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Properties

Label 211.4.l.a
Level $211$
Weight $4$
Character orbit 211.l
Analytic conductor $12.449$
Analytic rank $0$
Dimension $1248$
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] = [211,4,Mod(5,211)]
 
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([44]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 211.l (of order \(35\), 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: \(12.4494030112\)
Analytic rank: \(0\)
Dimension: \(1248\)
Relative dimension: \(52\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1248 q - 28 q^{2} - 22 q^{3} + 216 q^{4} - 25 q^{5} - q^{6} + 19 q^{7} + 95 q^{8} + 332 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1248 q - 28 q^{2} - 22 q^{3} + 216 q^{4} - 25 q^{5} - q^{6} + 19 q^{7} + 95 q^{8} + 332 q^{9} - q^{10} - 79 q^{11} + 402 q^{12} - 21 q^{13} - 208 q^{14} + 1298 q^{15} + 1736 q^{16} - 151 q^{17} + 123 q^{18} + 363 q^{19} - 287 q^{20} + 540 q^{21} + 431 q^{22} - 1382 q^{23} + 903 q^{24} + 1015 q^{25} + 468 q^{26} + 353 q^{27} - 1345 q^{28} + 1505 q^{29} - 1024 q^{30} + 842 q^{31} - 3898 q^{32} - 1342 q^{33} + 1816 q^{34} - 1785 q^{35} - 1436 q^{36} + 665 q^{37} - 2202 q^{38} + 2105 q^{39} + 3898 q^{40} + 1153 q^{41} - 2639 q^{42} + 3276 q^{43} - 2414 q^{44} - 2014 q^{45} + 108 q^{46} - 2341 q^{47} - 3972 q^{48} + 671 q^{49} + 269 q^{50} + 1212 q^{51} - 893 q^{52} + 1723 q^{53} + 307 q^{54} - 2248 q^{55} - 3698 q^{56} + 8146 q^{57} - 2626 q^{58} + 3130 q^{59} + 5466 q^{60} - 710 q^{61} + 1127 q^{62} - 6385 q^{63} + 4135 q^{64} + 1915 q^{65} - 190 q^{66} + 7023 q^{67} + 872 q^{68} + 342 q^{69} + 6644 q^{70} + 1810 q^{71} - 11689 q^{72} - 983 q^{73} - 8622 q^{74} + 5485 q^{75} - 5984 q^{76} - 2928 q^{77} + 1022 q^{78} - 1225 q^{79} - 5468 q^{80} - 446 q^{81} + 120 q^{82} + 1590 q^{83} - 12521 q^{84} + 8809 q^{85} - 3832 q^{86} + 4617 q^{87} + 2277 q^{88} + 3338 q^{89} - 2741 q^{90} - 3075 q^{91} + 5018 q^{92} - 3202 q^{93} - 4830 q^{94} - 1472 q^{95} + 4901 q^{96} + 897 q^{97} - 7204 q^{98} + 11745 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.15190 + 5.03463i 3.31698 + 0.298533i −15.1883 − 15.8857i −20.3983 + 1.83588i −8.64082 + 16.0573i 1.87361 + 4.38352i 71.6536 − 26.8920i −15.6529 − 2.84058i 34.6522 − 106.648i
5.2 −2.12566 + 4.97324i −4.71946 − 0.424759i −14.6861 − 15.3605i 0.340957 − 0.0306867i 12.1444 − 22.5681i −13.5513 − 31.7048i 67.1003 − 25.1832i −4.47321 − 0.811768i −0.572148 + 1.76089i
5.3 −2.09743 + 4.90718i 7.72887 + 0.695611i −14.1527 − 14.8026i 11.6182 − 1.04565i −19.6243 + 36.4680i −0.0437173 − 0.102282i 62.3526 − 23.4013i 32.6855 + 5.93154i −19.2371 + 59.2056i
5.4 −2.07007 + 4.84318i −1.86725 − 0.168056i −13.6427 − 14.2691i 8.54720 − 0.769262i 4.67927 − 8.69554i 13.8312 + 32.3598i 57.8997 − 21.7301i −23.1077 − 4.19343i −13.9677 + 42.9880i
5.5 −1.88024 + 4.39903i −8.59578 − 0.773634i −10.2877 − 10.7601i −5.12758 + 0.461491i 19.5653 − 36.3585i 3.30144 + 7.72410i 30.8455 − 11.5765i 46.7229 + 8.47895i 7.61095 − 23.4241i
