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Properties

Label 211.8.a.b
Level $211$
Weight $8$
Character orbit 211.a
Self dual yes
Analytic conductor $65.913$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [211,8,Mod(1,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 211.a (trivial)

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: yes
Analytic conductor: \(65.9132403771\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{2} + 148 q^{3} + 4352 q^{4} + 2249 q^{5} + 1976 q^{6} + 2243 q^{7} + 3072 q^{8} + 52842 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{2} + 148 q^{3} + 4352 q^{4} + 2249 q^{5} + 1976 q^{6} + 2243 q^{7} + 3072 q^{8} + 52842 q^{9} + 9680 q^{10} + 14713 q^{11} + 24633 q^{12} + 18464 q^{13} + 50203 q^{14} + 33312 q^{15} + 327204 q^{16} + 139153 q^{17} + 37348 q^{18} + 55095 q^{19} + 310040 q^{20} + 186788 q^{21} + 19961 q^{22} + 276801 q^{23} + 362882 q^{24} + 1107541 q^{25} + 587810 q^{26} + 280732 q^{27} + 634767 q^{28} + 1163176 q^{29} + 227203 q^{30} + 376830 q^{31} + 485777 q^{32} + 776065 q^{33} + 432070 q^{34} + 354369 q^{35} + 4288124 q^{36} + 976941 q^{37} + 797783 q^{38} + 689491 q^{39} + 1571268 q^{40} + 2434000 q^{41} + 1667353 q^{42} + 472931 q^{43} + 2361143 q^{44} + 3521410 q^{45} - 392609 q^{46} + 2038939 q^{47} + 3594145 q^{48} + 9930821 q^{49} + 513969 q^{50} + 1610956 q^{51} + 1877283 q^{52} + 6767362 q^{53} - 14768353 q^{54} - 4462008 q^{55} + 1358300 q^{56} - 6754455 q^{57} - 9320251 q^{58} + 3361016 q^{59} - 12287144 q^{60} + 10188363 q^{61} + 2721722 q^{62} + 6903810 q^{63} + 18574866 q^{64} + 11821177 q^{65} + 4047323 q^{66} + 3465499 q^{67} + 29545633 q^{68} + 24045665 q^{69} + 32655491 q^{70} + 17671915 q^{71} + 25148429 q^{72} + 16001008 q^{73} + 32221893 q^{74} + 47368751 q^{75} + 31261569 q^{76} + 32591086 q^{77} + 86706538 q^{78} + 20854575 q^{79} + 53395931 q^{80} + 77307908 q^{81} + 56234555 q^{82} + 16832176 q^{83} + 151350612 q^{84} + 38022616 q^{85} + 70318939 q^{86} + 49094584 q^{87} + 65426310 q^{88} + 78240466 q^{89} + 128108330 q^{90} + 54997298 q^{91} + 102782608 q^{92} + 55413745 q^{93} + 73786319 q^{94} + 66357575 q^{95} + 139119803 q^{96} + 40694440 q^{97} + 118942770 q^{98} + 81547853 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −22.3871 −50.9348 373.181 98.6100 1140.28 −104.865 −5488.88 407.350 −2207.59
1.2 −21.4919 −17.7852 333.902 379.685 382.237 977.071 −4425.22 −1870.69 −8160.15
1.3 −21.3928 93.3668 329.652 −294.242 −1997.38 705.091 −4313.90 6530.36 6294.67
1.4 −19.8360 −80.3821 265.468 163.724 1594.46 −1352.74 −2726.82 4274.29 −3247.63
1.5 −19.2725 −28.6521 243.427 −144.599 552.196 230.907 −2224.57 −1366.06 2786.77
1.6 −19.1507 25.0022 238.747 −422.668 −478.808 117.662 −2120.89 −1561.89 8094.36
1.7 −19.1253 40.9101 237.778 216.817 −782.419 507.352 −2099.55 −513.365 −4146.70
1.8 −18.9377 −25.6480 230.636 504.432 485.713 −1539.55 −1943.70 −1529.18 −9552.78
1.9 −17.9593 −45.6906 194.538 −467.972 820.574 948.642 −1194.98 −99.3668 8404.47
1.10 −17.5194 −6.72141 178.930 521.010 117.755 386.650 −892.256 −2141.82 −9127.78
1.11 −17.1457 77.1356 165.977 402.668 −1322.55 1381.99 −651.136 3762.90 −6904.04
1.12 −16.6257 −10.7640 148.415 −19.0335 178.960 −303.612 −339.418 −2071.14 316.445
1.13 −16.0251 70.0339 128.803 −345.283 −1122.30 −1260.84 −12.8718 2717.74 5533.19
1.14 −15.2734 72.9410 105.276 −171.475 −1114.05 199.734 347.073 3133.38 2619.00
1.15 −14.2171 −78.9477 74.1246 −345.121 1122.40 −874.421 765.949 4045.73 4906.60
1.16 −11.0057 18.5364 −6.87448 100.445 −204.006 −1145.94 1484.39 −1843.40 −1105.47
1.17 −10.9736 22.6129 −7.57961 −287.525 −248.145 242.435 1487.80 −1675.66 3155.19
1.18 −10.9537 80.5988 −8.01569 460.994 −882.858 −1409.85 1489.88 4309.17 −5049.60
1.19 −10.6228 22.3518 −15.1552 357.875 −237.440 78.0077 1520.72 −1687.40 −3801.65
1.20 −9.56685 −49.8787 −36.4753 −105.270 477.182 725.742 1573.51 300.883 1007.10
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(211\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.8.a.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.8.a.b ✓ 64 1.a even 1 1 trivial