# Properties

 Label 55.3.i.a Level $55$ Weight $3$ Character orbit 55.i Analytic conductor $1.499$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,3,Mod(6,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 9]))

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

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

chi := DirichletCharacter("55.6");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 55.i (of order $$10$$, degree $$4$$, 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: $$1.49864145398$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{2}+ \cdots - 7 \zeta_{10}^{3} q^{9} +O(q^{10})$$ q + (-z^3 + 2*z^2 - 2*z) * q^2 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^3 + (-4*z^2 + 3*z - 4) * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 + (-4*z^2 + 8*z - 8) * q^6 + (4*z^3 - 5*z^2 + 2*z + 1) * q^7 + (3*z^3 + z^2 + z + 3) * q^8 - 7*z^3 * q^9 $$q + ( - \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{2}+ \cdots + ( - 21 \zeta_{10}^{3} - 21 \zeta_{10}^{2} + \cdots + 49) q^{99}+O(q^{100})$$ q + (-z^3 + 2*z^2 - 2*z) * q^2 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^3 + (-4*z^2 + 3*z - 4) * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 + (-4*z^2 + 8*z - 8) * q^6 + (4*z^3 - 5*z^2 + 2*z + 1) * q^7 + (3*z^3 + z^2 + z + 3) * q^8 - 7*z^3 * q^9 + (2*z^3 - 4*z^2 + 6*z - 3) * q^10 + (-12*z^3 + 10*z^2 - 3*z + 6) * q^11 + (16*z^3 - 16*z^2 - 4) * q^12 + (-8*z^3 + 16*z^2 - 7*z + 9) * q^13 + (-7*z^3 + 18*z^2 - 18*z + 7) * q^14 + (-8*z^2 + 4*z - 8) * q^15 + (8*z^3 - 3*z^2 + 8*z) * q^16 + (16*z^3 - 2*z^2 - 12*z - 4) * q^17 + (14*z^3 - 14*z^2 + 7*z) * q^18 + (10*z^3 + 9*z^2 + 9*z + 10) * q^19 + (9*z^3 - 2*z + 2) * q^20 + (-4*z^3 + 20*z^2 - 24*z + 12) * q^21 + (15*z^3 - 29*z^2 + 23*z - 13) * q^22 + (z^3 - z^2 - 27) * q^23 + (-12*z^3 + 24*z^2 - 8*z + 16) * q^24 + (5*z^3 - 5*z^2 + 5*z - 5) * q^25 + (-23*z^2 + 29*z - 23) * q^26 + 8*z^2 * q^27 + (-7*z^3 - 10*z^2 + 27*z - 20) * q^28 + (4*z^3 - 20*z^2 + 2*z + 16) * q^29 + (12*z^3 - 4*z^2 - 4*z + 12) * q^30 + (-22*z^3 - 20*z + 20) * q^31 + (-24*z^3 + 22*z^2 - 46*z + 23) * q^32 + (-24*z^3 - 24*z^2 + 16*z + 12) * q^33 + (-40*z^3 + 40*z^2 - 10) * q^34 + (-7*z^3 + 14*z^2 - 11*z + 3) * q^35 + (7*z^3 + 21*z^2 - 21*z - 7) * q^36 + (-15*z^2 + 11*z - 15) * q^37 + (-21*z^3 + 13*z^2 - 21*z) * q^38 + (-36*z^3 + 4*z^2 + 28*z + 8) * q^39 + (-14*z^3 + 8*z^2 - 7*z + 6) * q^40 + (17*z^3 - 20*z^2 - 20*z + 17) * q^41 + (-28*z^3 + 44*z - 44) * q^42 + (16*z^3 + 28*z^2 - 12*z + 6) * q^43 + (14*z^3 + 3*z^2 - 46*z + 4) * q^44 + (14*z^3 - 14*z^2 - 7) * q^45 + (25*z^3 - 50*z^2 + 51*z + 1) * q^46 + (58*z^3 - 75*z^2 + 75*z - 58) * q^47 + (32*z^2 - 12*z + 32) * q^48 + (29*z^3 + 2*z^2 + 29*z) * q^49 + (5*z^2 - 10*z + 10) * q^50 + (16*z^3 + 48*z^2 + 8*z - 64) * q^51 + (20*z^3 - 33*z^2 - 33*z + 20) * q^52 + (-15*z^3 - 43*z + 43) * q^53 + (-16*z^2 + 16*z - 8) * q^54 + (5*z^3 - 28*z^2 + 4*z - 8) * q^55 + (17*z^3 - 17*z^2 + 9) * q^56 + (-40*z^3 + 80*z^2 - 4*z + 76) * q^57 + (-22*z^3 + 78*z^2 - 78*z + 22) * q^58 + (-19*z^2 - 71*z - 19) * q^59 + (-8*z^3 + 44*z^2 - 8*z) * q^60 + (-26*z^3 + 28*z^2 - 30*z + 56) * q^61 + (4*z^3 + 16*z^2 + 2*z - 20) * q^62 + (-21*z^3 + 14*z^2 + 14*z - 21) * q^63 + (-41*z^3 + 36*z - 36) * q^64 + (-11*z^3 - 23*z^2 + 12*z - 6) * q^65 + (52*z^3 + 8*z^2 - 64*z + 40) * q^66 + (-10*z^3 + 10*z^2 + 34) * q^67 + (34*z^3 - 68*z^2 + 92*z + 24) * q^68 + (108*z^3 - 104*z^2 + 104*z - 108) * q^69 + (-25*z^2 + 40*z - 25) * q^70 + (10*z^3 + 12*z^2 + 10*z) * q^71 + (-28*z^3 + 7*z^2 + 14*z + 14) * q^72 + (100*z^3 - 46*z^2 + 50*z - 54) * q^73 + (26*z^3 - 11*z^2 - 11*z + 26) * q^74 + 20*z^3 * q^75 + (-55*z^3 - 43*z^2 - 12*z + 6) * q^76 + (-39*z^3 + 60*z^2 - 29*z - 8) * q^77 + (92*z^3 - 92*z^2 + 24) * q^78 + (44*z^3 - 88*z^2 + 2*z - 86) * q^79 + (-21*z^3 + 19*z^2 - 19*z + 21) * q^80 + 95*z * q^81 + (-71*z^3 + 128*z^2 - 71*z) * q^82 + (56*z^3 - 82*z^2 + 108*z - 164) * q^83 + (80*z^3 - 108*z^2 + 40*z + 28) * q^84 + (-10*z^3 + 30*z^2 + 30*z - 10) * q^85 + (-50*z^3 + 40*z - 40) * q^86 + (-64*z^3 + 80*z^2 - 144*z + 72) * q^87 + (-22*z^3 + 44*z^2 + 22*z + 11) * q^88 + (-97*z^3 + 97*z^2 + 29) * q^89 + (-21*z^3 + 42*z^2 - 28*z + 14) * q^90 + (-29*z^3 + 81*z^2 - 81*z + 29) * q^91 + (105*z^2 - 74*z + 105) * q^92 + (-80*z^3 - 8*z^2 - 80*z) * q^93 + (17*z^3 + 75*z^2 - 167*z + 150) * q^94 + (-58*z^3 + 21*z^2 - 29*z + 37) * q^95 + (-92*z^3 - 4*z^2 - 4*z - 92) * q^96 + (-10*z^3 - 62*z + 62) * q^97 + (-29*z^3 + 25*z^2 - 54*z + 27) * q^98 + (-21*z^3 - 21*z^2 - 63*z + 49) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} + 4 q^{3} - 9 q^{4} - 5 q^{5} - 20 q^{6} + 15 q^{7} + 15 q^{8} - 7 q^{9}+O(q^{10})$$ 4 * q - 5 * q^2 + 4 * q^3 - 9 * q^4 - 5 * q^5 - 20 * q^6 + 15 * q^7 + 15 * q^8 - 7 * q^9 $$4 