# Properties

 Label 575.2.a.i Level $575$ Weight $2$ Character orbit 575.a Self dual yes Analytic conductor $4.591$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5744.1 Defining polynomial: $$x^{4} - 5x^{2} - 2x + 1$$ x^4 - 5*x^2 - 2*x + 1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} - 2x + 1$$ :

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

## 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
 2.37988 0.291367 −0.751024 −1.92022
−2.37988 1.95969 3.66382 0 −4.66382 −2.28394 −3.95969 0.840379 0
1.2 −0.291367 −3.14073 −1.91511 0 0.915105 1.20647 1.14073 6.86420 0
1.3 0.751024 0.580491 −1.43596 0 0.435963 −0.315061 −2.58049 −2.66303 0
1.4 1.92022 −1.39945 1.68725 0 −2.68725 −4.60747 −0.600553 −1.04155 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.i 4
3.b odd 2 1 5175.2.a.bv 4
4.b odd 2 1 9200.2.a.cq 4
5.b even 2 1 575.2.a.j 4
5.c odd 4 2 115.2.b.b 8
15.d odd 2 1 5175.2.a.bw 4
15.e even 4 2 1035.2.b.e 8
20.d odd 2 1 9200.2.a.ck 4
20.e even 4 2 1840.2.e.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 5.c odd 4 2
575.2.a.i 4 1.a even 1 1 trivial
575.2.a.j 4 5.b even 2 1
1035.2.b.e 8 15.e even 4 2
1840.2.e.d 8 20.e even 4 2
5175.2.a.bv 4 3.b odd 2 1
5175.2.a.bw 4 15.d odd 2 1
9200.2.a.ck 4 20.d odd 2 1
9200.2.a.cq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{4} - 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 5*T2^2 + 2*T2 + 1 $$T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 6T_{3} + 5$$ T3^4 + 2*T3^3 - 6*T3^2 - 6*T3 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5 T^{2} + 2 T + 1$$
$3$ $$T^{4} + 2 T^{3} - 6 T^{2} - 6 T + 5$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 6 T^{3} + 4 T^{2} - 12 T - 4$$
$11$ $$T^{4} - 2 T^{3} - 16 T^{2} + 44 T - 28$$
$13$ $$T^{4} + 14 T^{3} + 66 T^{2} + 118 T + 61$$
$17$ $$T^{4} + 14 T^{3} + 58 T^{2} + 64 T + 20$$
$19$ $$T^{4} + 4 T^{3} - 6 T^{2} - 28 T - 20$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} - 4 T^{3} - 22 T^{2} + 4 T + 5$$
$31$ $$T^{4} - 74 T^{2} - 256 T - 167$$
$37$ $$T^{4} + 2 T^{3} - 72 T^{2} - 380 T - 476$$
$41$ $$T^{4} + 8 T^{3} - 94 T^{2} + \cdots + 2485$$
$43$ $$T^{4} + 4 T^{3} - 118 T^{2} + \cdots + 1964$$
$47$ $$T^{4} + 2 T^{3} - 138 T^{2} + \cdots + 4513$$
$53$ $$T^{4} + 4 T^{3} - 104 T^{2} + \cdots + 2192$$
$59$ $$T^{4} - 20 T^{2} + 16 T + 16$$
$61$ $$T^{4} + 8 T^{3} - 102 T^{2} + \cdots - 2756$$
$67$ $$T^{4} - 2 T^{3} - 8 T^{2} + 12 T + 4$$
$71$ $$T^{4} + 24 T^{3} + 66 T^{2} + \cdots - 7435$$
$73$ $$T^{4} + 18 T^{3} - 22 T^{2} + \cdots - 8339$$
$79$ $$T^{4} - 24 T^{3} + 178 T^{2} + \cdots + 28$$
$83$ $$T^{4} - 6 T^{3} - 270 T^{2} + \cdots + 14948$$
$89$ $$T^{4} + 8 T^{3} - 86 T^{2} + \cdots + 2380$$
$97$ $$T^{4} + 34 T^{3} + 348 T^{2} + \cdots - 4676$$