# Properties

 Label 169.4.b.f 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (3 \beta_{3} + 1) q^{3} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + (6 \beta_{2} + \beta_1) q^{6} + ( - 5 \beta_{2} - 11 \beta_1) q^{7} + (2 \beta_{2} + 11 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9}+O(q^{10})$$ q + b1 * q^2 + (3*b3 + 1) * q^3 + (b3 + 3) * q^4 + (b2 + b1) * q^5 + (6*b2 + b1) * q^6 + (-5*b2 - 11*b1) * q^7 + (2*b2 + 11*b1) * q^8 + (15*b3 + 10) * q^9 $$q + \beta_1 q^{2} + (3 \beta_{3} + 1) q^{3} + (\beta_{3} + 3) q^{4} + (\beta_{2} + \beta_1) q^{5} + (6 \beta_{2} + \beta_1) q^{6} + ( - 5 \beta_{2} - 11 \beta_1) q^{7} + (2 \beta_{2} + 11 \beta_1) q^{8} + (15 \beta_{3} + 10) q^{9} + ( - \beta_{3} - 3) q^{10} + (17 \beta_{2} - 12 \beta_1) q^{11} + (13 \beta_{3} + 15) q^{12} + ( - \beta_{3} + 45) q^{14} + (10 \beta_{2} + 7 \beta_1) q^{15} + (15 \beta_{3} - 27) q^{16} + ( - 17 \beta_{3} - 1) q^{17} + (30 \beta_{2} + 10 \beta_1) q^{18} + (13 \beta_{2} - 32 \beta_1) q^{19} + (6 \beta_{2} + 5 \beta_1) q^{20} + ( - 86 \beta_{2} - 41 \beta_1) q^{21} + ( - 46 \beta_{3} + 94) q^{22} + ( - 12 \beta_{3} - 92) q^{23} + (74 \beta_{2} + 23 \beta_1) q^{24} + ( - 3 \beta_{3} + 120) q^{25} + (9 \beta_{3} + 163) q^{27} + ( - 42 \beta_{2} - 43 \beta_1) q^{28} + ( - 96 \beta_{3} + 26) q^{29} + ( - 13 \beta_{3} - 15) q^{30} + (13 \beta_{2} - 34 \beta_1) q^{31} + (46 \beta_{2} + 61 \beta_1) q^{32} + ( - 4 \beta_{2} + 90 \beta_1) q^{33} + ( - 34 \beta_{2} - \beta_1) q^{34} + (21 \beta_{3} + 43) q^{35} + (70 \beta_{3} + 90) q^{36} + (51 \beta_{2} - 5 \beta_1) q^{37} + ( - 58 \beta_{3} + 186) q^{38} + ( - 15 \beta_{3} - 37) q^{40} + (63 \beta_{2} + 22 \beta_1) q^{41} + (131 \beta_{3} + 33) q^{42} + (143 \beta_{3} - 215) q^{43} + (44 \beta_{2} - 2 \beta_1) q^{44} + (55 \beta_{2} + 40 \beta_1) q^{45} + ( - 24 \beta_{2} - 92 \beta_1) q^{46} + (139 \beta_{2} + 121 \beta_1) q^{47} + ( - 21 \beta_{3} + 153) q^{48} + ( - 99 \beta_{3} - 142) q^{49} + ( - 6 \beta_{2} + 120 \beta_1) q^{50} + ( - 71 \beta_{3} - 205) q^{51} + ( - 30 \beta_{3} - 44) q^{53} + (18 \beta_{2} + 163 \beta_1) q^{54} + ( - 22 \beta_{3} + 2) q^{55} + (33 \beta_{3} + 491) q^{56} + ( - 140 \beta_{2} + 46 \beta_1) q^{57} + ( - 192 \beta_{2} + 26 \beta_1) q^{58} + ( - 123 \beta_{2} - 124 \beta_1) q^{59} + (54 \beta_{2} + 41 \beta_1) q^{60} + (190 \beta_{3} - 624) q^{61} + ( - 60 \beta_{3} + 196) q^{62} + ( - 455 \beta_{2} - 260 \beta_1) q^{63} + (89 \beta_{3} - 429) q^{64} + (98 \beta_{3} - 458) q^{66} + ( - 75 \beta_{2} - 232 \beta_1) q^{67} + ( - 69 \beta_{3} - 71) q^{68} + ( - 324 \beta_{3} - 236) q^{69} + (42 \beta_{2} + 43 \beta_1) q^{70} + ( - 25 \beta_{2} - 231 \beta_1) q^{71} + (380 \beta_{2} + 170 \beta_1) q^{72} + (49 \beta_{2} - 260 \beta_1) q^{73} + ( - 107 \beta_{3} + 127) q^{74} + (348 \beta_{3} + 84) q^{75} + ( - 12 \beta_{2} - 70 \beta_1) q^{76} + (386 \beta_{3} - 574) q^{77} + ( - 40 \beta_{3} - 484) q^{79} + (18 \beta_{2} + 3 \beta_1) q^{80} + (120 \beta_{3} + 1) q^{81} + ( - 104 \beta_{3} + 16) q^{82} + ( - 535 \beta_{2} - 182 \beta_1) q^{83} + ( - 426 \beta_{2} - 295 \beta_1) q^{84} + ( - 52 \beta_{2} - 35 \beta_1) q^{85} + (286 \beta_{2} - 215 \beta_1) q^{86} + ( - 306 \beta_{3} - 1126) q^{87} + ( - 458 \beta_{3} + 850) q^{88} + ( - 83 \beta_{2} + 388 \beta_1) q^{89} + ( - 70 \beta_{3} - 90) q^{90} + ( - 140 \beta_{3} - 324) q^{92} + ( - 152 \beta_{2} + 44 \beta_1) q^{93} + ( - 157 \beta_{3} - 327) q^{94} + (6 \beta_{3} + 70) q^{95} + (550 \beta_{2} + 337 \beta_1) q^{96} + (359 \beta_{2} + 508 \beta_1) q^{97} + ( - 198 \beta_{2} - 142 \beta_1) q^{98} + (65 \beta_{2} + 390 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (3*b3 + 1) * q^3 + (b3 + 3) * q^4 + (b2 + b1) * q^5 + (6*b2 + b1) * q^6 + (-5*b2 - 11*b1) * q^7 + (2*b2 + 11*b1) * q^8 + (15*b3 + 10) * q^9 + (-b3 - 3) * q^10 + (17*b2 - 12*b1) * q^11 + (13*b3 + 15) * q^12 + (-b3 + 45) * q^14 + (10*b2 + 7*b1) * q^15 + (15*b3 - 27) * q^16 + (-17*b3 - 1) * q^17 + (30*b2 + 10*b1) * q^18 + (13*b2 - 32*b1) * q^19 + (6*b2 + 5*b1) * q^20 + (-86*b2 - 41*b1) * q^21 + (-46*b3 + 94) * q^22 + (-12*b3 - 92) * q^23 + (74*b2 + 23*b1) * q^24 + (-3*b3 + 120) * q^25 + (9*b3 + 163) * q^27 + (-42*b2 - 43*b1) * q^28 + (-96*b3 + 26) * q^29 + (-13*b3 - 15) * q^30 + (13*b2 - 34*b1) * q^31 + (46*b2 + 61*b1) * q^32 + (-4*b2 + 90*b1) * q^33 + (-34*b2 - b1) * q^34 + (21*b3 + 43) * q^35 + (70*b3 + 90) * q^36 + (51*b2 - 5*b1) * q^37 + (-58*b3 + 186) * q^38 + (-15*b3 - 37) * q^40 + (63*b2 + 22*b1) * q^41 + (131*b3 + 33) * q^42 + (143*b3 - 215) * q^43 + (44*b2 - 2*b1) * q^44 + (55*b2 + 40*b1) * q^45 + (-24*b2 - 92*b1) * q^46 + (139*b2 + 121*b1) * q^47 + (-21*b3 + 153) * q^48 + (-99*b3 - 142) * q^49 + (-6*b2 + 120*b1) * q^50 + (-71*b3 - 205) * q^51 + (-30*b3 - 44) * q^53 + (18*b2 + 163*b1) * q^54 + (-22*b3 + 2) * q^55 + (33*b3 + 491) * q^56 + (-140*b2 + 46*b1) * q^57 + (-192*b2 + 26*b1) * q^58 + (-123*b2 - 124*b1) * q^59 + (54*b2 + 41*b1) * q^60 + (190*b3 - 624) * q^61 + (-60*b3 + 196) * q^62 + (-455*b2 - 260*b1) * q^63 + (89*b3 - 429) * q^64 + (98*b3 - 458) * q^66 + (-75*b2 - 232*b1) * q^67 + (-69*b3 - 71) * q^68 + (-324*b3 - 236) * q^69 + (42*b2 + 43*b1) * q^70 + (-25*b2 - 231*b1) * q^71 + (380*b2 + 170*b1) * q^72 + (49*b2 - 260*b1) * q^73 + (-107*b3 + 127) * q^74 + (348*b3 + 84) * q^75 + (-12*b2 - 70*b1) * q^76 + (386*b3 - 574) * q^77 + (-40*b3 - 484) * q^79 + (18*b2 + 3*b1) * q^80 + (120*b3 + 1) * q^81 + (-104*b3 + 16) * q^82 + (-535*b2 - 182*b1) * q^83 + (-426*b2 - 295*b1) * q^84 + (-52*b2 - 35*b1) * q^85 + (286*b2 - 215*b1) * q^86 + (-306*b3 - 1126) * q^87 + (-458*b3 + 850) * q^88 + (-83*b2 + 388*b1) * q^89 + (-70*b3 - 90) * q^90 + (-140*b3 - 324) * q^92 + (-152*b2 + 44*b1) * q^93 + (-157*b3 - 327) * q^94 + (6*b3 + 70) * q^95 + (550*b2 + 337*b1) * q^96 + (359*b2 + 508*b1) * q^97 + (-198*b2 - 142*b1) * q^98 + (65*b2 + 390*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10})$$ 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9 $$4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100})$$ 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9 - 14 * q^10 + 86 * q^12 + 178 * q^14 - 78 * q^16 - 38 * q^17 + 284 * q^22 - 392 * q^23 + 474 * q^25 + 670 * q^27 - 88 * q^29 - 86 * q^30 + 214 * q^35 + 500 * q^36 + 628 * q^38 - 178 * q^40 + 394 * q^42 - 574 * q^43 + 570 * q^48 - 766 * q^49 - 962 * q^51 - 236 * q^53 - 36 * q^55 + 2030 * q^56 - 2116 * q^61 + 664 * q^62 - 1538 * q^64 - 1636 * q^66 - 422 * q^68 - 1592 * q^69 + 294 * q^74 + 1032 * q^75 - 1524 * q^77 - 2016 * q^79 + 244 * q^81 - 144 * q^82 - 5116 * q^87 + 2484 * q^88 - 500 * q^90 - 1576 * q^92 - 1622 * q^94 + 292 * q^95

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 2$$ (v^3 + 5*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 5\beta_1$$ 2*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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i −3.68466 1.43845 0.561553i 9.43845i 18.1771i 24.1771i −13.4233 −1.43845
168.2 1.56155i 8.68466 5.56155 3.56155i 13.5616i 27.1771i 21.1771i 48.4233 −5.56155
168.3 1.56155i 8.68466 5.56155 3.56155i 13.5616i 27.1771i 21.1771i 48.4233 −5.56155
168.4 2.56155i −3.68466 1.43845 0.561553i 9.43845i 18.1771i 24.1771i −13.4233 −1.43845
 $$n$$: 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.f 4
13.b even 2 1 inner 169.4.b.f 4
13.c even 3 2 169.4.e.f 8
13.d odd 4 1 13.4.a.b 2
13.d odd 4 1 169.4.a.g 2
13.e even 6 2 169.4.e.f 8
13.f odd 12 2 169.4.c.g 4
13.f odd 12 2 169.4.c.j 4
39.f even 4 1 117.4.a.d 2
39.f even 4 1 1521.4.a.r 2
52.f even 4 1 208.4.a.h 2
65.f even 4 1 325.4.b.e 4
65.g odd 4 1 325.4.a.f 2
65.k even 4 1 325.4.b.e 4
91.i even 4 1 637.4.a.b 2
104.j odd 4 1 832.4.a.s 2
104.m even 4 1 832.4.a.z 2
143.g even 4 1 1573.4.a.b 2
156.l odd 4 1 1872.4.a.bb 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 9T_{2}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$(T^{2} - 5 T - 32)^{2}$$
$5$ $$T^{4} + 13T^{2} + 4$$
$7$ $$T^{4} + 1069 T^{2} + 244036$$
$11$ $$T^{4} + 4424 T^{2} + 976144$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 19 T - 1138)^{2}$$
$19$ $$T^{4} + 12232 T^{2} + 6697744$$
$23$ $$(T^{2} + 196 T + 8992)^{2}$$
$29$ $$(T^{2} + 44 T - 38684)^{2}$$
$31$ $$T^{4} + 13524 T^{2} + 9388096$$
$37$ $$T^{4} + 22053 T^{2} + 116942596$$
$41$ $$T^{4} + 30564 T^{2} + 124724224$$
$43$ $$(T^{2} + 287 T - 66316)^{2}$$
$47$ $$T^{4} + 219061 T^{2} + 222546724$$
$53$ $$(T^{2} + 118 T - 344)^{2}$$
$59$ $$T^{4} + 198408 T^{2} + 991746064$$
$61$ $$(T^{2} + 1058 T + 126416)^{2}$$
$67$ $$T^{4} + \cdots + 51799939216$$
$71$ $$T^{4} + \cdots + 49503580036$$
$73$ $$T^{4} + \cdots + 55373619856$$
$79$ $$(T^{2} + 1008 T + 247216)^{2}$$
$83$ $$T^{4} + \cdots + 668574416896$$
$89$ $$T^{4} + \cdots + 260316284944$$
$97$ $$T^{4} + \cdots + 776999938576$$