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Properties

Label 170.2.o.a
Level $170$
Weight $2$
Character orbit 170.o
Analytic conductor $1.357$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [170,2,Mod(3,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([12, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.o (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: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{10} - 40 q^{15} + 16 q^{18} + 8 q^{20} - 8 q^{25} + 8 q^{26} - 72 q^{27} + 8 q^{28} + 8 q^{29} - 16 q^{31} - 64 q^{33} - 24 q^{34} + 32 q^{35} + 16 q^{37} + 32 q^{39} - 8 q^{40} + 16 q^{41} - 40 q^{42} + 48 q^{43} + 16 q^{44} + 24 q^{45} - 64 q^{47} + 16 q^{49} + 32 q^{50} + 32 q^{51} - 16 q^{52} - 24 q^{54} + 8 q^{55} + 8 q^{56} - 8 q^{57} - 16 q^{58} + 64 q^{59} - 48 q^{60} - 24 q^{61} - 24 q^{62} - 24 q^{63} - 16 q^{65} - 16 q^{67} - 16 q^{68} + 24 q^{70} + 8 q^{71} + 16 q^{73} - 8 q^{74} - 8 q^{75} + 40 q^{77} + 48 q^{78} - 72 q^{79} + 8 q^{80} + 48 q^{81} + 16 q^{82} + 16 q^{83} - 8 q^{85} - 64 q^{86} + 24 q^{87} + 16 q^{88} - 16 q^{89} + 48 q^{90} + 48 q^{91} + 8 q^{92} + 8 q^{93} - 8 q^{94} + 40 q^{95} + 16 q^{97} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.382683 + 0.923880i −0.800867 − 1.19858i −0.707107 − 0.707107i −0.598768 + 2.15441i 1.41382 − 0.281227i 0.660909 + 3.32261i 0.923880 − 0.382683i 0.352840 − 0.851831i −1.76128 − 1.37765i
3.2 −0.382683 + 0.923880i −0.572498 − 0.856804i −0.707107 − 0.707107i −1.01937 − 1.99020i 1.01067 − 0.201035i 0.326988 + 1.64388i 0.923880 − 0.382683i 0.741692 − 1.79060i 2.22880 − 0.180163i
3.3 −0.382683 + 0.923880i −0.189672 − 0.283864i −0.707107 − 0.707107i 1.79797 − 1.32941i 0.334840 − 0.0666039i −0.701012 − 3.52423i 0.923880 − 0.382683i 1.10345 − 2.66396i 0.540159 + 2.16985i
3.4 −0.382683 + 0.923880i 1.56304 + 2.33925i −0.707107 − 0.707107i −2.19349 − 0.434263i −2.75933 + 0.548865i 0.254312 + 1.27851i 0.923880 − 0.382683i −1.88095 + 4.54102i 1.24062 − 1.86034i
7.1 −0.923880 + 0.382683i −0.518138 − 2.60485i 0.707107 − 0.707107i 0.404835 − 2.19912i 1.47553 + 2.20829i −1.33574 + 0.892510i −0.382683 + 0.923880i −3.74516 + 1.55130i 0.467546 + 2.18664i
7.2 −0.923880 + 0.382683i −0.241738 − 1.21530i 0.707107 − 0.707107i 1.17123 + 1.90479i 0.688411 + 1.03028i −0.590918 + 0.394839i −0.382683 + 0.923880i 1.35312 − 0.560483i −1.81101 − 1.31159i
7.3 −0.923880 + 0.382683i 0.268393 + 1.34931i 0.707107 − 0.707107i −2.19088 + 0.447284i −0.764320 − 1.14389i −3.27846 + 2.19060i −0.382683 + 0.923880i 1.02305 − 0.423761i 1.85294 − 1.25165i
7.4 −0.923880 + 0.382683i 0.491482 + 2.47085i 0.707107 − 0.707107i 0.780722 + 2.09535i −1.39962 − 2.09468i 3.89855 − 2.60493i −0.382683 + 0.923880i −3.09190 + 1.28071i −1.52315 − 1.63708i
27.1 0.923880 − 0.382683i −3.22346 + 0.641185i 0.707107 − 0.707107i 2.10251 + 0.761213i −2.73271 + 1.82594i 1.37046 + 2.05104i 0.382683 − 0.923880i 7.20792 − 2.98562i 2.23377 − 0.101327i
