# Properties

 Label 650.6.a.a Level $650$ Weight $6$ Character orbit 650.a Self dual yes Analytic conductor $104.249$ 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] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 650.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: $$104.249482878$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + 170 q^{7} + 64 q^{8} - 243 q^{9}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + 170 * q^7 + 64 * q^8 - 243 * q^9 $$q + 4 q^{2} + 16 q^{4} + 170 q^{7} + 64 q^{8} - 243 q^{9} - 250 q^{11} + 169 q^{13} + 680 q^{14} + 256 q^{16} - 1062 q^{17} - 972 q^{18} - 78 q^{19} - 1000 q^{22} - 1576 q^{23} + 676 q^{26} + 2720 q^{28} + 2578 q^{29} - 8654 q^{31} + 1024 q^{32} - 4248 q^{34} - 3888 q^{36} - 10986 q^{37} - 312 q^{38} + 1050 q^{41} + 5900 q^{43} - 4000 q^{44} - 6304 q^{46} + 5962 q^{47} + 12093 q^{49} + 2704 q^{52} - 29046 q^{53} + 10880 q^{56} + 10312 q^{58} - 13922 q^{59} - 32882 q^{61} - 34616 q^{62} - 41310 q^{63} + 4096 q^{64} + 69566 q^{67} - 16992 q^{68} - 50542 q^{71} - 15552 q^{72} + 46750 q^{73} - 43944 q^{74} - 1248 q^{76} - 42500 q^{77} - 19348 q^{79} + 59049 q^{81} + 4200 q^{82} + 87438 q^{83} + 23600 q^{86} - 16000 q^{88} + 94170 q^{89} + 28730 q^{91} - 25216 q^{92} + 23848 q^{94} - 182786 q^{97} + 48372 q^{98} + 60750 q^{99}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + 170 * q^7 + 64 * q^8 - 243 * q^9 - 250 * q^11 + 169 * q^13 + 680 * q^14 + 256 * q^16 - 1062 * q^17 - 972 * q^18 - 78 * q^19 - 1000 * q^22 - 1576 * q^23 + 676 * q^26 + 2720 * q^28 + 2578 * q^29 - 8654 * q^31 + 1024 * q^32 - 4248 * q^34 - 3888 * q^36 - 10986 * q^37 - 312 * q^38 + 1050 * q^41 + 5900 * q^43 - 4000 * q^44 - 6304 * q^46 + 5962 * q^47 + 12093 * q^49 + 2704 * q^52 - 29046 * q^53 + 10880 * q^56 + 10312 * q^58 - 13922 * q^59 - 32882 * q^61 - 34616 * q^62 - 41310 * q^63 + 4096 * q^64 + 69566 * q^67 - 16992 * q^68 - 50542 * q^71 - 15552 * q^72 + 46750 * q^73 - 43944 * q^74 - 1248 * q^76 - 42500 * q^77 - 19348 * q^79 + 59049 * q^81 + 4200 * q^82 + 87438 * q^83 + 23600 * q^86 - 16000 * q^88 + 94170 * q^89 + 28730 * q^91 - 25216 * q^92 + 23848 * q^94 - 182786 * q^97 + 48372 * q^98 + 60750 * 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
4.00000 0 16.0000 0 0 170.000 64.0000 −243.000 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 650.6.a.a 1
5.b even 2 1 26.6.a.a 1
5.c odd 4 2 650.6.b.a 2
15.d odd 2 1 234.6.a.g 1
20.d odd 2 1 208.6.a.b 1
40.e odd 2 1 832.6.a.e 1
40.f even 2 1 832.6.a.d 1
65.d even 2 1 338.6.a.d 1
65.g odd 4 2 338.6.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.a 1 5.b even 2 1
208.6.a.b 1 20.d odd 2 1
234.6.a.g 1 15.d odd 2 1
338.6.a.d 1 65.d even 2 1
338.6.b.a 2 65.g odd 4 2
650.6.a.a 1 1.a even 1 1 trivial
650.6.b.a 2 5.c odd 4 2
832.6.a.d 1 40.f even 2 1
832.6.a.e 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(650))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 170$$
$11$ $$T + 250$$
$13$ $$T - 169$$
$17$ $$T + 1062$$
$19$ $$T + 78$$
$23$ $$T + 1576$$
$29$ $$T - 2578$$
$31$ $$T + 8654$$
$37$ $$T + 10986$$
$41$ $$T - 1050$$
$43$ $$T - 5900$$
$47$ $$T - 5962$$
$53$ $$T + 29046$$
$59$ $$T + 13922$$
$61$ $$T + 32882$$
$67$ $$T - 69566$$
$71$ $$T + 50542$$
$73$ $$T - 46750$$
$79$ $$T + 19348$$
$83$ $$T - 87438$$
$89$ $$T - 94170$$
$97$ $$T + 182786$$