# Properties

 Label 125.2.h.a Level $125$ Weight $2$ Character orbit 125.h Analytic conductor $0.998$ Analytic rank $0$ Dimension $240$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.h (of order $$50$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.998130025266$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$12$$ over $$\Q(\zeta_{50})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9}+O(q^{10})$$ 240 * q - 20 * q^2 - 20 * q^3 - 20 * q^4 - 25 * q^5 - 20 * q^6 - 25 * q^7 - 35 * q^8 - 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 25 q^{5} - 20 q^{6} - 25 q^{7} - 35 q^{8} - 20 q^{9} - 20 q^{10} - 25 q^{11} + 60 q^{12} - 20 q^{13} - 30 q^{14} - 40 q^{15} - 40 q^{16} - 15 q^{17} - 25 q^{18} - 10 q^{19} - 10 q^{20} - 35 q^{21} - 25 q^{22} + 70 q^{23} + 15 q^{24} + 35 q^{25} - 45 q^{26} - 20 q^{27} - 10 q^{28} - 10 q^{29} - 40 q^{30} - 30 q^{31} - 25 q^{32} - 35 q^{33} - 20 q^{34} - 40 q^{35} + 170 q^{36} - 55 q^{37} - 40 q^{38} - 35 q^{40} - 35 q^{41} - 10 q^{42} - 25 q^{43} + 15 q^{44} + 140 q^{45} - 40 q^{46} + 100 q^{47} + 5 q^{48} + 35 q^{49} - 10 q^{50} - 55 q^{51} - 15 q^{52} - 15 q^{53} + 30 q^{54} - 15 q^{55} + 65 q^{56} + 255 q^{58} + 5 q^{59} + 135 q^{60} - 40 q^{61} + 5 q^{62} - 35 q^{63} + 25 q^{64} - 30 q^{65} - 95 q^{66} + 105 q^{67} - 10 q^{69} - 55 q^{70} + 45 q^{71} - 30 q^{72} - 40 q^{73} + 35 q^{74} - 15 q^{75} - 65 q^{76} - 35 q^{77} + 100 q^{78} + 430 q^{80} - 95 q^{81} + 175 q^{82} + 20 q^{83} + 45 q^{84} - 10 q^{85} - 80 q^{86} - 5 q^{87} - 5 q^{88} + 30 q^{89} + 65 q^{91} - 55 q^{92} + 275 q^{93} + 60 q^{94} + 10 q^{95} - 135 q^{96} + 35 q^{97} - 15 q^{98} + 45 q^{99}+O(q^{100})$$ 240 * q - 20 * q^2 - 20 * q^3 - 20 * q^4 - 25 * q^5 - 20 * q^6 - 25 * q^7 - 35 * q^8 - 20 * q^9 - 20 * q^10 - 25 * q^11 + 60 * q^12 - 20 * q^13 - 30 * q^14 - 40 * q^15 - 40 * q^16 - 15 * q^17 - 25 * q^18 - 10 * q^19 - 10 * q^20 - 35 * q^21 - 25 * q^22 + 70 * q^23 + 15 * q^24 + 35 * q^25 - 45 * q^26 - 20 * q^27 - 10 * q^28 - 10 * q^29 - 40 * q^30 - 30 * q^31 - 25 * q^32 - 35 * q^33 - 20 * q^34 - 40 * q^35 + 170 * q^36 - 55 * q^37 - 40 * q^38 - 35 * q^40 - 35 * q^41 - 10 * q^42 - 25 * q^43 + 15 * q^44 + 140 * q^45 - 40 * q^46 + 100 * q^47 + 5 * q^48 + 35 * q^49 - 10 * q^50 - 55 * q^51 - 15 * q^52 - 15 * q^53 + 30 * q^54 - 15 * q^55 + 65 * q^56 + 255 * q^58 + 5 * q^59 + 135 * q^60 - 40 * q^61 + 5 * q^62 - 35 * q^63 + 25 * q^64 - 30 * q^65 - 95 * q^66 + 105 * q^67 - 10 * q^69 - 55 * q^70 + 45 * q^71 - 30 * q^72 - 40 * q^73 + 35 * q^74 - 15 * q^75 - 65 * q^76 - 35 * q^77 + 100 * q^78 + 430 * q^80 - 95 * q^81 + 175 * q^82 + 20 * q^83 + 45 * q^84 - 10 * q^85 - 80 * q^86 - 5 * q^87 - 5 * q^88 + 30 * q^89 + 65 * q^91 - 55 * q^92 + 275 * q^93 + 60 * q^94 + 10 * q^95 - 135 * q^96 + 35 * q^97 - 15 * q^98 + 45 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.69510 0.169561i 2.45880 + 1.15702i 5.25058 + 0.663302i 1.42401 1.72400i −6.43051 3.53521i −0.679948 + 0.935868i −8.73317 1.66594i 2.79471 + 3.37822i −4.13018 + 4.40490i
4.2 −2.48068 0.156071i −1.52292 0.716634i 4.14517 + 0.523657i −0.268080 + 2.21994i 3.66604 + 2.01542i 0.423640 0.583091i −5.31799 1.01446i −0.106538 0.128782i 1.01149 5.46511i
4.3 −1.80542 0.113587i −2.06171 0.970167i 1.26239 + 0.159477i 0.397505 2.20045i 3.61205 + 1.98574i −2.29756 + 3.16232i 1.29286 + 0.246625i 1.39715 + 1.68887i −0.967604 + 3.92758i
4.4 −1.58276 0.0995789i 0.718817 + 0.338250i 0.510990 + 0.0645531i −1.79303 1.33605i −1.10403 0.606948i 2.20198 3.03077i 2.31325 + 0.441277i −1.50999 1.82526i 2.70490 + 2.29320i
