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Properties

Label 784.2.m.l
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $0$
Dimension $48$
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] = [784,2,Mod(197,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(4))
 
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("784.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (of order \(4\), degree \(2\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} - 8 q^{11} + 32 q^{15} + 36 q^{16} + 20 q^{18} - 28 q^{22} - 16 q^{29} + 96 q^{30} + 40 q^{32} + 40 q^{36} - 16 q^{37} + 8 q^{43} + 4 q^{44} + 64 q^{46} - 28 q^{50} + 16 q^{53} - 20 q^{58} + 8 q^{60} + 44 q^{64} + 40 q^{67} - 196 q^{72} - 28 q^{74} + 56 q^{78} + 80 q^{79} - 48 q^{81} - 108 q^{86} - 100 q^{88} - 128 q^{95} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.40713 − 0.141372i −1.51592 + 1.51592i 1.96003 + 0.397858i −0.583934 − 0.583934i 2.34740 − 1.91879i 0 −2.70177 − 0.836931i − 1.59603i 0.739119 + 0.904223i
197.2 −1.40713 − 0.141372i 1.51592 − 1.51592i 1.96003 + 0.397858i 0.583934 + 0.583934i −2.34740 + 1.91879i 0 −2.70177 − 0.836931i − 1.59603i −0.739119 − 0.904223i
197.3 −1.25729 − 0.647469i −2.33911 + 2.33911i 1.16157 + 1.62811i 0.651614 + 0.651614i 4.45544 − 1.42644i 0 −0.406279 − 2.79910i − 7.94286i −0.397370 − 1.24117i
197.4 −1.25729 − 0.647469i 2.33911 − 2.33911i 1.16157 + 1.62811i −0.651614 − 0.651614i −4.45544 + 1.42644i 0 −0.406279 − 2.79910i − 7.94286i 0.397370 + 1.24117i
197.5 −1.22248 + 0.711010i −0.735728 + 0.735728i 0.988930 − 1.73839i 0.793005 + 0.793005i 0.376306 − 1.42253i 0 0.0270652 + 2.82830i 1.91741i −1.53327 − 0.405601i
197.6 −1.22248 + 0.711010i 0.735728 − 0.735728i 0.988930 − 1.73839i −0.793005 − 0.793005i −0.376306 + 1.42253i 0 0.0270652 + 2.82830i 1.91741i 1.53327 + 0.405601i
197.7 −0.725253 + 1.21409i −1.56851 + 1.56851i −0.948017 − 1.76104i 2.79594 + 2.79594i −0.766745 − 3.04188i 0 2.82561 + 0.126223i − 1.92048i −5.42227 + 1.36675i
197.8 −0.725253 + 1.21409i 1.56851 − 1.56851i −0.948017 − 1.76104i −2.79594 − 2.79594i 0.766745 + 3.04188i 0 2.82561 + 0.126223i − 1.92048i 5.42227 − 1.36675i
197.9 −0.435553 − 1.34547i −0.0761273 + 0.0761273i −1.62059 + 1.17205i 2.76470 + 2.76470i 0.135585 + 0.0692696i 0 2.28281 + 1.66997i 2.98841i 2.51565 − 4.92399i
197.10 −0.435553 − 1.34547i 0.0761273 − 0.0761273i −1.62059 + 1.17205i −2.76470 − 2.76470i −0.135585 − 0.0692696i 0 2.28281 + 1.66997i 2.98841i −2.51565 + 4.92399i
197.11 −0.0716322 + 1.41240i −1.24675 + 1.24675i −1.98974 − 0.202346i 0.459484 + 0.459484i −1.67160 − 1.85022i 0 0.428323 − 2.79581i − 0.108789i −0.681888 + 0.616061i
197.12 −0.0716322 + 1.41240i 1.24675 − 1.24675i −1.98974 − 0.202346i −0.459484 − 0.459484i 1.67160 + 1.85022i 0 0.428323 − 2.79581i − 0.108789i 0.681888 − 0.616061i
197.13 0.128885 − 1.40833i −0.954064 + 0.954064i −1.96678 − 0.363024i −2.03054 − 2.03054i 1.22067 + 1.46660i 0 −0.764745 + 2.72308i 1.17953i −3.12137 + 2.59796i
197.14 0.128885 − 1.40833i 0.954064 − 0.954064i −1.96678 − 0.363024i 2.03054 + 2.03054i −1.22067 − 1.46660i 0 −0.764745 + 2.72308i 1.17953i 3.12137 − 2.59796i
197.15 0.395893 + 1.35767i −2.06976 + 2.06976i −1.68654 + 1.07498i −2.64219 − 2.64219i −3.62946 − 1.99065i 0 −2.12716 − 1.86418i − 5.56783i 2.54120 − 4.63325i
197.16 0.395893 + 1.35767i 2.06976 − 2.06976i −1.68654 + 1.07498i 2.64219 + 2.64219i 3.62946 + 1.99065i 0 −2.12716 − 1.86418i − 5.56783i −2.54120 + 4.63325i
197.17 0.852488 − 1.12839i −1.49753 + 1.49753i −0.546529 − 1.92388i 0.568246 + 0.568246i 0.413172 + 2.96643i 0 −2.63679 − 1.02339i − 1.48520i 1.12563 − 0.156780i
197.18 0.852488 − 1.12839i 1.49753 − 1.49753i −0.546529 − 1.92388i −0.568246 − 0.568246i −0.413172 − 2.96643i 0 −2.63679 − 1.02339i − 1.48520i −1.12563 + 0.156780i
197.19 0.924781 + 1.06994i −0.242798 + 0.242798i −0.289561 + 1.97893i 1.67510 + 1.67510i −0.484316 − 0.0352455i 0 −2.38512 + 1.52026i 2.88210i −0.243163 + 3.34136i
197.20 0.924781 + 1.06994i 0.242798 − 0.242798i −0.289561 + 1.97893i −1.67510 − 1.67510i 0.484316 + 0.0352455i 0 −2.38512 + 1.52026i 2.88210i 0.243163 − 3.34136i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.l ✓ 48
7.b odd 2 1 inner 784.2.m.l ✓ 48
7.c even 3 2 784.2.x.p 96
7.d odd 6 2 784.2.x.p 96
16.e even 4 1 inner 784.2.m.l ✓ 48
112.l odd 4 1 inner 784.2.m.l ✓ 48
112.w even 12 2 784.2.x.p 96
112.x odd 12 2 784.2.x.p 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.l ✓ 48 1.a even 1 1 trivial
784.2.m.l ✓ 48 7.b odd 2 1 inner
784.2.m.l ✓ 48 16.e even 4 1 inner
784.2.m.l ✓ 48 112.l odd 4 1 inner
784.2.x.p 96 7.c even 3 2
784.2.x.p 96 7.d odd 6 2
784.2.x.p 96 112.w even 12 2
784.2.x.p 96 112.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{48} + 336 T_{3}^{44} + 43824 T_{3}^{40} + 2879232 T_{3}^{36} + 103720032 T_{3}^{32} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{5}^{48} + 832 T_{5}^{44} + 265808 T_{5}^{40} + 40927104 T_{5}^{36} + 3166604128 T_{5}^{32} + \cdots + 245635219456 \) Copy content Toggle raw display