Newspace parameters
Level: | \( N \) | \(=\) | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1950.y (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(15.5708283941\) |
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: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 78) |
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 \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(1301\) | \(1327\) |
\(\chi(n)\) | \(\zeta_{12}^{2}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
−0.500000 | − | 0.866025i | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0 | 0.866025 | + | 0.500000i | −1.36603 | + | 2.36603i | 1.00000 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||
49.2 | −0.500000 | − | 0.866025i | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0 | −0.866025 | − | 0.500000i | 0.366025 | − | 0.633975i | 1.00000 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||
199.1 | −0.500000 | + | 0.866025i | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0 | 0.866025 | − | 0.500000i | −1.36603 | − | 2.36603i | 1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
199.2 | −0.500000 | + | 0.866025i | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0 | −0.866025 | + | 0.500000i | 0.366025 | + | 0.633975i | 1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1950.2.y.b | 4 | |
5.b | even | 2 | 1 | 1950.2.y.g | 4 | ||
5.c | odd | 4 | 1 | 78.2.i.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 1950.2.bc.d | 4 | ||
13.e | even | 6 | 1 | 1950.2.y.g | 4 | ||
15.e | even | 4 | 1 | 234.2.l.c | 4 | ||
20.e | even | 4 | 1 | 624.2.bv.e | 4 | ||
60.l | odd | 4 | 1 | 1872.2.by.h | 4 | ||
65.f | even | 4 | 1 | 1014.2.e.i | 4 | ||
65.h | odd | 4 | 1 | 1014.2.i.a | 4 | ||
65.k | even | 4 | 1 | 1014.2.e.g | 4 | ||
65.l | even | 6 | 1 | inner | 1950.2.y.b | 4 | |
65.o | even | 12 | 1 | 1014.2.a.k | 2 | ||
65.o | even | 12 | 1 | 1014.2.e.g | 4 | ||
65.q | odd | 12 | 1 | 1014.2.b.e | 4 | ||
65.q | odd | 12 | 1 | 1014.2.i.a | 4 | ||
65.r | odd | 12 | 1 | 78.2.i.a | ✓ | 4 | |
65.r | odd | 12 | 1 | 1014.2.b.e | 4 | ||
65.r | odd | 12 | 1 | 1950.2.bc.d | 4 | ||
65.t | even | 12 | 1 | 1014.2.a.i | 2 | ||
65.t | even | 12 | 1 | 1014.2.e.i | 4 | ||
195.bc | odd | 12 | 1 | 3042.2.a.y | 2 | ||
195.bf | even | 12 | 1 | 234.2.l.c | 4 | ||
195.bf | even | 12 | 1 | 3042.2.b.i | 4 | ||
195.bl | even | 12 | 1 | 3042.2.b.i | 4 | ||
195.bn | odd | 12 | 1 | 3042.2.a.p | 2 | ||
260.be | odd | 12 | 1 | 8112.2.a.bp | 2 | ||
260.bg | even | 12 | 1 | 624.2.bv.e | 4 | ||
260.bl | odd | 12 | 1 | 8112.2.a.bj | 2 | ||
780.cw | odd | 12 | 1 | 1872.2.by.h | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
78.2.i.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
78.2.i.a | ✓ | 4 | 65.r | odd | 12 | 1 | |
234.2.l.c | 4 | 15.e | even | 4 | 1 | ||
234.2.l.c | 4 | 195.bf | even | 12 | 1 | ||
624.2.bv.e | 4 | 20.e | even | 4 | 1 | ||
624.2.bv.e | 4 | 260.bg | even | 12 | 1 | ||
1014.2.a.i | 2 | 65.t | even | 12 | 1 | ||
1014.2.a.k | 2 | 65.o | even | 12 | 1 | ||
1014.2.b.e | 4 | 65.q | odd | 12 | 1 | ||
1014.2.b.e | 4 | 65.r | odd | 12 | 1 | ||
1014.2.e.g | 4 | 65.k | even | 4 | 1 | ||
1014.2.e.g | 4 | 65.o | even | 12 | 1 | ||
1014.2.e.i | 4 | 65.f | even | 4 | 1 | ||
1014.2.e.i | 4 | 65.t | even | 12 | 1 | ||
1014.2.i.a | 4 | 65.h | odd | 4 | 1 | ||
1014.2.i.a | 4 | 65.q | odd | 12 | 1 | ||
1872.2.by.h | 4 | 60.l | odd | 4 | 1 | ||
1872.2.by.h | 4 | 780.cw | odd | 12 | 1 | ||
1950.2.y.b | 4 | 1.a | even | 1 | 1 | trivial | |
1950.2.y.b | 4 | 65.l | even | 6 | 1 | inner | |
1950.2.y.g | 4 | 5.b | even | 2 | 1 | ||
1950.2.y.g | 4 | 13.e | even | 6 | 1 | ||
1950.2.bc.d | 4 | 5.c | odd | 4 | 1 | ||
1950.2.bc.d | 4 | 65.r | odd | 12 | 1 | ||
3042.2.a.p | 2 | 195.bn | odd | 12 | 1 | ||
3042.2.a.y | 2 | 195.bc | odd | 12 | 1 | ||
3042.2.b.i | 4 | 195.bf | even | 12 | 1 | ||
3042.2.b.i | 4 | 195.bl | even | 12 | 1 | ||
8112.2.a.bj | 2 | 260.bl | odd | 12 | 1 | ||
8112.2.a.bp | 2 | 260.be | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 2T_{7}^{3} + 6T_{7}^{2} - 4T_{7} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + T + 1)^{2} \)
$3$
\( T^{4} - T^{2} + 1 \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \)
$11$
\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \)
$13$
\( (T^{2} + 5 T + 13)^{2} \)
$17$
\( T^{4} + 6 T^{3} - T^{2} - 78 T + 169 \)
$19$
\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \)
$23$
\( T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676 \)
$29$
\( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \)
$31$
\( T^{4} + 32T^{2} + 64 \)
$37$
\( T^{4} - 14 T^{3} + 159 T^{2} + \cdots + 1369 \)
$41$
\( T^{4} - 36 T^{3} + 539 T^{2} + \cdots + 11449 \)
$43$
\( T^{4} - 30 T^{3} + 374 T^{2} + \cdots + 5476 \)
$47$
\( (T^{2} - 6 T - 18)^{2} \)
$53$
\( T^{4} + 42T^{2} + 9 \)
$59$
\( T^{4} - 64T^{2} + 4096 \)
$61$
\( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \)
$67$
\( T^{4} + 2 T^{3} + 150 T^{2} + \cdots + 21316 \)
$71$
\( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \)
$73$
\( (T^{2} + 16 T + 61)^{2} \)
$79$
\( (T^{2} - 12 T + 24)^{2} \)
$83$
\( (T^{2} + 10 T - 2)^{2} \)
$89$
\( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \)
$97$
\( (T^{2} - 6 T + 36)^{2} \)
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