# Properties

 Label 575.2.a.h Level $575$ Weight $2$ Character orbit 575.a Self dual yes Analytic conductor $4.591$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^3 + (b2 + b1 + 1) * q^4 + (-b3 - 2*b1 + 1) * q^6 + (-b3 + b2 + b1 + 1) * q^7 + (-b3 - b2 - b1 - 2) * q^8 + (b2 + 2) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - 2 \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{8} + (\beta_{2} + 2) q^{9} + ( - 2 \beta_1 + 2) q^{11} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{12} + (2 \beta_{3} - \beta_{2} - 1) q^{13} + ( - 2 \beta_1 - 2) q^{14} + (2 \beta_{3} + 3 \beta_1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} - 3 \beta_1 + 1) q^{18} + (2 \beta_1 - 2) q^{19} + (2 \beta_{3} + 2) q^{21} + (2 \beta_{2} + 6) q^{22} + q^{23} + ( - 3 \beta_{2} - 4 \beta_1 - 7) q^{24} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{26} + ( - \beta_{2} + 3) q^{27} + (2 \beta_{3} + 2 \beta_1 + 4) q^{28} + (\beta_{3} + \beta_1 + 4) q^{29} + ( - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{31} + ( - 3 \beta_{2} - 3 \beta_1 - 5) q^{32} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{33} + ( - 2 \beta_{2} - 4) q^{34} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 5) q^{36} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{37} + ( - 2 \beta_{2} - 6) q^{38} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{39} + ( - \beta_{3} - \beta_1 + 4) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{42} + ( - 2 \beta_{3} + 2) q^{43} + ( - 2 \beta_{3} - 4 \beta_1 - 2) q^{44} - \beta_1 q^{46} + (\beta_{2} - 4 \beta_1 + 1) q^{47} + (\beta_{3} + 2 \beta_{2} + 10 \beta_1 + 1) q^{48} + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{49} + (4 \beta_1 - 4) q^{51} + ( - \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{52} + (\beta_{3} + \beta_{2} + 5 \beta_1 - 7) q^{53} + (\beta_{3} - 2 \beta_1 - 1) q^{54} + ( - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 2) q^{56} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{57} + ( - \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 3) q^{58} + (3 \beta_{3} - \beta_{2} - \beta_1 + 5) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{61} + (\beta_{3} + 4 \beta_{2} + 5) q^{62} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{63} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 6) q^{64} + (6 \beta_{2} + 14) q^{66} + (3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{67} + (2 \beta_{2} + 4 \beta_1) q^{68} + (\beta_{2} + 1) q^{69} + (3 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{71} + ( - \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 9) q^{72} + ( - 2 \beta_{3} + \beta_{2} + 9) q^{73} + (4 \beta_{3} + 4 \beta_1 - 6) q^{74} + (2 \beta_{3} + 4 \beta_1 + 2) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 - 11) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{79} - 7 q^{81} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{82} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{83} + (6 \beta_{2} + 8 \beta_1 + 6) q^{84} + (2 \beta_{3} + 2 \beta_{2}) q^{86} + (5 \beta_{2} + 4 \beta_1 + 5) q^{87} + (2 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{88} + ( - 4 \beta_{3} + 2 \beta_1) q^{89} + (6 \beta_{2} + 2 \beta_1 - 8) q^{91} + (\beta_{2} + \beta_1 + 1) q^{92} + (2 \beta_{3} - \beta_{2} - 8 \beta_1 + 