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Properties

Label 800.2.ba.f
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: yes
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
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 24 q^{12} + 16 q^{14} + 20 q^{16} + 40 q^{18} + 24 q^{22} - 8 q^{23} + 28 q^{24} - 48 q^{28} - 20 q^{32} + 20 q^{34} + 12 q^{36} + 16 q^{37} - 24 q^{39} - 40 q^{44} + 32 q^{46} + 80 q^{47} + 40 q^{48} + 16 q^{51} - 76 q^{54} + 48 q^{56} - 8 q^{58} - 32 q^{59} - 32 q^{61} - 44 q^{62} - 48 q^{64} + 16 q^{66} - 32 q^{68} + 32 q^{69} + 32 q^{71} + 96 q^{72} - 8 q^{74} + 52 q^{78} - 116 q^{82} - 56 q^{84} - 84 q^{86} + 20 q^{88} + 48 q^{91} + 88 q^{92} - 48 q^{93} - 32 q^{94} - 100 q^{96} - 72 q^{98} + 64 q^{99}+O(q^{100}) \) 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 −1.38149 − 0.302467i −1.26654 − 3.05770i 1.81703 + 0.835711i 0 0.824860 + 4.60727i −3.13874 + 3.13874i −2.25743 − 1.70412i −5.62411 + 5.62411i 0
149.2 −1.34730 + 0.429864i 0.718992 + 1.73580i 1.63043 − 1.15831i 0 −1.71486 − 2.02958i −2.76813 + 2.76813i −1.69877 + 2.26146i −0.374736 + 0.374736i 0
149.3 −1.33068 − 0.478839i −0.106495 − 0.257102i 1.54143 + 1.27436i 0 0.0186007 + 0.393114i 1.49162 − 1.49162i −1.44093 − 2.43387i 2.06656 − 2.06656i 0
149.4 −1.27009 + 0.621998i −0.210911 − 0.509183i 1.22624 − 1.57998i 0 0.584586 + 0.515521i 0.732076 − 0.732076i −0.574680 + 2.76943i 1.90654 − 1.90654i 0
149.5 −1.02078 − 0.978783i 1.07602 + 2.59773i 0.0839659 + 1.99824i 0 1.44425 − 3.70489i −1.18981 + 1.18981i 1.87013 − 2.12194i −3.46908 + 3.46908i 0
149.6 −0.799221 + 1.16672i −1.08855 − 2.62799i −0.722493 − 1.86494i 0 3.93614 + 0.830308i 2.39635 − 2.39635i 2.75330 + 0.647549i −3.60009 + 3.60009i 0
149.7 −0.461176 + 1.33691i 0.215852 + 0.521113i −1.57463 − 1.23310i 0 −0.796224 + 0.0482487i −0.916264 + 0.916264i 2.37472 − 1.53646i 1.89635 − 1.89635i 0
149.8 −0.0587773 − 1.41299i −0.645940 − 1.55944i −1.99309 + 0.166104i 0 −2.16550 + 1.00437i −2.63390 + 2.63390i 0.351851 + 2.80646i 0.106717 − 0.106717i 0
149.9 0.0509313 − 1.41330i 0.576853 + 1.39265i −1.99481 − 0.143962i 0 1.99760 − 0.744335i 1.85075 − 1.85075i −0.305059 + 2.81193i 0.514615 − 0.514615i 0
149.10 0.480692 + 1.33001i 1.19678 + 2.88928i −1.53787 + 1.27865i 0 −3.26750 + 2.98059i −0.134172 + 0.134172i −2.43987 − 1.43075i −4.79435 + 4.79435i 0
149.11 0.516460 + 1.31654i −0.735834 − 1.77646i −1.46654 + 1.35988i 0 1.95875 − 1.88622i −0.709913 + 0.709913i −2.54774 − 1.22843i −0.493042 + 0.493042i 0
149.12 0.807152 − 1.16125i −0.844082 − 2.03779i −0.697010 − 1.87461i 0 −3.04769 − 0.664619i 1.60593 − 1.60593i −2.73949 − 0.703695i −1.31881 + 1.31881i 0
149.13 1.16344 + 0.803988i 0.225021 + 0.543249i 0.707207 + 1.87079i 0 −0.174966 + 0.812955i 3.50918 − 3.50918i −0.681298 + 2.74515i 1.87683 − 1.87683i 0
149.14 1.22541 − 0.705948i 0.940874 + 2.27147i 1.00327 − 1.73016i 0 2.75650 + 2.11928i 0.807000 − 0.807000i 0.00802276 − 2.82842i −2.15301 + 2.15301i 0
149.15 1.33740 + 0.459751i 0.439222 + 1.06038i 1.57726 + 1.22974i 0 0.0999049 + 1.62008i −2.94684 + 2.94684i 1.54404 + 2.36979i 1.18984 − 1.18984i 0
149.16 1.38091 − 0.305097i −0.491256 − 1.18600i 1.81383 − 0.842624i 0 −1.04022 − 1.48787i −0.783571 + 0.783571i 2.24766 − 1.71698i 0.956065 − 0.956065i 0
349.1 −1.38149 + 0.302467i −1.26654 + 3.05770i 1.81703 − 0.835711i 0 0.824860 − 4.60727i −3.13874 − 3.13874i −2.25743 + 1.70412i −5.62411 − 5.62411i 0
349.2 −1.34730 − 0.429864i 0.718992 − 1.73580i 1.63043 + 1.15831i 0 −1.71486 + 2.02958i −2.76813 − 2.76813i −1.69877 − 2.26146i −0.374736 − 0.374736i 0
349.3 −1.33068 + 0.478839i −0.106495 + 0.257102i 1.54143 − 1.27436i 0 0.0186007 − 0.393114i 1.49162 + 1.49162i −1.44093 + 2.43387i 2.06656 + 2.06656i 0
349.4 −1.27009 − 0.621998i −0.210911 + 0.509183i 1.22624 + 1.57998i 0 0.584586 − 0.515521i 0.732076 + 0.732076i −0.574680 − 2.76943i 1.90654 + 1.90654i 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.f 64
5.b even 2 1 800.2.ba.h 64
5.c odd 4 1 800.2.y.d ✓ 64
5.c odd 4 1 800.2.y.e yes 64
32.g even 8 1 800.2.ba.h 64
160.v odd 8 1 800.2.y.e yes 64
160.z even 8 1 inner 800.2.ba.f 64
160.bb odd 8 1 800.2.y.d ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.y.d ✓ 64 5.c odd 4 1
800.2.y.d ✓ 64 160.bb odd 8 1
800.2.y.e yes 64 5.c odd 4 1
800.2.y.e yes 64 160.v odd 8 1
800.2.ba.f 64 1.a even 1 1 trivial
800.2.ba.f 64 160.z even 8 1 inner
800.2.ba.h 64 5.b even 2 1
800.2.ba.h 64 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 8 T_{3}^{59} - 16 T_{3}^{58} - 1080 T_{3}^{57} + 16768 T_{3}^{56} - 10944 T_{3}^{55} + \cdots + 1963464721 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display