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Properties

Label 89.8.a.b
Level $89$
Weight $8$
Character orbit 89.a
Self dual yes
Analytic conductor $27.802$
Analytic rank $0$
Dimension $29$
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] = [89,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(27.8022672681\)
Analytic rank: \(0\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q + 23 q^{2} + 27 q^{3} + 2239 q^{4} + 251 q^{5} + 831 q^{6} + 3772 q^{7} + 4479 q^{8} + 25670 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q + 23 q^{2} + 27 q^{3} + 2239 q^{4} + 251 q^{5} + 831 q^{6} + 3772 q^{7} + 4479 q^{8} + 25670 q^{9} + 5779 q^{10} + 11614 q^{11} - 2553 q^{12} + 16586 q^{13} + 7348 q^{14} + 32229 q^{15} + 176759 q^{16} + 38151 q^{17} + 82096 q^{18} + 127439 q^{19} + 9703 q^{20} + 85244 q^{21} + 133194 q^{22} + 33051 q^{23} - 432065 q^{24} + 495372 q^{25} - 217570 q^{26} + 152145 q^{27} + 528292 q^{28} + 19424 q^{29} + 1005733 q^{30} + 942683 q^{31} + 2380135 q^{32} + 1041146 q^{33} + 1389375 q^{34} + 1344732 q^{35} + 5084180 q^{36} + 1069166 q^{37} + 3154415 q^{38} + 2630598 q^{39} + 3455519 q^{40} + 923816 q^{41} + 7225192 q^{42} + 1975509 q^{43} + 4593402 q^{44} + 2613194 q^{45} + 6086675 q^{46} + 4361836 q^{47} + 2148383 q^{48} + 5161213 q^{49} + 6535478 q^{50} + 2160721 q^{51} + 5154206 q^{52} + 459121 q^{53} + 2263141 q^{54} + 5164426 q^{55} + 2427508 q^{56} + 3393385 q^{57} + 4742620 q^{58} + 283300 q^{59} + 3225141 q^{60} + 3145222 q^{61} - 3938617 q^{62} + 3999384 q^{63} + 15218383 q^{64} + 925486 q^{65} - 8402270 q^{66} + 8614844 q^{67} + 191091 q^{68} + 1438013 q^{69} + 922608 q^{70} + 4839286 q^{71} + 1765028 q^{72} + 2227971 q^{73} - 7457134 q^{74} - 5176340 q^{75} + 12509119 q^{76} - 12144824 q^{77} - 16841434 q^{78} + 26662698 q^{79} - 22414617 q^{80} + 24628949 q^{81} - 34680940 q^{82} - 3071644 q^{83} - 10555032 q^{84} + 21461305 q^{85} - 31279827 q^{86} - 10009240 q^{87} - 8314038 q^{88} - 20444101 q^{89} - 109231166 q^{90} + 1232888 q^{91} - 71974109 q^{92} - 78031563 q^{93} - 9159228 q^{94} - 45648119 q^{95} - 146392513 q^{96} - 57428667 q^{97} - 77042877 q^{98} - 3276068 q^{99}+O(q^{100}) \) 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 −21.7170 53.1609 343.629 −311.033 −1154.50 −782.650 −4682.81 639.081 6754.71
1.2 −19.6784 −17.0808 259.238 −254.425 336.122 1654.50 −2582.54 −1895.25 5006.67
1.3 −19.1062 86.0921 237.046 275.774 −1644.89 377.483 −2083.45 5224.84 −5268.99
1.4 −17.5949 −90.1738 181.582 −18.1650 1586.60 1358.90 −942.767 5944.31 319.611
1.5 −17.4334 −71.6731 175.925 50.7635 1249.51 −885.389 −835.501 2950.04 −884.984
1.6 −15.5597 10.5396 114.105 −322.997 −163.993 −1090.62 216.202 −2075.92 5025.75
1.7 −14.7319 42.2675 89.0301 132.115 −622.682 −1335.18 574.102 −400.462 −1946.31
1.8 −13.0544 −58.1176 42.4185 527.512 758.692 1100.02 1117.22 1190.65 −6886.38
1.9 −12.0427 −15.0279 17.0265 −68.1804 180.976 868.820 1336.42 −1961.16 821.075
1.10 −9.20983 −50.5633 −43.1790 212.030 465.680 126.779 1576.53 369.647 −1952.76
1.11 −7.96962 51.6825 −64.4852 273.127 −411.889 1187.57 1534.03 484.076 −2176.72
1.12 −2.04017 −5.49419 −123.838 −237.588 11.2091 −1487.26 513.791 −2156.81 484.720
1.13 −1.47282 53.7712 −125.831 −286.390 −79.1953 −682.425 373.847 704.343 421.801
1.14 −0.154918 6.52435 −127.976 −488.173 −1.01074 1473.03 39.6553 −2144.43 75.6268
1.15 2.80913 −42.6064 −120.109 484.594 −119.687 211.252 −696.970 −371.697 1361.29
1.16 3.29582 81.0331 −117.138 122.494 267.070 1164.92 −807.928 4379.36 403.716
1.17 5.29268 −25.4517 −99.9876 201.067 −134.707 −912.079 −1206.66 −1539.21 1064.19
1.18 7.84990 72.7093 −66.3791 442.566 570.761 −1121.58 −1525.86 3099.65 3474.10
1.19 8.53514 −71.1035 −55.1513 −207.527 −606.878 −590.234 −1563.22 2868.70 −1771.27
1.20 8.76826 −3.38039 −51.1176 −374.910 −29.6401 458.308 −1570.55 −2175.57 −3287.31
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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 89.8.a.b ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.8.a.b ✓ 29 1.a even 1 1 trivial