↑

Properties

Label 805.2.bp.b
Level $805$
Weight $2$
Character orbit 805.bp
Analytic conductor $6.428$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(61,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 55, 51]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.bp (of order \(66\), degree \(20\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 2 q^{2} + 34 q^{4} + 32 q^{5} - 3 q^{7} + 12 q^{8} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 2 q^{2} + 34 q^{4} + 32 q^{5} - 3 q^{7} + 12 q^{8} - 34 q^{9} - 2 q^{10} - 12 q^{11} + 12 q^{14} + 24 q^{16} - q^{17} + 18 q^{18} + 4 q^{19} - 68 q^{20} - 92 q^{21} + 14 q^{23} - 18 q^{24} + 32 q^{25} - 12 q^{26} - 2 q^{28} - 30 q^{29} + 6 q^{30} + 3 q^{31} - 312 q^{32} - 10 q^{33} + 16 q^{34} - 8 q^{35} - 54 q^{36} - 39 q^{37} - 77 q^{38} + 186 q^{39} + 16 q^{40} + 44 q^{41} + 50 q^{42} - 88 q^{43} + 59 q^{44} + 318 q^{45} - 16 q^{46} - 36 q^{47} + 352 q^{48} - 47 q^{49} - 18 q^{50} + 74 q^{51} - 33 q^{53} - 208 q^{54} - 51 q^{56} + 15 q^{58} - 27 q^{59} + 10 q^{61} - 230 q^{63} - 124 q^{64} + 2 q^{66} + 72 q^{68} - 150 q^{69} - 6 q^{70} + 22 q^{71} - 106 q^{72} - 18 q^{73} + 13 q^{74} - 104 q^{76} + 315 q^{77} + 140 q^{78} - 22 q^{79} + 46 q^{80} + 24 q^{81} + 102 q^{82} + 22 q^{83} - 274 q^{84} + 2 q^{85} + 148 q^{86} - 354 q^{87} - 324 q^{88} - 18 q^{89} - 36 q^{90} + 68 q^{91} + 156 q^{92} - 10 q^{93} - 455 q^{94} + 37 q^{95} - 60 q^{96} - 36 q^{97} - 117 q^{98} + 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −2.16272 − 1.70078i 1.88522 + 1.34246i 1.31317 + 5.41298i 0.981929 − 0.189251i −1.79397 − 6.10971i −2.64488 + 0.0677739i 4.08034 − 8.93470i 0.770660 + 2.22668i −2.44551 − 1.26075i
61.2 −2.12756 − 1.67313i −2.08619 − 1.48557i 1.25563 + 5.17578i 0.981929 − 0.189251i 1.95294 + 6.65111i −1.84978 + 1.89164i 3.73958 − 8.18853i 1.16406 + 3.36334i −2.40576 − 1.24025i
61.3 −1.96717 − 1.54700i 0.502179 + 0.357600i 1.00503 + 4.14279i 0.981929 − 0.189251i −0.434664 − 1.48033i 2.26808 + 1.36228i 2.35260 − 5.15148i −0.856898 − 2.47584i −2.22439 − 1.14675i
61.4 −1.84804 − 1.45332i −0.0114485 − 0.00815244i 0.831616 + 3.42797i 0.981929 − 0.189251i 0.00930925 + 0.0317044i −0.0200674 − 2.64568i 1.49175 − 3.26647i −0.981139 − 2.83482i −2.08969 − 1.07731i
61.5 −1.75490 − 1.38007i −2.05396 − 1.46262i 0.703565 + 2.90014i 0.981929 − 0.189251i 1.58598 + 5.40136i 1.65527 − 2.06400i 0.912832 − 1.99882i 1.09830 + 3.17334i −1.98437 − 1.02301i
61.6 −1.65297 − 1.29991i 2.69085 + 1.91615i 0.571027 + 2.35381i 0.981929 − 0.189251i −1.95708 − 6.66521i 1.18500 + 2.36554i 0.368715 − 0.807372i 2.58786 + 7.47714i −1.86911 − 0.963593i
61.7 −1.47765 − 1.16203i 0.783904 + 0.558215i 0.361599 + 1.49053i 0.981929 − 0.189251i −0.509667 − 1.73577i −1.88014 + 1.86148i −0.364088 + 0.797242i −0.678303 − 1.95983i −1.67086 − 0.861388i
61.8 −1.46939 − 1.15554i −1.27826 − 0.910243i 0.352319 + 1.45228i 0.981929 − 0.189251i 0.826438 + 2.81459i −2.50362 − 0.855499i −0.392619 + 0.859716i −0.175805 − 0.507954i −1.66153 − 0.856578i
