# Properties

 Label 3675.2.a.r 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: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
−2.61803 1.00000 4.85410 0 −2.61803 0 −7.47214 1.00000 0
1.2 −0.381966 1.00000 −1.85410 0 −0.381966 0 1.47214 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.r 2
5.b even 2 1 3675.2.a.bh 2
7.b odd 2 1 525.2.a.e 2
21.c even 2 1 1575.2.a.v 2
28.d even 2 1 8400.2.a.da 2
35.c odd 2 1 525.2.a.i yes 2
35.f even 4 2 525.2.d.e 4
105.g even 2 1 1575.2.a.l 2
105.k odd 4 2 1575.2.d.f 4
140.c even 2 1 8400.2.a.cy 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.e 2 7.b odd 2 1
525.2.a.i yes 2 35.c odd 2 1
525.2.d.e 4 35.f even 4 2
1575.2.a.l 2 105.g even 2 1
1575.2.a.v 2 21.c even 2 1
1575.2.d.f 4 105.k odd 4 2
3675.2.a.r 2 1.a even 1 1 trivial
3675.2.a.bh 2 5.b even 2 1
8400.2.a.cy 2 140.c even 2 1
8400.2.a.da 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{11}^{2} + 2 T_{11} - 19$$ $$T_{13}^{2} - 6 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-19 + 2 T + T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$-44 + 2 T + T^{2}$$
$19$ $$-4 - 2 T + T^{2}$$
$23$ $$( 5 + T )^{2}$$
$29$ $$-41 + 4 T + T^{2}$$
$31$ $$-20 + T^{2}$$
$37$ $$-19 + 2 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$31 + 12 T + T^{2}$$
$47$ $$20 - 10 T + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$4 - 6 T + T^{2}$$
$61$ $$-176 - 4 T + T^{2}$$
$67$ $$-29 + 8 T + T^{2}$$
$71$ $$5 + 10 T + T^{2}$$
$73$ $$-4 + 2 T + T^{2}$$
$79$ $$11 - 8 T + T^{2}$$
$83$ $$44 - 16 T + T^{2}$$
$89$ $$-44 + 2 T + T^{2}$$
$97$ $$44 - 16 T + T^{2}$$