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Properties

Label 381.2.i.a
Level $381$
Weight $2$
Character orbit 381.i
Analytic conductor $3.042$
Analytic rank $0$
Dimension $60$
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] = [381,2,Mod(4,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.i (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: \(3.04230031701\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 10 q^{3} + 4 q^{5} + q^{6} - 14 q^{7} + 4 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 10 q^{3} + 4 q^{5} + q^{6} - 14 q^{7} + 4 q^{8} - 10 q^{9} - 10 q^{10} + 9 q^{11} + 21 q^{12} + 2 q^{13} - q^{14} - 4 q^{15} + 6 q^{16} + 2 q^{17} - q^{18} + 18 q^{19} + 6 q^{20} + 14 q^{21} - 46 q^{22} - 20 q^{23} + 24 q^{24} + 2 q^{25} + 18 q^{26} + 10 q^{27} + 70 q^{28} - 34 q^{29} - 11 q^{30} + 6 q^{31} - 5 q^{32} + 12 q^{33} + 67 q^{34} - 19 q^{35} - 7 q^{36} - 78 q^{37} + 22 q^{38} - 2 q^{39} - 100 q^{40} - 23 q^{41} + 8 q^{42} - 13 q^{43} - 56 q^{44} + 4 q^{45} + 14 q^{46} - 9 q^{47} + 29 q^{48} - 4 q^{49} - 19 q^{50} - 2 q^{51} + 30 q^{52} - 30 q^{53} - 6 q^{54} - 24 q^{55} - 5 q^{56} - 11 q^{57} - 12 q^{58} - 32 q^{59} + 15 q^{60} + 35 q^{61} + 9 q^{62} - 8 q^{64} - 12 q^{65} - 24 q^{66} - 2 q^{67} - 4 q^{68} - 15 q^{69} + 6 q^{70} + 23 q^{71} + 4 q^{72} - 5 q^{73} - 17 q^{74} - 44 q^{75} + 101 q^{76} - 27 q^{77} + 10 q^{78} + 19 q^{79} + 33 q^{80} - 10 q^{81} - 70 q^{82} + 25 q^{83} + 32 q^{85} + 10 q^{86} - 22 q^{87} - 10 q^{88} - 52 q^{89} + 46 q^{90} + 6 q^{91} - 26 q^{92} + q^{93} + 12 q^{94} + 15 q^{95} + 5 q^{96} - 22 q^{97} - 131 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.44505 − 1.81204i 0.900969 + 0.433884i −0.750261 + 3.28711i −0.754129 − 3.30405i −0.515732 − 2.25957i −0.449612 − 1.96988i 2.86421 − 1.37933i 0.623490 + 0.781831i −4.89731 + 6.14103i
4.2 −1.36377 − 1.71011i 0.900969 + 0.433884i −0.619571 + 2.71452i 0.445584 + 1.95223i −0.486722 − 2.13247i −0.191093 − 0.837234i 1.54567 − 0.744354i 0.623490 + 0.781831i 2.73085 − 3.42438i
4.3 −0.788412 − 0.988638i 0.900969 + 0.433884i 0.0892312 − 0.390948i 0.287467 + 1.25948i −0.281381 − 1.23281i 0.875265 + 3.83479i −2.73543 + 1.31732i 0.623490 + 0.781831i 1.01852 − 1.27719i
4.4 −0.703755 − 0.882481i 0.900969 + 0.433884i 0.161540 − 0.707755i 0.256086 + 1.12198i −0.251167 − 1.10044i −0.886943 − 3.88595i −2.77218 + 1.33501i 0.623490 + 0.781831i 0.809908 − 1.01559i
4.5 −0.111953 − 0.140384i 0.900969 + 0.433884i 0.437868 − 1.91842i −0.531331 − 2.32792i −0.0399555 − 0.175056i −0.166781 − 0.730713i −0.641890 + 0.309118i 0.623490 + 0.781831i −0.267319 + 0.335207i
4.6 0.212206 + 0.266098i 0.900969 + 0.433884i 0.419265 − 1.83692i −0.0135529 − 0.0593793i 0.0757355 + 0.331819i 0.211225 + 0.925439i 1.19107 − 0.573587i 0.623490 + 0.781831i 0.0129247 − 0.0162071i
4.7 0.763777 + 0.957746i 0.900969 + 0.433884i 0.111120 − 0.486847i 0.716155 + 3.13768i 0.272589 + 1.19429i −0.281766 − 1.23450i 2.75853 − 1.32844i 0.623490 + 0.781831i −2.45812 + 3.08238i
