# Properties

 Label 5225.2.a.o Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8$$ x^8 - 2*x^7 - 9*x^6 + 12*x^5 + 28*x^4 - 17*x^3 - 28*x^2 + 6*x + 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1045) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (2 \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{6} + \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b6 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (2*b6 - b4 - b3 - b2 - b1) * q^6 + (-b7 - b6 - b5 + b3 + b2 + 1) * q^7 + (-b3 + b2 - b1 + 3) * q^8 + (b6 + b3 + b2 + 2) * q^9 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (2 \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{6}+ \cdots + (\beta_{6} + \beta_{3} + \beta_{2} + 2) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b6 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (2*b6 - b4 - b3 - b2 - b1) * q^6 + (-b7 - b6 - b5 + b3 + b2 + 1) * q^7 + (-b3 + b2 - b1 + 3) * q^8 + (b6 + b3 + b2 + 2) * q^9 + q^11 + (b7 + 3*b6 + b5 - b4 - b3 + 2) * q^12 + (b6 + b5 - b4 - b2 + 2) * q^13 + (-2*b7 - 2*b6 - b5 + b4 + 2*b3 + b2 - 2*b1 + 1) * q^14 + (b4 - 2*b3 - 3*b1 + 3) * q^16 + (b7 - b6 + b5 + b2 + b1 + 2) * q^17 + (2*b6 - 2*b4 - b3 - 2*b2 - 2*b1) * q^18 - q^19 + (-2*b7 + b6 - b5 + 2*b3 - 1) * q^21 + (-b1 + 1) * q^22 + (b7 - b5 - b3 + b1 + 1) * q^23 + (3*b7 + 3*b6 + 2*b5 - b4 - 2*b3 - b2 + 2) * q^24 + (b7 + 2*b6 + b5 - b4 + b3 - b2 - b1 + 2) * q^26 + (-b7 + 2*b3 + 3*b2 + b1 + 5) * q^27 + (-3*b7 - 4*b6 - b5 + 2*b4 + 2*b3 + 2*b2 - 2*b1 + 2) * q^28 + (-b7 - b6 - 2*b5 + 2*b3 + 2*b2 + 2*b1 - 1) * q^29 + (-b7 + b6 - b4 - b3 - 2*b1) * q^31 + (-b7 - b6 - b5 + 2*b4 - b3 + 3*b2 - 3*b1 + 6) * q^32 + (b6 + 1) * q^33 + (2*b7 - 2*b6 + b5 - b3 - b2 - 2*b1) * q^34 + (2*b7 + 4*b6 + 2*b5 - b4 - b3 - b2 + b1 + 5) * q^36 + (-b6 + b5 + 2*b4 - b2 - b1 + 2) * q^37 + (b1 - 1) * q^38 + (-2*b7 + 4*b6 - b4 - 3*b1 + 2) * q^39 + (b6 - b5 - b4 - 2*b3 + b1 - 1) * q^41 + (-4*b7 + b6 - 2*b5 - b4 + 3*b3 - b2 + b1 - 5) * q^42 + (b6 - b5 - b4 - b3 + 2*b1 + 2) * q^43 + (b2 - b1 + 2) * q^44 + (2*b7 + 2*b6 + b5 - 2*b3 - b2 - b1 + 1) * q^46 + (b7 - 2*b6 - b5 + 2*b2 + 2) * q^47 + (5*b7 + 2*b6 + 2*b5 - 2*b4 - 4*b3 - 4*b2) * q^48 + (-3*b7 + b6 - b5 - b4 + b3 + b2 - 3*b1 + 2) * q^49 + (2*b7 + 2*b6 + 2*b5 + b4 - 2*b3 + 2*b2 + 3*b1) * q^51 + (3*b7 + 3*b6 - 2*b4 - b2 + b1) * q^52 + (-b6 - b5 + b4 + b2 + 2*b1 + 2) * q^53 + (-2*b7 - b6 - b5 - b4 - 2*b2 - 7*b1 - 1) * q^54 + (-4*b7 - 8*b6 - 3*b5 + 3*b4 + b3 + 5*b2 - b1 + 2) * q^56 + (-b6 - 1) * q^57 + (-2*b7 - b6 - b5 + 2*b3 - 2*b2 - 7) * q^58 + (3*b7 - 3*b6 - b5 - b2 + b1 - 4) * q^59 + (4*b7 + b6 + b5 - b3 + b1 + 2) * q^61 + (-b7 + 2*b6 + b4 + 3*b2 - 2*b1 + 7) * q^62 + (-3*b7 + 2*b6 + 4*b3 - b2 - b1 + 1) * q^63 + (-4*b7 - 4*b6 - 3*b5 + b4 + 6*b2 - 5*b1 + 7) * q^64 + (2*b6 - b4 - b3 - b2 - b1) * q^66 + (-b7 + 2*b6 + b5 + b4 - b3 - b2 - b1 + 2) * q^67 + (2*b7 - b6 + b4 + b2 + 6) * q^68 + (3*b7 + b4 - 2*b3 - b2 + 3) * q^69 + (-b7 - 2*b6 - b5 - b4 + 2*b3 + 3*b1 - 4) * q^71 + (5*b7 + 5*b6 + 3*b5 - b4 - 3*b3 - 2*b2 + b1 + 1) * q^72 + (b7 - b4 - b3 - b2 + b1 + 2) * q^73 + (-2*b7 - 5*b6 - 2*b5 + b4 + 2*b2 - b1 + 1) * q^74 + (-b2 + b1 - 2) * q^76 + (-b7 - b6 - b5 + b3 + b2 + 1) * q^77 + (-3*b7 + 7*b6 - b5 - 2*b4 - b3 + b2 - 4*b1 + 7) * q^78 + (b7 - b6 + 3*b5 + b4 - b2) * q^79 + (-2*b7 + 2*b6 + b5 + b4 + 2*b3 + 3*b2 + 4*b1 + 4) * q^81 + (b7 + 4*b6 + b5 + b4 - 2*b3 - b1) * q^82 + (b7 - b6 + b5 + 2*b4 - 3*b2 - 3*b1 + 4) * q^83 + (-3*b7 - b6 - b5 + 4*b3 - b2 + 5*b1 - 10) * q^84 + (b7 + 4*b6 + b5 - b3 - 2*b2 - 3*b1 - 1) * q^86 + (-3*b7 - 2*b6 - 2*b5 + 2*b4 + 5*b3 + 3*b2 + 2*b1 + 2) * q^87 + (-b3 + b2 - b1 + 3) * q^88 + (2*b6 + 2*b5 - b4 + b1 - 2) * q^89 + (-3*b7 - 3*b5 - 3*b4 + 5*b3 + 2*b2 + 2*b1 - 1) * q^91 + (2*b7 + 5*b6 + 4*b5 - 2*b4 - 3*b3 - b2 - 2*b1 + 3) * q^92 + (2*b6 + b5 - 3*b4 - b3 - 3*b2 - 5*b1 - 2) * q^93 + (2*b7 - 2*b6 + b5 + b4 - b3 + b2 - 3*b1 + 6) * q^94 + (6*b7 + 3*b6 + 3*b5 - b4 - b3 - b2 + 5*b1 + 5) * q^96 + (-b7 - 4*b6 - 3*b5 - b2 + 3*b1 - 3) * q^97 + (-5*b7 + b6 - 2*b5 + b4 + 3*b3 + 4*b2 - 5*b1 + 9) * q^98 + (b6 + b3 + b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 11 q^{7} + 18 q^{8} + 11 q^{9}+O(q^{10})$$ 8 * q + 6 * q^2 + 7 * q^3 + 10 * q^4 + 11 * q^7 + 18 * q^8 + 11 * q^9 $$8 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 11 q^{7} + 18 q^{8} + 11 q^{9} + 8 q^{11} + 7 q^{12} + 17 q^{13} + 12 q^{14} + 18 q^{16} + 9 q^{17} + 2 q^{18} - 8 q^{19} + q^{21} + 6 q^{22} + 8 q^{23} + q^{24} + 10 q^{26} + 34 q^{27} + 22 q^{28} - 3 q^{29} - q^{31} + 37 q^{32} + 7 q^{33} - 8 q^{34} + 30 q^{36} + 17 q^{37} - 6 q^{38} + 14 q^{39} - 5 q^{41} - 15 q^{42} + 21 q^{43} + 10 q^{44} - 2 q^{46} + 8 q^{47} - 10 q^{48} + 19 q^{49} - 16 q^{51} - 9 q^{52} + 19 q^{53} - 3 q^{54} + 24 q^{56} - 7 q^{57} - 37 q^{58} - 33 q^{59} - q^{61} + 42 q^{62} + 20 q^{63} + 48 q^{64} + 18 q^{67} + 37 q^{68} + 16 q^{69} - 18 q^{71} - 13 q^{72} + 18 q^{73} + 15 q^{74} - 10 q^{76} + 11 q^{77} + 51 q^{78} - 5 