# Properties

 Label 169.4.b.e Level $169$ Weight $4$ Character orbit 169.b 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(168,169)]

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([1]))

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

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 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.

 $$f(q)$$ $$=$$ $$q + (3 \beta_{2} + \beta_1) q^{2} + (3 \beta_{3} + 1) q^{3} - 5 \beta_{3} q^{4} + (5 \beta_{2} - 5 \beta_1) q^{5} + (24 \beta_{2} + 10 \beta_1) q^{6} + (8 \beta_{2} + \beta_1) q^{7} + ( - 11 \beta_{2} - 7 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9}+O(q^{10})$$ q + (3*b2 + b1) * q^2 + (3*b3 + 1) * q^3 - 5*b3 * q^4 + (5*b2 - 5*b1) * q^5 + (24*b2 + 10*b1) * q^6 + (8*b2 + b1) * q^7 + (-11*b2 - 7*b1) * q^8 + (15*b3 + 10) * q^9 $$q + (3 \beta_{2} + \beta_1) q^{2} + (3 \beta_{3} + 1) q^{3} - 5 \beta_{3} q^{4} + (5 \beta_{2} - 5 \beta_1) q^{5} + (24 \beta_{2} + 10 \beta_1) q^{6} + (8 \beta_{2} + \beta_1) q^{7} + ( - 11 \beta_{2} - 7 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9} + 5 \beta_{3} q^{10} + (16 \beta_{2} + 15 \beta_1) q^{11} + ( - 20 \beta_{3} - 60) q^{12} + ( - 10 \beta_{3} - 18) q^{14} + ( - 40 \beta_{2} + 10 \beta_1) q^{15} + ( - 15 \beta_{3} + 36) q^{16} + (16 \beta_{3} - 43) q^{17} + (135 \beta_{2} + 55 \beta_1) q^{18} + (68 \beta_{2} - 5 \beta_1) q^{19} + (75 \beta_{2} - 25 \beta_1) q^{20} + (44 \beta_{2} + 25 \beta_1) q^{21} + ( - 46 \beta_{3} - 62) q^{22} + (33 \beta_{3} - 89) q^{23} + ( - 128 \beta_{2} - 40 \beta_1) q^{24} + (75 \beta_{3} - 75) q^{25} + (9 \beta_{3} + 163) q^{27} + ( - 60 \beta_{2} - 40 \beta_1) q^{28} + (60 \beta_{3} - 13) q^{29} + (20 \beta_{3} + 60) q^{30} + (20 \beta_{2} - 100 \beta_1) q^{31} + ( - 85 \beta_{2} - 65 \beta_1) q^{32} + (244 \beta_{2} + 63 \beta_1) q^{33} + ( - 17 \beta_{2} + 5 \beta_1) q^{34} + (30 \beta_{3} - 50) q^{35} + ( - 125 \beta_{3} - 300) q^{36} + ( - 117 \beta_{2} - 44 \beta_1) q^{37} + ( - 58 \beta_{3} - 126) q^{38} + (15 \beta_{3} - 100) q^{40} + ( - 279 \beta_{2} - 20 \beta_1) q^{41} + ( - 94 \beta_{3} - 138) q^{42} + ( - 97 \beta_{3} - 179) q^{43} + ( - 380 \beta_{2} - 80 \beta_1) q^{44} + ( - 175 \beta_{2} + 25 \beta_1) q^{45} + ( - 36 \beta_{2} + 10 \beta_1) q^{46} + ( - 40 \beta_{2} - 140 \beta_1) q^{47} + (48 \beta_{3} - 144) q^{48} + ( - 15 \beta_{3} + 290) q^{49} + (300 \beta_{2} + 150 \beta_1) q^{50} + ( - 65 \beta_{3} + 149) q^{51} + (165 \beta_{3} + 190) q^{53} + (552 \beta_{2} + 190 \beta_1) q^{54} + ( - 70 \beta_{3} + 290) q^{55} + (60 \beta_{3} + 56) q^{56} + (212 \beta_{2} + 199 \beta_1) q^{57} + (381 \beta_{2} + 167 \beta_1) q^{58} + ( - 432 \beta_{2} - 55 \beta_1) q^{59} + 200 \beta_1 q^{60} + ( - 200 \beta_{3} + 351) q^{61} + (180 \beta_{3} + 160) q^{62} + (260 \beta_{2} + 130 \beta_1) q^{63} + (95 \beta_{3} + 588) q^{64} + ( - 370 \beta_{3} - 614) q^{66} + (192 \beta_{2} - 91 \beta_1) q^{67} + (135 \beta_{3} - 320) q^{68} + ( - 135 \beta_{3} + 307) q^{69} + (60 \beta_{2} + 40 \beta_1) q^{70} + ( - 116 \beta_{2} - 105 \beta_1) q^{71} + ( - 695 \beta_{2} - 235 \beta_1) q^{72} + (335 \beta_{2} + 85 \beta_1) q^{73} + (205 \beta_{3} + 322) q^{74} + (75 \beta_{3} + 825) q^{75} + ( - 240 \beta_{2} - 340 \beta_1) q^{76} + ( - 121 \beta_{3} - 67) q^{77} + ( - 40 \beta_{3} + 140) q^{79} + (405 \beta_{2} - 255 \beta_1) q^{80} + (120 \beta_{3} + 1) q^{81} + (319 \beta_{3} + 598) q^{82} + ( - 80 \beta_{2} + 100 \beta_1) q^{83} + ( - 720 \beta_{2} - 220 \beta_1) q^{84} + ( - 455 \beta_{2} + 295 \beta_1) q^{85} + ( - 1216 \beta_{2} - 470 \beta_1) q^{86} + (201 \beta_{3} + 707) q^{87} + (172 \beta_{3} + 424) q^{88} + (398 \beta_{2} - 125 \beta_1) q^{89} + (125 \beta_{3} + 300) q^{90} + (280 \beta_{3} - 660) q^{92} + ( - 1120 \beta_{2} - 40 \beta_1) q^{93} + (320 \beta_{3} + 360) q^{94} + (390 \beta_{3} - 830) q^{95} + ( - 1120 \beta_{2} - 320 \beta_1) q^{96} + (442 \beta_{2} + 469 \beta_1) q^{97} + (765 \beta_{2} + 245 \beta_1) q^{98} + (1300 \beta_{2} + 390 \beta_1) q^{99}+O(q^{100})$$ q + (3*b2 + b1) * q^2 + (3*b3 + 1) * q^3 - 5*b3 * q^4 + (5*b2 - 5*b1) * q^5 + (24*b2 + 10*b1) * q^6 + (8*b2 + b1) * q^7 + (-11*b2 - 7*b1) * q^8 + (15*b3 + 10) * q^9 + 5*b3 * q^10 + (16*b2 + 15*b1) * q^11 + (-20*b3 - 60) * q^12 + (-10*b3 - 18) * q^14 + (-40*b2 + 10*b1) * q^15 + (-15*b3 + 36) * q^16 + (16*b3 - 43) * q^17 + (135*b2 + 55*b1) * q^18 + (68*b2 - 5*b1) * q^19 + (75*b2 - 25*b1) * q^20 + (44*b2 + 25*b1) * q^21 + (-46*b3 - 62) * q^22 + (33*b3 - 89) * q^23 + (-128*b2 - 40*b1) * q^24 + (75*b3 - 75) * q^25 + (9*b3 + 163) * q^27 + (-60*b2 - 40*b1) * q^28 + (60*b3 - 13) * q^29 + (20*b3 + 60) * q^30 + (20*b2 - 100*b1) * q^31 + (-85*b2 - 65*b1) * q^32 + (244*b2 + 63*b1) * q^33 + (-17*b2 + 5*b1) * q^34 + (30*b3 - 50) * q^35 + (-125*b3 - 300) * q^36 + (-117*b2 - 44*b1) * q^37 + (-58*b3 - 126) * q^38 + (15*b3 - 100) * q^40 + (-279*b2 - 20*b1) * q^41 + (-94*b3 - 138) * q^42 + (-97*b3 - 179) * q^43 + (-380*b2 - 80*b1) * q^44 + (-175*b2 + 25*b1) * q^45 + (-36*b2 + 10*b1) * q^46 + (-40*b2 - 140*b1) * q^47 + (48*b3 - 144) * q^48 + (-15*b3 + 290) * q^49 + (300*b2 + 150*b1) * q^50 + (-65*b3 + 149) * q^51 + (165*b3 + 190) * q^53 + (552*b2 + 190*b1) * q^54 + (-70*b3 + 290) * q^55 + (60*b3 + 56) * q^56 + (212*b2 + 199*b1) * q^57 + (381*b2 + 167*b1) * q^58 + (-432*b2 - 55*b1) * q^59 + 200*b1 * q^60 + (-200*b3 + 351) * q^61 + (180*b3 + 160) * q^62 + (260*b2 + 130*b1) * q^63 + (95*b3 + 588) * q^64 + (-370*b3 - 614) * q^66 + (192*b2 - 91*b1) * q^67 + (135*b3 - 320) * q^68 + (-135*b3 + 