# Properties

 Label 169.4.a.i Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 q^{3} - 5 q^{4} + \beta q^{5} + 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} +O(q^{10})$$ q + b * q^2 + 2 * q^3 - 5 * q^4 + b * q^5 + 2*b * q^6 - 8*b * q^7 - 13*b * q^8 - 23 * q^9 $$q + \beta q^{2} + 2 q^{3} - 5 q^{4} + \beta q^{5} + 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} + 3 q^{10} - 8 \beta q^{11} - 10 q^{12} - 24 q^{14} + 2 \beta q^{15} + q^{16} - 117 q^{17} - 23 \beta q^{18} + 66 \beta q^{19} - 5 \beta q^{20} - 16 \beta q^{21} - 24 q^{22} + 78 q^{23} - 26 \beta q^{24} - 122 q^{25} - 100 q^{27} + 40 \beta q^{28} - 141 q^{29} + 6 q^{30} - 90 \beta q^{31} + 105 \beta q^{32} - 16 \beta q^{33} - 117 \beta q^{34} - 24 q^{35} + 115 q^{36} + 83 \beta q^{37} + 198 q^{38} - 39 q^{40} + 157 \beta q^{41} - 48 q^{42} - 104 q^{43} + 40 \beta q^{44} - 23 \beta q^{45} + 78 \beta q^{46} + 174 \beta q^{47} + 2 q^{48} - 151 q^{49} - 122 \beta q^{50} - 234 q^{51} + 93 q^{53} - 100 \beta q^{54} - 24 q^{55} + 312 q^{56} + 132 \beta q^{57} - 141 \beta q^{58} - 164 \beta q^{59} - 10 \beta q^{60} + 145 q^{61} - 270 q^{62} + 184 \beta q^{63} + 307 q^{64} - 48 q^{66} - 454 \beta q^{67} + 585 q^{68} + 156 q^{69} - 24 \beta q^{70} - 610 \beta q^{71} + 299 \beta q^{72} + 265 \beta q^{73} + 249 q^{74} - 244 q^{75} - 330 \beta q^{76} + 192 q^{77} + 1276 q^{79} + \beta q^{80} + 421 q^{81} + 471 q^{82} + 456 \beta q^{83} + 80 \beta q^{84} - 117 \beta q^{85} - 104 \beta q^{86} - 282 q^{87} + 312 q^{88} + 564 \beta q^{89} - 69 q^{90} - 390 q^{92} - 180 \beta q^{93} + 522 q^{94} + 198 q^{95} + 210 \beta q^{96} - 116 \beta q^{97} - 151 \beta q^{98} + 184 \beta q^{99} +O(q^{100})$$ q + b * q^2 + 2 * q^3 - 5 * q^4 + b * q^5 + 2*b * q^6 - 8*b * q^7 - 13*b * q^8 - 23 * q^9 + 3 * q^10 - 8*b * q^11 - 10 * q^12 - 24 * q^14 + 2*b * q^15 + q^16 - 117 * q^17 - 23*b * q^18 + 66*b * q^19 - 5*b * q^20 - 16*b * q^21 - 24 * q^22 + 78 * q^23 - 26*b * q^24 - 122 * q^25 - 100 * q^27 + 40*b * q^28 - 141 * q^29 + 6 * q^30 - 90*b * q^31 + 105*b * q^32 - 16*b * q^33 - 117*b * q^34 - 24 * q^35 + 115 * q^36 + 83*b * q^37 + 198 * q^38 - 39 * q^40 + 157*b * q^41 - 48 * q^42 - 104 * q^43 + 40*b * q^44 - 23*b * q^45 + 78*b * q^46 + 174*b * q^47 + 2 * q^48 - 151 * q^49 - 122*b * q^50 - 234 * q^51 + 93 * q^53 - 100*b * q^54 - 24 * q^55 + 312 * q^56 + 132*b * q^57 - 141*b * q^58 - 164*b * q^59 - 10*b * q^60 + 145 * q^61 - 270 * q^62 + 184*b * q^63 + 307 * q^64 - 48 * q^66 - 454*b * q^67 + 585 * q^68 + 156 * q^69 - 24*b * q^70 - 610*b * q^71 + 299*b * q^72 + 265*b * q^73 + 249 * q^74 - 244 * q^75 - 330*b * q^76 + 192 * q^77 + 1276 * q^79 + b * q^80 + 421 * q^81 + 471 * q^82 + 456*b * q^83 + 80*b * q^84 - 117*b * q^85 - 104*b * q^86 - 282 * q^87 + 312 * q^88 + 564*b * q^89 - 69 * q^90 - 390 * q^92 - 180*b * q^93 + 522 * q^94 + 198 * q^95 + 210*b * q^96 - 116*b * q^97 - 151*b * q^98 + 184*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 10 q^{4} - 46 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9 $$2 q + 4 q^{3} - 10 q^{4} - 46 q^{9} + 6 q^{10} - 20 q^{12} - 48 q^{14} + 2 q^{16} - 234 q^{17} - 48 q^{22} + 156 q^{23} - 244 q^{25} - 200 q^{27} - 282 q^{29} + 12 q^{30} - 48 q^{35} + 230 q^{36} + 396 q^{38} - 78 q^{40} - 96 q^{42} - 208 q^{43} + 4 q^{48} - 302 q^{49} - 468 q^{51} + 186 q^{53} - 48 q^{55} + 624 q^{56} + 290 q^{61} - 540 q^{62} + 614 q^{64} - 96 q^{66} + 1170 q^{68} + 312 q^{69} + 498 q^{74} - 488 q^{75} + 384 q^{77} + 2552 q^{79} + 842 q^{81} + 942 q^{82} - 564 q^{87} + 624 q^{88} - 138 q^{90} - 780 q^{92} + 1044 q^{94} + 396 q^{95}+O(q^{100})$$ 2 * q + 4 * q^3 - 10 * q^4 - 46 * q^9 + 6 * q^10 - 20 * q^12 - 48 * q^14 + 2 * q^16 - 234 * q^17 - 48 * q^22 + 156 * q^23 - 244 * q^25 - 200 * q^27 - 282 * q^29 + 12 * q^30 - 48 * q^35 + 230 * q^36 + 396 * q^38 - 78 * q^40 - 96 * q^42 - 208 * q^43 + 4 * q^48 - 302 * q^49 - 468 * q^51 + 186 * q^53 - 48 * q^55 + 624 * q^56 + 290 * q^61 - 540 * q^62 + 614 * q^64 - 96 * q^66 + 1170 * q^68 + 312 * q^69 + 498 * q^74 - 488 * q^75 + 384 * q^77 + 2552 * q^79 + 842 * q^81 + 942 * q^82 - 564 * q^87 + 624 * q^88 - 138 * q^90 - 780 * q^92 + 1044 * q^94 + 396 * q^95

## 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
 −1.73205 1.73205
−1.73205 2.00000 −5.00000 −1.73205 −3.46410 13.8564 22.5167 −23.0000 3.00000
1.2 1.73205 2.00000 −5.00000 1.73205 3.46410 −13.8564 −22.5167 −23.0000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.a.i 2
3.b odd 2 1 1521.4.a.o 2
13.b even 2 1 inner 169.4.a.i 2
13.c even 3 2 169.4.c.h 4
13.d odd 4 2 169.4.b.d 2
13.e even 6 2 169.4.c.h 4
13.f odd 12 2 13.4.e.b 2
13.f odd 12 2 169.4.e.a 2
39.d odd 2 1 1521.4.a.o 2
39.k even 12 2 117.4.q.a 2
52.l even 12 2 208.4.w.b 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T - 2)^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$T^{2} - 192$$
$11$ $$T^{2} - 192$$
$13$ $$T^{2}$$
$17$ $$(T + 117)^{2}$$
$19$ $$T^{2} - 13068$$
$23$ $$(T - 78)^{2}$$
$29$ $$(T + 141)^{2}$$
$31$ $$T^{2} - 24300$$
$37$ $$T^{2} - 20667$$
$41$ $$T^{2} - 73947$$
$43$ $$(T + 104)^{2}$$
$47$ $$T^{2} - 90828$$
$53$ $$(T - 93)^{2}$$
$59$ $$T^{2} - 80688$$
$61$ $$(T - 145)^{2}$$
$67$ $$T^{2} - 618348$$
$71$ $$T^{2} - 1116300$$
$73$ $$T^{2} - 210675$$
$79$ $$(T - 1276)^{2}$$
$83$ $$T^{2} - 623808$$
$89$ $$T^{2} - 954288$$
$97$ $$T^{2} - 40368$$