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Properties

Label 381.2.i.b
Level $381$
Weight $2$
Character orbit 381.i
Analytic conductor $3.042$
Analytic rank $0$
Dimension $72$
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: \(72\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{2} - 12 q^{3} - 14 q^{4} - 4 q^{5} - 3 q^{6} + 16 q^{7} - 2 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{2} - 12 q^{3} - 14 q^{4} - 4 q^{5} - 3 q^{6} + 16 q^{7} - 2 q^{8} - 12 q^{9} - 6 q^{10} - q^{11} - 7 q^{12} - 18 q^{13} - 13 q^{14} - 4 q^{15} - 56 q^{16} - 10 q^{17} - 3 q^{18} + 14 q^{19} - 22 q^{20} + 16 q^{21} + 26 q^{22} + 10 q^{23} + 12 q^{24} - 24 q^{25} + 14 q^{26} - 12 q^{27} + 18 q^{28} + 6 q^{29} - 13 q^{30} - 26 q^{31} - 19 q^{32} + 6 q^{33} + 23 q^{34} + 61 q^{35} - 21 q^{36} + 22 q^{37} + 38 q^{38} - 18 q^{39} + 8 q^{40} + 25 q^{41} + 8 q^{42} - 29 q^{43} - 2 q^{44} - 4 q^{45} + 6 q^{46} + 3 q^{47} + 21 q^{48} - 16 q^{49} - 53 q^{50} - 10 q^{51} + 66 q^{52} + 2 q^{53} + 4 q^{54} + 40 q^{55} - 47 q^{56} - 21 q^{57} - 52 q^{58} - 44 q^{59} + 55 q^{60} - 5 q^{61} - 71 q^{62} - 12 q^{63} + 82 q^{64} + 42 q^{65} + 12 q^{66} - 34 q^{67} + 28 q^{68} + 3 q^{69} + 22 q^{70} - 47 q^{71} - 2 q^{72} + 33 q^{73} + 57 q^{74} + 88 q^{75} - 75 q^{76} + q^{77} - 14 q^{78} - 9 q^{79} - 35 q^{80} - 12 q^{81} - 6 q^{82} - 19 q^{83} - 52 q^{84} - 36 q^{85} + 26 q^{86} - 8 q^{87} - 6 q^{88} + 2 q^{89} + 22 q^{90} - 62 q^{91} - 18 q^{92} - 5 q^{93} - 68 q^{94} - 57 q^{95} - 19 q^{96} + 52 q^{97} - q^{98} - 8 q^{99}+O(q^{100}) \) 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 −1.71938 − 2.15603i −0.900969 − 0.433884i −1.24717 + 5.46422i 0.713056 + 3.12410i 0.613639 + 2.68853i −0.983642 − 4.30962i 8.95626 − 4.31311i 0.623490 + 0.781831i 5.50966 − 6.90889i
4.2 −1.48854 − 1.86657i −0.900969 − 0.433884i −0.823290 + 3.60707i −0.449797 − 1.97069i 0.531254 + 2.32757i 0.476978 + 2.08978i 3.65634 − 1.76080i 0.623490 + 0.781831i −3.00889 + 3.77303i
4.3 −1.15934 − 1.45376i −0.900969 − 0.433884i −0.324321 + 1.42094i 0.346755 + 1.51923i 0.413763 + 1.81281i 0.297667 + 1.30416i −0.908867 + 0.437687i 0.623490 + 0.781831i 1.80660 − 2.26540i
4.4 −0.535984 − 0.672102i −0.900969 − 0.433884i 0.280599 − 1.22938i 0.777226 + 3.40525i 0.191290 + 0.838098i −0.163559 − 0.716601i −2.52571 + 1.21632i 0.623490 + 0.781831i 1.87210 − 2.34753i
4.5 −0.420524 − 0.527321i −0.900969 − 0.433884i 0.343815 − 1.50635i −0.758665 − 3.32393i 0.150083 + 0.657558i −0.671533 − 2.94218i −2.15426 + 1.03744i 0.623490 + 0.781831i −1.43374 + 1.79785i
4.6 0.135959 + 0.170487i −0.900969 − 0.433884i 0.434461 − 1.90350i −0.483430 − 2.11805i −0.0485232 − 0.212594i 0.385724 + 1.68997i 0.776523 − 0.373954i 0.623490 + 0.781831i 0.295373 − 0.370386i
4.7 0.181756 + 0.227915i −0.900969 − 0.433884i 0.426132 − 1.86701i 0.402234 + 1.76230i −0.0648680 − 0.284205i 1.00822 + 4.41732i 1.02826 − 0.495185i 0.623490 + 0.781831i −0.328547 + 0.411984i
