Properties

Label 725.2.k.b
Level $725$
Weight $2$
Character orbit 725.k
Analytic conductor $5.789$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(146,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.k (of order \(5\), degree \(4\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 2 q^{3} - 26 q^{4} + 3 q^{5} + 6 q^{6} - 2 q^{7} + 9 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 2 q^{3} - 26 q^{4} + 3 q^{5} + 6 q^{6} - 2 q^{7} + 9 q^{8} - 28 q^{9} - 2 q^{10} + 9 q^{11} - 35 q^{12} - 2 q^{14} - 17 q^{15} - 2 q^{16} + 11 q^{17} - 8 q^{18} + 22 q^{19} - 2 q^{20} + 4 q^{21} - 23 q^{22} + 2 q^{23} - 72 q^{24} + 11 q^{25} - 26 q^{26} + 7 q^{27} - 14 q^{28} + 32 q^{29} + 24 q^{30} + 16 q^{31} + 8 q^{32} + 25 q^{33} + 22 q^{34} - q^{35} + 14 q^{36} - 32 q^{37} + 21 q^{38} + 10 q^{39} - 50 q^{40} + 2 q^{41} + 66 q^{42} - 10 q^{43} + 2 q^{44} - 56 q^{45} + 4 q^{46} - 52 q^{47} - 25 q^{48} + 22 q^{49} - 124 q^{51} + 62 q^{52} - 90 q^{53} + 87 q^{54} + 27 q^{55} + 30 q^{56} + 76 q^{57} + 25 q^{59} + 80 q^{60} + 24 q^{61} - 23 q^{62} + 72 q^{63} + 43 q^{64} - 56 q^{65} - 8 q^{66} + 15 q^{67} - 64 q^{68} + 32 q^{69} - 14 q^{70} + 10 q^{71} - 143 q^{72} - 38 q^{73} - 78 q^{74} + 33 q^{75} - 158 q^{76} + 4 q^{77} - 67 q^{78} + 26 q^{79} + 27 q^{80} + 27 q^{81} + 92 q^{82} + 72 q^{83} - 26 q^{84} + 27 q^{85} - 7 q^{86} - 3 q^{87} + 101 q^{88} + 5 q^{89} - 82 q^{90} + 48 q^{91} + 103 q^{92} + 50 q^{93} + 56 q^{94} - 62 q^{95} + 24 q^{96} - 30 q^{97} - 74 q^{98} - 172 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1 −0.821488 + 2.52828i 0.606668 + 0.440770i −4.09932 2.97833i −2.22416 + 0.230501i −1.61276 + 1.17174i 2.28822 6.59622 4.79243i −0.753283 2.31837i 1.24435 5.81264i
146.2 −0.786223 + 2.41975i −1.25903 0.914737i −3.61899 2.62935i 0.0415333 2.23568i 3.20331 2.32734i 0.860476 5.09096 3.69880i −0.178645 0.549813i 5.37713 + 1.85824i
146.3 −0.738518 + 2.27293i 1.38778 + 1.00828i −3.00275 2.18163i 0.399136 + 2.20016i −3.31665 + 2.40969i −1.73767 3.30933 2.40437i −0.0177458 0.0546160i −5.29556 0.717649i
146.4 −0.673796 + 2.07373i −1.10416 0.802222i −2.22832 1.61897i −2.23498 + 0.0697068i 2.40757 1.74920i −4.78538 1.33070 0.966812i −0.351434 1.08160i 1.36137 4.68172i
146.5 −0.661820 + 2.03687i −2.07844 1.51008i −2.09281 1.52051i 2.20276 + 0.384498i 4.45138 3.23412i 3.25449 1.01682 0.738763i 1.11254 + 3.42403i −2.24101 + 4.23228i
146.6 −0.654900 + 2.01557i 0.271999 + 0.197619i −2.01561 1.46443i 2.17315 0.526694i −0.576448 + 0.418814i 2.11417 0.842583 0.612172i −0.892121 2.74567i −0.361606 + 4.72508i
146.7 −0.566353 + 1.74306i 2.40988 + 1.75088i −1.09945 0.798799i 1.43640 1.71370i −4.41673 + 3.20894i 0.395038 −0.950433 + 0.690530i 1.81490 + 5.58568i 2.17356 + 3.47428i
