Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.f (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.790518980011\) |
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 33) |
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 \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-\zeta_{10}^{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
−0.309017 | − | 0.224514i | 0 | −0.572949 | − | 1.76336i | 1.30902 | − | 0.951057i | 0 | −0.309017 | − | 0.951057i | −0.454915 | + | 1.40008i | 0 | −0.618034 | ||||||||||||||||||||
64.1 | 0.809017 | − | 2.48990i | 0 | −3.92705 | − | 2.85317i | 0.190983 | + | 0.587785i | 0 | 0.809017 | + | 0.587785i | −6.04508 | + | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||
82.1 | 0.809017 | + | 2.48990i | 0 | −3.92705 | + | 2.85317i | 0.190983 | − | 0.587785i | 0 | 0.809017 | − | 0.587785i | −6.04508 | − | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||
91.1 | −0.309017 | + | 0.224514i | 0 | −0.572949 | + | 1.76336i | 1.30902 | + | 0.951057i | 0 | −0.309017 | + | 0.951057i | −0.454915 | − | 1.40008i | 0 | −0.618034 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.2.f.a | 4 | |
3.b | odd | 2 | 1 | 33.2.e.b | ✓ | 4 | |
9.c | even | 3 | 2 | 891.2.n.b | 8 | ||
9.d | odd | 6 | 2 | 891.2.n.c | 8 | ||
11.c | even | 5 | 1 | inner | 99.2.f.a | 4 | |
11.c | even | 5 | 1 | 1089.2.a.t | 2 | ||
11.d | odd | 10 | 1 | 1089.2.a.l | 2 | ||
12.b | even | 2 | 1 | 528.2.y.b | 4 | ||
15.d | odd | 2 | 1 | 825.2.n.c | 4 | ||
15.e | even | 4 | 2 | 825.2.bx.d | 8 | ||
33.d | even | 2 | 1 | 363.2.e.f | 4 | ||
33.f | even | 10 | 1 | 363.2.a.i | 2 | ||
33.f | even | 10 | 2 | 363.2.e.b | 4 | ||
33.f | even | 10 | 1 | 363.2.e.f | 4 | ||
33.h | odd | 10 | 1 | 33.2.e.b | ✓ | 4 | |
33.h | odd | 10 | 1 | 363.2.a.d | 2 | ||
33.h | odd | 10 | 2 | 363.2.e.k | 4 | ||
99.m | even | 15 | 2 | 891.2.n.b | 8 | ||
99.n | odd | 30 | 2 | 891.2.n.c | 8 | ||
132.n | odd | 10 | 1 | 5808.2.a.ci | 2 | ||
132.o | even | 10 | 1 | 528.2.y.b | 4 | ||
132.o | even | 10 | 1 | 5808.2.a.cj | 2 | ||
165.o | odd | 10 | 1 | 825.2.n.c | 4 | ||
165.o | odd | 10 | 1 | 9075.2.a.cb | 2 | ||
165.r | even | 10 | 1 | 9075.2.a.u | 2 | ||
165.v | even | 20 | 2 | 825.2.bx.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.2.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
33.2.e.b | ✓ | 4 | 33.h | odd | 10 | 1 | |
99.2.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
99.2.f.a | 4 | 11.c | even | 5 | 1 | inner | |
363.2.a.d | 2 | 33.h | odd | 10 | 1 | ||
363.2.a.i | 2 | 33.f | even | 10 | 1 | ||
363.2.e.b | 4 | 33.f | even | 10 | 2 | ||
363.2.e.f | 4 | 33.d | even | 2 | 1 | ||
363.2.e.f | 4 | 33.f | even | 10 | 1 | ||
363.2.e.k | 4 | 33.h | odd | 10 | 2 | ||
528.2.y.b | 4 | 12.b | even | 2 | 1 | ||
528.2.y.b | 4 | 132.o | even | 10 | 1 | ||
825.2.n.c | 4 | 15.d | odd | 2 | 1 | ||
825.2.n.c | 4 | 165.o | odd | 10 | 1 | ||
825.2.bx.d | 8 | 15.e | even | 4 | 2 | ||
825.2.bx.d | 8 | 165.v | even | 20 | 2 | ||
891.2.n.b | 8 | 9.c | even | 3 | 2 | ||
891.2.n.b | 8 | 99.m | even | 15 | 2 | ||
891.2.n.c | 8 | 9.d | odd | 6 | 2 | ||
891.2.n.c | 8 | 99.n | odd | 30 | 2 | ||
1089.2.a.l | 2 | 11.d | odd | 10 | 1 | ||
1089.2.a.t | 2 | 11.c | even | 5 | 1 | ||
5808.2.a.ci | 2 | 132.n | odd | 10 | 1 | ||
5808.2.a.cj | 2 | 132.o | even | 10 | 1 | ||
9075.2.a.u | 2 | 165.r | even | 10 | 1 | ||
9075.2.a.cb | 2 | 165.o | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \)
$7$
\( T^{4} - T^{3} + T^{2} - T + 1 \)
$11$
\( T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 121 \)
$13$
\( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \)
$17$
\( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \)
$19$
\( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \)
$23$
\( (T^{2} - 4 T - 1)^{2} \)
$29$
\( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \)
$31$
\( T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 961 \)
$37$
\( T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121 \)
$41$
\( T^{4} - 3 T^{3} + 19 T^{2} - 7 T + 1 \)
$43$
\( (T^{2} - 45)^{2} \)
$47$
\( T^{4} + 17 T^{3} + 114 T^{2} + \cdots + 121 \)
$53$
\( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \)
$59$
\( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \)
$61$
\( T^{4} + 21 T^{3} + 306 T^{2} + \cdots + 9801 \)
$67$
\( (T^{2} + 3 T - 9)^{2} \)
$71$
\( T^{4} + 15 T^{3} + 190 T^{2} + \cdots + 3025 \)
$73$
\( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \)
$79$
\( T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641 \)
$83$
\( T^{4} + 13 T^{3} + 69 T^{2} + \cdots + 121 \)
$89$
\( (T^{2} - 12 T + 31)^{2} \)
$97$
\( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \)
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