# Properties

 Label 546.2.bu.a 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 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 8 q^{6} + 24 q^{9} + 24 q^{10} - 8 q^{11} + 24 q^{13} - 4 q^{15} + 28 q^{16} + 4 q^{17} + 8 q^{19} + 4 q^{21} + 8 q^{23} + 8 q^{24} + 4 q^{26} - 24 q^{27} - 16 q^{30} - 8 q^{31} - 32 q^{33} - 24 q^{34} - 24 q^{35} - 12 q^{36} - 8 q^{37} - 60 q^{39} + 28 q^{41} - 8 q^{44} - 40 q^{45} - 20 q^{46} - 64 q^{50} + 8 q^{54} + 8 q^{55} + 28 q^{56} + 40 q^{57} + 4 q^{58} + 8 q^{59} + 4 q^{60} + 8 q^{61} + 32 q^{62} + 24 q^{65} + 32 q^{66} - 56 q^{69} + 112 q^{71} + 16 q^{72} + 8 q^{73} - 48 q^{74} - 40 q^{75} + 8 q^{76} - 8 q^{78} + 16 q^{79} + 12 q^{81} + 4 q^{83} - 4 q^{84} + 32 q^{85} + 16 q^{86} + 144 q^{87} - 88 q^{89} + 8 q^{90} - 8 q^{91} - 76 q^{93} - 8 q^{94} + 48 q^{95} - 4 q^{96} - 64 q^{97} - 40 q^{99}+O(q^{100})$$ 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^19 + 4 * q^21 + 8 * q^23 + 8 * q^24 + 4 * q^26 - 24 * q^27 - 16 * q^30 - 8 * q^31 - 32 * q^33 - 24 * q^34 - 24 * q^35 - 12 * q^36 - 8 * q^37 - 60 * q^39 + 28 * q^41 - 8 * q^44 - 40 * q^45 - 20 * q^46 - 64 * q^50 + 8 * q^54 + 8 * q^55 + 28 * q^56 + 40 * q^57 + 4 * q^58 + 8 * q^59 + 4 * q^60 + 8 * q^61 + 32 * q^62 + 24 * q^65 + 32 * q^66 - 56 * q^69 + 112 * q^71 + 16 * q^72 + 8 * q^73 - 48 * q^74 - 40 * q^75 + 8 * q^76 - 8 * q^78 + 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 - 76 * q^93 - 8 * q^94 + 48 * q^95 - 4 * q^96 - 64 * q^97 - 40 * 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 −1.68734 + 0.390990i 0.866025 0.500000i −2.95685 2.95685i 1.52865 0.814384i 0.258819 0.965926i −0.707107 + 0.707107i 2.69425 1.31947i 3.62139 + 2.09081i
71.2 −0.965926 + 0.258819i −1.35787 1.07526i 0.866025 0.500000i 0.489109 + 0.489109i 1.58990 + 0.687181i 0.258819 0.965926i −0.707107 + 0.707107i 0.687622 + 2.92013i −0.599034 0.345852i
71.3 −0.965926 + 0.258819i −0.847008 + 1.51082i 0.866025 0.500000i 1.36895 + 1.36895i 0.427118 1.67856i 0.258819 0.965926i −0.707107 + 0.707107i −1.56515 2.55935i −1.67662 0.967996i
71.4 −0.965926 + 0.258819i 0.504239 1.65703i 0.866025 0.500000i −2.02216 2.02216i −0.0581866 + 1.73107i 0.258819 0.965926i −0.707107 + 0.707107i −2.49149 1.67108i 2.47663 + 1.42988i
71.5 −0.965926 + 0.258819i 0.523037 + 1.65119i 0.866025 0.500000i −1.11112 1.11112i −0.932575 1.45956i 0.258819 0.965926i −0.707107 + 0.707107i −2.45286 + 1.72727i 1.36084 + 0.785680i
71.6 −0.965926 + 0.258819i 1.41231 1.00269i 0.866025 0.500000i 2.61115 + 2.61115i −1.10467 + 1.33405i 0.258819 0.965926i −0.707107 + 0.707107i 0.989233 2.83221i −3.19799 1.84636i
71.7 −0.965926 + 0.258819i 1.71146 0.266310i 0.866025 0.500000i −0.828570 0.828570i −1.58421 + 0.700193i 0.258819 0.965926i −0.707107 + 0.707107i 2.85816 0.911557i 1.01479 + 0.585888i
