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Properties

Label 23.9.d.a
Level $23$
Weight $9$
Character orbit 23.d
Analytic conductor $9.370$
Analytic rank $0$
Dimension $150$
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] = [23,9,Mod(5,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.5");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 23.d (of order \(22\), degree \(10\), 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: \(9.36970803141\)
Analytic rank: \(0\)
Dimension: \(150\)
Relative dimension: \(15\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 150 q - 3 q^{2} + 61 q^{3} - 1139 q^{4} - 11 q^{5} + 1666 q^{6} - 11 q^{7} + 2786 q^{8} - 25586 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 150 q - 3 q^{2} + 61 q^{3} - 1139 q^{4} - 11 q^{5} + 1666 q^{6} - 11 q^{7} + 2786 q^{8} - 25586 q^{9} - 11 q^{10} - 11 q^{11} + 53770 q^{12} - 33707 q^{13} - 11 q^{14} - 130218 q^{15} - 321563 q^{16} + 246004 q^{17} + 731730 q^{18} - 113894 q^{19} - 1571339 q^{20} - 1081058 q^{21} + 752336 q^{23} + 5136266 q^{24} + 2176012 q^{25} - 1126678 q^{26} - 2369906 q^{27} - 6167051 q^{28} - 4293000 q^{29} + 2511861 q^{30} + 5095318 q^{31} + 6052309 q^{32} - 5636796 q^{33} + 6374511 q^{34} - 5052083 q^{35} - 8703527 q^{36} + 9285749 q^{37} + 22442794 q^{38} + 8881135 q^{39} + 4234989 q^{40} - 10255539 q^{41} - 51900761 q^{42} - 21835627 q^{43} - 24763398 q^{44} + 28476045 q^{46} + 27380554 q^{47} + 59425082 q^{48} + 36123694 q^{49} + 49188504 q^{50} - 14740715 q^{51} - 128333848 q^{52} - 28488779 q^{53} - 66180294 q^{54} - 37618605 q^{55} - 99295328 q^{56} + 72285400 q^{57} + 191002739 q^{58} + 71014131 q^{59} + 67737659 q^{60} + 47508681 q^{61} + 23639570 q^{62} - 62854836 q^{63} - 91968238 q^{64} - 173044652 q^{65} - 172762656 q^{66} - 25488397 q^{67} + 41597646 q^{69} + 159474810 q^{70} + 115616508 q^{71} + 444412039 q^{72} + 53981543 q^{73} + 329575444 q^{74} - 119484746 q^{75} - 610693402 q^{76} - 545148410 q^{77} - 381690432 q^{78} - 108193503 q^{79} + 214101118 q^{80} + 924833330 q^{81} + 276669012 q^{82} + 421863970 q^{83} + 585479675 q^{84} - 98759833 q^{85} - 1051470365 q^{86} - 689527625 q^{87} - 546014403 q^{88} - 331520123 q^{89} - 592624846 q^{90} + 277395922 q^{92} + 518189756 q^{93} + 801676925 q^{94} + 1007053762 q^{95} + 1519625749 q^{96} - 137375018 q^{97} - 901313259 q^{98} - 653311670 q^{99}+O(q^{100}) \) 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 −26.8597 − 7.88673i −39.1259 + 45.1537i 443.883 + 285.266i −356.855 − 51.3080i 1407.02 − 904.240i 3710.07 − 1694.33i −4979.78 − 5746.98i 425.708 + 2960.87i 9180.37 + 4192.54i
5.2 −26.6935 − 7.83792i 56.6059 − 65.3267i 435.749 + 280.039i −19.0816 − 2.74352i −2023.03 + 1300.13i −2319.06 + 1059.08i −4772.81 − 5508.12i −129.620 − 901.529i 487.851 + 222.794i
5.3 −20.2860 − 5.95652i −80.8428 + 93.2976i 160.682 + 103.264i 1048.25 + 150.716i 2195.71 − 1411.10i −1321.02 + 603.288i 899.905 + 1038.55i −1235.15 − 8590.66i −20367.2 − 9301.38i
5.4 −16.7055 − 4.90519i 33.6170 − 38.7960i 39.6531 + 25.4835i 534.871 + 76.9029i −751.891 + 483.211i 1168.35 − 533.566i 2381.40 + 2748.28i 558.695 + 3885.81i −8558.09 − 3908.35i
