# Properties

 Label 3525.2.a.r 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + \beta q^{6} + ( 2 - \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + \beta q^{6} + ( 2 - \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} -2 q^{11} + ( 2 + \beta ) q^{12} + q^{13} + ( -4 + \beta ) q^{14} + 3 \beta q^{16} + ( 2 + \beta ) q^{17} + \beta q^{18} + ( -2 + \beta ) q^{19} + ( 2 - \beta ) q^{21} -2 \beta q^{22} + 5 q^{23} + ( 4 + \beta ) q^{24} + \beta q^{26} + q^{27} -\beta q^{28} + ( 6 - \beta ) q^{29} + ( -2 + 2 \beta ) q^{31} + ( 4 + \beta ) q^{32} -2 q^{33} + ( 4 + 3 \beta ) q^{34} + ( 2 + \beta ) q^{36} + ( 4 - \beta ) q^{38} + q^{39} -\beta q^{41} + ( -4 + \beta ) q^{42} + ( 3 - \beta ) q^{43} + ( -4 - 2 \beta ) q^{44} + 5 \beta q^{46} - q^{47} + 3 \beta q^{48} + ( 1 - 3 \beta ) q^{49} + ( 2 + \beta ) q^{51} + ( 2 + \beta ) q^{52} + ( 6 - 3 \beta ) q^{53} + \beta q^{54} + ( 4 - 3 \beta ) q^{56} + ( -2 + \beta ) q^{57} + ( -4 + 5 \beta ) q^{58} + ( 1 - 6 \beta ) q^{59} + ( -1 + 2 \beta ) q^{61} + 8 q^{62} + ( 2 - \beta ) q^{63} + ( 4 - \beta ) q^{64} -2 \beta q^{66} + ( 4 + 4 \beta ) q^{67} + ( 8 + 5 \beta ) q^{68} + 5 q^{69} + q^{71} + ( 4 + \beta ) q^{72} + ( -5 + 3 \beta ) q^{73} + \beta q^{76} + ( -4 + 2 \beta ) q^{77} + \beta q^{78} + ( -1 - 5 \beta ) q^{79} + q^{81} + ( -4 - \beta ) q^{82} + 8 q^{83} -\beta q^{84} + ( -4 + 2 \beta ) q^{86} + ( 6 - \beta ) q^{87} + ( -8 - 2 \beta ) q^{88} + ( 1 + \beta ) q^{89} + ( 2 - \beta ) q^{91} + ( 10 + 5 \beta ) q^{92} + ( -2 + 2 \beta ) q^{93} -\beta q^{94} + ( 4 + \beta ) q^{96} + ( 6 - 2 \beta ) q^{97} + ( -12 - 2 \beta ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} + 5q^{4} + q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} + 5q^{4} + q^{6} + 3q^{7} + 9q^{8} + 2q^{9} - 4q^{11} + 5q^{12} + 2q^{13} - 7q^{14} + 3q^{16} + 5q^{17} + q^{18} - 3q^{19} + 3q^{21} - 2q^{22} + 10q^{23} + 9q^{24} + q^{26} + 2q^{27} - q^{28} + 11q^{29} - 2q^{31} + 9q^{32} - 4q^{33} + 11q^{34} + 5q^{36} + 7q^{38} + 2q^{39} - q^{41} - 7q^{42} + 5q^{43} - 10q^{44} + 5q^{46} - 2q^{47} + 3q^{48} - q^{49} + 5q^{51} + 5q^{52} + 9q^{53} + q^{54} + 5q^{56} - 3q^{57} - 3q^{58} - 4q^{59} + 16q^{62} + 3q^{63} + 7q^{64} - 2q^{66} + 12q^{67} + 21q^{68} + 10q^{69} + 2q^{71} + 9q^{72} - 7q^{73} + q^{76} - 6q^{77} + q^{78} - 7q^{79} + 2q^{81} - 9q^{82} + 16q^{83} - q^{84} - 6q^{86} + 11q^{87} - 18q^{88} + 3q^{89} + 3q^{91} + 25q^{92} - 2q^{93} - q^{94} + 9q^{96} + 10q^{97} - 26q^{98} - 4q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 1.00000 0.438447 0 −1.56155 3.56155 2.43845 1.00000 0
1.2 2.56155 1.00000 4.56155 0 2.56155 −0.561553 6.56155 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.r 2
5.b even 2 1 705.2.a.h 2
15.d odd 2 1 2115.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.h 2 5.b even 2 1
2115.2.a.m 2 15.d odd 2 1
3525.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3525))$$:

 $$T_{2}^{2} - T_{2} - 4$$ $$T_{7}^{2} - 3 T_{7} - 2$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 - 3 T + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$2 - 5 T + T^{2}$$
$19$ $$-2 + 3 T + T^{2}$$
$23$ $$( -5 + T )^{2}$$
$29$ $$26 - 11 T + T^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-4 + T + T^{2}$$
$43$ $$2 - 5 T + T^{2}$$
$47$ $$( 1 + T )^{2}$$
$53$ $$-18 - 9 T + T^{2}$$
$59$ $$-149 + 4 T + T^{2}$$
$61$ $$-17 + T^{2}$$
$67$ $$-32 - 12 T + T^{2}$$
$71$ $$( -1 + T )^{2}$$
$73$ $$-26 + 7 T + T^{2}$$
$79$ $$-94 + 7 T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$-2 - 3 T + T^{2}$$
$97$ $$8 - 10 T + T^{2}$$