# Properties

 Label 546.2.i.i Level $546$ Weight $2$ Character orbit 546.i Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(79,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("546.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.i (of order $$3$$, 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: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} - q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + (-b3 - 2*b2 - b1) * q^5 - q^6 + (b3 + b2 - b1) * q^7 + q^8 + b2 * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} - q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + q^{8} + \beta_{2} q^{9} + (2 \beta_{2} + \beta_1 + 2) q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} - \beta_{2} q^{12} + q^{13} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{14} + ( - \beta_{3} + 2) q^{15} + \beta_{2} q^{16} + (\beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{3} - 2) q^{20} + ( - \beta_{3} - 2 \beta_1 - 1) q^{21} + ( - \beta_{3} + 1) q^{22} + ( - \beta_{3} - \beta_1) q^{23} + (\beta_{2} + 1) q^{24} + ( - \beta_{2} - 4 \beta_1 - 1) q^{25} + \beta_{2} q^{26} - q^{27} + (\beta_{3} + 2 \beta_1 + 1) q^{28} + ( - 2 \beta_{3} + 1) q^{29} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{30} - 6 \beta_1 q^{31} + ( - \beta_{2} - 1) q^{32} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{33} + ( - \beta_{3} - 1) q^{34} + (4 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{35} + q^{36} + ( - \beta_{3} - \beta_1) q^{37} + (5 \beta_{2} + 2 \beta_1 + 5) q^{38} + (\beta_{2} + 1) q^{39} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{40} - 7 \beta_{3} q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{42} + (6 \beta_{3} - 2) q^{43} + (\beta_{3} + \beta_{2} + \beta_1) q^{44} + (2 \beta_{2} + \beta_1 + 2) q^{45} + \beta_1 q^{46} + \beta_{2} q^{47} - q^{48} + ( - 4 \beta_{3} + 5 \beta_{2} + \cdots + 5) q^{49}+ \cdots + ( - \beta_{3} + 1) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 + 1) * q^3 + (-b2 - 1) * q^4 + (-b3 - 2*b2 - b1) * q^5 - q^6 + (b3 + b2 - b1) * q^7 + q^8 + b2 * q^9 + (2*b2 + b1 + 2) * q^10 + (-b2 - b1 - 1) * q^11 - b2 * q^12 + q^13 + (-2*b3 - b2 - b1 - 1) * q^14 + (-b3 + 2) * q^15 + b2 * q^16 + (b2 - b1 + 1) * q^17 + (-b2 - 1) * q^18 + (-2*b3 - 5*b2 - 2*b1) * q^19 + (b3 - 2) * q^20 + (-b3 - 2*b1 - 1) * q^21 + (-b3 + 1) * q^22 + (-b3 - b1) * q^23 + (b2 + 1) * q^24 + (-b2 - 4*b1 - 1) * q^25 + b2 * q^26 - q^27 + (b3 + 2*b1 + 1) * q^28 + (-2*b3 + 1) * q^29 + (b3 + 2*b2 + b1) * q^30 - 6*b1 * q^31 + (-b2 - 1) * q^32 + (-b3 - b2 - b1) * q^33 + (-b3 - 1) * q^34 + (4*b3 + 4*b2 + 3*b1) * q^35 + q^36 + (-b3 - b1) * q^37 + (5*b2 + 2*b1 + 5) * q^38 + (b2 + 1) * q^39 + (-b3 - 2*b2 - b1) * q^40 - 7*b3 * q^41 + (-b3 - b2 + b1) * q^42 + (6*b3 - 2) * q^43 + (b3 + b2 + b1) * q^44 + (2*b2 + b1 + 2) * q^45 + b1 * q^46 + b2 * q^47 - q^48 + (-4*b3 + 5*b2 - 2*b1 + 5) * q^49 + (-4*b3 + 1) * q^50 + (-b3 + b2 - b1) * q^51 + (-b2 - 1) * q^52 + (-b2 + 6*b1 - 1) * q^53 - b2 * q^54 + (3*b3 - 4) * q^55 + (b3 + b2 - b1) * q^56 + (-2*b3 + 5) * q^57 + (2*b3 + b2 + 2*b1) * q^58 + (5*b2 + 5*b1 + 5) * q^59 + (-2*b2 - b1 - 2) * q^60 + (b3 - 3*b2 + b1) * q^61 - 6*b3 * q^62 + (-2*b3 - b2 - b1 - 1) * q^63 + q^64 + (-b3 - 2*b2 - b1) * q^65 + (b2 + b1 + 1) * q^66 + (b2 + 2*b1 + 1) * q^67 + (b3 - b2 + b1) * q^68 - b3 * q^69 + (-b3 - 4*b2 - 4*b1 - 4) * q^70 - 5 * q^71 + b2 * q^72 + b1 * q^73 + b1 * q^74 + (-4*b3 - b2 - 4*b1) * q^75 + (2*b3 - 5) * q^76 + (4*b2 + 2*b1 + 3) * q^77 - q^78 + (4*b3 - 6*b2 + 4*b1) * q^79 + (2*b2 + b1 + 2) * q^80 + (-b2 - 1) * q^81 + (7*b3 + 7*b1) * q^82 + (4*b3 + 2) * q^83 + (2*b3 + b2 + b1 + 1) * q^84 + b3 * q^85 + (-6*b3 - 2*b2 - 6*b1) * q^86 + (b2 + 2*b1 + 1) * q^87 + (-b2 - b1 - 1) * q^88 + (b3 + 4*b2 + b1) * q^89 + (b3 - 2) * q^90 + (b3 + b2 - b1) * q^91 + b3 * q^92 + (-6*b3 - 6*b1) * q^93 + (-b2 - 1) * q^94 + (-14*b2 - 9*b1 - 14) * q^95 - b2 * q^96 + (5*b3 - 8) * q^97 + (2*b3 + 4*b1 - 5) * q^98 + (-b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 4 q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{13} - 2 q^{14} + 8 q^{15} - 2 q^{16} + 2 q^{17} - 2 q^{18} + 10 q^{19} - 8 q^{20} - 4 q^{21} + 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{26} - 4 q^{27} + 4 q^{28} + 4 q^{29} - 4 q^{30} - 2 q^{32} + 2 q^{33} - 4 q^{34} - 8 q^{35} + 4 q^{36} + 10 q^{38} + 2 q^{39} + 4 q^{40} + 2 q^{42} - 8 q^{43} - 2 q^{44} + 4 q^{45} - 2 q^{47} - 4 q^{48} + 10 q^{49} + 4 q^{50} - 2 q^{51} - 2 q^{52} - 2 q^{53} + 2 q^{54} - 16 q^{55} - 2 q^{56} + 20 q^{57} - 2 q^{58} + 10 q^{59} - 4 q^{60} + 6 q^{61} - 2 q^{63} + 4 q^{64} + 4 q^{65} + 2 q^{66} + 2 q^{67} + 2 q^{68} - 8 q^{70} - 20 q^{71} - 2 q^{72} + 2 q^{75} - 20 q^{76} + 4 q^{77} - 4 q^{78} + 12 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{83} + 2 q^{84} + 4 q^{86} + 2 q^{87} - 2 q^{88} - 8 q^{89} - 8 q^{90} - 2 q^{91} - 2 q^{94} - 28 q^{95} + 2 q^{96} - 32 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 + 4 * q^10 - 2 * q^11 + 2 * q^12 + 4 * q^13 - 2 * q^14 + 8 * q^15 - 2 * q^16 + 2 * q^17 - 2 * q^18 + 10 * q^19 - 8 * q^20 - 4 * q^21 + 4 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^26 - 4 * q^27 + 4 * q^28 + 4 * q^29 - 4 * q^30 - 2 * q^32 + 2 * q^33 - 4 * q^34 - 8 * q^35 + 4 * q^36 + 10 * q^38 + 2 * q^39 + 4 * q^40 + 2 * q^42 - 8 * q^43 - 2 * q^44 + 4 * q^45 - 2 * q^47 - 4 * q^48 + 10 * q^49 + 4 * q^50 - 2 * q^51 - 2 * q^52 - 2 * q^53 + 2 * q^54 - 16 * q^55 - 2 * q^56 + 20 * q^57 - 2 * q^58 + 10 * q^59 - 4 * q^60 + 6 * q^61 - 2 * q^63 + 4 * q^64 + 4 * q^65 + 2 * q^66 + 2 * q^67 + 2 * q^68 - 8 * q^70 - 20 * q^71 - 2 * q^72 + 2 * q^75 - 20 * q^76 + 4 * q^77 - 4 * q^78 + 12 * q^79 + 4 * q^80 - 2 * q^81 + 8 * q^83 + 2 * q^84 + 4 * q^86 + 2 * q^87 - 2 * q^88 - 8 * q^89 - 8 * q^90 - 2 * q^91 - 2 * q^94 - 28 * q^95 + 2 * q^96 - 32 * q^97 - 20 * q^98 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.292893 0.507306i −1.00000 1.62132 + 2.09077i 1.00000 −0.500000 + 0.866025i 0.292893 + 0.507306i
79.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.70711 2.95680i −1.00000 −2.62132 0.358719i 1.00000 −0.500000 + 0.866025i 1.70711 + 2.95680i
235.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.292893 + 0.507306i −1.00000 1.62132 2.09077i 1.00000 −0.500000 0.866025i 0.292893 0.507306i
235.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.70711 + 2.95680i −1.00000 −2.62132 + 0.358719i 1.00000 −0.500000 0.866025i 1.70711 2.95680i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.i.i 4
3.b odd 2 1 1638.2.j.m 4
7.c even 3 1 inner 546.2.i.i 4
7.c even 3 1 3822.2.a.bn 2
7.d odd 6 1 3822.2.a.bu 2
21.h odd 6 1 1638.2.j.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.i 4 1.a even 1 1 trivial
546.2.i.i 4 7.c even 3 1 inner
1638.2.j.m 4 3.b odd 2 1
1638.2.j.m 4 21.h odd 6 1
3822.2.a.bn 2 7.c even 3 1
3822.2.a.bu 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 4T_{5}^{3} + 14T_{5}^{2} - 8T_{5} + 4$$ T5^4 - 4*T5^3 + 14*T5^2 - 8*T5 + 4 $$T_{17}^{4} - 2T_{17}^{3} + 5T_{17}^{2} + 2T_{17} + 1$$ T17^4 - 2*T17^3 + 5*T17^2 + 2*T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4} - 4 T^{3} + \cdots + 4$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 49$$
$11$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$19$ $$T^{4} - 10 T^{3} + \cdots + 289$$
$23$ $$T^{4} + 2T^{2} + 4$$
$29$ $$(T^{2} - 2 T - 7)^{2}$$
$31$ $$T^{4} + 72T^{2} + 5184$$
$37$ $$T^{4} + 2T^{2} + 4$$
$41$ $$(T^{2} - 98)^{2}$$
$43$ $$(T^{2} + 4 T - 68)^{2}$$
$47$ $$(T^{2} + T + 1)^{2}$$
$53$ $$T^{4} + 2 T^{3} + \cdots + 5041$$
$59$ $$T^{4} - 10 T^{3} + \cdots + 625$$
$61$ $$T^{4} - 6 T^{3} + \cdots + 49$$
$67$ $$T^{4} - 2 T^{3} + \cdots + 49$$
$71$ $$(T + 5)^{4}$$
$73$ $$T^{4} + 2T^{2} + 4$$
$79$ $$T^{4} - 12 T^{3} + \cdots + 16$$
$83$ $$(T^{2} - 4 T - 28)^{2}$$
$89$ $$T^{4} + 8 T^{3} + \cdots + 196$$
$97$ $$(T^{2} + 16 T + 14)^{2}$$
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