# Properties

 Label 7650.2.a.cn 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 510) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 4 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 4 * q^7 + q^8 $$q + q^{2} + q^{4} + 4 q^{7} + q^{8} + 4 q^{11} + 2 q^{13} + 4 q^{14} + q^{16} + q^{17} - 4 q^{19} + 4 q^{22} - 4 q^{23} + 2 q^{26} + 4 q^{28} - 2 q^{29} + 4 q^{31} + q^{32} + q^{34} + 6 q^{37} - 4 q^{38} - 2 q^{41} + 12 q^{43} + 4 q^{44} - 4 q^{46} + 8 q^{47} + 9 q^{49} + 2 q^{52} - 2 q^{53} + 4 q^{56} - 2 q^{58} - 12 q^{59} + 2 q^{61} + 4 q^{62} + q^{64} - 4 q^{67} + q^{68} + 4 q^{71} + 14 q^{73} + 6 q^{74} - 4 q^{76} + 16 q^{77} - 12 q^{79} - 2 q^{82} - 4 q^{83} + 12 q^{86} + 4 q^{88} - 10 q^{89} + 8 q^{91} - 4 q^{92} + 8 q^{94} - 18 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 4 * q^7 + q^8 + 4 * q^11 + 2 * q^13 + 4 * q^14 + q^16 + q^17 - 4 * q^19 + 4 * q^22 - 4 * q^23 + 2 * q^26 + 4 * q^28 - 2 * q^29 + 4 * q^31 + q^32 + q^34 + 6 * q^37 - 4 * q^38 - 2 * q^41 + 12 * q^43 + 4 * q^44 - 4 * q^46 + 8 * q^47 + 9 * q^49 + 2 * q^52 - 2 * q^53 + 4 * q^56 - 2 * q^58 - 12 * q^59 + 2 * q^61 + 4 * q^62 + q^64 - 4 * q^67 + q^68 + 4 * q^71 + 14 * q^73 + 6 * q^74 - 4 * q^76 + 16 * q^77 - 12 * q^79 - 2 * q^82 - 4 * q^83 + 12 * q^86 + 4 * q^88 - 10 * q^89 + 8 * q^91 - 4 * q^92 + 8 * q^94 - 18 * q^97 + 9 * 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 4.00000 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.cn 1
3.b odd 2 1 2550.2.a.n 1
5.b even 2 1 1530.2.a.d 1
15.d odd 2 1 510.2.a.c 1
15.e even 4 2 2550.2.d.b 2
60.h even 2 1 4080.2.a.x 1
255.h odd 2 1 8670.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.c 1 15.d odd 2 1
1530.2.a.d 1 5.b even 2 1
2550.2.a.n 1 3.b odd 2 1
2550.2.d.b 2 15.e even 4 2
4080.2.a.x 1 60.h even 2 1
7650.2.a.cn 1 1.a even 1 1 trivial
8670.2.a.bb 1 255.h odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T - 2$$
$17$ $$T - 1$$
$19$ $$T + 4$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 4$$
$37$ $$T - 6$$
$41$ $$T + 2$$
$43$ $$T - 12$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T + 12$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T - 4$$
$73$ $$T - 14$$
$79$ $$T + 12$$
$83$ $$T + 4$$
$89$ $$T + 10$$
$97$ $$T + 18$$