# Properties

 Label 8450.2.a.y Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 + 3 * q^3 + q^4 + 3 * q^6 + q^7 + q^8 + 6 * q^9 $$q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{7} + q^{8} + 6 q^{9} + 2 q^{11} + 3 q^{12} + q^{14} + q^{16} + 3 q^{17} + 6 q^{18} - 6 q^{19} + 3 q^{21} + 2 q^{22} + 4 q^{23} + 3 q^{24} + 9 q^{27} + q^{28} + 2 q^{29} - 4 q^{31} + q^{32} + 6 q^{33} + 3 q^{34} + 6 q^{36} + 3 q^{37} - 6 q^{38} + 3 q^{42} + 5 q^{43} + 2 q^{44} + 4 q^{46} + 13 q^{47} + 3 q^{48} - 6 q^{49} + 9 q^{51} - 12 q^{53} + 9 q^{54} + q^{56} - 18 q^{57} + 2 q^{58} + 10 q^{59} - 8 q^{61} - 4 q^{62} + 6 q^{63} + q^{64} + 6 q^{66} - 2 q^{67} + 3 q^{68} + 12 q^{69} + 5 q^{71} + 6 q^{72} - 10 q^{73} + 3 q^{74} - 6 q^{76} + 2 q^{77} - 4 q^{79} + 9 q^{81} + 3 q^{84} + 5 q^{86} + 6 q^{87} + 2 q^{88} - 6 q^{89} + 4 q^{92} - 12 q^{93} + 13 q^{94} + 3 q^{96} + 14 q^{97} - 6 q^{98} + 12 q^{99}+O(q^{100})$$ q + q^2 + 3 * q^3 + q^4 + 3 * q^6 + q^7 + q^8 + 6 * q^9 + 2 * q^11 + 3 * q^12 + q^14 + q^16 + 3 * q^17 + 6 * q^18 - 6 * q^19 + 3 * q^21 + 2 * q^22 + 4 * q^23 + 3 * q^24 + 9 * q^27 + q^28 + 2 * q^29 - 4 * q^31 + q^32 + 6 * q^33 + 3 * q^34 + 6 * q^36 + 3 * q^37 - 6 * q^38 + 3 * q^42 + 5 * q^43 + 2 * q^44 + 4 * q^46 + 13 * q^47 + 3 * q^48 - 6 * q^49 + 9 * q^51 - 12 * q^53 + 9 * q^54 + q^56 - 18 * q^57 + 2 * q^58 + 10 * q^59 - 8 * q^61 - 4 * q^62 + 6 * q^63 + q^64 + 6 * q^66 - 2 * q^67 + 3 * q^68 + 12 * q^69 + 5 * q^71 + 6 * q^72 - 10 * q^73 + 3 * q^74 - 6 * q^76 + 2 * q^77 - 4 * q^79 + 9 * q^81 + 3 * q^84 + 5 * q^86 + 6 * q^87 + 2 * q^88 - 6 * q^89 + 4 * q^92 - 12 * q^93 + 13 * q^94 + 3 * q^96 + 14 * q^97 - 6 * 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 3.00000 1.00000 0 3.00000 1.00000 1.00000 6.00000 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.y 1
5.b even 2 1 338.2.a.a 1
13.b even 2 1 650.2.a.g 1
15.d odd 2 1 3042.2.a.l 1
20.d odd 2 1 2704.2.a.n 1
39.d odd 2 1 5850.2.a.bn 1
52.b odd 2 1 5200.2.a.c 1
65.d even 2 1 26.2.a.b 1
65.g odd 4 2 338.2.b.a 2
65.h odd 4 2 650.2.b.a 2
65.l even 6 2 338.2.c.c 2
65.n even 6 2 338.2.c.g 2
65.s odd 12 4 338.2.e.d 4
195.e odd 2 1 234.2.a.b 1
195.n even 4 2 3042.2.b.f 2
195.s even 4 2 5850.2.e.v 2
260.g odd 2 1 208.2.a.d 1
260.u even 4 2 2704.2.f.j 2
455.h odd 2 1 1274.2.a.o 1
455.bf odd 6 2 1274.2.f.a 2
455.bh even 6 2 1274.2.f.l 2
520.b odd 2 1 832.2.a.a 1
520.p even 2 1 832.2.a.j 1
585.be even 6 2 2106.2.e.h 2
585.bo odd 6 2 2106.2.e.t 2
715.c odd 2 1 3146.2.a.a 1
780.d even 2 1 1872.2.a.m 1
1040.be even 4 2 3328.2.b.g 2
1040.cb odd 4 2 3328.2.b.k 2
1105.h even 2 1 7514.2.a.i 1
1235.e odd 2 1 9386.2.a.f 1
1560.n even 2 1 7488.2.a.v 1
1560.y odd 2 1 7488.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 65.d even 2 1
208.2.a.d 1 260.g odd 2 1
234.2.a.b 1 195.e odd 2 1
338.2.a.a 1 5.b even 2 1
338.2.b.a 2 65.g odd 4 2
338.2.c.c 2 65.l even 6 2
338.2.c.g 2 65.n even 6 2
338.2.e.d 4 65.s odd 12 4
650.2.a.g 1 13.b even 2 1
650.2.b.a 2 65.h odd 4 2
832.2.a.a 1 520.b odd 2 1
832.2.a.j 1 520.p even 2 1
1274.2.a.o 1 455.h odd 2 1
1274.2.f.a 2 455.bf odd 6 2
1274.2.f.l 2 455.bh even 6 2
1872.2.a.m 1 780.d even 2 1
2106.2.e.h 2 585.be even 6 2
2106.2.e.t 2 585.bo odd 6 2
2704.2.a.n 1 20.d odd 2 1
2704.2.f.j 2 260.u even 4 2
3042.2.a.l 1 15.d odd 2 1
3042.2.b.f 2 195.n even 4 2
3146.2.a.a 1 715.c odd 2 1
3328.2.b.g 2 1040.be even 4 2
3328.2.b.k 2 1040.cb odd 4 2
5200.2.a.c 1 52.b odd 2 1
5850.2.a.bn 1 39.d odd 2 1
5850.2.e.v 2 195.s even 4 2
7488.2.a.v 1 1560.n even 2 1
7488.2.a.w 1 1560.y odd 2 1
7514.2.a.i 1 1105.h even 2 1
8450.2.a.y 1 1.a even 1 1 trivial
9386.2.a.f 1 1235.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{7} - 1$$ T7 - 1 $$T_{11} - 2$$ T11 - 2 $$T_{17} - 3$$ T17 - 3 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

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