# Properties

 Label 169.4.e.f Level $169$ Weight $4$ Character orbit 169.e Analytic conductor $9.971$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.1731891456.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} + \cdots - 11 \beta_1) q^{8}+ \cdots + (15 \beta_{7} - 15 \beta_{4} + \cdots - 25) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-3*b7 + b2) * q^3 + (b7 - b4 - 3*b2 - 4) * q^4 + (b6 + b5 - b3 - b1) * q^5 + (-6*b6 - b5) * q^6 + (5*b6 + 11*b5) * q^7 + (2*b6 + 11*b5 - 2*b3 - 11*b1) * q^8 + (15*b7 - 15*b4 - 10*b2 - 25) * q^9 $$q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} + \cdots - 11 \beta_1) q^{8}+ \cdots + (65 \beta_{6} + 390 \beta_{5} + \cdots - 390 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-3*b7 + b2) * q^3 + (b7 - b4 - 3*b2 - 4) * q^4 + (b6 + b5 - b3 - b1) * q^5 + (-6*b6 - b5) * q^6 + (5*b6 + 11*b5) * q^7 + (2*b6 + 11*b5 - 2*b3 - 11*b1) * q^8 + (15*b7 - 15*b4 - 10*b2 - 25) * q^9 + (b7 - 3*b2) * q^10 + (17*b3 - 12*b1) * q^11 + (13*b4 + 28) * q^12 + (-b4 + 44) * q^14 + (10*b3 + 7*b1) * q^15 + (-15*b7 - 27*b2) * q^16 + (-17*b7 + 17*b4 + b2 + 18) * q^17 + (30*b6 + 10*b5 - 30*b3 - 10*b1) * q^18 + (-13*b6 + 32*b5) * q^19 + (-6*b6 - 5*b5) * q^20 + (-86*b6 - 41*b5 + 86*b3 + 41*b1) * q^21 + (-46*b7 + 46*b4 - 94*b2 - 48) * q^22 + (12*b7 - 92*b2) * q^23 + (74*b3 + 23*b1) * q^24 + (-3*b4 + 117) * q^25 + (9*b4 + 172) * q^27 + (-42*b3 - 43*b1) * q^28 + (96*b7 + 26*b2) * q^29 + (-13*b7 + 13*b4 + 15*b2 + 28) * q^30 + (13*b6 - 34*b5 - 13*b3 + 34*b1) * q^31 + (-46*b6 - 61*b5) * q^32 + (4*b6 - 90*b5) * q^33 + (-34*b6 - b5 + 34*b3 + b1) * q^34 + (21*b7 - 21*b4 - 43*b2 - 64) * q^35 + (-70*b7 + 90*b2) * q^36 + (51*b3 - 5*b1) * q^37 + (-58*b4 + 128) * q^38 + (-15*b4 - 52) * q^40 + (63*b3 + 22*b1) * q^41 + (-131*b7 + 33*b2) * q^42 + (143*b7 - 143*b4 + 215*b2 + 72) * q^43 + (44*b6 - 2*b5 - 44*b3 + 2*b1) * q^44 + (-55*b6 - 40*b5) * q^45 + (24*b6 + 92*b5) * q^46 + (139*b6 + 121*b5 - 139*b3 - 121*b1) * q^47 + (-21*b7 + 21*b4 - 153*b2 - 132) * q^48 + (99*b7 - 142*b2) * q^49 + (-6*b3 + 120*b1) * q^50 + (-71*b4 - 276) * q^51 + (-30*b4 - 74) * q^53 + (18*b3 + 163*b1) * q^54 + (22*b7 + 2*b2) * q^55 + (33*b7 - 33*b4 - 491*b2 - 524) * q^56 + (-140*b6 + 46*b5 + 140*b3 - 46*b1) * q^57 + (192*b6 - 26*b5) * q^58 + (123*b6 + 124*b5) * q^59 + (54*b6 + 41*b5 - 54*b3 - 41*b1) * q^60 + (190*b7 - 190*b4 + 624*b2 + 434) * q^61 + (60*b7 + 196*b2) * q^62 + (-455*b3 - 260*b1) * q^63 + (89*b4 - 340) * q^64 + (98*b4 - 360) * q^66 + (-75*b3 - 232*b1) * q^67 + (69*b7 - 71*b2) * q^68 + (-324*b7 + 324*b4 + 236*b2 + 560) * q^69 + (42*b6 + 43*b5 - 42*b3 - 43*b1) * q^70 + (25*b6 + 231*b5) * q^71 + (-380*b6 - 170*b5) * q^72 + (49*b6 - 260*b5 - 49*b3 + 260*b1) * q^73 + (-107*b7 + 107*b4 - 127*b2 - 20) * q^74 + (-348*b7 + 84*b2) * q^75 + (-12*b3 - 70*b1) * q^76 + (386*b4 - 188) * q^77 + (-40*b4 - 524) * q^79 + (18*b3 + 3*b1) * q^80 + (-120*b7 + b2) * q^81 + (-104*b7 + 104*b4 - 16*b2 + 88) * q^82 + (-535*b6 - 182*b5 + 535*b3 + 182*b1) * q^83 + (426*b6 + 295*b5) * q^84 + (52*b6 + 35*b5) * q^85 + (286*b6 - 215*b5 - 286*b3 + 215*b1) * q^86 + (-306*b7 + 306*b4 + 1126*b2 + 1432) * q^87 + (458*b7 + 850*b2) * q^88 + (-83*b3 + 388*b1) * q^89 + (-70*b4 - 160) * q^90 + (-140*b4 - 464) * q^92 + (-152*b3 + 44*b1) * q^93 + (157*b7 - 327*b2) * q^94 + (6*b7 - 6*b4 - 70*b2 - 76) * q^95 + (550*b6 + 337*b5 - 550*b3 - 337*b1) * q^96 + (-359*b6 - 508*b5) * q^97 + (198*b6 + 142*b5) * q^98 + (65*b6 + 390*b5 - 65*b3 - 390*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{3} - 14 q^{4} - 70 q^{9}+O(q^{10})$$ 8 * q - 10 * q^3 - 14 * q^4 - 70 * q^9 $$8 q - 10 q^{3} - 14 q^{4} - 70 q^{9} + 14 q^{10} + 172 q^{12} + 356 q^{14} + 78 q^{16} + 38 q^{17} - 284 q^{22} + 392 q^{23} + 948 q^{25} + 1340 q^{27} + 88 q^{29} + 86 q^{30} - 214 q^{35} - 500 q^{36} + 1256 q^{38} - 356 q^{40} - 394 q^{42} + 574 q^{43} - 570 q^{48} + 766 q^{49} - 1924 q^{51} - 472 q^{53} + 36 q^{55} - 2030 q^{56} + 2116 q^{61} - 664 q^{62} - 3076 q^{64} - 3272 q^{66} + 422 q^{68} + 1592 q^{69} - 294 q^{74} - 1032 q^{75} - 3048 q^{77} - 4032 q^{79} - 244 q^{81} + 144 q^{82} + 5116 q^{87} - 2484 q^{88} - 1000 q^{90} - 3152 q^{92} + 1622 q^{94} - 292 q^{95}+O(q^{100})$$ 8 * q - 10 * q^3 - 14 * q^4 - 70 * q^9 + 14 * q^10 + 172 * q^12 + 356 * q^14 + 78 * q^16 + 38 * q^17 - 284 * q^22 + 392 * q^23 + 948 * q^25 + 1340 * q^27 + 88 * q^29 + 86 * q^30 - 214 * q^35 - 500 * q^36 + 1256 * q^38 - 356 * q^40 - 394 * q^42 + 574 * q^43 - 570 * q^48 + 766 * q^49 - 1924 * q^51 - 472 * q^53 + 36 * q^55 - 2030 * q^56 + 2116 * q^61 - 664 * q^62 - 3076 * q^64 - 3272 * q^66 + 422 * q^68 + 1592 * q^69 - 294 * q^74 - 1032 * q^75 - 3048 * q^77 - 4032 * q^79 - 244 * q^81 + 144 * q^82 + 5116 * q^87 - 2484 * q^88 - 1000 * q^90 - 3152 * q^92 + 1622 * q^94 - 292 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040$$ (9*v^6 - 65*v^4 + 585*v^2 - 1296) / 1040 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 181\nu ) / 130$$ (v^7 + 181*v) / 130 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 