Properties

Label 605.2.r.a
Level $605$
Weight $2$
Character orbit 605.r
Analytic conductor $4.831$
Analytic rank $0$
Dimension $1280$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(1280\)
Relative dimension: \(64\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1280q - 22q^{2} - 40q^{3} - 14q^{5} - 44q^{6} - 22q^{7} - 22q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1280q - 22q^{2} - 40q^{3} - 14q^{5} - 44q^{6} - 22q^{7} - 22q^{8} - 66q^{10} - 48q^{11} + 44q^{12} - 66q^{13} - 26q^{15} + 80q^{16} - 22q^{17} - 44q^{18} - 54q^{20} - 44q^{21} - 90q^{22} - 34q^{23} - 30q^{25} - 12q^{26} + 80q^{27} - 22q^{28} - 22q^{30} - 52q^{31} - 22q^{32} - 62q^{33} - 22q^{35} + 216q^{36} + 52q^{37} + 70q^{38} - 44q^{41} + 44q^{42} - 22q^{43} + 104q^{45} - 44q^{46} - 18q^{47} - 72q^{48} + 88q^{50} - 484q^{51} - 22q^{52} + 14q^{53} + 140q^{55} - 316q^{56} + 44q^{57} - 6q^{58} + 34q^{60} - 44q^{61} - 22q^{62} - 44q^{63} - 22q^{65} - 84q^{66} - 138q^{67} - 22q^{68} - 126q^{70} - 4q^{71} + 220q^{72} - 22q^{73} - 66q^{75} + 88q^{76} - 22q^{77} - 104q^{78} - 324q^{80} - 1056q^{81} - 58q^{82} - 22q^{83} - 110q^{85} - 52q^{86} + 44q^{87} - 2q^{88} + 176q^{90} + 104q^{91} - 278q^{92} - 128q^{93} - 22q^{95} - 44q^{96} + 46q^{97} - 22q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −1.65403 2.20953i −1.20427 + 1.20427i −1.58273 + 5.39029i 0.661016 + 2.13613i 4.65276 + 0.668966i −0.992826 + 4.56395i 9.35584 3.48955i 0.0994872i 3.62650 4.99377i
32.2 −1.63965 2.19032i 1.62268 1.62268i −1.54558 + 5.26376i 0.347156 2.20896i −6.21481 0.893555i 0.0527075 0.242292i 8.93645 3.33312i 2.26615i −5.40754 + 2.86154i
32.3 −1.60536 2.14451i −1.49754 + 1.49754i −1.45828 + 4.96643i 1.95274 1.08941i 5.61559 + 0.807400i 0.692711 3.18434i 7.97176 2.97331i 1.48527i −5.47109 2.43878i
32.4 −1.55810 2.08138i −0.762046 + 0.762046i −1.34100 + 4.56704i −2.23099 + 0.150681i 2.77346 + 0.398763i 0.945323 4.34558i 6.72311 2.50759i 1.83857i 3.78973 + 4.40876i
32.5 −1.55047 2.07118i 1.30510 1.30510i −1.32238 + 4.50362i −2.13842 + 0.653577i −4.72661 0.679584i −0.374253 + 1.72041i 6.52992 2.43554i 0.406559i 4.66923 + 3.41571i
32.6 −1.36166 1.81897i 0.545918 0.545918i −0.891058 + 3.03466i −0.445983 2.19114i −1.73637 0.249652i −0.413052 + 1.89877i 2.47545 0.923297i 2.40395i −3.37834 + 3.79483i
32.7 −1.34436 1.79586i −0.787015 + 0.787015i −0.854335 + 2.90960i −1.17893 1.90004i 2.47140 + 0.355334i −0.306060 + 1.40694i 2.17003 0.809380i 1.76121i −1.82729 + 4.67153i
32.8 −1.34419 1.79563i −0.436724 + 0.436724i −0.853977 + 2.90838i −0.335531 + 2.21075i 1.37124 + 0.197154i 0.125054 0.574863i 2.16709 0.808283i 2.61854i 4.42072 2.36919i
