# Properties

 Label 55.3.i.b 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + ( - 8 \zeta_{10}^{3} - 7 \zeta_{10} + 7) q^{9}+O(q^{10})$$ q + (-z^3 + 2*z^2 + 2) * q^2 + (4*z^3 - 3*z^2 + 3*z - 4) * q^3 + (4*z^2 - z + 4) * q^4 + (2*z^3 - z^2 + 2*z) * q^5 + (9*z^3 - 5*z^2 + z - 10) * q^6 + (-4*z^3 - 4*z^2 - 2*z + 8) * q^7 + (-z^3 + 3*z^2 + 3*z - 1) * q^8 + (-8*z^3 - 7*z + 7) * q^9 $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + ( - 111 \zeta_{10}^{3} + 17 \zeta_{10}^{2} + \cdots + 68) q^{99}+O(q^{100})$$ q + (-z^3 + 2*z^2 + 2) * q^2 + (4*z^3 - 3*z^2 + 3*z - 4) * q^3 + (4*z^2 - z + 4) * q^4 + (2*z^3 - z^2 + 2*z) * q^5 + (9*z^3 - 5*z^2 + z - 10) * q^6 + (-4*z^3 - 4*z^2 - 2*z + 8) * q^7 + (-z^3 + 3*z^2 + 3*z - 1) * q^8 + (-8*z^3 - 7*z + 7) * q^9 + (4*z^3 + 2*z^2 + 2*z - 1) * q^10 + (-7*z^3 + 9*z^2 + 3*z + 3) * q^11 + (15*z^3 - 15*z^2 - 16) * q^12 + (-6*z - 6) * q^13 + (-26*z^3 + 14*z^2 - 14*z + 26) * q^14 + (-7*z^2 + z - 7) * q^15 + (-8*z^3 + 5*z^2 - 8*z) * q^16 + (13*z^3 - 10*z^2 + 7*z - 20) * q^17 + (-30*z^3 + 7*z^2 - 15*z + 23) * q^18 + (-9*z^3 + 10*z^2 + 10*z - 9) * q^19 + (11*z^3 + 2*z - 2) * q^20 + (34*z^3 - 6*z^2 + 40*z - 20) * q^21 + (4*z^3 + 9*z^2 + 14*z + 14) * q^22 + (-4*z^3 + 4*z^2 - 2) * q^23 + (-11*z - 11) * q^24 + (5*z^3 - 5*z^2 + 5*z - 5) * q^25 + (-18*z^2 - 6*z - 18) * q^26 + (20*z^3 - 17*z^2 + 20*z) * q^27 + (-32*z^3 + 30*z^2 - 28*z + 60) * q^28 + (14*z^2 - 14) * q^29 + (-6*z^3 - 13*z^2 - 13*z - 6) * q^30 + (-10*z^3 - 36*z + 36) * q^31 + (-22*z^3 - 24*z^2 + 2*z - 1) * q^32 + (22*z^3 - 11*z - 33) * q^33 + (33*z^3 - 33*z^2 - 49) * q^34 + (2*z^3 - 4*z^2 + 16*z + 12) * q^35 + (-52*z^3 + 27*z^2 - 27*z + 52) * q^36 + (18*z^2 - 38*z + 18) * q^37 + (21*z^3 - 8*z^2 + 21*z) * q^38 + (-30*z^3 + 24*z^2 - 18*z + 48) * q^39 + (2*z^3 + 6*z^2 + z - 8) * q^40 + (8*z^3 + 9*z^2 + 9*z + 8) * q^41 + (116*z^3 + 62*z - 62) * q^42 + (37*z^3 - 19*z^2 + 56*z - 28) * q^43 + (18*z^3 + 2*z^2 + 52*z - 3) * q^44 + (-9*z^3 + 9*z^2 + 22) * q^45 + (2*z^3 - 4*z^2 + 4*z) * q^46 + (40*z^3 - 48*z^2 + 48*z - 40) * q^47 + (27*z^2 - 7*z + 27) * q^48 + (-16*z^3 - 43*z^2 - 16*z) * q^49 + (10*z^3 - 5*z^2 - 10) * q^50 + (-86*z^3 + 24*z^2 - 43*z + 62) * q^51 + (-24*z^3 - 18*z^2 - 18*z - 24) * q^52 + (-46*z^3 + 20*z - 20) * q^53 + (26*z^3 + 20*z^2 + 6*z - 3) * q^54 + (4*z^3 + 20*z^2 + 3*z - 8) * q^55 + (-44*z^3 + 44*z^2 + 22) * q^56 + (-17*z^3 + 34*z^2 - 48*z - 14) * q^57 + (42*z^3 - 28*z^2 + 28*z - 42) * q^58 + (77*z^2 - 14*z + 77) * q^59 + (-17*z^3 - 29*z^2 - 17*z) * q^60 + (54*z^3 - 16*z^2 - 22*z - 32) * q^61 + (-92*z^3 + 36*z^2 - 46*z + 56) * q^62 + (-20*z^3 - 58*z^2 - 58*z - 20) * q^63 + (-77*z^3 - 36*z + 36) * q^64 + (-18*z^3 + 6*z^2 - 24*z + 12) * q^65 + (66*z^3 - 77*z^2 + 11*z - 121) * q^66 + (-25*z^3 + 25*z^2 + 29) * q^67 + (37*z^3 - 