Properties

Label 770.2.bm.a
Level $770$
Weight $2$
Character orbit 770.bm
Analytic conductor $6.148$
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] = [770,2,Mod(61,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 25, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bm (of order \(30\), degree \(8\), 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: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{4} - 8 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{4} - 8 q^{6} - 22 q^{9} + 64 q^{10} + 6 q^{11} - 24 q^{13} + 4 q^{14} + 12 q^{15} + 16 q^{16} + 38 q^{17} + 10 q^{19} - 24 q^{21} + 4 q^{22} + 12 q^{23} - 4 q^{24} - 16 q^{25} + 20 q^{26} - 20 q^{29} - 2 q^{33} + 4 q^{35} - 24 q^{36} - 56 q^{37} + 12 q^{38} - 10 q^{39} + 16 q^{40} + 14 q^{41} - 60 q^{42} + 30 q^{46} + 42 q^{47} - 20 q^{49} + 60 q^{51} + 8 q^{52} + 12 q^{53} - 40 q^{54} + 4 q^{55} + 4 q^{58} + 60 q^{59} - 4 q^{60} + 4 q^{61} - 32 q^{62} - 36 q^{63} + 32 q^{64} - 30 q^{65} - 20 q^{66} - 16 q^{67} - 38 q^{68} + 64 q^{71} - 20 q^{72} - 78 q^{73} - 26 q^{74} + 20 q^{76} - 52 q^{77} - 96 q^{78} - 30 q^{79} + 30 q^{81} - 16 q^{82} - 80 q^{83} - 28 q^{84} + 158 q^{86} - 64 q^{87} + 4 q^{88} - 24 q^{89} - 44 q^{90} + 36 q^{91} - 36 q^{92} + 160 q^{93} + 46 q^{94} - 18 q^{95} - 6 q^{96} - 80 q^{97} + 64 q^{98} - 58 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −0.207912 + 0.978148i −2.43348 0.255769i −0.913545 0.406737i 0.743145 0.669131i 0.756130 2.32713i 0.853903 2.50417i 0.587785 0.809017i 2.92198 + 0.621086i 0.500000 + 0.866025i
61.2 −0.207912 + 0.978148i −2.03577 0.213968i −0.913545 0.406737i 0.743145 0.669131i 0.632553 1.94680i −2.45457 0.987455i 0.587785 0.809017i 1.16414 + 0.247446i 0.500000 + 0.866025i
61.3 −0.207912 + 0.978148i −1.02559 0.107794i −0.913545 0.406737i 0.743145 0.669131i 0.318671 0.980768i −2.44427 + 1.01269i 0.587785 0.809017i −1.89422 0.402630i 0.500000 + 0.866025i
61.4 −0.207912 + 0.978148i −0.303142 0.0318615i −0.913545 0.406737i 0.743145 0.669131i 0.0941921 0.289894i 0.780969 + 2.52786i 0.587785 0.809017i −2.84356 0.604418i 0.500000 + 0.866025i
61.5 −0.207912 + 0.978148i 0.647100 + 0.0680129i −0.913545 0.406737i 0.743145 0.669131i −0.201066 + 0.618819i 2.16378 1.52252i 0.587785 0.809017i −2.52033 0.535713i 0.500000 + 0.866025i
61.6 −0.207912 + 0.978148i 1.85945 + 0.195436i −0.913545 0.406737i 0.743145 0.669131i −0.577768 + 1.77819i −1.49259 + 2.18453i 0.587785 0.809017i 0.484926 + 0.103074i 0.500000 + 0.866025i
61.7 −0.207912 + 0.978148i 2.14847 + 0.225813i −0.913545 0.406737i 0.743145 0.669131i −0.667570 + 2.05457i −1.22395 2.34562i 0.587785 0.809017i 1.63048 + 0.346569i 0.500000 + 0.866025i
61.8 −0.207912 + 0.978148i 2.75214 + 0.289261i −0.913545 0.406737i 0.743145 0.669131i −0.855142 + 2.63186i 2.25103 + 1.39027i 0.587785 0.809017i 4.55615 + 0.968440i 0.500000 + 0.866025i
