# Properties

 Label 6525.2.a.ca Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10$$ x^9 - 2*x^8 - 12*x^7 + 21*x^6 + 48*x^5 - 68*x^4 - 73*x^3 + 66*x^2 + 40*x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^4 + b4 * q^7 + (-b3 - b2 - 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{4} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + (\beta_{5} - \beta_{4}) q^{11} - \beta_{5} q^{13} + (\beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots - 1) q^{14}+ \cdots + ( - 2 \beta_{8} + \beta_{5} + 4 \beta_{4} + \cdots - 5) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^4 + b4 * q^7 + (-b3 - b2 - 1) * q^8 + (b5 - b4) * q^11 - b5 * q^13 + (b6 - 2*b4 - b3 + b1 - 1) * q^14 + (-b8 + b6 + b5 + b3 + b2 + b1) * q^16 + (b7 - b4 - 1) * q^17 + (-b7 + b6 - b2 + 2*b1 - 1) * q^19 + (-b7 - 2*b6 + 2*b4 + b3 - b1) * q^22 + (b8 - 2*b6 + b4 + b3 + b2 - 2*b1 - 1) * q^23 + (b7 + b6 + 1) * q^26 + (-b8 + 2*b4 + b3 - b1 + 1) * q^28 + q^29 + (b8 - b6 - b5 + b4 + b3 + 1) * q^31 + (b8 - b6 - 2*b5 - b4 - b3 - 3*b2 + b1 - 2) * q^32 + (-b8 - b5 + b4 - b2 + 3*b1) * q^34 + (b8 + b7 - b5 - b4 - b3 - b2 + 2*b1) * q^37 + (b8 + b4 + b3 - b2 - 3) * q^38 + (-b8 - b7 - b6 + b4 + b3 - 2*b2 - 1) * q^41 + (2*b8 + b7 - b6 + 2*b3 + 3*b2 - 4*b1 + 1) * q^43 + (2*b8 - 2*b6 - b4 + b3 + 2*b2 - 2*b1 - 1) * q^44 + (b8 - b7 - 2*b6 + b5 - b4 + b3 + b2) * q^46 + (-2*b8 - 2*b7 + b6 + b5 + b4 - 3*b2 + 2*b1 - 4) * q^47 + (b8 + b7 - b6 - b5 - 2*b4 - b3 + 2*b1 - 1) * q^49 + (-b8 + 2*b6 - b4 - 2*b3 - 2*b2 + 3*b1) * q^52 + (-3*b8 + 2*b6 + b4 - b3 - b2 + 2*b1 - 3) * q^53 + (b8 + b7 - b6 - b5 - b4 - b3 - b2 + 3) * q^56 - b1 * q^58 + (-b8 - 2*b6 + b4 + b3 + 3*b2 - 2*b1 + 1) * q^59 + (b7 + b6 + b2 + 1) * q^61 + (b8 - b4 + b3 - b2 - 3) * q^62 + (b8 + b7 - b6 + 3*b4 + 4*b3 + 3*b2 + 2) * q^64 + (-2*b8 + 3*b6 - b4 - 2*b3 - b2 + 4*b1 - 2) * q^67 + (2*b6 - b4 - b3 - 2*b2 + 2*b1 - 5) * q^68 + (-b8 - 2*b7 + 2*b6 + 2*b5 + b4 + b3 - b2 - 1) * q^71 + (-2*b8 - 2*b7 + 2*b6 - 2*b3 - 2*b2) * q^73 + (-2*b8 + 2*b6 + 2*b4 + 2*b3 + 2*b1 - 4) * q^74 + (b8 + b7 - 2*b6 - b5 - b4 + b3 + b2 + 2*b1) * q^76 + (b8 + b7 + b5 + b3 + b2 - 2*b1 - 3) * q^77 + (2*b5 - 2*b4 - 2*b3 + 2*b1) * q^79 + (2*b8 + b7 - 2*b6 + b5 - 2*b4 + 2*b3 + 2*b2 + b1 + 1) * q^82 + (2*b6 - 2*b4 + 2*b1 - 6) * q^83 + (b8 - 2*b7 - 2*b6 - 2*b5 + b4 - b3 - 3*b2 + 3) * q^86 + (b8 + b5 + b3 - b1) * q^88 + (-b6 - 2*b4 - 2*b3 + b2) * q^89 + (-2*b8 - 2*b7 + b6 + 2*b4 - b2 - 3) * q^91 + (-2*b7 - 2*b6 + 2*b5 + 2*b4 + 2*b3 - 2*b2 - 2*b1 - 2) * q^92 + (2*b8 + b7 - b6 + b5 - 2*b4 + b3 + 2*b2 + 3*b1) * q^94 + (b7 + b6 + b2 + 2*b1 - 1) * q^97 + (-2*b8 + b5 + 4*b4 + 3*b3 - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10})$$ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 $$9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100})$$ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 2$$ v^3 - v^2 - 4*v + 2 $$\beta_{4}$$ $$=$$ $$( 2\nu^{8} - 3\nu^{7} - 19\nu^{6} + 26\nu^{5} + 44\nu^{4} - 75\nu^{3} + 5\nu^{2} + 76\nu - 25 ) / 13$$ (2*v^8 - 3*v^7 - 19*v^6 + 26*v^5 + 44*v^4 - 75*v^3 + 5*v^2 + 76*v - 25) / 13 $$\beta_{5}$$ $$=$$ $$( -2\nu^{8} + 3\nu^{7} + 19\nu^{6} - 13\nu^{5} - 57\nu^{4} - 29\nu^{3} + 47\nu^{2} + 106\nu + 25 ) / 13$$ (-2*v^8 + 3*v^7 + 19*v^6 - 13*v^5 - 57*v^4 - 29*v^3 + 47*v^2 + 106*v + 25) / 13 $$\beta_{6}$$ $$=$$ $$( 3\nu^{8} - 11\nu^{7} - 22\nu^{6} + 104\nu^{5} + 27\nu^{4} - 288\nu^{3} + 53\nu^{2} + 192\nu - 31 ) / 13$$ (3*v^8 - 11*v^7 - 22*v^6 + 104*v^5 + 27*v^4 - 288*v^3 + 53*v^2 + 192*v - 31) / 13 $$\beta_{7}$$ $$=$$ $$( -4\nu^{8} + 6\nu^{7} + 51\nu^{6} - 65\nu^{5} - 192\nu^{4} + 189\nu^{3} + 185\nu^{2} - 87\nu - 2 ) / 13$$ (-4*v^8 + 6*v^7 + 51*v^6 - 65*v^5 - 192*v^4 + 189*v^3 + 185*v^2 - 87*v - 2) / 13 $$\beta_{8}$$ $$=$$ $$( \nu^{8} - 8\nu^{7} - 3\nu^{6} + 91\nu^{5} - 43\nu^{4} - 304\nu^{3} + 178\nu^{2} + 259\nu - 71 ) / 13$$ (v^8 - 8*v^7 - 3*v^6 + 91*v^5 - 43*v^4 - 304*v^3 + 178*v^2 + 259*v - 71) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 14$$ -b8 + b6 + b5 + b3 + 7*b2 + b1 + 14 $$\nu^{5}$$ $$=$$ $$-\beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 19\beta _1 + 10$$ -b8 + b6 + 2*b5 + b4 + 9*b3 + 11*b2 + 19*b1 + 10 $$\nu^{6}$$ $$=$$ $$-9\beta_{8} + \beta_{7} + 9\beta_{6} + 10\beta_{5} + 3\beta_{4} + 14\beta_{3} + 49\beta_{2} + 10\beta _1 + 78$$ -9*b8 + b7 + 9*b6 + 10*b5 + 3*b4 + 14*b3 + 49*b2 + 10*b1 + 78 $$\nu^{7}$$ $$=$$ $$-13\beta_{8} + \beta_{7} + 11\beta_{6} + 24\beta_{5} + 16\beta_{4} + 71\beta_{3} + 97\beta_{2} + 98\beta _1 + 89$$ -13*b8 + b7 + 11*b6 + 24*b5 + 16*b4 + 71*b3 + 97*b2 + 98*b1 + 89 $$\nu^{8}$$ $$=$$ $$-70\beta_{8} + 11\beta_{7} + 67\beta_{6} + 83\beta_{5} + 46\beta_{4} + 138\beta_{3} + 349\beta_{2} + 85\beta _1 + 479$$ -70*b8 + 11*b7 + 67*b6 + 83*b5 + 46*b4 + 138*b3 + 349*b2 + 85*b1 + 479

## 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
 2.74387 2.18077 1.77820 1.23642 0.200649 −0.747618 −1.10241 −2.07907 −2.21081
−2.74387 0 5.52881 0 0 1.10432 −9.68257 0 0
