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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 $4$

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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.

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

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:

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 \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display