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Properties

Label 770.2.bc.a
Level $770$
Weight $2$
Character orbit 770.bc
Analytic conductor $6.148$
Analytic rank $0$
Dimension $80$
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(243,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 10, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.243");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bc (of order \(12\), 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: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 40 q^{11} - 88 q^{15} + 40 q^{16} + 60 q^{17} + 8 q^{18} + 40 q^{21} + 16 q^{25} - 12 q^{30} - 24 q^{31} + 36 q^{35} - 72 q^{36} - 44 q^{37} - 24 q^{38} - 52 q^{42} - 36 q^{45} + 20 q^{46} - 48 q^{47} + 16 q^{50} + 28 q^{51} + 36 q^{52} - 24 q^{53} - 4 q^{56} + 88 q^{57} + 24 q^{58} - 8 q^{60} + 108 q^{61} - 24 q^{63} - 8 q^{65} - 12 q^{66} - 16 q^{67} + 60 q^{68} + 16 q^{70} - 64 q^{71} + 8 q^{72} - 48 q^{73} - 108 q^{75} - 72 q^{78} + 64 q^{81} + 24 q^{82} - 72 q^{85} + 36 q^{86} + 96 q^{87} - 64 q^{91} + 28 q^{93} + 52 q^{95} - 12 q^{96} - 120 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
243.1 −0.258819 − 0.965926i −2.68132 − 0.718458i −0.866025 + 0.500000i 1.99691 − 1.00617i 2.77591i 2.58382 + 0.569079i 0.707107 + 0.707107i 4.07523 + 2.35283i −1.48872 − 1.66845i
243.2 −0.258819 − 0.965926i −1.70898 − 0.457920i −0.866025 + 0.500000i 0.926643 + 2.03503i 1.76927i −1.32061 − 2.29259i 0.707107 + 0.707107i 0.112848 + 0.0651528i 1.72585 − 1.42177i
243.3 −0.258819 − 0.965926i −1.42066 − 0.380664i −0.866025 + 0.500000i 2.23463 + 0.0802111i 1.47077i −1.20499 + 2.35542i 0.707107 + 0.707107i −0.724709 − 0.418411i −0.500887 − 2.17925i
243.4 −0.258819 − 0.965926i −0.732843 − 0.196365i −0.866025 + 0.500000i −2.03497 − 0.926773i 0.758695i −2.23125 + 1.42181i 0.707107 + 0.707107i −2.09958 − 1.21219i −0.368506 + 2.20549i
243.5 −0.258819 − 0.965926i −0.393765 − 0.105509i −0.866025 + 0.500000i −1.67163 − 1.48515i 0.407655i 2.56401 + 0.652575i 0.707107 + 0.707107i −2.45416 − 1.41691i −1.00189 + 1.99905i
243.6 −0.258819 − 0.965926i −0.0758760 − 0.0203309i −0.866025 + 0.500000i −1.36393 + 1.77192i 0.0785526i 2.43596 − 1.03252i 0.707107 + 0.707107i −2.59273 − 1.49691i 2.06455 + 0.858850i
243.7 −0.258819 − 0.965926i 0.961711 + 0.257690i −0.866025 + 0.500000i 1.58006 − 1.58222i − 0.995636i −2.53295 − 0.764309i 0.707107 + 0.707107i −1.73959 − 1.00435i −1.93726 − 1.11671i
243.8 −0.258819 − 0.965926i 2.43322 + 0.651979i −0.866025 + 0.500000i −1.86295 + 1.23669i − 2.51905i −1.49733 + 2.18129i 0.707107 + 0.707107i 2.89740 + 1.67281i 1.67672 + 1.47939i
243.9 −0.258819 − 0.965926i 2.45006 + 0.656490i −0.866025 + 0.500000i 1.29165 + 1.82528i − 2.53648i 2.15834 − 1.53021i 0.707107 + 0.707107i 2.97372 + 1.71688i 1.42878 − 1.72005i
