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Properties

Label 850.2.s.c
Level $850$
Weight $2$
Character orbit 850.s
Analytic conductor $6.787$
Analytic rank $0$
Dimension $32$
CM no
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(7,850)]
 
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, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("850.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 850.s (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: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{18} + 8 q^{26} + 72 q^{27} - 8 q^{28} + 8 q^{29} - 16 q^{31} + 64 q^{33} - 24 q^{34} - 16 q^{37} + 32 q^{39} + 16 q^{41} + 40 q^{42} - 48 q^{43} + 16 q^{44} + 64 q^{47} + 16 q^{49} + 32 q^{51} + 16 q^{52} - 24 q^{54} + 8 q^{56} + 8 q^{57} + 16 q^{58} + 64 q^{59} - 24 q^{61} + 24 q^{62} + 24 q^{63} + 16 q^{67} + 16 q^{68} + 8 q^{71} - 16 q^{73} - 8 q^{74} - 40 q^{77} - 48 q^{78} - 72 q^{79} + 48 q^{81} - 16 q^{82} - 16 q^{83} - 64 q^{86} - 24 q^{87} - 16 q^{88} - 16 q^{89} + 48 q^{91} - 8 q^{92} - 8 q^{93} - 8 q^{94} - 16 q^{97} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.923880 − 0.382683i −0.491482 − 2.47085i 0.707107 − 0.707107i 0 −1.39962 − 2.09468i −3.89855 + 2.60493i 0.382683 − 0.923880i −3.09190 + 1.28071i 0
7.2 0.923880 − 0.382683i −0.268393 − 1.34931i 0.707107 − 0.707107i 0 −0.764320 − 1.14389i 3.27846 − 2.19060i 0.382683 − 0.923880i 1.02305 − 0.423761i 0
7.3 0.923880 − 0.382683i 0.241738 + 1.21530i 0.707107 − 0.707107i 0 0.688411 + 1.03028i 0.590918 − 0.394839i 0.382683 − 0.923880i 1.35312 − 0.560483i 0
7.4 0.923880 − 0.382683i 0.518138 + 2.60485i 0.707107 − 0.707107i 0 1.47553 + 2.20829i 1.33574 − 0.892510i 0.382683 − 0.923880i −3.74516 + 1.55130i 0
57.1 0.382683 + 0.923880i −1.56304 + 2.33925i −0.707107 + 0.707107i 0 −2.75933 − 0.548865i −0.254312 + 1.27851i −0.923880 − 0.382683i −1.88095 − 4.54102i 0
57.2 0.382683 + 0.923880i 0.189672 − 0.283864i −0.707107 + 0.707107i 0 0.334840 + 0.0666039i 0.701012 − 3.52423i −0.923880 − 0.382683i 1.10345 + 2.66396i 0
57.3 0.382683 + 0.923880i 0.572498 − 0.856804i −0.707107 + 0.707107i 0 1.01067 + 0.201035i −0.326988 + 1.64388i −0.923880 − 0.382683i 0.741692 + 1.79060i 0
57.4 0.382683 + 0.923880i 0.800867 − 1.19858i −0.707107 + 0.707107i 0 1.41382 + 0.281227i −0.660909 + 3.32261i −0.923880 − 0.382683i 0.352840 + 0.851831i 0
243.1 0.923880 + 0.382683i −0.491482 + 2.47085i 0.707107 + 0.707107i 0 −1.39962 + 2.09468i −3.89855 − 2.60493i 0.382683 + 0.923880i −3.09190 − 1.28071i 0
243.2 0.923880 + 0.382683i −0.268393 + 1.34931i 0.707107 + 0.707107i 0 −0.764320 + 1.14389i 3.27846 + 2.19060i 0.382683 + 0.923880i 1.02305 + 0.423761i 0
243.3 0.923880 + 0.382683i 0.241738 − 1.21530i 0.707107 + 0.707107i 0 0.688411 − 1.03028i 0.590918 + 0.394839i 0.382683 + 0.923880i 1.35312 + 0.560483i 0
243.4 0.923880 + 0.382683i 0.518138 − 2.60485i 0.707107 + 0.707107i 0 1.47553 − 2.20829i 1.33574 + 0.892510i 0.382683 + 0.923880i −3.74516 − 1.55130i 0
343.1 0.382683 − 0.923880i −1.56304 − 2.33925i −0.707107 − 0.707107i 0 −2.75933 + 0.548865i −0.254312 − 1.27851i −0.923880 + 0.382683i −1.88095 + 4.54102i 0
343.2 0.382683 − 0.923880i 0.189672 + 0.283864i −0.707107 − 0.707107i 0 0.334840 − 0.0666039i 0.701012 + 3.52423i −0.923880 + 0.382683i 1.10345 − 2.66396i 0
343.3 0.382683 − 0.923880i 0.572498 + 0.856804i −0.707107 − 0.707107i 0 1.01067 − 0.201035i −0.326988 − 1.64388i −0.923880 + 0.382683i 0.741692 − 1.79060i 0
343.4 0.382683 − 0.923880i 0.800867 + 1.19858i −0.707107 − 0.707107i 0 1.41382 − 0.281227i −0.660909 − 3.32261i −0.923880 + 0.382683i 0.352840 − 0.851831i 0
643.1 −0.382683 + 0.923880i −1.70069 + 1.13636i −0.707107 − 0.707107i 0 −0.399038 − 2.00610i 1.27188 − 0.252993i 0.923880 − 0.382683i 0.452969 − 1.09356i 0
643.2 −0.382683 + 0.923880i −1.02222 + 0.683024i −0.707107 − 0.707107i 0 −0.239846 − 1.20579i −0.156756 + 0.0311808i 0.923880 − 0.382683i −0.569643 + 1.37524i 0
643.3 −0.382683 + 0.923880i 1.16272 − 0.776902i −0.707107 − 0.707107i 0 0.272812 + 1.37152i −3.77012 + 0.749924i 0.923880 − 0.382683i −0.399719 + 0.965007i 0
643.4 −0.382683 + 0.923880i 1.56019 − 1.04249i −0.707107 − 0.707107i 0 0.366072 + 1.84037i 3.19619 − 0.635763i 0.923880 − 0.382683i 0.199368 − 0.481317i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.o 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.s.c 32
5.b even 2 1 170.2.o.a ✓ 32
5.c odd 4 1 170.2.r.a yes 32
5.c odd 4 1 850.2.v.c 32
17.e odd 16 1 850.2.v.c 32
85.o even 16 1 inner 850.2.s.c 32
85.p odd 16 1 170.2.r.a yes 32
85.r even 16 1 170.2.o.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.a ✓ 32 5.b even 2 1
170.2.o.a ✓ 32 85.r even 16 1
170.2.r.a yes 32 5.c odd 4 1
170.2.r.a yes 32 85.p odd 16 1
850.2.s.c 32 1.a even 1 1 trivial
850.2.s.c 32 85.o even 16 1 inner
850.2.v.c 32 5.c odd 4 1
850.2.v.c 32 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 24 T_{3}^{29} - 48 T_{3}^{28} - 72 T_{3}^{27} + 72 T_{3}^{26} + 1336 T_{3}^{25} + \cdots + 3844 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display