# Properties

 Label 175.2.a.e Level $175$ Weight $2$ Character orbit 175.a Self dual yes Analytic conductor $1.397$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.39738203537$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -2 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + ( 5 - 4 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( -2 + 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + 2 q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + ( 5 - 4 \beta ) q^{9} + ( 1 + 2 \beta ) q^{11} + ( 4 - 2 \beta ) q^{12} -2 \beta q^{13} + \beta q^{14} -3 \beta q^{16} -4 \beta q^{17} + ( -4 + \beta ) q^{18} + ( -2 + 4 \beta ) q^{19} + ( -2 + 2 \beta ) q^{21} + ( 2 + 3 \beta ) q^{22} + ( 5 - 2 \beta ) q^{23} + ( -6 + 2 \beta ) q^{24} + ( -2 - 2 \beta ) q^{26} + ( -12 + 4 \beta ) q^{27} + ( -1 + \beta ) q^{28} + 5 q^{29} -6 \beta q^{31} + ( -5 + \beta ) q^{32} + ( 2 + 2 \beta ) q^{33} + ( -4 - 4 \beta ) q^{34} + ( -9 + 5 \beta ) q^{36} + 3 q^{37} + ( 4 + 2 \beta ) q^{38} -4 q^{39} + ( 6 + 2 \beta ) q^{41} + 2 q^{42} + ( 3 + 2 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( -2 + 3 \beta ) q^{46} -2 q^{47} -6 q^{48} + q^{49} -8 q^{51} -2 q^{52} + ( 6 - 4 \beta ) q^{53} + ( 4 - 8 \beta ) q^{54} + ( 1 - 2 \beta ) q^{56} + ( 12 - 4 \beta ) q^{57} + 5 \beta q^{58} + ( 8 - 6 \beta ) q^{59} + ( -6 + 6 \beta ) q^{61} + ( -6 - 6 \beta ) q^{62} + ( 5 - 4 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( 2 + 4 \beta ) q^{66} + ( -3 + 2 \beta ) q^{67} -4 q^{68} + ( -14 + 10 \beta ) q^{69} + ( 5 - 6 \beta ) q^{71} + ( 13 - 6 \beta ) q^{72} + ( -10 - 2 \beta ) q^{73} + 3 \beta q^{74} + ( 6 - 2 \beta ) q^{76} + ( 1 + 2 \beta ) q^{77} -4 \beta q^{78} + ( -5 + 10 \beta ) q^{79} + ( 17 - 12 \beta ) q^{81} + ( 2 + 8 \beta ) q^{82} + ( -4 + 6 \beta ) q^{83} + ( 4 - 2 \beta ) q^{84} + ( 2 + 5 \beta ) q^{86} + ( -10 + 10 \beta ) q^{87} + ( -3 - 4 \beta ) q^{88} + ( 16 - 2 \beta ) q^{89} -2 \beta q^{91} + ( -7 + 5 \beta ) q^{92} -12 q^{93} -2 \beta q^{94} + ( 12 - 10 \beta ) q^{96} + ( 4 - 2 \beta ) q^{97} + \beta q^{98} + ( -3 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q + q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 4 q^{17} - 7 q^{18} - 2 q^{21} + 7 q^{22} + 8 q^{23} - 10 q^{24} - 6 q^{26} - 20 q^{27} - q^{28} + 10 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{33} - 12 q^{34} - 13 q^{36} + 6 q^{37} + 10 q^{38} - 8 q^{39} + 14 q^{41} + 4 q^{42} + 8 q^{43} + 3 q^{44} - q^{46} - 4 q^{47} - 12 q^{48} + 2 q^{49} - 16 q^{51} - 4 q^{52} + 8 q^{53} + 20 q^{57} + 5 q^{58} + 10 q^{59} - 6 q^{61} - 18 q^{62} + 6 q^{63} + 4 q^{64} + 8 q^{66} - 4 q^{67} - 8 q^{68} - 18 q^{69} + 4 q^{71} + 20 q^{72} - 22 q^{73} + 3 q^{74} + 10 q^{76} + 4 q^{77} - 4 q^{78} + 22 q^{81} + 12 q^{82} - 2 q^{83} + 6 q^{84} + 9 q^{86} - 10 q^{87} - 10 q^{88} + 30 q^{89} - 2 q^{91} - 9 q^{92} - 24 q^{93} - 2 q^{94} + 14 q^{96} + 6 q^{97} + q^{98} - 8 q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
−0.618034 −3.23607 −1.61803 0 2.00000 1.00000 2.23607 7.47214 0
1.2 1.61803 1.23607 0.618034 0 2.00000 1.00000 −2.23607 −1.47214 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 175.2.a.e yes 2
3.b odd 2 1 1575.2.a.n 2
4.b odd 2 1 2800.2.a.bp 2
5.b even 2 1 175.2.a.d 2
5.c odd 4 2 175.2.b.c 4
7.b odd 2 1 1225.2.a.u 2
15.d odd 2 1 1575.2.a.s 2
15.e even 4 2 1575.2.d.k 4
20.d odd 2 1 2800.2.a.bh 2
20.e even 4 2 2800.2.g.s 4
35.c odd 2 1 1225.2.a.n 2
35.f even 4 2 1225.2.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 5.b even 2 1
175.2.a.e yes 2 1.a even 1 1 trivial
175.2.b.c 4 5.c odd 4 2
1225.2.a.n 2 35.c odd 2 1
1225.2.a.u 2 7.b odd 2 1
1225.2.b.k 4 35.f even 4 2
1575.2.a.n 2 3.b odd 2 1
1575.2.a.s 2 15.d odd 2 1
1575.2.d.k 4 15.e even 4 2
2800.2.a.bh 2 20.d odd 2 1
2800.2.a.bp 2 4.b odd 2 1
2800.2.g.s 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-1 - 4 T + T^{2}$$
$13$ $$-4 + 2 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$11 - 8 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$-36 + 6 T + T^{2}$$
$37$ $$( -3 + T )^{2}$$
$41$ $$44 - 14 T + T^{2}$$
$43$ $$11 - 8 T + T^{2}$$
$47$ $$( 2 + T )^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-20 - 10 T + T^{2}$$
$61$ $$-36 + 6 T + T^{2}$$
$67$ $$-1 + 4 T + T^{2}$$
$71$ $$-41 - 4 T + T^{2}$$
$73$ $$116 + 22 T + T^{2}$$
$79$ $$-125 + T^{2}$$
$83$ $$-44 + 2 T + T^{2}$$
$89$ $$220 - 30 T + T^{2}$$
$97$ $$4 - 6 T + T^{2}$$