# Properties

 Label 169.4.a.g Level $169$ Weight $4$ Character orbit 169.a Self dual yes Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10})$$ q - b * q^2 + (-3*b + 4) * q^3 + (b - 4) * q^4 + (-b + 2) * q^5 + (-b + 12) * q^6 + (-11*b + 10) * q^7 + (11*b - 4) * q^8 + (-15*b + 25) * q^9 $$q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} + ( - \beta + 4) q^{10} + ( - 12 \beta - 34) q^{11} + (13 \beta - 28) q^{12} + (\beta + 44) q^{14} + ( - 7 \beta + 20) q^{15} + ( - 15 \beta - 12) q^{16} + ( - 17 \beta + 18) q^{17} + ( - 10 \beta + 60) q^{18} + (32 \beta + 26) q^{19} + (5 \beta - 12) q^{20} + ( - 41 \beta + 172) q^{21} + (46 \beta + 48) q^{22} + ( - 12 \beta + 104) q^{23} + (23 \beta - 148) q^{24} + ( - 3 \beta - 117) q^{25} + ( - 9 \beta + 172) q^{27} + (43 \beta - 84) q^{28} + (96 \beta - 70) q^{29} + ( - 13 \beta + 28) q^{30} + (34 \beta + 26) q^{31} + ( - 61 \beta + 92) q^{32} + (90 \beta + 8) q^{33} + ( - \beta + 68) q^{34} + ( - 21 \beta + 64) q^{35} + (70 \beta - 160) q^{36} + ( - 5 \beta - 102) q^{37} + ( - 58 \beta - 128) q^{38} + (15 \beta - 52) q^{40} + ( - 22 \beta + 126) q^{41} + ( - 131 \beta + 164) q^{42} + (143 \beta + 72) q^{43} + (2 \beta + 88) q^{44} + ( - 40 \beta + 110) q^{45} + ( - 92 \beta + 48) q^{46} + (121 \beta - 278) q^{47} + (21 \beta + 132) q^{48} + ( - 99 \beta + 241) q^{49} + (120 \beta + 12) q^{50} + ( - 71 \beta + 276) q^{51} + (30 \beta - 74) q^{53} + ( - 163 \beta + 36) q^{54} + (22 \beta - 20) q^{55} + (33 \beta - 524) q^{56} + ( - 46 \beta - 280) q^{57} + ( - 26 \beta - 384) q^{58} + ( - 124 \beta + 246) q^{59} + (41 \beta - 108) q^{60} + ( - 190 \beta - 434) q^{61} + ( - 60 \beta - 136) q^{62} + ( - 260 \beta + 910) q^{63} + (89 \beta + 340) q^{64} + ( - 98 \beta - 360) q^{66} + (232 \beta - 150) q^{67} + (69 \beta - 140) q^{68} + ( - 324 \beta + 560) q^{69} + ( - 43 \beta + 84) q^{70} + (231 \beta - 50) q^{71} + (170 \beta - 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + (348 \beta - 432) q^{75} + ( - 70 \beta + 24) q^{76} + (386 \beta + 188) q^{77} + (40 \beta - 524) q^{79} + ( - 3 \beta + 36) q^{80} + ( - 120 \beta + 121) q^{81} + ( - 104 \beta + 88) q^{82} + (182 \beta - 1070) q^{83} + (295 \beta - 852) q^{84} + ( - 35 \beta + 104) q^{85} + ( - 215 \beta - 572) q^{86} + (306 \beta - 1432) q^{87} + ( - 458 \beta - 392) q^{88} + (388 \beta + 166) q^{89} + ( - 70 \beta + 160) q^{90} + (140 \beta - 464) q^{92} + ( - 44 \beta - 304) q^{93} + (157 \beta - 484) q^{94} + (6 \beta - 76) q^{95} + ( - 337 \beta + 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 142 \beta + 396) q^{98} + (390 \beta - 130) q^{99} +O(q^{100})$$ q - b * q^2 + (-3*b + 4) * q^3 + (b - 4) * q^4 + (-b + 2) * q^5 + (-b + 12) * q^6 + (-11*b + 10) * q^7 + (11*b - 4) * q^8 + (-15*b + 25) * q^9 + (-b + 4) * q^10 + (-12*b - 34) * q^11 + (13*b - 28) * q^12 + (b + 44) * q^14 + (-7*b + 20) * q^15 + (-15*b - 12) * q^16 + (-17*b + 18) * q^17 + (-10*b + 60) * q^18 + (32*b + 26) * q^19 + (5*b - 12) * q^20 + (-41*b + 172) * q^21 + (46*b + 48) * q^22 + (-12*b + 104) * q^23 + (23*b - 148) * q^24 + (-3*b - 117) * q^25 + (-9*b + 172) * q^27 + (43*b - 84) * q^28 + (96*b - 70) * q^29 + (-13*b + 28) * q^30 + (34*b + 26) * q^31 + (-61*b + 92) * q^32 + (90*b + 8) * q^33 + (-b + 68) * q^34 + (-21*b + 64) * q^35 + (70*b - 160) * q^36 + (-5*b - 102) * q^37 + (-58*b - 128) * q^38 + (15*b - 52) * q^40 + (-22*b + 126) * q^41 + (-131*b + 164) * q^42 + (143*b + 72) * q^43 + (2*b + 88) * q^44 + (-40*b + 110) * q^45 + (-92*b + 48) * q^46 + (121*b - 278) * q^47 + (21*b + 132) * q^48 + (-99*b + 241) * q^49 + (120*b + 12) * q^50 + (-71*b + 276) * q^51 + (30*b - 74) * q^53 + (-163*b + 