# Properties

 Label 169.4.b.a Level $169$ Weight $4$ Character orbit 169.b Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} - 7 q^{3} - 17 q^{4} + 7 i q^{5} - 35 i q^{6} - 13 i q^{7} - 45 i q^{8} + 22 q^{9} +O(q^{10})$$ q + 5*i * q^2 - 7 * q^3 - 17 * q^4 + 7*i * q^5 - 35*i * q^6 - 13*i * q^7 - 45*i * q^8 + 22 * q^9 $$q + 5 i q^{2} - 7 q^{3} - 17 q^{4} + 7 i q^{5} - 35 i q^{6} - 13 i q^{7} - 45 i q^{8} + 22 q^{9} - 35 q^{10} - 26 i q^{11} + 119 q^{12} + 65 q^{14} - 49 i q^{15} + 89 q^{16} - 77 q^{17} + 110 i q^{18} + 126 i q^{19} - 119 i q^{20} + 91 i q^{21} + 130 q^{22} + 96 q^{23} + 315 i q^{24} + 76 q^{25} + 35 q^{27} + 221 i q^{28} - 82 q^{29} + 245 q^{30} - 196 i q^{31} + 85 i q^{32} + 182 i q^{33} - 385 i q^{34} + 91 q^{35} - 374 q^{36} - 131 i q^{37} - 630 q^{38} + 315 q^{40} - 336 i q^{41} - 455 q^{42} + 201 q^{43} + 442 i q^{44} + 154 i q^{45} + 480 i q^{46} - 105 i q^{47} - 623 q^{48} + 174 q^{49} + 380 i q^{50} + 539 q^{51} - 432 q^{53} + 175 i q^{54} + 182 q^{55} - 585 q^{56} - 882 i q^{57} - 410 i q^{58} - 294 i q^{59} + 833 i q^{60} - 56 q^{61} + 980 q^{62} - 286 i q^{63} + 287 q^{64} - 910 q^{66} - 478 i q^{67} + 1309 q^{68} - 672 q^{69} + 455 i q^{70} - 9 i q^{71} - 990 i q^{72} + 98 i q^{73} + 655 q^{74} - 532 q^{75} - 2142 i q^{76} - 338 q^{77} + 1304 q^{79} + 623 i q^{80} - 839 q^{81} + 1680 q^{82} + 308 i q^{83} - 1547 i q^{84} - 539 i q^{85} + 1005 i q^{86} + 574 q^{87} - 1170 q^{88} - 1190 i q^{89} - 770 q^{90} - 1632 q^{92} + 1372 i q^{93} + 525 q^{94} - 882 q^{95} - 595 i q^{96} - 70 i q^{97} + 870 i q^{98} - 572 i q^{99} +O(q^{100})$$ q + 5*i * q^2 - 7 * q^3 - 17 * q^4 + 7*i * q^5 - 35*i * q^6 - 13*i * q^7 - 45*i * q^8 + 22 * q^9 - 35 * q^10 - 26*i * q^11 + 119 * q^12 + 65 * q^14 - 49*i * q^15 + 89 * q^16 - 77 * q^17 + 110*i * q^18 + 126*i * q^19 - 119*i * q^20 + 91*i * q^21 + 130 * q^22 + 96 * q^23 + 315*i * q^24 + 76 * q^25 + 35 * q^27 + 221*i * q^28 - 82 * q^29 + 245 * q^30 - 196*i * q^31 + 85*i * q^32 + 182*i * q^33 - 385*i * q^34 + 91 * q^35 - 374 * q^36 - 131*i * q^37 - 630 * q^38 + 315 * q^40 - 336*i * q^41 - 455 * q^42 + 201 * q^43 + 442*i * q^44 + 154*i * q^45 + 480*i * q^46 - 105*i * q^47 - 623 * q^48 + 174 * q^49 + 380*i * q^50 + 539 * q^51 - 432 * q^53 + 175*i * q^54 + 182 * q^55 - 585 * q^56 - 882*i * q^57 - 410*i * q^58 - 294*i * q^59 + 833*i * q^60 - 56 * q^61 + 980 * q^62 - 286*i * q^63 + 287 * q^64 - 910 * q^66 - 478*i * q^67 + 1309 * q^68 - 672 * q^69 + 455*i * q^70 - 9*i * q^71 - 990*i * q^72 + 98*i * q^73 + 655 * q^74 - 532 * q^75 - 2142*i * q^76 - 338 * q^77 + 1304 * q^79 + 623*i * q^80 - 839 * q^81 + 1680 * q^82 + 308*i * q^83 - 1547*i * q^84 - 539*i * q^85 + 1005*i * q^86 + 574 * q^87 - 1170 * q^88 - 1190*i * q^89 - 770 * q^90 - 1632 * q^92 + 