# Properties

 Label 65.2.f.a Level $65$ Weight $2$ Character orbit 65.f Analytic conductor $0.519$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,2,Mod(18,65)]

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([3, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("65.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.f (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: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + ( - i + 1) q^{3} + q^{4} + ( - i - 2) q^{5} + (i + 1) q^{6} - 2 q^{7} + 3 i q^{8} + i q^{9}+O(q^{10})$$ q + i * q^2 + (-i + 1) * q^3 + q^4 + (-i - 2) * q^5 + (i + 1) * q^6 - 2 * q^7 + 3*i * q^8 + i * q^9 $$q + i q^{2} + ( - i + 1) q^{3} + q^{4} + ( - i - 2) q^{5} + (i + 1) q^{6} - 2 q^{7} + 3 i q^{8} + i q^{9} + ( - 2 i + 1) q^{10} + (i - 1) q^{11} + ( - i + 1) q^{12} + ( - 3 i - 2) q^{13} - 2 i q^{14} + (i - 3) q^{15} - q^{16} + (i - 1) q^{17} - q^{18} + ( - 5 i + 5) q^{19} + ( - i - 2) q^{20} + (2 i - 2) q^{21} + ( - i - 1) q^{22} + ( - 3 i - 3) q^{23} + (3 i + 3) q^{24} + (4 i + 3) q^{25} + ( - 2 i + 3) q^{26} + (4 i + 4) q^{27} - 2 q^{28} + ( - 3 i - 1) q^{30} + (5 i + 5) q^{31} + 5 i q^{32} + 2 i q^{33} + ( - i - 1) q^{34} + (2 i + 4) q^{35} + i q^{36} + (5 i + 5) q^{38} + ( - i - 5) q^{39} + ( - 6 i + 3) q^{40} + ( - 7 i - 7) q^{41} + ( - 2 i - 2) q^{42} + (i + 1) q^{43} + (i - 1) q^{44} + ( - 2 i + 1) q^{45} + ( - 3 i + 3) q^{46} + 6 q^{47} + (i - 1) q^{48} - 3 q^{49} + (3 i - 4) q^{50} + 2 i q^{51} + ( - 3 i - 2) q^{52} + ( - 5 i + 5) q^{53} + (4 i - 4) q^{54} + ( - i + 3) q^{55} - 6 i q^{56} - 10 i q^{57} + (7 i + 7) q^{59} + (i - 3) q^{60} - 14 q^{61} + (5 i - 5) q^{62} - 2 i q^{63} - 7 q^{64} + (8 i + 1) q^{65} - 2 q^{66} - 4 i q^{67} + (i - 1) q^{68} - 6 q^{69} + (4 i - 2) q^{70} + (i + 1) q^{71} - 3 q^{72} + 10 i q^{73} + (i + 7) q^{75} + ( - 5 i + 5) q^{76} + ( - 2 i + 2) q^{77} + ( - 5 i + 1) q^{78} - 2 i q^{79} + (i + 2) q^{80} + 5 q^{81} + ( - 7 i + 7) q^{82} + 6 q^{83} + (2 i - 2) q^{84} + ( - i + 3) q^{85} + (i - 1) q^{86} + ( - 3 i - 3) q^{88} + ( - 5 i - 5) q^{89} + (i + 2) q^{90} + (6 i + 4) q^{91} + ( - 3 i - 3) q^{92} + 10 q^{93} + 6 i q^{94} + (5 i - 15) q^{95} + (5 i + 5) q^{96} + 2 i q^{97} - 3 i q^{98} + ( - i - 1) q^{99} +O(q^{100})$$ q + i * q^2 + (-i + 1) * q^3 + q^4 + (-i - 2) * q^5 + (i + 1) * q^6 - 2 * q^7 + 3*i * q^8 + i * q^9 + (-2*i + 1) * q^10 + (i - 1) * q^11 + (-i + 1) * q^12 + (-3*i - 2) * q^13 - 2*i * q^14 + (i - 3) * q^15 - q^16 + (i - 1) * q^17 - q^18 + (-5*i + 5) * q^19 + (-i - 2) * q^20 + (2*i - 2) * q^21 + (-i - 1) * q^22 + (-3*i - 3) * q^23 + (3*i + 3) * q^24 + (4*i + 3) * q^25 + (-2*i + 3) * q^26 + (4*i + 4) * q^27 - 2 * q^28 + (-3*i - 1) * q^30 + (5*i + 5) * q^31 + 5*i * q^32 + 2*i * q^33 + (-i - 1) * q^34 + (2*i + 4) * q^35 + i * q^36 + (5*i + 5) * q^38 + (-i - 5) * q^39 + (-6*i + 3) * q^40 + (-7*i - 7) * q^41 + (-2*i - 2) * q^42 + (i + 1) * q^43 + (i - 1) * q^44 + (-2*i + 1) * q^45 + (-3*i + 3) * q^46 + 6 * q^47 + (i - 1) * q^48 - 3 * q^49 + (3*i - 4) * q^50 + 2*i * q^51 + (-3*i - 2) * q^52 + (-5*i + 5) * q^53 + (4*i - 4) * q^54 + (-i + 3) * q^55 - 6*i * q^56 - 10*i * q^57 + (7*i + 7) * q^59 + (i - 3) * q^60 - 14 * q^61 + (5*i - 5) * q^62 - 2*i * q^63 - 7 * q^64 + (8*i + 1) * q^65 - 2 * q^66 - 4*i * q^67 + (i - 1) * q^68 - 6 * q^69 + (4*i - 2) * q^70 + (i + 1) * q^71 - 3 * q^72 + 10*i * q^73 + (i + 7) * q^75 + (-5*i + 5) * q^76 + (-2*i + 2) * q^77 + (-5*i + 1) * q^78 - 2*i * q^79 + (i + 2) * q^80 + 5 * q^81 + (-7*i + 7) * q^82 + 6 * q^83 + (2*i - 2) * q^84 + (-i + 3) * q^85 + (i - 1) * q^86 + (-3*i - 3) * q^88 + (-5*i - 5) * q^89 + (i + 2) * q^90 + (6*i + 4) * q^91 + (-3*i - 3) * q^92 + 10 * q^93 + 6*i * q^94 + (5*i - 15) * q^95 + (5*i + 5) * q^96 + 2*i * q^97 - 3*i * q^98 + (-i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 4 * q^7 $$2 q + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{7} + 2 q^{10} - 2 q^{11} + 2 q^{12} - 4 q^{13} - 6 q^{15} - 2 q^{16} - 2 q^{17} - 2 q^{18} + 10 q^{19} - 4 q^{20} - 4 q^{21} - 2 q^{22} - 6 q^{23} + 6 q^{24} + 6 q^{25} + 6 q^{26} + 8 q^{27} - 4 q^{28} - 2 q^{30} + 10 q^{31} - 2 q^{34} + 8 q^{35} + 10 q^{38} - 10 q^{39} + 6 q^{40} - 14 q^{41} - 4 q^{42} + 2 q^{43} - 2 q^{44} + 2 q^{45} + 6 q^{46} + 12 q^{47} - 2 q^{48} - 6 q^{49} - 8 q^{50} - 4 q^{52} + 10 q^{53} - 8 q^{54} + 6 q^{55} + 14 q^{59} - 6 q^{60} - 28 q^{61} - 10 q^{62} - 14 q^{64} + 2 q^{65} - 4 q^{66} - 2 q^{68} - 12 q^{69} - 4 q^{70} + 2 q^{71} - 6 q^{72} + 14 q^{75} + 10 q^{76} + 4 q^{77} + 2 q^{78} + 4 q^{80} + 10 q^{81} + 14 q^{82} + 12 q^{83} - 4 q^{84} + 6 q^{85} - 2 q^{86} - 6 q^{88} - 10 q^{89} + 4 q^{90} + 8 q^{91} - 6 q^{92} + 20 q^{93} - 30 q^{95} + 10 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 - 4 * q^7 + 2 * q^10 - 2 * q^11 + 2 * q^12 - 4 * q^13 - 6 * q^15 - 2 * q^16 - 2 * q^17 - 2 * q^18 + 10 * q^19 - 4 * q^20 - 4 * q^21 - 2 * q^22 - 6 * q^23 + 6 * q^24 + 6 * q^25 + 6 * q^26 + 8 * q^27 - 4 * q^28 - 2 * q^30 + 10 * q^31 - 2 * q^34 + 8 * q^35 + 10 * q^38 - 10 * q^39 + 6 * q^40 - 14 * q^41 - 4 * q^42 + 2 * q^43 - 2 * q^44 + 2 * q^45 + 6 * q^46 + 12 * q^47 - 2 * q^48 - 6 * q^49 - 8 * q^50 - 4 * q^52 + 10 * q^53 - 8 * q^54 + 6 * q^55 + 14 * q^59 - 6 * q^60 - 28 * q^61 - 10 * q^62 - 14 * q^64 + 2 * q^65 - 4 * q^66 - 2 * q^68 - 12 * q^69 - 4 * q^70 + 2 * q^71 - 6 * q^72 + 14 * q^75 + 10 * q^76 + 4 * q^77 + 2 * q^78 + 4 * q^80 + 10 * q^81 + 14 * q^82 + 12 * q^83 - 4 * q^84 + 6 * q^85 - 2 * q^86 - 6 * q^88 - 10 * q^89 + 4 * q^90 + 8 * q^91 - 6 * q^92 + 20 * q^93 - 30 * q^95 + 10 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$i$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 − 1.00000i 1.00000i