61.9 −1.29083 − 1.01512i −2.16675 − 1.54293i 0.164258 + 0.677081i 0.981929 − 0.189251i 1.23064 + 4.19118i 2.51050 + 0.835101i −0.889074 + 1.94680i 1.33294 + 3.85129i −1.45962 − 0.752486i
61.10 −1.03649 − 0.815106i 1.67497 + 1.19274i −0.0616010 − 0.253923i 0.981929 − 0.189251i −0.763882 − 2.60154i −0.215276 − 2.63698i −1.23866 + 2.71229i 0.401689 + 1.16060i −1.17202 − 0.604219i
61.11 −0.897332 − 0.705670i 1.36891 + 0.974793i −0.164283 − 0.677184i 0.981929 − 0.189251i −0.540480 − 1.84071i 1.83941 − 1.90172i −1.27890 + 2.80040i −0.0575238 − 0.166204i −1.01466 − 0.523096i
61.12 −0.802627 − 0.631193i 0.458195 + 0.326279i −0.225712 − 0.930399i 0.981929 − 0.189251i −0.161815 − 0.551090i 2.64479 − 0.0711593i −1.25445 + 2.74685i −0.877719 − 2.53600i −0.907576 − 0.467888i
61.13 −0.680844 − 0.535421i 1.10407 + 0.786205i −0.294646 − 1.21455i 0.981929 − 0.189251i −0.330748 − 1.12643i −0.833887 + 2.51090i −1.16931 + 2.56044i −0.380351 − 1.09895i −0.769869 − 0.396895i
61.14 −0.634767 − 0.499186i −1.09613 − 0.780549i −0.317776 − 1.30989i 0.981929 − 0.189251i 0.306146 + 1.04264i −2.64207 − 0.139573i −1.12309 + 2.45923i −0.388965 − 1.12384i −0.717767 − 0.370035i
61.15 −0.318021 − 0.250095i −0.947865 − 0.674971i −0.432928 − 1.78455i 0.981929 − 0.189251i 0.132634 + 0.451711i 0.286816 + 2.63016i −0.644764 + 1.41184i −0.538343 − 1.55544i −0.359605 − 0.185389i
61.16 0.0184191 + 0.0144850i −1.12103 − 0.798279i −0.471388 − 1.94309i 0.981929 − 0.189251i −0.00908528 − 0.0309416i 0.191958 − 2.63878i 0.0389314 − 0.0852478i −0.361754 − 1.04522i 0.0208276 + 0.0107374i
61.17 0.0534238 + 0.0420129i −2.26095 − 1.61002i −0.470429 − 1.93913i 0.981929 − 0.189251i −0.0531472 − 0.181003i 0.794489 + 2.52365i 0.112804 − 0.247006i 1.53855 + 4.44535i 0.0604094 + 0.0311432i
61.18 0.196709 + 0.154693i 2.26544 + 1.61321i −0.456754 − 1.88276i 0.981929 − 0.189251i 0.196078 + 0.667782i 2.64545 − 0.0400953i 0.409318 − 0.896280i 1.54857 + 4.47429i 0.222430 + 0.114670i
61.19 0.349679 + 0.274990i 0.869308 + 0.619031i −0.424862 − 1.75131i 0.981929 − 0.189251i 0.133751 + 0.455513i −2.40065 + 1.11216i 0.702625 − 1.53853i −0.608708 − 1.75875i 0.395402 + 0.203844i
61.20 0.391831 + 0.308139i 0.875714 + 0.623593i −0.412936 − 1.70215i 0.981929 − 0.189251i 0.150978 + 0.514185i −1.48444 − 2.19008i 0.776848 − 1.70106i −0.603197 − 1.74282i 0.443066 + 0.228416i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.bp.b yes 640
7.d odd 6 1 805.2.bp.a ✓ 640
23.d odd 22 1 805.2.bp.a ✓ 640
161.o even 66 1 inner 805.2.bp.b yes 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.bp.a ✓ 640 7.d odd 6 1
805.2.bp.a ✓ 640 23.d odd 22 1
805.2.bp.b yes 640 1.a even 1 1 trivial
805.2.bp.b yes 640 161.o even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{640} + 2 T_{2}^{639} - 47 T_{2}^{638} - 106 T_{2}^{637} + 980 T_{2}^{636} + 2628 T_{2}^{635} + \cdots + 19\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display