# Properties

 Label 7650.2.a.z Level $7650$ Weight $2$ Character orbit 7650.a Self dual yes Analytic conductor $61.086$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 2 * q^7 - q^8 $$q - q^{2} + q^{4} + 2 q^{7} - q^{8} + 6 q^{13} - 2 q^{14} + q^{16} - q^{17} + 4 q^{19} + 6 q^{23} - 6 q^{26} + 2 q^{28} + 4 q^{29} - 6 q^{31} - q^{32} + q^{34} + 4 q^{37} - 4 q^{38} + 10 q^{41} + 4 q^{43} - 6 q^{46} + 4 q^{47} - 3 q^{49} + 6 q^{52} - 2 q^{53} - 2 q^{56} - 4 q^{58} - 12 q^{59} - 4 q^{61} + 6 q^{62} + q^{64} + 12 q^{67} - q^{68} + 6 q^{71} - 2 q^{73} - 4 q^{74} + 4 q^{76} + 10 q^{79} - 10 q^{82} - 12 q^{83} - 4 q^{86} + 2 q^{89} + 12 q^{91} + 6 q^{92} - 4 q^{94} - 6 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^4 + 2 * q^7 - q^8 + 6 * q^13 - 2 * q^14 + q^16 - q^17 + 4 * q^19 + 6 * q^23 - 6 * q^26 + 2 * q^28 + 4 * q^29 - 6 * q^31 - q^32 + q^34 + 4 * q^37 - 4 * q^38 + 10 * q^41 + 4 * q^43 - 6 * q^46 + 4 * q^47 - 3 * q^49 + 6 * q^52 - 2 * q^53 - 2 * q^56 - 4 * q^58 - 12 * q^59 - 4 * q^61 + 6 * q^62 + q^64 + 12 * q^67 - q^68 + 6 * q^71 - 2 * q^73 - 4 * q^74 + 4 * q^76 + 10 * q^79 - 10 * q^82 - 12 * q^83 - 4 * q^86 + 2 * q^89 + 12 * q^91 + 6 * q^92 - 4 * q^94 - 6 * 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
 0
−1.00000 0 1.00000 0 0 2.00000 −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.z 1
3.b odd 2 1 2550.2.a.be 1
5.b even 2 1 306.2.a.d 1
15.d odd 2 1 102.2.a.a 1
15.e even 4 2 2550.2.d.q 2
20.d odd 2 1 2448.2.a.t 1
40.e odd 2 1 9792.2.a.b 1
40.f even 2 1 9792.2.a.a 1
60.h even 2 1 816.2.a.h 1
85.c even 2 1 5202.2.a.g 1
105.g even 2 1 4998.2.a.x 1
120.i odd 2 1 3264.2.a.bf 1
120.m even 2 1 3264.2.a.p 1
255.h odd 2 1 1734.2.a.h 1
255.i odd 4 2 1734.2.b.d 2
255.y odd 8 4 1734.2.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.a 1 15.d odd 2 1
306.2.a.d 1 5.b even 2 1
816.2.a.h 1 60.h even 2 1
1734.2.a.h 1 255.h odd 2 1
1734.2.b.d 2 255.i odd 4 2
1734.2.f.g 4 255.y odd 8 4
2448.2.a.t 1 20.d odd 2 1
2550.2.a.be 1 3.b odd 2 1
2550.2.d.q 2 15.e even 4 2
3264.2.a.p 1 120.m even 2 1
3264.2.a.bf 1 120.i odd 2 1
4998.2.a.x 1 105.g even 2 1
5202.2.a.g 1 85.c even 2 1
7650.2.a.z 1 1.a even 1 1 trivial
9792.2.a.a 1 40.f even 2 1
9792.2.a.b 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7650))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{19} - 4$$ T19 - 4 $$T_{23} - 6$$ T23 - 6 $$T_{29} - 4$$ T29 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 1$$
$19$ $$T - 4$$
$23$ $$T - 6$$
$29$ $$T - 4$$
$31$ $$T + 6$$
$37$ $$T - 4$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T - 4$$
$53$ $$T + 2$$
$59$ $$T + 12$$
$61$ $$T + 4$$
$67$ $$T - 12$$
$71$ $$T - 6$$
$73$ $$T + 2$$
$79$ $$T - 10$$
$83$ $$T + 12$$
$89$ $$T - 2$$
$97$ $$T + 6$$