71.8 0.965926 0.258819i −1.67570 + 0.438205i 0.866025 0.500000i −1.24412 1.24412i −1.50519 + 0.856977i −0.258819 + 0.965926i 0.707107 0.707107i 2.61595 1.46860i −1.52373 0.879726i
71.9 0.965926 0.258819i −1.39783 + 1.02278i 0.866025 0.500000i 2.04268 + 2.04268i −1.08548 + 1.34972i −0.258819 + 0.965926i 0.707107 0.707107i 0.907836 2.85934i 2.50176 + 1.44439i
71.10 0.965926 0.258819i −1.38152 1.04470i 0.866025 0.500000i 1.28362 + 1.28362i −1.60483 0.651543i −0.258819 + 0.965926i 0.707107 0.707107i 0.817188 + 2.88656i 1.57211 + 0.907660i
71.11 0.965926 0.258819i 0.442137 + 1.67467i 0.866025 0.500000i 0.359298 + 0.359298i 0.860508 + 1.50317i −0.258819 + 0.965926i 0.707107 0.707107i −2.60903 + 1.48087i 0.440048 + 0.254062i
71.12 0.965926 0.258819i 0.563184 1.63793i 0.866025 0.500000i −1.58301 1.58301i 0.120065 1.72788i −0.258819 + 0.965926i 0.707107 0.707107i −2.36565 1.84491i −1.93879 1.11936i
71.13 0.965926 0.258819i 1.59445 0.676548i 0.866025 0.500000i −0.0940280 0.0940280i 1.36502 1.06617i −0.258819 + 0.965926i 0.707107 0.707107i 2.08457 2.15745i −0.115160 0.0664879i
71.14 0.965926 0.258819i 1.59645 + 0.671817i 0.866025 0.500000i 1.68505 + 1.68505i 1.71593 + 0.235733i −0.258819 + 0.965926i 0.707107 0.707107i 2.09732 + 2.14505i 2.06376 + 1.19151i
197.1 −0.258819 + 0.965926i −1.73063 0.0700467i −0.866025 0.500000i 2.88717 + 2.88717i 0.515581 1.65353i 0.965926 0.258819i 0.707107 0.707107i 2.99019 + 0.242450i −3.53605 + 2.04154i
197.2 −0.258819 + 0.965926i −1.60917 0.640767i −0.866025 0.500000i −2.11813 2.11813i 1.03542 1.38849i 0.965926 0.258819i 0.707107 0.707107i 2.17884 + 2.06220i 2.59417 1.49774i
197.3 −0.258819 + 0.965926i −0.703834 + 1.58260i −0.866025 0.500000i −2.95010 2.95010i −1.34651 1.08946i 0.965926 0.258819i 0.707107 0.707107i −2.00924 2.22777i 3.61312 2.08604i
197.4 −0.258819 + 0.965926i 0.684122 + 1.59122i −0.866025 0.500000i 0.803802 + 0.803802i −1.71406 + 0.248973i 0.965926 0.258819i 0.707107 0.707107i −2.06396 + 2.17717i −0.984452 + 0.568374i
197.5 −0.258819 + 0.965926i 0.957713 1.44319i −0.866025 0.500000i −0.0544192 0.0544192i 1.14614 + 1.29860i 0.965926 0.258819i 0.707107 0.707107i −1.16557 2.76432i 0.0666496 0.0384802i
197.6 −0.258819 + 0.965926i 1.63803 + 0.562898i −0.866025 0.500000i 1.20435 + 1.20435i −0.967672 + 1.43653i 0.965926 0.258819i 0.707107 0.707107i 2.36629 + 1.84409i −1.47502 + 0.851606i
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.a 56
3.b odd 2 1 546.2.bu.b yes 56
13.f odd 12 1 546.2.bu.b yes 56
39.k even 12 1 inner 546.2.bu.a 56

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

## 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$.