Properties

Label 847.2.bc.a
Level $847$
Weight $2$
Character orbit 847.bc
Analytic conductor $6.763$
Analytic rank $0$
Dimension $6880$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,2,Mod(4,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(330))
 
chi = DirichletCharacter(H, H._module([220, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.bc (of order \(165\), degree \(80\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(6880\)
Relative dimension: \(86\) over \(\Q(\zeta_{165})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{165}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6880 q - 39 q^{2} - 30 q^{3} - 123 q^{4} - 43 q^{5} - 208 q^{6} - 81 q^{7} - 148 q^{8} + 784 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6880 q - 39 q^{2} - 30 q^{3} - 123 q^{4} - 43 q^{5} - 208 q^{6} - 81 q^{7} - 148 q^{8} + 784 q^{9} - 26 q^{10} - 28 q^{11} - 94 q^{12} - 172 q^{13} - 102 q^{14} - 174 q^{15} - 121 q^{16} - 33 q^{17} - 73 q^{18} - 37 q^{19} - 144 q^{20} - 103 q^{21} - 128 q^{22} - 68 q^{23} - 4 q^{24} - 145 q^{25} - 37 q^{26} - 258 q^{27} - 92 q^{28} - 200 q^{29} - 131 q^{30} - 35 q^{31} - 32 q^{32} - 84 q^{33} - 184 q^{34} - 69 q^{35} - 212 q^{36} - 18 q^{37} - 37 q^{38} - 44 q^{39} - 170 q^{40} - 238 q^{41} - 63 q^{42} - 132 q^{43} - 61 q^{44} - 105 q^{45} - 60 q^{46} - 61 q^{47} - 348 q^{48} + 27 q^{49} - 182 q^{50} - 197 q^{51} + 25 q^{52} + 47 q^{53} + 88 q^{54} - 180 q^{55} - 72 q^{56} - 262 q^{57} - 69 q^{58} - 43 q^{59} + 20 q^{60} - 62 q^{61} - 96 q^{62} - 275 q^{63} + 112 q^{64} - 34 q^{65} + 36 q^{66} - 52 q^{67} - 23 q^{68} - 124 q^{69} - 105 q^{70} - 196 q^{71} + 26 q^{72} - 70 q^{73} + 11 q^{74} - 91 q^{75} - 164 q^{76} + 144 q^{77} - 182 q^{78} - 30 q^{79} - 77 q^{80} + 772 q^{81} - 57 q^{82} - 148 q^{83} - 94 q^{84} + 142 q^{85} - 48 q^{86} - 77 q^{87} - 93 q^{88} + 81 q^{89} - 598 q^{90} - 157 q^{91} - 586 q^{92} - 97 q^{93} - 254 q^{94} - 154 q^{95} + 77 q^{96} - 198 q^{97} - 296 q^{98} - 510 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.23061 2.44389i −1.47523 1.63841i −3.26729 + 4.40817i 0.491572 0.781153i −2.18867 + 5.62155i −2.37186 + 1.17230i 9.40152 + 1.62700i −0.194496 + 1.85051i −2.51399 0.240057i
4.2 −1.21978 2.42238i 1.80004 + 1.99915i −3.18917 + 4.30277i 0.174931 0.277981i 2.64705 6.79890i −1.30506 2.30148i 8.96818 + 1.55200i −0.442859 + 4.21352i −0.886754 0.0846747i
4.3 −1.21689 2.41665i −1.18965 1.32124i −3.16845 + 4.27482i 1.15150 1.82984i −1.74530 + 4.48276i 2.34736 1.22061i 8.85416 + 1.53227i −0.0168215 + 0.160046i −5.82334 0.556061i
4.4 −1.21359 2.41009i 0.687247 + 0.763265i −3.14482 + 4.24293i −2.05054 + 3.25850i 1.00550 2.58261i 0.677642 + 2.55750i 8.72461 + 1.50985i 0.203320 1.93446i 10.3418 + 0.987520i
4.5 −1.15760 2.29891i −1.07578 1.19477i −2.75402 + 3.71568i −1.87138 + 2.97379i −1.50135 + 3.85618i 1.30507 2.30147i 6.65763 + 1.15215i 0.0434029 0.412951i 9.00279 + 0.859663i
4.6 −1.14692 2.27770i 1.16642 + 1.29544i −2.68158 + 3.61794i −0.208637 + 0.331544i 1.61284 4.14254i 2.29105 1.32329i 6.29053 + 1.08862i −0.00404607 + 0.0384958i 0.994450 + 0.0949585i
