Newspace parameters
Level: | \( N \) | \(=\) | \( 109 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 109.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.870369382032\) |
Analytic rank: | \(1\) |
Dimension: | \(3\) |
Coefficient field: | \(\Q(\zeta_{14})^+\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{3} - x^{2} - 2x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + 2 \) |
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 |
|
−2.24698 | 0.246980 | 3.04892 | −3.35690 | −0.554958 | 0.911854 | −2.35690 | −2.93900 | 7.54288 | |||||||||||||||||||||||||||
1.2 | −0.554958 | −1.44504 | −1.69202 | 1.04892 | 0.801938 | −4.85086 | 2.04892 | −0.911854 | −0.582105 | ||||||||||||||||||||||||||||
1.3 | 0.801938 | −2.80194 | −1.35690 | −3.69202 | −2.24698 | 2.93900 | −2.69202 | 4.85086 | −2.96077 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(109\) | \(1\) |
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 | 109.2.a.b | ✓ | 3 |
3.b | odd | 2 | 1 | 981.2.a.c | 3 | ||
4.b | odd | 2 | 1 | 1744.2.a.l | 3 | ||
5.b | even | 2 | 1 | 2725.2.a.h | 3 | ||
7.b | odd | 2 | 1 | 5341.2.a.d | 3 | ||
8.b | even | 2 | 1 | 6976.2.a.z | 3 | ||
8.d | odd | 2 | 1 | 6976.2.a.s | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
109.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
981.2.a.c | 3 | 3.b | odd | 2 | 1 | ||
1744.2.a.l | 3 | 4.b | odd | 2 | 1 | ||
2725.2.a.h | 3 | 5.b | even | 2 | 1 | ||
5341.2.a.d | 3 | 7.b | odd | 2 | 1 | ||
6976.2.a.s | 3 | 8.d | odd | 2 | 1 | ||
6976.2.a.z | 3 | 8.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(109))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 2T^{2} - T - 1 \)
$3$
\( T^{3} + 4 T^{2} + 3 T - 1 \)
$5$
\( T^{3} + 6 T^{2} + 5 T - 13 \)
$7$
\( T^{3} + T^{2} - 16 T + 13 \)
$11$
\( T^{3} + 13 T^{2} + 54 T + 71 \)
$13$
\( T^{3} + T^{2} - 16 T + 13 \)
$17$
\( T^{3} - 3 T^{2} - 4 T + 13 \)
$19$
\( T^{3} + 5 T^{2} - 8 T - 41 \)
$23$
\( T^{3} - T^{2} - 58 T - 13 \)
$29$
\( T^{3} + 6 T^{2} - 37 T - 181 \)
$31$
\( T^{3} + 7 T^{2} - 28 T + 7 \)
$37$
\( T^{3} - 7T - 7 \)
$41$
\( T^{3} + 6 T^{2} - 51 T + 71 \)
$43$
\( T^{3} - 9 T^{2} - 36 T + 351 \)
$47$
\( T^{3} + 10 T^{2} - 25 T - 125 \)
$53$
\( T^{3} - 9 T^{2} + 20 T - 13 \)
$59$
\( T^{3} + 25 T^{2} + 192 T + 461 \)
$61$
\( T^{3} + 10 T^{2} - 144 T - 1336 \)
$67$
\( T^{3} + 11 T^{2} - 25 T - 43 \)
$71$
\( T^{3} + 10 T^{2} - 11 T - 223 \)
$73$
\( T^{3} - 20 T^{2} + 131 T - 281 \)
$79$
\( T^{3} + 6 T^{2} - 79 T - 461 \)
$83$
\( T^{3} + 13 T^{2} - 2 T - 139 \)
$89$
\( T^{3} + 21 T^{2} + 84 T + 91 \)
$97$
\( T^{3} + 20 T^{2} + 75 T - 125 \)
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