# Properties

 Label 165.3.l.d 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: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + \beta_{2} q^{4} + (3 \beta_{2} + 4) q^{5} + 3 \beta_{3} q^{6} + 2 \beta_{3} q^{7} - 3 \beta_{3} q^{8} - 9 q^{9}+O(q^{10})$$ q + b1 * q^2 + 3*b2 * q^3 + b2 * q^4 + (3*b2 + 4) * q^5 + 3*b3 * q^6 + 2*b3 * q^7 - 3*b3 * q^8 - 9 * q^9 $$q + \beta_1 q^{2} + 3 \beta_{2} q^{3} + \beta_{2} q^{4} + (3 \beta_{2} + 4) q^{5} + 3 \beta_{3} q^{6} + 2 \beta_{3} q^{7} - 3 \beta_{3} q^{8} - 9 q^{9} + (3 \beta_{3} + 4 \beta_1) q^{10} + ( - 2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{11} - 3 q^{12} - 2 \beta_1 q^{13} - 10 q^{14} + (12 \beta_{2} - 9) q^{15} + 19 q^{16} - 14 \beta_1 q^{17} - 9 \beta_1 q^{18} + ( - 4 \beta_{3} + 4 \beta_1) q^{19} + (4 \beta_{2} - 3) q^{20} - 6 \beta_1 q^{21} + (9 \beta_{3} + 10 \beta_{2} + 10) q^{22} + (7 \beta_{2} + 7) q^{23} + 9 \beta_1 q^{24} + (24 \beta_{2} + 7) q^{25} - 10 \beta_{2} q^{26} - 27 \beta_{2} q^{27} - 2 \beta_1 q^{28} + ( - 6 \beta_{3} - 6 \beta_1) q^{29} + (12 \beta_{3} - 9 \beta_1) q^{30} + 20 q^{31} + 7 \beta_1 q^{32} + (6 \beta_{3} + 6 \beta_1 - 27) q^{33} - 70 \beta_{2} q^{34} + (8 \beta_{3} - 6 \beta_1) q^{35} - 9 \beta_{2} q^{36} + (7 \beta_{2} + 7) q^{37} + (20 \beta_{2} + 20) q^{38} - 6 \beta_{3} q^{39} + ( - 12 \beta_{3} + 9 \beta_1) q^{40} + ( - 22 \beta_{3} + 22 \beta_1) q^{41} - 30 \beta_{2} q^{42} - 14 \beta_1 q^{43} + (2 \beta_{3} + 2 \beta_1 - 9) q^{44} + ( - 27 \beta_{2} - 36) q^{45} + (7 \beta_{3} + 7 \beta_1) q^{46} + ( - 43 \beta_{2} + 43) q^{47} + 57 \beta_{2} q^{48} + 29 \beta_{2} q^{49} + (24 \beta_{3} + 7 \beta_1) q^{50} - 42 \beta_{3} q^{51} - 2 \beta_{3} q^{52} + ( - 17 \beta_{2} - 17) q^{53} - 27 \beta_{3} q^{54} + ( - 2 \beta_{3} + 36 \beta_{2} + 14 \beta_1 - 27) q^{55} + 30 \beta_{2} q^{56} + (12 \beta_{3} + 12 \beta_1) q^{57} + ( - 30 \beta_{2} + 30) q^{58} - 22 q^{59} + ( - 9 \beta_{2} - 12) q^{60} + (30 \beta_{3} + 30 \beta_1) q^{61} + 20 \beta_1 q^{62} - 18 \beta_{3} q^{63} - 41 \beta_{2} q^{64} + ( - 6 \beta_{3} - 8 \beta_1) q^{65} + (30 \beta_{2} - 27 \beta_1 - 30) q^{66} + ( - 47 \beta_{2} - 47) q^{67} - 14 \beta_{3} q^{68} + (21 \beta_{2} - 21) q^{69} + ( - 30 \beta_{2} - 40) q^{70} - 120 \beta_{2} q^{71} + 27 \beta_{3} q^{72} - 14 \beta_1 q^{73} + (7 \beta_{3} + 7 \beta_1) q^{74} + (21 \beta_{2} - 72) q^{75} + (4 \beta_{3} + 4 \beta_1) q^{76} + (20 \beta_{2} - 18 \beta_1 - 20) q^{77} + 30 q^{78} + (2 \beta_{3} - 2 \beta_1) q^{79} + (57 \beta_{2} + 76) q^{80} + 81 q^{81} + (110 \beta_{2} + 110) q^{82} + 38 \beta_{3} q^{83} - 6 \beta_{3} q^{84} + ( - 42 \beta_{3} - 56 \beta_1) q^{85} - 70 \beta_{2} q^{86} + ( - 18 \beta_{3} + 18 \beta_1) q^{87} + ( - 30 \beta_{2} + 27 \beta_1 + 30) q^{88} - 100 q^{89} + ( - 27 \beta_{3} - 36 \beta_1) q^{90} + 20 q^{91} + (7 \beta_{2} - 7) q^{92} + 60 \beta_{2} q^{93} + ( - 43 \beta_{3} + 43 \beta_1) q^{94} + ( - 4 \beta_{3} + 28 \beta_1) q^{95} + 21 \beta_{3} q^{96} + (43 \beta_{2} + 43) q^{97} + 29 \beta_{3} q^{98} + (18 \beta_{3} - 81 \beta_{2} - 18 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + 3*b2 * q^3 + b2 * q^4 + (3*b2 + 4) * q^5 + 3*b3 * q^6 + 2*b3 * q^7 - 3*b3 * q^8 - 9 * q^9 + (3*b3 + 4*b1) * q^10 + (-2*b3 + 9*b2 + 2*b1) * q^11 - 3 * q^12 - 2*b1 * q^13 - 10 * q^14 + (12*b2 - 9) * q^15 + 19 * q^16 - 14*b1 * q^17 - 9*b1 * q^18 + (-4*b3 + 4*b1) * q^19 + (4*b2 - 3) * q^20 - 6*b1 * q^21 + (9*b3 + 10*b2 + 10) * q^22 + (7*b2 + 7) * q^23 + 9*b1 * q^24 + (24*b2 + 7) * q^25 - 10*b2 * q^26 - 27*b2 * q^27 - 2*b1 * q^28 + (-6*b3 - 6*b1) * q^29 + (12*b3 - 9*b1) * q^30 + 20 * q^31 + 7*b1 * q^32 + (6*b3 + 6*b1 - 27) * q^33 - 70*b2 * q^34 + (8*b3 - 6*b1) * q^35 - 9*b2 * q^36 + (7*b2 + 7) * q^37 + (20*b2 + 20) * q^38 - 6*b3 * q^39 + (-12*b3 + 9*b1) * q^40 + (-22*b3 + 22*b1) * q^41 - 30*b2 * q^42 - 14*b1 * q^43 + (2*b3 + 2*b1 - 9) * q^44 + (-27*b2 - 36) * q^45 + (7*b3 + 7*b1) * q^46 + (-43*b2 + 43) * q^47 + 57*b2 * q^48 + 29*b2 * q^49 + (24*b3 + 7*b1) * q^50 - 42*b3 * q^51 - 2*b3 * q^52 + (-17*b2 - 17) * q^53 - 27*b3 * q^54 + (-2*b3 + 36*b2 + 14*b1 - 27) * q^55 + 30*b2 * q^56 + (12*b3 + 12*b1) * q^57 + (-30*b2 + 30) * q^58 - 22 * q^59 + (-9*b2 - 12) * q^60 + (30*b3 + 30*b1) * q^61 + 20*b1 * q^62 - 18*b3 * q^63 - 41*b2 * q^64 + (-6*b3 - 8*b1) * q^65 + (30*b2 - 27*b1 - 30) * q^66 + (-47*b2 - 47) * q^67 - 14*b3 * q^68 + (21*b2 - 21) * q^69 + (-30*b2 - 40) * q^70 - 120*b2 * q^71 + 27*b3 * q^72 - 14*b1 * q^73 + (7*b3 + 7*b1) * q^74 + (21*b2 - 72) * q^75 + (4*b3 + 4*b1) * q^76 + (20*b2 - 18*b1 - 20) * q^77 + 30 * q^78 + (2*b3 - 2*b1) * q^79 + (57*b2 + 76) * q^80 + 81 * q^81 + (110*b2 + 110) * q^82 + 38*b3 * q^83 - 6*b3 * q^84 + (-42*b3 - 56*b1) * q^85 - 70*b2 * q^86 + (-18*b3 + 18*b1) * q^87 + (-30*b2 + 27*b1 + 30) * q^88 - 100 * q^89 + (-27*b3 - 36*b1) * q^90 + 20 * q^91 + (7*b2 - 7) * q^92 + 60*b2 * q^93 + (-43*b3 + 43*b1) * q^94 + (-4*b3 + 28*b1) * q^95 + 21*b3 * q^96 + (43*b2 + 43) * q^97 + 29*b3 * q^98 + (18*b3 - 81*b2 - 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{5} - 36 q^{9}+O(q^{10})$$ 4 * q + 16 * q^5 - 36 * q^9 $$4 q + 16 q^{5} - 36 q^{9} - 12 q^{12} - 40 q^{14} - 36 q^{15} + 76 q^{16} - 12 q^{20} + 40 q^{22} + 28 q^{23} + 28 q^{25} + 80 q^{31} - 108 q^{33} + 28 q^{37} + 80 q^{38} - 36 q^{44} - 144 q^{45} + 172 q^{47} - 68 q^{53} - 108 q^{55} + 120 q^{58} - 88 q^{59} - 48 q^{60} - 120 q^{66} - 188 q^{67} - 84 q^{69} - 160 q^{70} - 288 q^{75} - 80 q^{77} + 120 q^{78} + 304 q^{80} + 324 q^{81} + 440 q^{82} + 120 q^{88} - 400 q^{89} + 80 q^{91} - 28 q^{92} + 172 q^{97}+O(q^{100})$$ 4 * q + 16 * q^5 - 36 * q^9 - 12 * q^12 - 40 * q^14 - 36 * q^15 + 76 * q^16 - 12 * q^20 + 40 * q^22 + 28 * q^23 + 28 * q^25 + 80 * q^31 - 108 * q^33 + 28 * q^37 + 80 * q^38 - 36 * q^44 - 144 * q^45 + 172 * q^47 - 68 * q^53 - 108 * q^55 + 120 * q^58 - 88 * q^59 - 48 * q^60 - 120 * q^66 - 188 * q^67 - 84 * q^69 - 160 * q^70 - 288 * q^75 - 80 * q^77 + 120 * q^78 + 304 * q^80 + 324 * q^81 + 440 * q^82 + 120 * q^88 - 400 * q^89 + 80 * q^91 - 28 * q^92 + 172 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*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
 −1.58114 − 1.58114i 1.58114 + 1.58114i −1.58114 + 1.58114i 1.58114 − 1.58114i
−1.58114 1.58114i 3.00000i 1.00000i 4.00000 + 3.00000i 4.74342 4.74342i 3.16228 3.16228i −4.74342 + 4.74342i −9.00000 −1.58114 11.0680i
32.2 1.58114 + 1.58114i 3.00000i 1.00000i 4.00000 + 3.00000i −4.74342 + 4.74342i −3.16228 + 3.16228i 4.74342 4.74342i −9.00000 1.58114 + 11.0680i
98.1 −1.58114 + 1.58114i 3.00000i 1.00000i 4.00000 3.00000i 4.74342 + 4.74342i 3.16228 + 3.16228i −4.74342 4.74342i −9.00000 −1.58114 + 11.0680i
98.2 1.58114 1.58114i 3.00000i 1.00000i 4.00000 3.00000i −4.74342 4.74342i −3.16228 3.16228i 4.74342 + 4.74342i −9.00000 1.58114 11.0680i
 $$n$$: 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.d yes 4
3.b odd 2 1 165.3.l.a 4
5.c odd 4 1 165.3.l.a 4
11.b odd 2 1 inner 165.3.l.d yes 4
15.e even 4 1 inner 165.3.l.d yes 4
33.d even 2 1 165.3.l.a 4
55.e even 4 1 165.3.l.a 4
165.l odd 4 1 inner 165.3.l.d yes 4

