Newspace parameters
Level: | \( N \) | \(=\) | \( 3520 = 2^{6} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 3520.cg (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.75670884447\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | \(\Q(\zeta_{20})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{10}\) |
Projective field: | Galois closure of 10.2.241453843558400000.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).
\(n\) | \(321\) | \(1541\) | \(2751\) | \(2817\) |
\(\chi(n)\) | \(-\zeta_{20}^{4}\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1249.1 |
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0 | 0 | 0 | 0.309017 | − | 0.951057i | 0 | −0.951057 | + | 0.690983i | 0 | −0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||||||||||
1249.2 | 0 | 0 | 0 | 0.309017 | − | 0.951057i | 0 | 0.951057 | − | 0.690983i | 0 | −0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||
1569.1 | 0 | 0 | 0 | −0.809017 | + | 0.587785i | 0 | −0.587785 | − | 1.80902i | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
1569.2 | 0 | 0 | 0 | −0.809017 | + | 0.587785i | 0 | 0.587785 | + | 1.80902i | 0 | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
1889.1 | 0 | 0 | 0 | −0.809017 | − | 0.587785i | 0 | −0.587785 | + | 1.80902i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
1889.2 | 0 | 0 | 0 | −0.809017 | − | 0.587785i | 0 | 0.587785 | − | 1.80902i | 0 | 0.809017 | − | 0.587785i | 0 | |||||||||||||||||||||||||||||||||||||
3489.1 | 0 | 0 | 0 | 0.309017 | + | 0.951057i | 0 | −0.951057 | − | 0.690983i | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||
3489.2 | 0 | 0 | 0 | 0.309017 | + | 0.951057i | 0 | 0.951057 | + | 0.690983i | 0 | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.e | odd | 2 | 1 | CM by \(\Q(\sqrt{-10}) \) |
4.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
40.f | even | 2 | 1 | inner |
44.g | even | 10 | 1 | inner |
440.ba | odd | 10 | 1 | inner |
440.bm | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3520.1.cg.a | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
5.b | even | 2 | 1 | 3520.1.cg.b | yes | 8 | |
8.b | even | 2 | 1 | 3520.1.cg.b | yes | 8 | |
8.d | odd | 2 | 1 | 3520.1.cg.b | yes | 8 | |
11.d | odd | 10 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
20.d | odd | 2 | 1 | 3520.1.cg.b | yes | 8 | |
40.e | odd | 2 | 1 | CM | 3520.1.cg.a | ✓ | 8 |
40.f | even | 2 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
44.g | even | 10 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
55.h | odd | 10 | 1 | 3520.1.cg.b | yes | 8 | |
88.k | even | 10 | 1 | 3520.1.cg.b | yes | 8 | |
88.p | odd | 10 | 1 | 3520.1.cg.b | yes | 8 | |
220.o | even | 10 | 1 | 3520.1.cg.b | yes | 8 | |
440.ba | odd | 10 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
440.bm | even | 10 | 1 | inner | 3520.1.cg.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3520.1.cg.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
3520.1.cg.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
3520.1.cg.a | ✓ | 8 | 11.d | odd | 10 | 1 | inner |
3520.1.cg.a | ✓ | 8 | 40.e | odd | 2 | 1 | CM |
3520.1.cg.a | ✓ | 8 | 40.f | even | 2 | 1 | inner |
3520.1.cg.a | ✓ | 8 | 44.g | even | 10 | 1 | inner |
3520.1.cg.a | ✓ | 8 | 440.ba | odd | 10 | 1 | inner |
3520.1.cg.a | ✓ | 8 | 440.bm | even | 10 | 1 | inner |
3520.1.cg.b | yes | 8 | 5.b | even | 2 | 1 | |
3520.1.cg.b | yes | 8 | 8.b | even | 2 | 1 | |
3520.1.cg.b | yes | 8 | 8.d | odd | 2 | 1 | |
3520.1.cg.b | yes | 8 | 20.d | odd | 2 | 1 | |
3520.1.cg.b | yes | 8 | 55.h | odd | 10 | 1 | |
3520.1.cg.b | yes | 8 | 88.k | even | 10 | 1 | |
3520.1.cg.b | yes | 8 | 88.p | odd | 10 | 1 | |
3520.1.cg.b | yes | 8 | 220.o | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{4} + 5T_{13}^{3} + 10T_{13}^{2} + 10T_{13} + 5 \)
acting on \(S_{1}^{\mathrm{new}}(3520, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$7$
\( T^{8} + 5 T^{6} + 10 T^{4} + 25 \)
$11$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$13$
\( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \)
$17$
\( T^{8} \)
$19$
\( T^{8} + 10 T^{4} + 25 T^{2} + 25 \)
$23$
\( (T^{4} + 3 T^{2} + 1)^{2} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \)
$41$
\( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \)
$43$
\( T^{8} \)
$47$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$53$
\( (T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2} \)
$59$
\( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \)
$61$
\( T^{8} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( T^{8} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{2} - T - 1)^{4} \)
$97$
\( T^{8} \)
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