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Properties

Label 211.2.j.a
Level $211$
Weight $2$
Character orbit 211.j
Analytic conductor $1.685$
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] = [211,2,Mod(34,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.j (of order \(21\), degree \(12\), 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: \(1.68484348265\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 9 q^{2} - 16 q^{3} + 11 q^{4} - 10 q^{5} - 9 q^{6} - 16 q^{7} - 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 9 q^{2} - 16 q^{3} + 11 q^{4} - 10 q^{5} - 9 q^{6} - 16 q^{7} - 4 q^{8} - q^{9} + 17 q^{10} - 6 q^{11} - 60 q^{12} - 18 q^{13} - 7 q^{14} + 23 q^{15} + 93 q^{16} - 9 q^{17} - 4 q^{18} - 12 q^{19} + 4 q^{20} - 48 q^{21} + 19 q^{22} - 76 q^{23} - 13 q^{24} - 60 q^{25} - 24 q^{26} - 58 q^{27} + 66 q^{28} - 6 q^{29} - 55 q^{30} + 10 q^{31} + 14 q^{32} - 31 q^{33} - 51 q^{34} - 87 q^{35} + 99 q^{36} - 15 q^{37} - 40 q^{38} - 50 q^{39} - 18 q^{40} - 15 q^{41} + 26 q^{42} + 17 q^{43} + 146 q^{44} + 41 q^{45} - 107 q^{46} - 3 q^{47} - 127 q^{48} - 17 q^{49} + 78 q^{50} + 100 q^{51} - 116 q^{52} + 8 q^{53} - 162 q^{54} - 48 q^{55} + 66 q^{56} - 65 q^{57} - 226 q^{58} + 61 q^{59} + 116 q^{60} + 51 q^{61} + 101 q^{62} + 14 q^{63} - 56 q^{64} + 54 q^{65} - 26 q^{66} + 26 q^{67} - 58 q^{68} - 58 q^{69} - 22 q^{70} - 6 q^{71} + 218 q^{72} + 117 q^{73} + 67 q^{74} + 57 q^{75} - 157 q^{76} - 18 q^{77} - 46 q^{78} + 42 q^{79} + 165 q^{80} + 49 q^{81} - 57 q^{82} + 32 q^{83} + 80 q^{84} - 158 q^{85} + 58 q^{86} + 118 q^{87} + 25 q^{88} - 40 q^{89} - 111 q^{90} + 120 q^{91} - 114 q^{92} + 42 q^{93} - 218 q^{94} + 136 q^{95} - 136 q^{96} + 4 q^{97} + 98 q^{98} + 142 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.53452 + 0.382018i 0.970920 − 0.146343i 4.36672 − 1.34696i −1.83349 + 2.29912i −2.40491 + 0.741818i −0.0538484 − 0.137204i −5.93436 + 2.85784i −1.94545 + 0.600092i 3.76871 − 6.52760i
34.2 −2.28371 + 0.344214i 0.351495 − 0.0529794i 3.18569 − 0.982657i 2.26526 − 2.84054i −0.784477 + 0.241979i −0.258865 − 0.659578i −2.77537 + 1.33655i −2.74598 + 0.847021i −4.19543 + 7.26671i
34.3 −2.26358 + 0.341180i −2.77231 + 0.417859i 3.09625 − 0.955067i 0.0173325 − 0.0217343i 6.13279 − 1.89172i −1.25488 − 3.19737i −2.55786 + 1.23180i 4.64440 − 1.43261i −0.0318182 + 0.0551108i
34.4 −1.58818 + 0.239379i −2.04144 + 0.307697i 0.553853 − 0.170841i −1.61165 + 2.02095i 3.16850 − 0.977353i 1.59575 + 4.06591i 2.05540 − 0.989828i 1.20607 − 0.372022i 2.07581 − 3.59541i
34.5 −1.41407 + 0.213137i 3.02510 − 0.455960i 0.0430174 − 0.0132691i −0.0500366 + 0.0627439i −4.18052 + 1.28952i −1.79686 − 4.57832i 2.51884 − 1.21301i 6.07661 − 1.87439i 0.0573821 − 0.0993888i
34.6 −1.18456 + 0.178543i −0.125810 + 0.0189628i −0.539852 + 0.166522i 0.502029 − 0.629525i 0.145643 − 0.0449250i 0.509781 + 1.29890i 2.76836 − 1.33317i −2.85125 + 0.879494i −0.482284 + 0.835341i
