Properties

Label 348.3.p.a
Level $348$
Weight $3$
Character orbit 348.p
Analytic conductor $9.482$
Analytic rank $0$
Dimension $360$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,3,Mod(7,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 348.p (of order \(14\), degree \(6\), 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: \(9.48231319974\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(60\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q + 4 q^{2} + 12 q^{4} + 8 q^{5} - 20 q^{8} + 180 q^{9} - 24 q^{13} - 28 q^{14} - 4 q^{16} - 40 q^{17} - 12 q^{18} - 64 q^{22} + 18 q^{24} - 140 q^{25} + 20 q^{26} + 252 q^{28} + 52 q^{29} - 48 q^{30} + 294 q^{32} + 48 q^{33} + 38 q^{34} - 36 q^{36} - 184 q^{37} - 112 q^{38} + 196 q^{40} - 200 q^{41} + 54 q^{42} - 38 q^{44} + 60 q^{45} + 376 q^{46} + 408 q^{48} + 340 q^{49} + 666 q^{50} - 4 q^{52} + 492 q^{53} - 380 q^{56} - 136 q^{58} - 180 q^{60} - 56 q^{61} + 280 q^{62} - 474 q^{64} - 804 q^{65} - 180 q^{66} - 834 q^{68} - 972 q^{70} - 150 q^{72} - 668 q^{73} - 446 q^{74} + 238 q^{76} - 288 q^{77} + 66 q^{78} - 148 q^{80} - 540 q^{81} + 790 q^{82} + 24 q^{84} + 16 q^{85} - 736 q^{86} + 224 q^{88} - 552 q^{89} - 678 q^{92} + 1176 q^{94} + 450 q^{96} + 916 q^{97} - 710 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.99911 + 0.0596572i −0.751509 + 1.56052i 3.99288 0.238523i −0.103650 0.454120i 1.40925 3.16449i 0.138351 0.287290i −7.96798 + 0.715037i −1.87047 2.34549i 0.234299 + 0.901653i
7.2 −1.99866 + 0.0730893i 0.751509 1.56052i 3.98932 0.292162i −0.987961 4.32854i −1.38796 + 3.17389i −2.47930 + 5.14832i −7.95195 + 0.875510i −1.87047 2.34549i 2.29097 + 8.57909i
7.3 −1.98520 + 0.242860i 0.751509 1.56052i 3.88204 0.964252i 0.211154 + 0.925126i −1.11291 + 3.28046i 3.86437 8.02445i −7.47244 + 2.85703i −1.87047 2.34549i −0.643859 1.78528i
7.4 −1.90359 0.613475i 0.751509 1.56052i 3.24730 + 2.33561i −0.704065 3.08471i −2.38791 + 2.50956i 4.78219 9.93031i −4.74868 6.43817i −1.87047 2.34549i −0.552142 + 6.30394i
7.5 −1.89547 + 0.638129i −0.751509 + 1.56052i 3.18558 2.41911i −1.33767 5.86070i 0.428643 3.43748i −0.670921 + 1.39318i −4.49446 + 6.61814i −1.87047 2.34549i 6.27538 + 10.2551i
7.6 −1.88944 + 0.655768i 0.751509 1.56052i 3.13994 2.47806i 0.548610 + 2.40362i −0.396586 + 3.44133i −5.75269 + 11.9456i −4.30768 + 6.74121i −1.87047 2.34549i −2.61278 4.18172i
7.7 −1.87351 + 0.699984i −0.751509 + 1.56052i 3.02005 2.62285i 1.93067 + 8.45881i 0.315614 3.44969i −4.21585 + 8.75430i −3.82212 + 7.02790i −1.87047 2.34549i −9.53814 14.4962i
7.8 −1.87071 0.707423i 0.751509 1.56052i 2.99911 + 2.64677i 0.521319 + 2.28405i −2.50980 + 2.38765i −2.29450 + 4.76459i −3.73807 7.07296i −1.87047 2.34549i 0.640552 4.64158i
