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Properties

Label 381.2.x.b
Level $381$
Weight $2$
Character orbit 381.x
Analytic conductor $3.042$
Analytic rank $0$
Dimension $1440$
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] = [381,2,Mod(14,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(126))
 
chi = DirichletCharacter(H, H._module([63, 61]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.x (of order \(126\), degree \(36\), 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: \(1440\)
Relative dimension: \(40\) over \(\Q(\zeta_{126})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{126}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1440 q - 36 q^{3} + 192 q^{4} - 57 q^{6} - 78 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1440 q - 36 q^{3} + 192 q^{4} - 57 q^{6} - 78 q^{7} - 36 q^{9} - 102 q^{10} - 39 q^{12} - 108 q^{13} - 54 q^{15} - 336 q^{16} - 60 q^{18} - 36 q^{19} + 48 q^{21} - 72 q^{22} + 24 q^{24} + 54 q^{25} - 60 q^{27} - 132 q^{28} - 72 q^{30} - 36 q^{31} - 33 q^{33} - 48 q^{34} - 3 q^{36} - 54 q^{37} + 33 q^{39} - 36 q^{40} + 15 q^{42} - 48 q^{43} - 135 q^{45} - 120 q^{46} - 24 q^{48} - 54 q^{49} - 33 q^{51} - 192 q^{52} - 177 q^{54} - 192 q^{55} + 36 q^{57} - 102 q^{58} + 114 q^{60} + 18 q^{61} - 42 q^{63} + 288 q^{64} - 51 q^{66} - 102 q^{67} - 18 q^{69} - 156 q^{70} + 27 q^{72} - 78 q^{73} - 276 q^{75} + 78 q^{76} - 111 q^{78} - 42 q^{79} + 156 q^{81} - 54 q^{82} - 252 q^{84} - 102 q^{85} - 132 q^{87} - 720 q^{88} - 288 q^{90} - 222 q^{91} - 114 q^{93} - 306 q^{94} - 15 q^{96} + 6 q^{97} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −1.17831 − 2.44680i −0.575707 + 1.63357i −3.35140 + 4.20253i −0.424359 + 1.08125i 4.67538 − 0.516226i 0.578202 − 0.946185i 8.93644 + 2.03968i −2.33712 − 1.88092i 3.14562 − 0.235732i
14.2 −1.12587 − 2.33789i −1.69470 − 0.357774i −2.95117 + 3.70065i 0.753526 − 1.91995i 1.07157 + 4.36482i −0.229868 + 0.376162i 6.91473 + 1.57824i 2.74400 + 1.21264i −5.33701 + 0.399954i
14.3 −1.11191 − 2.30891i 1.03517 − 1.38867i −2.84774 + 3.57095i 1.27177 − 3.24043i −4.35735 − 0.846038i 0.279577 − 0.457508i 6.41454 + 1.46408i −0.856832 − 2.87504i −8.89595 + 0.666659i
14.4 −1.05425 − 2.18918i 1.56488 + 0.742391i −2.43406 + 3.05222i 0.00928924 − 0.0236686i −0.0245554 − 4.20847i −1.91840 + 3.13932i 4.51019 + 1.02942i 1.89771 + 2.32351i −0.0616079 + 0.00461687i
14.5 −1.05087 − 2.18215i 1.67164 − 0.453435i −2.41047 + 3.02264i −1.31573 + 3.35243i −2.74614 − 3.17128i 2.02930 − 3.32081i 4.40639 + 1.00573i 2.58879 − 1.51596i 8.69816 − 0.651837i
14.6 −0.940661 − 1.95330i −1.73109 − 0.0577927i −1.68357 + 2.11113i −0.792016 + 2.01802i 1.51548 + 3.43570i −1.81827 + 2.97547i 1.48005 + 0.337812i 2.99332 + 0.200088i 4.68683 − 0.351230i
14.7 −0.921731 − 1.91399i −1.02218 − 1.39826i −1.56681 + 1.96471i 0.361993 − 0.922343i −1.73409 + 3.24528i 2.39642 − 3.92156i 1.06240 + 0.242486i −0.910283 + 2.85856i −2.09902 + 0.157300i
14.8 −0.846716 − 1.75822i 0.589552 + 1.62863i −1.12744 + 1.41377i 1.16134 − 2.95904i 2.36431 − 2.41555i 0.811741 − 1.32836i −0.364764 − 0.0832551i −2.30486 + 1.92032i −6.18597 + 0.463575i
