# Properties

 Label 1300.2.a.d Level $1300$ Weight $2$ Character orbit 1300.a Self dual yes Analytic conductor $10.381$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.3805522628$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{7} - 3 q^{9}+O(q^{10})$$ q + 2 * q^7 - 3 * q^9 $$q + 2 q^{7} - 3 q^{9} - 2 q^{11} + q^{13} - 6 q^{17} - 6 q^{19} - 8 q^{23} + 2 q^{29} + 10 q^{31} + 6 q^{37} - 6 q^{41} - 4 q^{43} + 2 q^{47} - 3 q^{49} - 6 q^{53} - 10 q^{59} - 2 q^{61} - 6 q^{63} - 10 q^{67} + 10 q^{71} - 2 q^{73} - 4 q^{77} - 4 q^{79} + 9 q^{81} + 6 q^{83} - 6 q^{89} + 2 q^{91} - 2 q^{97} + 6 q^{99}+O(q^{100})$$ q + 2 * q^7 - 3 * q^9 - 2 * q^11 + q^13 - 6 * q^17 - 6 * q^19 - 8 * q^23 + 2 * q^29 + 10 * q^31 + 6 * q^37 - 6 * q^41 - 4 * q^43 + 2 * q^47 - 3 * q^49 - 6 * q^53 - 10 * q^59 - 2 * q^61 - 6 * q^63 - 10 * q^67 + 10 * q^71 - 2 * q^73 - 4 * q^77 - 4 * q^79 + 9 * q^81 + 6 * q^83 - 6 * q^89 + 2 * q^91 - 2 * q^97 + 6 * q^99

## 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
 0
0 0 0 0 0 2.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## 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 1300.2.a.d 1
4.b odd 2 1 5200.2.a.q 1
5.b even 2 1 52.2.a.a 1
5.c odd 4 2 1300.2.c.c 2
15.d odd 2 1 468.2.a.b 1
20.d odd 2 1 208.2.a.c 1
35.c odd 2 1 2548.2.a.e 1
35.i odd 6 2 2548.2.j.f 2
35.j even 6 2 2548.2.j.e 2
40.e odd 2 1 832.2.a.f 1
40.f even 2 1 832.2.a.e 1
45.h odd 6 2 4212.2.i.i 2
45.j even 6 2 4212.2.i.d 2
55.d odd 2 1 6292.2.a.g 1
60.h even 2 1 1872.2.a.f 1
65.d even 2 1 676.2.a.c 1
65.g odd 4 2 676.2.d.c 2
65.l even 6 2 676.2.e.b 2
65.n even 6 2 676.2.e.c 2
65.s odd 12 4 676.2.h.c 4
80.k odd 4 2 3328.2.b.e 2
80.q even 4 2 3328.2.b.q 2
120.i odd 2 1 7488.2.a.bn 1
120.m even 2 1 7488.2.a.bw 1
195.e odd 2 1 6084.2.a.m 1
195.n even 4 2 6084.2.b.m 2
260.g odd 2 1 2704.2.a.g 1
260.u even 4 2 2704.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 5.b even 2 1
208.2.a.c 1 20.d odd 2 1
468.2.a.b 1 15.d odd 2 1
676.2.a.c 1 65.d even 2 1
676.2.d.c 2 65.g odd 4 2
676.2.e.b 2 65.l even 6 2
676.2.e.c 2 65.n even 6 2
676.2.h.c 4 65.s odd 12 4
832.2.a.e 1 40.f even 2 1
832.2.a.f 1 40.e odd 2 1
1300.2.a.d 1 1.a even 1 1 trivial
1300.2.c.c 2 5.c odd 4 2
1872.2.a.f 1 60.h even 2 1
2548.2.a.e 1 35.c odd 2 1
2548.2.j.e 2 35.j even 6 2
2548.2.j.f 2 35.i odd 6 2
2704.2.a.g 1 260.g odd 2 1
2704.2.f.f 2 260.u even 4 2
3328.2.b.e 2 80.k odd 4 2
3328.2.b.q 2 80.q even 4 2
4212.2.i.d 2 45.j even 6 2
4212.2.i.i 2 45.h odd 6 2
5200.2.a.q 1 4.b odd 2 1
6084.2.a.m 1 195.e odd 2 1
6084.2.b.m 2 195.n even 4 2
6292.2.a.g 1 55.d odd 2 1
7488.2.a.bn 1 120.i odd 2 1
7488.2.a.bw 1 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1300))$$:

 $$T_{3}$$ T3 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T + 2$$
$13$ $$T - 1$$
$17$ $$T + 6$$
$19$ $$T + 6$$
$23$ $$T + 8$$
$29$ $$T - 2$$
$31$ $$T - 10$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T - 2$$
$53$ $$T + 6$$
$59$ $$T + 10$$
$61$ $$T + 2$$
$67$ $$T + 10$$
$71$ $$T - 10$$
$73$ $$T + 2$$
$79$ $$T + 4$$
$83$ $$T - 6$$
$89$ $$T + 6$$
$97$ $$T + 2$$