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Properties

Label 784.2.w.g
Level $784$
Weight $2$
Character orbit 784.w
Analytic conductor $6.260$
Analytic rank $0$
Dimension $192$
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] = [784,2,Mod(19,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.w (of order \(12\), degree \(4\), not 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: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{4} - 16 q^{11} - 40 q^{16} + 40 q^{18} - 16 q^{22} + 32 q^{23} + 64 q^{29} + 80 q^{30} - 160 q^{36} - 32 q^{37} - 32 q^{43} + 8 q^{44} - 112 q^{50} - 32 q^{53} + 40 q^{58} - 176 q^{60} - 112 q^{64} - 80 q^{67} + 72 q^{72} - 168 q^{74} + 192 q^{78} + 96 q^{81} + 72 q^{86} - 40 q^{88} + 32 q^{92} - 160 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41387 + 0.0309690i −0.0367825 + 0.137274i 1.99808 − 0.0875726i 2.39619 − 0.642058i 0.0477546 − 0.195228i 0 −2.82232 + 0.185695i 2.58058 + 1.48990i −3.36803 + 0.981997i
19.2 −1.41387 + 0.0309690i 0.0367825 − 0.137274i 1.99808 − 0.0875726i −2.39619 + 0.642058i −0.0477546 + 0.195228i 0 −2.82232 + 0.185695i 2.58058 + 1.48990i 3.36803 − 0.981997i
19.3 −1.39991 − 0.200604i −0.750967 + 2.80265i 1.91952 + 0.561655i −0.297804 + 0.0797965i 1.61351 − 3.77282i 0 −2.57449 − 1.17133i −4.69280 − 2.70939i 0.432908 − 0.0519675i
19.4 −1.39991 − 0.200604i 0.750967 − 2.80265i 1.91952 + 0.561655i 0.297804 − 0.0797965i −1.61351 + 3.77282i 0 −2.57449 − 1.17133i −4.69280 − 2.70939i −0.432908 + 0.0519675i
19.5 −1.32158 − 0.503417i −0.650041 + 2.42599i 1.49314 + 1.33061i 4.12874 − 1.10629i 2.08036 − 2.87889i 0 −1.30346 − 2.51018i −2.86478 − 1.65398i −6.01338 − 0.616424i
19.6 −1.32158 − 0.503417i 0.650041 − 2.42599i 1.49314 + 1.33061i −4.12874 + 1.10629i −2.08036 + 2.87889i 0 −1.30346 − 2.51018i −2.86478 − 1.65398i 6.01338 + 0.616424i
19.7 −1.17181 + 0.791750i −0.589602 + 2.20042i 0.746264 − 1.85556i 0.545106 − 0.146061i −1.05129 − 3.04529i 0 0.594659 + 2.76521i −1.89616 − 1.09475i −0.523116 + 0.602743i
19.8 −1.17181 + 0.791750i 0.589602 − 2.20042i 0.746264 − 1.85556i −0.545106 + 0.146061i 1.05129 + 3.04529i 0 0.594659 + 2.76521i −1.89616 − 1.09475i 0.523116 − 0.602743i
19.9 −1.15509 − 0.815940i −0.431940 + 1.61202i 0.668484 + 1.88497i −1.77857 + 0.476568i 1.81424 − 1.50960i 0 0.765864 − 2.72277i 0.186033 + 0.107406i 2.44327 + 0.900730i
19.10 −1.15509 − 0.815940i 0.431940 − 1.61202i 0.668484 + 1.88497i 1.77857 − 0.476568i −1.81424 + 1.50960i 0 0.765864 − 2.72277i 0.186033 + 0.107406i −2.44327 − 0.900730i
19.11 −1.13635 − 0.841852i −0.256665 + 0.957888i 0.582572 + 1.91327i −3.25157 + 0.871257i 1.09806 − 0.872419i 0 0.948687 − 2.66458i 1.74640 + 1.00829i 4.42839 + 1.74729i
19.12 −1.13635 − 0.841852i 0.256665 − 0.957888i 0.582572 + 1.91327i 3.25157 − 0.871257i −1.09806 + 0.872419i 0 0.948687 − 2.66458i 1.74640 + 1.00829i −4.42839 − 1.74729i
