# Properties

 Label 169.4.c.h Level $169$ Weight $4$ Character orbit 169.c Analytic conductor $9.971$ Analytic rank $0$ Dimension $4$ CM no 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 5 \beta_1 + 5) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + (8 \beta_{3} - 8 \beta_{2}) q^{7} - 13 \beta_{3} q^{8} + ( - 23 \beta_1 + 23) q^{9}+O(q^{10})$$ q - b2 * q^2 - 2*b1 * q^3 + (-5*b1 + 5) * q^4 + b3 * q^5 + (-2*b3 + 2*b2) * q^6 + (8*b3 - 8*b2) * q^7 - 13*b3 * q^8 + (-23*b1 + 23) * q^9 $$q - \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 5 \beta_1 + 5) q^{4} + \beta_{3} q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{6} + (8 \beta_{3} - 8 \beta_{2}) q^{7} - 13 \beta_{3} q^{8} + ( - 23 \beta_1 + 23) q^{9} - 3 \beta_1 q^{10} + 8 \beta_{2} q^{11} - 10 q^{12} - 24 q^{14} - 2 \beta_{2} q^{15} - \beta_1 q^{16} + ( - 117 \beta_1 + 117) q^{17} - 23 \beta_{3} q^{18} + ( - 66 \beta_{3} + 66 \beta_{2}) q^{19} + (5 \beta_{3} - 5 \beta_{2}) q^{20} - 16 \beta_{3} q^{21} + ( - 24 \beta_1 + 24) q^{22} - 78 \beta_1 q^{23} + 26 \beta_{2} q^{24} - 122 q^{25} - 100 q^{27} - 40 \beta_{2} q^{28} + 141 \beta_1 q^{29} + (6 \beta_1 - 6) q^{30} - 90 \beta_{3} q^{31} + ( - 105 \beta_{3} + 105 \beta_{2}) q^{32} + (16 \beta_{3} - 16 \beta_{2}) q^{33} - 117 \beta_{3} q^{34} + ( - 24 \beta_1 + 24) q^{35} - 115 \beta_1 q^{36} - 83 \beta_{2} q^{37} + 198 q^{38} - 39 q^{40} - 157 \beta_{2} q^{41} + 48 \beta_1 q^{42} + ( - 104 \beta_1 + 104) q^{43} + 40 \beta_{3} q^{44} + (23 \beta_{3} - 23 \beta_{2}) q^{45} + ( - 78 \beta_{3} + 78 \beta_{2}) q^{46} + 174 \beta_{3} q^{47} + (2 \beta_1 - 2) q^{48} + 151 \beta_1 q^{49} + 122 \beta_{2} q^{50} - 234 q^{51} + 93 q^{53} + 100 \beta_{2} q^{54} + 24 \beta_1 q^{55} + (312 \beta_1 - 312) q^{56} + 132 \beta_{3} q^{57} + (141 \beta_{3} - 141 \beta_{2}) q^{58} + (164 \beta_{3} - 164 \beta_{2}) q^{59} - 10 \beta_{3} q^{60} + (145 \beta_1 - 145) q^{61} + 270 \beta_1 q^{62} - 184 \beta_{2} q^{63} + 307 q^{64} - 48 q^{66} + 454 \beta_{2} q^{67} - 585 \beta_1 q^{68} + (156 \beta_1 - 156) q^{69} - 24 \beta_{3} q^{70} + (610 \beta_{3} - 610 \beta_{2}) q^{71} + ( - 299 \beta_{3} + 299 \beta_{2}) q^{72} + 265 \beta_{3} q^{73} + (249 \beta_1 - 249) q^{74} + 244 \beta_1 q^{75} + 330 \beta_{2} q^{76} + 192 q^{77} + 1276 q^{79} - \beta_{2} q^{80} - 421 \beta_1 q^{81} + (471 \beta_1 - 471) q^{82} + 456 \beta_{3} q^{83} + ( - 80 \beta_{3} + 80 \beta_{2}) q^{84} + (117 \beta_{3} - 117 \beta_{2}) q^{85} - 104 \beta_{3} q^{86} + ( - 282 \beta_1 + 282) q^{87} - 312 \beta_1 q^{88} - 564 \beta_{2} q^{89} - 69 q^{90} - 390 q^{92} + 