# Properties

 Label 169.4.c.c 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.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + 9 q^{5} - 3 \zeta_{6} q^{6} + 15 \zeta_{6} q^{7} + 21 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10})$$ q + (-3*z + 3) * q^2 + (-z + 1) * q^3 - z * q^4 + 9 * q^5 - 3*z * q^6 + 15*z * q^7 + 21 * q^8 + 26*z * q^9 $$q + ( - 3 \zeta_{6} + 3) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + 9 q^{5} - 3 \zeta_{6} q^{6} + 15 \zeta_{6} q^{7} + 21 q^{8} + 26 \zeta_{6} q^{9} + ( - 27 \zeta_{6} + 27) q^{10} + (48 \zeta_{6} - 48) q^{11} - q^{12} + 45 q^{14} + ( - 9 \zeta_{6} + 9) q^{15} + ( - 71 \zeta_{6} + 71) q^{16} - 45 \zeta_{6} q^{17} + 78 q^{18} + 6 \zeta_{6} q^{19} - 9 \zeta_{6} q^{20} + 15 q^{21} + 144 \zeta_{6} q^{22} + ( - 162 \zeta_{6} + 162) q^{23} + ( - 21 \zeta_{6} + 21) q^{24} - 44 q^{25} + 53 q^{27} + ( - 15 \zeta_{6} + 15) q^{28} + ( - 144 \zeta_{6} + 144) q^{29} - 27 \zeta_{6} q^{30} - 264 q^{31} - 45 \zeta_{6} q^{32} + 48 \zeta_{6} q^{33} - 135 q^{34} + 135 \zeta_{6} q^{35} + ( - 26 \zeta_{6} + 26) q^{36} + ( - 303 \zeta_{6} + 303) q^{37} + 18 q^{38} + 189 q^{40} + (192 \zeta_{6} - 192) q^{41} + ( - 45 \zeta_{6} + 45) q^{42} - 97 \zeta_{6} q^{43} + 48 q^{44} + 234 \zeta_{6} q^{45} - 486 \zeta_{6} q^{46} - 111 q^{47} - 71 \zeta_{6} q^{48} + ( - 118 \zeta_{6} + 118) q^{49} + (132 \zeta_{6} - 132) q^{50} - 45 q^{51} - 414 q^{53} + ( - 159 \zeta_{6} + 159) q^{54} + (432 \zeta_{6} - 432) q^{55} + 315 \zeta_{6} q^{56} + 6 q^{57} - 432 \zeta_{6} q^{58} + 522 \zeta_{6} q^{59} - 9 q^{60} - 376 \zeta_{6} q^{61} + (792 \zeta_{6} - 792) q^{62} + (390 \zeta_{6} - 390) q^{63} + 433 q^{64} + 144 q^{66} + (36 \zeta_{6} - 36) q^{67} + (45 \zeta_{6} - 45) q^{68} - 162 \zeta_{6} q^{69} + 405 q^{70} + 357 \zeta_{6} q^{71} + 546 \zeta_{6} q^{72} + 1098 q^{73} - 909 \zeta_{6} q^{74} + (44 \zeta_{6} - 44) q^{75} + ( - 6 \zeta_{6} + 6) q^{76} - 720 q^{77} - 830 q^{79} + ( - 639 \zeta_{6} + 639) q^{80} + (649 \zeta_{6} - 649) q^{81} + 576 \zeta_{6} q^{82} + 438 q^{83} - 15 \zeta_{6} q^{84} - 405 \zeta_{6} q^{85} - 291 q^{86} - 144 \zeta_{6} q^{87} + (1008 \zeta_{6} - 1008) q^{88} + (438 \zeta_{6} - 438) q^{89} + 702 q^{90} - 162 q^{92} + (264 \zeta_{6} - 264) q^{93} + (333 \zeta_{6} - 333) q^{94} + 54 \zeta_{6} q^{95} - 45 q^{96} - 852 \zeta_{6} q^{97} - 354 \zeta_{6} q^{98} - 1248 q^{99} +O(q^{100})$$ q + (-3*z + 3) * q^2 + (-z + 1) * q^3 - z * q^4 + 9 * q^5 - 3*z * q^6 + 15*z * q^7 + 21 * q^8 + 26*z * q^9 + (-27*z + 27) * q^10 + (48*z - 48) * q^11 - q^12 + 45 * q^14 + (-9*z + 9) * q^15 + (-71*z + 71) * q^16 - 45*z * q^17 + 78 * q^18 + 6*z * q^19 - 9*z * q^20 + 15 * q^21 + 144*z * q^22 + (-162*z + 162) * q^23 + (-21*z + 21) * q^24 - 44 * q^25 + 53 * q^27 + (-15*z + 15) * q^28 + (-144*z + 144) * q^29 - 27*z * q^30 - 264 * q^31 - 