# Properties

 Label 605.2.o.a Level $605$ Weight $2$ Character orbit 605.o Analytic conductor $4.831$ Analytic rank $0$ Dimension $640$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.o (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$640q + 44q^{4} - 7q^{5} + 14q^{6} - 644q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q + 44q^{4} - 7q^{5} + 14q^{6} - 644q^{9} - 23q^{10} + 4q^{11} - 30q^{14} + 18q^{15} - 100q^{16} - 6q^{19} - 3q^{20} + 32q^{21} + 128q^{24} - 15q^{25} - 26q^{26} - 10q^{29} + 28q^{30} - 18q^{31} - 34q^{34} - 29q^{35} - 66q^{36} + 44q^{39} - 50q^{40} - 34q^{41} - 28q^{44} - 43q^{45} - 14q^{46} + 102q^{49} + 29q^{50} - 148q^{51} + 90q^{54} - 102q^{55} - 106q^{56} - 34q^{59} + 58q^{60} - 42q^{61} + 24q^{64} + 22q^{65} + 52q^{66} + 20q^{69} - 75q^{70} + 54q^{71} - 34q^{74} - 4q^{75} - 2q^{76} + 50q^{79} - 160q^{80} + 560q^{81} - 4q^{84} - 57q^{85} + 6q^{86} - 128q^{89} + 39q^{90} - 80q^{91} - 88q^{94} + 65q^{95} + 8q^{96} - 266q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1 −2.67094 + 0.384024i 1.90400i 5.06748 1.48795i 2.16316 0.566331i 0.731183 + 5.08549i −0.410154 + 0.638213i −8.05443 + 3.67833i −0.625232 −5.56020 + 2.34334i
34.2 −2.64166 + 0.379813i 3.02441i 4.91512 1.44321i 2.12227 + 0.704262i −1.14871 7.98946i 2.30183 3.58171i −7.58062 + 3.46195i −6.14705 −5.87379 1.05436i
34.3 −2.62964 + 0.378084i 1.51252i 4.85305 1.42498i −0.110182 + 2.23335i 0.571859 + 3.97737i 1.36099 2.11774i −7.38979 + 3.37480i 0.712295 −0.554657 5.91456i
34.4 −2.48873 + 0.357826i 1.71261i 4.14677 1.21760i −2.15526 0.595682i −0.612817 4.26224i 1.27993 1.99162i −5.31030 + 2.42513i 0.0669571 5.57703 + 0.711284i
34.5 −2.33583 + 0.335842i 0.0506944i 3.42433 1.00547i −0.467375 + 2.18668i −0.0170253 0.118413i −0.376776 + 0.586275i −3.36779 + 1.53802i 2.99743 0.357331 5.26467i
34.6 −2.32027 + 0.333604i 0.978501i 3.35338 0.984640i 2.21458 + 0.309217i −0.326432 2.27039i −1.21456 + 1.88989i −3.18766 + 1.45576i 2.04254 −5.24159 + 0.0213278i
34.7 −2.30133 + 0.330881i 2.79202i 3.26765 0.959469i −0.596989 2.15490i 0.923828 + 6.42537i 1.52237 2.36885i −2.97270 + 1.35758i −4.79539 2.08688 + 4.76161i
34.8 −2.21266 + 0.318132i 2.50454i 2.87566 0.844370i −1.55949 + 1.60249i −0.796776 5.54170i −1.48610 + 2.31242i −2.02743 + 0.925896i −3.27274 2.94082 4.04190i
34.9 −2.19868 + 0.316122i 2.73317i 2.81527 0.826637i −0.231277 2.22408i −0.864015 6.00936i −0.907380 + 1.41191i −1.88744 + 0.861966i −4.47020 1.21158 + 4.81691i
34.10 −2.18609 + 0.314312i 1.37149i 2.76121 0.810764i −2.22979 + 0.167423i 0.431075 + 2.99819i −2.66241 + 4.14280i −1.76344 + 0.805337i 1.11903 4.82190 1.06685i
34.11 −1.97057 + 0.283326i 3.23212i 1.88390 0.553163i 1.40068 + 1.74302i 0.915742 + 6.36913i −0.498267 + 0.775319i 0.0662218 0.0302425i −7.44660 −3.25398 3.03789i
34.12 −1.96271 + 0.282195i 0.434811i 1.85361 0.544269i 1.01315 1.99337i −0.122702 0.853408i 0.433841 0.675070i 0.122895 0.0561242i 2.81094 −1.42600 + 4.19831i
34.13 −1.95165 + 0.280604i 2.17977i 1.81120 0.531817i 0.330301 2.21154i 0.611653 + 4.25414i −1.97410 + 3.07175i 0.201476 0.0920112i −1.75139 −0.0240629 + 4.40883i
34.14 −1.76416 + 0.253647i 1.36174i 1.12892 0.331482i −2.12544 0.694632i 0.345401 + 2.40232i 1.80054 2.80170i 1.33495 0.609652i 1.14567 3.92580 + 0.686327i
34.15 −1.57721 + 0.226769i 0.629528i 0.517189 0.151860i 2.23567 + 0.0420345i 0.142757 + 0.992900i 2.30924 3.59325i 2.11759 0.967072i 2.60369 −3.53566 + 0.440684i
34.16 −1.51095 + 0.217241i 3.08747i 0.316777 0.0930142i 0.837700 + 2.07322i −0.670726 4.66500i −1.71177 + 2.66356i 2.31865 1.05889i −6.53248 −1.71611 2.95055i
34.17 −1.47735 + 0.212411i 1.29393i 0.218454 0.0641439i −1.31512 + 1.80844i −0.274845 1.91159i 1.96131 3.05186i 2.40622 1.09888i 1.32574 1.55876 2.95104i
34.18 −1.46552 + 0.210710i 2.35971i 0.184359 0.0541326i −1.20996 + 1.88042i 0.497215 + 3.45820i 0.230168 0.358148i 2.43481 1.11194i −2.56825 1.37700 3.01075i
34.19 −1.26524 + 0.181914i 1.21233i −0.351241 + 0.103134i −1.93141 1.12680i −0.220541 1.53389i −0.377024 + 0.586661i 2.75112 1.25640i 1.53025 2.64868 + 1.07432i
34.20 −1.19871 + 0.172349i 1.61947i −0.511781 + 0.150273i 1.81795 + 1.30195i 0.279113 + 1.94128i 0.386764 0.601816i 2.79078 1.27450i 0.377319 −2.40358 1.24734i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 584.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
121.e even 11 1 inner
605.o even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.o.a 640
5.b even 2 1 inner 605.2.o.a 640
121.e even 11 1 inner 605.2.o.a 640
605.o even 22 1 inner 605.2.o.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.o.a 640 1.a even 1 1 trivial
605.2.o.a 640 5.b even 2 1 inner
605.2.o.a 640 121.e even 11 1 inner
605.2.o.a 640 605.o even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(605, [\chi])$$.