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Properties

Label 89.12.a.a
Level $89$
Weight $12$
Character orbit 89.a
Self dual yes
Analytic conductor $68.383$
Analytic rank $1$
Dimension $37$
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] = [89,12,Mod(1,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 89.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: \(68.3825430698\)
Analytic rank: \(1\)
Dimension: \(37\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 37 q - 41 q^{2} - 729 q^{3} + 34239 q^{4} - 3719 q^{5} - 42657 q^{6} - 168134 q^{7} - 283137 q^{8} + 1803548 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 37 q - 41 q^{2} - 729 q^{3} + 34239 q^{4} - 3719 q^{5} - 42657 q^{6} - 168134 q^{7} - 283137 q^{8} + 1803548 q^{9} - 638701 q^{10} - 1455690 q^{11} - 1074777 q^{12} - 1109256 q^{13} - 2831868 q^{14} - 6899811 q^{15} + 20104695 q^{16} - 7046331 q^{17} - 14347952 q^{18} - 66775245 q^{19} + 4966583 q^{20} - 32414992 q^{21} - 79198758 q^{22} - 27807413 q^{23} - 158572059 q^{24} + 191644914 q^{25} - 506367006 q^{26} - 419433381 q^{27} - 568175090 q^{28} - 275771042 q^{29} - 768839081 q^{30} - 621644087 q^{31} - 114182237 q^{32} - 85491884 q^{33} + 159064131 q^{34} - 56340248 q^{35} + 2918996836 q^{36} - 280059010 q^{37} + 2658541173 q^{38} - 321665382 q^{39} + 2132855427 q^{40} + 43798702 q^{41} + 7987058908 q^{42} - 3784969269 q^{43} + 1236331038 q^{44} + 2599674406 q^{45} - 155537669 q^{46} - 1218152448 q^{47} - 5666321563 q^{48} + 6611837277 q^{49} - 1842444918 q^{50} - 10452565851 q^{51} - 10873955118 q^{52} - 5467642953 q^{53} - 13046781789 q^{54} - 16395016914 q^{55} - 20440678380 q^{56} - 18633477525 q^{57} - 10839598600 q^{58} - 15261626702 q^{59} - 61900264533 q^{60} - 28977124060 q^{61} - 26758629133 q^{62} - 47777545764 q^{63} - 53868564737 q^{64} - 27826027108 q^{65} - 79869041208 q^{66} - 29033457956 q^{67} - 98745704089 q^{68} - 35750062409 q^{69} - 119210144482 q^{70} - 72930463494 q^{71} - 181707520440 q^{72} - 67429732259 q^{73} - 152326094504 q^{74} - 145540760966 q^{75} - 179725754317 q^{76} - 145448891168 q^{77} - 174315066058 q^{78} - 209695168646 q^{79} - 216685904481 q^{80} - 101944659451 q^{81} - 298923081838 q^{82} - 133346324222 q^{83} - 461798311660 q^{84} - 173892684889 q^{85} - 206979684555 q^{86} - 323322704832 q^{87} - 363810133806 q^{88} + 206610199613 q^{89} - 530758490862 q^{90} - 197719344544 q^{91} - 188344789795 q^{92} + 132687813403 q^{93} - 328948807856 q^{94} - 295289767767 q^{95} - 670121904025 q^{96} - 231539736157 q^{97} - 619336089361 q^{98} - 576952741608 q^{99}+O(q^{100}) \) 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 −86.4522 643.407 5425.99 −2042.07 −55624.0 −45988.9 −292035. 236826. 176541.
1.2 −81.0072 −745.832 4514.17 5224.27 60417.8 59014.9 −199777. 379119. −423203.
1.3 −78.0740 −116.820 4047.55 −9914.16 9120.61 −61296.3 −156113. −163500. 774039.
1.4 −76.4735 383.668 3800.20 −1042.77 −29340.4 −13045.7 −133997. −29945.9 79744.6
1.5 −75.6317 346.326 3672.16 −884.973 −26193.2 57232.9 −122838. −57205.4 66932.1
1.6 −73.6896 −638.658 3382.15 −5463.13 47062.4 −21013.8 −98313.0 230737. 402576.
1.7 −63.9749 −288.439 2044.79 6247.72 18452.9 35288.6 205.195 −93950.0 −399698.
1.8 −62.1783 633.337 1818.14 12382.2 −39379.8 −78209.7 14292.4 223969. −769901.
1.9 −52.6842 −122.291 727.626 −3320.10 6442.81 75890.1 69562.9 −162192. 174917.
1.10 −50.7409 −610.955 526.644 11340.3 31000.4 26687.1 77195.1 196119. −575416.
1.11 −45.0862 146.548 −15.2381 −7697.39 −6607.28 −63866.3 93023.5 −155671. 347046.
1.12 −41.9412 604.757 −288.932 3211.00 −25364.3 −23019.1 98013.8 188584. −134673.
1.13 −40.8310 −358.881 −380.831 8006.15 14653.5 −63481.0 99171.5 −48351.5 −326899.
1.14 −31.0147 496.746 −1086.09 6204.65 −15406.5 52107.6 97202.9 69609.9 −192436.
1.15 −22.3406 −311.772 −1548.90 −13246.8 6965.19 −17578.7 80357.0 −79945.1 295942.
1.16 −9.83581 311.005 −1951.26 −10872.6 −3058.98 55522.0 39335.9 −80422.9 106941.
1.17 −7.33452 −469.654 −1994.20 −2424.85 3444.69 −25054.1 29647.6 43428.1 17785.1
1.18 −1.75787 −148.132 −2044.91 12812.2 260.398 39808.1 7194.82 −155204. −22522.2
1.19 −1.59325 658.366 −2045.46 −5265.82 −1048.94 20705.7 6521.92 256299. 8389.79
1.20 4.74316 307.908 −2025.50 2296.28 1460.45 −56726.4 −19321.3 −82339.9 10891.6
See all 37 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.37
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(89\) \(-1\)

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 89.12.a.a ✓ 37
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.12.a.a ✓ 37 1.a even 1 1 trivial