Properties

Label 23.8.c.a
Level $23$
Weight $8$
Character orbit 23.c
Analytic conductor $7.185$
Analytic rank $0$
Dimension $130$
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] = [23,8,Mod(2,23)]
 
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([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.2");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 23.c (of order \(11\), 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: \(7.18485558613\)
Analytic rank: \(0\)
Dimension: \(130\)
Relative dimension: \(13\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 130 q + 5 q^{2} + 17 q^{3} - 907 q^{4} - 399 q^{5} + 1196 q^{6} - 301 q^{7} + 2764 q^{8} - 11694 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 130 q + 5 q^{2} + 17 q^{3} - 907 q^{4} - 399 q^{5} + 1196 q^{6} - 301 q^{7} + 2764 q^{8} - 11694 q^{9} - 8253 q^{10} - 6281 q^{11} + 5268 q^{12} - 211 q^{13} - 10707 q^{14} + 33520 q^{15} + 31333 q^{16} - 47386 q^{17} - 194664 q^{18} - 25106 q^{19} - 110631 q^{20} + 196134 q^{21} + 437326 q^{22} + 236844 q^{23} + 381670 q^{24} - 296504 q^{25} - 372672 q^{26} - 721672 q^{27} - 788613 q^{28} + 337142 q^{29} + 1636599 q^{30} + 406606 q^{31} + 1420941 q^{32} - 1342768 q^{33} - 1323543 q^{34} + 1116223 q^{35} + 1269127 q^{36} + 1371401 q^{37} - 1405652 q^{38} - 3603575 q^{39} - 6885721 q^{40} - 1101077 q^{41} + 1599107 q^{42} + 574365 q^{43} + 4080374 q^{44} + 4316738 q^{45} + 7663133 q^{46} + 2731014 q^{47} + 933950 q^{48} - 1518106 q^{49} - 11451638 q^{50} - 8958977 q^{51} - 12817356 q^{52} - 3443395 q^{53} - 3116538 q^{54} + 7064567 q^{55} + 34080364 q^{56} + 13743720 q^{57} + 15924553 q^{58} - 5766789 q^{59} - 20975917 q^{60} - 8406297 q^{61} - 13356080 q^{62} - 11569690 q^{63} - 12561350 q^{64} + 8471546 q^{65} + 1049638 q^{66} + 4319109 q^{67} + 34088718 q^{68} + 17829810 q^{69} + 12244822 q^{70} + 8161534 q^{71} + 9858613 q^{72} - 6499685 q^{73} - 57676070 q^{74} - 41388544 q^{75} - 19806630 q^{76} + 17878346 q^{77} + 66490432 q^{78} + 12522417 q^{79} + 6099788 q^{80} - 21944640 q^{81} - 23663206 q^{82} + 5522262 q^{83} - 68134179 q^{84} - 62098631 q^{85} - 64031711 q^{86} - 16846735 q^{87} + 47287281 q^{88} + 23861509 q^{89} + 98245322 q^{90} + 60585012 q^{91} + 76780054 q^{92} + 95931468 q^{93} + 40265547 q^{94} - 3907254 q^{95} - 68953299 q^{96} - 15751494 q^{97} - 28065823 q^{98} - 23732378 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −18.0654 11.6099i −7.91347 55.0394i 138.396 + 303.044i −448.477 131.685i −496.044 + 1086.19i −507.891 + 586.137i 626.966 4360.65i −868.303 + 254.957i 6573.08 + 7585.74i
2.2 −15.1360 9.72735i 9.41538 + 65.4854i 81.3055 + 178.034i −34.9931 10.2749i 494.488 1082.78i 215.101 248.240i 173.405 1206.06i −2101.28 + 616.990i 429.710 + 495.912i
2.3 −14.3153 9.19986i −4.38914 30.5271i 67.1162 + 146.964i 504.542 + 148.147i −218.014 + 477.383i 166.777 192.471i 81.2827 565.334i 1185.77 348.173i −5859.71 6762.47i
2.4 −9.64791 6.20034i 2.52225 + 17.5426i 1.46493 + 3.20774i −111.583 32.7636i 84.4358 184.889i −461.777 + 532.919i −203.158 + 1412.99i 1797.03 527.655i 873.395 + 1007.95i
