# Properties

 Label 72.7.p.b Level $72$ Weight $7$ Character orbit 72.p Analytic conductor $16.564$ Analytic rank $0$ Dimension $136$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$16.5638940206$$ Analytic rank: $$0$$ Dimension: $$136$$ Relative dimension: $$68$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9}+O(q^{10})$$ 136 * q - 17 * q^2 + 42 * q^3 + 127 * q^4 - 561 * q^6 + 1774 * q^8 + 654 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9} - 132 q^{10} + 2336 q^{11} - 2796 q^{12} + 2346 q^{14} + 8191 q^{16} - 3460 q^{17} + 6306 q^{18} - 4972 q^{19} + 3588 q^{20} - 18577 q^{22} - 50553 q^{24} + 224998 q^{25} + 38352 q^{26} - 27792 q^{27} - 8196 q^{28} - 107130 q^{30} - 191717 q^{32} - 42852 q^{33} + 6125 q^{34} - 62508 q^{35} + 87339 q^{36} + 250223 q^{38} - 27126 q^{40} - 145084 q^{41} - 107478 q^{42} + 74912 q^{43} - 1118662 q^{44} + 22296 q^{46} + 690903 q^{48} + 1210102 q^{49} - 844727 q^{50} + 830778 q^{51} + 62952 q^{52} + 860925 q^{54} - 587718 q^{56} + 748686 q^{57} + 221136 q^{58} + 98960 q^{59} - 960810 q^{60} - 1062312 q^{62} - 784646 q^{64} - 62502 q^{65} + 802866 q^{66} + 596624 q^{67} + 681953 q^{68} + 86334 q^{70} - 2246001 q^{72} - 1186276 q^{73} - 782586 q^{74} + 3451098 q^{75} - 273235 q^{76} - 3340578 q^{78} + 2446320 q^{80} - 621690 q^{81} - 3119398 q^{82} - 179350 q^{83} - 227136 q^{84} + 917117 q^{86} - 1184629 q^{88} + 253712 q^{89} - 4940256 q^{90} - 470604 q^{91} + 1731894 q^{92} + 2055426 q^{94} + 8439666 q^{96} - 1822756 q^{97} + 6531850 q^{98} + 2509818 q^{99}+O(q^{100})$$ 136 * q - 17 * q^2 + 42 * q^3 + 127 * q^4 - 561 * q^6 + 1774 * q^8 + 654 * q^9 - 132 * q^10 + 2336 * q^11 - 2796 * q^12 + 2346 * q^14 + 8191 * q^16 - 3460 * q^17 + 6306 * q^18 - 4972 * q^19 + 3588 * q^20 - 18577 * q^22 - 50553 * q^24 + 224998 * q^25 + 38352 * q^26 - 27792 * q^27 - 8196 * q^28 - 107130 * q^30 - 191717 * q^32 - 42852 * q^33 + 6125 * q^34 - 62508 * q^35 + 87339 * q^36 + 250223 * q^38 - 27126 * q^40 - 145084 * q^41 - 107478 * q^42 + 74912 * q^43 - 1118662 * q^44 + 22296 * q^46 + 690903 * q^48 + 1210102 * q^49 - 844727 * q^50 + 830778 * q^51 + 62952 * q^52 + 860925 * q^54 - 587718 * q^56 + 748686 * q^57 + 221136 * q^58 + 98960 * q^59 - 960810 * q^60 - 1062312 * q^62 - 784646 * q^64 - 62502 * q^65 + 802866 * q^66 + 596624 * q^67 + 681953 * q^68 + 86334 * q^70 - 2246001 * q^72 - 1186276 * q^73 - 782586 * q^74 + 3451098 * q^75 - 273235 * q^76 - 3340578 * q^78 + 2446320 * q^80 - 621690 * q^81 - 3119398 * q^82 - 179350 * q^83 - 227136 * q^84 + 917117 * q^86 - 1184629 * q^88 + 253712 * q^89 - 4940256 * q^90 - 470604 * q^91 + 1731894 * q^92 + 2055426 * q^94 + 8439666 * q^96 - 1822756 * q^97 + 6531850 * q^98 + 2509818 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −7.99998 + 0.0188688i −26.9864 0.858100i 63.9993 0.301900i 186.544 107.701i 215.906 + 6.35558i 13.1755 + 7.60688i −511.987 + 3.62278i 727.527 + 46.3140i −1490.31 + 865.125i
43.2 −7.99923 + 0.111052i 14.5758 + 22.7277i 63.9753 1.77667i 100.062 57.7710i −119.119 180.185i −427.342 246.726i −511.556 + 21.3166i −304.094 + 662.547i −794.005 + 473.235i
43.3 −7.99373 0.316771i 11.6098 + 24.3765i 63.7993 + 5.06436i −110.205 + 63.6267i −85.0840 198.536i 340.379 + 196.518i −508.390 60.6929i −459.424 + 566.013i 901.101 473.705i
43.4 −7.92579 1.08711i −6.90289 26.1027i 61.6364 + 17.2324i 42.4034 24.4816i 26.3344 + 214.389i −225.391 130.130i −469.784 203.586i −633.700 + 360.368i −362.695 + 147.939i
