# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(1,575)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.59139811622$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{2} - T_{2} - 1$$ T2^2 - T2 - 1 $$T_{3}^{2} - 5$$ T3^2 - 5

## Hecke characteristic polynomials

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