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Properties

Label 663.2.bg.b
Level $663$
Weight $2$
Character orbit 663.bg
Analytic conductor $5.294$
Analytic rank $0$
Dimension $80$
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] = [663,2,Mod(196,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.196");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.bg (of order \(8\), degree \(4\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{5} + 8 q^{6} - 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 8 q^{5} + 8 q^{6} - 8 q^{7} - 24 q^{8} - 8 q^{11} - 96 q^{16} - 24 q^{17} + 8 q^{19} + 88 q^{20} - 8 q^{24} - 24 q^{25} + 32 q^{28} + 8 q^{29} + 8 q^{31} - 48 q^{32} - 48 q^{33} - 24 q^{34} - 32 q^{35} - 8 q^{36} + 16 q^{37} - 56 q^{40} - 8 q^{41} + 16 q^{42} - 24 q^{43} + 104 q^{46} - 16 q^{49} + 144 q^{50} + 96 q^{52} + 8 q^{54} + 184 q^{56} - 24 q^{58} - 24 q^{59} - 64 q^{60} + 16 q^{61} - 56 q^{62} + 8 q^{63} - 176 q^{67} - 152 q^{68} + 32 q^{69} - 72 q^{70} - 24 q^{71} + 88 q^{73} - 24 q^{74} + 16 q^{75} - 152 q^{76} + 112 q^{77} + 24 q^{79} - 48 q^{80} + 48 q^{82} - 96 q^{83} - 48 q^{84} - 8 q^{85} - 64 q^{86} + 120 q^{88} - 8 q^{91} - 112 q^{92} + 32 q^{93} + 120 q^{95} - 32 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1 −1.88021 + 1.88021i −0.923880 − 0.382683i − 5.07035i −1.48731 + 3.59068i 2.45661 − 1.01756i 0.338764 + 0.817849i 5.77289 + 5.77289i 0.707107 + 0.707107i −3.95478 − 9.54767i
196.2 −1.63202 + 1.63202i 0.923880 + 0.382683i − 3.32698i −1.18979 + 2.87240i −2.13234 + 0.883243i −1.59051 − 3.83983i 2.16566 + 2.16566i 0.707107 + 0.707107i −2.74606 − 6.62957i
196.3 −1.54567 + 1.54567i 0.923880 + 0.382683i − 2.77821i 0.421804 − 1.01832i −2.01952 + 0.836512i 0.400127 + 0.965991i 1.20286 + 1.20286i 0.707107 + 0.707107i 0.922026 + 2.22597i
196.4 −1.52739 + 1.52739i −0.923880 − 0.382683i − 2.66584i 1.22093 − 2.94758i 1.99563 − 0.826618i −1.27114 − 3.06881i 1.01700 + 1.01700i 0.707107 + 0.707107i 2.63728 + 6.36695i
196.5 −1.18260 + 1.18260i −0.923880 − 0.382683i − 0.797096i −1.04203 + 2.51568i 1.54514 − 0.640020i 0.605038 + 1.46069i −1.42256 − 1.42256i 0.707107 + 0.707107i −1.74274 − 4.20736i
196.6 −1.11331 + 1.11331i −0.923880 − 0.382683i − 0.478938i 1.28725 − 3.10769i 1.45462 − 0.602521i 0.854392 + 2.06269i −1.69342 − 1.69342i 0.707107 + 0.707107i 2.02672 + 4.89294i
196.7 −0.711644 + 0.711644i 0.923880 + 0.382683i 0.987126i 0.289764 − 0.699552i −0.929808 + 0.385139i −0.588952 − 1.42186i −2.12577 − 2.12577i 0.707107 + 0.707107i 0.291623 + 0.704041i
196.8 −0.604635 + 0.604635i 0.923880 + 0.382683i 1.26883i −1.15192 + 2.78099i −0.789994 + 0.327226i 1.55139 + 3.74540i −1.97645 − 1.97645i 0.707107 + 0.707107i −0.984990 − 2.37798i
