# Properties

 Label 208.3.bd.f Level $208$ Weight $3$ Character orbit 208.bd Analytic conductor $5.668$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,3,Mod(33,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("208.33");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 208.bd (of order $$12$$, degree $$4$$, 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: $$5.66758949869$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.612074651904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ x^8 - 74*x^6 + 2067*x^4 - 25778*x^2 + 121801 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 10 \beta_{3} + 1) q^{9}+O(q^{10})$$ q + (b4 + b1) * q^3 + (-b7 + 2*b5 + 2*b3 + b1) * q^5 + (-b7 + b6 - b5 + 2*b4 + 2*b3 - b2 - b1) * q^7 + (-b7 + 2*b6 - 4*b5 + 2*b4 - 10*b3 + 1) * q^9 $$q + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 10 \beta_{3} + 1) q^{9} + ( - \beta_{7} - 5 \beta_{5} - 5 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{11} + (2 \beta_{7} - \beta_{6} + 7 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + \beta_1 + 5) q^{13} + (\beta_{7} + \beta_{6} + 16 \beta_{5} - 17 \beta_{4} - 16 \beta_{3} + 2 \beta_{2} + \beta_1) q^{15} + (\beta_{7} - \beta_{5} + 8 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{17} + (3 \beta_{7} - \beta_{6} + 9 \beta_{5} - 6 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - \beta_1 - 10) q^{19} + ( - \beta_{7} + \beta_{6} + 3 \beta_{5} - 23 \beta_{4} + 20 \beta_{3} + \beta_{2} + \cdots - 22) q^{21}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{6} + 10 \beta_{5} - 50 \beta_{4} + 40 \beta_{3} + \cdots - 52) q^{99}+O(q^{100})$$ q + (b4 + b1) * q^3 + (-b7 + 2*b5 + 2*b3 + b1) * q^5 + (-b7 + b6 - b5 + 2*b4 + 2*b3 - b2 - b1) * q^7 + (-b7 + 2*b6 - 4*b5 + 2*b4 - 10*b3 + 1) * q^9 + (-b7 - 5*b5 - 5*b3 + b2 + b1 + 5) * q^11 + (2*b7 - b6 + 7*b5 - 9*b4 - 3*b3 + b1 + 5) * q^13 + (b7 + b6 + 16*b5 - 17*b4 - 16*b3 + 2*b2 + b1) * q^15 + (b7 - b5 + 8*b4 + 3*b3 - b2 - 2*b1 - 7) * q^17 + (3*b7 - b6 + 9*b5 - 6*b4 + 6*b3 - 3*b2 - b1 - 10) * q^19 + (-b7 + b6 + 3*b5 - 23*b4 + 20*b3 + b2 - 22) * q^21 + (-2*b7 + 2*b6 + 14*b5 - b4 + 4*b3 + 2*b2 + b1 + 4) * q^23 + (3*b6 + 11*b5 - 11*b4 - 25*b3 + 14) * q^25 + (-4*b7 + 2*b6 - 21*b5 - 18*b4 + 2*b3 - 3*b2 - 3*b1 - 8) * q^27 + (-b7 - b6 + b5 + 3*b4 + 2*b3 + 5*b1 - 2) * q^29 + (-3*b6 - 3*b5 + 2*b4 - b3 + 3*b2 + 3*b1 - 5) * q^31 + (-5*b7 + b6 + 11*b5 - 10*b4 - 10*b3 - 5*b2 - b1 + 12) * q^33 + (-3*b7 + 6*b6 + 7*b5 - 3*b4 + 21*b3 - b2 + 3) * q^35 + (-b7 - 5*b6 - 3*b5 + 12*b4 - 3*b3 + 6*b2 + b1 - 14) * q^37 + (6*b7 - 8*b6 - 24*b5 + 45*b4 - 22*b3 - 2*b2 + 5*b1 + 4) * q^39 + (2*b7 + 6*b6 - 8*b5 + 17*b4 + 8*b3 + 8*b2 + 2*b1 + 15) * q^41 + (6*b7 - b5 + 17*b4 - 12*b3 - b2 - 2*b1 + 18) * q^43 + (8*b7 - 7*b6 - 35*b5 + 7*b4 - 7*b3 - 8*b2 - 7*b1 + 28) * q^45 + (-2*b7 + b6 + 11*b5 - 4*b4 - 7*b3 + b2 - b1 - 3) * q^47 + (-b7 + b6 - 3*b5 + 2*b4 + 22*b3 - 4*b2 - 2*b1 + 22) * q^49 + (6*b6 - 9*b5 + 6*b4 + 26*b3 + 3*b2 - 3*b1 - 10) * q^51 + (-10*b7 + 5*b6 + 24*b5 + 26*b4 + 5*b3 - 2*b2 - 2*b1 - 14) * q^53 + (-5*b7 - 5*b6 + 8*b5 - 13*b4 - 38*b3 + 3*b1 + 38) * q^55 + (7*b7 - 7*b6 - 31*b5 + 39*b4 + 8*b3 + 7*b2 - 46) * q^57 + (-7*b7 + b6 - 23*b5 + 44*b4 + 44*b3 - 7*b2 - b1 - 22) * q^59 + (-4*b7 + 8*b6 + 6*b5 - 2*b4 + 11*b3 - 2*b2 + 4) * q^61 + (4*b7 - 14*b6 + 30*b5 - 30*b4 + 30*b3 + 10*b2 - 4*b1 - 14) * q^63 + (6*b7 - 14*b6 + 2*b5 + 10*b4 + 10*b3 - 4*b2 + 9*b1 - 51) * q^65 + (-b7 + 12*b6 - 9*b5 + 8*b4 + 9*b3 + 11*b2 - b1 + 11) * q^67 + (15*b7 + 2*b5 - 28*b4 - 29*b3 + 2*b2 + 4*b1 + 43) * q^69 + (5*b7 - 11*b6 - 21*b5 - 10*b4 + 10*b3 - 5*b2 - 11*b1 + 10) * q^71 + (-b7 - 2*b6 - 36*b5 + 8*b4 + 28*b3 - 2*b2 - 3*b1 + 6) * q^73 + (11*b7 - 11*b6 - 82*b5 + 14*b4 - 6*b3 - 28*b2 - 14*b1 - 6) * q^75 + (-4*b6 - 35*b5 + 28*b4 + 7*b2 - 7*b1 - 2) * q^77 + (40*b5 + 24*b4 + 16*b2 + 16*b1 + 12) * q^79 + (-12*b7 - 12*b6 + 24*b5 - 60*b4 + 9*b3 - 12*b1 - 9) * q^81 + (8*b7 - 3*b6 + 47*b5 - 12*b4 + 35*b3 + 3*b2 - 5*b1 + 9) * q^83 + (2*b7 + 11*b6 - 29*b5 + 40*b4 + 40*b3 + 2*b2 - 11*b1 - 18) * q^85 + (-4*b7 + 8*b6 + 19*b5 - 11*b4 - 88*b3 + 3*b2 + 4) * q^87 + (7*b7 - 6*b6 + 56*b5 - 66*b4 + 56*b3 - b2 - 7*b1 + 4) * q^89 + (-3*b7 + 9*b6 - 31*b5 - 40*b4 + 20*b3 + b2 + 7*b1 - 42) * q^91 + (-3*b7 + 5*b6 + 50*b5 + 7*b4 - 50*b3 + 2*b2 - 3*b1 + 62) * q^93 + (13*b7 + b5 + 31*b4 + 53*b3 + b2 + 2*b1 - 119) * q^95 + (3*b7 + 7*b6 + b5 + 21*b4 - 21*b3 - 3*b2 + 7*b1 + 6) * q^97 + (-2*b7 - 2*b6 + 10*b5 - 50*b4 + 40*b3 - 2*b2 - 4*b1 - 52) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{5} + 2 q^{7} - 42 q^{9}+O(q^{10})$$ 