5.5 −16.6745 − 4.89607i −54.3590 + 62.7336i 38.7061 + 24.8749i −923.892 − 132.836i 1213.56 − 779.906i −3518.69 + 1606.93i 2389.78 + 2757.96i −46.8798 − 326.056i 14755.1 + 6738.41i
5.6 −9.65515 − 2.83501i 80.9814 − 93.4575i −130.176 − 83.6592i −850.544 − 122.290i −1046.84 + 672.763i 859.623 − 392.577i 2706.66 + 3123.65i −1242.59 − 8642.43i 7865.44 + 3592.02i
5.7 −2.59503 − 0.761970i −2.96866 + 3.42601i −209.207 − 134.449i 338.182 + 48.6232i 10.3143 − 6.62859i −351.949 + 160.730i 893.862 + 1031.57i 930.803 + 6473.88i −840.544 − 383.863i
5.8 −1.96500 − 0.576975i −61.6217 + 71.1152i −211.833 − 136.137i −476.731 − 68.5436i 162.118 − 104.187i 2906.57 − 1327.39i 681.031 + 785.952i −326.413 − 2270.25i 897.228 + 409.750i
5.9 6.39810 + 1.87865i 98.2319 − 113.366i −177.955 − 114.364i 1173.54 + 168.730i 841.473 − 540.782i −1889.18 + 862.757i −2041.61 − 2356.14i −2268.54 − 15778.1i 7191.45 + 3284.22i
5.10 10.3135 + 3.02831i −84.1048 + 97.0621i −118.164 − 75.9394i 446.885 + 64.2523i −1161.35 + 746.351i −578.409 + 264.150i −2790.70 − 3220.64i −1413.71 − 9832.55i 4414.35 + 2015.97i
5.11 12.2626 + 3.60063i 21.2081 − 24.4754i −77.9536 − 50.0977i −596.272 − 85.7310i 348.193 − 223.770i −2808.50 + 1282.60i −2918.08 − 3367.64i 784.464 + 5456.07i −7003.17 − 3198.24i
5.12 16.5496 + 4.85939i 54.5482 − 62.9520i 34.9133 + 22.4374i −206.246 − 29.6536i 1208.66 − 776.757i 4210.20 − 1922.74i −2422.80 − 2796.06i −53.7186 − 373.621i −3269.18 − 1492.98i
5.13 23.8349 + 6.99854i −15.4997 + 17.8876i 303.760 + 195.214i 720.078 + 103.532i −494.619 + 317.873i 100.488 − 45.8913i 1709.38 + 1972.73i 854.002 + 5939.72i 16438.4 + 7507.16i
5.14 25.1127 + 7.37375i −72.5620 + 83.7410i 360.914 + 231.945i −987.165 − 141.933i −2439.71 + 1567.91i 480.973 − 219.653i 2965.48 + 3422.34i −813.587 − 5658.62i −23743.8 − 10843.4i
5.15 29.9650 + 8.79851i 84.0770 − 97.0300i 605.125 + 388.890i −346.386 − 49.8028i 3373.09 − 2167.75i −2622.95 + 1197.86i 9475.38 + 10935.2i −1412.16 − 9821.77i −9941.27 − 4540.03i
7.1 −20.5698 + 23.7388i 71.9245 − 46.2231i −103.982 − 723.209i −119.814 + 54.7170i −382.191 + 2658.20i 193.778 + 659.948i 12542.3 + 8060.44i 311.022 − 681.043i 1165.62 − 3969.74i
7.2 −17.1859 + 19.8336i −95.7570 + 61.5393i −61.5837 − 428.324i 581.686 − 265.647i 425.126 − 2956.82i −196.209 − 668.227i 3901.73 + 2507.49i 2656.79 − 5817.55i −4728.07 + 16102.3i
7.3 −13.7654 + 15.8861i −34.5821 + 22.2245i −26.4500 − 183.964i −651.397 + 297.483i 122.974 − 855.305i −506.694 − 1725.64i −1240.39 − 797.153i −2023.55 + 4430.95i 4240.89 − 14443.2i
7.4 −12.5163 + 14.4445i 29.6696 − 19.0675i −15.5552 − 108.189i 43.3474 − 19.7961i −95.9315 + 667.218i 892.234 + 3038.67i −2358.73 − 1515.86i −2208.82 + 4836.64i −256.602 + 873.906i
7.5 −10.0601 + 11.6100i 104.308 − 67.0348i 2.84657 + 19.7983i 541.178 − 247.148i −271.079 + 1885.40i −754.805 − 2570.63i −3566.92 − 2292.32i 3661.00 − 8016.46i −2574.93 + 8769.41i
See next 80 embeddings (of 150 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.9.d.a ✓ 150
23.d odd 22 1 inner 23.9.d.a ✓ 150
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.9.d.a ✓ 150 1.a even 1 1 trivial
23.9.d.a ✓ 150 23.d odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(23, [\chi])\).