Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.8556672451\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6469}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1617 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 144\sqrt{6469}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9997.93 | 1.59432e6 | −3.42591e7 | −1.80194e8 | −1.59399e10 | 1.89150e11 | 1.68442e12 | 2.54187e12 | 1.80157e12 | ||||||||||||||||||||||||
| 1.2 | 13165.9 | 1.59432e6 | 3.91241e7 | −4.72587e9 | 2.09908e10 | −3.40807e11 | −1.25200e12 | 2.54187e12 | −6.22205e13 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3.28.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 9.28.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 48.28.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.28.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 9.28.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 48.28.a.d | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3168T_{2} - 131632128 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3168 T - 131632128 \)
$3$
\( (T - 1594323)^{2} \)
$5$
\( T^{2} + 4906065060 T + 85\!\cdots\!00 \)
$7$
\( T^{2} + 151657089584 T - 64\!\cdots\!20 \)
$11$
\( T^{2} - 40854992127048 T - 14\!\cdots\!08 \)
$13$
\( T^{2} + 417397898638292 T - 39\!\cdots\!88 \)
$17$
\( T^{2} + \cdots + 13\!\cdots\!16 \)
$19$
\( T^{2} + \cdots + 90\!\cdots\!80 \)
$23$
\( T^{2} + \cdots - 42\!\cdots\!60 \)
$29$
\( T^{2} + \cdots + 31\!\cdots\!20 \)
$31$
\( T^{2} + \cdots - 42\!\cdots\!00 \)
$37$
\( T^{2} + \cdots - 23\!\cdots\!80 \)
$41$
\( T^{2} + \cdots - 34\!\cdots\!40 \)
$43$
\( T^{2} + \cdots + 31\!\cdots\!76 \)
$47$
\( T^{2} + \cdots - 12\!\cdots\!36 \)
$53$
\( T^{2} + \cdots + 30\!\cdots\!00 \)
$59$
\( T^{2} + \cdots + 43\!\cdots\!20 \)
$61$
\( T^{2} + \cdots - 28\!\cdots\!44 \)
$67$
\( T^{2} + \cdots - 20\!\cdots\!64 \)
$71$
\( T^{2} + \cdots + 42\!\cdots\!44 \)
$73$
\( T^{2} + \cdots - 68\!\cdots\!40 \)
$79$
\( T^{2} + \cdots - 13\!\cdots\!00 \)
$83$
\( T^{2} + \cdots + 13\!\cdots\!68 \)
$89$
\( T^{2} + \cdots + 61\!\cdots\!40 \)
$97$
\( T^{2} + \cdots - 18\!\cdots\!64 \)
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