# Properties

 Label 3675.2.a.y Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.23607 −1.00000 3.00000 0 2.23607 0 −2.23607 1.00000 0
1.2 2.23607 −1.00000 3.00000 0 −2.23607 0 2.23607 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 3675.2.a.y 2
5.b even 2 1 735.2.a.k 2
7.b odd 2 1 525.2.a.g 2
15.d odd 2 1 2205.2.a.w 2
21.c even 2 1 1575.2.a.r 2
28.d even 2 1 8400.2.a.cx 2
35.c odd 2 1 105.2.a.b 2
35.f even 4 2 525.2.d.c 4
35.i odd 6 2 735.2.i.k 4
35.j even 6 2 735.2.i.i 4
105.g even 2 1 315.2.a.d 2
105.k odd 4 2 1575.2.d.d 4
140.c even 2 1 1680.2.a.v 2
280.c odd 2 1 6720.2.a.cx 2
280.n even 2 1 6720.2.a.cs 2
420.o odd 2 1 5040.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 35.c odd 2 1
315.2.a.d 2 105.g even 2 1
525.2.a.g 2 7.b odd 2 1
525.2.d.c 4 35.f even 4 2
735.2.a.k 2 5.b even 2 1
735.2.i.i 4 35.j even 6 2
735.2.i.k 4 35.i odd 6 2
1575.2.a.r 2 21.c even 2 1
1575.2.d.d 4 105.k odd 4 2
1680.2.a.v 2 140.c even 2 1
2205.2.a.w 2 15.d odd 2 1
3675.2.a.y 2 1.a even 1 1 trivial
5040.2.a.bw 2 420.o odd 2 1
6720.2.a.cs 2 280.n even 2 1
6720.2.a.cx 2 280.c odd 2 1
8400.2.a.cx 2 28.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} - 5$$ $$T_{11}^{2} - 4 T_{11} - 16$$ $$T_{13}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-16 - 4 T + T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$-16 + 4 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$16 + 12 T + T^{2}$$
$37$ $$-76 + 4 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-80 + T^{2}$$
$47$ $$-64 - 8 T + T^{2}$$
$53$ $$44 - 16 T + T^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$80 - 20 T + T^{2}$$
$73$ $$44 + 16 T + T^{2}$$
$79$ $$-64 - 8 T + T^{2}$$
$83$ $$-16 + 16 T + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$-4 - 8 T + T^{2}$$