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Properties

Label 170.3.p.a
Level $170$
Weight $3$
Character orbit 170.p
Analytic conductor $4.632$
Analytic rank $0$
Dimension $48$
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] = [170,3,Mod(11,170)]
 
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([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 170.p (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: \(4.63216449413\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{3} + 16 q^{6} - 16 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{3} + 16 q^{6} - 16 q^{7} + 32 q^{9} + 48 q^{11} + 32 q^{12} - 48 q^{13} + 32 q^{14} - 16 q^{17} - 32 q^{18} - 128 q^{19} + 160 q^{21} - 144 q^{22} - 48 q^{23} - 64 q^{24} - 64 q^{27} + 144 q^{31} - 48 q^{34} + 64 q^{36} + 128 q^{37} + 96 q^{38} - 352 q^{39} + 240 q^{41} - 160 q^{42} + 96 q^{43} + 160 q^{45} + 160 q^{46} - 48 q^{47} + 64 q^{48} + 32 q^{49} + 192 q^{51} + 64 q^{53} + 112 q^{54} - 80 q^{55} + 176 q^{57} - 256 q^{58} - 160 q^{60} - 352 q^{61} + 192 q^{62} - 832 q^{63} - 400 q^{65} - 208 q^{66} + 64 q^{69} - 80 q^{70} + 16 q^{71} + 288 q^{72} - 192 q^{73} + 160 q^{74} + 160 q^{76} + 32 q^{77} - 160 q^{78} + 384 q^{79} - 256 q^{81} - 320 q^{82} + 144 q^{83} - 160 q^{85} - 32 q^{86} + 960 q^{87} + 64 q^{88} + 1056 q^{89} - 160 q^{90} - 544 q^{91} - 128 q^{92} + 176 q^{94} - 64 q^{96} + 96 q^{97} - 432 q^{98} - 992 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.30656 − 0.541196i −1.03104 − 5.18337i 1.41421 + 1.41421i −1.85922 − 1.24229i −1.45810 + 7.33039i −9.97529 + 6.66528i −1.08239 − 2.61313i −17.4893 + 7.24432i 1.75687 + 2.62934i
11.2 −1.30656 − 0.541196i −0.584205 − 2.93700i 1.41421 + 1.41421i −1.85922 − 1.24229i −0.826191 + 4.15354i 7.24051 − 4.83795i −1.08239 − 2.61313i 0.0302576 − 0.0125331i 1.75687 + 2.62934i
11.3 −1.30656 − 0.541196i −0.475109 − 2.38853i 1.41421 + 1.41421i 1.85922 + 1.24229i −0.671906 + 3.37790i 2.54976 − 1.70369i −1.08239 − 2.61313i 2.83555 − 1.17452i −1.75687 − 2.62934i
11.4 −1.30656 − 0.541196i −0.112730 − 0.566734i 1.41421 + 1.41421i 1.85922 + 1.24229i −0.159425 + 0.801483i −10.5747 + 7.06582i −1.08239 − 2.61313i 8.00644 − 3.31637i −1.75687 − 2.62934i
11.5 −1.30656 − 0.541196i 0.133606 + 0.671685i 1.41421 + 1.41421i −1.85922 − 1.24229i 0.188948 − 0.949906i −0.994766 + 0.664681i −1.08239 − 2.61313i 7.88161 − 3.26467i 1.75687 + 2.62934i
11.6 −1.30656 − 0.541196i 0.718320 + 3.61124i 1.41421 + 1.41421i 1.85922 + 1.24229i 1.01586 − 5.10706i 7.14141 − 4.77174i −1.08239 − 2.61313i −4.21015 + 1.74390i −1.75687 − 2.62934i
31.1 −1.30656 + 0.541196i −1.03104 + 5.18337i 1.41421 − 1.41421i −1.85922 + 1.24229i −1.45810 − 7.33039i −9.97529 − 6.66528i −1.08239 + 2.61313i −17.4893 − 7.24432i 1.75687 − 2.62934i
