Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.42514325043\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{19}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 19 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−3.35890 | −3.00000 | 3.28220 | 0 | 10.0767 | −30.4356 | 15.8466 | 9.00000 | 0 | ||||||||||||||||||||||||
1.2 | 5.35890 | −3.00000 | 20.7178 | 0 | −16.0767 | 4.43560 | 68.1534 | 9.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.4.a.e | yes | 2 |
3.b | odd | 2 | 1 | 225.4.a.j | 2 | ||
4.b | odd | 2 | 1 | 1200.4.a.bu | 2 | ||
5.b | even | 2 | 1 | 75.4.a.d | ✓ | 2 | |
5.c | odd | 4 | 2 | 75.4.b.c | 4 | ||
15.d | odd | 2 | 1 | 225.4.a.n | 2 | ||
15.e | even | 4 | 2 | 225.4.b.h | 4 | ||
20.d | odd | 2 | 1 | 1200.4.a.bl | 2 | ||
20.e | even | 4 | 2 | 1200.4.f.v | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.4.a.d | ✓ | 2 | 5.b | even | 2 | 1 | |
75.4.a.e | yes | 2 | 1.a | even | 1 | 1 | trivial |
75.4.b.c | 4 | 5.c | odd | 4 | 2 | ||
225.4.a.j | 2 | 3.b | odd | 2 | 1 | ||
225.4.a.n | 2 | 15.d | odd | 2 | 1 | ||
225.4.b.h | 4 | 15.e | even | 4 | 2 | ||
1200.4.a.bl | 2 | 20.d | odd | 2 | 1 | ||
1200.4.a.bu | 2 | 4.b | odd | 2 | 1 | ||
1200.4.f.v | 4 | 20.e | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} - 18 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T - 18 \)
$3$
\( (T + 3)^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 26T - 135 \)
$11$
\( T^{2} - 28T - 108 \)
$13$
\( T^{2} + 18T - 4783 \)
$17$
\( T^{2} - 68T - 6444 \)
$19$
\( T^{2} - 6T - 295 \)
$23$
\( T^{2} + 132T + 1620 \)
$29$
\( T^{2} - 92T - 12780 \)
$31$
\( T^{2} - 122T - 11175 \)
$37$
\( T^{2} - 284T + 9220 \)
$41$
\( T^{2} - 392T - 12960 \)
$43$
\( T^{2} + 690T + 118721 \)
$47$
\( T^{2} - 620T + 76644 \)
$53$
\( T^{2} - 848T + 164880 \)
$59$
\( T^{2} - 124T - 73980 \)
$61$
\( T^{2} - 750T + 81041 \)
$67$
\( T^{2} - 358T - 157959 \)
$71$
\( T^{2} - 824T + 162144 \)
$73$
\( T^{2} - 108T + 1700 \)
$79$
\( T^{2} + 880T - 292800 \)
$83$
\( T^{2} + 156T - 694332 \)
$89$
\( T^{2} + 864T - 207360 \)
$97$
\( (T + 521)^{2} \)
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