116 ) / 65$$ (v^6 + 116) / 65 $$\beta_{5}$$ $$=$$ $$( -9\nu^{7} + 65\nu^{5} - 585\nu^{3} + 1296\nu ) / 1040$$ (-9*v^7 + 65*v^5 - 585*v^3 + 1296*v) / 1040 $$\beta_{6}$$ $$=$$ $$( -29\nu^{7} + 325\nu^{5} - 1885\nu^{3} + 4176\nu ) / 2080$$ (-29*v^7 + 325*v^5 - 1885*v^3 + 4176*v) / 2080 $$\beta_{7}$$ $$=$$ $$( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040$$ (-29*v^6 + 325*v^4 - 1885*v^2 + 4176) / 1040
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} + 5\beta_{2} + 4$$ b7 - b4 + 5*b2 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{6} - 5\beta_{5} - 2\beta_{3} + 5\beta_1$$ 2*b6 - 5*b5 - 2*b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$9\beta_{7} + 29\beta_{2}$$ 9*b7 + 29*b2 $$\nu^{5}$$ $$=$$ $$18\beta_{6} - 29\beta_{5}$$ 18*b6 - 29*b5 $$\nu^{6}$$ $$=$$ $$65\beta_{4} - 116$$ 65*b4 - 116 $$\nu^{7}$$ $$=$$ $$130\beta_{3} - 181\beta_1$$ 130*b3 - 181*b1

## Character values

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

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

## 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
 −2.21837 + 1.28078i −1.35234 + 0.780776i 1.35234 − 0.780776i 2.21837 − 1.28078i −2.21837 − 1.28078i −1.35234 − 0.780776i 1.35234 + 0.780776i 2.21837 + 1.28078i
−2.21837 + 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i 0.561553i −8.17394 4.71922i −15.7418 9.08854i 24.1771i 6.71165 11.6249i 0.719224 + 1.24573i
23.2 −1.35234 + 0.780776i −4.34233 7.52113i −2.78078 + 4.81645i 3.56155i 11.7446 + 6.78078i −23.5360 13.5885i 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.3 1.35234 0.780776i −4.34233 7.52113i −2.78078 + 4.81645i 3.56155i −11.7446 6.78078i 23.5360 + 13.5885i 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.4 2.21837 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i 0.561553i 8.17394 + 4.71922i 15.7418 + 9.08854i 24.1771i 6.71165 11.6249i 0.719224 + 1.24573i
147.1 −2.21837 1.28078i 1.84233 3.19101i −0.719224 1.24573i 0.561553i −8.17394 + 4.71922i −15.7418 + 9.08854i 24.1771i 6.71165 + 11.6249i 0.719224 1.24573i
147.2 −1.35234 0.780776i −4.34233 + 7.52113i −2.78078 4.81645i 3.56155i 11.7446 6.78078i −23.5360 + 13.5885i 21.1771i −24.2116 41.9358i 2.78078 4.81645i
147.3 1.35234 + 0.780776i −4.34233 + 7.52113i −2.78078 4.81645i 3.56155i −11.7446 + 6.78078i 23.5360 13.5885i 21.1771i −24.2116 41.9358i 2.78078 4.81645i
147.4 2.21837 + 1.28078i 1.84233 3.19101i −0.719224 1.24573i 0.561553i 8.17394 4.71922i 15.7418 9.08854i 24.1771i 6.71165 + 11.6249i 0.719224 1.24573i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.4 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.f 8
