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Properties

Label 816.2.u.a
Level $816$
Weight $2$
Character orbit 816.u
Analytic conductor $6.516$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(205,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.205");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.u (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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 4 q^{4} + 12 q^{10} + 8 q^{11} - 8 q^{12} + 20 q^{14} + 24 q^{15} + 28 q^{16} - 64 q^{17} - 4 q^{18} - 48 q^{22} - 20 q^{26} + 16 q^{29} + 8 q^{30} - 24 q^{31} + 40 q^{32} - 24 q^{35} + 16 q^{37}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1 −1.41228 + 0.0739715i −0.707107 − 0.707107i 1.98906 − 0.208937i −2.01039 + 2.01039i 1.05094 + 0.946325i − 3.19495i −2.79364 + 0.442210i 1.00000i 2.69052 − 2.98794i
205.2 −1.40827 + 0.129521i 0.707107 + 0.707107i 1.96645 − 0.364802i −0.644405 + 0.644405i −1.08738 − 0.904212i − 1.28446i −2.72204 + 0.768437i 1.00000i 0.824032 − 0.990961i
205.3 −1.38749 − 0.273605i −0.707107 − 0.707107i 1.85028 + 0.759252i 1.87129 − 1.87129i 0.787638 + 1.17457i − 4.49580i −2.35952 − 1.55970i 1.00000i −3.10841 + 2.08441i
205.4 −1.32847 + 0.484949i 0.707107 + 0.707107i 1.52965 − 1.28848i −0.0879778 + 0.0879778i −1.28228 − 0.596458i 3.84329i −1.40724 + 2.45350i 1.00000i 0.0742109 − 0.159540i
205.5 −1.29863 + 0.559954i −0.707107 − 0.707107i 1.37290 − 1.45435i 0.593934 − 0.593934i 1.31422 + 0.522326i 1.08628i −0.968528 + 2.65743i 1.00000i −0.438728 + 1.10388i
205.6 −1.29298 − 0.572901i 0.707107 + 0.707107i 1.34357 + 1.48149i 2.98172 − 2.98172i −0.509169 − 1.31937i 1.17350i −0.888453 − 2.68527i 1.00000i −5.56351 + 2.14706i
205.7 −1.16815 − 0.797139i −0.707107 − 0.707107i 0.729140 + 1.86235i −0.639271 + 0.639271i 0.262343 + 1.38967i 2.85654i 0.632809 − 2.75673i 1.00000i 1.25635 − 0.237175i
205.8 −1.13900 − 0.838255i 0.707107 + 0.707107i 0.594656 + 1.90955i −0.565655 + 0.565655i −0.212661 − 1.39813i − 1.93055i 0.923376 − 2.67346i 1.00000i 1.11845 − 0.170119i
205.9 −0.860813 + 1.12205i 0.707107 + 0.707107i −0.518001 − 1.93175i 0.163974 − 0.163974i −1.40210 + 0.184723i − 4.22726i 2.61343 + 1.08166i 1.00000i 0.0428363 + 0.325138i
205.10 −0.850495 + 1.12989i −0.707107 − 0.707107i −0.553316 − 1.92194i 2.29734 − 2.29734i 1.40035 − 0.197564i 4.90695i 2.64218 + 1.00941i 1.00000i 0.641871 + 4.54962i
205.11 −0.780809 + 1.17913i −0.707107 − 0.707107i −0.780674 − 1.84134i −2.45315 + 2.45315i 1.38588 − 0.281652i − 3.71241i 2.78073 + 0.517226i 1.00000i −0.977128 − 4.80801i
205.12 −0.606058 − 1.27777i 0.707107 + 0.707107i −1.26539 + 1.54880i −2.11034 + 2.11034i 0.474971 − 1.33207i 1.09709i 2.74591 + 0.678206i 1.00000i 3.97552 + 1.41754i
205.13 −0.596675 − 1.28218i −0.707107 − 0.707107i −1.28796 + 1.53009i 0.102668 − 0.102668i −0.484724 + 1.32855i − 0.792539i 2.73033 + 0.738428i 1.00000i −0.192897 − 0.0703790i
205.14 −0.359024 + 1.36788i 0.707107 + 0.707107i −1.74220 − 0.982204i 2.64915 − 2.64915i −1.22111 + 0.713371i 1.00268i 1.96903 − 2.03050i 1.00000i 2.67262 + 4.57484i
205.15 0.0155033 + 1.41413i 0.707107 + 0.707107i −1.99952 + 0.0438473i −0.192924 + 0.192924i −0.988977 + 1.01090i − 4.31863i −0.0930049 − 2.82690i 1.00000i −0.275810 − 0.269828i
205.16 0.0612118 − 1.41289i 0.707107 + 0.707107i −1.99251 − 0.172971i −0.807173 + 0.807173i 1.04235 − 0.955780i 0.0213332i −0.366353 + 2.80460i 1.00000i 1.09104 + 1.18985i
205.17 0.0694285 − 1.41251i −0.707107 − 0.707107i −1.99036 − 0.196137i −2.79966 + 2.79966i −1.04789 + 0.949701i 3.41523i −0.415232 + 2.79778i 1.00000i 3.76017 + 4.14892i
205.18 0.158748 + 1.40528i −0.707107 − 0.707107i −1.94960 + 0.446169i −2.32351 + 2.32351i 0.881428 − 1.10593i 2.75370i −0.936484 − 2.66889i 1.00000i −3.63402 − 2.89632i
205.19 0.326049 + 1.37611i −0.707107 − 0.707107i −1.78738 + 0.897362i 2.69913 − 2.69913i 0.742509 − 1.20361i − 1.55925i −1.81765 − 2.16706i 1.00000i 4.59437 + 2.83427i
205.20 0.487868 − 1.32740i 0.707107 + 0.707107i −1.52397 − 1.29519i 0.905261 − 0.905261i 1.28359 − 0.593637i 5.09628i −2.46273 + 1.39103i 1.00000i −0.759993 − 1.64329i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.32
Significant digits:
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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 816.2.u.a ✓ 64
16.e even 4 1 inner 816.2.u.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.u.a ✓ 64 1.a even 1 1 trivial
816.2.u.a ✓ 64 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{64} + 16 T_{5}^{61} + 1056 T_{5}^{60} + 336 T_{5}^{59} + 128 T_{5}^{58} + 11536 T_{5}^{57} + \cdots + 982540877824 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display