# Properties

 Label 552.2.bf.a Level $552$ Weight $2$ Character orbit 552.bf Analytic conductor $4.408$ Analytic rank $0$ Dimension $920$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 552.bf (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.40774219157$$ Analytic rank: $$0$$ Dimension: $$920$$ Relative dimension: $$92$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$920q - 22q^{4} - 2q^{6} - 44q^{7} - 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$920q - 22q^{4} - 2q^{6} - 44q^{7} - 18q^{9} - 22q^{10} - 12q^{12} - 22q^{15} - 30q^{16} - 10q^{18} - 14q^{24} + 40q^{25} - 22q^{28} - 11q^{30} - 20q^{31} - 22q^{33} - 18q^{36} - 42q^{39} - 176q^{40} - 11q^{42} + 90q^{46} - 64q^{48} + 32q^{49} - 90q^{52} + 21q^{54} - 76q^{55} - 22q^{57} + 76q^{58} - 11q^{60} - 22q^{63} - 28q^{64} - 44q^{66} - 116q^{70} - 79q^{72} - 36q^{73} - 22q^{76} - 134q^{78} - 44q^{79} - 34q^{81} - 40q^{82} - 165q^{84} - 18q^{87} - 22q^{88} - 198q^{90} + 88q^{94} - 231q^{96} - 44q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.41419 + 0.00734365i 0.191591 1.72142i 1.99989 0.0207707i 3.15950 + 0.454268i −0.258305 + 2.43583i 0.966873 0.441556i −2.82808 + 0.0440603i −2.92659 0.659617i −4.47149 0.619221i
5.2 −1.41317 + 0.0543574i 0.873168 1.49585i 1.99409 0.153632i −1.97986 0.284661i −1.15262 + 2.16136i 2.90361 1.32603i −2.80963 + 0.325502i −1.47515 2.61226i 2.81335 + 0.294653i
5.3 −1.41053 0.101993i −1.59542 0.674260i 1.97919 + 0.287730i 3.02191 + 0.434486i 2.18162 + 1.11379i −3.87309 + 1.76878i −2.76237 0.607716i 2.09075 + 2.15146i −4.21819 0.921070i
5.4 −1.40980 0.111635i −1.53275 + 0.806651i 1.97508 + 0.314766i −0.920190 0.132303i 2.25092 0.966109i −0.322471 + 0.147268i −2.74932 0.664245i 1.69863 2.47278i 1.28251 + 0.289247i
5.5 −1.40646 0.147850i 1.72844 0.111718i 1.95628 + 0.415891i −2.37700 0.341761i −2.44751 0.0984225i −3.17635 + 1.45059i −2.68995 0.874172i 2.97504 0.386198i 3.29264 + 0.832114i
5.6 −1.39862 0.209415i 1.28405 + 1.16242i 1.91229 + 0.585785i 1.33695 + 0.192224i −1.55247 1.89469i 3.52408 1.60939i −2.55190 1.21976i 0.297559 + 2.98521i −1.82963 0.548825i
5.7 −1.36422 + 0.372693i −1.31009 1.13298i 1.72220 1.01687i −2.59161 0.372618i 2.20951 + 1.05738i 0.360822 0.164782i −1.97048 + 2.02909i 0.432691 + 2.96863i 3.67440 0.457543i
5.8 −1.35818 + 0.394141i 0.367339 + 1.69265i 1.68931 1.07063i −0.449826 0.0646752i −1.16605 2.15414i −0.674934 + 0.308232i −1.87240 + 2.11993i −2.73012 + 1.24355i 0.636436 0.0894541i
5.9 −1.34520 + 0.436397i 1.59574 + 0.673497i 1.61912 1.17408i 1.00923 + 0.145106i −2.44050 0.209609i 0.340247 0.155386i −1.66567 + 2.28595i 2.09280 + 2.14946i −1.42094 + 0.245230i
5.10 −1.32050 + 0.506245i −0.661683 + 1.60068i 1.48743 1.33699i 3.61745 + 0.520111i 0.0634149 2.44867i −0.429427 + 0.196113i −1.28731 + 2.51850i −2.12435 2.11829i −5.04014 + 1.14451i