74*z^2 - 5*z - 79) * q^68 + (-4*z^3 + 14*z^2 - 14*z + 4) * q^69 + (40*z^2 + 10*z + 40) * q^70 + (-4*z^3 - 46*z^2 - 4*z) * q^71 + (-37*z^3 + 17*z^2 + 3*z + 34) * q^72 + (26*z^3 + 33*z^2 + 13*z - 59) * q^73 + (-20*z^3 - 2*z^2 - 2*z - 20) * q^74 + (-20*z^3 - 5*z + 5) * q^75 + (43*z^3 - 55*z^2 + 98*z - 49) * q^76 + (-132*z^3 + 88*z^2 - 44*z + 66) * q^77 + (-78*z^3 + 78*z^2 + 114) * q^78 + (76*z^3 - 152*z^2 + 56*z - 96) * q^79 + (-19*z^3 + 21*z^2 - 21*z + 19) * q^80 + 40*z * q^81 + (35*z^3 + 25*z^2 + 35*z) * q^82 + (-9*z^3 + 37*z^2 - 65*z + 74) * q^83 + (244*z^3 - 86*z^2 + 122*z - 158) * q^84 + (-17*z^3 - 16*z^2 - 16*z - 17) * q^85 + (120*z^3 + 55*z - 55) * q^86 + (-56*z^3 + 28*z^2 - 84*z + 42) * q^87 + (43*z^3 + 6*z^2 + 2*z + 24) * q^88 + (-7*z^3 + 7*z^2 + 104) * q^89 + (-22*z^3 + 44*z^2 + 9*z + 53) * q^90 + (72*z^3 + 12*z^2 - 12*z - 72) * q^91 + (-12*z^2 + 22*z - 12) * q^92 + (118*z^3 - 42*z^2 + 118*z) * q^93 + (72*z^3 - 32*z^2 - 8*z - 64) * q^94 + (-16*z^3 + 37*z^2 - 8*z - 21) * q^95 + (20*z^3 + 91*z^2 + 91*z + 20) * q^96 + (-34*z^3 - 45*z + 45) * q^97 + (-134*z^3 - 16*z^2 - 118*z + 59) * q^98 + (-111*z^3 + 17*z^2 - 31*z + 68) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{2} - 6 q^{3} + 11 q^{4} + 5 q^{5} - 25 q^{6} + 30 q^{7} - 5 q^{8} + 13 q^{9}+O(q^{10})$$ 4 * q + 5 * q^2 - 6 * q^3 + 11 * q^4 + 5 * q^5 - 25 * q^6 + 30 * q^7 - 5 * q^8 + 13 * q^9 $$4 q + 5 q^{2} - 6 q^{3} + 11 q^{4} + 5 q^{5} - 25 q^{6} + 30 q^{7} - 5 q^{8} + 13 q^{9} - q^{11} - 34 q^{12} - 30 q^{13} + 50 q^{14} - 20 q^{15} - 21 q^{16} - 50 q^{17} + 40 q^{18} - 45 q^{19} + 5 q^{20} + 65 q^{22} - 16 q^{23} - 55 q^{24} - 5 q^{25} - 60 q^{26} + 57 q^{27} + 150 q^{28} - 70 q^{29} - 30 q^{30} + 98 q^{31} - 121 q^{33} - 130 q^{34} + 70 q^{35} + 102 q^{36} + 16 q^{37} + 50 q^{38} + 120 q^{39} - 35 q^{40} + 40 q^{41} - 70 q^{42} + 56 q^{44} + 70 q^{45} + 10 q^{46} - 24 q^{47} + 74 q^{48} + 11 q^{49} - 25 q^{50} + 95 q^{51} - 120 q^{52} - 106 q^{53} - 45 q^{55} - 155 q^{57} - 70 q^{58} + 217 q^{59} - 5 q^{60} - 80 q^{61} + 50 q^{62} - 100 q^{63} + 31 q^{64} - 330 q^{66} + 66 q^{67} - 210 q^{68} - 16 q^{69} + 130 q^{70} + 38 q^{71} + 85 q^{72} - 230 q^{73} - 100 q^{74} - 5 q^{75} + 300 q^{78} - 100 q^{79} + 15 q^{80} + 40 q^{81} + 45 q^{82} + 185 q^{83} - 180 q^{84} - 85 q^{85} - 45 q^{86} + 135 q^{88} + 402 q^{89} + 155 q^{90} - 240 q^{91} - 14 q^{92} + 278 q^{93} - 160 q^{94} - 145 q^{95} + 100 q^{96} + 101 q^{97} + 113 q^{99}+O(q^{100})$$ 4 * q + 5 * q^2 - 6 * q^3 + 11 * q^4 + 5 * q^5 - 25 * q^6 + 30 * q^7 - 5 * q^8 + 13 * q^9 - q^11 - 34 * q^12 - 30 * q^13 + 50 * q^14 - 20 * q^15 - 21 * q^16 - 50 * q^17 + 40 * q^18 - 45 * q^19 + 5 * q^20 + 65 * q^22 - 16 * q^23 - 55 * q^24 - 5 * q^25 - 60 * q^26 + 57 * q^27 + 150 * q^28 - 70 * q^29 - 30 * q^30 + 98 * q^31 - 