Newspace parameters
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.85028650086\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 |
|
−1.41421 | + | 1.41421i | −0.878680 | + | 5.12132i | − | 4.00000i | 0 | −6.00000 | − | 8.48528i | 18.0000 | + | 18.0000i | 5.65685 | + | 5.65685i | −25.4558 | − | 9.00000i | 0 | |||||||||||||||||
107.2 | 1.41421 | − | 1.41421i | −5.12132 | + | 0.878680i | − | 4.00000i | 0 | −6.00000 | + | 8.48528i | 18.0000 | + | 18.0000i | −5.65685 | − | 5.65685i | 25.4558 | − | 9.00000i | 0 | ||||||||||||||||||
143.1 | −1.41421 | − | 1.41421i | −0.878680 | − | 5.12132i | 4.00000i | 0 | −6.00000 | + | 8.48528i | 18.0000 | − | 18.0000i | 5.65685 | − | 5.65685i | −25.4558 | + | 9.00000i | 0 | |||||||||||||||||||
143.2 | 1.41421 | + | 1.41421i | −5.12132 | − | 0.878680i | 4.00000i | 0 | −6.00000 | − | 8.48528i | 18.0000 | − | 18.0000i | −5.65685 | + | 5.65685i | 25.4558 | + | 9.00000i | 0 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.4.e.a | ✓ | 4 |
3.b | odd | 2 | 1 | inner | 150.4.e.a | ✓ | 4 |
5.b | even | 2 | 1 | 150.4.e.b | yes | 4 | |
5.c | odd | 4 | 1 | inner | 150.4.e.a | ✓ | 4 |
5.c | odd | 4 | 1 | 150.4.e.b | yes | 4 | |
15.d | odd | 2 | 1 | 150.4.e.b | yes | 4 | |
15.e | even | 4 | 1 | inner | 150.4.e.a | ✓ | 4 |
15.e | even | 4 | 1 | 150.4.e.b | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.4.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
150.4.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
150.4.e.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
150.4.e.a | ✓ | 4 | 15.e | even | 4 | 1 | inner |
150.4.e.b | yes | 4 | 5.b | even | 2 | 1 | |
150.4.e.b | yes | 4 | 5.c | odd | 4 | 1 | |
150.4.e.b | yes | 4 | 15.d | odd | 2 | 1 | |
150.4.e.b | yes | 4 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 36T_{7} + 648 \)
acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 16 \)
$3$
\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 729 \)
$5$
\( T^{4} \)
$7$
\( (T^{2} - 36 T + 648)^{2} \)
$11$
\( (T^{2} + 2592)^{2} \)
$13$
\( T^{4} \)
$17$
\( T^{4} + 49787136 \)
$19$
\( (T^{2} + 15376)^{2} \)
$23$
\( T^{4} + 3111696 \)
$29$
\( (T^{2} - 64800)^{2} \)
$31$
\( (T - 88)^{4} \)
$37$
\( (T^{2} - 144 T + 10368)^{2} \)
$41$
\( (T^{2} + 2592)^{2} \)
$43$
\( (T^{2} + 684 T + 233928)^{2} \)
$47$
\( T^{4} + 41006250000 \)
$53$
\( T^{4} + 9475854336 \)
$59$
\( (T^{2} - 749088)^{2} \)
$61$
\( (T - 434)^{4} \)
$67$
\( (T^{2} + 36 T + 648)^{2} \)
$71$
\( (T^{2} + 259200)^{2} \)
$73$
\( (T^{2} - 720 T + 259200)^{2} \)
$79$
\( (T^{2} + 1048576)^{2} \)
$83$
\( T^{4} + 3662186256 \)
$89$
\( (T^{2} - 10368)^{2} \)
$97$
\( (T^{2} - 432 T + 93312)^{2} \)
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