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Properties

Label 89.10.g.a
Level $89$
Weight $10$
Character orbit 89.g
Analytic conductor $45.838$
Analytic rank $0$
Dimension $1340$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [89,10,Mod(5,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([35]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.5");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 89.g (of order \(44\), degree \(20\), minimal)

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: no
Analytic conductor: \(45.8381894186\)
Analytic rank: \(0\)
Dimension: \(1340\)
Relative dimension: \(67\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1340 q - 50 q^{2} + 274 q^{3} - 34322 q^{4} - 22 q^{5} - 500 q^{6} - 20 q^{7} - 22546 q^{8} - 221430 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1340 q - 50 q^{2} + 274 q^{3} - 34322 q^{4} - 22 q^{5} - 500 q^{6} - 20 q^{7} - 22546 q^{8} - 221430 q^{9} - 22 q^{10} + 230958 q^{11} + 243388 q^{12} + 223222 q^{13} - 3446048 q^{14} + 56154 q^{15} - 7705378 q^{16} - 22 q^{17} - 22 q^{18} + 496746 q^{19} - 22 q^{20} - 22 q^{21} - 5188434 q^{22} + 390640 q^{23} - 8135738 q^{24} + 56958084 q^{25} + 11246538 q^{26} + 30605092 q^{27} + 9375136 q^{28} - 3112586 q^{29} + 154269128 q^{30} + 3904884 q^{31} - 13629458 q^{32} + 9068754 q^{33} + 76718954 q^{35} - 22 q^{36} + 1906754 q^{37} + 144809570 q^{38} + 5623606 q^{39} + 266996488 q^{40} + 62676284 q^{41} - 22 q^{42} - 544303706 q^{43} + 163447790 q^{44} + 27352974 q^{45} + 93297052 q^{46} - 22 q^{47} - 915736054 q^{48} - 22 q^{49} - 23618850 q^{50} + 41085310 q^{51} + 331386258 q^{52} - 932747838 q^{53} - 8285342 q^{54} + 1518908716 q^{56} - 306235398 q^{57} - 476839934 q^{58} - 1152970482 q^{59} - 135449894 q^{60} - 749200466 q^{61} + 751394056 q^{62} - 21688200 q^{63} - 5537022434 q^{64} + 592075454 q^{65} - 550399144 q^{66} - 346446082 q^{67} + 11242 q^{68} + 2808628658 q^{69} - 147299144 q^{70} - 22 q^{71} + 8081798956 q^{72} + 1052214026 q^{73} + 2383177304 q^{74} - 3102927874 q^{75} - 5386316008 q^{76} - 841706550 q^{77} - 742269014 q^{78} - 130079598 q^{79} - 2168455190 q^{80} + 11905053080 q^{81} - 4326757736 q^{82} + 606324242 q^{83} + 6250359786 q^{84} - 6486556806 q^{85} - 989388660 q^{86} + 1854106510 q^{87} - 14048861240 q^{88} - 5063126928 q^{89} + 560413428 q^{90} - 2232334826 q^{91} + 5388274978 q^{92} + 7727885374 q^{93} + 6880151146 q^{94} - 7800533574 q^{95} + 21062479584 q^{96} + 2296519618 q^{97} - 13040402012 q^{98} - 6722268762 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −29.2600 + 33.7679i −181.160 + 135.615i −211.254 − 1469.31i 837.895 − 1303.79i 721.329 − 10085.5i −1035.09 − 4758.23i 36551.5 + 23490.2i 8882.33 − 30250.5i 19509.4 + 66442.8i
5.2 −29.0016 + 33.4696i −110.126 + 82.4392i −206.258 − 1434.55i −1210.68 + 1883.86i 434.617 − 6076.74i 2647.32 + 12169.6i 34920.5 + 22442.0i −213.849 + 728.303i −27940.3 − 95156.0i
5.3 −28.3982 + 32.7732i 98.7898 − 73.9531i −194.764 − 1354.61i 1296.89 − 2018.00i −381.767 + 5337.80i 1765.54 + 8116.04i 31247.7 + 20081.6i −1254.98 + 4274.06i 29307.1 + 99810.9i
5.4 −27.7555 + 32.0315i 63.7256 − 47.7044i −182.787 − 1271.31i 70.5208 − 109.733i −240.690 + 3365.29i −1323.90 − 6085.88i 27539.8 + 17698.8i −3760.10 + 12805.7i 1557.56 + 5304.57i