8 * q + 6 * q^5 + 2 * q^7 - 42 * q^9 $$8 q + 6 q^{5} + 2 q^{7} - 42 q^{9} + 18 q^{11} + 36 q^{13} - 66 q^{15} - 42 q^{17} - 46 q^{19} - 102 q^{21} + 36 q^{23} - 72 q^{27} - 6 q^{29} - 32 q^{31} + 42 q^{33} + 78 q^{35} - 106 q^{37} - 12 q^{39} + 132 q^{41} + 108 q^{43} + 240 q^{45} - 60 q^{47} + 258 q^{49} - 132 q^{53} + 162 q^{55} - 294 q^{57} - 18 q^{59} + 36 q^{61} + 72 q^{63} - 300 q^{65} + 74 q^{67} + 258 q^{69} + 174 q^{71} + 166 q^{73} - 6 q^{75} + 96 q^{79} - 12 q^{81} + 240 q^{83} - 24 q^{85} - 360 q^{87} + 294 q^{89} - 298 q^{91} + 270 q^{93} - 714 q^{95} - 58 q^{97} - 252 q^{99}+O(q^{100})$$ 8 * q + 6 * q^5 + 2 * q^7 - 42 * q^9 + 18 * q^11 + 36 * q^13 - 66 * q^15 - 42 * q^17 - 46 * q^19 - 102 * q^21 + 36 * q^23 - 72 * q^27 - 6 * q^29 - 32 * q^31 + 42 * q^33 + 78 * q^35 - 106 * q^37 - 12 * q^39 + 132 * q^41 + 108 * q^43 + 240 * q^45 - 60 * q^47 + 258 * q^49 - 132 * q^53 + 162 * q^55 - 294 * q^57 - 18 * q^59 + 36 * q^61 + 72 * q^63 - 300 * q^65 + 74 * q^67 + 258 * q^69 + 174 * q^71 + 166 * q^73 - 6 * q^75 + 96 * q^79 - 12 * q^81 + 240 * q^83 - 24 * q^85 - 360 * q^87 + 294 * q^89 - 298 * q^91 + 270 * q^93 - 714 * q^95 - 58 * q^97 - 252 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} - 37\nu^{2} + 4\nu + 349 ) / 8$$ (v^4 - 37*v^2 + 4*v + 349) / 8 $$\beta_{2}$$ $$=$$ $$( -\nu^{4} + 37\nu^{2} + 4\nu - 349 ) / 8$$ (-v^4 + 37*v^2 + 4*v - 349) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 74\nu^{5} + 1718\nu^{3} - 12865\nu + 1396 ) / 2792$$ (v^7 - 74*v^5 + 1718*v^3 - 12865*v + 1396) / 2792 $$\beta_{4}$$ $$=$$ $$( 40\nu^{7} - 349\nu^{6} - 2262\nu^{5} + 19195\nu^{4} + 44290\nu^{3} - 345859\nu^{2} - 296126\nu + 2024898 ) / 86552$$ (40*v^7 - 349*v^6 - 2262*v^5 + 19195*v^4 + 44290*v^3 - 345859*v^2 - 296126*v + 2024898) / 86552 $$\beta_{5}$$ $$=$$ $$( -40\nu^{7} - 349\nu^{6} + 2262\nu^{5} + 19195\nu^{4} - 44290\nu^{3} - 345859\nu^{2} + 296126\nu + 2024898 ) / 86552$$ (-40*v^7 - 349*v^6 + 2262*v^5 + 19195*v^4 - 44290*v^3 - 345859*v^2 + 296126*v + 2024898) / 86552 $$\beta_{6}$$ $$=$$ $$( \nu^{6} - 55\nu^{4} + 1053\nu^{2} - 6980 ) / 62$$ (v^6 - 55*v^4 + 1053*v^2 - 6980) / 62 $$\beta_{7}$$ $$=$$ $$( - 318 \nu^{7} + 698 \nu^{6} + 16901 \nu^{5} - 38390 \nu^{4} - 292601 \nu^{3} + 734994 \nu^{2} + 1626083 \nu - 4828764 ) / 86552$$ (-318*v^7 + 698*v^6 + 16901*v^5 - 38390*v^4 - 292601*v^3 + 734994*v^2 + 1626083*v - 4828764) / 86552