36) * q^54 + (22*b - 20) * q^55 + (33*b - 524) * q^56 + (-46*b - 280) * q^57 + (-26*b - 384) * q^58 + (-124*b + 246) * q^59 + (41*b - 108) * q^60 + (-190*b - 434) * q^61 + (-60*b - 136) * q^62 + (-260*b + 910) * q^63 + (89*b + 340) * q^64 + (-98*b - 360) * q^66 + (232*b - 150) * q^67 + (69*b - 140) * q^68 + (-324*b + 560) * q^69 + (-43*b + 84) * q^70 + (231*b - 50) * q^71 + (170*b - 760) * q^72 + (-260*b - 98) * q^73 + (107*b + 20) * q^74 + (348*b - 432) * q^75 + (-70*b + 24) * q^76 + (386*b + 188) * q^77 + (40*b - 524) * q^79 + (-3*b + 36) * q^80 + (-120*b + 121) * q^81 + (-104*b + 88) * q^82 + (182*b - 1070) * q^83 + (295*b - 852) * q^84 + (-35*b + 104) * q^85 + (-215*b - 572) * q^86 + (306*b - 1432) * q^87 + (-458*b - 392) * q^88 + (388*b + 166) * q^89 + (-70*b + 160) * q^90 + (140*b - 464) * q^92 + (-44*b - 304) * q^93 + (157*b - 484) * q^94 + (6*b - 76) * q^95 + (-337*b + 1100) * q^96 + (-508*b + 718) * q^97 + (-142*b + 396) * q^98 + (390*b - 130) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 $$2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99}+O(q^{100})$$ 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 + 7 * q^10 - 80 * q^11 - 43 * q^12 + 89 * q^14 + 33 * q^15 - 39 * q^16 + 19 * q^17 + 110 * q^18 + 84 * q^19 - 19 * q^20 + 303 * q^21 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 335 * q^27 - 125 * q^28 - 44 * q^29 + 43 * q^30 + 86 * q^31 + 123 * q^32 + 106 * q^33 + 135 * q^34 + 107 * q^35 - 250 * q^36 - 209 * q^37 - 314 * q^38 - 89 * q^40 + 230 * q^41 + 197 * q^42 + 287 * q^43 + 178 * q^44 + 180 * q^45 + 4 * q^46 - 435 * q^47 + 285 * q^48 + 383 * q^49 + 144 * q^50 + 481 * q^51 - 118 * q^53 - 91 * q^54 - 18 * q^55 - 1015 * q^56 - 606 * q^57 - 794 * q^58 + 368 * q^59 - 175 * q^60 - 1058 * q^61 - 332 * q^62 + 1560 * q^63 + 769 * q^64 - 818 * q^66 - 68 * q^67 - 211 * q^68 + 796 * q^69 + 125 * q^70 + 131 * q^71 - 1350 * q^72 - 456 * q^73 + 147 * q^74 - 516 * q^75 - 22 * q^76 + 762 * q^77 - 1008 * q^79 + 69 * q^80 + 122 * q^81 + 72 * q^82 - 1958 * q^83 - 1409 * q^84 + 173 * q^85 - 1359 * q^86 - 2558 * q^87 - 1242 * q^88 + 720 * q^89 + 250 * q^90 - 788 * q^92 - 652 * q^93 - 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 + 650 * q^98 + 130 * 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
 2.56155 −1.56155
−2.56155 −3.68466 −1.43845 −0.561553 9.43845 −18.1771 24.1771 −13.4233 1.43845
1.2 1.56155 8.68466 −5.56155 3.56155 13.5616 27.1771 −21.1771 48.4233 5.56155
 $$n$$: 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.g 2
3.b odd 2 1 1521.4.a.r 2
13.b even 2 1 13.4.a.b 2
13.c even 3 2 169.4.c.j 4
13.d odd 4 2 169.4.b.f 4
13.e even 6 2 169.4.c.g 4
13.f odd 12 4 169.4.e.f 8
39.d odd 2 1 117.4.a.d 2
52.b odd 2 1 208.4.a.h 2
65.d even 2 1 325.4.a.f 2
65.h odd 4 2 325.4.b.e 4
91.b odd 2 1 637.4.a.b 2
104.e even 2 1 832.4.a.s 2
104.h odd 2 1 832.4.a.z 2
143.d odd 2 1 1573.4.a.b 2
156.h even 2 1 1872.4.a.bb 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$T^{2} - 5T - 32$$
$5$ $$T^{2} - 3T - 2$$
$7$ $$T^{2} - 9T - 494$$
$11$ $$T^{2} + 80T + 988$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 19T - 1138$$
$19$ $$T^{2} - 84T - 2588$$
$23$ $$T^{2} - 196T + 8992$$
$29$ $$T^{2} + 44T - 38684$$
$31$ $$T^{2} - 86T - 3064$$
$37$ $$T^{2} + 209T + 10814$$
$41$ $$T^{2} - 230T + 11168$$
$43$ $$T^{2} - 287T - 66316$$
$47$ $$T^{2} + 435T - 14918$$
$53$ $$T^{2} + 118T - 344$$
$59$ $$T^{2} - 368T - 31492$$
$61$ $$T^{2} + 1058 T + 126416$$
$67$ $$T^{2} + 68T - 227596$$
$71$ $$T^{2} - 131T - 222494$$
$73$ $$T^{2} + 456T - 235316$$
$79$ $$T^{2} + 1008 T + 247216$$
$83$ $$T^{2} + 1958 T + 817664$$
$89$ $$T^{2} - 720T - 510212$$
$97$ $$T^{2} - 928T - 881476$$