Properties

 Label 169.4.a.j 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{2} + ( - 3 \beta + 4) q^{3} + ( - 5 \beta + 5) q^{4} + (5 \beta + 5) q^{5} + ( - 10 \beta + 24) q^{6} + (\beta - 8) q^{7} + ( - 7 \beta + 11) q^{8} + ( - 15 \beta + 25) q^{9}+O(q^{10})$$ q + (-b + 3) * q^2 + (-3*b + 4) * q^3 + (-5*b + 5) * q^4 + (5*b + 5) * q^5 + (-10*b + 24) * q^6 + (b - 8) * q^7 + (-7*b + 11) * q^8 + (-15*b + 25) * q^9 $$q + ( - \beta + 3) q^{2} + ( - 3 \beta + 4) q^{3} + ( - 5 \beta + 5) q^{4} + (5 \beta + 5) q^{5} + ( - 10 \beta + 24) q^{6} + (\beta - 8) q^{7} + ( - 7 \beta + 11) q^{8} + ( - 15 \beta + 25) q^{9} + (5 \beta - 5) q^{10} + (15 \beta - 16) q^{11} + ( - 20 \beta + 80) q^{12} + (10 \beta - 28) q^{14} + ( - 10 \beta - 40) q^{15} + (15 \beta + 21) q^{16} + (16 \beta + 27) q^{17} + ( - 55 \beta + 135) q^{18} + (5 \beta + 68) q^{19} + ( - 25 \beta - 75) q^{20} + (25 \beta - 44) q^{21} + (46 \beta - 108) q^{22} + (33 \beta + 56) q^{23} + ( - 40 \beta + 128) q^{24} + 75 \beta q^{25} + ( - 9 \beta + 172) q^{27} + (40 \beta - 60) q^{28} + ( - 60 \beta + 47) q^{29} + (20 \beta - 80) q^{30} + (100 \beta + 20) q^{31} + (65 \beta - 85) q^{32} + (63 \beta - 244) q^{33} + (5 \beta + 17) q^{34} + ( - 30 \beta - 20) q^{35} + ( - 125 \beta + 425) q^{36} + ( - 44 \beta + 117) q^{37} + ( - 58 \beta + 184) q^{38} + ( - 15 \beta - 85) q^{40} + (20 \beta - 279) q^{41} + (94 \beta - 232) q^{42} + ( - 97 \beta + 276) q^{43} + (80 \beta - 380) q^{44} + ( - 25 \beta - 175) q^{45} + (10 \beta + 36) q^{46} + ( - 140 \beta + 40) q^{47} + ( - 48 \beta - 96) q^{48} + ( - 15 \beta - 275) q^{49} + (150 \beta - 300) q^{50} + ( - 65 \beta - 84) q^{51} + ( - 165 \beta + 355) q^{53} + ( - 190 \beta + 552) q^{54} + (70 \beta + 220) q^{55} + (60 \beta - 116) q^{56} + ( - 199 \beta + 212) q^{57} + ( - 167 \beta + 381) q^{58} + ( - 55 \beta + 432) q^{59} + 200 \beta q^{60} + (200 \beta + 151) q^{61} + (180 \beta - 340) q^{62} + (130 \beta - 260) q^{63} + (95 \beta - 683) q^{64} + (370 \beta - 984) q^{66} + (91 \beta + 192) q^{67} + ( - 135 \beta - 185) q^{68} + ( - 135 \beta - 172) q^{69} + ( - 40 \beta + 60) q^{70} + (105 \beta - 116) q^{71} + ( - 235 \beta + 695) q^{72} + (85 \beta - 335) q^{73} + ( - 205 \beta + 527) q^{74} + (75 \beta - 900) q^{75} + ( - 340 \beta + 240) q^{76} + ( - 121 \beta + 188) q^{77} + (40 \beta + 100) q^{79} + (255 \beta + 405) q^{80} + ( - 120 \beta + 121) q^{81} + (319 \beta - 917) q^{82} + ( - 100 \beta - 80) q^{83} + (220 \beta - 720) q^{84} + (295 \beta + 455) q^{85} + ( - 470 \beta + 1216) q^{86} + ( - 201 \beta + 908) q^{87} + (172 \beta - 596) q^{88} + ( - 125 \beta - 398) q^{89} + (125 \beta - 425) q^{90} + ( - 280 \beta - 380) q^{92} + (40 \beta - 1120) q^{93} + ( - 320 \beta + 680) q^{94} + (390 \beta + 440) q^{95} + (320 \beta - 1120) q^{96} + ( - 469 \beta + 442) q^{97} + (245 \beta - 765) q^{98} + (390 \beta - 1300) q^{99}+O(q^{100})$$ q + (-b + 3) * q^2 + (-3*b + 4) * q^3 + (-5*b + 5) * q^4 + (5*b + 