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Properties

Label 207.3.n.a
Level $207$
Weight $3$
Character orbit 207.n
Analytic conductor $5.640$
Analytic rank $0$
Dimension $920$
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] = [207,3,Mod(2,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([11, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.n (of order \(66\), degree \(20\), 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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(920\)
Relative dimension: \(46\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 27 q^{2} - 20 q^{3} - 97 q^{4} - 45 q^{5} - 29 q^{6} - 9 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 920 q - 27 q^{2} - 20 q^{3} - 97 q^{4} - 45 q^{5} - 29 q^{6} - 9 q^{7} - 24 q^{9} - 20 q^{10} - 27 q^{11} - 42 q^{12} - 9 q^{13} + 45 q^{14} - 190 q^{15} + 143 q^{16} + 129 q^{18} - 36 q^{19} - 147 q^{20} + 111 q^{21} - 4 q^{22} + 15 q^{23} + 92 q^{24} - 205 q^{25} - 293 q^{27} - 84 q^{28} - 189 q^{29} - 252 q^{30} + 21 q^{31} - 15 q^{32} + 376 q^{33} - 5 q^{34} - q^{36} + 48 q^{37} + 177 q^{38} - 94 q^{39} + 259 q^{40} - 243 q^{41} - 92 q^{42} - 9 q^{43} + 304 q^{45} - 140 q^{46} + 12 q^{47} - 236 q^{48} + 445 q^{49} - 399 q^{50} - 82 q^{51} - 90 q^{52} - 300 q^{54} - 68 q^{55} - 2241 q^{56} - 225 q^{57} + 52 q^{58} - 243 q^{59} - 170 q^{60} + 87 q^{61} - 95 q^{63} + 818 q^{64} - 1026 q^{65} - 31 q^{66} - 87 q^{67} - 390 q^{68} + 33 q^{69} + 50 q^{70} + 20 q^{72} - 36 q^{73} + 1977 q^{74} + 997 q^{75} + 195 q^{76} + 603 q^{77} + 1751 q^{78} - 9 q^{79} - 1980 q^{81} - 218 q^{82} + 828 q^{83} + 2178 q^{84} - 59 q^{85} - 27 q^{86} + 604 q^{87} + 143 q^{88} - 392 q^{90} - 276 q^{91} - 81 q^{92} - 480 q^{93} - 30 q^{94} + 843 q^{95} - 1518 q^{96} - 468 q^{97} + 2048 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.77759 − 3.44804i 2.47579 + 1.69424i −6.40893 + 9.00008i −0.649626 + 0.157598i 1.44088 − 11.5483i −3.01864 − 8.72178i 27.0659 + 3.89149i 3.25908 + 8.38918i 1.69817 + 1.95979i
2.2 −1.76729 − 3.42807i −1.57094 − 2.55581i −6.30810 + 8.85849i 0.556081 − 0.134904i −5.98517 + 9.90214i 1.07630 + 3.10975i 26.2455 + 3.77354i −4.06429 + 8.03004i −1.44522 − 1.66787i
2.3 −1.66292 − 3.22561i −2.39070 + 1.81233i −5.31904 + 7.46954i −9.34257 + 2.26648i 9.82142 + 4.69770i 1.13235 + 3.27172i 18.5706 + 2.67005i 2.43089 − 8.66549i 22.8467 + 26.3665i
2.4 −1.50053 − 2.91061i 2.54144 − 1.59408i −3.89986 + 5.47659i 7.76731 − 1.88433i −8.45324 − 5.00519i 2.35348 + 6.79992i 8.82686 + 1.26911i 3.91783 − 8.10250i −17.1396 − 19.7801i
2.5 −1.49199 − 2.89406i −2.87464 + 0.858180i −3.82932 + 5.37753i 6.60550 − 1.60248i 6.77256 + 7.03897i −3.39793 − 9.81767i 8.38474 + 1.20554i 7.52705 − 4.93391i −14.4930 − 16.7258i
2.6 −1.48579 − 2.88203i 2.16781 − 2.07379i −3.77831 + 5.30590i −4.13538 + 1.00323i −9.19764 − 3.16648i −1.32661 − 3.83300i 8.06766 + 1.15995i 0.398802 − 8.99116i 9.03567 + 10.4277i
2.7 −1.42761 − 2.76918i 1.70156 + 2.47077i −3.31006 + 4.64833i 2.55375 − 0.619533i 4.41283 − 8.23923i 3.03327 + 8.76405i 5.26234 + 0.756610i −3.20937 + 8.40833i −5.36136 − 6.18734i
