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Properties

Label 668.6.a.b
Level $668$
Weight $6$
Character orbit 668.a
Self dual yes
Analytic conductor $107.136$
Analytic rank $0$
Dimension $34$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [668,6,Mod(1,668)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(668, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("668.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

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: yes
Analytic conductor: \(107.136391635\)
Analytic rank: \(0\)
Dimension: \(34\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 48 q^{3} + 46 q^{5} + 676 q^{7} + 2634 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 48 q^{3} + 46 q^{5} + 676 q^{7} + 2634 q^{9} + 297 q^{11} + 1609 q^{13} + 3103 q^{15} - 169 q^{17} + 2933 q^{19} + 1764 q^{21} + 10585 q^{23} + 23320 q^{25} + 14616 q^{27} + 2435 q^{29} + 9891 q^{31} + 1184 q^{33} + 14209 q^{35} + 31686 q^{37} + 4583 q^{39} + 3108 q^{41} + 35282 q^{43} + 1083 q^{45} + 16741 q^{47} + 70394 q^{49} - 73904 q^{51} - 21624 q^{53} - 36980 q^{55} - 76410 q^{57} + 29162 q^{59} + 53195 q^{61} + 155071 q^{63} - 653 q^{65} + 168276 q^{67} + 104992 q^{69} + 146465 q^{71} + 8755 q^{73} + 302277 q^{75} + 193884 q^{77} + 261316 q^{79} + 387678 q^{81} + 129810 q^{83} + 140155 q^{85} + 250527 q^{87} + 23713 q^{89} + 560226 q^{91} + 420698 q^{93} + 144950 q^{95} + 302150 q^{97} + 519955 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −27.9723 0 −83.9001 0 −27.7333 0 539.450 0
1.2 0 −27.3032 0 −6.87740 0 −98.8322 0 502.465 0
1.3 0 −26.2606 0 16.0300 0 −74.0506 0 446.622 0
1.4 0 −21.7833 0 −80.0291 0 244.325 0 231.513 0
1.5 0 −20.9622 0 95.7977 0 54.1932 0 196.415 0
1.6 0 −19.8061 0 17.1686 0 67.5474 0 149.280 0
1.7 0 −19.7100 0 −53.5374 0 85.1029 0 145.483 0
1.8 0 −17.7712 0 44.5581 0 163.940 0 72.8150 0
1.9 0 −14.7531 0 45.2769 0 −151.905 0 −25.3463 0
1.10 0 −14.5610 0 −36.6807 0 −88.1292 0 −30.9772 0
1.11 0 −8.50984 0 105.427 0 117.606 0 −170.583 0
1.12 0 −8.27091 0 −59.5629 0 81.6535 0 −174.592 0
1.13 0 −3.30262 0 −26.2538 0 73.4854 0 −232.093 0
1.14 0 −2.52148 0 35.6147 0 −169.819 0 −236.642 0
1.15 0 −2.10802 0 −76.2102 0 179.793 0 −238.556 0
1.16 0 −1.07784 0 67.1760 0 6.95570 0 −241.838 0
1.17 0 0.562570 0 93.6017 0 196.810 0 −242.684 0
1.18 0 2.33105 0 −9.84115 0 −151.482 0 −237.566 0
1.19 0 5.69918 0 −46.5526 0 −139.493 0 −210.519 0
1.20 0 6.59273 0 −25.8203 0 60.7695 0 −199.536 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(167\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.6.a.b ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.6.a.b ✓ 34 1.a even 1 1 trivial