Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(57.6840136504\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2305}) \) |
| Defining polynomial: |
\( x^{2} - x - 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 14) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2305}\). 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 |
|
0 | −233.052 | 0 | −356.781 | 0 | −2401.00 | 0 | 34630.3 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 247.052 | 0 | −2373.22 | 0 | −2401.00 | 0 | 41351.7 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
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 | 112.10.a.c | 2 | |
| 4.b | odd | 2 | 1 | 14.10.a.c | ✓ | 2 | |
| 12.b | even | 2 | 1 | 126.10.a.o | 2 | ||
| 20.d | odd | 2 | 1 | 350.10.a.j | 2 | ||
| 20.e | even | 4 | 2 | 350.10.c.j | 4 | ||
| 28.d | even | 2 | 1 | 98.10.a.e | 2 | ||
| 28.f | even | 6 | 2 | 98.10.c.h | 4 | ||
| 28.g | odd | 6 | 2 | 98.10.c.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 98.10.a.e | 2 | 28.d | even | 2 | 1 | ||
| 98.10.c.h | 4 | 28.f | even | 6 | 2 | ||
| 98.10.c.j | 4 | 28.g | odd | 6 | 2 | ||
| 112.10.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 126.10.a.o | 2 | 12.b | even | 2 | 1 | ||
| 350.10.a.j | 2 | 20.d | odd | 2 | 1 | ||
| 350.10.c.j | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 14T_{3} - 57576 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 14T - 57576 \)
$5$
\( T^{2} + 2730 T + 846720 \)
$7$
\( (T + 2401)^{2} \)
$11$
\( T^{2} + 44940 T - 2036361600 \)
$13$
\( T^{2} - 100282 T + 2397429256 \)
$17$
\( T^{2} + 870408 T + 183575251116 \)
$19$
\( T^{2} + 508774 T - 352577596856 \)
$23$
\( T^{2} + 79800 T - 115915968000 \)
$29$
\( T^{2} - 2006328 T - 31904129519604 \)
$31$
\( T^{2} + 2188732 T - 2367849772544 \)
$37$
\( T^{2} + 20723576 T + 60225113026444 \)
$41$
\( T^{2} + \cdots - 527816477266884 \)
$43$
\( T^{2} + \cdots - 271341247682336 \)
$47$
\( T^{2} - 74542524 T + 12\!\cdots\!44 \)
$53$
\( T^{2} + \cdots - 494746212296124 \)
$59$
\( T^{2} - 133642362 T + 15\!\cdots\!36 \)
$61$
\( T^{2} - 227801686 T + 11\!\cdots\!24 \)
$67$
\( T^{2} + 332930272 T + 21\!\cdots\!96 \)
$71$
\( T^{2} - 167985720 T - 85\!\cdots\!00 \)
$73$
\( T^{2} + 44684276 T - 83\!\cdots\!56 \)
$79$
\( T^{2} + 269642776 T + 68\!\cdots\!44 \)
$83$
\( T^{2} - 183105762 T - 76\!\cdots\!64 \)
$89$
\( T^{2} - 791657748 T + 50\!\cdots\!76 \)
$97$
\( T^{2} + 4169480 T - 45\!\cdots\!00 \)
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