5.11 −1.28981 0.579981i −1.33333 1.10555i 1.32724 + 1.49614i 1.33695 + 0.192224i 1.07855 + 2.19926i 3.52408 1.60939i −0.844167 2.69952i 0.555524 + 2.94812i −1.61293 1.02334i
5.12 −1.26748 + 0.627300i −1.72666 + 0.136498i 1.21299 1.59018i 1.70656 + 0.245367i 2.10288 1.25614i 4.42418 2.02045i −0.539916 + 2.77642i 2.96274 0.471373i −2.31695 + 0.759532i
5.13 −1.26313 0.636012i −0.135402 1.72675i 1.19098 + 1.60673i −2.37700 0.341761i −0.927205 + 2.26722i −3.17635 + 1.45059i −0.482454 2.78698i −2.96333 + 0.467611i 2.78509 + 1.94349i
5.14 −1.26067 + 0.640864i 0.0422748 1.73153i 1.17859 1.61584i 0.349508 + 0.0502517i 1.05638 + 2.20999i −2.03405 + 0.928918i −0.450281 + 2.79236i −2.99643 0.146400i −0.472820 + 0.160636i
5.15 −1.24635 0.668282i −0.580308 + 1.63194i 1.10680 + 1.66583i −0.920190 0.132303i 1.81387 1.64617i −0.322471 + 0.147268i −0.266214 2.81587i −2.32649 1.89406i 1.05847 + 0.779843i
5.16 −1.24176 0.676788i 0.894449 + 1.48323i 1.08392 + 1.68081i 3.02191 + 0.434486i −0.106857 2.44716i −3.87309 + 1.76878i −0.208404 2.82074i −1.39992 + 2.65334i −3.45842 2.58472i
5.17 −1.18573 0.770749i 1.67663 0.434625i 0.811892 + 1.82779i 3.15950 + 0.454268i −2.32301 0.776919i 0.966873 0.441556i 0.446090 2.79303i 2.62220 1.45741i −3.39618 2.97382i
5.18 −1.15945 0.809745i 1.35636 1.07716i 0.688626 + 1.87771i −1.97986 0.284661i −2.44486 + 0.150603i 2.90361 1.32603i 0.722042 2.73471i 0.679440 2.92205i 2.06503 + 1.93323i
5.19 −1.13324 + 0.846027i −1.52607 + 0.819214i 0.568475 1.91751i −4.20271 0.604259i 1.03633 2.21946i −3.19599 + 1.45956i 0.978044 + 2.65395i 1.65778 2.50036i 5.27391 2.87084i
5.20 −1.09564 + 0.894186i 1.69680 + 0.347656i 0.400861 1.95942i −3.74972 0.539128i −2.16996 + 1.13635i 1.49121 0.681011i 1.31288 + 2.50526i 2.75827 + 1.17981i 4.59043 2.76226i
See next 80 embeddings (of 920 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 389.92 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
8.b even 2 1 inner
23.d odd 22 1 inner
24.h odd 2 1 inner
69.g even 22 1 inner
184.m odd 22 1 inner
552.bf even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.2.bf.a 920
3.b odd 2 1 inner 552.2.bf.a 920
8.b even 2 1 inner 552.2.bf.a 920
23.d odd 22 1 inner 552.2.bf.a 920
24.h odd 2 1 inner 552.2.bf.a 920
69.g even 22 1 inner 552.2.bf.a 920
184.m odd 22 1 inner 552.2.bf.a 920
552.bf even 22 1 inner 552.2.bf.a 920

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.bf.a 920 1.a even 1 1 trivial
552.2.bf.a 920 3.b odd 2 1 inner
552.2.bf.a 920 8.b even 2 1 inner
552.2.bf.a 920 23.d odd 22 1 inner
552.2.bf.a 920 24.h odd 2 1 inner
552.2.bf.a 920 69.g even 22 1 inner
552.2.bf.a 920 184.m odd 22 1 inner
552.2.bf.a 920 552.bf even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(552, [\chi])$$.