# Properties

 Label 546.2.p.a Level $546$ Weight $2$ Character orbit 546.p Analytic conductor $4.360$ Analytic rank $0$ Dimension $8$ 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.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.45474709504.3 Defining polynomial: $$x^{8} + 38 x^{6} + 481 x^{4} + 2112 x^{2} + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 38 x^{6} + 481 x^{4} + 2112 x^{2} + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 6 \nu^{5} + 223 \nu^{3} + 1504 \nu$$$$)/1024$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 6 \nu^{6} - 82 \nu^{5} - 164 \nu^{4} - 611 \nu^{3} - 1094 \nu^{2} - 800 \nu - 192$$$$)/512$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} - 6 \nu^{6} + 82 \nu^{5} - 164 \nu^{4} + 611 \nu^{3} - 1094 \nu^{2} + 800 \nu - 192$$$$)/512$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 22 \nu^{6} - 82 \nu^{5} + 516 \nu^{4} - 611 \nu^{3} + 2902 \nu^{2} - 800 \nu + 704$$$$)/512$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{7} - 52 \nu^{6} - 170 \nu^{5} - 1336 \nu^{4} - 743 \nu^{3} - 8884 \nu^{2} + 2720 \nu - 6784$$$$)/1024$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{7} - 64 \nu^{6} + 334 \nu^{5} - 1664 \nu^{4} + 1965 \nu^{3} - 11072 \nu^{2} - 1120 \nu - 7168$$$$)/1024$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 10$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 11 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$13 \beta_{7} + 13 \beta_{6} + 7 \beta_{5} - 53 \beta_{4} - 47 \beta_{3} + 130$$ $$\nu^{5}$$ $$=$$ $$40 \beta_{7} - 40 \beta_{6} - 76 \beta_{4} + 36 \beta_{3} + 128 \beta_{2} + 137 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-173 \beta_{7} - 173 \beta_{6} - 9 \beta_{5} + 859 \beta_{4} + 695 \beta_{3} - 1762$$ $$\nu^{7}$$ $$=$$ $$-686 \beta_{7} + 686 \beta_{6} + 1348 \beta_{4} - 662 \beta_{3} - 2684 \beta_{2} - 1771 \beta_{1}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 − 3.49421i 3.90842i 0.742317i − 3.15653i 3.49421i − 3.90842i − 0.742317i 3.15653i
−0.707107 0.707107i 1.41421 + 1.00000i 1.00000i −2.76367 2.76367i −0.292893 1.70711i −0.707107 0.707107i 0.707107 0.707107i 1.00000 + 2.82843i 3.90842i
239.2 −0.707107 0.707107i 1.41421 + 1.00000i 1.00000i 2.47078 + 2.47078i −0.292893 1.70711i −0.707107 0.707107i 0.707107 0.707107i 1.00000 + 2.82843i 3.49421i
239.3 0.707107 + 0.707107i −1.41421 + 1.00000i 1.00000i −2.23200 2.23200i −1.70711 0.292893i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000 2.82843i 3.15653i
239.4 0.707107 + 0.707107i −1.41421 + 1.00000i 1.00000i 0.524897 + 0.524897i −1.70711 0.292893i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000 2.82843i 0.742317i
281.1 −0.707107 + 0.707107i 1.41421 1.00000i 1.00000i −2.76367 + 2.76367i −0.292893 + 1.70711i −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000 2.82843i 3.90842i
281.2 −0.707107 + 0.707107i 1.41421 1.00000i 1.00000i 2.47078 2.47078i −0.292893 + 1.70711i −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000 2.82843i 3.49421i
281.3 0.707107 0.707107i −1.41421 1.00000i 1.00000i −2.23200 + 2.23200i −1.70711 + 0.292893i 0.707107 0.707107i −0.707107 0.707107i 1.00000 + 2.82843i 3.15653i
281.4 0.707107 0.707107i −1.41421 1.00000i 1.00000i 0.524897 0.524897i −1.70711 + 0.292893i 0.707107 0.707107i −0.707107 0.707107i 1.00000 + 2.82843i 0.742317i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 281.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.p.a 8
3.b odd 2 1 546.2.p.b yes 8
13.d odd 4 1 546.2.p.b yes 8
39.f even 4 1 inner 546.2.p.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.p.a 8 1.a even 1 1 trivial
546.2.p.a 8 39.f even 4 1 inner
546.2.p.b yes 8 3.b odd 2 1
546.2.p.b yes 8 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 9 - 2 T^{2} + T^{4} )^{2}$$
$5$ $$1024 - 1536 T + 1152 T^{2} + 592 T^{3} + 161 T^{4} - 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$( 1 + T^{4} )^{2}$$
$11$ $$18496 + 25024 T + 16928 T^{2} + 4216 T^{3} + 561 T^{4} + 48 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$13$ $$28561 + 8788 T + 3718 T^{2} + 1508 T^{3} + 418 T^{4} + 116 T^{5} + 22 T^{6} + 4 T^{7} + T^{8}$$
$17$ $$( -68 + 44 T + 19 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$19$ $$3136 - 3136 T + 1568 T^{2} + 4312 T^{3} + 3361 T^{4} + 1084 T^{5} + 200 T^{6} + 20 T^{7} + T^{8}$$
$23$ $$( 2254 - 308 T - 89 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$29$ $$2116 + 2692 T^{2} + 717 T^{4} + 50 T^{6} + T^{8}$$
$31$ $$( 16 - 32 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$37$ $$16 + 640 T + 12800 T^{2} + 13184 T^{3} + 6881 T^{4} + 1832 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$41$ $$61504 + 59520 T + 28800 T^{2} - 1280 T^{3} + 1172 T^{4} + 760 T^{5} + 200 T^{6} + 20 T^{7} + T^{8}$$
$43$ $$3268864 + 480352 T^{2} + 17345 T^{4} + 230 T^{6} + T^{8}$$
$47$ $$( 1296 + T^{4} )^{2}$$
$53$ $$16192576 + 2362848 T^{2} + 52468 T^{4} + 404 T^{6} + T^{8}$$
$59$ $$7772944 - 1249024 T + 100352 T^{2} + 66304 T^{3} + 16328 T^{4} + 448 T^{5} + T^{8}$$
$61$ $$( -4738 + 2460 T - 201 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$67$ $$44782864 - 16489088 T + 3035648 T^{2} - 259168 T^{3} + 14984 T^{4} - 1504 T^{5} + 288 T^{6} - 24 T^{7} + T^{8}$$
$71$ $$107584 + 36736 T + 6272 T^{2} - 21504 T^{3} + 12756 T^{4} - 2968 T^{5} + 392 T^{6} - 28 T^{7} + T^{8}$$
$73$ $$28601104 - 21862624 T + 8355872 T^{2} - 956312 T^{3} + 56065 T^{4} - 680 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$79$ $$( -496 + 240 T + 10 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$83$ $$107584 - 36736 T + 6272 T^{2} + 21504 T^{3} + 12756 T^{4} + 2968 T^{5} + 392 T^{6} + 28 T^{7} + T^{8}$$
$89$ $$2347024 + 2745344 T + 1605632 T^{2} + 480256 T^{3} + 74888 T^{4} + 1792 T^{5} + T^{8}$$
$97$ $$( 784 - 448 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2}$$