# Properties

 Label 825.2.a.k Level $825$ Weight $2$ Character orbit 825.a Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ 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.

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
 0.311108 −1.48119 2.17009
−1.90321 −1.00000 1.62222 0 1.90321 4.42864 0.719004 1.00000 0
1.2 0.193937 −1.00000 −1.96239 0 −0.193937 −3.35026 −0.768452 1.00000 0
1.3 2.70928 −1.00000 5.34017 0 −2.70928 −1.07838 9.04945 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 825.2.a.k 3
3.b odd 2 1 2475.2.a.bb 3
5.b even 2 1 165.2.a.c 3
5.c odd 4 2 825.2.c.g 6
11.b odd 2 1 9075.2.a.cf 3
15.d odd 2 1 495.2.a.e 3
15.e even 4 2 2475.2.c.r 6
20.d odd 2 1 2640.2.a.be 3
35.c odd 2 1 8085.2.a.bk 3
55.d odd 2 1 1815.2.a.m 3
60.h even 2 1 7920.2.a.cj 3
165.d even 2 1 5445.2.a.z 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 5.b even 2 1
495.2.a.e 3 15.d odd 2 1
825.2.a.k 3 1.a even 1 1 trivial
825.2.c.g 6 5.c odd 4 2
1815.2.a.m 3 55.d odd 2 1
2475.2.a.bb 3 3.b odd 2 1
2475.2.c.r 6 15.e even 4 2
2640.2.a.be 3 20.d odd 2 1
5445.2.a.z 3 165.d even 2 1
7920.2.a.cj 3 60.h even 2 1
8085.2.a.bk 3 35.c odd 2 1
9075.2.a.cf 3 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1$$ T2^3 - T2^2 - 5*T2 + 1 $$T_{7}^{3} - 16T_{7} - 16$$ T7^3 - 16*T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 5T + 1$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 16T - 16$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$17$ $$T^{3} - 2 T^{2} - 52 T + 184$$
$19$ $$T^{3} - 8 T^{2} - 16 T + 160$$
$23$ $$T^{3} - 64T + 128$$
$29$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$31$ $$T^{3} - 8 T^{2} - 32 T + 128$$
$37$ $$(T - 2)^{3}$$
$41$ $$T^{3} + 14 T^{2} + 44 T + 8$$
$43$ $$T^{3} + 4 T^{2} - 80 T - 400$$
$47$ $$T^{3} - 8 T^{2} - 32 T + 128$$
$53$ $$T^{3} - 6 T^{2} - 52 T - 8$$
$59$ $$T^{3} - 12 T^{2} - 16 T + 320$$
$61$ $$T^{3} + 6 T^{2} - 52 T - 248$$
$67$ $$T^{3} - 4 T^{2} - 48 T + 64$$
$71$ $$T^{3} - 8 T^{2} - 32 T + 128$$
$73$ $$T^{3} - 14 T^{2} + 4 T + 344$$
$79$ $$T^{3} - 12 T^{2} - 64 T + 800$$
$83$ $$T^{3} - 120T - 16$$
$89$ $$T^{3} + 10 T^{2} - 52 T - 200$$
$97$ $$T^{3} + 22 T^{2} + 108 T + 8$$
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