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Properties

Label 211.8.a.a
Level $211$
Weight $8$
Character orbit 211.a
Self dual yes
Analytic conductor $65.913$
Analytic rank $1$
Dimension $58$
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: \(1\)
Dimension: \(58\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 58 q - 24 q^{2} - 122 q^{3} + 3392 q^{4} - 2251 q^{5} - 1480 q^{6} - 501 q^{7} - 4608 q^{8} + 30972 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 58 q - 24 q^{2} - 122 q^{3} + 3392 q^{4} - 2251 q^{5} - 1480 q^{6} - 501 q^{7} - 4608 q^{8} + 30972 q^{9} - 4320 q^{10} - 25217 q^{11} - 27207 q^{12} - 25476 q^{13} - 59557 q^{14} - 20688 q^{15} + 192036 q^{16} - 77019 q^{17} - 50132 q^{18} - 68367 q^{19} - 265960 q^{20} - 257740 q^{21} - 107815 q^{22} - 112543 q^{23} - 300670 q^{24} + 638791 q^{25} - 431598 q^{26} - 506588 q^{27} + 107919 q^{28} - 836722 q^{29} - 420797 q^{30} - 457318 q^{31} - 661103 q^{32} - 86423 q^{33} - 747050 q^{34} - 1189131 q^{35} + 788924 q^{36} - 846567 q^{37} - 2055561 q^{38} - 734165 q^{39} - 1116732 q^{40} - 2803996 q^{41} - 407111 q^{42} - 958195 q^{43} - 5305417 q^{44} - 6320090 q^{45} - 2728673 q^{46} - 2736919 q^{47} - 4147295 q^{48} + 3577775 q^{49} - 3861031 q^{50} - 2633876 q^{51} - 3747037 q^{52} - 3324265 q^{53} + 7822259 q^{54} + 2422957 q^{55} - 5263400 q^{56} + 1256083 q^{57} + 12621829 q^{58} - 3266953 q^{59} + 8441798 q^{60} - 6653955 q^{61} - 1729842 q^{62} + 2393804 q^{63} + 8112054 q^{64} - 5786386 q^{65} - 19432749 q^{66} - 4160923 q^{67} - 23371017 q^{68} - 23916119 q^{69} - 14938789 q^{70} - 10711031 q^{71} - 41368227 q^{72} - 21883123 q^{73} - 28060823 q^{74} - 21899053 q^{75} - 28877239 q^{76} - 36664508 q^{77} - 76772526 q^{78} - 24934567 q^{79} - 73072485 q^{80} - 11390634 q^{81} - 52358257 q^{82} - 47107171 q^{83} - 113923524 q^{84} - 41660796 q^{85} - 48122889 q^{86} - 27590480 q^{87} - 83560414 q^{88} - 49852146 q^{89} - 103538548 q^{90} - 52934460 q^{91} - 93764698 q^{92} - 54761439 q^{93} - 76668825 q^{94} - 41598916 q^{95} - 160466701 q^{96} - 49951344 q^{97} - 132887208 q^{98} - 102274817 q^{99}+O(q^{100}) \) 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 −21.5266 26.1610 335.393 −120.558 −563.155 1580.68 −4464.46 −1502.60 2595.20
1.2 −21.4843 39.5866 333.576 −306.874 −850.492 −601.131 −4416.67 −619.900 6592.98
1.3 −20.9759 33.5717 311.988 192.588 −704.195 −1014.33 −3859.30 −1059.94 −4039.70
1.4 −20.8425 −43.4253 306.411 −361.236 905.092 −867.676 −3718.55 −301.247 7529.07
1.5 −20.4468 73.9328 290.071 272.519 −1511.69 −720.952 −3313.84 3279.06 −5572.14
1.6 −20.2510 −90.7013 282.102 −355.455 1836.79 1666.02 −3120.71 6039.73 7198.31
1.7 −19.0517 −73.6263 234.967 260.754 1402.71 784.351 −2037.90 3233.84 −4967.81
1.8 −16.9383 35.6354 158.905 105.535 −603.602 984.054 −523.484 −917.121 −1787.58
1.9 −16.4659 3.02646 143.125 −185.024 −49.8334 −1667.87 −249.051 −2177.84 3046.59
1.10 −16.2171 −72.3797 134.993 157.477 1173.79 −527.385 −113.407 3051.82 −2553.81
1.11 −15.6493 −36.7170 116.900 −26.5289 574.595 1014.36 173.700 −838.862 415.159
1.12 −15.4580 47.7595 110.950 −374.247 −738.267 1338.97 263.554 93.9685 5785.11
1.13 −14.3935 −1.63422 79.1715 183.375 23.5220 −837.712 702.810 −2184.33 −2639.40
1.14 −13.9677 65.3553 67.0967 178.052 −912.864 −592.833 850.679 2084.32 −2486.98
1.15 −12.1211 −73.3849 18.9221 460.755 889.509 869.077 1322.15 3198.35 −5584.87
1.16 −12.1207 −20.1231 18.9103 170.593 243.905 1605.81 1322.24 −1782.06 −2067.70
1.17 −11.4463 −5.23583 3.01790 −527.801 59.9309 748.509 1430.58 −2159.59 6041.37
1.18 −11.3655 −52.0004 1.17395 −344.368 591.009 −719.922 1441.44 517.038 3913.90
1.19 −9.62210 80.3678 −35.4152 −65.4819 −773.308 1036.41 1572.40 4271.99 630.074
1.20 −9.58637 39.3145 −36.1015 470.050 −376.883 333.993 1573.14 −641.373 −4506.07
See all 58 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.58
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.a ✓ 58
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.8.a.a ✓ 58 1.a even 1 1 trivial