Newspace parameters
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.598878015160\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
\(n\) | \(26\) | \(52\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
− | 2.00000i | − | 1.00000i | −2.00000 | 0 | −2.00000 | 3.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||
49.2 | 2.00000i | 1.00000i | −2.00000 | 0 | −2.00000 | − | 3.00000i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.2.b.a | 2 | |
3.b | odd | 2 | 1 | 225.2.b.a | 2 | ||
4.b | odd | 2 | 1 | 1200.2.f.d | 2 | ||
5.b | even | 2 | 1 | inner | 75.2.b.a | 2 | |
5.c | odd | 4 | 1 | 75.2.a.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 75.2.a.c | yes | 1 | |
8.b | even | 2 | 1 | 4800.2.f.l | 2 | ||
8.d | odd | 2 | 1 | 4800.2.f.y | 2 | ||
12.b | even | 2 | 1 | 3600.2.f.p | 2 | ||
15.d | odd | 2 | 1 | 225.2.b.a | 2 | ||
15.e | even | 4 | 1 | 225.2.a.a | 1 | ||
15.e | even | 4 | 1 | 225.2.a.e | 1 | ||
20.d | odd | 2 | 1 | 1200.2.f.d | 2 | ||
20.e | even | 4 | 1 | 1200.2.a.c | 1 | ||
20.e | even | 4 | 1 | 1200.2.a.p | 1 | ||
35.f | even | 4 | 1 | 3675.2.a.b | 1 | ||
35.f | even | 4 | 1 | 3675.2.a.q | 1 | ||
40.e | odd | 2 | 1 | 4800.2.f.y | 2 | ||
40.f | even | 2 | 1 | 4800.2.f.l | 2 | ||
40.i | odd | 4 | 1 | 4800.2.a.bb | 1 | ||
40.i | odd | 4 | 1 | 4800.2.a.bq | 1 | ||
40.k | even | 4 | 1 | 4800.2.a.be | 1 | ||
40.k | even | 4 | 1 | 4800.2.a.br | 1 | ||
55.e | even | 4 | 1 | 9075.2.a.a | 1 | ||
55.e | even | 4 | 1 | 9075.2.a.s | 1 | ||
60.h | even | 2 | 1 | 3600.2.f.p | 2 | ||
60.l | odd | 4 | 1 | 3600.2.a.j | 1 | ||
60.l | odd | 4 | 1 | 3600.2.a.bk | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.2.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
75.2.a.c | yes | 1 | 5.c | odd | 4 | 1 | |
75.2.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
75.2.b.a | 2 | 5.b | even | 2 | 1 | inner | |
225.2.a.a | 1 | 15.e | even | 4 | 1 | ||
225.2.a.e | 1 | 15.e | even | 4 | 1 | ||
225.2.b.a | 2 | 3.b | odd | 2 | 1 | ||
225.2.b.a | 2 | 15.d | odd | 2 | 1 | ||
1200.2.a.c | 1 | 20.e | even | 4 | 1 | ||
1200.2.a.p | 1 | 20.e | even | 4 | 1 | ||
1200.2.f.d | 2 | 4.b | odd | 2 | 1 | ||
1200.2.f.d | 2 | 20.d | odd | 2 | 1 | ||
3600.2.a.j | 1 | 60.l | odd | 4 | 1 | ||
3600.2.a.bk | 1 | 60.l | odd | 4 | 1 | ||
3600.2.f.p | 2 | 12.b | even | 2 | 1 | ||
3600.2.f.p | 2 | 60.h | even | 2 | 1 | ||
3675.2.a.b | 1 | 35.f | even | 4 | 1 | ||
3675.2.a.q | 1 | 35.f | even | 4 | 1 | ||
4800.2.a.bb | 1 | 40.i | odd | 4 | 1 | ||
4800.2.a.be | 1 | 40.k | even | 4 | 1 | ||
4800.2.a.bq | 1 | 40.i | odd | 4 | 1 | ||
4800.2.a.br | 1 | 40.k | even | 4 | 1 | ||
4800.2.f.l | 2 | 8.b | even | 2 | 1 | ||
4800.2.f.l | 2 | 40.f | even | 2 | 1 | ||
4800.2.f.y | 2 | 8.d | odd | 2 | 1 | ||
4800.2.f.y | 2 | 40.e | odd | 2 | 1 | ||
9075.2.a.a | 1 | 55.e | even | 4 | 1 | ||
9075.2.a.s | 1 | 55.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 4 \)
$3$
\( T^{2} + 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 9 \)
$11$
\( (T - 2)^{2} \)
$13$
\( T^{2} + 1 \)
$17$
\( T^{2} + 4 \)
$19$
\( (T - 5)^{2} \)
$23$
\( T^{2} + 36 \)
$29$
\( (T + 10)^{2} \)
$31$
\( (T + 3)^{2} \)
$37$
\( T^{2} + 4 \)
$41$
\( (T + 8)^{2} \)
$43$
\( T^{2} + 1 \)
$47$
\( T^{2} + 4 \)
$53$
\( T^{2} + 16 \)
$59$
\( (T - 10)^{2} \)
$61$
\( (T - 7)^{2} \)
$67$
\( T^{2} + 9 \)
$71$
\( (T + 8)^{2} \)
$73$
\( T^{2} + 196 \)
$79$
\( T^{2} \)
$83$
\( T^{2} + 36 \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 289 \)
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