# Properties

 Label 169.4.c.a Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(22,169)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("169.22");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.c (of order $$3$$, degree $$2$$, 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: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (5 \zeta_{6} - 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} + 7 q^{5} + 35 \zeta_{6} q^{6} - 13 \zeta_{6} q^{7} + 45 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q + (5*z - 5) * q^2 + (-7*z + 7) * q^3 - 17*z * q^4 + 7 * q^5 + 35*z * q^6 - 13*z * q^7 + 45 * q^8 - 22*z * q^9 $$q + (5 \zeta_{6} - 5) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} - 17 \zeta_{6} q^{4} + 7 q^{5} + 35 \zeta_{6} q^{6} - 13 \zeta_{6} q^{7} + 45 q^{8} - 22 \zeta_{6} q^{9} + (35 \zeta_{6} - 35) q^{10} + (26 \zeta_{6} - 26) q^{11} - 119 q^{12} + 65 q^{14} + ( - 49 \zeta_{6} + 49) q^{15} + (89 \zeta_{6} - 89) q^{16} - 77 \zeta_{6} q^{17} + 110 q^{18} - 126 \zeta_{6} q^{19} - 119 \zeta_{6} q^{20} - 91 q^{21} - 130 \zeta_{6} q^{22} + ( - 96 \zeta_{6} + 96) q^{23} + ( - 315 \zeta_{6} + 315) q^{24} - 76 q^{25} + 35 q^{27} + (221 \zeta_{6} - 221) q^{28} + ( - 82 \zeta_{6} + 82) q^{29} + 245 \zeta_{6} q^{30} - 196 q^{31} - 85 \zeta_{6} q^{32} + 182 \zeta_{6} q^{33} + 385 q^{34} - 91 \zeta_{6} q^{35} + (374 \zeta_{6} - 374) q^{36} + (131 \zeta_{6} - 131) q^{37} + 630 q^{38} + 315 q^{40} + ( - 336 \zeta_{6} + 336) q^{41} + ( - 455 \zeta_{6} + 455) q^{42} + 201 \zeta_{6} q^{43} + 442 q^{44} - 154 \zeta_{6} q^{45} + 480 \zeta_{6} q^{46} + 105 q^{47} + 623 \zeta_{6} q^{48} + ( - 174 \zeta_{6} + 174) q^{49} + ( - 380 \zeta_{6} + 380) q^{50} - 539 q^{51} - 432 q^{53} + (175 \zeta_{6} - 175) q^{54} + (182 \zeta_{6} - 182) q^{55} - 585 \zeta_{6} q^{56} - 882 q^{57} + 410 \zeta_{6} q^{58} - 294 \zeta_{6} q^{59} - 833 q^{60} + 56 \zeta_{6} q^{61} + ( - 980 \zeta_{6} + 980) q^{62} + (286 \zeta_{6} - 286) q^{63} - 287 q^{64} - 910 q^{66} + ( - 478 \zeta_{6} + 478) q^{67} + (1309 \zeta_{6} - 1309) q^{68} - 672 \zeta_{6} q^{69} + 455 q^{70} + 9 \zeta_{6} q^{71} - 990 \zeta_{6} q^{72} - 98 q^{73} - 655 \zeta_{6} q^{74} + (532 \zeta_{6} - 532) q^{75} + (2142 \zeta_{6} - 2142) q^{76} + 338 q^{77} + 1304 q^{79} + (623 \zeta_{6} - 623) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1680 \zeta_{6} q^{82} + 308 q^{83} + 1547 \zeta_{6} q^{84} - 539 \zeta_{6} q^{85} - 1005 q^{86} - 574 \zeta_{6} q^{87} + (1170 \zeta_{6} - 1170) q^{88} + (1190 \zeta_{6} - 1190) q^{89} + 770 q^{90} - 1632 q^{92} + (1372 \zeta_{6} - 1372) q^{93} + (525 \zeta_{6} - 525) q^{94} - 882 \zeta_{6} q^{95} - 595 q^{96} + 70 \zeta_{6} q^{97} + 870 \zeta_{6} q^{98} + 572 q^{99} +O(q^{100})$$ q + (5*z - 5) * q^2 + (-7*z + 7) * q^3 - 17*z * q^4 + 7 * q^5 + 35*z * q^6 - 13*z * q^7 + 45 * q^8 - 22*z * q^9 + (35*z - 35) * q^10 + (26*z - 26) * q^11 - 119 * q^12 + 65 * q^14 + (-49*z + 49) * q^15 + (89*z - 89) * q^16 - 77*z * q^17 + 110 * q^18 - 126*z * q^19 - 119*z * q^20 - 91 * q^21 - 130*z * q^22 + (-96*z + 96) * q^23 + (-315*z + 315) * q^24 - 76 * q^25 + 35 * q^27 + (221*z - 221) * q^28 + (-82*z + 82) * q^29 + 245*z * q^30 - 196 * q^31 - 85*z * q^32 + 182*z * q^33 + 385 * q^34 - 91*z * q^35 + (374*z - 374) * q^36 + (131*z - 131) * q^37 + 630 * q^38 + 315 * q^40 + (-336*z + 336) * q^41 + (-455*z + 455) * q^42 + 201*z * q^43 + 442 * q^44 - 154*z * q^45 + 480*z * q^46 + 105 * q^47 + 623*z * q^48 + (-174*z + 174) * q^49 + (-380*z + 380) * q^50 - 539 * q^51 - 432 * q^53 + (175*z - 175) * q^54 + (182*z - 182) * q^55 - 585*z * q^56 - 882 * q^57 + 410*z * q^58 - 294*z * q^59 - 833 * q^60 + 56*z * q^61 + (-980*z + 980) * q^62 + (286*z - 286) * q^63 - 287 * q^64 - 910 * q^66 + (-478*z + 478) * q^67 + (1309*z - 1309) * q^68 - 672*z * q^69 + 455 * q^70 + 9*z * q^71 - 990*z * q^72 - 98 * q^73 - 655*z * q^74 + (532*z - 532) * q^75 + (2142*z - 2142) * q^76 + 338 * q^77 + 1304 * q^79 + (623*z - 623) * q^80 + (-839*z + 839) * q^81 + 1680*z * q^82 + 308 * q^83 + 1547*z * q^84 - 539*z * q^85 - 1005 * q^86 - 574*z * q^87 + (1170*z - 1170) * q^88 + (1190*z - 1190) * q^89 + 770 * q^90 - 1632 * q^92 + (1372*z - 1372) * q^93 + (525*z - 525) * q^94 - 882*z * q^95 - 595 * q^96 + 70*z * q^97 + 870*z * q^98 + 572 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{2} + 7 q^{3} - 17 q^{4} + 14 q^{5} + 35 q^{6} - 13 q^{7} + 90 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q - 5 * q^2 + 7 * q^3 - 17 * q^4 + 14 * q^5 + 35 * q^6 - 13 * q^7 + 90 * q^8 - 22 * q^9 $$2 q - 5 q^{2} + 7 q^{3} - 17 q^{4} + 14 q^{5} + 35 q^{6} - 13 q^{7} + 90 q^{8} - 22 q^{9} - 35 q^{10} - 26 q^{11} - 238 q^{12} + 130 q^{14} + 49 q^{15} - 89 q^{16} - 77 q^{17} + 220 q^{18} - 126 q^{19} - 119 q^{20} - 182 q^{21} - 130 q^{22} + 96 q^{23} + 315 q^{24} - 152 q^{25} + 70 q^{27} - 221 q^{28} + 82 q^{29} + 245 q^{30} - 392 q^{31} - 85 q^{32} + 182 q^{33} + 770 q^{34} - 91 q^{35} - 374 q^{36} - 131 q^{37} + 1260 q^{38} + 630 q^{40} + 336 q^{41} + 455 q^{42} + 201 q^{43} + 884 q^{44} - 154 q^{45} + 480 q^{46} + 210 q^{47} + 623 q^{48} + 174 q^{49} + 380 q^{50} - 1078 q^{51} - 864 q^{53} - 175 q^{54} - 182 q^{55} - 585 q^{56} - 1764 q^{57} + 410 q^{58} - 294 q^{59} - 1666 q^{60} + 56 q^{61} + 980 q^{62} - 286 q^{63} - 574 q^{64} - 1820 q^{66} + 478 q^{67} - 1309 q^{68} - 672 q^{69} + 910 q^{70} + 9 q^{71} - 990 q^{72} - 196 q^{73} - 655 