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

## Hecke kernels

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

 $$T_{2}^{4} + 25$$ T2^4 + 25 $$T_{23}^{2} - 14T_{23} + 98$$ T23^2 - 14*T23 + 98

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25$$
$3$ $$(T^{2} + 9)^{2}$$
$5$ $$(T^{2} - 8 T + 25)^{2}$$
$7$ $$T^{4} + 400$$
$11$ $$T^{4} + 82T^{2} + 14641$$
$13$ $$T^{4} + 400$$
$17$ $$T^{4} + 960400$$
$19$ $$(T^{2} - 160)^{2}$$
$23$ $$(T^{2} - 14 T + 98)^{2}$$
$29$ $$(T^{2} + 360)^{2}$$
$31$ $$(T - 20)^{4}$$
$37$ $$(T^{2} - 14 T + 98)^{2}$$
$41$ $$(T^{2} - 4840)^{2}$$
$43$ $$T^{4} + 960400$$
$47$ $$(T^{2} - 86 T + 3698)^{2}$$
$53$ $$(T^{2} + 34 T + 578)^{2}$$
$59$ $$(T + 22)^{4}$$
$61$ $$(T^{2} + 9000)^{2}$$
$67$ $$(T^{2} + 94 T + 4418)^{2}$$
$71$ $$(T^{2} + 14400)^{2}$$
$73$ $$T^{4} + 960400$$
$79$ $$(T^{2} - 40)^{2}$$
$83$ $$T^{4} + 52128400$$
$89$ $$(T + 100)^{4}$$
$97$ $$(T^{2} - 86 T + 3698)^{2}$$