# Properties

 Label 169.4.e.d Level $169$ Weight $4$ Character orbit 169.e 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(23,169)]

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([5]))

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

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

chi := DirichletCharacter("169.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, 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 $$\Q(\zeta_{6})$$ 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: $$3^{2}$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + \beta_{2} q^{4} + 3 \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{3} + 5 \beta_1) q^{7} - 7 \beta_{3} q^{8} + 26 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 + 1) * q^3 + b2 * q^4 + 3*b3 * q^5 + (-b3 + b1) * q^6 + (-5*b3 + 5*b1) * q^7 - 7*b3 * q^8 + 26*b2 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + \beta_{2} q^{4} + 3 \beta_{3} q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 5 \beta_{3} + 5 \beta_1) q^{7} - 7 \beta_{3} q^{8} + 26 \beta_{2} q^{9} + (27 \beta_{2} - 27) q^{10} + 16 \beta_1 q^{11} + q^{12} + 45 q^{14} + 3 \beta_1 q^{15} + ( - 71 \beta_{2} + 71) q^{16} + 45 \beta_{2} q^{17} + 26 \beta_{3} q^{18} + (2 \beta_{3} - 2 \beta_1) q^{19} + (3 \beta_{3} - 3 \beta_1) q^{20} - 5 \beta_{3} q^{21} + 144 \beta_{2} q^{22} + (162 \beta_{2} - 162) q^{23} - 7 \beta_1 q^{24} + 44 q^{25} + 53 q^{27} + 5 \beta_1 q^{28} + ( - 144 \beta_{2} + 144) q^{29} + 27 \beta_{2} q^{30} - 88 \beta_{3} q^{31} + ( - 15 \beta_{3} + 15 \beta_1) q^{32} + ( - 16 \beta_{3} + 16 \beta_1) q^{33} + 45 \beta_{3} q^{34} + 135 \beta_{2} q^{35} + (26 \beta_{2} - 26) q^{36} - 101 \beta_1 q^{37} - 18 q^{38} + 189 q^{40} - 64 \beta_1 q^{41} + ( - 45 \beta_{2} + 45) q^{42} + 97 \beta_{2} q^{43} + 16 \beta_{3} q^{44} + (78 \beta_{3} - 78 \beta_1) q^{45} + (162 \beta_{3} - 162 \beta_1) q^{46} + 37 \beta_{3} q^{47} - 71 \beta_{2} q^{48} + (118 \beta_{2} - 118) q^{49} + 44 \beta_1 q^{50} + 45 q^{51} - 414 q^{53} + 53 \beta_1 q^{54} + (432 \beta_{2} - 432) q^{55} - 315 \beta_{2} q^{56} + 2 \beta_{3} q^{57} + ( - 144 \beta_{3} + 144 \beta_1) q^{58} + ( - 174 \beta_{3} + 174 \beta_1) q^{59} + 3 \beta_{3} q^{60} - 376 \beta_{2} q^{61} + ( - 792 \beta_{2} + 792) q^{62} + 130 \beta_1 q^{63} - 433 q^{64} + 144 q^{66} - 12 \beta_1 q^{67} + (45 \beta_{2} - 45) q^{68} + 162 \beta_{2} q^{69} + 135 \beta_{3} q^{70} + (119 \beta_{3} - 119 \beta_1) q^{71} + ( - 182 \beta_{3} + 182 \beta_1) q^{72} - 366 \beta_{3} q^{73} - 909 \beta_{2} q^{74} + ( - 44 \beta_{2} + 44) q^{75} - 2 \beta_1 q^{76} + 720 q^{77} - 830 q^{79} + 213 \beta_1 q^{80} + (649 \beta_{2} - 649) q^{81} - 576 \beta_{2} q^{82} + 146 \beta_{3} q^{83} + ( - 5 \beta_{3} + 5 \beta_1) q^{84} + (135 \beta_{3} - 135 \beta_1) q^{85} + 97 \beta_{3} q^{86} - 144 \beta_{2} q^{87} + ( - 1008 \beta_{2} + 1008) q^{88} + 146 \beta_1 q^{89} - 702 q^{90} - 162 q^{92} - 88 \beta_1 q^{93} + (333 \beta_{2} - 333) q^{94} - 54 \beta_{2} q^{95} - 15 \beta_{3} q^{96} + ( - 