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Properties

Label 209.3.o.a
Level $209$
Weight $3$
Character orbit 209.o
Analytic conductor $5.695$
Analytic rank $0$
Dimension $204$
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] = [209,3,Mod(34,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.34");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 209.o (of order \(18\), 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: \(5.69483752513\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(34\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} + 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{4} + 36 q^{6} - 114 q^{10} - 144 q^{12} - 54 q^{13} + 90 q^{14} - 66 q^{15} - 12 q^{16} + 42 q^{17} + 36 q^{19} - 18 q^{21} + 192 q^{23} + 78 q^{24} + 36 q^{25} + 216 q^{27} + 60 q^{28} - 60 q^{29} - 294 q^{30} - 108 q^{31} - 228 q^{32} - 72 q^{34} - 174 q^{35} + 84 q^{36} - 360 q^{38} + 192 q^{39} - 456 q^{40} + 96 q^{41} + 372 q^{42} - 90 q^{43} - 198 q^{45} + 810 q^{46} + 342 q^{47} + 924 q^{48} - 798 q^{49} + 702 q^{50} - 180 q^{51} - 144 q^{52} - 144 q^{53} - 576 q^{54} - 210 q^{57} + 72 q^{58} + 54 q^{59} - 474 q^{60} - 768 q^{61} + 222 q^{62} - 186 q^{63} + 516 q^{64} + 810 q^{65} - 330 q^{66} + 306 q^{67} - 114 q^{68} - 864 q^{69} + 132 q^{70} + 48 q^{71} - 792 q^{72} + 546 q^{73} + 690 q^{74} + 384 q^{76} - 702 q^{78} + 138 q^{79} - 54 q^{80} - 360 q^{81} + 996 q^{82} + 120 q^{83} - 2160 q^{84} + 1344 q^{85} + 330 q^{86} - 384 q^{87} + 186 q^{89} + 690 q^{90} + 1116 q^{91} - 1362 q^{92} + 648 q^{93} - 90 q^{95} - 660 q^{96} - 384 q^{97} + 348 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −1.33704 + 3.67348i 4.97960 − 0.878038i −8.64261 − 7.25201i −0.680239 + 0.570788i −3.43246 + 19.4664i 6.11653 − 10.5941i 24.6536 − 14.2338i 15.5682 − 5.66637i −1.18727 − 3.26201i
34.2 −1.27131 + 3.49289i −3.53420 + 0.623175i −7.51990 − 6.30994i −3.39155 + 2.84585i 2.31638 − 13.1368i 1.59759 − 2.76711i 18.7238 − 10.8102i 3.64498 − 1.32666i −5.62854 − 15.4643i
34.3 −1.26598 + 3.47826i 1.26529 − 0.223106i −7.43142 − 6.23570i 4.99993 − 4.19544i −0.825823 + 4.68348i −6.55476 + 11.3532i 18.2751 − 10.5512i −6.90604 + 2.51359i 8.26301 + 22.7024i
34.4 −1.13663 + 3.12286i 1.39965 − 0.246795i −5.39616 − 4.52792i −2.60598 + 2.18667i −0.820170 + 4.65142i −1.83554 + 3.17924i 8.76130 − 5.05834i −6.55913 + 2.38733i −3.86665 − 10.6235i
34.5 −1.12473 + 3.09017i −3.79283 + 0.668778i −5.21993 − 4.38004i 4.37952 − 3.67486i 2.19927 − 12.4727i 1.00454 − 1.73992i 8.01442 − 4.62713i 5.48105 − 1.99494i 6.43014 + 17.6667i
34.6 −0.982540 + 2.69951i 3.09376 − 0.545513i −3.25777 − 2.73360i 4.88512 − 4.09910i −1.56713 + 8.88761i 0.634100 − 1.09829i 0.628737 − 0.363001i 0.816522 − 0.297190i 6.26573 + 17.2149i
