# Properties

 Label 1725.2.a.y Level $1725$ Weight $2$ Character orbit 1725.a Self dual yes Analytic conductor $13.774$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1725 = 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1725.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7741943487$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + 4 q^{4} -\beta q^{6} + q^{7} + 2 \beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} - q^{3} + 4 q^{4} -\beta q^{6} + q^{7} + 2 \beta q^{8} + q^{9} + \beta q^{11} -4 q^{12} + ( -2 - \beta ) q^{13} + \beta q^{14} + 4 q^{16} + ( 3 + \beta ) q^{17} + \beta q^{18} + ( 2 + \beta ) q^{19} - q^{21} + 6 q^{22} + q^{23} -2 \beta q^{24} + ( -6 - 2 \beta ) q^{26} - q^{27} + 4 q^{28} + ( 3 + 3 \beta ) q^{29} + ( 5 - 2 \beta ) q^{31} -\beta q^{33} + ( 6 + 3 \beta ) q^{34} + 4 q^{36} + ( 1 - 2 \beta ) q^{37} + ( 6 + 2 \beta ) q^{38} + ( 2 + \beta ) q^{39} + ( 3 + \beta ) q^{41} -\beta q^{42} -2 q^{43} + 4 \beta q^{44} + \beta q^{46} + ( -6 - \beta ) q^{47} -4 q^{48} -6 q^{49} + ( -3 - \beta ) q^{51} + ( -8 - 4 \beta ) q^{52} + ( -3 + \beta ) q^{53} -\beta q^{54} + 2 \beta q^{56} + ( -2 - \beta ) q^{57} + ( 18 + 3 \beta ) q^{58} + ( 3 + 3 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + ( -12 + 5 \beta ) q^{62} + q^{63} -8 q^{64} -6 q^{66} + 7 q^{67} + ( 12 + 4 \beta ) q^{68} - q^{69} + ( 3 - 3 \beta ) q^{71} + 2 \beta q^{72} + ( -2 + 3 \beta ) q^{73} + ( -12 + \beta ) q^{74} + ( 8 + 4 \beta ) q^{76} + \beta q^{77} + ( 6 + 2 \beta ) q^{78} -4 q^{79} + q^{81} + ( 6 + 3 \beta ) q^{82} + ( -3 - 5 \beta ) q^{83} -4 q^{84} -2 \beta q^{86} + ( -3 - 3 \beta ) q^{87} + 12 q^{88} + ( -12 - 2 \beta ) q^{89} + ( -2 - \beta ) q^{91} + 4 q^{92} + ( -5 + 2 \beta ) q^{93} + ( -6 - 6 \beta ) q^{94} + ( -8 + 2 \beta ) q^{97} -6 \beta q^{98} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 8q^{4} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 8q^{4} + 2q^{7} + 2q^{9} - 8q^{12} - 4q^{13} + 8q^{16} + 6q^{17} + 4q^{19} - 2q^{21} + 12q^{22} + 2q^{23} - 12q^{26} - 2q^{27} + 8q^{28} + 6q^{29} + 10q^{31} + 12q^{34} + 8q^{36} + 2q^{37} + 12q^{38} + 4q^{39} + 6q^{41} - 4q^{43} - 12q^{47} - 8q^{48} - 12q^{49} - 6q^{51} - 16q^{52} - 6q^{53} - 4q^{57} + 36q^{58} + 6q^{59} + 16q^{61} - 24q^{62} + 2q^{63} - 16q^{64} - 12q^{66} + 14q^{67} + 24q^{68} - 2q^{69} + 6q^{71} - 4q^{73} - 24q^{74} + 16q^{76} + 12q^{78} - 8q^{79} + 2q^{81} + 12q^{82} - 6q^{83} - 8q^{84} - 6q^{87} + 24q^{88} - 24q^{89} - 4q^{91} + 8q^{92} - 10q^{93} - 12q^{94} - 16q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
−2.44949 −1.00000 4.00000 0 2.44949 1.00000 −4.89898 1.00000 0
1.2 2.44949 −1.00000 4.00000 0 −2.44949 1.00000 4.89898 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 1725.2.a.y 2
3.b odd 2 1 5175.2.a.bl 2
5.b even 2 1 345.2.a.i 2
5.c odd 4 2 1725.2.b.m 4
15.d odd 2 1 1035.2.a.k 2
20.d odd 2 1 5520.2.a.bi 2
115.c odd 2 1 7935.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 5.b even 2 1
1035.2.a.k 2 15.d odd 2 1
1725.2.a.y 2 1.a even 1 1 trivial
1725.2.b.m 4 5.c odd 4 2
5175.2.a.bl 2 3.b odd 2 1
5520.2.a.bi 2 20.d odd 2 1
7935.2.a.t 2 115.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1725))$$:

 $$T_{2}^{2} - 6$$ $$T_{7} - 1$$ $$T_{11}^{2} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-6 + T^{2}$$
$13$ $$-2 + 4 T + T^{2}$$
$17$ $$3 - 6 T + T^{2}$$
$19$ $$-2 - 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-45 - 6 T + T^{2}$$
$31$ $$1 - 10 T + T^{2}$$
$37$ $$-23 - 2 T + T^{2}$$
$41$ $$3 - 6 T + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$30 + 12 T + T^{2}$$
$53$ $$3 + 6 T + T^{2}$$
$59$ $$-45 - 6 T + T^{2}$$
$61$ $$10 - 16 T + T^{2}$$
$67$ $$( -7 + T )^{2}$$
$71$ $$-45 - 6 T + T^{2}$$
$73$ $$-50 + 4 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-141 + 6 T + T^{2}$$
$89$ $$120 + 24 T + T^{2}$$
$97$ $$40 + 16 T + T^{2}$$