284 \beta_{3} + 284 \beta_1) q^{97} + (118 \beta_{3} - 118 \beta_1) q^{98} + 416 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 + 1) * q^3 + b2 * q^4 + 3*b3 * q^5 + (-b3 + b1) * q^6 + (-5*b3 + 5*b1) * q^7 - 7*b3 * q^8 + 26*b2 * q^9 + (27*b2 - 27) * q^10 + 16*b1 * q^11 + q^12 + 45 * q^14 + 3*b1 * q^15 + (-71*b2 + 71) * q^16 + 45*b2 * q^17 + 26*b3 * q^18 + (2*b3 - 2*b1) * q^19 + (3*b3 - 3*b1) * q^20 - 5*b3 * q^21 + 144*b2 * q^22 + (162*b2 - 162) * q^23 - 7*b1 * q^24 + 44 * q^25 + 53 * q^27 + 5*b1 * q^28 + (-144*b2 + 144) * q^29 + 27*b2 * q^30 - 88*b3 * q^31 + (-15*b3 + 15*b1) * q^32 + (-16*b3 + 16*b1) * q^33 + 45*b3 * q^34 + 135*b2 * q^35 + (26*b2 - 26) * q^36 - 101*b1 * q^37 - 18 * q^38 + 189 * q^40 - 64*b1 * q^41 + (-45*b2 + 45) * q^42 + 97*b2 * q^43 + 16*b3 * q^44 + (78*b3 - 78*b1) * q^45 + (162*b3 - 162*b1) * q^46 + 37*b3 * q^47 - 71*b2 * q^48 + (118*b2 - 118) * q^49 + 44*b1 * q^50 + 45 * q^51 - 414 * q^53 + 53*b1 * q^54 + (432*b2 - 432) * q^55 - 315*b2 * q^56 + 2*b3 * q^57 + (-144*b3 + 144*b1) * q^58 + (-174*b3 + 174*b1) * q^59 + 3*b3 * q^60 - 376*b2 * q^61 + (-792*b2 + 792) * q^62 + 130*b1 * q^63 - 433 * q^64 + 144 * q^66 - 12*b1 * q^67 + (45*b2 - 45) * q^68 + 162*b2 * q^69 + 135*b3 * q^70 + (119*b3 - 119*b1) * q^71 + (-182*b3 + 182*b1) * q^72 - 366*b3 * q^73 - 909*b2 * q^74 + (-44*b2 + 44) * q^75 - 2*b1 * q^76 + 720 * q^77 - 830 * q^79 + 213*b1 * q^80 + (649*b2 - 649) * q^81 - 576*b2 * q^82 + 146*b3 * q^83 + (-5*b3 + 5*b1) * q^84 + (135*b3 - 135*b1) * q^85 + 97*b3 * q^86 - 144*b2 * q^87 + (-1008*b2 + 1008) * q^88 + 146*b1 * q^89 - 702 * q^90 - 162 * q^92 - 88*b1 * q^93 + (333*b2 - 333) * q^94 - 54*b2 * q^95 - 15*b3 * q^96 + (-284*b3 + 284*b1) * q^97 + (118*b3 - 118*b1) * q^98 + 416*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 2 q^{4} + 52 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 2 * q^4 + 52 * q^9 $$4 q + 2 q^{3} + 2 q^{4} + 52 q^{9} - 54 q^{10} + 4 q^{12} + 180 q^{14} + 142 q^{16} + 90 q^{17} + 288 q^{22} - 324 q^{23} + 176 q^{25} + 212 q^{27} + 288 q^{29} + 54 q^{30} + 270 q^{35} - 52 q^{36} - 72 q^{38} + 756 q^{40} + 90 q^{42} + 194 q^{43} - 142 q^{48} - 236 q^{49} + 180 q^{51} - 1656 q^{53} - 864 q^{55} - 630 q^{56} - 752 q^{61} + 1584 q^{62} - 1732 q^{64} + 576 q^{66} - 90 q^{68} + 324 q^{69} - 1818 q^{74} + 88 q^{75} + 2880 q^{77} - 3320 q^{79} - 1298 q^{81} - 1152 q^{82} - 288 q^{87} + 2016 q^{88} - 2808 q^{90} - 648 q^{92} - 666 q^{94} - 108 q^{95}+O(q^{100})$$ 4 * q + 2 * q^3 + 2 * q^4 + 52 * q^9 - 54 * q^10 + 4 * q^12 + 180 * q^14 + 142 * q^16 + 90 * q^17 + 288 * q^22 - 324 * q^23 + 176 * q^25 + 212 * q^27 + 288 * q^29 + 54 * q^30 + 270 * q^35 - 52 * q^36 - 72 * q^38 + 756 * q^40 + 90 * q^42 + 194 * q^43 - 142 * q^48 - 236 * q^49 + 180 * q^51 - 1656 * q^53 - 864 * q^55 - 630 * q^56 - 752 * q^61 + 1584 * q^62 - 1732 * q^64 + 576 * q^66 - 90 * q^68 + 324 * q^69 - 1818 * q^74 + 88 * q^75 + 2880 * q^77 - 3320 * q^79 - 1298 * q^81 - 1152 * q^82 - 288 * q^87 + 2016 * q^88 - 2808 * q^90 - 648 * q^92 - 666 * q^94 - 108 * q^95

