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Properties

Label 946.2.j.e
Level $946$
Weight $2$
Character orbit 946.j
Analytic conductor $7.554$
Analytic rank $0$
Dimension $54$
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: \(54\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 5 q^{3} - 9 q^{4} + q^{5} + 2 q^{6} + 16 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{2} - 5 q^{3} - 9 q^{4} + q^{5} + 2 q^{6} + 16 q^{7} - 9 q^{8} - 4 q^{9} + q^{10} + 9 q^{11} - 5 q^{12} + 4 q^{13} - 5 q^{14} - 25 q^{15} - 9 q^{16} - 10 q^{17} - 11 q^{18} + 39 q^{19} + q^{20} + 43 q^{21} + 9 q^{22} - 21 q^{23} + 2 q^{24} - 30 q^{25} + 4 q^{26} + 4 q^{27} + 9 q^{28} - 17 q^{29} + 31 q^{30} + q^{31} - 9 q^{32} - 2 q^{33} - 3 q^{34} + 4 q^{35} + 52 q^{36} - 18 q^{37} - 24 q^{38} - 51 q^{39} + q^{40} + 53 q^{41} - 6 q^{42} - 13 q^{43} - 54 q^{44} - 31 q^{45} - 14 q^{46} + 16 q^{47} + 2 q^{48} + 10 q^{49} + 12 q^{50} - 4 q^{51} - 10 q^{52} + 15 q^{53} + 11 q^{54} - q^{55} - 12 q^{56} - 33 q^{57} + 4 q^{58} - 27 q^{59} + 31 q^{60} + 20 q^{61} + 29 q^{62} + 65 q^{63} - 9 q^{64} - 16 q^{65} - 2 q^{66} + 4 q^{67} - 3 q^{68} + 54 q^{69} + 4 q^{70} - 15 q^{71} - 11 q^{72} + q^{73} - 18 q^{74} + 30 q^{75} - 24 q^{76} - 9 q^{77} - 2 q^{78} + 76 q^{79} - 6 q^{80} - 90 q^{81} - 52 q^{82} - 14 q^{83} + 43 q^{84} - 14 q^{85} + 8 q^{86} + 70 q^{87} + 9 q^{88} + 26 q^{89} + 11 q^{90} + 15 q^{91} + 56 q^{92} - 58 q^{93} + 30 q^{94} - 18 q^{95} + 2 q^{96} - 42 q^{97} - 11 q^{98} + 11 q^{99}+O(q^{100}) \) 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 −2.14027 − 2.68381i −0.222521 − 0.974928i −0.693264 − 0.333858i −3.43272 −0.535611 −0.900969 − 0.433884i −1.95453 + 8.56334i −0.693264 + 0.333858i
133.2 0.623490 − 0.781831i −1.12902 − 1.41574i −0.222521 − 0.974928i 2.79610 + 1.34653i −1.81080 −0.714826 −0.900969 − 0.433884i −0.0620861 + 0.272017i 2.79610 − 1.34653i
133.3 0.623490 − 0.781831i −1.00568 − 1.26108i −0.222521 − 0.974928i 0.449257 + 0.216351i −1.61298 −3.19582 −0.900969 − 0.433884i 0.0886272 − 0.388301i 0.449257 − 0.216351i
133.4 0.623490 − 0.781831i −0.544107 − 0.682289i −0.222521 − 0.974928i −3.80968 − 1.83464i −0.872680 2.19146 −0.900969 − 0.433884i 0.498097 − 2.18231i −3.80968 + 1.83464i
133.5 0.623490 − 0.781831i −0.518443 − 0.650107i −0.222521 − 0.974928i 0.185845 + 0.0894982i −0.831518 0.830173 −0.900969 − 0.433884i 0.513707 − 2.25070i 0.185845 − 0.0894982i
133.6 0.623490 − 0.781831i 0.425909 + 0.534072i −0.222521 − 0.974928i 0.0137318 + 0.00661291i 0.683104 4.38471 −0.900969 − 0.433884i 0.563728 − 2.46985i 0.0137318 − 0.00661291i
133.7 0.623490 − 0.781831i 0.912714 + 1.14451i −0.222521 − 0.974928i −1.36295 − 0.656363i 1.46388 −2.05781 −0.900969 − 0.433884i 0.190713 − 0.835568i −1.36295 + 0.656363i
