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Properties

Label 770.2.y.b
Level $770$
Weight $2$
Character orbit 770.y
Analytic conductor $6.148$
Analytic rank $0$
Dimension $64$
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] = [770,2,Mod(41,770)]
 
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([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.y (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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 4 q^{6} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{4} + 4 q^{6} + 22 q^{9} + 64 q^{10} + 12 q^{13} + 6 q^{14} + 6 q^{15} - 16 q^{16} - 16 q^{17} - 20 q^{19} - 24 q^{21} - 4 q^{22} + 24 q^{23} - 4 q^{24} + 16 q^{25} - 10 q^{26} + 20 q^{29} + 16 q^{33} + 4 q^{35} - 12 q^{36} + 8 q^{37} + 10 q^{39} + 16 q^{40} + 2 q^{41} + 28 q^{42} - 60 q^{47} + 40 q^{49} + 30 q^{51} + 8 q^{52} + 24 q^{53} + 32 q^{54} + 4 q^{55} - 6 q^{56} + 8 q^{58} + 4 q^{60} + 4 q^{61} - 32 q^{62} - 16 q^{63} + 16 q^{64} + 76 q^{66} - 32 q^{67} + 16 q^{68} - 76 q^{71} - 40 q^{72} + 12 q^{73} - 40 q^{74} + 20 q^{76} - 64 q^{77} - 60 q^{79} + 6 q^{81} + 20 q^{82} - 32 q^{83} - 26 q^{84} - 8 q^{86} - 64 q^{87} - 16 q^{88} + 22 q^{90} + 62 q^{91} + 36 q^{92} - 76 q^{93} - 56 q^{94} - 6 q^{96} - 80 q^{97} - 32 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.587785 − 0.809017i −2.98727 + 0.970624i −0.309017 + 0.951057i −0.587785 + 0.809017i 2.54113 + 1.84624i −2.31887 − 1.27390i 0.951057 − 0.309017i 5.55464 − 4.03568i 1.00000
41.2 −0.587785 − 0.809017i −2.17094 + 0.705381i −0.309017 + 0.951057i −0.587785 + 0.809017i 1.84671 + 1.34171i 2.27532 + 1.35015i 0.951057 − 0.309017i 1.78836 − 1.29932i 1.00000
41.3 −0.587785 − 0.809017i −0.605754 + 0.196822i −0.309017 + 0.951057i −0.587785 + 0.809017i 0.515285 + 0.374377i 1.86850 − 1.87316i 0.951057 − 0.309017i −2.09885 + 1.52490i 1.00000
41.4 −0.587785 − 0.809017i −0.595210 + 0.193396i −0.309017 + 0.951057i −0.587785 + 0.809017i 0.506316 + 0.367860i −2.61553 − 0.398782i 0.951057 − 0.309017i −2.11018 + 1.53313i 1.00000
41.5 −0.587785 − 0.809017i 0.534460 − 0.173657i −0.309017 + 0.951057i −0.587785 + 0.809017i −0.454639 − 0.330314i 2.55440 − 0.689246i 0.951057 − 0.309017i −2.17156 + 1.57773i 1.00000
41.6 −0.587785 − 0.809017i 0.694373 − 0.225616i −0.309017 + 0.951057i −0.587785 + 0.809017i −0.590669 − 0.429146i −1.24255 + 2.33582i 0.951057 − 0.309017i −1.99580 + 1.45003i 1.00000
41.7 −0.587785 − 0.809017i 1.67490 − 0.544208i −0.309017 + 0.951057i −0.587785 + 0.809017i −1.42476 − 1.03515i −2.11004 − 1.59616i 0.951057 − 0.309017i 0.0820792 − 0.0596340i 1.00000
41.8 −0.587785 − 0.809017i 2.86766 − 0.931758i −0.309017 + 0.951057i −0.587785 + 0.809017i −2.43937 − 1.77231i 0.0499355 + 2.64528i 0.951057 − 0.309017i 4.92823 − 3.58057i 1.00000
