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Properties

Label 805.2.i.f
Level $805$
Weight $2$
Character orbit 805.i
Analytic conductor $6.428$
Analytic rank $0$
Dimension $34$
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] = [805,2,Mod(116,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.i (of order \(3\), degree \(2\), 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: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 5 q^{2} + 2 q^{3} - 23 q^{4} - 17 q^{5} + 8 q^{6} - q^{7} + 30 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 5 q^{2} + 2 q^{3} - 23 q^{4} - 17 q^{5} + 8 q^{6} - q^{7} + 30 q^{8} - 25 q^{9} - 5 q^{10} - 7 q^{11} - 6 q^{12} + 2 q^{13} + 11 q^{14} - 4 q^{15} - 27 q^{16} + 3 q^{17} - 53 q^{18} - 7 q^{19} + 46 q^{20} - 9 q^{21} + 30 q^{22} + 17 q^{23} - 4 q^{24} - 17 q^{25} - 6 q^{26} - 10 q^{27} + 13 q^{28} + 58 q^{29} - 4 q^{30} - 19 q^{31} - 50 q^{32} + 16 q^{33} + 4 q^{34} + 2 q^{35} + 70 q^{36} - 14 q^{37} + 22 q^{38} - 4 q^{39} - 15 q^{40} + 36 q^{41} - 52 q^{42} - 24 q^{43} - 45 q^{44} - 25 q^{45} + 5 q^{46} + 23 q^{47} + 104 q^{48} - 5 q^{49} + 10 q^{50} - 7 q^{51} + 31 q^{52} - 56 q^{53} + 16 q^{54} + 14 q^{55} - 98 q^{56} + 24 q^{57} - 12 q^{58} - 2 q^{59} - 6 q^{60} - 26 q^{61} + 12 q^{62} + 27 q^{63} + 98 q^{64} - q^{65} + 69 q^{66} - 7 q^{67} + 9 q^{68} + 4 q^{69} + 5 q^{70} + 74 q^{71} - 97 q^{72} + 19 q^{73} - 20 q^{74} + 2 q^{75} + 106 q^{76} + 24 q^{77} - 50 q^{78} - 12 q^{79} - 27 q^{80} - 49 q^{81} + 8 q^{82} + 22 q^{83} - 46 q^{84} - 6 q^{85} + 8 q^{86} + 24 q^{87} - 24 q^{88} - 2 q^{89} + 106 q^{90} - 4 q^{91} - 46 q^{92} - 53 q^{93} + 2 q^{94} - 7 q^{95} - 25 q^{96} - 36 q^{97} - 111 q^{98} - 78 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −1.38668 − 2.40180i −1.47972 + 2.56294i −2.84576 + 4.92899i −0.500000 − 0.866025i 8.20756 −2.61463 + 0.404636i 10.2379 −2.87912 − 4.98678i −1.38668 + 2.40180i
116.2 −1.36316 − 2.36106i 0.223592 − 0.387272i −2.71640 + 4.70495i −0.500000 − 0.866025i −1.21916 −0.202162 − 2.63802i 9.35892 1.40001 + 2.42489i −1.36316 + 2.36106i
116.3 −1.26084 − 2.18384i 1.61144 − 2.79109i −2.17943 + 3.77489i −0.500000 − 0.866025i −8.12707 −1.70010 + 2.02723i 5.94831 −3.69347 − 6.39728i −1.26084 + 2.18384i
116.4 −1.07546 − 1.86276i −1.27908 + 2.21543i −1.31325 + 2.27461i −0.500000 − 0.866025i 5.50243 1.95643 + 1.78111i 1.34755 −1.77210 − 3.06936i −1.07546 + 1.86276i
116.5 −0.903932 − 1.56566i −0.799461 + 1.38471i −0.634186 + 1.09844i −0.500000 − 0.866025i 2.89063 −2.60846 − 0.442658i −1.32268 0.221723 + 0.384036i −0.903932 + 1.56566i
116.6 −0.865269 − 1.49869i 1.62810 − 2.81996i −0.497380 + 0.861487i −0.500000 − 0.866025i −5.63499 2.62160 + 0.356690i −1.73961 −3.80145 − 6.58430i −0.865269 + 1.49869i
