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Properties

Label 189.6.s.a
Level $189$
Weight $6$
Character orbit 189.s
Analytic conductor $30.313$
Analytic rank $0$
Dimension $76$
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] = [189,6,Mod(17,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 189.s (of order \(6\), degree \(2\), 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: \(30.3125419447\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 3 q^{2} + 577 q^{4} + 6 q^{5} - 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 3 q^{2} + 577 q^{4} + 6 q^{5} - 30 q^{7} - 6 q^{10} - 543 q^{13} + 123 q^{14} - 8223 q^{16} - 801 q^{17} - 6 q^{19} + 96 q^{20} + 62 q^{22} + 37498 q^{25} + 10128 q^{26} + 860 q^{28} - 17904 q^{29} + 3249 q^{31} - 10299 q^{32} - 96 q^{34} + 3960 q^{35} + 2577 q^{37} - 29934 q^{38} + 28230 q^{41} - 9246 q^{43} - 69885 q^{44} - 9418 q^{46} + 28281 q^{47} + 2458 q^{49} + 67509 q^{50} + 25296 q^{53} - 27288 q^{56} + 9902 q^{58} + 29538 q^{59} + 4206 q^{61} + 79536 q^{62} - 198600 q^{64} + 173388 q^{65} - 622 q^{67} - 382992 q^{68} + 14178 q^{70} - 6 q^{73} + 2880 q^{76} + 238866 q^{77} - 29992 q^{79} + 243225 q^{80} + 90 q^{82} - 246930 q^{83} + 11973 q^{85} + 69502 q^{88} - 6345 q^{89} - 120111 q^{91} + 463488 q^{92} - 3 q^{94} + 267813 q^{95} + 104037 q^{97} - 646797 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −9.51481 − 5.49338i 0 44.3544 + 76.8242i 21.7045 0 −128.656 − 15.9572i − 623.047i 0 −206.515 − 119.231i
17.2 −8.91060 − 5.14454i 0 36.9326 + 63.9691i −83.0456 0 77.3170 − 104.063i − 430.754i 0 739.986 + 427.231i
17.3 −8.82016 − 5.09232i 0 35.8635 + 62.1174i −38.5112 0 −29.3317 + 126.280i − 404.606i 0 339.675 + 196.111i
17.4 −7.79850 − 4.50246i 0 24.5444 + 42.5121i −18.5089 0 125.924 + 30.8257i − 153.883i 0 144.342 + 83.3357i
17.5 −7.73828 − 4.46770i 0 23.9206 + 41.4317i 89.4521 0 −1.06043 − 129.637i − 141.548i 0 −692.205 − 399.645i
17.6 −7.65926 − 4.42208i 0 23.1095 + 40.0269i 81.2828 0 126.096 + 30.1117i − 125.755i 0 −622.566 − 359.439i
17.7 −7.60400 − 4.39017i 0 22.5472 + 39.0530i 45.3714 0 −128.638 + 16.1027i − 114.974i 0 −345.004 − 199.188i
17.8 −5.92898 − 3.42310i 0 7.43522 + 12.8782i −56.6004 0 −33.5012 + 125.238i 117.272i 0 335.583 + 193.749i
17.9 −5.65517 − 3.26501i 0 5.32062 + 9.21559i −52.8903 0 −11.4249 − 129.137i 139.473i 0 299.104 + 172.688i
17.10 −5.44894 − 3.14594i 0 3.79394 + 6.57129i −38.0355 0 29.5669 − 126.225i 153.598i 0 207.253 + 119.658i
17.11 −4.79535 − 2.76860i 0 −0.669744 − 1.16003i −79.4804 0 −129.640 − 0.699468i 184.607i 0 381.136 + 220.049i
17.12 −4.57657 − 2.64228i 0 −2.03668 − 3.52763i 36.6917 0 83.7515 + 98.9580i 190.632i 0 −167.922 − 96.9499i
17.13 −4.41896 − 2.55129i 0 −2.98183 − 5.16469i 74.3634 0 −112.650 + 64.1645i 193.713i 0 −328.609 − 189.723i
17.14 −2.76639 − 1.59718i 0 −10.8981 − 18.8760i 65.1776 0 97.2771 − 85.6981i 171.844i 0 −180.307 − 104.100i
17.15 −2.49682 − 1.44154i 0 −11.8439 − 20.5143i 21.5957 0 −100.688 − 81.6636i 160.553i 0 −53.9207 − 31.1311i
17.16 −2.36280 − 1.36416i 0 −12.2781 − 21.2663i 37.2949 0 56.1077 + 116.871i 154.304i 0 −88.1204 − 50.8763i
17.17 −1.43050 − 0.825900i 0 −14.6358 − 25.3499i −59.7795 0 127.627 + 22.7650i 101.208i 0 85.5146 + 49.3719i
17.18 −0.242271 − 0.139875i 0 −15.9609 − 27.6450i −19.3659 0 69.3827 − 109.513i 17.8822i 0 4.69181 + 2.70882i
17.19 −0.0675663 − 0.0390094i 0 −15.9970 − 27.7075i −92.3091 0 −119.454 + 50.3751i 4.99273i 0 6.23698 + 3.60092i
17.20 −0.0316775 − 0.0182890i 0 −15.9993 − 27.7117i −62.5776 0 87.2452 + 95.8920i 2.34094i 0 1.98230 + 1.14448i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.6.s.a 76
3.b odd 2 1 63.6.s.a yes 76
7.d odd 6 1 189.6.i.a 76
9.c even 3 1 63.6.i.a ✓ 76
9.d odd 6 1 189.6.i.a 76
21.g even 6 1 63.6.i.a ✓ 76
63.k odd 6 1 63.6.s.a yes 76
63.s even 6 1 inner 189.6.s.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.i.a ✓ 76 9.c even 3 1
63.6.i.a ✓ 76 21.g even 6 1
63.6.s.a yes 76 3.b odd 2 1
63.6.s.a yes 76 63.k odd 6 1
189.6.i.a 76 7.d odd 6 1
189.6.i.a 76 9.d odd 6 1
189.6.s.a 76 1.a even 1 1 trivial
189.6.s.a 76 63.s even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(189, [\chi])\).