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Properties

Label 170.2.o.b
Level $170$
Weight $2$
Character orbit 170.o
Analytic conductor $1.357$
Analytic rank $0$
Dimension $40$
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] = [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: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{10} - 8 q^{15} - 16 q^{18} - 16 q^{20} - 8 q^{25} - 8 q^{26} + 24 q^{27} + 8 q^{28} - 8 q^{29} - 16 q^{31} + 32 q^{33} - 8 q^{34} - 32 q^{35} + 16 q^{37} + 32 q^{39} + 8 q^{40} - 56 q^{41} - 8 q^{42} - 48 q^{43} - 16 q^{44} - 24 q^{45} - 96 q^{47} - 16 q^{49} - 32 q^{51} - 16 q^{52} - 40 q^{53} + 24 q^{54} + 8 q^{55} - 8 q^{56} - 8 q^{57} + 16 q^{58} + 24 q^{61} - 24 q^{62} - 24 q^{63} + 16 q^{65} - 16 q^{67} + 24 q^{68} + 8 q^{70} + 24 q^{71} + 16 q^{73} - 32 q^{74} + 184 q^{75} + 40 q^{77} + 16 q^{78} + 104 q^{79} + 8 q^{80} + 48 q^{81} + 56 q^{82} + 16 q^{83} - 8 q^{85} + 96 q^{86} - 8 q^{87} + 16 q^{88} + 16 q^{89} + 40 q^{90} + 48 q^{91} + 8 q^{92} + 136 q^{93} + 8 q^{94} + 8 q^{95} + 144 q^{97} - 160 q^{99}+O(q^{100}) \) 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 −1.65666 − 2.47936i −0.707107 − 0.707107i 0.488119 − 2.18214i −2.92460 + 0.581740i 0.420899 + 2.11600i −0.923880 + 0.382683i −2.25467 + 5.44325i −1.82924 − 1.28603i
3.2 0.382683 − 0.923880i −0.419918 − 0.628452i −0.707107 − 0.707107i 2.19553 + 0.423862i −0.741309 + 0.147456i −0.252926 − 1.27154i −0.923880 + 0.382683i 0.929430 − 2.24384i 1.23179 − 1.86620i
3.3 0.382683 − 0.923880i −0.288849 − 0.432293i −0.707107 − 0.707107i −2.21991 − 0.268322i −0.509925 + 0.101430i −0.618262 − 3.10821i −0.923880 + 0.382683i 1.04461 − 2.52190i −1.09742 + 1.94825i
3.4 0.382683 − 0.923880i 0.942312 + 1.41027i −0.707107 − 0.707107i −0.577195 + 2.16029i 1.66353 − 0.330896i 0.999670 + 5.02568i −0.923880 + 0.382683i 0.0471409 − 0.113808i 1.77496 + 1.35996i
3.5 0.382683 − 0.923880i 1.42311 + 2.12983i −0.707107 − 0.707107i 0.986476 − 2.00671i 2.51231 − 0.499730i −0.00818577 − 0.0411526i −0.923880 + 0.382683i −1.36290 + 3.29034i −1.47645 − 1.67932i
7.1 0.923880 − 0.382683i −0.664558 − 3.34096i 0.707107 − 0.707107i 1.48512 + 1.67165i −1.89250 − 2.83233i 0.923351 − 0.616963i 0.382683 − 0.923880i −7.94873 + 3.29247i 2.01179 + 0.976070i
7.2 0.923880 − 0.382683i −0.302197 − 1.51925i 0.707107 − 0.707107i −1.88843 − 1.19743i −0.860584 − 1.28795i −0.201119 + 0.134383i 0.382683 − 0.923880i 0.554854 − 0.229828i −2.20292 − 0.383608i
7.3 0.923880 − 0.382683i 0.140490 + 0.706293i 0.707107 − 0.707107i 1.66820 + 1.48900i 0.400083 + 0.598766i −2.55439 + 1.70679i 0.382683 − 0.923880i 2.29253 − 0.949595i 2.11103 + 0.737267i
7.4 0.923880 − 0.382683i 0.253568 + 1.27477i 0.707107 − 0.707107i 0.580724 − 2.15934i 0.722100 + 1.08070i 0.192645 − 0.128721i 0.382683 − 0.923880i 1.21089 − 0.501569i −0.289826 − 2.21721i
