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Properties

Label 850.2.v.a
Level $850$
Weight $2$
Character orbit 850.v
Analytic conductor $6.787$
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] = [850,2,Mod(107,850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(850, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("850.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.v (of order \(16\), degree \(8\), 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: \(6.78728417181\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{9} - 32 q^{13} - 16 q^{18} + 48 q^{27} + 16 q^{29} + 16 q^{31} - 8 q^{33} - 16 q^{34} + 16 q^{37} + 32 q^{39} + 48 q^{41} - 48 q^{42} + 16 q^{43} - 16 q^{44} + 32 q^{46} + 8 q^{48} + 16 q^{49}+ \cdots - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 0.382683 − 0.923880i −2.42161 − 0.481688i −0.707107 − 0.707107i 0 −1.37173 + 2.05294i 1.11717 − 1.67197i −0.923880 + 0.382683i 2.86054 + 1.18487i 0
107.2 0.382683 − 0.923880i −0.133168 − 0.0264888i −0.707107 − 0.707107i 0 −0.0754336 + 0.112894i −1.64260 + 2.45833i −0.923880 + 0.382683i −2.75461 − 1.14100i 0
107.3 0.382683 − 0.923880i 2.17210 + 0.432057i −0.707107 − 0.707107i 0 1.23039 − 1.84141i 2.37319 − 3.55173i −0.923880 + 0.382683i 1.75969 + 0.728887i 0
143.1 0.382683 + 0.923880i −2.42161 + 0.481688i −0.707107 + 0.707107i 0 −1.37173 − 2.05294i 1.11717 + 1.67197i −0.923880 − 0.382683i 2.86054 − 1.18487i 0
143.2 0.382683 + 0.923880i −0.133168 + 0.0264888i −0.707107 + 0.707107i 0 −0.0754336 − 0.112894i −1.64260 − 2.45833i −0.923880 − 0.382683i −2.75461 + 1.14100i 0
143.3 0.382683 + 0.923880i 2.17210 − 0.432057i −0.707107 + 0.707107i 0 1.23039 + 1.84141i 2.37319 + 3.55173i −0.923880 − 0.382683i 1.75969 − 0.728887i 0
193.1 −0.923880 + 0.382683i −2.30201 − 1.53815i 0.707107 − 0.707107i 0 2.71541 + 0.540128i −0.686850 − 0.136623i −0.382683 + 0.923880i 1.78528 + 4.31005i 0
193.2 −0.923880 + 0.382683i 1.23998 + 0.828527i 0.707107 − 0.707107i 0 −1.46265 − 0.290940i −3.43494 − 0.683252i −0.382683 + 0.923880i −0.296960 − 0.716925i 0
193.3 −0.923880 + 0.382683i 1.98591 + 1.32694i 0.707107 − 0.707107i 0 −2.34254 − 0.465960i 4.88716 + 0.972116i −0.382683 + 0.923880i 1.03501 + 2.49874i 0
207.1 −0.923880 − 0.382683i −2.30201 + 1.53815i 0.707107 + 0.707107i 0 2.71541 − 0.540128i −0.686850 + 0.136623i −0.382683 − 0.923880i 1.78528 − 4.31005i 0
207.2 −0.923880 − 0.382683i 1.23998 − 0.828527i 0.707107 + 0.707107i 0 −1.46265 + 0.290940i −3.43494 + 0.683252i −0.382683 − 0.923880i −0.296960 + 0.716925i 0
207.3 −0.923880 − 0.382683i 1.98591 − 1.32694i 0.707107 + 0.707107i 0 −2.34254 + 0.465960i 4.88716 − 0.972116i −0.382683 − 0.923880i 1.03501 − 2.49874i 0
507.1 0.923880 + 0.382683i −1.86340 − 2.78878i 0.707107 + 0.707107i 0 −0.654339 − 3.28959i −0.764874 − 3.84528i 0.382683 + 0.923880i −3.15696 + 7.62158i 0
507.2 0.923880 + 0.382683i 0.00766459 + 0.0114709i 0.707107 + 0.707107i 0 0.00269145 + 0.0135308i −0.407145 − 2.04686i 0.382683 + 0.923880i 1.14798 − 2.77146i 0
507.3 0.923880 + 0.382683i 0.931857 + 1.39462i 0.707107 + 0.707107i 0 0.327225 + 1.64507i 0.406652 + 2.04438i 0.382683 + 0.923880i 0.0714357 − 0.172461i 0
607.1 −0.382683 + 0.923880i −0.234689 + 1.17986i −0.707107 − 0.707107i 0 −1.00024 − 0.668337i −0.808827 − 0.540441i 0.923880 − 0.382683i 1.43465 + 0.594250i 0
607.2 −0.382683 + 0.923880i 0.180053 − 0.905187i −0.707107 − 0.707107i 0 0.767380 + 0.512747i −2.98491 − 1.99445i 0.923880 − 0.382683i 1.98469 + 0.822087i 0
607.3 −0.382683 + 0.923880i 0.437319 − 2.19855i −0.707107 − 0.707107i 0 1.86384 + 1.24538i 1.94597 + 1.30026i 0.923880 − 0.382683i −1.87075 − 0.774889i 0
793.1 0.923880 − 0.382683i −1.86340 + 2.78878i 0.707107 − 0.707107i 0 −0.654339 + 3.28959i −0.764874 + 3.84528i 0.382683 − 0.923880i −3.15696 − 7.62158i 0
793.2 0.923880 − 0.382683i 0.00766459 − 0.0114709i 0.707107 − 0.707107i 0 0.00269145 − 0.0135308i −0.407145 + 2.04686i 0.382683 − 0.923880i 1.14798 + 2.77146i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.r even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 850.2.v.a yes 24
5.b even 2 1 850.2.v.b yes 24
5.c odd 4 1 850.2.s.a ✓ 24
5.c odd 4 1 850.2.s.b yes 24
17.e odd 16 1 850.2.s.b yes 24
85.o even 16 1 850.2.v.b yes 24
85.p odd 16 1 850.2.s.a ✓ 24
85.r even 16 1 inner 850.2.v.a yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
850.2.s.a ✓ 24 5.c odd 4 1
850.2.s.a ✓ 24 85.p odd 16 1
850.2.s.b yes 24 5.c odd 4 1
850.2.s.b yes 24 17.e odd 16 1
850.2.v.a yes 24 1.a even 1 1 trivial
850.2.v.a yes 24 85.r even 16 1 inner
850.2.v.b yes 24 5.b even 2 1
850.2.v.b yes 24 85.o even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 4 T_{3}^{22} - 16 T_{3}^{21} + 58 T_{3}^{20} + 56 T_{3}^{19} - 356 T_{3}^{18} + 200 T_{3}^{17} + \cdots + 2 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display