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Properties

Label 117.2.h.a
Level $117$
Weight $2$
Character orbit 117.h
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(16,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.h (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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9} + 6 q^{11} - 3 q^{12} + 2 q^{14} + 11 q^{15} + 6 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} - 11 q^{20} - 25 q^{21} - 18 q^{22} + 17 q^{23} - 12 q^{24} - 6 q^{25} - 12 q^{26} + 2 q^{27} - 24 q^{29} - 8 q^{30} - 6 q^{31} - 38 q^{32} + 11 q^{33} + 18 q^{35} - 28 q^{36} - 3 q^{37} + 8 q^{38} + 3 q^{39} - 12 q^{40} + 5 q^{41} + 15 q^{42} - 3 q^{43} + 44 q^{44} + 19 q^{45} - 6 q^{46} + 21 q^{47} + 23 q^{48} + 3 q^{49} - 20 q^{50} + 7 q^{51} - 24 q^{52} - 20 q^{53} + 39 q^{54} + 3 q^{55} + 40 q^{56} + 9 q^{57} + 18 q^{58} + 38 q^{59} + 51 q^{60} - 6 q^{61} + 19 q^{62} + 13 q^{63} - 42 q^{64} - 2 q^{65} - 18 q^{66} - 6 q^{67} - 31 q^{69} + 27 q^{70} + 14 q^{71} - 18 q^{72} + 6 q^{73} + 29 q^{74} + 74 q^{75} - 15 q^{76} + 4 q^{77} + 80 q^{78} + 3 q^{79} - 16 q^{80} - 27 q^{81} - 9 q^{82} - 33 q^{83} + 5 q^{84} + 72 q^{86} - 32 q^{87} - 78 q^{88} - q^{89} + 21 q^{90} - 6 q^{91} - 10 q^{92} - 84 q^{93} - 9 q^{94} - 100 q^{95} - 79 q^{96} + 24 q^{97} - 61 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.65628 1.38934 + 1.03428i 5.05585 0.324360 + 0.561808i −3.69049 − 2.74733i −0.773958 − 1.34053i −8.11720 0.860545 + 2.87393i −0.861592 − 1.49232i
16.2 −2.28273 −0.409803 − 1.68287i 3.21086 −0.461458 − 0.799268i 0.935470 + 3.84155i −0.454185 − 0.786671i −2.76408 −2.66412 + 1.37929i 1.05338 + 1.82452i
16.3 −2.00790 −1.42488 + 0.984744i 2.03168 −1.37329 − 2.37860i 2.86102 − 1.97727i 1.11905 + 1.93825i −0.0636126 1.06056 − 2.80628i 2.75743 + 4.77600i
16.4 −1.20426 1.45393 − 0.941317i −0.549753 1.89177 + 3.27665i −1.75092 + 1.13359i −0.150228 − 0.260203i 3.07057 1.22785 − 2.73722i −2.27819 − 3.94594i
16.5 −0.867378 0.973656 + 1.43248i −1.24766 −0.0324057 − 0.0561283i −0.844528 − 1.24250i 1.96209 + 3.39845i 2.81694 −1.10399 + 2.78948i 0.0281080 + 0.0486844i
16.6 −0.163365 0.725052 − 1.57299i −1.97331 −1.55806 − 2.69863i −0.118448 + 0.256972i 0.0682144 + 0.118151i 0.649100 −1.94860 − 2.28100i 0.254532 + 0.440862i
16.7 0.216364 −1.67508 − 0.440592i −1.95319 −0.702153 − 1.21616i −0.362427 − 0.0953285i −1.65726 − 2.87046i −0.855329 2.61176 + 1.47605i −0.151921 − 0.263135i
