Properties

Label 211.10.a.b
Level $211$
Weight $10$
Character orbit 211.a
Self dual yes
Analytic conductor $108.673$
Analytic rank $0$
Dimension $82$
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] = [211,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(108.672561431\)
Analytic rank: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 82 q + 48 q^{2} + 331 q^{3} + 22272 q^{4} + 11249 q^{5} + 9392 q^{6} + 6632 q^{7} + 36864 q^{8} + 630129 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 82 q + 48 q^{2} + 331 q^{3} + 22272 q^{4} + 11249 q^{5} + 9392 q^{6} + 6632 q^{7} + 36864 q^{8} + 630129 q^{9} + 73920 q^{10} + 255733 q^{11} + 262929 q^{12} + 291015 q^{13} + 599651 q^{14} + 151566 q^{15} + 6088388 q^{16} + 1450152 q^{17} + 1803076 q^{18} + 1658423 q^{19} + 5677520 q^{20} + 4437026 q^{21} + 402001 q^{22} + 4279922 q^{23} + 8660330 q^{24} + 36692027 q^{25} + 14084490 q^{26} + 11692912 q^{27} + 1686063 q^{28} + 28121309 q^{29} + 22795315 q^{30} + 8118600 q^{31} + 13477081 q^{32} + 17174434 q^{33} + 19372534 q^{34} + 37542780 q^{35} + 215622860 q^{36} + 50413779 q^{37} + 58031367 q^{38} + 23234242 q^{39} + 58731852 q^{40} + 103434140 q^{41} + 15334009 q^{42} + 34281571 q^{43} + 183086279 q^{44} + 235979107 q^{45} + 149265183 q^{46} + 93000602 q^{47} + 114303721 q^{48} + 570480578 q^{49} + 222019553 q^{50} + 77929810 q^{51} + 79999859 q^{52} + 407531048 q^{53} + 1321436783 q^{54} + 449488153 q^{55} + 756123028 q^{56} + 335002024 q^{57} + 481532629 q^{58} + 570753278 q^{59} - 244569748 q^{60} + 694734563 q^{61} - 73212918 q^{62} - 540500692 q^{63} + 194355770 q^{64} + 53860661 q^{65} - 737702157 q^{66} - 162897225 q^{67} - 1298035011 q^{68} + 602016364 q^{69} - 3064991165 q^{70} + 323986036 q^{71} - 1711822491 q^{72} - 1077391483 q^{73} - 683958107 q^{74} - 1166367949 q^{75} - 478318455 q^{76} + 1584572500 q^{77} - 5134824150 q^{78} + 342296112 q^{79} + 1441041507 q^{80} + 5540929446 q^{81} - 1240907573 q^{82} + 1202053700 q^{83} - 1781018244 q^{84} + 312682194 q^{85} - 305650461 q^{86} + 1589558614 q^{87} + 1034360734 q^{88} + 3298871808 q^{89} + 1695581030 q^{90} + 1989076496 q^{91} + 3997408364 q^{92} + 2435977176 q^{93} + 2258199495 q^{94} + 4724645555 q^{95} + 9729772859 q^{96} + 2626833632 q^{97} + 7407181902 q^{98} + 6900390127 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −42.9522 −272.921 1332.89 2616.34 11722.6 −2274.16 −35259.0 54803.0 −112377.
1.2 −42.7495 173.421 1315.52 −921.781 −7413.66 −9584.90 −34350.2 10391.8 39405.7
1.3 −41.5078 198.276 1210.90 −1651.51 −8229.98 −7477.50 −29009.7 19630.2 68550.5
1.4 −41.4499 −168.991 1206.10 −173.857 7004.67 1282.71 −28770.2 8875.03 7206.38
1.5 −41.2529 209.636 1189.80 −331.195 −8648.07 2562.12 −27961.2 24264.1 13662.8
1.6 −40.7005 −229.764 1144.53 −2001.03 9351.50 −7882.09 −25744.4 33108.3 81443.0
1.7 −39.1065 −17.2889 1017.32 2451.89 676.107 9796.00 −19761.4 −19384.1 −95884.8
1.8 −38.7497 22.9806 989.541 −1153.36 −890.491 −1095.21 −18504.6 −19154.9 44692.5
1.9 −37.7791 61.3514 915.258 −812.282 −2317.80 9612.97 −15234.7 −15919.0 30687.2
1.10 −37.7547 255.082 913.417 2014.92 −9630.55 −1677.30 −15155.4 45383.9 −76072.6
1.11 −37.6864 −151.403 908.263 1470.80 5705.84 10788.4 −14933.7 3239.97 −55429.1
1.12 −37.1794 −114.877 870.305 583.545 4271.07 4499.03 −13321.6 −6486.18 −21695.8
1.13 −35.7948 125.776 769.270 −327.580 −4502.12 1820.37 −9208.93 −3863.47 11725.7
1.14 −32.1640 −36.2624 522.526 −2127.99 1166.35 −6745.62 −338.551 −18368.0 68444.9
1.15 −31.7734 −146.785 497.551 1486.83 4663.85 −4360.64 459.090 1862.72 −47241.8
1.16 −30.1558 −183.098 397.372 −2217.56 5521.45 −821.704 3456.69 13841.7 66872.4
1.17 −29.6607 −146.482 367.760 2338.75 4344.77 −2788.08 4278.27 1773.98 −69369.1
1.18 −29.1507 135.025 337.761 819.543 −3936.06 −12241.7 5079.18 −1451.32 −23890.2
1.19 −28.5483 −27.9042 303.004 1083.20 796.615 −5365.56 5966.48 −18904.4 −30923.6
1.20 −25.8744 201.470 157.486 2132.18 −5212.93 11367.3 9172.85 20907.2 −55169.0
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
Significant digits:
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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 211.10.a.b 82
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.10.a.b 82 1.a even 1 1 trivial