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Properties

Label 387.2.f.c
Level $387$
Weight $2$
Character orbit 387.f
Analytic conductor $3.090$
Analytic rank $0$
Dimension $38$
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(130,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.130");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.f (of order \(3\), degree \(2\), 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: \(38\)
Relative dimension: \(19\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 4 q^{2} - 7 q^{3} - 22 q^{4} - 9 q^{5} - 7 q^{7} + 24 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 4 q^{2} - 7 q^{3} - 22 q^{4} - 9 q^{5} - 7 q^{7} + 24 q^{8} - 3 q^{9} - 14 q^{10} - 5 q^{11} + 11 q^{12} + 5 q^{13} - 17 q^{14} - 5 q^{15} - 24 q^{16} + 42 q^{17} - 23 q^{18} - 8 q^{19} - 21 q^{20} + 20 q^{22} - 22 q^{23} - 14 q^{24} - 10 q^{25} + 34 q^{26} - 4 q^{27} - 2 q^{28} - 30 q^{29} + 63 q^{30} + 5 q^{31} - 48 q^{32} - q^{33} + 6 q^{34} + 106 q^{35} - 20 q^{36} - 2 q^{37} - 21 q^{38} + 25 q^{39} - 16 q^{40} - 29 q^{41} - 47 q^{42} - 19 q^{43} + 58 q^{44} - 37 q^{45} - 32 q^{47} + 45 q^{48} + 10 q^{49} + 11 q^{50} - 53 q^{51} - q^{52} + 76 q^{53} - 41 q^{54} + 4 q^{55} - 46 q^{56} + 23 q^{57} - 30 q^{58} - 30 q^{59} - 87 q^{60} + 10 q^{61} + 50 q^{62} + 21 q^{63} + 28 q^{64} - 8 q^{65} + 91 q^{66} - 3 q^{67} - 47 q^{68} - 40 q^{69} - 56 q^{70} + 42 q^{71} + 3 q^{72} + 16 q^{73} - 28 q^{74} + 19 q^{75} + 36 q^{76} - 49 q^{77} - 105 q^{78} - 4 q^{79} + 140 q^{80} + 77 q^{81} - 8 q^{82} - 29 q^{83} + 145 q^{84} + 4 q^{85} - 4 q^{86} - 24 q^{87} + 47 q^{88} + 108 q^{89} - 8 q^{90} + 8 q^{91} - 12 q^{92} - 4 q^{93} + 23 q^{94} - 33 q^{95} - 147 q^{96} + 4 q^{97} + 98 q^{98} + 85 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
130.1 −1.36225 − 2.35949i 1.72295 + 0.177285i −2.71146 + 4.69638i −1.18444 + 2.05150i −1.92880 − 4.30680i −1.93605 − 3.35333i 9.32575 2.93714 + 0.610907i 6.45400
130.2 −1.33284 − 2.30855i −0.499172 + 1.65856i −2.55293 + 4.42180i 0.659842 − 1.14288i 4.49418 − 1.05824i −0.548434 − 0.949916i 8.27921 −2.50165 − 1.65582i −3.51785
130.3 −1.26912 − 2.19819i −1.64245 − 0.549872i −2.22135 + 3.84749i −0.946835 + 1.63997i 0.875750 + 4.30826i 0.412403 + 0.714304i 6.20016 2.39528 + 1.80627i 4.80660
130.4 −1.09093 − 1.88955i 0.353034 − 1.69569i −1.38028 + 2.39071i −0.845931 + 1.46520i −3.58924 + 1.18281i −1.08430 − 1.87806i 1.65943 −2.75073 − 1.19727i 3.69142
130.5 −0.987453 − 1.71032i −1.70072 + 0.327948i −0.950127 + 1.64567i 1.66142 − 2.87766i 2.24028 + 2.58494i 2.32616 + 4.02902i −0.196989 2.78490 − 1.11550i −6.56230
130.6 −0.724508 − 1.25489i −0.714752 − 1.57770i −0.0498244 + 0.0862984i 1.28186 − 2.22025i −1.46199 + 2.03999i 0.350586 + 0.607232i −2.75364 −1.97826 + 2.25532i −3.71487
130.7 −0.651400 − 1.12826i 0.578738 + 1.63250i 0.151357 − 0.262158i −1.84461 + 3.19495i 1.46489 − 1.71638i −1.71786 − 2.97543i −2.99997 −2.33013 + 1.88958i 4.80630