q^{79} + 32 q^{81} - 12 q^{82} + 33 q^{83} - 51 q^{84} - 16 q^{86} + 26 q^{87} + 18 q^{88} - 20 q^{89} + 6 q^{91} + 3 q^{92} - 18 q^{93} + 30 q^{94} + 21 q^{96} + 69 q^{98} + 11 q^{99}+O(q^{100})$$ 8 * q + 6 * q^2 + 7 * q^3 + 10 * q^4 + 11 * q^7 + 18 * q^8 + 11 * q^9 + 8 * q^11 + 7 * q^12 + 17 * q^13 + 12 * q^14 + 18 * q^16 + 9 * q^17 + 2 * q^18 - 8 * q^19 + q^21 + 6 * q^22 + 8 * q^23 + q^24 + 10 * q^26 + 34 * q^27 + 22 * q^28 - 3 * q^29 - q^31 + 37 * q^32 + 7 * q^33 - 8 * q^34 + 30 * q^36 + 17 * q^37 - 6 * q^38 + 14 * q^39 - 5 * q^41 - 15 * q^42 + 21 * q^43 + 10 * q^44 - 2 * q^46 + 8 * q^47 - 10 * q^48 + 19 * q^49 - 16 * q^51 - 9 * q^52 + 19 * q^53 - 3 * q^54 + 24 * q^56 - 7 * q^57 - 37 * q^58 - 33 * q^59 - q^61 + 42 * q^62 + 20 * q^63 + 48 * q^64 + 18 * q^67 + 37 * q^68 + 16 * q^69 - 18 * q^71 - 13 * q^72 + 18 * q^73 + 15 * q^74 - 10 * q^76 + 11 * q^77 + 51 * q^78 - 5 * q^79 + 32 * q^81 - 12 * q^82 + 33 * q^83 - 51 * q^84 - 16 * q^86 + 26 * q^87 + 18 * q^88 - 20 * q^89 + 6 * q^91 + 3 * q^92 - 18 * q^93 + 30 * q^94 + 21 * q^96 + 69 * q^98 + 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 3$$ v^3 - 2*v^2 - 3*v + 3 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 4\nu^{2} + 5\nu + 2$$ v^4 - 2*v^3 - 4*v^2 + 5*v + 2 $$\beta_{5}$$ $$=$$ $$\nu^{6} - 2\nu^{5} - 8\nu^{4} + 10\nu^{3} + 21\nu^{2} - 8\nu - 12$$ v^6 - 2*v^5 - 8*v^4 + 10*v^3 + 21*v^2 - 8*v - 12 $$\beta_{6}$$ $$=$$ $$\nu^{7} - 3\nu^{6} - 6\nu^{5} + 18\nu^{4} + 11\nu^{3} - 29\nu^{2} - 4\nu + 11$$ v^7 - 3*v^6 - 6*v^5 + 18*v^4 + 11*v^3 - 29*v^2 - 4*v + 11 $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 2\nu^{6} + 9\nu^{5} - 13\nu^{4} - 24\nu^{3} + 19\nu^{2} + 12\nu - 6$$ -v^7 + 2*v^6 + 9*v^5 - 13*v^4 - 24*v^3 + 19*v^2 + 12*v - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 3$$ b3 + 2*b2 + 5*b1 + 3 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 8\beta_{2} + 9\beta _1 + 16$$ b4 + 2*b3 + 8*b2 + 9*b1 + 16 $$\nu^{5}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} + 9\beta_{3} + 19\beta_{2} + 31\beta _1 + 31$$ b7 + b6 + b5 + 3*b4 + 9*b3 + 19*b2 + 31*b1 + 31 $$\nu^{6}$$ $$=$$ $$2\beta_{7} + 2\beta_{6} + 3\beta_{5} + 14\beta_{4} + 24\beta_{3} + 61\beta_{2} + 71\beta _1 + 109$$ 2*b7 + 2*b6 + 3*b5 + 14*b4 + 24*b3 + 61*b2 + 71*b1 + 109 $$\nu^{7}$$ $$=$$ $$12\beta_{7} + 13\beta_{6} + 15\beta_{5} + 42\beta_{4} + 79\beta_{3} + 160\beta_{2} + 215\beta _1 + 268$$ 12*b7 + 13*b6 + 15*b5 + 42*b4 + 79*b3 + 160*b2 + 215*b1 + 268

