# Properties

 Label 169.4.c.i Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + 7 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + 8 \beta_{3} q^{5} + (14 \beta_{3} - 14 \beta_{2}) q^{6} + (13 \beta_{3} - 13 \beta_{2}) q^{7} - 8 \beta_{3} q^{8} + (22 \beta_1 - 22) q^{9}+O(q^{10})$$ q - 2*b2 * q^2 + 7*b1 * q^3 + (4*b1 - 4) * q^4 + 8*b3 * q^5 + (14*b3 - 14*b2) * q^6 + (13*b3 - 13*b2) * q^7 - 8*b3 * q^8 + (22*b1 - 22) * q^9 $$q - 2 \beta_{2} q^{2} + 7 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + 8 \beta_{3} q^{5} + (14 \beta_{3} - 14 \beta_{2}) q^{6} + (13 \beta_{3} - 13 \beta_{2}) q^{7} - 8 \beta_{3} q^{8} + (22 \beta_1 - 22) q^{9} - 48 \beta_1 q^{10} + 13 \beta_{2} q^{11} - 28 q^{12} - 78 q^{14} + 56 \beta_{2} q^{15} + 80 \beta_1 q^{16} + ( - 27 \beta_1 + 27) q^{17} + 44 \beta_{3} q^{18} + (51 \beta_{3} - 51 \beta_{2}) q^{19} + ( - 32 \beta_{3} + 32 \beta_{2}) q^{20} + 91 \beta_{3} q^{21} + ( - 78 \beta_1 + 78) q^{22} + 57 \beta_1 q^{23} - 56 \beta_{2} q^{24} + 67 q^{25} + 35 q^{27} + 52 \beta_{2} q^{28} + 69 \beta_1 q^{29} + ( - 336 \beta_1 + 336) q^{30} - 42 \beta_{3} q^{31} + (96 \beta_{3} - 96 \beta_{2}) q^{32} + ( - 91 \beta_{3} + 91 \beta_{2}) q^{33} - 54 \beta_{3} q^{34} + ( - 312 \beta_1 + 312) q^{35} - 88 \beta_1 q^{36} + 23 \beta_{2} q^{37} - 306 q^{38} - 192 q^{40} - 227 \beta_{2} q^{41} - 546 \beta_1 q^{42} + (85 \beta_1 - 85) q^{43} - 52 \beta_{3} q^{44} + ( - 176 \beta_{3} + 176 \beta_{2}) q^{45} + (114 \beta_{3} - 114 \beta_{2}) q^{46} + 198 \beta_{3} q^{47} + (560 \beta_1 - 560) q^{48} - 164 \beta_1 q^{49} - 134 \beta_{2} q^{50} + 189 q^{51} + 426 q^{53} - 70 \beta_{2} q^{54} + 312 \beta_1 q^{55} + (312 \beta_1 - 312) q^{56} + 357 \beta_{3} q^{57} + (138 \beta_{3} - 138 \beta_{2}) q^{58} + ( - 11 \beta_{3} + 11 \beta_{2}) q^{59} - 224 \beta_{3} q^{60} + ( - 17 \beta_1 + 17) q^{61} + 252 \beta_1 q^{62} + 286 \beta_{2} q^{63} + 64 q^{64} + 546 q^{66} + 95 \beta_{2} q^{67} + 108 \beta_1 q^{68} + (399 \beta_1 - 399) q^{69} - 624 \beta_{3} q^{70} + ( - 337 \beta_{3} + 337 \beta_{2}) q^{71} + (176 \beta_{3} - 176 \beta_{2}) q^{72} - 580 \beta_{3} q^{73} + ( - 138 \beta_1 + 138) q^{74} + 469 \beta_1 q^{75} + 204 \beta_{2} q^{76} + 507 q^{77} - 1244 q^{79} + 640 \beta_{2} q^{80} + 839 \beta_1 q^{81} + (1362 \beta_1 - 1362) q^{82} - 246 \beta_{3} q^{83} + ( - 364 \beta_{3} + 364 \beta_{2}) q^{84} + (216 \beta_{3} - 216 \beta_{2}) q^{85} + 170 \beta_{3} q^{86} + (483 \beta_1 - 483) q^{87} - 312 \beta_1 q^{88} - 177 \beta_{2} q^{89} + 1056 q^{90} - 228 q^{92} - 294 \beta_{2} q^{93} - 1188 \beta_1 q^{94} + ( - 1224 \beta_1 + 1224) q^{95} + 672 \beta_{3} q^{96} + ( - 713 \beta_{3} + 713 \beta_{2}) q^{97} + ( - 328 \beta_{3} + 328 \beta_{2}) q^{98} - 286 \beta_{3} q^{99}+O(q^{100})$$ q - 2*b2 * q^2 + 7*b1 * q^3 + (4*b1 - 4) * q^4 + 8*b3 * q^5 + (14*b3 - 14*b2) * q^6 + (13*b3 - 13*b2) * q^7 - 8*b3 * q^8 + (22*b1 - 22) * q^9 - 48*b1 * q^10 + 13*b2 * q^11 - 28 * q^12 - 78 * q^14 + 56*b2 * q^15 + 80*b1 * q^16 + (-27*b1 + 27) * q^17 + 44*b3 * q^18 + (51*b3 - 51*b2) * q^19 + (-32*b3 + 32*b2) * q^20 + 91*b3 * q^21 + (-78*b1 + 78) * q^22 + 57*b1 * q^23 - 56*b2 * q^24 + 67 * q^25 + 35 * q^27 + 52*b2 * q^28 + 69*b1 * q^29 + (-336*b1 + 336) * q^30 - 42*b3 * q^31 + (96*b3 - 96*b2) * q^32 + (-91*b3 + 91*b2) * q^33 - 54*b3 * q^34 + (-312*b1 + 312) * q^35 - 88*b1 * q^36 + 23*b2 * q^37 - 306 * q^38 - 192 * q^40 - 227*b2 * q^41 - 546*b1 * q^42 + (85*b1 - 85) * q^43 - 52*b3 * q^44 + (-176*b3 + 176*b2) * q^45 + (114*b3 - 114*b2) * q^46 + 198*b3 * q^47 + (560*b1 - 560) * q^48 - 164*b1 * q^49 - 134*b2 * q^50 + 189 * q^51 + 426 * q^53 - 70*b2 * q^54 + 312*b1 * q^55 + (312*b1 - 312) * q^56 + 357*b3 * q^57 + (138*b3 - 138*b2) * q^58 + (-11*b3 + 11*b2) * q^59 - 224*b3 * q^60 + (-17*b1 + 17) * q^61 + 252*b1 * q^62 + 286*b2 * q^63 + 64 * q^64 + 546 * q^66 + 95*b2 * q^67 + 108*b1 * q^68 + (399*b1 - 399) * q^69 - 624*b3 * q^70 + (-337*b3 + 337*b2) * q^71 + (176*b3 - 176*b2) * q^72 - 580*b3 * q^73 + (-138*b1 + 138) * q^74 + 469*b1 * q^75 + 204*b2 * q^76 + 507 * q^77 - 1244 * q^79 + 640*b2 * q^80 + 839*b1 * q^81 + (1362*b1 - 1362) * q^82 - 246*b3 * q^83 + (-364*b3 + 364*b2) * q^84 + (216*b3 - 216*b2) * q^85 + 170*b3 * q^86 + (483*b1 - 483) * q^87 - 312*b1 * q^88 - 177*b2 * q^89 + 1056 * q^90 - 228 * q^92 - 294*b2 * q^93 - 1188*b1 * q^94 + (-1224*b1 + 1224) * q^95 + 672*b3 * q^96 + (-713*b3 + 713*b2) * q^97 + (-328*b3 + 328*b2) * q^98 - 286*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{3} - 8 q^{4} - 44 q^{9}+O(q^{10})$$ 4 * q + 14 * q^3 - 8 * q^4 - 44 * q^9 $$4 q + 14 q^{3} - 8 q^{4} - 44 q^{9} - 96 q^{10} - 112 q^{12} - 312 q^{14} + 160 q^{16} + 54 q^{17} + 156 q^{22} + 114 q^{23} + 268 q^{25} + 140 q^{27} + 138 q^{29} + 672 q^{30} + 624 q^{35} - 176 q^{36} - 1224 q^{38} - 768 q^{40} - 1092 q^{42} - 170 q^{43} - 1120 q^{48} - 328 q^{49} + 756 q^{51} + 1704 q^{53} + 624 q^{55} - 624 q^{56} + 34 q^{61} + 504 q^{62} + 256 q^{64} + 2184 q^{66} + 216 q^{68} - 798 q^{69} + 276 q^{74} + 938 q^{75} + 2028 q^{77} - 4976 q^{79} + 1678 q^{81} - 2724 q^{82} - 966 q^{87} - 624 q^{88} + 4224 q^{90} - 912 q^{92} - 2376 q^{94} + 2448 q^{95}+O(q^{100})$$ 4 * q + 14 * q^3 - 8 * q^4 - 44 * q^9 - 96 * q^10 - 112 * q^12 - 312 * q^14 + 160 * q^16 + 54 * q^17 + 156 * q^22 + 114 * q^23 + 268 * q^25 + 140 * q^27 + 138 * q^29 + 672 * q^30 + 624 * q^35 - 176 * q^36 - 1224 * q^38 - 768 * q^40 - 1092 * q^42 - 170 * q^43 - 1120 * q^48 - 328 * q^49 + 756 * q^51 + 1704 * q^53 + 624 * q^55 - 624 * q^56 + 34 * q^61 + 504 * q^62 + 256 * q^64 + 2184 * q^66 + 216 * q^68 - 798 * q^69 + 276 * q^74 + 938 * q^75 + 2028 * q^77 - 4976 * q^79 + 1678 * q^81 - 2724 * q^82 - 966 * q^87 - 624 * q^88 + 4224 * q^90 - 912 * q^92 - 2376 * q^94 + 2448 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−1.73205 + 3.00000i 3.50000 6.06218i −2.00000 3.46410i 13.8564 12.1244 + 21.0000i 11.2583 + 19.5000i −13.8564 −11.0000 19.0526i −24.0000 + 41.5692i
22.2 1.73205 3.00000i 3.50000 6.06218i −2.00000 3.46410i −13.8564 −12.1244 21.0000i −11.2583 19.5000i 13.8564 −11.0000 19.0526i −24.0000 + 41.5692i
146.1 −1.73205 3.00000i 3.50000 + 6.06218i −2.00000 + 3.46410i 13.8564 12.1244 21.0000i 11.2583 19.5000i −13.8564 −11.0000 + 19.0526i −24.0000 41.5692i
146.2 1.73205 + 3.00000i 3.50000 + 6.06218i −2.00000 + 3.46410i −13.8564 −12.1244 + 21.0000i −11.2583 + 19.5000i 13.8564 −11.0000 + 19.0526i −24.0000 41.5692i
 $$n$$: 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
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.i 4
13.b even 2 1 inner 169.4.c.i 4
13.c even 3 1 169.4.a.h 2
13.c even 3 1 inner 169.4.c.i 4
13.d odd 4 1 13.4.e.a 2
13.d odd 4 1 169.4.e.b 2
13.e even 6 1 169.4.a.h 2
13.e even 6 1 inner 169.4.c.i 4
13.f odd 12 1 13.4.e.a 2
13.f odd 12 2 169.4.b.b 2
13.f odd 12 1 169.4.e.b 2
39.f even 4 1 117.4.q.c 2
39.h odd 6 1 1521.4.a.q 2
39.i odd 6 1 1521.4.a.q 2
39.k even 12 1 117.4.q.c 2
52.f even 4 1 208.4.w.a 2
52.l even 12 1 208.4.w.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 12T_{2}^{2} + 144$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 12T^{2} + 144$$
$3$ $$(T^{2} - 7 T + 49)^{2}$$
$5$ $$(T^{2} - 192)^{2}$$
$7$ $$T^{4} + 507 T^{2} + 257049$$
$11$ $$T^{4} + 507 T^{2} + 257049$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 27 T + 729)^{2}$$
$19$ $$T^{4} + 7803 T^{2} + 60886809$$
$23$ $$(T^{2} - 57 T + 3249)^{2}$$
$29$ $$(T^{2} - 69 T + 4761)^{2}$$
$31$ $$(T^{2} - 5292)^{2}$$
$37$ $$T^{4} + 1587 T^{2} + 2518569$$
$41$ $$T^{4} + \cdots + 23897140569$$
$43$ $$(T^{2} + 85 T + 7225)^{2}$$
$47$ $$(T^{2} - 117612)^{2}$$
$53$ $$(T - 426)^{4}$$
$59$ $$T^{4} + 363 T^{2} + 131769$$
$61$ $$(T^{2} - 17 T + 289)^{2}$$
$67$ $$T^{4} + 27075 T^{2} + 733055625$$
$71$ $$T^{4} + \cdots + 116081259849$$
$73$ $$(T^{2} - 1009200)^{2}$$
$79$ $$(T + 1244)^{4}$$
$83$ $$(T^{2} - 181548)^{2}$$
$89$ $$T^{4} + \cdots + 8833556169$$
$97$ $$T^{4} + \cdots + 2325951361449$$