7.5 0.923880 − 0.382683i 0.572697 + 2.87914i 0.707107 − 0.707107i −1.95326 + 1.08847i 1.63090 + 2.44082i 0.332949 − 0.222469i 0.382683 − 0.923880i −5.18983 + 2.14970i −1.38804 + 1.75310i
27.1 −0.923880 + 0.382683i −2.97990 + 0.592739i 0.707107 − 0.707107i −0.552685 + 2.16669i 2.52624 − 1.68798i −0.562808 − 0.842302i −0.382683 + 0.923880i 5.75684 − 2.38456i −0.318541 − 2.21326i
27.2 −0.923880 + 0.382683i −1.83338 + 0.364682i 0.707107 − 0.707107i −0.482408 − 2.18341i 1.55427 − 1.03853i 2.66660 + 3.99085i −0.382683 + 0.923880i 0.456659 − 0.189154i 1.28124 + 1.83260i
27.3 −0.923880 + 0.382683i 0.0897304 − 0.0178485i 0.707107 − 0.707107i −1.64234 − 1.51747i −0.0760698 + 0.0508282i −2.58379 − 3.86692i −0.382683 + 0.923880i −2.76391 + 1.14485i 2.09804 + 0.773458i
27.4 −0.923880 + 0.382683i 1.87274 − 0.372511i 0.707107 − 0.707107i 2.15442 − 0.598743i −1.58763 + 1.06082i −0.622065 − 0.930986i −0.382683 + 0.923880i 0.596748 − 0.247181i −1.76129 + 1.37763i
27.5 −0.923880 + 0.382683i 2.85081 − 0.567062i 0.707107 − 0.707107i −2.19776 + 0.412156i −2.41680 + 1.61486i 2.40863 + 3.60477i −0.382683 + 0.923880i 5.03395 − 2.08513i 1.87274 − 1.22183i
57.1 0.382683 + 0.923880i −1.65666 + 2.47936i −0.707107 + 0.707107i 0.488119 + 2.18214i −2.92460 − 0.581740i 0.420899 − 2.11600i −0.923880 − 0.382683i −2.25467 − 5.44325i −1.82924 + 1.28603i
57.2 0.382683 + 0.923880i −0.419918 + 0.628452i −0.707107 + 0.707107i 2.19553 − 0.423862i −0.741309 − 0.147456i −0.252926 + 1.27154i −0.923880 − 0.382683i 0.929430 + 2.24384i 1.23179 + 1.86620i
57.3 0.382683 + 0.923880i −0.288849 + 0.432293i −0.707107 + 0.707107i −2.21991 + 0.268322i −0.509925 − 0.101430i −0.618262 + 3.10821i −0.923880 − 0.382683i 1.04461 + 2.52190i −1.09742 − 1.94825i
57.4 0.382683 + 0.923880i 0.942312 − 1.41027i −0.707107 + 0.707107i −0.577195 − 2.16029i 1.66353 + 0.330896i 0.999670 − 5.02568i −0.923880 − 0.382683i 0.0471409 + 0.113808i 1.77496 − 1.35996i
57.5 0.382683 + 0.923880i 1.42311 − 2.12983i −0.707107 + 0.707107i 0.986476 + 2.00671i 2.51231 + 0.499730i −0.00818577 + 0.0411526i −0.923880 − 0.382683i −1.36290 − 3.29034i −1.47645 + 1.67932i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
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 170.2.o.b ✓ 40
5.b even 2 1 850.2.s.d 40
5.c odd 4 1 170.2.r.b yes 40
5.c odd 4 1 850.2.v.d 40
17.e odd 16 1 170.2.r.b yes 40
85.o even 16 1 inner 170.2.o.b ✓ 40
85.p odd 16 1 850.2.v.d 40
85.r even 16 1 850.2.s.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.b ✓ 40 1.a even 1 1 trivial
170.2.o.b ✓ 40 85.o even 16 1 inner
170.2.r.b yes 40 5.c odd 4 1
170.2.r.b yes 40 17.e odd 16 1
850.2.s.d 40 5.b even 2 1
850.2.s.d 40 85.r even 16 1
850.2.v.d 40 5.c odd 4 1
850.2.v.d 40 85.p odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 8 T_{3}^{37} - 48 T_{3}^{36} + 72 T_{3}^{35} + 136 T_{3}^{34} - 376 T_{3}^{33} + \cdots + 524288 \) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display