Newspace parameters
Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 650.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.19027613138\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 130) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(301\) |
\(\chi(n)\) | \(1\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 |
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−0.866025 | + | 0.500000i | −1.36603 | − | 2.36603i | 0.500000 | − | 0.866025i | 0 | 2.36603 | + | 1.36603i | 2.59808 | + | 1.50000i | 1.00000i | −2.23205 | + | 3.86603i | 0 | ||||||||||||||||||
101.2 | 0.866025 | − | 0.500000i | 0.366025 | + | 0.633975i | 0.500000 | − | 0.866025i | 0 | 0.633975 | + | 0.366025i | −2.59808 | − | 1.50000i | − | 1.00000i | 1.23205 | − | 2.13397i | 0 | ||||||||||||||||||
251.1 | −0.866025 | − | 0.500000i | −1.36603 | + | 2.36603i | 0.500000 | + | 0.866025i | 0 | 2.36603 | − | 1.36603i | 2.59808 | − | 1.50000i | − | 1.00000i | −2.23205 | − | 3.86603i | 0 | ||||||||||||||||||
251.2 | 0.866025 | + | 0.500000i | 0.366025 | − | 0.633975i | 0.500000 | + | 0.866025i | 0 | 0.633975 | − | 0.366025i | −2.59808 | + | 1.50000i | 1.00000i | 1.23205 | + | 2.13397i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 650.2.m.a | 4 | |
5.b | even | 2 | 1 | 130.2.l.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 650.2.n.a | 4 | ||
5.c | odd | 4 | 1 | 650.2.n.b | 4 | ||
13.e | even | 6 | 1 | inner | 650.2.m.a | 4 | |
13.f | odd | 12 | 1 | 8450.2.a.bf | 2 | ||
13.f | odd | 12 | 1 | 8450.2.a.bm | 2 | ||
15.d | odd | 2 | 1 | 1170.2.bs.c | 4 | ||
20.d | odd | 2 | 1 | 1040.2.da.a | 4 | ||
65.d | even | 2 | 1 | 1690.2.l.g | 4 | ||
65.g | odd | 4 | 1 | 1690.2.e.l | 4 | ||
65.g | odd | 4 | 1 | 1690.2.e.n | 4 | ||
65.l | even | 6 | 1 | 130.2.l.a | ✓ | 4 | |
65.l | even | 6 | 1 | 1690.2.d.f | 4 | ||
65.n | even | 6 | 1 | 1690.2.d.f | 4 | ||
65.n | even | 6 | 1 | 1690.2.l.g | 4 | ||
65.r | odd | 12 | 1 | 650.2.n.a | 4 | ||
65.r | odd | 12 | 1 | 650.2.n.b | 4 | ||
65.s | odd | 12 | 1 | 1690.2.a.j | 2 | ||
65.s | odd | 12 | 1 | 1690.2.a.m | 2 | ||
65.s | odd | 12 | 1 | 1690.2.e.l | 4 | ||
65.s | odd | 12 | 1 | 1690.2.e.n | 4 | ||
195.y | odd | 6 | 1 | 1170.2.bs.c | 4 | ||
260.w | odd | 6 | 1 | 1040.2.da.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
130.2.l.a | ✓ | 4 | 5.b | even | 2 | 1 | |
130.2.l.a | ✓ | 4 | 65.l | even | 6 | 1 | |
650.2.m.a | 4 | 1.a | even | 1 | 1 | trivial | |
650.2.m.a | 4 | 13.e | even | 6 | 1 | inner | |
650.2.n.a | 4 | 5.c | odd | 4 | 1 | ||
650.2.n.a | 4 | 65.r | odd | 12 | 1 | ||
650.2.n.b | 4 | 5.c | odd | 4 | 1 | ||
650.2.n.b | 4 | 65.r | odd | 12 | 1 | ||
1040.2.da.a | 4 | 20.d | odd | 2 | 1 | ||
1040.2.da.a | 4 | 260.w | odd | 6 | 1 | ||
1170.2.bs.c | 4 | 15.d | odd | 2 | 1 | ||
1170.2.bs.c | 4 | 195.y | odd | 6 | 1 | ||
1690.2.a.j | 2 | 65.s | odd | 12 | 1 | ||
1690.2.a.m | 2 | 65.s | odd | 12 | 1 | ||
1690.2.d.f | 4 | 65.l | even | 6 | 1 | ||
1690.2.d.f | 4 | 65.n | even | 6 | 1 | ||
1690.2.e.l | 4 | 65.g | odd | 4 | 1 | ||
1690.2.e.l | 4 | 65.s | odd | 12 | 1 | ||
1690.2.e.n | 4 | 65.g | odd | 4 | 1 | ||
1690.2.e.n | 4 | 65.s | odd | 12 | 1 | ||
1690.2.l.g | 4 | 65.d | even | 2 | 1 | ||
1690.2.l.g | 4 | 65.n | even | 6 | 1 | ||
8450.2.a.bf | 2 | 13.f | odd | 12 | 1 | ||
8450.2.a.bm | 2 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 9T^{2} + 81 \)
$11$
\( T^{4} - 9T^{2} + 81 \)
$13$
\( (T^{2} + 7 T + 13)^{2} \)
$17$
\( T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324 \)
$19$
\( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \)
$23$
\( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \)
$29$
\( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \)
$31$
\( T^{4} + 24T^{2} + 36 \)
$37$
\( T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81 \)
$41$
\( (T^{2} + 18 T + 108)^{2} \)
$43$
\( (T^{2} - 2 T + 4)^{2} \)
$47$
\( (T^{2} + 9)^{2} \)
$53$
\( (T^{2} - 6 T - 3)^{2} \)
$59$
\( (T^{2} - 18 T + 108)^{2} \)
$61$
\( T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676 \)
$67$
\( T^{4} \)
$71$
\( T^{4} - 36T^{2} + 1296 \)
$73$
\( T^{4} + 168T^{2} + 4356 \)
$79$
\( (T^{2} - 2 T - 26)^{2} \)
$83$
\( T^{4} + 72T^{2} + 324 \)
$89$
\( T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689 \)
$97$
\( T^{4} + 42 T^{3} + 726 T^{2} + \cdots + 19044 \)
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