Newspace parameters
Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 50.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.95009550029\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
Defining polynomial: |
\( x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 10) |
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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
\(n\) | \(27\) |
\(\chi(n)\) | \(-1\) |
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 | − | 8.00000i | −4.00000 | 0 | −16.0000 | 4.00000i | 8.00000i | −37.0000 | 0 | ||||||||||||||||||||||
49.2 | 2.00000i | 8.00000i | −4.00000 | 0 | −16.0000 | − | 4.00000i | − | 8.00000i | −37.0000 | 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 | 50.4.b.a | 2 | |
3.b | odd | 2 | 1 | 450.4.c.d | 2 | ||
4.b | odd | 2 | 1 | 400.4.c.c | 2 | ||
5.b | even | 2 | 1 | inner | 50.4.b.a | 2 | |
5.c | odd | 4 | 1 | 10.4.a.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 50.4.a.c | 1 | ||
15.d | odd | 2 | 1 | 450.4.c.d | 2 | ||
15.e | even | 4 | 1 | 90.4.a.a | 1 | ||
15.e | even | 4 | 1 | 450.4.a.q | 1 | ||
20.d | odd | 2 | 1 | 400.4.c.c | 2 | ||
20.e | even | 4 | 1 | 80.4.a.f | 1 | ||
20.e | even | 4 | 1 | 400.4.a.b | 1 | ||
35.f | even | 4 | 1 | 490.4.a.o | 1 | ||
35.f | even | 4 | 1 | 2450.4.a.b | 1 | ||
35.k | even | 12 | 2 | 490.4.e.a | 2 | ||
35.l | odd | 12 | 2 | 490.4.e.i | 2 | ||
40.i | odd | 4 | 1 | 320.4.a.m | 1 | ||
40.i | odd | 4 | 1 | 1600.4.a.d | 1 | ||
40.k | even | 4 | 1 | 320.4.a.b | 1 | ||
40.k | even | 4 | 1 | 1600.4.a.bx | 1 | ||
45.k | odd | 12 | 2 | 810.4.e.c | 2 | ||
45.l | even | 12 | 2 | 810.4.e.w | 2 | ||
55.e | even | 4 | 1 | 1210.4.a.b | 1 | ||
60.l | odd | 4 | 1 | 720.4.a.j | 1 | ||
65.h | odd | 4 | 1 | 1690.4.a.a | 1 | ||
80.i | odd | 4 | 1 | 1280.4.d.j | 2 | ||
80.j | even | 4 | 1 | 1280.4.d.g | 2 | ||
80.s | even | 4 | 1 | 1280.4.d.g | 2 | ||
80.t | odd | 4 | 1 | 1280.4.d.j | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
10.4.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
50.4.a.c | 1 | 5.c | odd | 4 | 1 | ||
50.4.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
50.4.b.a | 2 | 5.b | even | 2 | 1 | inner | |
80.4.a.f | 1 | 20.e | even | 4 | 1 | ||
90.4.a.a | 1 | 15.e | even | 4 | 1 | ||
320.4.a.b | 1 | 40.k | even | 4 | 1 | ||
320.4.a.m | 1 | 40.i | odd | 4 | 1 | ||
400.4.a.b | 1 | 20.e | even | 4 | 1 | ||
400.4.c.c | 2 | 4.b | odd | 2 | 1 | ||
400.4.c.c | 2 | 20.d | odd | 2 | 1 | ||
450.4.a.q | 1 | 15.e | even | 4 | 1 | ||
450.4.c.d | 2 | 3.b | odd | 2 | 1 | ||
450.4.c.d | 2 | 15.d | odd | 2 | 1 | ||
490.4.a.o | 1 | 35.f | even | 4 | 1 | ||
490.4.e.a | 2 | 35.k | even | 12 | 2 | ||
490.4.e.i | 2 | 35.l | odd | 12 | 2 | ||
720.4.a.j | 1 | 60.l | odd | 4 | 1 | ||
810.4.e.c | 2 | 45.k | odd | 12 | 2 | ||
810.4.e.w | 2 | 45.l | even | 12 | 2 | ||
1210.4.a.b | 1 | 55.e | even | 4 | 1 | ||
1280.4.d.g | 2 | 80.j | even | 4 | 1 | ||
1280.4.d.g | 2 | 80.s | even | 4 | 1 | ||
1280.4.d.j | 2 | 80.i | odd | 4 | 1 | ||
1280.4.d.j | 2 | 80.t | odd | 4 | 1 | ||
1600.4.a.d | 1 | 40.i | odd | 4 | 1 | ||
1600.4.a.bx | 1 | 40.k | even | 4 | 1 | ||
1690.4.a.a | 1 | 65.h | odd | 4 | 1 | ||
2450.4.a.b | 1 | 35.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 64 \)
acting on \(S_{4}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 4 \)
$3$
\( T^{2} + 64 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 16 \)
$11$
\( (T - 12)^{2} \)
$13$
\( T^{2} + 3364 \)
$17$
\( T^{2} + 4356 \)
$19$
\( (T - 100)^{2} \)
$23$
\( T^{2} + 17424 \)
$29$
\( (T - 90)^{2} \)
$31$
\( (T - 152)^{2} \)
$37$
\( T^{2} + 1156 \)
$41$
\( (T + 438)^{2} \)
$43$
\( T^{2} + 1024 \)
$47$
\( T^{2} + 41616 \)
$53$
\( T^{2} + 49284 \)
$59$
\( (T + 420)^{2} \)
$61$
\( (T - 902)^{2} \)
$67$
\( T^{2} + 1048576 \)
$71$
\( (T - 432)^{2} \)
$73$
\( T^{2} + 131044 \)
$79$
\( (T - 160)^{2} \)
$83$
\( T^{2} + 5184 \)
$89$
\( (T + 810)^{2} \)
$97$
\( T^{2} + 1223236 \)
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