243.10 −0.258819 − 0.965926i 2.84149 + 0.761376i −0.866025 + 0.500000i −1.09640 − 1.94882i − 2.94173i −1.92093 − 1.81935i 0.707107 + 0.707107i 4.89632 + 2.82689i −1.59865 + 1.56344i
243.11 0.258819 + 0.965926i −2.96319 − 0.793984i −0.866025 + 0.500000i 1.74042 + 1.40390i − 3.06772i −0.365110 − 2.62044i −0.707107 − 0.707107i 5.55200 + 3.20545i −0.905613 + 2.04447i
243.12 0.258819 + 0.965926i −2.58758 − 0.693341i −0.866025 + 0.500000i −1.24620 − 1.85660i − 2.67887i −2.46128 − 0.970627i −0.707107 − 0.707107i 3.61680 + 2.08816i 1.47080 − 1.68427i
243.13 0.258819 + 0.965926i −2.29840 − 0.615855i −0.866025 + 0.500000i 2.23602 − 0.0147898i − 2.37948i −0.0544853 + 2.64519i −0.707107 − 0.707107i 2.30530 + 1.33097i 0.593010 + 2.15600i
243.14 0.258819 + 0.965926i −0.993951 − 0.266328i −0.866025 + 0.500000i −2.16376 + 0.564048i − 1.02901i 0.573600 + 2.58282i −0.707107 − 0.707107i −1.68107 − 0.970565i −1.10485 − 1.94404i
243.15 0.258819 + 0.965926i −0.619494 − 0.165993i −0.866025 + 0.500000i 0.517292 − 2.17541i − 0.641347i 0.254577 + 2.63347i −0.707107 − 0.707107i −2.24186 − 1.29434i 2.23517 − 0.0633714i
243.16 0.258819 + 0.965926i 0.559722 + 0.149977i −0.866025 + 0.500000i 1.57371 − 1.58853i 0.579467i −0.0519464 − 2.64524i −0.707107 − 0.707107i −2.30728 − 1.33211i 1.94171 + 1.10895i
243.17 0.258819 + 0.965926i 0.860569 + 0.230589i −0.866025 + 0.500000i −0.985663 + 2.00710i 0.890926i −2.60577 − 0.458202i −0.707107 − 0.707107i −1.91067 − 1.10313i −2.19382 − 0.432600i
243.18 0.258819 + 0.965926i 1.09220 + 0.292653i −0.866025 + 0.500000i −1.95788 − 1.08015i 1.13073i 2.53750 − 0.749049i −0.707107 − 0.707107i −1.49083 − 0.860730i 0.536604 − 2.17073i
243.19 0.258819 + 0.965926i 2.03632 + 0.545631i −0.866025 + 0.500000i 1.89478 + 1.18735i 2.10816i 1.67718 + 2.04623i −0.707107 − 0.707107i 1.25083 + 0.722165i −0.656490 + 2.13753i
243.20 0.258819 + 0.965926i 3.24078 + 0.868363i −0.866025 + 0.500000i −1.60872 + 1.55307i 3.35510i 1.46166 − 2.20535i −0.707107 − 0.707107i 7.15050 + 4.12834i −1.91652 − 1.15194i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 243.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bc.a ✓ 80
5.c odd 4 1 inner 770.2.bc.a ✓ 80
7.d odd 6 1 inner 770.2.bc.a ✓ 80
35.k even 12 1 inner 770.2.bc.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bc.a ✓ 80 1.a even 1 1 trivial
770.2.bc.a ✓ 80 5.c odd 4 1 inner
770.2.bc.a ✓ 80 7.d odd 6 1 inner
770.2.bc.a ✓ 80 35.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 277 T_{3}^{76} - 108 T_{3}^{75} + 1008 T_{3}^{73} + 48505 T_{3}^{72} + 29916 T_{3}^{71} + \cdots + 937890625 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display