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Properties

Label 696.2.h.a
Level $696$
Weight $2$
Character orbit 696.h
Analytic conductor $5.558$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [696,2,Mod(637,696)]
 
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([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("696.637");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 696.h (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: \(30\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{3} + 2 q^{4} + 4 q^{7} + 3 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 30 q^{3} + 2 q^{4} + 4 q^{7} + 3 q^{8} + 30 q^{9} + 8 q^{10} - 2 q^{12} - 3 q^{14} + 2 q^{16} - 4 q^{21} - 7 q^{22} - 8 q^{23} - 3 q^{24} - 26 q^{25} - 25 q^{26} - 30 q^{27} - 19 q^{28} - 8 q^{30} + 10 q^{32} - 5 q^{34} + 2 q^{36} + 16 q^{37} + 24 q^{38} + 12 q^{40} + 3 q^{42} + 7 q^{44} - 24 q^{46} - 2 q^{48} + 30 q^{49} - 7 q^{52} - 28 q^{56} - 40 q^{58} + 16 q^{61} - 32 q^{62} + 4 q^{63} + 17 q^{64} - 16 q^{65} + 7 q^{66} - 17 q^{68} + 8 q^{69} - 52 q^{70} - 8 q^{71} + 3 q^{72} - 32 q^{74} + 26 q^{75} + 20 q^{76} + 36 q^{77} + 25 q^{78} + 6 q^{80} + 30 q^{81} + 10 q^{82} + 19 q^{84} + 56 q^{85} + 10 q^{86} - 12 q^{88} + 8 q^{90} + 10 q^{92} - 39 q^{94} - 10 q^{96} + 23 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
637.1 −1.41322 − 0.0530578i −1.00000 1.99437 + 0.149965i 2.52094i 1.41322 + 0.0530578i −0.00733597 −2.81052 − 0.317750i 1.00000 0.133755 − 3.56263i
637.2 −1.41322 + 0.0530578i −1.00000 1.99437 − 0.149965i − 2.52094i 1.41322 − 0.0530578i −0.00733597 −2.81052 + 0.317750i 1.00000 0.133755 + 3.56263i
637.3 −1.30544 − 0.543907i −1.00000 1.40833 + 1.42007i 1.58353i 1.30544 + 0.543907i −4.25470 −1.06610 − 2.61982i 1.00000 0.861294 − 2.06720i
637.4 −1.30544 + 0.543907i −1.00000 1.40833 − 1.42007i − 1.58353i 1.30544 − 0.543907i −4.25470 −1.06610 + 2.61982i 1.00000 0.861294 + 2.06720i
637.5 −1.23340 − 0.691902i −1.00000 1.04254 + 1.70678i − 2.03452i 1.23340 + 0.691902i 3.64451 −0.104946 − 2.82648i 1.00000 −1.40769 + 2.50937i
637.6 −1.23340 + 0.691902i −1.00000 1.04254 − 1.70678i 2.03452i 1.23340 − 0.691902i 3.64451 −0.104946 + 2.82648i 1.00000 −1.40769 − 2.50937i
637.7 −1.12376 − 0.858578i −1.00000 0.525688 + 1.92968i − 0.567497i 1.12376 + 0.858578i 1.73092 1.06603 − 2.61984i 1.00000 −0.487240 + 0.637732i
637.8 −1.12376 + 0.858578i −1.00000 0.525688 − 1.92968i 0.567497i 1.12376 − 0.858578i 1.73092 1.06603 + 2.61984i 1.00000 −0.487240 − 0.637732i
637.9 −0.877245 − 1.10925i −1.00000 −0.460883 + 1.94617i 4.22597i 0.877245 + 1.10925i 1.13566 2.56310 − 1.19603i 1.00000 4.68767 − 3.70721i
637.10 −0.877245 + 1.10925i −1.00000 −0.460883 − 1.94617i − 4.22597i 0.877245 − 1.10925i 1.13566 2.56310 + 1.19603i 1.00000 4.68767 + 3.70721i
637.11 −0.468259 − 1.33444i −1.00000 −1.56147 + 1.24973i − 3.26900i 0.468259 + 1.33444i −0.436072 2.39886 + 1.49849i 1.00000 −4.36229 + 1.53074i
637.12 −0.468259 + 1.33444i −1.00000 −1.56147 − 1.24973i 3.26900i 0.468259 − 1.33444i −0.436072 2.39886 − 1.49849i 1.00000 −4.36229 − 1.53074i
637.13 −0.400818 − 1.35622i −1.00000 −1.67869 + 1.08720i 1.23468i 0.400818 + 1.35622i −3.99820 2.14733 + 1.84091i 1.00000 1.67451 − 0.494883i
637.14 −0.400818 + 1.35622i −1.00000 −1.67869 − 1.08720i − 1.23468i 0.400818 − 1.35622i −3.99820 2.14733 − 1.84091i 1.00000 1.67451 + 0.494883i
637.15 −0.116251 − 1.40943i −1.00000 −1.97297 + 0.327694i − 0.560207i 0.116251 + 1.40943i 2.20020 0.691220 + 2.74267i 1.00000 −0.789571 + 0.0651244i
637.16 −0.116251 + 1.40943i −1.00000 −1.97297 − 0.327694i 0.560207i 0.116251 − 1.40943i 2.20020 0.691220 − 2.74267i 1.00000 −0.789571 − 0.0651244i
637.17 0.329678 − 1.37525i −1.00000 −1.78262 − 0.906780i 3.53785i −0.329678 + 1.37525i 2.00820 −1.83474 + 2.15261i 1.00000 4.86542 + 1.16635i
637.18 0.329678 + 1.37525i −1.00000 −1.78262 + 0.906780i − 3.53785i −0.329678 − 1.37525i 2.00820 −1.83474 − 2.15261i 1.00000 4.86542 − 1.16635i
637.19 0.701902 − 1.22774i −1.00000 −1.01467 − 1.72350i − 2.96858i −0.701902 + 1.22774i 4.86921 −2.82820 + 0.0360149i 1.00000 −3.64463 − 2.08365i
637.20 0.701902 + 1.22774i −1.00000 −1.01467 + 1.72350i 2.96858i −0.701902 − 1.22774i 4.86921 −2.82820 − 0.0360149i 1.00000 −3.64463 + 2.08365i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 637.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
232.g 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.h.a ✓ 30
4.b odd 2 1 2784.2.h.b 30
8.b even 2 1 696.2.h.b yes 30
8.d odd 2 1 2784.2.h.a 30
29.b even 2 1 696.2.h.b yes 30
116.d odd 2 1 2784.2.h.a 30
232.b odd 2 1 2784.2.h.b 30
232.g even 2 1 inner 696.2.h.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
696.2.h.a ✓ 30 1.a even 1 1 trivial
696.2.h.a ✓ 30 232.g even 2 1 inner
696.2.h.b yes 30 8.b even 2 1
696.2.h.b yes 30 29.b even 2 1
2784.2.h.a 30 8.d odd 2 1
2784.2.h.a 30 116.d odd 2 1
2784.2.h.b 30 4.b odd 2 1
2784.2.h.b 30 232.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{15} - 100 T_{11}^{13} + 24 T_{11}^{12} + 3786 T_{11}^{11} - 1024 T_{11}^{10} - 68984 T_{11}^{9} + \cdots - 2817024 \) acting on \(S_{2}^{\mathrm{new}}(696, [\chi])\). Copy content Toggle raw display