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Properties

Label 966.2.h.a
Level $966$
Weight $2$
Character orbit 966.h
Analytic conductor $7.714$
Analytic rank $0$
Dimension $24$
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] = [966,2,Mod(827,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("966.827");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 24 q^{4} - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 24 q^{4} - 4 q^{5} - 4 q^{9} - 4 q^{12} + 8 q^{13} + 24 q^{14} - 4 q^{15} + 24 q^{16} + 32 q^{17} + 4 q^{18} + 4 q^{20} - 8 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{30} - 16 q^{31} + 20 q^{33} + 4 q^{36} - 8 q^{39} + 4 q^{42} + 24 q^{45} - 4 q^{46} + 4 q^{48} - 24 q^{49} + 24 q^{51} - 8 q^{52} + 24 q^{53} - 12 q^{54} + 16 q^{55} - 24 q^{56} + 4 q^{57} + 4 q^{58} + 4 q^{60} - 4 q^{63} - 24 q^{64} - 12 q^{66} - 32 q^{68} - 24 q^{69} - 4 q^{70} - 4 q^{72} - 32 q^{73} - 16 q^{74} + 48 q^{75} + 12 q^{78} - 4 q^{80} - 8 q^{81} - 8 q^{82} + 16 q^{83} - 16 q^{85} + 16 q^{86} + 20 q^{87} + 24 q^{89} - 28 q^{90} + 8 q^{92} + 16 q^{93} + 8 q^{94} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
827.1 − 1.00000i −1.69648 − 0.349201i −1.00000 1.62492 −0.349201 + 1.69648i 1.00000i 1.00000i 2.75612 + 1.18483i − 1.62492i
827.2 − 1.00000i −1.33692 − 1.10121i −1.00000 −0.136871 −1.10121 + 1.33692i 1.00000i 1.00000i 0.574695 + 2.94444i 0.136871i
827.3 − 1.00000i −1.01359 + 1.40450i −1.00000 −1.51059 1.40450 + 1.01359i 1.00000i 1.00000i −0.945264 − 2.84719i 1.51059i
827.4 − 1.00000i −0.798999 + 1.53675i −1.00000 2.11859 1.53675 + 0.798999i 1.00000i 1.00000i −1.72320 − 2.45572i − 2.11859i
827.5 − 1.00000i −0.529264 − 1.64921i −1.00000 −2.93995 −1.64921 + 0.529264i 1.00000i 1.00000i −2.43976 + 1.74573i 2.93995i
827.6 − 1.00000i −0.0866531 − 1.72988i −1.00000 0.713996 −1.72988 + 0.0866531i 1.00000i 1.00000i −2.98498 + 0.299799i − 0.713996i
827.7 − 1.00000i 0.656415 + 1.60285i −1.00000 −0.252332 1.60285 − 0.656415i 1.00000i 1.00000i −2.13824 + 2.10426i 0.252332i
827.8 − 1.00000i 0.696580 + 1.58580i −1.00000 −3.16093 1.58580 − 0.696580i 1.00000i 1.00000i −2.02955 + 2.20928i 3.16093i
827.9 − 1.00000i 1.23024 − 1.21923i −1.00000 0.666555 −1.21923 − 1.23024i 1.00000i 1.00000i 0.0269708 − 2.99988i − 0.666555i
827.10 − 1.00000i 1.55095 − 0.771068i −1.00000 2.18892 −0.771068 − 1.55095i 1.00000i 1.00000i 1.81091 − 2.39178i − 2.18892i
827.11 − 1.00000i 1.59575 + 0.673481i −1.00000 2.44752 0.673481 − 1.59575i 1.00000i 1.00000i 2.09285 + 2.14942i − 2.44752i
827.12 − 1.00000i 1.73197 + 0.0164012i −1.00000 −3.75984 0.0164012 − 1.73197i 1.00000i 1.00000i 2.99946 + 0.0568128i 3.75984i
827.13 1.00000i −1.69648 + 0.349201i −1.00000 1.62492 −0.349201 − 1.69648i − 1.00000i − 1.00000i 2.75612 − 1.18483i 1.62492i
827.14 1.00000i −1.33692 + 1.10121i −1.00000 −0.136871 −1.10121 − 1.33692i − 1.00000i − 1.00000i 0.574695 − 2.94444i − 0.136871i
827.15 1.00000i −1.01359 − 1.40450i −1.00000 −1.51059 1.40450 − 1.01359i − 1.00000i − 1.00000i −0.945264 + 2.84719i − 1.51059i
827.16 1.00000i −0.798999 − 1.53675i −1.00000 2.11859 1.53675 − 0.798999i − 1.00000i − 1.00000i −1.72320 + 2.45572i 2.11859i
827.17 1.00000i −0.529264 + 1.64921i −1.00000 −2.93995 −1.64921 − 0.529264i − 1.00000i − 1.00000i −2.43976 − 1.74573i − 2.93995i
827.18 1.00000i −0.0866531 + 1.72988i −1.00000 0.713996 −1.72988 − 0.0866531i − 1.00000i − 1.00000i −2.98498 − 0.299799i 0.713996i
827.19 1.00000i 0.656415 − 1.60285i −1.00000 −0.252332 1.60285 + 0.656415i − 1.00000i − 1.00000i −2.13824 − 2.10426i − 0.252332i
827.20 1.00000i 0.696580 − 1.58580i −1.00000 −3.16093 1.58580 + 0.696580i − 1.00000i − 1.00000i −2.02955 − 2.20928i − 3.16093i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 827.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.h.a ✓ 24
3.b odd 2 1 966.2.h.b yes 24
23.b odd 2 1 966.2.h.b yes 24
69.c even 2 1 inner 966.2.h.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.h.a ✓ 24 1.a even 1 1 trivial
966.2.h.a ✓ 24 69.c even 2 1 inner
966.2.h.b yes 24 3.b odd 2 1
966.2.h.b yes 24 23.b odd 2 1