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Properties

Label 696.2.l.b
Level $696$
Weight $2$
Character orbit 696.l
Analytic conductor $5.558$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [696,2,Mod(347,696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("696.347");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 696.l (of order \(2\), degree \(1\), 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: \(5.55758798068\)
Analytic rank: \(0\)
Dimension: \(104\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 4 q^{4} - 12 q^{6} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 4 q^{4} - 12 q^{6} - 40 q^{9} + 12 q^{16} - 16 q^{22} - 12 q^{24} + 152 q^{25} - 44 q^{28} - 16 q^{30} - 16 q^{33} + 20 q^{34} - 16 q^{36} - 24 q^{42} - 16 q^{49} - 16 q^{51} + 104 q^{52} + 16 q^{54} + 8 q^{57} + 24 q^{58} - 4 q^{64} - 72 q^{67} - 76 q^{78} - 128 q^{81} + 4 q^{82} - 48 q^{88} + 120 q^{91} + 172 q^{94} - 12 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
347.1 −1.41311 − 0.0558043i 1.05754 − 1.37172i 1.99377 + 0.157715i 2.38708 −1.57097 + 1.87937i 4.52139i −2.80862 − 0.334131i −0.763207 − 2.90130i −3.37321 − 0.133209i
347.2 −1.41311 − 0.0558043i 1.05754 + 1.37172i 1.99377 + 0.157715i −2.38708 −1.41788 − 1.99740i 4.52139i −2.80862 − 0.334131i −0.763207 + 2.90130i 3.37321 + 0.133209i
347.3 −1.41311 + 0.0558043i 1.05754 − 1.37172i 1.99377 − 0.157715i −2.38708 −1.41788 + 1.99740i − 4.52139i −2.80862 + 0.334131i −0.763207 − 2.90130i 3.37321 − 0.133209i
347.4 −1.41311 + 0.0558043i 1.05754 + 1.37172i 1.99377 − 0.157715i 2.38708 −1.57097 − 1.87937i − 4.52139i −2.80862 + 0.334131i −0.763207 + 2.90130i −3.37321 + 0.133209i
347.5 −1.38233 − 0.298606i −0.311297 − 1.70385i 1.82167 + 0.825544i −2.63807 −0.0784634 + 2.44823i 0.0383815i −2.27163 − 1.68513i −2.80619 + 1.06081i 3.64669 + 0.787744i
347.6 −1.38233 − 0.298606i −0.311297 + 1.70385i 1.82167 + 0.825544i 2.63807 0.939094 − 2.26232i 0.0383815i −2.27163 − 1.68513i −2.80619 − 1.06081i −3.64669 − 0.787744i
347.7 −1.38233 + 0.298606i −0.311297 − 1.70385i 1.82167 − 0.825544i 2.63807 0.939094 + 2.26232i − 0.0383815i −2.27163 + 1.68513i −2.80619 + 1.06081i −3.64669 + 0.787744i
347.8 −1.38233 + 0.298606i −0.311297 + 1.70385i 1.82167 − 0.825544i −2.63807 −0.0784634 − 2.44823i − 0.0383815i −2.27163 + 1.68513i −2.80619 − 1.06081i 3.64669 − 0.787744i
347.9 −1.35650 − 0.399891i −1.36438 − 1.06699i 1.68017 + 1.08490i −0.296182 1.42410 + 1.99297i 1.53938i −1.84531 − 2.14356i 0.723060 + 2.91156i 0.401770 + 0.118441i
347.10 −1.35650 − 0.399891i −1.36438 + 1.06699i 1.68017 + 1.08490i 0.296182 2.27746 − 0.901768i 1.53938i −1.84531 − 2.14356i 0.723060 − 2.91156i −0.401770 − 0.118441i