307) * q^69 + (60*b2 + 40*b1) * q^70 + (-116*b2 - 105*b1) * q^71 + (-695*b2 - 235*b1) * q^72 + (335*b2 + 85*b1) * q^73 + (205*b3 + 322) * q^74 + (75*b3 + 825) * q^75 + (-240*b2 - 340*b1) * q^76 + (-121*b3 - 67) * q^77 + (-40*b3 + 140) * q^79 + (405*b2 - 255*b1) * q^80 + (120*b3 + 1) * q^81 + (319*b3 + 598) * q^82 + (-80*b2 + 100*b1) * q^83 + (-720*b2 - 220*b1) * q^84 + (-455*b2 + 295*b1) * q^85 + (-1216*b2 - 470*b1) * q^86 + (201*b3 + 707) * q^87 + (172*b3 + 424) * q^88 + (398*b2 - 125*b1) * q^89 + (125*b3 + 300) * q^90 + (280*b3 - 660) * q^92 + (-1120*b2 - 40*b1) * q^93 + (320*b3 + 360) * q^94 + (390*b3 - 830) * q^95 + (-1120*b2 - 320*b1) * q^96 + (442*b2 + 469*b1) * q^97 + (765*b2 + 245*b1) * q^98 + (1300*b2 + 390*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10})$$ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 $$4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100})$$ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
168.1
 − 1.56155i 2.56155i − 2.56155i 1.56155i
4.56155i 8.68466 −12.8078 2.80776i 39.6155i 9.56155i 21.9309i 48.4233 12.8078
168.2 0.438447i −3.68466 7.80776 17.8078i 1.61553i 5.43845i 6.93087i −13.4233 −7.80776
168.3 0.438447i −3.68466 7.80776 17.8078i 1.61553i 5.43845i 6.93087i −13.4233 −7.80776
168.4 4.56155i 8.68466 −12.8078 2.80776i 39.6155i 9.56155i 21.9309i 48.4233 12.8078
 $$n$$: 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.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.b.e 4
13.b even 2 1 inner 169.4.b.e 4
13.c even 3 2 169.4.e.g 8
13.d odd 4 1 169.4.a.f 2
13.d odd 4 1 169.4.a.j 2
13.e even 6 2 169.4.e.g 8
13.f odd 12 2 13.4.c.b 4
13.f odd 12 2 169.4.c.f 4
39.f even 4 1 1521.4.a.l 2
39.f even 4 1 1521.4.a.t 2
39.k even 12 2 117.4.g.d 4
52.l even 12 2 208.4.i.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 21T_{2}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 21T^{2} + 4$$
$3$ $$(T^{2} - 5 T - 32)^{2}$$
$5$ $$T^{4} + 325T^{2} + 2500$$
$7$ $$T^{4} + 121T^{2} + 2704$$
$11$ $$T^{4} + 2057 T^{2} + 781456$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 70 T + 137)^{2}$$
$19$ $$T^{4} + 10153 T^{2} + 23658496$$
$23$ $$(T^{2} + 145 T + 628)^{2}$$
$29$ $$(T^{2} - 34 T - 15011)^{2}$$
$31$ $$T^{4} + \cdots + 1413760000$$
$37$ $$T^{4} + 34506 T^{2} + 635209$$
$41$ $$T^{4} + \cdots + 4992976921$$
$43$ $$(T^{2} + 455 T + 11768)^{2}$$
$47$ $$T^{4} + \cdots + 6789760000$$
$53$ $$(T^{2} - 545 T - 41450)^{2}$$
$59$ $$T^{4} + \cdots + 22729783696$$
$61$ $$(T^{2} - 502 T - 106999)^{2}$$
$67$ $$T^{4} + 183201 T^{2} + 449948944$$
$71$ $$T^{4} + \cdots + 1833894976$$
$73$ $$T^{4} + \cdots + 3008522500$$
$79$ $$(T^{2} - 240 T + 7600)^{2}$$
$83$ $$T^{4} + 118800 T^{2} + 655360000$$
$89$ $$T^{4} + \cdots + 21215087716$$
$97$ $$T^{4} + \cdots + 795268001284$$