# Properties

 Label 3525.2.a.q 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 141) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + \beta q^{6} + ( -1 + \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + q^{3} + ( 2 + \beta ) q^{4} + \beta q^{6} + ( -1 + \beta ) q^{7} + ( 4 + \beta ) q^{8} + q^{9} + ( 3 + \beta ) q^{11} + ( 2 + \beta ) q^{12} + ( 4 - 2 \beta ) q^{13} + 4 q^{14} + 3 \beta q^{16} -2 \beta q^{17} + \beta q^{18} + 6 q^{19} + ( -1 + \beta ) q^{21} + ( 4 + 4 \beta ) q^{22} + ( 3 - 3 \beta ) q^{23} + ( 4 + \beta ) q^{24} + ( -8 + 2 \beta ) q^{26} + q^{27} + ( 2 + 2 \beta ) q^{28} + ( -7 - \beta ) q^{29} + ( 4 - 2 \beta ) q^{31} + ( 4 + \beta ) q^{32} + ( 3 + \beta ) q^{33} + ( -8 - 2 \beta ) q^{34} + ( 2 + \beta ) q^{36} + ( -5 - \beta ) q^{37} + 6 \beta q^{38} + ( 4 - 2 \beta ) q^{39} + ( 2 + 2 \beta ) q^{41} + 4 q^{42} + ( -8 + 2 \beta ) q^{43} + ( 10 + 6 \beta ) q^{44} -12 q^{46} - q^{47} + 3 \beta q^{48} + ( -2 - \beta ) q^{49} -2 \beta q^{51} -2 \beta q^{52} + ( 2 + 4 \beta ) q^{53} + \beta q^{54} + 4 \beta q^{56} + 6 q^{57} + ( -4 - 8 \beta ) q^{58} + ( 2 + 2 \beta ) q^{59} + 2 q^{61} + ( -8 + 2 \beta ) q^{62} + ( -1 + \beta ) q^{63} + ( 4 - \beta ) q^{64} + ( 4 + 4 \beta ) q^{66} -2 \beta q^{67} + ( -8 - 6 \beta ) q^{68} + ( 3 - 3 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + ( 4 + \beta ) q^{72} + ( -4 - 2 \beta ) q^{73} + ( -4 - 6 \beta ) q^{74} + ( 12 + 6 \beta ) q^{76} + ( 1 + 3 \beta ) q^{77} + ( -8 + 2 \beta ) q^{78} + ( -7 - \beta ) q^{79} + q^{81} + ( 8 + 4 \beta ) q^{82} + ( -2 - 2 \beta ) q^{83} + ( 2 + 2 \beta ) q^{84} + ( 8 - 6 \beta ) q^{86} + ( -7 - \beta ) q^{87} + ( 16 + 8 \beta ) q^{88} + ( 2 - 4 \beta ) q^{89} + ( -12 + 4 \beta ) q^{91} + ( -6 - 6 \beta ) q^{92} + ( 4 - 2 \beta ) q^{93} -\beta q^{94} + ( 4 + \beta ) q^{96} + ( -1 + 7 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} + ( 3 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} + 5q^{4} + q^{6} - q^{7} + 9q^{8} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} + 5q^{4} + q^{6} - q^{7} + 9q^{8} + 2q^{9} + 7q^{11} + 5q^{12} + 6q^{13} + 8q^{14} + 3q^{16} - 2q^{17} + q^{18} + 12q^{19} - q^{21} + 12q^{22} + 3q^{23} + 9q^{24} - 14q^{26} + 2q^{27} + 6q^{28} - 15q^{29} + 6q^{31} + 9q^{32} + 7q^{33} - 18q^{34} + 5q^{36} - 11q^{37} + 6q^{38} + 6q^{39} + 6q^{41} + 8q^{42} - 14q^{43} + 26q^{44} - 24q^{46} - 2q^{47} + 3q^{48} - 5q^{49} - 2q^{51} - 2q^{52} + 8q^{53} + q^{54} + 4q^{56} + 12q^{57} - 16q^{58} + 6q^{59} + 4q^{61} - 14q^{62} - q^{63} + 7q^{64} + 12q^{66} - 2q^{67} - 22q^{68} + 3q^{69} - 2q^{71} + 9q^{72} - 10q^{73} - 14q^{74} + 30q^{76} + 5q^{77} - 14q^{78} - 15q^{79} + 2q^{81} + 20q^{82} - 6q^{83} + 6q^{84} + 10q^{86} - 15q^{87} + 40q^{88} - 20q^{91} - 18q^{92} + 6q^{93} - q^{94} + 9q^{96} + 5q^{97} - 11q^{98} + 7q^{99} + O(q^{100})$$

## 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 −2.56155 2.43845 1.00000 0
1.2 2.56155 1.00000 4.56155 0 2.56155 1.56155 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.q 2
5.b even 2 1 141.2.a.f 2
15.d odd 2 1 423.2.a.h 2
20.d odd 2 1 2256.2.a.s 2
35.c odd 2 1 6909.2.a.m 2
40.e odd 2 1 9024.2.a.ca 2
40.f even 2 1 9024.2.a.cf 2
60.h even 2 1 6768.2.a.y 2
235.b odd 2 1 6627.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.a.f 2 5.b even 2 1
423.2.a.h 2 15.d odd 2 1
2256.2.a.s 2 20.d odd 2 1
3525.2.a.q 2 1.a even 1 1 trivial
6627.2.a.k 2 235.b odd 2 1
6768.2.a.y 2 60.h even 2 1
6909.2.a.m 2 35.c odd 2 1
9024.2.a.ca 2 40.e odd 2 1
9024.2.a.cf 2 40.f even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 4$$ $$T_{7}^{2} + T_{7} - 4$$ $$T_{11}^{2} - 7 T_{11} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 - T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 + T + T^{2}$$
$11$ $$8 - 7 T + T^{2}$$
$13$ $$-8 - 6 T + T^{2}$$
$17$ $$-16 + 2 T + T^{2}$$
$19$ $$( -6 + T )^{2}$$
$23$ $$-36 - 3 T + T^{2}$$
$29$ $$52 + 15 T + T^{2}$$
$31$ $$-8 - 6 T + T^{2}$$
$37$ $$26 + 11 T + T^{2}$$
$41$ $$-8 - 6 T + T^{2}$$
$43$ $$32 + 14 T + T^{2}$$
$47$ $$( 1 + T )^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$-8 - 6 T + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-16 + 2 T + T^{2}$$
$71$ $$-16 + 2 T + T^{2}$$
$73$ $$8 + 10 T + T^{2}$$
$79$ $$52 + 15 T + T^{2}$$
$83$ $$-8 + 6 T + T^{2}$$
$89$ $$-68 + T^{2}$$
$97$ $$-202 - 5 T + T^{2}$$