121 * q^33 - 130 * q^34 + 70 * q^35 + 102 * q^36 + 16 * q^37 + 50 * q^38 + 120 * q^39 - 35 * q^40 + 40 * q^41 - 70 * q^42 + 56 * q^44 + 70 * q^45 + 10 * q^46 - 24 * q^47 + 74 * q^48 + 11 * q^49 - 25 * q^50 + 95 * q^51 - 120 * q^52 - 106 * q^53 - 45 * q^55 - 155 * q^57 - 70 * q^58 + 217 * q^59 - 5 * q^60 - 80 * q^61 + 50 * q^62 - 100 * q^63 + 31 * q^64 - 330 * q^66 + 66 * q^67 - 210 * q^68 - 16 * q^69 + 130 * q^70 + 38 * q^71 + 85 * q^72 - 230 * q^73 - 100 * q^74 - 5 * q^75 + 300 * q^78 - 100 * q^79 + 15 * q^80 + 40 * q^81 + 45 * q^82 + 185 * q^83 - 180 * q^84 - 85 * q^85 - 45 * q^86 + 135 * q^88 + 402 * q^89 + 155 * q^90 - 240 * q^91 - 14 * q^92 + 278 * q^93 - 160 * q^94 - 145 * q^95 + 100 * q^96 + 101 * q^97 + 113 * 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
2.92705 0.951057i −3.73607 2.71441i 4.42705 3.21644i 0.690983 2.12663i −13.5172 4.39201i 6.38197 + 8.78402i 2.66312 3.66547i 3.80902 + 11.7229i 6.88191i
41.1 −0.427051 0.587785i 0.736068 + 2.26538i 1.07295 3.30220i 1.80902 + 1.31433i 1.01722 1.40008i 8.61803 + 2.80017i −5.16312 + 1.67760i 2.69098 1.95511i 1.62460i
46.1 2.92705 + 0.951057i −3.73607 + 2.71441i 4.42705 + 3.21644i 0.690983 + 2.12663i −13.5172 + 4.39201i 6.38197 8.78402i 2.66312 + 3.66547i 3.80902 11.7229i 6.88191i
51.1 −0.427051 + 0.587785i 0.736068 2.26538i 1.07295 + 3.30220i 1.80902 1.31433i 1.01722 + 1.40008i 8.61803 2.80017i −5.16312 1.67760i 2.69098 + 1.95511i 1.62460i
 $$n$$: 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.b 4
5.b even 2 1 275.3.x.a 4
5.c odd 4 2 275.3.q.a 8
11.c even 5 1 605.3.c.b 4
11.d odd 10 1 inner 55.3.i.b 4
11.d odd 10 1 605.3.c.b 4
55.h odd 10 1 275.3.x.a 4
55.l even 20 2 275.3.q.a 8

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5 T^{3} + \cdots + 5$$
$3$ $$T^{4} + 6 T^{3} + \cdots + 121$$
$5$ $$T^{4} - 5 T^{3} + \cdots + 25$$
$7$ $$T^{4} - 30 T^{3} + \cdots + 9680$$
$11$ $$T^{4} + T^{3} + \cdots + 14641$$
$13$ $$T^{4} + 30 T^{3} + \cdots + 6480$$
$17$ $$T^{4} + 50 T^{3} + \cdots + 18605$$
$19$ $$T^{4} + 45 T^{3} + \cdots + 59405$$
$23$ $$(T^{2} + 8 T - 4)^{2}$$
$29$ $$T^{4} + 70 T^{3} + \cdots + 192080$$
$31$ $$T^{4} - 98 T^{3} + \cdots + 2421136$$
$37$ $$T^{4} - 16 T^{3} + \cdots + 190096$$
$41$ $$T^{4} - 40 T^{3} + \cdots + 15125$$
$43$ $$T^{4} + 4325 T^{2} + 4560125$$
$47$ $$T^{4} + 24 T^{3} + \cdots + 3444736$$
$53$ $$T^{4} + 106 T^{3} + \cdots + 6948496$$
$59$ $$T^{4} - 217 T^{3} + \cdots + 46389721$$
$61$ $$T^{4} + 80 T^{3} + \cdots + 16128080$$
$67$ $$(T^{2} - 33 T - 509)^{2}$$
$71$ $$T^{4} - 38 T^{3} + \cdots + 5216656$$
$73$ $$T^{4} + 230 T^{3} + \cdots + 32385125$$
$79$ $$T^{4} + 100 T^{3} + \cdots + 237774080$$
$83$ $$T^{4} - 185 T^{3} + \cdots + 13138205$$
$89$ $$(T^{2} - 201 T + 10039)^{2}$$
$97$ $$T^{4} - 101 T^{3} + \cdots + 5755201$$