130.8 −0.451326 − 0.781720i 1.70234 + 0.319439i 0.592609 − 1.02643i 0.0415173 − 0.0719100i −0.518599 − 1.47492i −1.41989 − 2.45931i −2.87515 2.79592 + 1.08759i −0.0749514
130.9 −0.449049 − 0.777775i 1.72909 − 0.101233i 0.596711 − 1.03353i 0.818891 − 1.41836i −0.855182 − 1.29938i 0.930316 + 1.61135i −2.86800 2.97950 − 0.350083i −1.47089
130.10 −0.100298 − 0.173722i −1.21945 + 1.23002i 0.979881 − 1.69720i −0.316846 + 0.548794i 0.335989 + 0.0884752i 0.192248 + 0.332984i −0.794314 −0.0259022 − 2.99989i 0.127116
130.11 0.0180090 + 0.0311926i −1.41235 − 1.00263i 0.999351 − 1.73093i 0.638135 − 1.10528i 0.00583972 − 0.0621112i −1.94045 − 3.36095i 0.144026 0.989449 + 2.83214i 0.0459688
130.12 0.152464 + 0.264076i 0.0703844 + 1.73062i 0.953509 − 1.65153i −1.13542 + 1.96661i −0.446284 + 0.282444i 1.25241 + 2.16924i 1.19136 −2.99009 + 0.243617i −0.692446
130.13 0.326572 + 0.565640i −1.71918 + 0.210736i 0.786701 − 1.36261i −1.96622 + 3.40559i −0.680638 − 0.903617i −2.10690 − 3.64925i 2.33395 2.91118 − 0.724588i −2.56845
130.14 0.649321 + 1.12466i −1.24030 + 1.20899i 0.156764 − 0.271524i 1.38166 − 2.39311i −2.16505 − 0.609884i −0.516159 − 0.894013i 3.00445 0.0766735 − 2.99902i 3.58856
130.15 0.880687 + 1.52540i 0.427675 + 1.67842i −0.551220 + 0.954742i 0.398628 − 0.690444i −2.18361 + 2.13054i 0.958880 + 1.66083i 1.58094 −2.63419 + 1.43564i 1.40427
130.16 0.887526 + 1.53724i −0.0779340 − 1.73030i −0.575406 + 0.996632i −1.84963 + 3.20365i 2.59071 − 1.65549i 0.0448503 + 0.0776831i 1.50735 −2.98785 + 0.269698i −6.56638
130.17 1.09981 + 1.90493i 0.706662 − 1.58134i −1.41918 + 2.45809i 0.829860 − 1.43736i 3.78954 − 0.393032i 1.66958 + 2.89180i −1.84408 −2.00126 − 2.23494i 3.65077
130.18 1.12806 + 1.95386i 0.993572 + 1.41874i −1.54504 + 2.67608i −0.722147 + 1.25080i −1.65120 + 3.54172i −0.444392 − 0.769710i −2.45933 −1.02563 + 2.81923i −3.25850
130.19 1.27673 + 2.21136i −1.55815 − 0.756424i −2.26008 + 3.91458i −1.39974 + 2.42442i −0.316608 − 4.41138i 0.0769910 + 0.133352i −6.43513 1.85565 + 2.35724i −7.14837
259.1 −1.36225 + 2.35949i 1.72295 − 0.177285i −2.71146 − 4.69638i −1.18444 − 2.05150i −1.92880 + 4.30680i −1.93605 + 3.35333i 9.32575 2.93714 − 0.610907i 6.45400
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 130.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.f.c ✓ 38
3.b odd 2 1 1161.2.f.c 38
9.c even 3 1 inner 387.2.f.c ✓ 38
9.c even 3 1 3483.2.a.s 19
9.d odd 6 1 1161.2.f.c 38
9.d odd 6 1 3483.2.a.r 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.f.c ✓ 38 1.a even 1 1 trivial
387.2.f.c ✓ 38 9.c even 3 1 inner
1161.2.f.c 38 3.b odd 2 1
1161.2.f.c 38 9.d odd 6 1
3483.2.a.r 19 9.d odd 6 1
3483.2.a.s 19 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} + 4 T_{2}^{37} + 38 T_{2}^{36} + 112 T_{2}^{35} + 703 T_{2}^{34} + 1772 T_{2}^{33} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display