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Properties

Label 349.3.j.a
Level $349$
Weight $3$
Character orbit 349.j
Analytic conductor $9.510$
Analytic rank $0$
Dimension $3248$
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] = [349,3,Mod(6,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(116))
 
chi = DirichletCharacter(H, H._module([9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 349.j (of order \(116\), degree \(56\), 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: \(9.50956122617\)
Analytic rank: \(0\)
Dimension: \(3248\)
Relative dimension: \(58\) over \(\Q(\zeta_{116})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{116}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3248 q - 58 q^{2} - 58 q^{3} - 58 q^{4} - 58 q^{5} - 90 q^{6} - 40 q^{7} - 28 q^{8} + 294 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3248 q - 58 q^{2} - 58 q^{3} - 58 q^{4} - 58 q^{5} - 90 q^{6} - 40 q^{7} - 28 q^{8} + 294 q^{9} - 42 q^{10} - 66 q^{11} - 90 q^{12} - 80 q^{13} - 114 q^{14} - 90 q^{15} + 378 q^{16} - 58 q^{17} - 410 q^{18} - 6 q^{19} + 14 q^{20} + 820 q^{21} - 58 q^{22} - 258 q^{23} - 202 q^{24} + 350 q^{25} - 130 q^{26} - 58 q^{27} + 26 q^{28} - 58 q^{29} - 52 q^{30} - 126 q^{31} + 72 q^{32} - 204 q^{33} - 120 q^{34} - 578 q^{35} - 58 q^{36} - 58 q^{37} - 60 q^{38} - 122 q^{39} + 160 q^{40} + 610 q^{41} - 58 q^{42} + 122 q^{43} - 2410 q^{44} - 58 q^{45} - 180 q^{46} + 42 q^{47} - 58 q^{48} - 58 q^{49} - 114 q^{50} + 114 q^{51} - 3834 q^{52} + 130 q^{53} + 482 q^{54} + 2566 q^{55} - 58 q^{56} - 58 q^{57} + 82 q^{58} - 260 q^{59} - 58 q^{60} + 4230 q^{61} - 482 q^{62} - 420 q^{63} - 3712 q^{64} - 264 q^{65} + 210 q^{66} - 310 q^{67} + 346 q^{68} - 2610 q^{69} - 58 q^{70} - 70 q^{71} + 3976 q^{72} - 58 q^{73} + 10 q^{74} - 58 q^{75} - 58 q^{76} + 826 q^{77} - 58 q^{78} - 108 q^{79} - 58 q^{80} + 1674 q^{81} + 608 q^{82} - 58 q^{83} - 2564 q^{84} - 1298 q^{85} - 1682 q^{86} - 278 q^{87} - 118 q^{88} + 4766 q^{89} - 356 q^{90} - 58 q^{91} - 58 q^{92} - 58 q^{93} - 998 q^{94} - 58 q^{95} + 1530 q^{96} - 238 q^{97} - 762 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −3.79920 − 0.944819i −5.02131 − 0.272248i 10.0072 + 5.30549i −2.99343 − 6.47020i 18.8198 + 5.77856i −0.00813739 − 0.00584508i −21.3504 − 19.1543i 16.1922 + 1.76101i 5.25949 + 27.4098i
6.2 −3.67524 − 0.913992i 0.565453 + 0.0306580i 9.13799 + 4.84466i 0.478767 + 1.03484i −2.05016 − 0.629495i 5.19287 + 3.73003i −17.8804 − 16.0412i −8.62844 − 0.938400i −0.813752 − 4.24087i
6.3 −3.63797 − 0.904722i 5.77160 + 0.312927i 8.88224 + 4.70907i −3.79154 − 8.19528i −20.7138 − 6.36012i 5.88619 + 4.22804i −16.8913 − 15.1538i 24.2663 + 2.63911i 6.37905 + 33.2445i
6.4 −3.59855 − 0.894919i −2.93878 − 0.159336i 8.61462 + 4.56718i 3.51828 + 7.60465i 10.4327 + 3.20335i −5.53437 − 3.97533i −15.8722 − 14.2395i −0.336213 − 0.0365653i −5.85517 − 30.5143i
6.5 −3.42711 − 0.852285i 3.61216 + 0.195846i 7.48468 + 3.96812i 1.82540 + 3.94553i −12.2124 − 3.74978i −4.47249 − 3.21258i −11.7542 − 10.5451i 4.06212 + 0.441783i −2.89313 − 15.0775i
