# Properties

 Label 273.2.bw.b Level $273$ Weight $2$ Character orbit 273.bw Analytic conductor $2.180$ Analytic rank $0$ Dimension $128$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bw (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128 q - 4 q^{3} - 12 q^{4} - 4 q^{6} - 16 q^{7} - 16 q^{9}+O(q^{10})$$ 128 * q - 4 * q^3 - 12 * q^4 - 4 * q^6 - 16 * q^7 - 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$128 q - 4 q^{3} - 12 q^{4} - 4 q^{6} - 16 q^{7} - 16 q^{9} - 48 q^{12} - 16 q^{13} - 6 q^{15} + 32 q^{16} + 22 q^{18} - 16 q^{19} - 18 q^{21} - 8 q^{22} - 4 q^{24} - 40 q^{27} - 76 q^{28} - 4 q^{31} + 50 q^{33} - 48 q^{34} - 60 q^{36} + 28 q^{37} + 40 q^{39} + 44 q^{40} + 44 q^{42} - 144 q^{43} + 58 q^{45} + 48 q^{46} - 64 q^{48} + 24 q^{49} + 36 q^{51} - 22 q^{54} - 16 q^{55} + 40 q^{57} - 28 q^{58} - 4 q^{60} - 40 q^{61} + 20 q^{63} - 34 q^{66} + 96 q^{67} - 54 q^{69} + 64 q^{70} - 98 q^{72} + 48 q^{73} - 12 q^{75} + 144 q^{76} + 82 q^{78} - 24 q^{79} - 48 q^{81} + 4 q^{84} + 56 q^{85} - 2 q^{87} - 24 q^{91} + 10 q^{93} + 32 q^{94} - 54 q^{96} + 52 q^{97} - 10 q^{99}+O(q^{100})$$ 128 * q - 4 * q^3 - 12 * q^4 - 4 * q^6 - 16 * q^7 - 16 * q^9 - 48 * q^12 - 16 * q^13 - 6 * q^15 + 32 * q^16 + 22 * q^18 - 16 * q^19 - 18 * q^21 - 8 * q^22 - 4 * q^24 - 40 * q^27 - 76 * q^28 - 4 * q^31 + 50 * q^33 - 48 * q^34 - 60 * q^36 + 28 * q^37 + 40 * q^39 + 44 * q^40 + 44 * q^42 - 144 * q^43 + 58 * q^45 + 48 * q^46 - 64 * q^48 + 24 * q^49 + 36 * q^51 - 22 * q^54 - 16 * q^55 + 40 * q^57 - 28 * q^58 - 4 * q^60 - 40 * q^61 + 20 * q^63 - 34 * q^66 + 96 * q^67 - 54 * q^69 + 64 * q^70 - 98 * q^72 + 48 * q^73 - 12 * q^75 + 144 * q^76 + 82 * q^78 - 24 * q^79 - 48 * q^81 + 4 * q^84 + 56 * q^85 - 2 * q^87 - 24 * q^91 + 10 * q^93 + 32 * q^94 - 54 * q^96 + 52 * q^97 - 10 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.700434 2.61406i 1.08921 + 1.34671i −4.61063 + 2.66195i −2.04884 0.548984i 2.75745 3.79054i −0.908491 + 2.48488i 6.36068 + 6.36068i −0.627240 + 2.93370i 5.74030i
11.2 −0.679659 2.53652i −1.73115 0.0557742i −4.23995 + 2.44794i 2.98598 + 0.800092i 1.03512 + 4.42901i 2.50266 + 0.858313i 5.37724 + 5.37724i 2.99378 + 0.193107i 8.11779i
11.3 −0.614189 2.29218i −0.572173 1.63481i −3.14483 + 1.81567i 0.203686 + 0.0545776i −3.39587 + 2.31561i −2.22424 1.43274i 2.73738 + 2.73738i −2.34524 + 1.87079i 0.500408i
11.4 −0.590022 2.20199i 1.60162 0.659394i −2.76859 + 1.59845i −1.90854 0.511391i −2.39697 3.13771i 0.185390 2.63925i 1.92936 + 1.92936i 2.13040 2.11220i 4.50431i
11.5 −0.588963 2.19804i 1.52334 0.824281i −2.75245 + 1.58913i 3.96614 + 1.06272i −2.70899 2.86289i −1.45144 + 2.21209i 1.89590 + 1.89590i 1.64112 2.51132i 9.34364i
11.6 −0.548543 2.04719i −0.721270 + 1.57473i −2.15804 + 1.24594i −0.706275 0.189246i 3.61942 + 0.612770i 2.22477 1.43193i 0.737165 + 0.737165i −1.95954 2.27161i 1.54969i
11.7 −0.504901 1.88432i −1.22779 1.22170i −1.56368 + 0.902789i −3.20968 0.860032i −1.68215 + 2.93037i 0.850713 + 2.50525i −0.268188 0.268188i 0.0149148 + 2.99996i 6.48229i
11.8 −0.428681 1.59986i 1.25532 + 1.19339i −0.643729 + 0.371657i 0.428701 + 0.114870i 1.37112 2.51992i 2.29163 + 1.32228i −1.47180 1.47180i 0.151659 + 2.99616i 0.735103i