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−2.59808 + 1.50000i 0.500000 + 0.866025i 0.500000 0.866025i 9.00000i −2.59808 1.50000i −12.9904 7.50000i 21.0000i 13.0000 22.5167i −13.5000 23.3827i
23.2 2.59808 1.50000i 0.500000 + 0.866025i 0.500000 0.866025i 9.00000i 2.59808 + 1.50000i 12.9904 + 7.50000i 21.0000i 13.0000 22.5167i −13.5000 23.3827i
147.1 −2.59808 1.50000i 0.500000 0.866025i 0.500000 + 0.866025i 9.00000i −2.59808 + 1.50000i −12.9904 + 7.50000i 21.0000i 13.0000 + 22.5167i −13.5000 + 23.3827i
147.2 2.59808 + 1.50000i 0.500000 0.866025i 0.500000 + 0.866025i 9.00000i 2.59808 1.50000i 12.9904 7.50000i 21.0000i 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.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.e.d 4
13.b even 2 1 inner 169.4.e.d 4
13.c even 3 1 13.4.b.a 2
13.c even 3 1 inner 169.4.e.d 4
13.d odd 4 1 169.4.c.b 2
13.d odd 4 1 169.4.c.c 2
13.e even 6 1 13.4.b.a 2
13.e even 6 1 inner 169.4.e.d 4
13.f odd 12 1 169.4.a.b 1
13.f odd 12 1 169.4.a.c 1
13.f odd 12 1 169.4.c.b 2
13.f odd 12 1 169.4.c.c 2
39.h odd 6 1 117.4.b.a 2
39.i odd 6 1 117.4.b.a 2
39.k even 12 1 1521.4.a.d 1
39.k even 12 1 1521.4.a.i 1
52.i odd 6 1 208.4.f.b 2
52.j odd 6 1 208.4.f.b 2
65.l even 6 1 325.4.c.b 2
65.n even 6 1 325.4.c.b 2
65.q odd 12 1 325.4.d.a 2
65.q odd 12 1 325.4.d.b 2
65.r odd 12 1 325.4.d.a 2
65.r odd 12 1 325.4.d.b 2
104.n odd 6 1 832.4.f.c 2
104.p odd 6 1 832.4.f.c 2
104.r even 6 1 832.4.f.e 2
104.s even 6 1 832.4.f.e 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 9T_{2}^{2} + 81$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 9T^{2} + 81$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + 81)^{2}$$
$7$ $$T^{4} - 225 T^{2} + 50625$$
$11$ $$T^{4} - 2304 T^{2} + \cdots + 5308416$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 45 T + 2025)^{2}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$(T^{2} + 162 T + 26244)^{2}$$
$29$ $$(T^{2} - 144 T + 20736)^{2}$$
$31$ $$(T^{2} + 69696)^{2}$$
$37$ $$T^{4} - 91809 T^{2} + \cdots + 8428892481$$
$41$ $$T^{4} - 36864 T^{2} + \cdots + 1358954496$$
$43$ $$(T^{2} - 97 T + 9409)^{2}$$
$47$ $$(T^{2} + 12321)^{2}$$
$53$ $$(T + 414)^{4}$$
$59$ $$T^{4} - 272484 T^{2} + \cdots + 74247530256$$
$61$ $$(T^{2} + 376 T + 141376)^{2}$$
$67$ $$T^{4} - 1296 T^{2} + \cdots + 1679616$$
$71$ $$T^{4} - 127449 T^{2} + \cdots + 16243247601$$
$73$ $$(T^{2} + 1205604)^{2}$$
$79$ $$(T + 830)^{4}$$
$83$ $$(T^{2} + 191844)^{2}$$
$89$ $$T^{4} - 191844 T^{2} + \cdots + 36804120336$$
$97$ $$T^{4} - 725904 T^{2} + \cdots + 526936617216$$