196.9 −0.499440 + 0.499440i 0.923880 + 0.382683i 1.50112i 1.49961 − 3.62037i −0.652550 + 0.270295i −0.570804 − 1.37804i −1.74860 − 1.74860i 0.707107 + 0.707107i 1.05920 + 2.55712i
196.10 −0.308328 + 0.308328i −0.923880 − 0.382683i 1.80987i 0.271968 − 0.656588i 0.402851 − 0.166866i 1.83640 + 4.43346i −1.17469 − 1.17469i 0.707107 + 0.707107i 0.118589 + 0.286300i
196.11 0.105276 − 0.105276i −0.923880 − 0.382683i 1.97783i 0.911164 − 2.19974i −0.137549 + 0.0569748i −1.56256 − 3.77235i 0.418769 + 0.418769i 0.707107 + 0.707107i −0.135656 − 0.327503i
196.12 0.273403 − 0.273403i −0.923880 − 0.382683i 1.85050i −0.0397942 + 0.0960716i −0.357218 + 0.147965i −1.12265 − 2.71031i 1.05274 + 1.05274i 0.707107 + 0.707107i 0.0153864 + 0.0371461i
196.13 0.551850 − 0.551850i 0.923880 + 0.382683i 1.39092i −1.00609 + 2.42890i 0.721026 − 0.298659i −0.592082 − 1.42941i 1.87128 + 1.87128i 0.707107 + 0.707107i 0.785183 + 1.89560i
196.14 0.996044 − 0.996044i −0.923880 − 0.382683i 0.0157941i 1.62978 − 3.93463i −1.30139 + 0.539055i 1.06711 + 2.57624i 2.00782 + 2.00782i 0.707107 + 0.707107i −2.29573 − 5.54239i
196.15 1.03311 − 1.03311i 0.923880 + 0.382683i − 0.134613i 0.545318 − 1.31652i 1.34982 − 0.559113i 1.30846 + 3.15889i 1.92714 + 1.92714i 0.707107 + 0.707107i −0.796727 − 1.92347i
196.16 1.10573 − 1.10573i 0.923880 + 0.382683i − 0.445284i −0.0199355 + 0.0481285i 1.44471 − 0.598418i −1.62449 − 3.92186i 1.71910 + 1.71910i 0.707107 + 0.707107i 0.0311739 + 0.0752604i
196.17 1.48160 − 1.48160i −0.923880 − 0.382683i − 2.39029i −0.799587 + 1.93037i −1.93581 + 0.801837i 1.22063 + 2.94685i −0.578249 − 0.578249i 0.707107 + 0.707107i 1.67538 + 4.04471i
196.18 1.73094 − 1.73094i 0.923880 + 0.382683i − 3.99230i −1.69334 + 4.08808i 2.26158 − 0.936777i 1.35731 + 3.27683i −3.44854 − 3.44854i 0.707107 + 0.707107i 4.14516 + 10.0073i
196.19 1.84895 − 1.84895i −0.923880 − 0.382683i − 4.83727i 0.636632 − 1.53696i −2.41578 + 1.00065i −0.0210393 − 0.0507934i −5.24598 − 5.24598i 0.707107 + 0.707107i −1.66468 − 4.01888i
196.20 1.87835 − 1.87835i 0.923880 + 0.382683i − 5.05640i 0.301371 − 0.727575i 2.45418 − 1.01656i −0.766967 − 1.85162i −5.74100 − 5.74100i 0.707107 + 0.707107i −0.800560 − 1.93272i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 196.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.bg.b ✓ 80
17.d even 8 1 inner 663.2.bg.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.bg.b ✓ 80 1.a even 1 1 trivial
663.2.bg.b ✓ 80 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 8 T_{2}^{77} + 328 T_{2}^{76} + 48 T_{2}^{75} + 32 T_{2}^{74} + 2336 T_{2}^{73} + \cdots + 130896481 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display