2.5 −6.77816 4.35606i −7.10944 49.4473i −26.2049 57.3807i −177.765 52.1965i −167.206 + 366.131i 653.761 754.481i −219.106 + 1523.91i −296.076 + 86.9357i 977.549 + 1128.15i
2.6 −1.43748 0.923809i 6.16653 + 42.8892i −51.9602 113.777i −101.289 29.7411i 30.7572 67.3488i 679.427 784.101i −61.5435 + 428.044i 296.957 87.1944i 118.125 + 136.324i
2.7 −0.160791 0.103334i 9.57404 + 66.5889i −53.1579 116.400i 401.331 + 117.841i 5.34148 11.6962i −786.238 + 907.366i −6.96246 + 48.4250i −2244.01 + 658.900i −52.3533 60.4189i
2.8 −0.0532626 0.0342298i −10.0505 69.9027i −53.1715 116.429i 133.885 + 39.3123i −1.85744 + 4.06723i −916.475 + 1057.67i −2.30664 + 16.0430i −2686.96 + 788.963i −5.78544 6.67675i
2.9 7.29123 + 4.68579i 2.75598 + 19.1683i −21.9677 48.1025i −520.166 152.735i −69.7240 + 152.674i −684.009 + 789.388i 223.109 1551.76i 1738.58 510.494i −3076.97 3551.01i
2.10 7.91581 + 5.08718i −2.63237 18.3086i −16.3925 35.8946i 233.598 + 68.5907i 72.3016 158.318i 430.086 496.346i 224.249 1559.69i 1770.14 519.759i 1500.19 + 1731.31i
2.11 13.5156 + 8.68593i 11.4363 + 79.5415i 54.0522 + 118.358i −29.6143 8.69556i −536.323 + 1174.38i 538.265 621.191i −4.83843 + 33.6520i −4097.64 + 1203.18i −324.726 374.754i
2.12 13.7906 + 8.86268i −11.8811 82.6349i 58.4604 + 128.010i −324.875 95.3920i 568.519 1244.88i 482.425 556.748i −29.6934 + 206.522i −4588.96 + 1347.44i −3634.80 4194.78i
2.13 16.7731 + 10.7794i −0.358102 2.49065i 111.969 + 245.177i 118.801 + 34.8831i 20.8414 45.6362i −507.110 + 585.236i −401.607 + 2793.23i 2092.34 614.365i 1616.64 + 1865.71i
3.1 −2.99956 20.8624i −29.5319 + 64.6658i −303.428 + 89.0945i 64.3692 74.2860i 1437.67 + 422.138i 967.886 622.023i 1648.15 + 3608.95i −1877.36 2166.59i −1742.86 1120.07i
3.2 −2.78932 19.4002i 32.2381 70.5916i −245.771 + 72.1648i −273.530 + 315.670i −1459.41 428.522i 1047.63 673.270i 1043.37 + 2284.66i −2511.70 2898.66i 6887.02 + 4426.02i
3.3 −2.33422 16.2349i −1.38221 + 3.02662i −135.307 + 39.7298i −80.9070 + 93.3716i 52.3631 + 15.3752i −1369.85 + 880.347i 88.7099 + 194.247i 1424.93 + 1644.46i 1704.73 + 1095.56i
3.4 −2.00769 13.9638i 16.2316 35.5423i −68.1408 + 20.0079i 299.281 345.388i −528.892 155.297i 217.245 139.615i −333.941 731.228i 432.393 + 499.008i −5423.78 3485.65i
3.5 −1.38091 9.60444i −13.2015 + 28.9072i 32.4767 9.53603i −1.45769 + 1.68227i 295.868 + 86.8746i 778.129 500.073i −652.385 1428.52i 770.833 + 889.589i 18.1702 + 11.6773i
3.6 −0.505895 3.51858i −37.0467 + 81.1210i 110.691 32.5017i 33.4805 38.6385i 304.172 + 89.3130i −1023.97 + 658.067i −359.375 786.922i −3775.97 4357.70i −152.890 98.2566i
3.7 −0.266253 1.85183i −1.93441 + 4.23577i 119.457 35.0757i −325.785 + 375.976i 8.35896 + 2.45441i 537.138 345.198i −196.240 429.705i 1417.98 + 1636.44i 782.983 + 503.193i
See next 80 embeddings (of 130 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.8.c.a 130
23.c even 11 1 inner 23.8.c.a 130
23.c even 11 1 529.8.a.k 65
23.d odd 22 1 529.8.a.j 65
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.8.c.a 130 1.a even 1 1 trivial
23.8.c.a 130 23.c even 11 1 inner
529.8.a.j 65 23.d odd 22 1
529.8.a.k 65 23.c even 11 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(23, [\chi])\).