5) * q^5 + (-10*b + 24) * q^6 + (b - 8) * q^7 + (-7*b + 11) * q^8 + (-15*b + 25) * q^9 + (5*b - 5) * q^10 + (15*b - 16) * q^11 + (-20*b + 80) * q^12 + (10*b - 28) * q^14 + (-10*b - 40) * q^15 + (15*b + 21) * q^16 + (16*b + 27) * q^17 + (-55*b + 135) * q^18 + (5*b + 68) * q^19 + (-25*b - 75) * q^20 + (25*b - 44) * q^21 + (46*b - 108) * q^22 + (33*b + 56) * q^23 + (-40*b + 128) * q^24 + 75*b * q^25 + (-9*b + 172) * q^27 + (40*b - 60) * q^28 + (-60*b + 47) * q^29 + (20*b - 80) * q^30 + (100*b + 20) * q^31 + (65*b - 85) * q^32 + (63*b - 244) * q^33 + (5*b + 17) * q^34 + (-30*b - 20) * q^35 + (-125*b + 425) * q^36 + (-44*b + 117) * q^37 + (-58*b + 184) * q^38 + (-15*b - 85) * q^40 + (20*b - 279) * q^41 + (94*b - 232) * q^42 + (-97*b + 276) * q^43 + (80*b - 380) * q^44 + (-25*b - 175) * q^45 + (10*b + 36) * q^46 + (-140*b + 40) * q^47 + (-48*b - 96) * q^48 + (-15*b - 275) * q^49 + (150*b - 300) * q^50 + (-65*b - 84) * q^51 + (-165*b + 355) * q^53 + (-190*b + 552) * q^54 + (70*b + 220) * q^55 + (60*b - 116) * q^56 + (-199*b + 212) * q^57 + (-167*b + 381) * q^58 + (-55*b + 432) * q^59 + 200*b * q^60 + (200*b + 151) * q^61 + (180*b - 340) * q^62 + (130*b - 260) * q^63 + (95*b - 683) * q^64 + (370*b - 984) * q^66 + (91*b + 192) * q^67 + (-135*b - 185) * q^68 + (-135*b - 172) * q^69 + (-40*b + 60) * q^70 + (105*b - 116) * q^71 + (-235*b + 695) * q^72 + (85*b - 335) * q^73 + (-205*b + 527) * q^74 + (75*b - 900) * q^75 + (-340*b + 240) * q^76 + (-121*b + 188) * q^77 + (40*b + 100) * q^79 + (255*b + 405) * q^80 + (-120*b + 121) * q^81 + (319*b - 917) * q^82 + (-100*b - 80) * q^83 + (220*b - 720) * q^84 + (295*b + 455) * q^85 + (-470*b + 1216) * q^86 + (-201*b + 908) * q^87 + (172*b - 596) * q^88 + (-125*b - 398) * q^89 + (125*b - 425) * q^90 + (-280*b - 380) * q^92 + (40*b - 1120) * q^93 + (-320*b + 680) * q^94 + (390*b + 440) * q^95 + (320*b - 1120) * q^96 + (-469*b + 442) * q^97 + (245*b - 765) * q^98 + (390*b - 1300) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 $$2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9} - 5 q^{10} - 17 q^{11} + 140 q^{12} - 46 q^{14} - 90 q^{15} + 57 q^{16} + 70 q^{17} + 215 q^{18} + 141 q^{19} - 175 q^{20} - 63 q^{21} - 170 q^{22} + 145 q^{23} + 216 q^{24} + 75 q^{25} + 335 q^{27} - 80 q^{28} + 34 q^{29} - 140 q^{30} + 140 q^{31} - 105 q^{32} - 425 q^{33} + 39 q^{34} - 70 q^{35} + 725 q^{36} + 190 q^{37} + 310 q^{38} - 185 q^{40} - 538 q^{41} - 370 q^{42} + 455 q^{43} - 680 q^{44} - 375 q^{45} + 82 q^{46} - 60 q^{47} - 240 q^{48} - 565 q^{49} - 450 q^{50} - 233 q^{51} + 545 q^{53} + 914 q^{54} + 510 q^{55} - 172 q^{56} + 225 q^{57} + 595 q^{58} + 809 q^{59} + 200 q^{60} + 502 q^{61} - 500 q^{62} - 390 q^{63} - 1271 q^{64} - 1598 q^{66} + 475 q^{67} - 505 q^{68} - 479 q^{69} + 80 q^{70} - 127 q^{71} + 1155 q^{72} - 585 q^{73} + 849 q^{74} - 1725 q^{75} + 140 q^{76} + 255 q^{77} + 240 q^{79} + 1065 q^{80} + 122 q^{81} - 1515 q^{82} - 260 q^{83} - 1220 q^{84} + 1205 q^{85} + 1962 q^{86} + 1615 