} + 86700 \)
$53$
\( (T + 306)^{2} \)
$59$
\( T^{2} + 494508 \)
$61$
\( (T - 70)^{2} \)
$67$
\( T^{2} + 205932 \)
$71$
\( T^{2} + 1058508 \)
$73$
\( T^{2} + 192 \)
$79$
\( (T - 416)^{2} \)
$83$
\( T^{2} + 1160652 \)
$89$
\( T^{2} + 768 \)
$97$
\( T^{2} + 277248 \)
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