# Properties

 Label 84.12.b.a Level $84$ Weight $12$ Character orbit 84.b Analytic conductor $64.541$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 84.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.5408271670$$ Analytic rank: $$0$$ Dimension: $$44$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 23 q^{2} - 10692 q^{3} + 1541 q^{4} + 5589 q^{6} - 134 q^{7} - 18695 q^{8} + 2598156 q^{9}+O(q^{10})$$ 44 * q - 23 * q^2 - 10692 * q^3 + 1541 * q^4 + 5589 * q^6 - 134 * q^7 - 18695 * q^8 + 2598156 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$44 q - 23 q^{2} - 10692 q^{3} + 1541 q^{4} + 5589 q^{6} - 134 q^{7} - 18695 q^{8} + 2598156 q^{9} - 828424 q^{10} - 374463 q^{12} + 4198365 q^{14} - 4412759 q^{16} - 1358127 q^{18} + 6983052 q^{19} - 44417108 q^{20} + 32562 q^{21} - 17991462 q^{22} + 4542885 q^{24} - 453628696 q^{25} - 98243700 q^{26} - 631351908 q^{27} + 197028053 q^{28} - 77673208 q^{29} + 201307032 q^{30} + 363408768 q^{31} - 193132863 q^{32} + 244149820 q^{34} - 387281676 q^{35} + 90994509 q^{36} + 341371884 q^{37} - 414182420 q^{38} - 770410876 q^{40} - 1020202695 q^{42} - 880133666 q^{44} - 1775052978 q^{46} - 1792699304 q^{47} + 1072300437 q^{48} - 819617000 q^{49} + 2158233149 q^{50} - 4514138716 q^{52} + 1914807152 q^{53} + 330024861 q^{54} + 1785927092 q^{55} - 16112780167 q^{56} - 1696881636 q^{57} + 5084833750 q^{58} + 3729062416 q^{59} + 10793357244 q^{60} - 18353275776 q^{62} - 7912566 q^{63} - 12067108399 q^{64} - 5624825000 q^{65} + 4371925266 q^{66} + 885261520 q^{68} + 34763977928 q^{70} - 1103921055 q^{72} - 40922634778 q^{74} + 110231773128 q^{75} + 82420472236 q^{76} - 2093349112 q^{77} + 23873219100 q^{78} - 62176412340 q^{80} + 153418513644 q^{81} - 5083269068 q^{82} + 34205636344 q^{83} - 47877816879 q^{84} + 105152127628 q^{85} + 242964863598 q^{86} + 18874589544 q^{87} + 218894674838 q^{88} - 48917608776 q^{90} - 6678724880 q^{91} + 101339859530 q^{92} - 88308330624 q^{93} + 220283048096 q^{94} + 46931285709 q^{96} + 222198095305 q^{98}+O(q^{100})$$ 44 * q - 23 * q^2 - 10692 * q^3 + 1541 * q^4 + 5589 * q^6 - 134 * q^7 - 18695 * q^8 + 2598156 * q^9 - 828424 * q^10 - 374463 * q^12 + 4198365 * q^14 - 4412759 * q^16 - 1358127 * q^18 + 6983052 * q^19 - 44417108 * q^20 + 32562 * q^21 - 17991462 * q^22 + 4542885 * q^24 - 453628696 * q^25 - 98243700 * q^26 - 631351908 * q^27 + 197028053 * q^28 - 77673208 * q^29 + 201307032 * q^30 + 363408768 * q^31 - 193132863 * q^32 + 244149820 * q^34 - 387281676 * q^35 + 90994509 * q^36 + 341371884 * q^37 - 414182420 * q^38 - 770410876 * q^40 - 1020202695 * q^42 - 880133666 * q^44 - 1775052978 * q^46 - 1792699304 * q^47 + 1072300437 * q^48 - 819617000 * q^49 + 2158233149 * q^50 - 4514138716 * q^52 + 1914807152 * q^53 + 330024861 * q^54 + 1785927092 * q^55 - 16112780167 * q^56 - 1696881636 * q^57 + 5084833750 * q^58 + 3729062416 * q^59 + 10793357244 * q^60 - 18353275776 * q^62 - 7912566 * q^63 - 12067108399 * q^64 - 5624825000 * q^65 + 4371925266 * q^66 + 885261520 * q^68 + 34763977928 * q^70 - 1103921055 * q^72 - 40922634778 * q^74 + 110231773128 * q^75 + 82420472236 * q^76 - 2093349112 * q^77 + 23873219100 * q^78 - 62176412340 * q^80 + 153418513644 * q^81 - 5083269068 * q^82 + 34205636344 * q^83 - 47877816879 * q^84 + 105152127628 * q^85 + 242964863598 * q^86 + 18874589544 * q^87 + 218894674838 * q^88 - 48917608776 * q^90 - 6678724880 * q^91 + 101339859530 * q^92 - 88308330624 * q^93 + 220283048096 * q^94 + 46931285709 * q^96 + 222198095305 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1 −45.2221 1.72065i −243.000 2042.08 + 155.623i 479.067i 10989.0 + 418.118i 18444.0 40461.6i −92079.3 10551.3i 59049.0 824.307 21664.4i
