# Properties

 Label 65.3.h.a Level $65$ Weight $3$ Character orbit 65.h Analytic conductor $1.771$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,3,Mod(12,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([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("65.12");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 65.h (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: $$1.77112171834$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 4 q^{3}+O(q^{10})$$ 24 * q - 4 * q^3 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 4 q^{3} + 16 q^{10} + 72 q^{12} - 36 q^{13} - 104 q^{16} - 48 q^{17} + 8 q^{22} - 104 q^{23} - 88 q^{25} + 88 q^{26} + 56 q^{27} - 24 q^{30} - 64 q^{35} + 256 q^{36} + 124 q^{38} - 368 q^{40} + 216 q^{42} + 8 q^{43} + 196 q^{48} - 296 q^{51} + 16 q^{52} + 220 q^{53} + 332 q^{55} + 584 q^{56} - 8 q^{61} - 596 q^{62} + 420 q^{65} - 360 q^{66} - 640 q^{68} - 184 q^{75} + 388 q^{77} - 636 q^{78} - 224 q^{81} - 1004 q^{82} - 52 q^{87} + 780 q^{88} + 452 q^{90} - 512 q^{91} + 812 q^{92} - 136 q^{95}+O(q^{100})$$ 24 * q - 4 * q^3 + 16 * q^10 + 72 * q^12 - 36 * q^13 - 104 * q^16 - 48 * q^17 + 8 * q^22 - 104 * q^23 - 88 * q^25 + 88 * q^26 + 56 * q^27 - 24 * q^30 - 64 * q^35 + 256 * q^36 + 124 * q^38 - 368 * q^40 + 216 * q^42 + 8 * q^43 + 196 * q^48 - 296 * q^51 + 16 * q^52 + 220 * q^53 + 332 * q^55 + 584 * q^56 - 8 * q^61 - 596 * q^62 + 420 * q^65 - 360 * q^66 - 640 * q^68 - 184 * q^75 + 388 * q^77 - 636 * q^78 - 224 * q^81 - 1004 * q^82 - 52 * q^87 + 780 * q^88 + 452 * q^90 - 512 * q^91 + 812 * q^92 - 136 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1 −2.66395 + 2.66395i −2.12513 + 2.12513i 10.1932i −4.54019 + 2.09444i 11.3225i 2.95114 2.95114i 16.4984 + 16.4984i 0.0323960i 6.51533 17.6743i
12.2 −2.14254 + 2.14254i 1.85789 1.85789i 5.18097i 1.93662 4.60972i 7.96121i 5.84982 5.84982i 2.53029 + 2.53029i 2.09649i 5.72722 + 14.0258i
12.3 −1.78036 + 1.78036i 0.433767 0.433767i 2.33933i 3.33534 + 3.72498i 1.54452i −5.63522 + 5.63522i −2.95658 2.95658i 8.62369i −12.5699 0.693690i
12.4 −1.30775 + 1.30775i −3.70495 + 3.70495i 0.579580i 1.70825 4.69914i 9.69031i −2.51452 + 2.51452i −5.98895 5.98895i 18.4534i 3.91133 + 8.37926i
12.5 −0.474292 + 0.474292i −0.839739 + 0.839739i 3.55009i −4.99852 + 0.121682i 0.796563i −1.35251 + 1.35251i −3.58095 3.58095i 7.58968i 2.31304 2.42847i
12.6 −0.456148 + 0.456148i 3.37817 3.37817i 3.58386i −0.779969 + 4.93879i 3.08189i 7.82065 7.82065i −3.45936 3.45936i 13.8241i −1.89704 2.60860i
12.7 0.456148 0.456148i 3.37817 3.37817i 3.58386i 0.779969 4.93879i 3.08189i −7.82065 + 7.82065i 3.45936 + 3.45936i 13.8241i −1.89704 2.60860i