34.7 −0.333125 + 0.0502106i −0.575392 + 0.0867265i −1.80269 + 0.556058i 0.597837 − 0.749664i 0.187323 − 0.0577816i −1.00675 − 2.56516i 1.17965 − 0.568092i −2.54316 + 0.784462i −0.161514 + 0.279750i
34.8 −0.215343 + 0.0324577i 2.32503 − 0.350442i −1.86583 + 0.575531i 1.67617 − 2.10185i −0.489304 + 0.150930i 1.51530 + 3.86091i 0.775529 − 0.373475i 2.41625 − 0.745316i −0.292730 + 0.507023i
34.9 −0.152293 + 0.0229544i −3.00837 + 0.453439i −1.88848 + 0.582519i 1.29176 − 1.61982i 0.447745 − 0.138111i 0.193898 + 0.494044i 0.551752 − 0.265710i 5.97798 − 1.84396i −0.159544 + 0.276339i
34.10 −0.0842224 + 0.0126945i 1.95243 − 0.294281i −1.90421 + 0.587372i −2.63127 + 3.29951i −0.160703 + 0.0495702i 0.702130 + 1.78900i 0.306399 − 0.147554i 0.858659 − 0.264861i 0.179726 − 0.311295i
34.11 0.557194 − 0.0839835i −0.740325 + 0.111586i −1.60773 + 0.495920i −1.49600 + 1.87592i −0.403133 + 0.124350i −1.04909 − 2.67303i −1.86954 + 0.900323i −2.33109 + 0.719046i −0.676015 + 1.17089i
34.12 1.16660 − 0.175837i 2.31080 − 0.348297i −0.581100 + 0.179246i 0.657712 − 0.824745i 2.63455 − 0.812650i −0.422359 − 1.07615i −2.77229 + 1.33506i 2.35177 − 0.725425i 0.622269 − 1.07780i
34.13 1.43489 − 0.216275i −2.07678 + 0.313024i 0.100993 − 0.0311523i −1.41511 + 1.77449i −2.91226 + 0.898312i 0.743002 + 1.89314i −2.47661 + 1.19267i 1.34832 − 0.415900i −1.64675 + 2.85226i
34.14 1.80504 − 0.272066i −0.728178 + 0.109755i 1.27301 − 0.392671i 2.53820 − 3.18281i −1.28453 + 0.396225i −0.692092 − 1.76342i −1.09831 + 0.528920i −2.34852 + 0.724423i 3.71563 − 6.43566i
34.15 2.05505 − 0.309749i 0.856257 − 0.129060i 2.21614 − 0.683590i −0.0498308 + 0.0624859i 1.71967 − 0.530449i 0.632720 + 1.61214i 0.597644 − 0.287810i −2.15020 + 0.663249i −0.0830499 + 0.143847i
34.16 2.61212 − 0.393713i 0.261941 − 0.0394813i 4.75699 − 1.46734i −2.10245 + 2.63639i 0.668677 − 0.206259i −1.70049 − 4.33278i 7.08807 − 3.41344i −2.79966 + 0.863582i −4.45387 + 7.71433i
34.17 2.68901 − 0.405303i −2.73392 + 0.412072i 5.15535 − 1.59021i 0.866052 − 1.08599i −7.18452 + 2.21613i 0.979677 + 2.49618i 8.31810 − 4.00579i 4.43780 − 1.36888i 1.88866 − 3.27126i
43.1 −0.980584 − 2.49849i 0.808541 + 2.06013i −3.81479 + 3.53960i −1.10500 − 1.38563i 4.35436 − 4.04026i 2.88602 − 0.434998i 7.74793 + 3.73120i −1.39124 + 1.29088i −2.37843 + 4.11956i
43.2 −0.944044 − 2.40539i −0.289907 − 0.738670i −3.42856 + 3.18123i 0.134583 + 0.168761i −1.50310 + 1.39467i −4.48382 + 0.675827i 6.23258 + 3.00145i 1.73757 − 1.61223i 0.278884 − 0.483042i
43.3 −0.761572 − 1.94045i 0.0593356 + 0.151185i −1.71926 + 1.59524i 1.79982 + 2.25690i 0.248179 − 0.230276i 0.409818 − 0.0617701i 0.648605 + 0.312352i 2.17982 − 2.02258i 3.00872 − 5.21125i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.j even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.2.j.a ✓ 204
211.j even 21 1 inner 211.2.j.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.j.a ✓ 204 1.a even 1 1 trivial
211.2.j.a ✓ 204 211.j even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(211, [\chi])\).