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Properties

Label 387.2.y.c
Level $387$
Weight $2$
Character orbit 387.y
Analytic conductor $3.090$
Analytic rank $0$
Dimension $36$
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] = [387,2,Mod(10,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.y (of order \(21\), degree \(12\), 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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 10 q^{2} - 18 q^{4} + 17 q^{5} + 6 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 10 q^{2} - 18 q^{4} + 17 q^{5} + 6 q^{7} - 18 q^{8} - 7 q^{10} + 4 q^{11} - 18 q^{14} - 10 q^{16} + 10 q^{17} + 10 q^{19} + 3 q^{20} - 3 q^{22} - 4 q^{23} - 2 q^{25} + 15 q^{26} + 20 q^{28} - 9 q^{29} + 40 q^{31} - 48 q^{32} - 42 q^{34} - 11 q^{35} - 19 q^{37} + 21 q^{38} - 97 q^{40} + 28 q^{41} - 8 q^{43} - 14 q^{44} - 61 q^{46} + 30 q^{47} + 6 q^{49} + 3 q^{50} - 8 q^{52} + 24 q^{53} + 14 q^{55} - 39 q^{56} + 64 q^{58} + q^{59} - 14 q^{61} - 33 q^{62} + 48 q^{64} - 38 q^{65} + 66 q^{67} - 66 q^{68} + 47 q^{70} + 33 q^{71} + 29 q^{73} + 40 q^{74} - 39 q^{76} + 27 q^{77} - 17 q^{79} - 8 q^{80} - 54 q^{82} + 23 q^{83} - 56 q^{85} + 45 q^{86} - 17 q^{88} + 19 q^{89} - 13 q^{91} + 18 q^{92} + 44 q^{94} - q^{95} - 31 q^{97} + 5 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −0.581275 − 2.54673i 0 −4.34603 + 2.09294i −2.95419 + 0.445272i 0 −0.339884 − 0.588696i 4.59900 + 5.76696i 0 2.85118 + 7.26470i
10.2 −0.188565 − 0.826155i 0 1.15496 − 0.556200i 3.39819 − 0.512194i 0 −0.134521 − 0.232998i −1.73398 − 2.17435i 0 −1.06393 − 2.71085i
10.3 0.483758 + 2.11948i 0 −2.45624 + 1.18286i −0.0260188 + 0.00392170i 0 1.56464 + 2.71003i −0.984367 − 1.23436i 0 −0.0208988 − 0.0532492i
100.1 −0.993512 + 0.478450i 0 −0.488828 + 0.612971i 3.17479 − 0.979294i 0 1.23273 − 2.13515i 0.683135 − 2.99301i 0 −2.68565 + 2.49192i
100.2 0.982954 − 0.473366i 0 −0.504856 + 0.633069i −1.29085 + 0.398175i 0 −0.108163 + 0.187343i −0.682116 + 2.98855i 0 −1.08036 + 1.00243i
100.3 2.05399 − 0.989151i 0 1.99349 − 2.49975i 0.131558 − 0.0405802i 0 0.934721 − 1.61898i 0.607386 − 2.66113i 0 0.230079 − 0.213482i
109.1 0.0594739 + 0.0745779i 0 0.443017 − 1.94098i 0.284956 − 3.80248i 0 1.30981 + 2.26866i 0.342987 − 0.165174i 0 0.300528 − 0.204897i
109.2 0.651405 + 0.816837i 0 0.202149 − 0.885672i 0.0373507 − 0.498411i 0 −1.65334 − 2.86367i 2.73775 − 1.31843i 0 0.431451 − 0.294158i
109.3 1.72039 + 2.15730i 0 −1.24917 + 5.47296i −0.0684907 + 0.913945i 0 −0.971539 − 1.68276i −8.98382 + 4.32638i 0 −2.08949 + 1.42459i
154.1 −1.94880 − 0.938491i 0 1.67006 + 2.09419i 2.00127 − 1.85691i 0 1.01083 − 1.75082i −0.326607 − 1.43096i 0 −5.64276 + 1.74056i
154.2 −0.118393 − 0.0570152i 0 −1.23621 − 1.55016i 1.30537 − 1.21121i 0 −0.00749281 + 0.0129779i 0.116458 + 0.510236i 0 −0.223604 + 0.0689728i
154.3 1.06783 + 0.514239i 0 −0.371164 − 0.465425i −2.37710 + 2.20563i 0 −1.38418 + 2.39748i −0.684463 − 2.99883i 0 −3.67256 + 1.13283i
181.1 −1.06016 − 1.32939i 0 −0.198316 + 0.868879i 1.45700 + 0.993363i 0 −0.297283 + 0.514909i −1.69861 + 0.818009i 0 −0.224073 − 2.99004i
181.2 1.11243 + 1.39495i 0 −0.263330 + 1.15372i 0.492700 + 0.335917i 0 2.18154 − 3.77854i 1.31271 − 0.632167i 0 0.0795092 + 1.06098i
181.3 1.44188 + 1.80806i 0 −0.745019 + 3.26414i 2.09843 + 1.43068i 0 −1.09365 + 1.89425i −2.80884 + 1.35267i 0 0.438918 + 5.85695i
253.1 −0.377390 + 1.65345i 0 −0.789549 − 0.380227i −0.140805 − 0.358765i 0 1.74586 + 3.02391i −1.18819 + 1.48995i 0 0.646339 − 0.0974200i
253.2 0.178150 − 0.780524i 0 1.22446 + 0.589667i −0.511972 − 1.30448i 0 −2.37928 − 4.12103i 1.67671 − 2.10253i 0 −1.10939 + 0.167213i
253.3 0.515822 − 2.25996i 0 −3.03942 − 1.46371i 1.48781 + 3.79089i 0 1.38920 + 2.40617i −1.98513 + 2.48927i 0 9.33472 − 1.40698i
271.1 −0.581275 + 2.54673i 0 −4.34603 − 2.09294i −2.95419 − 0.445272i 0 −0.339884 + 0.588696i 4.59900 − 5.76696i 0 2.85118 − 7.26470i
271.2 −0.188565 + 0.826155i 0 1.15496 + 0.556200i 3.39819 + 0.512194i 0 −0.134521 + 0.232998i −1.73398 + 2.17435i 0 −1.06393 + 2.71085i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.y.c 36
3.b odd 2 1 43.2.g.a ✓ 36
12.b even 2 1 688.2.bg.c 36
43.g even 21 1 inner 387.2.y.c 36
129.n even 42 1 1849.2.a.o 18
129.o odd 42 1 43.2.g.a ✓ 36
129.o odd 42 1 1849.2.a.n 18
516.bb even 42 1 688.2.bg.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.g.a ✓ 36 3.b odd 2 1
43.2.g.a ✓ 36 129.o odd 42 1
387.2.y.c 36 1.a even 1 1 trivial
387.2.y.c 36 43.g even 21 1 inner
688.2.bg.c 36 12.b even 2 1
688.2.bg.c 36 516.bb even 42 1
1849.2.a.n 18 129.o odd 42 1
1849.2.a.o 18 129.n even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 10 T_{2}^{35} + 65 T_{2}^{34} - 296 T_{2}^{33} + 1093 T_{2}^{32} - 3288 T_{2}^{31} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display