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Properties

Label 770.2.ba.b
Level $770$
Weight $2$
Character orbit 770.ba
Analytic conductor $6.148$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(169,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.ba (of order \(10\), 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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 8 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 8 q^{6} + 12 q^{9} + 8 q^{10} + 8 q^{11} + 16 q^{14} + 14 q^{15} - 16 q^{16} + 16 q^{19} + 8 q^{21} - 8 q^{24} + 22 q^{25} + 24 q^{29} + 10 q^{30} - 16 q^{31} - 8 q^{34} - 2 q^{35} - 32 q^{36} + 48 q^{39} + 2 q^{40} + 28 q^{41} + 12 q^{44} - 12 q^{45} - 56 q^{46} + 16 q^{49} - 32 q^{50} - 36 q^{51} - 80 q^{54} - 46 q^{55} + 64 q^{56} - 20 q^{59} - 4 q^{60} - 36 q^{61} + 16 q^{64} + 12 q^{65} - 4 q^{66} + 16 q^{69} + 10 q^{70} + 8 q^{71} + 36 q^{74} - 142 q^{75} + 24 q^{76} + 72 q^{79} - 10 q^{80} - 4 q^{81} + 12 q^{84} + 30 q^{85} - 44 q^{86} + 144 q^{89} + 66 q^{90} + 88 q^{94} + 24 q^{95} - 12 q^{96} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1 −0.587785 + 0.809017i −3.11459 − 1.01199i −0.309017 − 0.951057i −2.22311 − 0.240388i 2.64943 − 1.92492i 0.951057 − 0.309017i 0.951057 + 0.309017i 6.24948 + 4.54051i 1.50119 − 1.65724i
169.2 −0.587785 + 0.809017i −2.24732 − 0.730198i −0.309017 − 0.951057i 1.88751 − 1.19888i 1.91168 − 1.38892i 0.951057 − 0.309017i 0.951057 + 0.309017i 2.09020 + 1.51862i −0.139530 + 2.23171i
169.3 −0.587785 + 0.809017i −1.15178 − 0.374237i −0.309017 − 0.951057i 2.04234 + 0.910416i 0.979765 − 0.711841i 0.951057 − 0.309017i 0.951057 + 0.309017i −1.24050 − 0.901277i −1.93700 + 1.11716i
169.4 −0.587785 + 0.809017i −0.784492 − 0.254897i −0.309017 − 0.951057i 0.884314 + 2.05377i 0.667329 − 0.484843i 0.951057 − 0.309017i 0.951057 + 0.309017i −1.87660 − 1.36343i −2.18132 − 0.491753i
169.5 −0.587785 + 0.809017i 0.320930 + 0.104276i −0.309017 − 0.951057i −2.23602 + 0.0145833i −0.272999 + 0.198346i 0.951057 − 0.309017i 0.951057 + 0.309017i −2.33493 − 1.69642i 1.30250 − 1.81755i
169.6 −0.587785 + 0.809017i 0.912267 + 0.296414i −0.309017 − 0.951057i −1.63040 − 1.53029i −0.776021 + 0.563812i 0.951057 − 0.309017i 0.951057 + 0.309017i −1.68268 − 1.22254i 2.19636 − 0.419542i
169.7 −0.587785 + 0.809017i 2.28080 + 0.741078i −0.309017 − 0.951057i −0.245880 + 2.22251i −1.94017 + 1.40961i 0.951057 − 0.309017i 0.951057 + 0.309017i 2.22582 + 1.61715i −1.65352 − 1.50528i
169.8 −0.587785 + 0.809017i 2.60861 + 0.847588i −0.309017 − 0.951057i −0.184564 − 2.22844i −2.21901 + 1.61221i 0.951057 − 0.309017i 0.951057 + 0.309017i 3.65938 + 2.65869i 1.91133 + 1.16053i
169.9 0.587785 − 0.809017i −2.60861 − 0.847588i −0.309017 − 0.951057i 2.06234 + 0.864156i −2.21901 + 1.61221i −0.951057 + 0.309017i −0.951057 − 0.309017i 3.65938 + 2.65869i 1.91133 − 1.16053i
