# Properties

 Label 165.3.l.b Level $165$ Weight $3$ Character orbit 165.l Analytic conductor $4.496$ 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] = [165,3,Mod(32,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("165.32");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 165.l (of order $$4$$, degree $$2$$, 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: $$4.49592436194$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 \beta_{2} q^{3} - \beta_{2} q^{4} + 5 \beta_{2} q^{5} - 3 \beta_{3} q^{6} - 4 \beta_{3} q^{7} - 5 \beta_{3} q^{8} - 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3*b2 * q^3 - b2 * q^4 + 5*b2 * q^5 - 3*b3 * q^6 - 4*b3 * q^7 - 5*b3 * q^8 - 9 * q^9 $$q + \beta_1 q^{2} - 3 \beta_{2} q^{3} - \beta_{2} q^{4} + 5 \beta_{2} q^{5} - 3 \beta_{3} q^{6} - 4 \beta_{3} q^{7} - 5 \beta_{3} q^{8} - 9 q^{9} + 5 \beta_{3} q^{10} + ( - 4 \beta_{3} - 5 \beta_{2} + 4 \beta_1) q^{11} - 3 q^{12} - 4 \beta_1 q^{13} + 12 q^{14} + 15 q^{15} + 11 q^{16} - 4 \beta_1 q^{17} - 9 \beta_1 q^{18} + ( - 8 \beta_{3} + 8 \beta_1) q^{19} + 5 q^{20} - 12 \beta_1 q^{21} + ( - 5 \beta_{3} + 12 \beta_{2} + 12) q^{22} + (7 \beta_{2} + 7) q^{23} - 15 \beta_1 q^{24} - 25 q^{25} - 12 \beta_{2} q^{26} + 27 \beta_{2} q^{27} - 4 \beta_1 q^{28} + (20 \beta_{3} + 20 \beta_1) q^{29} + 15 \beta_1 q^{30} - 48 q^{31} - 9 \beta_1 q^{32} + ( - 12 \beta_{3} - 12 \beta_1 - 15) q^{33} - 12 \beta_{2} q^{34} + 20 \beta_1 q^{35} + 9 \beta_{2} q^{36} + (37 \beta_{2} + 37) q^{37} + (24 \beta_{2} + 24) q^{38} + 12 \beta_{3} q^{39} + 25 \beta_1 q^{40} + (12 \beta_{3} - 12 \beta_1) q^{41} - 36 \beta_{2} q^{42} + 36 \beta_1 q^{43} + ( - 4 \beta_{3} - 4 \beta_1 - 5) q^{44} - 45 \beta_{2} q^{45} + (7 \beta_{3} + 7 \beta_1) q^{46} + (17 \beta_{2} - 17) q^{47} - 33 \beta_{2} q^{48} + \beta_{2} q^{49} - 25 \beta_1 q^{50} + 12 \beta_{3} q^{51} + 4 \beta_{3} q^{52} + ( - 43 \beta_{2} - 43) q^{53} + 27 \beta_{3} q^{54} + (20 \beta_{3} + 20 \beta_1 + 25) q^{55} - 60 \beta_{2} q^{56} + ( - 24 \beta_{3} - 24 \beta_1) q^{57} + (60 \beta_{2} - 60) q^{58} - 38 q^{59} - 15 \beta_{2} q^{60} + ( - 20 \beta_{3} - 20 \beta_1) q^{61} - 48 \beta_1 q^{62} + 36 \beta_{3} q^{63} - 71 \beta_{2} q^{64} - 20 \beta_{3} q^{65} + ( - 36 \beta_{2} - 15 \beta_1 + 36) q^{66} + ( - 13 \beta_{2} - 13) q^{67} + 4 \beta_{3} q^{68} + ( - 21 \beta_{2} + 21) q^{69} + 60 \beta_{2} q^{70} + 45 \beta_{3} q^{72} + 76 \beta_1 q^{73} + (37 \beta_{3} + 37 \beta_1) q^{74} + 75 \beta_{2} q^{75} + ( - 8 \beta_{3} - 8 \beta_1) q^{76} + ( - 48 \beta_{2} - 20 \beta_1 + 48) q^{77} - 36 q^{78} + (12 \beta_{3} - 12 \beta_1) q^{79} + 55 \beta_{2} q^{80} + 81 q^{81} + ( - 36 \beta_{2} - 36) q^{82} - 84 \beta_{3} q^{83} + 12 \beta_{3} q^{84} - 20 \beta_{3} q^{85} + 108 \beta_{2} q^{86} + ( - 60 \beta_{3} + 60 \beta_1) q^{87} + ( - 60 \beta_{2} - 25 \beta_1 + 60) q^{88} - 48 q^{89} - 45 \beta_{3} q^{90} - 48 q^{91} + ( - 7 \beta_{2} + 7) q^{92} + 144 \beta_{2} q^{93} + (17 \beta_{3} - 17 \beta_1) q^{94} + (40 \beta_{3} + 40 \beta_1) q^{95} + 27 \beta_{3} q^{96} + (47 \beta_{2} + 47) q^{97} + \beta_{3} q^{98} + (36 \beta_{3} + 45 \beta_{2} - 36 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - 3*b2 * q^3 - b2 * q^4 + 5*b2 * q^5 - 3*b3 * q^6 - 4*b3 * q^7 - 5*b3 * q^8 - 9 * q^9 + 5*b3 * q^10 + (-4*b3 - 5*b2 + 4*b1) * q^11 - 3 * q^12 - 4*b1 * q^13 + 12 * q^14 + 15 * q^15 + 11 * q^16 - 4*b1 * q^17 - 9*b1 * q^18 + (-8*b3 + 8*b1) * q^19 + 5 * q^20 - 12*b1 * q^21 + (-5*b3 + 12*b2 + 12) * q^22 + (7*b2 + 7) * q^23 - 15*b1 * q^24 - 25 * q^25 - 12*b2 * q^26 + 27*b2 * q^27 - 4*b1 * q^28 + (20*b3 + 20*b1) * q^29 + 15*b1 * q^30 - 48 * q^31 - 9*b1 * q^32 + (-12*b3 - 12*b1 - 15) * q^33 - 12*b2 * q^34 + 20*b1 * q^35 + 9*b2 * q^36 + (37*b2 + 37) * q^37 + (24*b2 + 24) * q^38 + 12*b3 * q^39 + 25*b1 * q^40 + (12*b3 - 12*b1) * q^41 - 36*b2 * q^42 + 36*b1 * q^43 + (-4*b3 - 4*b1 - 5) * q^44 - 45*b2 * q^45 + (7*b3 + 7*b1) * q^46 + (17*b2 - 17) * q^47 - 33*b2 * q^48 + b2 * q^49 - 25*b1 * q^50 + 12*b3 * q^51 + 4*b3 * q^52 + (-43*b2 - 43) * q^53 + 27*b3 * q^54 + (20*b3 + 20*b1 + 25) * q^55 - 60*b2 * q^56 + (-24*b3 - 24*b1) * q^57 + (60*b2 - 60) * q^58 - 38 * q^59 - 15*b2 * q^60 + (-20*b3 - 20*b1) * q^61 - 48*b1 * q^62 + 36*b3 * q^63 - 71*b2 * q^64 - 20*b3 * q^65 + (-36*b2 - 15*b1 + 36) * q^66 + (-13*b2 - 13) * q^67 + 4*b3 * q^68 + (-21*b2 + 21) * q^69 + 60*b2 * q^70 + 45*b3 * q^72 + 76*b1 * q^73 + (37*b3 + 37*b1) * q^74 + 75*b2 * q^75 + (-8*b3 - 8*b1) * q^76 + (-48*b2 - 20*b1 + 48) * q^77 - 36 * q^78 + (12*b3 - 12*b1) * q^79 + 55*b2 * q^80 + 81 * q^81 + (-36*b2 - 36) * q^82 - 84*b3 * q^83 + 12*b3 * q^84 - 20*b3 * q^85 + 108*b2 * q^86 + (-60*b3 + 60*b1) * q^87 + (-60*b2 - 25*b1 + 60) * q^88 - 48 * q^89 - 45*b3 * q^90 - 48 * q^91 + (-7*b2 + 7) * q^92 + 144*b2 * q^93 + (17*b3 - 17*b1) * q^94 + (40*b3 + 40*b1) * q^95 + 27*b3 * q^96 + (47*b2 + 47) * q^97 + b3 * q^98 + (36*b3 + 45*b2 - 36*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{9}+O(q^{10})$$ 4 * q - 36 * q^9 $$4 q - 36 q^{9} - 12 q^{12} + 48 q^{14} + 60 q^{15} + 44 q^{16} + 20 q^{20} + 48 q^{22} + 28 q^{23} - 100 q^{25} - 192 q^{31} - 60 q^{33} + 148 q^{37} + 96 q^{38} - 20 q^{44} - 68 q^{47} - 172 q^{53} + 100 q^{55} - 240 q^{58} - 152 q^{59} + 144 q^{66} - 52 q^{67} + 84 q^{69} + 