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Properties

Label 170.3.p.b
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
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{3} - 16 q^{6} + 16 q^{7} - 32 q^{9} - 48 q^{11} + 32 q^{12} + 144 q^{13} + 32 q^{14} - 16 q^{17} - 96 q^{18} + 32 q^{19} - 160 q^{21} - 48 q^{22} - 176 q^{23} - 64 q^{24} + 352 q^{27} - 80 q^{31} + 48 q^{34} - 64 q^{36} - 384 q^{37} + 96 q^{38} + 512 q^{39} + 624 q^{41} + 160 q^{42} - 128 q^{43} + 192 q^{44} + 160 q^{45} + 96 q^{46} + 48 q^{47} - 64 q^{48} + 32 q^{49} - 320 q^{51} - 448 q^{53} - 176 q^{54} - 240 q^{55} - 16 q^{57} - 256 q^{58} - 320 q^{59} - 160 q^{60} - 160 q^{61} - 192 q^{62} - 416 q^{63} - 80 q^{65} - 48 q^{66} - 192 q^{69} + 80 q^{70} + 272 q^{71} - 288 q^{72} + 192 q^{73} - 160 q^{74} - 160 q^{76} - 352 q^{77} + 160 q^{78} - 768 q^{79} + 320 q^{81} + 320 q^{82} + 144 q^{83} + 160 q^{85} - 32 q^{86} + 384 q^{87} - 64 q^{88} + 96 q^{89} + 160 q^{90} - 128 q^{91} + 128 q^{92} + 1024 q^{93} - 176 q^{94} + 64 q^{96} + 160 q^{97} + 432 q^{98} + 1888 q^{99}+O(q^{100}) \) 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 −1.15961 − 5.82976i 1.41421 + 1.41421i −1.85922 − 1.24229i 1.63994 − 8.24453i 8.10278 − 5.41411i 1.08239 + 2.61313i −24.3265 + 10.0764i −1.75687 − 2.62934i
11.2 1.30656 + 0.541196i −0.391450 − 1.96795i 1.41421 + 1.41421i 1.85922 + 1.24229i 0.553594 − 2.78311i 3.61392 − 2.41475i 1.08239 + 2.61313i 4.59531 − 1.90344i 1.75687 + 2.62934i
11.3 1.30656 + 0.541196i −0.208788 − 1.04965i 1.41421 + 1.41421i −1.85922 − 1.24229i 0.295271 − 1.48443i 2.87802 − 1.92303i 1.08239 + 2.61313i 7.25675 − 3.00584i −1.75687 − 2.62934i
11.4 1.30656 + 0.541196i 0.421292 + 2.11798i 1.41421 + 1.41421i 1.85922 + 1.24229i −0.595797 + 2.99527i −5.54990 + 3.70833i 1.08239 + 2.61313i 4.00657 − 1.65958i 1.75687 + 2.62934i
11.5 1.30656 + 0.541196i 0.804373 + 4.04386i 1.41421 + 1.41421i −1.85922 − 1.24229i −1.13755 + 5.71887i −10.0972 + 6.74676i 1.08239 + 2.61313i −7.39083 + 3.06138i −1.75687 − 2.62934i
11.6 1.30656 + 0.541196i 1.01825 + 5.11907i 1.41421 + 1.41421i 1.85922 + 1.24229i −1.44002 + 7.23946i 5.66553 − 3.78559i 1.08239 + 2.61313i −16.8532 + 6.98081i 1.75687 + 2.62934i
31.1 1.30656 − 0.541196i −1.15961 + 5.82976i 1.41421 − 1.41421i −1.85922 + 1.24229i 1.63994 + 8.24453i 8.10278 + 5.41411i 1.08239 − 2.61313i −24.3265 − 10.0764i −1.75687 + 2.62934i
31.2 1.30656 − 0.541196i −0.391450 + 1.96795i 1.41421 − 1.41421i 1.85922 − 1.24229i 0.553594 + 2.78311i 3.61392 + 2.41475i 1.08239 − 2.61313i 4.59531 + 1.90344i 1.75687 − 2.62934i
