# Properties

 Label 6525.2.a.bw Level $6525$ Weight $2$ Character orbit 6525.a Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $7$ 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: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18$$ x^7 - x^6 - 13*x^5 + 12*x^4 + 47*x^3 - 37*x^2 - 35*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1305) 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{5} - 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b5 - 1) * q^7 + (b3 + 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{5} - 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{6} + \beta_{2} + 1) q^{11} + (\beta_{4} - 1) q^{13} + (\beta_{6} + \beta_{4} - 2 \beta_1 + 1) q^{14} + (\beta_{4} + \beta_{2} + 3) q^{16} + (\beta_{4} + 1) q^{17} + ( - \beta_{6} + \beta_{3} + \beta_1 + 2) q^{19} + ( - 2 \beta_{6} - \beta_{5} + \beta_{3} + \cdots - 1) q^{22}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (b5 - 1) * q^7 + (b3 + 2*b1) * q^8 + (-b6 + b2 + 1) * q^11 + (b4 - 1) * q^13 + (b6 + b4 - 2*b1 + 1) * q^14 + (b4 + b2 + 3) * q^16 + (b4 + 1) * q^17 + (-b6 + b3 + b1 + 2) * q^19 + (-2*b6 - b5 + b3 + 3*b1 - 1) * q^22 + (b5 - b3 + b2 - b1 + 2) * q^23 + (2*b5 - b4 + b3 - b1 + 1) * q^26 + (2*b6 + b5 - b4 + b3 - 2*b2 + b1 - 4) * q^28 + q^29 + (b6 - b3 - b1 + 2) * q^31 + (2*b5 - b4 + b1 + 1) * q^32 + (2*b5 - b4 + b3 + b1 + 1) * q^34 + (-b4 - b3 + b2 + b1 - 2) * q^37 + (-2*b6 - b5 + b4 + 2*b2 + 2*b1 + 2) * q^38 + (b4 + b3 + b2 + b1 + 2) * q^41 + (-2*b5 + b4 + b3 + b2 + b1 - 2) * q^43 + (-3*b6 - 2*b5 + 2*b2 + 6) * q^44 + (b6 + b3 - 2*b2 + 3*b1 - 2) * q^46 + (-2*b6 + b4 + 1) * q^47 + (-2*b5 + b4 + 2) * q^49 + (2*b6 - 2*b5 + 2*b4 - b3 - b1 - 2) * q^52 + (2*b6 - b4 - b3 + b2 - b1) * q^53 + (3*b6 + b4 - 3*b3 + 2*b2 - 5*b1 + 3) * q^56 + b1 * q^58 + (2*b5 + 2) * q^59 + (-2*b6 - 2*b3 + 2*b1 + 2) * q^61 + (2*b6 + b5 - b4 - 2*b2 + 2*b1 - 2) * q^62 + (2*b6 - 2*b5 + b4 - b3 - b2 - b1 - 1) * q^64 + (-b5 + 4*b1 - 3) * q^67 + (2*b6 - 2*b5 + 2*b4 - b3 + 2*b2 - b1 + 2) * q^68 + (2*b6 - 2*b3 - 2*b1) * q^71 + (-2*b6 + 2*b5 - b4 + b3 + 3*b2 + b1 + 2) * q^73 + (-2*b5 + 4) * q^74 + (-3*b6 - 2*b4 + b3 + 2*b2 + 5*b1 + 2) * q^76 + (2*b6 - b4 - 4*b2 - 2*b1 - 5) * q^77 + (b6 + b3 - 2*b2 + b1) * q^79 + (2*b5 + 2*b3 + 2*b2 + 4*b1 + 4) * q^82 + (b5 + b3 + 3*b2 - 3*b1 + 4) * q^83 + (-2*b6 + 2*b5 - 2*b4 + 2*b3 + 2*b2 + 2*b1 + 2) * q^86 + (-4*b6 - b5 - 2*b4 + 6*b1 - 3) * q^88 + (2*b6 - 2*b5 - b4 - 2*b2 + 1) * q^89 + (-b4 - 2*b2 + 4*b1 + 1) * q^91 + (2*b6 - b5 + b4 + 2*b2 - 4*b1 + 8) * q^92 + (-4*b6 - b4 + b3 + b1 - 1) * q^94 + (-2*b6 + 2*b5 - b4 - b3 + b2 - b1) * q^97 + (-2*b6 + 2*b5 - 3*b4 + b3 + 4*b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 13 q^{4} - 10 q^{7}+O(q^{10})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 $$7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37} + 12 q^{38} + 13 q^{41} - 9 q^{43} + 37 q^{44} - 8 q^{46} + 2 q^{47} + 21 q^{49} + q^{52} + 5 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 14 q^{61} - 8 q^{62} + 8 q^{64} - 14 q^{67} + 27 q^{68} + 8 q^{71} - 3 q^{73} + 34 q^{74} + 4 q^{76} - 28 q^{77} + 4 q^{79} + 20 q^{82} + 17 q^{83} - 4 q^{86} - 26 q^{88} + 20 q^{89} + 12 q^{91} + 60 q^{92} - 21 q^{94} - 13 q^{97} - 20 q^{98}+O(q^{100})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 + 3 * q^11 - 6 * q^13 + 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 - 9 * q^22 + 11 * q^23 - 3 * q^26 - 25 * q^28 + 7 * q^29 + 18 * q^31 + q^32 - q^34 - 13 * q^37 + 12 * q^38 + 13 * q^41 - 9 * q^43 + 