↑

Properties

Label 334.2.c.a
Level $334$
Weight $2$
Character orbit 334.c
Analytic conductor $2.667$
Analytic rank $0$
Dimension $574$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [334,2,Mod(3,334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(334, base_ring=CyclotomicField(166))
 
chi = DirichletCharacter(H, H._module([94]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("334.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 334.c (of order \(83\), degree \(82\), 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: \(2.66700342751\)
Analytic rank: \(0\)
Dimension: \(574\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 574 q - 7 q^{2} - 2 q^{3} - 7 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} - 7 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 574 q - 7 q^{2} - 2 q^{3} - 7 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} - 7 q^{8} - 17 q^{9} - 2 q^{10} - 10 q^{11} - 2 q^{12} - 16 q^{13} - 6 q^{14} - 24 q^{15} - 7 q^{16} - 20 q^{17} - 17 q^{18} - 16 q^{19} - 2 q^{20} - 42 q^{21} - 10 q^{22} - 30 q^{23} - 2 q^{24} + 135 q^{25} - 16 q^{26} - 32 q^{27} - 6 q^{28} - 14 q^{29} - 24 q^{30} - 30 q^{31} - 7 q^{32} - 44 q^{33} - 20 q^{34} - 60 q^{35} - 17 q^{36} - 28 q^{37} + 67 q^{38} - 56 q^{39} - 2 q^{40} - 48 q^{41} - 42 q^{42} - 36 q^{43} - 10 q^{44} - 76 q^{45} - 30 q^{46} - 54 q^{47} - 2 q^{48} + 129 q^{49} - 31 q^{50} + 349 q^{51} - 16 q^{52} - 62 q^{53} - 32 q^{54} - 78 q^{55} - 6 q^{56} - 70 q^{57} - 14 q^{58} - 76 q^{59} - 24 q^{60} - 68 q^{61} - 30 q^{62} - 104 q^{63} - 7 q^{64} - 52 q^{65} - 44 q^{66} - 74 q^{67} - 20 q^{68} - 96 q^{69} - 60 q^{70} - 64 q^{71} - 17 q^{72} - 58 q^{73} - 28 q^{74} + 453 q^{75} - 16 q^{76} + 119 q^{77} - 56 q^{78} - 70 q^{79} - 2 q^{80} - 119 q^{81} - 48 q^{82} - 80 q^{83} - 42 q^{84} - 122 q^{85} - 36 q^{86} - 158 q^{87} + 156 q^{88} - 96 q^{89} + 339 q^{90} - 106 q^{91} - 30 q^{92} - 98 q^{93} - 54 q^{94} - 114 q^{95} - 2 q^{96} - 80 q^{97} - 37 q^{98} - 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.455676 + 0.890146i −1.93603 − 1.69516i −0.584719 − 0.811236i 0.451628 − 2.13800i 2.39114 − 0.950910i −0.391194 + 0.249507i 0.988561 − 0.150824i 0.478400 + 3.59004i 1.69734 + 1.37625i
3.2 −0.455676 + 0.890146i −0.847795 − 0.742314i −0.584719 − 0.811236i −0.252460 + 1.19515i 1.04709 − 0.416406i 0.151521 − 0.0966415i 0.988561 − 0.150824i −0.228543 − 1.71505i −0.948813 − 0.769326i
3.3 −0.455676 + 0.890146i −0.738577 − 0.646685i −0.584719 − 0.811236i −0.829988 + 3.92916i 0.912196 − 0.362762i −1.00704 + 0.642297i 0.988561 − 0.150824i −0.268974 − 2.01846i −3.11932 − 2.52923i
3.4 −0.455676 + 0.890146i 0.0878342 + 0.0769061i −0.584719 − 0.811236i 0.197051 − 0.932838i −0.108482 + 0.0431410i 2.33896 − 1.49181i 0.988561 − 0.150824i −0.394469 − 2.96020i 0.740570 + 0.600476i
3.5 −0.455676 + 0.890146i 0.859513 + 0.752574i −0.584719 − 0.811236i 0.904404 − 4.28144i −1.06156 + 0.422162i −3.55677 + 2.26854i 0.988561 − 0.150824i −0.223875 − 1.68002i 3.39899 + 2.75600i
3.6 −0.455676 + 0.890146i 1.42908 + 1.25128i −0.584719 − 0.811236i −0.201177 + 0.952371i −1.76502 + 0.701912i −2.69977 + 1.72194i 0.988561 − 0.150824i 0.0803052 + 0.602632i −0.756078 − 0.613050i
