# Properties

 Label 65.3.p.a Level $65$ Weight $3$ Character orbit 65.p Analytic conductor $1.771$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,3,Mod(6,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 5]))

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

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

chi := DirichletCharacter("65.6");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 65.p (of order $$12$$, 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: $$1.77112171834$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9}+O(q^{10})$$ 40 * q - 12 * q^6 - 40 * q^7 + 36 * q^8 - 72 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 12 q^{6} - 40 q^{7} + 36 q^{8} - 72 q^{9} - 12 q^{11} - 12 q^{13} + 48 q^{14} + 20 q^{15} + 128 q^{16} + 60 q^{17} - 136 q^{18} + 68 q^{19} - 80 q^{20} - 48 q^{21} - 48 q^{22} - 48 q^{23} - 56 q^{24} - 84 q^{26} + 24 q^{27} - 16 q^{28} + 28 q^{29} + 240 q^{30} + 128 q^{31} - 408 q^{32} + 136 q^{33} - 28 q^{34} + 40 q^{35} + 300 q^{36} + 56 q^{37} - 88 q^{39} + 68 q^{41} - 320 q^{42} - 372 q^{43} - 240 q^{44} - 40 q^{45} + 260 q^{46} + 152 q^{47} + 424 q^{48} - 132 q^{49} + 372 q^{52} - 288 q^{53} + 152 q^{54} - 40 q^{55} - 288 q^{56} + 252 q^{57} + 492 q^{58} + 492 q^{59} - 160 q^{60} - 100 q^{61} + 120 q^{62} + 844 q^{63} + 120 q^{65} - 456 q^{66} - 20 q^{67} + 72 q^{68} - 576 q^{69} + 120 q^{70} - 132 q^{71} - 780 q^{72} - 424 q^{73} - 160 q^{74} - 60 q^{75} - 992 q^{76} - 60 q^{78} - 248 q^{79} - 480 q^{80} + 600 q^{82} + 112 q^{83} + 1100 q^{84} - 120 q^{85} + 852 q^{86} - 160 q^{87} - 1188 q^{88} + 168 q^{89} - 160 q^{91} + 1192 q^{92} - 1008 q^{93} + 328 q^{94} + 120 q^{95} + 124 q^{96} - 1008 q^{97} - 636 q^{98} - 76 q^{99}+O(q^{100})$$ 40 * q - 12 * q^6 - 40 * q^7 + 36 * q^8 - 72 * q^9 - 12 * q^11 - 12 * q^13 + 48 * q^14 + 20 * q^15 + 128 * q^16 + 60 * q^17 - 136 * q^18 + 68 * q^19 - 80 * q^20 - 48 * q^21 - 48 * q^22 - 48 * q^23 - 56 * q^24 - 84 * q^26 + 24 * q^27 - 16 * q^28 + 28 * q^29 + 240 * q^30 + 128 * q^31 - 408 * q^32 + 136 * q^33 - 28 * q^34 + 40 * q^35 + 300 * q^36 + 56 * q^37 - 88 * q^39 + 68 * q^41 - 320 * q^42 - 372 * q^43 - 240 * q^44 - 40 * q^45 + 260 * q^46 + 152 * q^47 + 424 * q^48 - 132 * q^49 + 372 * q^52 - 288 * q^53 + 152 * q^54 - 40 * q^55 - 288 * q^56 + 252 * q^57 + 492 * q^58 + 492 * q^59 - 160 * q^60 - 100 * q^61 + 120 * q^62 + 844 * q^63 + 120 * q^65 - 456 * q^66 - 20 * q^67 + 72 * q^68 - 576 * q^69 + 120 * q^70 - 132 * q^71 - 780 * q^72 - 424 * q^73 - 160 * q^74 - 60 * q^75 - 992 * q^76 - 60 * q^78 - 248 * q^79 - 480 * q^80 + 600 * q^82 + 112 * q^83 + 1100 * q^84 - 120 * q^85 + 852 * q^86 - 160 * q^87 - 1188 * q^88 + 168 * q^89 - 160 * q^91 + 1192 * q^92 - 1008 * q^93 + 328 * q^94 + 120 * q^95 + 124 * q^96 - 1008 * q^97 - 636 * q^98 - 76 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −1.00018 3.73273i 1.63472 2.83142i −9.46878 + 5.46680i −1.58114 + 1.58114i −12.2039 3.27004i 2.30767 8.61234i 18.9464 + 18.9464i −0.844632 1.46295i 7.48338 + 4.32053i
6.2 −0.947644 3.53665i −2.34119 + 4.05506i −8.14580 + 4.70298i 1.58114 1.58114i 16.5600 + 4.43723i −2.82701 + 10.5506i 13.9961 + 13.9961i −6.46235 11.1931i −7.09030 4.09359i
6.3 −0.590063 2.20214i 1.05817 1.83281i −1.03716 + 0.598805i 1.58114 1.58114i −4.66050 1.24878i −0.314408 + 1.17338i −4.51768 4.51768i 2.26053 + 3.91536i −4.41487 2.54892i
6.4 −0.305985 1.14195i 2.30106 3.98555i 2.25367 1.30116i −1.58114 + 1.58114i −5.25539 1.40818i −3.05692 + 11.4086i −5.51932 5.51932i −6.08971 10.5477i 2.28939 + 1.32178i
