Newspace parameters
Level: | \( N \) | \(=\) | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1300.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.3805522628\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 260) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(651\) | \(677\) |
\(\chi(n)\) | \(i\) | \(1\) | \(-i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
57.1 |
|
0 | 2.00000 | − | 2.00000i | 0 | 0 | 0 | − | 4.00000i | 0 | − | 5.00000i | 0 | ||||||||||||||||||||
593.1 | 0 | 2.00000 | + | 2.00000i | 0 | 0 | 0 | 4.00000i | 0 | 5.00000i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1300.2.m.a | 2 | |
5.b | even | 2 | 1 | 260.2.m.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 260.2.r.a | yes | 2 | |
5.c | odd | 4 | 1 | 1300.2.r.a | 2 | ||
13.d | odd | 4 | 1 | 1300.2.r.a | 2 | ||
15.d | odd | 2 | 1 | 2340.2.u.c | 2 | ||
15.e | even | 4 | 1 | 2340.2.bp.b | 2 | ||
20.d | odd | 2 | 1 | 1040.2.bg.h | 2 | ||
20.e | even | 4 | 1 | 1040.2.cd.h | 2 | ||
65.f | even | 4 | 1 | 260.2.m.a | ✓ | 2 | |
65.g | odd | 4 | 1 | 260.2.r.a | yes | 2 | |
65.k | even | 4 | 1 | inner | 1300.2.m.a | 2 | |
195.n | even | 4 | 1 | 2340.2.bp.b | 2 | ||
195.u | odd | 4 | 1 | 2340.2.u.c | 2 | ||
260.l | odd | 4 | 1 | 1040.2.bg.h | 2 | ||
260.u | even | 4 | 1 | 1040.2.cd.h | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.m.a | ✓ | 2 | 5.b | even | 2 | 1 | |
260.2.m.a | ✓ | 2 | 65.f | even | 4 | 1 | |
260.2.r.a | yes | 2 | 5.c | odd | 4 | 1 | |
260.2.r.a | yes | 2 | 65.g | odd | 4 | 1 | |
1040.2.bg.h | 2 | 20.d | odd | 2 | 1 | ||
1040.2.bg.h | 2 | 260.l | odd | 4 | 1 | ||
1040.2.cd.h | 2 | 20.e | even | 4 | 1 | ||
1040.2.cd.h | 2 | 260.u | even | 4 | 1 | ||
1300.2.m.a | 2 | 1.a | even | 1 | 1 | trivial | |
1300.2.m.a | 2 | 65.k | even | 4 | 1 | inner | |
1300.2.r.a | 2 | 5.c | odd | 4 | 1 | ||
1300.2.r.a | 2 | 13.d | odd | 4 | 1 | ||
2340.2.u.c | 2 | 15.d | odd | 2 | 1 | ||
2340.2.u.c | 2 | 195.u | odd | 4 | 1 | ||
2340.2.bp.b | 2 | 15.e | even | 4 | 1 | ||
2340.2.bp.b | 2 | 195.n | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 4T + 8 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 16 \)
$11$
\( T^{2} + 8T + 32 \)
$13$
\( T^{2} - 6T + 13 \)
$17$
\( T^{2} + 2T + 2 \)
$19$
\( T^{2} \)
$23$
\( T^{2} - 8T + 32 \)
$29$
\( T^{2} + 4 \)
$31$
\( T^{2} + 4T + 8 \)
$37$
\( T^{2} + 4 \)
$41$
\( T^{2} + 14T + 98 \)
$43$
\( T^{2} + 4T + 8 \)
$47$
\( T^{2} + 64 \)
$53$
\( T^{2} + 2T + 2 \)
$59$
\( T^{2} \)
$61$
\( (T - 8)^{2} \)
$67$
\( (T - 12)^{2} \)
$71$
\( T^{2} - 4T + 8 \)
$73$
\( (T - 12)^{2} \)
$79$
\( T^{2} + 256 \)
$83$
\( T^{2} + 144 \)
$89$
\( T^{2} - 14T + 98 \)
$97$
\( (T - 2)^{2} \)
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