# Properties

 Label 169.4.c.g Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(22,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("169.22");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - 4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} - 12 \beta_{2} + \beta_1 + 12) q^{6} + (11 \beta_{3} - 10 \beta_{2} + \cdots + 10) q^{7}+ \cdots + ( - 15 \beta_{3} + 25 \beta_{2} + \cdots - 25) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-4*b2 + 3*b1) * q^3 + (b3 - 4*b2 + b1 + 4) * q^4 + (-b3 - 2) * q^5 + (b3 - 12*b2 + b1 + 12) * q^6 + (11*b3 - 10*b2 + 11*b1 + 10) * q^7 + (11*b3 + 4) * q^8 + (-15*b3 + 25*b2 - 15*b1 - 25) * q^9 $$q - \beta_1 q^{2} + ( - 4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} - 12 \beta_{2} + \beta_1 + 12) q^{6} + (11 \beta_{3} - 10 \beta_{2} + \cdots + 10) q^{7}+ \cdots + (390 \beta_{3} + 130) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-4*b2 + 3*b1) * q^3 + (b3 - 4*b2 + b1 + 4) * q^4 + (-b3 - 2) * q^5 + (b3 - 12*b2 + b1 + 12) * q^6 + (11*b3 - 10*b2 + 11*b1 + 10) * q^7 + (11*b3 + 4) * q^8 + (-15*b3 + 25*b2 - 15*b1 - 25) * q^9 + (-4*b2 + b1) * q^10 + (-34*b2 - 12*b1) * q^11 + (-13*b3 - 28) * q^12 + (-b3 + 44) * q^14 + (20*b2 - 7*b1) * q^15 + (12*b2 + 15*b1) * q^16 + (-17*b3 + 18*b2 - 17*b1 - 18) * q^17 + (-10*b3 - 60) * q^18 + (-32*b3 - 26*b2 - 32*b1 + 26) * q^19 + (-5*b3 + 12*b2 - 5*b1 - 12) * q^20 + (-41*b3 - 172) * q^21 + (46*b3 + 48*b2 + 46*b1 - 48) * q^22 + (-104*b2 + 12*b1) * q^23 + (-148*b2 + 23*b1) * q^24 + (3*b3 - 117) * q^25 + (9*b3 + 172) * q^27 + (-84*b2 + 43*b1) * q^28 + (70*b2 - 96*b1) * q^29 + (-13*b3 + 28*b2 - 13*b1 - 28) * q^30 + (34*b3 - 26) * q^31 + (61*b3 - 92*b2 + 61*b1 + 92) * q^32 + (-90*b3 - 8*b2 - 90*b1 + 8) * q^33 + (-b3 - 68) * q^34 + (-21*b3 + 64*b2 - 21*b1 - 64) * q^35 + (160*b2 - 70*b1) * q^36 + (-102*b2 - 5*b1) * q^37 + (58*b3 - 128) * q^38 + (-15*b3 - 52) * q^40 + (126*b2 - 22*b1) * q^41 + (-164*b2 + 131*b1) * q^42 + (143*b3 + 72*b2 + 143*b1 - 72) * q^43 + (2*b3 - 88) * q^44 + (40*b3 - 110*b2 + 40*b1 + 110) * q^45 + (92*b3 - 48*b2 + 92*b1 + 48) * q^46 + (121*b3 + 278) * q^47 + (21*b3 + 132*b2 + 21*b1 - 132) * q^48 + (-241*b2 + 99*b1) * q^49 + (12*b2 + 120*b1) * q^50 + (71*b3 + 276) * q^51 + (-30*b3 - 74) * q^53 + (36*b2 - 163*b1) * q^54 + (20*b2 - 22*b1) * q^55 + (33*b3 - 524*b2 + 33*b1 + 524) * q^56 + (-46*b3 + 280) * q^57 + (26*b3 + 384*b2 + 26*b1 - 384) * q^58 + (124*b3 - 246*b2 + 124*b1 + 246) * q^59 + (41*b3 + 108) * q^60 + (-190*b3 - 434*b2 - 190*b1 + 434) * q^61 + (136*b2 + 60*b1) * q^62 + (910*b2 - 260*b1) * q^63 + (-89*b3 + 340) * q^64 + (98*b3 - 360) * q^66 + (-150*b2 + 232*b1) * q^67 + (140*b2 - 69*b1) * q^68 + (-324*b3 + 560*b2 - 324*b1 - 560) * q^69 + (-43*b3 - 84) * q^70 + (-231*b3 + 50*b2 - 231*b1 - 50) * q^71 + (-170*b3 + 760*b2 - 170*b1 - 760) * q^72 + (-260*b3 + 98) * q^73 + (107*b3 + 20*b2 + 107*b1 - 20) * q^74 + (432*b2 - 348*b1) * q^75 + (24*b2 - 70*b1) * q^76 + (-386*b3 + 188) * q^77 + (-40*b3 - 524) * q^79 + (36*b2 - 3*b1) * q^80 + (-121*b2 + 120*b1) * q^81 + (-104*b3 + 88*b2 - 104*b1 - 88) * q^82 + (182*b3 + 1070) * q^83 + (-295*b3 + 852*b2 - 295*b1 - 852) * q^84 + (35*b3 - 104*b2 + 35*b1 + 104) * q^85 + (-215*b3 + 572) * q^86 + (306*b3 - 1432*b2 + 306*b1 + 1432) * q^87 + (392*b2 + 458*b1) * q^88 + (166*b2 + 388*b1) * q^89 + (70*b3 + 160) * q^90 + (-140*b3 - 464) * q^92 + (-304*b2 - 44*b1) * q^93 + (484*b2 - 157*b1) * q^94 + (6*b3 - 76*b2 + 6*b1 + 76) * q^95 + (-337*b3 - 1100) * q^96 + (508*b3 - 718*b2 + 508*b1 + 718) * q^97 + (142*b3 - 396*b2 + 142*b1 + 396) * q^98 + (390*b3 + 130) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 5 q^{3} + 7 q^{4} - 6 q^{5} + 23 q^{6} + 9 q^{7} - 6 q^{8} - 35 q^{9}+O(q^{10})$$ 4 * q - q^2 - 5 * q^3 + 7 * q^4 - 6 * q^5 + 23 * q^6 + 9 * q^7 - 6 * q^8 - 35 * q^9 $$4 q - q^{2} - 5 q^{3} + 7 q^{4} - 6 q^{5} + 23 q^{6} + 9 q^{7} - 6 q^{8} - 35 q^{9} - 7 q^{10} - 80 q^{11} - 86 q^{12} + 178 q^{14} + 33 q^{15} + 39 q^{16} - 19 q^{17} - 220 q^{18} + 84 q^{19} - 19 q^{20} - 606 q^{21} - 142 q^{22} - 196 q^{23} - 273 q^{24} - 474 q^{25} + 670 q^{27} - 125 q^{28} + 44 q^{29} - 43 q^{30} - 172 q^{31} + 123 q^{32} + 106 q^{33} - 270 q^{34} - 107 q^{35} + 250 q^{36} - 209 q^{37} - 628 q^{38} - 178 q^{40} + 230 q^{41} - 197 q^{42} - 287 q^{43} - 356 q^{44} + 180 q^{45} + 4 q^{46} + 870 q^{47} - 285 q^{48} - 383 q^{49} + 144 q^{50} + 962 q^{51} - 236 q^{53} - 91 q^{54} + 18 q^{55} + 1015 q^{56} + 1212 q^{57} - 794 q^{58} + 368 q^{59} + 350 q^{60} + 1058 q^{61} + 332 q^{62} + 1560 q^{63} + 1538 q^{64} - 1636 q^{66} - 68 q^{67} + 211 q^{68} - 796 q^{69} - 250 q^{70} + 131 q^{71} - 1350 q^{72} + 912 q^{73} - 147 q^{74} + 516 q^{75} - 22 q^{76} + 1524 q^{77} - 2016 q^{79} + 69 q^{80} - 122 q^{81} - 72 q^{82} + 3916 q^{83} - 1409 q^{84} + 173 q^{85} + 2718 q^{86} + 2558 q^{87} + 1242 q^{88} + 720 q^{89} + 500 q^{90} - 1576 q^{92} - 652 q^{93} + 811 q^{94} + 146 q^{95} - 3726 q^{96} + 928 q^{97} + 650 q^{98} - 260 q^{99}+O(q^{100})$$ 4 * q - q^2 - 5 * q^3 + 7 * q^4 - 6 * q^5 + 23 * q^6 + 9 * q^7 - 6 * q^8 - 35 * q^9 - 7 * q^10 - 80 * q^11 - 86 * q^12 + 178 * q^14 + 33 * q^15 + 39 * q^16 - 19 * q^17 - 220 * q^18 + 84 * q^19 - 19 * q^20 - 606 * q^21 - 142 * q^22 - 196 * q^23 - 273 * q^24 - 474 * q^25 + 670 * q^27 - 125 * q^28 + 44 * q^29 - 43 * q^30 - 172 * q^31 + 123 * q^32 + 106 * q^33 - 270 * q^34 - 107 * q^35 + 250 * q^36 - 209 * q^37 - 628 * q^38 - 178 * q^40 + 230 * q^41 - 197 * q^42 - 287 * q^43 - 356 * q^44 + 180 * q^45 + 4 * q^46 + 870 * q^47 - 285 * q^48 - 383 * q^49 + 144 * q^50 + 962 * q^51 - 236 * q^53 - 91 * q^54 + 18 * q^55 + 1015 * q^56 + 1212 * q^57 - 794 * q^58 + 368 * q^59 + 350 * q^60 + 1058 * q^61 + 332 * q^62 + 1560 * q^63 + 1538 * q^64 - 1636 * q^66 - 68 * q^67 + 211 * q^68 - 796 * q^69 - 250 * q^70 + 131 * q^71 - 1350 * q^72 + 912 * q^73 - 147 * q^74 + 516 * q^75 - 22 * q^76 + 1524 * q^77 - 2016 * q^79 + 69 * q^80 - 122 * q^81 - 72 * q^82 + 3916 * q^83 - 1409 * q^84 + 173 * q^85 + 2718 * q^86 + 2558 * q^87 + 1242 * q^88 + 720 * q^89 + 500 * q^90 - 1576 * q^92 - 652 * q^93 + 811 * q^94 + 146 * q^95 - 3726 * q^96 + 928 * q^97 + 650 * q^98 - 260 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
