Properties

 Label 3675.2.a.v 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: yes 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.

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

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
−1.61803 1.00000 0.618034 0 −1.61803 0 2.23607 1.00000 0
1.2 0.618034 1.00000 −1.61803 0 0.618034 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.v yes 2
5.b even 2 1 3675.2.a.ba yes 2
7.b odd 2 1 3675.2.a.u 2
35.c odd 2 1 3675.2.a.bc yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.2.a.u 2 7.b odd 2 1
3675.2.a.v yes 2 1.a even 1 1 trivial
3675.2.a.ba yes 2 5.b even 2 1
3675.2.a.bc yes 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2}^{2} + T_{2} - 1$$ $$T_{11}^{2} - 5$$ $$T_{13}^{2} + 4 T_{13} - 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$-16 + 4 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-16 - 4 T + T^{2}$$
$23$ $$-1 + 4 T + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-19 + 2 T + T^{2}$$
$41$ $$-80 + T^{2}$$
$43$ $$-41 - 4 T + T^{2}$$
$47$ $$16 + 12 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-16 - 4 T + T^{2}$$
$61$ $$( -12 + T )^{2}$$
$67$ $$-9 - 12 T + T^{2}$$
$71$ $$31 - 12 T + T^{2}$$
$73$ $$80 - 20 T + T^{2}$$
$79$ $$-29 - 8 T + T^{2}$$
$83$ $$-16 + 16 T + T^{2}$$
$89$ $$16 - 12 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$