Properties

Label 668.2.e.a
Level $668$
Weight $2$
Character orbit 668.e
Analytic conductor $5.334$
Analytic rank $0$
Dimension $1148$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.e (of order \(83\), degree \(82\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(1148\)
Relative dimension: \(14\) over \(\Q(\zeta_{83})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{83}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1148q - 2q^{5} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1148q - 2q^{5} - 14q^{9} + 2q^{11} + 4q^{13} + 14q^{15} + 2q^{17} + 2q^{19} + 14q^{23} - 6q^{25} + 2q^{29} - 2q^{31} + 16q^{33} - 2q^{35} + 10q^{37} + 6q^{39} + 4q^{41} + 4q^{43} - 2q^{45} + 2q^{47} - 30q^{49} - 2q^{51} - 6q^{55} - 4q^{57} + 6q^{59} + 2q^{61} + 14q^{63} + 22q^{65} + 12q^{67} - 14q^{69} - 8q^{71} - 18q^{73} - 26q^{75} - 2q^{79} - 6q^{81} - 22q^{83} + 34q^{85} + 2q^{87} + 14q^{89} - 6q^{91} + 32q^{93} - 8q^{95} + 44q^{97} + 2q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −0.426508 3.20063i 0 −3.18243 1.40730i 0 1.55798 3.35028i 0 −7.16683 + 1.94460i 0
9.2 0 −0.360076 2.70211i 0 2.66599 + 1.17892i 0 0.359959 0.774059i 0 −4.27644 + 1.16034i 0
9.3 0 −0.342745 2.57205i 0 −0.628586 0.277966i 0 −1.47454 + 3.17087i 0 −3.60268 + 0.977525i 0
9.4 0 −0.198520 1.48975i 0 −0.0282706 0.0125015i 0 2.06702 4.44493i 0 0.715369 0.194103i 0
9.5 0 −0.178407 1.33882i 0 −1.62557 0.718841i 0 −0.315447 + 0.678340i 0 1.13471 0.307883i 0
9.6 0 −0.0387535 0.290817i 0 −1.08122 0.478126i 0 −0.661531 + 1.42256i 0 2.81224 0.763053i 0
9.7 0 −0.0142693 0.107081i 0 2.37141 + 1.04866i 0 −0.709794 + 1.52635i 0 2.88405 0.782538i 0
9.8 0 0.0433273 + 0.325140i 0 3.41908 + 1.51194i 0 −1.67119 + 3.59374i 0 2.79147 0.757419i 0
9.9 0 0.174236 + 1.30752i 0 −3.98478 1.76210i 0 −0.546271 + 1.17470i 0 1.21607 0.329959i 0
9.10 0 0.189949 + 1.42543i 0 2.53109 + 1.11927i 0 0.927279 1.99403i 0 0.899535 0.244073i 0
9.11 0 0.200238 + 1.50264i 0 −0.247141 0.109288i 0 1.16135 2.49738i 0 0.677472 0.183820i 0
9.12 0 0.214009 + 1.60599i 0 −2.48108 1.09715i 0 1.06258 2.28498i 0 0.361919 0.0982005i 0
9.13 0 0.353640 + 2.65381i 0 −1.27217 0.562563i 0 −1.92428 + 4.13797i 0 −4.02235 + 1.09139i 0
9.14 0 0.383878 + 2.88073i 0 1.71454 + 0.758185i 0 0.166885 0.358871i 0 −5.25594 + 1.42611i 0
21.1 0 −3.30940 0.504914i 0 −2.26577 2.68874i 0 1.34830 1.87063i 0 7.83367 + 2.44733i 0
21.2 0 −2.93125 0.447219i 0 1.46272 + 1.73577i 0 −0.690896 + 0.958545i 0 5.52869 + 1.72722i 0
21.3 0 −1.90829 0.291147i 0 1.01478 + 1.20422i 0 2.44900 3.39773i 0 0.693279 + 0.216588i 0
21.4 0 −1.77371 0.270614i 0 −1.21708 1.44428i 0 −0.212405 + 0.294689i 0 0.209294 + 0.0653857i 0
21.5 0 −1.69992 0.259356i 0 −0.724427 0.859660i 0 −2.54038 + 3.52451i 0 −0.0410528 0.0128253i 0
21.6 0 −0.537235 0.0819657i 0 −1.19475 1.41779i 0 0.821536 1.13980i 0 −2.58161 0.806524i 0
See next 80 embeddings (of 1148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 653.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.c even 83 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.2.e.a 1148
167.c even 83 1 inner 668.2.e.a 1148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.e.a 1148 1.a even 1 1 trivial
668.2.e.a 1148 167.c even 83 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(668, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database