4.5 −0.921064 0.0579484i 1.90310 + 0.895532i −1.13923 0.143918i 1.22430 + 1.87112i −1.70099 0.935125i −0.820926 + 1.12991i 2.85404 + 0.544437i 0.907551 + 1.09704i −1.01923 1.79437i
4.6 −0.303433 0.0190904i −0.819682 0.385713i −1.89252 0.239081i 2.06070 0.868045i 0.241355 + 0.132686i 1.71112 2.35515i 1.16698 + 0.222614i −1.38917 1.67922i −0.641857 + 0.224054i
4.7 −0.0569284 0.00358163i −1.13144 0.532413i −1.98100 0.250259i −1.71095 + 1.43967i 0.0625039 + 0.0343618i −1.60389 + 2.20757i 0.223940 + 0.0427189i −0.915589 1.10676i 0.102558 0.0758299i
4.8 0.893875 + 0.0562379i 2.85666 + 1.34424i −1.18838 0.150127i −2.22874 0.180863i 2.47790 + 1.36224i 1.29134 1.77738i −2.81338 0.536680i 4.44123 + 5.36853i −1.98205 0.287009i
4.9 1.24554 + 0.0783628i 1.16976 + 0.550449i −0.438999 0.0554585i 1.46862 1.68617i 1.41385 + 0.777273i −0.902275 + 1.24187i −2.99424 0.571182i −0.846918 1.02375i 1.96135 1.98511i
4.10 1.33093 + 0.0837352i −2.88787 1.35893i −0.219857 0.0277744i −1.37708 1.76172i −3.72977 2.05046i 1.78965 2.46324i −2.91018 0.555146i 4.58082 + 5.53726i −1.68528 2.46004i
4.11 2.07048 + 0.130264i −0.974456 0.458544i 2.28570 + 0.288751i 0.707806 + 2.12109i −1.95786 1.07634i 1.45498 2.00261i 0.619230 + 0.118124i −1.17297 1.41788i 1.18920 + 4.48388i
4.12 2.31243 + 0.145486i 0.181456 + 0.0853865i 3.34195 + 0.422186i −2.20265 0.385146i 0.407181 + 0.223850i −1.85539 + 2.55372i 3.11468 + 0.594156i −1.88664 2.28055i −5.03744 1.21108i
9.1 −2.48947 1.17145i 0.834394 0.0524956i 3.55029 + 4.29156i −2.14236 + 0.640526i −2.13869 0.846768i −3.54615 + 1.15221i −2.44251 9.51294i −2.28289 + 0.288396i 6.08369 + 0.915114i
9.2 −1.67272 0.787121i −3.20210 + 0.201459i 0.903578 + 1.09224i −2.18650 0.468215i 5.51479 + 2.18346i 2.31306 0.751558i 0.267779 + 1.04293i 7.23652 0.914185i 3.28885 + 2.50423i
9.3 −1.61529 0.760100i 2.82599 0.177796i 0.756578 + 0.914545i 2.02990 0.937822i −4.69996 1.86084i −1.81397 + 0.589394i 0.360971 + 1.40589i 4.97829 0.628904i −3.99172 0.0280665i
9.4 −1.57955 0.743281i 1.06748 0.0671603i 0.667670 + 0.807075i 0.0877868 + 2.23434i −1.73606 0.687356i 4.47078 1.45264i 0.413537 + 1.61062i −1.84134 + 0.232615i 1.52208 3.59451i
9.5 −0.926191 0.435832i −1.75492 + 0.110410i −0.606968 0.733699i 1.67095 + 1.48591i 1.67351 + 0.662589i −3.38254 + 1.09905i 0.751522 + 2.92699i 0.0912001 0.0115212i −0.900008 2.10449i
9.6 −0.694701 0.326902i −0.172742 + 0.0108680i −0.899103 1.08683i −0.949236 2.02459i 0.123557 + 0.0489196i −2.28461 + 0.742313i 0.651196 + 2.53624i −2.94662 + 0.372245i −0.00240546 + 1.71679i
9.7 0.129181 + 0.0607882i 2.44002 0.153513i −1.26186 1.52532i −1.95252 1.08980i 0.324537 + 0.128493i 2.93692 0.954262i −0.141297 0.550317i 2.95378 0.373150i −0.185983 0.259472i
9.8 0.379582 + 0.178618i −1.61444 + 0.101572i −1.16267 1.40543i 1.99812 1.00375i −0.630955 0.249813i 4.50478 1.46369i −0.398949 1.55380i −0.380236 + 0.0480350i 0.937737 0.0241068i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 119.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.h even 50 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.h.a 240
5.b even 2 1 625.2.h.a 240
5.c odd 4 2 625.2.g.b 480
125.g even 25 1 625.2.h.a 240
125.h even 50 1 inner 125.2.h.a 240
125.i odd 100 2 625.2.g.b 480

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.h.a 240 1.a even 1 1 trivial
125.2.h.a 240 125.h even 50 1 inner
625.2.g.b 480 5.c odd 4 2
625.2.g.b 480 125.i odd 100 2
625.2.h.a 240 5.b even 2 1
625.2.h.a 240 125.g even 25 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(125, [\chi])$$.