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Properties

Label 211.3.n.a
Level $211$
Weight $3$
Character orbit 211.n
Analytic conductor $5.749$
Analytic rank $0$
Dimension $840$
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(8,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.8");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 211.n (of order \(70\), degree \(24\), 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: \(840\)
Relative dimension: \(35\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 16 q^{2} - 33 q^{3} - 116 q^{4} - 25 q^{5} - q^{6} - 48 q^{7} - 83 q^{8} - 154 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 16 q^{2} - 33 q^{3} - 116 q^{4} - 25 q^{5} - q^{6} - 48 q^{7} - 83 q^{8} - 154 q^{9} + 65 q^{10} - 13 q^{11} + 56 q^{12} - 47 q^{13} - 40 q^{14} + 524 q^{16} - 16 q^{17} + 7 q^{18} - 84 q^{19} - 119 q^{20} - 294 q^{21} + 173 q^{22} - 30 q^{23} - 45 q^{24} + 180 q^{25} - 28 q^{26} + 273 q^{27} - 561 q^{28} + 163 q^{29} - 152 q^{30} + 280 q^{31} - 588 q^{32} - 28 q^{33} - 152 q^{34} + 197 q^{35} + 44 q^{36} - 133 q^{37} + 371 q^{39} + 56 q^{40} + 75 q^{41} - 559 q^{42} - 72 q^{43} + 50 q^{44} + 227 q^{45} - 438 q^{46} - 512 q^{47} + 578 q^{48} - 399 q^{49} - 721 q^{50} + 229 q^{51} - 327 q^{52} + 436 q^{53} + 115 q^{54} - 190 q^{55} - 604 q^{56} + 364 q^{57} + 956 q^{58} - 138 q^{59} - 322 q^{60} + 490 q^{61} - 821 q^{62} - 350 q^{63} + 419 q^{64} - 67 q^{65} - 580 q^{66} + 322 q^{67} - 248 q^{68} + 67 q^{69} + 534 q^{70} - 467 q^{71} + 2373 q^{72} + 609 q^{73} - 220 q^{74} + 1477 q^{75} - 326 q^{76} - 1655 q^{77} - 378 q^{78} + 97 q^{79} - 158 q^{80} - 286 q^{81} - 2378 q^{82} - 337 q^{83} - 1127 q^{84} - 565 q^{85} + 1148 q^{86} - 241 q^{87} - 1757 q^{88} - 208 q^{89} - 157 q^{90} + 1260 q^{91} - 1164 q^{92} + 359 q^{93} - 574 q^{94} + 115 q^{95} + 647 q^{96} + 971 q^{97} + 584 q^{98} - 2656 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −3.86837 − 0.173729i 1.85813 − 4.95097i 10.9502 + 0.985540i 3.82411 − 1.43521i −8.04808 + 18.8294i −7.17414 + 0.322191i −26.8395 − 3.63566i −14.2818 − 12.4777i −15.0424 + 4.88758i
8.2 −3.71395 − 0.166794i −1.65961 + 4.42202i 9.78174 + 0.880373i −5.22706 + 1.96175i 6.90128 − 16.1464i 3.25933 − 0.146377i −21.4459 − 2.90504i −10.0223 − 8.75621i 19.7403 − 6.41400i
8.3 −3.63543 − 0.163267i 0.0413891 − 0.110281i 9.20576 + 0.828534i 5.55449 − 2.08463i −0.168472 + 0.394161i 12.5498 − 0.563614i −18.9070 − 2.56112i 6.76719 + 5.91232i −20.5333 + 6.67167i
8.4 −3.44220 − 0.154589i 0.820310 − 2.18571i 7.84095 + 0.705698i −8.80961 + 3.30630i −3.16156 + 7.39684i −0.762342 + 0.0342368i −13.2231 − 1.79119i 2.67323 + 2.33553i 30.8356 − 10.0191i
8.5 −3.26460 − 0.146614i −1.63175 + 4.34778i 6.65224 + 0.598713i 7.72660 − 2.89984i 5.96445 − 13.9545i −11.4574 + 0.514554i −8.67589 − 1.17523i −9.46293 − 8.26751i −25.6494 + 8.33400i
8.6 −3.16624 − 0.142196i 0.127809 − 0.340546i 6.02097 + 0.541897i 0.381145 − 0.143046i −0.453098 + 1.06008i −6.25308 + 0.280826i −6.42381 − 0.870164i 6.67801 + 5.83440i −1.22714 + 0.398721i
