Newspace parameters
Level: | \( N \) | \(=\) | \( 891 = 3^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 891.n (of order \(15\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.11467082010\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{15})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 33) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).
\(n\) | \(244\) | \(650\) |
\(\chi(n)\) | \(-\zeta_{15}^{2} - \zeta_{15}^{7}\) | \(-1 - \zeta_{15}^{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
136.1 |
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−0.348943 | + | 0.155360i | 0 | −1.24064 | + | 1.37787i | 1.47815 | + | 0.658114i | 0 | 0.978148 | + | 0.207912i | 0.454915 | − | 1.40008i | 0 | −0.618034 | ||||||||||||||||||||||||||||||||
190.1 | −0.348943 | − | 0.155360i | 0 | −1.24064 | − | 1.37787i | 1.47815 | − | 0.658114i | 0 | 0.978148 | − | 0.207912i | 0.454915 | + | 1.40008i | 0 | −0.618034 | |||||||||||||||||||||||||||||||||
379.1 | 2.56082 | + | 0.544320i | 0 | 4.43444 | + | 1.97434i | 0.604528 | − | 0.128496i | 0 | 0.104528 | + | 0.994522i | 6.04508 | + | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||||||||||||||
433.1 | 0.0399263 | − | 0.379874i | 0 | 1.81359 | + | 0.385489i | −0.169131 | − | 1.60917i | 0 | −0.669131 | + | 0.743145i | 0.454915 | − | 1.40008i | 0 | −0.618034 | |||||||||||||||||||||||||||||||||
460.1 | −1.75181 | − | 1.94558i | 0 | −0.507392 | + | 4.82751i | −0.413545 | + | 0.459289i | 0 | −0.913545 | + | 0.406737i | 6.04508 | − | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||||||||||||||
676.1 | −1.75181 | + | 1.94558i | 0 | −0.507392 | − | 4.82751i | −0.413545 | − | 0.459289i | 0 | −0.913545 | − | 0.406737i | 6.04508 | + | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||||||||||||||
757.1 | 2.56082 | − | 0.544320i | 0 | 4.43444 | − | 1.97434i | 0.604528 | + | 0.128496i | 0 | 0.104528 | − | 0.994522i | 6.04508 | − | 4.39201i | 0 | 1.61803 | |||||||||||||||||||||||||||||||||
784.1 | 0.0399263 | + | 0.379874i | 0 | 1.81359 | − | 0.385489i | −0.169131 | + | 1.60917i | 0 | −0.669131 | − | 0.743145i | 0.454915 | + | 1.40008i | 0 | −0.618034 | |||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
99.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 891.2.n.c | 8 | |
3.b | odd | 2 | 1 | 891.2.n.b | 8 | ||
9.c | even | 3 | 1 | 33.2.e.b | ✓ | 4 | |
9.c | even | 3 | 1 | inner | 891.2.n.c | 8 | |
9.d | odd | 6 | 1 | 99.2.f.a | 4 | ||
9.d | odd | 6 | 1 | 891.2.n.b | 8 | ||
11.c | even | 5 | 1 | inner | 891.2.n.c | 8 | |
33.h | odd | 10 | 1 | 891.2.n.b | 8 | ||
36.f | odd | 6 | 1 | 528.2.y.b | 4 | ||
45.j | even | 6 | 1 | 825.2.n.c | 4 | ||
45.k | odd | 12 | 2 | 825.2.bx.d | 8 | ||
99.h | odd | 6 | 1 | 363.2.e.f | 4 | ||
99.m | even | 15 | 1 | 33.2.e.b | ✓ | 4 | |
99.m | even | 15 | 1 | 363.2.a.d | 2 | ||
99.m | even | 15 | 2 | 363.2.e.k | 4 | ||
99.m | even | 15 | 1 | inner | 891.2.n.c | 8 | |
99.n | odd | 30 | 1 | 99.2.f.a | 4 | ||
99.n | odd | 30 | 1 | 891.2.n.b | 8 | ||
99.n | odd | 30 | 1 | 1089.2.a.t | 2 | ||
99.o | odd | 30 | 1 | 363.2.a.i | 2 | ||
99.o | odd | 30 | 2 | 363.2.e.b | 4 | ||
99.o | odd | 30 | 1 | 363.2.e.f | 4 | ||
99.p | even | 30 | 1 | 1089.2.a.l | 2 | ||
396.be | odd | 30 | 1 | 528.2.y.b | 4 | ||
