# Properties

 Label 1369.2.a.a Level $1369$ Weight $2$ Character orbit 1369.a Self dual yes Analytic conductor $10.932$ 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] = [1369,2,Mod(1,1369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1369, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1369.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1369 = 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1369.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: $$10.9315200367$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 37) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 3 * q^7 - 2 * q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 3 q^{7} - 2 q^{9} + 4 q^{10} + 3 q^{11} + 2 q^{12} - 6 q^{13} - 6 q^{14} - 2 q^{15} - 4 q^{16} + 2 q^{17} + 4 q^{18} + 6 q^{19} - 4 q^{20} + 3 q^{21} - 6 q^{22} + 4 q^{23} - q^{25} + 12 q^{26} - 5 q^{27} + 6 q^{28} + 4 q^{29} + 4 q^{30} + 8 q^{32} + 3 q^{33} - 4 q^{34} - 6 q^{35} - 4 q^{36} - 12 q^{38} - 6 q^{39} + 3 q^{41} - 6 q^{42} - 6 q^{43} + 6 q^{44} + 4 q^{45} - 8 q^{46} + 3 q^{47} - 4 q^{48} + 2 q^{49} + 2 q^{50} + 2 q^{51} - 12 q^{52} + 9 q^{53} + 10 q^{54} - 6 q^{55} + 6 q^{57} - 8 q^{58} - 4 q^{59} - 4 q^{60} - 6 q^{63} - 8 q^{64} + 12 q^{65} - 6 q^{66} + 12 q^{67} + 4 q^{68} + 4 q^{69} + 12 q^{70} - 3 q^{71} - 9 q^{73} - q^{75} + 12 q^{76} + 9 q^{77} + 12 q^{78} + 6 q^{79} + 8 q^{80} + q^{81} - 6 q^{82} + 9 q^{83} + 6 q^{84} - 4 q^{85} + 12 q^{86} + 4 q^{87} + 14 q^{89} - 8 q^{90} - 18 q^{91} + 8 q^{92} - 6 q^{94} - 12 q^{95} + 8 q^{96} + 12 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 3 * q^7 - 2 * q^9 + 4 * q^10 + 3 * q^11 + 2 * q^12 - 6 * q^13 - 6 * q^14 - 2 * q^15 - 4 * q^16 + 2 * q^17 + 4 * q^18 + 6 * q^19 - 4 * q^20 + 3 * q^21 - 6 * q^22 + 4 * q^23 - q^25 + 12 * q^26 - 5 * q^27 + 6 * q^28 + 4 * q^29 + 4 * q^30 + 8 * q^32 + 3 * q^33 - 4 * q^34 - 6 * q^35 - 4 * q^36 - 12 * q^38 - 6 * q^39 + 3 * q^41 - 6 * q^42 - 6 * q^43 + 6 * q^44 + 4 * q^45 - 8 * q^46 + 3 * q^47 - 4 * q^48 + 2 * q^49 + 2 * q^50 + 2 * q^51 - 12 * q^52 + 9 * q^53 + 10 * q^54 - 6 * q^55 + 6 * q^57 - 8 * q^58 - 4 * q^59 - 4 * q^60 - 6 * q^63 - 8 * q^64 + 12 * q^65 - 6 * q^66 + 12 * q^67 + 4 * q^68 + 4 * q^69 + 12 * q^70 - 3 * q^71 - 9 * q^73 - q^75 + 12 * q^76 + 9 * q^77 + 12 * q^78 + 6 * q^79 + 8 * q^80 + q^81 - 6 * q^82 + 9 * q^83 + 6 * q^84 - 4 * q^85 + 12 * q^86 + 4 * q^87 + 14 * q^89 - 8 * q^90 - 18 * q^91 + 8 * q^92 - 6 * q^94 - 12 * q^95 + 8 * q^96 + 12 * q^97 - 4 * q^98 - 6 * 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
−2.00000 1.00000 2.00000 −2.00000 −2.00000 3.00000 0 −2.00000 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$37$$ $$-1$$

## 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 1369.2.a.a 1
37.b even 2 1 1369.2.a.f 1
37.d odd 4 2 37.2.b.a 2
111.g even 4 2 333.2.c.a 2
148.g even 4 2 592.2.g.b 2
185.f even 4 2 925.2.d.d 2
185.j odd 4 2 925.2.c.b 2
185.k even 4 2 925.2.d.a 2
296.j even 4 2 2368.2.g.b 2
296.m odd 4 2 2368.2.g.f 2
444.j odd 4 2 5328.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.b.a 2 37.d odd 4 2
333.2.c.a 2 111.g even 4 2
592.2.g.b 2 148.g even 4 2
925.2.c.b 2 185.j odd 4 2
925.2.d.a 2 185.k even 4 2
925.2.d.d 2 185.f even 4 2
1369.2.a.a 1 1.a even 1 1 trivial
1369.2.a.f 1 37.b even 2 1
2368.2.g.b 2 296.j even 4 2
2368.2.g.f 2 296.m odd 4 2
5328.2.h.c 2 444.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1369))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T - 3$$
$11$ $$T - 3$$
$13$ $$T + 6$$
$17$ $$T - 2$$
$19$ $$T - 6$$
$23$ $$T - 4$$
$29$ $$T - 4$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 3$$
$43$ $$T + 6$$
$47$ $$T - 3$$
$53$ $$T - 9$$
$59$ $$T + 4$$
$61$ $$T$$
$67$ $$T - 12$$
$71$ $$T + 3$$
$73$ $$T + 9$$
$79$ $$T - 6$$
$83$ $$T - 9$$
$89$ $$T - 14$$
$97$ $$T - 12$$