5) q^{93} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 13) q^{94} + ( - 3 \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 14) q^{96} + ( - 2 \beta_{3} + 2 \beta_{2} + 6) q^{97} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{98} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^3 + (b2 + b1 + 1) * q^4 + (-b3 - 2*b1 + 1) * q^6 + (-b3 + b2 + b1 + 1) * q^7 + (-b3 - b2 - b1 - 2) * q^8 + (b2 + 2) * q^9 + (-2*b1 + 2) * q^11 + (b3 + b2 + 2*b1 + 4) * q^12 + (2*b3 - b2 - 1) * q^13 + (-2*b1 - 2) * q^14 + (2*b3 + 3*b1) * q^16 + (b3 - b2 + b1 - 1) * q^17 + (-b3 - 3*b1 + 1) * q^18 + (2*b1 - 2) * q^19 + (2*b3 + 2) * q^21 + (2*b2 + 6) * q^22 + q^23 + (-3*b2 - 4*b1 - 7) * q^24 + (-b3 - 2*b2 - 1) * q^26 + (-b2 + 3) * q^27 + (2*b3 + 2*b1 + 4) * q^28 + (b3 + b1 + 4) * q^29 + (-3*b3 + 2*b2 - b1 + 2) * q^31 + (-3*b2 - 3*b1 - 5) * q^32 + (-2*b3 + 2*b2 - 4*b1 + 4) * q^33 + (-2*b2 - 4) * q^34 + (b3 + 2*b2 + 3*b1 + 5) * q^36 + (-b3 - 3*b2 + b1 - 1) * q^37 + (-2*b2 - 6) * q^38 + (-2*b3 + b2 + 4*b1 - 1) * q^39 + (-b3 - b1 + 4) * q^41 + (-2*b3 - 2*b2 - 4*b1) * q^42 + (-2*b3 + 2) * q^43 + (-2*b3 - 4*b1 - 2) * q^44 - b1 * q^46 + (b2 - 4*b1 + 1) * q^47 + (b3 + 2*b2 + 10*b1 + 1) * q^48 + (b3 - 3*b2 + b1) * q^49 + (4*b1 - 4) * q^51 + (-b3 + 3*b2 + 4*b1) * q^52 + (b3 + b2 + 5*b1 - 7) * q^53 + (b3 - 2*b1 - 1) * q^54 + (-2*b3 - 4*b2 - 4*b1 - 2) * q^56 + (2*b3 - 2*b2 + 4*b1 - 4) * q^57 + (-b3 - 2*b2 - 6*b1 - 3) * q^58 + (3*b3 - b2 - b1 + 5) * q^59 + (-2*b2 + 2*b1 - 2) * q^61 + (b3 + 4*b2 + 5) * q^62 + (b3 + b2 + b1 + 3) * q^63 + (-b3 + 3*b2 + 5*b1 + 6) * q^64 + (6*b2 + 14) * q^66 + (3*b3 - b2 + b1 - 1) * q^67 + (2*b2 + 4*b1) * q^68 + (b2 + 1) * q^69 + (3*b3 - 4*b2 + b1 - 4) * q^71 + (-b3 - 4*b2 - 5*b1 - 9) * q^72 + (-2*b3 + b2 + 9) * q^73 + (4*b3 + 4*b1 - 6) * q^74 + (2*b3 + 4*b1 + 2) * q^76 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^77 + (b3 - 2*b2 - 2*b1 - 11) * q^78 + (-2*b3 + 2*b2 - 4*b1 + 4) * q^79 - 7 * q^81 + (b3 + 2*b2 - 2*b1 + 3) * q^82 + (b3 - 3*b2 + b1 + 3) * q^83 + (6*b2 + 8*b1 + 6) * q^84 + (2*b3 + 2*b2) * q^86 + (5*b2 + 4*b1 + 5) * q^87 + (2*b3 + 2*b2 + 8*b1) * q^88 + (-4*b3 + 2*b1) * q^89 + (6*b2 + 2*b1 - 8) * q^91 + (b2 + b1 + 1) * q^92 + (2*b3 - b2 - 8*b1 + 5) * q^93 + (-b3 + 4*b2 + 2*b1 + 13) * q^94 + (-3*b3 - 5*b2 - 6*b1 - 14) * q^96 + (-2*b3 + 2*b2 + 6) * q^97 + (2*b3 - 2*b2 + b1 - 6) * q^98 + (-2*b3 + 2*b2 - 6*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 + 4 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 6 * q^9 $$4 q - 2 q^{2} + 2 q^{3} + 4 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 19 q^{12} - 12 q^{14} + 8 q^{16} + q^{17} - 3 q^{18} - 4 q^{19} + 10 q^{21} + 20 q^{22} + 4 q^{23} - 30 q^{24} - q^{26} + 14 q^{27} + 22 q^{28} + 19 q^{29} - q^{31} - 20 q^{32} + 2 q^{33} - 12 q^{34} + 23 q^{36} + 3 q^{37} - 20 q^{38} + 13 q^{41} - 6 q^{42} + 6 q^{43} - 18 q^{44} - 2 q^{46} - 6 q^{47} + 21 q^{48} + 9 q^{49} - 8 q^{51} + q^{52} - 19 q^{53} - 7 q^{54} - 10 q^{56} - 2 q^{57} - 21 q^{58} + 23 q^{59} + 13 q^{62} + 13 q^{63} + 27 q^{64} + 44 q^{66} + 3 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{71} - 39 q^{72} + 32 q^{73} - 12 q^{74} + 18 q^{76} - 18 q^{77} - 43 q^{78} + 2 q^{79} - 28 q^{81} + 5 q^{82} + 21 q^{83} + 28 q^{84} - 2 q^{86} + 18 q^{87} + 14 q^{88} - 40 q^{91} + 4 q^{92} + 8 q^{93} + 47 q^{94} - 61 q^{96} + 18 q^{97} - 16 q^{98} + 6 