7.9 −1.81370 0.842913i −0.751509 + 1.56052i 2.57900 + 3.05758i −0.188089 0.824073i 2.67839 2.19686i −4.83870 + 10.0477i −2.10025 7.71939i −1.87047 2.34549i −0.353485 + 1.65316i
7.10 −1.80344 0.864632i −0.751509 + 1.56052i 2.50482 + 3.11863i −1.76951 7.75272i 2.70458 2.16454i 3.35862 6.97425i −1.82084 7.79003i −1.87047 2.34549i −3.51204 + 15.5116i
7.11 −1.73118 + 1.00150i −0.751509 + 1.56052i 1.99400 3.46756i 1.06534 + 4.66756i −0.261862 3.45419i 4.00714 8.32092i 0.0207712 + 7.99997i −1.87047 2.34549i −6.51886 7.01347i
7.12 −1.72275 1.01594i 0.751509 1.56052i 1.93573 + 3.50042i 2.09993 + 9.20041i −2.88006 + 1.92490i −0.318726 + 0.661841i 0.221455 7.99693i −1.87047 2.34549i 5.72942 17.9834i
7.13 −1.69008 + 1.06941i 0.751509 1.56052i 1.71274 3.61476i −1.98094 8.67907i 0.398723 + 3.44108i 1.05784 2.19662i 0.970976 + 7.94086i −1.87047 2.34549i 12.6294 + 12.5499i
7.14 −1.67048 1.09977i −0.751509 + 1.56052i 1.58101 + 3.67429i 1.77906 + 7.79457i 2.97160 1.78034i 4.70222 9.76426i 1.39982 7.87658i −1.87047 2.34549i 5.60034 14.9772i
7.15 −1.58178 + 1.22391i 0.751509 1.56052i 1.00407 3.87193i 1.38979 + 6.08909i 0.721225 + 3.38819i 2.55132 5.29787i 3.15070 + 7.35344i −1.87047 2.34549i −9.65087 7.93062i
7.16 −1.42923 + 1.39904i −0.751509 + 1.56052i 0.0853886 3.99909i −1.39231 6.10010i −1.10915 3.28173i −3.13087 + 6.50132i 5.47283 + 5.83507i −1.87047 2.34549i 10.5242 + 6.77055i
7.17 −1.37609 1.45134i 0.751509 1.56052i −0.212750 + 3.99434i −1.52239 6.67004i −3.29899 + 1.05673i −3.08524 + 6.40657i 6.08989 5.18780i −1.87047 2.34549i −7.58552 + 11.3881i
7.18 −1.34742 1.47799i −0.751509 + 1.56052i −0.368934 + 3.98295i 1.46187 + 6.40485i 3.31904 0.991951i −3.03117 + 6.29428i 6.38388 4.82141i −1.87047 2.34549i 7.49659 10.7906i
7.19 −1.10839 1.66477i −0.751509 + 1.56052i −1.54295 + 3.69044i −0.213954 0.937392i 3.43088 0.478576i 2.52620 5.24571i 7.85393 1.52179i −1.87047 2.34549i −1.32340 + 1.39518i
7.20 −1.09610 + 1.67289i −0.751509 + 1.56052i −1.59712 3.66732i −0.224107 0.981875i −1.78685 2.96769i −2.00246 + 4.15814i 7.88562 + 1.34796i −1.87047 2.34549i 1.88821 + 0.701331i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
29.d even 7 1 inner
116.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.3.p.a 360
4.b odd 2 1 inner 348.3.p.a 360
29.d even 7 1 inner 348.3.p.a 360
116.j odd 14 1 inner 348.3.p.a 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.3.p.a 360 1.a even 1 1 trivial
348.3.p.a 360 4.b odd 2 1 inner
348.3.p.a 360 29.d even 7 1 inner
348.3.p.a 360 116.j odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).