Properties

Label 712.2.b.d
Level $712$
Weight $2$
Character orbit 712.b
Analytic conductor $5.685$
Analytic rank $0$
Dimension $36$
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] = [712,2,Mod(357,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.357");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.b (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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} + 2 q^{6} - 16 q^{7} + 6 q^{8} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{4} + 2 q^{6} - 16 q^{7} + 6 q^{8} - 50 q^{9} - 2 q^{10} - 6 q^{12} - 2 q^{14} + 46 q^{15} - 2 q^{16} + 2 q^{17} - 6 q^{18} - 8 q^{20} + 10 q^{22} - 14 q^{23} + 10 q^{24} - 38 q^{25} - 8 q^{26} - 10 q^{28} - 16 q^{30} + 14 q^{31} - 50 q^{32} - 20 q^{33} + 10 q^{34} + 46 q^{36} - 36 q^{38} - 48 q^{39} - 40 q^{40} + 48 q^{42} + 32 q^{44} - 30 q^{46} + 8 q^{47} - 20 q^{48} + 60 q^{49} + 54 q^{50} - 4 q^{52} - 66 q^{54} - 60 q^{55} - 58 q^{56} + 62 q^{57} - 30 q^{58} + 30 q^{60} - 52 q^{62} + 84 q^{63} + 30 q^{64} - 4 q^{65} + 130 q^{66} - 4 q^{68} - 84 q^{70} - 28 q^{71} - 42 q^{72} + 34 q^{73} + 20 q^{74} + 24 q^{76} - 70 q^{78} + 132 q^{79} - 8 q^{80} + 76 q^{81} + 64 q^{82} + 76 q^{84} + 6 q^{86} - 64 q^{87} - 100 q^{88} - 36 q^{89} + 78 q^{90} + 64 q^{92} + 28 q^{94} + 126 q^{95} - 54 q^{96} - 26 q^{97} + 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
357.1 −1.41365 0.0398817i 1.27512i 1.99682 + 0.112758i 1.63611i 0.0508541 1.80258i 3.77754 −2.81831 0.239037i 1.37406 −0.0652507 + 2.31288i
357.2 −1.41365 + 0.0398817i 1.27512i 1.99682 0.112758i 1.63611i 0.0508541 + 1.80258i 3.77754 −2.81831 + 0.239037i 1.37406 −0.0652507 2.31288i
357.3 −1.39812 0.212747i 2.98776i 1.90948 + 0.594892i 2.44246i 0.635637 4.17724i −4.95507 −2.54312 1.23797i −5.92669 −0.519627 + 3.41485i
357.4 −1.39812 + 0.212747i 2.98776i 1.90948 0.594892i 2.44246i 0.635637 + 4.17724i −4.95507 −2.54312 + 1.23797i −5.92669 −0.519627 3.41485i
357.5 −1.39187 0.250382i 0.401361i 1.87462 + 0.697000i 2.41337i 0.100494 0.558643i 0.333867 −2.43471 1.43951i 2.83891 0.604264 3.35910i
357.6 −1.39187 + 0.250382i 0.401361i 1.87462 0.697000i 2.41337i 0.100494 + 0.558643i 0.333867 −2.43471 + 1.43951i 2.83891 0.604264 + 3.35910i
357.7 −1.08137 0.911395i 1.16415i 0.338718 + 1.97111i 4.23108i 1.06100 1.25888i 0.524743 1.43018 2.44020i 1.64475 −3.85618 + 4.57536i
357.8 −1.08137 + 0.911395i 1.16415i 0.338718 1.97111i 4.23108i 1.06100 + 1.25888i 0.524743 1.43018 + 2.44020i 1.64475 −3.85618 4.57536i
357.9 −1.02591 0.973398i 1.45392i 0.104992 + 1.99724i 0.334048i −1.41525 + 1.49160i −0.482217 1.83640 2.15119i 0.886105 0.325162 0.342704i
357.10 −1.02591 + 0.973398i 1.45392i 0.104992 1.99724i 0.334048i −1.41525 1.49160i −0.482217 1.83640 + 2.15119i 0.886105 0.325162 + 0.342704i
357.11 −0.697423 1.23028i 1.45749i −1.02720 + 1.71606i 4.29817i −1.79313 + 1.01649i −4.23390 2.82764 + 0.0669315i 0.875721 5.28797 2.99764i
357.12 −0.697423 + 1.23028i 1.45749i −1.02720 1.71606i 4.29817i −1.79313 1.01649i −4.23390 2.82764 0.0669315i 0.875721 5.28797 + 2.99764i
357.13 −0.599401 1.28091i 2.00329i −1.28144 + 1.53555i 2.02712i −2.56602 + 1.20077i 2.51595 2.73499 + 0.720987i −1.01317 −2.59654 + 1.21506i
357.14 −0.599401 + 1.28091i 2.00329i −1.28144 1.53555i 2.02712i −2.56602 1.20077i 2.51595 2.73499 0.720987i −1.01317 −2.59654 1.21506i
357.15 −0.588837 1.28580i 2.79991i −1.30654 + 1.51425i 3.56614i 3.60011 1.64869i −1.62232 2.71635 + 0.788304i −4.83950 4.58533 2.09987i
357.16 −0.588837 + 1.28580i 2.79991i −1.30654 1.51425i 3.56614i 3.60011 + 1.64869i −1.62232 2.71635 0.788304i −4.83950 4.58533 + 2.09987i
357.17 −0.456684 1.33845i 3.12650i −1.58288 + 1.22249i 2.53893i 4.18465 1.42782i 0.0880153 2.35912 + 1.56031i −6.77500 −3.39822 + 1.15949i
357.18 −0.456684 + 1.33845i 3.12650i −1.58288 1.22249i 2.53893i 4.18465 + 1.42782i 0.0880153 2.35912 1.56031i −6.77500 −3.39822 1.15949i
357.19 0.103239 1.41044i 3.25660i −1.97868 0.291224i 0.682759i −4.59324 0.336207i −1.89438 −0.615031 + 2.76075i −7.60545 −0.962991 0.0704872i
357.20 0.103239 + 1.41044i 3.25660i −1.97868 + 0.291224i 0.682759i −4.59324 + 0.336207i −1.89438 −0.615031 2.76075i −7.60545 −0.962991 + 0.0704872i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 357.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.b.d 36
4.b odd 2 1 2848.2.b.d 36
8.b even 2 1 inner 712.2.b.d 36
8.d odd 2 1 2848.2.b.d 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.b.d 36 1.a even 1 1 trivial
712.2.b.d 36 8.b even 2 1 inner
2848.2.b.d 36 4.b odd 2 1
2848.2.b.d 36 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(712, [\chi])\):

\( T_{3}^{36} + 79 T_{3}^{34} + 2827 T_{3}^{32} + 60679 T_{3}^{30} + 872187 T_{3}^{28} + 8879831 T_{3}^{26} + 66079399 T_{3}^{24} + 365769627 T_{3}^{22} + 1518590753 T_{3}^{20} + 4733929057 T_{3}^{18} + \cdots + 301088 \) Copy content Toggle raw display
\( T_{7}^{18} + 8 T_{7}^{17} - 46 T_{7}^{16} - 470 T_{7}^{15} + 624 T_{7}^{14} + 10606 T_{7}^{13} - 280 T_{7}^{12} - 117192 T_{7}^{11} - 54084 T_{7}^{10} + 667632 T_{7}^{9} + 402324 T_{7}^{8} - 1889052 T_{7}^{7} + \cdots - 10240 \) Copy content Toggle raw display