55.2 −45.2221 + 1.72065i −243.000 2042.08 155.623i 479.067i 10989.0 418.118i 18444.0 + 40461.6i −92079.3 + 10551.3i 59049.0 824.307 + 21664.4i
55.3 −44.4594 8.44758i −243.000 1905.28 + 751.149i 11354.1i 10803.6 + 2052.76i −41246.7 16614.4i −78362.1 49490.6i 59049.0 95914.4 504795.i
55.4 −44.4594 + 8.44758i −243.000 1905.28 751.149i 11354.1i 10803.6 2052.76i −41246.7 + 16614.4i −78362.1 + 49490.6i 59049.0 95914.4 + 504795.i
55.5 −40.8700 19.4330i −243.000 1292.72 + 1588.45i 1211.18i 9931.41 + 4722.22i −25032.4 + 36752.0i −21964.9 90041.5i 59049.0 −23536.9 + 49501.0i
55.6 −40.8700 + 19.4330i −243.000 1292.72 1588.45i 1211.18i 9931.41 4722.22i −25032.4 36752.0i −21964.9 + 90041.5i 59049.0 −23536.9 49501.0i
55.7 −40.1866 20.8095i −243.000 1181.93 + 1672.53i 11675.3i 9765.35 + 5056.70i −983.024 44456.3i −12693.5 91808.6i 59049.0 −242957. + 469190.i
55.8 −40.1866 + 20.8095i −243.000 1181.93 1672.53i 11675.3i 9765.35 5056.70i −983.024 + 44456.3i −12693.5 + 91808.6i 59049.0 −242957. 469190.i
55.9 −39.5815 21.9387i −243.000 1085.38 + 1736.73i 10289.8i 9618.30 + 5331.11i 43786.1 + 7752.48i −4859.36 92554.4i 59049.0 225745. 407286.i
55.10 −39.5815 + 21.9387i −243.000 1085.38 1736.73i 10289.8i 9618.30 5331.11i 43786.1 7752.48i −4859.36 + 92554.4i 59049.0 225745. + 407286.i
55.11 −32.5648 31.4250i −243.000 72.9330 + 2046.70i 4483.20i 7913.25 + 7636.29i 44244.4 + 4445.20i 61942.6 68942.3i 59049.0 −140885. + 145994.i
55.12 −32.5648 + 31.4250i −243.000 72.9330 2046.70i 4483.20i 7913.25 7636.29i 44244.4 4445.20i 61942.6 + 68942.3i 59049.0 −140885. 145994.i
55.13 −25.9452 37.0790i −243.000 −701.697 + 1924.04i 5436.30i 6304.67 + 9010.19i −42645.9 12595.7i 89547.0 23901.3i 59049.0 201572. 141046.i
55.14 −25.9452 + 37.0790i −243.000 −701.697 1924.04i 5436.30i 6304.67 9010.19i −42645.9 + 12595.7i 89547.0 + 23901.3i 59049.0 201572. + 141046.i
55.15 −23.3606 38.7593i −243.000 −956.568 + 1810.88i 7022.22i 5676.62 + 9418.51i −17991.9 + 40664.7i 92534.4 5227.27i 59049.0 −272177. + 164043.i
55.16 −23.3606 + 38.7593i −243.000 −956.568 1810.88i 7022.22i 5676.62 9418.51i −17991.9 40664.7i 92534.4 + 5227.27i 59049.0 −272177. 164043.i
55.17 −20.9262 40.1260i −243.000 −1172.19 + 1679.37i 6779.01i 5085.06 + 9750.61i 9419.93 43457.9i 91915.7 + 11892.5i 59049.0 272015. 141859.i
55.18 −20.9262 + 40.1260i −243.000 −1172.19 1679.37i 6779.01i 5085.06 9750.61i 9419.93 + 43457.9i 91915.7 11892.5i 59049.0 272015. + 141859.i
55.19 −8.52362 44.4449i −243.000 −1902.70 + 757.663i 11942.9i 2071.24 + 10800.1i −30635.0 32230.8i 49892.1 + 78107.1i 59049.0 −530802. + 101797.i
55.20 −8.52362 + 44.4449i −243.000 −1902.70 757.663i 11942.9i 2071.24 10800.1i −30635.0 + 32230.8i 49892.1 78107.1i 59049.0 −530802. 101797.i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.44 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.b.a 44
4.b odd 2 1 84.12.b.b yes 44
7.b odd 2 1 84.12.b.b yes 44
28.d even 2 1 inner 84.12.b.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.b.a 44 1.a even 1 1 trivial
84.12.b.a 44 28.d even 2 1 inner
84.12.b.b yes 44 4.b odd 2 1
84.12.b.b yes 44 7.b odd 2 1