1.2 −2.18077 0 2.75577 0 0 −4.62113 −1.64817 0 0
1.3 −1.77820 0 1.16198 0 0 2.84862 1.49016 0 0
1.4 −1.23642 0 −0.471260 0 0 3.27673 3.05552 0 0
1.5 −0.200649 0 −1.95974 0 0 −0.775135 0.794517 0 0
1.6 0.747618 0 −1.44107 0 0 −3.28783 −2.57260 0 0
1.7 1.10241 0 −0.784687 0 0 −0.259703 −3.06987 0 0
1.8 2.07907 0 2.32253 0 0 −0.602349 0.670572 0 0
1.9 2.21081 0 2.88766 0 0 1.31646 1.96245 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 6525.2.a.ca 9
3.b odd 2 1 6525.2.a.cc yes 9
5.b even 2 1 6525.2.a.cd yes 9
15.d odd 2 1 6525.2.a.cb yes 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6525.2.a.ca 9 1.a even 1 1 trivial
6525.2.a.cb yes 9 15.d odd 2 1
6525.2.a.cc yes 9 3.b odd 2 1
6525.2.a.cd yes 9 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6525))$$:

 $$T_{2}^{9} + 2T_{2}^{8} - 12T_{2}^{7} - 21T_{2}^{6} + 48T_{2}^{5} + 68T_{2}^{4} - 73T_{2}^{3} - 66T_{2}^{2} + 40T_{2} + 10$$ T2^9 + 2*T2^8 - 12*T2^7 - 21*T2^6 + 48*T2^5 + 68*T2^4 - 73*T2^3 - 66*T2^2 + 40*T2 + 10 $$T_{7}^{9} + T_{7}^{8} - 27T_{7}^{7} - 3T_{7}^{6} + 199T_{7}^{5} - 89T_{7}^{4} - 270T_{7}^{3} + 50T_{7}^{2} + 125T_{7} + 25$$ T7^9 + T7^8 - 27*T7^7 - 3*T7^6 + 199*T7^5 - 89*T7^4 - 270*T7^3 + 50*T7^2 + 125*T7 + 25 $$T_{11}^{9} - 2 T_{11}^{8} - 48 T_{11}^{7} + 124 T_{11}^{6} + 635 T_{11}^{5} - 2064 T_{11}^{4} + \cdots - 600$$ T11^9 - 2*T11^8 - 48*T11^7 + 124*T11^6 + 635*T11^5 - 2064*T11^4 - 1381*T11^3 + 8414*T11^2 - 5420*T11 - 600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} + 2 T^{8} + \cdots + 10$$
$3$ $$T^{9}$$
$5$ $$T^{9}$$
$7$ $$T^{9} + T^{8} + \cdots + 25$$
$11$ $$T^{9} - 2 T^{8} + \cdots - 600$$
$13$ $$T^{9} + T^{8} + \cdots - 5885$$
$17$ $$T^{9} + 12 T^{8} + \cdots + 1070$$
$19$ $$T^{9} + T^{8} + \cdots - 164800$$
$23$ $$T^{9} + 16 T^{8} + \cdots - 640$$
$29$ $$(T - 1)^{9}$$
$31$ $$T^{9} - 5 T^{8} + \cdots + 320$$
$37$ $$T^{9} - 148 T^{7} + \cdots - 11520$$
$41$ $$T^{9} + 10 T^{8} + \cdots + 28800$$
$43$ $$T^{9} + 3 T^{8} + \cdots - 1275840$$
$47$ $$T^{9} + 26 T^{8} + \cdots + 4977120$$
$53$ $$T^{9} + 22 T^{8} + \cdots - 3780480$$
$59$ $$T^{9} - 4 T^{8} + \cdots - 28656000$$
$61$ $$T^{9} - 7 T^{8} + \cdots + 46656$$
$67$ $$T^{9} + 5 T^{8} + \cdots - 2311495$$
$71$ $$T^{9} - 322 T^{7} + \cdots + 73200000$$
$73$ $$T^{9} - 10 T^{8} + \cdots + 29099520$$
$79$ $$T^{9} - 10 T^{8} + \cdots + 18312192$$
$83$ $$T^{9} + 46 T^{8} + \cdots - 1287680$$
$89$ $$T^{9} - 4 T^{8} + \cdots + 1503000$$
$97$ $$T^{9} + 7 T^{8} + \cdots - 88640$$