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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 11) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 |
|
−0.427051 | + | 0.587785i | 0.427051 | − | 1.31433i | 1.07295 | + | 3.30220i | −3.23607 | + | 2.35114i | 0.590170 | + | 0.812299i | −9.47214 | + | 3.07768i | −5.16312 | − | 1.67760i | 5.73607 | + | 4.16750i | − | 2.90617i | |||||||||||||
94.1 | 2.92705 | − | 0.951057i | −2.92705 | − | 2.12663i | 4.42705 | − | 3.21644i | 1.23607 | − | 3.80423i | −10.5902 | − | 3.44095i | −0.527864 | − | 0.726543i | 2.66312 | − | 3.66547i | 1.26393 | + | 3.88998i | − | 12.3107i | ||||||||||||||
112.1 | 2.92705 | + | 0.951057i | −2.92705 | + | 2.12663i | 4.42705 | + | 3.21644i | 1.23607 | + | 3.80423i | −10.5902 | + | 3.44095i | −0.527864 | + | 0.726543i | 2.66312 | + | 3.66547i | 1.26393 | − | 3.88998i | 12.3107i | |||||||||||||||
118.1 | −0.427051 | − | 0.587785i | 0.427051 | + | 1.31433i | 1.07295 | − | 3.30220i | −3.23607 | − | 2.35114i | 0.590170 | − | 0.812299i | −9.47214 | − | 3.07768i | −5.16312 | + | 1.67760i | 5.73607 | − | 4.16750i | 2.90617i | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 121.3.d.c | 4 | |
11.b | odd | 2 | 1 | 121.3.d.a | 4 | ||
11.c | even | 5 | 1 | 11.3.d.a | ✓ | 4 | |
11.c | even | 5 | 1 | 121.3.b.b | 4 | ||
11.c | even | 5 | 1 | 121.3.d.a | 4 | ||
11.c | even | 5 | 1 | 121.3.d.d | 4 | ||
11.d | odd | 10 | 1 | 11.3.d.a | ✓ | 4 | |
11.d | odd | 10 | 1 | 121.3.b.b | 4 | ||
11.d | odd | 10 | 1 | inner | 121.3.d.c | 4 | |
11.d | odd | 10 | 1 | 121.3.d.d | 4 | ||
33.f | even | 10 | 1 | 99.3.k.a | 4 | ||
33.f | even | 10 | 1 | 1089.3.c.e | 4 | ||
33.h | odd | 10 | 1 | 99.3.k.a | 4 | ||
33.h | odd | 10 | 1 | 1089.3.c.e | 4 | ||
44.g | even | 10 | 1 | 176.3.n.a | 4 | ||
44.h | odd | 10 | 1 | 176.3.n.a | 4 | ||
55.h | odd | 10 | 1 | 275.3.x.e | 4 | ||
55.j | even | 10 | 1 | 275.3.x.e | 4 | ||
55.k | odd | 20 | 2 | 275.3.q.d | 8 | ||
55.l | even | 20 | 2 | 275.3.q.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
11.3.d.a | ✓ | 4 | 11.c | even | 5 | 1 | |
11.3.d.a | ✓ | 4 | 11.d | odd | 10 | 1 | |
99.3.k.a | 4 | 33.f | even | 10 | 1 | ||
99.3.k.a | 4 | 33.h | odd | 10 | 1 | ||
121.3.b.b | 4 | 11.c | even | 5 | 1 | ||
121.3.b.b | 4 | 11.d | odd | 10 | 1 | ||
121.3.d.a | 4 | 11.b | odd | 2 | 1 | ||
121.3.d.a | 4 | 11.c | even | 5 | 1 | ||
121.3.d.c | 4 | 1.a | even | 1 | 1 | trivial | |
121.3.d.c | 4 | 11.d | odd | 10 | 1 | inner | |
121.3.d.d | 4 | 11.c | even | 5 | 1 | ||
121.3.d.d | 4 | 11.d | odd | 10 | 1 | ||
176.3.n.a | 4 | 44.g | even | 10 | 1 | ||
176.3.n.a | 4 | 44.h | odd | 10 | 1 | ||
275.3.q.d | 8 | 55.k | odd | 20 | 2 | ||
275.3.q.d | 8 | 55.l | even | 20 | 2 | ||
275.3.x.e | 4 | 55.h | odd | 10 | 1 | ||
275.3.x.e | 4 | 55.j | even | 10 | 1 | ||
1089.3.c.e | 4 | 33.f | even | 10 | 1 | ||
1089.3.c.e | 4 | 33.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 5T_{2} + 5 \)
acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5 \)
$3$
\( T^{4} + 5 T^{3} + 10 T^{2} + 25 \)
$5$
\( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \)
$7$
\( T^{4} + 20 T^{3} + 120 T^{2} + \cdots + 80 \)
$11$
\( T^{4} \)
$13$
\( T^{4} - 10T^{3} + 2000 \)
$17$
\( T^{4} - 65 T^{3} + 1690 T^{2} + \cdots + 142805 \)
$19$
\( T^{4} - 10 T^{3} + 90 T^{2} + \cdots + 605 \)
$23$
\( (T^{2} + 10 T + 20)^{2} \)
$29$
\( T^{4} - 50 T^{3} + 600 T^{2} + \cdots + 9680 \)
$31$
\( T^{4} - 32 T^{3} + 1384 T^{2} + \cdots + 55696 \)
$37$
\( T^{4} + 90 T^{3} + 4860 T^{2} + \cdots + 2624400 \)
$41$
\( T^{4} - 135 T^{3} + 5130 T^{2} + \cdots + 8405 \)
$43$
\( T^{4} + 1625 T^{2} + 581405 \)
$47$
\( T^{4} + 20 T^{3} + 640 T^{2} + \cdots + 384400 \)
$53$
\( T^{4} + 30 T^{3} + 5400 T^{2} + \cdots + 810000 \)
$59$
\( T^{4} + 52 T^{3} + 1054 T^{2} + \cdots + 5041 \)
$61$
\( T^{4} - 80 T^{3} + 2720 T^{2} + \cdots + 403280 \)
$67$
\( (T^{2} + 115 T + 2945)^{2} \)
$71$
\( T^{4} + 162 T^{3} + \cdots + 22619536 \)
$73$
\( T^{4} + 85 T^{3} - 2210 T^{2} + \cdots + 93787805 \)
$79$
\( T^{4} + 130 T^{3} + 4940 T^{2} + \cdots + 67280 \)
$83$
\( T^{4} + 10 T^{3} - 410 T^{2} + \cdots + 22281605 \)
$89$
\( (T^{2} - 61 T - 7681)^{2} \)
$97$
\( T^{4} - 170 T^{3} + \cdots + 31416025 \)
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