# Properties

 Label 546.2.c.a Level $546$ Weight $2$ Character orbit 546.c Analytic conductor $4.360$ 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] = [546,2,Mod(337,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("546.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.c (of order $$2$$, degree $$1$$, 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: $$4.35983195036$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 1.00000 1.00000
337.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 1.00000i 1.00000i 1.00000 1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.c.a 2
3.b odd 2 1 1638.2.c.f 2
4.b odd 2 1 4368.2.h.k 2
7.b odd 2 1 3822.2.c.e 2
13.b even 2 1 inner 546.2.c.a 2
13.d odd 4 1 7098.2.a.d 1
13.d odd 4 1 7098.2.a.s 1
39.d odd 2 1 1638.2.c.f 2
52.b odd 2 1 4368.2.h.k 2
91.b odd 2 1 3822.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 1.a even 1 1 trivial
546.2.c.a 2 13.b even 2 1 inner
1638.2.c.f 2 3.b odd 2 1
1638.2.c.f 2 39.d odd 2 1
3822.2.c.e 2 7.b odd 2 1
3822.2.c.e 2 91.b odd 2 1
4368.2.h.k 2 4.b odd 2 1
4368.2.h.k 2 52.b odd 2 1
7098.2.a.d 1 13.d odd 4 1
7098.2.a.s 1 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{11}^{2} + 1$$ T11^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 1$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T - 3)^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 81$$
$41$ $$T^{2} + 64$$
$43$ $$(T - 7)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 11)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 121$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 144$$
$97$ $$T^{2} + 4$$