# Properties

 Label 460.2.o.b Level $460$ Weight $2$ Character orbit 460.o Analytic conductor $3.673$ Analytic rank $0$ Dimension $640$ 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.o (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$64$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$640 q - 6 q^{4} - 22 q^{5} + 2 q^{6} - 100 q^{9}+O(q^{10})$$ 640 * q - 6 * q^4 - 22 * q^5 + 2 * q^6 - 100 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$640 q - 6 q^{4} - 22 q^{5} + 2 q^{6} - 100 q^{9} - 11 q^{10} - 22 q^{14} + 2 q^{16} - 11 q^{20} - 44 q^{21} - 8 q^{24} - 38 q^{25} - 46 q^{26} - 52 q^{29} - 11 q^{30} - 44 q^{34} + 18 q^{36} - 11 q^{40} - 76 q^{41} - 154 q^{44} - 30 q^{46} - 100 q^{49} + 21 q^{50} + 178 q^{54} + 242 q^{56} - 11 q^{60} - 44 q^{61} + 6 q^{64} - 22 q^{65} + 44 q^{66} - 12 q^{69} + 22 q^{70} - 22 q^{74} + 88 q^{76} - 110 q^{80} - 252 q^{81} - 374 q^{84} + 94 q^{85} - 198 q^{86} - 44 q^{89} - 242 q^{90} - 62 q^{94} + 372 q^{96}+O(q^{100})$$ 640 * q - 6 * q^4 - 22 * q^5 + 2 * q^6 - 100 * q^9 - 11 * q^10 - 22 * q^14 + 2 * q^16 - 11 * q^20 - 44 * q^21 - 8 * q^24 - 38 * q^25 - 46 * q^26 - 52 * q^29 - 11 * q^30 - 44 * q^34 + 18 * q^36 - 11 * q^40 - 76 * q^41 - 154 * q^44 - 30 * q^46 - 100 * q^49 + 21 * q^50 + 178 * q^54 + 242 * q^56 - 11 * q^60 - 44 * q^61 + 6 * q^64 - 22 * q^65 + 44 * q^66 - 12 * q^69 + 22 * q^70 - 22 * q^74 + 88 * q^76 - 110 * q^80 - 252 * q^81 - 374 * q^84 + 94 * q^85 - 198 * q^86 - 44 * q^89 - 242 * q^90 - 62 * q^94 + 372 * q^96

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41365 0.0397971i 2.67003 0.783991i 1.99683 + 0.112519i 1.97695 + 1.04482i −3.80569 + 1.00203i 0.790909 0.113716i −2.81835 0.238531i 3.99064 2.56463i −2.75315 1.55569i
19.2 −1.41287 + 0.0615925i 2.08651 0.612653i 1.99241 0.174045i −0.546640 2.16822i −2.91023 + 0.994114i −1.11477 + 0.160279i −2.80430 + 0.368620i 1.45440 0.934688i 0.905878 + 3.02975i
19.3 −1.41242 0.0712068i −1.44577 + 0.424517i 1.98986 + 0.201148i −0.632143 2.14485i 2.07226 0.496647i 4.69086 0.674443i −2.79619 0.425797i −0.613721 + 0.394415i 0.740123 + 3.07445i
19.4 −1.40349 + 0.173829i −0.723511 + 0.212442i 1.93957 0.487934i 1.78262 1.34991i 0.978512 0.423927i −1.71228 + 0.246189i −2.63735 + 1.02196i −2.04542 + 1.31451i −2.26724 + 2.20446i
19.5 −1.40029 + 0.197940i 1.48939 0.437323i 1.92164 0.554348i −2.20194 + 0.389205i −1.99901 + 0.907190i −0.230848 + 0.0331909i −2.58113 + 1.15662i −0.496740 + 0.319236i 3.00632 0.980851i
19.6 −1.39440 + 0.235922i −2.94595 + 0.865009i 1.88868 0.657939i −1.40285 + 1.74127i 3.90375 1.90118i 1.84920 0.265875i −2.47835 + 1.36301i 5.40662 3.47462i 1.54533 2.75898i
19.7 −1.34217 0.445613i −0.610817 + 0.179352i 1.60286 + 1.19618i −1.58117 + 1.58110i 0.899744 + 0.0314662i 0.165059 0.0237318i −1.61828 2.31973i −2.18283 + 1.40282i 2.82677 1.41752i
19.8 −1.33042 + 0.479557i −0.820997 + 0.241066i 1.54005 1.27603i 0.912951 + 2.04121i 0.976668 0.714435i −3.61150 + 0.519256i −1.43699 + 2.43620i −1.90784 + 1.22609i −2.19348 2.27786i
19.9 −1.27975 0.601871i −2.51556 + 0.738636i 1.27550 + 1.54049i 2.08492 + 0.808149i 3.66385 + 0.568779i −0.0121874 + 0.00175228i −0.705145 2.73912i 3.25872 2.09425i −2.18177 2.28908i
