# Properties

 Label 169.4.a.a Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ 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] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 17 q^{5} - 8 q^{6} - 20 q^{7} - 23 q^{9}+O(q^{10})$$ q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 17 * q^5 - 8 * q^6 - 20 * q^7 - 23 * q^9 $$q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 17 q^{5} - 8 q^{6} - 20 q^{7} - 23 q^{9} + 68 q^{10} + 32 q^{11} + 16 q^{12} + 80 q^{14} - 34 q^{15} - 64 q^{16} - 13 q^{17} + 92 q^{18} - 30 q^{19} - 136 q^{20} - 40 q^{21} - 128 q^{22} + 78 q^{23} + 164 q^{25} - 100 q^{27} - 160 q^{28} + 197 q^{29} + 136 q^{30} + 74 q^{31} + 256 q^{32} + 64 q^{33} + 52 q^{34} + 340 q^{35} - 184 q^{36} + 227 q^{37} + 120 q^{38} + 165 q^{41} + 160 q^{42} - 156 q^{43} + 256 q^{44} + 391 q^{45} - 312 q^{46} + 162 q^{47} - 128 q^{48} + 57 q^{49} - 656 q^{50} - 26 q^{51} + 93 q^{53} + 400 q^{54} - 544 q^{55} - 60 q^{57} - 788 q^{58} + 864 q^{59} - 272 q^{60} + 145 q^{61} - 296 q^{62} + 460 q^{63} - 512 q^{64} - 256 q^{66} - 862 q^{67} - 104 q^{68} + 156 q^{69} - 1360 q^{70} - 654 q^{71} - 215 q^{73} - 908 q^{74} + 328 q^{75} - 240 q^{76} - 640 q^{77} - 76 q^{79} + 1088 q^{80} + 421 q^{81} - 660 q^{82} - 628 q^{83} - 320 q^{84} + 221 q^{85} + 624 q^{86} + 394 q^{87} + 266 q^{89} - 1564 q^{90} + 624 q^{92} + 148 q^{93} - 648 q^{94} + 510 q^{95} + 512 q^{96} - 238 q^{97} - 228 q^{98} - 736 q^{99}+O(q^{100})$$ q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 17 * q^5 - 8 * q^6 - 20 * q^7 - 23 * q^9 + 68 * q^10 + 32 * q^11 + 16 * q^12 + 80 * q^14 - 34 * q^15 - 64 * q^16 - 13 * q^17 + 92 * q^18 - 30 * q^19 - 136 * q^20 - 40 * q^21 - 128 * q^22 + 78 * q^23 + 164 * q^25 - 100 * q^27 - 160 * q^28 + 197 * q^29 + 136 * q^30 + 74 * q^31 + 256 * q^32 + 64 * q^33 + 52 * q^34 + 340 * q^35 - 184 * q^36 + 227 * q^37 + 120 * q^38 + 165 * q^41 + 160 * q^42 - 156 * q^43 + 256 * q^44 + 391 * q^45 - 312 * q^46 + 162 * q^47 - 128 * q^48 + 57 * q^49 - 656 * q^50 - 26 * q^51 + 93 * q^53 + 400 * q^54 - 544 * q^55 - 60 * q^57 - 788 * q^58 + 864 * q^59 - 272 * q^60 + 145 * q^61 - 296 * q^62 + 460 * q^63 - 512 * q^64 - 256 * q^66 - 862 * q^67 - 104 * q^68 + 156 * q^69 - 1360 * q^70 - 654 * q^71 - 215 * q^73 - 908 * q^74 + 328 * q^75 - 240 * q^76 - 640 * q^77 - 76 * q^79 + 1088 * q^80 + 421 * q^81 - 660 * q^82 - 628 * q^83 - 320 * q^84 + 221 * q^85 + 624 * q^86 + 394 * q^87 + 266 * q^89 - 1564 * q^90 + 624 * q^92 + 148 * q^93 - 648 * q^94 + 510 * q^95 + 512 * q^96 - 238 * q^97 - 228 * q^98 - 736 * 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 2.00000 8.00000 −17.0000 −8.00000 −20.0000 0 −23.0000 68.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$1$$

## 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 169.4.a.a 1
3.b odd 2 1 1521.4.a.k 1
13.b even 2 1 169.4.a.d 1
13.c even 3 2 169.4.c.d 2
13.d odd 4 2 169.4.b.c 2
13.e even 6 2 13.4.c.a 2
13.f odd 12 4 169.4.e.c 4
39.d odd 2 1 1521.4.a.b 1
39.h odd 6 2 117.4.g.c 2
52.i odd 6 2 208.4.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 13.e even 6 2
117.4.g.c 2 39.h odd 6 2
169.4.a.a 1 1.a even 1 1 trivial
169.4.a.d 1 13.b even 2 1
169.4.b.c 2 13.d odd 4 2
169.4.c.d 2 13.c even 3 2
169.4.e.c 4 13.f odd 12 4
208.4.i.b 2 52.i odd 6 2
1521.4.a.b 1 39.d odd 2 1
1521.4.a.k 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T - 2$$
$5$ $$T + 17$$
$7$ $$T + 20$$
$11$ $$T - 32$$
$13$ $$T$$
$17$ $$T + 13$$
$19$ $$T + 30$$
$23$ $$T - 78$$
$29$ $$T - 197$$
$31$ $$T - 74$$
$37$ $$T - 227$$
$41$ $$T - 165$$
$43$ $$T + 156$$
$47$ $$T - 162$$
$53$ $$T - 93$$
$59$ $$T - 864$$
$61$ $$T - 145$$
$67$ $$T + 862$$
$71$ $$T + 654$$
$73$ $$T + 215$$
$79$ $$T + 76$$
$83$ $$T + 628$$
$89$ $$T - 266$$
$97$ $$T + 238$$