q - 5 q^{2} + 4 q^{3} - 9 q^{4} - 5 q^{5} - 20 q^{6} + 15 q^{7} + 15 q^{8} - 7 q^{9} - q^{11} + 16 q^{12} + 5 q^{13} - 15 q^{14} - 20 q^{15} + 19 q^{16} - 10 q^{17} + 35 q^{18} + 50 q^{19} + 15 q^{20} + 15 q^{22} - 106 q^{23} + 20 q^{24} - 5 q^{25} - 40 q^{26} - 8 q^{27} - 50 q^{28} + 90 q^{29} + 60 q^{30} + 38 q^{31} + 64 q^{33} - 120 q^{34} - 20 q^{35} - 63 q^{36} - 34 q^{37} - 55 q^{38} + 20 q^{39} - 5 q^{40} + 85 q^{41} - 160 q^{42} - 19 q^{44} + 130 q^{46} - 24 q^{47} + 84 q^{48} + 56 q^{49} + 25 q^{50} - 280 q^{51} + 100 q^{52} + 114 q^{53} + 5 q^{55} + 70 q^{56} + 180 q^{57} - 90 q^{58} - 128 q^{59} - 60 q^{60} + 140 q^{61} - 90 q^{62} - 105 q^{63} - 149 q^{64} + 140 q^{66} + 116 q^{67} + 290 q^{68} - 116 q^{69} - 35 q^{70} + 8 q^{71} + 35 q^{72} - 20 q^{73} + 130 q^{74} + 20 q^{75} - 160 q^{77} + 280 q^{78} - 210 q^{79} + 25 q^{80} + 95 q^{81} - 270 q^{82} - 410 q^{83} + 340 q^{84} - 50 q^{85} - 170 q^{86} - 78 q^{89} - 35 q^{90} - 75 q^{91} + 241 q^{92} - 152 q^{93} + 375 q^{94} + 40 q^{95} - 460 q^{96} + 176 q^{97} + 133 q^{99}+O(q^{100})$$ 4 * q - 5 * q^2 + 4 * q^3 - 9 * q^4 - 5 * q^5 - 20 * q^6 + 15 * q^7 + 15 * q^8 - 7 * q^9 - q^11 + 16 * q^12 + 5 * q^13 - 15 * q^14 - 20 * q^15 + 19 * q^16 - 10 * q^17 + 35 * q^18 + 50 * q^19 + 15 * q^20 + 15 * q^22 - 106 * q^23 + 20 * q^24 - 5 * q^25 - 40 * q^26 - 8 * q^27 - 50 * q^28 + 90 * q^29 + 60 * q^30 + 38 * q^31 + 64 * q^33 - 120 * q^34 - 20 * q^35 - 63 * q^36 - 34 * q^37 - 55 * q^38 + 20 * q^39 - 5 * q^40 + 85 * q^41 - 160 * q^42 - 19 * q^44 + 130 * q^46 - 24 * q^47 + 84 * q^48 + 56 * q^49 + 25 * q^50 - 280 * q^51 + 100 * q^52 + 114 * q^53 + 5 * q^55 + 70 * q^56 + 180 * q^57 - 90 * q^58 - 128 * q^59 - 60 * q^60 + 140 * q^61 - 90 * q^62 - 105 * q^63 - 149 * q^64 + 140 * q^66 + 116 * q^67 + 290 * q^68 - 116 * q^69 - 35 * q^70 + 8 * q^71 + 35 * q^72 - 20 * q^73 + 130 * q^74 + 20 * q^75 - 160 * q^77 + 280 * q^78 - 210 * q^79 + 25 * q^80 + 95 * q^81 - 270 * q^82 - 410 * q^83 + 340 * q^84 - 50 * q^85 - 170 * q^86 - 78 * q^89 - 35 * q^90 - 75 * q^91 + 241 * q^92 - 152 * q^93 + 375 * q^94 + 40 * q^95 - 460 * q^96 + 176 * q^97 + 133 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$1$$ $$\zeta_{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0.809017 − 0.587785i −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i
