# Properties

 Label 7650.2.a.bt Level $7650$ Weight $2$ Character orbit 7650.a Self dual yes Analytic conductor $61.086$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^7 + q^8 $$q + q^{2} + q^{4} - q^{7} + q^{8} + 3 q^{11} + 2 q^{13} - q^{14} + q^{16} + q^{17} - 7 q^{19} + 3 q^{22} - 6 q^{23} + 2 q^{26} - q^{28} - 6 q^{29} - 7 q^{31} + q^{32} + q^{34} - 7 q^{37} - 7 q^{38} + 6 q^{41} - q^{43} + 3 q^{44} - 6 q^{46} - 9 q^{47} - 6 q^{49} + 2 q^{52} + 3 q^{53} - q^{56} - 6 q^{58} - 10 q^{61} - 7 q^{62} + q^{64} + 5 q^{67} + q^{68} + 6 q^{71} - 16 q^{73} - 7 q^{74} - 7 q^{76} - 3 q^{77} + 17 q^{79} + 6 q^{82} + 6 q^{83} - q^{86} + 3 q^{88} + 12 q^{89} - 2 q^{91} - 6 q^{92} - 9 q^{94} - 10 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^7 + q^8 + 3 * q^11 + 2 * q^13 - q^14 + q^16 + q^17 - 7 * q^19 + 3 * q^22 - 6 * q^23 + 2 * q^26 - q^28 - 6 * q^29 - 7 * q^31 + q^32 + q^34 - 7 * q^37 - 7 * q^38 + 6 * q^41 - q^43 + 3 * q^44 - 6 * q^46 - 9 * q^47 - 6 * q^49 + 2 * q^52 + 3 * q^53 - q^56 - 6 * q^58 - 10 * q^61 - 7 * q^62 + q^64 + 5 * q^67 + q^68 + 6 * q^71 - 16 * q^73 - 7 * q^74 - 7 * q^76 - 3 * q^77 + 17 * q^79 + 6 * q^82 + 6 * q^83 - q^86 + 3 * q^88 + 12 * q^89 - 2 * q^91 - 6 * q^92 - 9 * q^94 - 10 * q^97 - 6 * 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
 0
1.00000 0 1.00000 0 0 −1.00000 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.bt 1
3.b odd 2 1 2550.2.a.j 1
5.b even 2 1 7650.2.a.v 1
15.d odd 2 1 2550.2.a.w yes 1
15.e even 4 2 2550.2.d.c 2

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

 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 3$$ T11 - 3 $$T_{13} - 2$$ T13 - 2 $$T_{19} + 7$$ T19 + 7 $$T_{23} + 6$$ T23 + 6 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T - 2$$
$17$ $$T - 1$$
$19$ $$T + 7$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T + 7$$
$37$ $$T + 7$$
$41$ $$T - 6$$
$43$ $$T + 1$$
$47$ $$T + 9$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T - 5$$
$71$ $$T - 6$$
$73$ $$T + 16$$
$79$ $$T - 17$$
$83$ $$T - 6$$
$89$ $$T - 12$$
$97$ $$T + 10$$