# Properties

 Label 2925.2.a.p Level $2925$ Weight $2$ Character orbit 2925.a Self dual yes Analytic conductor $23.356$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + 4 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} + 4 q^{7} - 3 q^{8} - 4 q^{11} - q^{13} + 4 q^{14} - q^{16} + 2 q^{17} - 4 q^{22} - q^{26} - 4 q^{28} + 10 q^{29} + 4 q^{31} + 5 q^{32} + 2 q^{34} + 2 q^{37} - 6 q^{41} + 12 q^{43} + 4 q^{44} + 9 q^{49} + q^{52} + 6 q^{53} - 12 q^{56} + 10 q^{58} - 12 q^{59} - 2 q^{61} + 4 q^{62} + 7 q^{64} + 8 q^{67} - 2 q^{68} - 2 q^{73} + 2 q^{74} - 16 q^{77} + 8 q^{79} - 6 q^{82} + 4 q^{83} + 12 q^{86} + 12 q^{88} + 2 q^{89} - 4 q^{91} - 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^4 + 4 * q^7 - 3 * q^8 - 4 * q^11 - q^13 + 4 * q^14 - q^16 + 2 * q^17 - 4 * q^22 - q^26 - 4 * q^28 + 10 * q^29 + 4 * q^31 + 5 * q^32 + 2 * q^34 + 2 * q^37 - 6 * q^41 + 12 * q^43 + 4 * q^44 + 9 * q^49 + q^52 + 6 * q^53 - 12 * q^56 + 10 * q^58 - 12 * q^59 - 2 * q^61 + 4 * q^62 + 7 * q^64 + 8 * q^67 - 2 * q^68 - 2 * q^73 + 2 * q^74 - 16 * q^77 + 8 * q^79 - 6 * q^82 + 4 * q^83 + 12 * q^86 + 12 * q^88 + 2 * q^89 - 4 * q^91 - 10 * q^97 + 9 * q^98

## 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
1.00000 0 −1.00000 0 0 4.00000 −3.00000 0 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 2925.2.a.p 1
3.b odd 2 1 975.2.a.f 1
5.b even 2 1 117.2.a.a 1
5.c odd 4 2 2925.2.c.e 2
15.d odd 2 1 39.2.a.a 1
15.e even 4 2 975.2.c.f 2
20.d odd 2 1 1872.2.a.h 1
35.c odd 2 1 5733.2.a.e 1
40.e odd 2 1 7488.2.a.by 1
40.f even 2 1 7488.2.a.bl 1
45.h odd 6 2 1053.2.e.b 2
45.j even 6 2 1053.2.e.d 2
60.h even 2 1 624.2.a.i 1
65.d even 2 1 1521.2.a.e 1
65.g odd 4 2 1521.2.b.b 2
105.g even 2 1 1911.2.a.f 1
120.i odd 2 1 2496.2.a.q 1
120.m even 2 1 2496.2.a.e 1
165.d even 2 1 4719.2.a.c 1
195.e odd 2 1 507.2.a.a 1
195.n even 4 2 507.2.b.a 2
195.x odd 6 2 507.2.e.a 2
195.y odd 6 2 507.2.e.b 2
195.bh even 12 4 507.2.j.e 4
780.d even 2 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 15.d odd 2 1
117.2.a.a 1 5.b even 2 1
507.2.a.a 1 195.e odd 2 1
507.2.b.a 2 195.n even 4 2
507.2.e.a 2 195.x odd 6 2
507.2.e.b 2 195.y odd 6 2
507.2.j.e 4 195.bh even 12 4
624.2.a.i 1 60.h even 2 1
975.2.a.f 1 3.b odd 2 1
975.2.c.f 2 15.e even 4 2
1053.2.e.b 2 45.h odd 6 2
1053.2.e.d 2 45.j even 6 2
1521.2.a.e 1 65.d even 2 1
1521.2.b.b 2 65.g odd 4 2
1872.2.a.h 1 20.d odd 2 1
1911.2.a.f 1 105.g even 2 1
2496.2.a.e 1 120.m even 2 1
2496.2.a.q 1 120.i odd 2 1
2925.2.a.p 1 1.a even 1 1 trivial
2925.2.c.e 2 5.c odd 4 2
4719.2.a.c 1 165.d even 2 1
5733.2.a.e 1 35.c odd 2 1
7488.2.a.bl 1 40.f even 2 1
7488.2.a.by 1 40.e odd 2 1
8112.2.a.s 1 780.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2925))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T - 4$$
$89$ $$T - 2$$
$97$ $$T + 10$$