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{5} + 2\beta_{4} + 19$$ b6 + 2*b5 + 2*b4 + 19 $$\nu^{3}$$ $$=$$ $$4\beta_{7} - 2\beta_{6} - 19\beta_{5} + 19\beta_{4} - 8\beta_{3} + 18\beta_{2} + 18\beta _1 + 2$$ 4*b7 - 2*b6 - 19*b5 + 19*b4 - 8*b3 + 18*b2 + 18*b1 + 2 $$\nu^{4}$$ $$=$$ $$37\beta_{6} + 74\beta_{5} + 74\beta_{4} - 4\beta_{2} + 4\beta _1 + 354$$ 37*b6 + 74*b5 + 74*b4 - 4*b2 + 4*b1 + 354 $$\nu^{5}$$ $$=$$ $$140\beta_{7} - 70\beta_{6} - 727\beta_{5} + 727\beta_{4} - 440\beta_{3} + 317\beta_{2} + 317\beta _1 + 150$$ 140*b7 - 70*b6 - 727*b5 + 727*b4 - 440*b3 + 317*b2 + 317*b1 + 150 $$\nu^{6}$$ $$=$$ $$1044\beta_{6} + 1964\beta_{5} + 1964\beta_{4} - 220\beta_{2} + 220\beta _1 + 6443$$ 1044*b6 + 1964*b5 + 1964*b4 - 220*b2 + 220*b1 + 6443 $$\nu^{7}$$ $$=$$ $$3488 \beta_{7} - 1744 \beta_{6} - 21156 \beta_{5} + 21156 \beta_{4} - 16024 \beta_{3} + 5399 \beta_{2} + 5399 \beta _1 + 6268$$ 3488*b7 - 1744*b6 - 21156*b5 + 21156*b4 - 16024*b3 + 5399*b2 + 5399*b1 + 6268

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −4.71318 − 0.500000i 4.71318 − 0.500000i −3.90972 + 0.500000i 3.90972 + 0.500000i −4.71318 + 0.500000i 4.71318 + 0.500000i −3.90972 − 0.500000i 3.90972 − 0.500000i
0 −1.92358 + 3.33174i 0 3.77418 + 3.77418i 0 9.91095 + 2.65563i 0 −2.90031 5.02349i 0
33.2 0 2.78960 4.83174i 0 0.323893 + 0.323893i 0 −7.67890 2.05755i 0 −11.0638 19.1630i 0
97.1 0 −2.38787 + 4.13592i 0 −5.88981 + 5.88981i 0 −0.0922225 + 0.344179i 0 −6.90386 11.9578i 0
97.2 0 1.52185 2.63592i 0 4.79174 4.79174i 0 −1.13983 + 4.25390i 0 −0.132034 0.228689i 0
145.1 0 −1.92358 3.33174i 0 3.77418 3.77418i 0 9.91095 2.65563i 0 −2.90031 + 5.02349i 0
145.2 0 2.78960 + 4.83174i 0 0.323893 0.323893i 0 −7.67890 + 2.05755i 0 −11.0638 + 19.1630i 0
193.1 0 −2.38787 4.13592i 0 −5.88981 5.88981i 0 −0.0922225 0.344179i 0 −6.90386 + 11.9578i 0
193.2 0 1.52185 + 2.63592i 0 4.79174 + 4.79174i 0 −1.13983 4.25390i 0 −0.132034 + 0.228689i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.2 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.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.3.bd.f 8
4.b odd 2 1 26.3.f.b 8
12.b even 2 1 234.3.bb.f 8