27.2 0.923880 − 0.382683i 0.0811867 − 0.0161490i 0.707107 − 0.707107i 1.07264 − 1.96200i 0.0688268 − 0.0459886i 0.143301 + 0.214466i 0.382683 − 0.923880i −2.76531 + 1.14543i 0.240162 − 2.22313i
27.3 0.923880 − 0.382683i 0.917782 − 0.182558i 0.707107 − 0.707107i 0.304091 + 2.21529i 0.778058 − 0.519882i 1.08548 + 1.62454i 0.382683 − 0.923880i −1.96264 + 0.812953i 1.12870 + 1.93029i
27.4 0.923880 − 0.382683i 2.22449 − 0.442478i 0.707107 − 0.707107i −2.23094 + 0.151404i 1.88583 − 1.26007i −1.29268 − 1.93464i 0.382683 − 0.923880i 1.98092 − 0.820524i −2.00318 + 0.993622i
57.1 −0.382683 − 0.923880i −0.800867 + 1.19858i −0.707107 + 0.707107i −0.598768 − 2.15441i 1.41382 + 0.281227i 0.660909 − 3.32261i 0.923880 + 0.382683i 0.352840 + 0.851831i −1.76128 + 1.37765i
57.2 −0.382683 − 0.923880i −0.572498 + 0.856804i −0.707107 + 0.707107i −1.01937 + 1.99020i 1.01067 + 0.201035i 0.326988 − 1.64388i 0.923880 + 0.382683i 0.741692 + 1.79060i 2.22880 + 0.180163i
57.3 −0.382683 − 0.923880i −0.189672 + 0.283864i −0.707107 + 0.707107i 1.79797 + 1.32941i 0.334840 + 0.0666039i −0.701012 + 3.52423i 0.923880 + 0.382683i 1.10345 + 2.66396i 0.540159 − 2.16985i
57.4 −0.382683 − 0.923880i 1.56304 − 2.33925i −0.707107 + 0.707107i −2.19349 + 0.434263i −2.75933 − 0.548865i 0.254312 − 1.27851i 0.923880 + 0.382683i −1.88095 − 4.54102i 1.24062 + 1.86034i
63.1 0.923880 + 0.382683i −3.22346 − 0.641185i 0.707107 + 0.707107i 2.10251 − 0.761213i −2.73271 − 1.82594i 1.37046 − 2.05104i 0.382683 + 0.923880i 7.20792 + 2.98562i 2.23377 + 0.101327i
63.2 0.923880 + 0.382683i 0.0811867 + 0.0161490i 0.707107 + 0.707107i 1.07264 + 1.96200i 0.0688268 + 0.0459886i 0.143301 − 0.214466i 0.382683 + 0.923880i −2.76531 − 1.14543i 0.240162 + 2.22313i
63.3 0.923880 + 0.382683i 0.917782 + 0.182558i 0.707107 + 0.707107i 0.304091 − 2.21529i 0.778058 + 0.519882i 1.08548 − 1.62454i 0.382683 + 0.923880i −1.96264 − 0.812953i 1.12870 − 1.93029i
63.4 0.923880 + 0.382683i 2.22449 + 0.442478i 0.707107 + 0.707107i −2.23094 − 0.151404i 1.88583 + 1.26007i −1.29268 + 1.93464i 0.382683 + 0.923880i 1.98092 + 0.820524i −2.00318 − 0.993622i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
Significant digits:
Format:

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 170.2.o.a ✓ 32
5.b even 2 1 850.2.s.c 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 1 170.2.r.a yes 32
85.o even 16 1 inner 170.2.o.a ✓ 32
85.p odd 16 1 850.2.v.c 32
85.r even 16 1 850.2.s.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.a ✓ 32 1.a even 1 1 trivial
170.2.o.a ✓ 32 85.o even 16 1 inner
170.2.r.a yes 32 5.c odd 4 1
170.2.r.a yes 32 17.e odd 16 1
850.2.s.c 32 5.b even 2 1
850.2.s.c 32 85.r even 16 1
850.2.v.c 32 5.c odd 4 1
850.2.v.c 32 85.p 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 \(S_{2}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display