13.b even 2 1 inner 169.4.e.f 8
13.c even 3 1 169.4.b.f 4
13.c even 3 1 inner 169.4.e.f 8
13.d odd 4 1 169.4.c.g 4
13.d odd 4 1 169.4.c.j 4
13.e even 6 1 169.4.b.f 4
13.e even 6 1 inner 169.4.e.f 8
13.f odd 12 1 13.4.a.b 2
13.f odd 12 1 169.4.a.g 2
13.f odd 12 1 169.4.c.g 4
13.f odd 12 1 169.4.c.j 4
39.k even 12 1 117.4.a.d 2
39.k even 12 1 1521.4.a.r 2
52.l even 12 1 208.4.a.h 2
65.o even 12 1 325.4.b.e 4
65.s odd 12 1 325.4.a.f 2
65.t even 12 1 325.4.b.e 4
91.bc even 12 1 637.4.a.b 2
104.u even 12 1 832.4.a.z 2
104.x odd 12 1 832.4.a.s 2
143.o even 12 1 1573.4.a.b 2
156.v odd 12 1 1872.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 13.f odd 12 1
117.4.a.d 2 39.k even 12 1
169.4.a.g 2 13.f odd 12 1
169.4.b.f 4 13.c even 3 1
169.4.b.f 4 13.e even 6 1
169.4.c.g 4 13.d odd 4 1
169.4.c.g 4 13.f odd 12 1
169.4.c.j 4 13.d odd 4 1
169.4.c.j 4 13.f odd 12 1
169.4.e.f 8 1.a even 1 1 trivial
169.4.e.f 8 13.b even 2 1 inner
169.4.e.f 8 13.c even 3 1 inner
169.4.e.f 8 13.e even 6 1 inner
208.4.a.h 2 52.l even 12 1
325.4.a.f 2 65.s odd 12 1
325.4.b.e 4 65.o even 12 1
325.4.b.e 4 65.t even 12 1
637.4.a.b 2 91.bc even 12 1
832.4.a.s 2 104.x odd 12 1
832.4.a.z 2 104.u even 12 1
1521.4.a.r 2 39.k even 12 1
1573.4.a.b 2 143.o even 12 1
1872.4.a.bb 2 156.v odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 9 T^{6} + \cdots + 256$$
$3$ $$(T^{4} + 5 T^{3} + \cdots + 1024)^{2}$$
$5$ $$(T^{4} + 13 T^{2} + 4)^{2}$$
$7$ $$T^{8} + \cdots + 59553569296$$
$11$ $$T^{8} + \cdots + 952857108736$$
$13$ $$T^{8}$$
$17$ $$(T^{4} - 19 T^{3} + \cdots + 1295044)^{2}$$
$19$ $$T^{8} + \cdots + 44859774689536$$
$23$ $$(T^{4} - 196 T^{3} + \cdots + 80856064)^{2}$$
$29$ $$(T^{4} - 44 T^{3} + \cdots + 1496451856)^{2}$$
$31$ $$(T^{4} + 13524 T^{2} + 9388096)^{2}$$
$37$ $$T^{8} + \cdots + 13\!\cdots\!16$$
$41$ $$T^{8} + \cdots + 15\!\cdots\!76$$
$43$ $$(T^{4} - 287 T^{3} + \cdots + 4397811856)^{2}$$
$47$ $$(T^{4} + 219061 T^{2} + 222546724)^{2}$$
$53$ $$(T^{2} + 118 T - 344)^{4}$$
$59$ $$T^{8} + \cdots + 98\!\cdots\!96$$
$61$ $$(T^{4} - 1058 T^{3} + \cdots + 15981005056)^{2}$$
$67$ $$T^{8} + \cdots + 26\!\cdots\!56$$
$71$ $$T^{8} + \cdots + 24\!\cdots\!96$$
$73$ $$(T^{4} + 678568 T^{2} + 55373619856)^{2}$$
$79$ $$(T^{2} + 1008 T + 247216)^{4}$$
$83$ $$(T^{4} + 2198436 T^{2} + 668574416896)^{2}$$
$89$ $$T^{8} + \cdots + 67\!\cdots\!36$$
$97$ $$T^{8} + \cdots + 60\!\cdots\!76$$