34.7 −0.875944 + 2.40664i 5.33255 − 0.940272i −1.96044 − 1.64501i −4.73023 + 3.96913i −2.40812 + 13.6571i −4.94553 + 8.56591i −3.19570 + 1.84504i 19.0947 − 6.94991i −5.40884 − 14.8607i
34.8 −0.825960 + 2.26931i 2.00192 − 0.352992i −1.40336 − 1.17756i −5.18329 + 4.34930i −0.852457 + 4.83452i 1.84651 − 3.19825i −4.53425 + 2.61785i −4.57416 + 1.66486i −5.58869 − 15.3548i
34.9 −0.800651 + 2.19977i −4.24218 + 0.748012i −1.13377 − 0.951343i −4.17175 + 3.50051i 1.75105 − 9.93072i −4.75405 + 8.23425i −5.10879 + 2.94956i 8.97937 − 3.26822i −4.36021 − 11.9796i
34.10 −0.733077 + 2.01411i −2.75456 + 0.485702i −0.455071 − 0.381850i 3.38442 − 2.83986i 1.04104 − 5.90404i −2.16566 + 3.75104i −6.32218 + 3.65011i −1.10556 + 0.402392i 3.23877 + 8.89844i
34.11 −0.522090 + 1.43443i 4.47833 − 0.789651i 1.27917 + 1.07335i 4.59665 − 3.85705i −1.20539 + 6.83613i −0.762044 + 1.31990i −7.49539 + 4.32747i 10.9747 − 3.99446i 3.13280 + 8.60729i
34.12 −0.448968 + 1.23353i −2.79200 + 0.492305i 1.74416 + 1.46352i −6.84571 + 5.74423i 0.646246 − 3.66504i 3.59304 − 6.22333i −7.13567 + 4.11978i −0.904326 + 0.329148i −4.01217 − 11.0234i
34.13 −0.387904 + 1.06576i 0.952321 − 0.167920i 2.07881 + 1.74433i −0.892093 + 0.748555i −0.190447 + 1.08008i −2.35224 + 4.07421i −6.59424 + 3.80718i −7.57852 + 2.75835i −0.451731 − 1.24112i
34.14 −0.344484 + 0.946461i −5.75502 + 1.01476i 2.28706 + 1.91907i 2.95265 − 2.47756i 1.02207 − 5.79647i 2.55851 − 4.43146i −6.09323 + 3.51793i 23.6332 − 8.60179i 1.32778 + 3.64805i
34.15 −0.259709 + 0.713544i 0.0269140 − 0.00474567i 2.62248 + 2.20052i 1.48847 − 1.24898i −0.00360356 + 0.0204368i −5.14216 + 8.90648i −4.88167 + 2.81843i −8.45653 + 3.07793i 0.504631 + 1.38646i
34.16 −0.174617 + 0.479757i 4.67680 − 0.824646i 2.86450 + 2.40360i −1.92332 + 1.61386i −0.421020 + 2.38773i 4.72905 − 8.19096i −3.42192 + 1.97565i 12.7352 − 4.63522i −0.438414 − 1.20453i
34.17 0.0156971 − 0.0431274i −1.93725 + 0.341589i 3.06256 + 2.56980i 0.206438 − 0.173222i −0.0156773 + 0.0889103i 6.06722 − 10.5087i 0.317888 − 0.183532i −4.82100 + 1.75470i −0.00423013 − 0.0116222i
34.18 0.0758641 − 0.208435i 0.584440 − 0.103052i 3.02649 + 2.53952i 7.17947 − 6.02429i 0.0228582 − 0.129636i 3.00192 − 5.19948i 1.52731 − 0.881790i −8.12628 + 2.95773i −0.711008 − 1.95348i
34.19 0.0816678 − 0.224380i −3.87679 + 0.683582i 3.02050 + 2.53450i −0.992912 + 0.833152i −0.163226 + 0.925701i −0.340014 + 0.588922i 1.64253 − 0.948315i 6.10494 − 2.22202i 0.105854 + 0.290832i
34.20 0.206392 − 0.567058i −0.0157724 + 0.00278109i 2.78522 + 2.33708i −5.21918 + 4.37941i −0.00167825 + 0.00951784i −1.43057 + 2.47782i 3.99052 − 2.30393i −8.45699 + 3.07809i 1.40618 + 3.86345i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.3.o.a ✓ 204
19.f odd 18 1 inner 209.3.o.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.3.o.a ✓ 204 1.a even 1 1 trivial
209.3.o.a ✓ 204 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(209, [\chi])\).