14.9 −0.817104 − 1.69673i −0.526329 − 1.65014i −0.964265 + 1.20915i −0.890125 + 2.26800i −2.36979 + 2.24138i −0.263857 + 0.431782i −0.832520 − 0.190017i −2.44596 + 1.73704i 4.57552 − 0.342888i
14.10 −0.700725 − 1.45507i −1.14677 + 1.29804i −0.379235 + 0.475545i −0.597756 + 1.52306i 2.69232 + 0.759061i 1.24338 − 2.03470i −2.19134 − 0.500159i −0.369832 − 2.97712i 2.63502 − 0.197467i
14.11 −0.638738 − 1.32635i 0.645158 + 1.60741i −0.104246 + 0.130720i −1.52820 + 3.89378i 1.71991 − 1.88242i −0.169055 + 0.276646i −2.63049 − 0.600393i −2.16754 + 2.07407i 6.14065 − 0.460178i
14.12 −0.609110 − 1.26483i −0.527325 − 1.64983i 0.0181979 − 0.0228195i 1.35399 − 3.44991i −1.76555 + 1.67190i −2.37559 + 3.88748i −2.77726 − 0.633893i −2.44386 + 1.73999i −5.18828 + 0.388808i
14.13 −0.590103 − 1.22536i 1.71190 − 0.263461i 0.0936913 − 0.117485i 0.692063 − 1.76335i −1.33303 − 1.94222i 2.19674 − 3.59480i −2.85115 − 0.650756i 2.86118 − 0.902035i −2.56913 + 0.192529i
14.14 −0.508676 − 1.05628i −0.453534 + 1.67162i 0.390011 − 0.489058i 0.305776 − 0.779104i 1.99639 − 0.371255i −2.53499 + 4.14834i −3.00094 − 0.684945i −2.58861 − 1.51627i −0.978490 + 0.0733277i
14.15 −0.497818 − 1.03373i 1.69768 + 0.343333i 0.426206 − 0.534445i 0.310728 − 0.791723i −0.490222 − 1.92586i −0.582189 + 0.952710i −3.00182 − 0.685145i 2.76424 + 1.16574i −0.973114 + 0.0729248i
14.16 −0.440103 − 0.913882i 1.22177 − 1.22771i 0.605489 − 0.759259i −0.101161 + 0.257753i −1.65969 − 0.576240i −0.419740 + 0.686875i −2.93815 − 0.670614i −0.0145360 − 2.99996i 0.280077 − 0.0209889i
14.17 −0.342527 − 0.711264i −1.73202 − 0.00959217i 0.858408 − 1.07641i 1.11804 − 2.84873i 0.586442 + 1.23521i 0.464707 − 0.760459i −2.59894 − 0.593191i 2.99982 + 0.0332277i −2.40916 + 0.180541i
14.18 −0.163390 − 0.339284i −1.48274 − 0.895256i 1.15856 − 1.45279i −0.793278 + 2.02124i −0.0614802 + 0.649345i −1.47896 + 2.42021i −1.41648 − 0.323301i 1.39703 + 2.65486i 0.815387 − 0.0611048i
14.19 −0.157594 − 0.327247i 0.282438 − 1.70887i 1.16472 − 1.46052i −0.885590 + 2.25645i −0.603732 + 0.176880i 1.23202 − 2.01612i −1.36972 − 0.312630i −2.84046 − 0.965298i 0.877979 − 0.0657954i
14.20 −0.0617412 − 0.128207i 1.26881 + 1.17903i 1.23435 − 1.54783i −0.575587 + 1.46657i 0.0728216 − 0.235465i −0.129962 + 0.212673i −0.552115 − 0.126017i 0.219778 + 2.99194i 0.223562 − 0.0167536i
See next 80 embeddings (of 1440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
127.l odd 126 1 inner
381.x even 126 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.2.x.b ✓ 1440
3.b odd 2 1 inner 381.2.x.b ✓ 1440
127.l odd 126 1 inner 381.2.x.b ✓ 1440
381.x even 126 1 inner 381.2.x.b ✓ 1440
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.x.b ✓ 1440 1.a even 1 1 trivial
381.2.x.b ✓ 1440 3.b odd 2 1 inner
381.2.x.b ✓ 1440 127.l odd 126 1 inner
381.2.x.b ✓ 1440 381.x even 126 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1440} - 336 T_{2}^{1438} + 57426 T_{2}^{1436} - 6654508 T_{2}^{1434} + 588015945 T_{2}^{1432} + \cdots + 53\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display