41.9 0.587785 + 0.809017i −2.97250 + 0.965824i −0.309017 + 0.951057i 0.587785 − 0.809017i −2.52856 − 1.83711i 1.11826 + 2.39781i −0.951057 + 0.309017i 5.47590 − 3.97847i 1.00000
41.10 0.587785 + 0.809017i −2.19924 + 0.714577i −0.309017 + 0.951057i 0.587785 − 0.809017i −1.87079 − 1.35921i 2.46081 − 0.971816i −0.951057 + 0.309017i 1.89900 − 1.37970i 1.00000
41.11 0.587785 + 0.809017i −1.23892 + 0.402550i −0.309017 + 0.951057i 0.587785 − 0.809017i −1.05389 − 0.765695i −1.80532 − 1.93412i −0.951057 + 0.309017i −1.05417 + 0.765901i 1.00000
41.12 0.587785 + 0.809017i −0.822856 + 0.267362i −0.309017 + 0.951057i 0.587785 − 0.809017i −0.699963 − 0.508553i 1.95014 − 1.78801i −0.951057 + 0.309017i −1.82144 + 1.32336i 1.00000
41.13 0.587785 + 0.809017i 0.440669 − 0.143182i −0.309017 + 0.951057i 0.587785 − 0.809017i 0.374855 + 0.272348i −2.38551 + 1.14426i −0.951057 + 0.309017i −2.25336 + 1.63716i 1.00000
41.14 0.587785 + 0.809017i 2.01943 − 0.656153i −0.309017 + 0.951057i 0.587785 − 0.809017i 1.71783 + 1.24808i −0.588220 − 2.57953i −0.951057 + 0.309017i 1.22051 − 0.886753i 1.00000
41.15 0.587785 + 0.809017i 2.42269 − 0.787179i −0.309017 + 0.951057i 0.587785 − 0.809017i 2.06086 + 1.49730i 2.39333 + 1.12781i −0.951057 + 0.309017i 2.82272 − 2.05082i 1.00000
41.16 0.587785 + 0.809017i 2.93852 − 0.954783i −0.309017 + 0.951057i 0.587785 − 0.809017i 2.49965 + 1.81610i −1.60465 + 2.10360i −0.951057 + 0.309017i 5.29623 − 3.84794i 1.00000
321.1 −0.951057 − 0.309017i −1.35126 − 1.85985i 0.809017 + 0.587785i −0.951057 + 0.309017i 0.710398 + 2.18638i 2.24806 − 1.39508i −0.587785 − 0.809017i −0.706079 + 2.17309i 1.00000
321.2 −0.951057 − 0.309017i −1.34071 − 1.84532i 0.809017 + 0.587785i −0.951057 + 0.309017i 0.704851 + 2.16931i 2.51081 + 0.834169i −0.587785 − 0.809017i −0.680674 + 2.09490i 1.00000
321.3 −0.951057 − 0.309017i −1.13356 − 1.56021i 0.809017 + 0.587785i −0.951057 + 0.309017i 0.595948 + 1.83414i −2.62185 − 0.354851i −0.587785 − 0.809017i −0.222252 + 0.684022i 1.00000
321.4 −0.951057 − 0.309017i 0.0123397 + 0.0169841i 0.809017 + 0.587785i −0.951057 + 0.309017i −0.00648734 − 0.0199660i −2.40924 − 1.09343i −0.587785 − 0.809017i 0.926915 − 2.85275i 1.00000
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.l 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.y.b yes 64
7.b odd 2 1 770.2.y.a ✓ 64
11.d odd 10 1 770.2.y.a ✓ 64
77.l even 10 1 inner 770.2.y.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.y.a ✓ 64 7.b odd 2 1
770.2.y.a ✓ 64 11.d odd 10 1
770.2.y.b yes 64 1.a even 1 1 trivial
770.2.y.b yes 64 77.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 35 T_{3}^{62} + 710 T_{3}^{60} - 11062 T_{3}^{58} + 180 T_{3}^{57} + 149833 T_{3}^{56} + \cdots + 92416 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display