16.8 0.697564 −1.39192 + 1.03081i −1.51340 1.44568 + 2.50399i −0.970952 + 0.719053i 1.58641 + 2.74774i −2.45082 0.874877 − 2.86960i 1.00846 + 1.74670i
16.9 1.13584 1.69295 + 0.365956i −0.709859 −0.0587384 − 0.101738i 1.92293 + 0.415669i −0.424723 − 0.735641i −3.07798 2.73215 + 1.23909i −0.0667177 − 0.115558i
16.10 1.80162 −0.336410 − 1.69907i 1.24582 1.73153 + 2.99909i −0.606082 − 3.06106i −1.62239 − 2.81005i −1.35875 −2.77366 + 1.14317i 3.11954 + 5.40321i
16.11 2.05086 −0.160613 + 1.72459i 2.20604 −0.737604 − 1.27757i −0.329395 + 3.53689i −0.582175 − 1.00836i 0.422561 −2.94841 − 0.553983i −1.51272 − 2.62012i
16.12 2.27968 −1.33623 − 1.10203i 3.19692 −1.46964 − 2.54549i −3.04618 − 2.51228i 2.42914 + 4.20740i 2.72859 0.571039 + 2.94515i −3.35030 − 5.80289i
22.1 −2.65628 1.38934 − 1.03428i 5.05585 0.324360 − 0.561808i −3.69049 + 2.74733i −0.773958 + 1.34053i −8.11720 0.860545 − 2.87393i −0.861592 + 1.49232i
22.2 −2.28273 −0.409803 + 1.68287i 3.21086 −0.461458 + 0.799268i 0.935470 − 3.84155i −0.454185 + 0.786671i −2.76408 −2.66412 − 1.37929i 1.05338 − 1.82452i
22.3 −2.00790 −1.42488 − 0.984744i 2.03168 −1.37329 + 2.37860i 2.86102 + 1.97727i 1.11905 − 1.93825i −0.0636126 1.06056 + 2.80628i 2.75743 − 4.77600i
22.4 −1.20426 1.45393 + 0.941317i −0.549753 1.89177 − 3.27665i −1.75092 − 1.13359i −0.150228 + 0.260203i 3.07057 1.22785 + 2.73722i −2.27819 + 3.94594i
22.5 −0.867378 0.973656 − 1.43248i −1.24766 −0.0324057 + 0.0561283i −0.844528 + 1.24250i 1.96209 − 3.39845i 2.81694 −1.10399 − 2.78948i 0.0281080 − 0.0486844i
22.6 −0.163365 0.725052 + 1.57299i −1.97331 −1.55806 + 2.69863i −0.118448 − 0.256972i 0.0682144 − 0.118151i 0.649100 −1.94860 + 2.28100i 0.254532 − 0.440862i
22.7 0.216364 −1.67508 + 0.440592i −1.95319 −0.702153 + 1.21616i −0.362427 + 0.0953285i −1.65726 + 2.87046i −0.855329 2.61176 − 1.47605i −0.151921 + 0.263135i
22.8 0.697564 −1.39192 − 1.03081i −1.51340 1.44568 − 2.50399i −0.970952 − 0.719053i 1.58641 − 2.74774i −2.45082 0.874877 + 2.86960i 1.00846 − 1.74670i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.h.a yes 24
3.b odd 2 1 351.2.h.a 24
9.c even 3 1 117.2.f.a ✓ 24
9.d odd 6 1 351.2.f.a 24
13.c even 3 1 117.2.f.a ✓ 24
39.i odd 6 1 351.2.f.a 24
117.h even 3 1 inner 117.2.h.a yes 24
117.k odd 6 1 351.2.h.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.f.a ✓ 24 9.c even 3 1
117.2.f.a ✓ 24 13.c even 3 1
117.2.h.a yes 24 1.a even 1 1 trivial
117.2.h.a yes 24 117.h even 3 1 inner
351.2.f.a 24 9.d odd 6 1
351.2.f.a 24 39.i odd 6 1
351.2.h.a 24 3.b odd 2 1
351.2.h.a 24 117.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(117, [\chi])\).