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 + 4 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 6 * q^9 + 4 * q^11 + 19 * q^12 - 12 * q^14 + 8 * q^16 + q^17 - 3 * q^18 - 4 * q^19 + 10 * q^21 + 20 * q^22 + 4 * q^23 - 30 * q^24 - q^26 + 14 * q^27 + 22 * q^28 + 19 * q^29 - q^31 - 20 * q^32 + 2 * q^33 - 12 * q^34 + 23 * q^36 + 3 * q^37 - 20 * q^38 + 13 * q^41 - 6 * q^42 + 6 * q^43 - 18 * q^44 - 2 * q^46 - 6 * q^47 + 21 * q^48 + 9 * q^49 - 8 * q^51 + q^52 - 19 * q^53 - 7 * q^54 - 10 * q^56 - 2 * q^57 - 21 * q^58 + 23 * q^59 + 13 * q^62 + 13 * q^63 + 27 * q^64 + 44 * q^66 + 3 * q^67 + 4 * q^68 + 2 * q^69 - 3 * q^71 - 39 * q^72 + 32 * q^73 - 12 * q^74 + 18 * q^76 - 18 * q^77 - 43 * q^78 + 2 * q^79 - 28 * q^81 + 5 * q^82 + 21 * q^83 + 28 * q^84 - 2 * q^86 + 18 * q^87 + 14 * q^88 - 40 * q^91 + 4 * q^92 + 8 * q^93 + 47 * q^94 - 61 * q^96 + 18 * q^97 - 16 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 1$$ v^3 - v^2 - 4*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 2$$ b3 + b2 + 5*b1 + 2

## 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
 2.69353 1.32973 −0.329727 −1.69353
−2.69353 2.56155 5.25508 0 −6.89961 2.74252 −8.76763 3.56155 0
1.2 −1.32973 −1.56155 −0.231826 0 2.07644 3.50407 2.96772 −0.561553 0
1.3 0.329727 −1.56155 −1.89128 0 −0.514886 −4.06562 −1.28306 −0.561553 0
1.4 1.69353 2.56155 0.868028 0 4.33805 0.819031 −1.91702 3.56155 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.h 4
3.b odd 2 1 5175.2.a.bx 4
4.b odd 2 1 9200.2.a.cl 4
5.b even 2 1 115.2.a.c 4
5.c odd 4 2 575.2.b.e 8
15.d odd 2 1 1035.2.a.o 4
20.d odd 2 1 1840.2.a.u 4
35.c odd 2 1 5635.2.a.v 4
40.e odd 2 1 7360.2.a.cg 4
40.f even 2 1 7360.2.a.cj 4
115.c odd 2 1 2645.2.a.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.a.c 4 5.b even 2 1
575.2.a.h 4 1.a even 1 1 trivial
575.2.b.e 8 5.c odd 4 2
1035.2.a.o 4 15.d odd 2 1
1840.2.a.u 4 20.d odd 2 1
2645.2.a.m 4 115.c odd 2 1
5175.2.a.bx 4 3.b odd 2 1
5635.2.a.v 4 35.c odd 2 1
7360.2.a.cg 4 40.e odd 2 1
7360.2.a.cj 4 40.f even 2 1
9200.2.a.cl 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 2$$ T2^4 + 2*T2^3 - 4*T2^2 - 5*T2 + 2 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} - 4 T^{2} - 5 T + 2$$
$3$ $$(T^{2} - T - 4)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 3 T^{3} - 14 T^{2} + 52 T - 32$$
$11$ $$T^{4} - 4 T^{3} - 16 T^{2} + 40 T + 32$$
$13$ $$T^{4} - 41T^{2} + 212$$
$17$ $$T^{4} - T^{3} - 18 T^{2} + 24 T + 32$$
$19$ $$T^{4} + 4 T^{3} - 16 T^{2} - 40 T + 32$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} - 19 T^{3} + 117 T^{2} + \cdots + 202$$
$31$ $$T^{4} + T^{3} - 101 T^{2} + 11 T + 2144$$
$37$ $$T^{4} - 3 T^{3} - 116 T^{2} + \cdots + 2008$$
$41$ $$T^{4} - 13 T^{3} + 45 T^{2} - 3 T - 94$$
$43$ $$T^{4} - 6 T^{3} - 36 T^{2} + 16 T + 128$$
$47$ $$T^{4} + 6 T^{3} - 83 T^{2} - 548 T - 128$$
$53$ $$T^{4} + 19 T^{3} - 34 T^{2} + \cdots - 8776$$
$59$ $$T^{4} - 23 T^{3} + 100 T^{2} + \cdots - 3136$$
$61$ $$T^{4} - 56 T^{2} + 136 T - 32$$
$67$ $$T^{4} - 3 T^{3} - 98 T^{2} + \cdots + 2032$$
$71$ $$T^{4} + 3 T^{3} - 149 T^{2} - 535 T - 8$$
$73$ $$T^{4} - 32 T^{3} + 343 T^{2} + \cdots + 1684$$
$79$ $$T^{4} - 2 T^{3} - 140 T^{2} + \cdots + 512$$
$83$ $$T^{4} - 21 T^{3} + 96 T^{2} + \cdots - 1216$$
$89$ $$T^{4} - 216 T^{2} - 1496 T - 2752$$
$97$ $$T^{4} - 18 T^{3} + 72 T^{2} + \cdots - 1072$$