192 q^{77} - 144 q^{78} + 324 q^{81} - 144 q^{82} + 240 q^{88} - 192 q^{89} - 192 q^{91} + 28 q^{92} + 188 q^{97}+O(q^{100})$$ 4 * q - 36 * q^9 - 12 * q^12 + 48 * q^14 + 60 * q^15 + 44 * q^16 + 20 * q^20 + 48 * q^22 + 28 * q^23 - 100 * q^25 - 192 * q^31 - 60 * q^33 + 148 * q^37 + 96 * q^38 - 20 * q^44 - 68 * q^47 - 172 * q^53 + 100 * q^55 - 240 * q^58 - 152 * q^59 + 144 * q^66 - 52 * q^67 + 84 * q^69 + 192 * q^77 - 144 * q^78 + 324 * q^81 - 144 * q^82 + 240 * q^88 - 192 * q^89 - 192 * q^91 + 28 * q^92 + 188 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$67$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.22474 1.22474i 3.00000i 1.00000i 5.00000i −3.67423 + 3.67423i −4.89898 + 4.89898i −6.12372 + 6.12372i −9.00000 6.12372 6.12372i
32.2 1.22474 + 1.22474i 3.00000i 1.00000i 5.00000i 3.67423 3.67423i 4.89898 4.89898i 6.12372 6.12372i −9.00000 −6.12372 + 6.12372i
98.1 −1.22474 + 1.22474i 3.00000i 1.00000i 5.00000i −3.67423 3.67423i −4.89898 4.89898i −6.12372 6.12372i −9.00000 6.12372 + 6.12372i
98.2 1.22474 1.22474i 3.00000i 1.00000i 5.00000i 3.67423 + 3.67423i 4.89898 + 4.89898i 6.12372 + 6.12372i −9.00000 −6.12372 6.12372i
 $$n$$: 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.b odd 2 1 inner
15.e even 4 1 inner
165.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.3.l.b 4
3.b odd 2 1 165.3.l.f yes 4
5.c odd 4 1 165.3.l.f yes 4
11.b odd 2 1 inner 165.3.l.b 4
15.e even 4 1 inner 165.3.l.b 4
33.d even 2 1 165.3.l.f yes 4
55.e even 4 1 165.3.l.f yes 4
165.l odd 4 1 inner 165.3.l.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.l.b 4 1.a even 1 1 trivial
165.3.l.b 4 11.b odd 2 1 inner
165.3.l.b 4 15.e even 4 1 inner
165.3.l.b 4 165.l odd 4 1 inner
165.3.l.f yes 4 3.b odd 2 1
165.3.l.f yes 4 5.c odd 4 1
165.3.l.f yes 4 33.d even 2 1
165.3.l.f yes 4 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(165, [\chi])$$:

 $$T_{2}^{4} + 9$$ T2^4 + 9 $$T_{23}^{2} - 14T_{23} + 98$$ T23^2 - 14*T23 + 98

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$T^{4} + 2304$$
$11$ $$T^{4} - 142 T^{2} + 14641$$
$13$ $$T^{4} + 2304$$
$17$ $$T^{4} + 2304$$
$19$ $$(T^{2} - 384)^{2}$$
$23$ $$(T^{2} - 14 T + 98)^{2}$$
$29$ $$(T^{2} + 2400)^{2}$$
$31$ $$(T + 48)^{4}$$
$37$ $$(T^{2} - 74 T + 2738)^{2}$$
$41$ $$(T^{2} - 864)^{2}$$
$43$ $$T^{4} + 15116544$$
$47$ $$(T^{2} + 34 T + 578)^{2}$$
$53$ $$(T^{2} + 86 T + 3698)^{2}$$
$59$ $$(T + 38)^{4}$$
$61$ $$(T^{2} + 2400)^{2}$$
$67$ $$(T^{2} + 26 T + 338)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 300259584$$
$79$ $$(T^{2} - 864)^{2}$$
$83$ $$T^{4} + 448084224$$
$89$ $$(T + 48)^{4}$$
$97$ $$(T^{2} - 94 T + 4418)^{2}$$