11.9 −0.375380 1.40094i −1.57757 + 0.715025i −0.0896663 + 0.0517688i 0.283932 + 0.0760792i 1.59390 + 1.94168i −2.00629 + 1.72476i −1.94493 1.94493i 1.97748 2.25601i 0.426329i
11.10 −0.329383 1.22927i 0.639267 1.60976i 0.329428 0.190195i 1.27845 + 0.342559i −2.18940 0.255606i 2.63584 + 0.228814i −2.14209 2.14209i −2.18268 2.05814i 1.68440i
11.11 −0.305471 1.14003i 1.47664 + 0.905280i 0.525688 0.303506i 1.64653 + 0.441185i 0.580979 1.95996i −1.12852 2.39300i −2.17571 2.17571i 1.36093 + 2.67355i 2.01186i
11.12 −0.250255 0.933963i 0.138617 + 1.72650i 0.922391 0.532543i −4.14513 1.11068i 1.57779 0.561527i −2.32521 1.26230i −2.09562 2.09562i −2.96157 + 0.478645i 4.14935i
11.13 −0.248591 0.927753i −1.67503 0.440774i 0.933123 0.538739i 2.67091 + 0.715669i 0.00746719 + 1.66358i −1.47717 2.19498i −2.09011 2.09011i 2.61144 + 1.47662i 2.65586i
11.14 −0.179441 0.669682i 0.594756 1.62673i 1.31578 0.759664i −1.48044 0.396684i −1.19612 0.106395i −2.58806 + 0.549511i −1.72532 1.72532i −2.29253 1.93502i 1.06261i
11.15 −0.0262469 0.0979549i −0.748033 + 1.56219i 1.72314 0.994858i 2.20563 + 0.590996i 0.172658 + 0.0322707i 1.71000 2.01889i −0.286094 0.286094i −1.88089 2.33714i 0.231564i
11.16 −0.00573734 0.0214121i −1.43178 0.974676i 1.73163 0.999754i 2.15869 + 0.578419i −0.0126552 + 0.0362495i 1.44048 + 2.21924i −0.0626912 0.0626912i 1.10001 + 2.79105i 0.0495405i
11.17 0.00573734 + 0.0214121i −1.43178 + 0.974676i 1.73163 0.999754i −2.15869 0.578419i −0.0290845 0.0250654i 1.44048 + 2.21924i 0.0626912 + 0.0626912i 1.10001 2.79105i 0.0495405i
11.18 0.0262469 + 0.0979549i −0.748033 1.56219i 1.72314 0.994858i −2.20563 0.590996i 0.133391 0.114276i 1.71000 2.01889i 0.286094 + 0.286094i −1.88089 + 2.33714i 0.231564i
11.19 0.179441 + 0.669682i 0.594756 + 1.62673i 1.31578 0.759664i 1.48044 + 0.396684i −0.982671 + 0.690199i −2.58806 + 0.549511i 1.72532 + 1.72532i −2.29253 + 1.93502i 1.06261i
11.20 0.248591 + 0.927753i −1.67503 + 0.440774i 0.933123 0.538739i −2.67091 0.715669i −0.825325 1.44444i −1.47717 2.19498i 2.09011 + 2.09011i 2.61144 1.47662i 2.65586i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 254.32 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
91.bd odd 12 1 inner
273.bw even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bw.b yes 128
3.b odd 2 1 inner 273.2.bw.b yes 128
7.c even 3 1 273.2.bv.b 128
13.f odd 12 1 273.2.bv.b 128
21.h odd 6 1 273.2.bv.b 128
39.k even 12 1 273.2.bv.b 128
91.bd odd 12 1 inner 273.2.bw.b yes 128
273.bw even 12 1 inner 273.2.bw.b yes 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bv.b 128 7.c even 3 1
273.2.bv.b 128 13.f odd 12 1
273.2.bv.b 128 21.h odd 6 1
273.2.bv.b 128 39.k even 12 1
273.2.bw.b yes 128 1.a even 1 1 trivial
273.2.bw.b yes 128 3.b odd 2 1 inner
273.2.bw.b yes 128 91.bd odd 12 1 inner
273.2.bw.b yes 128 273.bw even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{128} + 6 T_{2}^{126} - 208 T_{2}^{124} - 1320 T_{2}^{122} + 25278 T_{2}^{120} + 163512 T_{2}^{118} - 2083130 T_{2}^{116} - 13706802 T_{2}^{114} + 129577261 T_{2}^{112} + 859384398 T_{2}^{110} + \cdots + 269517889$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.