180 \beta_{2} q^{93} - 522 \beta_1 q^{94} + (198 \beta_1 - 198) q^{95} + 210 \beta_{3} q^{96} + (116 \beta_{3} - 116 \beta_{2}) q^{97} + (151 \beta_{3} - 151 \beta_{2}) q^{98} + 184 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^2 - 2*b1 * q^3 + (-5*b1 + 5) * q^4 + b3 * q^5 + (-2*b3 + 2*b2) * q^6 + (8*b3 - 8*b2) * q^7 - 13*b3 * q^8 + (-23*b1 + 23) * q^9 - 3*b1 * q^10 + 8*b2 * q^11 - 10 * q^12 - 24 * q^14 - 2*b2 * q^15 - b1 * q^16 + (-117*b1 + 117) * q^17 - 23*b3 * q^18 + (-66*b3 + 66*b2) * q^19 + (5*b3 - 5*b2) * q^20 - 16*b3 * q^21 + (-24*b1 + 24) * q^22 - 78*b1 * q^23 + 26*b2 * q^24 - 122 * q^25 - 100 * q^27 - 40*b2 * q^28 + 141*b1 * q^29 + (6*b1 - 6) * q^30 - 90*b3 * q^31 + (-105*b3 + 105*b2) * q^32 + (16*b3 - 16*b2) * q^33 - 117*b3 * q^34 + (-24*b1 + 24) * q^35 - 115*b1 * q^36 - 83*b2 * q^37 + 198 * q^38 - 39 * q^40 - 157*b2 * q^41 + 48*b1 * q^42 + (-104*b1 + 104) * q^43 + 40*b3 * q^44 + (23*b3 - 23*b2) * q^45 + (-78*b3 + 78*b2) * q^46 + 174*b3 * q^47 + (2*b1 - 2) * q^48 + 151*b1 * q^49 + 122*b2 * q^50 - 234 * q^51 + 93 * q^53 + 100*b2 * q^54 + 24*b1 * q^55 + (312*b1 - 312) * q^56 + 132*b3 * q^57 + (141*b3 - 141*b2) * q^58 + (164*b3 - 164*b2) * q^59 - 10*b3 * q^60 + (145*b1 - 145) * q^61 + 270*b1 * q^62 - 184*b2 * q^63 + 307 * q^64 - 48 * q^66 + 454*b2 * q^67 - 585*b1 * q^68 + (156*b1 - 156) * q^69 - 24*b3 * q^70 + (610*b3 - 610*b2) * q^71 + (-299*b3 + 299*b2) * q^72 + 265*b3 * q^73 + (249*b1 - 249) * q^74 + 244*b1 * q^75 + 330*b2 * q^76 + 192 * q^77 + 1276 * q^79 - b2 * q^80 - 421*b1 * q^81 + (471*b1 - 471) * q^82 + 456*b3 * q^83 + (-80*b3 + 80*b2) * q^84 + (117*b3 - 117*b2) * q^85 - 104*b3 * q^86 + (-282*b1 + 282) * q^87 - 312*b1 * q^88 - 564*b2 * q^89 - 69 * q^90 - 390 * q^92 + 180*b2 * q^93 - 522*b1 * q^94 + (198*b1 - 198) * q^95 + 210*b3 * q^96 + (116*b3 - 116*b2) * q^97 + (151*b3 - 151*b2) * q^98 + 184*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 10 q^{4} + 46 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 10 * q^4 + 46 * q^9 $$4 q - 4 q^{3} + 10 q^{4} + 46 q^{9} - 6 q^{10} - 40 q^{12} - 96 q^{14} - 2 q^{16} + 234 q^{17} + 48 q^{22} - 156 q^{23} - 488 q^{25} - 400 q^{27} + 282 q^{29} - 12 q^{30} + 48 q^{35} - 230 q^{36} + 792 q^{38} - 156 q^{40} + 96 q^{42} + 208 q^{43} - 4 q^{48} + 302 q^{49} - 936 q^{51} + 372 q^{53} + 48 q^{55} - 624 q^{56} - 290 q^{61} + 540 q^{62} + 1228 q^{64} - 192 q^{66} - 1170 q^{68} - 312 q^{69} - 498 q^{74} + 488 q^{75} + 768 q^{77} + 5104 q^{79} - 842 q^{81} - 942 q^{82} + 564 q^{87} - 624 q^{88} - 276 q^{90} - 1560 q^{92} - 1044 q^{94} - 396 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 + 10 * q^4 + 46 * q^9 - 6 * q^10 - 40 * q^12 - 96 * q^14 - 2 * q^16 + 234 * q^17 + 48 * q^22 - 156 * q^23 - 488 * q^25 - 400 * q^27 + 282 * q^29 - 12 * q^30 + 48 * q^35 - 230 * q^36 + 792 * q^38 - 156 * q^40 + 96 * q^42 + 208 * q^43 - 4 * q^48 + 302 * q^49 - 936 * q^51 + 372 * q^53 + 48 * q^55 - 624 * q^56 - 290 * q^61 + 540 * q^62 + 1228 * q^64 - 192 * q^66 - 1170 * q^68 - 312 * q^69 - 498 * q^74 + 488 * q^75 + 768 * q^77 + 5104 * q^79 - 842 * q^81 - 942 * q^82 + 564 * q^87 - 624 * q^88 - 276 * q^90 - 1560 * q^92 - 1044 * q^94 - 396 * 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
−0.866025 + 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i 1.73205 −1.73205 3.00000i 6.92820 + 12.0000i −22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i 2.50000 + 4.33013i −1.73205 1.73205 + 3.00000i −6.92820 12.0000i 22.5167 11.5000 + 19.9186i −1.50000 + 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i 2.50000 4.33013i 1.73205 −1.73205 + 3.00000i 6.92820 12.0000i −22.5167 11.5000 19.9186i −1.50000 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i 2.50000 4.33013i −1.73205 1.73205 3.00000i −6.92820 + 12.0000i 22.5167 11.5000 19.9186i −1.50000 2.59808i
 $$n$$: 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.h 4
13.b even 2 1 inner 169.4.c.h 4
13.c even 3 1 169.4.a.i 2
13.c even 3 1 inner 169.4.c.h 4
13.d odd 4 1 13.4.e.b 2
13.d odd 4 1 169.4.e.a 2
13.e even 6 1 169.4.a.i 2
13.e even 6 1 inner 169.4.c.h 4
13.f odd 12 1 13.4.e.b 2
13.f odd 12 2 169.4.b.d 2
13.f odd 12 1 169.4.e.a 2
39.f even 4 1 117.4.q.a 2
39.h odd 6 1 1521.4.a.o 2
39.i odd 6 1 1521.4.a.o 2
39.k even 12 1 117.4.q.a 2
52.f even 4 1 208.4.w.b 2
52.l even 12 1 208.4.w.b 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$(T^{2} + 2 T + 4)^{2}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4} + 192 T^{2} + 36864$$
$11$ $$T^{4} + 192 T^{2} + 36864$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 117 T + 13689)^{2}$$
$19$ $$T^{4} + 13068 T^{2} + 170772624$$
$23$ $$(T^{2} + 78 T + 6084)^{2}$$
$29$ $$(T^{2} - 141 T + 19881)^{2}$$
$31$ $$(T^{2} - 24300)^{2}$$
$37$ $$T^{4} + 20667 T^{2} + 427124889$$
$41$ $$T^{4} + \cdots + 5468158809$$
$43$ $$(T^{2} - 104 T + 10816)^{2}$$
$47$ $$(T^{2} - 90828)^{2}$$
$53$ $$(T - 93)^{4}$$
$59$ $$T^{4} + \cdots + 6510553344$$
$61$ $$(T^{2} + 145 T + 21025)^{2}$$
$67$ $$T^{4} + \cdots + 382354249104$$
$71$ $$T^{4} + \cdots + 1246125690000$$
$73$ $$(T^{2} - 210675)^{2}$$
$79$ $$(T - 1276)^{4}$$
$83$ $$(T^{2} - 623808)^{2}$$
$89$ $$T^{4} + \cdots + 910665586944$$
$97$ $$T^{4} + \cdots + 1629575424$$