5.5 −26.3802 + 30.4443i 188.464 − 141.082i −158.079 − 1099.47i −745.715 + 1160.36i −676.551 + 9459.42i 465.193 + 2138.46i 20291.6 + 13040.6i 10069.0 − 34292.0i −15654.2 − 53313.2i
5.6 −25.0569 + 28.9172i 65.6751 − 49.1638i −135.491 − 942.357i −730.402 + 1136.53i −223.935 + 3131.03i −739.980 − 3401.63i 14164.6 + 9103.04i −3649.20 + 12428.0i −14563.6 − 49599.0i
5.7 −24.6181 + 28.4109i −89.3340 + 66.8746i −128.258 − 892.057i −189.103 + 294.249i 299.273 − 4184.39i −962.631 − 4425.14i 12309.5 + 7910.82i −2036.99 + 6937.34i −3704.52 − 12616.4i
5.8 −23.6748 + 27.3221i −24.8614 + 18.6110i −113.139 − 786.900i 148.864 − 231.638i 80.0954 − 1119.88i 1415.04 + 6504.82i 8606.73 + 5531.21i −5273.62 + 17960.3i 2804.50 + 9551.26i
5.9 −23.5096 + 27.1315i 222.061 − 166.233i −110.553 − 768.910i 882.148 − 1372.65i −710.416 + 9932.91i −1684.61 − 7744.01i 7997.74 + 5139.84i 16132.4 − 54942.0i 16503.1 + 56204.4i
5.10 −23.1506 + 26.7172i −155.595 + 116.477i −104.993 − 730.245i −1261.43 + 1962.83i 490.176 − 6853.55i −2053.84 − 9441.35i 6713.92 + 4314.77i 5097.56 − 17360.7i −23238.3 − 79142.3i
5.11 −22.2988 + 25.7342i −124.832 + 93.4482i −92.1463 − 640.891i 826.882 − 1286.65i 378.794 − 5296.23i 945.796 + 4347.75i 3880.97 + 2494.15i 1305.17 − 4444.99i 14672.5 + 49969.9i
5.12 −21.2709 + 24.5479i −47.3422 + 35.4399i −77.2842 − 537.523i 1412.73 − 2198.26i 137.034 − 1915.99i −1669.37 − 7673.99i 848.473 + 545.280i −4560.05 + 15530.1i 23912.5 + 81438.5i
5.13 −20.2442 + 23.3630i −209.574 + 156.885i −63.1388 − 439.140i −9.20156 + 14.3179i 577.340 − 8072.27i 1043.06 + 4794.85i −1777.38 − 1142.25i 13762.9 − 46872.1i −148.231 − 504.830i
5.14 −19.8480 + 22.9058i 111.025 − 83.1122i −57.8674 − 402.477i 530.546 − 825.546i −299.869 + 4192.72i 1674.19 + 7696.12i −2687.01 − 1726.84i −126.451 + 430.653i 8379.50 + 28538.0i
5.15 −19.6704 + 22.7009i 129.443 − 96.8997i −55.5389 − 386.281i −1164.31 + 1811.70i −346.487 + 4844.52i 1729.34 + 7949.67i −3076.43 − 1977.10i 1820.56 − 6200.27i −18224.8 − 62067.8i
5.16 −17.9207 + 20.6816i 90.6053 − 67.8263i −33.7119 − 234.471i 901.628 − 1402.96i −220.956 + 3089.36i −1073.17 − 4933.27i −6333.62 − 4070.37i −1936.42 + 6594.85i 12857.7 + 43789.2i
5.17 −17.1191 + 19.7565i 165.967 − 124.242i −24.3905 − 169.640i 393.328 − 612.030i −386.633 + 5405.84i 461.093 + 2119.61i −7490.73 − 4814.00i 6563.83 − 22354.3i 5358.16 + 18248.2i
5.18 −16.7792 + 19.3642i −43.2012 + 32.3400i −20.5664 − 143.043i −1210.15 + 1883.04i 98.6421 − 1379.20i 426.397 + 1960.11i −7921.18 − 5090.64i −4724.87 + 16091.4i −16158.1 − 55029.5i
5.19 −14.7816 + 17.0589i 127.426 − 95.3896i 0.355601 + 2.47326i −901.276 + 1402.41i −256.315 + 3583.75i −2229.33 − 10248.1i −9769.76 − 6278.64i 1592.76 − 5424.44i −10601.3 − 36104.7i
5.20 −13.8407 + 15.9730i −151.786 + 113.625i 9.29281 + 64.6329i −719.656 + 1119.81i 285.881 − 3997.13i 23.9153 + 109.937i −10264.5 − 6596.56i 4582.85 − 15607.7i −7926.15 − 26994.0i
See next 80 embeddings (of 1340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.67
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.g even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.10.g.a ✓ 1340
89.g even 44 1 inner 89.10.g.a ✓ 1340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.10.g.a ✓ 1340 1.a even 1 1 trivial
89.10.g.a ✓ 1340 89.g even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(89, [\chi])\).