396.be | odd | 30 | 1 | 5808.2.a.cj | 2 | ||
396.bf | even | 30 | 1 | 5808.2.a.ci | 2 | ||
495.bl | even | 30 | 1 | 825.2.n.c | 4 | ||
495.bl | even | 30 | 1 | 9075.2.a.cb | 2 | ||
495.br | odd | 30 | 1 | 9075.2.a.u | 2 | ||
495.bt | odd | 60 | 2 | 825.2.bx.d | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.2.e.b | ✓ | 4 | 9.c | even | 3 | 1 | |
33.2.e.b | ✓ | 4 | 99.m | even | 15 | 1 | |
99.2.f.a | 4 | 9.d | odd | 6 | 1 | ||
99.2.f.a | 4 | 99.n | odd | 30 | 1 | ||
363.2.a.d | 2 | 99.m | even | 15 | 1 | ||
363.2.a.i | 2 | 99.o | odd | 30 | 1 | ||
363.2.e.b | 4 | 99.o | odd | 30 | 2 | ||
363.2.e.f | 4 | 99.h | odd | 6 | 1 | ||
363.2.e.f | 4 | 99.o | odd | 30 | 1 | ||
363.2.e.k | 4 | 99.m | even | 15 | 2 | ||
528.2.y.b | 4 | 36.f | odd | 6 | 1 | ||
528.2.y.b | 4 | 396.be | odd | 30 | 1 | ||
825.2.n.c | 4 | 45.j | even | 6 | 1 | ||
825.2.n.c | 4 | 495.bl | even | 30 | 1 | ||
825.2.bx.d | 8 | 45.k | odd | 12 | 2 | ||
825.2.bx.d | 8 | 495.bt | odd | 60 | 2 | ||
891.2.n.b | 8 | 3.b | odd | 2 | 1 | ||
891.2.n.b | 8 | 9.d | odd | 6 | 1 | ||
891.2.n.b | 8 | 33.h | odd | 10 | 1 | ||
891.2.n.b | 8 | 99.n | odd | 30 | 1 | ||
891.2.n.c | 8 | 1.a | even | 1 | 1 | trivial | |
891.2.n.c | 8 | 9.c | even | 3 | 1 | inner | |
891.2.n.c | 8 | 11.c | even | 5 | 1 | inner | |
891.2.n.c | 8 | 99.m | even | 15 | 1 | inner | |
1089.2.a.l | 2 | 99.p | even | 30 | 1 | ||
1089.2.a.t | 2 | 99.n | odd | 30 | 1 | ||
5808.2.a.ci | 2 | 396.bf | even | 30 | 1 | ||
5808.2.a.cj | 2 | 396.be | odd | 30 | 1 | ||
9075.2.a.u | 2 | 495.br | odd | 30 | 1 | ||
9075.2.a.cb | 2 | 495.bl | even | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{7} - 5T_{2}^{6} - 14T_{2}^{5} + 39T_{2}^{4} + 26T_{2}^{3} + 10T_{2}^{2} + 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{7} - 5 T^{6} - 14 T^{5} + \cdots + 1 \)
$3$
\( T^{8} \)
$5$
\( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \)
$7$
\( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \)
$11$
\( T^{8} - 11 T^{7} + 70 T^{6} + \cdots + 14641 \)
$13$
\( T^{8} + 7 T^{7} + 30 T^{6} + 127 T^{5} + \cdots + 1 \)
$17$
\( (T^{4} - 12 T^{3} + 54 T^{2} + 27 T + 81)^{2} \)
$19$
\( (T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25)^{2} \)
$23$
\( (T^{4} - 4 T^{3} + 17 T^{2} + 4 T + 1)^{2} \)
$29$
\( T^{8} + 6 T^{7} - 216 T^{5} + \cdots + 1679616 \)
$31$
\( T^{8} - 12 T^{7} + 50 T^{6} + \cdots + 923521 \)
$37$
\( (T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121)^{2} \)
$41$
\( T^{8} - 3 T^{7} - 10 T^{6} - 43 T^{5} + \cdots + 1 \)
$43$
\( (T^{4} + 45 T^{2} + 2025)^{2} \)
$47$
\( T^{8} + 17 T^{7} + 175 T^{6} + \cdots + 14641 \)
$53$
\( (T^{4} - 4 T^{3} + 6 T^{2} + T + 1)^{2} \)
$59$
\( T^{8} - 6 T^{7} - 40 T^{6} + \cdots + 25411681 \)
$61$
\( T^{8} - 21 T^{7} + 135 T^{6} + \cdots + 96059601 \)
$67$
\( (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} \)
$71$
\( (T^{4} - 15 T^{3} + 190 T^{2} - 1100 T + 3025)^{2} \)
$73$
\( (T^{4} - 14 T^{3} + 136 T^{2} - 704 T + 1936)^{2} \)
$79$
\( T^{8} - 11 T^{7} + \cdots + 214358881 \)
$83$
\( T^{8} + 13 T^{7} + 100 T^{6} + \cdots + 14641 \)
$89$
\( (T^{2} + 12 T + 31)^{4} \)
$97$
\( T^{8} + 3 T^{7} - 45 T^{6} + \cdots + 6561 \)
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