# Properties

 Label 5220.2.a.y Level $5220$ Weight $2$ Character orbit 5220.a Self dual yes Analytic conductor $41.682$ 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] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5220.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: $$41.6819098551$$ 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: $$x^{7} - 3x^{6} - 22x^{5} + 38x^{4} + 81x^{3} - 75x^{2} - 42x + 18$$ x^7 - 3*x^6 - 22*x^5 + 38*x^4 + 81*x^3 - 75*x^2 - 42*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta_{2} q^{7}+O(q^{10})$$ q - q^5 + b2 * q^7 $$q - q^{5} + \beta_{2} q^{7} - \beta_{5} q^{11} - \beta_{6} q^{13} + ( - \beta_{4} - 1) q^{17} + (\beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{5} - \beta_{3}) q^{23} + q^{25} - q^{29} + (\beta_{3} + \beta_{2}) q^{31} - \beta_{2} q^{35} + ( - \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 2) q^{37}+ \cdots + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 3) q^{97}+O(q^{100})$$ q - q^5 + b2 * q^7 - b5 * q^11 - b6 * q^13 + (-b4 - 1) * q^17 + (b2 + b1 + 1) * q^19 + (-b5 - b3) * q^23 + q^25 - q^29 + (b3 + b2) * q^31 - b2 * q^35 + (-b6 - b5 - b3 - b2 + 2) * q^37 + (b6 + b5 + b3 + b2) * q^41 + (b6 + b5 + b2 - b1 + 1) * q^43 + (-b6 + 2*b4 + b3 - b1 - 1) * q^47 + (b6 - b3 - b1 + 4) * q^49 + (-b6 + b5 + b2 + b1 - 3) * q^53 + b5 * q^55 + (b3 + b1 + 1) * q^59 + (b3 - b1 + 1) * q^61 + b6 * q^65 + (2*b6 + b2 + 4) * q^67 + (-b3 + b1 + 1) * q^71 + (b5 - b4 - b2 + b1 + 4) * q^73 + (-2*b6 + b4 - 2*b2 - 2*b1 - 1) * q^77 + (-2*b6 - 2*b5 + 2*b4 - b2 - b1 + 1) * q^79 + (2*b6 + b5 + 2*b2 - b1 - 1) * q^83 + (b4 + 1) * q^85 + (b6 + 2*b5 - 2*b2) * q^89 + (b6 - 2*b5 - 2*b4 - b3 + b1 + 5) * q^91 + (-b2 - b1 - 1) * q^95 + (-b5 - b4 + b3 - b2 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 7 q^{5}+O(q^{10})$$ 7 * q - 7 * q^5 $$7 q - 7 q^{5} - 3 q^{11} + 2 q^{13} - 8 q^{17} + 6 q^{19} - q^{23} + 7 q^{25} - 7 q^{29} - 2 q^{31} + 15 q^{37} - q^{41} + 9 q^{43} - 4 q^{47} + 29 q^{49} - 17 q^{53} + 3 q^{55} + 4 q^{59} + 6 q^{61} - 2 q^{65} + 24 q^{67} + 8 q^{71} + 29 q^{73} + 8 q^{79} - 7 q^{83} + 8 q^{85} + 4 q^{89} + 26 q^{91} - 6 q^{95} + 15 q^{97}+O(q^{100})$$ 7 * q - 7 * q^5 - 3 * q^11 + 2 * q^13 - 8 * q^17 + 6 * q^19 - q^23 + 7 * q^25 - 7 * q^29 - 2 * q^31 + 15 * q^37 - q^41 + 9 * q^43 - 4 * q^47 + 29 * q^49 - 17 * q^53 + 3 * q^55 + 4 * q^59 + 6 * q^61 - 2 * q^65 + 24 * q^67 + 8 * q^71 + 29 * q^73 + 8 * q^79 - 7 * q^83 + 8 * q^85 + 4 * q^89 + 26 * q^91 - 6 * q^95 + 15 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 22x^{5} + 38x^{4} + 81x^{3} - 75x^{2} - 42x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( 51\nu^{6} - 76\nu^{5} - 1207\nu^{4} - 390\nu^{3} + 4613\nu^{2} + 10576\nu - 3248 ) / 1517$$ (51*v^6 - 76*v^5 - 1207*v^4 - 390*v^3 + 4613*v^2 + 10576*v - 3248) / 1517 $$\beta_{2}$$ $$=$$ $$( -173\nu^{6} + 585\nu^{5} + 3083\nu^{4} - 6619\nu^{3} - 597\nu^{2} + 3864\nu - 16437 ) / 4551$$ (-173*v^6 + 585*v^5 + 3083*v^4 - 6619*v^3 - 597*v^2 + 3864*v - 16437) / 4551 $$\beta_{3}$$ $$=$$ $$( 643\nu^{6} - 2148\nu^{5} - 13195\nu^{4} + 28100\nu^{3} + 39837\nu^{2} - 48402\nu - 16827 ) / 4551$$ (643*v^6 - 2148*v^5 - 13195*v^4 + 28100*v^3 + 39837*v^2 - 48402*v - 16827) / 4551 $$\beta_{4}$$ $$=$$ $$( -643\nu^{6} + 2148\nu^{5} + 13195\nu^{4} - 28100\nu^{3} - 39837\nu^{2} + 57504\nu + 12276 ) / 4551$$ (-643*v^6 + 2148*v^5 + 13195*v^4 - 28100*v^3 - 39837*v^2 + 57504*v + 12276) / 4551 $$\beta_{5}$$ $$=$$ $$( -1313\nu^{6} + 2730\nu^{5} + 31580\nu^{4} - 22798\nu^{3} - 128697\nu^{2} + 11964\nu + 65892 ) / 9102$$ (-1313*v^6 + 2730*v^5 + 31580*v^4 - 22798*v^3 - 128697*v^2 + 11964*v + 65892) / 9102 $$\beta_{6}$$ $$=$$ $$( 802\nu^{6} - 2028\nu^{5} - 18475\nu^{4} + 21530\nu^{3} + 70638\nu^{2} - 20070\nu - 24633 ) / 4551$$ (802*v^6 - 2028*v^5 - 18475*v^4 + 21530*v^3 + 70638*v^2 - 20070*v - 24633) / 4551
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} + 1 ) / 2$$ (b4 + b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -3\beta_{6} - 2\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} + \beta _1 + 16 ) / 2$$ (-3*b6 - 2*b5 + b4 + 3*b3 + 2*b2 + b1 + 16) / 2 $$\nu^{3}$$ $$=$$ $$( -8\beta_{6} - 8\beta_{5} + 21\beta_{4} + 24\beta_{3} - 5\beta _1 + 36 ) / 2$$ (-8*b6 - 8*b5 + 21*b4 + 24*b3 - 5*b1 + 36) / 2 $$\nu^{4}$$ $$=$$ $$-43\beta_{6} - 32\beta_{5} + 26\beta_{4} + 49\beta_{3} + 12\beta_{2} + 5\beta _1 + 164$$ -43*b6 - 32*b5 + 26*b4 + 49*b3 + 12*b2 + 5*b1 + 164 $$\nu^{5}$$ $$=$$ $$( -322\beta_{6} - 300\beta_{5} + 503\beta_{4} + 626\beta_{3} + 10\beta_{2} - 105\beta _1 + 1198 ) / 2$$ (-322*b6 - 300*b5 + 503*b4 + 626*b3 + 10*b2 - 105*b1 + 1198) / 2 $$\nu^{6}$$ $$=$$ $$( -2305\beta_{6} - 1842\beta_{5} + 1843\beta_{4} + 2957\beta_{3} + 402\beta_{2} + 11\beta _1 + 8296 ) / 2$$ (-2305*b6 - 1842*b5 + 1843*b4 + 2957*b3 + 402*b2 + 11*b1 + 8296) / 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
 −0.654294 1.04504 0.317654 5.35563 −3.73648 2.39884 −1.72639
0 0 0 −1.00000 0 −3.71027 0 0 0
1.2 0 0 0 −1.00000 0 −3.60894 0 0 0
1.3 0 0 0 −1.00000 0 −3.39461 0 0 0
1.4 0 0 0 −1.00000 0 0.432270 0 0 0
1.5 0 0 0 −1.00000 0 2.23366 0 0 0
1.6 0 0 0 −1.00000 0 2.99302 0 0 0
1.7 0 0 0 −1.00000 0 5.05487 0 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 5220.2.a.y 7
3.b odd 2 1 5220.2.a.z yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5220.2.a.y 7 1.a even 1 1 trivial
5220.2.a.z yes 7 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5220))$$:

