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Properties

Label 784.2.m.j
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
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] = [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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{10} + 4 q^{11} - 2 q^{12} - 12 q^{13} - 20 q^{15} - 16 q^{16} - 8 q^{17} - 18 q^{18} + 4 q^{19} - 8 q^{20} - 18 q^{24} + 10 q^{26} - 12 q^{27} + 12 q^{29} + 4 q^{30} - 28 q^{31} - 16 q^{32} - 16 q^{33} - 22 q^{34} - 36 q^{36} + 24 q^{37} - 20 q^{38} - 26 q^{40} - 20 q^{43} - 6 q^{44} + 28 q^{45} + 14 q^{46} + 20 q^{47} + 28 q^{48} + 28 q^{50} - 24 q^{51} + 16 q^{52} + 16 q^{53} - 64 q^{54} + 6 q^{58} + 20 q^{59} - 46 q^{60} - 8 q^{61} + 12 q^{62} + 40 q^{64} - 8 q^{65} + 20 q^{66} - 48 q^{67} - 20 q^{69} + 32 q^{72} + 8 q^{74} + 4 q^{75} - 18 q^{76} + 58 q^{78} + 36 q^{79} + 28 q^{80} - 2 q^{82} - 4 q^{83} + 20 q^{86} + 42 q^{88} - 10 q^{90} + 38 q^{92} - 8 q^{93} + 72 q^{94} + 4 q^{95} + 120 q^{96} - 24 q^{97} - 12 q^{99}+O(q^{100}) \) 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 −1.38357 − 0.292823i 0.0398054 − 0.0398054i 1.82851 + 0.810281i −0.920778 − 0.920778i −0.0667293 + 0.0434174i 0 −2.29259 − 1.65651i 2.99683i 1.00433 + 1.54358i
197.2 −1.33465 + 0.467669i −2.13390 + 2.13390i 1.56257 − 1.24835i 0.545748 + 0.545748i 1.85005 − 3.84597i 0 −1.50167 + 2.39687i − 6.10706i −0.983611 − 0.473152i
197.3 −0.913051 − 1.07997i 1.80111 − 1.80111i −0.332676 + 1.97214i −1.37283 − 1.37283i −3.58965 − 0.300642i 0 2.43360 − 1.44138i − 3.48800i −0.229153 + 2.73608i
197.4 −0.669670 − 1.24561i −0.608827 + 0.608827i −1.10308 + 1.66829i 1.48423 + 1.48423i 1.16607 + 0.350647i 0 2.81674 + 0.256804i 2.25866i 0.854824 − 2.84271i
197.5 −0.604218 + 1.27864i −0.853080 + 0.853080i −1.26984 − 1.54515i −0.718099 − 0.718099i −0.575337 − 1.60623i 0 2.74296 − 0.690063i 1.54451i 1.35208 − 0.484303i
197.6 0.350694 + 1.37004i 1.17747 − 1.17747i −1.75403 + 0.960931i −1.42676 − 1.42676i 2.02612 + 1.20025i 0 −1.93164 − 2.06610i 0.227125i 1.45437 − 2.45508i
197.7 0.700256 − 1.22867i 2.22611 − 2.22611i −1.01928 − 1.72077i 1.37091 + 1.37091i −1.17632 − 4.29401i 0 −2.82803 + 0.0473857i − 6.91115i 2.64439 − 0.724414i
197.8 0.844695 + 1.13424i 0.614312 − 0.614312i −0.572980 + 1.91617i 2.31562 + 2.31562i 1.21568 + 0.177868i 0 −2.65738 + 0.968682i 2.24524i −0.670466 + 4.58245i
197.9 1.08920 − 0.902016i −0.606021 + 0.606021i 0.372734 − 1.96496i −3.00784 − 3.00784i −0.113440 + 1.20672i 0 −1.36644 − 2.47646i 2.26548i −5.98927 − 0.563034i
197.10 1.19460 + 0.756916i −1.42954 + 1.42954i 0.854157 + 1.80843i −0.702089 − 0.702089i −2.78978 + 0.625694i 0 −0.348448 + 2.80688i − 1.08719i −0.307296 − 1.37014i
197.11 1.32935 − 0.482511i −1.83801 + 1.83801i 1.53437 − 1.28286i 2.13175 + 2.13175i −1.55651 + 3.33023i 0 1.42072 − 2.44572i − 3.75657i 3.86245 + 1.80526i
