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Properties

Label 714.2.w.e
Level $714$
Weight $2$
Character orbit 714.w
Analytic conductor $5.701$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(83,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.w (of order \(8\), degree \(4\), 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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{7} - 16 q^{14} - 16 q^{15} - 80 q^{16} + 16 q^{18} - 32 q^{21} - 16 q^{23} - 32 q^{25} + 64 q^{29} - 16 q^{30} - 48 q^{37} + 16 q^{39} - 24 q^{42} + 16 q^{43} - 16 q^{44} - 16 q^{46} + 16 q^{49} + 104 q^{51} - 80 q^{53} - 16 q^{57} - 16 q^{58} - 48 q^{60} + 56 q^{63} + 80 q^{65} - 144 q^{67} + 8 q^{70} + 16 q^{71} - 32 q^{74} - 24 q^{77} + 32 q^{78} + 32 q^{79} - 16 q^{84} + 208 q^{85} + 16 q^{88} - 8 q^{91} + 16 q^{92} + 48 q^{93} - 16 q^{95} - 16 q^{98} + 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 0.707107 − 0.707107i −1.72967 + 0.0907356i − 1.00000i −1.23141 + 0.510066i −1.15890 + 1.28722i 0.516019 − 2.59494i −0.707107 − 0.707107i 2.98353 − 0.313886i −0.510066 + 1.23141i
83.2 0.707107 − 0.707107i −1.72704 + 0.131707i − 1.00000i 0.103371 − 0.0428177i −1.12807 + 1.31433i −0.593072 + 2.57842i −0.707107 − 0.707107i 2.96531 − 0.454927i 0.0428177 − 0.103371i
83.3 0.707107 − 0.707107i −1.70634 + 0.297343i − 1.00000i 2.47278 − 1.02426i −0.996310 + 1.41682i 2.40380 + 1.10533i −0.707107 − 0.707107i 2.82317 − 1.01473i 1.02426 − 2.47278i
83.4 0.707107 − 0.707107i −1.47024 + 0.915636i − 1.00000i −2.44731 + 1.01371i −0.392165 + 1.68707i −2.61269 + 0.416962i −0.707107 − 0.707107i 1.32322 − 2.69241i −1.01371 + 2.44731i
83.5 0.707107 − 0.707107i −1.39247 − 1.03006i − 1.00000i 2.17160 − 0.899506i −1.71299 + 0.256259i 0.880980 − 2.49477i −0.707107 − 0.707107i 0.877938 + 2.86866i 0.899506 − 2.17160i
83.6 0.707107 − 0.707107i −1.25039 − 1.19855i − 1.00000i 3.41749 − 1.41557i −1.73166 + 0.0366532i −2.56408 − 0.652293i −0.707107 − 0.707107i 0.126942 + 2.99731i 1.41557 − 3.41749i
83.7 0.707107 − 0.707107i −0.923536 − 1.46529i − 1.00000i −1.11174 + 0.460496i −1.68916 − 0.383079i −2.37717 + 1.16149i −0.707107 − 0.707107i −1.29416 + 2.70650i −0.460496 + 1.11174i
83.8 0.707107 − 0.707107i −0.498655 + 1.65872i − 1.00000i 0.0389656 − 0.0161401i 0.820288 + 1.52549i −2.40704 − 1.09824i −0.707107 − 0.707107i −2.50269 − 1.65425i 0.0161401 − 0.0389656i
83.9 0.707107 − 0.707107i −0.223046 + 1.71763i − 1.00000i 2.82761 − 1.17124i 1.05683 + 1.37226i −0.153600 + 2.64129i −0.707107 − 0.707107i −2.90050 − 0.766222i 1.17124 − 2.82761i
83.10 0.707107 − 0.707107i −0.0756553 + 1.73040i − 1.00000i −2.32780 + 0.964207i 1.17008 + 1.27707i 1.59075 + 2.11412i −0.707107 − 0.707107i −2.98855 − 0.261828i −0.964207 + 2.32780i
