# Properties

 Label 546.2.c.d Level $546$ Weight $2$ Character orbit 546.c Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

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 3.00000i 1.00000i 1.00000i 1.00000i 1.00000 3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i 1.00000 3.00000
 $$n$$: 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.d 2
3.b odd 2 1 1638.2.c.g 2
4.b odd 2 1 4368.2.h.b 2
7.b odd 2 1 3822.2.c.a 2
13.b even 2 1 inner 546.2.c.d 2
13.d odd 4 1 7098.2.a.p 1
13.d odd 4 1 7098.2.a.x 1
39.d odd 2 1 1638.2.c.g 2
52.b odd 2 1 4368.2.h.b 2
91.b odd 2 1 3822.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.d 2 1.a even 1 1 trivial
546.2.c.d 2 13.b even 2 1 inner
1638.2.c.g 2 3.b odd 2 1
1638.2.c.g 2 39.d odd 2 1
3822.2.c.a 2 7.b odd 2 1
3822.2.c.a 2 91.b odd 2 1
4368.2.h.b 2 4.b odd 2 1
4368.2.h.b 2 52.b odd 2 1
7098.2.a.p 1 13.d odd 4 1
7098.2.a.x 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$:

 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{11}^{2} + 25$$ T11^2 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 25$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T - 5)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T - 14)^{2}$$
$59$ $$T^{2} + 196$$
$61$ $$(T + 3)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2} + 100$$
$73$ $$T^{2} + 121$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} + 4$$