6.5 0.0928988 + 0.346703i −0.295143 + 0.511202i 3.35253 1.93558i 1.58114 1.58114i −0.204654 0.0548368i −0.0877254 + 0.327396i 1.99774 + 1.99774i 4.32578 + 7.49247i 0.695072 + 0.401300i
6.6 0.107580 + 0.401493i −2.08519 + 3.61165i 3.31448 1.91361i −1.58114 + 1.58114i −1.67438 0.448649i −2.68917 + 10.0361i 2.30053 + 2.30053i −4.19603 7.26773i −0.804916 0.464718i
6.7 0.312366 + 1.16577i 1.19180 2.06426i 2.20266 1.27171i −1.58114 + 1.58114i 2.77872 + 0.744556i 2.00836 7.49531i 5.58415 + 5.58415i 1.65922 + 2.87385i −2.33713 1.34934i
6.8 0.715386 + 2.66986i −2.11281 + 3.65949i −3.15226 + 1.81996i 1.58114 1.58114i −11.2818 3.02294i 0.943772 3.52221i 0.703773 + 0.703773i −4.42790 7.66935i 5.35254 + 3.09029i
6.9 0.729421 + 2.72224i 2.39711 4.15192i −3.41442 + 1.97132i 1.58114 1.58114i 13.0510 + 3.49701i −1.27402 + 4.75472i 0.114321 + 0.114321i −6.99232 12.1111i 5.45755 + 3.15092i
6.10 0.886220 + 3.30742i −0.0164891 + 0.0285599i −6.68953 + 3.86220i −1.58114 + 1.58114i −0.109072 0.0292259i 0.185607 0.692695i −9.01754 9.01754i 4.49946 + 7.79329i −6.63073 3.82825i
11.1 −1.00018 + 3.73273i 1.63472 + 2.83142i −9.46878 5.46680i −1.58114 1.58114i −12.2039 + 3.27004i 2.30767 + 8.61234i 18.9464 18.9464i −0.844632 + 1.46295i 7.48338 4.32053i
11.2 −0.947644 + 3.53665i −2.34119 4.05506i −8.14580 4.70298i 1.58114 + 1.58114i 16.5600 4.43723i −2.82701 10.5506i 13.9961 13.9961i −6.46235 + 11.1931i −7.09030 + 4.09359i
11.3 −0.590063 + 2.20214i 1.05817 + 1.83281i −1.03716 0.598805i 1.58114 + 1.58114i −4.66050 + 1.24878i −0.314408 1.17338i −4.51768 + 4.51768i 2.26053 3.91536i −4.41487 + 2.54892i
11.4 −0.305985 + 1.14195i 2.30106 + 3.98555i 2.25367 + 1.30116i −1.58114 1.58114i −5.25539 + 1.40818i −3.05692 11.4086i −5.51932 + 5.51932i −6.08971 + 10.5477i 2.28939 1.32178i
11.5 0.0928988 0.346703i −0.295143 0.511202i 3.35253 + 1.93558i 1.58114 + 1.58114i −0.204654 + 0.0548368i −0.0877254 0.327396i 1.99774 1.99774i 4.32578 7.49247i 0.695072 0.401300i
11.6 0.107580 0.401493i −2.08519 3.61165i 3.31448 + 1.91361i −1.58114 1.58114i −1.67438 + 0.448649i −2.68917 10.0361i 2.30053 2.30053i −4.19603 + 7.26773i −0.804916 + 0.464718i
11.7 0.312366 1.16577i 1.19180 + 2.06426i 2.20266 + 1.27171i −1.58114 1.58114i 2.77872 0.744556i 2.00836 + 7.49531i 5.58415 5.58415i 1.65922 2.87385i −2.33713 + 1.34934i
11.8 0.715386 2.66986i −2.11281 3.65949i −3.15226 1.81996i 1.58114 + 1.58114i −11.2818 + 3.02294i 0.943772 + 3.52221i 0.703773 0.703773i −4.42790 + 7.66935i 5.35254 3.09029i
11.9 0.729421 2.72224i 2.39711 + 4.15192i −3.41442 1.97132i 1.58114 + 1.58114i 13.0510 3.49701i −1.27402 4.75472i 0.114321 0.114321i −6.99232 + 12.1111i 5.45755 3.15092i
11.10 0.886220 3.30742i −0.0164891 0.0285599i −6.68953 3.86220i −1.58114 1.58114i −0.109072 + 0.0292259i 0.185607 + 0.692695i −9.01754 + 9.01754i 4.49946 7.79329i −6.63073 + 3.82825i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.10 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 65.3.p.a 40
5.b even 2 1 325.3.t.d 40
5.c odd 4 1 325.3.w.e 40
5.c odd 4 1 325.3.w.f 40
13.f odd 12 1 inner 65.3.p.a 40
65.o even 12 1 325.3.w.e 40
65.s odd 12 1 325.3.t.d 40
65.t even 12 1 325.3.w.f 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.p.a 40 1.a even 1 1 trivial
65.3.p.a 40 13.f odd 12 1 inner
325.3.t.d 40 5.b even 2 1
325.3.t.d 40 65.s odd 12 1
325.3.w.e 40 5.c odd 4 1
325.3.w.e 40 65.o even 12 1
325.3.w.f 40 5.c odd 4 1
325.3.w.f 40 65.t even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(65, [\chi])$$.