q^{74} - 532 q^{75} - 2142 q^{76} + 676 q^{77} + 2608 q^{79} - 623 q^{80} + 839 q^{81} + 1680 q^{82} + 616 q^{83} + 1547 q^{84} - 539 q^{85} - 2010 q^{86} - 574 q^{87} - 1170 q^{88} - 1190 q^{89} + 1540 q^{90} - 3264 q^{92} - 1372 q^{93} - 525 q^{94} - 882 q^{95} - 1190 q^{96} + 70 q^{97} + 870 q^{98} + 1144 q^{99}+O(q^{100})$$ 2 * q - 5 * q^2 + 7 * q^3 - 17 * q^4 + 14 * q^5 + 35 * q^6 - 13 * q^7 + 90 * q^8 - 22 * q^9 - 35 * q^10 - 26 * q^11 - 238 * q^12 + 130 * q^14 + 49 * q^15 - 89 * q^16 - 77 * q^17 + 220 * q^18 - 126 * q^19 - 119 * q^20 - 182 * q^21 - 130 * q^22 + 96 * q^23 + 315 * q^24 - 152 * q^25 + 70 * q^27 - 221 * q^28 + 82 * q^29 + 245 * q^30 - 392 * q^31 - 85 * q^32 + 182 * q^33 + 770 * q^34 - 91 * q^35 - 374 * q^36 - 131 * q^37 + 1260 * q^38 + 630 * q^40 + 336 * q^41 + 455 * q^42 + 201 * q^43 + 884 * q^44 - 154 * q^45 + 480 * q^46 + 210 * q^47 + 623 * q^48 + 174 * q^49 + 380 * q^50 - 1078 * q^51 - 864 * q^53 - 175 * q^54 - 182 * q^55 - 585 * q^56 - 1764 * q^57 + 410 * q^58 - 294 * q^59 - 1666 * q^60 + 56 * q^61 + 980 * q^62 - 286 * q^63 - 574 * q^64 - 1820 * q^66 + 478 * q^67 - 1309 * q^68 - 672 * q^69 + 910 * q^70 + 9 * q^71 - 990 * q^72 - 196 * q^73 - 655 * q^74 - 532 * q^75 - 2142 * q^76 + 676 * q^77 + 2608 * q^79 - 623 * q^80 + 839 * q^81 + 1680 * q^82 + 616 * q^83 + 1547 * q^84 - 539 * q^85 - 2010 * q^86 - 574 * q^87 - 1170 * q^88 - 1190 * q^89 + 1540 * q^90 - 3264 * q^92 - 1372 * q^93 - 525 * q^94 - 882 * q^95 - 1190 * q^96 + 70 * q^97 + 870 * q^98 + 1144 * q^99

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.5 + 0.866025i 0.5 − 0.866025i
−2.50000 + 4.33013i 3.50000 6.06218i −8.50000 14.7224i 7.00000 17.5000 + 30.3109i −6.50000 11.2583i 45.0000 −11.0000 19.0526i −17.5000 + 30.3109i
146.1 −2.50000 4.33013i 3.50000 + 6.06218i −8.50000 + 14.7224i 7.00000 17.5000 30.3109i −6.50000 + 11.2583i 45.0000 −11.0000 + 19.0526i −17.5000 30.3109i
 $$n$$: 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 5T_{2} + 25$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 5T + 25$$
$3$ $$T^{2} - 7T + 49$$
$5$ $$(T - 7)^{2}$$
$7$ $$T^{2} + 13T + 169$$
$11$ $$T^{2} + 26T + 676$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 77T + 5929$$
$19$ $$T^{2} + 126T + 15876$$
$23$ $$T^{2} - 96T + 9216$$
$29$ $$T^{2} - 82T + 6724$$
$31$ $$(T + 196)^{2}$$
$37$ $$T^{2} + 131T + 17161$$
$41$ $$T^{2} - 336T + 112896$$
$43$ $$T^{2} - 201T + 40401$$
$47$ $$(T - 105)^{2}$$
$53$ $$(T + 432)^{2}$$
$59$ $$T^{2} + 294T + 86436$$
$61$ $$T^{2} - 56T + 3136$$
$67$ $$T^{2} - 478T + 228484$$
$71$ $$T^{2} - 9T + 81$$
$73$ $$(T + 98)^{2}$$
$79$ $$(T - 1304)^{2}$$
$83$ $$(T - 308)^{2}$$
$89$ $$T^{2} + 1190 T + 1416100$$
$97$ $$T^{2} - 70T + 4900$$