2.8 −1.35044 − 2.61949i −0.490983 + 2.95955i −2.71780 + 3.81661i −0.602504 + 0.146166i 8.41554 − 2.71057i −1.83383 − 5.29850i 1.99936 + 0.287465i −8.51787 − 2.90618i 1.19653 + 1.38086i
2.9 −1.25358 − 2.43161i −2.99890 − 0.0810497i −2.02103 + 2.83813i 1.33216 − 0.323177i 3.56229 + 7.39377i 2.58596 + 7.47163i −1.39676 − 0.200824i 8.98686 + 0.486121i −2.45581 − 2.83415i
2.10 −1.19656 − 2.32100i 2.84217 + 0.960237i −1.63506 + 2.29612i −7.18146 + 1.74220i −1.17211 − 7.74566i 3.27346 + 9.45804i −3.05308 − 0.438966i 7.15589 + 5.45832i 12.6367 + 14.5835i
2.11 −1.16426 − 2.25835i −0.921940 − 2.85483i −1.42440 + 2.00029i 4.06347 − 0.985787i −5.37381 + 5.40581i −0.513623 − 1.48402i −3.88401 − 0.558436i −7.30005 + 5.26395i −6.95717 − 8.02900i
2.12 −1.15234 − 2.23523i −2.11792 − 2.12471i −1.34812 + 1.89317i −7.73246 + 1.87587i −2.30864 + 7.18243i −4.54495 − 13.1318i −4.17158 − 0.599783i −0.0287896 + 8.99995i 13.1034 + 15.1221i
2.13 −0.904747 − 1.75496i −0.386323 + 2.97502i 0.0588971 − 0.0827094i −1.75933 + 0.426808i 5.57058 − 2.01366i 0.254144 + 0.734302i −8.01586 − 1.15251i −8.70151 − 2.29864i 2.34078 + 2.70140i
2.14 −0.841542 − 1.63236i 2.10810 + 2.13446i 0.363808 − 0.510897i 9.00019 − 2.18342i 1.71016 − 5.23742i −2.74936 − 7.94375i −8.41144 − 1.20938i −0.111829 + 8.99931i −11.1382 − 12.8541i
2.15 −0.784229 − 1.52119i 0.00280928 − 3.00000i 0.621216 − 0.872376i −6.14232 + 1.49011i −4.56578 + 2.34841i 4.03111 + 11.6471i −8.59033 − 1.23510i −8.99998 − 0.0168557i 7.08374 + 8.17507i
2.16 −0.756582 − 1.46756i 2.96326 − 0.468093i 0.738899 − 1.03764i 1.23916 − 0.300618i −2.92890 − 3.99462i −1.85257 − 5.35265i −8.61905 − 1.23923i 8.56178 − 2.77416i −1.37870 − 1.59111i
2.17 −0.651548 − 1.26383i 0.834379 − 2.88163i 1.14749 − 1.61142i 3.40329 − 0.825630i −4.18552 + 0.823011i −1.38999 − 4.01612i −8.41386 − 1.20973i −7.60762 − 4.80875i −3.26086 − 3.76324i
2.18 −0.500955 − 0.971717i −2.78775 − 1.10835i 1.62695 − 2.28473i −4.07537 + 0.988675i 0.319532 + 3.26414i 1.14460 + 3.30712i −7.36362 − 1.05873i 6.54311 + 6.17962i 3.00229 + 3.46483i
2.19 −0.480226 − 0.931508i −2.55882 + 1.56602i 1.68314 − 2.36364i −1.54067 + 0.373762i 2.68757 + 1.63152i 0.119722 + 0.345913i −7.15940 − 1.02937i 4.09516 − 8.01434i 1.08803 + 1.25565i
2.20 −0.415073 − 0.805130i −1.74884 + 2.43753i 1.84428 − 2.58993i 9.06925 − 2.20018i 2.68843 + 0.396293i 2.87841 + 8.31663i −6.43716 − 0.925524i −2.88309 − 8.52571i −5.53583 − 6.38869i
See next 80 embeddings (of 920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
23.c even 11 1 inner
207.n odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.n.a ✓ 920
9.d odd 6 1 inner 207.3.n.a ✓ 920
23.c even 11 1 inner 207.3.n.a ✓ 920
207.n odd 66 1 inner 207.3.n.a ✓ 920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.3.n.a ✓ 920 1.a even 1 1 trivial
207.3.n.a ✓ 920 9.d odd 6 1 inner
207.3.n.a ✓ 920 23.c even 11 1 inner
207.3.n.a ✓ 920 207.n odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(207, [\chi])\).