Properties

Label 750.2.o.a
Level $750$
Weight $2$
Character orbit 750.o
Analytic conductor $5.989$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [750,2,Mod(19,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.o (of order \(50\), degree \(20\), 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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 10 q^{11} + 20 q^{13} - 10 q^{17} + 10 q^{19} + 60 q^{23} - 60 q^{24} + 60 q^{25} + 10 q^{28} + 20 q^{29} + 10 q^{30} - 30 q^{31} - 10 q^{34} - 10 q^{35} + 20 q^{37} - 90 q^{38} - 20 q^{41} - 10 q^{42} - 10 q^{46} + 140 q^{47} + 50 q^{49} + 20 q^{50} + 10 q^{51} - 80 q^{52} + 30 q^{53} - 10 q^{55} - 30 q^{58} + 20 q^{59} + 40 q^{60} - 100 q^{61} - 10 q^{62} - 10 q^{63} - 20 q^{65} - 40 q^{66} - 10 q^{67} - 10 q^{69} + 10 q^{70} + 80 q^{71} + 20 q^{73} - 80 q^{75} + 10 q^{76} + 20 q^{77} + 140 q^{79} + 100 q^{82} - 70 q^{83} - 20 q^{85} + 20 q^{86} + 30 q^{87} - 10 q^{88} + 40 q^{89} + 10 q^{90} + 170 q^{91} + 120 q^{92} + 50 q^{93} - 30 q^{94} + 220 q^{95} - 60 q^{97} - 120 q^{98} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i −2.22361 0.235684i 0.187381 + 0.982287i 1.43168 + 1.97054i 0.125333 0.992115i −0.0627905 + 0.998027i 1.75117 + 1.39047i
19.2 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i −1.72567 + 1.42199i 0.187381 + 0.982287i −2.37884 3.27419i 0.125333 0.992115i −0.0627905 + 0.998027i 2.21897 0.275964i
19.3 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i −0.372997 + 2.20474i 0.187381 + 0.982287i 1.88483 + 2.59424i 0.125333 0.992115i −0.0627905 + 0.998027i 1.49629 1.66166i
19.4 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i 1.00957 1.99519i 0.187381 + 0.982287i 0.151095 + 0.207964i 0.125333 0.992115i −0.0627905 + 0.998027i −1.92148 + 1.14364i
19.5 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i 1.96917 + 1.05942i 0.187381 + 0.982287i 0.897283 + 1.23500i 0.125333 0.992115i −0.0627905 + 0.998027i −1.09496 1.94963i
19.6 −0.844328 0.535827i −0.684547 0.728969i 0.425779 + 0.904827i 2.05215 0.888075i 0.187381 + 0.982287i −1.25325 1.72495i 0.125333 0.992115i −0.0627905 + 0.998027i −2.20854 0.349771i
19.7 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i −2.13539 0.663420i 0.187381 + 0.982287i −2.59386 3.57014i −0.125333 + 0.992115i −0.0627905 + 0.998027i −1.44749 1.70434i
19.8 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i −1.34723 1.78465i 0.187381 + 0.982287i 2.76889 + 3.81104i −0.125333 + 0.992115i −0.0627905 + 0.998027i −0.181243 2.22871i
19.9 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i −0.551051 2.16710i 0.187381 + 0.982287i 0.177081 + 0.243730i −0.125333 + 0.992115i −0.0627905 + 0.998027i 0.695925 2.12501i
19.10 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i −0.461651 + 2.18789i 0.187381 + 0.982287i −0.0291323 0.0400972i −0.125333 + 0.992115i −0.0627905 + 0.998027i −1.56212 + 1.59993i
19.11 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i 2.23162 + 0.141047i 0.187381 + 0.982287i −1.07198 1.47545i −0.125333 + 0.992115i −0.0627905 + 0.998027i 1.80864 + 1.31485i
19.12 0.844328 + 0.535827i 0.684547 + 0.728969i 0.425779 + 0.904827i 2.23606 0.00611834i 0.187381 + 0.982287i 2.65738 + 3.65757i −0.125333 + 0.992115i −0.0627905 + 0.998027i 1.89125 + 1.19297i
79.1 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i −2.22361 + 0.235684i 0.187381 0.982287i 1.43168 1.97054i 0.125333 + 0.992115i −0.0627905 0.998027i 1.75117 1.39047i
79.2 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i −1.72567 1.42199i 0.187381 0.982287i −2.37884 + 3.27419i 0.125333 + 0.992115i −0.0627905 0.998027i 2.21897 + 0.275964i
79.3 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i −0.372997 2.20474i 0.187381 0.982287i 1.88483 2.59424i 0.125333 + 0.992115i −0.0627905 0.998027i 1.49629 + 1.66166i
79.4 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i 1.00957 + 1.99519i 0.187381 0.982287i 0.151095 0.207964i 0.125333 + 0.992115i −0.0627905 0.998027i −1.92148 1.14364i
79.5 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i 1.96917 1.05942i 0.187381 0.982287i 0.897283 1.23500i 0.125333 + 0.992115i −0.0627905 0.998027i −1.09496 + 1.94963i
79.6 −0.844328 + 0.535827i −0.684547 + 0.728969i 0.425779 0.904827i 2.05215 + 0.888075i 0.187381 0.982287i −1.25325 + 1.72495i 0.125333 + 0.992115i −0.0627905 0.998027i −2.20854 + 0.349771i
79.7 0.844328 0.535827i 0.684547 0.728969i 0.425779 0.904827i −2.13539 + 0.663420i 0.187381 0.982287i −2.59386 + 3.57014i −0.125333 0.992115i −0.0627905 0.998027i −1.44749 + 1.70434i
79.8 0.844328 0.535827i 0.684547 0.728969i 0.425779 0.904827i −1.34723 + 1.78465i 0.187381 0.982287i 2.76889 3.81104i −0.125333 0.992115i −0.0627905 0.998027i −0.181243 + 2.22871i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.h even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.o.a 240
125.h even 50 1 inner 750.2.o.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.2.o.a 240 1.a even 1 1 trivial
750.2.o.a 240 125.h even 50 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{240} - 235 T_{7}^{238} - 50 T_{7}^{237} + 29150 T_{7}^{236} + 11750 T_{7}^{235} - 2537660 T_{7}^{234} - 1450850 T_{7}^{233} + 174300620 T_{7}^{232} + 125000200 T_{7}^{231} - 10050921831 T_{7}^{230} + \cdots + 11\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display