# Properties

 Label 546.2.bu.b Level $546$ Weight $2$ Character orbit 546.bu Analytic conductor $4.360$ Analytic rank $0$ Dimension $56$ 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.bu (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 8 q^{6} - 24 q^{9}+O(q^{10})$$ 56 * q + 8 * q^6 - 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 8 q^{6} - 24 q^{9} + 24 q^{10} + 8 q^{11} + 24 q^{13} + 4 q^{15} + 28 q^{16} - 4 q^{17} - 8 q^{18} + 8 q^{19} + 4 q^{21} - 8 q^{23} - 8 q^{24} - 4 q^{26} - 24 q^{27} - 8 q^{30} - 8 q^{31} + 8 q^{33} - 24 q^{34} + 24 q^{35} - 12 q^{36} - 8 q^{37} - 20 q^{39} - 28 q^{41} + 8 q^{44} + 72 q^{45} - 20 q^{46} + 64 q^{50} + 16 q^{54} + 8 q^{55} - 28 q^{56} + 4 q^{58} - 8 q^{59} + 20 q^{60} + 8 q^{61} - 32 q^{62} - 16 q^{63} - 24 q^{65} + 32 q^{66} - 16 q^{69} - 112 q^{71} + 8 q^{73} + 48 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{79} + 12 q^{81} - 4 q^{83} - 4 q^{84} + 32 q^{85} - 16 q^{86} - 144 q^{87} + 88 q^{89} - 8 q^{90} - 8 q^{91} + 52 q^{93} - 8 q^{94} - 48 q^{95} + 4 q^{96} - 64 q^{97} + 56 q^{99}+O(q^{100})$$ 56 * q + 8 * q^6 - 24 * q^9 + 24 * q^10 + 8 * q^11 + 24 * q^13 + 4 * q^15 + 28 * q^16 - 4 * q^17 - 8 * q^18 + 8 * q^19 + 4 * q^21 - 8 * q^23 - 8 * q^24 - 4 * q^26 - 24 * q^27 - 8 * q^30 - 8 * q^31 + 8 * q^33 - 24 * q^34 + 24 * q^35 - 12 * q^36 - 8 * q^37 - 20 * q^39 - 28 * q^41 + 8 * q^44 + 72 * q^45 - 20 * q^46 + 64 * q^50 + 16 * q^54 + 8 * q^55 - 28 * q^56 + 4 * q^58 - 8 * q^59 + 20 * q^60 + 8 * q^61 - 32 * q^62 - 16 * q^63 - 24 * q^65 + 32 * q^66 - 16 * q^69 - 112 * q^71 + 8 * q^73 + 48 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^79 + 12 * q^81 - 4 * q^83 - 4 * q^84 + 32 * q^85 - 16 * q^86 - 144 * q^87 + 88 * q^89 - 8 * q^90 - 8 * q^91 + 52 * q^93 - 8 * q^94 - 48 * q^95 + 4 * q^96 - 64 * q^97 + 56 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1 −0.965926 + 0.258819i −1.67137 + 0.454432i 0.866025 0.500000i −0.359298 0.359298i 1.49681 0.871531i −0.258819 + 0.965926i −0.707107 + 0.707107i 2.58698 1.51905i 0.440048 + 0.254062i
71.2 −0.965926 + 0.258819i −1.38004 1.04666i 0.866025 0.500000i −1.68505 1.68505i 1.60391 + 0.653816i −0.258819 + 0.965926i −0.707107 + 0.707107i 0.809005 + 2.88886i 2.06376 + 1.19151i
71.3 −0.965926 + 0.258819i −0.211319 1.71911i 0.866025 0.500000i 0.0940280 + 0.0940280i 0.649057 + 1.60584i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.91069 + 0.726562i −0.115160 0.0664879i
71.4 −0.965926 + 0.258819i −0.186842 + 1.72194i 0.866025 0.500000i −2.04268 2.04268i −0.265197 1.71163i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.93018 0.643461i 2.50176 + 1.44439i
71.5 −0.965926 + 0.258819i 0.458354 + 1.67030i 0.866025 0.500000i 1.24412 + 1.24412i −0.875042 1.49476i −0.258819 + 0.965926i −0.707107 + 0.707107i −2.57982 + 1.53118i −1.52373 0.879726i
71.6 −0.965926 + 0.258819i 1.13690 1.30670i 0.866025 0.500000i 1.58301 + 1.58301i −0.759962 + 1.55642i −0.258819 + 0.965926i −0.707107 + 0.707107i −0.414918 2.97117i −1.93879 1.11936i
71.7 −0.965926 + 0.258819i 1.59550 + 0.674078i 0.866025 0.500000i −1.28362 1.28362i −1.71560 0.238164i −0.258819 + 0.965926i −0.707107 + 0.707107i 2.09124 + 2.15098i 1.57211 + 0.907660i
