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Properties

Label 209.3.s.a
Level $209$
Weight $3$
Character orbit 209.s
Analytic conductor $5.695$
Analytic rank $0$
Dimension $304$
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] = [209,3,Mod(7,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 209.s (of order \(30\), degree \(8\), 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: \(304\)
Relative dimension: \(38\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 5 q^{2} + q^{3} - 75 q^{4} - 5 q^{5} + 15 q^{6} + 10 q^{7} - 20 q^{8} + 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 5 q^{2} + q^{3} - 75 q^{4} - 5 q^{5} + 15 q^{6} + 10 q^{7} - 20 q^{8} + 87 q^{9} + 14 q^{11} - 4 q^{12} - 5 q^{13} - 51 q^{14} - 51 q^{15} + 117 q^{16} + 70 q^{17} + 60 q^{18} - 40 q^{19} - 72 q^{20} - 46 q^{22} + 56 q^{23} + 95 q^{24} + 159 q^{25} + 202 q^{26} - 32 q^{27} - 5 q^{28} - 40 q^{29} - 230 q^{30} + 36 q^{31} + 210 q^{33} + 102 q^{34} - 165 q^{35} - 532 q^{36} - 204 q^{37} + 213 q^{38} + 190 q^{39} - 60 q^{40} + 55 q^{41} - 64 q^{42} - 404 q^{44} - 364 q^{45} + 20 q^{46} - 47 q^{47} - 384 q^{48} + 846 q^{49} + 60 q^{50} + 65 q^{51} - 390 q^{52} - 18 q^{53} + 373 q^{55} - 1192 q^{56} + 900 q^{57} - 128 q^{58} + 295 q^{59} - 420 q^{60} - 160 q^{61} - 25 q^{62} + 130 q^{63} + 332 q^{64} + 294 q^{66} - 140 q^{67} - 1020 q^{68} + 612 q^{69} - 148 q^{70} - 51 q^{71} - 185 q^{72} + 155 q^{73} - 655 q^{74} + 80 q^{75} - 192 q^{77} - 218 q^{78} + 85 q^{79} + 709 q^{80} + 187 q^{81} + 115 q^{82} - 690 q^{83} - 1160 q^{84} - 390 q^{85} - 613 q^{86} - 338 q^{88} - 62 q^{89} - 1395 q^{90} + 39 q^{91} + 598 q^{92} + 631 q^{93} + 3240 q^{94} - 1210 q^{95} - 170 q^{96} - 271 q^{97} - 15 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.77625 − 0.396900i −3.88049 + 0.824823i 10.1899 + 2.16594i 2.67194 + 1.18963i 14.9811 − 1.57457i 4.78850 − 1.55588i −23.1753 − 7.53010i 6.15595 − 2.74081i −9.61777 − 5.55282i
7.2 −3.60188 − 0.378572i −0.631842 + 0.134302i 8.91761 + 1.89550i −8.93451 − 3.97790i 2.32666 − 0.244542i 3.93250 − 1.27775i −17.6247 − 5.72662i −7.84072 + 3.49091i 30.6751 + 17.7103i
7.3 −3.57964 − 0.376235i 5.30246 − 1.12707i 8.75968 + 1.86193i 1.89193 + 0.842339i −19.4049 + 2.03954i 9.38203 − 3.04841i −16.9632 − 5.51167i 18.6238 − 8.29187i −6.45549 − 3.72708i
7.4 −3.52844 − 0.370854i 2.35165 − 0.499858i 8.39979 + 1.78543i 2.73499 + 1.21770i −8.48303 + 0.891602i −9.47737 + 3.07938i −15.4791 − 5.02947i −2.94152 + 1.30965i −9.19867 − 5.31085i
7.5 −3.23732 − 0.340256i −2.81675 + 0.598720i 6.45189 + 1.37139i −0.142393 − 0.0633977i 9.32246 − 0.979830i −8.17642 + 2.65668i −8.03688 − 2.61134i −0.646270 + 0.287738i 0.439402 + 0.253689i
7.6 −2.98790 − 0.314041i 3.21428 − 0.683217i 4.91631 + 1.04499i −5.90172 − 2.62762i −9.81850 + 1.03197i −2.25794 + 0.733648i −2.93202 − 0.952672i 1.64291 − 0.731472i 16.8086 + 9.70443i
7.7 −2.91678 − 0.306566i 0.705174 − 0.149889i 4.50103 + 0.956723i 6.30498 + 2.80716i −2.10279 + 0.221012i 11.3839 − 3.69885i −1.67800 − 0.545216i −7.74711 + 3.44923i −17.5297 − 10.1208i
