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Properties

Label 117.2.z.b
Level $117$
Weight $2$
Character orbit 117.z
Analytic conductor $0.934$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(5,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.z (of order \(12\), degree \(4\), 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: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{3} - 14 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 2 q^{3} - 14 q^{6} - 14 q^{9} - 6 q^{11} + 4 q^{13} - 36 q^{14} + 4 q^{15} + 4 q^{18} + 12 q^{19} - 18 q^{20} + 22 q^{21} - 28 q^{22} - 24 q^{24} + 4 q^{27} - 16 q^{28} + 8 q^{31} + 6 q^{32} - 14 q^{33} - 12 q^{34} - 16 q^{37} + 16 q^{39} + 36 q^{40} + 36 q^{41} + 20 q^{42} - 56 q^{45} - 12 q^{46} - 18 q^{47} - 8 q^{48} + 120 q^{50} - 8 q^{52} + 52 q^{54} - 64 q^{55} - 8 q^{57} - 40 q^{58} - 24 q^{59} + 8 q^{60} + 2 q^{61} + 64 q^{63} - 42 q^{65} - 4 q^{66} - 36 q^{68} + 20 q^{70} + 18 q^{72} + 48 q^{73} + 216 q^{74} - 26 q^{76} + 22 q^{78} - 22 q^{79} + 10 q^{81} + 30 q^{83} + 94 q^{84} - 18 q^{85} - 120 q^{86} + 68 q^{87} - 72 q^{92} - 62 q^{93} + 32 q^{94} + 16 q^{96} - 4 q^{97} - 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.600405 + 2.24074i 0.724368 − 1.57331i −2.92839 − 1.69071i 3.78248 − 1.01351i 3.09046 + 2.56774i −0.608850 + 2.27226i 2.26599 − 2.26599i −1.95058 − 2.27931i 9.08409i
5.2 −0.535189 + 1.99735i 1.67588 + 0.437507i −1.97094 − 1.13792i −2.39559 + 0.641897i −1.77077 + 3.11318i −0.185908 + 0.693817i 0.403326 − 0.403326i 2.61718 + 1.46642i − 5.12838i
5.3 −0.337526 + 1.25966i 0.425920 + 1.67887i 0.259225 + 0.149663i 2.31922 − 0.621434i −2.25856 − 0.0301452i 0.940006 − 3.50815i −2.12029 + 2.12029i −2.63718 + 1.43012i 3.13119i
5.4 −0.267684 + 0.999012i −1.73168 − 0.0358377i 0.805681 + 0.465160i 1.80337 − 0.483213i 0.499346 − 1.72038i −1.07064 + 3.99567i −2.14302 + 2.14302i 2.99743 + 0.124119i 1.93094i
5.5 −0.126147 + 0.470787i −0.933667 + 1.45886i 1.52632 + 0.881223i −3.78110 + 1.01314i −0.569032 − 0.623589i 0.139526 − 0.520717i −1.29669 + 1.29669i −1.25653 − 2.72417i − 1.90790i
5.6 −0.120326 + 0.449064i −0.453962 − 1.67150i 1.54487 + 0.891932i 0.207401 − 0.0555730i 0.805235 − 0.00273231i 0.845006 − 3.15361i −1.24390 + 1.24390i −2.58784 + 1.51760i 0.0998232i
5.7 0.0389963 − 0.145536i 1.36970 − 1.06015i 1.71239 + 0.988649i −1.38592 + 0.371357i −0.100877 − 0.240683i −0.436407 + 1.62869i 0.423741 − 0.423741i 0.752165 − 2.90418i 0.216183i
5.8 0.359097 − 1.34017i −0.694508 + 1.58671i 0.0649546 + 0.0375016i 1.79870 − 0.481960i 1.87706 + 1.50054i −0.653846 + 2.44019i 2.03572 − 2.03572i −2.03532 − 2.20397i − 2.58363i
5.9 0.407254 − 1.51989i 1.33706 + 1.10103i −0.412171 − 0.237967i −1.74067 + 0.466411i 2.21797 − 1.58380i 0.556891 − 2.07835i 1.69574 − 1.69574i 0.575484 + 2.94429i 2.83558i