−0.690983 + 0.224514i 3.23607 + 2.35114i −2.80902 + 2.04087i −0.690983 + 2.12663i −2.76393 0.898056i −0.163119 0.224514i 3.19098 4.39201i 2.16312 + 6.65740i 1.62460i
41.1 −1.80902 2.48990i −1.23607 3.80423i −1.69098 + 5.20431i −1.80902 1.31433i −7.23607 + 9.95959i 7.66312 + 2.48990i 4.30902 1.40008i −5.66312 + 4.11450i 6.88191i
46.1 −0.690983 0.224514i 3.23607 2.35114i −2.80902 2.04087i −0.690983 2.12663i −2.76393 + 0.898056i −0.163119 + 0.224514i 3.19098 + 4.39201i 2.16312 6.65740i 1.62460i
51.1 −1.80902 + 2.48990i −1.23607 + 3.80423i −1.69098 5.20431i −1.80902 + 1.31433i −7.23607 9.95959i 7.66312 2.48990i 4.30902 + 1.40008i −5.66312 4.11450i 6.88191i
 $$n$$: 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
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.3.i.a 4
5.b even 2 1 275.3.x.d 4
5.c odd 4 2 275.3.q.c 8
11.c even 5 1 605.3.c.a 4
11.d odd 10 1 inner 55.3.i.a 4
11.d odd 10 1 605.3.c.a 4
55.h odd 10 1 275.3.x.d 4
55.l even 20 2 275.3.q.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.i.a 4 1.a even 1 1 trivial
55.3.i.a 4 11.d odd 10 1 inner
275.3.q.c 8 5.c odd 4 2
275.3.q.c 8 55.l even 20 2
275.3.x.d 4 5.b even 2 1
275.3.x.d 4 55.h odd 10 1
605.3.c.a 4 11.c even 5 1
605.3.c.a 4 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 5T_{2}^{3} + 15T_{2}^{2} + 15T_{2} + 5$$ acting on $$S_{3}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + \cdots + 5$$
$3$ $$T^{4} - 4 T^{3} + \cdots + 256$$
$5$ $$T^{4} + 5 T^{3} + \cdots + 25$$
$7$ $$T^{4} - 15 T^{3} + \cdots + 5$$
$11$ $$T^{4} + T^{3} + \cdots + 14641$$
$13$ $$T^{4} - 5 T^{3} + \cdots + 25205$$
$17$ $$T^{4} + 10 T^{3} + \cdots + 242000$$
$19$ $$T^{4} - 50 T^{3} + \cdots + 59405$$
$23$ $$(T^{2} + 53 T + 701)^{2}$$
$29$ $$T^{4} - 90 T^{3} + \cdots + 403280$$
$31$ $$T^{4} - 38 T^{3} + \cdots + 126736$$
$37$ $$T^{4} + 34 T^{3} + \cdots + 72361$$
$41$ $$T^{4} - 85 T^{3} + \cdots + 1017005$$
$43$ $$T^{4} + 2600 T^{2} + 1682000$$
$47$ $$T^{4} + 24 T^{3} + \cdots + 16491721$$
$53$ $$T^{4} - 114 T^{3} + \cdots + 5148361$$
$59$ $$T^{4} + 128 T^{3} + \cdots + 36348841$$
$61$ $$T^{4} - 140 T^{3} + \cdots + 3494480$$
$67$ $$(T^{2} - 58 T + 716)^{2}$$
$71$ $$T^{4} - 8 T^{3} + \cdots + 26896$$
$73$ $$T^{4} + 20 T^{3} + \cdots + 26083280$$
$79$ $$T^{4} + 210 T^{3} + \cdots + 20402000$$
$83$ $$T^{4} + 410 T^{3} + \cdots + 334562000$$
$89$ $$(T^{2} + 39 T - 11381)^{2}$$
$97$ $$T^{4} - 176 T^{3} + \cdots + 19044496$$