1372*i * q^93 + 525 * q^94 - 882 * q^95 - 595*i * q^96 - 70*i * q^97 + 870*i * q^98 - 572*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 $$2 q - 14 q^{3} - 34 q^{4} + 44 q^{9} - 70 q^{10} + 238 q^{12} + 130 q^{14} + 178 q^{16} - 154 q^{17} + 260 q^{22} + 192 q^{23} + 152 q^{25} + 70 q^{27} - 164 q^{29} + 490 q^{30} + 182 q^{35} - 748 q^{36} - 1260 q^{38} + 630 q^{40} - 910 q^{42} + 402 q^{43} - 1246 q^{48} + 348 q^{49} + 1078 q^{51} - 864 q^{53} + 364 q^{55} - 1170 q^{56} - 112 q^{61} + 1960 q^{62} + 574 q^{64} - 1820 q^{66} + 2618 q^{68} - 1344 q^{69} + 1310 q^{74} - 1064 q^{75} - 676 q^{77} + 2608 q^{79} - 1678 q^{81} + 3360 q^{82} + 1148 q^{87} - 2340 q^{88} - 1540 q^{90} - 3264 q^{92} + 1050 q^{94} - 1764 q^{95}+O(q^{100})$$ 2 * q - 14 * q^3 - 34 * q^4 + 44 * q^9 - 70 * q^10 + 238 * q^12 + 130 * q^14 + 178 * q^16 - 154 * q^17 + 260 * q^22 + 192 * q^23 + 152 * q^25 + 70 * q^27 - 164 * q^29 + 490 * q^30 + 182 * q^35 - 748 * q^36 - 1260 * q^38 + 630 * q^40 - 910 * q^42 + 402 * q^43 - 1246 * q^48 + 348 * q^49 + 1078 * q^51 - 864 * q^53 + 364 * q^55 - 1170 * q^56 - 112 * q^61 + 1960 * q^62 + 574 * q^64 - 1820 * q^66 + 2618 * q^68 - 1344 * q^69 + 1310 * q^74 - 1064 * q^75 - 676 * q^77 + 2608 * q^79 - 1678 * q^81 + 3360 * q^82 + 1148 * q^87 - 2340 * q^88 - 1540 * q^90 - 3264 * q^92 + 1050 * q^94 - 1764 * q^95

## 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
 − 1.00000i 1.00000i
5.00000i −7.00000 −17.0000 7.00000i 35.0000i 13.0000i 45.0000i 22.0000 −35.0000
168.2 5.00000i −7.00000 −17.0000 7.00000i 35.0000i 13.0000i 45.0000i 22.0000 −35.0000
 $$n$$: 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.a 2
13.b even 2 1 inner 169.4.b.a 2
13.c even 3 2 169.4.e.e 4
13.d odd 4 1 13.4.a.a 1
13.d odd 4 1 169.4.a.e 1
13.e even 6 2 169.4.e.e 4
13.f odd 12 2 169.4.c.a 2
13.f odd 12 2 169.4.c.e 2
39.f even 4 1 117.4.a.b 1
39.f even 4 1 1521.4.a.a 1
52.f even 4 1 208.4.a.g 1
65.f even 4 1 325.4.b.b 2
65.g odd 4 1 325.4.a.d 1
65.k even 4 1 325.4.b.b 2
91.i even 4 1 637.4.a.a 1
104.j odd 4 1 832.4.a.r 1
104.m even 4 1 832.4.a.a 1
143.g even 4 1 1573.4.a.a 1
156.l odd 4 1 1872.4.a.k 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 25$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$(T + 7)^{2}$$
$5$ $$T^{2} + 49$$
$7$ $$T^{2} + 169$$
$11$ $$T^{2} + 676$$
$13$ $$T^{2}$$
$17$ $$(T + 77)^{2}$$
$19$ $$T^{2} + 15876$$
$23$ $$(T - 96)^{2}$$
$29$ $$(T + 82)^{2}$$
$31$ $$T^{2} + 38416$$
$37$ $$T^{2} + 17161$$
$41$ $$T^{2} + 112896$$
$43$ $$(T - 201)^{2}$$
$47$ $$T^{2} + 11025$$
$53$ $$(T + 432)^{2}$$
$59$ $$T^{2} + 86436$$
$61$ $$(T + 56)^{2}$$
$67$ $$T^{2} + 228484$$
$71$ $$T^{2} + 81$$
$73$ $$T^{2} + 9604$$
$79$ $$(T - 1304)^{2}$$
$83$ $$T^{2} + 94864$$
$89$ $$T^{2} + 1416100$$
$97$ $$T^{2} + 4900$$