19.13 −1.06575 + 0.929606i −0.103307 + 0.385547i 0.271667 − 1.98146i −2.48567 + 0.666032i −0.248306 − 0.506933i 0 1.55245 + 2.36430i 2.46010 + 1.42034i 2.02996 − 3.02052i
19.14 −1.06575 + 0.929606i 0.103307 − 0.385547i 0.271667 − 1.98146i 2.48567 − 0.666032i 0.248306 + 0.506933i 0 1.55245 + 2.36430i 2.46010 + 1.42034i −2.02996 + 3.02052i
19.15 −1.01893 − 0.980703i −0.772777 + 2.88404i 0.0764421 + 1.99854i 0.719511 − 0.192792i 3.61580 − 2.18078i 0 1.88208 − 2.11134i −5.12245 − 2.95745i −0.922204 − 0.509185i
19.16 −1.01893 − 0.980703i 0.772777 − 2.88404i 0.0764421 + 1.99854i −0.719511 + 0.192792i −3.61580 + 2.18078i 0 1.88208 − 2.11134i −5.12245 − 2.95745i 0.922204 + 0.509185i
19.17 −0.970652 + 1.02851i −0.875817 + 3.26859i −0.115671 − 1.99665i −2.44706 + 0.655686i −2.51167 − 4.07345i 0 2.16586 + 1.81909i −7.31856 − 4.22537i 1.70086 − 3.15327i
19.18 −0.970652 + 1.02851i 0.875817 − 3.26859i −0.115671 − 1.99665i 2.44706 − 0.655686i 2.51167 + 4.07345i 0 2.16586 + 1.81909i −7.31856 − 4.22537i −1.70086 + 3.15327i
19.19 −0.588616 + 1.28590i −0.531125 + 1.98218i −1.30706 − 1.51380i 0.851849 − 0.228252i −2.23626 − 1.84972i 0 2.71595 − 0.789701i −1.04889 − 0.605575i −0.207903 + 1.22974i
19.20 −0.588616 + 1.28590i 0.531125 − 1.98218i −1.30706 − 1.51380i −0.851849 + 0.228252i 2.23626 + 1.84972i 0 2.71595 − 0.789701i −1.04889 − 0.605575i 0.207903 − 1.22974i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
16.f odd 4 1 inner
112.j even 4 1 inner
112.u odd 12 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.w.g 192
7.b odd 2 1 inner 784.2.w.g 192
7.c even 3 1 784.2.j.b ✓ 96
7.c even 3 1 inner 784.2.w.g 192
7.d odd 6 1 784.2.j.b ✓ 96
7.d odd 6 1 inner 784.2.w.g 192
16.f odd 4 1 inner 784.2.w.g 192
112.j even 4 1 inner 784.2.w.g 192
112.u odd 12 1 784.2.j.b ✓ 96
112.u odd 12 1 inner 784.2.w.g 192
112.v even 12 1 784.2.j.b ✓ 96
112.v even 12 1 inner 784.2.w.g 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.j.b ✓ 96 7.c even 3 1
784.2.j.b ✓ 96 7.d odd 6 1
784.2.j.b ✓ 96 112.u odd 12 1
784.2.j.b ✓ 96 112.v even 12 1
784.2.w.g 192 1.a even 1 1 trivial
784.2.w.g 192 7.b odd 2 1 inner
784.2.w.g 192 7.c even 3 1 inner
784.2.w.g 192 7.d odd 6 1 inner
784.2.w.g 192 16.f odd 4 1 inner
784.2.w.g 192 112.j even 4 1 inner
784.2.w.g 192 112.u odd 12 1 inner
784.2.w.g 192 112.v even 12 1 inner

Hecke kernels

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

\( T_{3}^{192} - 672 T_{3}^{188} + 250880 T_{3}^{184} - 64295808 T_{3}^{180} + 12490001408 T_{3}^{176} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
\( T_{5}^{192} - 1584 T_{5}^{188} + 1418224 T_{5}^{184} - 863898112 T_{5}^{180} + 395776314464 T_{5}^{176} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display