83.11 0.707107 − 0.707107i 0.0756553 − 1.73040i − 1.00000i 2.32780 − 0.964207i −1.17008 − 1.27707i 2.61974 − 0.370078i −0.707107 − 0.707107i −2.98855 − 0.261828i 0.964207 − 2.32780i
83.12 0.707107 − 0.707107i 0.223046 − 1.71763i − 1.00000i −2.82761 + 1.17124i −1.05683 − 1.37226i 1.75906 − 1.97628i −0.707107 − 0.707107i −2.90050 − 0.766222i −1.17124 + 2.82761i
83.13 0.707107 − 0.707107i 0.498655 − 1.65872i − 1.00000i −0.0389656 + 0.0161401i −0.820288 − 1.52549i −2.47861 − 0.925461i −0.707107 − 0.707107i −2.50269 − 1.65425i −0.0161401 + 0.0389656i
83.14 0.707107 − 0.707107i 0.923536 + 1.46529i − 1.00000i 1.11174 − 0.460496i 1.68916 + 0.383079i −0.859618 − 2.50221i −0.707107 − 0.707107i −1.29416 + 2.70650i 0.460496 − 1.11174i
83.15 0.707107 − 0.707107i 1.25039 + 1.19855i − 1.00000i −3.41749 + 1.41557i 1.73166 − 0.0366532i −2.27432 − 1.35184i −0.707107 − 0.707107i 0.126942 + 2.99731i −1.41557 + 3.41749i
83.16 0.707107 − 0.707107i 1.39247 + 1.03006i − 1.00000i −2.17160 + 0.899506i 1.71299 − 0.256259i −1.14112 + 2.38702i −0.707107 − 0.707107i 0.877938 + 2.86866i −0.899506 + 2.17160i
83.17 0.707107 − 0.707107i 1.47024 − 0.915636i − 1.00000i 2.44731 − 1.01371i 0.392165 − 1.68707i −1.55261 − 2.14229i −0.707107 − 0.707107i 1.32322 − 2.69241i 1.01371 − 2.44731i
83.18 0.707107 − 0.707107i 1.70634 − 0.297343i − 1.00000i −2.47278 + 1.02426i 0.996310 − 1.41682i 2.48133 + 0.918153i −0.707107 − 0.707107i 2.82317 − 1.01473i −1.02426 + 2.47278i
83.19 0.707107 − 0.707107i 1.72704 − 0.131707i − 1.00000i −0.103371 + 0.0428177i 1.12807 − 1.31433i 1.40386 − 2.24259i −0.707107 − 0.707107i 2.96531 − 0.454927i −0.0428177 + 0.103371i
83.20 0.707107 − 0.707107i 1.72967 − 0.0907356i − 1.00000i 1.23141 − 0.510066i 1.15890 − 1.28722i −1.47002 + 2.19978i −0.707107 − 0.707107i 2.98353 − 0.313886i 0.510066 − 1.23141i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
51.g odd 8 1 inner
357.w even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.w.e ✓ 80
3.b odd 2 1 714.2.w.f yes 80
7.b odd 2 1 inner 714.2.w.e ✓ 80
17.d even 8 1 714.2.w.f yes 80
21.c even 2 1 714.2.w.f yes 80
51.g odd 8 1 inner 714.2.w.e ✓ 80
119.l odd 8 1 714.2.w.f yes 80
357.w even 8 1 inner 714.2.w.e ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.w.e ✓ 80 1.a even 1 1 trivial
714.2.w.e ✓ 80 7.b odd 2 1 inner
714.2.w.e ✓ 80 51.g odd 8 1 inner
714.2.w.e ✓ 80 357.w even 8 1 inner
714.2.w.f yes 80 3.b odd 2 1
714.2.w.f yes 80 17.d even 8 1
714.2.w.f yes 80 21.c even 2 1
714.2.w.f yes 80 119.l odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{80} + 16 T_{5}^{78} + 128 T_{5}^{76} - 2240 T_{5}^{74} + 60610 T_{5}^{72} + 792240 T_{5}^{70} + \cdots + 5473632256 \) Copy content Toggle raw display
\( T_{11}^{40} + 20 T_{11}^{38} + 168 T_{11}^{37} + 200 T_{11}^{36} + 24 T_{11}^{35} + 4532 T_{11}^{34} + \cdots + 112659579904 \) Copy content Toggle raw display