Newspace parameters
Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 121.d (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.29701119876\) |
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: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 11) |
Sato-Tate group: | $\mathrm{U}(1)[D_{10}]$ |
$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(\zeta_{10}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
40.1 |
|
0 | −1.54508 | + | 4.75528i | 1.23607 | + | 3.80423i | 0.809017 | − | 0.587785i | 0 | 0 | 0 | −12.9443 | − | 9.40456i | 0 | ||||||||||||||||||||||
94.1 | 0 | 4.04508 | + | 2.93893i | −3.23607 | + | 2.35114i | −0.309017 | + | 0.951057i | 0 | 0 | 0 | 4.94427 | + | 15.2169i | 0 | |||||||||||||||||||||||
112.1 | 0 | 4.04508 | − | 2.93893i | −3.23607 | − | 2.35114i | −0.309017 | − | 0.951057i | 0 | 0 | 0 | 4.94427 | − | 15.2169i | 0 | |||||||||||||||||||||||
118.1 | 0 | −1.54508 | − | 4.75528i | 1.23607 | − | 3.80423i | 0.809017 | + | 0.587785i | 0 | 0 | 0 | −12.9443 | + | 9.40456i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 121.3.d.b | 4 | |
11.b | odd | 2 | 1 | CM | 121.3.d.b | 4 | |
11.c | even | 5 | 1 | 11.3.b.a | ✓ | 1 | |
11.c | even | 5 | 3 | inner | 121.3.d.b | 4 | |
11.d | odd | 10 | 1 | 11.3.b.a | ✓ | 1 | |
11.d | odd | 10 | 3 | inner | 121.3.d.b | 4 | |
33.f | even | 10 | 1 | 99.3.c.a | 1 | ||
33.h | odd | 10 | 1 | 99.3.c.a | 1 | ||
44.g | even | 10 | 1 | 176.3.h.a | 1 | ||
44.h | odd | 10 | 1 | 176.3.h.a | 1 | ||
55.h | odd | 10 | 1 | 275.3.c.a | 1 | ||
55.j | even | 10 | 1 | 275.3.c.a | 1 | ||
55.k | odd | 20 | 2 | 275.3.d.a | 2 | ||
55.l | even | 20 | 2 | 275.3.d.a | 2 | ||
77.j | odd | 10 | 1 | 539.3.c.a | 1 | ||
77.l | even | 10 | 1 | 539.3.c.a | 1 | ||
88.k | even | 10 | 1 | 704.3.h.a | 1 | ||
88.l | odd | 10 | 1 | 704.3.h.a | 1 | ||
88.o | even | 10 | 1 | 704.3.h.b | 1 | ||
88.p | odd | 10 | 1 | 704.3.h.b | 1 | ||
132.n | odd | 10 | 1 | 1584.3.j.a | 1 | ||
132.o | even | 10 | 1 | 1584.3.j.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.3.b.a | ✓ | 1 | 11.c | even | 5 | 1 | |
11.3.b.a | ✓ | 1 | 11.d | odd | 10 | 1 | |
99.3.c.a | 1 | 33.f | even | 10 | 1 | ||
99.3.c.a | 1 | 33.h | odd | 10 | 1 | ||
121.3.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
121.3.d.b | 4 | 11.b | odd | 2 | 1 | CM | |
121.3.d.b | 4 | 11.c | even | 5 | 3 | inner | |
121.3.d.b | 4 | 11.d | odd | 10 | 3 | inner | |
176.3.h.a | 1 | 44.g | even | 10 | 1 | ||
176.3.h.a | 1 | 44.h | odd | 10 | 1 | ||
275.3.c.a | 1 | 55.h | odd | 10 | 1 | ||
275.3.c.a | 1 | 55.j | even | 10 | 1 | ||
275.3.d.a | 2 | 55.k | odd | 20 | 2 | ||
275.3.d.a | 2 | 55.l | even | 20 | 2 | ||
539.3.c.a | 1 | 77.j | odd | 10 | 1 | ||
539.3.c.a | 1 | 77.l | even | 10 | 1 | ||
704.3.h.a | 1 | 88.k | even | 10 | 1 | ||
704.3.h.a | 1 | 88.l | odd | 10 | 1 | ||
704.3.h.b | 1 | 88.o | even | 10 | 1 | ||
704.3.h.b | 1 | 88.p | odd | 10 | 1 | ||
1584.3.j.a | 1 | 132.n | odd | 10 | 1 | ||
1584.3.j.a | 1 | 132.o | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 5 T^{3} + 25 T^{2} - 125 T + 625 \)
$5$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( (T - 35)^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} - 37 T^{3} + 1369 T^{2} + \cdots + 1874161 \)
$37$
\( T^{4} - 25 T^{3} + 625 T^{2} + \cdots + 390625 \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} + 50 T^{3} + 2500 T^{2} + \cdots + 6250000 \)
$53$
\( T^{4} - 70 T^{3} + 4900 T^{2} + \cdots + 24010000 \)
$59$
\( T^{4} + 107 T^{3} + \cdots + 131079601 \)
$61$
\( T^{4} \)
$67$
\( (T - 35)^{4} \)
$71$
\( T^{4} - 133 T^{3} + \cdots + 312900721 \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( (T + 97)^{4} \)
$97$
\( T^{4} + 95 T^{3} + 9025 T^{2} + \cdots + 81450625 \)
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