347.11 −1.35650 + 0.399891i −1.36438 − 1.06699i 1.68017 − 1.08490i 0.296182 2.27746 + 0.901768i − 1.53938i −1.84531 + 2.14356i 0.723060 + 2.91156i −0.401770 + 0.118441i
347.12 −1.35650 + 0.399891i −1.36438 + 1.06699i 1.68017 − 1.08490i −0.296182 1.42410 − 1.99297i − 1.53938i −1.84531 + 2.14356i 0.723060 − 2.91156i 0.401770 − 0.118441i
347.13 −1.30342 − 0.548731i 1.37036 − 1.05930i 1.39779 + 1.43045i 1.96366 −2.36742 + 0.628752i − 2.07610i −1.03697 − 2.63148i 0.755764 − 2.90324i −2.55947 − 1.07752i
347.14 −1.30342 − 0.548731i 1.37036 + 1.05930i 1.39779 + 1.43045i −1.96366 −1.20488 − 2.13267i − 2.07610i −1.03697 − 2.63148i 0.755764 + 2.90324i 2.55947 + 1.07752i
347.15 −1.30342 + 0.548731i 1.37036 − 1.05930i 1.39779 − 1.43045i −1.96366 −1.20488 + 2.13267i 2.07610i −1.03697 + 2.63148i 0.755764 − 2.90324i 2.55947 − 1.07752i
347.16 −1.30342 + 0.548731i 1.37036 + 1.05930i 1.39779 − 1.43045i 1.96366 −2.36742 − 0.628752i 2.07610i −1.03697 + 2.63148i 0.755764 + 2.90324i −2.55947 + 1.07752i
347.17 −1.16250 − 0.805357i 1.54080 − 0.791154i 0.702799 + 1.87245i −4.01902 −2.42834 − 0.321183i 3.86754i 0.690990 − 2.74272i 1.74815 − 2.43803i 4.67209 + 3.23674i
347.18 −1.16250 − 0.805357i 1.54080 + 0.791154i 0.702799 + 1.87245i 4.01902 −1.15402 − 2.16061i 3.86754i 0.690990 − 2.74272i 1.74815 + 2.43803i −4.67209 − 3.23674i
347.19 −1.16250 + 0.805357i 1.54080 − 0.791154i 0.702799 − 1.87245i 4.01902 −1.15402 + 2.16061i − 3.86754i 0.690990 + 2.74272i 1.74815 − 2.43803i −4.67209 + 3.23674i
347.20 −1.16250 + 0.805357i 1.54080 + 0.791154i 0.702799 − 1.87245i −4.01902 −2.42834 + 0.321183i − 3.86754i 0.690990 + 2.74272i 1.74815 + 2.43803i 4.67209 − 3.23674i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 347.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner
232.b odd 2 1 inner
696.l even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 696.2.l.b ✓ 104
3.b odd 2 1 inner 696.2.l.b ✓ 104
8.d odd 2 1 inner 696.2.l.b ✓ 104
24.f even 2 1 inner 696.2.l.b ✓ 104
29.b even 2 1 inner 696.2.l.b ✓ 104
87.d odd 2 1 inner 696.2.l.b ✓ 104
232.b odd 2 1 inner 696.2.l.b ✓ 104
696.l even 2 1 inner 696.2.l.b ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
696.2.l.b ✓ 104 1.a even 1 1 trivial
696.2.l.b ✓ 104 3.b odd 2 1 inner
696.2.l.b ✓ 104 8.d odd 2 1 inner
696.2.l.b ✓ 104 24.f even 2 1 inner
696.2.l.b ✓ 104 29.b even 2 1 inner
696.2.l.b ✓ 104 87.d odd 2 1 inner
696.2.l.b ✓ 104 232.b odd 2 1 inner
696.2.l.b ✓ 104 696.l even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{26} - 84 T_{5}^{24} + 3072 T_{5}^{22} - 64408 T_{5}^{20} + 857778 T_{5}^{18} - 7591676 T_{5}^{16} + \cdots - 723136 \) acting on \(S_{2}^{\mathrm{new}}(696, [\chi])\). Copy content Toggle raw display