## 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
 2.87791 2.25600 1.06639 0.714778 −0.649219 −0.865980 −1.57959 −1.82030
−1.87791 3.37956 1.52654 0 −6.34651 0.818685 0.889109 8.42145 0
1.2 −1.25600 −0.772194 −0.422456 0 0.969878 3.10169 3.04261 −2.40372 0
1.3 −0.0663929 0.255194 −1.99559 0 −0.0169431 −2.56056 0.265279 −2.93488 0
1.4 0.285222 1.61587 −1.91865 0 0.460881 1.06724 −1.11768 −0.388965 0
1.5 1.64922 2.98027 0.719922 0 4.91512 5.09280 −2.11113 5.88203 0
1.6 1.86598 −2.01288 1.48188 0 −3.75600 2.04136 −0.966800 1.05170 0
1.7 2.57959 2.53274 4.65426 0 6.53343 −2.87748 6.84689 3.41479 0
1.8 2.82030 −0.978567 5.95409 0 −2.75985 4.31627 11.1517 −2.04241 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$11$$ $$-1$$
$$19$$ $$+1$$

## 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 5225.2.a.o 8
5.b even 2 1 1045.2.a.i 8
15.d odd 2 1 9405.2.a.bf 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.i 8 5.b even 2 1
5225.2.a.o 8 1.a even 1 1 trivial
9405.2.a.bf 8 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{8} - 6T_{2}^{7} + 5T_{2}^{6} + 28T_{2}^{5} - 47T_{2}^{4} - 21T_{2}^{3} + 60T_{2}^{2} - 11T_{2} - 1$$ T2^8 - 6*T2^7 + 5*T2^6 + 28*T2^5 - 47*T2^4 - 21*T2^3 + 60*T2^2 - 11*T2 - 1 $$T_{7}^{8} - 11T_{7}^{7} + 23T_{7}^{6} + 120T_{7}^{5} - 489T_{7}^{4} + 47T_{7}^{3} + 1792T_{7}^{2} - 2384T_{7} + 896$$ T7^8 - 11*T7^7 + 23*T7^6 + 120*T7^5 - 489*T7^4 + 47*T7^3 + 1792*T7^2 - 2384*T7 + 896

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 6 T^{7} + \cdots - 1$$
$3$ $$T^{8} - 7 T^{7} + \cdots - 16$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 11 T^{7} + \cdots + 896$$
$11$ $$(T - 1)^{8}$$
$13$ $$T^{8} - 17 T^{7} + \cdots - 13392$$
$17$ $$T^{8} - 9 T^{7} + \cdots + 8344$$
$19$ $$(T + 1)^{8}$$
$23$ $$T^{8} - 8 T^{7} + \cdots + 1504$$
$29$ $$T^{8} + 3 T^{7} + \cdots + 84152$$
$31$ $$T^{8} + T^{7} + \cdots - 35840$$
$37$ $$T^{8} - 17 T^{7} + \cdots - 280640$$
$41$ $$T^{8} + 5 T^{7} + \cdots + 30440$$
$43$ $$T^{8} - 21 T^{7} + \cdots - 18944$$
$47$ $$T^{8} - 8 T^{7} + \cdots + 29984$$
$53$ $$T^{8} - 19 T^{7} + \cdots + 10528$$
$59$ $$T^{8} + 33 T^{7} + \cdots + 6694912$$
$61$ $$T^{8} + T^{7} + \cdots + 25298072$$
$67$ $$T^{8} - 18 T^{7} + \cdots + 56432$$
$71$ $$T^{8} + 18 T^{7} + \cdots - 833024$$
$73$ $$T^{8} - 18 T^{7} + \cdots - 22472$$
$79$ $$T^{8} + 5 T^{7} + \cdots - 197248$$
$83$ $$T^{8} - 33 T^{7} + \cdots + 2043584$$
$89$ $$T^{8} + 20 T^{7} + \cdots + 216$$
$97$ $$T^{8} - 419 T^{6} + \cdots + 9664$$