4.8 1.05062 + 1.31744i 0.900969 + 0.433884i −0.186794 + 0.818396i −0.847867 − 3.71475i 0.374962 + 1.64282i −1.04904 − 4.59616i 1.76195 − 0.848509i 0.623490 + 0.781831i 4.00316 − 5.01980i
4.9 1.05255 + 1.31986i 0.900969 + 0.433884i −0.189117 + 0.828576i −0.608174 − 2.66458i 0.375651 + 1.64583i 1.15948 + 5.08003i 1.74930 − 0.842420i 0.623490 + 0.781831i 2.87673 − 3.60731i
4.10 1.48777 + 1.86561i 0.900969 + 0.433884i −0.821980 + 3.60133i 0.247824 + 1.08579i 0.530979 + 2.32637i −0.308880 − 1.35329i −3.64180 + 1.75380i 0.623490 + 0.781831i −1.65695 + 2.07774i
16.1 −0.510695 + 2.23750i −0.623490 − 0.781831i −2.94366 − 1.41759i −3.46603 + 1.66915i 2.06776 − 0.995781i −0.169116 + 0.0814417i 1.81330 − 2.27380i −0.222521 + 0.974928i −1.96464 − 8.60767i
16.2 −0.468062 + 2.05071i −0.623490 − 0.781831i −2.18440 − 1.05195i 0.449126 − 0.216288i 1.89514 − 0.912653i −1.14352 + 0.550690i 0.556724 − 0.698110i −0.222521 + 0.974928i 0.233325 + 1.02227i
16.3 −0.294904 + 1.29206i −0.623490 − 0.781831i 0.219495 + 0.105703i 0.102481 − 0.0493523i 1.19404 − 0.575019i 3.28442 − 1.58169i −1.85391 + 2.32473i −0.222521 + 0.974928i 0.0335439 + 0.146966i
16.4 −0.205825 + 0.901777i −0.623490 − 0.781831i 1.03110 + 0.496552i 3.01080 − 1.44992i 0.833367 − 0.401328i −4.30513 + 2.07324i −1.81342 + 2.27396i −0.222521 + 0.974928i 0.687811 + 3.01350i
16.5 −0.0319709 + 0.140073i −0.623490 − 0.781831i 1.78334 + 0.858811i −2.75606 + 1.32725i 0.129447 − 0.0623386i −0.342029 + 0.164712i −0.356472 + 0.447002i −0.222521 + 0.974928i −0.0977988 − 0.428484i
16.6 0.173200 − 0.758839i −0.623490 − 0.781831i 1.25610 + 0.604905i 1.37578 − 0.662540i −0.701273 + 0.337715i 1.88139 − 0.906027i 1.64717 − 2.06549i −0.222521 + 0.974928i −0.264477 − 1.15875i
16.7 0.215609 − 0.944645i −0.623490 − 0.781831i 0.956071 + 0.460419i 0.735892 − 0.354387i −0.872983 + 0.420407i −0.823458 + 0.396556i 1.84932 − 2.31897i −0.222521 + 0.974928i −0.176105 − 0.771566i
16.8 0.430954 − 1.88813i −0.623490 − 0.781831i −1.57738 − 0.759626i −2.30795 + 1.11145i −1.74490 + 0.840297i −4.53018 + 2.18162i 0.300959 − 0.377391i −0.222521 + 0.974928i 1.10394 + 4.83669i
16.9 0.456538 − 2.00022i −0.623490 − 0.781831i −1.99052 − 0.958586i 1.64558 − 0.792467i −1.84848 + 0.890183i 2.89319 − 1.39329i −0.267753 + 0.335752i −0.222521 + 0.974928i −0.833844 − 3.65331i
16.10 0.556707 − 2.43909i −0.623490 − 0.781831i −3.83731 − 1.84795i 3.45736 − 1.66498i −2.25406 + 1.08550i −3.59642 + 1.73194i −3.52387 + 4.41879i −0.222521 + 0.974928i −2.13630 − 9.35974i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.2.i.a ✓ 60
127.e even 7 1 inner 381.2.i.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.i.a ✓ 60 1.a even 1 1 trivial
381.2.i.a ✓ 60 127.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 6 T_{2}^{59} + 28 T_{2}^{58} - 96 T_{2}^{57} + 313 T_{2}^{56} - 889 T_{2}^{55} + \cdots + 241081 \) acting on \(S_{2}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display