169.10 0.587785 − 0.809017i −2.28080 − 0.741078i −0.309017 − 0.951057i −2.18971 − 0.452947i −1.94017 + 1.40961i −0.951057 + 0.309017i −0.951057 − 0.309017i 2.22582 + 1.61715i −1.65352 + 1.50528i
169.11 0.587785 − 0.809017i −0.912267 − 0.296414i −0.309017 − 0.951057i 0.951570 + 2.02349i −0.776021 + 0.563812i −0.951057 + 0.309017i −0.951057 − 0.309017i −1.68268 − 1.22254i 2.19636 + 0.419542i
169.12 0.587785 − 0.809017i −0.320930 − 0.104276i −0.309017 − 0.951057i −0.704838 + 2.12208i −0.272999 + 0.198346i −0.951057 + 0.309017i −0.951057 − 0.309017i −2.33493 − 1.69642i 1.30250 + 1.81755i
169.13 0.587785 − 0.809017i 0.784492 + 0.254897i −0.309017 − 0.951057i −1.67999 − 1.47568i 0.667329 − 0.484843i −0.951057 + 0.309017i −0.951057 − 0.309017i −1.87660 − 1.36343i −2.18132 + 0.491753i
169.14 0.587785 − 0.809017i 1.15178 + 0.374237i −0.309017 − 0.951057i −0.234740 − 2.22371i 0.979765 − 0.711841i −0.951057 + 0.309017i −0.951057 − 0.309017i −1.24050 − 0.901277i −1.93700 − 1.11716i
169.15 0.587785 − 0.809017i 2.24732 + 0.730198i −0.309017 − 0.951057i 1.72348 − 1.42465i 1.91168 − 1.38892i −0.951057 + 0.309017i −0.951057 − 0.309017i 2.09020 + 1.51862i −0.139530 − 2.23171i
169.16 0.587785 − 0.809017i 3.11459 + 1.01199i −0.309017 − 0.951057i −0.458356 + 2.18859i 2.64943 − 1.92492i −0.951057 + 0.309017i −0.951057 − 0.309017i 6.24948 + 4.54051i 1.50119 + 1.65724i
379.1 −0.951057 + 0.309017i −1.93699 + 2.66604i 0.809017 − 0.587785i 2.23336 − 0.109978i 1.01834 − 3.13412i −0.587785 − 0.809017i −0.587785 + 0.809017i −2.42879 − 7.47504i −2.09007 + 0.794742i
379.2 −0.951057 + 0.309017i −1.72160 + 2.36958i 0.809017 − 0.587785i −2.03284 − 0.931427i 0.905097 − 2.78560i −0.587785 − 0.809017i −0.587785 + 0.809017i −1.72394 − 5.30574i 2.22117 + 0.257657i
379.3 −0.951057 + 0.309017i −1.26251 + 1.73770i 0.809017 − 0.587785i −1.53795 + 1.62318i 0.663741 − 2.04278i −0.587785 − 0.809017i −0.587785 + 0.809017i −0.498605 − 1.53455i 0.961088 − 2.01899i
379.4 −0.951057 + 0.309017i −0.670045 + 0.922238i 0.809017 − 0.587785i 0.666104 − 2.13455i 0.352264 − 1.08416i −0.587785 − 0.809017i −0.587785 + 0.809017i 0.525488 + 1.61729i 0.0261093 + 2.23592i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.ba.b ✓ 64
5.b even 2 1 inner 770.2.ba.b ✓ 64
11.c even 5 1 inner 770.2.ba.b ✓ 64
55.j even 10 1 inner 770.2.ba.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.ba.b ✓ 64 1.a even 1 1 trivial
770.2.ba.b ✓ 64 5.b even 2 1 inner
770.2.ba.b ✓ 64 11.c even 5 1 inner
770.2.ba.b ✓ 64 55.j even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 30 T_{3}^{62} + 595 T_{3}^{60} - 9690 T_{3}^{58} + 138835 T_{3}^{56} - 1637806 T_{3}^{54} + \cdots + 3574462890625 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display