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Properties

Label 69.8.e.a
Level $69$
Weight $8$
Character orbit 69.e
Analytic conductor $21.555$
Analytic rank $0$
Dimension $140$
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] = [69,8,Mod(4,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.e (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: \(21.5545667584\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9} + 1322 q^{10} + 7892 q^{11} - 19980 q^{12} + 39810 q^{13} - 26296 q^{14} + 135 q^{15} - 142164 q^{16} - 97081 q^{17} - 11664 q^{18} + 120905 q^{19} + 124755 q^{20} + 26406 q^{21} - 837266 q^{22} - 224688 q^{23} - 60264 q^{24} - 180624 q^{25} + 385108 q^{26} - 275562 q^{27} + 1121760 q^{28} - 375654 q^{29} + 35694 q^{30} + 505029 q^{31} - 661528 q^{32} - 360423 q^{33} - 1872759 q^{34} - 964610 q^{35} - 1269189 q^{36} - 28802 q^{37} - 1459868 q^{38} + 1074870 q^{39} + 5371901 q^{40} - 1557480 q^{41} + 883116 q^{42} + 1160786 q^{43} - 4521825 q^{44} - 2009124 q^{45} - 4123580 q^{46} + 811422 q^{47} - 1563111 q^{48} - 10960262 q^{49} + 16636695 q^{50} + 4413852 q^{51} - 4843791 q^{52} + 4291672 q^{53} - 98415 q^{54} + 2093539 q^{55} - 12548382 q^{56} - 4007907 q^{57} - 7414367 q^{58} + 2955424 q^{59} + 7289973 q^{60} + 4919292 q^{61} + 16083211 q^{62} + 712962 q^{63} - 1673596 q^{64} - 13246221 q^{65} + 7219449 q^{66} + 148565 q^{67} - 23039568 q^{68} - 657018 q^{69} + 12781104 q^{70} - 7247645 q^{71} - 600696 q^{72} + 32596055 q^{73} + 32550658 q^{74} - 10061874 q^{75} + 12150569 q^{76} - 43349150 q^{77} - 17201700 q^{78} + 4815410 q^{79} + 36529590 q^{80} - 7440174 q^{81} + 20862470 q^{82} + 35160800 q^{83} - 5346540 q^{84} + 48578363 q^{85} + 76972708 q^{86} + 1505088 q^{87} - 35912212 q^{88} + 1401264 q^{89} + 963738 q^{90} - 91970208 q^{91} - 74293537 q^{92} + 555606 q^{93} - 32520253 q^{94} + 79436789 q^{95} + 41519736 q^{96} + 18296816 q^{97} - 38956823 q^{98} - 9731421 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −8.97624 − 19.6552i −25.9063 + 7.60678i −221.933 + 256.124i 196.683 + 126.400i 382.054 + 440.914i −227.971 − 1585.58i 4372.52 + 1283.89i 613.274 − 394.127i 718.955 − 5000.44i
4.2 −8.25539 − 18.0768i −25.9063 + 7.60678i −174.796 + 201.726i −370.312 − 237.985i 351.373 + 405.506i 104.114 + 724.126i 2648.91 + 777.790i 613.274 − 394.127i −1244.94 + 8658.72i
4.3 −7.20815 − 15.7837i −25.9063 + 7.60678i −113.344 + 130.806i 267.037 + 171.614i 306.799 + 354.065i 122.562 + 852.436i 750.547 + 220.380i 613.274 − 394.127i 783.856 − 5451.84i
4.4 −5.02622 − 11.0059i −25.9063 + 7.60678i −12.0443 + 13.8999i 97.9555 + 62.9522i 213.930 + 246.888i −58.9625 − 410.093i −1272.45 − 373.626i 613.274 − 394.127i 200.498 − 1394.50i
4.5 −3.93501 − 8.61648i −25.9063 + 7.60678i 25.0628 − 28.9240i −404.695 − 260.082i 167.485 + 193.288i −168.093 − 1169.11i −1511.21 − 443.731i 613.274 − 394.127i −648.508 + 4510.47i
