Newspace parameters
| Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 176.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(90.6463071648\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{889}) \) |
| Defining polynomial: |
\( x^{2} - x - 222 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{889})\). 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 | −123.672 | 0 | 1185.58 | 0 | −1174.66 | 0 | −4388.12 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 144.672 | 0 | −1706.58 | 0 | 8664.66 | 0 | 1247.12 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
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 | 176.10.a.e | 2 | |
| 4.b | odd | 2 | 1 | 22.10.a.d | ✓ | 2 | |
| 12.b | even | 2 | 1 | 198.10.a.n | 2 | ||
| 44.c | even | 2 | 1 | 242.10.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.10.a.d | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 176.10.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 198.10.a.n | 2 | 12.b | even | 2 | 1 | ||
| 242.10.a.e | 2 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 21T_{3} - 17892 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(176))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 21T - 17892 \)
$5$
\( T^{2} + 521 T - 2023290 \)
$7$
\( T^{2} - 7490 T - 10178000 \)
$11$
\( (T + 14641)^{2} \)
$13$
\( T^{2} - 150314 T + 5595212424 \)
$17$
\( T^{2} - 690472 T + 110050366092 \)
$19$
\( T^{2} + 511212 T - 235841735968 \)
$23$
\( T^{2} + 874751 T - 251963912952 \)
$29$
\( T^{2} + 2951058 T - 11964147063840 \)
$31$
\( T^{2} - 5818705 T - 21505055373504 \)
$37$
\( T^{2} - 2658905 T - 34431390323214 \)
$41$
\( T^{2} - 13427994 T + 33438413932128 \)
$43$
\( T^{2} + \cdots - 200309296545160 \)
$47$
\( T^{2} + \cdots - 168165989315712 \)
$53$
\( T^{2} + \cdots + 866157473672220 \)
$59$
\( T^{2} - 119136183 T - 11\!\cdots\!48 \)
$61$
\( T^{2} + 90424326 T - 25\!\cdots\!60 \)
$67$
\( T^{2} - 295944891 T + 15\!\cdots\!68 \)
$71$
\( T^{2} - 322953267 T + 26\!\cdots\!60 \)
$73$
\( T^{2} + 255975514 T + 15\!\cdots\!08 \)
$79$
\( T^{2} - 889658 T - 87\!\cdots\!20 \)
$83$
\( T^{2} - 277699042 T - 34\!\cdots\!84 \)
$89$
\( T^{2} - 1363672217 T + 41\!\cdots\!62 \)
$97$
\( T^{2} - 1398434043 T + 13\!\cdots\!02 \)
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