133.8 0.623490 − 0.781831i 1.30510 + 1.63654i −0.222521 − 0.974928i 1.78531 + 0.859762i 2.09322 −1.03917 −0.900969 − 0.433884i −0.307424 + 1.34691i 1.78531 − 0.859762i
133.9 0.623490 − 0.781831i 1.91631 + 2.40298i −0.222521 − 0.974928i 1.53661 + 0.739994i 3.07353 4.04874 −0.900969 − 0.433884i −1.43449 + 6.28493i 1.53661 − 0.739994i
441.1 0.623490 + 0.781831i −2.14027 + 2.68381i −0.222521 + 0.974928i −0.693264 + 0.333858i −3.43272 −0.535611 −0.900969 + 0.433884i −1.95453 − 8.56334i −0.693264 − 0.333858i
441.2 0.623490 + 0.781831i −1.12902 + 1.41574i −0.222521 + 0.974928i 2.79610 − 1.34653i −1.81080 −0.714826 −0.900969 + 0.433884i −0.0620861 − 0.272017i 2.79610 + 1.34653i
441.3 0.623490 + 0.781831i −1.00568 + 1.26108i −0.222521 + 0.974928i 0.449257 − 0.216351i −1.61298 −3.19582 −0.900969 + 0.433884i 0.0886272 + 0.388301i 0.449257 + 0.216351i
441.4 0.623490 + 0.781831i −0.544107 + 0.682289i −0.222521 + 0.974928i −3.80968 + 1.83464i −0.872680 2.19146 −0.900969 + 0.433884i 0.498097 + 2.18231i −3.80968 − 1.83464i
441.5 0.623490 + 0.781831i −0.518443 + 0.650107i −0.222521 + 0.974928i 0.185845 − 0.0894982i −0.831518 0.830173 −0.900969 + 0.433884i 0.513707 + 2.25070i 0.185845 + 0.0894982i
441.6 0.623490 + 0.781831i 0.425909 − 0.534072i −0.222521 + 0.974928i 0.0137318 − 0.00661291i 0.683104 4.38471 −0.900969 + 0.433884i 0.563728 + 2.46985i 0.0137318 + 0.00661291i
441.7 0.623490 + 0.781831i 0.912714 − 1.14451i −0.222521 + 0.974928i −1.36295 + 0.656363i 1.46388 −2.05781 −0.900969 + 0.433884i 0.190713 + 0.835568i −1.36295 − 0.656363i
441.8 0.623490 + 0.781831i 1.30510 − 1.63654i −0.222521 + 0.974928i 1.78531 − 0.859762i 2.09322 −1.03917 −0.900969 + 0.433884i −0.307424 − 1.34691i 1.78531 + 0.859762i
441.9 0.623490 + 0.781831i 1.91631 − 2.40298i −0.222521 + 0.974928i 1.53661 − 0.739994i 3.07353 4.04874 −0.900969 + 0.433884i −1.43449 − 6.28493i 1.53661 + 0.739994i
551.1 −0.222521 + 0.974928i −0.610827 − 2.67621i −0.900969 − 0.433884i 1.53398 − 1.92355i 2.74503 −4.28839 0.623490 − 0.781831i −4.08608 + 1.96775i 1.53398 + 1.92355i
551.2 −0.222521 + 0.974928i −0.583932 − 2.55837i −0.900969 − 0.433884i 0.818315 − 1.02613i 2.62416 2.99829 0.623490 − 0.781831i −3.50138 + 1.68618i 0.818315 + 1.02613i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.9
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.e ✓ 54
43.e even 7 1 inner 946.2.j.e ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.j.e ✓ 54 1.a even 1 1 trivial
946.2.j.e ✓ 54 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{54} + 5 T_{3}^{53} + 28 T_{3}^{52} + 93 T_{3}^{51} + 401 T_{3}^{50} + 1180 T_{3}^{49} + \cdots + 262144 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display