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Properties

Label 115.3.f.a
Level $115$
Weight $3$
Character orbit 115.f
Analytic conductor $3.134$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(47,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.f (of order \(4\), 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: \(3.13352304014\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} - 8 q^{5} - 16 q^{7} + 12 q^{8} - 4 q^{10} - 24 q^{11} + 48 q^{12} + 4 q^{13} + 60 q^{15} - 224 q^{16} + 24 q^{17} - 88 q^{18} - 56 q^{20} + 8 q^{21} - 48 q^{22} + 84 q^{25} + 56 q^{26} - 132 q^{27}+ \cdots + 780 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −2.74685 − 2.74685i 3.72491 − 3.72491i 11.0904i 4.51806 − 2.14175i −20.4635 −2.87693 − 2.87693i 19.4763 − 19.4763i − 18.7498i −18.2935 − 6.52737i
47.2 −2.67250 − 2.67250i −2.93217 + 2.93217i 10.2845i −1.24988 + 4.84126i 15.6725 −7.73999 − 7.73999i 16.7954 − 16.7954i − 8.19524i 16.2786 − 9.59798i
47.3 −2.33506 − 2.33506i 1.23717 − 1.23717i 6.90499i −4.37439 + 2.42172i −5.77773 9.05038 + 9.05038i 6.78332 − 6.78332i 5.93882i 15.8693 + 4.55958i
47.4 −2.11083 − 2.11083i −0.751612 + 0.751612i 4.91121i −1.47659 − 4.77700i 3.17305 −3.30771 − 3.30771i 1.92342 − 1.92342i 7.87016i −6.96661 + 13.2003i
47.5 −1.95143 − 1.95143i −0.210269 + 0.210269i 3.61618i 4.62528 + 1.89915i 0.820651 2.37158 + 2.37158i −0.749007 + 0.749007i 8.91157i −5.31987 − 12.7320i
47.6 −1.46415 − 1.46415i 3.27120 − 3.27120i 0.287471i −4.76493 + 1.51507i −9.57906 −5.58012 − 5.58012i −5.43570 + 5.43570i − 12.4015i 9.19486 + 4.75829i
47.7 −1.45626 − 1.45626i −3.93198 + 3.93198i 0.241409i −4.31815 − 2.52063i 11.4520 5.37896 + 5.37896i −5.47350 + 5.47350i − 21.9209i 2.61766 + 9.95907i
47.8 −1.06417 − 1.06417i −3.13999 + 3.13999i − 1.73510i 4.95922 + 0.637278i 6.68293 −2.16662 − 2.16662i −6.10310 + 6.10310i − 10.7190i −4.59926 − 5.95560i
47.9 −0.888159 − 0.888159i 1.93967 − 1.93967i − 2.42235i 1.78108 − 4.67202i −3.44547 0.600352 + 0.600352i −5.70407 + 5.70407i 1.47535i −5.73138 + 2.56762i
47.10 −0.616305 − 0.616305i −1.36474 + 1.36474i − 3.24034i −1.00227 + 4.89852i 1.68219 2.04545 + 2.04545i −4.46226 + 4.46226i 5.27497i 3.63668 − 2.40128i
47.11 −0.306853 − 0.306853i 3.69350 − 3.69350i − 3.81168i 2.97062 + 4.02187i −2.26672 8.01534 + 8.01534i −2.39704 + 2.39704i − 18.2839i 0.322578 − 2.14567i
47.12 −0.275364 − 0.275364i −0.547869 + 0.547869i − 3.84835i −4.58859 + 1.98617i 0.301727 −4.59988 − 4.59988i −2.16115 + 2.16115i 8.39968i 1.81045 + 0.716611i
47.13 0.605975 + 0.605975i 1.58126 − 1.58126i − 3.26559i 4.51316 + 2.15207i 1.91641 −8.09437 − 8.09437i 4.40277 − 4.40277i 3.99921i 1.43076 + 4.03896i
47.14 0.694377 + 0.694377i −1.86203 + 1.86203i − 3.03568i 3.90318 − 3.12494i −2.58590 7.40287 + 7.40287i 4.88542 − 4.88542i 2.06569i 4.88016 + 0.540392i
47.15 0.701445 + 0.701445i −2.49569 + 2.49569i − 3.01595i −1.50802 − 4.76717i −3.50118 −7.98271 − 7.98271i 4.92130 − 4.92130i − 3.45692i 2.28611 − 4.40170i
47.16 1.03401 + 1.03401i 2.52164 − 2.52164i − 1.86167i −4.44006 − 2.29910i 5.21478 1.69314 + 1.69314i 6.06099 − 6.06099i − 3.71736i −2.21377 − 6.96833i
47.17 1.56641 + 1.56641i 0.265917 − 0.265917i 0.907288i −0.310831 + 4.99033i 0.833069 3.53378 + 3.53378i 4.84446 − 4.84446i 8.85858i −8.30380 + 7.33002i
47.18 1.71185 + 1.71185i −3.25485 + 3.25485i 1.86083i −4.79289 + 1.42415i −11.1436 2.88120 + 2.88120i 3.66193 − 3.66193i − 12.1881i −10.6426 − 5.76677i
47.19 2.13032 + 2.13032i 1.62695 − 1.62695i 5.07649i 2.93890 − 4.04510i 6.93185 −0.365078 − 0.365078i −2.29326 + 2.29326i 3.70603i 14.8781 − 2.35655i
47.20 2.29778 + 2.29778i −2.60211 + 2.60211i 6.55958i 4.64561 + 1.84886i −11.9581 −4.55541 − 4.55541i −5.88134 + 5.88134i − 4.54191i 6.42633 + 14.9229i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.f.a ✓ 44
5.c odd 4 1 inner 115.3.f.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.f.a ✓ 44 1.a even 1 1 trivial
115.3.f.a ✓ 44 5.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(115, [\chi])\).