# Properties

 Label 460.2.w.a Level $460$ Weight $2$ Character orbit 460.w Analytic conductor $3.673$ Analytic rank $0$ Dimension $1360$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.w (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$1360$$ Relative dimension: $$68$$ over $$\Q(\zeta_{44})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1360q - 18q^{2} - 36q^{5} - 44q^{6} - 18q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1360q - 18q^{2} - 36q^{5} - 44q^{6} - 18q^{8} - 18q^{10} - 6q^{12} - 36q^{13} - 44q^{16} - 36q^{17} - 38q^{18} - 14q^{20} - 88q^{21} - 28q^{22} - 36q^{25} - 36q^{26} - 34q^{28} + 2q^{30} + 2q^{32} - 60q^{33} - 36q^{36} - 36q^{37} - 10q^{38} + 2q^{40} - 56q^{41} - 202q^{42} - 120q^{45} - 44q^{46} - 50q^{48} - 34q^{50} - 250q^{52} - 84q^{53} - 84q^{56} - 44q^{57} - 58q^{58} - 42q^{60} - 136q^{61} - 82q^{62} - 68q^{65} - 84q^{66} - 4q^{68} + 16q^{70} - 104q^{72} - 36q^{73} + 4q^{76} - 44q^{77} - 50q^{78} + 204q^{80} + 80q^{81} - 10q^{82} - 252q^{85} + 44q^{86} - 50q^{88} + 264q^{90} - 22q^{92} - 88q^{93} + 52q^{96} - 36q^{97} + 30q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.41421 0.00122474i 0.959477 + 2.57246i 2.00000 + 0.00346408i 1.75727 1.38275i −1.35375 3.63918i 0.698270 + 0.151899i −2.82842 0.00734842i −3.42969 + 2.97185i −2.48685 + 1.95335i
3.2 −1.40653 + 0.147185i −0.487655 1.30745i 1.95667 0.414043i 0.154780 2.23070i 0.878342 + 1.76720i −0.318907 0.0693739i −2.69118 + 0.870359i 0.795620 0.689408i 0.110624 + 3.16034i
3.3 −1.40026 + 0.198151i 0.585996 + 1.57112i 1.92147 0.554927i −2.01963 0.959740i −1.13187 2.08386i 4.26825 + 0.928502i −2.58061 + 1.15779i 0.142232 0.123245i 3.01818 + 0.943697i
3.4 −1.39894 0.207265i −0.826183 2.21508i 1.91408 + 0.579903i −2.13268 + 0.672078i 0.696675 + 3.27001i −2.06386 0.448965i −2.55750 1.20797i −1.95677 + 1.69555i 3.12279 0.498170i
3.5 −1.37680 + 0.323160i 0.184673 + 0.495127i 1.79114 0.889851i −1.56442 + 1.59768i −0.414262 0.622010i −2.72912 0.593685i −2.17846 + 1.80397i 2.05620 1.78171i 1.63758 2.70524i
3.6 −1.37121 + 0.346108i −0.338395 0.907273i 1.76042 0.949171i 1.81802 + 1.30185i 0.778024 + 1.12694i 1.65566 + 0.360166i −2.08538 + 1.91080i 1.55862 1.35055i −2.94346 1.15588i
3.7 −1.35719 0.397537i 0.361954 + 0.970436i 1.68393 + 1.07907i −0.0598253 2.23527i −0.105456 1.46096i −3.20911 0.698099i −1.85644 2.13392i 1.45651 1.26208i −0.807408 + 3.05746i
3.8 −1.32376 0.497658i 0.189809 + 0.508896i 1.50467 + 1.31756i 0.243658 + 2.22275i 0.00199564 0.768116i 3.80918 + 0.828636i −1.33613 2.49294i 2.04430 1.77140i 0.783626 3.06365i
3.9 −1.29922 0.558604i −0.594892 1.59497i 1.37592 + 1.45149i 2.23464 0.0798607i −0.118062 + 2.40452i −1.24509 0.270853i −0.976810 2.65440i 0.0772235 0.0669146i −2.94789 1.14452i
