# Properties

 Label 72.7.j.a Level $72$ Weight $7$ Character orbit 72.j Analytic conductor $16.564$ Analytic rank $0$ Dimension $140$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 72.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.5638940206$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$70$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 140 * q - 3 * q^2 - q^4 + 317 * q^6 - 2 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9} + 124 q^{10} - 6946 q^{12} - 7428 q^{14} + 1454 q^{15} - q^{16} + 20878 q^{18} - 35346 q^{20} + 127 q^{22} - 6 q^{23} + 50209 q^{24} - 193752 q^{25} + 8188 q^{28} - 21344 q^{30} - 2 q^{31} - 180093 q^{32} + 45534 q^{33} + 20061 q^{34} + 39831 q^{36} + 274239 q^{38} + 125738 q^{39} - 58376 q^{40} + 31314 q^{41} - 213688 q^{42} - 22560 q^{46} - 6 q^{47} - 114411 q^{48} - 974808 q^{49} + 723249 q^{50} - 79338 q^{52} - 231173 q^{54} + 62492 q^{55} + 1294542 q^{56} + 125032 q^{57} - 221138 q^{58} - 1156344 q^{60} + 275294 q^{63} - 804478 q^{64} - 6 q^{65} + 1782258 q^{66} + 2100195 q^{68} - 86464 q^{70} + 1113091 q^{72} - 8 q^{73} - 1886976 q^{74} + 432081 q^{76} + 1658428 q^{78} - 2 q^{79} + 597124 q^{81} - 965382 q^{82} + 4997288 q^{84} - 1041879 q^{86} - 1337970 q^{87} - 536717 q^{88} - 4261994 q^{90} - 737184 q^{92} - 72534 q^{94} + 93744 q^{95} + 1053622 q^{96} - 2 q^{97}+O(q^{100})$$ 140 * q - 3 * q^2 - q^4 + 317 * q^6 - 2 * q^7 - 4 * q^9 + 124 * q^10 - 6946 * q^12 - 7428 * q^14 + 1454 * q^15 - q^16 + 20878 * q^18 - 35346 * q^20 + 127 * q^22 - 6 * q^23 + 50209 * q^24 - 193752 * q^25 + 8188 * q^28 - 21344 * q^30 - 2 * q^31 - 180093 * q^32 + 45534 * q^33 + 20061 * q^34 + 39831 * q^36 + 274239 * q^38 + 125738 * q^39 - 58376 * q^40 + 31314 * q^41 - 213688 * q^42 - 22560 * q^46 - 6 * q^47 - 114411 * q^48 - 974808 * q^49 + 723249 * q^50 - 79338 * q^52 - 231173 * q^54 + 62492 * q^55 + 1294542 * q^56 + 125032 * q^57 - 221138 * q^58 - 1156344 * q^60 + 275294 * q^63 - 804478 * q^64 - 6 * q^65 + 1782258 * q^66 + 2100195 * q^68 - 86464 * q^70 + 1113091 * q^72 - 8 * q^73 - 1886976 * q^74 + 432081 * q^76 + 1658428 * q^78 - 2 * q^79 + 597124 * q^81 - 965382 * q^82 + 4997288 * q^84 - 1041879 * q^86 - 1337970 * q^87 - 536717 * q^88 - 4261994 * q^90 - 737184 * q^92 - 72534 * q^94 + 93744 * q^95 + 1053622 * q^96 - 2 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −7.98824 0.433656i 19.9076 18.2398i 63.6239 + 6.92830i 54.0624 93.6388i −166.936 + 137.070i −182.538 316.165i −505.238 82.9358i 63.6224 726.218i −472.470 + 724.565i
5.2 −7.97469 0.635864i −7.07752 + 26.0559i 63.1914 + 10.1416i 48.9289 84.7473i 73.0091 203.287i 6.45730 + 11.1844i −497.483 121.058i −628.817 368.822i −444.080 + 644.721i
5.3 −7.95659 + 0.832232i 24.0024 + 12.3647i 62.6148 13.2435i 0.173829 0.301081i −201.268 78.4053i 167.050 + 289.340i −487.179 + 157.483i 423.229 + 593.564i −1.13252 + 2.54024i
5.4 −7.93602 1.00978i −16.2129 21.5903i 61.9607 + 16.0272i 92.8213 160.771i 106.864 + 187.713i 288.471 + 499.646i −475.537 189.758i −203.285 + 700.083i −898.975 + 1182.15i
5.5 −7.83990 + 1.59247i −26.3328 + 5.96513i 58.9281 24.9696i −104.839 + 181.586i 196.947 88.7002i 128.782 + 223.056i −422.227 + 289.600i 657.834 314.158i 532.756 1590.57i
