# Properties

 Label 7650.2.a.df Level $7650$ Weight $2$ Character orbit 7650.a Self dual yes Analytic conductor $61.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7650,2,Mod(1,7650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7650, 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("7650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7650.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: $$61.0855575463$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2550) 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - \beta q^{7} + q^{8} +O(q^{10})$$ q + q^2 + q^4 - b * q^7 + q^8 $$q + q^{2} + q^{4} - \beta q^{7} + q^{8} + (\beta - 2) q^{11} - 2 q^{13} - \beta q^{14} + q^{16} - q^{17} + \beta q^{19} + (\beta - 2) q^{22} + (\beta + 1) q^{23} - 2 q^{26} - \beta q^{28} + (2 \beta - 4) q^{29} + ( - 3 \beta + 2) q^{31} + q^{32} - q^{34} + q^{37} + \beta q^{38} + ( - \beta - 1) q^{41} + ( - \beta - 6) q^{43} + (\beta - 2) q^{44} + (\beta + 1) q^{46} + ( - \beta - 4) q^{47} + (\beta + 1) q^{49} - 2 q^{52} + ( - 2 \beta + 1) q^{53} - \beta q^{56} + (2 \beta - 4) q^{58} + (3 \beta - 3) q^{59} + (3 \beta - 1) q^{61} + ( - 3 \beta + 2) q^{62} + q^{64} + (\beta + 2) q^{67} - q^{68} + ( - \beta - 1) q^{71} - 8 q^{73} + q^{74} + \beta q^{76} + (\beta - 8) q^{77} + (3 \beta - 4) q^{79} + ( - \beta - 1) q^{82} + ( - \beta - 1) q^{83} + ( - \beta - 6) q^{86} + (\beta - 2) q^{88} + ( - 2 \beta - 2) q^{89} + 2 \beta q^{91} + (\beta + 1) q^{92} + ( - \beta - 4) q^{94} + ( - 2 \beta + 8) q^{97} + (\beta + 1) q^{98} +O(q^{100})$$ q + q^2 + q^4 - b * q^7 + q^8 + (b - 2) * q^11 - 2 * q^13 - b * q^14 + q^16 - q^17 + b * q^19 + (b - 2) * q^22 + (b + 1) * q^23 - 2 * q^26 - b * q^28 + (2*b - 4) * q^29 + (-3*b + 2) * q^31 + q^32 - q^34 + q^37 + b * q^38 + (-b - 1) * q^41 + (-b - 6) * q^43 + (b - 2) * q^44 + (b + 1) * q^46 + (-b - 4) * q^47 + (b + 1) * q^49 - 2 * q^52 + (-2*b + 1) * q^53 - b * q^56 + (2*b - 4) * q^58 + (3*b - 3) * q^59 + (3*b - 1) * q^61 + (-3*b + 2) * q^62 + q^64 + (b + 2) * q^67 - q^68 + (-b - 1) * q^71 - 8 * q^73 + q^74 + b * q^76 + (b - 8) * q^77 + (3*b - 4) * q^79 + (-b - 1) * q^82 + (-b - 1) * q^83 + (-b - 6) * q^86 + (b - 2) * q^88 + (-2*b - 2) * q^89 + 2*b * q^91 + (b + 1) * q^92 + (-b - 4) * q^94 + (-2*b + 8) * q^97 + (b + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - q^{7} + 2 q^{8} - 3 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} + q^{19} - 3 q^{22} + 3 q^{23} - 4 q^{26} - q^{28} - 6 q^{29} + q^{31} + 2 q^{32} - 2 q^{34} + 2 q^{37} + q^{38} - 3 q^{41} - 13 q^{43} - 3 q^{44} + 3 q^{46} - 9 q^{47} + 3 q^{49} - 4 q^{52} - q^{56} - 6 q^{58} - 3 q^{59} + q^{61} + q^{62} + 2 q^{64} + 5 q^{67} - 2 q^{68} - 3 q^{71} - 16 q^{73} + 2 q^{74} + q^{76} - 15 q^{77} - 5 q^{79} - 3 q^{82} - 3 q^{83} - 13 q^{86} - 3 q^{88} - 6 q^{89} + 2 q^{91} + 3 q^{92} - 9 q^{94} + 14 q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - q^7 + 2 * q^8 - 3 * q^11 - 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 + q^19 - 3 * q^22 + 3 * q^23 - 4 * q^26 - q^28 - 6 * q^29 + q^31 + 2 * q^32 - 2 * q^34 + 2 * q^37 + q^38 - 3 * q^41 - 13 * q^43 - 3 * q^44 + 3 * q^46 - 9 * q^47 + 3 * q^49 - 4 * q^52 - q^56 - 6 * q^58 - 3 * q^59 + q^61 + q^62 + 2 * q^64 + 5 * q^67 - 2 * q^68 - 3 * q^71 - 16 * q^73 + 2 * q^74 + q^76 - 15 * q^77 - 5 * q^79 - 3 * q^82 - 3 * q^83 - 13 * q^86 - 3 * q^88 - 6 * q^89 + 2 * q^91 + 3 * q^92 - 9 * q^94 + 14 * q^97 + 3 * q^98

## 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
 3.37228 −2.37228
1.00000 0 1.00000 0 0 −3.37228 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 2.37228 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$17$$ $$1$$

## 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 7650.2.a.df 2
3.b odd 2 1 2550.2.a.bg 2
5.b even 2 1 7650.2.a.cv 2
15.d odd 2 1 2550.2.a.bm yes 2
15.e even 4 2 2550.2.d.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.a.bg 2 3.b odd 2 1
2550.2.a.bm yes 2 15.d odd 2 1
2550.2.d.v 4 15.e even 4 2
7650.2.a.cv 2 5.b even 2 1
7650.2.a.df 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7650))$$:

 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{11}^{2} + 3T_{11} - 6$$ T11^2 + 3*T11 - 6 $$T_{13} + 2$$ T13 + 2 $$T_{19}^{2} - T_{19} - 8$$ T19^2 - T19 - 8 $$T_{23}^{2} - 3T_{23} - 6$$ T23^2 - 3*T23 - 6 $$T_{29}^{2} + 6T_{29} - 24$$ T29^2 + 6*T29 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 8$$
$11$ $$T^{2} + 3T - 6$$
$13$ $$(T + 2)^{2}$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2} - T - 8$$
$23$ $$T^{2} - 3T - 6$$
$29$ $$T^{2} + 6T - 24$$
$31$ $$T^{2} - T - 74$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 3T - 6$$
$43$ $$T^{2} + 13T + 34$$
$47$ $$T^{2} + 9T + 12$$
$53$ $$T^{2} - 33$$
$59$ $$T^{2} + 3T - 72$$
$61$ $$T^{2} - T - 74$$
$67$ $$T^{2} - 5T - 2$$
$71$ $$T^{2} + 3T - 6$$
$73$ $$(T + 8)^{2}$$
$79$ $$T^{2} + 5T - 68$$
$83$ $$T^{2} + 3T - 6$$
$89$ $$T^{2} + 6T - 24$$
$97$ $$T^{2} - 14T + 16$$