61.9 0.207912 0.978148i −3.23113 0.339605i −0.913545 0.406737i −0.743145 + 0.669131i −1.00397 + 3.08991i 1.62925 2.08460i −0.587785 + 0.809017i 7.39042 + 1.57088i 0.500000 + 0.866025i
61.10 0.207912 0.978148i −2.32343 0.244202i −0.913545 0.406737i −0.743145 + 0.669131i −0.721933 + 2.22188i 0.323542 + 2.62589i −0.587785 + 0.809017i 2.40424 + 0.511036i 0.500000 + 0.866025i
61.11 0.207912 0.978148i −2.06773 0.217327i −0.913545 0.406737i −0.743145 + 0.669131i −0.642483 + 1.97736i 0.222301 2.63640i −0.587785 + 0.809017i 1.29383 + 0.275013i 0.500000 + 0.866025i
61.12 0.207912 0.978148i −0.299068 0.0314333i −0.913545 0.406737i −0.743145 + 0.669131i −0.0929262 + 0.285997i −2.61672 0.390839i −0.587785 + 0.809017i −2.84599 0.604934i 0.500000 + 0.866025i
61.13 0.207912 0.978148i −0.0646054 0.00679031i −0.913545 0.406737i −0.743145 + 0.669131i −0.0200742 + 0.0617819i 2.36588 + 1.18432i −0.587785 + 0.809017i −2.93032 0.622858i 0.500000 + 0.866025i
61.14 0.207912 0.978148i 1.36793 + 0.143775i −0.913545 0.406737i −0.743145 + 0.669131i 0.425042 1.30815i −2.53657 + 0.752196i −0.587785 + 0.809017i −1.08388 0.230385i 0.500000 + 0.866025i
61.15 0.207912 0.978148i 2.20564 + 0.231822i −0.913545 0.406737i −0.743145 + 0.669131i 0.685335 2.10925i 1.90345 1.83762i −0.587785 + 0.809017i 1.87667 + 0.398900i 0.500000 + 0.866025i
61.16 0.207912 0.978148i 2.80322 + 0.294630i −0.913545 0.406737i −0.743145 + 0.669131i 0.871013 2.68070i 0.274572 + 2.63147i −0.587785 + 0.809017i 4.83677 + 1.02809i 0.500000 + 0.866025i
101.1 −0.207912 0.978148i −2.43348 + 0.255769i −0.913545 + 0.406737i 0.743145 + 0.669131i 0.756130 + 2.32713i 0.853903 + 2.50417i 0.587785 + 0.809017i 2.92198 0.621086i 0.500000 0.866025i
101.2 −0.207912 0.978148i −2.03577 + 0.213968i −0.913545 + 0.406737i 0.743145 + 0.669131i 0.632553 + 1.94680i −2.45457 + 0.987455i 0.587785 + 0.809017i 1.16414 0.247446i 0.500000 0.866025i
101.3 −0.207912 0.978148i −1.02559 + 0.107794i −0.913545 + 0.406737i 0.743145 + 0.669131i 0.318671 + 0.980768i −2.44427 1.01269i 0.587785 + 0.809017i −1.89422 + 0.402630i 0.500000 0.866025i
101.4 −0.207912 0.978148i −0.303142 + 0.0318615i −0.913545 + 0.406737i 0.743145 + 0.669131i 0.0941921 + 0.289894i 0.780969 2.52786i 0.587785 + 0.809017i −2.84356 + 0.604418i 0.500000 0.866025i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bm.a 128
7.d odd 6 1 770.2.bm.b yes 128
11.d odd 10 1 770.2.bm.b yes 128
77.n even 30 1 inner 770.2.bm.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bm.a 128 1.a even 1 1 trivial
770.2.bm.a 128 77.n even 30 1 inner
770.2.bm.b yes 128 7.d odd 6 1
770.2.bm.b yes 128 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{128} + 35 T_{3}^{126} + 506 T_{3}^{124} - 72 T_{3}^{123} + 2243 T_{3}^{122} - 1440 T_{3}^{121} - 36362 T_{3}^{120} + 4548 T_{3}^{119} - 800392 T_{3}^{118} + 582418 T_{3}^{117} - 7613782 T_{3}^{116} + \cdots + 38\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display