5.6 −7.83675 1.60790i 6.22898 + 26.2717i 58.8293 + 25.2014i −96.7705 + 167.611i −6.57280 215.900i −278.012 481.531i −420.509 292.089i −651.400 + 327.291i 1027.87 1157.93i
5.7 −7.59729 + 2.50624i −12.5996 23.8799i 51.4376 38.0812i −40.6546 + 70.4159i 155.572 + 149.845i −158.685 274.850i −295.345 + 418.229i −411.498 + 601.756i 132.386 636.860i
5.8 −7.59716 2.50663i 12.6861 23.8341i 51.4336 + 38.0866i −106.929 + 185.206i −156.122 + 149.272i 267.810 + 463.860i −295.280 418.275i −407.126 604.723i 1276.60 1139.01i
5.9 −7.46395 2.87915i −26.9713 + 1.24421i 47.4210 + 42.9797i −7.54460 + 13.0676i 204.895 + 68.3678i −156.472 271.018i −230.202 457.330i 725.904 67.1160i 93.9362 75.8140i
5.10 −7.32138 + 3.22450i −26.7644 + 3.55934i 43.2052 47.2156i 100.826 174.637i 184.475 112.361i −187.285 324.386i −164.075 + 484.998i 703.662 190.527i −175.073 + 1603.70i
5.11 −7.03719 + 3.80499i −10.4599 + 24.8916i 35.0441 53.5529i −8.24990 + 14.2892i −21.1036 214.967i 142.080 + 246.089i −42.8438 + 510.204i −510.180 520.728i 3.68567 131.947i
5.12 −6.81381 + 4.19189i 10.4599 24.8916i 28.8561 57.1255i 8.24990 14.2892i 33.0707 + 213.453i 142.080 + 246.089i 42.8438 + 510.204i −510.180 520.728i 3.68567 + 131.947i
5.13 −6.72146 4.33843i −13.8891 23.1537i 26.3561 + 58.3211i −24.0551 + 41.6647i −7.09581 + 215.883i −101.031 174.991i 75.8702 506.347i −343.188 + 643.166i 342.445 175.687i
5.14 −6.53010 4.62145i −21.8009 + 15.9287i 21.2844 + 60.3571i 17.6699 30.6052i 215.975 3.26399i 194.799 + 337.401i 139.948 492.502i 221.556 694.517i −256.827 + 118.194i
5.15 −6.45319 + 4.72825i 26.7644 3.55934i 19.2873 61.0246i −100.826 + 174.637i −155.886 + 149.518i −187.285 324.386i 164.075 + 484.998i 703.662 190.527i −175.073 1603.70i
5.16 −6.38046 4.82594i 27.0000 0.0492591i 17.4206 + 61.5835i −22.8039 + 39.4975i −172.510 129.986i −41.8302 72.4520i 186.046 477.002i 728.995 2.65999i 336.112 141.962i
5.17 −6.34800 4.86856i 15.8845 + 21.8331i 16.5943 + 61.8112i 122.026 211.355i 5.46045 215.931i −28.5227 49.4028i 195.591 473.168i −224.364 + 693.615i −1803.62 + 747.594i
5.18 −5.96911 + 5.32633i 12.5996 + 23.8799i 7.26052 63.5868i 40.6546 70.4159i −202.401 75.4318i −158.685 274.850i 295.345 + 418.229i −411.498 + 601.756i 132.386 + 636.860i
5.19 −5.29907 + 5.99332i 26.3328 5.96513i −7.83975 63.5180i 104.839 181.586i −103.788 + 189.431i 128.782 + 223.056i 422.227 + 289.600i 657.834 314.158i 532.756 + 1590.57i
5.20 −5.06626 6.19136i 1.22852 + 26.9720i −12.6659 + 62.7342i −68.8118 + 119.186i 160.770 144.254i 221.876 + 384.301i 452.579 239.408i −725.981 + 66.2712i 1086.54 177.787i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.70 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.d odd 6 1 inner
72.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.j.a 140
8.b even 2 1 inner 72.7.j.a 140
9.d odd 6 1 inner 72.7.j.a 140
72.j odd 6 1 inner 72.7.j.a 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.j.a 140 1.a even 1 1 trivial
72.7.j.a 140 8.b even 2 1 inner
72.7.j.a 140 9.d odd 6 1 inner
72.7.j.a 140 72.j odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(72, [\chi])$$.