4.7 −1.09197 2.16858i 0.240894 + 0.267540i −2.31940 + 3.12929i −0.922980 + 1.46670i 0.317131 0.814543i −2.60006 0.489558i 4.53398 + 0.784636i 0.300038 2.85467i 4.18852 + 0.399955i
4.8 −1.08702 2.15874i 2.06298 + 2.29117i −2.28762 + 3.08642i 0.177774 0.282499i 2.70354 6.94398i 1.43027 + 2.22583i 4.38632 + 0.759082i −0.679993 + 6.46970i −0.803085 0.0766853i
4.9 −1.07561 2.13607i 0.177504 + 0.197138i −2.21495 + 2.98837i 1.97862 3.14421i 0.230177 0.591204i −1.28890 2.31057i 4.05264 + 0.701336i 0.306230 2.91358i −8.84447 0.844544i
4.10 −1.02809 2.04171i −0.754573 0.838039i −1.92068 + 2.59135i 0.372021 0.591175i −0.935261 + 2.40220i 0.0111053 + 2.64573i 2.76048 + 0.477720i 0.180658 1.71884i −1.58948 0.151777i
4.11 −1.01908 2.02380i 0.492008 + 0.546430i −1.86635 + 2.51805i 1.64901 2.62043i 0.604475 1.55258i 2.29599 + 1.31469i 2.53257 + 0.438278i 0.257071 2.44587i −6.98370 0.666863i
4.12 −1.00780 2.00142i −1.79161 1.98979i −1.79909 + 2.42731i −1.69658 + 2.69602i −2.17681 + 5.59109i −2.62261 0.349202i 2.25516 + 0.390271i −0.435794 + 4.14630i 7.10569 + 0.678511i
4.13 −0.934263 1.85538i −2.28606 2.53893i −1.37865 + 1.86005i 0.103820 0.164979i −2.57488 + 6.61352i 2.41879 + 1.07214i 0.645316 + 0.111676i −0.906493 + 8.62471i −0.403093 0.0384907i
4.14 −0.899031 1.78541i 0.525828 + 0.583992i −1.18850 + 1.60350i −1.01707 + 1.61622i 0.569926 1.46384i 1.93192 1.80767i −0.00801139 0.00138643i 0.249035 2.36941i 3.79999 + 0.362855i
4.15 −0.860197 1.70829i 1.38309 + 1.53608i −0.987379 + 1.33215i 0.622437 0.989109i 1.43433 3.68405i −1.91129 + 1.82947i −0.644208 0.111485i −0.133012 + 1.26553i −2.22510 0.212471i
4.16 −0.860059 1.70801i −1.04358 1.15901i −0.986678 + 1.33121i −1.34289 + 2.13397i −1.08207 + 2.77927i 1.52260 + 2.16372i −0.646320 0.111850i 0.0593327 0.564513i 4.79982 + 0.458327i
4.17 −0.848040 1.68414i −1.50427 1.67066i −0.926242 + 1.24967i 2.04012 3.24193i −1.53795 + 3.95020i −1.36153 + 2.26853i −0.825864 0.142921i −0.214697 + 2.04271i −7.18998 0.686560i
4.18 −0.844935 1.67798i −0.215561 0.239404i −0.910766 + 1.22879i 0.977338 1.55308i −0.219580 + 0.563987i −1.33462 2.28447i −0.870953 0.150724i 0.302737 2.88035i −3.43182 0.327699i
4.19 −0.800529 1.58979i 1.78562 + 1.98313i −0.695664 + 0.938578i −0.801920 + 1.27432i 1.72332 4.42631i −0.643792 2.56623i −1.45875 0.252447i −0.430786 + 4.09865i 2.66787 + 0.254751i
4.20 −0.792818 1.57448i −0.885258 0.983179i −0.659493 + 0.889776i −0.557106 + 0.885292i −0.846142 + 2.17330i 2.54101 + 0.737059i −1.55021 0.268275i 0.130627 1.24283i 1.83555 + 0.175274i
See next 80 embeddings (of 6880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
121.g even 55 1 inner
847.bc even 165 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.bc.a 6880
7.c even 3 1 inner 847.2.bc.a 6880
121.g even 55 1 inner 847.2.bc.a 6880
847.bc even 165 1 inner 847.2.bc.a 6880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.bc.a 6880 1.a even 1 1 trivial
847.2.bc.a 6880 7.c even 3 1 inner
847.2.bc.a 6880 121.g even 55 1 inner
847.2.bc.a 6880 847.bc even 165 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).