# Properties

 Label 273.2.bn.c Level $273$ Weight $2$ Character orbit 273.bn Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bn (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q + 32 q^{4} - 4 q^{7} - 10 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64 q + 32 q^{4} - 4 q^{7} - 10 q^{9} - 18 q^{15} - 16 q^{16} + 8 q^{18} - 44 q^{21} - 32 q^{22} + 48 q^{25} + 40 q^{28} - 40 q^{30} - 6 q^{36} + 12 q^{37} - 12 q^{42} + 40 q^{43} - 36 q^{46} - 24 q^{49} - 28 q^{51} - 68 q^{57} - 116 q^{60} + 4 q^{63} - 48 q^{64} - 28 q^{67} - 64 q^{70} + 62 q^{72} - 2 q^{78} + 56 q^{79} + 10 q^{81} + 28 q^{85} + 80 q^{88} - 16 q^{91} + 46 q^{93} - 32 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1 −2.30434 + 1.33041i −0.643938 + 1.60790i 2.54000 4.39941i −3.38484 −0.655319 4.56186i −1.66594 + 2.05540i 8.19533i −2.17069 2.07078i 7.79984 4.50324i
146.2 −2.30434 + 1.33041i 0.643938 1.60790i 2.54000 4.39941i 3.38484 0.655319 + 4.56186i −0.947057 + 2.47044i 8.19533i −2.17069 2.07078i −7.79984 + 4.50324i
146.3 −2.01386 + 1.16270i −1.06546 1.36557i 1.70376 2.95099i −0.993521 3.73345 + 1.51126i −0.593544 2.57831i 3.27304i −0.729580 + 2.90993i 2.00081 1.15517i
146.4 −2.01386 + 1.16270i 1.06546 + 1.36557i 1.70376 2.95099i 0.993521 −3.73345 1.51126i 2.52966 0.775133i 3.27304i −0.729580 + 2.90993i −2.00081 + 1.15517i
146.5 −1.88922 + 1.09074i −1.60830 0.642938i 1.37944 2.38926i −0.155426 3.73972 0.539589i 0.959965 + 2.46545i 1.65550i 2.17326 + 2.06808i 0.293635 0.169530i
146.6 −1.88922 + 1.09074i 1.60830 + 0.642938i 1.37944 2.38926i 0.155426 −3.73972 + 0.539589i −2.61513 + 0.401373i 1.65550i 2.17326 + 2.06808i −0.293635 + 0.169530i
146.7 −1.43804 + 0.830256i −1.23680 + 1.21257i 0.378649 0.655839i −1.88889 0.771823 2.77059i 0.896028 2.48940i 2.06352i 0.0593312 2.99941i 2.71631 1.56826i
146.8 −1.43804 + 0.830256i 1.23680 1.21257i 0.378649 0.655839i 1.88889 −0.771823 + 2.77059i 1.70787 2.02068i 2.06352i 0.0593312 2.99941i −2.71631 + 1.56826i
146.9 −1.29226 + 0.746087i −1.56373 + 0.744804i 0.113292 0.196227i 2.73189 1.46506 2.12916i 1.84288 + 1.89837i 2.64625i 1.89053 2.32935i −3.53032 + 2.03823i
146.10 −1.29226 + 0.746087i 1.56373 0.744804i 0.113292 0.196227i −2.73189 −1.46506 + 2.12916i −2.56547 0.646794i 2.64625i 1.89053 2.32935i 3.53032 2.03823i
146.11 −0.828363 + 0.478255i −1.28843 1.15756i −0.542544 + 0.939713i 3.79208 1.62090 + 0.342680i −2.21378 1.44885i 2.95092i 0.320110 + 2.98287i −3.14122 + 1.81358i
146.12 −0.828363 + 0.478255i 1.28843 + 1.15756i −0.542544 + 0.939713i −3.79208 −1.62090 0.342680i 2.36163 + 1.19277i 2.95092i 0.320110 + 2.98287i 3.14122 1.81358i
146.13 −0.755524 + 0.436202i −0.0385029 + 1.73162i −0.619456 + 1.07293i 0.294723 −0.726247 1.32508i −0.863899 + 2.50074i 2.82564i −2.99704 0.133345i −0.222671 + 0.128559i
146.14 −0.755524 + 0.436202i 0.0385029 1.73162i −0.619456 + 1.07293i −0.294723 0.726247 + 1.32508i −1.73375 + 1.99853i 2.82564i −2.99704 0.133345i 0.222671 0.128559i
146.15 −0.265128 + 0.153072i −1.51585 0.837970i −0.953138 + 1.65088i −2.83438 0.530164 0.00986484i 2.54431 0.725603i 1.19588i 1.59561 + 2.54048i 0.751474 0.433863i
146.16 −0.265128 + 0.153072i 1.51585 + 0.837970i −0.953138 + 1.65088i 2.83438 −0.530164 + 0.00986484i −0.643764 2.56624i 1.19588i 1.59561 + 2.54048i −0.751474 + 0.433863i
146.17 0.265128 0.153072i −0.0322229 + 1.73175i −0.953138 + 1.65088i 2.83438 0.256539 + 0.464068i 2.54431 0.725603i 1.19588i −2.99792 0.111604i 0.751474 0.433863i
146.18 0.265128 0.153072i 0.0322229 1.73175i −0.953138 + 1.65088i −2.83438 −0.256539 0.464068i −0.643764 2.56624i 1.19588i −2.99792 0.111604i −0.751474 + 0.433863i
146.19 0.755524 0.436202i −1.51888 0.832467i −0.619456 + 1.07293i −0.294723 −1.51067 + 0.0335901i −0.863899 + 2.50074i 2.82564i 1.61400 + 2.52884i −0.222671 + 0.128559i
146.20 0.755524 0.436202i 1.51888 + 0.832467i −0.619456 + 1.07293i 0.294723 1.51067 0.0335901i −1.73375 + 1.99853i 2.82564i 1.61400 + 2.52884i 0.222671 0.128559i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 230.32 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 273.2.bn.c 64
3.b odd 2 1 inner 273.2.bn.c 64
7.b odd 2 1 inner 273.2.bn.c 64
13.c even 3 1 inner 273.2.bn.c 64
21.c even 2 1 inner 273.2.bn.c 64
39.i odd 6 1 inner 273.2.bn.c 64
91.n odd 6 1 inner 273.2.bn.c 64
273.bn even 6 1 inner 273.2.bn.c 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bn.c 64 1.a even 1 1 trivial
273.2.bn.c 64 3.b odd 2 1 inner
273.2.bn.c 64 7.b odd 2 1 inner
273.2.bn.c 64 13.c even 3 1 inner
273.2.bn.c 64 21.c even 2 1 inner
273.2.bn.c 64 39.i odd 6 1 inner
273.2.bn.c 64 91.n odd 6 1 inner
273.2.bn.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 $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{32} - \cdots$$ $$13\!\cdots\!38$$$$T_{19}^{18} +$$$$22\!\cdots\!33$$$$T_{19}^{16} -$$$$27\!\cdots\!00$$$$T_{19}^{14} +$$$$24\!\cdots\!69$$$$T_{19}^{12} -$$$$16\!\cdots\!28$$$$T_{19}^{10} +$$$$80\!\cdots\!96$$$$T_{19}^{8} -$$$$27\!\cdots\!60$$$$T_{19}^{6} +$$$$68\!\cdots\!92$$$$T_{19}^{4} -$$$$10\!\cdots\!96$$$$T_{19}^{2} +$$$$98\!\cdots\!56$$">$$T_{19}^{32} - \cdots$$