# Properties

 Label 575.2.a.f Level $575$ Weight $2$ Character orbit 575.a Self dual yes Analytic conductor $4.591$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.59139811622$$ Analytic rank: $$1$$ 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 23) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 - 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + ( -2 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + 2 q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 - 2 \beta ) q^{3} + ( -1 + \beta ) q^{4} + ( -2 - \beta ) q^{6} + ( -2 + 2 \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + 2 q^{9} + ( -4 + 2 \beta ) q^{11} + ( -3 + \beta ) q^{12} -3 q^{13} + 2 q^{14} -3 \beta q^{16} + ( -2 - 2 \beta ) q^{17} + 2 \beta q^{18} -2 q^{19} + ( -6 + 2 \beta ) q^{21} + ( 2 - 2 \beta ) q^{22} - q^{23} + 5 q^{24} -3 \beta q^{26} + ( -1 + 2 \beta ) q^{27} + ( 4 - 2 \beta ) q^{28} -3 q^{29} + ( 3 - 6 \beta ) q^{31} + ( -5 + \beta ) q^{32} + ( -8 + 6 \beta ) q^{33} + ( -2 - 4 \beta ) q^{34} + ( -2 + 2 \beta ) q^{36} -2 \beta q^{37} -2 \beta q^{38} + ( -3 + 6 \beta ) q^{39} + ( -1 + 4 \beta ) q^{41} + ( 2 - 4 \beta ) q^{42} + ( 6 - 4 \beta ) q^{44} -\beta q^{46} + ( 1 - 2 \beta ) q^{47} + ( 6 + 3 \beta ) q^{48} + ( 1 - 4 \beta ) q^{49} + ( 2 + 6 \beta ) q^{51} + ( 3 - 3 \beta ) q^{52} + ( 2 + 4 \beta ) q^{53} + ( 2 + \beta ) q^{54} + ( -6 + 2 \beta ) q^{56} + ( -2 + 4 \beta ) q^{57} -3 \beta q^{58} + ( 4 - 4 \beta ) q^{59} + ( -2 + 8 \beta ) q^{61} + ( -6 - 3 \beta ) q^{62} + ( -4 + 4 \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( 6 - 2 \beta ) q^{66} + ( 4 + 2 \beta ) q^{67} -2 \beta q^{68} + ( -1 + 2 \beta ) q^{69} + ( 11 - 2 \beta ) q^{71} + ( 2 - 4 \beta ) q^{72} + ( -9 - 4 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} + ( 2 - 2 \beta ) q^{76} + ( 12 - 8 \beta ) q^{77} + ( 6 + 3 \beta ) q^{78} + ( -6 + 8 \beta ) q^{79} -11 q^{81} + ( 4 + 3 \beta ) q^{82} + ( 10 + 2 \beta ) q^{83} + ( 8 - 6 \beta ) q^{84} + ( -3 + 6 \beta ) q^{87} + ( -8 + 6 \beta ) q^{88} + ( -8 + 4 \beta ) q^{89} + ( 6 - 6 \beta ) q^{91} + ( 1 - \beta ) q^{92} + 15 q^{93} + ( -2 - \beta ) q^{94} + ( -7 + 9 \beta ) q^{96} + ( -14 + 6 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} + ( -8 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 5q^{6} - 2q^{7} + 4q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 5q^{6} - 2q^{7} + 4q^{9} - 6q^{11} - 5q^{12} - 6q^{13} + 4q^{14} - 3q^{16} - 6q^{17} + 2q^{18} - 4q^{19} - 10q^{21} + 2q^{22} - 2q^{23} + 10q^{24} - 3q^{26} + 6q^{28} - 6q^{29} - 9q^{32} - 10q^{33} - 8q^{34} - 2q^{36} - 2q^{37} - 2q^{38} + 2q^{41} + 8q^{44} - q^{46} + 15q^{48} - 2q^{49} + 10q^{51} + 3q^{52} + 8q^{53} + 5q^{54} - 10q^{56} - 3q^{58} + 4q^{59} + 4q^{61} - 15q^{62} - 4q^{63} + 4q^{64} + 10q^{66} + 10q^{67} - 2q^{68} + 20q^{71} - 22q^{73} - 6q^{74} + 2q^{76} + 16q^{77} + 15q^{78} - 4q^{79} - 22q^{81} + 11q^{82} + 22q^{83} + 10q^{84} - 10q^{88} - 12q^{89} + 6q^{91} + q^{92} + 30q^{93} - 5q^{94} - 5q^{96} - 22q^{97} - 11q^{98} - 12q^{99} + O(q^{100})$$

## 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 2.23607 −1.61803 0 −1.38197 −3.23607 2.23607 2.00000 0
1.2 1.61803 −2.23607 0.618034 0 −3.61803 1.23607 −2.23607 2.00000 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
$$5$$ $$1$$
$$23$$ $$1$$

## 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 575.2.a.f 2
3.b odd 2 1 5175.2.a.be 2
4.b odd 2 1 9200.2.a.bt 2
5.b even 2 1 23.2.a.a 2
5.c odd 4 2 575.2.b.d 4
15.d odd 2 1 207.2.a.d 2
20.d odd 2 1 368.2.a.h 2
35.c odd 2 1 1127.2.a.c 2
40.e odd 2 1 1472.2.a.s 2
40.f even 2 1 1472.2.a.t 2
55.d odd 2 1 2783.2.a.c 2
60.h even 2 1 3312.2.a.ba 2
65.d even 2 1 3887.2.a.i 2
85.c even 2 1 6647.2.a.b 2
95.d odd 2 1 8303.2.a.e 2
115.c odd 2 1 529.2.a.a 2
115.i odd 22 10 529.2.c.n 20
115.j even 22 10 529.2.c.o 20
345.h even 2 1 4761.2.a.w 2
460.g even 2 1 8464.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 5.b even 2 1
207.2.a.d 2 15.d odd 2 1
368.2.a.h 2 20.d odd 2 1
529.2.a.a 2 115.c odd 2 1
529.2.c.n 20 115.i odd 22 10
529.2.c.o 20 115.j even 22 10
575.2.a.f 2 1.a even 1 1 trivial
575.2.b.d 4 5.c odd 4 2
1127.2.a.c 2 35.c odd 2 1
1472.2.a.s 2 40.e odd 2 1
1472.2.a.t 2 40.f even 2 1
2783.2.a.c 2 55.d odd 2 1
3312.2.a.ba 2 60.h even 2 1
3887.2.a.i 2 65.d even 2 1
4761.2.a.w 2 345.h even 2 1
5175.2.a.be 2 3.b odd 2 1
6647.2.a.b 2 85.c even 2 1
8303.2.a.e 2 95.d odd 2 1
8464.2.a.bb 2 460.g even 2 1
9200.2.a.bt 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{3}^{2} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$-5 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 + 2 T + T^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$4 + 6 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$-45 + T^{2}$$
$37$ $$-4 + 2 T + T^{2}$$
$41$ $$-19 - 2 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$-5 + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-16 - 4 T + T^{2}$$
$61$ $$-76 - 4 T + T^{2}$$
$67$ $$20 - 10 T + T^{2}$$
$71$ $$95 - 20 T + T^{2}$$
$73$ $$101 + 22 T + T^{2}$$
$79$ $$-76 + 4 T + T^{2}$$
$83$ $$116 - 22 T + T^{2}$$
$89$ $$16 + 12 T + T^{2}$$
$97$ $$76 + 22 T + T^{2}$$