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Properties

Label 946.2.j.d
Level $946$
Weight $2$
Character orbit 946.j
Analytic conductor $7.554$
Analytic rank $0$
Dimension $48$
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] = [946,2,Mod(133,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.j (of order \(7\), degree \(6\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{2} - 3 q^{3} - 8 q^{4} + q^{5} - 10 q^{6} - 4 q^{7} - 8 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{2} - 3 q^{3} - 8 q^{4} + q^{5} - 10 q^{6} - 4 q^{7} - 8 q^{8} - 19 q^{9} + q^{10} - 8 q^{11} - 3 q^{12} + 5 q^{13} + 3 q^{14} - q^{15} - 8 q^{16} + 4 q^{17} + 2 q^{18} - 9 q^{19} + q^{20} - 3 q^{21} - 8 q^{22} - 8 q^{23} + 4 q^{24} + 9 q^{25} + 5 q^{26} + 18 q^{27} + 3 q^{28} + 9 q^{29} - 15 q^{30} - 15 q^{31} - 8 q^{32} + 4 q^{33} + 11 q^{34} + 44 q^{35} + 30 q^{36} + 34 q^{37} + 12 q^{38} - 7 q^{39} + q^{40} - 23 q^{41} - 10 q^{42} + 17 q^{43} + 48 q^{44} + 25 q^{45} - 8 q^{46} + 32 q^{47} + 4 q^{48} - 44 q^{49} + 2 q^{50} + 30 q^{51} - 2 q^{52} - 39 q^{53} + 25 q^{54} + q^{55} - 4 q^{56} + 75 q^{57} + 30 q^{58} - 7 q^{59} - 15 q^{60} + 46 q^{61} + 6 q^{62} - 5 q^{63} - 8 q^{64} + 12 q^{65} + 4 q^{66} + 44 q^{67} + 11 q^{68} - 32 q^{69} + 44 q^{70} + 2 q^{72} - 19 q^{73} + 6 q^{74} + 2 q^{75} + 12 q^{76} + 3 q^{77} + 42 q^{78} - 92 q^{79} - 6 q^{80} + 95 q^{81} - 2 q^{82} + 44 q^{83} - 3 q^{84} + 130 q^{85} - 53 q^{86} - 114 q^{87} - 8 q^{88} + 12 q^{89} + 67 q^{90} - 9 q^{91} - 22 q^{92} + 26 q^{93} - 3 q^{94} + 28 q^{95} + 4 q^{96} + 24 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 0.623490 − 0.781831i −1.89751 − 2.37940i −0.222521 − 0.974928i 3.20746 + 1.54463i −3.04337 2.04561 −0.900969 − 0.433884i −1.39345 + 6.10511i 3.20746 − 1.54463i
133.2 0.623490 − 0.781831i −1.51352 − 1.89790i −0.222521 − 0.974928i −1.86095 − 0.896187i −2.42750 −2.37916 −0.900969 − 0.433884i −0.643704 + 2.82025i −1.86095 + 0.896187i
133.3 0.623490 − 0.781831i −1.43290 − 1.79680i −0.222521 − 0.974928i 0.581709 + 0.280137i −2.29820 1.55199 −0.900969 − 0.433884i −0.507728 + 2.22450i 0.581709 − 0.280137i
133.4 0.623490 − 0.781831i −0.210429 − 0.263870i −0.222521 − 0.974928i −2.54265 − 1.22448i −0.337502 −3.55976 −0.900969 − 0.433884i 0.642216 − 2.81373i −2.54265 + 1.22448i
133.5 0.623490 − 0.781831i 0.171525 + 0.215085i −0.222521 − 0.974928i 2.85852 + 1.37659i 0.275104 3.47659 −0.900969 − 0.433884i 0.650722 − 2.85100i 2.85852 − 1.37659i
133.6 0.623490 − 0.781831i 0.374201 + 0.469233i −0.222521 − 0.974928i 0.192689 + 0.0927944i 0.600172 −1.73386 −0.900969 − 0.433884i 0.587409 − 2.57361i 0.192689 − 0.0927944i
