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Properties

Label 850.2.v.c
Level $850$
Weight $2$
Character orbit 850.v
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(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: \(32\)
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
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 16 q^{18} + 8 q^{26} - 24 q^{27} + 8 q^{28} - 8 q^{29} - 16 q^{31} - 64 q^{33} + 24 q^{34} + 32 q^{37} - 32 q^{39} + 16 q^{41} + 24 q^{42} + 16 q^{43} - 16 q^{44} - 16 q^{49} + 32 q^{51} + 16 q^{52} - 16 q^{53} + 24 q^{54} + 8 q^{56} + 24 q^{57} + 16 q^{58} - 64 q^{59} - 24 q^{61} + 40 q^{62} + 24 q^{63} - 16 q^{67} + 8 q^{71} + 16 q^{72} - 32 q^{73} + 8 q^{74} - 24 q^{77} - 32 q^{78} + 72 q^{79} + 48 q^{81} - 48 q^{82} - 16 q^{83} - 64 q^{86} - 40 q^{87} - 32 q^{88} + 16 q^{89} + 48 q^{91} - 24 q^{92} + 8 q^{93} + 8 q^{94} + 48 q^{97} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.382683 + 0.923880i −2.60485 − 0.518138i −0.707107 − 0.707107i 0 1.47553 − 2.20829i −0.892510 + 1.33574i 0.923880 − 0.382683i 3.74516 + 1.55130i 0
107.2 −0.382683 + 0.923880i −1.21530 − 0.241738i −0.707107 − 0.707107i 0 0.688411 − 1.03028i −0.394839 + 0.590918i 0.923880 − 0.382683i −1.35312 − 0.560483i 0
107.3 −0.382683 + 0.923880i 1.34931 + 0.268393i −0.707107 − 0.707107i 0 −0.764320 + 1.14389i −2.19060 + 3.27846i 0.923880 − 0.382683i −1.02305 − 0.423761i 0
107.4 −0.382683 + 0.923880i 2.47085 + 0.491482i −0.707107 − 0.707107i 0 −1.39962 + 2.09468i 2.60493 − 3.89855i 0.923880 − 0.382683i 3.09190 + 1.28071i 0
143.1 −0.382683 − 0.923880i −2.60485 + 0.518138i −0.707107 + 0.707107i 0 1.47553 + 2.20829i −0.892510 − 1.33574i 0.923880 + 0.382683i 3.74516 − 1.55130i 0
143.2 −0.382683 − 0.923880i −1.21530 + 0.241738i −0.707107 + 0.707107i 0 0.688411 + 1.03028i −0.394839 − 0.590918i 0.923880 + 0.382683i −1.35312 + 0.560483i 0
143.3 −0.382683 − 0.923880i 1.34931 − 0.268393i −0.707107 + 0.707107i 0 −0.764320 − 1.14389i −2.19060 − 3.27846i 0.923880 + 0.382683i −1.02305 + 0.423761i 0
143.4 −0.382683 − 0.923880i 2.47085 − 0.491482i −0.707107 + 0.707107i 0 −1.39962 − 2.09468i 2.60493 + 3.89855i 0.923880 + 0.382683i 3.09190 − 1.28071i 0
193.1 0.923880 − 0.382683i −2.33925 − 1.56304i 0.707107 − 0.707107i 0 −2.75933 − 0.548865i 1.27851 + 0.254312i 0.382683 − 0.923880i 1.88095 + 4.54102i 0
193.2 0.923880 − 0.382683i 0.283864 + 0.189672i 0.707107 − 0.707107i 0 0.334840 + 0.0666039i −3.52423 − 0.701012i 0.382683 − 0.923880i −1.10345 − 2.66396i 0
193.3 0.923880 − 0.382683i 0.856804 + 0.572498i 0.707107 − 0.707107i 0 1.01067 + 0.201035i 1.64388 + 0.326988i 0.382683 − 0.923880i −0.741692 − 1.79060i 0
193.4 0.923880 − 0.382683i 1.19858 + 0.800867i 0.707107 − 0.707107i 0 1.41382 + 0.281227i 3.32261 + 0.660909i 0.382683 − 0.923880i −0.352840 − 0.851831i 0
207.1 0.923880 + 0.382683i −2.33925 + 1.56304i 0.707107 + 0.707107i 0 −2.75933 + 0.548865i 1.27851 − 0.254312i 0.382683 + 0.923880i 1.88095 − 4.54102i 0
207.2 0.923880 + 0.382683i 0.283864 − 0.189672i 0.707107 + 0.707107i 0 0.334840 − 0.0666039i −3.52423 + 0.701012i 0.382683 + 0.923880i −1.10345 + 2.66396i 0
207.3 0.923880 + 0.382683i 0.856804 − 0.572498i 0.707107 + 0.707107i 0 1.01067 − 0.201035i 1.64388 − 0.326988i 0.382683 + 0.923880i −0.741692 + 1.79060i 0
207.4 0.923880 + 0.382683i 1.19858 − 0.800867i 0.707107 + 0.707107i 0 1.41382 − 0.281227i 3.32261 − 0.660909i 0.382683 + 0.923880i −0.352840 + 0.851831i 0
507.1 −0.923880 − 0.382683i −1.04249 − 1.56019i 0.707107 + 0.707107i 0 0.366072 + 1.84037i 0.635763 + 3.19619i −0.382683 − 0.923880i −0.199368 + 0.481317i 0
507.2 −0.923880 − 0.382683i −0.776902 − 1.16272i 0.707107 + 0.707107i 0 0.272812 + 1.37152i −0.749924 − 3.77012i −0.382683 − 0.923880i 0.399719 − 0.965007i 0
507.3 −0.923880 − 0.382683i 0.683024 + 1.02222i 0.707107 + 0.707107i 0 −0.239846 − 1.20579i −0.0311808 − 0.156756i −0.382683 − 0.923880i 0.569643 − 1.37524i 0
507.4 −0.923880 − 0.382683i 1.13636 + 1.70069i 0.707107 + 0.707107i 0 −0.399038 − 2.00610i 0.252993 + 1.27188i −0.382683 − 0.923880i −0.452969 + 1.09356i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 8 T_{3}^{29} - 48 T_{3}^{28} - 168 T_{3}^{27} + 248 T_{3}^{26} + 472 T_{3}^{25} + \cdots + 3844 \) acting on \(S_{2}^{\mathrm{new}}(850, [\chi])\). Copy content Toggle raw display