6.6 −3.23713 − 0.805039i 0.503631 + 0.0273061i 6.29689 + 3.33840i −1.03662 − 2.24062i −1.60834 − 0.493836i 5.17675 + 3.71845i −7.76448 − 6.96583i −8.69434 − 0.945567i 1.55190 + 8.08771i
6.7 −3.15517 − 0.784655i 4.16711 + 0.225934i 5.80535 + 3.07780i 3.24834 + 7.02117i −12.9706 − 3.98260i 4.87003 + 3.49814i −6.22149 − 5.58154i 8.36649 + 0.909911i −4.73986 − 24.7018i
6.8 −3.15111 − 0.783647i −2.78715 − 0.151115i 5.78137 + 3.06509i −0.244499 − 0.528475i 8.66420 + 2.66032i −2.36731 − 1.70044i −6.14788 − 5.51551i −1.20188 − 0.130712i 0.356305 + 1.85689i
6.9 −3.12060 − 0.776058i 0.384614 + 0.0208532i 5.60181 + 2.96989i −3.26511 − 7.05741i −1.18404 − 0.363557i −8.35223 − 5.99940i −5.60190 − 5.02569i −8.79975 − 0.957030i 4.71212 + 24.5572i
6.10 −2.97880 − 0.740793i −5.14849 − 0.279143i 4.79040 + 2.53971i 1.83466 + 3.96554i 15.1295 + 4.64548i 10.0277 + 7.20291i −3.24899 − 2.91480i 17.4818 + 1.90126i −2.52742 − 13.1716i
6.11 −2.84452 − 0.707401i 4.52957 + 0.245586i 4.05685 + 2.15081i −0.343766 − 0.743039i −12.7108 − 3.90280i −6.96552 − 5.00333i −1.29105 − 1.15825i 11.5095 + 1.25173i 0.452225 + 2.35677i
6.12 −2.31393 − 0.575450i −0.131926 − 0.00715284i 1.48910 + 0.789473i 2.76732 + 5.98146i 0.301153 + 0.0924682i −9.07508 − 6.51862i 4.10799 + 3.68544i −8.92989 − 0.971184i −2.96136 − 15.4332i
6.13 −2.28413 − 0.568038i −3.01785 − 0.163623i 1.36054 + 0.721312i 1.56524 + 3.38322i 6.80022 + 2.08799i 3.01152 + 2.16318i 4.31002 + 3.86669i 0.133417 + 0.0145099i −1.65342 − 8.61683i
6.14 −2.19519 − 0.545920i 1.84025 + 0.0997752i 0.986796 + 0.523166i −2.09301 − 4.52396i −3.98522 − 1.22365i 6.28981 + 4.51797i 4.85446 + 4.35513i −5.57069 − 0.605849i 2.12483 + 11.0736i
6.15 −2.18439 − 0.543234i −3.97731 − 0.215643i 0.942422 + 0.499641i −0.995320 − 2.15135i 8.57086 + 2.63166i −5.70743 − 4.09964i 4.91473 + 4.40920i 6.82524 + 0.742290i 1.00548 + 5.24008i
6.16 −2.07461 − 0.515932i 0.899012 + 0.0487430i 0.503773 + 0.267083i 3.54259 + 7.65718i −1.83995 − 0.564952i 8.61373 + 6.18724i 5.45777 + 4.89638i −8.14139 − 0.885430i −3.39890 − 17.7134i
6.17 −1.97853 − 0.492039i −3.65262 − 0.198039i 0.138443 + 0.0733979i −4.08791 − 8.83588i 7.12939 + 2.18906i 6.29688 + 4.52304i 5.83253 + 5.23260i 4.35518 + 0.473654i 3.74047 + 19.4935i
6.18 −1.97109 − 0.490188i 3.28489 + 0.178101i 0.110866 + 0.0587772i −2.18998 − 4.73356i −6.38751 − 1.96127i 0.633095 + 0.454752i 5.85778 + 5.25525i 1.81152 + 0.197014i 1.99631 + 10.4038i
6.19 −1.92468 − 0.478646i 4.47103 + 0.242412i −0.0587572 − 0.0311511i −0.134781 − 0.291325i −8.48928 − 2.60661i −3.70540 − 2.66159i 6.00328 + 5.38579i 10.9841 + 1.19460i 0.119969 + 0.625220i
6.20 −1.91239 − 0.475591i −5.75888 − 0.312237i −0.102983 − 0.0545983i −1.13384 − 2.45075i 10.8648 + 3.33599i −7.41329 − 5.32496i 6.03839 + 5.41728i 24.1200 + 2.62321i 1.00279 + 5.22604i
See next 80 embeddings (of 3248 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.j odd 116 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.3.j.a ✓ 3248
349.j odd 116 1 inner 349.3.j.a ✓ 3248
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.3.j.a ✓ 3248 1.a even 1 1 trivial
349.3.j.a ✓ 3248 349.j odd 116 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(349, [\chi])\).