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Properties

Label 800.2.ba.g
Level $800$
Weight $2$
Character orbit 800.ba
Analytic conductor $6.388$
Analytic rank $0$
Dimension $64$
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] = [800,2,Mod(149,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.ba (of order \(8\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 24 q^{12} + 16 q^{14} - 40 q^{16} - 40 q^{18} + 48 q^{22} - 16 q^{23} - 32 q^{24} + 48 q^{28} - 40 q^{32} - 40 q^{34} - 24 q^{36} + 32 q^{37} + 48 q^{39} + 80 q^{44} - 64 q^{46} - 80 q^{47} + 80 q^{48} + 16 q^{51} + 56 q^{54} - 72 q^{56} - 16 q^{58} - 32 q^{59} - 32 q^{61} - 88 q^{62} - 48 q^{64} + 16 q^{66} - 64 q^{68} + 32 q^{69} + 32 q^{71} + 72 q^{72} - 8 q^{74} + 104 q^{78} + 8 q^{82} - 56 q^{84} + 72 q^{86} - 80 q^{88} - 96 q^{91} - 16 q^{92} - 96 q^{93} + 40 q^{94} + 104 q^{96} + 72 q^{98} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −1.40592 + 0.152916i −1.10663 − 2.67164i 1.95323 − 0.429976i 0 1.96437 + 3.58690i 3.58127 − 3.58127i −2.68034 + 0.903193i −3.79172 + 3.79172i 0
149.2 −1.34874 − 0.425313i 0.567426 + 1.36989i 1.63822 + 1.14728i 0 −0.182680 − 2.08896i −0.427148 + 0.427148i −1.72158 − 2.24414i 0.566703 − 0.566703i 0
149.3 −1.32540 + 0.493282i 1.26808 + 3.06141i 1.51335 − 1.30759i 0 −3.19084 − 3.43206i −1.47530 + 1.47530i −1.36077 + 2.47958i −5.64289 + 5.64289i 0
149.4 −1.09419 + 0.895966i −0.390768 − 0.943398i 0.394491 − 1.96071i 0 1.27283 + 0.682139i −3.31153 + 3.31153i 1.32508 + 2.49883i 1.38402 − 1.38402i 0
149.5 −1.05001 + 0.947356i −0.137709 − 0.332460i 0.205033 − 1.98946i 0 0.459554 + 0.218626i 1.44201 − 1.44201i 1.66944 + 2.28319i 2.02975 − 2.02975i 0
149.6 −0.992681 − 1.00727i 0.504015 + 1.21680i −0.0291697 + 1.99979i 0 0.725315 − 1.71557i 2.78045 − 2.78045i 2.04327 − 1.95577i 0.894749 − 0.894749i 0
149.7 −0.555169 + 1.30069i 0.787169 + 1.90039i −1.38357 − 1.44420i 0 −2.90883 − 0.0311798i 2.82718 − 2.82718i 2.64657 − 0.997820i −0.870540 + 0.870540i 0
149.8 −0.349733 − 1.37029i −0.0553729 − 0.133682i −1.75537 + 0.958469i 0 −0.163817 + 0.122630i −0.648440 + 0.648440i 1.92729 + 2.07016i 2.10652 − 2.10652i 0
149.9 −0.154683 + 1.40573i −1.03874 − 2.50775i −1.95215 − 0.434886i 0 3.68589 − 1.07228i 0.525640 − 0.525640i 0.913296 − 2.67692i −3.08849 + 3.08849i 0
149.10 0.318190 − 1.37795i 0.393957 + 0.951096i −1.79751 − 0.876901i 0 1.43592 − 0.240225i −2.40285 + 2.40285i −1.78028 + 2.19787i 1.37194 − 1.37194i 0
149.11 0.845782 − 1.13343i −0.393611 − 0.950262i −0.569307 − 1.91726i 0 −1.40996 − 0.357585i 1.84130 − 1.84130i −2.65458 − 0.976318i 1.37325 − 1.37325i 0
149.12 0.919667 + 1.07434i −0.121369 − 0.293011i −0.308424 + 1.97608i 0 0.203175 − 0.399864i 1.60956 − 1.60956i −2.40663 + 1.48598i 2.05020 − 2.05020i 0
149.13 1.08820 + 0.903228i −1.13424 − 2.73829i 0.368359 + 1.96579i 0 1.23902 − 4.00428i −1.77906 + 1.77906i −1.37470 + 2.47188i −4.09041 + 4.09041i 0
149.14 1.13661 − 0.841492i −0.932074 − 2.25023i 0.583782 − 1.91290i 0 −2.95295 − 1.77330i −1.83918 + 1.83918i −0.946158 − 2.66548i −2.07343 + 2.07343i 0
149.15 1.18879 + 0.766020i 0.960659 + 2.31924i 0.826427 + 1.82127i 0 −0.634562 + 3.49296i −0.669347 + 0.669347i −0.412682 + 2.79816i −2.33467 + 2.33467i 0
149.16 1.36507 − 0.369574i 0.829213 + 2.00190i 1.72683 − 1.00899i 0 1.87178 + 2.42627i 0.773883 − 0.773883i 1.98435 − 2.01553i −1.19868 + 1.19868i 0
349.1 −1.40592 − 0.152916i −1.10663 + 2.67164i 1.95323 + 0.429976i 0 1.96437 − 3.58690i 3.58127 + 3.58127i −2.68034 − 0.903193i −3.79172 − 3.79172i 0
349.2 −1.34874 + 0.425313i 0.567426 − 1.36989i 1.63822 − 1.14728i 0 −0.182680 + 2.08896i −0.427148 − 0.427148i −1.72158 + 2.24414i 0.566703 + 0.566703i 0
349.3 −1.32540 − 0.493282i 1.26808 − 3.06141i 1.51335 + 1.30759i 0 −3.19084 + 3.43206i −1.47530 − 1.47530i −1.36077 − 2.47958i −5.64289 − 5.64289i 0
349.4 −1.09419 − 0.895966i −0.390768 + 0.943398i 0.394491 + 1.96071i 0 1.27283 − 0.682139i −3.31153 − 3.31153i 1.32508 − 2.49883i 1.38402 + 1.38402i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.z even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.ba.g 64
5.b even 2 1 800.2.ba.e 64
5.c odd 4 1 160.2.x.a ✓ 64
5.c odd 4 1 800.2.y.c 64
20.e even 4 1 640.2.x.a 64
32.g even 8 1 800.2.ba.e 64
160.u even 8 1 640.2.x.a 64
160.v odd 8 1 800.2.y.c 64
160.z even 8 1 inner 800.2.ba.g 64
160.bb odd 8 1 160.2.x.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.x.a ✓ 64 5.c odd 4 1
160.2.x.a ✓ 64 160.bb odd 8 1
640.2.x.a 64 20.e even 4 1
640.2.x.a 64 160.u even 8 1
800.2.y.c 64 5.c odd 4 1
800.2.y.c 64 160.v odd 8 1
800.2.ba.e 64 5.b even 2 1
800.2.ba.e 64 32.g even 8 1
800.2.ba.g 64 1.a even 1 1 trivial
800.2.ba.g 64 160.z even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 16 T_{3}^{59} + 32 T_{3}^{58} - 720 T_{3}^{57} + 16672 T_{3}^{56} - 4512 T_{3}^{55} + \cdots + 1597696 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display