32.9 −1.31487 1.75646i 0.251483 0.251483i −0.792800 + 2.70003i 2.23442 + 0.0859194i −0.772387 0.111053i 0.518840 2.38507i 1.67342 0.624152i 2.87351i −2.78705 4.03763i
32.10 −1.29735 1.73306i 2.35487 2.35487i −0.756913 + 2.57781i 0.140101 + 2.23167i −7.13625 1.02604i −0.147963 + 0.680177i 1.39275 0.519471i 8.09087i 3.68587 3.13808i
32.11 −1.29475 1.72958i −2.08349 + 2.08349i −0.751609 + 2.55974i −2.21633 0.296468i 6.30115 + 0.905969i −0.317966 + 1.46166i 1.35184 0.504211i 5.68187i 2.35682 + 4.21716i
32.12 −1.20651 1.61170i 1.27231 1.27231i −0.578464 + 1.97007i 2.15875 + 0.582930i −3.58564 0.515537i −1.05425 + 4.84632i 0.100425 0.0374565i 0.237553i −1.66503 4.18257i
32.13 −1.18351 1.58098i 2.15121 2.15121i −0.535343 + 1.82321i 1.95281 1.08928i −5.94699 0.855048i 0.546359 2.51157i −0.184700 + 0.0688894i 6.25540i −4.03329 1.79819i
32.14 −1.17119 1.56452i −2.21909 + 2.21909i −0.512590 + 1.74572i 1.83519 + 1.27754i 6.07079 + 0.872848i 0.0862130 0.396315i −0.330664 + 0.123331i 6.84872i −0.150612 4.36743i
32.15 −1.05562 1.41014i 1.43490 1.43490i −0.310697 + 1.05814i −2.23306 0.115999i −3.53811 0.508704i 0.943957 4.33930i −1.48074 + 0.552288i 1.11789i 2.19367 + 3.27137i
32.16 −1.02585 1.37037i −1.69135 + 1.69135i −0.262091 + 0.892598i 1.54512 1.61636i 4.05284 + 0.582710i −0.361306 + 1.66090i −1.71570 + 0.639922i 2.72132i −3.80007 0.459245i
32.17 −0.989614 1.32197i −0.0346243 + 0.0346243i −0.204802 + 0.697492i 0.766483 + 2.10060i 0.0800369 + 0.0115076i 0.0122172 0.0561614i −1.96972 + 0.734666i 2.99760i 2.01840 3.09205i
32.18 −0.832083 1.11153i 1.00692 1.00692i 0.0203215 0.0692087i −1.62222 + 1.53896i −1.95707 0.281384i −0.335672 + 1.54306i −2.69570 + 1.00545i 0.972218i 3.06043 + 0.522603i
32.19 −0.826501 1.10408i −1.13965 + 1.13965i 0.0275847 0.0939450i −1.95581 + 1.08388i 2.20018 + 0.316338i −1.02748 + 4.72324i −2.71093 + 1.01113i 0.402407i 2.81317 + 1.26353i
32.20 −0.823163 1.09962i 1.35528 1.35528i 0.0319051 0.108659i −1.65144 1.50756i −2.60591 0.374674i 0.0674781 0.310192i −2.71972 + 1.01440i 0.673588i −0.298338 + 3.05692i
See next 80 embeddings (of 1280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
121.f odd 22 1 inner
605.r even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.r.a 1280
5.c odd 4 1 inner 605.2.r.a 1280
121.f odd 22 1 inner 605.2.r.a 1280
605.r even 44 1 inner 605.2.r.a 1280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.r.a 1280 1.a even 1 1 trivial
605.2.r.a 1280 5.c odd 4 1 inner
605.2.r.a 1280 121.f odd 22 1 inner
605.2.r.a 1280 605.r even 44 1 inner

Hecke kernels

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