# Properties

 Label 8450.2.a.f Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ 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] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8450 = 2 \cdot 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8450.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: $$67.4735897080$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 - 4 * q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 3 q^{9} + 4 q^{11} + 4 q^{14} + q^{16} - 3 q^{17} + 3 q^{18} - 4 q^{22} + 4 q^{23} - 4 q^{28} - q^{29} + 4 q^{31} - q^{32} + 3 q^{34} - 3 q^{36} - 3 q^{37} - 9 q^{41} + 8 q^{43} + 4 q^{44} - 4 q^{46} + 8 q^{47} + 9 q^{49} + 9 q^{53} + 4 q^{56} + q^{58} - 4 q^{59} + 7 q^{61} - 4 q^{62} + 12 q^{63} + q^{64} - 4 q^{67} - 3 q^{68} - 8 q^{71} + 3 q^{72} - 11 q^{73} + 3 q^{74} - 16 q^{77} - 4 q^{79} + 9 q^{81} + 9 q^{82} - 8 q^{86} - 4 q^{88} - 6 q^{89} + 4 q^{92} - 8 q^{94} - 2 q^{97} - 9 q^{98} - 12 q^{99}+O(q^{100})$$ q - q^2 + q^4 - 4 * q^7 - q^8 - 3 * q^9 + 4 * q^11 + 4 * q^14 + q^16 - 3 * q^17 + 3 * q^18 - 4 * q^22 + 4 * q^23 - 4 * q^28 - q^29 + 4 * q^31 - q^32 + 3 * q^34 - 3 * q^36 - 3 * q^37 - 9 * q^41 + 8 * q^43 + 4 * q^44 - 4 * q^46 + 8 * q^47 + 9 * q^49 + 9 * q^53 + 4 * q^56 + q^58 - 4 * q^59 + 7 * q^61 - 4 * q^62 + 12 * q^63 + q^64 - 4 * q^67 - 3 * q^68 - 8 * q^71 + 3 * q^72 - 11 * q^73 + 3 * q^74 - 16 * q^77 - 4 * q^79 + 9 * q^81 + 9 * q^82 - 8 * q^86 - 4 * q^88 - 6 * q^89 + 4 * q^92 - 8 * q^94 - 2 * q^97 - 9 * q^98 - 12 * q^99

## 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 −3.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$13$$ $$+1$$

## 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 8450.2.a.f 1
5.b even 2 1 338.2.a.e 1
13.b even 2 1 8450.2.a.s 1
13.c even 3 2 650.2.e.c 2
15.d odd 2 1 3042.2.a.e 1
20.d odd 2 1 2704.2.a.h 1
65.d even 2 1 338.2.a.c 1
65.g odd 4 2 338.2.b.b 2
65.l even 6 2 338.2.c.e 2
65.n even 6 2 26.2.c.a 2
65.q odd 12 4 650.2.o.c 4
65.s odd 12 4 338.2.e.b 4
195.e odd 2 1 3042.2.a.k 1
195.n even 4 2 3042.2.b.e 2
195.x odd 6 2 234.2.h.c 2
260.g odd 2 1 2704.2.a.i 1
260.u even 4 2 2704.2.f.g 2
260.v odd 6 2 208.2.i.b 2
455.y odd 6 2 1274.2.e.m 2
455.ba even 6 2 1274.2.e.n 2
455.bm even 6 2 1274.2.h.b 2
455.bp odd 6 2 1274.2.g.a 2
455.bw odd 6 2 1274.2.h.a 2
520.bv even 6 2 832.2.i.e 2
520.bx odd 6 2 832.2.i.f 2
780.br even 6 2 1872.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a 2 65.n even 6 2
208.2.i.b 2 260.v odd 6 2
234.2.h.c 2 195.x odd 6 2
338.2.a.c 1 65.d even 2 1
338.2.a.e 1 5.b even 2 1
338.2.b.b 2 65.g odd 4 2
338.2.c.e 2 65.l even 6 2
338.2.e.b 4 65.s odd 12 4
650.2.e.c 2 13.c even 3 2
650.2.o.c 4 65.q odd 12 4
832.2.i.e 2 520.bv even 6 2
832.2.i.f 2 520.bx odd 6 2
1274.2.e.m 2 455.y odd 6 2
1274.2.e.n 2 455.ba even 6 2
1274.2.g.a 2 455.bp odd 6 2
1274.2.h.a 2 455.bw odd 6 2
1274.2.h.b 2 455.bm even 6 2
1872.2.t.k 2 780.br even 6 2
2704.2.a.h 1 20.d odd 2 1
2704.2.a.i 1 260.g odd 2 1
2704.2.f.g 2 260.u even 4 2
3042.2.a.e 1 15.d odd 2 1
3042.2.a.k 1 195.e odd 2 1
3042.2.b.e 2 195.n even 4 2
8450.2.a.f 1 1.a even 1 1 trivial
8450.2.a.s 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}$$ T3 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 4$$ T11 - 4 $$T_{17} + 3$$ T17 + 3 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 4$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T$$
$23$ $$T - 4$$
$29$ $$T + 1$$
$31$ $$T - 4$$
$37$ $$T + 3$$
$41$ $$T + 9$$
$43$ $$T - 8$$
$47$ $$T - 8$$
$53$ $$T - 9$$
$59$ $$T + 4$$
$61$ $$T - 7$$
$67$ $$T + 4$$
$71$ $$T + 8$$
$73$ $$T + 11$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 2$$