146.8 −0.510818 + 1.57214i 1.14366 + 0.830919i −0.592644 0.430581i −1.61380 1.54779i −1.89052 + 1.37354i −2.26486 −1.69501 + 1.23150i −0.309515 0.952590i 3.25770 1.74647i
146.9 −0.439625 + 1.35303i −2.45011 1.78011i −0.0193748 0.0140766i 0.652515 + 2.13874i 3.48566 2.53248i −2.48626 −2.27434 + 1.65241i 1.90720 + 5.86975i −3.18064 0.0573756i
146.10 −0.427668 + 1.31623i 2.27433 + 1.65239i 0.0684815 + 0.0497547i −1.22085 + 1.87337i −3.14758 + 2.28685i 2.95234 −2.33407 + 1.69580i 1.51510 + 4.66300i −1.94366 2.40810i
146.11 −0.423967 + 1.30484i −1.66657 1.21084i 0.0951865 + 0.0691571i −1.21250 + 1.87879i 2.28652 1.66125i 3.54864 −2.35051 + 1.70775i 0.384293 + 1.18273i −1.93745 2.37865i
146.12 −0.177522 + 0.546357i 1.18418 + 0.860357i 1.35104 + 0.981589i 2.21483 0.307420i −0.680280 + 0.494253i 1.69340 −1.70566 + 1.23923i −0.264984 0.815536i −0.225221 + 1.26467i
146.13 −0.164067 + 0.504945i 0.467794 + 0.339872i 1.38998 + 1.00988i −0.974508 + 2.01254i −0.248367 + 0.180449i −2.75589 −1.59705 + 1.16032i −0.823733 2.53519i −0.856341 0.822265i
146.14 −0.150328 + 0.462661i −2.09623 1.52300i 1.42658 + 1.03647i 1.98698 1.02563i 1.01975 0.740895i −2.47509 −1.48111 + 1.07609i 1.14760 + 3.53195i 0.175818 + 1.07348i
146.15 −0.0645949 + 0.198803i 0.220238 + 0.160012i 1.58268 + 1.14989i −2.00487 0.990194i −0.0460371 + 0.0334479i 2.87866 −0.669057 + 0.486098i −0.904150 2.78269i 0.326358 0.334613i
146.16 −0.0471777 + 0.145198i −0.898143 0.652539i 1.59918 + 1.16187i 1.78971 + 1.34050i 0.137120 0.0996232i 1.00896 −0.491172 + 0.356857i −0.546197 1.68102i −0.279072 + 0.196621i
146.17 0.132214 0.406913i 1.10805 + 0.805042i 1.46994 + 1.06797i 2.22805 + 0.189245i 0.474081 0.344440i −4.03143 1.32120 0.959907i −0.347379 1.06912i 0.371585 0.881599i
146.18 0.157218 0.483866i −2.53550 1.84215i 1.40862 + 1.02343i −1.87103 1.22444i −1.28998 + 0.937225i 2.75056 1.53987 1.11878i 2.10820 + 6.48836i −0.886623 + 0.712826i
146.19 0.178826 0.550370i 2.41017 + 1.75109i 1.34711 + 0.978730i −2.20556 + 0.368129i 1.39475 1.01334i −1.61383 1.71591 1.24668i 1.81554 + 5.58765i −0.191804 + 1.27970i
146.20 0.206285 0.634880i 0.530961 + 0.385766i 1.25751 + 0.913638i −0.382080 2.20318i 0.354444 0.257519i −3.38525 1.91958 1.39466i −0.793947 2.44352i −1.47758 0.211909i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.k.b 128
25.d even 5 1 inner 725.2.k.b 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.k.b 128 1.a even 1 1 trivial
725.2.k.b 128 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} + 45 T_{2}^{126} - 3 T_{2}^{125} + 1116 T_{2}^{124} - 127 T_{2}^{123} + 20215 T_{2}^{122} - 2948 T_{2}^{121} + 299503 T_{2}^{120} - 50370 T_{2}^{119} + 3812817 T_{2}^{118} - 715856 T_{2}^{117} + 42829347 T_{2}^{116} + \cdots + 1679616 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display