# Properties

 Label 605.2.w.a Level $605$ Weight $2$ Character orbit 605.w Analytic conductor $4.831$ Analytic rank $0$ Dimension $5120$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.w (of order $$220$$, degree $$80$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$5120$$ Relative dimension: $$64$$ over $$\Q(\zeta_{220})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{220}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$5120q - 78q^{2} - 60q^{3} - 86q^{5} - 156q^{6} - 88q^{7} - 78q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$5120q - 78q^{2} - 60q^{3} - 86q^{5} - 156q^{6} - 88q^{7} - 78q^{8} - 44q^{10} - 152q^{11} - 134q^{12} - 34q^{13} - 94q^{15} - 280q^{16} - 88q^{17} - 56q^{18} - 96q^{20} - 176q^{21} - 10q^{22} - 56q^{23} - 110q^{25} - 188q^{26} - 180q^{27} - 138q^{28} - 118q^{30} - 148q^{31} - 88q^{32} - 88q^{33} - 78q^{35} - 496q^{36} - 152q^{37} - 230q^{38} - 60q^{40} - 216q^{41} - 144q^{42} - 88q^{43} - 194q^{45} - 236q^{46} - 52q^{47} + 12q^{48} - 148q^{50} + 244q^{51} - 38q^{52} - 84q^{53} - 200q^{55} + 136q^{56} - 184q^{57} - 14q^{58} - 114q^{60} - 116q^{61} - 188q^{62} - 36q^{63} - 88q^{65} - 76q^{66} + 48q^{67} - 58q^{68} - 14q^{70} - 196q^{71} - 410q^{72} - 138q^{73} - 114q^{75} - 308q^{76} - 158q^{77} + 14q^{78} + 124q^{80} + 836q^{81} - 82q^{82} - 178q^{83} - 30q^{85} - 268q^{86} - 154q^{87} - 258q^{88} - 266q^{90} - 344q^{91} + 188q^{92} - 32q^{93} - 48q^{95} - 176q^{96} - 86q^{97} - 88q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.86668 1.97647i −0.223065 + 1.40838i −0.307761 + 5.38212i −1.72207 + 1.42635i 3.20000 2.18810i −1.00869 1.52134i 7.05246 5.93681i 0.919406 + 0.298733i 6.03369 + 0.741082i
2.2 −1.86517 1.97487i 0.517775 3.26910i −0.307079 + 5.37020i 1.45529 1.69769i −7.42179 + 5.07489i −0.0523717 0.0789887i 7.02195 5.91113i −7.56577 2.45827i −6.06707 + 0.292469i
2.3 −1.82550 1.93287i −0.499211 + 3.15189i −0.289350 + 5.06016i 2.23158 0.141650i 7.00350 4.78887i −1.28817 1.94286i 6.24096 5.25368i −6.83205 2.21987i −4.34753 4.05476i
2.4 −1.79170 1.89708i −0.0393495 + 0.248443i −0.274545 + 4.80124i 1.52452 + 1.63580i 0.541818 0.370486i 1.79950 + 2.71407i 5.60768 4.72058i 2.79299 + 0.907499i 0.371772 5.82299i
2.5 −1.73635 1.83847i 0.335328 2.11718i −0.250899 + 4.38772i −1.16194 + 1.91047i −4.47462 + 3.05966i 0.600946 + 0.906367i 4.63314 3.90021i −1.51682 0.492845i 5.52988 1.18104i
2.6 −1.66957 1.76777i −0.214500 + 1.35430i −0.223358 + 3.90609i −1.67153 1.48525i 2.75221 1.88191i −0.126444 0.190707i 3.55757 2.99479i 1.06505 + 0.346056i 0.165164 + 5.43462i
2.7 −1.63020 1.72608i −0.0446050 + 0.281625i −0.207627 + 3.63097i 1.61903 1.54231i 0.558823 0.382113i −1.68230 2.53730i 2.97316 2.50283i 2.77585 + 0.901927i −5.30150 0.280303i
2.8 −1.55775 1.64937i 0.240669 1.51952i −0.179660 + 3.14189i −1.15878 1.91239i −2.88116 + 1.97009i −2.21948 3.34750i 1.99079 1.67586i 0.602146 + 0.195649i −1.34914 + 4.89029i