19.10 −1.27640 0.608942i 1.81640 0.533344i 1.25838 + 1.55450i 0.889872 + 2.05137i −2.64323 0.425324i −4.44634 + 0.639287i −0.659592 2.75044i 0.491100 0.315611i 0.113335 3.16025i
19.11 −1.23367 + 0.691424i 0.820997 0.241066i 1.04387 1.70597i 0.912951 + 2.04121i −0.846157 + 0.865052i 3.61150 0.519256i −0.108234 + 2.82636i −1.90784 + 1.22609i −2.53761 1.88693i
19.12 −1.21441 0.724711i −0.381027 + 0.111880i 0.949588 + 1.76019i −1.73128 1.41516i 0.543804 + 0.140267i −2.86375 + 0.411745i 0.122443 2.82578i −2.39110 + 1.53666i 1.07690 + 2.97326i
19.13 −1.19551 0.755479i 1.52020 0.446372i 0.858502 + 1.80637i 1.91991 1.14628i −2.15465 0.614838i 4.13344 0.594300i 0.338324 2.80812i −0.411991 + 0.264771i −3.16126 0.0800530i
19.14 −1.09143 + 0.899318i 2.94595 0.865009i 0.382455 1.96309i −1.40285 + 1.74127i −2.43739 + 3.59345i −1.84920 + 0.265875i 1.34802 + 2.48653i 5.40662 3.47462i −0.0348340 3.16209i
19.15 −1.09031 0.900680i 2.64511 0.776673i 0.377553 + 1.96404i −2.22046 + 0.263774i −3.58352 1.53558i 2.67954 0.385260i 1.35732 2.48147i 3.86960 2.48684i 2.65856 + 1.71232i
19.16 −1.06659 + 0.928648i −1.48939 + 0.437323i 0.275227 1.98097i −2.20194 + 0.389205i 1.18245 1.84956i 0.230848 0.0331909i 1.54607 + 2.36847i −0.496740 + 0.319236i 1.98713 2.45994i
19.17 −1.05046 + 0.946853i 0.723511 0.212442i 0.206938 1.98927i 1.78262 1.34991i −0.558869 + 0.908221i 1.71228 0.246189i 1.66616 + 2.28559i −2.04542 + 1.31451i −0.594407 + 3.10591i
19.18 −0.971783 + 1.02744i −2.08651 + 0.612653i −0.111277 1.99690i −0.546640 2.16822i 1.39816 2.73913i 1.11477 0.160279i 2.15984 + 1.82622i 1.45440 0.934688i 2.75894 + 1.54540i
19.19 −0.925935 1.06895i −0.832655 + 0.244489i −0.285289 + 1.97955i 0.527840 + 2.17287i 1.03233 + 0.663681i 3.68197 0.529388i 2.38019 1.52797i −1.89022 + 1.21477i 1.83394 2.57617i
19.20 −0.921514 1.07276i −2.25339 + 0.661656i −0.301623 + 1.97713i 0.592624 2.15611i 2.78633 + 1.80762i −0.146802 + 0.0211070i 2.39893 1.49838i 2.11623 1.36002i −2.85909 + 1.35114i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.64 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.b even 2 1 inner
20.d odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner
115.i odd 22 1 inner
460.o even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.o.b 640
4.b odd 2 1 inner 460.2.o.b 640
5.b even 2 1 inner 460.2.o.b 640
20.d odd 2 1 inner 460.2.o.b 640
23.d odd 22 1 inner 460.2.o.b 640
92.h even 22 1 inner 460.2.o.b 640
115.i odd 22 1 inner 460.2.o.b 640
460.o even 22 1 inner 460.2.o.b 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.o.b 640 1.a even 1 1 trivial
460.2.o.b 640 4.b odd 2 1 inner
460.2.o.b 640 5.b even 2 1 inner
460.2.o.b 640 20.d odd 2 1 inner
460.2.o.b 640 23.d odd 22 1 inner
460.2.o.b 640 92.h even 22 1 inner
460.2.o.b 640 115.i odd 22 1 inner
460.2.o.b 640 460.o even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{320} + 73 T_{3}^{318} + 2903 T_{3}^{316} + 82945 T_{3}^{314} + 1899638 T_{3}^{312} + 36991014 T_{3}^{310} + 635641327 T_{3}^{308} + 9891386579 T_{3}^{306} + 142043845913 T_{3}^{304} + \cdots + 58\!\cdots\!04$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.