1.00000i 1.00000 + 1.00000i 1.00000 −2.00000 + 1.00000i 1.00000 1.00000i −2.00000 3.00000i 1.00000i 1.00000 + 2.00000i
47.1 1.00000i 1.00000 1.00000i 1.00000 −2.00000 1.00000i 1.00000 + 1.00000i −2.00000 3.00000i 1.00000i 1.00000 2.00000i
 $$n$$: 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
65.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.f.a 2
3.b odd 2 1 585.2.n.c 2
4.b odd 2 1 1040.2.cd.b 2
5.b even 2 1 325.2.f.a 2
5.c odd 4 1 65.2.k.a yes 2
5.c odd 4 1 325.2.k.a 2
13.b even 2 1 845.2.f.a 2
13.c even 3 2 845.2.t.a 4
13.d odd 4 1 65.2.k.a yes 2
13.d odd 4 1 845.2.k.a 2
13.e even 6 2 845.2.t.b 4
13.f odd 12 2 845.2.o.a 4
13.f odd 12 2 845.2.o.b 4
15.e even 4 1 585.2.w.b 2
20.e even 4 1 1040.2.bg.a 2
39.f even 4 1 585.2.w.b 2
52.f even 4 1 1040.2.bg.a 2
65.f even 4 1 inner 65.2.f.a 2
65.g odd 4 1 325.2.k.a 2
65.h odd 4 1 845.2.k.a 2
65.k even 4 1 325.2.f.a 2
65.k even 4 1 845.2.f.a 2
65.o even 12 2 845.2.t.b 4
65.q odd 12 2 845.2.o.a 4
65.r odd 12 2 845.2.o.b 4
65.t even 12 2 845.2.t.a 4
195.u odd 4 1 585.2.n.c 2
260.l odd 4 1 1040.2.cd.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 1.a even 1 1 trivial
65.2.f.a 2 65.f even 4 1 inner
65.2.k.a yes 2 5.c odd 4 1
65.2.k.a yes 2 13.d odd 4 1
325.2.f.a 2 5.b even 2 1
325.2.f.a 2 65.k even 4 1
325.2.k.a 2 5.c odd 4 1
325.2.k.a 2 65.g odd 4 1
585.2.n.c 2 3.b odd 2 1
585.2.n.c 2 195.u odd 4 1
585.2.w.b 2 15.e even 4 1
585.2.w.b 2 39.f even 4 1
845.2.f.a 2 13.b even 2 1
845.2.f.a 2 65.k even 4 1
845.2.k.a 2 13.d odd 4 1
845.2.k.a 2 65.h odd 4 1
845.2.o.a 4 13.f odd 12 2
845.2.o.a 4 65.q odd 12 2
845.2.o.b 4 13.f odd 12 2
845.2.o.b 4 65.r odd 12 2
845.2.t.a 4 13.c even 3 2
845.2.t.a 4 65.t even 12 2
845.2.t.b 4 13.e even 6 2
845.2.t.b 4 65.o even 12 2
1040.2.bg.a 2 20.e even 4 1
1040.2.bg.a 2 52.f even 4 1
1040.2.cd.b 2 4.b odd 2 1
1040.2.cd.b 2 260.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$T^{2} - 10T + 50$$
$23$ $$T^{2} + 6T + 18$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 10T + 50$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 14T + 98$$
$43$ $$T^{2} - 2T + 2$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2} - 14T + 98$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2} - 2T + 2$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2} + 4$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 10T + 50$$
$97$ $$T^{2} + 4$$