31.3 1.30656 − 0.541196i −0.208788 + 1.04965i 1.41421 − 1.41421i −1.85922 + 1.24229i 0.295271 + 1.48443i 2.87802 + 1.92303i 1.08239 − 2.61313i 7.25675 + 3.00584i −1.75687 + 2.62934i
31.4 1.30656 − 0.541196i 0.421292 − 2.11798i 1.41421 − 1.41421i 1.85922 − 1.24229i −0.595797 − 2.99527i −5.54990 − 3.70833i 1.08239 − 2.61313i 4.00657 + 1.65958i 1.75687 − 2.62934i
31.5 1.30656 − 0.541196i 0.804373 − 4.04386i 1.41421 − 1.41421i −1.85922 + 1.24229i −1.13755 − 5.71887i −10.0972 − 6.74676i 1.08239 − 2.61313i −7.39083 − 3.06138i −1.75687 + 2.62934i
31.6 1.30656 − 0.541196i 1.01825 − 5.11907i 1.41421 − 1.41421i 1.85922 − 1.24229i −1.44002 − 7.23946i 5.66553 + 3.78559i 1.08239 − 2.61313i −16.8532 − 6.98081i 1.75687 − 2.62934i
41.1 −0.541196 + 1.30656i −2.70375 + 4.04645i −1.41421 − 1.41421i 0.436235 + 2.19310i −3.82369 − 5.72255i −2.13409 + 10.7288i 2.61313 − 1.08239i −5.61935 − 13.5663i −3.10152 − 0.616930i
41.2 −0.541196 + 1.30656i −2.63648 + 3.94576i −1.41421 − 1.41421i −0.436235 − 2.19310i −3.72854 − 5.58015i 2.43058 − 12.2193i 2.61313 − 1.08239i −5.17390 − 12.4909i 3.10152 + 0.616930i
41.3 −0.541196 + 1.30656i −0.311420 + 0.466073i −1.41421 − 1.41421i −0.436235 − 2.19310i −0.440414 − 0.659126i −0.717353 + 3.60638i 2.61313 − 1.08239i 3.32391 + 8.02463i 3.10152 + 0.616930i
41.4 −0.541196 + 1.30656i 0.0947611 − 0.141820i −1.41421 − 1.41421i 0.436235 + 2.19310i 0.134012 + 0.200564i −0.124636 + 0.626588i 2.61313 − 1.08239i 3.43302 + 8.28804i −3.10152 − 0.616930i
41.5 −0.541196 + 1.30656i 0.600732 − 0.899059i −1.41421 − 1.41421i −0.436235 − 2.19310i 0.849563 + 1.27146i −0.448363 + 2.25407i 2.61313 − 1.08239i 2.99672 + 7.23473i 3.10152 + 0.616930i
41.6 −0.541196 + 1.30656i 2.16345 − 3.23783i −1.41421 − 1.41421i 0.436235 + 2.19310i 3.05958 + 4.57898i 1.91147 − 9.60962i 2.61313 − 1.08239i −2.35888 − 5.69485i −3.10152 − 0.616930i
61.1 0.541196 − 1.30656i −3.05605 − 2.04199i −1.41421 − 1.41421i −2.19310 + 0.436235i −4.32191 + 2.88781i −2.80539 − 0.558026i −2.61313 + 1.08239i 1.72560 + 4.16596i −0.616930 + 3.10152i
61.2 0.541196 − 1.30656i −1.29925 − 0.868131i −1.41421 − 1.41421i 2.19310 − 0.436235i −1.83742 + 1.22772i −6.29247 − 1.25165i −2.61313 + 1.08239i −2.50975 − 6.05908i 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.b ✓ 48
17.e odd 16 1 inner 170.3.p.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.3.p.b ✓ 48 1.a even 1 1 trivial
170.3.p.b ✓ 48 17.e odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 16 T_{3}^{47} + 144 T_{3}^{46} - 1152 T_{3}^{45} + 7916 T_{3}^{44} - 43872 T_{3}^{43} + \cdots + 213488977232644 \) acting on \(S_{3}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display