q^{87} - 1020 q^{88} - 921 q^{89} - 725 q^{90} - 1040 q^{92} - 2200 q^{93} + 1040 q^{94} + 1270 q^{95} - 1920 q^{96} + 415 q^{97} - 1285 q^{98} - 2210 q^{99}+O(q^{100})$$ 2 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 + 15 * q^5 + 38 * q^6 - 15 * q^7 + 15 * q^8 + 35 * q^9 - 5 * q^10 - 17 * q^11 + 140 * q^12 - 46 * q^14 - 90 * q^15 + 57 * q^16 + 70 * q^17 + 215 * q^18 + 141 * q^19 - 175 * q^20 - 63 * q^21 - 170 * q^22 + 145 * q^23 + 216 * q^24 + 75 * q^25 + 335 * q^27 - 80 * q^28 + 34 * q^29 - 140 * q^30 + 140 * q^31 - 105 * q^32 - 425 * q^33 + 39 * q^34 - 70 * q^35 + 725 * q^36 + 190 * q^37 + 310 * q^38 - 185 * q^40 - 538 * q^41 - 370 * q^42 + 455 * q^43 - 680 * q^44 - 375 * q^45 + 82 * q^46 - 60 * q^47 - 240 * q^48 - 565 * q^49 - 450 * q^50 - 233 * q^51 + 545 * q^53 + 914 * q^54 + 510 * q^55 - 172 * q^56 + 225 * q^57 + 595 * q^58 + 809 * q^59 + 200 * q^60 + 502 * q^61 - 500 * q^62 - 390 * q^63 - 1271 * q^64 - 1598 * q^66 + 475 * q^67 - 505 * q^68 - 479 * q^69 + 80 * q^70 - 127 * q^71 + 1155 * q^72 - 585 * q^73 + 849 * q^74 - 1725 * q^75 + 140 * q^76 + 255 * q^77 + 240 * q^79 + 1065 * q^80 + 122 * q^81 - 1515 * q^82 - 260 * q^83 - 1220 * q^84 + 1205 * q^85 + 1962 * q^86 + 1615 * q^87 - 1020 * q^88 - 921 * q^89 - 725 * q^90 - 1040 * q^92 - 2200 * q^93 + 1040 * q^94 + 1270 * q^95 - 1920 * q^96 + 415 * q^97 - 1285 * q^98 - 2210 * 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
0.438447 −3.68466 −7.80776 17.8078 −1.61553 −5.43845 −6.93087 −13.4233 7.80776
1.2 4.56155 8.68466 12.8078 −2.80776 39.6155 −9.56155 21.9309 48.4233 −12.8078
 $$n$$: 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.j 2
3.b odd 2 1 1521.4.a.l 2
13.b even 2 1 169.4.a.f 2
13.c even 3 2 169.4.c.f 4
13.d odd 4 2 169.4.b.e 4
13.e even 6 2 13.4.c.b 4
13.f odd 12 4 169.4.e.g 8
39.d odd 2 1 1521.4.a.t 2
39.h odd 6 2 117.4.g.d 4
52.i odd 6 2 208.4.i.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 13.e even 6 2
117.4.g.d 4 39.h odd 6 2
169.4.a.f 2 13.b even 2 1
169.4.a.j 2 1.a even 1 1 trivial
169.4.b.e 4 13.d odd 4 2
169.4.c.f 4 13.c even 3 2
169.4.e.g 8 13.f odd 12 4
208.4.i.e 4 52.i odd 6 2
1521.4.a.l 2 3.b odd 2 1
1521.4.a.t 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(169))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5T + 2$$
$3$ $$T^{2} - 5T - 32$$
$5$ $$T^{2} - 15T - 50$$
$7$ $$T^{2} + 15T + 52$$
$11$ $$T^{2} + 17T - 884$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 70T + 137$$
$19$ $$T^{2} - 141T + 4864$$
$23$ $$T^{2} - 145T + 628$$
$29$ $$T^{2} - 34T - 15011$$
$31$ $$T^{2} - 140T - 37600$$
$37$ $$T^{2} - 190T + 797$$
$41$ $$T^{2} + 538T + 70661$$
$43$ $$T^{2} - 455T + 11768$$
$47$ $$T^{2} + 60T - 82400$$
$53$ $$T^{2} - 545T - 41450$$
$59$ $$T^{2} - 809T + 150764$$
$61$ $$T^{2} - 502T - 106999$$
$67$ $$T^{2} - 475T + 21212$$
$71$ $$T^{2} + 127T - 42824$$
$73$ $$T^{2} + 585T + 54850$$
$79$ $$T^{2} - 240T + 7600$$
$83$ $$T^{2} + 260T - 25600$$
$89$ $$T^{2} + 921T + 145654$$
$97$ $$T^{2} - 415T - 891778$$