# 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 $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 69.e (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$140q - 16q^{2} - 378q^{3} - 740q^{4} - 6q^{5} + 1647q^{6} + 978q^{7} - 824q^{8} - 10206q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q - 16q^{2} - 378q^{3} - 740q^{4} - 6q^{5} + 1647q^{6} + 978q^{7} - 824q^{8} - 10206q^{9} + 1322q^{10} + 7892q^{11} - 19980q^{12} + 39810q^{13} - 26296q^{14} + 135q^{15} - 142164q^{16} - 97081q^{17} - 11664q^{18} + 120905q^{19} + 124755q^{20} + 26406q^{21} - 837266q^{22} - 224688q^{23} - 60264q^{24} - 180624q^{25} + 385108q^{26} - 275562q^{27} + 1121760q^{28} - 375654q^{29} + 35694q^{30} + 505029q^{31} - 661528q^{32} - 360423q^{33} - 1872759q^{34} - 964610q^{35} - 1269189q^{36} - 28802q^{37} - 1459868q^{38} + 1074870q^{39} + 5371901q^{40} - 1557480q^{41} + 883116q^{42} + 1160786q^{43} - 4521825q^{44} - 2009124q^{45} - 4123580q^{46} + 811422q^{47} - 1563111q^{48} - 10960262q^{49} + 16636695q^{50} + 4413852q^{51} - 4843791q^{52} + 4291672q^{53} - 98415q^{54} + 2093539q^{55} - 12548382q^{56} - 4007907q^{57} - 7414367q^{58} + 2955424q^{59} + 7289973q^{60} + 4919292q^{61} + 16083211q^{62} + 712962q^{63} - 1673596q^{64} - 13246221q^{65} + 7219449q^{66} + 148565q^{67} - 23039568q^{68} - 657018q^{69} + 12781104q^{70} - 7247645q^{71} - 600696q^{72} + 32596055q^{73} + 32550658q^{74} - 10061874q^{75} + 12150569q^{76} - 43349150q^{77} - 17201700q^{78} + 4815410q^{79} + 36529590q^{80} - 7440174q^{81} + 20862470q^{82} + 35160800q^{83} - 5346540q^{84} + 48578363q^{85} + 76972708q^{86} + 1505088q^{87} - 35912212q^{88} + 1401264q^{89} + 963738q^{90} - 91970208q^{91} - 74293537q^{92} + 555606q^{93} - 32520253q^{94} + 79436789q^{95} + 41519736q^{96} + 18296816q^{97} - 38956823q^{98} - 9731421q^{99} + O(q^{100})$$

## 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.

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 64.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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