# Properties

 Label 546.2.bq.c Level $546$ Weight $2$ Character orbit 546.bq Analytic conductor $4.360$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bq (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{4} + 16q^{7} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{4} + 16q^{7} + 16q^{9} - 12q^{15} - 32q^{16} - 24q^{18} + 16q^{21} - 16q^{28} + 8q^{30} - 16q^{36} - 48q^{39} + 20q^{42} - 8q^{43} + 8q^{46} + 8q^{49} - 8q^{51} + 8q^{57} + 16q^{58} - 24q^{60} + 32q^{63} - 64q^{64} + 24q^{67} + 48q^{70} - 12q^{72} - 12q^{78} + 96q^{79} + 68q^{81} + 8q^{84} - 112q^{85} - 16q^{91} - 56q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1 −0.866025 + 0.500000i −1.73204 0.00466741i 0.500000 0.866025i 0.915374 1.50233 0.861980i −2.64454 + 0.0800636i 1.00000i 2.99996 + 0.0161683i −0.792737 + 0.457687i
419.2 −0.866025 + 0.500000i −1.73041 + 0.0753636i 0.500000 0.866025i −2.24165 1.46090 0.930472i −0.301604 2.62850i 1.00000i 2.98864 0.260820i 1.94133 1.12083i
419.3 −0.866025 + 0.500000i −1.69217 + 0.369559i 0.500000 0.866025i −3.03607 1.28068 1.16613i 2.25111 + 1.39014i 1.00000i 2.72685 1.25071i 2.62932 1.51804i
419.4 −0.866025 + 0.500000i −1.66375 0.481611i 0.500000 0.866025i 4.01418 1.68165 0.414785i 1.77617 1.96093i 1.00000i 2.53610 + 1.60256i −3.47639 + 2.00709i
419.5 −0.866025 + 0.500000i −1.22965 + 1.21982i 0.500000 0.866025i 1.88020 0.454992 1.67122i −1.36777 + 2.26478i 1.00000i 0.0240567 2.99990i −1.62830 + 0.940099i
419.6 −0.866025 + 0.500000i −0.998793 1.41507i 0.500000 0.866025i 1.30549 1.57251 + 0.726086i 2.26007 + 1.37552i 1.00000i −1.00482 + 2.82672i −1.13058 + 0.652743i
419.7 −0.866025 + 0.500000i −0.344709 + 1.69740i 0.500000 0.866025i 1.64213 −0.550175 1.64235i −1.55066 2.14370i 1.00000i −2.76235 1.17022i −1.42213 + 0.821065i
419.8 −0.866025 + 0.500000i −0.211712 + 1.71906i 0.500000 0.866025i −0.932541 −0.676184 1.59461i 2.18650 1.48970i 1.00000i −2.91036 0.727891i 0.807604 0.466271i
419.9 −0.866025 + 0.500000i 0.211712 1.71906i 0.500000 0.866025i 0.932541 0.676184 + 1.59461i 0.196871 2.63842i 1.00000i −2.91036 0.727891i −0.807604 + 0.466271i
419.10 −0.866025 + 0.500000i 0.344709 1.69740i 0.500000 0.866025i −1.64213 0.550175 + 1.64235i 2.63183 + 0.271064i 1.00000i −2.76235 1.17022i 1.42213 0.821065i
419.11 −0.866025 + 0.500000i 0.998793 + 1.41507i 0.500000 0.866025i −1.30549 −1.57251 0.726086i −2.32128 1.26952i 1.00000i −1.00482 + 2.82672i 1.13058 0.652743i
419.12 −0.866025 + 0.500000i 1.22965 1.21982i 0.500000 0.866025i −1.88020 −0.454992 + 1.67122i −1.27747 + 2.31691i 1.00000i 0.0240567 2.99990i 1.62830 0.940099i
419.13 −0.866025 + 0.500000i 1.66375 + 0.481611i 0.500000 0.866025i −4.01418 −1.68165 + 0.414785i 0.810131 2.51867i 1.00000i 2.53610 + 1.60256i 3.47639 2.00709i
419.14 −0.866025 + 0.500000i 1.69217 0.369559i 0.500000 0.866025i 3.03607 −1.28068 + 1.16613i −2.32945 1.25445i 1.00000i 2.72685 1.25071i −2.62932 + 1.51804i
419.15 −0.866025 + 0.500000i 1.73041 0.0753636i 0.500000 0.866025i 2.24165 −1.46090 + 0.930472i 2.42715 1.05306i 1.00000i 2.98864 0.260820i −1.94133 + 1.12083i
419.16 −0.866025 + 0.500000i 1.73204 + 0.00466741i 0.500000 0.866025i −0.915374 −1.50233 + 0.861980i 1.25293 + 2.33027i 1.00000i 2.99996 + 0.0161683i 0.792737 0.457687i
419.17 0.866025 0.500000i −1.67122 + 0.454992i 0.500000 0.866025i −1.88020 −1.21982 + 1.22965i −1.36777 + 2.26478i 1.00000i 2.58596 1.52079i −1.62830 + 0.940099i
419.18 0.866025 0.500000i −1.64235 0.550175i 0.500000 0.866025i −1.64213 −1.69740 + 0.344709i −1.55066 2.14370i 1.00000i 2.39462 + 1.80716i −1.42213 + 0.821065i
419.19 0.866025 0.500000i −1.59461 0.676184i 0.500000 0.866025i 0.932541 −1.71906 + 0.211712i 2.18650 1.48970i 1.00000i 2.08555 + 2.15650i 0.807604 0.466271i
419.20 0.866025 0.500000i −1.16613 + 1.28068i 0.500000 0.866025i 3.03607 −0.369559 + 1.69217i 2.25111 + 1.39014i 1.00000i −0.280280 2.98688i 2.62932 1.51804i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 503.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.c even 3 1 inner
21.c even 2 1 inner
39.i odd 6 1 inner
91.n odd 6 1 inner
273.bn even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bq.c 64
3.b odd 2 1 inner 546.2.bq.c 64
7.b odd 2 1 inner 546.2.bq.c 64
13.c even 3 1 inner 546.2.bq.c 64
21.c even 2 1 inner 546.2.bq.c 64
39.i odd 6 1 inner 546.2.bq.c 64
91.n odd 6 1 inner 546.2.bq.c 64
273.bn even 6 1 inner 546.2.bq.c 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bq.c 64 1.a even 1 1 trivial
546.2.bq.c 64 3.b odd 2 1 inner
546.2.bq.c 64 7.b odd 2 1 inner
546.2.bq.c 64 13.c even 3 1 inner
546.2.bq.c 64 21.c even 2 1 inner
546.2.bq.c 64 39.i odd 6 1 inner
546.2.bq.c 64 91.n odd 6 1 inner
546.2.bq.c 64 273.bn even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{16} - \cdots$$ $$18\!\cdots\!30$$$$T_{61}^{14} +$$$$42\!\cdots\!08$$$$T_{61}^{12} -$$$$51\!\cdots\!78$$$$T_{61}^{10} +$$$$45\!\cdots\!02$$$$T_{61}^{8} -$$$$19\!\cdots\!44$$$$T_{61}^{6} + 563862866604 T_{61}^{4} - 4511601402 T_{61}^{2} + 35153041$$">$$T_{61}^{32} - \cdots$$