4.8 0.717385 + 0.899573i −0.900969 − 0.433884i 0.150452 − 0.659175i 0.178642 + 0.782683i −0.256032 − 1.12175i −0.970771 − 4.25323i 2.77421 − 1.33599i 0.623490 + 0.781831i −0.575925 + 0.722187i
4.9 1.08550 + 1.36118i −0.900969 − 0.433884i −0.229445 + 1.00526i 0.334796 + 1.46684i −0.387411 − 1.69736i 0.112062 + 0.490975i 1.51979 − 0.731891i 0.623490 + 0.781831i −1.63320 + 2.04797i
4.10 1.20197 + 1.50722i −0.900969 − 0.433884i −0.381949 + 1.67343i −0.582811 − 2.55346i −0.428979 − 1.87948i −0.260170 − 1.13988i 0.492477 − 0.237165i 0.623490 + 0.781831i 3.14812 − 3.94762i
4.11 1.68391 + 2.11156i −0.900969 − 0.433884i −1.17808 + 5.16153i −0.568743 − 2.49183i −0.600982 − 2.63308i 0.889852 + 3.89870i −8.01602 + 3.86031i 0.623490 + 0.781831i 4.30393 − 5.39696i
4.12 1.71824 + 2.15461i −0.900969 − 0.433884i −1.24494 + 5.45443i 0.892675 + 3.91107i −0.613234 − 2.68675i −0.614791 − 2.69357i −8.92540 + 4.29825i 0.623490 + 0.781831i −6.89298 + 8.64353i
16.1 −0.615598 + 2.69711i 0.623490 + 0.781831i −5.09351 − 2.45290i −1.47989 + 0.712678i −2.49250 + 1.20033i −0.813215 + 0.391624i 6.30157 − 7.90191i −0.222521 + 0.974928i −1.01115 − 4.43015i
16.2 −0.517772 + 2.26851i 0.623490 + 0.781831i −3.07610 − 1.48137i 1.94763 − 0.937930i −2.09642 + 1.00958i 3.82634 − 1.84267i 2.05168 − 2.57273i −0.222521 + 0.974928i 1.11927 + 4.90385i
16.3 −0.358451 + 1.57048i 0.623490 + 0.781831i −0.535967 − 0.258108i −1.23361 + 0.594074i −1.45134 + 0.698927i −2.99429 + 1.44197i −1.41124 + 1.76964i −0.222521 + 0.974928i −0.490791 − 2.15030i
16.4 −0.342526 + 1.50070i 0.623490 + 0.781831i −0.332853 − 0.160293i −2.90504 + 1.39899i −1.38686 + 0.667876i 2.34731 − 1.13040i −1.56491 + 1.96233i −0.222521 + 0.974928i −1.10442 − 4.83880i
16.5 −0.245771 + 1.07679i 0.623490 + 0.781831i 0.702860 + 0.338479i 2.92416 − 1.40820i −0.995106 + 0.479218i −0.423122 + 0.203765i −1.91448 + 2.40069i −0.222521 + 0.974928i 0.797666 + 3.49480i
16.6 0.0461365 − 0.202137i 0.623490 + 0.781831i 1.76321 + 0.849116i 1.28248 − 0.617608i 0.186803 − 0.0899595i −0.848177 + 0.408461i 0.511529 − 0.641437i −0.222521 + 0.974928i −0.0656726 − 0.287731i
16.7 0.0633407 − 0.277514i 0.623490 + 0.781831i 1.72894 + 0.832612i −1.42862 + 0.687989i 0.256461 − 0.123505i −4.38834 + 2.11331i 0.695526 − 0.872162i −0.222521 + 0.974928i 0.100436 + 0.440040i
16.8 0.0756200 − 0.331313i 0.623490 + 0.781831i 1.69789 + 0.817660i −0.103527 + 0.0498562i 0.306179 − 0.147448i 3.73142 − 1.79696i 0.823061 − 1.03209i −0.222521 + 0.974928i 0.00868925 + 0.0380701i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
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.b ✓ 72
127.e even 7 1 inner 381.2.i.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.i.b ✓ 72 1.a even 1 1 trivial
381.2.i.b ✓ 72 127.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 4 T_{2}^{71} + 27 T_{2}^{70} - 82 T_{2}^{69} + 394 T_{2}^{68} - 1055 T_{2}^{67} + \cdots + 6036849 \) acting on \(S_{2}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display