197.12 1.39634 + 0.224104i 1.61057 − 1.61057i 1.89956 + 0.625852i −1.69986 − 1.69986i 2.60985 − 1.88798i 0 2.51218 + 1.29960i − 2.18789i −1.99265 − 2.75454i
589.1 −1.38357 + 0.292823i 0.0398054 + 0.0398054i 1.82851 − 0.810281i −0.920778 + 0.920778i −0.0667293 − 0.0434174i 0 −2.29259 + 1.65651i − 2.99683i 1.00433 − 1.54358i
589.2 −1.33465 − 0.467669i −2.13390 − 2.13390i 1.56257 + 1.24835i 0.545748 − 0.545748i 1.85005 + 3.84597i 0 −1.50167 − 2.39687i 6.10706i −0.983611 + 0.473152i
589.3 −0.913051 + 1.07997i 1.80111 + 1.80111i −0.332676 − 1.97214i −1.37283 + 1.37283i −3.58965 + 0.300642i 0 2.43360 + 1.44138i 3.48800i −0.229153 − 2.73608i
589.4 −0.669670 + 1.24561i −0.608827 − 0.608827i −1.10308 − 1.66829i 1.48423 − 1.48423i 1.16607 − 0.350647i 0 2.81674 − 0.256804i − 2.25866i 0.854824 + 2.84271i
589.5 −0.604218 − 1.27864i −0.853080 − 0.853080i −1.26984 + 1.54515i −0.718099 + 0.718099i −0.575337 + 1.60623i 0 2.74296 + 0.690063i − 1.54451i 1.35208 + 0.484303i
589.6 0.350694 − 1.37004i 1.17747 + 1.17747i −1.75403 − 0.960931i −1.42676 + 1.42676i 2.02612 − 1.20025i 0 −1.93164 + 2.06610i − 0.227125i 1.45437 + 2.45508i
589.7 0.700256 + 1.22867i 2.22611 + 2.22611i −1.01928 + 1.72077i 1.37091 − 1.37091i −1.17632 + 4.29401i 0 −2.82803 − 0.0473857i 6.91115i 2.64439 + 0.724414i
589.8 0.844695 − 1.13424i 0.614312 + 0.614312i −0.572980 − 1.91617i 2.31562 − 2.31562i 1.21568 − 0.177868i 0 −2.65738 − 0.968682i − 2.24524i −0.670466 − 4.58245i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 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.j 24
7.b odd 2 1 784.2.m.k 24
7.c even 3 2 112.2.w.c ✓ 48
7.d odd 6 2 784.2.x.o 48
16.e even 4 1 inner 784.2.m.j 24
28.g odd 6 2 448.2.ba.c 48
56.k odd 6 2 896.2.ba.e 48
56.p even 6 2 896.2.ba.f 48
112.l odd 4 1 784.2.m.k 24
112.u odd 12 2 448.2.ba.c 48
112.u odd 12 2 896.2.ba.e 48
112.w even 12 2 112.2.w.c ✓ 48
112.w even 12 2 896.2.ba.f 48
112.x odd 12 2 784.2.x.o 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.c ✓ 48 7.c even 3 2
112.2.w.c ✓ 48 112.w even 12 2
448.2.ba.c 48 28.g odd 6 2
448.2.ba.c 48 112.u odd 12 2
784.2.m.j 24 1.a even 1 1 trivial
784.2.m.j 24 16.e even 4 1 inner
784.2.m.k 24 7.b odd 2 1
784.2.m.k 24 112.l odd 4 1
784.2.x.o 48 7.d odd 6 2
784.2.x.o 48 112.x odd 12 2
896.2.ba.e 48 56.k odd 6 2
896.2.ba.e 48 112.u odd 12 2
896.2.ba.f 48 56.p even 6 2
896.2.ba.f 48 112.w even 12 2

Hecke kernels

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

\( T_{3}^{24} + 4 T_{3}^{21} + 162 T_{3}^{20} + 24 T_{3}^{19} + 8 T_{3}^{18} + 468 T_{3}^{17} + 7855 T_{3}^{16} + \cdots + 441 \) Copy content Toggle raw display
\( T_{5}^{24} + 4 T_{5}^{23} + 8 T_{5}^{22} - 4 T_{5}^{21} + 266 T_{5}^{20} + 908 T_{5}^{19} + \cdots + 2660161 \) Copy content Toggle raw display