2.9 −1.52867 1.61858i 0.256577 1.61996i −0.168790 + 2.95180i 1.47700 + 1.67883i −3.01426 + 2.06110i −1.20402 1.81594i 1.62933 1.37158i 0.294721 + 0.0957606i 0.459474 4.95701i
2.10 −1.49444 1.58234i 0.322560 2.03656i −0.156261 + 2.73269i −1.96968 1.05847i −3.70458 + 2.53313i 1.84358 + 2.78055i 1.22742 1.03325i −1.19037 0.386775i 1.26872 + 4.69852i
2.11 −1.48817 1.57569i −0.399974 + 2.52534i −0.153994 + 2.69305i 0.336311 2.21063i 4.57439 3.12789i 2.61648 + 3.94627i 1.15643 0.973491i −3.36419 1.09309i −3.98377 + 2.75987i
2.12 −1.34858 1.42789i −0.353163 + 2.22978i −0.106045 + 1.85451i −2.16972 + 0.540671i 3.66016 2.50276i 0.961173 + 1.44967i −0.214051 + 0.180190i −1.99404 0.647904i 3.69805 + 2.36899i
2.13 −1.34797 1.42725i 0.193438 1.22132i −0.105846 + 1.85104i 2.15715 0.588819i −2.00388 + 1.37021i 1.76243 + 2.65815i −0.219176 + 0.184504i 1.39897 + 0.454553i −3.74816 2.28508i
2.14 −1.26609 1.34056i −0.0572426 + 0.361416i −0.0799287 + 1.39779i −0.182529 + 2.22861i 0.556972 0.380848i −2.44095 3.68153i −0.846282 + 0.712406i 2.72583 + 0.885674i 3.21867 2.57693i
2.15 −1.18560 1.25533i −0.529186 + 3.34115i −0.0560322 + 0.979891i −0.519117 + 2.17498i 4.82164 3.29695i −0.359230 0.541803i −1.34541 + 1.13258i −8.03007 2.60913i 3.34577 1.92698i
2.16 −1.10482 1.16980i 0.0340061 0.214706i −0.0336278 + 0.588084i −2.09072 + 0.793034i −0.288733 + 0.197431i 0.954022 + 1.43889i −1.73683 + 1.46208i 2.80823 + 0.912448i 3.23755 + 1.56956i
2.17 −1.05807 1.12030i −0.180139 + 1.13735i −0.0213845 + 0.373973i 2.23461 0.0806053i 1.46477 1.00159i 0.333391 + 0.502832i −1.91617 + 1.61304i 1.59205 + 0.517289i −2.45468 2.41815i
2.18 −1.01696 1.07677i 0.243561 1.53779i −0.0110539 + 0.193311i 0.540109 2.16986i −1.90353 + 1.30160i 0.222191 + 0.335117i −2.04674 + 1.72296i 0.547705 + 0.177960i −2.88570 + 1.62508i
2.19 −0.967000 1.02388i 0.464579 2.93323i 0.000947099 0.0165629i −2.23310 + 0.115139i −3.45251 + 2.36077i −2.25769 3.40513i −2.17269 + 1.82899i −5.53486 1.79839i 2.27730 + 2.17508i
2.20 −0.886874 0.939036i −0.0632831 + 0.399554i 0.0189344 0.331126i −0.878832 2.05613i 0.431319 0.294929i 0.577165 + 0.870500i −2.30400 + 1.93952i 2.69753 + 0.876481i −1.15136 + 2.64878i
See next 80 embeddings (of 5120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 568.64 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.c odd 4 1 inner
121.h odd 110 1 inner
605.w even 220 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.w.a 5120
5.c odd 4 1 inner 605.2.w.a 5120
121.h odd 110 1 inner 605.2.w.a 5120
605.w even 220 1 inner 605.2.w.a 5120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.w.a 5120 1.a even 1 1 trivial
605.2.w.a 5120 5.c odd 4 1 inner
605.2.w.a 5120 121.h odd 110 1 inner
605.2.w.a 5120 605.w even 220 1 inner

## Hecke kernels

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