# Properties

 Label 6897.2.a.f Level $6897$ Weight $2$ Character orbit 6897.a Self dual yes Analytic conductor $55.073$ 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] = [6897,2,Mod(1,6897)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6897, 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("6897.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6897 = 3 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6897.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: $$55.0728222741$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 5 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^2 - q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 5 * q^7 + q^9 $$q + 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 5 q^{7} + q^{9} - 6 q^{10} - 2 q^{12} - 2 q^{13} + 10 q^{14} + 3 q^{15} - 4 q^{16} + q^{17} + 2 q^{18} + q^{19} - 6 q^{20} - 5 q^{21} - 4 q^{23} + 4 q^{25} - 4 q^{26} - q^{27} + 10 q^{28} + 2 q^{29} + 6 q^{30} - 6 q^{31} - 8 q^{32} + 2 q^{34} - 15 q^{35} + 2 q^{36} + 2 q^{38} + 2 q^{39} - 10 q^{42} + q^{43} - 3 q^{45} - 8 q^{46} - 9 q^{47} + 4 q^{48} + 18 q^{49} + 8 q^{50} - q^{51} - 4 q^{52} + 10 q^{53} - 2 q^{54} - q^{57} + 4 q^{58} - 8 q^{59} + 6 q^{60} + q^{61} - 12 q^{62} + 5 q^{63} - 8 q^{64} + 6 q^{65} + 8 q^{67} + 2 q^{68} + 4 q^{69} - 30 q^{70} - 12 q^{71} + 11 q^{73} - 4 q^{75} + 2 q^{76} + 4 q^{78} - 16 q^{79} + 12 q^{80} + q^{81} - 12 q^{83} - 10 q^{84} - 3 q^{85} + 2 q^{86} - 2 q^{87} - 6 q^{89} - 6 q^{90} - 10 q^{91} - 8 q^{92} + 6 q^{93} - 18 q^{94} - 3 q^{95} + 8 q^{96} - 10 q^{97} + 36 q^{98}+O(q^{100})$$ q + 2 * q^2 - q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 5 * q^7 + q^9 - 6 * q^10 - 2 * q^12 - 2 * q^13 + 10 * q^14 + 3 * q^15 - 4 * q^16 + q^17 + 2 * q^18 + q^19 - 6 * q^20 - 5 * q^21 - 4 * q^23 + 4 * q^25 - 4 * q^26 - q^27 + 10 * q^28 + 2 * q^29 + 6 * q^30 - 6 * q^31 - 8 * q^32 + 2 * q^34 - 15 * q^35 + 2 * q^36 + 2 * q^38 + 2 * q^39 - 10 * q^42 + q^43 - 3 * q^45 - 8 * q^46 - 9 * q^47 + 4 * q^48 + 18 * q^49 + 8 * q^50 - q^51 - 4 * q^52 + 10 * q^53 - 2 * q^54 - q^57 + 4 * q^58 - 8 * q^59 + 6 * q^60 + q^61 - 12 * q^62 + 5 * q^63 - 8 * q^64 + 6 * q^65 + 8 * q^67 + 2 * q^68 + 4 * q^69 - 30 * q^70 - 12 * q^71 + 11 * q^73 - 4 * q^75 + 2 * q^76 + 4 * q^78 - 16 * q^79 + 12 * q^80 + q^81 - 12 * q^83 - 10 * q^84 - 3 * q^85 + 2 * q^86 - 2 * q^87 - 6 * q^89 - 6 * q^90 - 10 * q^91 - 8 * q^92 + 6 * q^93 - 18 * q^94 - 3 * q^95 + 8 * q^96 - 10 * q^97 + 36 * 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
2.00000 −1.00000 2.00000 −3.00000 −2.00000 5.00000 0 1.00000 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## 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 6897.2.a.f 1
11.b odd 2 1 57.2.a.a 1
33.d even 2 1 171.2.a.d 1
44.c even 2 1 912.2.a.g 1
55.d odd 2 1 1425.2.a.j 1
55.e even 4 2 1425.2.c.b 2
77.b even 2 1 2793.2.a.b 1
88.b odd 2 1 3648.2.a.bh 1
88.g even 2 1 3648.2.a.r 1
132.d odd 2 1 2736.2.a.v 1
143.d odd 2 1 9633.2.a.o 1
165.d even 2 1 4275.2.a.b 1
209.d even 2 1 1083.2.a.e 1
231.h odd 2 1 8379.2.a.p 1
627.b odd 2 1 3249.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 11.b odd 2 1
171.2.a.d 1 33.d even 2 1
912.2.a.g 1 44.c even 2 1
1083.2.a.e 1 209.d even 2 1
1425.2.a.j 1 55.d odd 2 1
1425.2.c.b 2 55.e even 4 2
2736.2.a.v 1 132.d odd 2 1
2793.2.a.b 1 77.b even 2 1
3249.2.a.b 1 627.b odd 2 1
3648.2.a.r 1 88.g even 2 1
3648.2.a.bh 1 88.b odd 2 1
4275.2.a.b 1 165.d even 2 1
6897.2.a.f 1 1.a even 1 1 trivial
8379.2.a.p 1 231.h odd 2 1
9633.2.a.o 1 143.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6897))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{5} + 3$$ T5 + 3 $$T_{7} - 5$$ T7 - 5 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 1$$
$5$ $$T + 3$$
$7$ $$T - 5$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 1$$
$19$ $$T - 1$$
$23$ $$T + 4$$
$29$ $$T - 2$$
$31$ $$T + 6$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 9$$
$53$ $$T - 10$$
$59$ $$T + 8$$
$61$ $$T - 1$$
$67$ $$T - 8$$
$71$ $$T + 12$$
$73$ $$T - 11$$
$79$ $$T + 16$$
$83$ $$T + 12$$
$89$ $$T + 6$$
$97$ $$T + 10$$