Newspace parameters
Level: | \( N \) | \(=\) | \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2340.gu (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.16781212956\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{12}\) |
Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).
\(n\) | \(937\) | \(1081\) | \(1171\) | \(2081\) |
\(\chi(n)\) | \(\zeta_{24}^{6}\) | \(-\zeta_{24}^{8}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
647.1 |
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−0.258819 | + | 0.965926i | 0 | −0.866025 | − | 0.500000i | 0.965926 | − | 0.258819i | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 1.00000i | ||||||||||||||||||||||||||||||||||
647.2 | 0.258819 | − | 0.965926i | 0 | −0.866025 | − | 0.500000i | −0.965926 | + | 0.258819i | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||
1187.1 | −0.965926 | + | 0.258819i | 0 | 0.866025 | − | 0.500000i | 0.258819 | − | 0.965926i | 0 | 0 | −0.707107 | + | 0.707107i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||
1187.2 | 0.965926 | − | 0.258819i | 0 | 0.866025 | − | 0.500000i | −0.258819 | + | 0.965926i | 0 | 0 | 0.707107 | − | 0.707107i | 0 | 1.00000i | |||||||||||||||||||||||||||||||||||
1583.1 | −0.965926 | − | 0.258819i | 0 | 0.866025 | + | 0.500000i | 0.258819 | + | 0.965926i | 0 | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
1583.2 | 0.965926 | + | 0.258819i | 0 | 0.866025 | + | 0.500000i | −0.258819 | − | 0.965926i | 0 | 0 | 0.707107 | + | 0.707107i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
2123.1 | −0.258819 | − | 0.965926i | 0 | −0.866025 | + | 0.500000i | 0.965926 | + | 0.258819i | 0 | 0 | 0.707107 | + | 0.707107i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
2123.2 | 0.258819 | + | 0.965926i | 0 | −0.866025 | + | 0.500000i | −0.965926 | − | 0.258819i | 0 | 0 | −0.707107 | − | 0.707107i | 0 | − | 1.00000i | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
3.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
65.r | odd | 12 | 1 | inner |
195.bf | even | 12 | 1 | inner |
260.bg | even | 12 | 1 | inner |
780.cw | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2340.1.gu.b | yes | 8 |
3.b | odd | 2 | 1 | inner | 2340.1.gu.b | yes | 8 |
4.b | odd | 2 | 1 | CM | 2340.1.gu.b | yes | 8 |
5.c | odd | 4 | 1 | 2340.1.gu.a | ✓ | 8 | |
12.b | even | 2 | 1 | inner | 2340.1.gu.b | yes | 8 |
13.e | even | 6 | 1 | 2340.1.gu.a | ✓ | 8 | |
15.e | even | 4 | 1 | 2340.1.gu.a | ✓ | 8 | |
20.e | even | 4 | 1 | 2340.1.gu.a | ✓ | 8 | |
39.h | odd | 6 | 1 | 2340.1.gu.a | ✓ | 8 | |
52.i | odd | 6 | 1 | 2340.1.gu.a | ✓ | 8 | |
60.l | odd | 4 | 1 | 2340.1.gu.a | ✓ | 8 | |
65.r | odd | 12 | 1 | inner | 2340.1.gu.b | yes | 8 |
156.r | even | 6 | 1 | 2340.1.gu.a | ✓ | 8 | |
195.bf | even | 12 | 1 | inner | 2340.1.gu.b | yes | 8 |
260.bg | even | 12 | 1 | inner | 2340.1.gu.b | yes | 8 |
780.cw | odd | 12 | 1 | inner | 2340.1.gu.b | yes | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2340.1.gu.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
2340.1.gu.a | ✓ | 8 | 13.e | even | 6 | 1 | |
2340.1.gu.a | ✓ | 8 | 15.e | even | 4 | 1 | |
2340.1.gu.a | ✓ | 8 | 20.e | even | 4 | 1 | |
2340.1.gu.a | ✓ | 8 | 39.h | odd | 6 | 1 | |
2340.1.gu.a | ✓ | 8 | 52.i | odd | 6 | 1 | |
2340.1.gu.a | ✓ | 8 | 60.l | odd | 4 | 1 | |
2340.1.gu.a | ✓ | 8 | 156.r | even | 6 | 1 | |
2340.1.gu.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
2340.1.gu.b | yes | 8 | 3.b | odd | 2 | 1 | inner |
2340.1.gu.b | yes | 8 | 4.b | odd | 2 | 1 | CM |
2340.1.gu.b | yes | 8 | 12.b | even | 2 | 1 | inner |
2340.1.gu.b | yes | 8 | 65.r | odd | 12 | 1 | inner |
2340.1.gu.b | yes | 8 | 195.bf | even | 12 | 1 | inner |
2340.1.gu.b | yes | 8 | 260.bg | even | 12 | 1 | inner |
2340.1.gu.b | yes | 8 | 780.cw | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{37}^{4} - 4T_{37}^{3} + 5T_{37}^{2} - 2T_{37} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2340, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{4} + 1 \)
$3$
\( T^{8} \)
$5$
\( T^{8} - T^{4} + 1 \)
$7$
\( T^{8} \)
$11$
\( T^{8} \)
$13$
\( (T^{2} - T + 1)^{4} \)
$17$
\( T^{8} - T^{4} + 1 \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$31$
\( T^{8} \)
$37$
\( (T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1)^{2} \)
$41$
\( T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1 \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( (T^{4} + 1)^{2} \)
$59$
\( T^{8} \)
$61$
\( (T^{4} + 3 T^{2} + 9)^{2} \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( (T^{4} - 2 T^{3} + 2 T^{2} + 2 T + 1)^{2} \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$97$
\( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \)
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