Newspace parameters
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.g (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.726638658394\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).
\(n\) | \(15\) | \(66\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 |
|
−0.500000 | − | 0.866025i | −3.00000 | 0.500000 | − | 0.866025i | −1.50000 | + | 2.59808i | 1.50000 | + | 2.59808i | −2.50000 | + | 0.866025i | −3.00000 | 6.00000 | 3.00000 | ||||||||||||||
81.1 | −0.500000 | + | 0.866025i | −3.00000 | 0.500000 | + | 0.866025i | −1.50000 | − | 2.59808i | 1.50000 | − | 2.59808i | −2.50000 | − | 0.866025i | −3.00000 | 6.00000 | 3.00000 | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.g | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.2.g.a | ✓ | 2 |
3.b | odd | 2 | 1 | 819.2.n.c | 2 | ||
7.b | odd | 2 | 1 | 637.2.g.a | 2 | ||
7.c | even | 3 | 1 | 91.2.h.a | yes | 2 | |
7.c | even | 3 | 1 | 637.2.f.b | 2 | ||
7.d | odd | 6 | 1 | 637.2.f.a | 2 | ||
7.d | odd | 6 | 1 | 637.2.h.a | 2 | ||
13.c | even | 3 | 1 | 91.2.h.a | yes | 2 | |
13.c | even | 3 | 1 | 1183.2.e.a | 2 | ||
13.e | even | 6 | 1 | 1183.2.e.c | 2 | ||
21.h | odd | 6 | 1 | 819.2.s.a | 2 | ||
39.i | odd | 6 | 1 | 819.2.s.a | 2 | ||
91.g | even | 3 | 1 | inner | 91.2.g.a | ✓ | 2 |
91.g | even | 3 | 1 | 8281.2.a.i | 1 | ||
91.h | even | 3 | 1 | 637.2.f.b | 2 | ||
91.h | even | 3 | 1 | 1183.2.e.a | 2 | ||
91.k | even | 6 | 1 | 1183.2.e.c | 2 | ||
91.m | odd | 6 | 1 | 637.2.g.a | 2 | ||
91.m | odd | 6 | 1 | 8281.2.a.j | 1 | ||
91.n | odd | 6 | 1 | 637.2.h.a | 2 | ||
91.p | odd | 6 | 1 | 8281.2.a.g | 1 | ||
91.u | even | 6 | 1 | 8281.2.a.c | 1 | ||
91.v | odd | 6 | 1 | 637.2.f.a | 2 | ||
273.bm | odd | 6 | 1 | 819.2.n.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
91.2.g.a | ✓ | 2 | 91.g | even | 3 | 1 | inner |
91.2.h.a | yes | 2 | 7.c | even | 3 | 1 | |
91.2.h.a | yes | 2 | 13.c | even | 3 | 1 | |
637.2.f.a | 2 | 7.d | odd | 6 | 1 | ||
637.2.f.a | 2 | 91.v | odd | 6 | 1 | ||
637.2.f.b | 2 | 7.c | even | 3 | 1 | ||
637.2.f.b | 2 | 91.h | even | 3 | 1 | ||
637.2.g.a | 2 | 7.b | odd | 2 | 1 | ||
637.2.g.a | 2 | 91.m | odd | 6 | 1 | ||
637.2.h.a | 2 | 7.d | odd | 6 | 1 | ||
637.2.h.a | 2 | 91.n | odd | 6 | 1 | ||
819.2.n.c | 2 | 3.b | odd | 2 | 1 | ||
819.2.n.c | 2 | 273.bm | odd | 6 | 1 | ||
819.2.s.a | 2 | 21.h | odd | 6 | 1 | ||
819.2.s.a | 2 | 39.i | odd | 6 | 1 | ||
1183.2.e.a | 2 | 13.c | even | 3 | 1 | ||
1183.2.e.a | 2 | 91.h | even | 3 | 1 | ||
1183.2.e.c | 2 | 13.e | even | 6 | 1 | ||
1183.2.e.c | 2 | 91.k | even | 6 | 1 | ||
8281.2.a.c | 1 | 91.u | even | 6 | 1 | ||
8281.2.a.g | 1 | 91.p | odd | 6 | 1 | ||
8281.2.a.i | 1 | 91.g | even | 3 | 1 | ||
8281.2.a.j | 1 | 91.m | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T + 1 \)
$3$
\( (T + 3)^{2} \)
$5$
\( T^{2} + 3T + 9 \)
$7$
\( T^{2} + 5T + 7 \)
$11$
\( (T + 3)^{2} \)
$13$
\( T^{2} + 2T + 13 \)
$17$
\( T^{2} - 2T + 4 \)
$19$
\( (T + 1)^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} + 7T + 49 \)
$31$
\( T^{2} + 3T + 9 \)
$37$
\( T^{2} + 2T + 4 \)
$41$
\( T^{2} + 3T + 9 \)
$43$
\( T^{2} - 7T + 49 \)
$47$
\( T^{2} + T + 1 \)
$53$
\( T^{2} + 3T + 9 \)
$59$
\( T^{2} - 4T + 16 \)
$61$
\( (T + 13)^{2} \)
$67$
\( (T + 3)^{2} \)
$71$
\( T^{2} + 13T + 169 \)
$73$
\( T^{2} - 13T + 169 \)
$79$
\( T^{2} - 3T + 9 \)
$83$
\( T^{2} \)
$89$
\( T^{2} + 6T + 36 \)
$97$
\( T^{2} - 5T + 25 \)
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