12.8 0.474292 0.474292i −0.839739 + 0.839739i 3.55009i 4.99852 0.121682i 0.796563i 1.35251 1.35251i 3.58095 + 3.58095i 7.58968i 2.31304 2.42847i
12.9 1.30775 1.30775i −3.70495 + 3.70495i 0.579580i −1.70825 + 4.69914i 9.69031i 2.51452 2.51452i 5.98895 + 5.98895i 18.4534i 3.91133 + 8.37926i
12.10 1.78036 1.78036i 0.433767 0.433767i 2.33933i −3.33534 3.72498i 1.54452i 5.63522 5.63522i 2.95658 + 2.95658i 8.62369i −12.5699 0.693690i
12.11 2.14254 2.14254i 1.85789 1.85789i 5.18097i −1.93662 + 4.60972i 7.96121i −5.84982 + 5.84982i −2.53029 2.53029i 2.09649i 5.72722 + 14.0258i
12.12 2.66395 2.66395i −2.12513 + 2.12513i 10.1932i 4.54019 2.09444i 11.3225i −2.95114 + 2.95114i −16.4984 16.4984i 0.0323960i 6.51533 17.6743i
38.1 −2.66395 2.66395i −2.12513 2.12513i 10.1932i −4.54019 2.09444i 11.3225i 2.95114 + 2.95114i 16.4984 16.4984i 0.0323960i 6.51533 + 17.6743i
38.2 −2.14254 2.14254i 1.85789 + 1.85789i 5.18097i 1.93662 + 4.60972i 7.96121i 5.84982 + 5.84982i 2.53029 2.53029i 2.09649i 5.72722 14.0258i
38.3 −1.78036 1.78036i 0.433767 + 0.433767i 2.33933i 3.33534 3.72498i 1.54452i −5.63522 5.63522i −2.95658 + 2.95658i 8.62369i −12.5699 + 0.693690i
38.4 −1.30775 1.30775i −3.70495 3.70495i 0.579580i 1.70825 + 4.69914i 9.69031i −2.51452 2.51452i −5.98895 + 5.98895i 18.4534i 3.91133 8.37926i
38.5 −0.474292 0.474292i −0.839739 0.839739i 3.55009i −4.99852 0.121682i 0.796563i −1.35251 1.35251i −3.58095 + 3.58095i 7.58968i 2.31304 + 2.42847i
38.6 −0.456148 0.456148i 3.37817 + 3.37817i 3.58386i −0.779969 4.93879i 3.08189i 7.82065 + 7.82065i −3.45936 + 3.45936i 13.8241i −1.89704 + 2.60860i
38.7 0.456148 + 0.456148i 3.37817 + 3.37817i 3.58386i 0.779969 + 4.93879i 3.08189i −7.82065 7.82065i 3.45936 3.45936i 13.8241i −1.89704 + 2.60860i
38.8 0.474292 + 0.474292i −0.839739 0.839739i 3.55009i 4.99852 + 0.121682i 0.796563i 1.35251 + 1.35251i 3.58095 3.58095i 7.58968i 2.31304 + 2.42847i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 38.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
13.b even 2 1 inner
65.h odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.3.h.a 24
5.b even 2 1 325.3.h.b 24
5.c odd 4 1 inner 65.3.h.a 24
5.c odd 4 1 325.3.h.b 24
13.b even 2 1 inner 65.3.h.a 24
65.d even 2 1 325.3.h.b 24
65.h odd 4 1 inner 65.3.h.a 24
65.h odd 4 1 325.3.h.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.h.a 24 1.a even 1 1 trivial
65.3.h.a 24 5.c odd 4 1 inner
65.3.h.a 24 13.b even 2 1 inner
65.3.h.a 24 65.h odd 4 1 inner
325.3.h.b 24 5.b even 2 1
325.3.h.b 24 5.c odd 4 1
325.3.h.b 24 65.d even 2 1
325.3.h.b 24 65.h odd 4 1

## Hecke kernels

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