# Properties

 Label 3675.2.a.w Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{3} + ( 1 + \beta ) q^{4} -\beta q^{6} -3 q^{8} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + q^{3} + ( 1 + \beta ) q^{4} -\beta q^{6} -3 q^{8} + q^{9} -3 q^{11} + ( 1 + \beta ) q^{12} + ( 2 - 2 \beta ) q^{13} + ( -2 + \beta ) q^{16} + 2 \beta q^{17} -\beta q^{18} + ( -2 - 2 \beta ) q^{19} + 3 \beta q^{22} + ( -3 + 4 \beta ) q^{23} -3 q^{24} + 6 q^{26} + q^{27} + ( -3 - 2 \beta ) q^{29} + ( -2 + 4 \beta ) q^{31} + ( 3 + \beta ) q^{32} -3 q^{33} + ( -6 - 2 \beta ) q^{34} + ( 1 + \beta ) q^{36} + ( -5 + 4 \beta ) q^{37} + ( 6 + 4 \beta ) q^{38} + ( 2 - 2 \beta ) q^{39} + ( -5 - 2 \beta ) q^{43} + ( -3 - 3 \beta ) q^{44} + ( -12 - \beta ) q^{46} + ( 6 + 2 \beta ) q^{47} + ( -2 + \beta ) q^{48} + 2 \beta q^{51} + ( -4 - 2 \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} -\beta q^{54} + ( -2 - 2 \beta ) q^{57} + ( 6 + 5 \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( -8 + 4 \beta ) q^{61} + ( -12 - 2 \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + 3 \beta q^{66} + ( -11 - 2 \beta ) q^{67} + ( 6 + 4 \beta ) q^{68} + ( -3 + 4 \beta ) q^{69} -3 q^{71} -3 q^{72} + ( -4 + 2 \beta ) q^{73} + ( -12 + \beta ) q^{74} + ( -8 - 6 \beta ) q^{76} + 6 q^{78} + ( -1 - 6 \beta ) q^{79} + q^{81} + ( 6 - 4 \beta ) q^{83} + ( 6 + 7 \beta ) q^{86} + ( -3 - 2 \beta ) q^{87} + 9 q^{88} + ( 6 - 6 \beta ) q^{89} + ( 9 + 5 \beta ) q^{92} + ( -2 + 4 \beta ) q^{93} + ( -6 - 8 \beta ) q^{94} + ( 3 + \beta ) q^{96} + ( -10 + 4 \beta ) q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 2q^{3} + 3q^{4} - q^{6} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + 2q^{3} + 3q^{4} - q^{6} - 6q^{8} + 2q^{9} - 6q^{11} + 3q^{12} + 2q^{13} - 3q^{16} + 2q^{17} - q^{18} - 6q^{19} + 3q^{22} - 2q^{23} - 6q^{24} + 12q^{26} + 2q^{27} - 8q^{29} + 7q^{32} - 6q^{33} - 14q^{34} + 3q^{36} - 6q^{37} + 16q^{38} + 2q^{39} - 12q^{43} - 9q^{44} - 25q^{46} + 14q^{47} - 3q^{48} + 2q^{51} - 10q^{52} + 8q^{53} - q^{54} - 6q^{57} + 17q^{58} + 14q^{59} - 12q^{61} - 26q^{62} - 4q^{64} + 3q^{66} - 24q^{67} + 16q^{68} - 2q^{69} - 6q^{71} - 6q^{72} - 6q^{73} - 23q^{74} - 22q^{76} + 12q^{78} - 8q^{79} + 2q^{81} + 8q^{83} + 19q^{86} - 8q^{87} + 18q^{88} + 6q^{89} + 23q^{92} - 20q^{94} + 7q^{96} - 16q^{97} - 6q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
−2.30278 1.00000 3.30278 0 −2.30278 0 −3.00000 1.00000 0
1.2 1.30278 1.00000 −0.302776 0 1.30278 0 −3.00000 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.w 2
5.b even 2 1 3675.2.a.bb 2
7.b odd 2 1 525.2.a.f 2
21.c even 2 1 1575.2.a.t 2
28.d even 2 1 8400.2.a.df 2
35.c odd 2 1 525.2.a.h yes 2
35.f even 4 2 525.2.d.d 4
105.g even 2 1 1575.2.a.o 2
105.k odd 4 2 1575.2.d.g 4
140.c even 2 1 8400.2.a.cw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.a.f 2 7.b odd 2 1
525.2.a.h yes 2 35.c odd 2 1
525.2.d.d 4 35.f even 4 2
1575.2.a.o 2 105.g even 2 1
1575.2.a.t 2 21.c even 2 1
1575.2.d.g 4 105.k odd 4 2
3675.2.a.w 2 1.a even 1 1 trivial
3675.2.a.bb 2 5.b even 2 1
8400.2.a.cw 2 140.c even 2 1
8400.2.a.df 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} + T_{2} - 3$$ $$T_{11} + 3$$ $$T_{13}^{2} - 2 T_{13} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$-12 - 2 T + T^{2}$$
$17$ $$-12 - 2 T + T^{2}$$
$19$ $$-4 + 6 T + T^{2}$$
$23$ $$-51 + 2 T + T^{2}$$
$29$ $$3 + 8 T + T^{2}$$
$31$ $$-52 + T^{2}$$
$37$ $$-43 + 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$23 + 12 T + T^{2}$$
$47$ $$36 - 14 T + T^{2}$$
$53$ $$-36 - 8 T + T^{2}$$
$59$ $$36 - 14 T + T^{2}$$
$61$ $$-16 + 12 T + T^{2}$$
$67$ $$131 + 24 T + T^{2}$$
$71$ $$( 3 + T )^{2}$$
$73$ $$-4 + 6 T + T^{2}$$
$79$ $$-101 + 8 T + T^{2}$$
$83$ $$-36 - 8 T + T^{2}$$
$89$ $$-108 - 6 T + T^{2}$$
$97$ $$12 + 16 T + T^{2}$$