4.6 −3.75287 − 8.21765i −25.9063 + 7.60678i 30.3765 − 35.0564i 42.6731 + 27.4244i 159.733 + 184.342i 138.669 + 964.463i −1511.60 − 443.845i 613.274 − 394.127i 65.2168 − 453.593i
4.7 −0.464659 − 1.01746i −25.9063 + 7.60678i 83.0029 − 95.7904i −317.877 − 204.287i 19.7772 + 22.8241i 233.295 + 1622.60i −273.405 − 80.2789i 613.274 − 394.127i −60.1498 + 418.351i
4.8 0.479194 + 1.04929i −25.9063 + 7.60678i 82.9508 − 95.7303i 438.544 + 281.835i −20.3958 − 23.5381i 54.8910 + 381.775i 281.869 + 82.7642i 613.274 − 394.127i −85.5787 + 595.213i
4.9 0.514925 + 1.12753i −25.9063 + 7.60678i 82.8160 − 95.5748i −8.00752 − 5.14612i −21.9167 − 25.2932i −142.750 − 992.847i 302.642 + 88.8637i 613.274 − 394.127i 1.67912 − 11.6786i
4.10 4.39581 + 9.62547i −25.9063 + 7.60678i 10.4956 − 12.1125i −374.109 − 240.425i −187.098 − 215.923i −55.3147 − 384.722i 1462.32 + 429.376i 613.274 − 394.127i 669.696 − 4657.84i
4.11 5.46303 + 11.9624i −25.9063 + 7.60678i −29.4313 + 33.9655i −2.23868 − 1.43871i −232.522 − 268.345i 238.756 + 1660.58i 1048.02 + 307.727i 613.274 − 394.127i 4.98044 − 34.6397i
4.12 6.33397 + 13.8695i −25.9063 + 7.60678i −68.4208 + 78.9618i 90.7130 + 58.2977i −269.592 − 311.126i −159.891 − 1112.07i 344.070 + 101.028i 613.274 − 394.127i −233.984 + 1627.40i
4.13 7.08490 + 15.5138i −25.9063 + 7.60678i −106.659 + 123.091i 333.068 + 214.050i −301.553 − 348.011i 19.0468 + 132.473i −570.665 − 167.562i 613.274 − 394.127i −960.967 + 6683.67i
4.14 8.74099 + 19.1401i −25.9063 + 7.60678i −206.116 + 237.871i −231.772 − 148.951i −372.041 − 429.359i 49.7343 + 345.910i −3770.32 − 1107.06i 613.274 − 394.127i 825.017 − 5738.12i
13.1 −14.6942 − 16.9580i 22.7138 + 14.5973i −53.4383 + 371.672i −73.6346 + 161.237i −86.2206 − 599.677i −399.535 − 117.314i 4671.84 − 3002.41i 302.838 + 663.122i 3816.27 − 1120.56i
13.2 −11.6508 − 13.4458i 22.7138 + 14.5973i −26.8307 + 186.612i 71.8962 − 157.431i −68.3631 − 475.476i −54.4916 − 16.0002i 905.962 − 582.226i 302.838 + 663.122i −2954.43 + 867.498i
13.3 −10.1647 − 11.7307i 22.7138 + 14.5973i −16.0715 + 111.780i 55.0130 − 120.462i −59.6429 − 414.826i 655.137 + 192.366i −196.790 + 126.469i 302.838 + 663.122i −1972.28 + 579.115i
13.4 −10.0698 − 11.6212i 22.7138 + 14.5973i −15.4345 + 107.349i −210.391 + 460.693i −59.0862 − 410.954i 1311.31 + 385.036i −252.861 + 162.504i 302.838 + 663.122i 7472.40 − 2194.09i
13.5 −6.24555 − 7.20775i 22.7138 + 14.5973i 5.27156 − 36.6645i 130.697 − 286.187i −36.6468 − 254.884i −922.772 − 270.950i −1324.16 + 850.988i 302.838 + 663.122i −2879.04 + 845.362i
13.6 −3.09608 − 3.57307i 22.7138 + 14.5973i 15.0352 − 104.572i −55.7726 + 122.125i −18.1668 − 126.353i 157.866 + 46.3536i −929.290 + 597.219i 302.838 + 663.122i 609.038 − 178.830i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
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 69.8.e.a ✓ 140
23.c even 11 1 inner 69.8.e.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.e.a ✓ 140 1.a even 1 1 trivial
69.8.e.a ✓ 140 23.c even 11 1 inner