# Properties

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

 $$f(q)$$ $$=$$ $$q - 3 q^{2} - q^{3} + q^{4} + 9 q^{5} + 3 q^{6} - 15 q^{7} + 21 q^{8} - 26 q^{9}+O(q^{10})$$ q - 3 * q^2 - q^3 + q^4 + 9 * q^5 + 3 * q^6 - 15 * q^7 + 21 * q^8 - 26 * q^9 $$q - 3 q^{2} - q^{3} + q^{4} + 9 q^{5} + 3 q^{6} - 15 q^{7} + 21 q^{8} - 26 q^{9} - 27 q^{10} + 48 q^{11} - q^{12} + 45 q^{14} - 9 q^{15} - 71 q^{16} + 45 q^{17} + 78 q^{18} - 6 q^{19} + 9 q^{20} + 15 q^{21} - 144 q^{22} - 162 q^{23} - 21 q^{24} - 44 q^{25} + 53 q^{27} - 15 q^{28} - 144 q^{29} + 27 q^{30} - 264 q^{31} + 45 q^{32} - 48 q^{33} - 135 q^{34} - 135 q^{35} - 26 q^{36} - 303 q^{37} + 18 q^{38} + 189 q^{40} + 192 q^{41} - 45 q^{42} + 97 q^{43} + 48 q^{44} - 234 q^{45} + 486 q^{46} - 111 q^{47} + 71 q^{48} - 118 q^{49} + 132 q^{50} - 45 q^{51} - 414 q^{53} - 159 q^{54} + 432 q^{55} - 315 q^{56} + 6 q^{57} + 432 q^{58} - 522 q^{59} - 9 q^{60} + 376 q^{61} + 792 q^{62} + 390 q^{63} + 433 q^{64} + 144 q^{66} + 36 q^{67} + 45 q^{68} + 162 q^{69} + 405 q^{70} - 357 q^{71} - 546 q^{72} + 1098 q^{73} + 909 q^{74} + 44 q^{75} - 6 q^{76} - 720 q^{77} - 830 q^{79} - 639 q^{80} + 649 q^{81} - 576 q^{82} + 438 q^{83} + 15 q^{84} + 405 q^{85} - 291 q^{86} + 144 q^{87} + 1008 q^{88} + 438 q^{89} + 702 q^{90} - 162 q^{92} + 264 q^{93} + 333 q^{94} - 54 q^{95} - 45 q^{96} + 852 q^{97} + 354 q^{98} - 1248 q^{99}+O(q^{100})$$ q - 3 * q^2 - q^3 + q^4 + 9 * q^5 + 3 * q^6 - 15 * q^7 + 21 * q^8 - 26 * q^9 - 27 * q^10 + 48 * q^11 - q^12 + 45 * q^14 - 9 * q^15 - 71 * q^16 + 45 * q^17 + 78 * q^18 - 6 * q^19 + 9 * q^20 + 15 * q^21 - 144 * q^22 - 162 * q^23 - 21 * q^24 - 44 * q^25 + 53 * q^27 - 15 * q^28 - 144 * q^29 + 27 * q^30 - 264 * q^31 + 45 * q^32 - 48 * q^33 - 135 * q^34 - 135 * q^35 - 26 * q^36 - 303 * q^37 + 18 * q^38 + 189 * q^40 + 192 * q^41 - 45 * q^42 + 97 * q^43 + 48 * q^44 - 234 * q^45 + 486 * q^46 - 111 * q^47 + 71 * q^48 - 118 * q^49 + 132 * q^50 - 45 * q^51 - 414 * q^53 - 159 * q^54 + 432 * q^55 - 315 * q^56 + 6 * q^57 + 432 * q^58 - 522 * q^59 - 9 * q^60 + 376 * q^61 + 792 * q^62 + 390 * q^63 + 433 * q^64 + 144 * q^66 + 36 * q^67 + 45 * q^68 + 162 * q^69 + 405 * q^70 - 357 * q^71 - 546 * q^72 + 1098 * q^73 + 909 * q^74 + 44 * q^75 - 6 * q^76 - 720 * q^77 - 830 * q^79 - 639 * q^80 + 649 * q^81 - 576 * q^82 + 438 * q^83 + 15 * q^84 + 405 * q^85 - 291 * q^86 + 144 * q^87 + 1008 * q^88 + 438 * q^89 + 702 * q^90 - 162 * q^92 + 264 * q^93 + 333 * q^94 - 54 * q^95 - 45 * q^96 + 852 * q^97 + 354 * q^98 - 1248 * 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
−3.00000 −1.00000 1.00000 9.00000 3.00000 −15.0000 21.0000 −26.0000 −27.0000
 $$n$$: 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.b 1
3.b odd 2 1 1521.4.a.i 1
13.b even 2 1 169.4.a.c 1
13.c even 3 2 169.4.c.c 2
13.d odd 4 2 13.4.b.a 2
13.e even 6 2 169.4.c.b 2
13.f odd 12 4 169.4.e.d 4
39.d odd 2 1 1521.4.a.d 1
39.f even 4 2 117.4.b.a 2
52.f even 4 2 208.4.f.b 2
65.f even 4 2 325.4.d.b 2
65.g odd 4 2 325.4.c.b 2
65.k even 4 2 325.4.d.a 2
104.j odd 4 2 832.4.f.e 2
104.m even 4 2 832.4.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.d odd 4 2
117.4.b.a 2 39.f even 4 2
169.4.a.b 1 1.a even 1 1 trivial
169.4.a.c 1 13.b even 2 1
169.4.c.b 2 13.e even 6 2
169.4.c.c 2 13.c even 3 2
169.4.e.d 4 13.f odd 12 4
208.4.f.b 2 52.f even 4 2
325.4.c.b 2 65.g odd 4 2
325.4.d.a 2 65.k even 4 2
325.4.d.b 2 65.f even 4 2
832.4.f.c 2 104.m even 4 2
832.4.f.e 2 104.j odd 4 2
1521.4.a.d 1 39.d odd 2 1
1521.4.a.i 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T + 1$$
$5$ $$T - 9$$
$7$ $$T + 15$$
$11$ $$T - 48$$
$13$ $$T$$
$17$ $$T - 45$$
$19$ $$T + 6$$
$23$ $$T + 162$$
$29$ $$T + 144$$
$31$ $$T + 264$$
$37$ $$T + 303$$
$41$ $$T - 192$$
$43$ $$T - 97$$
$47$ $$T + 111$$
$53$ $$T + 414$$
$59$ $$T + 522$$
$61$ $$T - 376$$
$67$ $$T - 36$$
$71$ $$T + 357$$
$73$ $$T - 1098$$
$79$ $$T + 830$$
$83$ $$T - 438$$
$89$ $$T - 438$$
$97$ $$T - 852$$