43.5 −7.72302 2.08685i 25.9765 7.36361i 55.2901 + 32.2336i −144.957 + 83.6907i −215.984 + 2.66019i −59.7430 34.4926i −359.740 364.323i 620.555 382.561i 1294.15 343.842i
43.6 −7.72049 + 2.09618i 26.9984 0.294271i 55.2120 32.3671i 109.916 63.4600i −207.824 + 58.8655i 509.750 + 294.305i −358.417 + 365.625i 728.827 15.8897i −715.581 + 720.346i
43.7 −7.70909 + 2.13776i −20.4110 + 17.6746i 54.8600 32.9603i −67.6940 + 39.0831i 119.566 179.888i 139.025 + 80.2661i −352.459 + 371.371i 104.220 721.512i 438.308 446.009i
43.8 −7.58109 + 2.55481i 14.1249 23.0106i 50.9459 38.7364i −17.1790 + 9.91831i −48.2949 + 210.532i −281.285 162.400i −287.262 + 423.822i −329.973 650.045i 104.896 119.081i
43.9 −7.44575 + 2.92588i −21.4315 16.4223i 46.8784 43.5708i −135.303 + 78.1175i 207.623 + 59.5707i −19.7572 11.4068i −221.562 + 461.578i 189.614 + 703.909i 778.873 977.525i
43.10 −7.43575 2.95121i 4.82015 26.5663i 46.5807 + 43.8890i 29.7357 17.1679i −114.244 + 183.315i 464.017 + 267.900i −216.836 463.817i −682.532 256.107i −271.773 + 39.8998i
43.11 −7.36656 3.11990i −18.7030 + 19.4730i 44.5325 + 45.9658i −116.445 + 67.2296i 198.530 85.0981i −587.496 339.191i −184.643 477.547i −29.3990 728.407i 1067.55 131.955i
43.12 −7.01582 3.84426i −24.8393 10.5835i 34.4433 + 53.9412i −89.5082 + 51.6776i 133.582 + 169.740i 227.154 + 131.147i −34.2842 510.851i 504.981 + 525.771i 826.635 18.4677i
43.13 −6.67611 4.40790i −5.67293 + 26.3973i 25.1408 + 58.8553i 99.0560 57.1900i 154.230 151.225i 231.955 + 133.919i 91.5860 503.742i −664.636 299.500i −913.396 54.8228i
43.14 −6.67572 4.40849i 25.5449 8.74402i 25.1304 + 58.8597i 171.291 98.8948i −209.079 54.2419i −265.898 153.516i 91.7187 503.718i 576.084 446.730i −1579.47 94.9401i
43.15 −6.06984 + 5.21124i −7.87021 + 25.8275i 9.68594 63.2628i 110.415 63.7480i −86.8224 197.782i 81.2276 + 46.8968i 270.886 + 434.471i −605.120 406.536i −337.994 + 962.339i
43.16 −6.00254 + 5.28862i 26.0855 + 6.96739i 8.06090 63.4903i 17.5470 10.1308i −193.427 + 96.1346i −225.997 130.480i 287.391 + 423.734i 631.911 + 363.496i −51.7487 + 153.610i
43.17 −5.89620 + 5.40692i −9.33665 25.3343i 5.53046 63.7606i 118.367 68.3392i 192.031 + 98.8937i 261.548 + 151.005i 312.140 + 405.848i −554.654 + 473.075i −328.411 + 1042.94i
43.18 −5.86870 + 5.43676i 12.5085 + 23.9277i 4.88334 63.8134i −166.194 + 95.9519i −203.498 72.4189i −147.339 85.0661i 318.279 + 401.052i −416.073 + 598.602i 453.673 1466.67i
43.19 −5.16108 6.11255i 23.0486 + 14.0627i −10.7264 + 63.0947i −44.3287 + 25.5932i −32.9969 213.465i 23.6171 + 13.6353i 441.029 260.072i 333.480 + 648.253i 385.224 + 138.873i
43.20 −4.97818 + 6.26240i 18.3439 19.8117i −14.4354 62.3508i −162.167 + 93.6271i 32.7494 + 213.503i 318.591 + 183.939i 462.328 + 219.993i −56.0038 726.846i 220.965 1481.65i
See next 80 embeddings (of 136 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.68 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.d odd 2 1 inner
9.c even 3 1 inner
72.p 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.p.b 136
8.d odd 2 1 inner 72.7.p.b 136
9.c even 3 1 inner 72.7.p.b 136
72.p odd 6 1 inner 72.7.p.b 136

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.p.b 136 1.a even 1 1 trivial
72.7.p.b 136 8.d odd 2 1 inner
72.7.p.b 136 9.c even 3 1 inner
72.7.p.b 136 72.p odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{136} - 643749 T_{5}^{134} + 218075496876 T_{5}^{132} + \cdots + 74\!\cdots\!00$$ acting on $$S_{7}^{\mathrm{new}}(72, [\chi])$$.