7.8 −2.51494 − 0.264331i −4.47269 + 0.950699i 2.34247 + 0.497908i −0.851023 − 0.378900i 11.4999 − 1.20868i 3.17437 − 1.03142i 3.86053 + 1.25436i 10.8792 − 4.84373i 2.04012 + 1.17786i
7.9 −2.46871 − 0.259471i −0.266471 + 0.0566401i 2.11459 + 0.449471i 8.76802 + 3.90377i 0.672534 − 0.0706862i −4.28209 + 1.39134i 4.33956 + 1.41001i −8.15411 + 3.63044i −20.6327 − 11.9123i
7.10 −2.29518 − 0.241233i 3.44467 − 0.732188i 1.29708 + 0.275703i −2.50795 − 1.11661i −8.08278 + 0.849535i 0.279230 − 0.0907274i 5.86896 + 1.90694i 3.10777 − 1.38367i 5.48683 + 3.16783i
7.11 −2.10774 − 0.221533i −0.671743 + 0.142783i 0.480910 + 0.102221i −1.48575 − 0.661498i 1.44749 − 0.152138i 7.71710 − 2.50744i 7.07151 + 2.29767i −7.79106 + 3.46880i 2.98503 + 1.72341i
7.12 −1.98262 − 0.208381i −2.88678 + 0.613603i −0.0252471 − 0.00536644i −4.28152 − 1.90625i 5.85123 − 0.614989i −11.2499 + 3.65532i 7.63280 + 2.48005i −0.264943 + 0.117960i 8.09137 + 4.67156i
7.13 −1.69309 − 0.177951i 5.31536 − 1.12981i −1.07770 − 0.229072i 4.09556 + 1.82346i −9.20043 + 0.967004i −7.92397 + 2.57466i 8.26026 + 2.68392i 18.7546 − 8.35009i −6.60966 − 3.81609i
7.14 −1.17730 − 0.123739i −5.41059 + 1.15006i −2.54187 − 0.540291i 7.17909 + 3.19634i 6.51219 − 0.684458i 1.24305 − 0.403893i 7.42906 + 2.41385i 19.7300 − 8.78434i −8.05641 − 4.65137i
7.15 −0.885908 − 0.0931127i 1.27261 − 0.270502i −3.13643 − 0.666668i −3.46241 − 1.54156i −1.15260 + 0.121143i 1.10495 − 0.359019i 6.10527 + 1.98372i −6.67554 + 2.97214i 2.92384 + 1.68808i
7.16 −0.871198 − 0.0915666i 1.40899 − 0.299491i −3.16199 − 0.672101i 1.39824 + 0.622537i −1.25494 + 0.131899i −2.81114 + 0.913394i 6.02567 + 1.95786i −6.32634 + 2.81667i −1.16114 − 0.670386i
7.17 −0.428941 − 0.0450835i −2.83773 + 0.603178i −3.73063 − 0.792970i −8.67019 − 3.86022i 1.24441 − 0.130793i 8.83639 − 2.87112i 3.20525 + 1.04145i −0.533023 + 0.237317i 3.54497 + 2.04669i
7.18 −0.414010 − 0.0435142i 4.13153 − 0.878184i −3.74308 − 0.795616i 5.67099 + 2.52489i −1.74871 + 0.183796i 6.98727 − 2.27030i 3.09871 + 1.00683i 8.07642 − 3.59585i −2.23798 − 1.29210i
7.19 −0.278051 − 0.0292243i −2.30495 + 0.489933i −3.83613 − 0.815395i 4.09727 + 1.82422i 0.655212 − 0.0688655i −1.46316 + 0.475409i 2.10640 + 0.684412i −3.14913 + 1.40208i −1.08594 − 0.626966i
7.20 −0.0624107 − 0.00655963i 5.14343 − 1.09327i −3.90874 − 0.830828i −8.19647 − 3.64930i −0.328176 + 0.0344927i −6.83606 + 2.22117i 0.477229 + 0.155061i 17.0377 − 7.58567i 0.487609 + 0.281521i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
19.c even 3 1 inner
209.s odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.3.s.a ✓ 304
11.d odd 10 1 inner 209.3.s.a ✓ 304
19.c even 3 1 inner 209.3.s.a ✓ 304
209.s odd 30 1 inner 209.3.s.a ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.3.s.a ✓ 304 1.a even 1 1 trivial
209.3.s.a ✓ 304 11.d odd 10 1 inner
209.3.s.a ✓ 304 19.c even 3 1 inner
209.3.s.a ✓ 304 209.s odd 30 1 inner

Hecke kernels

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