5.10 0.496218 − 1.85191i −0.584991 − 1.63027i −1.45130 − 0.837906i 0.863788 − 0.231451i −3.30940 + 0.274380i −0.426917 + 1.59328i 0.439499 − 0.439499i −2.31557 + 1.90739i − 1.71451i
5.11 0.685712 − 2.55911i −1.63413 + 0.574122i −4.34680 − 2.50963i −1.47168 + 0.394337i 0.348700 + 4.57561i 0.901137 − 3.36309i −5.65626 + 5.65626i 2.34077 − 1.87638i 4.03661i
47.1 −0.600405 − 2.24074i 0.724368 + 1.57331i −2.92839 + 1.69071i 3.78248 + 1.01351i 3.09046 − 2.56774i −0.608850 − 2.27226i 2.26599 + 2.26599i −1.95058 + 2.27931i − 9.08409i
47.2 −0.535189 − 1.99735i 1.67588 − 0.437507i −1.97094 + 1.13792i −2.39559 − 0.641897i −1.77077 − 3.11318i −0.185908 − 0.693817i 0.403326 + 0.403326i 2.61718 − 1.46642i 5.12838i
47.3 −0.337526 − 1.25966i 0.425920 − 1.67887i 0.259225 − 0.149663i 2.31922 + 0.621434i −2.25856 + 0.0301452i 0.940006 + 3.50815i −2.12029 − 2.12029i −2.63718 − 1.43012i − 3.13119i
47.4 −0.267684 − 0.999012i −1.73168 + 0.0358377i 0.805681 − 0.465160i 1.80337 + 0.483213i 0.499346 + 1.72038i −1.07064 − 3.99567i −2.14302 − 2.14302i 2.99743 − 0.124119i − 1.93094i
47.5 −0.126147 − 0.470787i −0.933667 − 1.45886i 1.52632 − 0.881223i −3.78110 − 1.01314i −0.569032 + 0.623589i 0.139526 + 0.520717i −1.29669 − 1.29669i −1.25653 + 2.72417i 1.90790i
47.6 −0.120326 − 0.449064i −0.453962 + 1.67150i 1.54487 − 0.891932i 0.207401 + 0.0555730i 0.805235 + 0.00273231i 0.845006 + 3.15361i −1.24390 − 1.24390i −2.58784 − 1.51760i − 0.0998232i
47.7 0.0389963 + 0.145536i 1.36970 + 1.06015i 1.71239 − 0.988649i −1.38592 − 0.371357i −0.100877 + 0.240683i −0.436407 − 1.62869i 0.423741 + 0.423741i 0.752165 + 2.90418i − 0.216183i
47.8 0.359097 + 1.34017i −0.694508 − 1.58671i 0.0649546 − 0.0375016i 1.79870 + 0.481960i 1.87706 − 1.50054i −0.653846 − 2.44019i 2.03572 + 2.03572i −2.03532 + 2.20397i 2.58363i
47.9 0.407254 + 1.51989i 1.33706 − 1.10103i −0.412171 + 0.237967i −1.74067 − 0.466411i 2.21797 + 1.58380i 0.556891 + 2.07835i 1.69574 + 1.69574i 0.575484 − 2.94429i − 2.83558i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.d odd 4 1 inner
117.z even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.z.b ✓ 44
3.b odd 2 1 351.2.bc.b 44
9.c even 3 1 351.2.bc.b 44
9.d odd 6 1 inner 117.2.z.b ✓ 44
13.d odd 4 1 inner 117.2.z.b ✓ 44
39.f even 4 1 351.2.bc.b 44
117.y odd 12 1 351.2.bc.b 44
117.z even 12 1 inner 117.2.z.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.z.b ✓ 44 1.a even 1 1 trivial
117.2.z.b ✓ 44 9.d odd 6 1 inner
117.2.z.b ✓ 44 13.d odd 4 1 inner
117.2.z.b ✓ 44 117.z even 12 1 inner
351.2.bc.b 44 3.b odd 2 1
351.2.bc.b 44 9.c even 3 1
351.2.bc.b 44 39.f even 4 1
351.2.bc.b 44 117.y odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} - 62 T_{2}^{40} - 30 T_{2}^{39} + 180 T_{2}^{37} + 2883 T_{2}^{36} + 1860 T_{2}^{35} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display