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Properties

Label 211.3.m.a
Level $211$
Weight $3$
Character orbit 211.m
Analytic conductor $5.749$
Analytic rank $0$
Dimension $408$
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] = [211,3,Mod(26,211)]
 
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([29]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 211.m (of order \(42\), 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: \(5.74933357800\)
Analytic rank: \(0\)
Dimension: \(408\)
Relative dimension: \(34\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 408 q - 4 q^{2} - 14 q^{3} - 90 q^{4} - 10 q^{5} + 29 q^{6} + 10 q^{7} + 56 q^{8} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 408 q - 4 q^{2} - 14 q^{3} - 90 q^{4} - 10 q^{5} + 29 q^{6} + 10 q^{7} + 56 q^{8} - 96 q^{9} - 21 q^{10} - 18 q^{11} - 182 q^{12} - 46 q^{13} - 19 q^{14} - 21 q^{15} + 178 q^{16} - 58 q^{17} - 14 q^{18} + 85 q^{19} + 4 q^{20} + 256 q^{21} + 215 q^{22} - 214 q^{24} - 350 q^{25} - 56 q^{26} - 266 q^{27} - 42 q^{28} - 136 q^{29} + 31 q^{30} + 210 q^{31} - 111 q^{32} + 103 q^{33} + 344 q^{34} - 179 q^{35} - 58 q^{36} - q^{37} - 19 q^{38} - 286 q^{39} + 1246 q^{40} + 159 q^{41} - 202 q^{43} + 27 q^{44} - 405 q^{45} + 145 q^{46} + 243 q^{47} - 689 q^{48} - 130 q^{49} - 617 q^{50} + 881 q^{51} - 118 q^{52} + 406 q^{53} + 267 q^{54} - 104 q^{55} + 124 q^{56} - 613 q^{57} + 56 q^{58} - 280 q^{59} + 182 q^{60} - 243 q^{61} + 733 q^{62} - 336 q^{63} - 646 q^{64} - 128 q^{65} - 1110 q^{66} + 175 q^{67} - 14 q^{68} - 18 q^{69} - 1172 q^{70} - 206 q^{71} - 1143 q^{72} + 761 q^{73} + 309 q^{74} + 213 q^{75} + 242 q^{76} + 918 q^{77} + 6 q^{78} + 146 q^{79} - 379 q^{80} - 18 q^{81} - 165 q^{82} - 133 q^{83} + 2248 q^{84} - 462 q^{85} + 994 q^{86} - 396 q^{87} - 231 q^{88} - 14 q^{89} + 1267 q^{90} - 180 q^{91} + 838 q^{92} + 12 q^{93} - 842 q^{94} - 90 q^{95} - 640 q^{96} + 350 q^{97} + 182 q^{98} + 204 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.20624 + 3.23596i 1.02670 − 1.50589i −4.14258 − 10.5551i −8.21122 − 3.95432i 2.60786 + 6.64471i 7.16034 − 0.536594i 28.0223 + 6.39592i 2.07448 + 5.28568i 30.9120 − 17.8470i
26.2 −2.02526 + 2.97051i 3.23049 − 4.73826i −3.26088 − 8.30858i 6.83953 + 3.29374i 7.53246 + 19.1924i 6.31968 − 0.473594i 17.2645 + 3.94051i −8.72699 − 22.2360i −23.6359 + 13.6462i
26.3 −1.96122 + 2.87658i −2.45597 + 3.60225i −2.96698 − 7.55974i −2.26878 − 1.09259i −5.54546 − 14.1296i −3.54794 + 0.265881i 13.9881 + 3.19270i −3.65633 − 9.31618i 7.59249 − 4.38353i
26.4 −1.94988 + 2.85995i −1.39490 + 2.04594i −2.91592 − 7.42963i 8.68522 + 4.18258i −3.13140 − 7.97867i −0.479680 + 0.0359471i 13.4356 + 3.06658i 1.04795 + 2.67012i −28.8971 + 16.6838i
26.5 −1.86868 + 2.74085i 0.745781 − 1.09386i −2.55893 − 6.52004i 0.665283 + 0.320383i 1.60448 + 4.08814i −8.97156 + 0.672325i 9.71589 + 2.21759i 2.64773 + 6.74631i −2.12132 + 1.22475i
26.6 −1.74962 + 2.56622i −0.889598 + 1.30480i −2.06295 − 5.25631i 1.15517 + 0.556301i −1.79195 − 4.56581i 13.1924 − 0.988633i 4.98607 + 1.13804i 2.37695 + 6.05637i −3.44870 + 1.99111i