5.6 −1.77086 + 4.14312i 5.35462 + 0.481924i −8.50103 − 8.89138i 2.12774 − 0.191500i −11.4789 + 21.3314i −3.01805 − 7.06107i 18.1449 − 6.80991i 1.87358 + 0.340005i −2.97451 + 9.15459i
5.7 −1.74842 + 4.09064i 6.60415 + 0.594384i −8.14784 − 8.52197i −9.90943 + 0.891865i −13.9783 + 25.9760i −7.11455 − 16.6453i 15.7865 − 5.92476i 16.6954 + 3.02977i 13.6776 − 42.0952i
5.8 −1.71480 + 4.01197i −0.154065 − 0.0138661i −7.62684 − 7.97705i 15.9005 − 1.43107i 0.319821 − 0.594328i −2.01434 − 4.71277i 12.4033 − 4.65502i −26.5426 − 4.81676i −21.5248 + 66.2464i
5.9 −1.62109 + 3.79272i −1.41647 − 0.127485i −6.22830 − 6.51429i 9.49769 − 0.854808i 2.77973 − 5.16561i −6.69855 − 15.6720i 3.91045 − 1.46762i −24.5760 − 4.45988i −12.1545 + 37.4078i
5.10 −1.61071 + 3.76845i −2.37212 − 0.213494i −6.07830 − 6.35740i −13.2962 + 1.19668i 4.62534 − 8.59532i 4.94242 + 11.5634i 3.05259 − 1.14566i −20.9847 − 3.80817i 16.9067 − 52.0335i
5.11 −1.45576 + 3.40593i 9.82109 + 0.883914i −3.95259 − 4.13408i −5.30310 + 0.477287i −17.3077 + 32.1631i 11.3869 + 26.6409i −7.90807 + 2.96795i 69.1064 + 12.5410i 6.09444 − 18.7568i
5.12 −1.45039 + 3.39336i −8.50455 − 0.765423i −3.88274 − 4.06102i 17.7734 − 1.59964i 14.9323 − 27.7488i 0.0254445 + 0.0595304i −8.22811 + 3.08806i 45.1753 + 8.19811i −20.3503 + 62.6317i
5.13 −1.27923 + 2.99292i 2.89559 + 0.260608i −1.79260 − 1.87491i −5.84596 + 0.526146i −4.48412 + 8.33289i 6.73165 + 15.7495i −16.4737 + 6.18270i −18.2496 − 3.31181i 5.90363 − 18.1695i
5.14 −1.21994 + 2.85419i −5.98891 − 0.539012i −1.12964 − 1.18151i −20.7144 + 1.86433i 8.84455 − 16.4359i −10.6771 − 24.9803i −18.4980 + 6.94243i 9.01046 + 1.63516i 19.9491 − 61.3971i
5.15 −1.07765 + 2.52128i −6.57740 − 0.591977i 0.332964 + 0.348253i −0.295766 + 0.0266194i 8.58066 − 15.9455i −1.76787 − 4.13613i −21.7736 + 8.17177i 16.3456 + 2.96630i 0.251616 − 0.774396i
5.16 −1.03111 + 2.41240i 4.07226 + 0.366510i 0.772033 + 0.807483i −8.47692 + 0.762936i −5.08310 + 9.44599i −10.6230 − 24.8537i −22.3938 + 8.40454i −10.1171 − 1.83599i 6.90011 − 21.2363i
5.17 −1.00734 + 2.35678i 4.75197 + 0.427685i 0.988820 + 1.03423i 16.7085 − 1.50379i −5.79479 + 10.7685i 8.08264 + 18.9103i −22.6303 + 8.49330i −4.16781 − 0.756345i −13.2869 + 40.8930i
5.18 −0.969225 + 2.26761i 9.55604 + 0.860059i 1.32582 + 1.38670i 18.4117 − 1.65708i −11.2122 + 20.8358i −14.2232 − 33.2768i −22.9000 + 8.59453i 64.0122 + 11.6165i −14.0874 + 43.3567i
5.19 −0.776426 + 1.81654i −4.63162 − 0.416853i 2.83152 + 2.96154i −5.06849 + 0.456173i 4.35334 − 8.08986i 7.13472 + 16.6925i −22.3746 + 8.39732i −5.28800 − 0.959630i 3.10666 − 9.56130i
5.20 −0.604032 + 1.41320i −2.45076 − 0.220572i 3.89621 + 4.07512i 8.89907 − 0.800931i 1.79205 − 3.33019i −8.79508 − 20.5771i −19.6234 + 7.36480i −20.6085 − 3.73990i −4.24344 + 13.0600i
See next 80 embeddings (of 1248 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.l even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.4.l.a ✓ 1248
211.l even 35 1 inner 211.4.l.a ✓ 1248
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.4.l.a ✓ 1248 1.a even 1 1 trivial
211.4.l.a ✓ 1248 211.l even 35 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(211, [\chi])\).