3.7 −0.455676 + 0.890146i 1.62178 + 1.42000i −0.584719 − 0.811236i 0.143898 − 0.681211i −2.00302 + 0.796561i 3.33426 − 2.12662i 0.988561 − 0.150824i 0.217498 + 1.63216i 0.540806 + 0.438501i
7.1 0.206677 + 0.978409i −2.75072 + 1.75443i −0.914569 + 0.404430i −1.54527 + 1.98128i −2.28507 − 2.32873i −2.58605 + 1.02842i −0.584719 − 0.811236i 3.22341 − 6.93165i −2.25788 − 1.10242i
7.2 0.206677 + 0.978409i −1.58357 + 1.01002i −0.914569 + 0.404430i 1.35353 − 1.73545i −1.31550 − 1.34063i −2.54097 + 1.01049i −0.584719 − 0.811236i 0.222570 − 0.478617i 1.97773 + 0.965632i
7.3 0.206677 + 0.978409i −1.30492 + 0.832292i −0.914569 + 0.404430i 1.73730 − 2.22750i −1.08402 − 1.10473i 1.66409 − 0.661777i −0.584719 − 0.811236i −0.254883 + 0.548102i 2.53847 + 1.23942i
7.4 0.206677 + 0.978409i −0.0190366 + 0.0121417i −0.914569 + 0.404430i −1.72954 + 2.21755i −0.0158140 − 0.0161162i 3.95293 − 1.57200i −0.584719 − 0.811236i −1.26478 + 2.71979i −2.52713 − 1.23388i
7.5 0.206677 + 0.978409i 0.651555 − 0.415568i −0.914569 + 0.404430i −1.30866 + 1.67792i 0.541257 + 0.551599i −2.78532 + 1.10767i −0.584719 − 0.811236i −1.01317 + 2.17872i −1.91216 − 0.933619i
7.6 0.206677 + 0.978409i 1.09270 − 0.696934i −0.914569 + 0.404430i 0.847829 − 1.08706i 0.907723 + 0.925067i 1.54263 − 0.613475i −0.584719 − 0.811236i −0.556721 + 1.19718i 1.23881 + 0.604854i
7.7 0.206677 + 0.978409i 2.77939 − 1.77272i −0.914569 + 0.404430i 1.87480 − 2.40380i 2.30888 + 2.35300i −3.43178 + 1.36475i −0.584719 − 0.811236i 3.31746 − 7.13388i 2.73938 + 1.33751i
9.1 −0.584719 − 0.811236i −0.434654 − 3.26177i −0.316208 + 0.948690i −0.230379 − 0.101875i −2.39191 + 2.25982i −1.60897 + 3.45994i 0.954504 − 0.298198i −7.55488 + 2.04989i 0.0520618 + 0.246460i
9.2 −0.584719 − 0.811236i −0.275864 − 2.07016i −0.316208 + 0.948690i −3.13794 − 1.38762i −1.51809 + 1.43426i 1.28338 − 2.75979i 0.954504 − 0.298198i −1.31417 + 0.356576i 0.709121 + 3.35697i
9.3 −0.584719 − 0.811236i −0.246093 − 1.84675i −0.316208 + 0.948690i 1.76693 + 0.781353i −1.35426 + 1.27947i 0.958875 − 2.06197i 0.954504 − 0.298198i −0.454622 + 0.123354i −0.399298 − 1.89027i
9.4 −0.584719 − 0.811236i −0.133520 − 1.00197i −0.316208 + 0.948690i 1.06886 + 0.472659i −0.734766 + 0.694190i −0.931203 + 2.00246i 0.954504 − 0.298198i 1.90919 − 0.518026i −0.241544 − 1.14347i
9.5 −0.584719 − 0.811236i 0.179729 + 1.34874i −0.316208 + 0.948690i −1.05953 − 0.468534i 0.989055 − 0.934437i 1.80326 − 3.87774i 0.954504 − 0.298198i 1.10852 − 0.300777i 0.239437 + 1.13349i
9.6 −0.584719 − 0.811236i 0.260524 + 1.95504i −0.316208 + 0.948690i 2.20170 + 0.973611i 1.43367 − 1.35450i −0.657934 + 1.41483i 0.954504 − 0.298198i −0.859005 + 0.233076i −0.497548 − 2.35539i
See next 80 embeddings (of 574 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.7
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 334.2.c.a ✓ 574
167.c even 83 1 inner 334.2.c.a ✓ 574
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
334.2.c.a ✓ 574 1.a even 1 1 trivial
334.2.c.a ✓ 574 167.c even 83 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{574} + 2 T_{3}^{573} + 21 T_{3}^{572} + 54 T_{3}^{571} + 348 T_{3}^{570} + 1066 T_{3}^{569} + \cdots + 68\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(334, [\chi])\). Copy content Toggle raw display