26.7 −1.68558 + 2.47229i 1.69445 − 2.48531i −1.80969 − 4.61101i −0.369155 − 0.177776i 3.28827 + 8.37837i −5.10987 + 0.382932i 2.78134 + 0.634822i −0.0175036 − 0.0445984i 1.06175 − 0.613004i
26.8 −1.41630 + 2.07733i 2.61118 − 3.82990i −0.848034 − 2.16075i −3.74206 − 1.80208i 4.25775 + 10.8486i 2.89384 − 0.216864i −4.11499 − 0.939220i −4.56179 − 11.6232i 9.04340 − 5.22121i
26.9 −1.12594 + 1.65145i −0.650586 + 0.954235i 0.00182044 + 0.00463842i −1.26806 − 0.610664i −0.843349 − 2.14882i 5.61292 − 0.420630i −7.80427 − 1.78127i 2.80077 + 7.13624i 2.43623 − 1.40656i
26.10 −1.05786 + 1.55160i −0.848222 + 1.24411i 0.172977 + 0.440739i −7.66785 − 3.69264i −1.03306 − 2.63220i 0.375004 − 0.0281027i −8.19013 − 1.86934i 2.45973 + 6.26730i 13.8410 − 7.99112i
26.11 −0.985456 + 1.44540i −3.10281 + 4.55098i 0.343311 + 0.874743i −2.41014 − 1.16066i −3.52030 − 8.96958i 2.41789 − 0.181196i −8.42471 − 1.92289i −7.79596 − 19.8638i 4.05270 − 2.33983i
26.12 −0.914409 + 1.34119i 1.10872 − 1.62620i 0.498713 + 1.27070i 6.42408 + 3.09367i 1.16721 + 2.97402i 1.68140 − 0.126004i −8.49048 − 1.93790i 1.87282 + 4.77187i −10.0234 + 5.78704i
26.13 −0.843474 + 1.23715i −1.58954 + 2.33142i 0.642275 + 1.63649i 3.50960 + 1.69013i −1.54358 − 3.93299i −11.1293 + 0.834026i −8.40546 − 1.91849i 0.379170 + 0.966108i −5.05120 + 2.91631i
26.14 −0.736841 + 1.08075i 2.42558 − 3.55767i 0.836285 + 2.13082i −5.35663 − 2.57962i 2.05767 + 5.24287i −6.25866 + 0.469022i −8.02003 − 1.83052i −3.48550 − 8.88091i 6.73490 − 3.88839i
26.15 −0.260133 + 0.381546i 2.46468 − 3.61502i 1.38346 + 3.52499i 0.847562 + 0.408164i 0.738150 + 1.88078i 12.0434 − 0.902527i −3.50566 − 0.800144i −3.70567 − 9.44190i −0.376212 + 0.217206i
26.16 −0.171986 + 0.252257i 0.555556 − 0.814851i 1.42731 + 3.63673i −5.08464 − 2.44864i 0.110004 + 0.280286i 3.70697 − 0.277799i −2.35348 − 0.537167i 2.93273 + 7.47247i 1.49217 − 0.861507i
26.17 −0.0135867 + 0.0199281i −2.37435 + 3.48254i 1.46115 + 3.72295i 4.44471 + 2.14046i −0.0371406 − 0.0946327i 10.8466 − 0.812844i −0.188101 − 0.0429328i −3.20244 − 8.15968i −0.103044 + 0.0594927i
26.18 0.181990 − 0.266930i 0.432890 − 0.634932i 1.42323 + 3.62634i −2.78631 − 1.34182i −0.0907012 − 0.231103i −11.6124 + 0.870228i 2.48686 + 0.567610i 3.07232 + 7.82815i −0.865253 + 0.499554i
26.19 0.202891 − 0.297587i −1.58660 + 2.32712i 1.41397 + 3.60274i 4.28339 + 2.06277i 0.370612 + 0.944304i 2.50542 − 0.187755i 2.76357 + 0.630766i 0.389890 + 0.993425i 1.48291 − 0.856161i
26.20 0.337652 − 0.495245i −2.39277 + 3.50954i 1.33011 + 3.38905i −5.77371 − 2.78047i 0.930160 + 2.37001i −3.63460 + 0.272376i 4.46500 + 1.01911i −3.30349 − 8.41717i −3.32652 + 1.92057i
See next 80 embeddings (of 408 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.m odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.3.m.a ✓ 408
211.m odd 42 1 inner 211.3.m.a ✓ 408
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.3.m.a ✓ 408 1.a even 1 1 trivial
211.3.m.a ✓ 408 211.m odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(211, [\chi])\).