31.2 −1.30656 + 0.541196i −0.584205 + 2.93700i 1.41421 − 1.41421i −1.85922 + 1.24229i −0.826191 − 4.15354i 7.24051 + 4.83795i −1.08239 + 2.61313i 0.0302576 + 0.0125331i 1.75687 − 2.62934i
31.3 −1.30656 + 0.541196i −0.475109 + 2.38853i 1.41421 − 1.41421i 1.85922 − 1.24229i −0.671906 − 3.37790i 2.54976 + 1.70369i −1.08239 + 2.61313i 2.83555 + 1.17452i −1.75687 + 2.62934i
31.4 −1.30656 + 0.541196i −0.112730 + 0.566734i 1.41421 − 1.41421i 1.85922 − 1.24229i −0.159425 − 0.801483i −10.5747 − 7.06582i −1.08239 + 2.61313i 8.00644 + 3.31637i −1.75687 + 2.62934i
31.5 −1.30656 + 0.541196i 0.133606 − 0.671685i 1.41421 − 1.41421i −1.85922 + 1.24229i 0.188948 + 0.949906i −0.994766 − 0.664681i −1.08239 + 2.61313i 7.88161 + 3.26467i 1.75687 − 2.62934i
31.6 −1.30656 + 0.541196i 0.718320 − 3.61124i 1.41421 − 1.41421i 1.85922 − 1.24229i 1.01586 + 5.10706i 7.14141 + 4.77174i −1.08239 + 2.61313i −4.21015 − 1.74390i −1.75687 + 2.62934i
41.1 0.541196 − 1.30656i −2.35318 + 3.52178i −1.41421 − 1.41421i 0.436235 + 2.19310i 3.32790 + 4.98055i 1.02634 − 5.15977i −2.61313 + 1.08239i −3.42135 − 8.25987i 3.10152 + 0.616930i
41.2 0.541196 − 1.30656i −2.16846 + 3.24533i −1.41421 − 1.41421i −0.436235 − 2.19310i 3.06667 + 4.58960i 1.24487 − 6.25838i −2.61313 + 1.08239i −2.38581 − 5.75986i −3.10152 − 0.616930i
41.3 0.541196 − 1.30656i −1.47019 + 2.20030i −1.41421 − 1.41421i −0.436235 − 2.19310i 2.07916 + 3.11169i −1.76874 + 8.89206i −2.61313 + 1.08239i 0.764311 + 1.84521i −3.10152 − 0.616930i
41.4 0.541196 − 1.30656i −0.523937 + 0.784127i −1.41421 − 1.41421i 0.436235 + 2.19310i 0.740959 + 1.10892i −2.10901 + 10.6027i −2.61313 + 1.08239i 3.10381 + 7.49325i 3.10152 + 0.616930i
41.5 0.541196 − 1.30656i 1.90462 − 2.85046i −1.41421 − 1.41421i −0.436235 − 2.19310i −2.69353 − 4.03116i 0.871125 − 4.37944i −2.61313 + 1.08239i −1.05341 − 2.54315i −3.10152 − 0.616930i
41.6 0.541196 − 1.30656i 3.04470 − 4.55672i −1.41421 − 1.41421i 0.436235 + 2.19310i −4.30586 − 6.44417i −0.182189 + 0.915927i −2.61313 + 1.08239i −8.04931 − 19.4328i 3.10152 + 0.616930i
61.1 −0.541196 + 1.30656i −4.35941 − 2.91286i −1.41421 − 1.41421i −2.19310 + 0.436235i 6.16513 − 4.11941i −8.11564 − 1.61430i 2.61313 − 1.08239i 7.07551 + 17.0818i 0.616930 − 3.10152i
61.2 −0.541196 + 1.30656i −3.70059 − 2.47266i −1.41421 − 1.41421i 2.19310 − 0.436235i 5.23343 − 3.49687i 1.23433 + 0.245523i 2.61313 − 1.08239i 4.13621 + 9.98569i −0.616930 + 3.10152i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.3.p.a ✓ 48
17.e odd 16 1 inner 170.3.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.3.p.a ✓ 48 1.a even 1 1 trivial
170.3.p.a ✓ 48 17.e odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 16 T_{3}^{47} + 112 T_{3}^{46} + 352 T_{3}^{45} - 212 T_{3}^{44} - 7104 T_{3}^{43} + \cdots + 56\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display