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Properties

Label 770.2.bc.b
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
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 40 q^{11} + 40 q^{15} + 40 q^{16} + 12 q^{17} + 8 q^{18} + 16 q^{21} - 16 q^{25} - 12 q^{30} - 24 q^{31} - 12 q^{35} - 72 q^{36} + 44 q^{37} - 24 q^{38} + 4 q^{42} - 64 q^{43} - 36 q^{45} + 4 q^{46} + 48 q^{47} - 16 q^{50} - 28 q^{51} - 36 q^{52} + 40 q^{53} - 4 q^{56} - 8 q^{57} + 8 q^{58} - 8 q^{60} - 84 q^{61} + 160 q^{63} - 8 q^{65} + 12 q^{66} + 12 q^{68} - 32 q^{70} + 8 q^{72} - 48 q^{73} + 60 q^{75} + 40 q^{78} + 24 q^{82} - 24 q^{85} - 60 q^{86} - 24 q^{87} + 128 q^{91} - 36 q^{93} - 68 q^{95} - 12 q^{96} + 8 q^{98}+O(q^{100}) \) 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 −2.45037 − 0.656574i −0.866025 + 0.500000i 0.00830523 − 2.23605i 2.53681i 0.960558 + 2.46522i 0.707107 + 0.707107i 2.97513 + 1.71769i −2.16201 + 0.570711i
243.2 −0.258819 − 0.965926i −2.27992 − 0.610903i −0.866025 + 0.500000i −1.60395 + 1.55800i 2.36035i −2.28291 + 1.33728i 0.707107 + 0.707107i 2.22675 + 1.28562i 1.92004 + 1.14606i
243.3 −0.258819 − 0.965926i −1.50233 − 0.402548i −0.866025 + 0.500000i 0.106512 − 2.23353i 1.55533i −2.25956 − 1.37638i 0.707107 + 0.707107i −0.503125 − 0.290479i −2.18499 + 0.475197i
243.4 −0.258819 − 0.965926i −0.932340 − 0.249820i −0.866025 + 0.500000i 2.22694 − 0.201809i 0.965229i 1.53896 − 2.15212i 0.707107 + 0.707107i −1.79123 − 1.03417i −0.771308 − 2.09883i
243.5 −0.258819 − 0.965926i −0.0972358 − 0.0260543i −0.866025 + 0.500000i 1.14831 + 1.91869i 0.100666i 1.33903 + 2.28189i 0.707107 + 0.707107i −2.58930 − 1.49493i 1.55611 − 1.60578i
243.6 −0.258819 − 0.965926i 0.601995 + 0.161304i −0.866025 + 0.500000i −0.719798 + 2.11705i − 0.623231i −2.51263 − 0.828679i 0.707107 + 0.707107i −2.26170 − 1.30579i 2.23121 + 0.147340i
243.7 −0.258819 − 0.965926i 1.21500 + 0.325558i −0.866025 + 0.500000i −2.18065 + 0.494758i − 1.25786i −0.384831 − 2.61761i 0.707107 + 0.707107i −1.22784 − 0.708895i 1.04229 + 1.97829i
243.8 −0.258819 − 0.965926i 1.71242 + 0.458843i −0.866025 + 0.500000i 1.18966 − 1.89333i − 1.77283i 2.20200 − 1.46670i 0.707107 + 0.707107i 0.123784 + 0.0714667i −2.13673 − 0.659090i
243.9 −0.258819 − 0.965926i 2.58436 + 0.692477i −0.866025 + 0.500000i 1.99714 + 1.00569i − 2.67552i −2.15241 + 1.53855i 0.707107 + 0.707107i 3.60131 + 2.07922i 0.454526 − 2.18938i
243.10 −0.258819 − 0.965926i 2.82145 + 0.756005i −0.866025 + 0.500000i −2.17248 − 0.529460i − 2.92098i 2.58587 + 0.559726i 0.707107 + 0.707107i 4.79095 + 2.76606i 0.0508601 + 2.23549i
243.11 0.258819 + 0.965926i −3.27757 − 0.878223i −0.866025 + 0.500000i −1.90044 + 1.17828i − 3.39319i 2.42459 + 1.05893i −0.707107 − 0.707107i 7.37313 + 4.25688i −1.63000 − 1.53072i
243.12 0.258819 + 0.965926i −1.80035 − 0.482404i −0.866025 + 0.500000i 2.14684 − 0.625373i − 1.86386i 2.62623 + 0.320790i −0.707107 − 0.707107i 0.410488 + 0.236995i 1.15971 + 1.91183i
243.13 0.258819 + 0.965926i −1.78700 − 0.478824i −0.866025 + 0.500000i −2.23596 − 0.0221814i − 1.85003i −2.63406 + 0.248471i −0.707107 − 0.707107i 0.366003 + 0.211312i −0.557283 − 2.16551i
243.14 0.258819 + 0.965926i −1.42493 − 0.381809i −0.866025 + 0.500000i −0.366022 + 2.20591i − 1.47520i −0.224782 − 2.63619i −0.707107 − 0.707107i −0.713430 − 0.411899i −2.22548 + 0.217380i
243.15 0.258819 + 0.965926i −1.03899 − 0.278396i −0.866025 + 0.500000i −1.00597 − 1.99701i − 1.07564i 1.62427 − 2.08848i −0.707107 − 0.707107i −1.59608 − 0.921498i 1.66860 − 1.48855i
243.16 0.258819 + 0.965926i 0.0856795 + 0.0229577i −0.866025 + 0.500000i 2.13253 + 0.672552i 0.0887019i −2.64574 − 0.00612082i −0.707107 − 0.707107i −2.59126 − 1.49607i −0.0976965 + 2.23393i
243.17 0.258819 + 0.965926i 0.150223 + 0.0402521i −0.866025 + 0.500000i 0.212336 − 2.22596i 0.155522i −1.91229 + 1.82843i −0.707107 − 0.707107i −2.57713 − 1.48791i 2.20507 − 0.371021i
243.18 0.258819 + 0.965926i 2.02820 + 0.543454i −0.866025 + 0.500000i −0.830326 + 2.07619i 2.09974i −0.685308 + 2.55546i −0.707107 − 0.707107i 1.22017 + 0.704464i −2.22035 − 0.264676i
243.19 0.258819 + 0.965926i 2.52168 + 0.675681i −0.866025 + 0.500000i 2.01604 + 0.967265i 2.61063i 0.291362 − 2.62966i −0.707107 − 0.707107i 3.30424 + 1.90770i −0.412518 + 2.19769i
243.20 0.258819 + 0.965926i 2.87003 + 0.769022i −0.866025 + 0.500000i −0.169030 − 2.22967i 2.97127i 2.10165 + 1.60719i −0.707107 − 0.707107i 5.04760 + 2.91423i 2.10995 − 0.740351i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 243.20
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} - 261 T_{3}^{76} - 36 T_{3}^{75} - 192 T_{3}^{73} + 43369 T_{3}^{72} + 9396 T_{3}^{71} + \cdots + 2825761 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display