Newspace parameters
Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 121.c (of order \(5\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.966189864457\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{10})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} + x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 11) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 |
|
1.61803 | − | 1.17557i | −0.309017 | − | 0.951057i | 0.618034 | − | 1.90211i | −0.809017 | − | 0.587785i | −1.61803 | − | 1.17557i | −0.618034 | + | 1.90211i | 0 | 1.61803 | − | 1.17557i | −2.00000 | ||||||||||||||||
9.1 | −0.618034 | + | 1.90211i | 0.809017 | − | 0.587785i | −1.61803 | − | 1.17557i | 0.309017 | + | 0.951057i | 0.618034 | + | 1.90211i | 1.61803 | + | 1.17557i | 0 | −0.618034 | + | 1.90211i | −2.00000 | |||||||||||||||||
27.1 | −0.618034 | − | 1.90211i | 0.809017 | + | 0.587785i | −1.61803 | + | 1.17557i | 0.309017 | − | 0.951057i | 0.618034 | − | 1.90211i | 1.61803 | − | 1.17557i | 0 | −0.618034 | − | 1.90211i | −2.00000 | |||||||||||||||||
81.1 | 1.61803 | + | 1.17557i | −0.309017 | + | 0.951057i | 0.618034 | + | 1.90211i | −0.809017 | + | 0.587785i | −1.61803 | + | 1.17557i | −0.618034 | − | 1.90211i | 0 | 1.61803 | + | 1.17557i | −2.00000 | |||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 3 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 8T_{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \)
$3$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$5$
\( T^{4} + T^{3} + T^{2} + T + 1 \)
$7$
\( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$17$
\( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \)
$19$
\( T^{4} \)
$23$
\( (T + 1)^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401 \)
$37$
\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \)
$41$
\( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \)
$43$
\( (T + 6)^{4} \)
$47$
\( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \)
$53$
\( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$59$
\( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \)
$61$
\( T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736 \)
$67$
\( (T + 7)^{4} \)
$71$
\( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \)
$73$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$79$
\( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 10000 \)
$83$
\( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$89$
\( (T - 15)^{4} \)
$97$
\( T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401 \)
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