−1.28078 + 2.21837i 1.84233 3.19101i 0.719224 + 1.24573i 0.561553 4.71922 + 8.17394i −9.08854 15.7418i −24.1771 6.71165 + 11.6249i −0.719224 + 1.24573i
22.2 0.780776 1.35234i −4.34233 + 7.52113i 2.78078 + 4.81645i −3.56155 6.78078 + 11.7446i 13.5885 + 23.5360i 21.1771 −24.2116 41.9358i −2.78078 + 4.81645i
146.1 −1.28078 2.21837i 1.84233 + 3.19101i 0.719224 1.24573i 0.561553 4.71922 8.17394i −9.08854 + 15.7418i −24.1771 6.71165 11.6249i −0.719224 1.24573i
146.2 0.780776 + 1.35234i −4.34233 7.52113i 2.78078 4.81645i −3.56155 6.78078 11.7446i 13.5885 23.5360i 21.1771 −24.2116 + 41.9358i −2.78078 4.81645i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.g 4
13.b even 2 1 169.4.c.j 4
13.c even 3 1 13.4.a.b 2
13.c even 3 1 inner 169.4.c.g 4
13.d odd 4 2 169.4.e.f 8
13.e even 6 1 169.4.a.g 2
13.e even 6 1 169.4.c.j 4
13.f odd 12 2 169.4.b.f 4
13.f odd 12 2 169.4.e.f 8
39.h odd 6 1 1521.4.a.r 2
39.i odd 6 1 117.4.a.d 2
52.j odd 6 1 208.4.a.h 2
65.n even 6 1 325.4.a.f 2
65.q odd 12 2 325.4.b.e 4
91.n odd 6 1 637.4.a.b 2
104.n odd 6 1 832.4.a.z 2
104.r even 6 1 832.4.a.s 2
143.k odd 6 1 1573.4.a.b 2
156.p even 6 1 1872.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.c even 3 1
117.4.a.d 2 39.i odd 6 1
169.4.a.g 2 13.e even 6 1
169.4.b.f 4 13.f odd 12 2
169.4.c.g 4 1.a even 1 1 trivial
169.4.c.g 4 13.c even 3 1 inner
169.4.c.j 4 13.b even 2 1
169.4.c.j 4 13.e even 6 1
169.4.e.f 8 13.d odd 4 2
169.4.e.f 8 13.f odd 12 2
208.4.a.h 2 52.j odd 6 1
325.4.a.f 2 65.n even 6 1
325.4.b.e 4 65.q odd 12 2
637.4.a.b 2 91.n odd 6 1
832.4.a.s 2 104.r even 6 1
832.4.a.z 2 104.n odd 6 1
1521.4.a.r 2 39.h odd 6 1
1573.4.a.b 2 143.k odd 6 1
1872.4.a.bb 2 156.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + \cdots + 16$$
$3$ $$T^{4} + 5 T^{3} + \cdots + 1024$$
$5$ $$(T^{2} + 3 T - 2)^{2}$$
$7$ $$T^{4} - 9 T^{3} + \cdots + 244036$$
$11$ $$T^{4} + 80 T^{3} + \cdots + 976144$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 19 T^{3} + \cdots + 1295044$$
$19$ $$T^{4} - 84 T^{3} + \cdots + 6697744$$
$23$ $$T^{4} + 196 T^{3} + \cdots + 80856064$$
$29$ $$T^{4} + \cdots + 1496451856$$
$31$ $$(T^{2} + 86 T - 3064)^{2}$$
$37$ $$T^{4} + 209 T^{3} + \cdots + 116942596$$
$41$ $$T^{4} - 230 T^{3} + \cdots + 124724224$$
$43$ $$T^{4} + \cdots + 4397811856$$
$47$ $$(T^{2} - 435 T - 14918)^{2}$$
$53$ $$(T^{2} + 118 T - 344)^{2}$$
$59$ $$T^{4} - 368 T^{3} + \cdots + 991746064$$
$61$ $$T^{4} + \cdots + 15981005056$$
$67$ $$T^{4} + \cdots + 51799939216$$
$71$ $$T^{4} + \cdots + 49503580036$$
$73$ $$(T^{2} - 456 T - 235316)^{2}$$
$79$ $$(T^{2} + 1008 T + 247216)^{2}$$
$83$ $$(T^{2} - 1958 T + 817664)^{2}$$
$89$ $$T^{4} + \cdots + 260316284944$$
$97$ $$T^{4} + \cdots + 776999938576$$