 $$T_{7}^{7} - 39T_{7}^{5} - 10T_{7}^{4} + 448T_{7}^{3} + 24T_{7}^{2} - 1628T_{7} + 664$$ T7^7 - 39*T7^5 - 10*T7^4 + 448*T7^3 + 24*T7^2 - 1628*T7 + 664 $$T_{11}^{7} + 3T_{11}^{6} - 57T_{11}^{5} - 115T_{11}^{4} + 904T_{11}^{3} + 840T_{11}^{2} - 2340T_{11} - 2028$$ T11^7 + 3*T11^6 - 57*T11^5 - 115*T11^4 + 904*T11^3 + 840*T11^2 - 2340*T11 - 2028 $$T_{19}^{7} - 6T_{19}^{6} - 80T_{19}^{5} + 480T_{19}^{4} + 836T_{19}^{3} - 6104T_{19}^{2} - 1648T_{19} + 18464$$ T19^7 - 6*T19^6 - 80*T19^5 + 480*T19^4 + 836*T19^3 - 6104*T19^2 - 1648*T19 + 18464

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7}$$
$5$ $$(T + 1)^{7}$$
$7$ $$T^{7} - 39 T^{5} + \cdots + 664$$
$11$ $$T^{7} + 3 T^{6} + \cdots - 2028$$
$13$ $$T^{7} - 2 T^{6} + \cdots + 960$$
$17$ $$T^{7} + 8 T^{6} + \cdots + 36000$$
$19$ $$T^{7} - 6 T^{6} + \cdots + 18464$$
$23$ $$T^{7} + T^{6} + \cdots + 144$$
$29$ $$(T + 1)^{7}$$
$31$ $$T^{7} + 2 T^{6} + \cdots + 1536$$
$37$ $$T^{7} - 15 T^{6} + \cdots + 20480$$
$41$ $$T^{7} + T^{6} + \cdots - 49344$$
$43$ $$T^{7} - 9 T^{6} + \cdots - 124928$$
$47$ $$T^{7} + 4 T^{6} + \cdots - 113664$$
$53$ $$T^{7} + 17 T^{6} + \cdots - 1435968$$
$59$ $$T^{7} - 4 T^{6} + \cdots - 6144$$
$61$ $$T^{7} - 6 T^{6} + \cdots - 21248$$
$67$ $$T^{7} - 24 T^{6} + \cdots - 250840$$
$71$ $$T^{7} - 8 T^{6} + \cdots + 138240$$
$73$ $$T^{7} - 29 T^{6} + \cdots - 811584$$
$79$ $$T^{7} - 8 T^{6} + \cdots + 345600$$
$83$ $$T^{7} + 7 T^{6} + \cdots - 2967408$$
$89$ $$T^{7} - 4 T^{6} + \cdots + 7111008$$
$97$ $$T^{7} - 15 T^{6} + \cdots + 8457408$$