45*z * q^32 + 48*z * q^33 - 135 * q^34 + 135*z * q^35 + (-26*z + 26) * q^36 + (-303*z + 303) * q^37 + 18 * q^38 + 189 * q^40 + (192*z - 192) * q^41 + (-45*z + 45) * q^42 - 97*z * q^43 + 48 * q^44 + 234*z * q^45 - 486*z * q^46 - 111 * q^47 - 71*z * q^48 + (-118*z + 118) * q^49 + (132*z - 132) * q^50 - 45 * q^51 - 414 * q^53 + (-159*z + 159) * q^54 + (432*z - 432) * q^55 + 315*z * q^56 + 6 * q^57 - 432*z * q^58 + 522*z * q^59 - 9 * q^60 - 376*z * q^61 + (792*z - 792) * q^62 + (390*z - 390) * q^63 + 433 * q^64 + 144 * q^66 + (36*z - 36) * q^67 + (45*z - 45) * q^68 - 162*z * q^69 + 405 * q^70 + 357*z * q^71 + 546*z * q^72 + 1098 * q^73 - 909*z * q^74 + (44*z - 44) * q^75 + (-6*z + 6) * q^76 - 720 * q^77 - 830 * q^79 + (-639*z + 639) * q^80 + (649*z - 649) * q^81 + 576*z * q^82 + 438 * q^83 - 15*z * q^84 - 405*z * q^85 - 291 * q^86 - 144*z * q^87 + (1008*z - 1008) * q^88 + (438*z - 438) * q^89 + 702 * q^90 - 162 * q^92 + (264*z - 264) * q^93 + (333*z - 333) * q^94 + 54*z * q^95 - 45 * q^96 - 852*z * q^97 - 354*z * q^98 - 1248 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + q^{3} - q^{4} + 18 q^{5} - 3 q^{6} + 15 q^{7} + 42 q^{8} + 26 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + q^3 - q^4 + 18 * q^5 - 3 * q^6 + 15 * q^7 + 42 * q^8 + 26 * q^9 $$2 q + 3 q^{2} + q^{3} - q^{4} + 18 q^{5} - 3 q^{6} + 15 q^{7} + 42 q^{8} + 26 q^{9} + 27 q^{10} - 48 q^{11} - 2 q^{12} + 90 q^{14} + 9 q^{15} + 71 q^{16} - 45 q^{17} + 156 q^{18} + 6 q^{19} - 9 q^{20} + 30 q^{21} + 144 q^{22} + 162 q^{23} + 21 q^{24} - 88 q^{25} + 106 q^{27} + 15 q^{28} + 144 q^{29} - 27 q^{30} - 528 q^{31} - 45 q^{32} + 48 q^{33} - 270 q^{34} + 135 q^{35} + 26 q^{36} + 303 q^{37} + 36 q^{38} + 378 q^{40} - 192 q^{41} + 45 q^{42} - 97 q^{43} + 96 q^{44} + 234 q^{45} - 486 q^{46} - 222 q^{47} - 71 q^{48} + 118 q^{49} - 132 q^{50} - 90 q^{51} - 828 q^{53} + 159 q^{54} - 432 q^{55} + 315 q^{56} + 12 q^{57} - 432 q^{58} + 522 q^{59} - 18 q^{60} - 376 q^{61} - 792 q^{62} - 390 q^{63} + 866 q^{64} + 288 q^{66} - 36 q^{67} - 45 q^{68} - 162 q^{69} + 810 q^{70} + 357 q^{71} + 546 q^{72} + 2196 q^{73} - 909 q^{74} - 44 q^{75} + 6 q^{76} - 1440 q^{77} - 1660 q^{79} + 639 q^{80} - 649 q^{81} + 576 q^{82} + 876 q^{83} - 15 q^{84} - 405 q^{85} - 582 q^{86} - 144 q^{87} - 1008 q^{88} - 438 q^{89} + 1404 q^{90} - 324 q^{92} - 264 q^{93} - 333 q^{94} + 54 q^{95} - 90 q^{96} - 852 q^{97} - 354 q^{98} - 2496 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + q^3 - q^4 + 18 * q^5 - 3 * q^6 + 15 * q^7 + 42 * q^8 + 26 * q^9 + 27 * q^10 - 48 * q^11 - 2 * q^12 + 90 * q^14 + 9 * q^15 + 71 * q^16 - 45 * q^17 + 156 * q^18 + 6 * q^19 - 9 * q^20 + 30 * q^21 + 144 * q^22 + 162 * q^23 + 21 * q^24 - 88 * q^25 + 106 * q^27 + 15 * q^28 + 144 * q^29 - 27 * q^30 - 528 * q^31 - 45 * q^32 + 48 * q^33 - 270 * q^34 + 135 * q^35 + 26 * q^36 + 303 * q^37 + 36 * q^38 + 378 * q^40 - 192 * q^41 + 45 * q^42 - 97 * q^43 + 96 * q^44 + 234 * q^45 - 486 * q^46 - 222 * q^47 - 71 * q^48 + 118 * q^49 - 132 * q^50 - 90 * q^51 - 828 * q^53 + 159 * q^54 - 432 * q^55 + 315 * q^56 + 12 * q^57 - 432 * q^58 + 522 * q^59 - 18 * q^60 - 376 * q^61 - 792 * q^62 - 390 * q^63 + 866 * q^64 + 288 * q^66 - 36 * q^67 - 45 * q^68 - 162 * q^69 + 810 * q^70 + 357 * q^71 + 546 * q^72 + 2196 * q^73 - 909 * q^74 - 44 * q^75 + 6 * q^76 - 1440 * q^77 - 1660 * q^79 + 639 * q^80 - 649 * q^81 + 576 * q^82 + 876 * q^83 - 15 * q^84 - 405 * q^85 - 582 * q^86 - 144 * q^87 - 1008 * q^88 - 438 * q^89 + 1404 * q^90 - 324 * q^92 - 264 * q^93 - 333 * q^94 + 54 * q^95 - 90 * q^96 - 852 * q^97 - 354 * q^98 - 2496 * 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
1.50000 2.59808i 0.500000 0.866025i −0.500000 0.866025i 9.00000 −1.50000 2.59808i 7.50000 + 12.9904i 21.0000 13.0000 + 22.5167i 13.5000 23.3827i
146.1 1.50000 + 2.59808i 0.500000 + 0.866025i −0.500000 + 0.866025i 9.00000 −1.50000 + 2.59808i 7.50000 12.9904i 21.0000 13.0000 22.5167i 13.5000 + 23.3827i
 $$n$$: 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.c 2
13.b even 2 1 169.4.c.b 2
13.c even 3 1 169.4.a.b 1
13.c even 3 1 inner 169.4.c.c 2
13.d odd 4 2 169.4.e.d 4
13.e even 6 1 169.4.a.c 1
13.e even 6 1 169.4.c.b 2
13.f odd 12 2 13.4.b.a 2
13.f odd 12 2 169.4.e.d 4
39.h odd 6 1 1521.4.a.d 1
39.i odd 6 1 1521.4.a.i 1
39.k even 12 2 117.4.b.a 2
52.l even 12 2 208.4.f.b 2
65.o even 12 2 325.4.d.a 2
65.s odd 12 2 325.4.c.b 2
65.t even 12 2 325.4.d.b 2
104.u even 12 2 832.4.f.c 2
104.x odd 12 2 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.f odd 12 2
117.4.b.a 2 39.k even 12 2
169.4.a.b 1 13.c even 3 1
169.4.a.c 1 13.e even 6 1
169.4.c.b 2 13.b even 2 1
169.4.c.b 2 13.e even 6 1
169.4.c.c 2 1.a even 1 1 trivial
169.4.c.c 2 13.c even 3 1 inner
169.4.e.d 4 13.d odd 4 2
169.4.e.d 4 13.f odd 12 2
208.4.f.b 2 52.l even 12 2
325.4.c.b 2 65.s odd 12 2
325.4.d.a 2 65.o even 12 2
325.4.d.b 2 65.t even 12 2
832.4.f.c 2 104.u even 12 2
832.4.f.e 2 104.x odd 12 2
1521.4.a.d 1 39.h odd 6 1
1521.4.a.i 1 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T - 9)^{2}$$
$7$ $$T^{2} - 15T + 225$$
$11$ $$T^{2} + 48T + 2304$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 45T + 2025$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} - 162T + 26244$$
$29$ $$T^{2} - 144T + 20736$$
$31$ $$(T + 264)^{2}$$
$37$ $$T^{2} - 303T + 91809$$
$41$ $$T^{2} + 192T + 36864$$
$43$ $$T^{2} + 97T + 9409$$
$47$ $$(T + 111)^{2}$$
$53$ $$(T + 414)^{2}$$
$59$ $$T^{2} - 522T + 272484$$
$61$ $$T^{2} + 376T + 141376$$
$67$ $$T^{2} + 36T + 1296$$
$71$ $$T^{2} - 357T + 127449$$
$73$ $$(T - 1098)^{2}$$
$79$ $$(T + 830)^{2}$$
$83$ $$(T - 438)^{2}$$
$89$ $$T^{2} + 438T + 191844$$
$97$ $$T^{2} + 852T + 725904$$