# Properties

 Label 3525.2.a.s Level $3525$ Weight $2$ Character orbit 3525.a Self dual yes Analytic conductor $28.147$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3525 = 3 \cdot 5^{2} \cdot 47$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.1472667125$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 705) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 3 q^{7} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 3 q^{7} + ( 3 + \beta ) q^{8} + q^{9} + 2 \beta q^{11} + ( -1 - 2 \beta ) q^{12} + ( 5 + \beta ) q^{13} + ( 3 + 3 \beta ) q^{14} + 3 q^{16} + ( 3 - 3 \beta ) q^{17} + ( 1 + \beta ) q^{18} + ( -3 - 3 \beta ) q^{19} -3 q^{21} + ( 4 + 2 \beta ) q^{22} + ( 3 + 2 \beta ) q^{23} + ( -3 - \beta ) q^{24} + ( 7 + 6 \beta ) q^{26} - q^{27} + ( 3 + 6 \beta ) q^{28} + ( -3 - 2 \beta ) q^{29} + 2 q^{31} + ( -3 + \beta ) q^{32} -2 \beta q^{33} -3 q^{34} + ( 1 + 2 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( -9 - 6 \beta ) q^{38} + ( -5 - \beta ) q^{39} + ( 1 + 4 \beta ) q^{41} + ( -3 - 3 \beta ) q^{42} + ( 2 + 2 \beta ) q^{43} + ( 8 + 2 \beta ) q^{44} + ( 7 + 5 \beta ) q^{46} + q^{47} -3 q^{48} + 2 q^{49} + ( -3 + 3 \beta ) q^{51} + ( 9 + 11 \beta ) q^{52} + ( -3 + 5 \beta ) q^{53} + ( -1 - \beta ) q^{54} + ( 9 + 3 \beta ) q^{56} + ( 3 + 3 \beta ) q^{57} + ( -7 - 5 \beta ) q^{58} + ( 5 - \beta ) q^{59} -3 q^{61} + ( 2 + 2 \beta ) q^{62} + 3 q^{63} + ( -7 - 2 \beta ) q^{64} + ( -4 - 2 \beta ) q^{66} + ( 2 - 10 \beta ) q^{67} + ( -9 + 3 \beta ) q^{68} + ( -3 - 2 \beta ) q^{69} + ( -9 - 3 \beta ) q^{71} + ( 3 + \beta ) q^{72} + 6 \beta q^{73} + ( -6 - 2 \beta ) q^{74} + ( -15 - 9 \beta ) q^{76} + 6 \beta q^{77} + ( -7 - 6 \beta ) q^{78} -4 q^{79} + q^{81} + ( 9 + 5 \beta ) q^{82} + ( 4 + 4 \beta ) q^{83} + ( -3 - 6 \beta ) q^{84} + ( 6 + 4 \beta ) q^{86} + ( 3 + 2 \beta ) q^{87} + ( 4 + 6 \beta ) q^{88} + ( -4 - 8 \beta ) q^{89} + ( 15 + 3 \beta ) q^{91} + ( 11 + 8 \beta ) q^{92} -2 q^{93} + ( 1 + \beta ) q^{94} + ( 3 - \beta ) q^{96} + ( -4 + 6 \beta ) q^{97} + ( 2 + 2 \beta ) q^{98} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 6q^{8} + 2q^{9} - 2q^{12} + 10q^{13} + 6q^{14} + 6q^{16} + 6q^{17} + 2q^{18} - 6q^{19} - 6q^{21} + 8q^{22} + 6q^{23} - 6q^{24} + 14q^{26} - 2q^{27} + 6q^{28} - 6q^{29} + 4q^{31} - 6q^{32} - 6q^{34} + 2q^{36} + 4q^{37} - 18q^{38} - 10q^{39} + 2q^{41} - 6q^{42} + 4q^{43} + 16q^{44} + 14q^{46} + 2q^{47} - 6q^{48} + 4q^{49} - 6q^{51} + 18q^{52} - 6q^{53} - 2q^{54} + 18q^{56} + 6q^{57} - 14q^{58} + 10q^{59} - 6q^{61} + 4q^{62} + 6q^{63} - 14q^{64} - 8q^{66} + 4q^{67} - 18q^{68} - 6q^{69} - 18q^{71} + 6q^{72} - 12q^{74} - 30q^{76} - 14q^{78} - 8q^{79} + 2q^{81} + 18q^{82} + 8q^{83} - 6q^{84} + 12q^{86} + 6q^{87} + 8q^{88} - 8q^{89} + 30q^{91} + 22q^{92} - 4q^{93} + 2q^{94} + 6q^{96} - 8q^{97} + 4q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
−0.414214 −1.00000 −1.82843 0 0.414214 3.00000 1.58579 1.00000 0
1.2 2.41421 −1.00000 3.82843 0 −2.41421 3.00000 4.41421 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
$$3$$ $$1$$
$$5$$ $$1$$
$$47$$ $$-1$$

## 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 3525.2.a.s 2
5.b even 2 1 705.2.a.g 2
15.d odd 2 1 2115.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.g 2 5.b even 2 1
2115.2.a.n 2 15.d odd 2 1
3525.2.a.s 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{7} - 3$$ $$T_{11}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -3 + T )^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$23 - 10 T + T^{2}$$
$17$ $$-9 - 6 T + T^{2}$$
$19$ $$-9 + 6 T + T^{2}$$
$23$ $$1 - 6 T + T^{2}$$
$29$ $$1 + 6 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$-31 - 2 T + T^{2}$$
$43$ $$-4 - 4 T + T^{2}$$
$47$ $$( -1 + T )^{2}$$
$53$ $$-41 + 6 T + T^{2}$$
$59$ $$23 - 10 T + T^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$-196 - 4 T + T^{2}$$
$71$ $$63 + 18 T + T^{2}$$
$73$ $$-72 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-16 - 8 T + T^{2}$$
$89$ $$-112 + 8 T + T^{2}$$
$97$ $$-56 + 8 T + T^{2}$$