71.8 0.965926 0.258819i −1.69149 + 0.372632i 0.866025 0.500000i 1.11112 + 1.11112i −1.53741 + 0.797725i 0.258819 0.965926i 0.707107 0.707107i 2.72229 1.26061i 1.36084 + 0.785680i
71.9 0.965926 0.258819i −0.884905 + 1.48894i 0.866025 0.500000i −1.36895 1.36895i −0.469386 + 1.66724i 0.258819 0.965926i 0.707107 0.707107i −1.43389 2.63514i −1.67662 0.967996i
71.10 0.965926 0.258819i −0.625096 1.61532i 0.866025 0.500000i 0.828570 + 0.828570i −1.02187 1.39849i 0.258819 0.965926i 0.707107 0.707107i −2.21851 + 2.01946i 1.01479 + 0.585888i
71.11 0.965926 0.258819i 0.162199 1.72444i 0.866025 0.500000i −2.61115 2.61115i −0.289646 1.70766i 0.258819 0.965926i 0.707107 0.707107i −2.94738 0.559404i −3.19799 1.84636i
71.12 0.965926 0.258819i 0.505064 + 1.65678i 0.866025 0.500000i 2.95685 + 2.95685i 0.916660 + 1.46960i 0.258819 0.965926i 0.707107 0.707107i −2.48982 + 1.67356i 3.62139 + 2.09081i
71.13 0.965926 0.258819i 1.18291 1.26520i 0.866025 0.500000i 2.02216 + 2.02216i 0.815146 1.52825i 0.258819 0.965926i 0.707107 0.707107i −0.201451 2.99323i 2.47663 + 1.42988i
71.14 0.965926 0.258819i 1.61014 + 0.638319i 0.866025 0.500000i −0.489109 0.489109i 1.72048 + 0.199834i 0.258819 0.965926i 0.707107 0.707107i 2.18510 + 2.05556i −0.599034 0.345852i
197.1 −0.258819 + 0.965926i −1.64483 + 0.542708i −0.866025 0.500000i 1.01262 + 1.01262i −0.0985019 1.72925i −0.965926 + 0.258819i 0.707107 0.707107i 2.41094 1.78532i −1.24020 + 0.716030i
197.2 −0.258819 + 0.965926i −1.34996 1.08518i −0.866025 0.500000i 0.429580 + 0.429580i 1.39760 1.02310i −0.965926 + 0.258819i 0.707107 0.707107i 0.644780 + 2.92989i −0.526126 + 0.303759i
197.3 −0.258819 + 0.965926i −1.12111 + 1.32027i −0.866025 0.500000i −0.381662 0.381662i −0.985117 1.42462i −0.965926 + 0.258819i 0.707107 0.707107i −0.486221 2.96034i 0.467439 0.269876i
197.4 −0.258819 + 0.965926i −0.413071 1.68207i −0.866025 0.500000i −2.27299 2.27299i 1.73167 + 0.0363565i −0.965926 + 0.258819i 0.707107 0.707107i −2.65874 + 1.38963i 2.78383 1.60724i
197.5 −0.258819 + 0.965926i 0.984178 1.42527i −0.866025 0.500000i −1.05727 1.05727i 1.12198 + 1.31953i −0.965926 + 0.258819i 0.707107 0.707107i −1.06279 2.80544i 1.29489 0.747605i
197.6 −0.258819 + 0.965926i 1.01587 + 1.40286i −0.866025 0.500000i −1.99319 1.99319i −1.61798 + 0.618168i −0.965926 + 0.258819i 0.707107 0.707107i −0.936020 + 2.85024i 2.44115 1.40940i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.14 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.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bu.b yes 56
3.b odd 2 1 546.2.bu.a 56
13.f odd 12 1 546.2.bu.a 56
39.k even 12 1 inner 546.2.bu.b yes 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bu.a 56 3.b odd 2 1
546.2.bu.a 56 13.f odd 12 1
546.2.bu.b yes 56 1.a even 1 1 trivial
546.2.bu.b yes 56 39.k even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{56} + 16 T_{5}^{53} + 860 T_{5}^{52} + 224 T_{5}^{51} + 128 T_{5}^{50} + 10400 T_{5}^{49} + 287904 T_{5}^{48} + 140800 T_{5}^{47} + 81408 T_{5}^{46} + 2525712 T_{5}^{45} + 48653252 T_{5}^{44} + \cdots + 19932921856$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.