116.7 −0.651700 − 1.12878i 0.364518 − 0.631364i 0.150575 − 0.260804i −0.500000 − 0.866025i −0.950226 −1.04030 + 2.43265i −2.99932 1.23425 + 2.13779i −0.651700 + 1.12878i
116.8 −0.333824 − 0.578200i −1.33250 + 2.30796i 0.777123 − 1.34602i −0.500000 − 0.866025i 1.77929 1.86986 − 1.87180i −2.37299 −2.05113 − 3.55266i −0.333824 + 0.578200i
116.9 −0.278674 − 0.482677i 0.977577 − 1.69321i 0.844682 − 1.46303i −0.500000 − 0.866025i −1.08970 1.84408 − 1.89720i −2.05626 −0.411313 − 0.712415i −0.278674 + 0.482677i
116.10 −0.104861 − 0.181625i 0.786337 − 1.36198i 0.978008 − 1.69396i −0.500000 − 0.866025i −0.329825 0.259831 − 2.63296i −0.829665 0.263347 + 0.456131i −0.104861 + 0.181625i
116.11 0.201075 + 0.348272i −1.44847 + 2.50882i 0.919138 − 1.59199i −0.500000 − 0.866025i −1.16500 0.183370 + 2.63939i 1.54356 −2.69611 − 4.66979i 0.201075 − 0.348272i
116.12 0.501983 + 0.869460i 0.415180 − 0.719113i 0.496026 − 0.859142i −0.500000 − 0.866025i 0.833654 −2.59346 + 0.523404i 3.00392 1.15525 + 2.00095i 0.501983 − 0.869460i
116.13 0.757569 + 1.31215i 1.51823 − 2.62966i −0.147821 + 0.256033i −0.500000 − 0.866025i 4.60066 −1.93160 − 1.80801i 2.58234 −3.11006 − 5.38679i 0.757569 − 1.31215i
116.14 0.818107 + 1.41700i 0.428011 − 0.741337i −0.338597 + 0.586466i −0.500000 − 0.866025i 1.40064 0.648321 + 2.56509i 2.16439 1.13361 + 1.96347i 0.818107 − 1.41700i
116.15 1.00708 + 1.74431i −0.672543 + 1.16488i −1.02841 + 1.78125i −0.500000 − 0.866025i −2.70921 2.61166 − 0.423386i −0.114431 0.595371 + 1.03121i 1.00708 − 1.74431i
116.16 1.15298 + 1.99702i 0.550296 − 0.953141i −1.65872 + 2.87299i −0.500000 − 0.866025i 2.53792 1.70719 + 2.02126i −3.03798 0.894348 + 1.54906i 1.15298 − 1.99702i
116.17 1.28561 + 2.22675i −0.491518 + 0.851333i −2.30560 + 3.99341i −0.500000 − 0.866025i −2.52760 −1.51163 − 2.17140i −6.71396 1.01682 + 1.76119i 1.28561 − 2.22675i
576.1 −1.38668 + 2.40180i −1.47972 − 2.56294i −2.84576 − 4.92899i −0.500000 + 0.866025i 8.20756 −2.61463 − 0.404636i 10.2379 −2.87912 + 4.98678i −1.38668 − 2.40180i
576.2 −1.36316 + 2.36106i 0.223592 + 0.387272i −2.71640 − 4.70495i −0.500000 + 0.866025i −1.21916 −0.202162 + 2.63802i 9.35892 1.40001 − 2.42489i −1.36316 − 2.36106i
576.3 −1.26084 + 2.18384i 1.61144 + 2.79109i −2.17943 − 3.77489i −0.500000 + 0.866025i −8.12707 −1.70010 − 2.02723i 5.94831 −3.69347 + 6.39728i −1.26084 − 2.18384i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.i.f ✓ 34
7.c even 3 1 inner 805.2.i.f ✓ 34
7.c even 3 1 5635.2.a.bm 17
7.d odd 6 1 5635.2.a.bn 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.i.f ✓ 34 1.a even 1 1 trivial
805.2.i.f ✓ 34 7.c even 3 1 inner
5635.2.a.bm 17 7.c even 3 1
5635.2.a.bn 17 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + 5 T_{2}^{33} + 41 T_{2}^{32} + 140 T_{2}^{31} + 747 T_{2}^{30} + 2124 T_{2}^{29} + \cdots + 24336 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display