3.10 −1.28874 0.582374i 1.09678 + 2.94057i 1.32168 + 1.50105i 0.110052 + 2.23336i 0.299057 4.42835i −1.66107 0.361343i −0.829123 2.70417i −5.17679 + 4.48572i 1.15882 2.94230i
3.11 −1.19585 + 0.754944i 0.865316 + 2.32000i 0.860120 1.80560i −0.780937 + 2.09527i −2.78626 2.12111i −0.367846 0.0800199i 0.334552 + 2.80857i −2.36639 + 2.05049i −0.647923 3.09519i
3.12 −1.18443 + 0.772733i −1.12425 3.01422i 0.805766 1.83050i 1.84293 + 1.26634i 3.66078 + 2.70140i −3.65616 0.795348i 0.460113 + 2.79075i −5.55434 + 4.81286i −3.16137 0.0757989i
3.13 −1.13266 0.846813i −0.616030 1.65164i 0.565816 + 1.91829i −1.87330 1.22097i −0.700881 + 2.39240i 2.66779 + 0.580342i 0.983562 2.65191i −0.0811762 + 0.0703396i 1.08787 + 2.96927i
3.14 −1.13162 0.848200i 0.616030 + 1.65164i 0.561114 + 1.91967i −1.87330 1.22097i 0.703812 2.39154i −2.66779 0.580342i 0.993302 2.64827i −0.0811762 + 0.0703396i 1.08423 + 2.97060i
3.15 −1.11682 + 0.867590i −0.106315 0.285041i 0.494574 1.93788i 1.46736 1.68726i 0.366034 + 0.226102i 3.49451 + 0.760183i 1.12894 + 2.59336i 2.19730 1.90397i −0.174928 + 3.15744i
3.16 −1.06351 + 0.932176i 0.317081 + 0.850128i 0.262098 1.98275i 2.06623 0.854811i −1.12969 0.608542i −4.65012 1.01157i 1.56953 + 2.35299i 1.64507 1.42546i −1.40062 + 2.83519i
3.17 −1.04207 + 0.956077i −0.816083 2.18800i 0.171833 1.99260i −2.19626 + 0.420065i 2.94232 + 1.49982i 3.93457 + 0.855914i 1.72602 + 2.24073i −1.85412 + 1.60660i 1.88705 2.53753i
3.18 −1.03221 + 0.966716i 1.10382 + 2.95946i 0.130921 1.99571i −1.01263 1.99363i −4.00033 1.98770i −1.74183 0.378912i 1.79415 + 2.18656i −5.27271 + 4.56883i 2.97253 + 1.07892i
3.19 −0.921863 1.07246i −1.09678 2.94057i −0.300338 + 1.97732i 0.110052 + 2.23336i −2.14257 + 3.88705i 1.66107 + 0.361343i 2.39747 1.50072i −5.17679 + 4.48572i 2.29373 2.17688i
3.20 −0.902008 1.08921i 0.594892 + 1.59497i −0.372763 + 1.96495i 2.23464 0.0798607i 1.20066 2.08664i 1.24509 + 0.270853i 2.47649 1.36639i 0.0772235 0.0669146i −2.10265 2.36196i
See next 80 embeddings (of 1360 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.68 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner
23.c even 11 1 inner
92.g odd 22 1 inner
115.k odd 44 1 inner
460.w even 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.w.a 1360
4.b odd 2 1 inner 460.2.w.a 1360
5.c odd 4 1 inner 460.2.w.a 1360
20.e even 4 1 inner 460.2.w.a 1360
23.c even 11 1 inner 460.2.w.a 1360
92.g odd 22 1 inner 460.2.w.a 1360
115.k odd 44 1 inner 460.2.w.a 1360
460.w even 44 1 inner 460.2.w.a 1360

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.w.a 1360 1.a even 1 1 trivial
460.2.w.a 1360 4.b odd 2 1 inner
460.2.w.a 1360 5.c odd 4 1 inner
460.2.w.a 1360 20.e even 4 1 inner
460.2.w.a 1360 23.c even 11 1 inner
460.2.w.a 1360 92.g odd 22 1 inner
460.2.w.a 1360 115.k odd 44 1 inner
460.2.w.a 1360 460.w even 44 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(460, [\chi])$$.