# Properties

 Label 3750.2.a.b Level $3750$ Weight $2$ Character orbit 3750.a Self dual yes Analytic conductor $29.944$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3750.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9439007580$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) 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.

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

## 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
 1.61803 −0.618034
−1.00000 −1.00000 1.00000 0 1.00000 2.00000 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 2.00000 −1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 3750.2.a.b 2
5.b even 2 1 3750.2.a.g 2
5.c odd 4 2 3750.2.c.c 4
25.d even 5 2 150.2.g.b 4
25.e even 10 2 750.2.g.a 4
25.f odd 20 4 750.2.h.a 8
75.j odd 10 2 450.2.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 25.d even 5 2
450.2.h.b 4 75.j odd 10 2
750.2.g.a 4 25.e even 10 2
750.2.h.a 8 25.f odd 20 4
3750.2.a.b 2 1.a even 1 1 trivial
3750.2.a.g 2 5.b even 2 1
3750.2.c.c 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3750))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$-9 + 3 T + T^{2}$$
$17$ $$9 - 9 T + T^{2}$$
$19$ $$20 + 10 T + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$5 + 5 T + T^{2}$$
$31$ $$-36 + 6 T + T^{2}$$
$37$ $$19 + 11 T + T^{2}$$
$41$ $$-31 + T + T^{2}$$
$43$ $$-4 - 2 T + T^{2}$$
$47$ $$44 - 14 T + T^{2}$$
$53$ $$-99 + 3 T + T^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$19 + 11 T + T^{2}$$
$67$ $$-36 + 6 T + T^{2}$$
$71$ $$-116 + 6 T + T^{2}$$
$73$ $$31 + 13 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$5 - 5 T + T^{2}$$
$97$ $$99 + 21 T + T^{2}$$