133.7 0.623490 − 0.781831i 0.959339 + 1.20297i −0.222521 − 0.974928i −2.40558 − 1.15847i 1.53866 1.16727 −0.900969 − 0.433884i 0.140750 − 0.616664i −2.40558 + 1.15847i
133.8 0.623490 − 0.781831i 1.52484 + 1.91209i −0.222521 − 0.974928i 0.869777 + 0.418862i 2.44566 0.233254 −0.900969 − 0.433884i −0.663390 + 2.90650i 0.869777 − 0.418862i
441.1 0.623490 + 0.781831i −1.89751 + 2.37940i −0.222521 + 0.974928i 3.20746 − 1.54463i −3.04337 2.04561 −0.900969 + 0.433884i −1.39345 − 6.10511i 3.20746 + 1.54463i
441.2 0.623490 + 0.781831i −1.51352 + 1.89790i −0.222521 + 0.974928i −1.86095 + 0.896187i −2.42750 −2.37916 −0.900969 + 0.433884i −0.643704 − 2.82025i −1.86095 − 0.896187i
441.3 0.623490 + 0.781831i −1.43290 + 1.79680i −0.222521 + 0.974928i 0.581709 − 0.280137i −2.29820 1.55199 −0.900969 + 0.433884i −0.507728 − 2.22450i 0.581709 + 0.280137i
441.4 0.623490 + 0.781831i −0.210429 + 0.263870i −0.222521 + 0.974928i −2.54265 + 1.22448i −0.337502 −3.55976 −0.900969 + 0.433884i 0.642216 + 2.81373i −2.54265 − 1.22448i
441.5 0.623490 + 0.781831i 0.171525 − 0.215085i −0.222521 + 0.974928i 2.85852 − 1.37659i 0.275104 3.47659 −0.900969 + 0.433884i 0.650722 + 2.85100i 2.85852 + 1.37659i
441.6 0.623490 + 0.781831i 0.374201 − 0.469233i −0.222521 + 0.974928i 0.192689 − 0.0927944i 0.600172 −1.73386 −0.900969 + 0.433884i 0.587409 + 2.57361i 0.192689 + 0.0927944i
441.7 0.623490 + 0.781831i 0.959339 − 1.20297i −0.222521 + 0.974928i −2.40558 + 1.15847i 1.53866 1.16727 −0.900969 + 0.433884i 0.140750 + 0.616664i −2.40558 − 1.15847i
441.8 0.623490 + 0.781831i 1.52484 − 1.91209i −0.222521 + 0.974928i 0.869777 − 0.418862i 2.44566 0.233254 −0.900969 + 0.433884i −0.663390 − 2.90650i 0.869777 + 0.418862i
551.1 −0.222521 + 0.974928i −0.721226 − 3.15990i −0.900969 − 0.433884i −1.55454 + 1.94933i 3.24116 −2.11728 0.623490 − 0.781831i −6.76188 + 3.25635i −1.55454 − 1.94933i
551.2 −0.222521 + 0.974928i −0.342611 − 1.50108i −0.900969 − 0.433884i −0.884852 + 1.10957i 1.53968 −0.629341 0.623490 − 0.781831i 0.567051 − 0.273078i −0.884852 − 1.10957i
551.3 −0.222521 + 0.974928i −0.167745 − 0.734939i −0.900969 − 0.433884i 0.104994 − 0.131658i 0.753839 4.58494 0.623490 − 0.781831i 2.19091 − 1.05509i 0.104994 + 0.131658i
551.4 −0.222521 + 0.974928i −0.134822 − 0.590695i −0.900969 − 0.433884i 2.67879 − 3.35910i 0.605886 −0.882038 0.623490 − 0.781831i 2.37216 − 1.14237i 2.67879 + 3.35910i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.j.d ✓ 48
43.e even 7 1 inner 946.2.j.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.j.d ✓ 48 1.a even 1 1 trivial
946.2.j.d ✓ 48 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 3 T_{3}^{47} + 26 T_{3}^{46} + 55 T_{3}^{45} + 297 T_{3}^{44} + 370 T_{3}^{43} + \cdots + 262144 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display