8.7 −2.70258 − 0.121373i −1.09687 + 2.92261i 3.30530 + 0.297482i −1.44341 + 0.541721i 3.31912 − 7.76546i 1.84507 − 0.0828622i 1.82655 + 0.247423i −0.560887 − 0.490032i 3.96668 − 1.28885i
8.8 −2.64637 − 0.118848i 1.28509 − 3.42412i 3.00524 + 0.270476i 0.649283 − 0.243680i −3.80778 + 8.90874i 6.44542 − 0.289464i 2.57943 + 0.349407i −3.29547 − 2.87917i −1.74720 + 0.567700i
8.9 −2.03244 − 0.0912771i 1.78729 − 4.76222i 0.138595 + 0.0124738i 2.06112 − 0.773549i −4.06725 + 9.51581i −6.05615 + 0.271982i 7.78377 + 1.05438i −12.7067 − 11.1015i −4.25971 + 1.38406i
8.10 −1.96770 − 0.0883694i −1.63258 + 4.35000i −0.119875 − 0.0107890i 2.67685 − 1.00464i 3.59683 − 8.41520i 10.2642 − 0.460964i 8.04234 + 1.08941i −9.47950 − 8.28199i −5.35600 + 1.74027i
8.11 −1.93151 − 0.0867442i 0.443199 − 1.18090i −0.260696 − 0.0234631i 7.65968 − 2.87473i −0.958479 + 2.24247i −3.43111 + 0.154091i 8.16533 + 1.10607i 5.57955 + 4.87470i −15.0441 + 4.88813i
8.12 −1.70663 − 0.0766448i −0.0443782 + 0.118245i −1.07719 − 0.0969486i −4.06135 + 1.52425i 0.0848000 − 0.198399i 7.47184 − 0.335561i 8.60248 + 1.16529i 6.76563 + 5.91095i 7.04804 − 2.29005i
8.13 −1.58687 − 0.0712666i −1.02121 + 2.72100i −1.47081 − 0.132375i −6.14022 + 2.30446i 1.81445 − 4.24511i −11.2371 + 0.504660i 8.62094 + 1.16779i 0.416652 + 0.364018i 9.90798 − 3.21930i
8.14 −1.22945 − 0.0552146i 1.54007 − 4.10350i −2.47540 − 0.222790i −5.85754 + 2.19837i −2.12001 + 4.96001i −1.54840 + 0.0695389i 7.90928 + 1.07138i −7.68929 − 6.71793i 7.32293 − 2.37936i
8.15 −0.657115 − 0.0295111i −1.07419 + 2.86216i −3.55297 − 0.319773i 5.39002 − 2.02291i 0.790331 − 1.84907i 1.90151 − 0.0853968i 4.93257 + 0.668162i −0.260455 − 0.227553i −3.60157 + 1.17022i
8.16 −0.185653 − 0.00833770i 0.314791 − 0.838759i −3.94950 − 0.355461i 1.01425 − 0.380654i −0.0654354 + 0.153094i −9.48559 + 0.425998i 1.46691 + 0.198706i 6.17322 + 5.39338i −0.191472 + 0.0622132i
8.17 −0.179433 − 0.00805834i 0.307460 − 0.819225i −3.95177 − 0.355665i −4.68512 + 1.75835i −0.0617701 + 0.144518i 6.35184 − 0.285262i 1.41816 + 0.192103i 6.20105 + 5.41769i 0.854834 − 0.277753i
8.18 −0.165800 − 0.00744609i 1.45798 − 3.88479i −3.95646 − 0.356088i 5.78223 − 2.17011i −0.270661 + 0.633242i 11.1753 − 0.501882i 1.31119 + 0.177613i −6.18821 − 5.40647i −0.974853 + 0.316749i
8.19 0.100980 + 0.00453502i −1.58506 + 4.22337i −3.97372 − 0.357641i 3.07308 − 1.15335i −0.179212 + 0.419288i −6.11974 + 0.274838i −0.800313 − 0.108410i −8.54680 − 7.46711i 0.315550 − 0.102528i
8.20 0.175209 + 0.00786866i −1.95336 + 5.20471i −3.95326 − 0.355800i −8.20648 + 3.07994i −0.383201 + 0.896543i 8.02358 − 0.360339i −1.38504 − 0.187617i −16.4957 − 14.4119i −1.46209 + 0.475061i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.n odd 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.3.n.a ✓ 840
211.n odd 70 1 inner 211.3.n.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.3.n.a ✓ 840 1.a even 1 1 trivial
211.3.n.a ✓ 840 211.n odd 70 1 inner

Hecke kernels

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