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^52 + 5 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 - 14 * q^67 + 27 * q^68 + 8 * q^71 - 3 * q^73 + 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 20 * q^82 + 17 * q^83 - 4 * q^86 - 26 * q^88 + 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 - 13 * q^97 - 20 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 6\nu$$ v^3 - 6*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 7\nu^{2} + 5$$ v^4 - 7*v^2 + 5 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + \nu^{4} - 8\nu^{3} - 7\nu^{2} + 11\nu + 4 ) / 2$$ (v^5 + v^4 - 8*v^3 - 7*v^2 + 11*v + 4) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + \nu^{5} - 10\nu^{4} - 7\nu^{3} + 25\nu^{2} + 6\nu - 12 ) / 2$$ (v^6 + v^5 - 10*v^4 - 7*v^3 + 25*v^2 + 6*v - 12) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 6\beta_1$$ b3 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 7\beta_{2} + 23$$ b4 + 7*b2 + 23 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 8\beta_{3} + 37\beta _1 + 1$$ 2*b5 - b4 + 8*b3 + 37*b1 + 1 $$\nu^{6}$$ $$=$$ $$2\beta_{6} - 2\beta_{5} + 11\beta_{4} - \beta_{3} + 45\beta_{2} - \beta _1 + 141$$ 2*b6 - 2*b5 + 11*b4 - b3 + 45*b2 - b1 + 141

## 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
 −2.66072 −2.16940 −0.897436 0.431560 1.28209 2.43890 2.57501
−2.66072 0 5.07943 0 0 −4.68271 −8.19351 0 0
1.2 −2.16940 0 2.70628 0 0 0.484966 −1.53221 0 0
1.3 −0.897436 0 −1.19461 0 0 −3.83036 2.86696 0 0
1.4 0.431560 0 −1.81376 0 0 2.42505 −1.64586 0 0
1.5 1.28209 0 −0.356256 0 0 −3.04834 −3.02092 0 0
1.6 2.43890 0 3.94822 0 0 −3.59688 4.75150 0 0
1.7 2.57501 0 4.63069 0 0 2.24826 6.77405 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.bw 7
3.b odd 2 1 6525.2.a.bv 7
5.b even 2 1 1305.2.a.s 7
15.d odd 2 1 1305.2.a.t yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.a.s 7 5.b even 2 1
1305.2.a.t yes 7 15.d odd 2 1
6525.2.a.bv 7 3.b odd 2 1
6525.2.a.bw 7 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{7} - T_{2}^{6} - 13T_{2}^{5} + 12T_{2}^{4} + 47T_{2}^{3} - 37T_{2}^{2} - 35T_{2} + 18$$ T2^7 - T2^6 - 13*T2^5 + 12*T2^4 + 47*T2^3 - 37*T2^2 - 35*T2 + 18 $$T_{7}^{7} + 10T_{7}^{6} + 15T_{7}^{5} - 114T_{7}^{4} - 280T_{7}^{3} + 400T_{7}^{2} + 956T_{7} - 520$$ T7^7 + 10*T7^6 + 15*T7^5 - 114*T7^4 - 280*T7^3 + 400*T7^2 + 956*T7 - 520 $$T_{11}^{7} - 3T_{11}^{6} - 41T_{11}^{5} + 51T_{11}^{4} + 480T_{11}^{3} - 168T_{11}^{2} - 1588T_{11} + 12$$ T11^7 - 3*T11^6 - 41*T11^5 + 51*T11^4 + 480*T11^3 - 168*T11^2 - 1588*T11 + 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - T^{6} + \cdots + 18$$
$3$ $$T^{7}$$
$5$ $$T^{7}$$
$7$ $$T^{7} + 10 T^{6} + \cdots - 520$$
$11$ $$T^{7} - 3 T^{6} + \cdots + 12$$
$13$ $$T^{7} + 6 T^{6} + \cdots - 1408$$
$17$ $$T^{7} - 8 T^{6} + \cdots + 384$$
$19$ $$T^{7} - 10 T^{6} + \cdots + 6656$$
$23$ $$T^{7} - 11 T^{6} + \cdots - 4752$$
$29$ $$(T - 1)^{7}$$
$31$ $$T^{7} - 18 T^{6} + \cdots - 1408$$
$37$ $$T^{7} + 13 T^{6} + \cdots + 3328$$
$41$ $$T^{7} - 13 T^{6} + \cdots + 2496$$
$43$ $$T^{7} + 9 T^{6} + \cdots + 171008$$
$47$ $$T^{7} - 2 T^{6} + \cdots - 23040$$
$53$ $$T^{7} - 5 T^{6} + \cdots + 734208$$
$59$ $$T^{7} - 8 T^{6} + \cdots - 49152$$
$61$ $$T^{7} - 14 T^{6} + \cdots + 2422656$$
$67$ $$T^{7} + 14 T^{6} + \cdots - 366536$$
$71$ $$T^{7} - 8 T^{6} + \cdots - 245760$$
$73$ $$T^{7} + 3 T^{6} + \cdots - 2022656$$
$79$ $$T^{7} - 4 T^{6} + \cdots + 27712$$
$83$ $$T^{7} - 17 T^{6} + \cdots - 52800$$
$89$ $$T^{7} - 20 T^{6} + \cdots + 26400$$
$97$ $$T^{7} + 13 T^{6} + \cdots - 249856$$