Newspace parameters
Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 750.h (of order \(10\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.98878015160\) |
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: | \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 150) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(251\) |
\(\chi(n)\) | \(\zeta_{20}^{2}\) | \(1\) |
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 |
|
−0.587785 | + | 0.809017i | −0.951057 | + | 0.309017i | −0.309017 | − | 0.951057i | 0 | 0.309017 | − | 0.951057i | 2.00000i | 0.951057 | + | 0.309017i | 0.809017 | − | 0.587785i | 0 | ||||||||||||||||||||||||||||||
49.2 | 0.587785 | − | 0.809017i | 0.951057 | − | 0.309017i | −0.309017 | − | 0.951057i | 0 | 0.309017 | − | 0.951057i | − | 2.00000i | −0.951057 | − | 0.309017i | 0.809017 | − | 0.587785i | 0 | ||||||||||||||||||||||||||||||
199.1 | −0.587785 | − | 0.809017i | −0.951057 | − | 0.309017i | −0.309017 | + | 0.951057i | 0 | 0.309017 | + | 0.951057i | − | 2.00000i | 0.951057 | − | 0.309017i | 0.809017 | + | 0.587785i | 0 | ||||||||||||||||||||||||||||||
199.2 | 0.587785 | + | 0.809017i | 0.951057 | + | 0.309017i | −0.309017 | + | 0.951057i | 0 | 0.309017 | + | 0.951057i | 2.00000i | −0.951057 | + | 0.309017i | 0.809017 | + | 0.587785i | 0 | |||||||||||||||||||||||||||||||
349.1 | −0.951057 | + | 0.309017i | 0.587785 | + | 0.809017i | 0.809017 | − | 0.587785i | 0 | −0.809017 | − | 0.587785i | − | 2.00000i | −0.587785 | + | 0.809017i | −0.309017 | + | 0.951057i | 0 | ||||||||||||||||||||||||||||||
349.2 | 0.951057 | − | 0.309017i | −0.587785 | − | 0.809017i | 0.809017 | − | 0.587785i | 0 | −0.809017 | − | 0.587785i | 2.00000i | 0.587785 | − | 0.809017i | −0.309017 | + | 0.951057i | 0 | |||||||||||||||||||||||||||||||
649.1 | −0.951057 | − | 0.309017i | 0.587785 | − | 0.809017i | 0.809017 | + | 0.587785i | 0 | −0.809017 | + | 0.587785i | 2.00000i | −0.587785 | − | 0.809017i | −0.309017 | − | 0.951057i | 0 | |||||||||||||||||||||||||||||||
649.2 | 0.951057 | + | 0.309017i | −0.587785 | + | 0.809017i | 0.809017 | + | 0.587785i | 0 | −0.809017 | + | 0.587785i | − | 2.00000i | 0.587785 | + | 0.809017i | −0.309017 | − | 0.951057i | 0 | ||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
25.e | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 750.2.h.a | 8 | |
5.b | even | 2 | 1 | inner | 750.2.h.a | 8 | |
5.c | odd | 4 | 1 | 150.2.g.b | ✓ | 4 | |
5.c | odd | 4 | 1 | 750.2.g.a | 4 | ||
15.e | even | 4 | 1 | 450.2.h.b | 4 | ||
25.d | even | 5 | 1 | inner | 750.2.h.a | 8 | |
25.d | even | 5 | 1 | 3750.2.c.c | 4 | ||
25.e | even | 10 | 1 | inner | 750.2.h.a | 8 | |
25.e | even | 10 | 1 | 3750.2.c.c | 4 | ||
25.f | odd | 20 | 1 | 150.2.g.b | ✓ | 4 | |
25.f | odd | 20 | 1 | 750.2.g.a | 4 | ||
25.f | odd | 20 | 1 | 3750.2.a.b | 2 | ||
25.f | odd | 20 | 1 | 3750.2.a.g | 2 | ||
75.l | even | 20 | 1 | 450.2.h.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.2.g.b | ✓ | 4 | 5.c | odd | 4 | 1 | |
150.2.g.b | ✓ | 4 | 25.f | odd | 20 | 1 | |
450.2.h.b | 4 | 15.e | even | 4 | 1 | ||
450.2.h.b | 4 | 75.l | even | 20 | 1 | ||
750.2.g.a | 4 | 5.c | odd | 4 | 1 | ||
750.2.g.a | 4 | 25.f | odd | 20 | 1 | ||
750.2.h.a | 8 | 1.a | even | 1 | 1 | trivial | |
750.2.h.a | 8 | 5.b | even | 2 | 1 | inner | |
750.2.h.a | 8 | 25.d | even | 5 | 1 | inner | |
750.2.h.a | 8 | 25.e | even | 10 | 1 | inner | |
3750.2.a.b | 2 | 25.f | odd | 20 | 1 | ||
3750.2.a.g | 2 | 25.f | odd | 20 | 1 | ||
3750.2.c.c | 4 | 25.d | even | 5 | 1 | ||
3750.2.c.c | 4 | 25.e | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$3$
\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \)
$5$
\( T^{8} \)
$7$
\( (T^{2} + 4)^{4} \)
$11$
\( (T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16)^{2} \)
$13$
\( T^{8} - 36 T^{6} + 486 T^{4} + \cdots + 6561 \)
$17$
\( T^{8} + 36 T^{6} + 3726 T^{4} + \cdots + 6561 \)
$19$
\( (T^{4} + 40 T^{2} + 200 T + 400)^{2} \)
$23$
\( T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616 \)
$29$
\( (T^{4} + 10 T^{2} + 25 T + 25)^{2} \)
$31$
\( (T^{4} + 12 T^{3} + 144 T^{2} + 648 T + 1296)^{2} \)
$37$
\( T^{8} + 41 T^{6} + 5806 T^{4} + \cdots + 130321 \)
$41$
\( (T^{4} + 12 T^{3} + 94 T^{2} + 403 T + 961)^{2} \)
$43$
\( (T^{4} + 12 T^{2} + 16)^{2} \)
$47$
\( T^{8} - 124 T^{6} + 5856 T^{4} + \cdots + 3748096 \)
$53$
\( T^{8} - 171 T^{6} + \cdots + 96059601 \)
$59$
\( (T^{4} + 20 T^{3} + 240 T^{2} + 1600 T + 6400)^{2} \)
$61$
\( (T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361)^{2} \)
$67$
\( T^{8} + 36 T^{6} + 7776 T^{4} + \cdots + 1679616 \)
$71$
\( (T^{4} - 28 T^{3} + 384 T^{2} + \cdots + 13456)^{2} \)
$73$
\( T^{8} + 44 T^{6} + 8566 T^{4} + \cdots + 923521 \)
$79$
\( T^{8} \)
$83$
\( T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616 \)
$89$
\( (T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2} \)
$97$
\( T^{8} + 36 T^{6} + 29646 T^{4} + \cdots + 96059601 \)
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