13.f odd 12 1 inner 208.3.bd.f 8
52.b odd 2 1 338.3.f.i 8
52.f even 4 1 338.3.f.h 8
52.f even 4 1 338.3.f.j 8
52.i odd 6 1 338.3.d.f 8
52.i odd 6 1 338.3.f.j 8
52.j odd 6 1 338.3.d.g 8
52.j odd 6 1 338.3.f.h 8
52.l even 12 1 26.3.f.b 8
52.l even 12 1 338.3.d.f 8
52.l even 12 1 338.3.d.g 8
52.l even 12 1 338.3.f.i 8
156.v odd 12 1 234.3.bb.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b 8 4.b odd 2 1
26.3.f.b 8 52.l even 12 1
208.3.bd.f 8 1.a even 1 1 trivial
208.3.bd.f 8 13.f odd 12 1 inner
234.3.bb.f 8 12.b even 2 1
234.3.bb.f 8 156.v odd 12 1
338.3.d.f 8 52.i odd 6 1
338.3.d.f 8 52.l even 12 1
338.3.d.g 8 52.j odd 6 1
338.3.d.g 8 52.l even 12 1
338.3.f.h 8 52.f even 4 1
338.3.f.h 8 52.j odd 6 1
338.3.f.i 8 52.b odd 2 1
338.3.f.i 8 52.l even 12 1
338.3.f.j 8 52.f even 4 1
338.3.f.j 8 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 39T_{3}^{6} + 24T_{3}^{5} + 1209T_{3}^{4} + 468T_{3}^{3} + 12312T_{3}^{2} - 3744T_{3} + 97344$$ acting on $$S_{3}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 39 T^{6} + 24 T^{5} + \cdots + 97344$$
$5$ $$T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 19044$$
$7$ $$T^{8} - 2 T^{7} - 127 T^{6} + \cdots + 16384$$
$11$ $$T^{8} - 18 T^{7} + 105 T^{6} + \cdots + 389376$$
$13$ $$T^{8} - 36 T^{7} + \cdots + 815730721$$
$17$ $$T^{8} + 42 T^{7} + 570 T^{6} + \cdots + 471969$$
$19$ $$T^{8} + 46 T^{7} + \cdots + 1228362304$$
$23$ $$T^{8} - 36 T^{7} + \cdots + 2508807744$$
$29$ $$T^{8} + 6 T^{7} + \cdots + 25455883401$$
$31$ $$T^{8} + 32 T^{7} + \cdots + 8111524096$$
$37$ $$T^{8} + 106 T^{7} + \cdots + 321419829721$$
$41$ $$T^{8} - 132 T^{7} + \cdots + 326485389321$$
$43$ $$T^{8} - 108 T^{7} + \cdots + 325666531584$$
$47$ $$T^{8} + 60 T^{7} + \cdots + 1853819136$$
$53$ $$(T^{4} + 66 T^{3} - 5319 T^{2} + \cdots - 5234376)^{2}$$
$59$ $$T^{8} + 18 T^{7} + \cdots + 70410089309184$$
$61$ $$T^{8} - 36 T^{7} + \cdots + 313453297161$$
$67$ $$T^{8} - 74 T^{7} + \cdots + 2456391674944$$
$71$ $$T^{8} - 174 T^{7} + \cdots + 950999436864$$
$73$ $$T^{8} - 166 T^{7} + \cdots + 3554348548804$$
$79$ $$(T^{4} - 48 T^{3} - 14880 T^{2} + \cdots + 2312448)^{2}$$
$83$ $$T^{8} + \cdots + 154848357540864$$
$89$ $$T^{8} - 294 T^{7} + \cdots + 14950765690884$$
$97$ $$T^{8} + 58 T^{7} + \cdots + 9988090235236$$